Perturbation method for the Rigorous Coupled Wave Analysis of grating diffraction

Kofi Edee; Jean-Pierre Plumey; Gérard Granet; Jérome Hazart

doi:10.1364/OE.18.026274

1. Introduction

For many years the electromagnetic wave diffraction by surface relief gratings has been widely studied. Most of the methods which deal with this subject are based on a differential approach and use the staircase approximation of the grating profile. This technique of considering the grating as a stack of lamellar gratings was first suggested by Peng et al. [1] and was later used in the very popular Rigorous Coupled Wave Analysis (RCWA) [2]. In all the methods employing this technique, the electromagnetic field is written as a modal expansion in each layer and the expansions in the different layers are connected by boundary conditions. Numerical difficulties which may be encountered are linked, on the one hand, to the way in which the field is calculated inside the lamellar gratings, and, on the other hand, to the way in which the field expansions are propagated through the layers.

The field components in a lamellar grating may be thought of in terms of modal field expansions. The classical modal method [3, 4] is a rigorous theory since it uses basis functions which constitute exact solutions of Maxwell’s equations and satisfy the pseudo-periodic boundary conditions. The Fourier modal method (FMM) was introduced by Knop [5] in the case of dielectric gratings. In this technique, each mode is approximated by a truncated Fourier series. The application of this method to a metallic grating in TM polarization was much questioned until 1996 when several authors [6–8] proposed a dramatic improvement based on an efficient Fourier expansion of the periodic permittivity function. In contrast with the classical modal method, the FMM leads to a numerical scheme which permits straightforward implementation.

This feature explains the popularity of the RCWA which combines the multilayer approach with the FMM, whereas the numerical implementation of the classical modal method requires more sophisticated mathematics in order to overcome numerical difficulties [9,10]. In all cases, convergence with respect to the number of layers depends on the way in which the modal field expansion is propagated through the layers [8, 11, 12]. Many studies have been devoted to this subject up to now [?]. In contrast to the so-called differential method [14] and the coordinate transformation method (C-method ) [15–17], all the above mentioned methods rely on the assumption that the staircase approximation is valid. It has been shown that it is not the case for highly conducting gratings. Although the RCWA fails in such cases, it remains very efficient in most cases thanks to both its versatility and simplicity.

The staircase approximation entails solving as many eigenvalue problems as there are layers. The aim of this paper is to show that the number of eigenvalue problems can be drastically reduced by means of a perturbation method (PM). This approach is based on the fact that the permittivity function varies very little from one layer to the adjacent one.

2. Staircase approximation

Let us consider the grating depicted in Fig. 1. This structure is invariant along the z-direction and it is illuminated by an incident wave, under incidence angle θ, that can be either TE (the only non nul components of the field are E_z, H_x and H_y) or TM (the only non nul components of the field are H_z, E_x and E_y) polarized. It is well known that in these cases of polarization, all the field components can be expressed in terms of the E_z-component in the TE case or in terms of the H_z-component in the TM case. Let us add that throughout this paper, the time dependence of the fields will be held by the term exp(iωt) where ω denotes the circular frequency of the monochromatic incident wave. The incident medium is vacuum and the substrate is a dielectric medium with or without losses. E_z and H_z satisfy the propagation equations:

TE : (\partial_{x} \partial_{x} + \partial_{y} \partial_{y} + k^{2} ν^{2} (x, y)) E_{z} = 0,

TM : (\partial_{x} \frac{1}{ν^{2} (x, y)} \partial_{x} + \partial_{y} \frac{1}{ν^{2} (x, y)} \partial_{y} + k^{2}) H_{z} = 0 .

k = 2π/λ denotes the vacuum wave number and ν is the refractive index.

Fig. 1 The grating configuration: two periods are depicted.

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The grating surface is approximated by a staircase profile (see Fig. 2) so that the structure is replaced by a stack of layers, each of them is characterized by its thickness h_j and its refractive index ν_j which only depends on the x variable:

ν_{j} (x) = {\begin{array}{l} n_{1}, & x \in [\frac{d - l_{j}}{2}, \frac{d + l_{j}}{2}], \\ n_{2}, & x \in [0, \frac{d - l_{j}}{2}] \cup [\frac{d + l_{j}}{2}, d] . \end{array}

Fig. 2 The staircase approximation: the grating is replaced by a stack of lamellar gratings.

