Stochastic modeling of 2D photonic crystals

Due to the fabrication processes, inaccurate manufacturing of the photonic crystals (PCs) might occur which affect their performance. In this paper, we examine the effects of tolerance variations of the radii of the rods and the permittivity of the material of the two-dimensional PCs on their performance. The presented stochastic analysis relies on plane wave expansion method and Mote Carlo simulations. We focus on two structures, namely Si-Rods PCs and Air-Holes PCs. Numerical results show—for both structures—that uncertainties in the dimensions of the PCs have higher impact on its photonic gap than do the uncertainties in the permittivity of the Si material. In addition, Air-Holes PCs could be a good candidate with least alteration in the photonic gap considering deviations that might occur in the permittivity of Si due to impurities up to 5%.


Introduction
Photonic Crystals (PCs) are promising candidates for many applications, and they might act as a waveguide or microcavity laser (Painter et al. 1999;Baba et al. 1999) because PCs are engineered to provide controlled optical properties. Such optical properties are not available in homogeneous materials. It is known that PCs are designed to have a periodic structure of different materials which can control the electromagnetic waves propagation through the photonic crystal (Gong and Hu 2013a). The structures of the PCs exhibit a photonic bandgap (PBG), where the electromagnetic wave cannot propagate in this range of frequencies due to the total reflection (Joannopoulos 2008). In many applications, a large PBG of PCs is considered as an important feature (Susa 2002). The PCs' designs and fabrications are challenging due to the diversity of periodic arrangement and layer dimensions and materials which the designed PC needs. For two-dimensional PCs, the embedding of circular rods into square array and changing the radius of one or more rods results in defect modes. In Lee (2020), varying of an air-rod radius that is placed at the center of the array leads to local defects with tunable frequencies those depend on the radius of these air rods.
The widely used fabrication approaches of photonic crystals are mainly microfabrication etching or thin film fabrication (Gong and Hu 2013b). In these fabrication processes, it is too difficult for exact realization of dimensions of the photonic crystal and the permittivities of its materials. This fabrication inaccuracies affect the operating performance of the PCs. In the presented work, we investigate the effects of deviations-that may arise due to fabrication issues-in the photonic crystals' components on their photonic band gap. We consider deviations that might occur in the dimensions as well as the permittivity of the PC's material using Monte Carlo simulation.
We have applied Monte Carlo technique for the stochastic analysis part in this work because it is widely used technique to determine tolerance (Graham and Talay 2013;Vidal-Codina et al. 2019). In addition, Monte Carlo modeling is accurate, and easy to implement approach. In Graham and Talay (2013), Monte Carlo modeling of imperfections in the materials of 2D-PCs is applied. In Vidal-Codina et al. 2019), a multiscale continuous Galerkin method has been used providing high accuracy and fact stochastic simulation of time-harmonic wave propagation through PCs. The technique is used in Zhang and Liu (2016) to study the properties of photonic band gaps in 2D plasma-dielectric PCs using the modified plane wave expansion method. Monte Carlo has been used (Liu and Oliver 2003) to investigate the impacts of the randomness on the optical featured of waveguides embedded in disordered 2D-PCs for TM-polarized radiation. In our previous work (Wafa et al. 2020), a stochastic analysis of one dimensional PC has been studied showing the impacts of the tolerances in PC's dimensions on the operating optical properties for defect free PC and for PC with a defect air-layer.
In this work, we consider two types of 2D Photonic Crystals; the first type has rods with high refractive index placed in background of lower refractive index, whereas the second type is constructed of rods with lower refractive index compared to the background of the crystal. For these structures, we focus our attention on the stochastic analysis of 2D-PCs where the effects of the tolerances of both the dimensions of the PC and permittivities of its materials on the normalized gap-width are modeled and studied. In this stochastic analysis, the effects of the uncertainties of the radii of the rods and the lattice constant of the crystal on its functionality are considered. We have also studied the effects of stochastic changes that might occur in the relative permittivity of the PC rods that may results from the impurities. Our work is based on the plane wave expansion and Monte Carlo simulation. Our results revealed that the uncertainties in the dimensions of the PCs have serious consequences on the photonic gap than do the uncertainties in the permittivity of the Si material.
First, the proposed 2D-PCs structures are detailed described and the presented stochastic modeling is presented and discussed in Sect. 2. After that, the numerical results of the stochastic analyses are presented in Sect. 3, where both 2D-PCs with Si rods in air background and 2D-PCs with air-Holes in Si background are studied. In both structures, the impacts of the tolerances in the crystal dimensions and permittivities on the operating photonic bandgap are investigated. Finally, the conclusions are summarized. Figure 1 shows a schematic of 2D photonic crystal where the rods (of radius r) are embedded into the background of square lattice with a lattice spacing of a. The rods of 2D-PC are parallel having length much greater than r and these rods are placed along the z-direction. The relative permittivities of the rods and the background are ε a and ε b , respectively. In the presented modeling, transverse-electric field (TE) is considered. It is defined as:

