1. Introduction

A periodic structure consists of any number of randomly shaped parts that are arranged periodically. Although they have been known mostly as gratings for over a century [1–3], they remain of interest to researchers owing to their extensive utilization in optical elements [4–12]. Being illuminated by an incident wave, these structures exhibit specific resonant response under certain conditions, which refers to an abrupt change in the reflection or transmission spectra. The resonant response of the periodic structures makes them suitable for a wide range of applications in optics, including filters, sensors, lasers, polarizers, and spectral analysis [4–8,11–27]. Regarding wavelength to periodicity ratio, the grating diffraction regimes are classified as: diffraction, resonant-subwavelength, and deep-subwavelength. If the incident wavelength is significantly shorter (longer) than the periodicity, it falls in the diffraction (deep-subwavelength) regime [15,28] and the behavior of the structure is well known in these two regimes. The spectral resonance that emerges in the resonant-subwavelength regime is a result of excitation of the higher-order guided modes in the grating region (known as guided-mode resonance (GMR)) [29,30]. This paper aims at investigating the resonance effect and also homogenization of the periodic structures in the resonant-subwavelength regime, i.e. the periodic lattice is replaced by a homogenized layer with an equivalent refractive index.

In general, most analysis methods for periodic structures such as rigorous coupled-wave analysis (RCWA) [1,31] intend to have a complex mathematical nature and they fail to provide a qualitative model for interpreting the structure’s behavior. In this regard, the homogenization methods can be helpful. Thus far, most of the proposed methods for homogenization of periodic structures have been limited to the deep-subwavelength regime, and they have not been capable of performing in the resonant-subwavelength regime. Besides, they have been also limited to the simple case of binary gratings with no consideration of incident angle effect on equivalent refractive index [10,30,32]. Aside from the homogenization methods, it is also necessary to determine the dispersion relation of periodic structures in order to find the propagative modes in the structure, thereby advancing the understanding of the resonance. Rytov’s and Botten's seminal works paved the way for establishing a dispersion relation for the periodic structures [33,34]. They achieved tangential and sinusoidal equations, which are known as Rytov and Botten equations, respectively.

Most of the homogenization methods have been based on assigning an equivalent refractive index to a periodic structure and treating the structure as a dielectric layer/waveguide. These methods can be categorized into two groups as follows. The first group equated the periodic structure with a layer with the average refractive index of that media. On the other hand, they considered the equivalent refractive index to be a constant value independent of incident wavelength (λ) and incident angle [10,30,32]. This average refractive index actually is the response of the Rytov equation, under the condition of λ→∞. Afterward, the second-order approximation of the tangent has been applied to the tangential Rytov equation in order to obtain a more accurate equivalent refractive index. The method is called the second-order effective medium theory [9,35]. These methods were restricted to the low contrast structures (i.e., an almost homogeneous medium). However, their response still shows error that renders these methods ineffective.

Second group accurately solved the Rytov equation and obtained a wavelength-dependent equivalent refractive index [36]. Both groups incorporated the achieved equivalent refractive index in the dielectric waveguide characteristic equation. Without considering higher-order responses of Rytov equation, it cannot be utilized for the resonant-subwavelength regime, so their homogenization is therefore limited to the deep-subwavelength regime. Further, the proposed model was restricted to symmetric binary gratings. Besides the homogenization methods, other groups of researchers found that the Rytov equation has higher-order responses as a result of its tangential form and asserted that this equation can be used in the resonant-subwavelength regime [15,37,38]. They also found the dispersion relation of some periodic structures. The dispersion relation provides the vertical component of refractive index.

The objective of this paper is to utilize the dispersion relation in order to homogenize the periodic structures in the resonant-subwavelength regime, and to present an efficient analytical-physical model without preceding limitations. In fact, we propose a generalized homogenization method by finding both vertical and horizontal components of the equivalent refractive index. In contrast to prior models that considered only the symmetric profiles, the aim of this model is to consider asymmetric cases as well and to investigate the effect of profile asymmetry on the behavior of the structures. An asymmetric case may consist of oblique incidence and/or an asymmetric grating profile.

This paper is structured as follows. In section 2, the theoretical background is presented, in which the procedure to obtain the dispersion relation will be demonstrated, followed by a discussion of how to utilize the results to homogenize the structure. The grating under investigation in this section is a comprehensive case with an arbitrary number of rectangular parts arranged periodically and illuminated obliquely. In the third section, by providing some examples, the performance of this model is examined and it will be explained how to utilize this model in design of the optical elements. Finally, in the last section, this model is compared with other homogenization models and also with other simulation methods, and advantages and disadvantages of each one will be discussed.