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3. Eigenvalue problem

In this section, we solve the propagation equation Eq. (1) in any j^th layer, therefore the subscript index j is omitted. The equations Eq. (1) can be rewritten in a concise form:

ℒ F (x, y) = - \frac{1}{k^{2}} \partial_{y}^{2} F (x, y),

where F(x,y) denotes E_z or H_z. ℒ is an operator which is different according to the polarization:

TE : ℒ = \frac{1}{k^{2}} \partial_{x}^{2} + ν^{2} (x),

TM : ℒ = \frac{1}{k^{2}} ν^{2} (x) \partial_{x} \frac{1}{ν^{2} (x)} \partial_{x} + ν^{2} (x) .

Since the operator ℒ depends only on the x-variable, the separation of variables method leads to

F (x, y) = ψ (x) e^{\pm i k r y},

where ψ is the eigenfunction of the ℒ-operator:

ℒ ψ (x) = r^{2} ψ (x),

that satisfies the pseudo-periodic condition:

ψ (x + d) = e^{- i k α d} ψ (x) .

α is determined by the incident wave: α = sinθ. The square root of the eigenvalue r² is chosen in such a way that

Im (r) < \begin{matrix} 0 & or & r \end{matrix} > 0 if r is real .

Let us recall briefly some important properties of the ℒ-operator [3, 4] that will be useful for the perturbation scheme that will be developed in section (5). For that purpose, we define an inner product for any pseudo-periodic functions f and g:

TE : < f, g >_{E} = \frac{1}{d} \int_{x_{0}}^{x_{0} + d} \bar{f} (x) g (x) d x,

TM : < f, g >_{H} = \frac{1}{d} \int_{x_{0}}^{x_{0} + d} \frac{1}{ν^{2} (x)} \bar{f} (x) g (x) d x .

f denotes the complex conjugate of f.

By using the pseudo-periodicity, it can be established that the operator ℒ^† defined by:

TE : ℒ^{†} = \frac{1}{k^{2}} \frac{d^{2}}{d x^{2}} + \bar{ν^{2}} (x),

TM : ℒ^{†} = \frac{1}{k^{2}} \bar{ν^{2}} (x) \frac{d}{d x} \frac{1}{\bar{ν^{2}} (x)} \frac{d}{d x} + \bar{ν^{2}} (x),

is the adjoint of the operator ℒ. The sets of eigenfunctions ψ_m and

ψ_{m}^{†}

are orthogonal with respect to the convenient inner product. The completeness of eigenfunctions allows to expand any pseudoperiodic function f(x) as follows:

f (x) = \sum_{m = 1}^{\infty} a_{m} ψ_{m} (x), where a_{m} = < ψ_{m}^{†}, f > .

F(x,y) (i.e. E_z or H_z) may be expanded in the eigenfunction series:

F (x, y) = \sum_{m = 1}^{\infty} (A_{m} e^{{i k r}_{m} y} + B_{m} e^{- {i k r}_{m} y}) ψ_{m} (x) .

According to Eq. (8)A_m is associated with downgoing waves and B_m to upgoing waves. When ν(x) is a real function ℒ^† and ℒ are identical i.e. ℒ is self-adjoint.

4. Fourier modal expansion

The eigenfunctions can be calculated exactly by solving a transcendental equation [3, 4]. Another way for implementing the modal method [2, 5] consists in expanding any eigenfunction ψ_m in the Fourier series:

ψ_{m} (x) = \sum_{n = - \infty}^{n = \infty} ψ_{m n} e_{n} (x),

where

e_{n} (x) = e^{- i k α_{n} x}, α_{n} = sin θ + n \frac{λ}{d} .

Numerically only a finite sequence of N Fourier coefficients is kept. Equation (6) involves products of functions of the x-variables which are represented by truncated Fourier series. Li described the procedures for correctly converting such products in discrete Fourier space [8]. Solving Eq. (6) leads to the research of the eigenvalues and eigenfunctions of a (2N + 1) × (2N + 1) matrix:

TE : L_{E} = ℰ - α α,

TM : L_{H} = {\tilde{ℰ}}^{- 1} [I - α ℰ^{- 1} α] .

The ε_mn and ε̃_mn elements of ℰ and ℰ̃ respectively are the (m – n)^th Fourier coefficients of ν²(x) and 1/ν²(x). Each eigenvalue of L_E and L_H is associated with an approximate r_m eigenvalue which is computed according to Eq. (8). Finally, the modal expansion of Eq. (5) is approximated as follows:

F (x, y) = \sum_{m = 1}^{m = 2 N + 1} (A_{m} e^{{i k r}_{m} y} + B_{m} e^{- {i k r}_{m} y}) ψ_{m} (x),

with

ψ_{m} (x) = \sum_{n = - N}^{n = N} ψ_{n m} e_{n} (x) .