Stochastic modeling and methods
where ω and r are the angular frequency and, respectively. For TE, both the dielectric function and the electric field can be expanded in a Fourier series due to the periodicity in the x-y plane, resulting in the following eigenvalue problem, where k and G are the two dimensional direct and reciprocal-lattice vectors. Equation (1) is solved numerically by substituting G = 2 a n 1 , n 2 for the square lattice where n 1 and n 2 are integers and the upper limit for both n 1 and n 2 is set to be ten, while the total number of plane waves of 441 are used which ensures adequate convergence (El-Naggar 2012; Aly et al. 2014). In the presented numerical simulations, a normalized frequency of ωa/2π c = 1 has been used and we have mainly focused on the normalized bandgap-width (Δω/ω g ), where Δω is the frequency of the photonic bandgap that results between the first and the second bands, and ω g is at the middle of this frequency range.
In the presented stochastic model, Monte Carlo technique use the deterministic simulation code as a black-box that relates between inputs which express the design parameters of the 2D-PCs and the associated outputs those express the operating performance. The technique generates a large number of samples (M) and evaluates the deterministic model for each of them, resulting in the response vector. According to the law of large numbers, x and are defined as the mean and the standard deviation of these samples, respectively and they can be determined as following: It is worth mentioning that increasing the number of samples enhances the degree of accuracy of the probability distribution function PDF of the output response and leads to a decrease in the standard deviation σ.

Modeling results
In these simulations, we consider two types of 2D photonic crystals (2D-PCs). The first type is composed of Si rods embedded in air background and we refer to this type as Si-Rods PCs. In this type, we set r/a = 0.2, ε a = 12.4 and ε b = 1. The second type of the studied 2D-PCs is composed of air holes in a Si background and we refer to this type as Air-Holes PCs. In this type, we set r/a = 0.45, ε a = 1 and ε b = 12.4. In order to held comparison between photonic band gaps of the two structures, we calculate the normalized photonic bandgap (Δω∕ω g ), where Δω is the frequency of the bandgap width that results between the first and second band and ω g is the central frequency of the bandgap. The design parameters of the 2D-PC are r, a and ε a and ε b . The photonic bands of these 2D-PCs are shown in Fig. 2, and they are denoted by "deterministic case" where there is no uncertainty either in the dimensions or in the permittivities of the photonic crystals' materials. In Fig. 2a, the photonic band gap of Si-Rods PCs is shown where it extends from 0.2853 to 0.4283 in units of 2πc/a, which gives Δω∕ω g = 40.08%. The photonic band gap of Air-Holes PCs structure is shown in Fig. 2b, where it extends from 0.2287 to 0.2693 in units of 2πc/a, which gives Δω∕ω g = 16.31%. In the following sections, we introduce uncertainty in our design parameters; namely r, a and ε of the Silicon and examine the effects of these uncertainties on the normalized photonic band gap Δω∕ω g for each structure.

Si-rods 2D-PCs
First, we consider uncertainties in both a and r by applying Monte Carlo technique with 10,000 sample set size. In Fig. 3, the photonic band gap is shown for uncertainties of 1%, 2%, 5% and 10% in both a and r, respectively. As shown in this figure, there is a large tolerance of the photonic bandgap width and its upper and lower limits. Also, the uncertainty of the bandgap width increases by increasing the tolerances of the rod radius and the lattice constant. The proven gap between the first and the second frequency bands for all given uncertainties of the PC stochastic parameters is defined as confident photonic bandgap (Wafa et al. 2020). The confident PBG is simulated for different uncertainties of both r and a and listed in Table 1, where it can be noticed easily that increasing these tolerances leads (a) (b) Fig. 2 The photonic band structure for a Si-Rods PCs and b Air-Holes PCs to a decrease of the width of the confident bandgap. Further increase in the uncertainties of the stochastic parameters of 2D-PC makes the confident bandgap disappears completely as in the case of 10% tolerance. In such case, the PC might not work as designed and losses its major feature. We also plot mean and the standard deviation of the upper and the lower normalized frequencies for the band structure of the first PBG in Fig. 4. As shown in these figures, by increasing the uncertainties in both a and r, the mean of the upper and the lower normalized frequencies of this band structure mainly are not affected by the uncertainties in Fig. 3 The Normalized frequency and Normalized frequency mean for tolerances of r and a of a 1%, b 2%, c 5% and d 10% these design parameters while the standard deviations increases clearly by increasing these uncertainties as shown in Fig. 4-b. The probability distribution function (PDF) of the normalized band gap for different tolerances are plotted in Fig. 5(a) while the PDF of the normalized first and second bands for these tolerances are shown in Fig. 5b. As shown in these figures, when the uncertainties in a and r increases from 1 to 10%, the PDF of the normalized bandgap extends over wider range, while the mean of the normalized frequency gap slightly changes. Also, the PDF of the normalized first and second bands broaden for larger uncertainties. The extension of the PDF over wide range reflects the increase in the standard deviation as well as the high uncertainty of both the bandgap width and its upper and lower edges. The standard deviations σ for the PBGs for various uncertainties of r and a are shown in Table 2. In this table, it can be easily noted that the standard deviation increases more than 12 times as the uncertainty increases from 1 to 10%. In addition, the ratio between the standard Next, the effects of the impurities of the material of rods of the Si-Rods PCs on its performance have been investigated. We consider uncertainties of 1%, 2% and 5% in the rod's permittivity ε a and we plot the PDF of the bandgap and that of the first and second band frequency limits as shown in Fig. 6a, b, respectively. The standard deviations for the PBGs for various uncertainties of the permittivity are shown in Table 3. We observe that the effects of the uncertainties in the permittivity on the band gap of the crystal are small compared to the case when considering the uncertainties of the crystal dimensions. In Table 3, we note that increasing the tolerance from 1 to 5% leads to a change in σ/Mean ratio from 0.326 to 1.61%, while corresponding values in Table 2 vary from 0.7% to 3.8%. We note from these percentages that the impact on σ/Mean due to deviations in the dimensions-r and a-is almost double its value due to deviations in the permittivity up to 5%.