2. Theoretical framework

In this section, we present the theoretical background of the proposed homogenization method for periodic photonic lattices. The proposed method herein is capable of being applied to sophisticated multi-part period gratings rather than the conventional binary two-part period gratings. The procedure of homogenization method is portrayed in Fig. 1(a). This figure shows a periodic layer with period Λ and thickness d_g. Each period is made of “p” parts with refractive indices of n₁, n₂, …, and n_p and the filling factors of F₁, F₂, …, and F_p. This layer is located between two media of cover (with a refractive index of n_c) and substrate (with a refractive index of n_s). The illumination is not limited to the normal incidence (θ_c represents the incident angle) and both TE polarization (the electric field is in the y direction) and TM polarization (the magnetic field is in the y direction) are treated. The constituent materials are lossless, the structure is transversely infinite and infinitely periodic along the x-axis. Here, the aim is to transfer the multi-part grating of Fig. 1(a) to the homogenized one of Fig. 1(b).

 

Fig. 1. The procedure of homogenization. a) Schematic view of a multi-part grating with period of Λ and thickness of d_g. “I” denotes the incidence plane wave and “R₀” and “T₀” represent the reflected wave to the cover medium and transmitted wave to the substrate, respectively. Each period comprises “p” parts with a refractive index of n_i and filling factors of F_i (i = 1, 2, …, p). Each “i” part in one cell of the periodic lattice have the horizontal propagation constant of α_i,m for the mode of “m”. b) The equivalent homogeneous structure with the refractive index of ${\rm n}_{\rm m}^{\textrm{EHS}}$ and horizontal propagation constant of β_m for the mode of “m”. (The vertical propagation constant, γ_m, is taken to be equal in both (a) and (b)).

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2.1 Vertical characteristic equation (VCE)

As the structure consists of “p” parts along the x-axis (Fig. 1(a)), it is recommended to use a characteristic matrix in order to find the characteristic equation. The characteristic matrix for the case of multi-part periodic lattices, which are illuminated obliquely, is as follows [39,40]:

Solving this equation enables us to find the dispersion relation between the refractive index n (appears in α_i) and λ. Since “n” is the vertical refractive index (n = n_⊥), this dispersion equation is named the vertical characteristic equation (VCE). Equations (3) and (4) present the closed-form expressions of Eq. (2) for the conventional case of binary gratings. Equation (3) aims at the asymmetric case (oblique illumination), while Eq. (4) is utilized for the symmetric case (normal illumination), respectively.

Higher-order modes (m≠0) are classified into two groups of even and odd modes. Asymmetric grating profiles and oblique incidence make the odd modes emerge in addition to the even ones. However, in a symmetric structure, the odd modes will not be excited since the grating profile possesses even symmetry [33,41]. In this paper, the mode that can be excited for both symmetric and asymmetric structures is named “even mode”, and the one that is vanished in case of the symmetric structures is termed “odd mode”.

Cut-off wavelength occurs at n_m(λ_c,m) = 0 [15], which results in α_i= (2π/λ_c)n_i. The exact location of cut-off wavelength can be seen on the dispersion diagrams. Applying this condition to Eq. (2), (or alternatively in Eqs. (3) and (4)), reveals that for asymmetric structures $\mathrm{\lambda }_{\textrm{c},\textrm{m}}^{\textrm{TE}} = \mathrm{\lambda }_{\textrm{c},\textrm{m}}^{\textrm{TM}}$ (an example of this result is available in section 3.1 Fig. 4). The symmetry situation (normal incidence and symmetric profile) divides the VCE equation of (3) into two equations for the odd and even modes, separately (see Eq. (4)). Applying the cut-off wavelength condition n_m(λ_c,m) = 0 to the odd/even VCE, separately, does not meet the equivalency of cut-off wavelength for both polarizations and thereby for the symmetric cases, which only the even modes can be excited, $\mathrm{\lambda }_{\textrm{c},\textrm{m}}^{\textrm{TE}} \ne \mathrm{\lambda }_{\textrm{c},\textrm{m}}^{\textrm{TM}}$ (an example of this fact can be seen in Fig. 9(a) and 9(c)).