ψ_nm denotes n^th Fourier coefficients of the eigenfunction ψ_m(x).

5. Perturbations

Perturbation theory is a mathematical method used to find an approximate solution to a problem from the exact solution of a related problem. This method is applicable if the problem at hand can be formulated by adding a “small” term to the mathematical description of the exactly solvable problem. Historically, the perturbation method has its roots in early celestial mechanics and it was first used to solve algebraic equations, before being applied to the operator theory, especially in classical quantum mechanics [19, 20].

Assuming that the eigenvalues and eigenfunctions of Eq. (6) are exactly known, we introduce the superscript index (0) to refer to this case which is characterized by the refractive index function ν⁽⁰⁾(x). In accordance with this notation Eq. (6) becomes:

ℒ^{(0)} ψ_{p}^{(0)} (x) = λ_{p}^{(0)} ψ_{p}^{(0)} (x)

for any eigenvalue labeled by the subscript index p. In any layer, the refractive index function ν(x) is slightly different from ν⁽⁰⁾(x), thus the operator ℒ may be expressed using the unperturbated operator ℒ₀ plus a correction term:

\begin{matrix} ℒ - ℒ^{(0)} = 𝒫 = ζ \tilde{𝒫}, & ζ ≪ 1 \end{matrix}

where ζ is presumed to be “small”. It must be emphasized that the only operator P is known and ζ is a parameter useful only for analytical calculus. The eigenvalues

λ_{p} = r_{p}^{2}

and eigen-functions ψ_p of the operator are expressed in terms of power series of ζ:

\begin{matrix} λ_{p} = \sum_{n} ζ^{n} λ_{p}^{(n)}, & ψ_{p} = \sum_{n} ζ^{n} ψ_{p}^{(n)} \end{matrix} .

For the 0^th order approximation

λ_{p} = λ_{p}^{(0)}

,

ψ_{p} = ψ_{p}^{(0)}

.

To calculate the higher order perturbation correction terms, we put the equations Eq. (19) and Eq. (20) into Eq. (6):

(ℒ^{(0)} + ζ \tilde{𝒫}) \sum_{n} ζ^{n} ψ_{p}^{(n)} = \sum_{m, n} ζ^{m + n} λ_{p}^{(m)} ψ_{p}^{(n)} .

By successively identifying the 0, 1 and 2 rank coefficients of ζ we get:

ℒ^{(0)} ψ_{p}^{(0)} = λ_{p}^{(0)} ψ_{p}^{(0)},

ℒ^{(0)} ψ_{p}^{(1)} + \tilde{𝒫} ψ_{p}^{(0)} = λ_{p}^{(0)} ψ_{p}^{(1)} + λ_{p}^{(1)} ψ_{p}^{(0)},

ℒ^{(0)} ψ_{p}^{(2)} + \tilde{𝒫} ψ_{p}^{(1)} = λ_{p}^{(0)} ψ_{p}^{(2)} + λ_{p}^{(1)} ψ_{p}^{(1)} + λ_{p}^{(2)} ψ_{p}^{(0)} .

Afterwards we consider the case where the operator is self-adjoint which corresponds to a real ν(x) index. The eigenvalues are therefore real. The complex eigenvectors are normalized by setting conditions on both the modulus and the phase:

\begin{matrix} < ψ_{p}, ψ_{p} > = 1 & < ψ_{p}^{(0)}, ψ_{p} > real . \end{matrix}

By applying this normalization to 0, 1 and 2 orders we get:

< ψ_{p}^{(0)}, ψ_{p}^{(0)} > = 1,

< ψ_{p}^{(0)}, ψ_{p}^{(1)} > = < ψ_{p}^{(1)}, ψ_{p}^{(0)} > = 0,

< ψ_{p}^{(0)}, ψ_{p}^{(2)} > = < ψ_{p}^{(2)}, ψ_{p}^{(0)} > = \frac{1}{2} < ψ_{p}^{(1)}, ψ_{p}^{(1)} > .

