### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Coupled Differential-Equation Set
- 3. Constitutive Relations in Fourier Space
- 4. Numerical Examples
- 5. Conclusion
- References

[1] Li's Fourier factorization rules have contributed greatly to the differential theory that is one of the most commonly used approaches in the analysis of diffraction gratings. This paper gives a differential formulation for nonsmooth profiled anisotropic gratings by taking into account Li's remark, but a coupled first-order differential-equation set is derived using only the Laurent rule. The present formulation is applied to sinusoidal and echelette gratings, and numerical results show that the present formulation provides significant improvement of convergence.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Coupled Differential-Equation Set
- 3. Constitutive Relations in Fourier Space
- 4. Numerical Examples
- 5. Conclusion
- References

[2] There have been extensive theoretical investigations on the electromagnetic scattering by the diffraction gratings and many numerical approaches have been proposed. The original idea of differential method [*Petit*, 1966; *Cerutti-Maori et al.*, 1969; *Vincent*, 1980] was proposed in 1960s and, now, it becomes one of the most commonly used approaches in the analysis of various diffraction gratings, such as stack of gratings [*Nevière*, 1994], gratings covered with dielectric layers [*Hutley et al.*, 1975], grating couplers [*Nevière et al.*, 1973a, 1973b], gratings made of anisotropic materials [*Tayeb et al.*, 1987; *Tayeb*, 1990] because of its simplicity and wide applicability. However, the difficulty concerning to the convergence was criticized for deep and highly conducting gratings [*Nevière et al.*, 1974] and this approach was felt to be limited in its application range during about 30 years.

[3] The deadlock was broken by Li's recent work [*Li*, 1996a], which clarified the origin of the poor convergence. He pointed out that the Laurent rule, which had been always adopted for factorizing the truncated Fourier series corresponding to products of two periodic functions, is valid only when two functions have no concurrent jump discontinuities (type 1 products in Li's terminology). Also, he showed another Fourier factorization rule called “inverse rule.” In most cases, it can be applied for the truncated Fourier factorization of products of two functions that have only pairwise complementary jump discontinuities (type 2 products). And products of two functions that have concurrent but not complementary jump discontinuities (type 3 products) can be Fourier factorized by neither the Laurent rule nor the inverse rule. Consequently, much interest has been shown in obtaining new formulations by avoiding type 3 products.

[4] Li applied these factorization rules to the rigorous coupled-wave method, in which each grating profile is represented as a staircase approximation of several or more layers of lamellar gratings, and proposed novel formulations of lamella profiled gratings made with isotropic and anisotropic dielectric materials [*Li*, 1996a, 1998]. The rigorous coupled-wave method is, of course, well suited for lamellar profiled gratings or lossless dielectric gratings, but a recent paper [*Popov et al.*, 2002] showed that, for smooth profiled gratings made of conducting materials, the profile ridges introduced by the staircase approximation cause sharp maxima of the fields and they make the convergence worse.

[5] Another approach of the differential method for arbitrary profiled gratings is the use of the shooting method [*Vincent*, 1980]. This approach does not use the staircase approximation and, therefore, formulations not only for lamellar profiled gratings but also for arbitrary profiled ones are necessary. *Popov and Nevière* [2000] considered the continuation of the tangential and normal components of the field and proposed a novel formulation of isotropic gratings based on the Laurent rule and the inverse rule. This approach was named the fast Fourier factorization method by *Popov and Nevière* [2001]. *Watanabe et al.* [2002] investigated smooth profiled gratings made of anisotropic materials with electric and/or magnetic properties by using intermediary functions continuous everywhere. The intermediary functions make it possible to get rid of not only type 3 products but type 2 ones in the formulation, and therefore only the Laurent rule is used. This formulation was generalized by *Popov and Nevière* [2001] to the scattering problems by periodic surface, which is assumed to have normal vector everywhere and separates two anisotropic media.