Further explanation about odd and even modes will be presented in section 3 via examples. Although lots of efforts have been done to achieve dispersion relation of periodic structures [28,34,37,38,42], far less attention has been paid to the higher modes [15,28,43,44], which were also limited to the modes of simple cases of symmetric binary gratings. Also, in some cases [41,45,46], the emergence of odd modes was studied, which was restricted to the dispersion relation of binary or 4-part period gratings. What set this report apart from the previous ones is the attention given to the higher modes, particularly the emergence of both odd and even modes. To elucidate the origin of each resonance in the subwavelength regime, it is vital to distinguish odd and even higher modes from other higher-order modes. Some previous works have ignored this fact and referred to odd/even modes as higher modes [28,37,39,42,43,47]. Therefore, this paper aims to clarify the behavior of all modes of periodic lattices and also formalize them, moreover, utilizing them in order to homogenize the grating profile.

2.2 Homogenization of periodic lattices

The subject of this section is to find the equivalent homogeneous structure of periodic photonic lattices (see Fig. 1). To do that, the vertical component of propagation γ_m (γ_m= k₀ n_m, where n_m is the vertical refractive index along the z-axis) is assumed to be the same in both structures (Fig. 1(a) and 1(b)), and we seek to identify $n_m^{EHS}$, which indicates the refractive index of the Equivalent Homogeneous Structure (EHS). The homogeneous structure of Fig. 1(b) is actually a periodic structure that consists of one part with the filling factor of F₁= 1 and a refractive index of $ n_m^{EHS}$. Applying these conditions to the VCE (Eq. (2)) results in:

As n_m denotes the vertical component of $n_m^{EHS}$ and $n_m^{EHS}\; = \sqrt {n_ \bot ^2 + {n_\parallel }^2} $, hence the horizontal refractive index and consequently the horizontal propagation constant are n_||=n_csinθ_c-mλ/Λ and β_m= k₀n_||, respectively. The sign of “m” (positive or negative) is opposite for the odd and even modes.

The key feature of this homogenization method is that not only the equivalent refractive index depends on the incident wavelength λ, but also depends on “m”, meaning that it is exclusive to each propagation mode “m”. Hence, it is incorrect to assume a constant refractive index for the equivalent homogeneous structure of periodic lattices. These propagative modes behave independently, since each mode has its own unique equivalent structure.

2.3 Horizontal characteristic equation (HCE)

So far, the periodic grating has been investigated and consequently homogenized (see Fig. 1). Now it is time to consider the outer regions of the grating. When a periodic structure is illuminated by a plane wave, diffraction orders emerge. For each of these orders, there is a Rayleigh wavelength boundary (λ_R,m) which determines the status of propagation/evanescence of the order in the cover or substrate media. If λ>λ_R,m (λ<λ_R,m), the order “m” is evanescent (propagative) in the outer regions of cover and substrate. The Rayleigh wavelength is achieved when the phase-matching condition at the grazing angle is satisfied (max{n_c,n_s}=n_||=|n_csinθ_c-mλ_R,m/Λ|) [30]. Since there is no negative wavelength, the sign of “m” should be positive. Therefore:

By comparing Eqs. (6), λ_R,m > 0> λ_R,m < 0 . For each mode, longer λ_R,m results in longer λ_c,m. Thus, for example for |m|=1, λ_c,1> λ_c,-1. Table 1 provides a brief summary of each diffraction order’s status. Except for the m = 0 order, all diffraction orders in the regime of λ_R,m<λ< λ_c,m propagate within the grating region, while they are evanescent outside of it [15]. On the other hand, the orders m≠0 is guided through the grating layer, so this layer acts as a dielectric waveguide for this order. In the case of m = 0, since λ_R,0 → ∞, this order propagates in all regions of the cover, the grating, and the substrate, regardless of wavelength spectrum limitations.

Table 1. The status of propagation/evanescence of waves

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Figure 2 illustrates three regimes of gratings versus incident wavelength λ. Previously, the working regimes of periodic gratings were reported in [15] for symmetric structures. Here in Fig. 2, a generalized graph incorporating the asymmetry case is presented. When all higher orders become evanescent in the outer regions, the structure is no longer in the diffraction regime but enters the subwavelength regime. Thus, the maximum amount of λ_R,m among all orders, which is actually λ_R,1, determines the transition wavelength between the two regimes of subwavelength and diffraction. The diffraction regime is out of the scope of this article since the aim herein is to investigate the resonance effect only in the resonant-subwavelength regime. In the subwavelength regime, higher-order diffraction orders vanish in the outer regions (cover and substrate) and only m = 0 propagates. This regime is divided into two sub-regimes of resonant-subwavelength and deep-subwavelength. The resonant-subwavelength regime corresponds to the regime where at least one higher-order mode is guided through the grating layer [30,48]. Considering that λ_c,1 is the largest cut-off wavelength among all modes, the regime of λ_R,1<λ<λ_c,1 is the boundary of the resonant-subwavelength regime. Furthermore, in the deep-subwavelength regime, all higher orders are evanescent, and only m = 0 remains in every region.