In order to arrive at the first order approximation the equation Eq. (22b) is projected on $ψ_{p}^{(0)}$ . By using Eq. (24a) and Eq. (24b) as well as the self adjoint property of the operator we get the expressions of the eigenvalues

λ_{p} = λ_{p}^{(0)} + < ψ_{p}^{(0)}, 𝒫 ψ_{p}^{(0)} >,

and the eigenvectors

ψ_{p} = ψ_{p}^{(0)} + \underset{q \neq p}{Σ} \frac{< ψ_{q}^{(0)}, 𝒫 ψ_{p}^{(0)} >}{λ_{p}^{(0)} - λ_{q}^{(0)}} .

By combining Eq. (22c) and Eq. (24c) we obtain the second order approximation of the eigenvalues

λ_{p} = λ_{p}^{(0)} + < ψ_{p}^{(0)}, 𝒫 ψ_{p}^{(0)} > + \underset{q \neq p}{Σ} \frac{< ψ_{q}^{(0)}, 𝒫 ψ_{p}^{(0)} > < ψ_{p}^{(0)}, 𝒫 ψ_{q}^{(0)} >}{λ_{p}^{(0)} - λ_{q}^{(0)}}

and of the eigenvectors

\begin{array}{l} ψ_{p} = ψ_{p}^{(0)} + \underset{q \neq p}{Σ} \frac{< ψ_{q}^{(0)}, 𝒫 ψ_{p}^{(0)} >}{λ_{p}^{(0)} - λ_{q}^{(0)}} ψ_{q}^{(0)} \\ + \underset{q \neq p}{Σ} \underset{l \neq p}{Σ} \frac{< ψ_{l}^{(0)}, 𝒫 ψ_{p}^{(0)} > < ψ_{q}^{(0)}, 𝒫 ψ_{l}^{(0)} >}{(λ_{p}^{(0)} - λ_{l}^{(0)}) (λ_{p}^{0} - λ_{q}^{(0)})} ψ_{q}^{(0)} \\ - \underset{q \neq p}{Σ} \frac{< ψ_{p}^{(0)}, 𝒫 ψ_{p}^{(0)} > < ψ_{q}^{(0)}, 𝒫 ψ_{p}^{(0)} >}{{(λ_{p}^{(0)} - λ_{q}^{(0)})}^{2}} ψ_{q}^{(0)} \\ - \frac{1}{2} \underset{q \neq p}{Σ} \frac{< ψ_{p}^{(0)}, 𝒫 ψ_{q}^{(0)} > < ψ_{q}^{(0)}, 𝒫 ψ_{p}^{(0)} >}{{(λ_{p}^{(0)} - λ_{q}^{(0)})}^{2}} ψ_{q}^{(0)} . \end{array}

6. Numerical results

In this section, the perturbation method (PM) is applied to a surface grating made of a stack of two dielectric trapezoidal elements. This grating is characterized by six geometrical parameters indicated on Fig. (3): d = 480 nm, b₁ = 240 nm, b₂ = 120 nm, b₃ = 80 nm, h₁ = 80 nm, h₂ = 200 nm. The direction of the incident plane wave is given by θ = 75° and the refractive index of the grating is ν = n₁ = 1.51309 for the wavelength λ = 240 nm.

Fig. 3 Surface grating made of a stack of two dielectric trapezoidal elements. Numerical parameters: λ = 240 nm, d = 480 nm, b₁ = 240 nm, b₂ = 120 nm, b₃ = 80 nm, h₁ = 80 nm, h₂ = 200 nm, θ = 75°, n₁ = n₃ = 1.51309, n₂ = 1.

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First, we investigate the convergence of the RCWA with respect to the truncation order M (Table 1) and the number of layers N_c (Table 2) for the zero-order reflected efficiency. These results show that, for the studied case, the convergence of the RCWA is slow, with respect to M and to N_c, in TE polarization as well as in TM polarization. In order to assure an accuracy equal to 10⁻³ for the considered case, we will use throughout this paper the following parameters: Nc = 220 and M = 30.

Table 1. Convergence of the RCWA with respect of the truncation order M Results for the grating described by Figure 3 with (Nc = 220)

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Table 2. Convergence of the RCWA with respect of the number of layers Nc Results for the grating described by Figure 3 with 30 Fourier harmonics (M = 30)

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Secondly, these results are taken as reference for evaluating the performance of the perturbation method. This is why numerical results are given with 15 digits even if only 3 digits are significant. In the perturbation method, the number of layers Nc is preserved but the eigenvalue problem is only solved for n_p-number of layers smaller than N_c. The solutions for the remaining layers are obtained from Eqs. (25)–(28) at different perturbation orders. We define an error function ξ as:

ξ (n_{p}) = - Int ({log}_{10} | {eff}_{RCWA} - {eff}_{PM} (n_{p}) |),

Here Int(x) provides the integer part of x, eff_RCWA and eff_PM refer to the zero-order reflected efficiency with RCWA and perturbation method respectively. We must note that the solving procedure of equations deduced from boundary conditions is the same as the one intervening in the RCWA. The first column of Tables 3–8 list the number of equations n_p that have been solved and the last column presents the time-saving ts(n_p) = 1 – t_PM(n_p)/t_RCWA. t_PM is the computation time of the PM and t_RCWA is referred to the RCWA method.

