**Entropy and Random Walk Trails Water Confinement and Non-Thermal Equilibrium in Photon-Induced Nanocavities**

**Vassilios Gavriil 1,2, Margarita Chatzichristidi 3, Dimitrios Christofilos 2, Gerasimos A. Kourouklis 2, Zoe Kollia 1, Evangelos Bakalis 1,4, Alkiviadis-Constantinos Cefalas <sup>1</sup> and Evangelia Sarantopoulou 1,\***


Received: 29 April 2020; Accepted: 22 May 2020; Published: 2 June 2020

**Abstract:** Molecules near surfaces are regularly trapped in small cavitations. Molecular confinement, especially water confinement, shows intriguing and unexpected behavior including surface entropy adjustment; nevertheless, observations of entropic variation during molecular confinement are scarce. An experimental assessment of the correlation between surface strain and entropy during molecular confinement in tiny crevices is difficult because strain variances fall in the nanometer scale. In this work, entropic variations during water confinement in 2D nano/micro cavitations were observed. Experimental results and random walk simulations of water molecules inside different size nanocavitations show that the mean escaping time of molecular water from nanocavities largely deviates from the mean collision time of water molecules near surfaces, crafted by 157 nm vacuum ultraviolet laser light on polyacrylamide matrixes. The mean escape time distribution of a few molecules indicates a non-thermal equilibrium state inside the cavity. The time differentiation inside and outside nanocavities reveals an additional state of ordered arrangements between nanocavities and molecular water ensembles of fixed molecular length near the surface. The configured number of microstates correctly counts for the experimental surface entropy deviation during molecular water confinement. The methodology has the potential to identify confined water molecules in nanocavities with life science importance.

**Keywords:** nanocavities; non-thermal equilibrium; water; entropy; nanothermodynamics; nanoindentation; AFM; electric dipole interactions; VUV irradiation; random walk

#### **1. Introduction**

Confined molecular water in nanocavities shows intriguing and unexpected behavior. The dynamic evolution of confined molecular water swings between bulk response, molecular collective actions and interface binding reactions [1]. Translational and rotational motions of confined water point to different stretching dynamics from its bulk counterpart [2]. It is also known that confined water builds tight hydrogen-bonded (H-bonded) networks, and its flow response is diverging by orders of magnitude from macroscopic hydrodynamics [3]. Possible lack of H-bonding of water molecules in small volumes counts for de-wetting, cavity expulsion [4], water self-dissociation [5] and a diverging dielectric constant [6]. It is plausible; therefore, that diverging behaviors of the biological and geological evolution of molecular enclosures in small systems [7–13] also imply a nanothermodynamic approach [14,15].

The central element of any thermodynamic theory of small systems is based on the hypothesis that nanometer-sized configurations pullout an additional physical component to the free energy of the associated macroscopic system from interactions among nanostructure entities. Moreover, the confinement of a relatively large number of molecules in nanocavities, restraints the molecular degrees of freedom (translational, vibration or rotational), and finally the system evolves through different entropic states before equilibration. Most interesting, the confinement of a small number of molecules in a large number of distinguishable tiny spaces might well indicate a thermodynamic entropic collective behavior [13], space and time local heterogeneities, not-extensive fluctuations and intriguing surface-boundary effects. The reduction of the translational degrees of freedom of molecules in tiny spaces and the deviation of the molecular trapping time inside a cavity from the mean molecular collision time outside, highlight the presence of an entropic barrier that separates the molecular motions inside and outside the cavities.

Today, both theoretical [14–19] and experimental advancements [20,21] gradually disclose the intriguing issues of thermodynamics of small systems, with major impacts on colloids, liquids, surfaces, interphases, chemical sensors, micro/nanofluidics, nanoporous media, proteins and DNA folding [10,22–27]. In cell biology, the presence of different nano-sized molecular scaffolds in the extracellular matrix environment implies a vast diversity of cellular activities and responses, including uncorrelated diverging drug delivery efficiencies [28].

Because thermodynamic potential variations and fluctuations allow for volume and surface stressing, any experimental verification of local volume and surface stress might well point to entropic fluctuations during molecular confinement [13,29]. Commonly, bulk and surface stressing go along with self-assembled structures, translational symmetry breaking, non-linearity, bifurcations, chaos, instability and morphological and shape nano configurations [30,31]. In the non-equilibrium state, rapidly changing thermodynamic potentials across phase boundaries usually force tiny systems to pass from different morphological progressions and physical states by tracing minimum energy and maximum entropy production pathways. This universal principle appears everywhere in Nature; from self-assembled bio and macromolecular structures and folding of large protein molecules [32] to nano/micro flower-like artificial structures [33,34].

The confinement of molecules within nano-size cavitations, usually on the surface of a matrix, is linked to system's entropy diversity before and after trapping [13,27,35,36]. It is also known that for the same translational entropy, any confined molecular state attains a small variation of its rotational entropy compared to the non-confined molecular state. Likewise, rotational restriction affects surface molecular bonding and sorption/desorption kinetics [35]. Specific response of nanoentropic potentials from molecular confinement within photon-induced nanocavitations in PDMS matrixes underlines an inherent correlation between internal stressing and 2D entropy diversion [13].

Commonly, photon-processing of surfaces reconfigures their physicochemical properties, including thermodynamic potentials [37–40]. Irradiation of a polymeric matrix with vacuum ultraviolet (VUV) light in the spectral range from 110 to 180 nm entails an extensive modification of topological and thus of physical features, because of bond breaking and formation of new bonds. Any 2D topological transform is accompanied by a diversion of surface characteristics, such as porosity, sensing efficiency, chemical stability and extensive nanocavitation [41–46]. The adsorption of various molecules on 2D nanostructured surfaces [47–49], might well boost a plethora of surfactant effects along with molecular sensing [43,50], gas separation and storage [51–54], and also applications with particular emphasis on nanomedicine [55], bio-engineering [56,57] and drug delivery systems [58,59]. Among other polymeric matrixes, polyacrylamide (PAM) is a hydrophilic low toxic, biocompatible, water-soluble, synthetic linear or cross-linked molecule, modified accordingly for a wide range of

applications, including oil recuperation, wastewater treatment, soil conditioner, cosmetics food and biomedical industries [60–62]. A diverging number of physical and chemical methods are currently applied to optimize the biocompatibility level of different polymers (e.g., PDMS, PET, PTFEMA, PEG), for biomedical applications, biosensors, tissue engineering and artificial organs [46,63,64]. Well established methods of surface functionalization through photon irradiation with UV, VUV and EUV (extreme ultraviolet) light sources and plasma treatment at various wavelengths and electron energies, aim to optimize chemical instability and surface modification for controlling a plethora of surface functionalities [65].

Today, several methods exist to improve the strength and the physicochemical properties of PAM matrixes by blending the matrix with chitosan, starch or other polymers [66]. While functionalization of pure PAM polymeric surfaces is mostly done via sunlight exposure at standard environmental conditions, a limited number of studies include plasma processing [67–71]. However, no data exist for VUV processing of PAM surfaces, preventing thus precise tailoring of PAM's physicochemical surface characteristics (surface roughness, structure size, elasticity, chemical composition, etc.) and the formation of controlled micro/nanopatterns and cavitations for different applications [37,42,43,63,64].

The current work establishes the link between entropy variation and molecular water confinement in small nanocavities fabricated by 157 nm laser photons in polymeric PAM matrixes. The work follows a line of a rational evolution. First, the correlation between 157 nm molecular photodissociation (laser fluence or a number of laser pulses) and surface topological features, including nanocavitations, is established from fractal and surface analysis by using atomic force microscopy (AFM). Next, the correlation between surface strain and 157 nm molecular photodissociation is revealed by applying AFM nanoindentation (AFM-NI), contact angle (CA) wetting and white light reflection spectroscopy (WLRS). Random walk simulations of water molecules inside cavitations differentiate the escape time of confined molecular water and the mean collision time of water molecules near the PAM surface. The different time scales inside and outside the nanocavities point to an additional state of ordered arrangements between nanocavities and the molecular water ensembles of fixed molecular length near the surface. The configured number of microstates properly counts for the experimental surface entropy deviation during molecular water confinement, in agreement with the experimental results. Finally, the mean time distribution for a small number of water molecules for different runs reveals a non-equilibrium state inside tiny cavities. The experimental method has the potential to identify confined water molecules in nanocavities via entropy variation. The proposed roadmap of analysis may be used in applications related to life science.

#### **2. Materials and Methods**

#### *2.1. Materials*

PAM (typical Mn = 150 K, Mw 400 K) purchased from Sigma-Aldrich (St. Louis, MO, USA) used to prepare solution 5% *w*/*w* in water. Thin layers (426 ± 1 nm) on Si wafer substrates were made by spin-coating for 60 s at 2500 rpm, and finally, cured at 110 ◦C for 15 min at a temperature rate of 0.37 ◦C s−<sup>1</sup> and then left to cool at room temperature. WLRS measures the thickness of PAM films coated on Si wafers.

#### *2.2. 157 nm Laser*

PAM layers irradiated with a high power pulse discharged molecular fluorine laser at 157 nm (Lambda Physik 250 (LPFTM 200), Lambda Physik AG (Coherent), Göttingen, Germany), under continuous nitrogen flow (99.999%) at 105 Pa and room temperature. The layers were mounted into a computer-controlled *X-Y-Z-*θ translation-rotation motorized stage, placed inside a 316 stainless-steel chamber. The laser temporal pulse duration at FWHM, the energy of an unfocused laser beam per laser pulse, the photon fluence per laser pulse and laser repetition rate were set up at 15 ns, 28 mJ, 250 J m−<sup>2</sup> and 10 Hz. For dipping the amount of oxygen inside the stainless-steel chamber, nitrogen purging of the chamber was applied for 10 min before the irradiating stage.

#### *2.3. AFM Imaging and AFM-NI*

An AFM system (*diInnova*, Veeco Instruments Inc. (SPM Bruker), Santa Barbara, CA, USA) used for surface imaging of exposed/non exposed areas and the AFM-NI measurements. The imaging carried out in a tapping mode at a scanning rate of 0.5 Hz, using phosphorus-(n)-doped silicon cantilever (MPP-11123-10), having a spring constant of 40 nN nm−<sup>1</sup> and tip radius of 8 nm, operating at a resonance frequency of 300 kHz at ambient conditions. The surface parameters of the samples were also evaluated.

The force versus distance (F-D) response from ten different points on each non-exposed and exposed areas was also recorded with the same cantilever. The elastic modulus (Young's modulus) was calculated using the SPIP force curve analysis software by fitting a Hertz model to the force-distance curve. The hysteresis between approach and retract curves were corrected by the same software. Calculations performed with a Poisson's ratio value of 0.3 [72].

#### *2.4. Fractal Analysis*

The fractal characteristics of the exposed and non-exposed areas were quantified through the fractal dimensionality D*<sup>f</sup>* that describes the topology and the cavitation of a surface quantitatively. D*<sup>f</sup>* was derived from AFM images by four different algorithms, the cube counting, triangulation, variance and power spectrum methods, besides an algorithm provided by the AFM's "lake pattern" software (diSPMLab Vr.5.01). A detailed description of the concept and the specific methodologies of the different algorithms can be found in [27]. The D*<sup>f</sup>* was calculated for the four different methods using "Gwyddion, SPM data visualisation and analysis tool" [73]. The D*<sup>f</sup>* calculated with the four different algorithms follow the same trend, despite small dimensionality divergences coming up from systematic errors, because of the different converging speed of the fractal analytical approaches.

#### *2.5. Water Contact Angle (CA)*

The chemical modification of PAM surfaces following PAM surface laser irradiation was monitored by water CA surface measurements under ambient atmospheric conditions. Distilled water droplets with a volume of 0.5 μL were gently deposited onto the sample surface using a microsyringe. Water CAs on samples before and after irradiation and at different time intervals were measured using a CA measurement system (Digidrop, GBX, Romans sur Isere, Drôme, France) equipped with a CCD camera to capture lateral snapshots of a droplet deposited on top of the preselected area, suitable for both static and dynamic CA measurements. Droplet images captured at a speed of 50 frames/s. CA values were obtained via the Digidrop software analysis, approximating the tangent of the drop profile at the triple point (three-phase contact point). Three different CA measurements were taken from each sample at different sample positions to calculate the average values.

#### *2.6. White Light Reflectance Spectroscopy (WLRS)*

The WLRS measurements were performed by an FR-Basic, ThetaMetrisis™ (ThetaMetrisis SA, Athens, Greece) equipped with a VIS–NIR spectrometer (Theta Metrisis SA, Athens, Greece) having 2048 pixels detector and optical resolution of 0.35 nm. The beam of the light source comes from a white light halogen lamp, with a uniquely designed stable power supply and soft-start circuit, ensuring stable operation over time that is necessary for long time duration experiments. Software controls the instrument, performing the data acquisition and film thickness calculations. The PAM films were spin-coated on native oxide Si wafers and SiO2 layer on the top with a thickness of 2–3 nm.

#### *2.7. Random Walk Model*

The mean escape time of a water molecule confined in nanocavities was computed by applying different 3D random walk models with diverging numbers of water molecules, variable spherical size nanocavities, and entrance-escape hole sizes. Two different models of non-interactive and interactive water molecules inside the cavities were used. The first model, the non-interactive random walk model, uses molecular masses of zero volume and elastic collisions of the water molecule with the cavity wall and it records the sequence of positions of water molecules inside the spherical cavity until it gets back to the entrance-escape hole. The collision angle was varied randomly with a uniform distribution. The model calculates the total distance that molecules travel in the cavity before they escape from the entrance-escape hole. The mean escape time was calculated by considering that the molecule attains its kinetic energy after an elastic collision with the walls of the cavity. Therefore the kinetic energy transfer from the wall to the molecule should be equal with the thermal energy of the wall <sup>3</sup> <sup>2</sup> *kBT*, where *kB* is Boltzmann's constant.

The escape time from the entrance-escape hole for a non-interactive water molecule in the cavity is given by the equation:

$$t\_c = \frac{\sum\_{i=0}^{n} \left(\sqrt{2R^2(1-\sin\theta\_{i+1}\sin\theta\_i\cos(q\_{i+1}-q\_i) + \cos\theta\_{i+1}\cos\theta\_i)}\right) - 2R + (R\_0 + R\_n)}{\sqrt{\frac{3k\_BT\_A}{M\mu\_2O}}}\tag{1}$$

where *n* is the number of collisions in each run, *R* is the radius of the spherical cavity, (*R*, θ*i*,ϕ*i*) is the position of the molecule in the *i*th collision. The entrance and the exit point in the cavity wall are given in spherical coordinates (*R*0, θ0,ϕ*o*) and (*R*0, θ*n*,ϕ*n*), accordingly, and *MH*2*<sup>O</sup>* is the molecular mass of water.

The interactive random walk model records the sequence of positions of a specific molecule that enters a spherical cavity through the entrance-escape hole, alongside with the locations of a variable number of neighboring molecules trapped in the cavity, until it gets back to the entrance-escape hole. At first, because of non-thermal equilibrium between water molecules within the cavity, the molecules are placed inside the cavity in random positions with random velocities of uniform distribution between 0 and 3*kBTNA MH*2*<sup>O</sup>* m/s. The position of each molecule was recorded every 10<sup>−</sup>14s. The collision of each water molecule with the cavity wall and its neighbouring molecules is considered to be elastic. The collision angle was varied randomly with a uniform distribution. Contrary to the non-interactive model of zero-size molecules, the interactive model uses a spherical molecular diameter of 0.3 nm.

For every pair of the cavity size and entrance-escape hole, the random walk was run 102 times and the mean escape time was calculated. In addition, the mean-escape time distribution for different cavities and number of molecules was used to evaluate the thermodynamic state inside the cavity. The model was designed and run in MATLAB. 9.4.0.813654 (R2018a), The MathWorks Inc.; Natick, MA, USA.

#### **3. Results**

#### *3.1. Surface Analysis*

Commonly, four surface parameters, the surface roughness histogram, the area roughness, the area root mean square (RMS) and the maximum range characterize a surface and mean area values are plotted as functions of the laser pulse number or the laser fluence, Figure 1. The surface parameters are extracted from AFM images, Figure 2a–e.

**Figure 1.** Surface parameters of irradiated polyacrylamide (PAM) layers for a 2 μm × 2 μm area: (**a**) Mean *z-*height; (**b**) area roughness (Ra); (**c**) area RMS; (**d**) maximum range. The area roughness and area RMS parameters show an increment with laser pulses up to ~ 200 lp followed by a dip at 103 lp.

**Figure 2.** Atomic force microscopy (AFM) surface image of polyacrylamide (PAM) layers. Scan area <sup>2</sup> <sup>×</sup> <sup>2</sup> <sup>μ</sup>m2, laser fluence 250 J m−<sup>2</sup> per pulse: (**a**) non-irradiated PAM layer; (**b**) irradiated PAM layer with 100 laser pulses (lp), 25 kJ m−*2*; (**c**) 200 lp, 50 kJ m<sup>−</sup>2; (**d**) 103 lp, 250 kJm<sup>−</sup>2; (**e**) scan area 2.3 <sup>×</sup> 2.3 <sup>μ</sup>m2, 103 lp, 250 kJ m<sup>−</sup>2. The surface morphology is area size-dependent.

The surface roughness histogram, or average *z*-height, is the arithmetic mean defined as the sum of all height values divided by the number of data points <sup>|</sup>*Z*<sup>|</sup> = <sup>1</sup> *N* -*N i*=1 *Zi*. Next, the *Ra* (area roughness or roughness average) is the arithmetic mean of the height deviation from the image's mean value, *Ra* = <sup>1</sup> *n n i*=1 *Z<sup>i</sup>* <sup>−</sup> *<sup>Z</sup>* . The area RMS (*Rrms*) is the value defined as the square root of the mean 

*N*

value of the squares of the distance of the points from the image mean value: *Rrms* = 1 *N i*=1 (*Z* − *Zi*) 2 . Finally, the maximum range of *Zmax* is defined as the maximum value of *z-*heights. The surface parameter values (*z*-height, area roughness, area RMS, and maximum range) of photon exposed areas were more considerable compared to the non-irradiated ones. However, because surface parameters are area size-dependent (Figure 2d,e), they are utilized only for a comparative qualitative evaluation of area modification under 157 nm laser irradiation.

#### *3.2. Fractal Analysis of 157 nm Photon Processed PAM Polymeric Matrixes*

Because of statistical self-similarity between matrix space topology during a scaling-down route, there is a strong correlation between porosity, stage of cavitations and fractal dimensionality. Furthermore, in porous materials, the linear, area and volumetric porosities are alike, and therefore the 3D fractal dimensionality is similar to the area one. The dimensionality of a surface is equal to two for an ideal solid (Euclidean surfaces) and equal to three for completely porous surfaces with a fractal character. Areas with *Zi* values above a threshold *Z* height are known as "islands", while those with *Z*i's below the threshold height value are named as "lakes". AFM "island-lake structure" of non-irradiated and VUV irradiated 2 μm × 2 μm areas are shown in Figure 3.

**Figure 3.** Atomic force microscopy (AFM) image of "lake" (grey) and "island" (orange) for a fractal area of 2 <sup>×</sup> <sup>2</sup> <sup>μ</sup>m*2*: (**a**) Non-irradiated area; (**b**) irradiated area with 10<sup>3</sup> laser pulses.

The mean *Zi* heights of non-irradiated and the irradiated regions (103 laser pulses) were set at 0.75 and 1.94 nm respectively, and the irradiated areas show a diverging surface topology, in agreement with previous results [13,16,56,58]. Following a standard procedure, two parameters, the fractal dimensionality *Df* (which is a dimensionless number) and the "periphery to the area ratio" (PAR) are used to describe a set of "islands" or "lakes". Both parameters are linked to the surface roughness, cavitations and topological entropy [27,74]. PAR is the ratio of logarithms of the perimeter Π to the area *A*, where Π = α 1 + *Df A*(1−*Df*)/2. For assessing the state of cavitations, the fractal dimensionality is calculated by the partitioning, the cube counting, the triangulation, and the power spectrum algorithms [58]. Results are compared with those derived directly from the AFM "lake" pattern software, Figure 4a.

**Figure 4.** Fractal dimensionality, "lake" surface area and lake number vs. laser fluence. (**a**) Surface -fractal dimensionality calculated with four different fractal analytical methodologies. Colors, symbols and lines to assist the eye: blue squares for partitioning (PA), purple spheres for power spectrum (PS), dark yellow stars for cube counting (CC), green triangles for triangulation (TR), and brown pentagons for atomic force microscopy (AFM) "lake" pattern (LA). The five methods show a similar fractality trend vs. laser fluence; (**b**) "lake" surface area vs. the number of lakes at different number of laser pulses (laser fluence).

AFM images of 2 μm × 2 μm laser-irradiated areas were digitized to a 512 px × 512 px matrix, and then they processed with four different fractal algorithms. It is unveiled that fractal dimensionality, and thus cavitations, are functions of the laser photon fluence. All algorithms exhibit a similar trend of fractal dimensionality with the number of laser pulses (laser fluence), although the fractal dimensionality derived with the power spectra methodology seems slightly different, as expected [13,25]. The fractal dimensionality initially dips, attaining its minimum value around 500 laser pulses and then rises again with a small gradient up to 10<sup>3</sup> laser pulses, Figure 4a. For a constant "lake" surface area the number of "lakes", and thus the number of cavities, is a function of the laser pulses (laser fluence), Figure 4b. The number of "lakes" within a given surface area vs. the number of laser pulses is shown in Figure 5a. The number of "lake" areas rises almost exponentially with the number of laser pulses and small area "lakes" prevail over larger ones The fractal dimensionality vs. laser fluence has a non-monotonous complex structure. Small size features (1–10<sup>2</sup> nm2) are associated with nanocavity-like structures, Figure 5b. It is also confirmed that below 10<sup>3</sup> laser pulses small size features contribute to a high cavitation state because small size features have a higher dimensionality than large size structures, Figure 5c. On the contrary, large size features are prominent at 103 laser pulses, indicating the complexity of the associated processes. In addition, for the same number of laser pulses, small size cavitation prevails over larger ones, Figure 5a. The experimental results indicate that water confinement is rather associated with small cavitations, in agreement with WLRS measurements (vide infra).

**Figure 5.** Fractal parameters of polyacrylamide (PAM) at different laser fluence: (**a**) Number of "lakes" for different fractal size vs. the number of laser pulses *(n)*; (**b**) fractal dimensionality vs. the number of laser pulses *(n)* for different fractal size. The concentration of small size nanocavities increases at higher laser fluence; (**c**) fractal dimension vs. fractal size at a different number of laser pulses (lp).

#### *3.3. AFM-NI*

The mechanical response of 426 nm-thick PAM polymers was evaluated with nanoscale resolution via the F-D curves at different laser fluence, Figure 6a–d. Young's modulus and adhesion forces were also evaluated. Major non-monotonic modifications were recorded indicating substantial conformational changes of the surface energy of the PAM layers, Figure 6a. A diverging Young's modulus is attributed to accelerated ageing because of molecular bond breaks, accompanied by the formation of new carbon and carbonyl bonds [75–77]. A nonlinear alteration of the elastic modulus of PAM gel formulations during ten days ageing was also reported, revealing substantial changes of PAM's mechanical properties during irradiation [78].

The approach and retract curves follow different paths in all irradiating conditions, describing thus a system evolving out of equilibrium. The elastic modulus of the dry state hydrogels is significantly reduced after immersion to water, e.g., from 18 GPa to 3.3 MPa [79]. The Young's modulus of the non-irradiated PAM hydrogels depends on the hydration conditions, e.g., it decreases from 295 MPa in the dried state to 266 kPa in the fully hydrated state [78,80]. A Young's modulus of 2.84 GPa of uncured PAM hydrogel was recently attributed to the presence of pre-polymerized PAM oligomers [81]. Moreover, enhancement of Young's modulus to 4.84 GPa was predicted via an extension of the 3D polymeric networks at higher cross-linking states [81].

In this work, the non-irradiated PAM surfaces were thermally cured after being spin-coated on a silicon wafer; therefore, their mechanical properties are expected to deviate from those in the gel state. The average Young's modulus prior to and post-irradiation with 500 and 103 laser pulses was 2.0 ± 0.8 and 1.6 ± 0.42 and 2.55 ± 1.29 GPa, respectively, Figure 7a. The significant errors of Young's moduli at different points in the same sample are credited to various morphological heterogeneities and a progressive phase transformation to a relatively high carbonized state. Young's moduli follow a similar trend with fractal dimensionality vs. laser fluence, Figures 4a and 5b.

**Figure 6.** Typical force-distance (F-D) curves of polyacrylamide (PAM) thin layer surfaces (426 nm) irradiated with a different number of laser pulses (lp) (250 J m−<sup>2</sup> per laser pulse): (**a**) F-D curves of the non-irradiated layer; (**b**) F-D curves with 200 lp; (**c**) F-D curves with 300 lp; (**d**) F-D curves with 400 lp.

**Figure 7.** Young's modulus and adhesion force of irradiated polyacrylamide (PAM) surfaces showing enhanced carbonization at higher laser fluence. (**a**) Young's modulus; (**b**) adhesion force of a PAM surface irradiated at different laser fluence up to 250 J m−*2*; (**c**) Young's modulus and adhesion force column charts of PAM vs. laser fluence.

Additionally, the adhesive force, as it is measured during the penetrating state of AFM's tip, follows a similar trend with Young's modulus, Figure 7b,c. Because of diverging surface carbonization, the adhesive force drops from 130 to 26 nN between 0–400 laser pulses and then it rises again to ~ 150 nN for 103 laser pulses.

#### *3.4. Water Contact Angle (CA)*

Water CAs of PAM matrixes were recorded for varying photon fluence. The average CAs rise from 20◦ ± 2◦ to a saturated "plateau" at ~65◦± 7◦ after 200 laser pulses, Figure 8a.

**Figure 8.** Water contact angle (CA) vs. the number of laser pulses: (**a**) Water CA vs. the number of laser pulses. (**b**) column chart diagram of CA and fractal dimensionality vs. laser fluence. The mean correlation factor is −0.833; (**c**) column chart diagram of CA and area RMS of PAM vs. laser fluence. The mean correlation factor is 0.768 pointing to a strong correlation between fractal dimensionality, CA, and area RMS over a wide range of laser fluence; (**d**) column chart diagram of water CA at different time intervals. The almost similar slopes point to a uniform surface response at different VUV photon fluence. From Figures 4a, 5a,b, 7a,b and 8a the surface chemical modification is saturated at ~500 laser pulses, because of the low penetrating depth of the 157 nm laser photons, indicating the strong correlation between fractal dimensionality, CA, area RMS, Young's modulus and surface modification.

VUV photon processed PAM matrixes attain higher CA values, displaying thus a hydrophobic state, affirming that VUV irradiation has a primary effect on the surface wettability by altering both the material's physicochemical properties and surface nano/micro features, Figure 8a,b. In addition, the mean correlation factors of –0.833 and 0.768 between CA, Df and area RMS indicate a secure interconnection between surface morphology and *Df*, Figure 8b,c.

The wetting behavior was also analyzed with time, Figure 8d. The CAs of non-irradiated and irradiated with 100 laser pulses matrixes decrease consistently for 5 min. The dynamic CA of irradiated samples exhibits similar slope values, suggesting similar diffusion constants for different porosities, a fact that stresses out a picture of molecular water confinement in nanocavitations.

#### *3.5. White Light Reflectance Spectroscopy (WLRS)*

WLRS uses a broad-band light source and a spectrometer. The white light emitted from the light source is guided to a reflection probe through a number of optical fibers that incident vertically onto a sample. The sample consists of a stack of transparent and semi-transparent films placed over a reflective substrate. A reflection probe collects the reflected light through a fiber, directing it to the spectrometer. The light source beam interacts with the sample and generates a reflectance signal that is constantly recorded by the spectrometer. The number and the shape of interference fringes, registered in the CCD

of the spectrometer, depend on the thickness and the refractive index of the film(s). The fitting of the experimental spectrum is performed by using the Levenberg-Marquardt algorithm.

Water confinement is a source of volume strain and the relative surface deformation of the PAM polymeric matrixes caused by molecular water confinement is monitored by WLRS, Figure 9. The layer's thickness during water confinement and the relative surface deformation of the PAM layer prior and after water confinement in the irradiated surfaces is calculated from the phase shift and the superposition of amplitudes of the reflected light beam on the PAM surfaces. The white light beam records the surface strain within a cylindrical volume ~ *V* = 4.09 <sup>×</sup> 10−<sup>14</sup> m3, defined by the cross-sectional diameter of the white light beam of 3.5 <sup>×</sup> 10−<sup>4</sup> m and the thickness of the polymeric layer of 426 nm.

**Figure 9.** Principle of operation of white light reflectance spectroscopy (WLRS). (**a**) White light beam reflection in PAM surfaces. (**b**) experimental details and geometry of the reflected beams (**c**) surface strain response during water confinement in nanocavities. The contribution of the SiO2 layer at the interference pattern is negligible.

#### *3.6. Random Walk Model*

The comparison between calculations with the diffusion model [82] (water vapour diffusion coefficient in the air at normal pressure at 293 K is ~ 2.42 <sup>×</sup> <sup>10</sup>−<sup>5</sup> <sup>m</sup><sup>2</sup> <sup>s</sup><sup>−</sup>1) and the current non-interactive random walk model for 10<sup>3</sup> runs is shown in Tables A1 and A2. There is a noticeable difference between the two models for small size nanocavities because the diffusion constant for small size nanocavities is undetermined. The mean escape time from random walk models with the interactive model for the different number of confined molecules, cavity and the entrance-escape hole size is given in Figure 10 and Appendix A, Tables A3–A6.

**Figure 10.** Non-interactive random walk of one water molecule in a nanocavity. (**a**) The water molecule enters the cavity (yellow arrow) and then it collides with the inside walls of the spherical cavity (10 nm) several times (A–I points and blue lines) before escaping from the entrance-escape hole (3 nm, red line); (**b**) mean escape time for 10<sup>3</sup> different random walk runs in 1 nm (green), 10 nm (red), 102 nm (blue), and 10<sup>3</sup> nm (magenta) spherical cavities for different entrance-escape hole diameters (0.3 nm–500 nm). The y-axis represents a logarithmic time scale.

#### **4. Discussion**

#### *4.1. 157 nm Molecular Photodissociation of PAM Polymeric Chains*

Initially, surface and fractal analytical methods were used to typify surface cavitations crafted by 157 nm laser photons on PAM surfaces. Diverging texture morphologies of 2 μm × 2 μm PAM areas irradiated with 157 nm with a different number of laser pulses (photon fluence) are shown in Figure 2. Major conformational changes of photon processed PAM surfaces are evident through a diversity of fractal dimensionalities and surface parameters. Specifically, irradiated areas exhibit either a uniform or heterogeneous surface structural networks, according to the laser fluence (Figure 2b–e). Different size nano/microstructures including "hills and lakes" and fractal dimensionality diversity, nano aggregations (1–10<sup>3</sup> nm) and cavitations are shreds of evidence of significant photochemical topological matrix alterations (Figure 2c–e). Similar structures were previously observed on PAM hydrogel surfaces by cross-link concentration variations [78].

The energy of 157 nm laser photons is used to excite a molecular site in the polymeric chain from an electronic ground state (A) vibrational level to an excited electronic state (B) vibrational level, Figure 11a. The excitation is followed by a rapid internal transition to a dissociative (repulsive) state (Γ), and the parent molecule is disintegrated fast to a number of smaller size photo-fragments, Figure 11b,c. Consequently, surface irradiation with 157 nm laser photons modifies the morphology of the PAM matrix by creating defective molecular sites (DE) and micro/nano cavitations, Figure 11c. The volatile compounds, such as carbon-hydrogen monomers, ions, or larger polymer fragments, are moving away from the matrix at high velocities [37,41,42]. Carbon cluster (CL) formation (Figure 11d) also appears on the surface from re-deposited photo-dissociated products on the matrix (Figure 11e) and the photo-dissociated cycle profoundly modifies the chemical and the morphological features of the exposed polymeric surface. Because each 157 nm laser photon destroys via photo-dissociation one chemical bond of the polymeric matrix, Figure 11, it is reasonable to accept diverging cavitations and local nano-matrix volume diversities [37] in agreement with surface and fractal analysis results.

**Figure 11.** Nanocavitation by 157 nm laser photodissociation of polyacrylamide (PAM) matrixes: (**a**) Molecular photodissociation at 157 nm. Vertical arrows indicate photon transitions between two vibrational levels of the ground (A) and an excited electronic state (B). A transition from the excited to a repulsive electronic state (red curve) through an avoided crossing via a vibration state at the point (Γ) is very fast (< 1 ps) and breaks a molecular bond in the polymeric chain; (**b**) PAM surface irradiation with 157 nm photons; (**c**) (DE): a bond break is followed by molecular decomposition and nanocavitation; (**d**) (CL): possible recombination of carbon dissociative products, and cluster formation in the gas phase; (**e**) carbon cluster deposition on the polymeric matrix and possible structure of carbon nanocavitation.

#### *4.2. Trapping of Water Molecules in Nanocavities*

Water molecular confinement is a complex issue having great importance in life sciences [8,83,84]. Because the nature of the H-bond undergoes a diverging number of structural conformations at surface boundaries, and also, inside tiny spaces, the water confinement is hindered with the long-range fluctuations of both the water networks [85] and single molecules, Figure 12a. The dynamics and the time scale of interactions in confined spaces are notably diverging with the spatial scale length and the local geometries. For example, terahertz spectroscopy of water molecules in gemstone nanocavities identify quantum water molecular tunnelling through a six-well potential caused by the interaction of the water molecule with the cavity walls [86]. The length and directionality of H-bonds are highly susceptible to the type of confining surfaces and the degree of confinement [87]. In addition, atomistic molecular dynamics simulations of dipolar fluids confined to spherical nanocavities of radii ranging from 1 to 4 nm reveal a surprisingly small Kirkwood correlation factor in water, but not so in a dipolar fluid because of ultrafast relaxation of the total dipole moment time correlation function of water [6]. The static dielectric constant of confined water exhibits a strong dependence on the size too, with a remarkably low value even at 3 nm and a slow convergence to the bulk value because of surface-induced long-range orientation correlations [6]. The trapped water experiences peculiar thermodynamic properties and under confinement unexpectedly shows high pressures (GPa) [88]. Because the mean escape time is independent of the number of molecules, inside the cavity, Figure 12, the average mechanical pressure exerted on the walls of the cavity is independent of the number of molecules. Therefore, the molecular state inside the cavity deviates from an equilibrium thermodynamic state because the escape time in "equilibrium thermodynamic" cavities should be pressure dependent. In addition, the extensive thermodynamic properties of confined molecules in tiny spaces might be disproportional to the volume of the system, and instead, they could be higher-order functions of size and shape [89–92].

It is also known that for tiny empty spaces, equal or below the atomic dimensions, stressing fields are emerging from electromagnetic vacuum fluctuations. The repulsive Casimir stress σ*c*(*R*, *t*) within a conductive spherical cavity of radius *<sup>R</sup>* at time *<sup>t</sup>* was calculated to be 0.09*c* <sup>8</sup>π*R*4(*t*) [93]. For balancing the Casimir stress with the atmospheric pressure, a 10 nm spherical cavity has the proper size, if the equation of ideal gases is used. On the other hand, for a spherical cavity in thermal equilibrium with the matrix that bears a small hole on its surface connecting the inside with the outside volume space of the cavity, in the case of a pressure balance outside and inside the cavity, four molecules are confined in the cavity, if the equation of ideal gases is used as a first approximation. However, for an average molecular thermal energy of *Ekin* <sup>∼</sup> *kT* and a spherical volume *V* of 5.34 <sup>×</sup> 10−<sup>25</sup> m3, the volume stress exerted on the walls of the cavity from the collisions of a molecule with the walls of a cavity should be of the order of <sup>∼</sup> *kT <sup>V</sup>* = 7.9 <sup>×</sup> 104 Pa, a value that almost matches the atmospheric pressure outside the cavity. By increasing the number of molecules inside a small cavity, the volume stress should be increased proportionally to the number of molecules because of mechanical collisions with the cavity walls. Consequently, extremely high pressures should be developed inside small cavities, in agreement with [88]. In addition, for small size cavities, there is rather an entropic than an energy barrier that balances the flow kinetics of molecules in and out the cavity [94,95]. Previous studies indicated that in the case of elastic collisions in the cavity, the molecular dynamics depends on the number of molecules inside the cavity and is either frictionless (inertial dynamics), moderately frictional (Langevin dynamics), or strongly frictional (Brownian dynamics) [96], where the noise term should be properly taken into account. For small entrance-escape holes, the number correlation function generally decays exponentially with time. The transition rate in the frictionless limit is given by a microcanonical ensemble. As the strength of the friction is increased, the rate of collisions approaches the diffusive limit without a Kramers turnover. In this work, random-walk calculations of non-interactive and interactive molecules in the cavity for 103 and 102 runs, point to variable escape times of water molecules from different size nanocavities (1–10<sup>3</sup> nm) and entrance-escape holes (0.3–5 <sup>×</sup> 102 nm), Figure 10 and Appendix A, Tables A1 and A2. For the same cavity size the mean

escape time falls with large entrance-escape hole size, extended over a wide dynamic range of 10 orders of magnitude. The mean escape time for the interactive model is independent of the number of molecules inside small cavities and interestingly, the mean escape time fluctuates a great deal inside tiny cavities, Figure 12a–d and Tables A3–A6; suggesting that the system is in non-thermal equilibrium, a state that dominates the statistics and the dynamics of molecules inside small cavities.

