1. INTRODUCTION

Rigorous coupled-wave theory (RCWT) can be applied to various types of diffraction grating problems. The solutions are always rigorous and no special assumptions are needed, except for periodicity. The history of RCWT may be traced back to the theory provided by Tamir et al. [1]. They analyzed the properties of sinusoidally stratified dielectric volume gratings. Then Knop [2] applied RCWT to rectangular surface-relief gratings. In the following decades, RCWT has been applied to various types of diffraction gratings, including dielectric planar gratings [3], dielectric surface-relief gratings [4], metallic gratings [5, 6, 7], and anisotropic gratings [8, 9, 10], for example. The difficulty of the slow convergence for the TM mode diffraction problem was solved by Lalanne and Morris [11], Granet and Guizal [12], and Li [13]. A good review of the RCWT formulation for isotropic gratings can be found in [14, 15], and that for anisotropic gratings in [16]. While much literature exists for gratings in nongyrotropic media, few papers for gratings made with gyrotropic materials can be found. Some related work has been done by Rokushima and Yamakita [17], Lakhtakia et al. [18], and Wang and Lakhtakia [19, 20].

If asymmetric constitutive relations [21, 22, 23, 24] are used, the diffraction gratings made with gyrotropic materials can be modeled by the previously developed RCWT [25, 26], in which only the permittivity tensor should be modified according to the gyrotropy of the materials. However, as shown by Silverman [27, 28] and Peterson [29], the asymmetric and symmetric constitutive relations [30, 31, 32, 33, 34] are not equivalent and have differences in many aspects. Thus, it is necessary and useful to have an RCWT algorithm that can deal with the symmetric constitutive relations. However, no such RCWT algorithms can be found in literature to the best of the authors’ knowledge.

Here, we provide an RCWT formulation that adopts the symmetric constitutive relations. The present algorithm is applicable to gratings made with either gyrotropic or nongyrotropic materials. The incident, exiting, and grating materials can be isotropic, uniaxial, or biaxial, with or without optical activity. The principal values of the electric permittivity tensor, the magnetic permeability tensor, and the gyrotropic tensor can take arbitrary values and the principal axes may be arbitrarily and independently oriented. The present RCWT formulation can also deal with the asymmetric constitutive relations as a special case. The scattering matrix method [35, 36, 37] is employed to solve the boundary condition equations for numerical stability. As examples, the diffraction efficiency profiles of single-layer bi-isotropic gratings and multilayer biaxial gyrotropic gratings are investigated. Distinctive polarization coupling effects due to optical activity are observed in both cases. The convergence property of the present RCWT formulation is demonstrated. As a special case, diffraction efficiencies for a multilayer grating made with nongyrotropic uniaxial material are shown to exhibit good agreement with simulation data available in literature.

2. FORMULATION

2A. Description of Grating Diffraction Problem

The geometry of the grating diffraction problem analyzed here is depicted in Fig. 1 using a single-layer binary grating as an example. The media in regions I and II and the grating region may be birefringent, gyrotropic, or both. An electromagnetic wave is obliquely incident on the grating structure with an angle of incidence ${\theta }_{\text{in}}$ and an azimuthal angle ${\phi }_{\text{in}}$. The relative electric permittivity tensor, the relative magnetic permeability tensor, and the gyrotropic tensor of the medium in region I are ${\mathit{\epsilon }}_{\mathrm{I}}$, ${\mathit{\mu }}_{\mathrm{I}}$, and ${\mathbf{G}}_{\mathrm{I}}$, and those in region II are ${\mathit{\epsilon }}_{\mathrm{II}}$, ${\mathit{\mu }}_{\mathrm{II}}$, and ${\mathbf{G}}_{\mathrm{II}}$, respectively. The surface normal of the mean grating plane is parallel to the z direction and the grating vector is parallel to the x direction. The depth of the grating groove is h. In the coordinate system depicted in Fig. 1, the incident electric field of unit amplitude is given by

Floquet’s theorem requires the wave vectors of the diffracted waves to satisfy the following conditions:

2B. Electromagnetic Waves in Bianisotropic Media

Gyrotropic materials may have chiral structure, which lacks both mirror and inversion symmetry. The constitutive relations for gyrotropic media are discussed in detail by Silverman [27, 28], Peterson [29], and McClain et al. [39, 40]. According to their papers, the constitutive relations proposed for the gyrotropic materials may be categorized into two groups: symmetric constitutive relations and asymmetric constitutive relations. The constitutive relations proposed by Condon [31], Fedorov [32], and Post [33] can be classified as symmetric constitutive relations. While those constitutive relations do not have the same mathematical form, it is shown by Laktakia et al. [34] that Fedorov’s and Post’s forms are equivalent to Condon’s form for time harmonic fields in source free regions.