Table 3. Results for the grating described by Figure 3 with 220 layers (N_c = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

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Table 4. Results for the grating described by Figure 3 with 220 layers (N_c = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

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Table 5. Results for the grating described by Figure 3 with 220 layers (N_c = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

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Table 6. Results for the grating described by Figure 3 with 220 layers (N_c = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

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Table 7. Results for the grating described by Figure 3 with 220 layers (N_c = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

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Table 8. Results for the grating described by Figure 3 with 220 layers (N_c = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

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The results obtained by approximating the eigenvalues and eigenvectors at first order, by means of Eq. (25) and Eq. (26), are presented in Tables 3 and 4. We notice that the 10⁻³ accuracy,as defined by Eq. (29) is obtained for a value of n_p = 4 i.e. by solving only 4 eigenvalue problems instead of 220 for both polarizations. Let us emphasize that the 10⁻³ accuracy is the one of RCWA reference method. The corresponding time-saving amounts to 25%. In addition, we can study the intrinsic efficiency of the perturbation method in relation to the RCWA method by supposing that the results provided by the latter method are accurate. We can remark that the PM converges with respect to n_p and that 10⁻⁴ accuracy is achieved for n_p = 20 and with a time-saving amounting to 25%.

The results obtained with the second-order approximation of both he eigenvalues ans eigenvectors i.e. by means of Eq. (27) and Eq. (28) are given in Tables 5 and 6. These results show that the second-order PM does not provide any significant improvement over the first-order one. For the TE polarization, we get 10⁻³ accuracy by solving fewer eigenvalue equations than the first order (2 instead of 4) but with a time-saving of only 13% instead of 25%. In others words, second order approximation is formally better than first-order one but this amelioration is numerically annihilated by the complexity of the second order approximation formulas. The situation worsens for TM polarization. By comparing the results in Table 6 with those in Table 4, we can notice that the second order approximation does not provide a better accuracy than first order one. The comparison of the operators in TE polarization [Eq. (15a)] and TM [Eq. (15b)] can explain these results. Only the l_j parameter (Cf. Fig. 2) differs from one layer to another. This parameter has influence through only the ν²(x)-function on the TE operator while it has influence more through the function 1/ν²(x) on the TM operator. Equations (26) and (28) show that the computation of the corrective term of second order eigenvectors increases the computation time. This observation has led us to numerically test a method that uses the correction of eigenvalue at second order Eq. (27) combined with the correction of eigenvectors with first order Eq. (26). These results are shown in Tables 7 et 8. Overall the results do not differ much from those obtained with the first order method as far as accuracy and computation time are concerned.

7. Conclusion

We have shown that the perturbation method allows to improve the run time of codes based on RCWA. Indeed, in many layers modal expansions may be deduced from those calculated in a reference layer thus avoiding to solve eigenvalue problems which is the most consuming part of the RCWA. Time saving may seem modest and certainly it is if one single point is to be calculated. However, the situation becomes completely different if the code has to be run many times when it is used to build libraries for solving inverse problems.

The other important case of interest is that of crossed gratings where the size of the matrices is squared compared to that of one-dimensional problems. Therefore, within the framework of the RCWA, avoiding solving eigenvalues problems is a significant improvement. Furthermore there is no theoretical difficulty in extending the perturbation method to this latter case. Hence, numerically speaking, we believe that the perturbation method applied to crossed gratings will provide high acceleration ratios.

Acknowledgments

This work was supported by ANR project LORS.

