**Two-Layer Erbium-Doped Air-Core Circular Photonic Crystal Fiber Amplifier for Orbital Angular Momentum Mode Division Multiplexing System**

**Hu Zhang 1,2,\*, Di Han 1, Lixia Xi 1,\*, Zhuo Zhang 3, Xiaoguang Zhang 1, Hui Li <sup>1</sup> and Wenbo Zhang 1,4**


Received: 25 February 2019; Accepted: 12 March 2019; Published: 15 March 2019

**Abstract:** Orbital angular momentum (OAM) mode-division multiplexing (MDM) has recently been under intense investigations as a new way to increase the capacity of fiber communication. In this paper, a two-layer Erbium-doped fiber amplifier (EDFA) for an OAM multiplexing system is proposed. The amplifier is based on the circular photonic crystal fiber (C-PCF), which can maintain a stable transmission for 14 OAM modes by a large index difference between the fiber core and the cladding. Further, the two-layer doped region can balance the amplification performance of different modes. The relationship between the performance and the parameters of the amplifier is analyzed numerically to optimize the amplifier design. The optimized amplifier can amplify 18 modes (14 OAM modes) simultaneously over the C-band with a differential mode gain (DMG) lower than 0.1 dB while keeping the modal gain over 23 dB and noise figure below 4 dB. Finally, the fabrication tolerance and feasibility are discussed. The result shows a relatively large fabrication tolerance in the OAM EDFA parameters.

**Keywords:** mode-division multiplexing; Erbium-doped fiber amplifier; photonic crystal fibers; orbital angular momentum

#### **1. Introduction**

Optical communication technology has developed rapidly in the past decades. All kinds of multiplexing and high order modulation technological have greatly increased the capacity of single fibers. The transmission capacity is gradually reaching the limit in that a standard single-mode fiber (SMF) cannot carry more than about 100 Tbit·s−<sup>1</sup> of data in the C + L band [1]. To meet the stupendously increasing demands for transmission capacity, mode-division multiplexing (MDM), which is one of space-division multiplexing (SDM), has been proposed. MDM utilizing the orthogonality among different orbital angular momentum (OAM) states as the multiplex method has exhibited promising prospects in recent years. An OAM beam is characterized by a helical phase front exp (ilϕ) (in polar coordinates, l is the topological charge, and ϕ is the azimuthal angle), which is an optical vortex beam [2–4]. Theoretically, l can be any integer value (that is, OAM has an infinite number of orthogonal eigenstates), which means that OAM has great potential to increase the transmission capacity in a MDM system [5]. For communications using OAM beams, there are two key problems

to be solved. One is the optical vortex generation, which is the most fundamental technique to implement OAM multiplexing. The optical vortex can be generated by a variety of schemes, such as spatial light modulator [6] and modified interference of different modes [7]. A liquid-crystal spatial light modulator is another promising scheme for generating OAM modes, which exhibits some good properties including simplicity, high efficiency, and reconfiguration [8–10]. Another is the optical vortex beam transmission, which needs the fiber supporting OAM modes. In current research, the ring fiber was designed mostly to transmit OAM modes [11–15]. The circular photonic crystal fiber (C-PCF) was also proposed as a potential OAM fiber structure [16–19]. One fiber can transmit several dozens of mutually orthogonal modes. To date, terabit data transmission based on an OAM fiber has been demonstrated [20]. The transmission distance of the OAM mode in fibers has already reached 50 km [21]. However, many techniques are still needed for handling the implementation of long-haul MDM systems based on OAM modes. OAM carrier signal amplification is one of the critical techniques.

The Erbium-doped fiber amplifier (EDFA) not only allows the light signal to be amplified online directly but also possesses the same range between amplification wavelength and the low loss wavelength of optical fibers, which is suitable for MDM systems based on OAM modes. However, unlike the single-mode fiber amplifier, there are tens (even dozens) of modes in the OAM fiber amplifier. The transverse distribution of each mode is different. The biggest difference of these mode gains was defined as differential mode gain (DMG) [22]. DMG should be low enough to ensure approximately equal amplification of the multimode in the fiber.

In recent studies, most designs of OAM-EDFA are based on the ordinary circular air-core fiber with a one-layer doped region [23–26]. These OAM-EDFAs perform well in the modal gains but the DMG still needs to be improved. A new design of a two-layer Erbium-doped fiber amplifier transmitting OAM modes was introduced in reference [27] and provides a feasible way to deal with the DMG. They reduce DMG by adjusting the concentrations in the two doped regions. In addition, a one-layer doped region OAM fiber amplifier based on the C-PCF has also been presented by our research team. The amplifier can provide DMG lower than 0.2 dB, which is lower than that of the ordinary circular air-core fiber amplifier [28]. Naturally, the C-PCF structure combined with a two-layer doped region would be a more optimal design for OAM-EDFA. Compared with previous work, this OAM-EDFA can provide better performance in both modal gain and DMG. In addition, our design exhibits a higher tolerance for application.

