Spectral element method with modified Legendre polynomials for modal analysis of lamellar gratings

Gérard Granet

doi:10.1364/JOSAA.409666

Spectral element method with modified Legendre polynomials for modal analysis of lamellar gratings

Gérard Granet

Journal of the Optical Society of America A
Vol. 38,
Issue 1,
pp. 52-59
(2021)
•https://doi.org/10.1364/JOSAA.409666

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Gérard Granet, "Spectral element method with modified Legendre polynomials for modal analysis of lamellar gratings," J. Opt. Soc. Am. A 38, 52-59 (2021)

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Abstract

We report on the derivation of a spectral element method whose originality comes from the use of a hierarchical basis built with modified Legendre polynomials. We restrict our work to TM polarization, which is the most challenging. The validation and convergence are carefully checked for metallic dielectric gratings. The method is shown to be highly efficient and remains stable for huge truncation numbers. All the necessary information is given so that non-specialists can implement the method.

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References

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Author
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Publication

M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[Crossref]
R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[Crossref]
K. Edee, “Modal method based on sub-sectional Gegenbauer polynomial expansion for lamellar grating,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[Crossref]
K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2012).
[Crossref]
M. H. Randriamihaja, G. Granet, K. Edee, and K. Raniriharinosy, “Polynomial modal analysis of lamellar diffraction gratings in conical mounting,” J. Opt. Soc. Am. A 33, 1679–1686 (2016).
[Crossref]
E. Popov, ed., Gratings: Theory and Numeric Applications (Institut Fresnel, CNRS, AMU, 2012).
A. C. Polycarpou, Introduction to the Finite Element Method in Electromagnetics, Synthesis Lectures on Computational Electromagnetics (Morgan and Claypool, 2006).
J. L. Volakis, A. Chatterje, and L. C. Kempel, Finite Element Method for Electromagnetics, Antennas, Microwave Circuits, and Scattering Applications (IEEE, 1998).
C. Pozirikidis, Introduction to Finite and Spectral Element Methods Using MATLAB (CRC Press, 2014).
E. Jorgensen, J. L. Volakis, P. Meincke, and O. Breinbjerg, “Higher order hierarchical Legendre basis functions for electromagnetic modeling,” IEEE Trans. Antennas Propag. 52, 2985–2994 (2004).
[Crossref]
G. Granet, “Fourier-matching pseudospectral modal method for diffraction gratings: comment,” J. Opt. Soc. Am. A 29, 1843–1845 (2012).
[Crossref]
S. Peng and G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[Crossref]
P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[Crossref]
Y.-P. Chiou, W.-E. Yeh, and N.-Y. Shih, “Analysis of highly conducting lamellar gratings with multidomain pseudospectral method,”J. Lightwave Technol. 27, 5151–5159 (2009).
[Crossref]
D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudospectral modal method for diffraction gratings,” J. Opt. Soc. Am. A 28, 613–620 (2011).
[Crossref]
M. Foresti, L. Menez, and A. V. Tishchenko, “Modal method in deep metal-dielectric gratings: the decisive role of hidden modes,” J. Opt. Soc. Am. A 23, 2501–2509 (2006).
[Crossref]
G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[Crossref]
L. Li and G. Granet, “Field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings,” J. Opt. Soc. Am. A 28, 738–746 (2011).
[Crossref]

2016 (1)

M. H. Randriamihaja, G. Granet, K. Edee, and K. Raniriharinosy, “Polynomial modal analysis of lamellar diffraction gratings in conical mounting,” J. Opt. Soc. Am. A 33, 1679–1686 (2016).
[Crossref]

2012 (2)

G. Granet, “Fourier-matching pseudospectral modal method for diffraction gratings: comment,” J. Opt. Soc. Am. A 29, 1843–1845 (2012).
[Crossref]

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2012).
[Crossref]

2004 (1)

E. Jorgensen, J. L. Volakis, P. Meincke, and O. Breinbjerg, “Higher order hierarchical Legendre basis functions for electromagnetic modeling,” IEEE Trans. Antennas Propag. 52, 2985–2994 (2004).
[Crossref]

1999 (1)

G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[Crossref]

1996 (1)

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[Crossref]

1995 (2)

S. Peng and G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[Crossref]

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[Crossref]

1982 (1)

M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[Crossref]

Breinbjerg, O.

Chatterje, A.

J. L. Volakis, A. Chatterje, and L. C. Kempel, Finite Element Method for Electromagnetics, Antennas, Microwave Circuits, and Scattering Applications (IEEE, 1998).

Chiou, Y.-P.

Y.-P. Chiou, W.-E. Yeh, and N.-Y. Shih, “Analysis of highly conducting lamellar gratings with multidomain pseudospectral method,”J. Lightwave Technol. 27, 5151–5159 (2009).
[Crossref]

Edee, K.

K. Edee, “Modal method based on sub-sectional Gegenbauer polynomial expansion for lamellar grating,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[Crossref]

Fenniche, I.

Foresti, M.

M. Foresti, L. Menez, and A. V. Tishchenko, “Modal method in deep metal-dielectric gratings: the decisive role of hidden modes,” J. Opt. Soc. Am. A 23, 2501–2509 (2006).
[Crossref]

Gaylord, T. K.

