*2.2. E*ff*ect of Refractive Index Contrast on Resonant Interaction*

The refractive index of the sub-wavelength dielectric particle plays a critical role in determining the light scattering strength and hence its resonance characteristics. Figure 5 shows the scattering efficiency spectra for varying refractive indices for isolated dielectric sub-wavelength disk. A constant refractive index across the spectral range of interest and lossless dielectric medium are assumed in these simulations. The dimensions of the disk are same as in Figure 3. It is found that the contrast or sharpness of the resonances increases with increasing refractive index. This is associated with the higher quality factor of the resonance and stronger field concentration in the dielectric structure with higher refractive index. Furthermore, the scattering spectra is found to shift to longer wavelengths with increasing refractive index. This can be associated with the increased optical path length with increasing refractive index of the dielectric structure and its comparison relative to the wavelength range over which scattering is observed.

**Figure 5.** Scattering efficiency spectrum from isolated sub-wavelength dielectric sub-wavelength disk with dimensions same as in Figure 3 as a function of varying refractive index. Refractive index of 1.9 (blue curve), 2.2 (red curve), 3.5 (black curve) and n = 4.2 (green curve) are shown. The refractive index is assumed to be constant across the spectral range shown for each curve.

A detailed comparison of the scattering spectra of plasmonic and dielectric sub-wavelength structures as a function of increasing dielectric constant is reported in ref. [47].

In the case of 1D sub-wavelength periodic grating structures, the role of the refractive index of the grating material on the resonant spectra are shown in Figure 6 [48,49]. A schematic view of the 1D grating structures simulated is shown in Figure 6a. The transmission spectra contour maps of fully-etched, high-index contrast grating structure of silicon and medium-index contrast grating structure of silicon nitride are compared for varying grating heights. The spectra are shown for incident transverse electric (TE) polarization, oriented parallel to the gratings. It is found that, for silicon high contrast grating structures, the transmission spectra show prominent resonance features [50], as evident from the checkerboard type patterns in Figure 6b. In contrast to this, the silicon nitride medium contrast gratings show poor contrast resonance features, as shown in Figure 6c with a reduced wavelength range over which the resonance features are observed. This is a direct consequence of the reduced spectral window between the zeroth order diffraction and first order diffraction into the glass substrate for the case of silicon nitride when compared to silicon. The wavelength range across which zeroth order diffraction occurs, also called as the dual-mode resonance is denoted by the black arrows in each of the contour maps is found to be reduced for the silicon nitride structures. Thus, high index contrast periodic structures are generally better suited for engineering optical resonances for nonlinear optical applications. However, they invariably end up being highly lossy in the visible and near infrared wavelength region due to enhanced absorption above their energy bandgap. The medium index contrast materials such as silicon nitride, titanium oxide etc. do offer certain benefits in terms of extended low-loss transmission window when compared to high-index contrast structures. For example, in the case of silicon nitride, the low-loss optical window extends from close to 300 nm to 5 μm. This is particularly beneficial to realize high quality resonances both at the fundamental excitation and nonlinear signal wavelengths. Thus, there is still interest in research into alternate structures based on guided-mode resonances, partially etched zero contrast grating structures etc. to obtain prominent resonances using medium index contrast material systems [48].

**Figure 6.** (**a**) Schematic of fully-etched dielectric one-dimensional grating structure. The simulated transmission spectra for (**b**) silicon high contrast grating and (**c**) silicon nitride medium contrast grating as a function of varying height for fixed incident TE polarization. The dimensions and wavelength are normalized by the pitch of the grating structure. (Figure is reproduced with permission from ref. [49]).

#### **3. Physical Mechanisms behind Resonances in Arrayed Structures**

In this section, we discuss the underlying physical mechanisms that lead to resonances in the arrayed structures. Though the resonances in isolated and arrayed structures can be accurately modelled and designed based on scattering expansion or coupled-wave analysis methods as discussed in Section 2, these techniques do not give much physical insight into the working of the arrayed resonant structures and how incident light field interacts with the structures. Thus, it is instructive to describe the underlying working mechanism of the arrayed structures used for resonant nonlinear optical studies. Here, the resonance phenomenon studied are broadly classified as guided-mode resonances, EIT- like resonances and bound-states-in-continuum resonances. Though this list is not exhaustive [17,51], a majority of the resonant structures studied fall into these categories.