With the choice of increasing phase sign convention, the symmetric constitutive relations for gyrotropic media can be expressed as

Note that rotation matrices ${\mathbf{R}}_{\epsilon }$, ${\mathbf{R}}_{\mu }$, and ${\mathbf{R}}_G$ may not be the same if the material properties diagonalize in different directions.

On the other hand, the asymmetric form of the constitutive relations can be represented by

A set of asymmetric constitutive relations may be obtained from the symmetric constitutive relations by employing the electric permittivity tensor described by Eq. (17). Silverman [27, 28] and Peterson [29] demonstrated that the asymmetric constitutive relations do not yield the same results as those predicted by the symmetric constitutive relations. It is also discussed by Silverman that the asymmetric constitutive relation set is not invariant under a duality transformation, but the symmetric set is invariant, while Maxwell’s equations are invariant under the duality transformation. For the reasons described previously, the symmetric constitutive relations described by Eqs. (10, 11) are adopted for our RCWT formulation.

Substituting Eqs. (10, 11) into Maxwell’s equations, the following equation is obtained:

This equation relates the wave vector k to the material constants ε, μ, and G. The electromagnetic field vectors D, E, B, and H for the eigenpolarization states that correspond to each distinct wave vector k can then be obtained from Eq. (19).

For isotropic, uniaxial, and biaxial nongyrotropic media, the dispersion relationship (21) can be reduced into a simpler form. The explicit forms of the dispersion relationship for those materials are well described in literature [41, 42, 43]. In the case of gyrotropic materials, the wave vector of the incident wave is obtained by solving the dispersion relationship (21). The algebraic solutions of Eq. (21) have a simple form when $\mu =1$ and the principal axes of ε, μ, and G are aligned with the global coordinate axes $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, and $\hat{\mathbf{z}}$ so that ε, μ, and G become diagonal matrices. The explicit form of the algebraic solutions for gyrotropic materials are given in Appendix A.

If the principal axes are not aligned with the global axes, the wave vectors should be transformed from the global coordinates to the principal coordinates. Then, the solutions in the principal coordinates are given by the same equations. If the principal axes of ε, μ, and G are not the same or the relative magnetic permeability is not a scalar constant, a numerical approach to the dispersion relationship (21) may be easier rather than obtaining algebraic solutions. Then, the eigen polarization states are obtained by solving Eq. (19).

2C. Coupled-Wave Equations

With the symmetric constitutive relations (10, 11), Maxwell’s equations can be expressed as

In the coordinate system depicted in Fig. 1, these equations can be expressed by

To set up the coupled-wave equations, Eqs. (25, 26, 27, 28, 29, 30) are expressed in Fourier space by expanding ${\epsilon }_{pq}$, ${\mu }_{pq}$, and $G_{pq}$ as well as the electromagnetic field vectors E and ${\mathbf{H}}^{\prime }$. In doing the Fourier expansion, special care must be taken. As demonstrated by Li [13], the use of Laurent’s rule and the inverse rule should be chosen according to the type of products. In each of Eqs. (25, 26, 27), the two products, ${\mu }_{qx}{H}_x^{\prime }$ and $G_{qx}{E}_x$ ($q=x$, y, z), which are referred to as “Type-3” product in [13], may be Fourier factorized neither by Laurent’s rule nor the inverse rule. Similarly, in each of Eqs. (28, 29, 30), the two products, ${\epsilon }_{qx}{E}_x$ and $G_{qx}{H}_x^{\prime }$ ($q=x,y,z$) are also Type-3 products. So we rewrite Eqs. (25, 26, 27, 28, 29, 30) into another form so that Laurent’s rule and the inverse rule can be applied. The details of the Fourier expansion procedures are described in Appendix B. The results after the Fourier expansion of Maxwell’s equations (25, 26, 27, 28, 29, 30) become

By eliminating ${\mathbf{S}}_z$ and ${\mathbf{U}}_z$ from Eqs. (31, 32), the coupled-wave equations are obtained as

By denoting ${\mathbf{V}}^T=[{\mathbf{S}}_x,{\mathbf{S}}_y,{\mathbf{U}}_x,{\mathbf{U}}_y{]}^T$, Eq. (33) can be written in a simple form