[6] In this paper, we consider the electromagnetic scattering by nonsmooth profiled gratings made of anisotropic materials with electric and/or magnetic properties, and present a formulation by taking into account Li's Fourier factorization rules. We introduce the intermediary functions as shown by *Watanabe et al.* [2002] but they are discontinuous in this paper. The formulation avoids type 2 and 3 products and is based on the Laurent rule only. We give some examples of computation for sinusoidal and echelette gratings made of anisotropic and highly conducting materials to validate the formulation.

### 2. Coupled Differential-Equation Set

- Top of page
- Abstract
- 1. Introduction
- 2. Coupled Differential-Equation Set
- 3. Constitutive Relations in Fourier Space
- 4. Numerical Examples
- 5. Conclusion
- References

[7] We investigate the diffraction problem on a grating ruled on an anisotropic and homogeneous substrate schematically shown in Figure 1. The structure is uniform in the *z*-direction and the grating profile is characterized by a known periodic and continuous function *p*(*x*) with a piecewise-continuous and bounded derivative. Let *x*_{j} (*j* = 1, 2, …) be the abscissas of the discontinuities of the derivative of *p*(*x*) for later explanation. We denote the grating period by *d* and the grating depth by *h*. The region *y* > *p*(*x*) is filled with a homogeneous and isotropic material described by the relative permittivity ε_{1} and the relative permeability μ_{1}, and the homogeneous and anisotropic material which fills the region *y* < *p*(*x*) is described by the relative permittivity tensor and the relative permeability tensor . We consider only time harmonic fields assuming a time dependence in exp(−*i*ω*t*), and deal with the plane incident wave propagating in the direction of polar angle θ and azimuth angle ϕ. To make expressions simple, we normalize the fields, namely, *E* field by , *H* field by , *D* field by , and *B* field by , where ε_{0} and μ_{0} denote the permittivity and the permeability in free space, respectively.

[8] The *x*, *y*, and *z* components of electromagnetic fields are pseudoperiodic functions of *x* and can be approximately expanded in truncated generalized Fourier series [*Vincent*, 1980]; for example *E*_{x} can be written as

where *N* is the truncation order and *k*_{0} and λ_{0} are the wave number and the wavelength in free space. In the present paper, all series will be symmetrically truncated with the same truncation order. *Popov and Nevière* [2001] showed that the coupled differential-equation set which describes the electromagnetic fields in the groove region can be derived if we obtain the relations between the generalized Fourier coefficients of *D*_{x}, *D*_{y}, *D*_{z}, *B*_{x}, *B*_{y}, *B*_{z} and those of *E*_{x}, *E*_{y}, *E*_{z}, *H*_{x}, *H*_{y}, *H*_{z}. We assume these “constitutive relations in Fourier space” as follows:

where [*f*] gives the (2*N* + 1) × 1 column matrix constructed with the generalized Fourier coefficients of *f*(*x*). As shown by *Popov and Nevière* [2001], substituting equations (4) and (5) into Maxwell's curl equations and eliminating [*E*_{y}] and [*H*_{y}], we may obtain the following coupled differential-equation set:

with

If electromagnetic parameters of anisotropic substrate are physically adequate [*Tayeb*, 1990], applying the shooting method [*Vincent*, 1980] and the S-matrix propagation algorithm [*Li*, 1996b; *Montiel et al.*, 1998] can numerically solve this coupled differential-equation set.

### 3. Constitutive Relations in Fourier Space

- Top of page
- Abstract
- 1. Introduction
- 2. Coupled Differential-Equation Set
- 3. Constitutive Relations in Fourier Space
- 4. Numerical Examples
- 5. Conclusion
- References

[9] Now, the diffraction problem has been reduced how to obtain the constitutive relations in Fourier space. Let ε_{2,νμ} and μ_{2,νμ} (ν, μ = *x*, *y*, *z*) be respectively the (ν, μ) entries of the relative permittivity and the relative permeability tensors of the region *y* < *p*(*x*), and *w*(*x*, *y*) and (*x*, *y*) be the window functions defined by