In this paper, we propose and design a new OAM-EDFA with two-layer Erbium doped based on the C-PCF. The intensity distribution of OAM modes is studied to adjust the doped regions reasonably. The widths and the doped concentrations of the two layers are considered. The performance of the OAM-EDFA with different parameters is analyzed to get an optimal design. All 14 OAM modes are amplified as equally as possible to achieve long-haul transmission.

#### **2. Theory**

Erbium ion has a unique three-level system, as shown in Figure 1. It includes a ground state, metastable state, and excited state, which is suitable to realize the population inversion between the metastable state and ground state to amplify a signal light whose wavelength is around the 1550 nm window (C-band). We used the Giles and Desurvire model to carry out the analyses upon the population conversion, which is widely used in simulations for fiber amplifiers [29–33]. Owing to the lifetime of the Erbium ion in the metastable state, which is much larger than that in the excited state, the particles of the excited state can be ignored; therefore, the model is considered as a two-level system. Then, we can describe the changing rate of the Erbium ion concentration in the metastable state by the following equations:

$$\frac{dn\_2(r,\varphi,z)}{dt} = \sum\_k \frac{P\_k i\_k \sigma\_{ak}}{h\nu\_k} n\_1(r,\varphi,z) - \sum\_k \frac{P\_k i\_k \sigma\_{ek}}{h\nu\_k} n\_2(r,\varphi,z) - \frac{n\_2(r,\varphi,z)}{\tau} \tag{1}$$

$$n\_l(r, \varphi, z) = n\_1(r, \varphi, z) + n\_2(r, \varphi, z) \tag{2}$$

where *n*<sup>1</sup> and *n*<sup>2</sup> are the Erbium ion concentrations of the ground state and the metastable state, respectively; *Pk* is the power of light; *k* = *s,p,a* corresponds to the light power of the signal and pump as well as the amplifier spontaneous emission (ASE) noise, respectively; *ik* is the normalized light intensity; *σak* and *σek* are the absorption and emission cross-sections (which are the attributes of Er3+) respectively; *h* is Planck constant; and *τ* is the lifetime of the metastable state. Equation (2) denotes the particle conservation in the two-level system, where *nt* is the total Erbium ion concentration. When the EDFA is in a stable state, the number of particles in the metastable state remains unchanged, and Equation (1) is equal to zero. Then, the particle inversion can be given by:

$$m\_2(r, \varphi, z) = m\_l \frac{\sum\_k \frac{\tau P\_k i\_k \sigma\_{ak}}{h \nu\_k}}{1 + \sum\_k \frac{\tau P\_k i\_k (\sigma\_{ak} + \sigma\_{ck})}{h \nu\_k}} \tag{3}$$

The transmission equation in the optical fiber can be expressed by the following propagation equation:

$$\begin{cases} \frac{dP\_k}{dz} = \sigma\_{ek}(P\_k(z) + mh\nu\_k \Delta\nu\_k) \int\_0^{2\pi} \int\_0^\infty i\_k(r, \Phi) \ n\_2(r, \Phi, z) r dr d\Phi\\ -\sigma\_{dk} P\_k(z) \int\_0^{2\pi} \int\_0^\infty i\_k(r, \Phi) \ n\_1(r, \Phi, z) r dr d\Phi \end{cases} \tag{4}$$

**Figure 1.** Three-level diagram of Erbium ion.

In the EDFA, the amplification occurs in the process of energy exchange between the pump light and Erbium ions. Therefore, the overlap of the pump light and the doped region determines mainly the performance of EDFA. Therefore, we define the overlap factor as the dimensionless integral overlap between the normalized optical intensity distribution and the normalized Erbium ion distribution:

$$\Gamma\_k = \frac{\int\_0^{2\pi} \int\_a^b i\_k(r, \varphi) n\_2(r, \varphi, z) r dr d\varphi}{\overline{n\_2}} \tag{5}$$

where *a* and *b* are the inner and outer radius of the fiber doped region. To obtain efficient amplification, it is necessary to make the signal light and the doped area perfectly overlapped, and the signal light and pump light matched greatly. Thus, a correction factor *η* is presented to evaluate the overlap between the signal light and the pump light [28]:

$$\eta = \frac{\left| \iint E\_s^\* \cdot E\_p dx dy \right|^2}{\left| \iint E\_s \cdot E\_s^\* dx dy \right| \left| \iint E\_p \cdot E\_p^\* dx dy \right|} \tag{6}$$

where *E*<sup>s</sup> and *E*<sup>p</sup> is the signal and pump electric field, respectively. For an accurate description, the overlap factor is modified as Γ*k'* = Γ*<sup>k</sup>* × *η*. The noise figure is another property of EDFA and can be calculated by:

$$F\_{\rm tr} = 2n\_{\rm sp}(G-1)/G \tag{7}$$

where *nsp* and *G* represent the spontaneous emission factor and the mode gain, respectively. The Equations (1)–(7) are combined to analyze the performance of the amplifier based on the OAM fiber.