The expressions of the $M\times M$ component block matrices ${\mathbf{\Gamma }}_{ij}$ are given in Appendix C. The general solution of Eq. (34) is

2D. Extension to Multilayer Grating

The above discussion can be easily extended to the multilayer grating diffraction problem. Suppose that we have a multilayer grating consisting of L layers. The electric field and the magnetic field in each grating layer may be also expanded into a Floquet–Fourier series as

Equation (34) and its solution [Eq. (35)] can be also used for each layer of the grating with appropriate constitutive relations. For the lth layer of the grating, the coupled-wave equations and the general solution can be expressed by

2E. Boundary Conditions

The unknown constants $c_{l,m}$ are determined from the boundary conditions by matching the tangential components of the electric and magnetic fields at the incident interface, exiting interface, and the boundaries between the grating layers. The boundary conditions for the incident interface are given by

In Eqs. (2, 3), the electric fields for the ith forward and backward diffracted waves can be expressed by the products of the normalized component vectors and the amplitude coefficients as

The $M\times 1$ vector quantities ${\mathbf{r}}_O$, ${\mathbf{r}}_E$, ${\mathbf{t}}_O$, and ${\mathbf{t}}_E$ are defined such that their ith elements are the electric field amplitude of the ith diffracted waves, $r_{O,i}$, $r_{E,i}$, $t_{O,i}$, and $t_{E,i}$, respectively. The $M\times M$ diagonal matrices ${\mathbf{r}}_{Oq}$, ${\mathbf{r}}_{Eq}$, ${\mathbf{t}}_{Oq}$, and ${\mathbf{t}}_{Eq}$ ($q=x$, y) are defined such that their ith diagonal elements are the q components of the normalized electric fields of the O and E mode eigenpolarization states of the ith backward and forward diffracted waves, respectively.

By using Eqs. (39, 44), the boundary condition Eqs. (40, 41, 42, 43) can be rewritten in matrix form as

The boundary condition for the interface between the lth and ($l+1$)th layer is given by

The reflected and transmitted amplitudes of the diffracted waves, ${\mathbf{r}}_{O,i}$, ${\mathbf{r}}_{E,i},$, ${\mathbf{t}}_{O,i}$, and ${\mathbf{t}}_{E,i}$, can be obtained simultaneously from Eqs. (49, 50, 54).

The solution to this class of equations can be developed in several ways, such as the transfer matrix method [36], the enhanced transmittance matrix method [15], and the scattering matrix method [35], for example. We adopted the scattering matrix approach to our formulation for numerical stability. The details about the scattering matrix method are not described in this paper since it is a well-established technique and well documented in literature [16, 35, 36, 37].

2F. Diffraction Efficiencies

The diffraction efficiencies of the propagating orders are given by

The numerators and the denominators in Eqs. (55, 56) are the z components of the Poynting vectors for the ith-order diffracted waves and the incident wave, respectively, which together represent cosine factors.

3. CONVERGENCE PROPERTY

To demonstrate the convergence properties of the present algorithm, the diffraction efficiencies of a binary grating corrugated on a gyrotropic biaxial substrate are calculated as a function of the truncation order. The principal values of the relative electric permittivity tensor of the substrate are ${\epsilon }_x=1.44$, ${\epsilon }_y=2.89$, and ${\epsilon }_z=6.25$, and those of the gyrotropic tensor are $G_x=0.10$, $G_y=0.90$, and $G_z=1.80$ with identical principal axes for ε and G. The principal axes are aligned first with the global coordinate axes $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, and $\hat{\mathbf{z}}$ depicted in Fig. 1 and then rotated $60°$ around the x axis. The incident medium is isotropic and nongyrotropic with a relative electric permittivity of 1.0. The grating vector is parallel with the x direction. The grating period is $1.5\lambda$ with a duty cycle of 0.5 and the grating depth is $55.0\lambda$. The magnetic permeability is 1.0 for both media. An O mode (TE mode) wave is obliquely incident on the grating with an angle of incidence ${\theta }_{\text{in}}=30°$. The plane of incidence is parallel to the $x–z$ plane. These parameter values are chosen such that the materials have large anisotropy and gyrotropy, and the RCWT calculations exhibit relatively slow convergence, although those values are highly artificial.