Air-holes 2D-PCs
In this section, we examine the effects of tolerances of the 2D-PC's stochastic parameters on the PBG in the case of Air-Holes 2D-PC's structure. The presented analysis starts by considering uncertainties of 1%, 2%, 5% and 10% in r. The mean of the lower and upper bands are plotted in Fig. 7a for these values in addition to the deterministic case while the standard deviation of both the lower and the upper normalized frequencies for the band structure of the first PBG are shown in Fig. 7b. As shown in these figures, by increasing the tolerances in r, the mean of the upper and the lower normalized frequencies of this band structure mainly are not affected by the uncertainties in r. However, the standard deviations for these frequencies shown in Fig. 7b increase by increasing the tolerance from 1 to 10%. The standard deviations for the PBGs for various uncertainties of r are shown in Table 4. Comparison between Tables 2 and 4 reveals that the standard deviations in Air-Holes PCs structure have much higher corresponding values for Si-Rods PCs structure. After that, the confident PBG width for different uncertainties of r for this PC structure have been evaluated and listed in Table 5. As shown in this table, the normalized confident PBG Δω∕ω g decreases to 6.59% for a tolerance 1% in r while no confident gap is guaranteed for any further increase in this tolerance because the standard deviation is very high in this case.
Finally, we have investigated the effects of uncertainty in the permittivity of the background material for the Air-Hoes 2D-PCs structure. We have considered uncertainties in ε b of 1%, 2%, and 5% that may result from the impurities within the background material. The PDF of the bandgap and of the first and second band frequency limits are shown Fig. 8a, b, respectively. The standard deviations for the PBGs for this PC structure for different tolerances of the permittivity for this case are presented in Table 6. By comparing the values of σ/Mean ratios in both Tables 3 and 6 for the two structures of the 2D-PCs, we observe that the values of σ/Mean for the Air-Holes PCs are almost one-third there corresponding values for Si-Rods PCs structure up to uncertainty 5% in the permittivity of the Si. These values suggest that the Air-Holes PCs could be a good candidate with least alteration in the photonic gap considering deviations that might occur in the permittivity.

Conclusions
In this work, an intensive stochastic analysis for 2D-PC is presented that is based on the plane wave expansion and Monte Carlo technique. We have investigated the effects of the tolerances of the stochastic parameters of 2D-PCs on their optical operating properties. Our analysis is of great value during designing the PCs by keeping in mind the tolerances those may arise during the fabrication and their effects on the functionality of the PCs. We have examined the effects of tolerances of the stochastic parameters-the radius of the rods and the permittivities of the materials-of two-dimensional photonic crystals (2D-PCs) on their performance. In these analyses, we have studied two structures of 2D-PCs; namely Si-Rods PCs and Air-Holes PCs. Numerical results show that Si-Rods PCs structure has photonic gap that is less affected by uncertainty in radius of the rods in comparison to (a) (b) Fig. 6 a PDF of the normalized frequency gap for different tolerances of ε a b PDF of The highest normalized frequency for the first band and the lowest Normalized frequency for the second band for different tolerances of ε a Air-Holes PCs structure. Our analysis revealed that the effects of uncertainty in the permittivity of Si material-up to 5%-on the photonic gap are less than those of uncertainty in dimensions for both structures. However, higher standard deviations are noticed with the Si-Rods structure in comparison with Air-Holes structure in this case.   Fig. 8 a PDF of the normalized frequency gap for different tolerances of ε b for Air-Holes PCs structure b PDF of the highest normalized frequency for the first band and the lowest Normalized frequency for the second band for different tolerances of ε b for Air-Holes PCs structure Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.