#### **3. Modeling of the OAM-EDFA**

The light intensity distributions of OAM modes transmitted in fiber are ring-shaped. Therefore, the circular fiber structure, such as C-PCF, is a good choice for an OAM fiber. The OAM modes are formed by linearly combining orthogonal even and odd vector modes with a π/2 phase shift by the following equations:

$$\begin{cases} OAM\_{\pm l,m}^{\pm} = HE\_{l+1,m}^{\text{even}} \pm jHE\_{i+1,m}^{\text{odd}}\\ OAM\_{\pm l,m}^{\mp} = EH\_{l-1,m}^{\text{even}} \pm jHE\_{i-1,m}^{\text{odd}}\\ OAM\_{\pm 1,m}^{\pm} = HE\_{2,m}^{\text{even}} \pm jHE\_{2,m}^{\text{odd}}\\ OAM\_{\pm 1,m}^{\mp} = TM\_{0m} \pm jTE\_{0m} \end{cases} \tag{8}$$

where *l* is called the topological charge representing the number of the azimuthal period, and m is the radial order giving the number of concentric rings. The superscript "±" denotes the polarization state of the OAM mode. OAM modes combined of eigenmode HE possess a circular polarization in the same direction as the OAM rotation, while OAM modes formed of eigenmode EH exhibit a circular polarization in the opposite direction as the OAM rotation [12]. For a given topological charge (*l* > 1) and radial order, four OAM modes form an OAM family. The OAM1,1 family composed of even and odd mode of HE2,1 have two OAM modes because the OAM1,1 composed by TE0,m and TM0,m mode is unstable [34].

The C-PCF fiber structure without the two doped regions in Figure 2a was proposed by our group [17]. It supports the 14 OAM mode transmission and exhibits some good features, such as wide bandwidth, flat dispersion, and low confinement loss. The C-PCF is formed by a large air-hole located at the fiber center, solid circular ring region, and four rings of air-hole arrays as the photonic crystal cladding. The substrate material is pure silica with a refractive index of 1.444 (at 1.55 μm), and the air region contributes to the refractive index of 1. Thus, the refractive index of the cladding constituted by the period photonic crystal structure is determined by the weight of the two materials, which can be changed by tailoring the air-filling fraction of the photonic crystal cladding [35]. Between the central air-hole and the cladding air-hole arrays is a ring-shaped area, which acts as the high index fiber ring-core to confine the OAM modes well within it. If we arrange the two-layer Erbium-doped regions as shown in Figure 2a, it would be a good OAM-EDFA structure for both transmission and amplification. The spatial lattice positions on the *x–y* plane are given by:

$$\alpha = \Lambda N \cos \left(\frac{2n\pi}{6N}\right), \ y = \Lambda N \sin \left(\frac{2n\pi}{6N}\right), \ n = 1 - 6N \tag{9}$$

where Λ and *N* are the lattice constant and number of concentric lattice periods, respectively; *r* denotes the inner radius of the ring-shaped area with a high index (that is the radius of a large air-hole in the fiber center), and *d*<sup>2</sup> to *d*<sup>5</sup> are the diameters of the cladding air holes, ϕ is the azimuthal angle. We set the lattice constant of Λ = 2 μm, the diameter of a large air hole of *d*<sup>0</sup> = 2.4 μm, and parameters *dn*/Λ = 0.8, *d*<sup>2</sup> = *d*<sup>3</sup> = *d*<sup>4</sup> = *d*<sup>5</sup> = 1.6 μm.

**Figure 2.** (**a**) Schematic of a cross-section and the Erbium-doped region of the circular photonic crystal fiber (C-PCF); (**b**) the refractive index and Erbium concentration profile of the fiber; (**c**) normalized intensity profile of the modes in the C-PCF.

The schematic of the two doping regions of EDFA is shown in Figure 2a covered with a red grid. The refractive index and Erbium concentration profile of the fiber is shown in Figure 2b and the two doping regions correspond to the two grey areas in Figure 2c. The normalized intensity distribution of the 14 OAM modes is shown in Figure 2c. We can see that the low order OAM modes tend to distribute in the inner area of the high index region, while the high order OAM modes fall in the outer area. According to Equation (5), the amplification of different OAM modes is determined by the overlapping region between the doped region and the mode field distributions, which will cause different DMGs.