**Figure 12.** Mean escape time for a different number of non-interactive molecules calculated for 103 runs. (**a**) Fluctuations of mean molecular escape time are prominent for small size cavities, while remains constant for a large number of molecules; (**b**–**d**) mean escape time for 5, 10, 15 and 20 nm spherical cavities with different entrance-escape holes and number of molecules in the cavity. The mean escape time is independent of the number of molecules and is a function only of the cavity geometry (diameter and entrance-escape hole).

On the contrary, for larger size cavities the volume stress is diminished and the state of molecules inside the cavities approaches the thermodynamic limit, in agreement with [86]. he gradient of the mean escape time, the mean escape time distribution and the mean distance a molecule travels inside a cavity before its escape through the entrance-escape holes is diverging for very small size cavities (1 nm) cavities, Figures 12a, 13 and 14a.

The mean escape time and the mean travelling distance retain a constant ratio for large size cavities (10–103 nm), while the ratio deviates for small ones, suggesting again a non-equilibrium thermal state and large fluctuations inside small size cavities, Figure 14. Most interesting, the local fluctuations of mean molecular escape time are prominent for small size cavities and small number of molecules, while the mean molecular escape time remains steadier for a larger number of molecules, Figure 12a–d, in agreement with molecular dynamic results [6,85,89–92,96] and general nanothermodynamic considerations [31]. In addition, the mean escape time distribution of molecules for both the non-interactive and interactive models (1 and 150 molecules) inside different size small cavities reveals a rather non-thermal distribution and the absence of a thermal equilibrium state inside the cavities, Figures 13 and 15.

**Figure 13.** Distribution of mean escape times for 10<sup>2</sup> random walk runs of a water molecule (non-interactive model) for different cavity geometries (cavity size D and entrance-escape hole size h). (**a**) D = 15 nm, h = 1 nm; (**b**) D = 10 nm, h = 1 nm; (**c**) D = 5 nm, h = 1 nm; (**d**) D = 15 nm, h = 2 nm; (**e**) D = 10 nm, h = 2 nm (**f**) D = 5 nm, h = 1 nm. Distributions are non-normal and besides that skewness and long tails indicate non-equilibrium processes inside the cavities.

**Figure 14.** Mean escape time and mean travelling distance of a molecule within cavities of different geometries. (**a**,**b**) Gradient of the mean escape time and the mean distance that a molecule travels in the cavity before it escapes with different entrance-escape holes is diverging for very small size cavities (1, 10 nm); (**c**,**d**) mean escape time and the travelling distance gradients are constant for large size cavities (102, 103 nm), suggesting a non-thermal equilibrium state and large fluctuations for small size cavities.

**Figure 15.** (**a**–**c**) Mean escape time distribution of 150 interactive molecules for different cavity and entrance-escape hole size for 10<sup>2</sup> runs; (**d**) best-fitting of Figure 15a is for a log-normal distribution; (**e**) mean escape time vs. the cavity ratio *<sup>h</sup> <sup>D</sup>* . The time differentiation of molecular movements inside and outside cavities is provided by the dependence of the ratio <sup>h</sup> <sup>D</sup> on the waiting time τ, which for <sup>D</sup>=10<sup>2</sup> and 10<sup>3</sup> nm cavities goes as a power law of the waiting time with exponent <sup>−</sup>0.5.

The mean escape time of water molecules in the cavity is diverging from the mean collision time (~70 ns) and the thermal de Broglie time outside the cavity by many orders of magnitude, according to the geometry of the cavity. Therefore, the "molecular time" inside the cavities "flows differently" than the physical time of the events on the PAM surface. This spatial "time differentiation" across a thin layer in the PAM surface is responsible for the excess entropic potential, arising from a state of ordered arrangements between nanocavities and the water molecular ensemble of fixed molecular length near the PAM surface after 157 nm irradiation. A further sign of time differentiation of molecular movements inside and outside cavities is provided by the dependence of the ratio *<sup>h</sup> <sup>D</sup>* on the waiting time. It goes as a power law of the waiting time with exponent −0.5, Figure 15. Finally, the configured number of microstates from confinement properly counts for the experimental surface entropy deviation during the trapping of water molecules (vide infra).

#### *4.3. Stress-Strain Response in Polymeric Matrixes-A Relation between Physics and Mechanics*

During the interaction of a system with a thermal bath, the exchange of energy appears in the form of heat or work. The first law of thermodynamics states that the infinitesimal change of the heat absorbed by a system per unit volume *Q* is equal to the increase of the differential of its internal energy change *U* minus the infinitesimal change of the work done on the system *W*:

$$
\delta Q = d\mathcal{U} - \delta \mathcal{W} \tag{2}
$$

The departure from a state of equilibrium will be governed by the second law of thermodynamics and the direction of entropy change. Any thermodynamic system is described by its extensive and intensive variables, *U*, *S*, σ*ij*, *eij*, *T*, where *U* is the internal energy, *S* is the entropy, σ*ij* and ε*ij* stand for second-rank stress and strain tensors acting on the volume element *dV* and *T* is the temperature of the system. Following Cauchy's theory of stress, the equilibrium of elements requires the balance of forces acting on a volume element, [97]

$$t\_i^u = \frac{d\overrightarrow{F}\_i}{dA} = \sigma\_{ji} n\_j \tag{3}$$

*t n <sup>i</sup>* is the *i th* component of a traction force *d* → *F* along the *i-*axis, along a unit vector <sup>→</sup> *n* perpendicular on an infinitesimal surface area *d* → *A*, σ*ij* is the (*i*, *j*) component of the stress tensor, and *nj* is the *jth* component of the <sup>→</sup> *n* vector that is perpendicular to the surface area *d* → *A*.

The force component *Fi* acting on the volume element *dV* and bounded by the surface area *A* is given by the equation

$$F\_i = \int\_A t\_i^\eta \, dA + \int\_V X\_i dV \tag{4}$$

where *Xi* are the body forces (e.g., the weight of the volume element *dV*). From Equation (4) and the Gauss theorem, the surface integral of the components of the traction forces is transformed into a volume integral

$$F\_i = \int\_V \left(\frac{\partial \mathbf{t}\_i^n}{\partial \mathbf{x}\_i} + X\_i\right) dV = \int\_V \left(\frac{\partial \sigma\_{ji}}{\partial \mathbf{x}\_j} + X\_i\right) dV \tag{5}$$

For the infinitesimal theory of elasticity the strain tensor ε*ij* is reduced to a linear form

$$
\varepsilon\_{ij} = \left(\frac{\partial \mathbf{u}\_i}{\partial \mathbf{x}\_j} + \frac{\partial \mathbf{u}\_j}{\partial \mathbf{x}\_i}\right) \tag{6}
$$

where the displacement gradient of the volume element *dV* along one axis <sup>∂</sup>*ui* <sup>∂</sup>*xj* is a second-order tensor, and the derivative of *ui* is a second rank tensor

$$\frac{\partial u\_i}{\partial \mathbf{x}\_j} = \frac{1}{2} \left( \frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i} \right) + \frac{1}{2} \left( \frac{\partial u\_i}{\partial \mathbf{x}\_j} - \frac{\partial u\_j}{\partial \mathbf{x}\_i} \right) = \frac{1}{2} \varepsilon\_{ij} - \frac{1}{2} \omega v\_{ij} \tag{7}$$

where <sup>ω</sup>*ij* = <sup>∂</sup>*ui* ∂*xj* <sup>−</sup> <sup>∂</sup>*uj* <sup>∂</sup>*xi* is the rotational skew-symmetric tensor.

The infinitesimal displacement *dui* along the *j* direction and for ω*ij* = 0 is

$$du\_{i} = \frac{\partial u\_{i}}{\partial \mathbf{x}\_{j}} d\mathbf{x}\_{j} = \frac{1}{2} \varepsilon\_{ij} d\mathbf{x}\_{j} \tag{8}$$

Using Gauss' theorem and Equations (4) and (8) the total mechanical work is done on the system by the traction and the body forces

$$\mathcal{W} = \int\_{A} (t\_i^n \, du\_i) dA + \int\_{V} (\mathcal{X}\_i du\_i) dV = \int\_{A} \sigma\_{j\bar{l}} n\_j d\mu\_i dA + \int\_{V} \mathcal{X}\_i d\mu\_i dV \tag{9}$$

For an isothermal and isobaric process during sorption, the infinitesimal mechanical work δ*W* along the direction *ni* outwards the boundaries of a surface *A* enclosing the volume *dV* = *dxidA* is equal with

$$
\delta \delta \mathcal{W} = (\frac{1}{2} \sigma\_{i\bar{j}} n\_{\bar{j}} \varepsilon\_{i\bar{j}} + X\_i d u\_i) dV \tag{10}
$$

Neglecting the body forces *Xi* , the mechanical work is

$$
\delta \mathcal{W} = \frac{1}{2} \sigma\_{ij} n\_j \varepsilon\_{ij} dV \tag{11}
$$

A superposition of the three normal stress components uniformly distributed over the volume *dV* is used to calculate the strain in a given direction, say the *z*-axis:

$$
\varepsilon\_z = \frac{1}{E(n)} (\sigma\_x + \sigma\_y + \sigma\_z) \tag{12}
$$

where *E*(*n*) is the modulus of elasticity in tension or Young's modulus.

$$
\varepsilon\_x = -\nu \frac{\sigma\_z}{(n)}, \quad \varepsilon\_y = -\nu \frac{\sigma\_z}{(n)} \tag{13}
$$

in which ν is a constant called Poisson's ratio, equal to ~ 0.3. Using Equations (12) and (13) we obtain the strain equations along the principal axes

$$
\varepsilon\_{\mathcal{X}} = \frac{1}{E(n)} [\sigma\_{\mathcal{X}} - \nu(\sigma\_{\mathcal{Y}} + \sigma\_z)] \tag{14}
$$

$$
\varepsilon\_{\mathcal{Y}} = \frac{1}{E(n)} [\sigma\_{\mathcal{Y}} - \nu(\sigma\_z + \sigma\_x)]
$$

$$
\varepsilon\_z = \frac{1}{E(n)} [\sigma\_z - \nu(\sigma\_x + \sigma\_y)]
$$

For relatively thick and isotropic layers and for small linear and homogeneous elastic deformations along the axes, the normal stresses are equal and Equation (14) reads as

$$
\varepsilon\_z = \frac{1 - 2\nu}{E(n)} \sigma\_z \tag{15}
$$

Because a contraction of a volume element in the *z*-direction in an elastic medium is accompanied by lateral extensions

$$
\varepsilon\_{\overline{z}} = -\varepsilon\_{\overline{x}} = -\varepsilon\_{\overline{y}} \tag{16}
$$

and using Equation (16) in Equation (11) the mechanical work along the principal axes is

$$
\delta \mathcal{W} = \frac{1}{2} \sigma\_{\overline{z}} \varepsilon\_{\overline{z}} dV \tag{17}
$$

From the first and second law of thermodynamics, the mechanical work *W* done on a system is

$$\mathcal{W} = T\,\mathcal{S} - \mathcal{U} + \mu\_i \mathcal{N}\_i \, + \,\forall \mathcal{A} \tag{18}$$

and the infinitesimal mechanical work per unit volume before and after sorption is

$$\frac{(n)}{2(1-2\nu)}\epsilon\_z^2 dV = -\delta \mathcal{U} + T\delta \mathcal{S} + \mu\_l \delta \mathcal{N}\_l + \delta(\Psi \mathcal{A}) \tag{19}$$

where δ*U*, δ*S*, and δ*Ni* stand for the variations of the internal energy, entropy and the number of bind water molecules on active polymeric sites prior and after sorption, *ez* is the strain of the volume element along the *z-*axis from the confinement of water molecules, μ*<sup>i</sup>* is the chemical potential of δ*Ni* absorbed particle on the polymeric matrix. The term δ(Ψ(*n*)) = δ[γ(*n*) + (*n*)*<sup>s</sup>* - *nkdAk*] is the algebraic sum of the surface energy δ(γ(*n*)) plus the elastic energy strain (*n*)*<sup>s</sup>* of the nanocavities per unit area, from surface irradiation with some *n* laser pulses at 157 nm. The last term is zero under isothermal and isobaric sorption, δ(Ψ(*n*)) = 0, [97]. The term δ*Ni* is relatively negligible because of a small number of absorbed water molecules. Finally, the strain of a volume *dV* along the *z*-axis before and after water confinement is given by the equation

$$
\varepsilon\_z = \left(\frac{2(1-2\nu)(T\delta S - dII)}{(n)dV}\right)^{1/2} \tag{20}
$$

#### *4.4. Internal Energy Variation during Molecular Water Confinement*

Besides confinement, molecular water molecules are attached to polymeric sites via electric dipole interactions. The internal energy variation is the outcome of the photon-escalating number of active dipole binding sites spawn via VUV matrix irradiation, Figure 11.

For surface irradiation with *n* laser pulses, the internal energy variation δ*Ub* is given by the relation [13,98]

$$
\delta l l\_b = -\lambda l \text{N}\_b(n) < \Phi > = -\lambda l \text{N}\_b(n) \frac{5d\_{xy}^4}{64\pi^4 \varepsilon\_0^2 \varepsilon\_1^2 k\_B T r^6} \tag{21}
$$

*Nb*(*n*) is the number of water molecules attached on the active sites, λ is the probability that a water molecule will overcome an energy threshold barrier and bind in a photon-activated dipole binding site and *l* is the average number of adsorbed water molecules on each photon-activated dipole binding site. *dxy*= *er* is the *x, y* component of the electric dipole moment between a water molecule and a photo-activated dipole binding site. *<sup>e</sup>* = 1.6 <sup>×</sup> <sup>10</sup>−<sup>19</sup> <sup>C</sup> is the electron charge and *<sup>r</sup>* <sup>∼</sup> 0.1 nm is a mean separating distance between a water molecule and a photon-activated dipole binding site, ε<sup>0</sup> is the vacuum permittivity equal to 8.85 <sup>×</sup> 10−<sup>12</sup> Fm−1, <sup>ε</sup><sup>1</sup> <sup>∼</sup> 80 is the relative electric permittivity of the polymer-water system, *kB* = 1.38 <sup>×</sup> <sup>10</sup>−<sup>23</sup> J K−<sup>1</sup> is Boltzmann's constant and T <sup>=</sup> 300 K is the absolute temperature. Because the energy of each laser pulse at 157 nm is 28 mJ, the number of photons carried in one laser pulse is *<sup>n</sup>* = 2.26 <sup>×</sup> 1016 photons/laser pulse, and this number equals to the number of photon-activated dipole binding sites. Each VUV photon at 157 nm dissociates one molecular bond and creates one active site on the polymeric matrix, Figure 11. For a 1.12 <sup>×</sup> 10−<sup>4</sup> m2 cross-sectional area of the 157 nm laser beam and 426 nm layer thickness, it is found that 4.73 <sup>×</sup> <sup>10</sup><sup>26</sup> photon-activated dipole binding sites are generated within 1 m<sup>3</sup> per laser pulse. For a cross-section area of the WLRS beam of 4.90 <sup>×</sup> <sup>10</sup>−<sup>8</sup> m2 and 426 nm matrix thickness, the volume *dV* of the polymeric matrix occupied by the white light beam is 4.09 <sup>×</sup> 10−<sup>14</sup> m3 and thus the total number of active binding sites per laser pulse within the volume occupied by the white beam is *Nb* = 2.31 <sup>×</sup> 1013. From Equation (21) <sup>&</sup>lt; <sup>Φ</sup> <sup>&</sup>gt; <sup>≈</sup> 1.51 x 10−<sup>23</sup> J for <sup>λ</sup>*<sup>l</sup>* = 0.05 (vide infra) and finally <sup>δ</sup>*Ud* = 1.43 <sup>×</sup> <sup>10</sup>−<sup>11</sup> J.

#### *4.5. Entropic Energy Variation during Molecular Confinement*

Photon-induced nanocavitations are also responsible for surface entropic variation at the boundary between inside and outside nanocavity areas. The entropic variation at the interphase has its origin from time differentiation between the inside and outside areas of nanocavities. Actually, the mean collision time (~70 ns) of water molecules outside the nanocavities within the matrix or near the surface is specified by the laws of ideal gases. On the contrary, the mean escape time of water molecules inside the nanocavities is specified by the hole geometry and the interplay between entrance-escape hole size with cavity diameter. The waiting times follow an inverse power law behavior because thermal equilibrium does not apply in tiny spaces, Figures 12–16 and Tables A1 and A2. In addition, VUV laser irradiation locally ablates the polymeric material, crafting photon-guided "hill-lake" morphologies. The total number of lakes (cavities) vs. the surface area follows a power-law behavior. In this dependency, the number of laser pulses is present through a pre-factor term, Figure 4b. A schematic layout of this modified interphase between photon processed PAM surface and water vapor domain is illustrated in Figure 16.

**Figure 16.** Schematic layout of the interphase between the photon processed polyacrylamide PAM surface and the water vapor domain. (**a**) Thermal equilibrium domain. Reference time and space scales are determined by the mean collision time *tcol* between the water and air molecules and the entropy of the ideal gases and the mean collision distance. The entropy *S1* is given by the Sackur–Tetrode equation for the ideal gases [99]; (**b**) local fluctuations domain. Nanocavitations on the surface with confined molecules. The time scale is determined by the mean escape time τ of water molecules. The entropy *S2* in this domain is determined by the number of microstates Ω*(Nb(n), Nc(n), E*α*)*, which specify a state of ordered arrangements between nanocavities in one hand and molecular water ensembles of fixed molecular length near the surface on the other; (**c**) volume matrix domain.

Random movements in such complex landscapes could be modeled in the frame of continuous time random walk [100,101] by also taking into account the fractal properties of the modified polymeric material [102]. We leave this challenging task for future work where both analytical and extensive numerical calculations combined with experimental results will be presented. Because different water molecules enter and escape the nanocavitations, the number of different microstates Ω*(Nb(n), Nc(n), E*α*)* per unit time is specified by the frequency of water molecules confined in the nanocavities. The rate of visits is regulated by the mean escape time of water molecules. *n* and *Nc*(*n*) is the number of laser pulses and nanocavities, respectively, *Na* is the number of water molecules outside the nanocavities with energy *E*<sup>α</sup> and *Nc(n)* is the number of nanocavities. The number of microstates is equal to the number of indistinguishable permutations *Na*(*n*), *Nb* (*n*) + *Nc*(*n*) between the number of water molecules *Na* and the number of nanocavities *Nc*(*n*) and the photon-induced dipole binding sites *Nb*(*n*)

$$\begin{array}{c} \Omega(\_{b}(n), N\_{\mathfrak{c}}(n), \, \_{a}) = \{N\_{\mathfrak{a}}(n), N\_{\mathfrak{b}}\left(n\right) + N\_{\mathfrak{c}}(n)\} \\ = \frac{N\_{\mathfrak{b}}!}{(N\_{\mathfrak{b}}\left(n\right) + N\_{\mathfrak{c}}\left(n\right))! (N\_{\mathfrak{a}} - (N\_{\mathfrak{b}}\left(n\right) + N\_{\mathfrak{c}}\left(n\right))!} \text{ for } \ N\_{\mathfrak{b}}\left(n\right) + N\_{\mathfrak{c}}\left(n\right) < N\_{\mathfrak{a}} \end{array} \tag{22}$$

$$\begin{array}{ll} \Omega\_{\text{b}}(n), \ N\_{\text{c}}(n)\_{\text{'}\text{ a}}) = \{\mathcal{N}\_{\text{b}}\,(n) + \mathcal{N}\_{\text{c}}(n), \mathcal{N}\_{\text{d}}(n)\} \\ = \frac{(\mathcal{N}\_{\text{b}}\,(n) + \mathcal{N}\_{\text{c}}(n))!}{\mathcal{N}\_{\text{a}}!((\mathcal{N}\_{\text{b}}\,(n) + \mathcal{N}\_{\text{c}}(n)) - \mathcal{N}\_{\text{d}})!} \quad \text{for} \; \mathcal{N}\_{\text{b}}\,(n) + \mathcal{N}\_{\text{c}}(n) > \mathcal{N}\_{\text{a}} \end{array} \tag{23}$$

To arrive in Equations (22) and (23) it is considered that only one water molecule per unit time is either trapped to a specific nanocavity or attached a photon-induced polar binding site. An escalating number of nanocavities is building up in the matrix after each laser pulse, and the ratio of the sum of the number of dipole binding sites and nanocavities to the number of water molecules near the surface is a function of the number of laser pulses

$$\mathbf{x}(n) = \frac{\mathbf{N}\_b(n) + \mathbf{N}\_c(n)}{\mathbf{N}\_a} \tag{24}$$

From Equations (22)–(24), the entropy from the confinement and the attachment of water molecules in nanocavities and photon-induced polar adhesion binding sites is [13,26]

$$\delta S = k\_{\rm B} \ln \Omega(\mathbf{n}), \ N\_{\rm c}(\mathbf{n}), \ n) = k\_{\rm B} \ln \langle N\_{\rm d}(\mathbf{n}), N\_{\rm b}(\mathbf{n}) + N\_{\rm c}(\mathbf{n}) \rangle, \ x(n) < 1 \tag{25}$$

$$\text{VoS} = k\_{\text{B}} \ln \Omega(\text{s}(n), \text{N}\_{\text{c}}(n), \text{ }\_{\text{a}}) = k\_{\text{B}} \ln \text{[N}\_{\text{b}}\text{(n)} + \text{N}\_{\text{c}}(n), \text{N}\_{\text{a}}(n)], \text{ } \text{x}(n) > 1 \tag{26}$$

Using Equation (22), Equation (25) read as

$$\delta S = k\_B \left[ \ln(\mathcal{N}\_b!) - \ln(\left( \mathcal{N}\_b \left( n \right) + \mathcal{N}\_c(n) \right))! - \ln(\mathcal{N}\_b - \left( \mathcal{N}\_b \left( n \right) + \mathcal{N}\_c(n) \right))! \right. \tag{27}$$

By using Stirling's formula

$$
\ln \text{N!}! = \text{N!} \ln \text{N} - \text{N} \tag{28}
$$

Equation (27) transforms to

$$\begin{aligned} \delta S &= k\_{\rm B} \left[ N\_{\rm d} \ln(N\_{\rm d}) - N\_{\rm d} - \left( N\_{\rm b} \left( n \right) + N\_{\rm c}(n) \right) \ln(N\_{\rm b} \left( n \right) + N\_{\rm c}(n) \right) + \left( N\_{\rm b} \left( n \right) + N\_{\rm c}(n) \right) \\ &- \left( N\_{\rm d} - \left( N\_{\rm b} \left( n \right) + N\_{\rm c}(n) \right) \ln(N\_{\rm d} - \left( N\_{\rm b} \left( n \right) + N\_{\rm c}(n) + \left( N\_{\rm d} - \left( N\_{\rm b} \left( n \right) + N\_{\rm c}(n) \right) \right) \right) \right. \\ &= k\_{\rm B} \left[ N\_{\rm d} \ln(N\_{\rm a}) - \left( N\_{\rm b} \left( n \right) + N\_{\rm c}(n) \right) \ln(N\_{\rm b} \left( n \right) + N\_{\rm c}(n) ) \right. \\ &- \left( N\_{\rm d} - \left( N\_{\rm b} \left( n \right) + N\_{\rm c}(n) \right) \ln(N\_{\rm d} - \left( N\_{\rm b} \left( n \right) + N\_{\rm c}(n) \right)) \right. \end{aligned} \tag{29}$$

Using Equation (24), Equation (29) becomes

$$\delta S = k\_B(N\_b(n) + N\_c(n)) \left\{ \ln \left( \frac{1 - \mathbf{x}(n)}{\mathbf{x}(n)} \right) - \frac{1}{\mathbf{x}(n)} \ln(1 - \mathbf{x}(n)) \right\}, \ \mathbf{x}(n) < 1 \tag{30}$$

Similarly from Equations (23) and (24)

$$\delta S = k\_B(N\_b(n) + N\_t(n)) \left\{ \ln \left( \frac{\mathbf{x}(n)}{\mathbf{x}(n) - 1} \right) + \frac{1}{\mathbf{x}(n)} \ln(\mathbf{x}(n) - 1) \right\}, \qquad \mathbf{x}(n) > 1 \tag{31}$$

In the case of a constant attachment of water molecules in the photon-induced binding sites, Equations (30) and (31) are modified accordingly

$$\delta S = k\_B N\_c(n) \left\{ \ln \left( \frac{1 - y(n)}{y(n)} \right) - \frac{1}{y(n)} \ln(1 - y(n)) \right\}, \ y(n) < 1 \tag{32}$$

$$\delta S = k\_B N\_b(n) \beta(n) \left\{ \ln \left( \frac{y(n)}{y(n) - 1} \right) + \frac{1}{y(n)} \ln(y(n) - 1) \right\}, \qquad y(n) > 1 \tag{33}$$

where

$$\text{N}\_{\text{c}}(n) = \frac{\text{N}\_{\text{c}}(n)}{\text{N}\_{\text{a}}}, \text{ N}\_{\text{c}}(n) = \beta(n)\text{N}\_{\text{b}}(n) \tag{34}$$

In the case where some nanocavities are not visited by the water molecules, then y(n) > 1 . This condition is fulfilled under the current experimental configuration, Figure 14. For β(*n*) ∼ 0.2, *Nb* = 2.31 <sup>×</sup> 1013, *<sup>y</sup>*(*n*) = 2, the entropic energy at 300 K is *kBT*δ*<sup>S</sup>* = 1.31 <sup>×</sup> 10−<sup>8</sup> J, which is almost three orders of magnitude larger than δ*Ud*. Equations (32), (33) properly reflect the extensive variable character of the entropy as it should be.

#### *4.6. Surface Strain from the Confinement of Water Molecules*

Using Equations (20) and (32)–(34) the surface strain following 157 nm laser irradiation takes the form

$$\varepsilon\_{z} = \left(\frac{N\_b(n)}{E(n)dV}\right)^{\frac{1}{2}} [-\lambda l < \Phi > + k\_B T \beta(n)] \ln\left(\frac{1 - y(n)}{y(n)}\right) - \frac{1}{y(n)} \ln(1 - y(n))]^{\frac{1}{2}} \quad y(n) < 1\tag{35}$$

*Nanomaterials* **2020**, *10*, 1101

$$\varepsilon\_{z} = \left(\frac{N\_b(n)}{E(n)dV}\right)^{\frac{1}{2}} [-\lambda l < \Phi > + k\_B T \beta(n)] \ln \frac{y(n)}{y(n) - 1} + \frac{1}{y(n)} \ln(y(n) - 1)]^{\frac{1}{2}} \quad y(n) > 1 \tag{36}$$

Equations (35) and (36) shape the main result. The equations relate the surface strain ε*<sup>z</sup>* and Young's modulus *E*(*n*) with the number of nanocavities, the photon-induced dipole binding sites in the matrix, and the water vapor molecules near nanocavities. For the current experimental configuration *y*(*n*) > 1. From Equation (36), the strain at 400 laser pulses is ~ 0.1 in agreement with the experimental results of Figure 17.

**Figure 17.** Relative surface deformation (strain) of the 426 nm polyacrylamide (PAM) layers measured with white light reflectance spectroscopy (WLRS) at different 157 nm irradiating conditions of the PAM matrix and relative humidity (RH). The solid lines at different RH represent the best fit of Equation (36) to the experimental data. The black lines at 80 % RH are the fittings for different λl values of 0, 0.05, 0.1 and 0.2. The best fit (orange line) respectively, is for 0 ≤ λl < 0.05 suggesting a small contribution to the relative surface deformation from electric dipole attachment of water molecules to active binding sites in the PAM matrix and a substantial contribution from the confinement of water molecules in nanocavities, Equation (36).

By fitting Equation (36) to the experimental data of Figure 17, the functional dependence of *y(n)* on the number of photons *n* is determined at different relative humidity (RH) values. Because *y(n)* is proportional to the number of dipole binding sites and the number of nanocavities *Nc (n)*, *y(n)* is a measure of the surface carbonization. By using a linear functional for both *y(n)* and *E(n),* the best fit of Equation (36) to the experimental data of Figure 17 for relative humidity 80% is for β(*n*) = 0.2 and 0 ≤ λ*l* < 0.05. The above fitting values suggest a small and large contribution from the electric dipole interactions and the entropic variation in surface strain, respectively. From Equation (36), the surface strain is proportional to the square root of the number of nanocavities and the concentration of the water molecules (RH) and inversely proportional to the square root of Young's modulus of the surface, in agreement with the experimental results of Figure 17. Finally, the entropic jump, probed by WLRS, trails the confinement of water molecules in nanocavities, while the deep physical root of surface entropy variation originates from the different "time flow and scales" and the validity and invalidity of thermal equilibrium outside and inside the nanocavities, respectively, Figures 15–17.

The experimental approach permits to monitor water confinement on surfaces, including biological ones.

#### **5. Conclusions**

Water molecules confined inside laser photon crafted nanocavities on PAM polymeric matrixes are in a state of non-thermal equilibrium. The mean escape time of water molecules from the nanocavities diverges from the mean collision time of water molecules outside the nanocavities (ideal gas state). The time differentiation inside and outside the nanocavities reveals an additional state of ordered arrangements between nanocavities and molecular water ensembles of fixed molecular length near the surface. The configured number of microstates correctly counts for the experimental surface entropy deviation during molecular water confinement.

**Author Contributions:** Conceptualization, V.G., E.S. and A.-C.C.; methodology, E.S., V.G. and A.-C.C.; software, V.G.; validation, V.G., E.S., A.-C.C., Z.K., E.B., D.C., G.A.K. and M.C.; data curation, V.G., E.S., Z.K. and M.C.; visualization, V.G., A.-C.C. and E.S.; formal analysis, V.G., E.B., E.S. and A.-C.C.; resources, E.S., V.G., Z.K., A.C.-C. and M.C.; investigation, V.G.,E.S., Z.K. and M.C.; writing-original draft preparation, V.G. and E.S.; writing-review and editing, V.G., E.S., E.B., D.C. and A.-C.C.; supervision, E.S.; project administration, E.S.; funding acquisition, A.-C.C. and E.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work was funded under the frame of the projects "ELI – LASERLAB Europe Synergy, HiPER and IPERION-CH.gr" (MIS 5002735) which is implemented under the Action "Reinforcement of the Research and Innovation Infrastructure", funded by the Operational Programme "Competitiveness, Entrepreneurship and Innovation" (NSRF 2014–2020) and co-financed by Greece and the European Union (European Regional Development Fund and "Advanced Materials and Devices" (MIS 5002409) which is implemented under the "Action for the Strategic Development on the Research and Technological Sector", funded by the Operational Programme "Competitiveness, Entrepreneurship and Innovation" (NSRF 2014–2020) and co-financed by Greece and the European Union (European Regional Development Fund).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**


**Table A1.** Mean escape time (τ) of a given molecule, errors and total traveling distance of the confined molecule in a spherical cavity (cavity diameter D of 1 nm and 10 nm) for different entrance–escape hole diameters (h) calculated by the diffusion and the non-interactive random walk models.

**Table A2.** Mean escape time (τ) of a given molecule, errors and total traveling distance of the confined molecule in a spherical cavity (cavity diameter 10<sup>2</sup> nm and 103 nm) for different entrance–escape hole diameters (h) calculated by the diffusion and the non-interactive random walk models.



**Table A3.** Mean escape time (τ) and associated errors of different number molecules confined in a 5-nm spherical cavity for different entrance–escape hole diameters (h) calculated by the interactive random walk models for 102 runs.

**Table A4.** Mean escape time (τ) and associated errors of different number molecules confined in a 10-nm spherical cavity for different entrance–escape hole diameters (h) calculated by the interactive random walk models for 102 runs.



**Table A5.** Mean escape time (τ) and associated errors of different number molecules confined in a 15-nm spherical cavity for different entrance–escape hole diameters (h) calculated by the interactive random walk models for 102 runs.

**Table A6.** Mean escape time (τ) and associated errors of different number molecules confined in a 20-nm spherical cavity for different entrance–escape hole diameters (h) calculated by the interactive random walk models for 102 runs.


#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Quantum Characteristics of a Nanomechanical Resonator Coupled to a Superconducting LC Resonator in Quantum Computing Systems**

#### **Jeong Ryeol Choi \* and Sanghyun Ju**

Department of Physics, Kyonggi University, Yeongtong-gu, Suwon, Gyeonggi-do 16227, Korea; shju@kyonggi.ac.kr

**\*** Correspondence: jrchoi@kyonggi.ac.kr or choiardor@hanmail.net; Tel.: +82-31-249-1320

Received: 26 November 2018; Accepted: 21 December 2018; Published: 24 December 2018

**Abstract:** The mechanical and quantum properties of a nanomechanical resonator can be improved by connecting it to a superconducting resonator in a way that the resonator exhibits new phenomena that are possibly available to novel quantum technologies. The quantum characteristics of a nanomechanical resonator coupled to a superconducting resonator have been investigated on the basis of rigorous quantum solutions of the combined system. The solutions of the Schrödinger equation for the coupled system have been derived using the unitary transformation approach. The analytic formula of the wave functions has been obtained by applying the adiabatic condition for time evolution of the coupling parameter. The behavior of the quantum wave functions has been analyzed for several different values of parameters. The probability densities depicted in the plane of the two resonator coordinates are distorted and rotated due to the coupling between the resonators. In addition, we have shown that there are squeezing effects in the wave packet along one of the two resonator coordinates or along both the two depending on the magnitude of several parameters, such as mass, inductance, and angular frequencies.

**Keywords:** nanomechanical resonator; superconducting resonator; wave function; unitary transformation; Hamiltonian; probability density; adiabatic condition; quantum solution

#### **1. Introduction**

A rapidly developing field in nano-based science and technology is optomechanics which deals with the interaction of light with a mechanical motion [1]. Especially, optomechanics combined with nanomechanical resonators can be practically applied to a broad scientific domain such as quantum information processing [2], biological sensing [3], wave detections [4], measurements of mechanical displacement [5], and quantum metrologies [6]. Such optomechanics developed in the quantum regime with low phonon occupation states will play a major role in future-oriented quantum technologies with nanodevices.

The investigation of nanodevices regarding their application in quantum information science, including quantum computing, is a promising research topic. In particular, research into nanomechanical resonators in which the parameters are dependent on time is quite necessary for the advancement of the quantum information technology [7–11]. Now, it is possible to design quantum computing devices with a reliable architecture for multi-qubit operations in the GHz-frequency range by coupling mechanical resonators to Josephson phase qubits [12]. Onchip-integrated hybrid systems, i.e., mechanical resonators combined with phase qubits which are composed of Josephson-junction superconducting circuits, can be used as a compact quantum information storage with a high quality factor [13]. Recent advances in nanotechnology in the past decade enabled the fabrication of nanomechanical resonators, of which quality factors are high in the desired frequency ranges.

By connecting nanomechanical resonators to the superconducting resonators, it is possible to enhance the properties of nanomechanical resonators. Thanks to this, many relevant experiments have been developed and renewed, such as, for example, preparing quantum ground state in resonators [10], a frequency up/down conversion [11], and maintaining longer coherence times [14]. In particular, coherent feedback control of the nanoresonators can be used for cooling them to the zero-point temperature. The cooling of a mechanical system to the ground-state has already been achieved in several laboratories [15,16].