 

Fig. 2. Schematic illustration of grating regimes in terms of incident wavelength.

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The grating structure can be regarded as a Fabry-Perot cavity for all modes of m = 0 and m≠0 [49]. It enables us to find the phase selection rule in which the modes in the grating region interfere constructively. One approach to compute the HCE of the structure is to use the phase selection rule. Thus, the focus of the next two sub-sections is on finding the phase accumulation after one round trip. This phase consists of two components, one related to traveling through the grating, and the other one is due to the reflection coefficients from interfaces of cover-grating (z = 0) and grating-substrate (z = -d_g). The reflection coefficients can be attained by Fresnel relations [50].

Phase selection rule for m = 0: The reflection of m = 0 at the interfaces of z = 0 and z = -d_g does not add any extra phase (for the case of a lossless dielectric structure) [51]. Consequently, all the phase accumulation (ψ_m = 0) for one round trip of m = 0 is pertinent to passing the thickness of the grating. The constructive interference for m = 0 requires that:

The noticeable feature of this method is that it distinguishes between the behavior of m = 0 and higher-order modes. This fact was ignored in the previous works since they assumed that the grating behavior is the same for all modes (including m = 0 and higher-order modes) and it operates as a waveguide for m = 0 as well and hence they used m = 0 in the transcendental equation of the dielectric waveguide [9,10,30,32,35,36]. As presented here, the waveguiding effect occurs just for higher-order modes. Additionally, in some previous works [37,38,42,52], they assigned m = 1 for the first dispersion relation (Eq. (2)) and continued numbering the modes in the form of (m = 1,2,…). This type of numbering is somehow misleading and different behavior of the first mode (m = 0) and other higher-order modes is undistinguishable. It is therefore important to begin numbering from “0” rather than “1”.

According to the HCEs (Eqs. (7) and (10)), these equations depict the dependence of d_g on the other parameters. Therefore, for a certain grating, it is possible to draw the diagram of d_g(λ). In order to better illustrate the process, a flowchart is provided (see Fig. 3). By solving VCE (Eq. (2), or alternatively using its closed-form solution in Eqs. (3) and (4) for the desired structure), the dispersion relation of n_m(λ) is determined in the first step. By substituting n_m(λ) into HCEs, the diagram of d_g(λ) is obtained. Suppose the reflectance map of the grating based on the variation of λ and d_g (R(λ,d_g)) is calculated by RCWA. Overlaying the diagram of d_g(λ) to the R(λ,d_g) reveals that the Eq. (10) predicts the location of GMRs. Additionally, solving these HCEs facilitates design of the grating structures. In the next section, some examples are employed to elucidate these matters further.

 

Fig. 3. The flowchart showing the steps involved in finding the diagram for d_g(λ), or alternatively, the locations of GMRs.

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3. Results

The aim of this section is to present the applicability of the proposed method in section 2. During solving the VCE and HCE, it will be explained how this method can predict the physical behavior of the structure and, furthermore, how it paves the way for design of the grating structures. Following the steps in Fig. 3 results in finding the lines of d_g(λ). These obtained lines are called resonance lines because they can predict the resonance locations. Consequently, these resonance lines are helpful in the design of the optical devices based on gratings such as wideband reflectors, narrowband reflectors, polarizers, and so on. The proposed method in section 2 can be applied to any multi-part rectangular grating that is illuminated obliquely. However, in this section, only two structures have been selected as examples to illustrate how dispersion relations and resonance lines can be determined.

To examine the emergence of even and odd modes, an asymmetric structure is considered. This structure consists of a single layer two-part period binary grating under oblique incidence with Λ=1.0 µm, n₁= 3.48, n₂= n_c= 1, n_s= 1.4, F₁= 0.7, F₂= 1- F₁ and θ_c= 40^° is considered.