To reduce this difference, we arranged two-layer doped Erbium regions at the inner and outer sides of the high index region. The inner doped region overlaps more with the low order OAM modes, while the outer doped region overlaps more with the high order OAM modes. Therefore, this two-layer Erbium-doped arrangement balances the amplification difference between the lower and higher order OAM modes, and hence minimizes the DMG. Then, the doped widths and Erbium ion concentrations in the two regions should be optimized to obtain the optimal performance.

In the one-layer structure, the design can only be adjusted at the boundary of the doped region where the difference among the intensity of modes is the biggest, which means that the tolerance for application is low. Little doped boundary unconformity will cause a big difference for the performance. However, in the two-layer structure, the design is adjusted where intensities of different modes are very close. The tolerance of our design will be much higher.

#### **4. Analysis and Optimization**

To design an OAM-EDFA with good features, we need to study the influence of the parameters on the performance of the amplifier. The equilibrium amplification of different OAM modes and the conversion efficiency should be considered to realize low DMG and high gain. Figure 2a shows the structural parameters of the proposed OAM-EDFA, where *w*<sup>1</sup> and *w*<sup>2</sup> are the widths of the inner and outer ring doped region, respectively, which defines the performance of the amplifier. *d* denotes the spacing of the two-layer doped region, which provides the total effective amplifier area of the EDFA, and, furthermore, determines the conversion efficiency. Next, we investigate the selection of the parameters. First, *w*<sup>1</sup> and *w*<sup>2</sup> are set as equal, and *d* is swept from 0.8 μm to 0.2 μm with a step of 0.2 μm. Figure 3 shows the gains of different order modes versus *d* at 1550 nm wavelength. When *d* is equal to 0.4 μm, the gains of different order modes are almost identical, so the DMG is the lowest.

**Figure 3.** Gain as a function of spacing *d* at 1550 nm.

Figure 2c shows that the overlap between the field distribution of the lower order modes and the doped region is larger than that of higher order modes when *w*<sup>1</sup> is increased. Thus, the gain of lower order modes will increase faster than that of higher order modes. On the contrary, when *w*<sup>2</sup> is increased, the increases in the gain of lower order modes will be slower than that of higher order modes. We can balance the amplification of different modes by increasing *w*<sup>1</sup> and decreasing *w2*; accordingly, the DMG can be reduced. The final parameters are tailored to *w*<sup>1</sup> = 0.9 μm and *w*<sup>2</sup> = 0.7 μm while *d* = 0.4 μm through further optimization, where the *w*<sup>1</sup> is nearly at the point of the intersection of different modes in Figure 2c.

Besides the widths of the doped regions, the doping concentrations and the length of the EDFA also affect the performance. The relationship between the two concentrations of the doped regions and DMG (Erbium-doped profile is assumed uniform in each area) is shown in Figure 4a. The concentration in the circle can be set with an acceptable DMG. However, when the doped concentration is higher than the value of 1.8 × <sup>10</sup><sup>25</sup> <sup>m</sup><sup>−</sup>3, the noise figure is higher than 4dB, as shown in Figure 4b. Considering to balance the DMG and noise figure, *<sup>N</sup>*<sup>1</sup> is set at 1.5 × <sup>10</sup><sup>25</sup> <sup>m</sup>−<sup>3</sup> and *<sup>N</sup>*<sup>2</sup> is set at 1.6 × 1025 <sup>m</sup>−<sup>3</sup> as shown in Figure 2b, providing a great result for both low DMG and noise. The length of EDFA is swept from 0 m to 15m as shown in Figure 4c. The mode gains increase rapidly and then gradually saturates with the increase in fiber length, while the noise figure (NF) increases slowly and then rises rapidly. To compromise the mode gains and the noise figure, the length is set to 7 m.

**Figure 4.** (**a**) Differential mode gain (DMG) as a function of the doping concentration *N*<sup>1</sup> and *N*2, (**b**) Gain and noise figure (NF) as a function of the doping concentration *N*; (**c**) Gain and NF versus fiber length.

When the parameters of the structure are settled, the mode gains and DMG as functions of the pump power and signal power are shown in Figure 5. The signal wavelength and the pump wavelength are set to 1550 nm and 980 nm, respectively. The pump power is swept from 50 mW to 300 mW. The gain first increases rapidly and then gradually becomes saturated, and the DMG changes similarly, as shown in Figure 5a. To get a relatively high gain and low DMG, the pump power is chosen as 150 mW by a trade-off method where the gain is close to saturation and the DMG is low enough. Figure 5b shows the gain and DMG as the function of input signal power. Considering the trade-off between gain and DMG, we can take the value of the input signal power to be −15 dBm, which provides the gain is larger than 20 dB, and DMG is less than 0.05 dB.