As the size of the resonator reaches below a critical value that is the limiting one from a quantum mechanical point of view, there emerge distinguishing quantum features that are totally absent in the classical world [5]. Quantum behaviors of a nanomechanical resonator were observed through its sideband laser-cooling over the quantum ground state [17]. A deeper understanding of quantum characteristics of nanomechanical resonators, where the device-size is within the Heisenberg uncertainty principle limit, is crucial for utilizing them in quantum computing systems. We investigate quantum properties of a nanomechanical resonator coupled to a superconducting resonator. This is important for providing theoretical knowledge as background information for manipulating quantum computing processes. Furthermore, this research may help to achieve robust quantum computations on the basis of the restriction of the decoherence and noise.

#### **2. Materials and Methods**

Cooling of nanomechanical resonators can be achieved by the methods of sideband cooling or measurement-based feedback cooling [18]. The former is classified as coherent feedback controlling for a nanomechanical system, that can be carried out by coupling it to an auxiliary one such as an optical cavity and a superconducting oscillator, whereas the systems in the letter case are controlled by means of the feedback technique with the data which are continuously measured from homodyne detection. In this work, we consider sideband cooling using a high-frequency superconducting LC oscillator as an auxiliary mode in order to control the motion of the nanomechanical resonator. For more details of the nanomechanical-resonator design and the related mechanics, refer to Refs. [10,11].

To see quantum features of the system, we first need to derive relevant quantum solutions in a rigorous way. We introduce the Hamiltonian for describing the quantum dynamics of the nanomechanical resonator coupled to the superconducting resonator. The Hamiltonian involves a coupling term which is associated with the coupling of the two resonators. Due to not only such a coupling term but the time-dependence of the parameters as well, it may be not an easy task to solve the Schrödinger equation on the basis of the conventional separation of variables method. For this reason, we will derive quantum solutions by making use of another method which is the unitary transformation method [7,8,19].

It may be convenient that we unfold quantum theory after mathematically transforming the system described by a time-dependent Hamiltonian into a simple one that can be easily managed. By introducing a unitary operator, we will transform the original system to a system composed of two decoupled harmonic oscillators of which quantum solutions are well known. By inversely transforming the quantum solutions associated to the transformed system, it is possible to obtain the complete quantum solutions in the (original) system. This is the main strategy used in this work for deriving quantum solutions of the system whose Hamiltonian is a fairly complicated form. We will analyze quantum properties of the system in detail on the basis of the quantum solutions evaluated in such a way.

#### **3. Results and Discussion**

#### *3.1. Hamiltonian and the Unitary Transformation*

Superconducting systems can be used to improve the properties of other quantum systems by connecting them to the target systems [20]. From there, we can explore new phenomena which

could possibly bring about the deveopment of novel quantum technologies for quantum information processing. For instance, a sideband cooling of a nanoresonator is possible through its modulation via coupling it to a superconducting LC oscillator. In this case, we can regard the nanoresonator as the target device and the superconducting LC resonator as an auxiliary one.

The Hamiltonian that describes the nanomechanical resonator combined with the superconducting LC resonator is given by [10,11]

$$
\hat{H} = \hbar\omega(\hat{a}^\dagger\hat{a} + 1/2) + \hbar\Omega(\hat{b}^\dagger\hat{b} + 1/2) + \hbar\lambda(t)(\hat{a} + \hat{a}^\dagger)(\hat{b} + \hat{b}^\dagger),
\tag{1}
$$

where *a*ˆ and ˆ *b* are annihilation operators in the nanomechanical resonator and the superconducting resonator, respectively, and *ω* is the frequency of the nanoresonator, while Ω is the frequency of the superconducting resonator. Here, we assume that the parameter *λ*(*t*) is a slowly varying function so that we can apply the adiabatic theorem. Although we are interested in the quantum description of the nanomechanical resonator coupled to the superconducting resonator with Equation (1), the Hamiltonian in Equation (1) can also be used for other purposes such as the gauge field theory of two superconducting resonators which are tunably coupled to each other [21] and quantum simulation of bosonic modes utilizing superconducting circuits [22].

Because the operators *a*ˆ and ˆ *b* are represented in the form

$$\mathfrak{A}\_{\parallel} = \sqrt{\frac{m\omega}{2\hbar}}\mathfrak{X} + \frac{i\mathfrak{P}\_{\parallel}}{\sqrt{2\hbar m\omega}}\mathfrak{X} \tag{2}$$

$$
\hat{\mathfrak{H}}\_{\parallel} = \sqrt{\frac{L\Omega}{2\hbar}}\hat{\mathfrak{H}} + \frac{i\mathfrak{H}\_{\parallel}}{\sqrt{2\hbar L\Omega}}\,,\tag{3}
$$

with *p*ˆ*<sup>x</sup>* = −*ih*¯ *∂*/*∂x* and *p*ˆ*<sup>q</sup>* = −*ih*¯ *∂*/*∂q*, Equation (1) can be rewritten as

$$
\hat{H} = \frac{\not p\_x^2}{2m} + \frac{1}{2}m\omega^2 \hat{\mathfrak{x}}^2 + \frac{\not p\_q^2}{2L} + \frac{1}{2}L\Omega^2 \hat{\mathfrak{y}}^2 + 2\lambda(t)\sqrt{mL\omega\Omega}\hat{\mathfrak{x}}\hat{\mathfrak{y}}.\tag{4}
$$

Due to the coupling term (the last term) in the above equation, it may be somewhat difficult to treat this system from a quantum mechanical point of view. In order to overcome this, we will mathematically decouple the two sub-systems by means of the unitary transformation method [7,8,19]. For this purpose, we introduce a unitary operator of the form

$$\begin{split} \mathcal{U} &= \ \exp\left[\frac{i}{4\hbar}(\not p\_x \pounds + \not p\_y)\ln\left(\sqrt{m/L}\right)\right] \\ &\times \exp\left[\frac{i}{4\hbar}(\not p\_q \not p + \not p\_q)\ln\left(\sqrt{L/m}\right)\right] \\ &\times \exp\left[-\frac{i\theta(t)}{\hbar}(\not p\_x \not p - \not p\_q \not t)\right], \end{split} \tag{5}$$

where

$$\theta(t) = \frac{1}{2} \tan^{-1} \left( \frac{4\lambda(t)\sqrt{\omega \Omega}}{\Omega^2 - \omega^2} \right). \tag{6}$$

By transforming *H*ˆ in Equation (4) using this operator:

$$
\hat{H}' = \hat{\mathcal{U}}^{-1}\hat{H}\hat{\mathcal{U}} - i\hbar\hat{\mathcal{U}}^{-1}\frac{\partial\hat{\mathcal{U}}}{\partial t} \tag{7}
$$

*Nanomaterials* **2019**, *9*, 20

we have the Hamiltonian in the transformed system as

$$\hat{H}' = \frac{\hat{p}\_x^2 + \hat{p}\_q^2}{2\sqrt{mL}} + \frac{1}{2}\sqrt{mL}\omega\_1^2(t)\hat{\mathfrak{x}}^2 + \frac{1}{2}\sqrt{mL}\omega\_2^2(t)\hat{q}^2 + \dot{\lambda}(t)\beta(t)(\hat{p}\_x\hat{q} - \hat{p}\_q\hat{x}),\tag{8}$$

where

$$
\omega\_1^2(t) = -\omega^2 \cos^2 \theta(t) + \Omega^2 \sin^2 \theta(t) - 4\lambda(t)\sqrt{\omega \Omega} \cos \theta(t) \sin \theta(t), \tag{9}
$$

$$
\omega\_2^2(t) = -\omega^2 \sin^2 \theta(t) + \Omega^2 \cos^2 \theta(t) + 4\lambda(t)\sqrt{\omega \Omega} \cos \theta(t) \sin \theta(t), \tag{10}
$$

$$\beta(t) = -\frac{2\sqrt{\omega\Omega}}{\Omega^2 - \omega^2 + 16\omega\Omega\lambda^2(t)/(\Omega^2 - \omega^2)}.\tag{11}$$

According to the adiabatic condition which is that *λ*(*t*) is a sufficiently slowly time-varying function, we can neglect the last term in Equation (8), leading to

$$
\hat{H}' \simeq \frac{\not p\_x^2 + \not p\_q^2}{2\mu} + \frac{1}{2}\mu\mathbf{w}\_1^2(t)\hat{\mathfrak{x}}^2 + \frac{1}{2}\mu\mathbf{w}\_2^2(t)\hat{\mathfrak{q}}^2,\tag{12}
$$

where *<sup>μ</sup>* <sup>=</sup> <sup>√</sup>*mL*. Thus, the two sub-systems are decoupled as can be confirmed from the above Hamiltonian. The quantum treatment of the system relying on the transformed Hamiltonian, Equation (12), may be much simpler than that relying on the original Hamiltonian given in Equation (4). Two independent classical equations of motion that correspond to *H*ˆ are given by

$$\frac{d^2\mathbf{x}}{dt^2} + \mathbf{w}\_1^2(t)\mathbf{x} = \mathbf{0},\tag{13}$$

$$\frac{d^2q}{dt^2} + \mathbf{w}\_2^2(t)q = 0.\tag{14}$$

In the next section, we will derive the quantum wave solutions associated to the Hamiltonian given in Equation (12). By using the unitary relation between the wave functions in the transformed system and those in the original system, the quantum wave solutions in the original system will be obtained and analyzed.

#### *3.2. Quantum Wave Solutions*

From the knowledge of the formulae of quantum wave functions in the transformed system, we can obtain the wave functions in the original system because the two systems are connected by a unitary operator. If we denote the wave functions in the transformed system associated with Equations (13) and (14) as *ψ <sup>n</sup>*(*x*, *t*) and *ψ*˜ *l* (*q*, *t*), respectively, they can be divided into kernel and phase parts such that

$$\psi\_n'(\mathbf{x}, t) \quad = \quad \phi\_n'(\mathbf{x}, t) \exp[i\alpha\_n(t)],\tag{15}$$

$$\begin{array}{rcl}\tilde{\psi}\_l'(q,t) &=& \tilde{\phi}\_l'(q,t)\exp[i\tilde{\alpha}\_l(t)], \end{array} \tag{16}$$

where *αn*(*t*) and *α*˜*l*(*t*) are time-dependent phases.

Let us write the corresponding Schrödinger equation as

$$i\hbar \frac{\partial \Psi\_{n,l}'(\mathbf{x}, q, t)}{\partial t} = \hat{H}' \Psi\_{n,l}'(\mathbf{x}, q, t), \tag{17}$$

where Ψ *n*,*l* (*x*, *q*, *t*) are wave functions in the transformed system, that are of the form

$$
\Psi\_{n,l}'(\mathbf{x}, q, t) = \psi\_n'(\mathbf{x}, t)\bar{\psi}\_l'(q, t). \tag{18}
$$

*Nanomaterials* **2019**, *9*, 20

Because the transformed system is composed of the two decoupled harmonic oscillators, we can easily identify the corresponding quantum solutions, Equation (18). Then, the wave functions Ψ*n*,*l*(*x*, *q*, *t*) in the original system are obtained from such solutions through the use of the unitary relation Ψ*n*,*l*(*x*, *q*, *t*) = *U*ˆ Ψ *n*,*l* (*x*, *q*, *t*). According to this, the wave functions in the original system are represented as

$$
\Psi\_{n,l}(\mathbf{x}, q, t) = \psi\_n(\mathbf{x}, q, t)\tilde{\psi}\_l(\mathbf{x}, q, t),
\tag{19}
$$

where

$$\psi\_n(\mathbf{x}, q, t) \quad = \ \phi\_n(\mathbf{x}, q, t) \exp[i\alpha\_n(t)],\tag{20}$$

$$\bar{\psi}\_l(\mathbf{x}, q, t) \quad = \quad \bar{\phi}\_l(\mathbf{x}, q, t) \exp[i\mathbb{I}\_l(t)],\tag{21}$$

while

$$
\phi\_n(\mathbf{x}, q, t) = \hat{\mathcal{U}} \phi'\_n(\mathbf{x}, t), \tag{22}
$$

$$
\tilde{\Phi}\_l(\mathbf{x}, q, \mathbf{t}) = \hat{\mathcal{U}} \tilde{\Phi}\_l'(q, \mathbf{t}).\tag{23}
$$

Let us now further see for the case that the coupling parameter is a positive real constant, *λ*(*t*) = *λ*0. In this case, w*<sup>i</sup>* (*i* = 1, 2) and *θ* become constants. As a consequence, the corresponding quantum solutions are easily identified to be

$$\mathfrak{a}\_n(t) \quad = \ -(n+1/2)\mathfrak{w}\_1 t,\tag{24}$$

$$\mathbf{u}\_l(t) \quad = \ -(l+1/2)\mathbf{w}\_2 t,\tag{25}$$

$$\phi\_n'(\mathbf{x}) \quad = \left(\frac{\sqrt{k\_1/\pi}}{2^n n!}\right)^{1/2} H\_\mathbb{II}\left(\sqrt{k\_1}\mathbf{x}\right) \exp\left[-k\_1 \mathbf{x}^2/2\right],\tag{26}$$

$$\left|\vec{\phi}\_{l}^{\prime}(q)\right| = \left(\frac{\sqrt{k\_{2}/\pi}}{2^{l}l!}\right)^{1/2} H\_{l}\left(\sqrt{k\_{2}q}\right) \exp\left[-k\_{2}q^{2}/2\right],\tag{27}$$

where *ki* = *μ*w*i*/¯*h*, and *Hn*(*l*) are Hermite polynomials. The probability density, |Ψ *n*,*l* (*x*, *q*, *t*)| 2, in the transformed system is illustrated in Figure 1 with the choice of (*n*, *l*) = (3, 5) under the limit that the coupling parameter is constant. This density is not deformed because the Hamiltonian of the transformed system does not involve the coupling term.

**Figure 1.** The probability density, |Ψ *n*,*l* (*x*, *q*, *t*)| 2, in the transformed system with the choice of (*m*, *L*) = (1.0, 1.0) under the limit *λ*(*t*) = *λ*0. This is associated to the wave functions, Equation (18) with Equations (15), (16), and (24)–(27). We used (*n*, *l*) = (3, 5), (*ω*, Ω) = (0.5, 0.49), *h*¯ = 1, and *λ*<sup>0</sup> = 0.1.

*Nanomaterials* **2019**, *9*, 20

According to the relations, Equations (22) and (23), the eigenfunctions in the original system can be derived to be

$$\phi\_{\rm n}(\mathbf{x}, q, t) \quad = \left(\frac{\sqrt{k\_1/\pi}}{2^n n!}\right)^{1/2} H\_{\rm n}\left(\sqrt{k\_1}Q\_1\right) \exp\left[-k\_1 Q\_1^2/2\right],\tag{28}$$

$$\tilde{\Phi}(\mathbf{x}, q, t) \quad = \left(\frac{\sqrt{k\_2/\pi}}{2^l l!}\right)^{1/2} H\_l\left(\sqrt{k\_2}Q\_2\right) \exp\left[-k\_2 Q\_2^2/2\right],\tag{29}$$

where

$$Q\_1 \quad = \left(\frac{m}{L}\right)^{1/4} \cos\theta(t)\mathbf{x} - \left(\frac{L}{m}\right)^{1/4} \sin\theta(t)q\_\prime \tag{30}$$

$$Q\_2 \quad = \left(\frac{m}{L}\right)^{1/4} \sin \theta(t) \mathbf{x} + \left(\frac{L}{m}\right)^{1/4} \cos \theta(t) q. \tag{31}$$

Notice that the wave packets in the original system in the *x*-*p* coordinate have more or less been deformed depending on the values of parameters such as *m* and *L*, and rotated in proportion to *θ* from those in the transformed system. If *θ* is positive, the direction of rotation in the *x*-*p* plane is clockwise. The probability densities, |Ψ*n*,*l*(*x*, *q*, *t*)| 2, are illustrated in Figures 2–4. By comparing these figures with that in the transformed system represented in Figure 1, we can confirm the effects of the coupling on the behavior of the wave functions. Figure 2 is for several different values of *ω* with the choice of (*m*, *L*) = (1.0, 0.5) under the condition *ω* > Ω. The rotation angle *θ* can be evaluated from Equation (6) and is given by −0.76 rad for Figure 2A, −0.18 rad for Figure 2B, and −0.05 rad for Figure 2C.

We see from Figure 2 that the uncertainty of *x* is relatively small than that of *q*, which means that the wave packet is squeezed along the *x* coordinate. Such a squeezing effect becomes large as the difference *ω* − Ω increases. On the other hand, Figure 3, which is for the case that *ω* − Ω is negative, shows the squeeze of waves along the *q*-coordinate.

For the case of Figure 4 which is depicted under (*m*, *L*) = (0.5, 2.0), the wave packets exhibit *q*-squeezing. However, a weak *x*-squeezing for the waves also takes place as *ω* increases (see Figure 4C).

For the case that *λ*(*t*) is not a constant, it is necessary to use the quantum theory of time-dependent harmonic oscillators [9,23,24] in order to manage the Schrödinger equation, Equation (17) with Equation (12). According to that theory, the phases and the eigenfunctions in the transformed system are given in terms of time functions as [24]

$$a\_n(t) \quad = \ -(n+1/2)\gamma\_1(t),\tag{32}$$

$$
\Phi\_l(t) = -(l+1/2)\gamma\_2(t),
\tag{33}
$$

$$\phi\_n'(\mathbf{x}, t) \quad = \left(\frac{\sqrt{\kappa\_1(t)/\pi}}{2^n n!}\right)^{1/2} H\_\mathbb{R}\left(\sqrt{\kappa\_1(t)}\mathbf{x}\right) \exp\left[-\frac{\kappa\_1(t)}{2}\left(1 - \frac{i\dot{s}\_1(t)}{\dot{\gamma}\_1(t)s\_1(t)}\right)\mathbf{x}^2\right],\tag{34}$$

$$\left|\tilde{\phi}\right|(q,t) = \left(\frac{\sqrt{\kappa\_2(t)/\pi}}{2^l l!}\right)^{1/2} H\_l\left(\sqrt{\kappa\_2(t)}q\right) \exp\left[-\frac{\kappa\_2(t)}{2}\left(1-\frac{i\dot{\kappa}\_2(t)}{\dot{\gamma}\_2(t)s\_2(t)}\right)q^2\right],\tag{35}$$

where *κ<sup>i</sup>* = *μγ*˙ *<sup>i</sup>*/¯*h*, while the time functions *si*(*t*) and *γi*(*t*) satisfy

$$\ddot{s}\_i + \mathbf{w}\_i^2(t)s\_i - \mathbf{C}\_i^2 / s\_i^3 = 0,\tag{36}$$

$$\dot{\gamma}\_i = \mathcal{C}\_i / s\_i^2 \, \tag{37}$$

where *Ci* are arbitrary real constants.

**Figure 2.** The probability density |Ψ*n*,*l*(*x*, *q*, *t*)| <sup>2</sup> in the original system for several different values of *ω* with the choice of (*m*, *L*) = (1.0, 0.5) under the limit *λ*(*t*) = *λ*0. This is associated to the wave functions given in Equation (19) with Equations (20), (21), (24), (25), (28), and (29). The values of (*ω*, Ω) are (0.5, 0.49) for (**A**), (1.0, 0.49) for (**B**), and (2.0, 0.49) for (**C**). We used (*n*, *l*) = (3, 5), ¯*h* = 1, and *λ*<sup>0</sup> = 0.1.

**Figure 3.** This figure is the same as Figure 2C, but for a different choice of angular frequencies. The values of (*ω*, Ω) are (0.49, 2.0).

From a minor evaluation through Equations (22) and (23), we have the wave solutions in the original system:

$$\phi\_n(\mathbf{x}, q, t) \quad = \left(\frac{\sqrt{\kappa\_1(t)/\pi}}{2^n n!}\right)^{1/2} H\_\text{il}\left(\sqrt{\kappa\_1(t)}Q\_1\right) \exp\left[-\frac{\kappa\_1(t)}{2}\left(1 - \frac{i\mathbf{s}\_1(t)}{\dot{\gamma}\_1(t)\mathbf{s}\_1(t)}\right)Q\_1^2\right],\tag{38}$$

$$\Phi\_l(\mathbf{x}, q, t) \quad = \left(\frac{\sqrt{\kappa\_2(t)/\pi}}{2^l l!}\right)^{1/2} H\_l\left(\sqrt{\kappa\_2(t)} Q\_2\right) \exp\left[-\frac{\kappa\_2(t)}{2} \left(1 - \frac{i\dot{s}\_2(t)}{\dot{\gamma}\gamma(t)s\_2(t)}\right) Q\_2^2\right],\tag{39}$$

where *Qi* are given by Equations (30) and (31), but in terms of *λ*(*t*) which is not a constant.

The overall wave function is given by

$$\Psi(\mathbf{x}, q, \mathbf{t}) = \sum\_{n=0}^{\infty} \sum\_{l=0}^{\infty} c\_{n,l} \Psi\_{n,l}(\mathbf{x}, q, \mathbf{t}), \tag{40}$$

where *cn*,*<sup>l</sup>* are complex numbers that yield ∑<sup>∞</sup> *<sup>n</sup>*=<sup>0</sup> <sup>∑</sup><sup>∞</sup> *<sup>l</sup>*=<sup>0</sup> |*cn*,*l*| <sup>2</sup> = 1. Thus, we have obtained the full wave function of the system which is given by Equation (40) with Equations (19)–(21), (32), (33), (38), and (39). This wave function is crucial in unfolding quantum theory of the system and can be used to investigate diverse quantum characteristics that the system exhibits. We can use the wave function in estimating various quantum variables, such as energy eigenvalues, expectation values and fluctuations of the canonical variables, uncertainty products, Wigner distribution functions, and so on.

**Figure 4.** The probability density |Ψ*n*,*l*(*x*, *q*, *t*)| <sup>2</sup> in the original system for several different values of *ω* with the choice of (*m*, *L*) = (0.5, 2.0) under the limit *λ*(*t*) = *λ*0. This is associated to the wave functions given in Equation (19) with Equations (20), (21), (24), (25), (28), and (29). The values of (*ω*, Ω) are (0.5, 0.49) for (**A**), (1.0, 0.49) for (**B**), and (2.0, 0.49) for (**C**). We used (*n*, *l*) = (3, 5), ¯*h* = 1, and *λ*<sup>0</sup> = 0.1.

#### **4. Conclusions**

The quantum properties of the nanomechanical resonator coupled to a superconducting resonator via a small time-varying coupling constant have been investigated. We have used an adiabatic condition under the assumption that the time-variation of the coupling parameter is sufficiently slow. Due to the presence of the coupling term in the Hamiltonian, we cannot develop quantum theory for *x* and *q* coordinates independently in the original system. Hence, by means of the unitary transformation approach, we have decoupled *x* and *q* coordinates from each other in the expression of the Hamiltonian.

Through the transformation performed using a unitary operator, the Hamiltonian became a manageable one. Not only did the coupling term no longer appear in the transformed Hamiltonian, but the transformed system has also become much more simplified compared to the original one. More precisely speaking, the transformed system is composed of two independent harmonic oscillators with time-dependent angular frequencies. We accordingly have easily identified quantum solutions in the transformed system. By inverse transformation of those quantum solutions, we have obtained complete quantum solutions in the original system.

The exact wave functions derived here can be used for evaluating various quantum quantities of the coupled resonators, such as expectation values and fluctuations of the canonical variables, uncertainties, and energy eigenvalues, which are necessary for identifying the quantum characteristics of the system. It is also possible to derive Wigner distribution function of the system, which serves in describing signal processing, from the wave functions given in the text. Wigner distribution function can be used not only in estimating quantum corrections from classical statistical mechanics but also in demonstrating nonclassicalities of the system through quantum probability distribution. Terraneo et al. used Wigner distribution functions for the purpose of developing an efficient way for extracting information from the wave functions of quantum algorithms associated with quantum computation [25].

The probability densities which are the absolute square of the wave functions were illustrated in detail in the limit that *λ* is a constant. The wave packets in the *x*-*q* plane are rotated and deformed compared to those in the transformed system. The rotation direction is counterclockwise when *ω* − Ω is positive. Due to the coupling of the two sub-systems, the wave packets of the system underwent *x*-squeezing, *q*-squeezing, or both depending on the values of *m*, *L*, and the frequency difference *ω* − Ω. For the situation of *x*-squeezing (*q*-squeezing), the wave amplitude of the nanomechanical (the superconducting) resonator is small and, as a consequence, the nanomechanical (the superconducting) resonator has less energy. The energy of the nanomechanical resonator flows to the superconducting resonator, and vice versa [10,18]. From this process, the superconducting resonator extracts information about the nanomechanical resonator in order to control the nanoresonator.

The techniques for sideband cooling, frequency conversion, and state-swapping through coherent control of the resonator are important in quantum information systems, especially, in the realization of quantum computers. For the purpose of developing such techniques, the use of nanomechanical resonators is more effective than the use of nanophotonic resonators [11]. It is also noticeable that quantum data bus in quantum computers could be designed using nanomechanical resonators through their controllable coupling with qubits [2,26].

Our analysis regarding the wave functions of the system is useful for understanding the quantum behavior of the combined system. If we consider that theoretical analysis of a quantum system starts from the wave functions rigorously derived from the Schrödinger equation, the wave functions developed in this work are the basic tool for elucidating quantum properties of the system. Exact analyses of the quantum behaviors of nanoresonators based on such clarified quantum properties are necessary for realizing efficient quantum information processing.

The advantage of information processing and physical simulations using quantum computing devices rather than classical computers is that they make it possible to solve time-consuming problems within a polynomial time, such as factorizing large numbers which is classically very difficult to solve. The theoretical research for demonstrating the characteristics of quantum devices and their operations are important stepping stones for the development of quantum information science. While the system we treated here is a completely solvable one using our method, we also expect that quantum characteristics of more complicated namomecanical systems, including nanoresonators described by higher-order

nonlinearity such as Duffing nonlinearity [27] and/or newly designed nanophotonic resonators [28,29], would be analyzed in the near future using the same method. Nanomechanical resonators are used for the detection of quantum states, spins, thermal fluctuations, etc., whereas nanophotonic resonators as passive optical components are mainly used for detecting light–matter interactions.

The coupled system investigated in this work can be used as a basic component of quantum computing systems. Quantum computation as a next generation technology should be developed together with other quantum information science such as quantum communication and quantum cryptography. Moreover recent development of neural computation through neural network with multi-core optical fibers [30,31] enhances the performance of the technology in quantum information science.

As a next task, the investigation of geometric phases that appear in the wave functions when parameters vary over time may be a good research topic which can be fulfilled on the basis of the quantum theory developed in the present work. Geometric phase can be applied not only to fundamental physics [23] but also to various next generation quantum technologies, such as quantum computing [32], interferometric imaging of microstructures [33], and beam steering in virtual/augmented reality displays [34].

**Author Contributions:** J.R.C. proposed the idea of the research. J.R.C. and S.J. co-wrote the manuscript. The graphics in the text have been prepared by J.R.C. All authors read and approved the final manuscript.

**Funding:** This research was supported by the Basic Science Research Program of the year 2018 through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No.: NRF-2016R1D1A1A09919503).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


c 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **On-Chip Real-Time Chemical Sensors Based on Water-Immersion-Objective Pumped Whispering-Gallery-Mode Microdisk Laser**

#### **Qijing Lu 1, Xiaogang Chen 1, Liang Fu 1, Shusen Xie 1,\* and Xiang Wu 2,\***


Received: 21 February 2019; Accepted: 19 March 2019; Published: 24 March 2019

**Abstract:** Optical whispering-gallery-mode (WGM) microresonator-based sensors with high sensitivity and low detection limit down to single unlabeled biomolecules show high potential for disease diagnosis and clinical application. However, most WGM microresonator-based sensors, which are packed in a microfluidic cell, are a "closed" sensing configuration that prevents changing and sensing the surrounding liquid refractive index (RI) of the microresonator immediately. Here, we present an "open" sensing configuration in which the WGM microdisk laser is directly covered by a water droplet and pumped by a water-immersion-objective (WIO). This allows monitoring the chemical reaction progress in the water droplet by tracking the laser wavelength. A proof-of-concept demonstration of chemical sensor is performed by observing the process of salt dissolution in water and diffusion of two droplets with different RI. This WIO pumped sensing configuration provides a path towards an on-chip chemical sensor for studying chemical reaction kinetics in real time.

**Keywords:** whispering-gallery-mode (WGM); sensor; water-immersion-objective (WIO); microdisk; laser

#### **1. Introduction**

Light whispering-gallery-modes (WGMs) in microresonators, which are the analogy of acoustic waves travelling in a closed concave surface such as St. Paul's Cathedral, support ultrasmall mode volumes and ultrahigh quality (Q) factors. WGM microresonators with high local field intensities make them an excellent platform for enhancing the interaction between light and matter, based on which many applications have been demonstrated over the past two decades, such as low-threshold lasers [1,2], opto-mechanics [3,4], integrated optical devices [5,6], non-linear optics processes [7–9], and optical sensors [10–12]. Particularly, the WGM microresonator with strong evanescent waves has been proven as a versatile label-free biochemical sensor [13–17] with high sensitivity and low detection limit down to single molecules and nanoparticles by monitoring the shift [18], split [16,19–21], or broadening [22] of the resonance spectra. Various types of WGM microresonators including microsphere [18], microtoroid [16], microbubble [23,24], microdisk/ring [25], and microcapillaries [26,27] have been used in biosensing experiments; they can be catalogued into passive resonators and active resonators. In passive resonator-based sensor configuration, the resonator is driven by a waveguide, such as a relatively fragile tapered fiber or prism. The relative position between them must be adjusted precisely by a nano-positioning platform that makes the sensing

system costly and cumbersome. While in active resonator sensor configuration, the fluorescence or laser spectrum is collected by free-space optics that makes the sensing system relatively practical.

In most WGM microresonator (both for passive and active) based biochemical sensor configurations, the microcavity is immersed in water and sealed in a microfluidic cell. This sensing configuration is a closed system. To change the refractive index (RI) of liquid around the microresonator, a micro syringe pump is used to pump the biochemical sample under test through the microfluidic channel. Thus, this sensing configuration cannot change the RI of liquid around the microresonator and sensing it immediately because sensing samples must be crossed through the microfluidic channel. Here, we report on the demonstration of a biochemical sensor based on a water-immersion-objective (WIO) pumped on-chip microdisk laser. This WIO pumped configuration is an open sensing system. The microdisk is directly immersed in the water environment that is what the WIO requires. By monitoring the laser wavelength, any feeble disturbance of the RI of the water can be detected in real time. Thus, the WIO pumped configuration provides an excellent platform for monitoring the RI change of the water that is induced by chemical reactions. In this paper, we demonstrate that the salt dissolution in water process and diffusion process of two droplets with different RI can be tracked by the lasing mode wavelength in real time. We believe that this work opens up a novel on-chip biochemical sensor based on a WIO pumped configuration, which provides potential accesses to study chemical reaction kinetics.

#### **2. Laser Operation**

#### *2.1. Optical System Setup*

The optical system setup is shown in Figure 1a, which is a typical optically pumped microcavity laser setup, except that a water-immersion objective (WIO) instead of conventional air objective is used. The pump laser is focused on the active microdisk resonator, which is directly immersed in water, through this WIO (LUMPlanFLN 40× W, Olympus, Tokyo, Japan). In this study, an active SU-8 microdisk resonator doped with Rhodamine B (RhB) is used for the experiment. The microdisk laser array (Figure 1b) is fabricated on silica-on-silicon chip and the detailed fabrication process can be found in previous studies [28,29]. The diameter and thickness of the microdisk are controlled to 40 μm and 1 μm, respectively. To avoid thermal wavelength drift induced by the fluctuation of the lab temperature, the sample chip is mounted on a home-made TEC (TZET Co. Ltd., Tianjin, China). The TEC is connected to a temperature controller (TED4015, Thorlabs Inc., Newton, NJ, USA) with a controlling precision of 2 mK. The pump light used here is an ns-pulsed OPO laser with a repetition rate of 10 Hz. The wavelength of pump laser is tuned to 532 nm for matching the absorption band of RhB. The laser emitted from the active microdisk is collected through free-space and recorded by the monochromator (HORIBA iHR550, Kyoto, Japan) equipped with a cooled CCD detector. A 1200 mm−<sup>1</sup> grating is installed in the monochromator.

**Figure 1.** Optical system setup. (**a**) Schematic of lasing chemical sensor pumped by water-immersion-objective (WIO). LP: linear polarizer; AT: attenuator; L: lens; M: mirror; DM: dichroic mirror; BS: beam splitter; PMF: polarization-maintaining fiber. (**b**) Optical images of microdisk array and single microdisk, the scale bar is 40 μm. (**c**) Close-up image of the microdisk laser-based chemical sensor under WIO. The silicon chip is mounted on a home-made TEC which is connected to a temperature controller (Thorlabs, TED4015).

#### *2.2. Threshold Measurement and Spectral Identification*

The threshold of the laser emission is firstly measured by tuning the power of pump light and the light emission form the active microdisk is recorded correspondingly. Both the air objective (LMPlanFLN 20×, Olympus, Tokyo, Japan) pumped configuration and WIO pumped configuration are performed for comparison; the result is shown in Figure 2. Under both configurations, there are obvious kinks (Figure 2a,b), which indicate that the lasing thresholds are about 5.2 μJ/mm2 and 13.9 μJ/mm2, respectively. The linewidth of the laser peak above the threshold is about 0.08 nm. The corresponding Q factor is about 7800. However, this is not the real Q factor which is limited by the grating of the monochromator. The laser threshold pumped by WIO is higher than that pumped by the air objective. This is attributed to that the RI contrast is lower, which increases radiation loss in the WIO pumped configuration.

To verify this, the eigenmodes of a microdisk surrounded by air and water are calculated by Finite Element Method (FEM) with COMSOL Multiphysics 3.5a [30]. The perfectly matched layers (PMLs) are introduced to accurately calculate the radiation-loss-related Qrad. The Qrad of the fundamental radial mode (radial quantum number *<sup>p</sup>* =1) is 1.5×108 when the microdisk is exposed in air, while it drops to 2.4×10<sup>7</sup> when the microdisk is immersed in water.

The Qrad and field distribution of different radial modes are calculated and plotted in Figure 3. The Qrad decreases exponentially with the increment of *p*. Thus, we can conclude that the laser peaks shown in Figure 2c,d are the fundamental modes with adjacent azimuthal mode numbers. The free spectral range (FSR) is slightly larger in the WIO pumped configuration, because of the RI increment of surrounding materials of the microdisk. Little discrepancy between the calculated FSR and measured FSR may result from the inaccuracy of the microdisk diameter during the fabrication process.

**Figure 2.** Comparisons of the output laser intensity as a function of the energy density (light-light curve) around the single mode lasing wavelength pumped by (**a**) air objective and (**b**) WIO, respectively. Typical laser spectrum pumped by air objective (**c**) and WIO (**d**), respectively. Calculated free spectral range (FSR) and measured FSR are both denoted.

**Figure 3.** (**a**) The field distributions for different radial mode numbers. (**b**) Calculated radiation-lossrelated Qrad for different radial mode numbers.

#### **3. Basic Element Sensing**

#### *3.1. Bulk Refractive Index Sensitivity (BRIS)*

Under the WIO pumped configuration, the microdisk is directly immersed in a water environment, so the lasing spectra will be quite sensitive to the RI of the water. The BRIS of the microdisk laser is tested by directly changing the RI of the water droplet between the WIO and the silicon chip. Different RI solutions are obtained by dissolving different masses of NaCl into deionized water. Five NaCl solutions with concentrations of 1, 2.05, 3, 4.03, and 4.94 mol/liter are prepared for testing as shown in Figure 4a. To change in advance each droplet between the WIO and silicon chip, the sample is washed by the deionized water at least twice to eliminate the error of the droplet RI.

**Figure 4.** (**a**) Lasing spectrum shift by slightly increasing the refractive index (RI) of the water droplet. Insets show the RI of the NaCl solutions. (**b**) Measured wavelengths of the lasing modes as a function of the NaCl concentration (top *x*-axis) and RI of the water droplet (bottom *x*-axis); the measured data is linear fitted. (**c**) The calculated RI sensitivity.

The lasing spectra are recorded for each droplet, respectively, which are shown in Figure 4a. The central lasing wavelength is obtained by fitting the lasing peaks with Lorentz function. The lasing wavelength shift is plotted as a function of RI (Figure 4b). By linear fitting the measured wavelength data, the slope, i.e., the BRIS is 11.3 nm/RIU.