3.1 Analyzing the structure

Based on Fig. 3, the first step in analyzing this structure is to solve Eq. (3) in order to determine the vertical refractive indices (n_m) (see Fig. 4). n₀ corresponds to the vertical refractive index of m = 0. $n_m^{o/e}$ indicates the vertical refractive index of m=±1 for odd (blue lines) and even (red lines) modes, respectively. GMR occurs in the resonant-subwavelength regime (λ_R,1<λ<λ_c,1) which is 2.042µm < λ < 3.612µm for this example for both TE and TM polarizations. The cut-off wavelength for TE/TM polarizations is the same, except that the odd and even modes were substituted with each other. To identify which mode is odd/even, we plot the diagram of θ_c (the incident angle) versus λ_c (the cut-off wavelength) (see Fig. 5). This figure was produced using VCE (Eq. (3)) under the condition of n = 0. In the case of θ_c= 0^°, since the structure becomes a symmetric binary structure, it is possible to use equations (4). Note that adding the condition θ_c= 0^° to Eq. (3) simplifies this dispersion relation into two equations of (4) for odd and even modes separately. According to equations (4), for TE, the cut-off wavelength for the odd mode is $\lambda _{c,1}^o = 3.2\mu m$, while the cut-off wavelength for the even mode is $\lambda _{c, - 1}^e = 2.5\mu m$. Thus, in the case of TE, the right (left) branch in Fig. 5 belongs to the odd (even) mode. By applying Eqs. (4) to the TM case and repeating this procedure, it is revealed that the TM case is converse to the TE case.

 

Fig. 4. Dispersion relation for structure with parameters of Λ=1.0 µm, n₁= 3.48, n₂= n_c= 1, n_s= 1.4, F₁= 0.7, F₂= 1- F₁, θ_c= 40^°. The vertical refractive indices n_m(λ) for (a) TE and (b) TM.

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Fig. 5. The dependence of cut-off wavelength of odd/even (m=±1) modes to incident angle.

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As noted earlier, in some previous works, odd/even modes were referred to as higher-order modes [28,37,39,42,43,47]. This means that the modes were numbered in the form of m = 0,1,2,3, …, which is somehow incorrect. In fact, in the case of asymmetry, they ignored the appearance of negativity for the “m”. Lastly, the correct format of numbering is m = 0, ± 1, ± 2, …, which addresses all previous issues regarding mode numbering (starting at “0” and considering the negative amounts as well). As shown in Fig. 5, breaking the symmetry of the structure will not cause the symmetric mode of m = 1 (or alternatively even mode) to split into two modes of m = 1 and m = -1. In fact, asymmetry leads to the emergence of the odd mode, which does not correspond to the even mode and is not the split form of that (in contrast to [30]). This issue can be seen on Fig. 5 that by deflecting from normal incidence, the even mode doesn't split into two modes. Additionally, field distributions of odd and even modes are shown in Fig. 7.

The next step is to solve HCEs of (7) and (10) and achieve the diagram of d_g(λ) which is called resonance lines here. The contour maps of reflectance R(λ, d_g), which have been acquired by RCWA, are illustrated in Fig. 6(a) and 6(c) for TE and TM polarizations, respectively. The subwavelength regime is shown in this figure and λ_c,1 indicate the border of resonant-subwavelength and deep-subwavelength regimes. As well, in Fig. 6(b) and 6(d) the achieved resonance lines are overlaid on the simulated reflectance map R(λ, d_g).

 

Fig. 6. The reflectance map of R(λ, d_g), acquired by RCWA, for the binary grating with parameters of Λ=1µm, n₁= 3.48, n₂= n_c =1, n_s= 1.4, F₁= 0.7, F₂= 1- F₁, θ_c= 40^°, when illuminated by (a) TE and (c) TM polarized waves. The overlaid resonance lines of d_g(λ) on the reflectance map of R(λ, d_g) are shown in (b) and (d). Black lines represent the resonance lines of m = 0 (Eq. (7)). For the guided modes of |m|=1, blue and red lines demonstrate the resonance lines of odd and even modes, respectively (Eq. (10)). In (d), solid white lines indicate high reflected sections, corresponding to Δψ=qπ (q = 1,3,…). Conversely, the dashed white lines demonstrate low reflected sections, corresponding to Δψ=qπ (q = 0,2,4,…).

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Fig. 7. Field distribution of E_y for (a) the odd mode $\mathrm{\lambda }_\textrm{r}^\textrm{o}$=3.367µm and (b) the even mode $\mathrm{\lambda }_\textrm{r}^\textrm{e}$=2.116µm related to the example of Fig. 6(b) at d_g= 1µm (TE). Note that, first mode of this structure shows sine distribution (odd mode).

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Black lines in this figure illustrate resonance lines for m = 0 (Eq. (7)). Also, the guided waves are denoted by the labels of $\textrm{TE}/\textrm{TM}_{m,N}^{o/e}$. Where the mode's order along the x-axis and z-axis is represented by “m” and “N”, respectively. The order determined by “m” is that of the mode corresponding to the VCE, while the order determined by “N” is that considered to be the HCE (Eq. (10)). For this example, only m=±1 appears in the subwavelength regime and higher orders are evanescent. Also, the blue and red lines belong to the resonance lines of odd/even modes, respectively.