**Figure 5.** (**a**) Gain and DMG versus pump power; (**b**) Gain and DMG versus signal power.

Figure 6 exhibits the performance of the OAM-EDFA with optimal parameters. The parameters within the doped region are set to *<sup>w</sup>*<sup>1</sup> = 0.9 <sup>μ</sup>m, *<sup>w</sup>*<sup>2</sup> = 0.7 <sup>μ</sup>m, *<sup>d</sup>* = 0.4 <sup>μ</sup>m, *N1* = 1.5 × <sup>10</sup><sup>25</sup> <sup>m</sup><sup>−</sup>3, and *N2* <sup>=</sup> 1.6 × 1025 <sup>m</sup>−3. The length of OAM-EDFA is selected as 7 m. The pump power and signal power are set to 150 mW and −15 dBm, respectively. The DMG of 14 OAM modes is lower than 0.1 dB, and the noise figures are below 4 dB over the whole C-band, as shown in Figure 6a. As well, over 23 dB gain across the C-band is obtained for the OAM-EDFA, as shown in Figure 6b, which can meet the need of MDM systems well.

**Figure 6.** (**a**) DMG and NF versus wavelength; (**b**) Gain versus wavelength.

In Figure 4a, we can see that the DMG would be acceptable when the doping concentrations are in the marked area, which means a large range for concentration tolerance. To obtain a balance between avoiding the radially higher order modes and achieving a good quality of OAM modes, a value of d0 can be selected from 2.2 μm to 3.6 μm, whose large range indicates a large fabrication tolerance [17]. Besides the concentration and the diameter of a large air-hole, the doped width tolerance should also be considered. Owing to the two-layer structure, OAM-EDFA can equalize the gain of the high order OAM modes and the lower order OAM modes. Therefore, the width between *w1* and *w2* can be tolerated within a larger range. At present, the fabrication of PCF and the doping process of the ion have been matured. A fiber similar to the C-PCF structure has been drawn and experimented on [36–38]. Therefore, it is feasible to manufacture the OAM-EDFA based on C-PCF.

#### **5. Conclusions**

We have presented a new design of OAM-EDFA with a two-layer doped profile to stably maintain 18 eigenmodes (14 OAM modes) for the MDM system. The amplifier is based on the C-PCF supporting OAM modes and adopts the core-pumping scheme. Parameters that affect the amplifier performance, such as the width of the two doped regions, the doping concentrations, the length of the amplifier fiber, and the pump power and signal power, are optimized by a trade-off scheme. The fabrication tolerance of the EDFA is also discussed, and the results show a relatively large tolerance. The two-layer doped region can balance the amplification performance of different modes to minimize the DMG of 14 OAM modes below 0.08 dB. The optimal OAM-EDFA designed will theoretically achieve a gain over 23 dB and a noise figure less than 4 dB for all 14 OAM modes across the full C-band.

**Author Contributions:** H.Z. and L.X. conceived and designed the manuscript, H.Z. wrote the manuscript, and L.X. modified the manuscript and confirmed the final version to be submitted; D.H. and H.L. calculated and analyzed the data; X.Z., Z.Z., and W.Z. investigated the literature and presented formal analyses.

**Funding:** This research was funded partly by the National Natural Science Foundation of China (NSFC), grant number 61571057, 61527820, 61575082.

**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* **Flexible Photonic Nanojet Formed by Cylindrical Graded-Index Lens**

#### **Cheng-Yang Liu**

Department of Biomedical Engineering, National Yang-Ming University, Taipei City 11221, Taiwan; cyliu66@ym.edu.tw; Tel.: +886-2-28267020

Received: 18 March 2019; Accepted: 5 April 2019; Published: 7 April 2019

**Abstract:** Photonic nanojets formed in the vicinity of the cylindrical graded-index lens with different types of index grading are numerically investigated based on the finite-difference time-domain method. The cylindrical lens with 1600 nm diameter is assembled by eighty-seven hexagonally arranged close-contact nanofibers with 160 nm diameter. Simulation and analysis results show that it is possible to engineer and elongate the photonic nanojet. Using differently graded-index nanofibers as building elements to compose this lens, the latitudinal and longitudinal sizes of the produced photonic nanojet can be flexibly adjusted. At an incident wavelength of 532 nm, the cylindrical lens with index grading = 2 can generate a photonic nanojet with a waist about 173 nm (0.32 wavelength). This lens could potentially contribute to the development of a novel device for breaking the diffraction limit in the field of optical nano-scope and bio-photonics.