The wavelengths of the eigenmodes are calculated by changing the RI of the surrounding environment of the microdisk in the simulation model, and are also plotted over RI in Figure 4c. As shown, the calculated BRIS is 11.7 nm/RIU. The measured BRIS matches very well with this calculated value. The relatively high BRIS guarantees the performance of microdisk laser as a chemical sensor.

#### *3.2. Thermal Sensing*

SU-8 material exhibits high thermal optics (TO) coefficient of −3.5 × <sup>10</sup>−<sup>4</sup> <sup>K</sup>−<sup>1</sup> [31], which is higher than that of materials that are commonly used to fabricate WGM resonators, such as silica (1 × <sup>10</sup>−<sup>5</sup> <sup>K</sup><sup>−</sup>1) [32] and silicon (1.8 × <sup>10</sup>−<sup>4</sup> <sup>K</sup>−1) [33]. Combining with on-chip integration, ease of laser probing and readout based on free-space optics against passive WGM thermal sensors using relatively fragile tapered microfiber [34,35], the fabricated all-polymer microdisk laser provides a good platform for ultrasensitive thermal sensing. Thus, the thermal sensing of the SU-8 microdisk laser is performed.

The laser emission spectra are recorded in real time when changing the TEC temperature by a step of 1 K. At each temperature, the spectra are recorded for about 2 mins. A home-made code is written to track the spectral positions of the laser peaks over time. The result of spectral shift over time with different temperatures is shown in Figure 5a, where the spectral shift (triangle dots) exhibits step change to the temperature. However, at each constant temperature, the lasing wavelength minor shifts linearly which can be seen from the enlarged view of spectral shift at 20 ◦C (inset in Figure 5a). This blue shift of laser wavelength is attributed to the decreased RI of the polymer-dye composites due to photobleaching [36–39]. As the spectral shift is linear over time [38], the spectral shift can be corrected by eliminating its slope (red fitting line in the inset of Figure 5a). The corrected spectral shift (circular dots) is almost flat over time at each constant temperature.

**Figure 5.** Temperature dependence of wavelengths of lasing mode. (**a**) Lasing emission wavelength shift with photobleaching and corrected data for photobleaching when changing the TEC temperature at a periodic interval. Inset shows the enlarged view at 20 ◦C. Brown and red line are the linear fitting curve. (**b**) Tracked laser wavelength with error bar vs temperature.

The average values of laser wavelength with error bar for each temperature is plotted in Figure 5b. The linear fit result indicates the thermal sensitivity is as high as 120.6 pm/K and has benefited from the high TO coefficient. Here the thermal expansion effect is ignored since its coefficient of ~ 10−<sup>6</sup> K−<sup>1</sup> is much lower than TO coefficient [40].

For bio-chemical sensing, the temperature fluctuation and photobleaching will cause the resonance wavelength drift and influence the quantification of the proposed analytes. To eliminate the effect of temperature fluctuation, the sensor chip is mounted on a TEC for stabilizing the temperature. The wavelength drift caused by the photobleaching can be corrected by eliminating its slope. Recently, S. F. Wondimu also reported a novel scheme that allows for simultaneous compensation of temperature drift and photobleaching by using microdisk laser arrays with on-chip references [39].

#### **4. Chemical Sensor**

As the microdisk is directly immersed in the water droplet and the microdisk sensor exhibits high sensitivities, so the lasing mode can quickly sense the local change (e.g., RI) of the water droplet in real time. This provides an excellent platform for studying some chemical reactions in water, e.g., dissolution and diffusion process, by monitoring the laser wavelength. Thus in the following study, a proof-of-principle chemical sensor is demonstrated based on a WIO pumped configuration by adding salt crystals or NaCl solution into the water droplet.

#### *4.1. Salt Dissolution Process in Water*

In this section, adding a certain quality of salt crystal into the water droplet between the WIO and silicon chip is performed. The water droplet with a fixed volume of 0.2 mL is firstly injected into the space between the WIO and silicon chip by a pipette. By tuning one microdisk just under the focus of pump light, the laser is generated above the threshold and the emission spectrum is automatically recorded with integration time of 1 s. Then salt crystal with a mass of 14 mg is cast into the edge of the droplet with a tweezer. In this process, the acquisition of emission spectra is ongoing for about 7 mins.

After data acquisition, one laser peak is monitored and its spectral wavelength shift is shown in Figure 6a (triangle dots). Slow linear change of wavelength induced by photobleaching is observed, after correction, and the spectral shift is denoted by circular dots. Before casting salt into the droplet, the lasing wavelength is quiet, which can be seen from Figure 6a (circular dots). After casting salt into the droplet, the lasing wavelength is red shifted dramatically. Then the lasing wavelength blue shifts slowly and finally approaches to equilibrium over 400 s. The whole process is explained below.

**Figure 6.** Monitoring the process of dissolution of salt in water using the lasing chemical sensor. (**a**) Measured lasing wavelength and corrected lasing wavelength shifts when adding the 14 mg salt in water under WIO. (**b**) Close-up view of corrected lasing wavelength and Langevin fitting. Inset shows the schematic diagram of operation process. (**c**) Dynamic change of lasing spectra in the process of dissolution of salt in water. The blue line (initial position) to red line (end position) is guided by the green dotted line.

Once the salt is cast into the droplet, the salt will dissolve and diffuse in the water, which induces the increment of RI of the droplet. As a result, the spectrum shifts to longer wavelength dramatically. Since the salt crystal is cast in one side of the droplet, there exists a gradient in concentration of NaCl solution from this side to the other side. Therefore, the RI in the surrounding volume of the lasing microdisk first increases and then decreases with the progress of diffusion. Consequently, the spectrum starts to shift slowly to shorter wavelength and finally no longer changes at the end of diffusion process (> 400 s), as shown in Figure 6a.

A close-up view of the dramatic change of laser wavelength is shown in Figure 6b. The trajectory of spectral shift is fitted well by Langevin function, which is written as: L(t) = coth(t) − 1/t, where the "coth" is the hyperbolic cotangent, defined as coth(t) = (e<sup>t</sup> + e−<sup>t</sup> )/(e<sup>t</sup> − <sup>e</sup>−<sup>t</sup> ). The dynamic change of the laser spectra is also plotted in Figure 6c (guided by the dotted line).

We found that the spectral does not shift as soon as the salt is cast into the droplet. From adding the salt into the droplet, the laser wavelength is almost stable for a certain time until the wavelength shifts obviously. Here, this time is named as dead time. In order to measure the dead time, the spectral data are recorded separately before and after adding the salt (8 mg) into the droplet, which is shown in Figure 7a. The time to add the salt into the droplet is set to 0 s. The dead time is about 6.5 s, as shown in Figure 7a (denoted as a yellow region). After the dead time, the laser wavelength drifts steeply. Dead time as a function of salt mass is also investigated and plotted in Figure 7b. As shown, the dead time decrease exponentially with the salt mass. This is attributed to the heavier salt possessing a larger surface area and the dissolution process is shorter.

Adding salt into the water actually contains two processes: dissolution of salt and diffusion of NaCl solution. The onset of the spectral shift means the RI of the surrounding volume of the lasing microdisk starts to change due to the diffusion of NaCl solution. The dead time is the sum of salt dissolution time and diffusion time of NaCl solution from one side of the droplet to the middle (where the lasing microdisk locates) of the droplet. Therefore, we perform next experiment by injecting NaCl solution directly into the water droplet to investigate the diffusion time.

**Figure 7.** (**a**) Response of lasing wavelength recorded separately before (green dots) and after (blue dots) adding the salt (8 mg) into the water. The yellow region represents the dissolution process (dead time). (**b**) Dead time as function of the mass of the salt.

#### *4.2. Diffusion Process of Two Different Solutions*

In this section, the diffusion process of two different RI solutions is investigated. The water droplet with a fixed volume of 0.1 mL is firstly injected to the volume between the WIO and silicon chip. Then 0.1 mL NaCl solution with RI of 1.3516 is quickly injected into the water droplet by a pipette tip.

One laser peak was monitored and its spectral shift is shown in Figure 8. The spectral data are recorded separately before and after injecting NaCl solution into the droplet. The spectral response is different from that in adding salt into the droplet. As soon as the NaCl solution is injected into the droplet, the laser wavelength shifts immediately. This indicates the RI of the surrounding volume of the lasing microdisk changes immediately due to the diffusion of NaCl solution, and this diffusion process is very fast (less than 1 s as shown in Figure 8). Therefore, it can be confirmed that the dead time in Figure 7a is the reaction time of salt dissolution in water.

**Figure 8.** Monitoring the process of mixing NaCl solution with water using the lasing chemical sensor. The yellow line represents the diffusion process. Inset shows the schematic diagram of operation process.

#### **5. Conclusion**

We introduced and systematically studied on-chip versatile chemical sensors based on a WIO pumped WGM microlaser in which the microdisk is directly immersed in the water droplet. Laser properties including threshold and spectrum are firstly investigated and also compared with the microdisk pumped by WIO and air objective, respectively. Lasing mode identification is also performed by using FEM. Then, basic physical elements of RI and temperature sensing are tested, which shows that BRIS is as high as 11.3 nm/RIU and the thermal sensitivity can reach 120.6 pm/K. Finally, a proof-of-concept demonstration of the chemical sensor is carried out by locally changing the droplet RI with the help of typical chemical reactions. The tracking of the progression of dissolution and the diffusion process in real time are realized by monitoring the laser spectral shift.

We anticipate that our demonstration can be further extended to study various biochemical reactions, such as endothermic and exothermic reactions. The application of the WIO pumped configuration can be also broadened to multifunctional biological monitoring and treatment by directly immersing cells or tissues into the WIO droplet.

**Author Contributions:** Conceptualization, Q.L., X.C., X.W., and S.X.; methodology, Q.L., X.C., and L.F.; validation, X.C., L.F., and Q.L.; writing—original draft preparation, Q.L. and X.C.; writing—review and editing, X.W. and S.X.; funding acquisition, S.X. All authors discussed the results and commented on the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China, grant number 61705039; National Key Basic Research Program of China (973 project), grant number 2015CB352006; China Postdoctoral Science Foundation, grant number 2017M610389; Fujian Provincial Program for Distinguished Young Scientists in University; Fujian Provincial Key Project of Natural Science Foundation for Young Scientists in University, grant number JZ160423; Program for Changjiang Scholars and Innovative Research Team in University, grant number IRT\_15R10; Special Funds of the Central Government Guiding Local Science and Technology Development, grant number 2017L3009.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Spectral Modulation of Optofluidic Coupled-Microdisk Lasers in Aqueous Media**

**Zhihe Guo 1, Haotian Wang 2,\*, Chenming Zhao 1, Lin Chen 1, Sheng Liu 1, Jinliang Hu 1, Yi Zhou <sup>1</sup> and Xiang Wu 1,\***


Received: 2 September 2019; Accepted: 9 October 2019; Published: 11 October 2019

**Abstract:** We present the spectral modulation of an optofluidic microdisk device and investigate the mechanism and characteristics of the microdisk laser in aqueous media. The optofluidic microdisk device combines a solid-state dye-doped polymer microdisk with a microfluidic channel device, whose optical field can interact with the aqueous media. Interesting phenomena, such as mode splitting and single-mode lasing in the laser spectrum, can be observed in two coupled microdisks under the pump laser. We modulated the spectra by changing the gap of the two coupled microdisks, the refractive indices of the aqueous media, and the position of a pump light, namely, selective pumping schemes. This optofluidic microlaser provides a method to modulate the laser spectra precisely and flexibly, which will help to further understand spectral properties of coupled microcavity laser systems and develop potential applications in photobiology and photomedicine.

**Keywords:** microdisk laser; spectral modulation; optofluidic; coupled microdisk

#### **1. Introduction**

Microcavity lasers based on whispering-gallery mode (WGM), possessing small size and strong optical confinement, have been attracting great interest in fundamental research and for practical applications [1–4]. Compared to the laser wavelength, the size of the microcavity is usually large enough to cause multimode lasing, resulting in a noisy spectrum, which severely degrades the performance of microlasers in applications such as optical communication and ultrasensitive optical sensors [5,6]. Coupled microcavity configuration is a compact and effective method to purify and regulate the laser spectra of microcavities. For example, a single-mode laser can be realized, based on the Vernier effect, in two coupled cavities with slightly different sizes [6–11]. Non-Hermitian optical systems, which have emerged recently, for lasers are, in essence, open and non-conservative systems. This means the most common gains and losses of physical quantities can be distributed differently with artificial purposes in such coupled microcavities [12]. In particular, if the optical microcavity is tuned to the exceptional point (EP) [13], where the resonance frequencies of the coupled modes coalesce both in their real and imaginary parts and become degenerate, some unique phenomena appear, such as parity–time (PT) symmetry breaking [14–18], chiral modes [19], and enhanced perturbation at higher-order EPs [20,21]. Therefore, this principle provides an alternative approach to modulate the laser spectrum by controlling the gain and loss elements. By managing the gain and loss distribution for the

coupled modes in microcavities, some studies have successfully achieved single-mode lasing in WGM microcavities [22–24].

One of the key points in such coupled microcavity systems is the flexible method for controlling the system parameters, for example, the coupling strength between the coupled laser modes. The coupling strength can be regulated by controlling the gap between different microcavities, which is predetermined for the desired modes in the fabrication process, or precisely controlled by the nano-positioning stages in real-time experiments [14,25]. However, these mechanisms all lack efficiency and flexibility, and are sensitive to external noise sources, such as acoustic waves. Moreover, the coupled cavities fabricated on chip have the fixed coupling gaps. They need a reliable and flexible means for controlling the coupling strengths on chip. Optofluidic devices can integrate some functional fluids into optical or photonics devices [26,27]. Such fluids usually have high transmittance and tunable optical properties, e.g., the refractive index [28,29]. Because the optofluidic devices have a good compatibility with aqueous media, they are widely used in biology and medical science [30–34]. Therefore, optofluidic devices have found great potential in on-chip microlaser systems for their highly integrated level and, more importantly, the microlaser cavity is given a liquid environment, which provides an additional means for tuning the microlaser.

In this study, solid-state dye-doped polymer microdisks are fabricated and integrated with a microfluidic channel, in order to investigate the mechanism and characteristics of the microdisk laser in aqueous media. Some phenomena, such as single-frequency laser and mode splitting in the coupled microcavities, were studied theoretically and experimentally in aqueous media for the first time. We observe that the resonance detuning of mode splitting is affected by changes in the gap between two coupled microdisks. Mode splitting and single-mode lasing in the laser spectrum can also be modulated by changing the refractive indices of the aqueous media and the position of the pump light. In addition, the aqueous media not only have a spectral modulation function but also a mode purification ability because they decrease the refractive index difference between the cavity and the background media, which eliminates some high-order WGM with larger reflection angles at the cavity surface. The results presented in this paper show that optofluidic microlasers possess potential for several promising applications, such as tunable single-mode lasers on chip and bio-photonic sensors.

#### **2. Coupled-Mode Analysis**

The coupled-mode theory could be applied to analyze such a coupled-microdisk laser system. We assume the two coupled modes in two respective cavities have amplitudes *a* and *b*. Considering the sinusoidally varying fields in the time domain, *e*−*i*ω*nt* , *a* and *b* obey the coupled differential equation [20]:

$$i\frac{d}{dt}\begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} i g\_a & \kappa \\ \kappa & i g\_b \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} \tag{1}$$

where *ga* and *gb* represent the gain (*g* > 0) or loss (*g* < 0) for modes *a* and *b*, respectively. κ denotes the coupling strength between the two microcavities. It leads to an eigenvalue problem: *H*0*Vn* = ω*nVn*, where the eigenvalue ω*<sup>n</sup>* and eigenvector *Vn* satisfy *Vn e*−*i*ω*nt* = (*a*, *b*) T, and the Hamiltonian *H*<sup>0</sup> of the system is

$$H0 = \left(\begin{array}{cc} i\text{g}\_a & \kappa \\ \kappa & i\text{g}\_b \end{array}\right) \tag{2}$$

To determine the eigenfrequency, we solve the equation det(*H*<sup>0</sup> – ω*nI*) = 0 (where *I* is the identity matrix), which results in two eigenfrequencies:

$$
\omega\_{\pm} = \frac{i(g\_a + g\_b)}{2} \pm \sqrt{\kappa^2 - \frac{\left(g\_a - g\_b\right)^2}{4}},\tag{3}
$$

The difference between the two eigenfrequencies is

$$
\Delta\omega \equiv \omega\_{+} - \omega\_{-} = 2\sqrt{\kappa^2 - \left(\mathcal{g}\_a - \mathcal{g}\_b\right)^2 / 4},\tag{4}
$$

where Δ*g*˜ = (*ga* − *gb*)/2κ is the normalization gain difference between the two modes. If the two modes experience the same gain or loss, i.e., Δ*g*˜ = 0, the frequency difference equals 2κ, which is the well-known mode-splitting effect, as shown in Figure 1a. However, the two modes have different gains or losses in more general situations. In particular, such frequency difference changes into a purely imaginary number when Δ*g*˜ > 1. In this case, the difference between the two eigenfrequencies no longer shows two different resonance wavelengths in the spectrum, but instead, introduces two modes with different growth or decay rates, which often indicate different net gains in a laser system. Moreover, at the threshold where Δ*g*˜ = 1 two eigenfrequencies coalesce to a single one, with ω = *i*(*ga* + *gb*)/2.

**Figure 1.** (**a**) Real and imaginary parts of Δω/2κ as a function of Δ*g*˜. The difference between the eigenfrequencies changes from purely real to purely imaginary at Δ*g*˜= 0. (**b**) Coupling coefficient *K* as a function of refractive index under different gaps.

The coupling strength κ could be related to the coupling coefficient *K*; defined in the coupling between two parallel waveguides:

$$\kappa = \frac{\text{Kcl}\_{eff}}{n\_{eff}2\pi\text{R}'} \tag{5}$$

where *Le*ff and *ne*ff are the effective interaction length and effective index of the cavity modes, respectively, and *R* is the radius of the microdisk. The coupling coefficient *K*; can be expressed by the overlap integral between the two coupling modes:

$$K = -\frac{i\omega}{4} \int\_{-\infty}^{+\infty} \int\_{-\infty}^{+\infty} (n\_0^2 - n^2) E\_{\mathbf{a}} \bullet E\_{\mathbf{b}} dx dy,\tag{6}$$

where *Ea* and *Eb* are the normalized field distributions in the two respective cavities, and *n*<sup>0</sup> and *n* are the refractive indices for the microcavities and the background medium, respectively. The coupling strength *K* in Equation (6) can be calculated numerically, and the relationships between the coupling strength *K* and refractive index *n* under different gaps are shown in Figure 1b. *K* grows with an increase in *n*, or with a decrease in the coupling gap. Because the increased *n* reduces the difference of the refractive indices between the microdisk and background medium, more of the optical field will leak into the coupling gap. This results in an improvement in the overlap between the optical fields in two microdisks, and accordingly increases the value of *K*. Similarly, a smaller coupling gap also improves the overlap of the optical fields and provides greater coupling strength.

#### **3. Optofluidic Microdisk Lasers**

#### *3.1. Fabrication of the Optofluidic Microdisk Device and Experimental Setup*

The experimental setup of the optofluidic microcavity laser is shown in Figure 2a. The optofluidic microdisk device consists of a microfluidic channel and solid-state dye-doped polymer microdisks as shown in Figure 2b,c. Rhodamine B dye (RhB, CAS: 81-88-9, J&K Scientific Ltd., Beijing, China)-doped SU-8 photoresist (SU-8 2002, MicroChem Corp. MA, USA) in a ratio of 1.5 mg:1 mL was used as gain media and mixed by a magnetic stirrer for 2 h. During stirring, attention should be paid to shading the photoresist, in order to avoid quality deterioration. A 2.4 μm active layer of the gain media was then deposited by a spin coater (Specialty Coating Systems Inc., Indianapolis, USA) onto a silicon wafer with a 2 μm thick thermal oxide layer. The geometric shapes of the single and coupled microdisks were pre-designed on the chromium mask, and the gap between each pair of coupled microdisks was increased from 0.2 to 1 μm, and 0.1 μm for each pair. Therefore, the solid-state dye-doped polymer microdisk arrays were fabricated via single-mask standard lithography (Karl Suss MJB 3 Mask Aligner, SUSS MicroTec SE, Garching, Germany). The geometric sizes of the microdisks were measured via scanning electron microscopy (SEM), as shown in Figure 2c and Figure S1 (details can be found in the Supplementary Materials).

**Figure 2.** (**a**) Experimental setup schematic of the optofluidic microcavity laser. (**b**) 3-D schematic of the optofluidic microdisk device. (**c**) Scanning electron microscope (SEM) image of typical coupled microdisks. The scale bar is 10 μm.

The optofluidic microdisk device consists of a microfluidic channel device and solid-state dye-doped polymer microdisks (details can be found in Supplementary Materials). This microfluidic channel is a device that combines a clean glass slide and patterned polydimethylsiloxane (PDMS) leaf. The fabrication process of the optofluidic microdisk device is described in Figure S2 (in Supplementary Materials). The shape of the glass mold was spliced with glass sheets, and the convex structure is shown as in Figure S2a. After the PDMS leaf was formed from the glass mold, the inlet and outlet holes were punched by a needle. The solid-state dye-doped polymer microdisks were sandwiched between the microfluidic channel (PDMS leaf) and a glass slide, as shown in Figure S2d,e. Finally, an optofluidic microdisk device, consisting of a microfluidic channel and solid-state dye-doped polymer microdisks, was fabricated, as shown in Figure 2b and Figure S2e.

The laser spectra were measured using the experimental setup as shown in Figure 2a. The pump laser was a picosecond pulse laser (532 nm, PL2143A, EKSPLA, Vilnius, Lithuanian), and the pulse duration of the pump laser was 30 ps, with a repetition rate of 10 Hz. A monochromator (Acton SpectraPro 2750, Princeton Instruments, NJ, USA) was employed to collect the laser spectra. The spectra were measured by a monochromator with a 1200 grooves/mm grating and the spectral resolution was 0.01 nm. The size of the pump laser was controlled via tunable beam-shaping optics (BSO), and the pump laser energy was adjusted by a variable attenuator (VA). Firstly, the laser beam spot was coupled into the microscope

system and further focused to nearly 50 μm under a 20× objective lens. Through the microscope with a camera, the pump position of the laser beam spot was able to be marked on a screen. The microcavity sample was then moved to the pump position using a 3-D stage. The output laser passed through a Glan–Taylor prism (GTP) and was collected by an optical multi-channel fiber bundle (core diameter 400 μm). The GTP was used as the polarization controller. The laser light was then coupled into the input slit (30 μm) of a monochromator (f = 0.75 m) and detected by an electron-multiplying charged coupled device (EMCCD, DV401A, Andor iDus, OX, UK). Finally, the laser spectra were recorded by a computer.

#### *3.2. Modulation of the Microdisk Laser in Aqueous Media*

A single microdisk was firstly investigated and the appearance of the microdisk sample was measured via SEM, as shown in Figure 3a. In this case, the diameter of the microdisk was set to 20.34 μm and the thickness was 2.4 μm, as measured using scanning electron microscopy (SEM) and a surface profiler, respectively. Compared to air cladding, the microdisk surrounded by water represents a much clearer spectrum. The water environment decreases the refractive index (RI) difference between the microdisk and the background media, where high-order WGMs with larger reflection angles at the cavity surface suffer from degradation of *Q* (quality) values and larger radiation losses. Therefore, they have a higher laser threshold and only fundamental-order WGMs with high *Q* values will exist in the laser spectra, which are shown in Figure 3b. The free spectral range (FSR) of this microdisk was 3.81 nm, agreeing well with the FSR calculated from *FSR* <sup>=</sup> <sup>λ</sup>2/(2π*Rne*ff) <sup>≈</sup> 3.87 nm, where <sup>λ</sup> <sup>=</sup> 615 nm, *ne*ff = 1.53, and *R* = 10.17 μm were the lasing wavelength, effective refractive index of the microdisk, and radius of the microdisk, respectively. The laser spectra were relatively clear in a water environment, which was conducive to sensing, single-frequency lasers, and other applications. Figure 3c,d show the simulation results of the electric field distribution of a single microdisk using the finite element method (FEM). To reduce consumption of computer memory, we divided the full 3-D model into two 2-D models, as shown in the top view and side view in Figure 3c,d, respectively. The refractive index of the microdisk in the simulation shown in Figure 3c was obtained by calculating the effective index of the three-layer planar waveguide of water-SU8-SiO2. As shown in Figure 3d, the electric field distribution in the cross section of the microdisk was calculated using an axially symmetric model [35]. All the simulation regions for the two 2-D models were surrounded by the perfect matching layer (PML) to avoid reflecting from the boundary. The microdisk was immersed in water, and the polarization of the fundamental-order-radial modes was transverse electric (TE) polarization.

**Figure 3.** (**a**) SEM image of the microdisk. The scale bar is 10 μm. (**b**) Laser spectra of a microdisk dipped in water. The free spectral range (FSR) was 3.81 nm. (**c**) and (**d**) Field distributions of whispering-gallery modes (WGMs) in fundamental-order-radial mode with transverse electric (TE) polarization in the top view and side view. The direction indicated by the red arrow was the direction of electric field propagation. Finite element method (FEM) simulations were performed with the same parameters for the experimental data.

The long-term stability of the microdisk laser was determined by pumping the microdisk every 20 s over a period of 1800 s, with the laser spectra being collected simultaneously. As shown in Figure 4a,b, the laser wavelength maintained the same position during this period. The standard deviation of the curve in Figure 4b is 5.84 pm, which is much smaller than the resolution of the monochromator (0.01 nm). These results indicate that an optofluidic microdisk laser does not have a large wavelength shift over a long term and that the system stability is good enough for our next experiments.

**Figure 4.** (**a**) Laser spectra of optofluidic microdisk laser at different times. (**b**) Measured wavelengths of the lasers as a function of time. Standard deviation was 5.84 pm. (**c**) Laser spectra shifts when the refractive index of the DMSO solution is slightly increased. (**d**) Mean wavelength shifts of the lasers as a function of the refractive indices of DMSO solutions. Violet dots and pink line represent the experimental data and FEM simulation data, respectively. Insert: fundamental-order-radial mode with TE polarization.

The spectral modulation of the optofluidic microdisk laser can be achieved by changing the refractive index of the external environment, similar to the method for measuring the bulk refractive index sensitivity (*S*) of a sensor. Because this is an optofluidic device, it is essential to first characterize the sensitivity according to the changes in the refractive index of the solvent on the sensing surface [36]. In Figure 4c,d, the optofluidic microdisk laser was tested by flowing progressively higher concentrations of dimethyl sulfoxide (DMSO, CAS: 67-68-5, Lingfeng Chemical Reagent Co. Ltd., Shanghai, China) solutions over the sensing surface. The DMSO was diluted with deionized water (DI water) into concentrations of 2%, 4%, 6%, 8%, and 10%. The refractive indices of the DMSO solutions were proportional to the volume ratios of the DMSO (details can be found in Supplementary Materials) [37]. Prior to use, the DI water and DMSO–water mixtures were kept at room temperature for a longer period, to minimize temperature-induced spectrum changes. A redshift of the wavelength in response to an increasing refractive index of the solution was observed from the laser spectra measurement and can be explained by *m*λ = 2π*ne*ff *R* (see details in Supplementary Materials) [38]. *Sexp* and *Ssim* represent the *S* from the experiment and from the FEM simulation, respectively. *Sexp* was obtained by directly measuring the wavelength shift, which was 18.14 nm/RIU. The process of detecting the *S* of the microdisk was also simulated using FEM, based on a 2-D axisymmetric model. Results show that the *Ssim* of the fundamental and second-order radial modes with TE polarization were 18.12 and 28.90 nm/RIU, respectively, and that the *Ssim* of the fundamental and second-order radial modes with transverse magnetic (TM) polarization were 22.98 and 37.65 nm/RIU,

respectively. When the GTP was used to observe the experiment, the polarization state of the output laser was determined to be TE polarization. Therefore, the *S* of the measured microdisk was determined to be fundamental-order-radial mode with TE polarization.

#### *3.3. Mode Splitting from the Coupled-Microdisk Laser in Aqueous Media*

The method of purifying and adjusting the microcavity laser spectrum using the coupled microcavity structure is compact and effective. Normally, a typical WGM propagates along a spherical equatorial plane, whose evanescent field component allows interaction with the surrounding environment. Therefore, when two WGM microcavities are close to each other, their WGMs can be evanescently coupled and form mode splitting [31]. A large array of dye-doped polymer microlasers can be fabricated in parallel via a single step of deep-ultraviolet (DUV) lithography [39]. The coupled microdisks were investigated by immersing in water. At this time, the two modes experience the same gain or loss (*ga* = *gb*), and the frequency difference equals 2κ. Using the gap between the coupled microdisk of 0.3 μm, the diameters of these two coupled microdisks were 19.71 and 19.63 μm, and their SEM image is plotted in Figure 5a. Under evenly pumping schemes, as shown in the insert of Figure 5b, the mode splitting of the laser spectrum can be observed. Δλ represents the distance between two laser peaks under mode splitting and is 0.183 nm in this case. Figure 5c shows the field distributions of WGMs with the fundamental-order-radial mode in a symmetric coupled-microdisk system. The wavelengths of the supermodes are 620.619 and 620.437 nm, respectively. Δλ of the FEM simulation is 0.182 nm, which is very close to the experimental result of 0.183 nm. Different gaps between coupled microdisks were investigated under the same pumping conditions. Because the optical field decays exponentially outside the microdisk, the coupling strengths κ between the two cavities will decrease. Δλ will also decrease, which is shown in Figure 5d. The experimental data and FEM simulation results were well-fitted by the exponential function. As the gap exceeded 0.6 μm, the coupling strengths *K* of the supermodes became weak and the Δλ value stayed at a level of no more than a few dozen nanometers. Because of the limited resolution of the monochromator, narrower mode splitting cannot be observed in experiments.

**Figure 5.** (**a**) SEM image of the coupled microdisks. Gap between two microdisks is approximately 0.3 μm. Scale baris 10μm. (**b**) Laser spectra of coupledmicrodisks with 0.3μm gap. Δλdenotes the distance between two peaks of mode splitting. Insert: schematic of evenly pumping schemes. (**c**) Field distributions ofWGMs

with fundamental-order-radial mode. (**d**) Δλ of the experimental results and FEM simulations as a function of the gap. Purple dots represent the experimentally detected data; cerulean dots and red curves refer to the FEM simulation results and exponential fitting, respectively. FEM simulations were performed with the same parameters for the experimental data.

Furthermore, when we change the refractive index of the aqueous medium *n*, it will also affect the coupling strengths of the supermodes *K*. Here, we used coupled microdisks with a gap of 0.4 μm. As shown in Figure 6a, when the refractive index of the DMSO solution was changed, the resonance detuning of the mode splitting was affected. Figure 6b shows that Δλ of the experimental data was increased, which agreed well with the results of the FEM simulation. The increased background refractive index *n* will extract a greater part of the optical field into the coupling gap, leading to enhanced coupling strength and a larger difference in resonance frequencies of the coupled modes. The coupling coefficient *K* is as a function of *ne*ff, which is given in Equation (5), and the experimental phenomenon is consistent with the FEM results in Figure 1b.

**Figure 6.** (**a**) Mode splitting in a coupled-microdisk resonator laser with resonance detuning. (**b**) Δλ of the experimental data and FEM simulation, shown as a function of refractive index. Pink dots represent the experimentally detected data, while the green curves represent the FEM simulation results. FEM simulations were performed using the same parameters for the experimental data.

In order to increase the difference in the gains/losses between the two microcavities, we used selective pumping schemes in which we changed the position of the pump beam spot gradually from the left side of the coupled microdisks to the right side by 8 μm per step. Figure 6 demonstrates the laser spectra of coupled microdisks with a 0.3 μm gap, with the coupled microdisks immersed in water. We focused on the wavelengths of the spectra from 612 to 615 nm, which are marked in red in Figure 7a,c. The laser spectra in Figure 7a,c correspond to the pump positions plotted in Figure 7b,d. The laser emerged when one of the coupled microdisks was pumped, and the splitting mode arose when the coupled microdisks were evenly pumped. When the pump beam spot gradually moved to the other side of the coupled microdisks, the splitting mode disappeared, and the laser finally vanished. This result is due to the selective pumping process, which changed the gain/loss status between the two coupled microdisks. The modulation gain branches of various supermodes will lead to a switchable single-frequency laser or mode splitting [9]. When only one of the coupled microdisks is pumped, the eigenfrequencies difference will become purely imaginary because in this case Δ*g*˜ is large enough that the value under the square root in Equation (4) is smaller than zero. This will lead to the emission of a single-mode laser when the other laser modes are also below the threshold. Under even pumping, the gain difference between the two microcavities is very small and Δω is real, so it shows doublet peaks in the laser spectrum, i.e., mode splitting.

**Figure 7.** Laser spectra from different positions of the pump beam spot. (**a**) and (**c**) spectra during the moving of the pump laser from left to right. (**b**) and (**d**) schematics of the position of the pump beam spot.

#### *3.4. Single-Frequency Lasing from the Coupled Microdisk in Aqueous Media*

Because of their high beam quality and spectral purity, lasers with a single-frequency laser emission feature are indispensable to many scientific and industrial applications, such as laser spectroscopy, laser metrology, and biomolecular sensing [5,6,39]. Normally, WGM lasers are usually multimode because of the lack of mode selection [9,40]. Methods such as microcavity size reduction, Vernier effect, and PT symmetry effect of coupled microcavities are possible strategies to realize single-frequency laser [9,14,40–42]. These methods need to control the gap size or thermal modulation of the two microcavities in order to control the coupling strength between them, which often requires a complex implementation system and results in low accuracy.

The results of the previous section indicate that selective pumping schemes would change the spectrum of the coupled microdisks. In order to overcome the major drawback of a single-microcavity resonator laser, which essentially involves multimode laser emission, we used selective pumping schemes in a coupled-microdisk system, in order to realize a single-frequency laser emission. The coupled-microdisk resonator was dipped in water. In this case, the gap of the coupled microdisks was 0.5 μm. The position of the pump laser was "left pumping", and the simulated mode field of the single-frequency laser is plotted in the inserts of Figure 8b. In the range 580–610 nm (much larger than one FSR and covers the tuning range of RhB in the ethanol solvent), only one laser peak has been collected by the spectrometer. Therefore, it is determined to be a single-frequency laser. The single-frequency laser spectra of the coupled microdisks at different pumping intensities are plotted in Figure 8a. The central wavelength of the single-frequency laser was 594.39 nm. When the pump laser energy was below the threshold, the spectrum was actually the fluorescence, which increases slowly with the pump energy. Moreover, the fluorescence efficiency of the gain media was very low. When the pump laser energy was higher than the laser threshold, the laser intensity will increase with the pump energy with a higher slope efficiency. On the curve in Figure 8b, there is a sudden change point (laser threshold) between the fluorescence and laser as the pump energy increases. The corresponding laser threshold curve shows that the laser threshold was 83.88 μJ/mm2, as shown in Figure 8b. Selective breaking of PT symmetry can systematically improve the effective amplification of single-mode operation [23].

**Figure 8.** (**a**) Single-frequency emissions of the coupled microdisks under different pump energies. Insert: the energy of the pump laser. (**b**) Lasing intensity as a function of the pump energy intensity extracted from (**a**). Lasing threshold is approximately 0.60 μW (84.88 μJ/mm2) in a water environment. Insert: the position of the pump laser and the mode field of the single-frequency laser.