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M	TM	TE
10	0.08948867133941	0.20019490807133
15	0.08901067373154	0.19996856756561
20	0.08876516369415	0.19991502468408
25	0.08861117263702	0.19989563523817
30	0.08850709081867	0.19988689438072
35	0.08843212500310	0.19988238562554
40	0.08837571161222	0.19987984217835
45	0.08833133851717	0.19987832861513
50	0.08829587380656	0.19987738271599
55	0.08826686327190	0.19987675417602
60	0.08824257778239	0.19987631039014
65	0.08822176832110	0.19987598636793
70	0.08820390386685	0.19987574702571
75	0.08818848029206	0.19987556855067
80	0.08817501287529	0.19987543306324

Nc	TM	TE
200	0.088417155912	0.200028293914
210	0.088464267967	0.199954266677
220	0.088507090819	0.199886894381
230	0.088546184675	0.199825318344
240	0.088582016260	0.199768821501
250	0.088614977536	0.199716800368
260	0.088645400094	0.199668743414
270	0.088673566344	0.199624214187
280	0.088699718311	0.199582838036
290	0.088724064600	0.199544291539
300	0.088746785974	0.199508294043
310	0.088768039847	0.199474600818
320	0.088787963938	0.199442997490
330	0.088806679258	0.199413295476
340	0.088824292567	0.199385328216
350	0.088840898411	0.199358948052

M	TM	TE
10	0.08948867133941	0.20019490807133
15	0.08901067373154	0.19996856756561
20	0.08876516369415	0.19991502468408
25	0.08861117263702	0.19989563523817
30	0.08850709081867	0.19988689438072
35	0.08843212500310	0.19988238562554
40	0.08837571161222	0.19987984217835
45	0.08833133851717	0.19987832861513
50	0.08829587380656	0.19987738271599
55	0.08826686327190	0.19987675417602
60	0.08824257778239	0.19987631039014
65	0.08822176832110	0.19987598636793
70	0.08820390386685	0.19987574702571
75	0.08818848029206	0.19987556855067
80	0.08817501287529	0.19987543306324

Nc	TM	TE
200	0.088417155912	0.200028293914
210	0.088464267967	0.199954266677
220	0.088507090819	0.199886894381
230	0.088546184675	0.199825318344
240	0.088582016260	0.199768821501
250	0.088614977536	0.199716800368
260	0.088645400094	0.199668743414
270	0.088673566344	0.199624214187
280	0.088699718311	0.199582838036
290	0.088724064600	0.199544291539
300	0.088746785974	0.199508294043
310	0.088768039847	0.199474600818
320	0.088787963938	0.199442997490
330	0.088806679258	0.199413295476
340	0.088824292567	0.199385328216
350	0.088840898411	0.199358948052

Perturbation method for the Rigorous Coupled Wave Analysis of grating diffraction

Abstract

1. Introduction

2. Staircase approximation

3. Eigenvalue problem

4. Fourier modal expansion

5. Perturbations

6. Numerical results

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Tables (8)

Equations (38)

Optics Express

TE polarization, first order
n_p	RCWA	PM	ξ	ts(%)
1	0.19988689438072	0.204225695934385	2	25
2	-	0.201547549962947	2	25
4	-	0.200450748279565	3	25
5	-	0.200182489885528	3	24
10	-	0.199968166043854	4	23
11	-	0.199954199519059	4	23
20	-	0.199907262335079	4	21

TM polarization, first order
n_p	RCWA	PM	ξ	ts(%)
1	0.08850709081867	0.08984815731847	2	25
2	-	0.09104029308725	2	24
4	-	0.08903950505849	3	24
5	-	0.08891545481287	3	24
10	-	0.08862913389169	3	23
11	-	0.08861450138594	3	22
20	-	0.08853540436365	4	20

TE polarization, second order
n_p	RCWA	PM	ξ	ts(%)
1	0.19988689438072	0.169407724349512	1	13
2	-	0.200647125677975	3	13
4	-	0.199797794707405	4	13
5	-	0.199707800945216	3	13
10	-	0.199850793286191	4	12
11	-	0.199856094447670	4	11
20	-	0.199877706221066	5	10

TM polarization, second order
n_p	RCWA	PM	ξ	ts(%)
1	0.08850709081867	0.13446018793204	1	11
2	-	0.10270514590623	1	11
4	-	0.09332250791522	2	11
5	-	0.09153836150588	2	11
10	-	0.08933363383712	3	10
11	-	0.08920165911004	3	9
20	-	0.08871110661100	3	9

TM polarization, first order
n_p	RCWA	PM	ξ	ts(%)
1	0.08850709081867	0.08984815731847	2	25
2	-	0.09104029308725	2	24
4	-	0.08903950505849	3	24
5	-	0.08891545481287	3	24
10	-	0.08862913389169	3	23
11	-	0.08861450138594	3	22
20	-	0.08853540436365	4	20