**Keywords:** cylindrical lens; photonic nanojet; graded-index

#### **1. Introduction**

Optical super-resolution has become significant for many applications including optical imaging [1,2], optical trapping and manipulation [3,4], nano-patterning and lithography [5,6], spectroscopy [7], and data storage [8]. Because the traditional objective lens has a diffraction-limited light spot, many investigations have been devoted to finding a practical way to obtain a small focusing spot beyond the diffraction limit [9]. One of the practical ways is the photonic nanojet (PNJ). The PNJ generated by an illuminated dielectric microcylinder is introduced and numerically demonstrated by Chen et al. in 2004 [10]. The mechanism of super-resolution imaging by dielectric microcylinders and microspheres have been increasingly attractive to researchers [11–15]. The PNJ is a high-intensity narrow focusing spot in the near-field of transparent microcylinder. When the diameter of transparent microcylinder is larger than the incident wavelength, the PNJ is generated due to the interferences between the scattering and illuminating fields. The main property of the PNJ is that it is a non-resonant phenomenon with low divergence and a small waist on the sub-wavelength scale. To generate a PNJ, it has been investigated that the refractive index contrast between the single microcylinder and its surrounding medium performs a critical role in the characters of PNJ [16]. This feature of microcylinder-based PNJ restricts the selection of transparent materials.

In order to optimize key parameters (focal length, waist, and intensity) of the PNJ, several studies indicate that the PNJ distributions depend on the geometric shape and refractive index of the microcylinder [17–24]. Moreover, the microcylinder or microsphere consisted of a concentric core-shell structure with different refractive indices for adjusting the propagation length and width of the PNJ [25–33]. The PNJ phenomenon can be changed significantly by applying shell materials with refractive index > 2. The PNJ length formed by the core-shell microcylinder is increased to approximately 20 wavelengths. However, the price for this PNJ elongation is the waist widening and the intensity attenuation. The fabrication process of layered inhomogeneous core-shell microcylinder is very difficult and rather costly. Therefore, the new and simple procedure is an interesting research issue for PNJ shaping. The transparent medium in other geometries, such as micro-cuboids, micro-axicons, nanofibers, and optical fiber tips, are presented for the formation of PNJ [34–39]. These novel structures of transparent medium cause special features of PNJ-like intensity distributions and are highly probable to develop new applications.

In this paper, the combination of the metamaterial concept with the PNJ by plane wave illumination is proposed and numerically investigated. The graded-index nanofibers are used as building blocks to assemble the artificial cylindrical lens. By varying the graded-index type of the compositional nanofibers, the focusing properties of the lens are able to modulate according to our requirements. Using the finite-difference time-domain (FDTD) method, we simulated the optical field propagation of a plane wave passing through the cylindrical lens assembled by hexagonally arranged nanofibers in the air medium. The physical modeling is given in Section 2 for cylindrical graded-index lens. The effects of the graded-index types on the shape, focal length, full-width at half-maximum (FWHM), and intensity of PNJs are presented and discussed in Section 3. Finally, the conclusions of this investigation are summarized in Section 4.

#### **2. Physical Modeling**

Several numerical methods have theoretically studied optical intensity distribution in the vicinity of a transparent core-shell microcylinder or microsphere illuminated by a light source [26–29]. These studies suggest that the refractive index contrast between different shells plays a critical role in the formation of a PNJ. In order to verify the influence of the graded-index nanofibers, we performed FDTD calculations for modeling computational electromagnetics [40]. The schematic diagram of a PNJ generated by the cylindrical graded-index lens is shown in Figure 1. Geometrically, this cylindrical lens is constructed by multiple hexagonally arranged close-contact nanofibers which fully fill a cylindrical area with a particular diameter of 1600 nm. The number and diameter of these nanofibers are 87 and 160 nm. The proposed cylindrical lens is normally illuminated by a transverse electric plane wave propagating along *x* direction with the electric field polarized along the *z* direction. The length of this lens along the *z* direction is defined as infinitely long for guaranteeing the accuracy and speed of FDTD calculation. The grid size of the FDTD mesh is chosen to be 10 nm after the convergence verification. The boundary conditions at the *x* and *y* directions added enough space to deliver the power flow distributions of optical beam in the background medium. The background medium is air with a refractive index of 1.