An interesting phenomenon was observed when we changed the surrounding refractive index of the single-frequency laser. Another coupled-microdisk resonator was dipped in water under left pumping, with the gap at 0.6 μm. As shown in Figure 9a, the wavelength of the original single-frequency laser was 606.994 nm. With the increase in the liquid refractive index, the wavelength of the single-frequency laser gradually shifted to longer wavelengths, and the intensity decreased. Another hopped laser mode occurred at 603.131 nm. If the liquid refractive index is further increased, the original laser will disappear and only one laser peak can be seen in the spectrum. Figure 9c shows that *ln*(*Ihopped*/*Ioriginal*) changed linearly with the liquid refractive index. Here, *Ioriginal* and *Ihopped* are the light intensity of the original laser line and the hopped laser line, respectively. The hopped laser mode resulted from the slightly asymmetric structure of the coupled microdisks, which responded slightly differently to the change in the surrounding refractive index [30]. Therefore, the wavelength shifts of the coupled-microdisk resonances did not shift synchronously. Furthermore, the slight change of refractive index will also change the coupling efficiency of two coupled microdisks. As shown in Figure 9b,c, the wavelength shift sensitivities (*Soriginal* and *Shopped*) were 14.04 and 14.65 nm/RIU, respectively, and the sensitivity of the intensity change (*SI*) was 149.87 RIU<sup>−</sup>1. Considering standard deviation (σ = 10 a.u.) of the background signal intensity as the lowest detectable lasing intensity, the detection limit of *SI* was 5.34 <sup>×</sup> 10−<sup>5</sup> RIU, based on the equation in a previous article [30]. The resolution of the monochromator can reach 0.01 nm. The detection limits of *Soriginal* and *Shopped* were 7.12 <sup>×</sup> 10−<sup>4</sup> and 6.83 <sup>×</sup> 10−<sup>4</sup> RIU, respectively. These results indicate that *SI* was 13 times more sensitive compared with *Soriginal* and *Shopped*. Because of the Vernier effect, the coupling of the WGMs in the two resonators can produce a large amplification sensitivity. It was noted that the attenuation of light intensity was due to the bleaching of dye, caused by long-time laser pumping. The relative intensities of the original and hopped laser peaks were not affected.

**Figure 9.** (**a**) Single-frequency emissions of the coupled-microdisk resonator laser. (**b**) Wavelength shifts of single-frequency lasers as a function of refractive index. (**c**) Intensity ratio [*ln*(*Ihopped*/*Ioriginal*)] of the two lasing modes, as a function of refractive index. Insert: spectra at the two refractive indices values are given.

#### **4. Conclusions**

In this study, the spectral characteristics of coupled microdisks in aqueous media were examined. The spectral characteristics and modulation mechanism of mode splitting and single-frequency lasers from coupled microdisks were investigated. The resonance detuning of mode splitting was studied by changing the gap between two microdisks, the refractive index of the aqueous media, and the position of the pump laser, namely, selective pumping schemes. The variation of the single-frequency laser was studied by changing the intensity of the pump light and the refractive index of the aqueous media. For single-frequency lasers, the sensitivity can be amplified by changing the light intensity for sensing. The results of this study will help to deepen the understanding of the spectral characteristics and the modulation mechanism of microcavity lasers. Such optofluidic microcavity lasers have broad application prospects in tunable single-mode on-chip lasers and biosensors.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2079-4991/9/10/1439/s1, Figure S1. 3-D schematic of the solid-state dye-doped polymer microdisks. Figure S2. 3-D schematic diagram shows the fabrication process of the optofluidic microdisk device.

**Author Contributions:** X.W. originally conceived the idea. Z.G., C.Z. and L.C. fabricated the devices. Z.G., S.L. and J.H. finished the experiment. Z.G., H.W. and X.W. wrote the manuscript. Z.G. and Y.Z. carried out SEM measurements. Z.G. and H.W. carried out the COMSOL Multiphysics. All authors discussed the progress of the research and reviewed the manuscript.

**Funding:** This research was funded by Special Project of National Key Technology R & D Program of the Ministry of Science and Technology of China, grant number 2016YFC0201401; Shanghai Science and Technology Committee Grant, grant number 18JC1411500; National Natural Science Foundation of China (NSFC), grant number 61805112; National Key Basic Research Program of China (Program 973), grant number 2015CB352006; Natural Science Foundation of Jiangsu Province, grant number BK20181003.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Guided Mode Resonance Sensors with Optimized Figure of Merit**

#### **Yi Zhou, Bowen Wang, Zhihe Guo and Xiang Wu \***

Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Department of Optical Science and Engineering, Shanghai Engineering Research Center of Ultra Precision Optical Manufacturing, Fudan University, Shanghai 200433, China; 18110720008@fudan.edu.cn (Y.Z.); 16210720012@fudan.edu.cn (B.W.); 17110720004@fudan.edu.cn (Z.G.)

**\*** Correspondence: wuxiang@fudan.edu.cn

Received: 6 May 2019; Accepted: 22 May 2019; Published: 1 June 2019

**Abstract:** The guided mode resonance (GMR) effect is widely used in biosensing due to its advantages of narrow linewidth and high efficiency. However, the optimization of a figure of merit (FOM) has not been considered for most GMR sensors. Aimed at obtaining a higher FOM of GMR sensors, we proposed an effective design method for the optimization of FOM. Combining the analytical model and numerical simulations, the FOM of "grating–waveguide" GMR sensors for the wavelength and angular shift detection schemes were investigated systematically. In contrast with previously reported values, higher FOM values were obtained using this method. For the "waveguide–grating" GMR sensors, a linear relationship between the grating period and groove depth was obtained, which leads to excellent FOM values for both the angular and wavelength resonance. Such higher performance GMR sensors will pave the way to lower detection limits in biosensing.

**Keywords:** guided mode resonance; figure of merit; biosensor; detection limits

#### **1. Introduction**

Guided mode resonance (GMR) is a physical effect that occurs in thin-film structures that contain diffractive elements and a waveguide layer ("grating–waveguide" GMR structure) [1] or a planar dielectric-layer diffraction grating ("waveguide–grating" GMR structure) [2]. Light can be coupled into the waveguide modes by different grating diffraction orders. Due to periodic modulation in the waveguide, part of the guided light will leak, and thus, guided modes cannot be sustained on the waveguide grating and will interfere with the noncoupled reflected or transmitted waves [3]. The GMR effect occurs within a narrow spectral band at a particular wavelength, angle, and polarization [4].

Since Magnusson and Wang suggested the application of the GMR effect for sensing purposes due to its narrow, controllable linewidth and high efficiency [5], many researchers have shown great interest in GMR sensors, especially in biosensing [6–13]. To date, there are four main detection schemes for GMR sensors, including wavelength detection [14–17], angular shift detection [18–20], intensity shift detection [21–25], and phase shift detection [26–29]. The wavelength and angular shift detection schemes are used most often as the incident light is totally reflected with highly angular and spectral selectivity at resonance [30]. To reduce the cost of detection systems, for example, by eliminating the need for an expensive spectrometer and laser source, several intensity detection schemes have been proposed [21–24]. In addition, the sensing performance can also be enhanced by implementing phase detection schemes [26–29].

Although different detection methods result in different detection effects, the structural parameters and optical properties of GMR sensors are also important because these factors directly determine the sensors' performance [31–36]. In a label-free biosensing system, a figure of merit (FOM), which combines the sensitivity (S) and the full width at half maximum (FWHM) of the resonance, is the accepted marker to compare the performance of biosensors, which can be defined as FOM = S/FWHM [37]. This value can simultaneously reflect the effects of the magnitude of a measured quantity change and the ability to measure small wavelength shifts [38]. In addition, a FOM can be considered as a good indicator for the expected performance of a sensor [39]. However, a narrow resonant linewidth will result in a sacrifice in the sensitivity, as resonant modes are mostly confined within the solid dielectric medium, rather than the sensing medium [40]. In contrast, highly resonant sensitivity may result in a broadened resonant linewidth, which limits the capability to measure small resonant wavelength shifts with higher accuracy [37]. To solve this problem, Ge et al. proposed an external cavity laser (ECL) GMR sensor using an active resonator laser that maintains the high resolution of laser biosensors without sacrificing sensitivity [41–43]. However, although the properties of sensitivity and linewidth have been investigated widely by varying structural parameters [31–36], the optimization of a FOM has not been considered for most GMR sensors. Most recently, Lan et al. investigated the FOM of a "waveguide-grating" GMR sensor by varying the geometric parameters and incident angle [44]. However, their work only investigated one GMR sensor for wavelength resonance rather than providing a design guideline that is suitable for all GMR sensors. Therefore, a convenient and effective design method to achieve an optimized FOM of GMR sensors for both angular and wavelength resonance is necessary.

In this work, by combining an analytical model and numerical simulation, we provide a convenient and efficient method as a guideline for parametric design to achieve the optimized FOM of "grating–waveguide" GMR sensors for both angular and wavelength resonance. Focusing on "waveguide–grating" GMR sensors, a linear relationship between the grating period and grating groove depth was found as a guideline to achieve optimized FOM.

#### **2. Analytical Model for GMR Sensors**

Lin et al. proposed a model for calculating the wavelength sensitivity of GMR sensors [35]. Briefly, as the incident light λ passes through the grating with an incident angle θ*i*, the diffraction grating equation can be expressed as follows [17]:

$$
\Lambda \left[ n\_{w\mathcal{g}} \sin(\theta\_d) - n\_c \sin(\theta\_i) \right] = m\_{\mathcal{g}} \lambda\_\prime m\_{\mathcal{g}} = 0, \pm 1, \pm 2, \dots, \tag{1}
$$

where Λ is the grating period, *nwg* is the refractive index of the waveguide layer, *nc* is the refractive index of the surrounding medium, θ*<sup>d</sup>* is the diffraction angle and *mg* is the order of the diffracted wave. The grating diffracted wave will couple into the waveguide layer, once the diffracted wave is phase-matched to the waveguide mode, and the diffraction angle θ*<sup>d</sup>* is the propagation angle in the waveguide layer because the grating groove depth (*dg*) is assumed to be extremely thin, and thus, other influences are ignored. The guided wave condition of the planar waveguide can be defined as follows:

*k*0*nwgdwg* cos(θ*d*) − *m*π = φ*<sup>t</sup>* + φ*d*, *m* = 0, 1, 2, ... , (2)

where *k*<sup>0</sup> = 2π/λ is the wavenumber in a vacuum, *dwg* is the thickness of the waveguide layer, *m* is a positive integer number that stands for the mode number of a waveguide, ϕ*<sup>t</sup>* and ϕ*<sup>d</sup>* represent phase shifts that occur due to the total internal Fresnel reflection at the waveguide grating interface and the waveguide substrate interface, respectively. For a given waveguide structure, we can use Equation (2) to calculate the θ*d*, and thus the incident angle θ*<sup>i</sup>* in Equation (1) will be obtained. When the surrounding refractive index *nc* is changed, the different incident angles can be solved using Equations (1) and (2), and thus the angular sensitivity *Sa* = Δθ*i*/Δ*nc* can be solved, where Δθ*<sup>i</sup>* is the shift in the resonant angle induced by a change in the refractive index of the surrounding medium Δ*nc*.

As Equation (2) only stands for a single wavelength, it needs to be modified as follows to be suitable for multiple wavelengths (suitable for normal incidence only) [35]:

$$\frac{2\pi}{\Lambda}m\_{\&}d\_{\&\|}\cot(\theta\_d) - m\pi = \phi\_l + \phi\_{d\!\!/}m = 0, 1, 2, \dots, \tag{3}$$

Using Equations (1) and (3), the wavelength sensitivity *Sw* = Δλ/Δ*nc* can be solved, where Δλ is the shift in the resonant wavelength induced by a change in the refractive index of the surrounding medium Δ*nc*.

#### **3. Simulation Results and Analysis**

The RSoft 7.1 (RSoft Design Group, Inc., Ossining, NY, USA) based on rigorous coupled-wave analysis (RCWA) was used to simulate the GMR effect [36]. To calculate the field distributions, we used COMSOL Multiphysics 5.2a (COMSOL Inc., Stockholm, Sweden), which is based on the finite element (FEA) analysis method [45]. In our case, we simulated the GMR sensors made of a high-index material, such as silicon nitride (Si3N4, *ng* = *nwg* = 2.00) immersed in water (*nc* = 1.333) and with silicon dioxide (SiO2, *ns* = 1.45) as a substrate. The filling factor (FF) of the grating is fixed at 0.5 and the total thickness of the grating and waveguide layer *d* was also kept constant.

#### *3.1. Angular Shift Detection Scheme for "Grating–Waveguide" GMR Sensors*

For an angular interrogation technique, a monochromatic light source, such as a He–Ne laser, is typically used to measure the GMR effect in this case. The incident wavelength λ is set at 633 nm, and Λ is set at 280 nm. The total thickness of the grating and waveguide layer *d* is fixed at 100 nm. *dg* and *dwg* are the thicknesses of the grating and waveguide layer, respectively.

The schematic geometries of the angular GMR sensors are depicted in Figure 1a. The incident light can excite transverse electric (TE) or transverse magnetic (TM) modes, depending on the electric field *Ez* or magnetic field *Hz* perpendicular to the plane of incidence (*x*–*y* plane). When *nc* varies, the resonant angle θ*<sup>i</sup>* will change according to Equation (1). The angular position of the resonant peak will shift as illustrated in Figure 1b. Figure 1c shows the sensitivity and FWHM versus *dg* for TE polarization. *Sa* increases and the resonant linewidth broadens as *dg* increases for TE polarization. Consequently, the angular FOM cannot achieve a higher value with higher sensitivity, as shown in Figure 1d. However, *Sa* increases and the resonant linewidth decreases as *dg* decreases for TM polarization when *dg* is below 60 nm, as shown in Figure 1e. Figure 1f shows that, for TM polarization, the angular FOM increases as the *dg* decreases. A maximum FOM of 5709 was achieved when *dg* was 10 nm and *dwg* was 90 nm, which is higher than previously reported values [18]. The higher FOM not only arises from a lower *dg* resulting in a lower resonance linewidth, but also generates a higher sensitivity. The maximum FOM is 16.3 times higher than the minimum FOM of 350, as shown in Figure 1f.

**Figure 1.** (**a**) Schematic structure of the "grating–waveguide" guided mode resonance (GMR) structure for angular resonance. (**b**) An example of reflection spectra for oblique incidence at the resonance of a monochromatic light (*dg* = *dwg* = 50 nm for transverse magnetic (TM) polarization). Calculated sensitivity, resonant linewidth and FOM versus *dg* for transverse electric (TE) polarization (**c**,**d**) and TM polarization (**e**,**f**). Electric field distribution of the angular resonance of the GMR structure with different *dg* values of 50 nm (*dwg* of 50 nm), 10 nm (*dwg* of 90 nm) and for TE (**g**,**h**) and TM (**i**,**j**). (*d* is 100 nm, Λ is 280 nm, and the filling factor (FF) is 0.5).

TE and TM polarization have different performances when *d* is fixed at 100 nm. To investigate the different performances between TE and TM polarization, the COMSOL Multiphysics 5.2a (COMSOL Inc., Stockholm, Sweden) software was used to calculate the field distributions of resonant positions. Figure 1g,h, respectively, show the electric field distributions of TE polarization when *dg* is 50 nm and 10 nm. The electric fields are mainly stored in the waveguide layer. Therefore, a thick waveguide layer confines more electric energy, which results in a narrower resonant linewidth but reduces the sensitivity. Therefore, the value is mainly attributed to a narrow linewidth of 0.016 degrees at the cost of sensitivity, although a higher FOM of 881 was obtained for TE polarization. For the TM case, most electric energy is not confined in the waveguide layer and is distributed in the cover medium and substrate, as shown in Figure 1i,j. In the case where *dg* = 50 nm, up to 70.65% of the total resonance energy is distributed in the substrate, which is 82.56% of the total evanescent energy, and only 19.55% of the total resonance energy is distributed in the cover medium, which is 17.44% of the total evanescent energy. In the *dg* = 10 nm case, 62% and 25% of the total resonance energy is distributed in the substrate and cover medium, respectively, which respectively contain 69.72% and 30.38% of the total evanescent energy. Therefore, the evanescent energy increases by almost 13%, which leads to a higher angular sensitivity. Meanwhile, an obvious increase in the maximum electric intensity from 9.27 <sup>×</sup> 105 <sup>V</sup>/m (*dg* <sup>=</sup> 50 nm) to 4.44 <sup>×</sup> <sup>10</sup><sup>6</sup> <sup>V</sup>/m (*dg* <sup>=</sup> 10 nm) is observed for TM polarization, as shown in Figure 1i,j. This indicates that more energy was confined in the structure, thus decreasing the resonant linewidth. This result can also be proven by referencing [46], because the grating groove depth controls the coupling-loss coefficient in an exponential manner, thus the coupling-loss coefficient tends to be smaller as *dg* decreases. Therefore, when *d* is fixed at 100 nm for TM polarization, a shallow *dg* facilitates a narrow linewidth, and a relatively thick *dwg* (near 90 nm) also results in a higher sensitivity. A much higher FOM will be obtained thanks to the higher sensitivity and narrower linewidth.

To explain this phenomenon, we used the previous angular model to calculate *Sa*. The red curve in Figure 2a depicts the *Sa* of TE polarization, and a higher sensitivity region is located near *dwg* = 50 nm. The curved sensitivity line of TM polarization has the largest sensitivity region near *dwg* = 100 nm, as shown in Figure 2b. The highest sensitivity is 25.9 degrees/RIU (refractive index unit), which is very close to the highest sensitivity value in Figure 1f for TM polarization. Moreover, this largest sensitivity region *dwg* = 100 nm is the same as the value of the total depth of the grating and waveguide layer *d* that we set in Section 3.1. Combined with these results, the extremely high FOM accompanied by a high sensitivity and narrow linewidth shown in Figure 1f can be explained as follows: a shallower *dg* facilitates a narrower linewidth and a thicker *dwg*, approaching the highest sensitivity region (100 nm) and resulting in higher sensitivity.

**Figure 2.** The red curved line represents the analytical model of the *Sa* calculation for the transverse electric (TE) (**a**) and transverse magnetic (TM) (**b**) modes. Blue, green, and magenta marked symbols respectively represent calculation results of 10 nm, 25 nm and 50 nm *dg* of the rigorous coupled-wave analysis (RCWA) simulation method.

To further verify the analytical result, an RCWA method was used to compare the results. In Figure 2, blue, green, and magenta marked symbols represent the numerical results for cases with a *dg* of 10 nm, 25 nm, and 50 nm, respectively. The results display evident differences when *dg* is relatively high because the influence of the grating on both the phase shift and total internal reflection cannot be ignored. However, this model is still useful for calculating the angular sensitivity when the grating depth is small.

For optimizing the FOM of "grating–waveguide" GMR sensors, a high value region of sensitivity should be evaluated first and then *d* should be maintained near this value. Finally, by fabricating a shallower grating depth, higher sensitivity and narrower linewidth will be achieved, resulting in a higher FOM. This is a design guideline for "grating–waveguide" GMR sensors to achieve optimized FOM.

However, the TE polarization has its highest sensitivity region near *dwg* = 50 nm, as shown in Figure 2a. Based on the previous deduction, we set *d* = 50 nm and simultaneously varied *dg* and *dwg* (more details are shown in Table 1). A maximum FOM (with a narrower linewidth and higher sensitivity) of 1158 was achieved when *dg* = 10 nm and *dwg* = 40 nm. The missing values in Table 1 indicate the instances when the phase-matched condition was not satisfied [17,35].

**Table 1.** Linewidth (FWHM), sensitivity (S) and figure of merit (FOM) of angular resonance for *d* = 50 nm in transverse electric (TE) mode.


#### *3.2. Wavelength Shift Detection Scheme for "Grating–Waveguide" GMR Sensors*

The most common detection technique for GMR sensors is the wavelength shift-tracking method. A broadband light source is used, such as an LED light and white light source. To investigate the wavelength detection scheme, a normal incidence light was used and Λ was set at 410 nm. Similarly, we set *d* at 100 nm and *dg* and *dwg* were simultaneously varied. The schematic geometry of the GMR sensors is depicted in Figure 3a. When *nc* is varied, the resonant wavelength λ changes according to Equation (1). Thus, the resonant peak position of the wavelength will shift, as illustrated in Figure 3b.

Figure 3c,d demonstrates the relationship among resonant linewidth, sensitivity, and FOM for TE polarization. As *dg* decreases, a narrow linewidth and low *Sw* is obtained, and thus the FOM cannot achieve a relatively high value combined with high sensitivity. On the other hand, for TM polarization, *Sw* increases and the resonant linewidth decreases when *dg* decreases (*dwg* increases), as shown in Figure 3e. A maximum FOM of 2154 was achieved when the *dg* was 10 nm (*dwg* was 90 nm), as shown in Figure 3f, which is higher than those of common "grating–waveguide" GMR sensors [39].

**Figure 3.** (**a**) Schematic representation of the "grating–waveguide" structure for wavelength resonance. (**b**) An example of reflection spectra for normal incidence at the resonance of a broadband light (*dg* = *dwg* = 50 nm for transverse magnetic (TM) polarization). Calculated sensitivity, resonant linewidth and figure of merit (FOM) versus dg for transverse electric (TE) polarization (**c**,**d**) and TM polarization (**e**,**f**). Electric field distribution of the wavelength resonance of the guided mode resonance (GMR) structure with different *dg* of 50 nm (*dwg* of 50 nm), 10 nm (*dwg* of 90 nm) and for TE (**g**,**h**) and TM (**i**,**j**), for which the resonance wavelengths are 628.91, 651.51, 601.76 and 611.26 nm, respectively. *d* is set at 100 nm, Λ is 410 nm, and filling factor (FF) is 0.5.

The electric field distributions for both TE and TM polarization were simulated to further explain the previous results and are shown in Figure 3g–j. Compared to Figure 1g–j, similar results can be achieved here. For the TE mode, most of the evanescent field energy exists in the waveguide layer, and thus the sensing performances were affected. TM cases have a larger light-matter interaction region in the cover medium, and more light is strongly confined (the maximum electric intensity up to 3.61 <sup>×</sup> 106 <sup>V</sup>/m), making it more suitable to achieve better performances in biosensing. The sensitivity curve of the TM polarization has the largest sensitivity region near *dwg* = 100 nm, which is the same as the value of *d* that we set here. On the other side, the TE polarization has the largest sensitivity region near the 50 nm of *dwg*, and *d* = 50 nm is maintained. Table 2 shows the difference in performance caused by varying *dg* and *dwg*. As shown in Table 2, a higher FOM is obtained with a narrower linewidth and higher sensitivity.

**Table 2.** Linewidth (FWHM), sensitivity (S) and figure of merit (FOM) of wavelength resonance for *d* = 50 nm in transverse electric (TE) mode.


Combining these results with those of Section 3.1, TM polarization has better results for both angular and wavelength resonance, although different detection methods were used. This phenomenon can be explained in two respects: first, TM polarized modes have a larger phase shift length at the resonance point compared to TE modes, resulting in higher sensitivity, as shown in Figure 4. The red and blue curves represent the TM polarization phase shift in the waveguide layer at different surrounding *nc*. Yellow and green curves represent the TE polarization phase shift at different surrounding *nc*. The label *dwg* = 60 nm (purple curve) and *dwg* = 90 nm (gray curve) represent the calculated results coming from the left side of Equations (2) and (3) (called structure relation [35]). The intersection points of the structure relation curve and phase shift curve represent the points of resonant condition, where the GMR effect occurs. The length between two intersection points is proportional to resonant angle or wavelength shift. Therefore, a larger phase change length at different *nc*, results in greater sensitivity. In Figure 4, the TM polarization has a larger phase change length compared to the TE mode, when the *dwg* was 90 nm. Second, the coupling-loss coefficient for a TM polarization was smaller than that for the TE polarization, resulting in a narrow linewidth [46]. Furthermore, the phase shift length of *dwg* = 90 nm (purple curve) exceeds that of the *dwg* = 60 nm (gray curve) condition for TM polarization. This also explains why a higher sensitivity was obtained for the case where *dwg* = 90 nm, as shown in Figures 1f and 3f.

**Figure 4.** Phase shift curve in waveguide layer at different propagation angle, (**a**) angular resonance, (**b**) wavelength resonance.

#### *3.3. Optimized FOM for "Waveguide–Grating" GMR Sensors*

The schematic geometries of angular and wavelength resonance of the "waveguide–grating" GMR sensors are depicted in Figure 5a,b, respectively. Aimed at "waveguide–grating" GMR sensors, different *dg* values were swept by a "MOST Optimize/Scanner" (DiffractMod, RSoft 7.1, RSoft Design Group, Inc., Ossining, NY, USA) under certain Λ values to investigate whether a minimum resonant linewidth and a higher sensitivity exist for both the TE and TM modes. The chosen parameters are similar to those presented in Section 3.1 for the angular shift and Section 3.2 for the wavelength shift detection schemes, except *d* (*dg* and *dwg*).

For angular resonance, a better result was obtained under TM polarization; the series of results for different *dg* (from 150 nm to 400 nm) are shown in Figure 5c,d. Figure 5c shows the resonant linewidth and *Sa* versus *dg* for TM polarization. A minimum resonant linewidth and a higher sensitivity occur for a value of *dg* of 300 nm. A higher FOM, almost up to 106, was achieved, as shown in Figure 5d, which is 168 times greater than the highest FOM value presented in Figure 1f. For wavelength resonance, a superior result was obtained under TE polarization, and the series of results for different *dg* values (from 100 nm to 500 nm) are shown in Figure 5e,f. Figure 5c shows the resonant linewidth and *Sw* versus *dg* for TE polarization. A minimum resonant linewidth and a higher sensitivity will occur under 380 nm of *dg*. Thus, a high FOM of 1618 will be achieved as shown in Figure 5d. The electric energy distribution helps us to further understand the mechanism, as shown in Figure 5a,b. For both angular and wavelength resonance, the electric field will ascend from the substrate to the grating and the surrounding medium as *dg* increases. Therefore, the total evanescent energy will increase in the surrounding medium, thus increasing the sensitivity. Meanwhile, an obvious increase in the maximum electric intensity from 7.44 <sup>×</sup> 105 V/m (*dg* = 200 nm) to 3.89 <sup>×</sup> 107 V/m (*dg* = 300 nm) was observed for angular resonance under TM polarization, as shown in Figure 4g,h. For wavelength resonance, the maximum electric intensity increased from 1.84 <sup>×</sup> 105 V/m (*dg* = 200 nm) to 2.31 <sup>×</sup> 106 V/m (*dg* = 380 nm) for TE polarization, as shown in Figure 5i,j. These results indicate that more energy was confined in the GMR structure, and thus, the linewidth was decreased. Combining these two points, a higher FOM value was achieved, accompanied by high sensitivity and a narrow linewidth.

**Figure 5.** Schematic structure of the "waveguide–grating" structure for angular (**a**) and wavelength (**b**) resonance. Calculated sensitivity, resonant linewidth and figure of merit (FOM) versus *dg* for angular resonance under transverse magnetic (TM) polarization (**c**,**d**), and for wavelength resonance under transverse electric (TE) polarization (**e**,**f**). Electric field distribution of the "waveguide–grating" guided mode resonance (GMR) structure of the wavelength resonance with *dg* values of 200 nm and 300 nm for TM (**g**,**h**) and angular resonance with *dg* values of 200 nm, 380 nm for TE (**i**,**j**). Λ is 280 and 410 nm for the angular and wavelength resonance respectively and filling factor (FF) is 0.5.

Different Λ were chosen to determine the *dg* value that lead to a superior FOM. For wavelength resonance, different Λ varying from 330 nm to 550 nm, were chosen to calculate the position at which there is a minimum resonant linewidth and higher sensitivity for TE polarization, as shown in Table 3. The relationship among Λ, *dg*, and the corresponding FOM is depicted in Figure 6a.

**Table 3.** Linewidth (FWHM), sensitivity (S) and figure of merit (FOM) of the wavelength resonance

under different Λ and corresponding *dg* for transverse electric (TE) modes. **Λ (nm)** *dg* **(nm) FWHM (nm)** *S* **(nm**/**RIU) FOM** 330 306.44 0.12 195 1625 410 380.00 0.15 242.75 1618.333 450 417.50 0.16 266.00 1662.500 550 509.74 0.20 325.25 1626.250

**Figure 6.** The best figure of merit (FOM) value for wavelength (**a**) and angular resonance (**b**) under different Λ and corresponding *dg* values. Λ as a function of *dg* for wavelength (**c**) and angular (**d**) resonance.

For angular resonance, different Λ, varying from 260 nm to 320 nm, were chosen to determine the best FOM under TM polarization, as shown in Table 4. The angular sensitivity is inversely proportional to Λ. The relationship between Λ, *dg*, and the corresponding FOM is depicted in a three-dimensional schematic diagram in Figure 6b. All FOM values are higher than those of already reported values [18,39]. Λ as a function of *dg* for the wavelength and angular resonance are respectively shown in Figure 6c,d. The ratio of Λ to *dg* for the wavelength resonance was 0.925. A linear relationship was obtained between Λ and *dg* for the angular resonance, as shown in Figure 6d, and the formula of the best-fit line is *y* = 1097.411 − 2.808*x*. We can apply the linear relationship to design "waveguide–grating" GMR sensors with optimized FOM.


**Table 4.** Linewidth (FWHM), sensitivity (S) and figure of merit (FOM) of the angular resonance under different Λ and corresponding *dg* for transverse magnetic (TM) modes.

The point of these optimized FOM results is that a narrow resonant linewidth occurred abruptly, as shown in Figure 5c,e. Norton et al. investigated linewidth of "waveguide–grating" GMR structure through coupled-mode theory [47]. Briefly, the angular and wavelength linewidth is proportional to the coupling loss coefficient, and this coefficient is determined by the overlap of the bound mode and the radiation mode. At higher grating depths, the orthogonality of the two modes decrease the overall magnitude of the coupling loss, resulting in a narrow resonant linewidth. In our work, similar resonant linewidth phenomenon can be found in Figure 5c,e. Although Norton et al. just investigated one grating period, it can be predicted that different grating periods have their own grating depths, where the narrow resonant linewidth occurs abruptly.

#### **4. Conclusions**

In summary, with the aim of achieving a higher angular and wavelength FOM for GMR sensors, we systematically presented a parametric analysis elucidating the influence of structural design factors on the performance of GMR sensors. Combining an analytical model and numerical algorithm, we determine an effective and convenient method to achieve higher angular and wavelength FOM of "grating–waveguide" GMR sensors. A suitable fixed depth of the grating and waveguide will facilitate a higher FOM. This method is suitable for both angular and wavelength resonance. A linear relationship to design "waveguide–grating" GMR sensors with optimized FOMs was determined using numerical stimulations. These high FOM values can facilitate the performance of GMR sensors to achieve lower detection limits.

**Author Contributions:** Conceptualization, Y.Z. and X.W.; methodology, Y.Z. and X.W.; software, Y.Z., Z.G. and B.W.; validation, X.W., Z.G. and B.W.; writing—original draft preparation, Y.Z.; writing—review and editing, X.W.; funding acquisition, X.W.; All authors discussed the results and commented on the manuscript.

**Funding:** This work was supported by the Special Project of National Key Technology R&D Program of the Ministry of Science and Technology of China (2016YFC0201401), the National Key Basic Research Program of China (973 program-No. 2015CB352006) and the National Natural Science Foundation of China (NSFC) (61378080, 61327008, 61505032, and 61705039).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **High-Performance Ultraviolet Light Detection Using Nano-Scale-Fin Isolation AlGaN/GaN Heterostructures with ZnO Nanorods**

#### **Fasihullah Khan, Waqar Khan and Sam-Dong Kim \***

Division of Electronics and Electrical Engineering, Dongguk University, Seoul 100-715, Korea; fasihullah.khan@dongguk.edu (F.K.); waqarkyz@gmail.com (W.K.) **\*** Correspondence: samdong@dongguk.edu; Tel.: +822-2260-3800

Received: 24 February 2019; Accepted: 8 March 2019; Published: 15 March 2019

**Abstract:** Owing to their intrinsic wide bandgap properties ZnO and GaN materials are widely used for fabricating passive-type visible-blind ultraviolet (UV) photodetectors (PDs). However, most of these PDs have a very low spectral responsivity *R*, which is not sufficient for detecting very low-level UV signals. We demonstrate an active type UV PD with a ZnO nanorod (NR) structure for the floating gate of AlGaN/GaN high electron mobility transistor (HEMT), where the AlGaN/GaN epitaxial layers are isolated by the nano-scale fins (NFIs) of two different fin widths (70 and 80 nm). In the dark condition, oxygen adsorbed at the surface of the ZnO NRs generates negative gate potential. Upon UV light illumination, the negative charge on the ZnO NRs is reduced due to desorption of oxygen, and this reversible process controls the source-drain carrier transport property of HEMT based PDs. The NFI PDs of a 70 nm fin width show the highest *<sup>R</sup>* of a ~3.2 × 107 A/W at 340 nm wavelength among the solid-state UV PDs reported to date. We also compare the performances of NFI PDs with those of conventional mesa isolation (MI, 40 × <sup>100</sup> <sup>μ</sup>m2). NFI devices show ~100 times enhanced *<sup>R</sup>* and on-off current ratio than those of MI devices. Due to the volume effect of the small active region, a much faster response speed (rise-up and fall-off times of 0.21 and 1.05 s) is also obtained from the NFI PDs with a 70 nm fin width upon the UV on-off transient.

**Keywords:** high-responsivity; ultraviolet photodetectors; nano-scale fin isolation; wide-band gap semiconductors; ZnO nanorods; two-dimensional electron gas; visible-blind

#### **1. Introduction**

Ultraviolet (UV) light detection with high responsivity is of great interest, due to its promising applications in pathology [1], water treatment, safety, and defense-technologies [2]. In most commercialized systems, UV detection has been realized by photomultiplier tubes (PMTs), thermal detectors, Si or Ge based photodetectors (PDs), and charge-coupled devices (CCDs). However, these detection methods pose various problems for practical applications. For example, PMTs exhibit fragile vacuum-tube construction and require a high-voltage bias. Moreover, PMTs are sensitive to the magnetic-field; therefore, they need to be shielded from ambient light. Si based PDs or CCDs are one of the most commonly used solutions, due to their highly matured process technology [3]. Recently, n-Si/p+-B photodiodes of a very high sensitivity and stability have been demonstrated, but they still exhibit responsivity in the order of ~10−<sup>1</sup> A/W [4]. Many attempts have been made to improve the limited responsivity of Si PDs working in linear mode. Among them, avalanche-mode PDs are now very popular, and single-photon avalanche diodes are also used to obtain high-gain responsivity [5]. However, the use of stop-band filters for visible light reduces the quantum efficiency of these devices in the UV range.

In recent years, the nanostructures of wide bandgap (WBG) materials, such as diamond, ZnO, III-nitride, and SiC based UV PDs have attracted tremendous research interest for their many advantages [6]. For instance, they are capable of room temperature (RT) operation and have intrinsic optical transparency (visibly-blindness) in the visible spectral range. Furthermore, they have a low thermal conductivity, high breakdown field, and stability at elevated temperatures. In spite of these promising advantages, the WBG semiconductors have very low electron mobility. Even though noble heterostructures with a very high electron mobility of ~10<sup>6</sup> cm2/V·s using a material system such as MgZnO/ZnO [7] have been reported, most of the passive PDs fabricated using conventional WBG semiconductors have very low spectral responsivity [8]. Moreover, the response speeds of PDs based on ZnO or GaN are very slow in general because the photoresponse characteristics depend on the well-known bottle-neck chemisorption process of oxygen at the surface of such materials [9].

Zinc oxide (ZnO) nanorod (NR) based UV PDs [10–15] have shown promising results in terms of response speed and spectral responsivity [16]. Significant research effort has been made to improve the performance of the PDs, either by improving the crystalline quality of NRs or by utilizing the composite coaxial structure of ZnO with other materials, such as copper oxide [17] and graphene [18]. However, most of the passive type PDs reported to date exhibit very slow response speed (tens of seconds) and low responsivity (~hundreds of A/W). Recently, an active-type UV PD using the ZnO NR-gated AlGaN/GaN high electron mobility transistor (HEMT) structure has been attempted as a part of an effort to attain a breakthrough in responsivity (~105 A/W) and to obtain a relatively fast response speed [19]. However, due to the very high dark current of the device, the HEMT based PD showed a relatively low on-off current ratio of ~3.

In this study, we demonstrate high-responsivity UV PDs based on the ZnO NR-gated AlGaN/GaN HEMT structure with nano-scale fin isolation (NFI). ZnO NRs act as the floating gate while the UV driven chemisorption process of oxygen at the surface of ZnO NRs [20] controls the conduction of the underlying two-dimensional electron gas (2-DEG) channel. The 2-DEG present at the interface of the AlGaN (barrier) and GaN (channel) layers is due to polarization induced electric potential in the heterostructure [21]. In the NFI PD structure, ZnO NRs surround the channel in the gate area. Consequently, the carriers in the 2-DEG channel are confined along the channel and in perpendicular direction to the interface. Therefore, under dark conditions, the 2-DEG channel is fully depleted due to negative surface potential generated via oxygen adsorption at the surface of ZnO NRs [9]. This structure resembles enhancement-mode (normally-off) AlGaN/GaN fin-shaped field-effect transistors (FINFETs) [22], demonstrating extremely broad transconductance and excellent off-state characteristics. In this study, the performance of the NFI PDs is compared to conventional mesa isolation (MI) AlGaN/GaN HEMT based PD of ~100 × <sup>40</sup> <sup>μ</sup>m2 active area. We examine various photoresponse characteristics of NFI and MI devices and investigate how structural differences influence their performance.