#### *3.1. Guided-Mode Resonances*

Guided-mode resonances arise in dielectric grating-waveguide coupled structures due to the evanescent diffraction orders from the grating coupling the incident electromagnetic wave into guided modes of the waveguiding layer. In other words, the optical resonances can also be described by the wavelength selective in- and out-coupling of electromagnetic wave into the waveguide through interaction with the grating structure. The waveguide can be either a separate high index layer located close to the grating structures or can be the grating structure itself which is fully or partially etched. Few examples of such structures which support guided-mode resonances in various waveguide-grating arrangement are shown in Figure 7. Here, we briefly outline some of the properties of periodic grating structures and demonstrated applications. Recent review articles provide a comprehensive overview of the recent advances in resonant waveguide grating structures [17,52].

**Figure 7.** Cross-section view of various guided-mode resonance based grating structures with waveguide and grating layer refractive indices nH and nG respectively. (**a**) The grating and waveguide are made of different materials with the waveguide of higher index below the fully-etched grating structure. (**b**) Fully etched grating structures which can act as the effective waveguide. (**c**) Partially etched grating structures with the waveguide layer made of same material as the etched gratings. (**d**) Substrate grating structures is coated with a high-index waveguiding layer on top.

Wood anomaly in diffraction gratings has been studied in the past to explain the occurrence of sharp spectral orders in the diffracted light [53]. In particular, resonance type anomalies are explained based on leaky guided-modes supported in the waveguide grating structures. A basic waveguide-grating model used to describe the diffraction effect from the periodically index modulated structure leading to coupling of waveguide modes is shown in Figure 8a [38,54]. This model is adapted from the seminal paper by R. Magnusson et.al. [38], which explained the working mechanism and applications of the guided-mode resonant structures. Under weak perturbation, the supported guided mode resonances can be understood based on the frequency-selective excitation of the modes of an effective waveguide formed by the periodically index modulated structure with the grating providing the required phase matching to couple the free-space incident light to the waveguide mode. Typical resonant filter characteristics obtained at normal incidence is shown in shown in Figure 8b [38]. Under weak index modulation, the filter reflection spectra are found to be strongly angle sensitive, as shown in Figure 8b. This offers a mechanism to angle-tune the filter characteristics. Such resonant response from waveguide-grating structures have been utilized as band-pass and band-stop filters working across different electromagnetic spectral regions. They have also been proposed to be used as intra-cavity narrow-band high-reflection mirrors, photorefractive tunable filters, and as electro-optic switches [38]. Guided-mode resonances have also found applications as bio-sensors in immune-assays in which local reaction between the functionalized antibody and the target antigen leads to a shift in the guided-mode resonance position [48,50,52,55]. Such assays have been proposed to be used in bright-field (colorimetric), fluorescence and also in wide-field imaging modes. In the context of nonlinear optics, the guided-mode resonance grating structure have been utilized as the nonlinear media in which the nonlinear optical interaction is enhanced due to the local electric-field built-up in the structure close to the resonance wavelengths. Some of these examples are discussed in Section 4 below.

**Figure 8.** (**a**) A cross-section schematic of the guided-mode resonance structure showing the resonant coupling of incident light into the waveguide region through interaction with the grating structure. (**b**) Simulated filter response for the guided-mode resonance structure for parameters: pitch = 330 nm, height = 330 nm, dielectric constant difference (normalized), Δε/εavg = 0.05 and center wavelength of 547 nm and dependence of the filter response on the angle of incidence. (Figure b is reproduced with permission from ref. [38]).