The zeroth- and $-1$st-order diffraction efficiencies for both the O and E mode (left and right elliptically polarized) transmitted waves calculated by the present RCWT algorithm are plotted in Fig. 2a. Those calculated by RCWT using only Laurent’s rule are shown in Fig. 2b. The horizontal axis is the order of Fourier terms N, where $2N+1$ terms, ($-N$)th- to ($+N$)th-order Fourier terms, are retained. The diffraction efficiencies are calculated up to $N=100$. The convergence test given in Fig. 2 shows that the RCWT formulation pres ented in this paper has good convergence.

4. EXAMPLES OF SINGLE-LAYER GRATINGS IN BI-ISOTROPIC MEDIA

Single-layer binary gratings corrugated on isotropic gyrotropic substrates are investigated in this section. The grating configurations are identical, except that the gyrotropic constant of the substrates and the grating period are varied. The refractive index of the incident medium is $n=1.0$ and the substrate is $n=1.5$. The relative magnetic permeability is unity for both the incident medium and the substrate. The grating depth is $1.0\lambda$ and the grating period is varied from zero to $10\lambda$ with a grating duty cycle of 0.5, where λ is the wavelength of the incident wave in free space. The O mode (TE mode) electromagnetic wave, whose E field is parallel to the grating grooves, is normally incident on the gratings. The truncation order is $N=25$ for all sample calculations demonstrated in this section.

The diffraction efficiencies of the transmitted waves are plotted as a function of the grating period with various gyrotropic constants. Figure 3 shows the zeroth- and first-order diffraction efficiencies of the O and E mode (left and right circularly polarized) transmitted waves with the gyrotopic constant (a) $G=0.02$ and (b) $G=0.1$, respectively. The diffraction efficiencies of the E mode transmitted waves are not zero, although the incident wave includes the O mode (TE mode) only. It is because both the O and E modes exist in the bi-isotropic medium and the incident electromagnetic energy is separated into those two modes due to polarization coupling.

For comparison, the zeroth- and first-order diffraction efficiencies of the O and E mode (TE and TM mode) transmitted waves of the same grating, corrugated on an isotropic nongyrotropic medium with the same refractive index, are also plotted in Fig. 4. The configuration of the grating is the same as those for the previous examples. This case can be understood as a limiting case for the gyrotropic constant G approaching zero. Note that the scale of the vertical axis in Fig. 4 is doubled. In the isotropic nongyrotropic grating, the E mode (TM mode) wave does not appear and polarization coupling is not observed. The diffraction efficiency profiles plotted in Fig. 3 approach the profiles plotted in Fig. 4 as the gyrotropic constant G becomes small, except that the values of the diffraction efficiencies are one half of those plotted in Fig. 4.

5. EXAMPLES OF MULTILAYER GRATINGS IN GYROTROPIC BIAXIAL MEDIUM

In this section, four-step sawtooth multilayer gratings in biaxial gyrotropic media are investigated. The grating structure is depicted in Fig. 5. The incident medium is assumed to be air. The principal relative electric permittivity of the grating me dium and the substrate are ${\epsilon }_x=1.50$, ${\epsilon }_y=1.55$, and ${\epsilon }_z=1.60$, and the principal gyrotropic constants are $G_x=0.10$, $G_y=0.12$, and $G_z=0.14$, with identical principal axes for ε and G. The principal axes are aligned first with the global coordinate axes $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, and $\hat{\mathbf{z}}$ depicted in Fig. 1 and then rotated $90°$ around the z axis, followed by $30°$ rotation around the x axis. The relative permeability is unity. The grating period is $1.5\lambda$. The duty cycle of the first, second, third, and fourth layers are 0.125, 0.375, 0.625, and 0.875, respectively, with the thickness of each layer being the same. The total grating depth is varied from zero up to $20\lambda$. An O mode (TE mode) wave is obliquely incident on the grating with an angle of incidence ${\theta }_{\text{in}}=30°$. The plane of incidence is parallel to the grating vector. These parameter values are arbitrarily chosen since no measurement data are available in literature, to the best of authors’ knowledge. The truncation order is $N=20$ for the sample calculations demonstrated in this section.

The first-order diffraction efficiencies for the O and E mode (left and right elliptically polarized) transmitted waves are plotted in Fig. 6 as a function of the grating depth varying from zero to $20\lambda$. The diffraction efficiency profiles of the O mode and E mode behave quite differently. The diffraction efficiencies for the E mode transmitted waves are not zero due to polarization coupling between the two modes. Note that this configuration does not yield E mode transmitted and reflected orders when the principal gyrotropic constants are zero. Some diffraction efficiency values are listed in Table 1 as a reference.