#### **2. Device Fabrication and Characterization**

Figure 1 shows schematic illustrations of two different HEMT based PD structures examined in this study. The interface of the GaN (channel) and AlGaN (barrier) layers have a confinement of 2-DEG, which acts as channel of very high electron concentration and mobility.

Figure 2 illustrates the essential process steps for our gateless NFI and MI HEMT photodetectors (PDs). The epitaxial layers of AlGaN/GaN were deposited on a 6 inch Si (111) by a metal-organic chemical vapor deposition system at NTT-AT (NTT-Advanced Technology Corporation, Kanagawa, Japan). As shown in the schematics, the undoped GaN buffers and channel layers of 3000–4000 nm thickness were grown first, followed by subsequent growths of the barrier layer of ~20 nm Al0.25GaN0.75 and the GaN cap layer of ~1.2 nm. The measured electron sheet carrier concentration and Hall mobility of the epitaxial layer were ~5 × 1012/cm2 and ~1750 cm2/V·s, respectively. The active area (100 × <sup>40</sup> <sup>μ</sup>m2) for the gateless HEMT was defined by a mesa etching of 100 nm depth in the case of conventional MI structures, as shown in Figure 1. The NFI structure, on the other hand, was fabricated

with 10 nano-fin-shaped isolations (10 gate fingers) with the same mesa depth of 100 nm and two different fin widths (*Wfin*) of 70 and 80 nm. By opening the 100 nm Si3N4 passivation layers deposited on the active region by plasma-enhanced chemical vapor deposition (PECVD), the gate area of a 2 μm gate length was defined. The ZnO NRs were then selectively grown as an active element for the UV light detection.

**Figure 1.** Schematics of the NFI (nanoscale fin isolation) and MI (mesa isolation) structures for the AlGaN/GaN HEMT (high electron mobility transistor)-based UV photodetectors.

**Figure 2.** Schematic illustration of the major process steps used for the fabrication of the NFI and MI devices.

The epitaxial layers were cleaned and agitated with acetone and ethanol in bath sonication to remove dust and surface contamination, followed by rinsing in deionized (DI) water, then drying with nitrogen (N2) gas. Optical lithography (Karl Suss, Garching, Germany, MA6 mask aligner, 365 nm) was used to define the active MI regions, including all the device patterns used in this experiment. However, patterning for the NFIs was performed by an electron beam lithography tool (Jeol, Tokyo,

Japan, JBX-9300FS, 100 keV) with a 70 nm PECVD SiO2 hard mask to avoid the mask pattern erosion during the mesa etching. After the pattern development of the active regions, mesa etching was performed by removing 100 nm of thickness from the peripheral areas in a reactive ion etching system (RIE, STS Multiplex ICP) using BCl3 and Cl2 gases. Ohmic contacts were achieved by depositing the metal stack of Ti/Ni/Au (20/30/80 nm), by using an electron beam evaporation system, and a pattern lift-off method, using image reversal photoresist. Ohmic metals were then subjected to a subsequent rapid thermal alloy process at 900 ◦C for 35 s in N2. The samples were then passivated by a 100 nm silicon nitride (Si3N4) layer deposited in a PECVD system at 200 ◦C and RF power of 1 kW using a NH3/SiH4 gas flow rate ratio of 1.5. The gate areas (2 × 100 μm) of two different structures were opened by etching the Si3N4 passivation using RIE in CF4 plasma at a gas flow rate of 110 sccm and a chamber pressure of 40 mTorr (at an etching rate of ~9.4 nm/min) under the RF power of 100 W.

ZnO NRs were then grown in the gate area by using the hydrothermal synthesis method. Prior to the growth of NRs, a 20 nm thick seed layer (SL) (as shown in the top-right inset of Figure 3c) was deposited by spin coating the seed solution (3000 rpm and prebaking at 120 ◦C for 60 s) repeatedly 15 times. After that, the crystalline quality of the SL was improved by annealing the samples at ~350 ◦C on a hotplate for 1 h. The seed solution used in this study was prepared by dispersing 0.66 g of zinc acetate-dehydrate (C4H6O4Zn·2H2O) salt in 30 mL of 1-propanol (CH3CH2CH2OH). In this work, a 20 mM zinc-acetate-dehydrate concentration for seed solution was selected for the desired SL film quality, assuring the ZnO NR crystalline characteristics to be used for this PD application [23]. Various attempts, such as vacuum annealing and O2 plasma post-treatment methods [10,24] to improve the SL film quality, which is the key to the high-quality NR crystallites, are also underway in our laboratory. The growth of ZnO NRs was limited to the gate area, which was done by etching the SL from all other areas except the gate region. The samples were then placed in a growth solution for 6 h on a hotplate at ~90 ◦C to grow ZnO NRs. The growth solution was prepared by mixing 0.25 mole equimolar concentration of zinc-nitrate-hexahydrate (Zn(NO3)2·6H2O, 99%) and hexamethylenetetramine (HMTA) (C6H12N4, 99.5%) in deionized (DI) water. After growth of the the NRs, the samples were carefully cleaned with acetone, ethanol, and DI water, sequentially.

Sloped etching profiles for the NFI structure, the growth morphology of NRs, and the processed device structures were examined by plane-view scanning electron microscopy (SEM, 10 kV S-4800S-Hitachi, Tokyo, Japan). As shown in the top-right inset of Figure 3a, the measured bottom dimensions of the trapezoidal fins were 170 nm after RIE for the *Wfin* of 70 nm. The transmission electron microscope (TEM, 9500-Hitachi) was used to characterize the cross-sectional view and crystalline quality of the as-grown NRs. The RT photo-luminescence (PL) emission spectra from the as grown NRs was obtained by using a He-Cd laser illumination source with a 325 nm wavelength. Gate areas opened with a trench-shaped pattern inside the dark-shaded Si3N4 passivation and the NRs grown on them are shown in the plane-view SEM micrographs in Figure 3a,b. An inset (bottom-right) of Figure 3a shows the morphology of the ZnO NRs grown around the underlying nano-fins in the gate area of the NFI structures at a magnification of ×3000. Across the gate region, 10 fingers of NFIs are running in parallel, whereas the NRs are grown all the way vertically along the gate area in the MI structure, as shown in Figure 3b. The SEM image in Figure 3c shows that the NRs exhibit an average diameter of ~85 nm and a length of ~1.4 μm. Figure 3d shows a PL spectrum measured at RT from the as-grown ZnO NRs employed as a light absorbing structure in this work. PL characterization at RT is one of the most efficient tools to evaluate the crystalline quality for the WBG ZnO NRs of a direct band gap property. The spectrum exhibits a strong near band edge emission peak at a wavelength of ~380 nm, which is mainly associated with the band-to-band excitonic recombination of ZnO [25]. The near band edge emission intensity was increased about two times with the increase of the NR aspect ratio (AR, length/diameter) from ~8 to ~16. For this reason, NRs with an AR of ~16 were used in this experiment. Emissions in a visible range (420–650 nm) are due to various form of intrinsic defects, such as oxygen vacancy, zinc vacancy, and hydrogen and oxygen interstitials [10]. Despite the inevitable intrinsic defects and consequent visible emissions, as observed from most of

the nano-crystallites grown through the aqueous solution based growth methods, our PL spectrum reveals fairly good optical properties in our ZnO crystallites compared to those of the NRs grown using similar methods [2,26,27].

**Figure 3.** Plane-view SEM images of (**a**) 70 nm NFI PDs (photodetectors) (×1000) with an enlarged top-view of the NFI (*Wfin* = 70 nm) (top-right inset, ×5000) and ZnO NR (nanorod) profiles (bottom-right inset, ×3000) and (**b**) MI PDs (×500) with enlarged ZnO NRs profile (bottom-right inset, ×3000). (**c**) Cross-sectional SEM image of the as-grown NRs. Cross-sectional TEM image of the NRs and SL (seed layer) (top-right inset). Top-view of the NRs (bottom-right inset). (**d**) RT (room temperature) PL (photo-luminescence) spectrum of the as-grown ZnO NRs.

A wide band (300–700 nm) Xenon (Xe) lamp was employed as a light source to measure the photoresponse characteristics of the fabricated devices under optical intensities ranging from 0.5 to 16.5 μW/cm2 (100–300 W lamp power). The transient characteristic measurements, according to the UV light (370 nm) on-and-off transient, were carried out by a programmable light shutter controlled in our measurement set-up. The transients of the drain current (*Ids*) for the HEMT PDs as a function of time were recorded in a Keithley source measurement unit with a floating gate configuration at a drain voltage (*Vds*) of 4 V. The spectral response of the PDs was measured by a focused illumination of a monochromatic light from a wide band (300–1100 nm) Xe-lamp (Ushio UXL-75XE, Ushio Inc., Tokyo, Japan) light source of a 16.5 μW incident optical power. In this measurement set-up, the responsivity was measured at a light chopping frequency of 30 Hz using a lock-in amplifier in a series configuration with drain-source probes for the detection of an amplified change in current. The change of *Ids* as a function of incident light wavelength varying from 300 to 800 nm was recorded under a floating gate configuration at a *Vds* of 4 V. To obtain a monochromatic light from the Xe-lamp, a spectral optics monochromator (CM110 1/8 m) with 2400 lines/mm grating was used.

#### **3. Sensing Mechanism**

As illustrated in the schematics of Figure 4, the UV sensing mechanism of the HEMT based PDs depends on the chemisorption of oxygen at the surface of NRs and the consequent change in carrier concentration in underlying 2-DEG. The as-grown crystalline ZnO-NRs are n-type in nature due to a large number of donor defects, such as hydrogen interstitial and Zn interstitial [28]. Under dark conditions (no UV light illumination), oxygen molecules (O2) transported from the ambient air to the ZnO NR surface tend to trap electrons from the conduction band of ZnO and leave behind positively charged ionized donors in the surface space charge region, while the negatively charged O2 molecules are fixed to the surface of NRs as adsorbed oxygen ions (O− 2ads ):

$$\text{O}\_2 + \text{e}^- \rightarrow \text{O}\_{2\_{\text{ads}}}^-. \tag{1}$$

This reaction leads to the expansion of the space charge region near the surface of the NRs due to the depletion of the surface electron states by O− 2ads , as depicted in the left of Figure 4a. As a consequence, the adsorption process gives rise to a negative potential at the gate of our HEMT based PDs (as shown in the left of Figure 4b,c), thereby dropping the carrier concentration in the 2-DEG channel at the interface of AlGaN/GaN. This process will eventually reduce the conductance of channel and drain-to-source current (*Ids*) in dark conditions. The Schottky gate (Ni/Au) HEMTs (2 μm gate length, 100 μm gate width) fabricated in our group using the same epitaxial structure and Si3N4 passivation showed a threshold voltage (*VTH*) of ~−3V[29]. This result demonstrates that our HEMT devices with an Au/Ni gate electrode are normally on at zero gate voltage *Vgs* (depletion-mode). We suppose that the *VTH* of the HEMTs is given by [30]

$$
\left(\oslash\!\_{\text{b}} - \Delta E\_{\text{c}} - \left(\ulcorner \text{en}\_{\text{s}} \text{d}^{2}\right) / \text{2}\varepsilon\right) / \text{2}\varepsilon \tag{2}
$$

where ∅*<sup>b</sup>* is the Schottky barrier height, Δ*Ec* is the conduction band offset, *e* is the electron charge, *ns* is the 2-DEG sheet carrier density, *d* is the barrier layer thickness, and *ε* is the dielectric constant. Because the difference between the estimated ∅*<sup>b</sup>* of Ni at the interface with GaN (~1.1 eV) [31] and the GaN surface band bending (~1.0 eV) [32] in ambient air (fully saturated by O− 2ads in the dark state) is quite small, it can be reasonably assumed that a significant amount of dark current (*Idark*) through the 2-DEG channel is unavoidable from our gateless MI HEMT devices under a floating gate condition due to their normally-on characteristics as shown in Figure 4c.

On the other hand, a different mode of operation is expected for the NFI structure because the 2-DEG channel region inside the nano-fins is three-dimensionally surrounded by a free surface, whereon the ZnO NRs are grown with many O− 2ads around the surface in dark conditions, as shown in Figure 4b. This makes the NFI PDs operate more closely to a normally-off mode (enhancement mode), thereby exhibiting a much lower *Idark*, because the carriers in the 2-DEG region can be highly depleted, even under the floating gate conditions, due to the surface depletion of the NRs by the O− 2ads surrounding the extremely small volume of the active region. The same phenomenon was also observed in the AlGaN/GaN heterojunction FinFETs [33]. With the decrease of *Wfin* from 200 to 60 nm, it was observed from the HEMT devices that electron density in the 2-DEG channel at zero *Vgs* rapidly drops with a positive shift in *VTH* because of the fringing-field from the side gates depleting the 2-DEG channel. As a result, the fin-HEMT showed a change in the conduction mechanism from normally-on to normally-off modes.

Due to light absorption, UV light illumination generates electron-hole pairs near the surface of the ZnO NRs [34]. The generated holes recombine with the electrons trapped by O− 2ads at the surface. In this way, the O2 molecules start to desorb from the surface, as shown in the right of Figure 4a. This phenomenon gives rise to a reduction of negative charge in the gate region, thereby increasing the carrier concentration in the 2-DEG channel and the drain to the source current (*Iphoto*) under UV illumination, as shown in the right parts of Figure 4b,c.

**Figure 4.** Schematic illustrations of (**a**) oxygen adsorption (left: in the dark) and desorption (right: under UV illumination) at the surface of ZnO-NRs. Oxygen adsorption (left: in the dark) and desorption (right: under UV illumination) processes are taking place at the surface of ZnO NRs grown on the gate areas of (**b**) NFI and (**c**) MI PDs. Cross-sectional schematics of each PD structure viewed in y-axis direction are illustrated; 2-Deg (two-dimensional electron gas).

As long as the response speed of PDs is accounted for, it can be assumed that the response kinetics upon the UV light on-and-off transient are not controlled by the drift motion of electrons in the 2-DEG channel but by the adsorption and desorption reaction of O2 on the ZnO NR surface. This result occurs because the mobility of the electrons confined two-dimensionally in the channel region (~1750 cm<sup>2</sup> V<sup>−</sup>1s−1) [33,35] is so fast that the carrier channel transit time cannot be a bottleneck parameter controlling the whole sensor response speed. As proposed in our previous study, the rate of charge change *dQ*/*dt* can be given by [35]:

$$d\mathbf{Q}/dt = -\mathfrak{a}e^{\mathfrak{f}\mathbf{Q}}\tag{3}$$

where *α* and *β* are the constants. By numerically solving Equation (3), it can be determined that one critical parameter affecting the response time is the gate area. From the calculations based on this model, the response (or recovery) time increases with the increase of gate area caused by the consequent increase of the total gate charge. We have much smaller gate capacitance in the NFI PD structure than in the MI structure; therefore, a faster response speed upon UV transient illumination can be expected from the NFI structure.

Spectral responsivity *R*, which can be defined as a ratio of *Iphoto* − *Idark* to the incident optical power *Pi*, is one of the key measures to evaluate the performance of PDs. As reported in our previous studies [19,35], a significant enhancement in *R* was shown from the NR-gated PDs due to a vast surface area of the ZnO nanostructure and a much higher surface-to-volume ratio than that of the planar ZnO thin film gate structure. In our HEMT-based PDs, the *R* can be also significantly influenced by the gain characteristic for the HEMT, which can be expressed by the change of *Iphoto* according to the change of light power irradiating on the gate area.

#### **4. Results and Discussion**

Figure 5a shows the equivalent electrical circuit diagram of both MI and NFI devices, where MI PD is illustrated as a normally-on transistor while the NFI PD is a normally-off transistor. To examine the characteristic difference of UV responses from the two different PD structures, we first measured *Ids* in dark conditions (*Idark*) and under UV illumination (*Iphoto*) at drain voltage (*Vds*), ranging from 0–5 V. UV light exposure was provided from an Xe lamp, operating at 300 W power, with a monochromatic light filter at 370 nm. This UV source produced an incident light intensity of 16 μW/cm2, as measured by a power meter. Figure 5b demonstrates that our MI PD exhibits a high *Idark* of ~10 mA/mm at a *Vds* of 4 V due to the enhancement-mode operation, as discussed in the previous section. In the case of NFI PDs, significantly reduced values of *Idark* (~0.19 and ~0.27 mA/mm for *Wfin* of 70 and 80 nm, respectively) were measured at a *Vds* of 4 V.

**Figure 5.** Photoresponse of MI and NFI PDs. (**a**) Equivalent electrical circuits of MI (left) and NFI devices (right). (**b**) Measured *Ids* versus *Vds* and (**c**) on-off current ratio (*Iphoto/Idark*) of MI and NFI PDs. Time resolved photoresponse characteristics of (**d**) NFI and (**e**) MI PDs (*Vds* = 4 V). (**f**) Magnified views of transient characteristics of NFI (left) and MI (right) PDs.

The performance of PDs can be evaluated by few important parameters, such as spectral responsivity *R*, photoconductive gain *G*, specific detectivity *D*\*, and sensitivity *S* [36]. Despite the high *R* nature of our MI HEMT-based PDs, this high *Idark* of the MI structure can critically deteriorate the photo-sensitivity performance associated with *S,* which is given by (*Iphoto* − *Idark*)/*Idark*), or the on-off current ratio (*Iphoto*/*Idark*), as well as *D*\*. As shown in Figure 5c, much improved on-off current ratios (290~340) were recorded from the NFI PDs compared to those of the MI devices (~4). This result is mainly due to the suppression of *Idark* caused by the fully depleted 2-DEG channel of the NFI devices in dark conditions.

The high-speed transient characteristic of the UV PD is one of the key factors for real-time application. Specific cases, such as the non-invasive assessment of cancer cells by optical biopsy [37], require ultra-fast PDs with response and recovery time in the order of a few milliseconds. To assess the photocurrent transient of the fabricated devices, the change in *Ids* as a function of the on-and-off of the UV light exposure time was measured using a programmable shutter. Figure 5d,e shows that both the NFI and MI PDs produce sharp increases in *Ids* upon UV illumination and a slower fall-off upon termination of UV exposure. The NFI device with a 70 nm *Wfin* showed the fastest rise-up time (or UV response time) and fall-off time (or recovery time) of 0.21 and 1.05 s, respectively. We hereafter define the rise-up and fall-off time as the time intervals for *Ids* to ramp up to 90% of the maximum saturation value after UV turn-on and to ramp down by 90% from the maximum value after UV-off. On the other hand, the MI device showed much slower response and recovery times of ~0.71 s and ~1.84 s, respectively. This significant improvement in response speed of the NFI device is due to the minimized dimension of gate area where the light absorption takes place; therefore, less time is required to complete the O2 adsorption-desorption process in the smaller area of the NFI PDs than in the larger area of the MI device, as discussed in the previous section. The ZnO NR based photoconductive PDs [26,38] still show a long recovery time on a seconds scale, even though a great amount of research effort has been made on enhancing the response speed of these passive PDs. UV PDs based on ZnO nanowire networks with Pt contacts have been fabricated on glass substrates by exhibiting a fast recovery time of 0.2 s with a high photosensitivity (~5 × <sup>10</sup>3) at 365 nm [39]. The fastest UV PDs of GaN-based metal-semiconductor-metal, p-i-n, or metal Schottky barrier devices [6] have shown extremely high speed (from microseconds to picoseconds) and low-noise capabilities. However, PDs of these GaN-based structures, developed specially to improve the UV response characteristics, exhibit a very low spectral responsivity of less than 1 A/W [8].

Figure 6a shows the schematic illustration of PD before and after the growth of the ZnO NRs. Shown in Figure 6b,c are the measured *Idark* values of the two different device structures before and after NR growths on the gate region. For the MI PDs, the dark currents were significantly reduced by the attachment of NRs on the gate region. However, the *Idark* values measured over the entire drain voltage range were still very high (~10 mA/mm) due to the incomplete depletion of the 2-DEG channel in a large volume of the active region. The measured *Idark* of the NFI PDs before the NR growth were very high, reaching up to ~560 mA/mm, but they rapidly dropped to a very low level of ~hundreds of μA/mm by the attachment of NRs. This result reveals that both the surface depletion effect in the small volume of an active region and the attachment of the NRs with many O− 2ads on their surfaces lead to the formation of a fully-depleted 2-DEG channel in the NFI structure.

**Figure 6.** (**a**) Schematic illustration of NR-gated PD and Gateless PD. Measured *Idark* of MI (**b**) and NFI (**c**) device structure before (gateless) and after NR growths (NR-gated).

The performance of the PDs can be also assessed by spectral responsivity *R* and specific detectivity *D*\*, which are expressed in the following equations [36]:

$$R = \frac{I\_{photo} - I\_{dark}}{P\_l} \tag{4}$$

$$D\* = \sqrt{A} \text{R} / \sqrt{2qI\_{dark}}\tag{5}$$

where *A* is the active area of device, *q* is the electron charge, and *Pi* is the radiant light power incident on the active area of the device. The responsivities of our PDs were measured using an incident light intensity of 16 <sup>μ</sup>W/cm2 with an effective area of 2.4 × 2.4 mm2. NFI PDs showed an extremely high responsivity of ~3.2 × 107 A/W at 340 nm, which is the best performance among any solid-state PDs reported to date [40–44], and even ~100 times higher than that (~2 × 105 A/W at 340 nm) of the MI device, as shown in the right of Figure 7a. This improvement in responsivity of the NFI PD is attributed to a higher photonic transconductance (hereafter, we call it *gm,photo*) characteristic of our HEMT-based PDs, which represents the ratio of the photocurrent change at the drain terminal to the change in incident optical power at the gate terminal. We define the *gm,photo* as following.

$$\mathcal{g}\_{m,photon} = \frac{\Delta I\_{photo}}{\Delta P\_l} (A/W). \tag{6}$$

**Figure 7.** (**a**) Specific detectivity (left) and spectral responsivity (right) as functions of radiant light wavelength measured from NFI and MI UV detectors. (**b**) *Iphoto* versus *Pi* of the NFI (left) and MI (right) PDs measured at *Vds* of 5 V.

To calculate the *gm,photo* values of the two different device structures, the *Iphoto* values were measured at various optical intensities, varying from 0.5 to 16 μW/cm<sup>2</sup> at a *Vds* of 5 V. Shown in Figure 7b is the measured *Iphoto* versus *Pi* from the MI and NFI PDs. The estimated *gm,photo* values for the NFI and MI PDs were 3.63 × <sup>10</sup><sup>8</sup> and 2.58 × <sup>10</sup><sup>6</sup> A/W, respectively. Even though this *gm,photo* parameter does not directly represent the gain (*gm* = Δ*Ids*/Δ*Vgs*) characteristics of the field-effect transistors, it can be used as a performance measure to estimate the gain characteristic of our gateless HEMT PDs, because the change of incident optical power is directly associated with the virtual Δ*Vgs* induced by the change in numbers of O− <sup>2</sup> ads on the ZnO NR surface under the illumination of UV light.

FinFET technology has recently shown a major increase in adoption of use within Si integrated circuits. The advantages of a FINFET structure, even though there are a number of subtly different forms, can be numerous, but they are basically based on "channel controllability" in a nano-scale channel-length regime of FETs [45]. Furthermore, Si-based FINFETs have shown a significant enhancement in the dependence of *Ids* on *Vgs* at any applied bias in the sub- and near-threshold regimes by the superior electrostatics of the device architecture [46]. Normally-off Al2O3/GaN metal-insulator FINFETs (*Wfin* = 50 nm, 1 μm gate length) also showed very high maximum *Ids*, *gm*, and maximum field-effect mobility of 1.51 A/mm, 580 mS/mm, and 293 cm2V−1s−1, respectively, due to the more effective increase of 2-DEG electron concentration and higher electron mobility by enhanced gate controllability than the planar devices [47,48]. The superior performance of *R* in our NFI PDs is most likely attributed to this enhancement of "gate controllability" screening the field lines effectively from the interface traps or the defects near the 2-DEG channel, thereby reducing the electron scattering in the GaN channel.

The *D*\* of a PD, as defined in Equation (5), is also an important figure of merits used to describe performance. The NFI PDs of 70 nm *Wfin* exhibited a maximum *<sup>D</sup>*\* of ~3.2 × <sup>10</sup><sup>12</sup> Jones (cm·Hz<sup>−</sup>1/2/W) at 340 nm, which is four orders of magnitude higher than that of MI PDs (~4 × <sup>10</sup><sup>8</sup> Jones), as shown in the left of Figure 7a, which is mainly due to the very low *Idark* of the NFI structure. A very high *D*\* up to a value of 1.4 × <sup>10</sup><sup>15</sup> Jones [14] was reported from the ZnO NRs based UV PDs. However, the *<sup>R</sup>* achieved by this device was much lower (~10<sup>3</sup> A/W) than that of PDs.

#### **5. Conclusions**

Most of the performance parameters, such as *R*, *D*\*, on-off current ratio, and response speed, were all significantly improved by employing the NFI structure for the AlGaN/GaN HEMT based UV PDs with a ZnO NR UV-absorbing structure. The NFI PD, especially, exhibited an extremely high *<sup>R</sup>* of ~3.2 × 107 A/W. This performance enhancement was due to the subsequent characteristic change of the gateless HEMTs, induced by the reduction of *Wfin* to 70–80 nm. The NFI structure significantly improved the gain characteristics caused by enhanced gate controllability in nano-fin channels beyond the inherent high performance in conversion efficiency of the photon to electron-hole pair generation due to the large surface-to-volume ratio of the ZnO NRs grown in the active region. As the width of the 2-DEG channel is reduced by the NFI profile, the side-wall surface depletion in nano-fins and the attachment of NRs with numerous O− 2ads on the surface of the NRs lead to the formation of a fully-depleted 2-DEG channel and pushed the *VTH* to a positive value. The measured high on-off current ratio and *D*\* are mainly due to this normally-off operation of the NFI PD structure. An improvement in response speed of the PDs is associated with the minimized dimension of the gate area and the resulting gate capacitance of the NFI structure, where much less time for charging or discharging is required for the O2 adsorption-desorption process. The fabricated PDs also showed a linear dependence of photocurrent on the input light intensity in a range of 0.5–16.5 μW/cm2, regardless of device structure. The measured *gm,photo* value for the NFI PDs of 70 nm *Wfin* was 3.63 × <sup>10</sup><sup>8</sup> A/W, which was ~100 times greater than that of the MI PDs.

**Author Contributions:** Conceptualization, F.K.; methodology, F.K.; validation, F.K.; formal analysis, F.K.; investigation, F.K.; resources, S.-D.K.; data curation, F.K.; writing—original draft preparation, F.K.; writing—review and editing, F.K. and S.-D.K.; visualization, S.-D.K.; supervision, S.-D.K. and W.K.; project administration, S.-D.K.; funding acquisition, S.-D.K.

**Funding:** All the experiments performed in this work are financially sponsored by National Research Foundation (NRF) grant (2016006533) under the program: "Mid-career Researchers" by the Ministry of Science and Technology, Republic of Korea. Similarly, the materials characterization was achieved with the financial support from NRF, Republic of Korea under the program: "Nano-materials Technology Development" (2009-0082580) managed by Ministry of Science, ICT and Future Planning.

**Acknowledgments:** The authors would like to thank Jihyun Kim, Hafiz Muhammad Salman Ajmal, Kiyun Nam and Noor ul Huda from Dongguk University for the continued encouragement, help, and support in carrying out the experiments and preparation of results.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Porous Si-SiO2 UV Microcavities to Modulate the Responsivity of a Broadband Photodetector**

#### **María R. Jimenéz-Vivanco 1, Godofredo García 1, Jesús Carrillo 1, Francisco Morales-Morales 2, Antonio Coyopol 1, Miguel Gracia 3, Rafael Doti 4, Jocelyn Faubert <sup>4</sup> and J. Eduardo Lugo 4,\***


Received: 28 November 2019; Accepted: 9 January 2020; Published: 28 January 2020

**Abstract:** Porous Si-SiO2 UV microcavities are used to modulate a broad responsivity photodetector (GVGR-T10GD) with a detection range from 300 to 510 nm. The UV microcavity filters modified the responsivity at short wavelengths, while in the visible range the filters only attenuated the responsivity. All microcavities had a localized mode close to 360 nm in the UV-A range, and this meant that porous Si-SiO2 filters cut off the photodetection range of the photodetector from 300 to 350 nm, where microcavities showed low transmission. In the short-wavelength range, the photons were absorbed and did not contribute to the photocurrent. Therefore, the density of recombination centers was very high, and the photodetector sensitivity with a filter was lower than the photodetector without a filter. The maximum transmission measured at the localized mode (between 356 and 364 nm) was dominant in the UV-A range and enabled the flow of high energy photons. Moreover, the filters favored light transmission with a wavelength from 390 nm to 510 nm, where photons contributed to the photocurrent. Our filters made the photodetector more selective inside the specific UV range of wavelengths. This was a novel result to the best of our knowledge.

**Keywords:** photonic nanoscience; nanotechnology; porous Si-SiO2; UV filters; responsivity

#### **1. Introduction**

Porous silicon (PS) is a promising material for many different applications, such as solar cells, specifically as anti-reflection coating [1], chemical sensing [2,3], biomedical applications [4], biosensing [5,6], as a photodetector [7], or light-emitting diode [8]. The anti-reflection coating (ARC) reduces reflection losses thereby increasing the photocurrent and efficiency in solar cells [9–12]. A microcavity filter (MF) obtained in the IR range has been used as the ARC, where the maximum transmission at the localized mode wavelength was 95.2%, with a minimum reflectance of 4.8% at the same wavelength. The PS ARC was deposited onto a solar cell, where the entire surface of the solar cell was covered, and its external quantum efficiency (EQE) was measured, with and without the MF. The EQE at different wavelengths matched the shape of the MF transmission spectra [9]. ARCs are used to convert down the higher energy solar radiation into a wide range of light spectra, which is absorbed more efficiently into bulk Si [10]. In another interesting application, silver nanoparticles were infiltrated within a porous silicon photonic crystal to detect the trace of explosives (Rhodamine 6G dye

and Picric acid explosives) using surface-enhanced Raman scattering [13]. Recently, porous silicon nanoparticles (PSN) have been used for the tunable delivery of camptothecin, a small therapeutic molecule, where PSN acted as a therapeutics nanocarrier into the electrospun composite of poly fibers [14]. Additionally, mesoporous Si has been investigated for applications in biology and medicine, and an example is a porous silicon container employed to enclose the drug and release it in a controlled manner [15]. Previously, PS MFs have been fabricated to work as red-infrared filters coupled with a silicon photodetector, where the localized mode can be tuned by changing the optical path of the defect layer [16]. The reflectivity at the localized mode wavelength also increases. This effect occurs because PS has less absorption at long wavelengths [16,17]. This kind of filter can be lifted off from the Si substrate creating a free-standing PS MF. This MF can be transferred to another substrate such as quartz and glass [18,19]. Infrared long-wave pass and short-wave pass filters based in macroporous Si have been investigated [20]; they can be used together to suppress the radiation from shorter wavelengths. Consequently, the signal-to-noise performance of detectors may improve [18]. Moreover, luminescent silicon quantum dots embedded in free-standing PS and PS MFs infiltered with CdSe/ZnS and AgInS2/ZnS quantum dots have been reported [21–23]. The localized mode of the PS MF was tuned experimentally to match the emission spectra of CdSe/Zn and AgInS2/ZnS quantum dots to achieve enhanced photoluminescence. The authors mentioned that the localized mode of the MF modulated the photoluminescence of the quantum dots [22,23]; where electrical and thermal tuning of localized modes could also be achieved by infiltrating liquid crystals in PS MFs [24]. PS was also utilized as an anode for fast charge-discharge in lithium-ion batteries [25,26], and it was found that laser carbonization and wet oxidation PS structures had memristive properties. The first memristor showed properties of plasticity and short/long term memory, whereas the second exhibited strong filamentary-type resistance switching; and they have been used as two terminal resistive memory cells [27].

As the reader already noticed, PS is a versatile material, and herein, we will focus on the optical properties of PS, which allow the design of various kinds of interference filters in a broad range of wavelengths. Among these designs, we found rugate filters [28,29], Fibonacci filters [30], Bragg reflectors filters (BRF) [31,32], and MFs [33]. However, many drawbacks have been found in these filters, such as high chemical instability, high photon losses due to light absorption, and scattering in the visible and UV ranges. Some solutions to these problems have been identified by carrying out dry oxidation in PS structures [32,34], and recently it has been possible to manufacture porous Si-SiO2 filters (BRF and MF) in the UV range [34–36]. Our specific goal is integrating a PS MF in a photodetector to enlarge the responsivity spectrum bandwidth. In the past, the integration of different PS bandpass filters in Si photodetectors was attainable to achieve different filtering effects. It was possible to tune the responsivity spectrum from green to the near-infrared range obtaining all-Si color-sensitive photodetectors in that wavelength interval but with no below the green range [16,31,37]. It has been reported that UV filters based in multilayer stacks show substantial drawbacks in the deep UV spectral range due to the limited number of transparent materials within that range. They have been proposed as an alternative to extending optoelectronic technology towards the UV spectral range; for instance, PS filters based on macroporous silicon and low-pressure chemical vapor deposition were tested for many solar-blind applications such as electrical spark imaging and non-line of sight (NLOS) UV optical communications [18,38]. Following the same line of thought of extending silicon-based selective filters towards the UV spectral range; in this work, porous Si-SiO2 microcavities were used to modulate the responsivity of a broad photodetector with a detection range from 300 to 510 nm. The porous Si-SiO2 microcavities made the photodetector more selective in the UV range; and all microcavities had a localized mode close to 360 nm in the UV-A range. The maximum transmission at the localized mode wavelength (360 nm) was dominant in the UV-A range, allowing high energy photons to pass and produce a maximum peak in the photodetector.

This work is divided as follows: In Section 2, we give some technical details for materials and methods. Section 3 describes the results and their discussion. Finally, in Section 4, we present some conclusions.

#### **2. Materials and Methods**

#### *2.1. Porous Silicon Microcavities Filters*

PS microcavities filters were fabricated using p-type Si wafers, with (100) orientation and electrical resistivity between the range 0.01–0.02 Ω cm. The microcavities were etched anodically in an aqueous electrolyte based in HF and ethanol with a volume ratio of 1:1. Before etching, the Si wafers were cleaned with HF and ethanol for 5 min to remove the native oxide. A Keithley 2460 current source controlled by a laptop was employed to deliver a current profile for the microcavities formation, where the current profile consisted of switching two different current pulses, with each current pulse producing low/high porosity layers that respectively corresponded to the low/high refractive index layers. This process produced a stack of layers with specific refractive indexes, while anodization time determined the thickness of the layers. A pause of 3 s was introduced between each current pulse to prevent any porosity gradient.

Anodization etching included low/high current pulses of 5 mA/cm2 and 80 mA/cm2 to obtain layers with porosities of 39% and 64%, respectively. Meanwhile, the anodization times of 4.1 and 1.1 s were applied to form the first and second porous layers, and finally a third porous layer (defect) was built-in using a current pulse of 80 mA/cm<sup>2</sup> for 2.2 s. The PS microcavities filters were detached from the Si substrates using the same aqueous electrolyte before mentioned, but a high current density (450 mA/cm2) was applied for 2 s. This process created a free-standing PS microcavity filter, which was transferred to a quartz substrate. All samples were rinsed with ethanol and dried at room temperature after the anodization process.

#### *2.2. Dry Oxidation in PS Microcavities Filters*

Microcavities filters were systematically oxidized using two stages of dry oxidation. In the first stage, a temperature of 350 ◦C for 30 min was used. In the second stage the oxidation temperature was increased to 900 ◦C for 1 h. During the oxidation process, the oxygen flow was changed from 1.15 to 4.52 SLPM (standard liter per minute) to observe any change in the microcavity optical response. The reasons behind the two dry oxidation steps were: the first step was a low-temperature pre-oxidation needed to equilibrate the silicon structure. That is, this step prevented the aggregation of the pores during further treatments at higher temperatures. The second step was applied to grow an oxide layer of greater thickness than that obtained in the last oxidation step. The layer thickness is higher than the natural native oxide grown in the environment resulting in the consolidation of SiO2.