**Figure 1.** Schematic diagram of the cylindrical graded-index lens for photonic nanojet.

Since the location and the intensity of PNJ depend on the refractive index contrast between each nanofiber layer, we consider a micrometer size cylindrical lens consisting of several concentric nanofibers with equal diameter. Figure 2a shows the graded-index model of the hexagonally arranged nanofibers. Every nanofiber layer with a number *s* is a homogeneous material and is defined by the refractive index *ns* (*s* = 0 to N). In order to specify the refractive index variation from layer to layer, the refractive index contrast of the cylindrical lens is expressed as *ns*/*n*<sup>0</sup> = (*n*N/*n*0) (*s*/N)*<sup>t</sup>* [26]. The index grading type parameter is *t* and the dielectric central core is *s* = 0. The *t* > 0 indicates that the refractive index grading starts from the central nanofiber and terminates in the outermost nanofiber, which has the lowest value of the refractive index. Figure 2b shows the different index grading types of refractive index *ns* in the graded-index lens. The *t* value defines the variety of refractive index grading including linear (*t* = 1), concave (*t* = 0.2 and *t* = 0.5), and convex (*t* = 1.5 and *t* = 2) types. In the present lens, the refractive index grading is realized with N = 5 distinct concentric nanofibers. When the *t* value is 1, the refractive index grading is the linear layer-by-layer variation with the constant contrast. The maximum value of refractive index at the central nanofiber is 1.5 and it decreases in the radial direction to a minimum value of 1.05 at the outer nanofiber. This choice of index values is based on the practicability of modern micro-scale coating technology of objects with a thin film that has adjustable refractive indices [41,42]. The refractive indices in the range of 1.05 to 2.0 for material synthesis have been realized by controlling the porosity of silica glass. This manufacturing process is also possible to use for the graded-index photonic crystal structure and fiber.

**Figure 2.** (**a**) Graded-index model of the hexagonally arranged nanofibers; (**b**) Different index grading types of refractive index *ns* in the graded-index lens.

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

The PNJ produced by a microcylinder has been found to present several important properties. First, the PNJ intensity is several hundred times higher than the incident light power. Second, the PNJ has a smaller waist than the classical diffraction limit. Using high-resolution FDTD calculation, we have simulated the intensity distributions of the cylindrical lens at different index grading types. The incident beam is linearly polarized with a wavelength of 532 nm. Figure 3 shows the spatial intensity distributions of PNJs formed in the vicinity of cylindrical graded-index lens with homogeneous material (*n* = 1.5), *t* = 0.2, *t* = 0.5, *t* = 1, *t* = 1.5, and *t* = 2. Figure 3a represents the reference model at the same modeling conditions, which are the cylindrical lens with all homogeneous nanofibers. It demonstrated that an optical beam propagates from the top of the lens and the significant near-field focusing effect is observed at the bottom of the lens. Accordingly, an intensity peak of the electric field

is known as the PNJ. The intensity peak in Figure 3a is 3.6 compared to the incident intensity of 1, and the FWHM of PNJ is 165 nm smaller than the incident wavelength of 532 nm. The similar intensity distributions representing a cylindrical lens with nanofibers of five different index grading types are shown in Figure 3b–f. We could see that the PNJ gradually shifts from the outside to the inside of the lens when the index grading type parameter *t* increases from 0.2 to 2. The focusing effect plays a significant role in the propagation of the light wave in the lens and the PNJ is located at the interior of the nanofibers. Compared to the homogeneous model, PNJs created by different graded-index nanofibers assembled lens have stronger modulation of intensity peak and FWHM.

**Figure 3.** Spatial intensity distributions of photonic nanojets formed in the vicinity of cylindrical graded-index lens with (**a**) homogeneous material (*n* = 1.5), (**b**) *t* = 0.2, (**c**) *t* = 0.5, (**d**) *t* = 1, (**e**) *t* = 1.5, and (**f**) *t* = 2.

Figure 4a shows the normalized intensity distributions of PNJ for cylindrical graded-index lens along the propagation axis (*x* axis). The longitudinal profile in Figure 4a is acquired as a two-dimensional cross-section of the intensity distribution by the straight line located at the center of the lens. The dashed line is the edge of the lens. According to Figure 4a, the position of intensity peak for PNJ decreases from 1035 nm to 603 nm as the index grading type parameter increases. The transversal profiles at the highest intensity peak are plotted along the *y* axis in Figure 4b. The FWHMs are 326 nm, 261 nm, 213 nm, 177 nm, and 173 nm corresponding to *t* = 0.2, 0.5, 1, 1.5, and 2, respectively. The FWHM of the PNJ monotonically increases with the growth of the index grading type parameter as well. These indicate that the graded-index nanofibers are able to focus a light spot smaller than the Abbe diffraction limit. The smallest FWHM (173 nm) of the PNJ achieved by the model at *t* = 2 is 35% smaller than the half of incident wavelength. Meanwhile, the highest intensity peak of the PNJ

is delivered by the same graded-index nanofibers at *t* = 2. These graded-index lens can be used in combination with a traditional objective lens for super-resolution imaging applications.

**Figure 4.** Normalized intensity distributions of photonic nanojet for cylindrical graded-index lens along (**a**) the propagation axis (*x* axis) and (**b**) the transversal axis (*y* axis). The dashed line is the edge of the lens.