Fully etched high index contrast gratings, also termed as high-contrast gratings [50] are also used to create high quality factor resonances. A cross-section schematic view of such a structure is shown in Figure 7b above. The resonances in these structures can be understood either based on guided mode resonances with the grating coupling light into strongly-modulated effective waveguide created by the air/silicon periodic structure or based on optical modes supported in the periodic air-high index structure along the longitudinal propagation direction (i.e., along the optical axis) with energy exchanged with the interface due to reflection [56]. The resonant characteristics achieved in the high contrast grating structures have been classified as crossing and anti-crossing resonances based on the phase difference between the interacting longitudinal modes being odd and even multiple of π respectively [50,56]. The classification of the resonance spectra as crossing and anti-crossing features is shown in Figure 9. The anti-crossing resonances exhibit highly asymmetric Fano-like line shape with strong-coupling between the interacting modes. This results in strong field enhancement within the structure as shown in Figure 9c with high quality factor. In contrast, the crossing resonances exhibit more symmetric spectral shape, however with lower quality factor and reduced field enhancement, as shown in Figure 9d. The high contrast gratings have been used to build both narrow-band and broad-band reflectors with reflectivity greater than 95% for use as end-facet mirrors in vertical cavity surface-emitting lasers (VCSELs) [50]. They have also been used as bio-sensors based on refractive index changes altering the resonance characteristics [55]. The strong field-concentration has also been utilized for Raman scattering enhancement with suitable metal nanostructures patterned in regions of strong dielectric field concentration [57]. In the context of nonlinear optics, the strong nonlinear optical properties of the high index materials, such as Aluminum Gallium Arsenide and Silicon are leveraged in enhancing second-harmonic generation and four-wave mixing due to strong field enhancement near anti-crossing resonances [58,59]. This is further discussed in Section 4 below.

**Figure 9.** (**a**) Reflection spectra contour for free-standing silicon high contrast gratings with dimensions: ngrating = 3.48, duty cycle = 70% and thickness varied. The various regions of operation of the grating are also shown. (**b**) The reflection spectra overlapped with the solutions to the eigen-mode equations of the resonant modes (white curves). The overlap regions of the white curves result in anti-crossing and crossing type resonance. (**c**) Field intensity profile at anti-crossing resonance. (**d**) Field intensity profile at crossing resonance. (Figures are reproduced with permission from ref. [50]).

In comparison to fully-etched, high contrast grating structures, partially etched structures, also termed as zero-contrast gratings [60,61], offer an additional degree of freedom for designing the grating characteristics based on the chosen etch depth. Such structures can achieve robust spectral characteristics comparable to, if not better than that of high contrast grating structures. As an example, the realization of broadband reflective filter using zero-contrast gratings is shown in Figure 10 [60]. The structures shown here have been optimized using an inverse design approach in which particle swarm search algorithm is utilized to quickly search a wide design space with the performance optimized based on achieving a desired value of figure-of-merit. In this example, the figure-of-merit is chosen as follows [60]:

$$Figure\ of\ merit = \left\{\frac{1}{M} \sum\_{i=1}^{M} \left[R\_{desired}(\lambda\_i) - R\_{design}(\lambda\_i)\right] \right\}^{1/2} \tag{4}$$

The figure of merit compares the root-mean square error between the desired and designed reflection spectra with the goal of minimizing the difference between the two through optimized design. With multiple iterations this process helps optimize the device design to achieve the designed spectral response as close to the desired one as possible. A comparison of broadband reflector performance of zero-contrast gratings with high contrast gratings can also be found in ref. [60]. Such inverse design

approaches are most promising to search a wide design space and at the same time achieve close to the ideal response for the resonant structure. Another emerging direction is the use of neural networks for optical metasurface design optimization [62,63] to achieve optimal specifications with often non-intuitive, but effective meta-atom shapes which can be fabricated with present-day advanced nanolithography tools.

The angle sensitivity of the guided-mode resonance filter is another useful characteristic for filter response tuning or for wide field of view application. This is also found to be a strong function of the effective index of the waveguide-grating structure, with higher index resulting in stronger field confinement in the unit-cells and hence less angular sensitivity [50]. In this context, high contrast gratings with highly confined field profiles within the etched structures are found to be more angle insensitive than the partially etched zero-contrast gratings [50,64], which support diffused field-profiles extending across the unetched high-index slab region. Furthermore, conical mounting of partially etched gratings is also found to result in reduced angle sensitivity when compared to conventional mounting [65]. One-dimensional grating structures discussed above are inherently polarization selective and polarization independence is achieved by using two-dimensional symmetric meta-atoms, such as square or hexagonal arrangement of circular features. Such polarization independent structures are particularly useful for realizing optical filters for unpolarized light [66] or for realizing resonantly enhanced fluorescence sensors [48].

**Figure 10.** The simulated zeroth order reflection (R0) and transmission (T0) spectra for silicon zero-contrast gratings. The inset shows the cross-section of the zero-contrast gratings with the dimensions optimized using particle swarm combined with inverse-design algorithm. The optimized dimensions of the structure obtained are: etched grating height of 490 nm, unetched slab thickness of 255 nm, pitch of 827 nm, and fill factor of 0.643. (Figure reproduced with permission from ref. [60]).