Then, the original constitutive relations can be written as

where *I* denotes the identity matrix. The relations (26) and (27) are transformed into truncated Fourier space and expressed as follows:

[10] We need the relations between the generalized Fourier coefficients of *D* and *B* fields and those of *E* and *H* fields to derive the coupled differential-equation set (6). The conventional formulation [*Tayeb*, 1990] apply the intuitive Laurent rule [*Li*, 1996a] to equations (26) and (27) directly, and use the following expressions for the constitutive relations in Fourier space:

where ν, μ = *x*, *y*, *z* and the notation expresses a (2*N* + 1) × (2*N* + 1) square Toeplitz matrix generated by the generalized Fourier coefficients of *f*(*x*) in such a way that its (*n*, *m*) entries are the (*n*–*m*)th coefficients. This approach is very powerful and widely used for many types of gratings but as noted before, the difficulty concerning to the convergence has been criticized for several cases [*Nevière et al.*, 1974]. Li's Fourier factorization rules [*Li*, 1996a] claim that the Fourier coefficients of products: [*w E*_{x}], [*w E*_{y}], [*E*_{x}], [*E*_{y}], [*w H*_{x}], [*w H*_{y}], [*H*_{x}], and [*H*_{y}] can not be factorized by the Laurent rule because *w*(*x*, *y*), (*x*, *y*) and the *x*, *y* components of *E* and *H* fields are concurrently discontinuous on the grating profile: *y* = *p*(*x*).

[11] To avoid this difficulty, we introduce four functions *D*_{n}, *E*_{t}, *B*_{n}, and *H*_{t} which were also used by *Watanabe et al.* [2002]. On the grating profile, they give the normal component of *D* field, the tangential component of *E* field, the normal component of *B* field, and the tangential component of *H* field, respectively, and the definitions are given by

where ψ(*x*) = tan^{−1}(*dp*/*dx*) for ψ ∈ (−π/2,π/2), and *E*_{z} and *H*_{z} are also expressed for convenience sake. As mentioned by *Watanabe et al.* [2002], these four functions are defined not only on the grating profile but everywhere and they are pseudoperiodic functions of *x*. The formulation of *Watanabe et al.* [2002] was limited to smooth profiled gratings and the introduced functions were continuous everywhere. However, we have to note that the present functions defined by equations (32) and (33) are discontinuous on *x* = *x*_{1}, *x*_{2}, …, and the aim of these functions is to make two sets of functions. The first set *P* is the set of the functions continuous except for the grating surface *y* = *p*(*x*), and the second set *Q* is the set of the functions continuous except for *x* = *x*_{1}, *x*_{2}, …. Namely, the functions *E*_{x}, *E*_{y}, *H*_{x}, *H*_{y}, *w*, , etc. belong to the set *P*, and the functions *D*_{n}, *E*_{t}, *B*_{n}, *H*_{t}, sin ψ, cos ψ, etc. belong to the set *Q*. Following Li's Fourier factorization rules, a product of an element of *P* and an element of *Q* can be Fourier factorized by the Laurent rule for almost every *y* (except for *y* = *p*(*x*_{1}), *p*(*x*_{2}), …), because they have no concurrent jump discontinuities. However, if two functions belong to the same set, their product cannot be Fourier factorized by the Laurent rule.

[12] Multiplying equation (32) by *w*(*x*, *y*) and (*x*, *y*), and using the relations: *w*^{2} = *w*, ^{2} = , and *w* = 0, we obtain

The equations (36) and (37) above are transformed into the Fourier space with careful use of the Laurent rule, and we obtain the following relations:

Then, after the inversion of matrices, we get

with

Besides, considering the relation: [*E*_{ν}] = [*w E*_{ν}] + [*E*_{ν}] (ν = *x*, *y*, *z*), we obtain

From equations (40), (41), and (44), we have the following relations:

These relations are substituted into equation (28), and the matrix *Q*^{(e)} which gives the electric constitutive relation in Fourier space (4) is derived as

Similarly, from equation (33), we obtain the matrix *Q*^{(h)} for the magnetic constitutive relation in Fourier space as

with

The required coupled differential-equation set (6) is obtained by substituting equations (47) and (48) into equations (7)–(22).