If the PNJ is focused inside the lens, a magnified real image is formed as in the cases of Figure 3d–f. It can be noted that the cylindrical graded-index lens produces a one-dimensional super-resolution image along the nanofiber axis. Therefore, we may obtain a complete two-dimensional super-resolution image in a large area by rotating the cylindrical lens in a circular mode. It is clear that the improved PNJ properties in the proposed graded-index lens originate from the introduced inhomogeneity of the refractive index. This graded-index lens is essentially a compact compound scattering media with altering refractive index along the propagation direction. Figure 5 shows the focal length and FWHM as a function of the index grading type parameter for cylindrical lens. Apparently, decreasing grading parameter *t* results in elongated PNJ, accompanied by an expanded FWHM and decreased peak intensity. The evolution of the FWHM with the intensity peak with grading parameter t increasing from 326 nm to 173 nm. Therefore, the key way to manipulate PNJ is to find an optimum graded-index configuration. If the grading parameter is *t* > 1, the optical contrast of the serial nanofiber layers increases with their layer number. The central nanofiber plays the major function in the transformation of the PNJ inside the lens. When the grading parameter is *t* >> 1, the graded-index lens according to its optical properties becomes similar to a homogeneous lens with high refractive index.

**Figure 5.** Focal length and full-width half-maximum as a function of the index grading type parameter for a cylindrical graded-index lens.

It can be seen from Figure 5 that the PNJ length decreases as *t* value increases. The intensity peak is placed inside the nanofibers at the value of *t* = 2. Combining basic properties of the PNJ, a modified quality criterion Q is expressed as Q = (L × I) / FWHM [25]. The effective length, maximum peak intensity, and FWHM of the PNJ are L, I, and FWHM, respectively. The usability of a PNJ can be estimated by using this quality criterion in the solution of practical problems. When the Q value is high, the peak intensity of nanojet is high, its FWHM is small, and the effective length is long. Figure 6 shows the quality criterion as a function of the index grading type parameter for cylindrical graded-index lens. At the value of *t* = 0.2, the cylindrical graded-index lens optimally combines the high spatial localization with high intensity. The super-resolution and the relationship between the nanofibers and the light beam in the graded-index lens may have a physical connection with photonic crystal [43]. An individual nanofiber operates like a single nanolens. Due to the hexagonal arrangement, the grading refractive index is capable of guiding the intensity flow to the bottom nanofiber and generating a strong focus with a high-intensity peak. The nanofiber-assembled graded-index lens has some singular points which could be used to focus more power on the same phase. Therefore, this graded-index lens with the selected refractive index is suitable for nano-scale imaging.

**Figure 6.** Quality criterion as a function of the index grading type parameter for the cylindrical graded-index lens.

In order to compare the PNJ properties of our lens assembly to those of a single uniform-index microcylinder, we also performed FDTD calculation for a single microcylinder. The refractive index of a single microcylinder is 1.5. Spatial intensity distribution, normalized intensity distributions along the propagation axis and the transversal axis for PNJ formed by a single microcylinder with 1600 nm diameter are shown in Figure 7. In comparison with PNJ formed by cylindrical graded-index lens, it is noted that maximal intensity for the graded-index lens along the propagation axis is larger than peak intensity for uniform single microcylinder. Moreover, the FWHM for uniform single microcylinder is 277 nm, but the FWHM for the graded-index lens at *t* = 0.2 is 173 nm. The focal length for uniform single microcylinder is 65 nm and the normalized peak intensity is 0.76. The location of PNJ is close to the surface of the microcylinder.

**Figure 7.** PNJ formed by a single microcylinder with 1600 nm diameter: (**a**) spatial intensity distribution, (**b**) normalized intensity distribution along the propagation axis, and (**c**) normalized intensity distribution along the transversal axis. The dashed line is the edge of the microcylinder.

#### **4. Conclusions**

In conclusion, the cylindrical graded-index lens assembled by hexagonally arranged transparent nanofibers is reported. The effective refractive index of the nanofibers can be changed by tuning the index grading type parameter. We are able to modulate the PNJ by varying the graded-index type of the compositional nanofibers. Using high-resolution FDTD calculation, we indicate that the PNJ is dynamically switched by the graded-index lens. Moreover, we present an optimization demonstration which pursues better focusing characters of the PNJ. The cylindrical graded-index lens can successfully achieve lateral resolution beyond the diffraction limit under the plane wave illumination of 532 nm wavelength. The hexagonal arrangement of the graded-index nanofibers leads to an alternating change of refractive index that effectively collects evanescent waves accompanied by near-field coupling of scattering light. Such a mechanism for PNJ manipulation may bring about new applications for optical imaging with super-resolution.

**Funding:** This research was funded by Ministry of Science and Technology of Taiwan, grant number MOST 107-2221-E-032-033.

**Acknowledgments:** The author would like to thank Igor V. Minin for his invaluable discussion.

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

#### **References**


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