Transmittance and reflectance spectra measurements were carried out before and after dry oxidation with a Varian (Agilent Technologies, CA, USA) UV-Vis-NIR spectrophotometer at normal incidence, and 20◦ from 200 to 800 nm. The cross-section SEM images of an oxidized microcavity was obtained using a JEOL-JSM7600F (Jeol, MA, USA).

#### *2.3. Photocurrent Modulation by Porous Si-SiO2 Microcavities Filters*

The photodetector employed to modulate its responsivity was a GVGR-T10GD (Electro Optical Components Inc., CA, USA) based in indium gallium nitride with a spectral detection range from 300 to 510 nm. In the experimental process, an optical chopper SR540 (Stanford Research Systems Inc., CA, USA) with a frequency of 287 Hz was used to cut off the light coming from the Xenon lamp 6254 (Oriel Corporation, CT, USA), concentrated at the input of a monochromator. A lens was used to focus the output light from the chopper, on the photodetector. The PS microcavity filter was placed before the photodetector, which was polarized with 5 volts using a Keithley 2460 source (Keithley, OHIO, USA), and then the photodetector photocurrent inputted a Look-in amplifier SR-530 (Stanford

Research Systems, CA, USA)). The filtered and amplified photocurrent versus light wavelength (from 276 to 536 nm) was displayed in a PC. The experimental set-up is shown in Figure 1.

**Figure 1.** Experimental set-up used to obtain the responsivity of a commercial photodetector with different porous silicon (PS) microcavity filters (up panel). The bottom panel shows the scheme and cross-section SEM images of an oxidized microcavity filter with a total thickness of 1.92 μm.

#### *2.4. Theoretical Mechanism to Modulate the Responsivity of a Broadband Photodetector in the UV Optical Range with Porous Si-SiO2 Microcavities*

In an ideal photodetector, its responsivity is proportional to the input light power being the proportional factor, the so-called quantum efficiency η, which is a function of the light reflection *R* at the surface, the fraction of electron-hole pairs that contribute effectively to the photocurrent ζ, the light absorption in the bulk of the material α, the photodetector depth *d*, and the light wavelength.

Herein, we proposed to modulate the responsivity of a commercial photodetector by changing *R*. This was done by adding a microcavity filter, where the spectral reflection response of the microcavity filter modifies the original reflection spectral response at the surface of the commercial photodetector completely. The microcavity filter reflection depends on the high and low refractive indexes and thickness values, as well as the refractive index and thickness of the defect layer values. Initially, the microcavities are made of PS to filter out blue light. This means that the layers' refractive index values, controlled by the porosities and in which the crystalline silicon nanostructures remains, were chosen to respond in that particular region of the light spectrum. However, our primary goal was to achieve responsivity modulation within the UV region using a silicon-based filter. We cannot use the sensitive blue light microcavities for that purpose because the crystalline silicon nanostructures strongly absorb light within the UV and blue regions. This absorption process is equivalent to having a 100% reflection at the photodetector surface. Thus, the quantum efficiency, and consequently, the responsivity, tends to zero. The oxidation process helps to avoid this problem in two ways: first, since there is a phase change, from crystalline silicon to silicon dioxide, the layers' refractive index values decrease, pushing the filter response towards the UV region. Second, there is much less light absorption in the UV band. The oxygen flow value determines the concentration of silicon dioxide, followed by the layer's refractive index values, and consequently, the spectral response of the filter. More specifically, the filters spectral response contains a unique photonic state known as a localized state represented as a

resonant peak in the reflection spectrum. It is a state that drastically modifies the responsivity, and therefore, it is the flow of oxygen that determines the final position of the localized state along with its reflection amplitude.

#### **3. Results and Discussion**

#### *3.1. Porous Silicon Microcavities Filters in the Blue Range*

The averaged theoretical (red line) and experimental (black line) reflection (broken line) and transmission (solid) spectra from the PS microcavities filters are depicted in Figure 2a. This corresponded to the average measurements of transmittance and reflectance spectrum from four microcavities. All four MCs were designed to show a localized mode in the same wavelength (590 nm). Experimental transmission and reflection measurements of the microcavities were taken using a UV-VIS-NIR spectrophotometer in the wavelength range from 200 to 800 nm. The transmission spectrum was measured at normal incidence and reflectance spectrum of the same microcavities was obtained at 20◦. Meanwhile, theoretical transmission and reflection spectrum were obtained using the matrix method. This method is well known (Supplementary Materials) for obtaining the theoretical spectrums that we considered the angle measurements taken with the UV-VIS-NIR spectrophotometer.

**Figure 2.** Experimental averaged transmission and reflection spectra of four microcavities compared to its theoretical spectra. (**a**) The theoretical (red line) and experimental (black line) averaged reflection and transmission spectra for four unoxidized microcavities obtained in the blue optical range; (**b**) the averaged band structure of the same microcavities. (**c**,**d**) are the comparison of the averaged theoretical and experimental defect mode locations. In (**d**), two angles of incidence (zero and 20 degrees) were considered.

The theoretical reflection spectrum of the microcavity predicts more reflection in the UV range than its experimental counterpart, while the theoretical and experimental transmission spectra matched very well. Moreover, the light transmission was low in the UV-VIS range due to strong absorption of Si. The localized mode of the microcavity was located at 490 nm and exhibited a maximum transmission peak of 4%, while that in the visible range from 555 to 800 nm showed high light transmission. This result was because PS had small absorption losses in this range of the electromagnetic spectrum, where the primary photon loss was light scattering. When the reflection spectrum of the microcavity was taken at 20◦, the position of the localized mode changed, and it shifted towards high energies (483 nm). It could also be observed in the theoretical reflection spectrum. Figure 2b shows the averaged photonic bandgap (PBG) of an unoxidized microcavity. It was obtained using the dispersion relation

inside the first Broullin zone. We calculated it only in one dimension and both measurement angles were considered, which were mentioned before.

We obtained the defect frequency can be obtained using a combination of the transfer matrix technique and variational methods. The defect modes are represented by the maximum peak of transmission in the microcavities [39,40]. When the symmetry of the photonic structure is broken up by a defect layer, it is possible to have a confined state, where photons are trapped and cannot escape, being confined inside the bandgap. We calculated the defect mode frequency for an antisymmetric state because our photonic structures were built as follows:

$$(HL)\_8L(HL)\_7$$

Our antisymmetric photonic structures based on PS and porous Si-SiO2 consisted of 31 layers, alternating between high refractive and low refractive indexes, with a low refractive index defect layer, and where the structure was surrounded by air.

The averaged theoretical defect mode frequency location is shown in Figure 2d, where it is compared with the averaged transmission and reflection spectra (Figure 2c). The defect mode is shown as a transmission maximum or as a reflection minimum at a specific wavelength between the photonic bandgap (PBG) edges. The variational method defect mode location was 18 nm off, concerning the experimental result from the reflection measurements, which was taken at 20 degrees. In this case, the defect mode location is depicted in Figure 2d (red broken line). Using the same variational method, the defect mode location was 16 nm off for the experimental transmission spectrum, which was taken at normal incidence. The localized mode is shown in Figure 2d (black broken line). The transfer matrix method predicted the same position as the experimental result. Both theoretical methods showed good agreement with the experiments.

#### *3.2. Porous Si-SO2 Microcavities Filters in the UV*

Averaged transmission (solid line) and reflection (broken line) spectra from porous Si-SiO2 microcavities filters are depicted in Figure 3a. The theoretical (red line) and experimental (black line) spectrum showed a good fit. The theoretical spectra were obtained by the transfer matrix (Supplementary Materials) considering both measurement angles, where one was 0◦, which corresponded to the transmission measurements, and the other was 20◦, considered for the reflection measurements. The microcavity to the left was first oxidized at 350 ◦C for 30 min and then at 900 ◦C for 1 hr, and the oxygen flow was 1.15 SLPM (standard liter per minute). The microcavity shown to the right was oxidized using the same temperature and oxidation time mentioned above. In this case, an oxidation flow of 2.21 SLPM was applied. The microcavity MF2 exhibited a maximum transmission amplitude of 70%, and the microcavity MF3 showed a maximum transmission amplitude of 57% in the localized mode wavelength, which indicated that the microcavity MF3 had more absorption losses than the microcavity MF3. Porous Si-SiO2 microcavities filters in Figure 3a show a transmission maximum peak (356 and 364 nm) and a reflection minimum peak (349 nm) in the UV range. We observed a mismatch between both peaks because the measurements were taken at different incidence angles.

Figure 3c shows the averaged transmission and reflection spectrum from UV filters compared to the defect mode location obtained with the variational method (Figure 3d). The defect mode location corresponding to the reflection spectrum was 4 nm off using the variational method, which is shown in Figure 3d (red broken line). The position of the defect mode obtained from the variational method (black broken line, Figure 3d) was 9 nm off for the transmission spectrum, as compared to what was predicted using the transfer matrix method.

**Figure 3.** The averaged theoretical and experimental transmission and reflection spectra of two UV microcavities filters, with the averaged photonic bandgap and defect mode locations. (**a**) The averaged theoretical (red line) and experimental (black line) reflection (broken line) and transmission (solid line) spectra of oxidized microcavities obtained in the UV range; (**b**) the averaged photonic bandgap structure (empty box) of the microcavities. (**c**,**d**) show the comparison of the averaged theoretical and experimental defect mode locations. In (**d**), two angles of incidence (zero and 20 degrees) were considered.

Figure 4a depicts the averaged theoretical (red line) and experimental (black line) transmission (solid line) and reflection (broken line) spectra of two oxidized MF. In this case, the porous Si-SiO2 microcavity filter MF4 showed a maximum transmission peak (368 nm) in the UV range, with an amplitude of 42%. Meanwhile, the microcavity MF5 displayed a maximum transmission peak at 369 nm, with an amplitude of 21% (red and black solid lines). A reflection minimum peak (349 nm) is also shown in both microcavities (red and black broken lines). The maximum transmission peaks were less intense than the microcavities shown in Figure 3. Here, the same temperature and oxidation time were employed to obtain the UV filters, but the oxygen flow applied for sample MF4 was 3.39 SLPM, while for sample MF5, it was 4.52 SLPM. The oxygen flow was increased gradually.

**Figure 4.** The averaged theoretical and experimental reflection and transmission spectrum of another two UV microcavities filters. Additionally, its averaged photonic bandgap and the defect mode locations are depicted. (**a**) The averaged theoretical (red line) and experimental (black line) reflection (broken line) and transmission (solid line) spectra of oxidized microcavities obtained in the UV range; (**b**) the averaged photonic bandgap structure (empty box) of the microcavities. (**c**,**d**) are the comparison between the averaged theoretical and experimental defect mode locations. In (**d**), two angles of incidence (zero and 20 degrees) were considered.

Figure 4c,d shows the averaged theoretical (red line) and experimental (black line) reflection (broken line) and transmission (solid line) spectra near to the defect mode location. The variational method was employed to obtain the defect mode location between the PBG edges. The broken red plots (Figure 4d) corresponded to the defect mode position of the microcavities measured at 20◦. The defect mode locations of microcavities measured at normal incidence are depicted as broken black lines in Figure 4d. There was a mismatch between the defect mode position calculated using the transfer matrix and the variational method.

#### *3.3. Refractive Index of Porous Silicon and Porous Si-SiO2*

The complex refractive index (refractive index and extinction coefficient) of the PS was obtained using the effective medium approximation Maxwell–Garnett. For porous Si-SiO2, we used a three-component model developed from the J.E. Lugo model. Figure 5a,b shows the refractive index values, and Figure 5c,d depicts the extinction coefficient values of PS and Porous SiO2, respectively, obtained from both models. The continuous lines in Figure 5a correspond to a high refractive index (red line) and a low refractive index (blue line) for PS. Their extinction coefficients are displayed with dotted lines (red and blue) in Figure 5c, where these values were used to fabricate PS microcavities filters in the blue range. The continuous lines of different colors in Figure 5b between 1.45 and 1.6, and the green line with values between 1.35 and 1.5, represent high and low refractive indexes for the porous Si-SiO2 layers (see Figure 5b). Their corresponding extinction coefficients are shown below the value of 0.1 (dotted lines) in Figure 5d. The vertical black dotted lines that intersect the complex refractive index components (Figure 5) indicate the exact values utilized to design MF in the blue and UV range.

**Figure 5.** Theoretical values of the complex refractive index for PS layers and porous Si-SiO2 layers. (**a**,**b**) shows the refractive index values for PS and porous Si-SiO2, and (**c**,**d**) depicts the extinction coefficient values of PS and Porous Si-SiO2, respectively, obtained from both models.

Figure 5 shows the refractive index and extinction coefficient are high when the PS is not oxidized. Meanwhile, the complex refractive index component values for porous Si-SiO2 are lower than the PS values. The growth of SiO2 inside PS layers decreased the refractive index and extinction coefficient. Moreover, during the PS layers' oxidation process, some air and Si fractions were replaced by SiO2, causing a lattice expansion and a decrement on the layer's porosity, as shown in Table 1.


**Table 1.** Theoretical values of porosity before and after dry oxidation, thickness, Si fraction, and oxide fraction of five microcavities.

Knowing the refractive index of each layer that made up the microcavity allowed estimation of the theoretical thickness of each layer by fitting the theoretical transmission and the reflection spectrum (applying the matrix method) with its experimental result. The thickness values displayed in Table 1 correspond to theoretical thickness for different microcavities. On the other hand, an increase in the thickness for the oxidized microcavities (MF2, MF3, MF4, and MF5) was shown after dry oxidation compared to the thickness of the unoxidized microcavity (MF2).

Additionally, the thickness of an oxidized microcavity was obtained using SEM measurements. Figure 1 (down panel) shows where the porous Si-SiO2 UV filter has a total thickness of 1.92 μm and it can observe a defect between two BRF. The light gray thickness (*dH* = 48 nm) corresponds to layers with high refractive index (low porosity) and the dark gray thickness (*dH* = 75 nm) depicts the layers with low refractive index (high porosity). Meanwhile, the defect in the microcavity had twice the thickness of the high porosity layer. The thickness found using both methods showed a difference of a few nanometers, and both results give an approximate thickness value for each layer of an oxidized microcavity filter.

Furthermore, the optical path was modified by decreasing the refractive index and increasing the physical thickness. Finally, the vertical black dotted lines that intersected with the complex refractive index components (Figure 5) indicated the exact values utilized to design MF in the blue and UV range.

#### *3.4. Optical Losses Due to Light Absorption*

Optical losses in the VIS and UV range due to light absorption are found frequently in PS. This absorption loss impedes the fabrication of PS filters in the UV range, but these losses can be decreased if the PS structure is thermally oxidized in an oxygen environment [35,36]. Figure 6 (MF1) shows the theoretical (a) and experimental (b) absorbance spectrum of five microcavities. As observed, the absorption was dominant in the VIS and UV range, while in the infrared range, the absorption did not play an important role. It has been reported that optical losses are mainly due to dispersion in the infrared range [41,42]. Moreover, Figure 6a (MF2-MF5) shows the theoretical absorption spectra of four oxidized PS MCs. Figure 6b displays the experimental absorption spectra of the same four oxidized PS MCs filters using different oxygen flows. These plots showed an absorption increase when the oxygen flow builds up. The average absorbance spectra of the oxidized microcavities showed less optical losses due to absorption. Their amplitude decreased more than 70% in the UV range and almost disappeared within the VIS range after dry oxidation. We obtained a good fit between the theoretical and experimental absorption spectra in the VIS and UV range.

These results showed that the defect mode of the PS microcavities filters had a wavelength shift of 134 nm to lower wavelengths when the PS microcavities were oxidized. It also had a wavelength shift of 13 nm to lower wavelengths when a maximum oxygen flow of 4.52 SLPM was applied. Increasing the oxygen flow made the porous Si-SiO2 microcavities exhibit less amplitude in the transmission and reflection spectrum inside the UV range. High oxygen flow did not allow oxygen particles to incorporate into the PS structure. Thus, there was less SiO2 formation and consequently, optical absorption increased. Besides, a decrease of the PBG bandwidth was achieved by incorporating SiO2 within the PS microcavities. This bandwidth decrease happened because there was less contrast between the high refractive index and the low refractive index of the porous Si-SiO2 layers.

**Figure 6.** Theoretical (**a**) and experimental (**b**) absorption spectra of MFs in the blue and UV range. (**c-up**) Four unoxidized PS microcavities on a quartz substrate. (**c-bottom**) Four oxidized porous SiO2 microcavities.

#### *3.5. Photocurrent and Responsivity Measurements of a Commercial Photodetector without and with Filters*

Porous Si-SiO2 UV microcavities filters were used to modulate the photocurrent of a broad photodetector (GVGR-T10GD) with a detection range from 300 to 510 nm. Figure 7 depicts the measured photocurrent of a commercial photodetector (gray line) without and with filters (black, red, blue, pink lines). The MF produced a photocurrent maximum in the UV range, which corresponded to the localized mode of the porous Si-SiO2 microcavity filters positioned between 356 and 364 nm. This meant that porous Si-SiO2 filters cut off the photocurrent range from 300 to 350 nm. Therefore, in the short-wavelength range, the photons were absorbed and did not contribute to the photocurrent, whereas in the VIS range, photons with wavelength light from 390 to 510 nm were allowed to pass and they contributed to increasing the photocurrent. Besides, all photocurrent spectra modulated with porous Si-SiO2 microcavities showed a decline in the UV range, which was attributed to the transmission amplitude of the microcavity as it decreased when the oxygen flow increased.

Figure 7b shows the normalized photocurrents of a commercial photodetector modulated with porous Si-SiO2 microcavities filters. We observed that the photocurrent peak at the localized mode wavelength decreased when the filters with less transmission amplitude were used.

Spectra responsivities were obtained using a 150 W Xenon arc lamp as a light source and a monochromator with a light wavelength from 300 to 510 nm at 5 V applied bias. Thus, we calculated the responsivity from:

$$\mathcal{R} = \frac{\mathcal{I}\_{\text{ph}}}{\mathcal{P}\_{\text{inc}}} \tag{1}$$

where R is the responsivity of the modulate photodetector with porous Si-SiO2 microcavities filters, Iph is the photocurrent in Ampers, and Pinc is the power of the Xenon lamp in watts. The photodetector responsivities without and with porous Si-SiO2 microcavities filters were found in the range from 300 to 510 nm. The result is shown in Figure 8.

**Figure 7.** Photocurrents spectra of a commercial photodetector with and without porous Si-SiO2 microcavity filters. (**a**) The photocurrent spectra of a commercial photodetector (green line) and its modulated photocurrent with porous Si-SiO2 microcavities filters (black, red, blue, and pink lines) and (**b**) the normalized photocurrent.

**Figure 8.** Spectral responsivity comparison of a commercial photodetector without (line green) and with porous Si-SiO2 filters (black, red, blue, and pink lines).

The spectral responsivity of a commercial photodetector was modified using different porous Si-SiO2 filters. It decreased when porous Si-SiO2 filters with low transmission in the UV range were placed on the commercial photodetector. However, a maximum peak of detection corresponding to the localized mode of the microcavity was observed in the UV-A range. In the VIS range, an increase of the responsivity was displayed using the MF5 filter. It was due to an increase in its transmission. The spectral responsivity shape was the same as the photocurrent. We also observed that the sensitivity of the photodetector with a filter was lower than the photodetector without a filter.

PS Bragg reflectors and microcavities were used to filter incident light reaching the Si photosensitive wafer. To tailor its spectral response, they were designed to different wavelengths and integrated above the p-n junction of a silicon photodiode. In this way, they were converted in an array of PS detectors sensitive to the color [16,31], where the sensitivity peak could be tuned along from green to near-infrared, and where sharp peaks in the spectral responsivity were achieved using microcavities and color-sensitivity by Bragg-reflectors [16,43]. The transmittance of the PS filters mainly modulates the spectral responsivity of the silicon photodiode. Therefore, when a Bragg reflector is used to modulate the spectral response of the Si photodiode, its reflectance maximum is nearly insensitive [16,37].

The study on the responsivity has focused on the range visible and infrared placing different kinds of filters on Si photodetectors. A narrowing in the responsivity spectrum of the Si photodetector was observed from the green to infrared range. Moreover, a decrease in the responsivity spectrum of the PS detectors sensitive to the color was displayed, which was compared to the responsivity spectrum of the Si photodetector without filters. The difference between both results was approximately a one magnitude order [2,37]. On the other hand, MF and BRF are used to modify the responsivity of a Si photodetector in the orange range, where the photodetector's responsivity with a filter was stronger. It reduced the undesired effects in the responsivity spectrum [16,43].

Furthermore, it has been reported that below 500 nm, no photocurrent is detected due to the low responsivity of a silicon photodetector. Its modulation using UV filters cannot be achieved because the absorption on PS is dominant from the blue to UV range [37]. Thus, it is not possible to modulate the responsivity of broadband photodetectors in that part of the electromagnetic spectrum. To address this limitation, we employed dry oxidation to decrease the absorption losses in the UV-A range. The transmission improvement in the porous Si-SiO2 UV filters allowed us to modulate the responsivity of a commercial broadband photodetector.

#### **4. Conclusions**

In this work, porous Si-SiO2 UV filters were manufactured based on Si and SiO2. We fabricated both symmetric and antisymmetric filters in the blue region, where the antisymmetric structures gave us better results. This result was because we tried to reduce the number of layers to obtain porous silicon MFs as close as possible to the short wavelength region. Furthermore, we noticed during the dry oxidation that the number of layers had an enormous influence on the growth of SiO2 into the porous silicon structure. This is because large structures can be transformed entirely in porous SiO2. Besides, the presence of a higher quantity of silicon would cause absorption losses in the porous silicon structure; therefore, it could not be possible to obtain porous SiO2 MF in the UV range.

The filters were placed on quartz substrates to obtain their transmission and reflection spectra. First, we observed the onset of a localized mode in the blue region in unoxidized MFs, and second, we observed an optical shift of the localized mode location towards small wavelengths (UV region). The shift was due to the oxidation process. The ultimate consequence of the oxidation procedure was the phase change of the PS skeleton, which was converted into SiO2. Consequently, the refractive index decreased and the extinction values also decreased, although more drastically. The refractive index decrease is the origin of the localized mode location shift, whereas the extinction value decrease is related to a severe light absorption decrement, which means less optical losses.

The filters on quartz substrates were placed on a commercial photodetector. The light coming from a Xenon lamp was filtered out by the filters and changed the reflection at the top photodetector surface, achieving modulation of the photodetector responsivity. As a result, the spectral responsivity obtained matched the UV filter transmission spectrum. The filters' spectral responses were modified by applying different oxidation flows in the oxidation process. When a high flow was used, the transmission in the localized mode wavelength decayed. We have shown that the photodetector became more selective in the UV range using porous UV Si-SiO2 filters, where an average responsivity peak arose due to the localized mode of the microcavity. The position and amplitude of the localized mode microcavity can be manipulated and tuned in the UV range, by applying different oxygen flows during dry oxidation. In this work, we demonstrated that the responsivity of a commercial photodetector could be modulated using porous Si-SiO2 UV filters. Moreover, the responsivity peak may be fully tunable, depending on the filter design from UV-A to near-infrared. In the future, the modulated responsivity could be enhanced using Rugate filters, and these filters could be implemented or integrated into Si-based photodetectors, thereby achieving better sensitivity to UV light. Rugate filters could exhibit a unique maximum peak in all the detection ranges of the UV photodetector. It has been reported that Rugate filters have a narrow PBG and the sidelobes on each side of these filters are smaller than other kinds of filters [29].

Other possible applications of these MFs are to embed quantum dots and liquid crystals inside porous Si-SiO2 UV filters to match their emission spectra with the microcavity localized mode. In this way, we would enhance and modulate the photoluminescence in the UV range by experimentally tuning the localized mode of the microcavity. Porous Si-SiO2 UV filters may be used as antireflection coatings [9,44] to enhance the efficiency and photocurrent in solar cells. This is because the localized mode of the microcavity shows a maximum amplitude of 70% in the UV range, whereas, in the VIS range, more than 80% of the light is transmitted. Decreasing reflection and increasing transmission in the UV filters could be raised by the efficiency of solar cells in the high-energy range. It could be achieved if a UV filter attains high transmission close to 100%.

One significant contribution of this work is that our microcavities are based on silicon solely. This contribution is important because up to now, silicon photodiodes and solar cells can only work in the UV region. However, silicon photodetectors have a broad response, and our UV microcavities can make them more selective in that range. In the future, the possibility of integrating these UV filters along with porous silicon photodetectors may be a reality, as well as UV light-emitting devices based only on silicon and silicon dioxide technology. Moreover, as shown above, similar approaches to modulate the responsivity of a photodetector have been proposed. Nevertheless, to the best of our knowledge, this is the first time that it has been done within the UV range.

What is the advantage of using porous SiO2 compared to other materials with similar optical properties? Some authors have reported that several fluorides with bandgap energies around 10 eV are suitable for filtering applications in the UV wavelength range, even below 200 nm. An example is a mixture of lanthanum fluoride (LaF3) and magnesium fluoride (MgF2), which have been used as high reflectance coatings. They showed strong absorption in the VIS and UV range, as well as a maximum reflectance higher than 90% at 180 nm, while their transmittance was lower than 2%. Their refractive indexes were found in the range from 1.41 to 1.80. High reflectance coatings have been produced by ion-assisted deposition, ion-beam sputtering, and electron-beam evaporation. However, several difficulties exist in these materials due to optical losses, mechanical stress of the thin films, small refractive index difference between both materials, and the optical inhomogeneity of LaF3 films [45,46].

On the other hand, many oxides are an essential class of coating materials, which react at high temperatures. HfO2 (hafnium oxide) and ZrO2 (zirconium oxide) are examples of transparent oxides, where HfO2 films are produced by electron-beam evaporation, reactive sputtering, and ion-assisted deposition. ZrO2 films are prepared by electron-beam evaporation, which showed strong inhomogeneity of the refractive index. HfO2 and ZrO2 are common materials used to manufacture UV filters due to their transparency, low-absorption, and high refractive index in that optical range. A long pass edge filter composed of 91 layers of HfO2/SiO2 deposited on a fused silica substrate by ion-assisted deposition (IAD) has been reported. The total thickness of the coating was around 3 μm [47–49]. The 50% transmission cut-on edge of this particular filter was at 256 nm, and the transmission in the passband average above 90% was from 260 nm to 1200 nm. Moisture stable HfO2 films and stacks containing HfO2 and SiO2 can be deposited at lower substrate temperatures (100 ◦C to 130 ◦C) using an argon oxygen gas mixture IAD process. Most thin-films deposited without ion-assistance are porous and sensitive to moisture going in and out of the voids, thereby causing an apparent shift in the refractive index depending on the relative humidity [50]. HfO2 films obtained by ion-assisted electron-beam evaporation showed low-absorption, with a refractive index of 2.19. In non-ion-assisted, the refractive index was 2.06, and the absorption coefficient was 0.006 and 0.003 for ion-assisted and non-ion-assisted [51]. ZrO2 films have been deposited by magnetron sputtering at different argon partial pressure values, where the films of ZrO2 showed a maximum transmission of 80% from 300 to 800 nm. This material has a high refractive index and low absorption above 240 nm to the IR range (below 8 mm) [52]. Oxide films are also widely used in multilayers systems, such as cold light mirrors, heat-reflecting filters, color separators, narrow band interference filters, and laser coatings. ZrO2 and HfO2 have disadvantages; for instance, they form uneven surfaces during evaporation, which often cause inhomogeneous thickness distributions and inferior thickness reproducibility. It has been investigated that no pure oxides exist with refractive indices of about 2.2. Available mixtures of silver (Ag)–silicon dioxide (SiO2) can be found, which have been used as thin-film bandpass filters

for the UV range obtained by radiofrequency sputtering. In this filter, the suppression of undesired visible and infrared parts of the spectrum was achieved. Some structures applicable for bandpass filtering in the UV have high transmission in the passband but a limited range of out-band blocking. An additional blocking component such as monolithic deposited silver has been used to remove unwanted out-of-band radiation. However, this also reduces the overall transmission through the filter. Some pairs of Ag/SiO2 layers and a layer of silica were added to serve as an antireflection coating, where the band-pass filter was designed with the maximum transmission in the UV-A range and with a resonance at 320 nm. The use of transparent metals can ensure a transmission decrease of several orders of magnitude in the visible and infrared wavelength range, at the same time, fully preserving transparency in the ultraviolet spectrum [53].

Solar-blind deep-UV band-pass filters based on a mixture of aluminum (Al) and SiO2 have been developed. The filters showed a 27% transmission peak at 290 nm, a band-pass from 250 to 350 nm, and a rejection ratio to visible light of 20 dB, where the peak of transmission could be tuned by adjusting the metal nano-grind dimensions [54]. An alternative approach to solar-blind UV detection is to integrate Si-based photodetectors with solar-blind UV-pass filters to reject visible and IR light in the solar spectrum. Moreover, the designed and fabricated filters are fully applicable for the enhancement of UV silicon detectors since one of the problems with the use of silicon photodetectors is the avoidance of the visible and infrared components of radiation. Under standard conditions, it often exceeds the UV components by three to four orders of magnitude, thereby completely masking the useful signal. A straightforward way to avoid the problem is to use bandpass filters for the UV strong rejection of unwanted visible and infrared radiation in the range of the detector sensitivity. However, a disadvantage of Ag/SiO2 and Al/SiO2 structures is that they present reduced transmission (less than 50%) in the UV range due to the absorption of the Ag and Al.

Moreover, dielectric interference filters have also been demonstrated for solar-blind UV applications. However, this type of filter requires a thick stack of multiple dielectric materials layers that must be deposited with precise thickness and uniformity control, with high quality over a very long deposition time. This inevitably increases production costs and reduces the uniform area. The filters also need accurate control of pressure and temperature.

In our work, dielectric interference filters in the UV were achieved by electrochemical etching, followed by two-stage dry oxidation. Our method was easy, cheap, and fast to fabricate compared to the methods mentioned before. The thickness of each layer that made up the UV dielectric microcavity was precise and uniform, and the fabrication time of these dielectric microcavities was shorter than other methods. Besides, physical vapor deposition infrastructure is expensive, and ion-assisted deposition is a costly and slow technique. Additionally, dielectric oxide films do not show sufficient thermal stability because the structure of the oxide films is easily converted from amorphous to polycrystalline and it reacts with the Si substrate.

To conclude, the porous Si-SiO2 UV filters (MF2, MF3, and MF4) can be applied as UV hot mirrors or UV bandpass filters since they display more than 70% of the transmitted light in the UV-VIS range. Meanwhile, the MF5 filter can be used as a UV blocking filter since it cuts off more UV light than other filters, and it shows a maximum transmission amplitude of 80% in the VIS-IR.

#### **Supplementary Materials:** The following are available online at http://www.mdpi.com/2079-4991/10/2/222/s1.

**Author Contributions:** Conceptualization, M.R.J.-V., G.G., and J.E.L.; Formal analysis, M.R.J.-V. and J.E.L.; Funding acquisition, G.G., J.C., M.G., and J.F.; Investigation, M.R.J.-V.; Methodology, M.R.J.-V., G.G., and J.E.L.; Writing—original draft, M.R.J.-V. and J.E.L.; Writing—review & editing, G.G., J.C., F.M.-M., A.C., M.G., R.D., J.F., and J.E.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by a National Sciences and Engineering Research Council of Canada Discovery operating grant.

**Acknowledgments:** R.J. wants to give thanks to CONACYT for the scholarship granted during her doctorate studies. The authors acknowledge D.L. Serrano for the technical assistance provided in the UV-VIS-NIR measurements.

**Conflicts of Interest:** The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Deep Subwavelength-Scale Light Focusing and Confinement in Nanohole-Structured Mesoscale Dielectric Spheres**

#### **Yinghui Cao 1, Zhenyu Liu 2, Oleg V. Minin 3,4 and Igor V. Minin 4,\***


Received: 28 December 2018; Accepted: 30 January 2019; Published: 1 February 2019

**Abstract:** One of the most captivating properties of dielectric mesoscale particles is their ability to form a sub-diffraction limited-field localization region, near their shadow surfaces. However, the transverse size of the field localization region of a dielectric mesoscale particle is usually larger than λ/3. In this present paper, for the first time, we present numerical simulations to demonstrate that the size of the electromagnetic field that forms in the localized region of the dielectric mesoscale sphere can be significantly reduced by introducing a nanohole structure at its shadow surface, which improves the spatial resolution up to λ/40 and beyond the solid immersion diffraction limit of λ/2*n*. The proposed nanohole-structured microparticles can be made from common natural optical materials, such as glass, and are important for advancing the particle-lens-based super-resolution technologies, including sub-diffraction imaging, interferometry, surface fabrication, enhanced Raman scattering, nanoparticles synthesis, optical tweezer, etc.

**Keywords:** nanohole; microsphere; subwavelength-scale light focusing

#### **1. Introduction**

Photonic Nanojet (PNJ) is the phenomenon of subwavelength-scale light focusing that is generated by dielectric microparticles. It has been widely applied in different areas, including laser cleaning, nanolithography, super-resolution imaging, enhanced Raman scattering, non-linear fluorescence enhancement, etc. [1–4]. However, one limitation for PNJ is its minimum beam width of λ/3 [1] (where λ is the wavelength of the incident light) and, thus, more efforts are needed to further reduce the focal spot sizes of the PNJ.

Over the past decade, many studies have been focused on decreasing the width of the PNJ to be as small as possible. If this could be achieved, PNJs would be able to provide new pathways for the trapping of nanoparticles and cells [5,6], subwavelength optical imaging [7], ultrafast all-optical switching [8], etc. Multilayer [9] and a two-layer [10] graded refractive index dielectric particles have also been considered. By using the anomalously intensity-enhanced apodization effect, PNJ with a Full Width at Half Maximum (FWHM) focal spot size that is less than 0.3λ have been attained in previous studies [11,12]. PNJ with a FWHM focal spot size of 0.29λ, which was generated by overstepping the upper refractive index limit has also been reported in the literature [13].

Nanohole-structured dielectric objects have also been considered in the literature. For example, as shown in Reference [14], the effects of the total light far-field scattering of deep holes in the spherical

particles with a refractive index near 1.05, were investigated, which showed that these structural details had a negligible influence. In Reference [15], it was shown that the PNJ beam sizes shrank by nearly 28%, due to the introducing of concentric rings on the illumination side of microspheres. To control the unidirectional scattering by the spectral overlapping of the Mie-type resonant modes, a dielectric high-index nanocylinder (n ~ 3.4), with an axial nanohole, has been considered in some studies [16]. Nevertheless, although several methods of light localization have been investigated, deep subwavelength localization, based on a single dielectric mesoscale particle, has not yet been realized.

In this work, a nanohole-structured dielectric microsphere is proposed for deep subwavelength-scale light focusing and strong light confinement, well below the diffraction limit. The field enhancement from the nanostructured mesoscale dielectric microsphere is due to the permittivity contrast between the material of the microsphere and the nanohole. The proposed nanohole-structured microparticles can be made from common dielectric materials, such as glass or latex, and has the advantage of the ability to tailor the spatial region of light confinement and enhancement by choosing the proper geometry, shape, and size of the nanohole. The approach provides a wide platform for deep subwavelength focusing and imaging, which offers the capability of sub-diffraction techniques for microscopy systems, light–matter interactions, interferometry, surface fabrication, enhanced Raman scattering, optical tweezer, and information processing.

This paper has been organized as follows. In Section 2, microspheres without a nanohole have been numerically modeled and analyzed. In Section 3, nanohole-structured microspheres have been simulated and analyzed. Finally, in Section 4, conclusions have been drawn.

#### **2. Dielectric Microspheres without Hole**

This section describes the dielectric microspheres with a fixed refractive index of *n* = 1.5 and different sphere diameters of *Ds* = 1.5λ, 2.5λ, 3.5λ and 4.5λ, which were numerically modeled and simulated. For λ = 600 nm, the simulated dielectric spheres had a diameter of *Ds* = 0.9, 1.5, 2.1, and 2.7 μm, respectively. We selected a refractive index of *n* =1.5, because in an optical band, many commonly used dielectric materials have a refractive index nearly equal to 1.5, such as glass, PMMA, fused silica, etc. [17]. We chose the popular spherical shape of a microparticle for the photonic nanojet formation [4]. The dielectric microspheres were modeled and simulated by using the commercial software COMSOL Multiphysics, which is based on the finite elements method (FEM). In this simulation, a non-uniform mesh was employed to reduce the computational cost and the Perfect Matched Layer (PML) was applied as the boundary condition. The incident light was assumed to be a plane wave that propagates along the z-axis, with a linear polarization along the y-axis. A schematic diagram for the simulated dielectric microsphere is given in Figure 1a.