### 4. Numerical Examples

- Top of page
- Abstract
- 1. Introduction
- 2. Coupled Differential-Equation Set
- 3. Constitutive Relations in Fourier Space
- 4. Numerical Examples
- 5. Conclusion
- References

[13] In order to confirm the validity, the formulation is applied to a sinusoidal grating made of anisotropic and conducting materials (cobalt) and compared with the formulation obtained by the previous studies [*Tayeb*, 1990; *Watanabe et al.*, 2002]. Figure 2 shows the efficiencies of −1st and zeroth-order diffraction waves as functions of the truncation order *N* which truncates the Fourier series expansion from −*N*th to *N*th order. The values of grating parameters are chosen as λ_{0} = 0.6328 μm, θ = 30°, ϕ = 0°, *d* = 0.6 μm, *h* = 0.5 μm, *p*(*x*) = (*h*/2)[1 + cos (2π *x*/*d*)], ε_{1} = μ_{1} = 1, ε_{2,xx} = ε_{2,yy} = ε_{2,zz} = −8.19 + *i* 16.38, ε_{2,xz} = −ε_{2,zx} = −0.495 − *i* 0.106, ε_{2,xy} = ε_{2,yx} = ε_{2,yz} = ε_{2,zy} = 0, = *I*, and the incident polarization is TM (the *y* component of *H* field is zero). We divided the groove region into 10 layers in the S-matrix propagation algorithm, and 10 steps of integration per layer are used for the shooting method with the help of fourth-order Runge–Kutta method. The solid lines are obtained by the present formulation, the dotted lines are obtained by the conventional one which is given by equations (30) and (31), and the dashed lines are obtained by the formulation of *Watanabe et al.* [2002]. Comparing with the conventional formulation, it is observed that the present one provides a significant improvement of convergence. Also, the present formulation is in good agreement with the formulation of *Watanabe et al.* [2002] which was confirmed its validity by comparing the numerical results with the integral method [*Maystre*, 1980], though the convergence of the present formulation is a little slower than that of the formulation of *Watanabe et al.* [2002].

[14] Next, we consider echelette gratings. The profile is nonsmooth and illustrated in Figure 3, which introduces a new notation *d*_{1}. *Tayeb* [1990] gave numerical results of a lossless anisotropic grating by the integral method, and we use the same parameter to check our results in absence of experimental measurements. The parameters are λ_{0} = 0.6 μm, θ = 20°, ϕ = 0°, *d* = 0.5 μm, *h* = 0.2 μm, *d*_{1} = *h*, ε_{1} = μ_{1} = 1, ε_{2,xx} = 6.31, ε_{2,yy} = 6.81, ε_{2,zz} = 7.34, the off-diagonal entries of are zero, = *I*, and the incident polarization is TM. This grating does not generate the TE diffraction waves, and the efficiencies of TM diffraction waves in −1st and zeroth-order computed by the present formulation with 51 Fourier components (from −25 to 25) are respectively 0.1165 and 0.0143. The numerical results of the integral method are 0.1166 and 0.0143 for the −1st and zeroth-order, which are in good agreement with our results. Also, the present formulation is applied to a deep grating made of highly conducting material, which is one of the most difficult problems for the differential method. The value of geometry parameters are chosen as λ_{0} = 0.6328 μm, θ = 30°, ϕ = 20°, *d* = 0.6 μm, *h* = 0.5 μm, *d*_{1} = 0.4 μm, and the permittivities and the permeabilities of the materials are the same as the previous sinusoidal grating. The incident polarization is TM and solid lines in Figure 4 shows the −1st and zeroth-order diffraction efficiencies obtained by the present formulation. The numerical results are compared with those of the conventional formulation drawn by the dotted lines. It is clearly observed that the present formulation provides significant improvement of convergence.