Here, the double-mode regime, in which only the first two modes propagate in the grating region, is examined. This double-mode regime is depicted in Fig. 6(d) for TM polarization (for the sake of brevity, TE polarization is not shown). In this regime, the resonance lines of m = 0 and m = -1 (for TM) divide the reflectance pattern of R(λ,d_g) into multiple and repetitive sections with high and low reflectance. The phase difference between these two modes (Δψ) determines high and low reflection sections. Based on the Eqs. (7) and (9), $ \Delta \psi = ({{\gamma_0} - {\gamma_1}} ){d_g} - \measuredangle r_1^c - \measuredangle r_1^s$. Upon accumulation of π phase difference after the half-round trip in the grating region, they interfere destructively with each other, canceling out transmission and finally reflecting the power [52]. Hence, Δψ=qπ (q = 1,3,…), which is illustrated by solid white lines on Fig. 6(d), determines the high reflected sections. Conversely, Δψ=qπ (q = 0,2,4,…), the dashed white lines, indicates low reflected sections. The concept of phase difference has been explored in [14,15]. However, they only considered vertical refractive indices in the computation of phase differences (Δψ=(n₀-n₁)d_g), and the presence of phase difference due to $\textrm{r}_{\textrm{m} \ne 0}^{\textrm{c},\textrm{s}}$ (TIRs) was utterly ignored. Due to the nature of Fabry-Perot resonances, they manifest themselves as abrupt changes in reflectivity/transmittivity spectra [48,49], therefore, if the resonance occurs in the low (high) reflectivity/transmittivity background, the spectrum reaches high (low) at this wavelength. This concept can be seen in Fig. 6(a) and 6(c), where m = -1 produce an abrupt change for TE and TM polarizations, respectively.

In Fig. 7, which is computed by RCWA, the field distribution of odd ($\mathrm{\lambda }_\textrm{r}^\textrm{o} = 3.367{\mathrm{\mu} \mathrm{m}}$) and even ($\mathrm{\lambda }_\textrm{r}^\textrm{e} = 2.116{\mathrm{\mu} \mathrm{m}}$) modes are shown at d_g= 1µm for the example of Fig. 6(b) (TE). As seen, sine (cosine) distribution along the x-axis (periodic axis) prevails for the odd (even) mode. This rule applies to the transverse components of the fields (E_y for TE and H_y for TM). Assume that the structure is modified in a way to become a symmetric profile (θ_c= 0), so the transverse component of the fields obtains purely even (odd) symmetry with respect to the middle line of each part of the grating for the even (odd) mode. Finally and according to Fig. 6, the proposed homogenization method based on solving VCE and HCE can predict the resonance locations, and consequently, the pattern of R(λ,d_g). So far, the ability of the proposed method to analyze an exemplar structure has been demonstrated. Resonance lines, the origin of each mode, and the phase difference between modes have been considered. The next section describes how this method is used in the design of optical elements.

3.2 Design method

Another positive aspect of the proposed method is the utilization of these resonance lines in designing optical elements. Adjusting these resonance lines in a desired manner can be helpful in designing elements such as wideband reflectors, narrowband filters, polarizers [52]. This section demonstrates how to employ these resonance lines in order to achieve optical elements, for instance, wideband reflectors, narrowband reflectors, and polarizers.

Wideband reflector: One of the practical optical elements is the wideband reflector based on periodic structures [5,7,13,24–26]. They can provide high reflectivity over a wide wavelength range. Since Δψ =qπ lines (q = odd integers) indicate the high reflection sections, the structure which produces an approximately straight Δψ lines (regrading to λ, means $\frac{{\Delta \psi }}{{\Delta \lambda }} \approx 0$) can provide a wideband reflector. Since the emergence of other modes manifest itself as abrupt variation in the reflectance spectrum, they cause zero reflection within the bandwidth. Thereby, the most suitable wavelength range for a wideband reflector is the double-mode regime. This fact can be seen in Fig. 6(d), which indicates that the above-mentioned example could act as a wideband reflector for the TM polarized incident wave. In this step, one of the Δψ=qπ lines (q = odd) must be selected by choosing the appropriate amount of grating thickness (d_g). In this example, d_g= 2.53µm has been chosen (indicated by the purple line in Fig. 6(d)). Following this, we have resonance lines (HCEs) that determine bandwidth, as indicated by the solid purple line in Fig. 6(d), for this case. The solid part of the purple line at d_g= 2.53µm demonstrate that this structure is capable of providing a wideband reflector covering approximately the range of 2.3µm to 2.89µm. To verify this claim, the simulated reflectance spectrum of this grating structure using RCWA is provided in Fig. 8(a). Finally, if wideband reflectors are to provide more bandwidth, the higher-order and odd/even modes should be kept as far apart as possible from the m = 1 mode.