The diffraction efficiency profiles of the first-order E mode transmitted wave of the same grating are plotted in Fig. 7 with various values of principal gyrotropic constant, $0.05\times \mathbf{G}$, $0.10\times \mathbf{G}$, $0.20\times \mathbf{G}$, and $0.50\times \mathbf{G}$, where G is the principal gyrotropic tensor previously used ($G_x=0.10$, $G_y=0.12$, and $G_z=0.14$). The diffraction efficiency increases as the gyrotropic constants become large. In contrast, for gyrotropic isotropic gratings, the first-order E mode transmitted wave receives a considerable fraction of energy, about half of total energy of the first-order transmitted waves, even when the gyrotropic constants are very small. This behavior is due to the difference of the polarization coupling between bi-isotropic and biaxial gyrotropic media.

The two orthogonal eigenpolarization states in gyrotropic isotropic media are left and right circular polarization regardless the value of gyrotropic constant. Since the incident wave is in the TE mode, the polarization coupling effect can be large even for a very small value of the gyrotropic constant. At the limit of G approaching zero, the directions of the wave vectors of the left and right circular polarization states become parallel, and these two modes are combined, resulting in the TE mode and causing the diffraction efficiencies to be doubled.

On the other hand, in gyrotropic biaxial media with very small gyrotropic constants, the two orthogonal eigenpolarization states are almost linearly polarized. The parameter values are determined such that one of the eigenpolarization states is parallel to the incident polarization state and the other eigenpolarization is orthogonal, when $G=0$. Thus, the polarization coupling effect is very weak when the gyrotropic constants are very small. However, the polarization coupling effect becomes larger as the gyrotropic constants become larger, which makes the eigenpolarization states more elliptical.

6. NONGYROTROPIC SPECIAL CASE

The present RCWT formulation is also applicable to the gratings made with nongyrotropic materials as a special case. The diffraction efficiency curves of an isotropic dielectric lamellar grating coated with an anisotropic layer are calculated and compared with the corresponding results described in [10]. The grating geometries are the same as those depicted in Fig. 1 in [10]. The grating duty cycle is 0.5. The relative electric permittivity of the incident material is ${\epsilon }_u=1.0$, and that of the substrate is ${\epsilon }_s=2.25$. The anisotropic layer is uniaxial with the relative electric permittivity given by $n_o^2=2.465$ and $n_e^2=2.161$. The optical axis is in the $y–z$ plane rotated $30°$ from the z axis. The thickness of the anisotropic layer is $0.2\lambda$ and the period of the grating is $1.0\lambda$, where λ is the wave length of the incident wave in free space. The truncation order is $N=10$.

Figure 8 shows the zeroth- and $-1$st-order TE and TM mode transmission diffraction efficiencies as the grating thickness varies from zero to $60\lambda$. The angle of incidence is $32.189°$. Figure 9 plots the zeroth- and $-1$st-order TE and TM mode transmission diffraction efficiencies as a function of the incident angle for the grating depth of $50\lambda$. It is seen that both plots match the corresponding plots (Fig. 3 and Fig. 4 shown in [10]).

It can be verified that our formulation reduces to the RCWT formulation for anisotropic nongyrotropic gratings by setting the gyrotropic constants to be zero. Maxwell’s equations in Fourier space and the coupled-wave equations for nongyrotropic gratings can be obtained from our formulation, which are the same as those described in [16].

7. SUMMARY

An RCWT formulation that can deal with the symmetric constitutive relations is presented. Our formulation applies to diffraction gratings in bianisotropic media that exhibit linear birefringence and/or optical activity. All of the incident, exiting, and grating materials can be isotropic, uniaxial, or biaxial, with or without optical activity. The principal values of the electric permittivity tensor, the magnetic permeability tensor, and the gyrotropic tensor of the materials can take arbitrary values, and the principal axes may be arbitrarily and independently oriented. The coupled-wave equations for bianisotropic gratings are derived from Maxwell’s equations along with appropriate procedures for Fourier expansion. Both the symmetric and asymmetric constitutive relations are briefly reviewed and discussed. It is demonstrated that the present algorithm exhibits fast convergence. As examples, the diffraction efficiency profiles for single-layer bi-isotropic gratings and multilayer biaxial gyrotropic gratings are calculated. Distinctive polarization coupling effects due to optical activity are observed in both cases. The polarization coupling effect demonstrated in this paper reflects the inherent characteristics of the materials. As a special case, diffraction efficiencies for a multilayer grating made with nongyrotropic uniaxial material are shown to exhibit good agreement with simulation data available in literature.