**Figure 1.** Schematic diagrams for the simulated dielectric sphere (**a**) without a nanohole, (**b**) with a through hole, and (**c**) with a blind hole. The incident light is a plan wave that propagates along the z-axis, **k** is the wave vector. The incident field **E** is polarized along the y-axis. The oval-shaped zones in dark red color indicate the focal spots, and dash-dotted lines indicate the symmetrical axes for the spheres.

Figure 2 shows the light intensity distribution around the simulated dielectric spheres. The light focusing properties of the simulated microspheres are shown in Table 1, including the FWHM focal spot sizes along the *x*, *y*, and *z*-axis (*Sx*, *Sy*, *Sz*), the focal volumes *V* (which were obtained by volume integration inside the FWHM focal spot) and the maximum light intensity *Imax*, in the focal spot. After combining Figure 2 and Table 1, we found that, with an increase in the sphere diameter from *Ds* = 1.5λ to 3.5λ, the focal volume *V* decreased from 0.057λ<sup>3</sup> to 0.051λ3, while the maximum light intensity *Imax* increased from 16.1*I*<sup>0</sup> to 61.1*I*<sup>0</sup> (*I*<sup>0</sup> is the light intensity of the incident light). Furthermore, when the sphere diameter was increased to *Ds* = 4.5λ, the focal spot split into two parts, with the major part of the focal spot expanding into the surrounding medium, and a smaller part staying inside the microsphere [2,4]. Thus, for the microspheres with a refractive index of *n* = 1.5, when the sphere had a smaller diameter (*Ds* < 4.5λ), the light incident on the sphere was focused on the shadow surface (the surface of the particle that is opposite to the irradiated side [4]) of the sphere. When the sphere was large enough (*Ds* ≥ 4.5λ), PNJ formed beyond the shadow surface of the sphere [2,4], and the second focal spot formed inside the sphere and close to its shadow surface, due to light reflection at the inner side of the shadow surface [4]. It could be noted that, in this case, the maximal field intensity was slightly reduced (Table 1).

**Table 1.** Light focusing properties of the microspheres with a fixed refractive index of *n* = 1.5 and different sphere diameters (*Ds*).


**Figure 2.** Light intensity distribution of a simulated microsphere with a sphere diameter of (**a**,**b**) *Ds* = 1.5λ, (**c**,**d**) *Ds* = 2.5λ, (**e**,**f**) *Ds* = 3.5λ, and (**g**,**h**) *Ds* = 4.5λ. Subfigures (**a**,**c**,**e**,**g**) are plotted in the zx-plane, which is perpendicular to the plane of polarization; subfigures (**b**,**d**,**g**), and (**h**) are plotted in the zx-plane, which is the plane of polarization. The Full Width at Half Maximum (FWHM) focal spot of the simulated microspheres are indicated by the contour lines at the value of half maximum light intensity 0.5*Imax*, which are plotted by the solid green lines.

#### **3. Nanohole-Structured Dielectric Microspheres**

This section describes the nanohole-structured microspheres with a sphere diameter of *Ds* = 3.5λ and a refractive index of *n* = 1.5, which were simulated and analyzed. To make the structure clear, schematic diagrams for the simulated microspheres with a through hole and a blind hole are depicted in Figure 1b,c, respectively.

#### *3.1. Microspheres with a through Hole*

First, the nanohole-structured microsphere with a through hole of diameter *dh* = λ/5, λ/10, and λ/15 were simulated; light intensity around the simulated microspheres are shown in Figure 3. The corresponding focal spot properties, including the FWHM focal spot sizes (*Sx*, *Sy*, *Sz*), the focal volumes *V*, and the maximum light intensity *Imax*, in the focal spot, are shown in Table 2. After comparing Figure 3a,b with Figure 2e,f, we found that the focal spot sizes (*Sx,y,z*) and focal volume *V* of the λ/5-nanohole-structured microsphere were even larger than that of the microspheres without a nanohole, which could be explained as the weakening of the light focusing capability of the dielectric microsphere, due to the comparatively large λ/5-sized hole. When the hole size was reduced to be smaller than λ/10, the focal spot sizes and focal volume were reduced considerably, as shown in Figure 3c–h and Table 2. Finally, the features of the FWHM focal spot of the microspheres with λ/15-sized hole are plotted in Figure 3g, where the green solid lines, the green dashed lines, and the gray solid lines represent the contour lines with a value of 0.5*Imax*, 0.8*Imax*, and 0.9*Imax*, respectively. A comparison of the data in Tables 1 and 2 shows that, with a hole of λ/15 diameter, the maximum field intensity near the shadow surface of the particle is increased by nearly two times, with a significant decrease in the field localization volume, compared to the unstructured particle.

**Table 2.** Light focusing properties of the simulated nanohole-structured microspheres with a sphere diameter of *Ds* = 3.5λ, refractive index of *n* = 1.5, and a through hole of *dh* = λ/15 in diameter.

**Figure 3.** Light intensity (|E|2) of the simulated dielectric microspheres with a nanohole of size (**a**,**b**) *dh* = λ/5, (**c**,**d**) *dh* = λ/10, and (**e**–**h**) *dh* = λ/15. The sphere diameter and refractive index are set as *Ds* = 3.5λ and *n* = 1.5. Features of the focal spot in (**f**) are plotted in (**g**), where the green solid lines, the gray dashed lines, and the gray solid lines indicate the contour lines at values 0.5*Imax*, 0.8*Imax*, and 0.9*Imax,* respectively. For a sphere with a hole size of *dh* = λ/15, light intensity along a plane that is 0.001λ from the shadow surface is plotted in (**h**), with a circle plotted in gray dashed lines, indicating the projection of the hole interface.

#### *3.2. Microspheres with a Blind Hole*

After this, dielectric microspheres with a blind hole at the shadow surface were also simulated and the results are shown in Figure 4. In the simulation, the hole diameters were set to be *dh* = λ/5, λ/10, and λ/40, respectively, with a hole depth of 3*dh*. The corner radius at the opening of the hole, as well as the fillet radius at the blind end of the hole, were both set to be *r* = *dh*/2. From Figure 4a,b, we can see that for a sphere with a hole size of *dh* = λ/5, the focusing capability of the sphere is weakened by the hole, compared to a sphere without a hole in Figure 1e,f. When the hole sizes are decreased below λ/10, the focused light spot was mainly confined to the blind hole, as shown in Figure 4c–h. "Hot spots" can be observed in Figure 4e,h, which were located inside the hole and near the opening of the hole in the polarization plane (z–y plane), as indicated by the contour lines at the value of 0.85*Imax* and 0.9*Imax*.

**Figure 4.** Light intensity (|E|2) of the simulated dielectric microspheres with a blind nanohole of the diameter (**a**–**c**) *dh* = λ/10, (**d**–**f**) *dh* = λ/10, and (**g**–**i**) *dh* = λ/40. The sphere diameter and refractive index are set as *Ds* = 3.5λ and *n* = 1.5, with a hole depth of 3*dh*. Subfigures (**a**,**d**,**g**) are plotted in the zx-plane, which is perpendicular to the polarization plane. Subfigures (**b**,**e**,**h**) are plotted in the zy-plane, which is the polarization plane. To indicate the "hot spots" in subfigures (**e**,**h**,**i**), contour lines at the value of 0.5*Imax*, 0.85*Imax*, and 0.9*Imax* are plotted by the green solid lines, the green dashed lines, and the gray solid lines, respectively. Subfigures (**c**,**f**,**i**) show the light intensity along a plane that is 0.001λ from the shadow surface of the sphere and the circles, which are plotted by gray dashed lines, indicating the projection of the hole interface in the shadow surface.

#### **4. Discussion**

The effect of 3D-field localization inside and near the open boundary of a hole, which is clearly visible in Figures 3 and 4, is similar to the horizontal slot wave-guide in 2D dimensions [18], as a result

of this slot concentrating on the optical energy in the low refractive index region of the wave-guide [19]. It should be noticed that in Reference [18], the refractive index contrast was found to be more than 2. Moreover, in our case, the hole was surrounded by dielectric material with axial symmetry, and the boundaries of such a three-dimensional hole structure were essential for the field localization and enhancement, as is shown in Figures 3 and 4. Furthermore, the high E-field confinement, and the large discontinuity in the air hole, is clearly visible in Figure 5. The E-field confinement in the air hole is dictated by the dielectric refractive index contrast [18,19]. It must be mentioned that, in our case, unlike the slot-wave-guides, the field became localized only near the shadow surface of the particle and not along the entire hole. The observed field enhancement effect was analogous to the lens effect. This allowed us to choose the length of the hole, which was not necessary to be equal to the diameter of the particle, but only a part of it (not the through hole). For example, this could be the length of the field localization region inside the particle, near the shadow surface.

**Figure 5.** Light intensity along the two imaginary cutting lines L1 and L2, which lie on the shadow surface of the simulated microspheres. L1 is defined as *y* = 0 and *z* = *Ds*/2 + 0.001λ, which is parallel to the x-axis and perpendicular to the polarization direction. L2 is defined as *x* = 0 and *z* = *Ds*/2 + 0.001λ, which is parallel to the y-axis, as well as the polarization direction. The gray circles, the blue dashed lines, and the red solid lines indicate the light intensity of the 3.5λ-diameter microsphere without hole, the microsphere with a through hole of λ/15 (see Figure 3e–h), and the microsphere with a blind hole of λ/40 (see Figure 4g–i), respectively.

In addition, the proposed nanohole-structured sphere had several unique properties. For example, it could produce a high-field intensity and optical power in relatively low-index materials (air), at levels that cannot be achieved through a conventional PNJ produced by spheres with the same diameter [2–4]. This property allowed for highly efficient interactions between materials and the fields, in the hole. Moreover, it could produce a strong field confinement which is localized in a nanometer-sized hole, obtaining a resolution that is comparable to the nanohole size, beyond the diffraction limit. Thus, such nanohole-structured mesoscale particles can be used to significantly enhance the efficiency of the near-field probes, and to increase the sensitivity of the compact optical sensing devices.

#### **5. Conclusions**

In this work, a nanohole-structured dielectric mesoscale microsphere for a subwavelength-scale light focusing and confinement has been proposed. Numerical simulations showed that light can be focused and confined, considerably, in the nanohole that is located at the shadow surface of the dielectric microparticles, achieving a resolution beyond the solid immersion diffraction limit (λ/2*n*) [20], with a plane wave illumination. The introduction of a nanohole on the shadow surface of a dielectric particle, allows us to "compress" the field localization characteristic of a photonic nanojet, to the size of this nanohole. The proposed nanohole-structured microparticles are important for advancing the particle-lens-based super-resolution technologies, including sub-diffraction imaging, interferometry, surface fabrication, enhanced Raman scattering, optical tweezer, and so on.

**Author Contributions:** I.V.M. and O.V.M., initiation and supervision; Y.C., simulation and visualization of the results; Z.L., methodology and verification of simulation; all authors wrote the paper and discussed the results.

**Funding:** This research was funded by the National Natural Science Foundation of China, Grant Number 51675506.

**Acknowledgments:** The work by Igor V. Minin was carried out within the framework of the Tomsk Polytechnic University Competitiveness Enhancement Program.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Nanopillar Di**ff**raction Gratings by Two-Photon Lithography**

#### **Julia Purtov 1,2,\*, Peter Rogin 1, Andreas Verch 1, Villads Egede Johansen <sup>3</sup> and René Hensel 1,\***


Received: 7 September 2019; Accepted: 15 October 2019; Published: 19 October 2019

**Abstract:** Two-dimensional photonic structures such as nanostructured pillar gratings are useful for various applications including wave coupling, diffractive optics, and security features. Two-photon lithography facilitates the generation of such nanostructured surfaces with high precision and reproducibility. In this work, we report on nanopillar diffraction gratings fabricated by two-photon lithography with various laser powers close to the polymerization threshold of the photoresist. As a result, defect-free arrays of pillars with diameters down to 184 nm were fabricated. The structure sizes were analyzed by scanning electron microscopy and compared to theoretical predictions obtained from Monte Carlo simulations. The optical reflectivities of the nanopillar gratings were analyzed by optical microscopy and verified by rigorous coupled-wave simulations.

**Keywords:** nanostructures; optical pillar gratings; photonic crystals; two-photon lithography; direct laser writing

#### **1. Introduction**

Two-dimensional photonic structures such as periodical pillar gratings are applicable for light coupling devices [1], solar devices [2], sensors [3], encoders [4], holographic structures [5], or security features [6]. The optical characteristics of such gratings are very sensitive to the shape, diameter, and height of the nanostructures, as well as the pitch and periodicity of the array [7–11]. Therefore, these features require strict tolerances to ensure uniformity with virtually no defects within the grating. Simultaneously, there is a desire to enable flexible manufacturing of optical elements, as each application has its own requirements regarding design and feature size. In this context, two-photon lithography (TPL) is a promising candidate for the fabrication of nanostructured gratings with tunable optical properties.

In recent years, TPL has been established as a suitable technique for the fabrication of complex three-dimensional structures with submicron resolution [12–15]. Its versatility has been demonstrated by various applications ranging from microfluidic devices [16], micropatterned adhesives [17,18], biological and mechano-sensitive scaffolds [19–22], and optical devices, such as lenses [23] and photonic crystals [24,25]. In a typical TPL process, a focused, femtosecond-pulsed, near-infrared laser (λ = 780 nm) exposes a photoresist, that is composed of reactive oligomers and a photoinitiator. However, the photoreaction is only initiated when two photons excite the initiator concertedly. As a result, the initiator molecule decomposes into radicals, which induce a cross-linking reaction of the oligomers. Such a two-photon event is very rare, so that the probability of a two-photon excitation, and thus the start of polymerization, is only given in the focal region where the photon density is highest [15]. The polymerization reaction is in most cases terminated by oxygen quenching of

radicals [26]. The size of the voxel, i.e., the volume element at which the polymerization occurs, is a function of the beam [15,27,28], the exposure parameters [29,30], and the chemical and physical properties of the photoresist [26,29,31,32]. In order to estimate the lateral voxel size *S*, Sun et al. [33] empirically derived an equation based on the diffraction limit *ldi f f* , a material constant α, the applied laser power *Ere*, and the threshold laser power *Eth* required for the cross-linking reaction:

$$S = l\_{diff} \alpha \sqrt{\frac{\ln(E\_{\rm tr}/E\_{\rm th})}{4k \ln 2}} \tag{1}$$

where *k* equals 1 or 2 for a single or two-photon excitation, respectively. According to Equation (1), *S* decreases with *Ere* and reaches 0 for *Ere* = *Eth*. From this it can be concluded that the smallest lateral feature sizes are achieved for laser powers close to the polymerization threshold of the photoresist. Small feature sizes, however, are highly susceptible to deformations and collapse due to their low mechanical resistance [34–36]. In particular, nanostructures with high aspect ratios and low mechanical stiffness are prone to such defects [37,38].

In the present study, we report on the fabrication of nanopillar diffraction gratings. These are fabricated via TPL and laser energies close to the polymerization threshold of the photoresist (10–23 mW) combined with a recently reported, improved development routine [36]. The obtained structures are analyzed by scanning electron microscopy. To enable a prediction of the size of the nanopillars for further investigations, numerical simulations based on a Monte Carlo algorithm were performed. The optical properties of the gratings with pillar diameters between 120 and 430 nm and heights ranging from 330 to 1315 nm were corroborated by rigorous coupled-wave analysis (RCWA) simulations.

#### **2. Experimental**

#### *2.1. Two-Photon Lithography (TPL)*

Diffraction gratings were fabricated from a negative tone photoresist IP-Dip (Nanoscribe, Eggenstein-Leopoldshafen, Germany) on fused silica substrates using the Professional GT two-photon lithography system (Nanoscribe, Eggenstein-Leopoldshafen, Germany). The system consisted of a 63× objective (NA = 1.4, Carl Zeiss, Oberkochen, Germany) and a femtosecond pulsed IR-laser (λ = 780 nm, 80 MHz laser repetition rate, and 100–200 fs pulse length, TOPTICA Photonics AG, Graefelfing, Germany). The system was operated in 'dip-in' mode, where the objective is immersed into the photoresist. The nanopillars were arranged with a center-to-center distance of 1 μm in a square lattice of 50 × 50 μm. Each pillar consisted of four vertically stacked voxels, whose focal points were separated by 300 nm, whereby the lowest voxel was placed 200 nm below the substrate-resist interface to ensure appropriate attachment to the fused silica substrate. The exposure and settling times were set to 0.1 and 2 ms, respectively. All diffraction gratings were fabricated on the same substrate with altered laser powers ranging from 10 to 23 mW for different gratings. To improve the mechanical stability of the nanopillars, the development was performed according to Purtov et al. [36]. Structures were developed for 20 min in PGMEA (Sigma-Aldrich, Steinheim, Germany), after which 70% of the solution was carefully replaced with isopropanol (Sigma-Aldrich, Steinheim, Germany) without exposing the structures to air. Such a solvent exchange was repeated three times, separated by a residence time of 10 min. Subsequently, a UV-post-crosslinking was applied (*t* = 300 s, λ*UV* = 365 nm, 350 mW, OmniCure S1500A, igb-tech, Friedelsheim, Germany) before structures were removed from the liquid and air-dried.

#### *2.2. Scanning Electron Microscopy (SEM)*

Samples were fixed on a metallic sample holder and investigated at tilt angles of 0◦ and 40◦ using a Quanta 250 FEG (FEI, Eindhoven, The Netherlands) equipped with an Everhart-Thornley-Detector (ETD) in high-vacuum mode. Copper tape was placed close to the nanostructures to avoid charging, as no conductive coating was applied to preserve the optical properties of the arrays. The spot size and the acceleration voltage were set to 2.0 and 2 kV, respectively. The measured pillar heights in micrographs were corrected for the sample tilt.

#### *2.3. Optical Microscopy*

The optical reflection characteristics of the pillar arrays were investigated using an optical microscope (Eclipse LV100ND, Nikon, Tokyo, Japan) equipped with a 20× color-corrected objective (NA = 0.45). The microscope was operated in bright field mode with a fully opened illumination aperture upon a white balance using a white sheet of paper.

#### **3. Numerical Simulations**

#### *3.1. Voxel Sizes*

In order to gain a better understanding of the size- and shape-changing effects during the TPL-process, the obtained nanostructures were analyzed and compared to theoretical pillar sizes derived from simulations considering the different laser powers applied. Since the Gaussian beam formalism based on the paraxial approximation is not appropriate to describe the tightly focused beam used in the experiment, the electric field distribution around the focal spot had to be determined by numerical integration. This integration was performed using the Huygens' principle, i.e., by assuming the field distribution to be the result of superimposing fields originating from an ensemble of emitting elementary sources. This approach was implemented in a self-written software executing a Monte Carlo algorithm.

The coordinate system of the simulation was defined with the light incident from the +*z*-direction. The origin of the coordinate system was set to the focal spot of the focussed beam. The elementary emitters were assumed to be dipole oscillators distributed in a planar arrangement parallel to the *x*-*y* focal plane with a normal distance *z*0. The plane of the dipole oscillators can be regarded as the exit pupil of the focusing objective with *z*<sup>0</sup> being the working distance. The phase of the dipoles ϕ(*r*) as a function of distance *r* from the *z*-axis was adjusted to result in a constant phase at the origin of the coordinate system (i.e., the center of the focal spot), which gives focusing:

$$
\varphi = -\frac{2\pi}{\lambda} \cdot \sqrt{z\_0^2 + r^2} \tag{2}
$$

The amplitude of the dipole strength per unit area <sup>→</sup> *P* followed a Gaussian radial profile, while the polarization was assumed to be circular in order to result in a rotationally symmetric field distribution around the focal spot. By arbitrarily setting all constant factors to 1, the full description of the radial distribution of the dipole strength is given by:

$$\overrightarrow{P}(r) = e^{-\left(r/w\right)^{2}} \cdot \begin{pmatrix} 1 \\ i \\ 0 \end{pmatrix} \cdot e^{iqr(r)}\tag{3}$$

The width *w* of the Gaussian function was calculated from the half divergence angle α, (experimentally determined to be 31.4 degrees by analyzing the beam profile as a function of the *z*-coordinate) and the working distance *z*0, as

$$w = z\_0 \cdot \tan(\alpha) \tag{4}$$

The distribution described above was then cut-off at a finite maximum radius *rmax* representing the finite opening of the focusing lens, which is defined by

$$
\arctan(r\_{\text{max}}/z\_0) = \arcsin(\frac{NA}{n})\tag{5}
$$

where *NA* is the numerical aperture of the lens and *n* the refractive index of the medium.

The field distribution near the focal spot was calculated as the superposition of elementary waves emerging from the emitter distribution (Equation (3)). To do this, the field in the volume surrounding the focal spot was mapped to a two-dimensional array of pixels addressed by axial and radial coordinates in the *x*-*z*-plane, taking advantage of the deliberately introduced rotational symmetry of the source. The resolution of this field map was chosen to be λ/50, where λ was set to 780/1.52 = 513 nm (the wavelength of the laser divided by the refractive index of the photoresist). The Monte Carlo algorithm repeatedly picked a randomly selected pair of a pixel <sup>→</sup> *x <sup>f</sup>* in this map and a point <sup>→</sup> *x <sup>s</sup>* in the source distribution in order to calculate the contribution of the source emitter to the selected pixel. The latter was taken to be the field of an elementary dipole equivalent to the emitter strength <sup>→</sup> *<sup>P</sup>* <sup>=</sup> <sup>→</sup> *P* → *x s* . Omitting constant factors, the elementary dipole field is given by:

$$\overrightarrow{E}\left(\overrightarrow{\mathbf{x}}\_{f}\right) = \left\{ \frac{\left(\overrightarrow{n} \times \overrightarrow{P}\right) \times \overrightarrow{n}}{R} + \left[\overrightarrow{3} \cdot \overrightarrow{n} \cdot \left(\overrightarrow{n} \cdot \overrightarrow{P}\right) - \overrightarrow{P}\right] \cdot \left(\mathbf{R}^{-3} + i \cdot \mathbf{R}^{-2}\right) \right\} \cdot e^{i\mathbf{R}} \tag{6}$$

where *R* = <sup>2</sup><sup>π</sup> λ → *<sup>x</sup> <sup>f</sup>* <sup>−</sup> <sup>→</sup> *x s* is the distance between the source and the pixel in the field map scaled by the wavelength, and <sup>→</sup> *n* = → *x <sup>f</sup>* − → *x s* → *x <sup>f</sup>* − → *x s* is the normalized vector pointing from the source point to the pixel. All contributions to one pixel originating from different source points (typically 10<sup>5</sup> contributions per pixel) were averaged into one field vector approximating the electrical field with full phase information at the center of each pixel. The square of the above field vector, *<sup>I</sup>* <sup>=</sup> <sup>→</sup> *E* 2 , is a relative intensity proportional to the physical intensity; the factor of proportionality arising from all the constants explicitly or implicitly omitted above. To calculate voxel sizes as a function of the laser power, this intensity distribution needs to be correlated to the laser power. We start by expressing the latter as a factor *f* times the threshold laser power (determined to be 9.3 mW by the analysis of the observed pillar diameters according to Equation (1); see Figure 1b). On the other hand, the above intensity distribution has a maximum *Imax* at the focal spot. At the threshold laser power (*f* = 1), this maximum is equivalent to the polymerization threshold *Ith*, *Imax* = *Ith*. At a higher laser power, the whole intensity distribution is multiplied by *f*, and polymerization is initiated wherever the scaled intensity exceeds the threshold. The theoretical height of a voxel is thus determined as the size of the interval in a longitudinal section through the intensity map where *f*·*I* > *Ith*. To derive the width of the voxel, the same analysis is applied to the cross-section.

The resulting theoretical voxel sizes were used to calculate the theoretical height *h* of the pillars as follows:

$$h = (a - 1)b + c + d/2\tag{7}$$

where *a* = 4 is the number of stacked voxels, *b* = 300 nm is the vertical center-to-center distance between voxels, *c* = −200 nm is the centre distance of the first voxel from the substrate interface, and *d* is the height of an individual voxel obtained from the numerical simulations. The resulting theoretical pillar heights and diameters were further used to calculate the initial aspect ratios of pillars and to estimate the shrinkage by comparing the theoretical with the experimental values (see Table S1 in the Supplementary Materials).

**Figure 1.** Sizes of nanopillars as a function of the applied laser power. (**a**) Scanning electron micrographs of nanopillars fabricated with different laser powers. The scale bar is 500 nm. (**b**) Diameters (black symbols) and heights (red symbols) of the nanopillars obtained from two-photon lithography (open circles) compared to numerical simulations (filled stars). The dashed line shows the fit of the pillar diameters using Equation (1) to estimate the threshold laser power of the photo resist. (**c**) Defect rates of optical gratings expressed as fractions of upright pillars in dependence on the applied laser power. The values were obtained from Scanning Electron Microscopy (SEM)-images as shown for 15 mW in the insert. The scale bar is 10 μm.

#### *3.2. Optical Spectra*

Although no reflection spectra were recorded due to instrumental limitations, an attempt was made to correlate the observed colors with simulated optical spectra. These were calculated from simulated diffraction efficiencies taking into account the different diffraction orders and the finite aperture angle of the microscope objective used. Diffraction was simulated by rigorous coupled-wave analysis (RCWA) using the electromagnetic solver, S4, developed at the Stanford University [39,40]. In these simulations, the pillar shape was approximated as a cylinder with an ellipsoidal tip. The radii of the cylinders *rc* were set to the experimentally determined radii of pillars fabricated at different laser powers. The cylinder heights *hc* were calculated with *hc* = *hp* − *he*, where *hp* was the measured pillar height. The height of the ellipsoidal tip *he* was assigned to half the voxel height evaluated by Monte Carlo simulations described above, as this parameter was hard to determine experimentally. Using this set of parameters, the pillar envelope function can be described as follows:

$$y(z) = \begin{cases} r\_c & \text{for } z \le h\_c \\ r\_c \sqrt{1 - \left(\frac{z - h\_c}{h\_c}\right)} & \text{for } h\_c < z \le h\_p \end{cases} \tag{8}$$

where the *z*-axis is assumed to be normal to the substrate surface, and thus corresponds to the structure expansion in the vertical direction. The pillars were discretized in slices of 10 nm. A periodic boundary condition was applied with a box size of 1 × 1 μm (in accordance with the pillar center-to-center distance) as well as an incidence angle of 0◦. The refractive index and extinction coefficient of the cured photoresist and the fused silica substrate were taken from ellipsometry measurements (see Figure S1 in the Supplementary Materials).

In experiments, the observed colors originate from reflected light that is diffracted by the grating. Reflection occurred at two different interfaces, i.e., the pillar-substrate interface and the backside of the 1 mm fused silica substrate. Due to the opening angle of the objective with NA = 0.45, only diffraction angles of equal to or below 27◦ were collected. To quantify the contribution of different diffraction orders to the coloration, an overlap factor *OF* between the incident light and the light cones of different diffraction orders were numerically calculated using the given geometrical parameters of the fabricated nanostructures and the imaging system (angle of incidence, collection angle). In the case of the 1st diffraction order, *OF* was found to range from 0.45 at a wavelength of 400 nm to 0.045 at 800 nm. Overlap factors for higher diffraction orders were zero except for negligible values at the shortest wavelengths and, therefore, were not considered in further calculations. The reflection at the pillar-substrate interface was calculated by summing 100% of the reflectivity values obtained for the 0th diffraction order and the values of the 1st diffraction order multiplied by 4*OF*λ, where 4 is the number of contributing 1st order diffraction cones, and *OF*<sup>λ</sup> the overlap for the respective wavelength. The reflection at the backside of the substrate was calculated in the same way but using transmissive diffraction efficiencies. For each grating, both contributions were added to a final reflection spectrum.

#### **4. Results and Discussion**

#### *4.1. Nanopillar Sizes*

Sizes of the nanopillars fabricated with laser powers varying from 10 to 23 mW were obtained from SEM-micrographs (Figure 1a). The pillar diameters ranged from 120 nm to 430 nm (measured at the bottom of the pillars). The heights of the pillars extended from 330 to 1315 nm. The structures exhibited an almost constant aspect ratio (height over diameter) of about 3 (Figure 1b). Evaluation of the defect rate revealed high quality of the obtained optical gratings with 100% upright standing pillars for all gratings with pillar diameters down to 184 nm (Figure 1c). Gratings fabricated with the lowest laser power of 10 mW and thus, closest to the polymerization threshold, exhibited 30% freestanding pillars and 70% collapsed structures. With a diameter of 120 nm, these pillars were the smallest high aspect ratio pillars fabricated with TPL to our best knowledge so far. The collapse of the pillars is most likely induced by capillary forces during drying upon development and post-curing. These collapses occur when the capillary forces exceed the elastic restoring forces of the pillars [36]. As the latter decrease with pillar diameter, smaller structures tend to collapse more easily. For the sake of completeness and to demonstrate the importance of using a UV-post-curing during development,

similar pillar structures were fabricated without the additional UV-exposure, exhibiting much more defects even for larger structures (see Figure S2 in the Supplementary Materials).

Theoretical voxel sizes obtained from numerical simulations ranged from 144 to 503 nm in diameter and from 531 to 1873 nm in height with a mean aspect ratio of 3.6 ± 0.1. As a result, the theoretical height of the nanopillars (i.e., four vertically stacked voxels in accordance to Equation (7)) ranges from 965 to 1637 nm as shown in Figure 1b. All values are summarized in Table S1 in the Supplementary Materials. For the diameter, the numerical simulations overestimate the experimental data by 18 ± 3% for all applied laser powers. For the heights, the numerical values overestimate the experimental data by 65% for 10 mW, but only 20% for 23 mW. These discrepancies are most likely related to the shrinkage of the nanopillars, their adhesion to the substrate, and surface tension [36,41]. The degree of shrinkage is mostly related to the removal of unreacted molecules during development. With the reduction of the laser power, the amount of unreacted and not covalently bound oligomers and fragments increased, which in turn did not contribute to the formation of the nanostructures. Furthermore, the distance of the stacked voxels was kept constant for all laser powers, so that voxels overlapped less at lower laser powers, which amplified the effect of an incomplete cross-linking reaction. These unreacted and non-crosslinked fragments were removed during development, which explains higher shrinkage with lower laser power. Upon development, the nanostructures remained in isopropanol and were exposed to UV again. This post-curing led to an additional crosslinking reaction providing enhanced mechanical stability and higher resistance against a capillary forces during drying as is evident from the comparison with structures developed without the additional cross-linking (Figure S2 in the Supplementary Materials). Furthermore, the shrinkage close to the substrate is limited by the adhesion of the nanopillars to the rigid substrate, which led to low shrinkage of the nanopillar diameters. The shrinkage of the pillars heights, in contrast, is not constrained and thus much stronger. This anisotropic shrinkage as well as surface tension effects are assumed to lead to the conical shape of the nanopillars.

#### *4.2. Optical Properties*

Figure 2 shows optical micrographs of the nanopillar gratings. The divergence angle of the illuminating light as well as the collection angle of the microscope were both 27◦. The optical micrographs revealed a colored reflection of the gratings, which is based mainly on diffraction and interference effects. Absorption can be neglected due to the low extinction coefficients of all materials involved (Figure S1 in the Supplementary Materials). The color of the gratings changed with the laser power from slight brownish (11 mW) to blue-green (23 mW). For 10 mW, the surface appeared colorless (Figure 2a). To quantify the optical appearance of the gratings, the optical micrographs were compared with numerical simulations that provide expected optical spectra for the experimental set-up used (Figure 3). The simulations were performed for six gratings corresponding to 10 mW (colorless), 11 mW (slightly yellow), 14 mw (brown), 17 mW (blue-brown), 20 mW (blue), and 23 mW (blue-green).

The obtained spectra did not fully agree with the colors recorded by optical microscopy. Minor deviations can be expected from using diffraction efficiencies for normal incidence to approximate the whole cone of incident light. Moreover, considerable differences could be attributed to variations of the pillar shape. In the fabricated gratings, the shape of the pillars varied between cylinders and cones (compare Figure 1a), whereas cylindrical pillars with elliptical tips were assumed in the simulations. The shape, though, is important for the choice of an appropriate pillar diameter for simulations. This argument was confirmed by simulations with 20% smaller pillar diameters, which led to a significant blue-shift of the spectra (dashed lines in Figure 3) and a better correspondence with the colors observed. From this, it can be concluded that even small variations in the shape of the pillars dramatically affect the optical appearance of the gratings, and that it is therefore not sufficient to evaluate diameters at the pillars' bases only. This outcome supports our arguments that good understanding of occurring effects, such a shrinkage, surface tension, and related mechanisms, as well as their influence on the

feature shape and sizes are important for a precise prediction of the optical properties of pillar gratings fabricated via TPL.

**Figure 2.** Optical appearance of the nanopillar gratings in dependence on the laser power. (**a**) Optical micrographs of 50 × 50 μm nanopillar gratings on a fused silica substrate. Scale bar is 25 μm. (**b**) Scanning electron micrographs showing the corresponding nanopillars. Scale bar is 1 μm.

**Figure 3.** Optical properties of the nanopillar gratings. Reflectivity in dependence on the wavelength obtained from numerical simulations for structure sizes as measured by SEM (solid line) and pillars assuming 20% smaller diameters (dashed line) (left) compared to optical micrographs of nanopillar gratings fabricated with different laser powers (right). The scale bar is 25 μm.

The total reflectivities shown in Figure 3 comprise the reflected diffraction at the pillar-substrate interface as well as the transmissive diffraction of light reflected at the backside of the substrate. As these two are expected to differ strongly in their minima due to variation in the optical path, minima as low as 1% predicted by simulation were surprising. Nevertheless, their presence is confirmed by the intense colors of the gratings. We assume that this could be caused by Mie resonances [42]. This assumption is supported by the strong dependence of the spectra on the pillar diameter. Small pillars obtained at low laser powers interact predominantly with ultraviolet to blue light, leading to a brownish hue. As the pillar size increases with increasing laser power, the resonance shifts towards red wavelengths, resulting in a blueish hue. This variability in color due to the size and shape of nanopillars gratings allows for efficient diffractive color filters.

#### **5. Conclusions**

Optical pillar gratings were successfully fabricated via TPL at different laser powers close to the polymerization threshold of the photoresist and investigated with respect to their sizes and optical properties using imaging techniques and numerical simulations. The following conclusions can be drawn:

(1) Defect-free nanopillar gratings were fabricated down to pillar diameters of 184 nm and aspect ratios about 3. The smallest pillar diameters achieved were 120 nm, but on imperfect arrays, and therefore would require further optimization in fabrication.

(2) Simulations of the voxel sizes overestimated experimental pillar sizes by 20% in lateral and up to 65% in the vertical direction. This effect can be rationalized by shrinkage that differs due to varying amounts of unreacted oligomers, different overlaps between adjacent voxels, substrate adhesion, and probably surface tension effects.

(3) The nanopillar gratings interfered with visible wavelengths and varied in their optical properties depending on the pillar sizes tuned by TPL. The simulation of the optical spectra confirmed that the coloration originates indeed from the diffraction of reflected light, but also prompts the notion that the size and shape of the nanopillars strongly influence the optical appearance.

In summary, optical gratings based on different pillar sizes can be manufactured by varying the laser power in TPL in a single process step on one surface. However, the fabrication of precise optical gratings close to the polymerization threshold of the photoresist requires a deep understanding of the involved processes.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2079-4991/9/10/1495/s1, Table S1: Summary of pillar dimensions as a function of the laser power. Figure S1: Optical properties of the cured photoresist and the fused silica substrate obtained by ellipsometry. Figure S2: Optical pillar gratings when fabricated without UV-post-curing.

**Author Contributions:** Conceptualization, J.P. and R.H.; methodology, J.P. and P.R.; validation, A.V., P.R., R.H.; formal analysis, J.P., P.R., A.V., V.E.J.; investigation, J.P.; resources, R.H.; data curation, J.P. and P.R.; writing, J.P., P.R., A.V., and R.H.; visualization, J.P. and R.H.; project administration, J.P. and R.H.; funding acquisition, V.E.J. and R.H.

**Funding:** This research was funded by the European Commission (Marie Curie Fellowship Looking Through Disorder (LODIS), 701455) as well as the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013)/ERC Advanced Grant no. 340929.

**Acknowledgments:** The authors thank Reza Hosseinabadi for his help with the TPL-fabrication and the optical imaging as well as Bruno Schäfer for his support during the ellipsometry measurements.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*