 

Fig. 8. The reflectance spectra of (a) a wideband and (b) a narrowband reflector according to Fig. 6(d) with a grating thickness of d_g= 2.53µm and d_g= 3.33µm, respectively (for TM polarization).

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Narrowband reflector: There are many applications for narrowband reflectors in optics, such as in GMR-based sensors. The key feature of these sensors is their FWHM (full width half maximum), meaning sensors with narrower resonance linewidth (lowest FWHM) perform better [6,8,17–23]. Thus, it is essential to design sensors with the narrowest resonance linewidth. Contrary to wideband reflectors, Δψ=qπ (q = even) lines are used for narrowband reflectors. These lines determine the low reflection sections, then the emergence of GMRs manifest itself as narrowband resonance. For this example, by choosing the thickness of d_g= 3.33µm one of the dashed white lines has been chosen (Fig. 6(d)). The predicted resonance wavelength based on this homogenization method is λ_r= 3.475µm (the right-first red resonance line on Fig. 6(d)). The reflectance spectrum of this structure has been plotted in Fig. 8(b) using RCWA. As seen, this structure can generate a resonance at the λ_r= 3.485µm with FWHM (full width half maximum) of 0.4nm. Since the objective of narrowband structures is the appearance of only one resonance in the desired wavelength band, the higher waveguide mode (the higher red line in Fig. 6(d)) represents the minimum operation wavelength (the proposed method: λ_min= 3.15 µm, RCWA: λ_min= 3.14 µm). As the resonance wavelength (λ_r= 3.475µm) has been chosen on the first waveguide mode ($\textrm{TM}_{1,0}^\textrm{o}$), for the range of λ>λ_r, no waveguide mode exists, or on the other hand, there is no resonance, and the structure enters the deep-subwavelength regime (Fig. 6(d)). Thus, there is no upper limit on the wavelength band (${\lambda _{max}} \to \infty $).

Polarizer: Thus far, it has been observed that the proposed method enables us to use just one example to design two different optical elements of narrowband and wideband reflectors. In addition to the above-mentioned applications, it would be possible to design another critical optical element, a polarizer. The polarizers convert an unpolarized incident wave into a polarized one. The linear resonant polarizers are devices that exhibit high reflection only for one specific polarization of TE or TM, while the other one's reflectance is suppressed [13–16]. Hence, in the desired wavelength spectrum, the emergence of higher modes (m≠0), or on the other hand the occurrence of the GMR effect, should occur only for one polarization; thereby, for the other polarization, the cut-off wavelengths of higher modes (m≠0) should be lower than the desired wavelength spectrum range. Therefore, the cut-off wavelengths of TE and TM polarizations determine the wavelength boundary for a polarizer. As stated in section 2.1, the VCE equations are identical for TE and TM polarizations at the cut-off wavelength (n_m= 0). VCEs that are entirely similar lead to λ_c|_TE = λ_c|_TM. It is impossible for the same mode to have analogous cut-off wavelengths for both TE/TM. In fact, this condition implies that the cut-off wavelength of the odd (even) mode for TE polarization is identical to the even (odd) one in TM. To elucidate this issue, consider VCEs of a symmetric binary grating case (Eq. (4)). Where, in the condition of cut-off wavelength, x = n₁/n₂ for TE polarization and x = n₂/n₁ for TM polarization. In fact, x|_TE= 1/x|_TM. Substitution of this situation in Eq. (4) reveals that, in the cut-off wavelength condition, VCE|_TE= VCE|_TM or $ \lambda _c^o{|_{TE}} = \lambda _c^e{|_{TM}}\; $ ($ \lambda _c^e{|_{TE}} = \lambda _c^o{|_{TM}}\; $).

It is therefore impossible to find a wavelength range, for asymmetric structures, in which higher modes emerge only for one polarization. Consequently, asymmetric structures are not suitable as polarizers, whereas symmetric structures qualify. Due to the fact that the odd modes will not be excited in the symmetric profiles, only the even ones emerge $({\lambda_c^e{|_{TE}} \ne \lambda_c^e{|_{TM}}} )$, thus it is possible to identify a wavelength range in which GMR is observed only for one polarization. Therefore, it is imperative to note that during the implementation of linear polarizers, the normal incidence is essential for preserving symmetry. The polarizer presented in [15] is examined here as an example. It is a binary structure having the following parameters: Λ=0.86µm, n₁= 3.48, n₂= n_c= n_s= 1.5, F₁= 0.1, F₂= 1- F₁, θ_c= 0°. Figure 9(a) and 9(c) illustrate the dispersion relations for TE and TM polarizations, respectively. According to the results, $\lambda _c^e{|_{TE}} = \lambda _c^o{|_{TM}} = 1.682\; \mu m$ and $\lambda _c^o{|_{TE}} = \lambda _c^e{|_{TM}} = 1.311\; \mu m$. Since odd modes do not exist in the symmetric structures, the dispersion relations for these modes are depicted by dashed lines in Fig. 9(a) and 9(c). Figure 9(b) and 9(d) illustrate the reflectance spectra obtained by RCWA for TE and TM polarizations. These R(λ,d_g) maps confirm the results that the desired wavelength range for this polarizer is 1.311µm<λ<1.682µm, in which only an even mode exists in the TE state, while no guided mode exists in the TM state. Since the operation wavelength regime for TM, herein, is the deep-subwavelength regime, only m = 0 appears in the grating layer. Based on Fig. 9(c), in this regime, n₀≈1.58, which is so close to the refractive index of the outer regions (n_c= n_s= 1.5). Due to the lack of refractive index contrast between the grating and outer regions, the reflectivity of this structure in the deep-subwavelength regime is close to zero [53], making it challenging to observe the HCE behavior of m = 0 in Fig. 9(d).

 

Fig. 9. Analysis of the polarizer with parameters of Λ=0.86µm, n₁= 3.48, n₂= n_c= n_s= 1.5, F₁= 0.1, F₂= 1- F₁, θ_c= 0°. (a) and (b) are dispersion relation and R(λ,d_g) map correspond to the TE, respectively, (c) the dispersion relation of the TM state, besides, the related R(λ,d_g) map is provided in (d). The shaded area on (a) and (c) demonstrate the desired wavelength range for this polarizer.

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4. Conclusion

The method we propose in this paper predicts the spectral behavior of rectangular periodic structures. This method is based upon utilizing dispersion relation in order to homogenize the periodic structures. As stated in the introduction section, previous homogenization methods were limited to the deep-subwavelength regime, and they were not capable of functioning in the resonant-subwavelength regime. Besides, they have been also limited to the simple case of symmetric binary gratings [10,30,32]. As compared to previous methods, this homogenization method has no limitations and can be applied to any rectangular structure under oblique incidence. The features of this method include: 1) In contrast to previous works, in this method the equivalent refractive index is dependent on wavelength as well as being different for each mode, including odd/even and higher-order modes. 2) In previous works, the calculated equivalent refractive index was inserted into the dielectric waveguide characteristic equation, which is ineffective. The first response of the dispersion relation belongs to m = 0, which does not have a cut-off wavelength. Therefore, the refractive index of m = 0 (which is almost equal to the average refractive index in conjunction with λ→∞) should not be incorporated into the dielectric waveguide characteristic equation. They are higher-order modes that have a cut-off wavelength and can be guided and propagated in a periodic structure, and the structure serves as a waveguide. As a result, another advantage of this technique is the ability to separate the performance of gratings for zero-order modes from higher-order modes.

The analysis methods, in which the famous one is RCWA [1,31], are based on electromagnetic fields expansion in all regions. These methods rely on solving complex electromagnetic equations, and the simulation process can take a great deal of time. Additionally, they do not provide a suitable physical insight for analyzing periodic structures. In other words, they do not identify the origin of each mode and do not determine the cause of resonances. However, this method is capable of homogenization, it is a fully analytical method capable of identifying the origins of resonances, that is, which modes contributed to the occurrence of which GMR. In addition, it does not depend on field expansion, hence, there is no need to compute complex electromagnetic equations. Just solving the dispersion relation of Eq. (2) and also using the HCEs of Eqs. (7) and (10) is sufficient. This results in a much shorter simulation time. For instance, Fig. 6(b) has been chosen to compare the simulation time. The R(λ,d_g) can be calculated by RCWA in about two minutes and a half, while the HCEs can be achieved by the proposed method in about 13 seconds. A computer with a core i7-4790K@4 GHz processor and 32GB of RAM was used for the simulation. It should be noted that this is a homogenization method and regarded as an approximation method. But unlike the prior homogenization works, it is much more accurate, which makes it well suited for utilization in the simulation and design of periodic structures.

In summary, a homogenization method for periodic structures has been proposed. This method is based on utilizing dispersion relations in a manner to convert a periodic structure to the homogeneous slab media. It was ascertained that having found the resonance lines, it is possible to analyze the structure, determine the origin of each mode, and finally, use them to design optical components.