M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[Crossref]

Granet, G.

G. Granet, “Fourier-matching pseudospectral modal method for diffraction gratings: comment,” J. Opt. Soc. Am. A 29, 1843–1845 (2012).
[Crossref]

G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[Crossref]

Guizal, B.

Jorgensen, E.

Kempel, L. C.

J. L. Volakis, A. Chatterje, and L. C. Kempel, Finite Element Method for Electromagnetics, Antennas, Microwave Circuits, and Scattering Applications (IEEE, 1998).

Lalanne, P.

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[Crossref]

Li, L.

Lu, Y. Y.

D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudospectral modal method for diffraction gratings,” J. Opt. Soc. Am. A 28, 613–620 (2011).
[Crossref]

Meincke, P.

Menez, L.

M. Foresti, L. Menez, and A. V. Tishchenko, “Modal method in deep metal-dielectric gratings: the decisive role of hidden modes,” J. Opt. Soc. Am. A 23, 2501–2509 (2006).
[Crossref]

Moharam, M. G.

M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[Crossref]

Morf, R. H.

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[Crossref]

Morris, G. M.

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[Crossref]

S. Peng and G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[Crossref]

Peng, S.

S. Peng and G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[Crossref]

Polycarpou, A. C.

A. C. Polycarpou, Introduction to the Finite Element Method in Electromagnetics, Synthesis Lectures on Computational Electromagnetics (Morgan and Claypool, 2006).

Pozirikidis, C.

C. Pozirikidis, Introduction to Finite and Spectral Element Methods Using MATLAB (CRC Press, 2014).

Randriamihaja, M. H.

Raniriharinosy, K.

Shih, N.-Y.

Y.-P. Chiou, W.-E. Yeh, and N.-Y. Shih, “Analysis of highly conducting lamellar gratings with multidomain pseudospectral method,”J. Lightwave Technol. 27, 5151–5159 (2009).
[Crossref]

Song, D.

D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudospectral modal method for diffraction gratings,” J. Opt. Soc. Am. A 28, 613–620 (2011).
[Crossref]

Tishchenko, A. V.

M. Foresti, L. Menez, and A. V. Tishchenko, “Modal method in deep metal-dielectric gratings: the decisive role of hidden modes,” J. Opt. Soc. Am. A 23, 2501–2509 (2006).
[Crossref]

Volakis, J. L.

J. L. Volakis, A. Chatterje, and L. C. Kempel, Finite Element Method for Electromagnetics, Antennas, Microwave Circuits, and Scattering Applications (IEEE, 1998).

Yeh, W.-E.

Y.-P. Chiou, W.-E. Yeh, and N.-Y. Shih, “Analysis of highly conducting lamellar gratings with multidomain pseudospectral method,”J. Lightwave Technol. 27, 5151–5159 (2009).
[Crossref]

Yuan, L.

D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudospectral modal method for diffraction gratings,” J. Opt. Soc. Am. A 28, 613–620 (2011).
[Crossref]

IEEE Trans. Antennas Propag. (1)

J. Lightwave Technol. (1)

Y.-P. Chiou, W.-E. Yeh, and N.-Y. Shih, “Analysis of highly conducting lamellar gratings with multidomain pseudospectral method,”J. Lightwave Technol. 27, 5151–5159 (2009).
[Crossref]

J. Opt. Soc. Am. A (11)

D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudospectral modal method for diffraction gratings,” J. Opt. Soc. Am. A 28, 613–620 (2011).
[Crossref]

M. Foresti, L. Menez, and A. V. Tishchenko, “Modal method in deep metal-dielectric gratings: the decisive role of hidden modes,” J. Opt. Soc. Am. A 23, 2501–2509 (2006).
[Crossref]

G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[Crossref]

G. Granet, “Fourier-matching pseudospectral modal method for diffraction gratings: comment,” J. Opt. Soc. Am. A 29, 1843–1845 (2012).
[Crossref]

S. Peng and G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[Crossref]

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[Crossref]

M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[Crossref]

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[Crossref]

K. Edee, “Modal method based on sub-sectional Gegenbauer polynomial expansion for lamellar grating,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[Crossref]

Prog. Electromagn. Res. (1)

Other (4)

E. Popov, ed., Gratings: Theory and Numeric Applications (Institut Fresnel, CNRS, AMU, 2012).

A. C. Polycarpou, Introduction to the Finite Element Method in Electromagnetics, Synthesis Lectures on Computational Electromagnetics (Morgan and Claypool, 2006).

J. L. Volakis, A. Chatterje, and L. C. Kempel, Finite Element Method for Electromagnetics, Antennas, Microwave Circuits, and Scattering Applications (IEEE, 1998).

C. Pozirikidis, Introduction to Finite and Spectral Element Methods Using MATLAB (CRC Press, 2014).

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Fig. 1. Typical lamellar grating.

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Fig. 2. Illustration of the first four modified Legendre polynomials.

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Fig. 3. Example of assembly process of three elementary matrices for the solution to eigenvector Eq. (6) with pseudo-periodic boundary conditions. The cardinality of elements is 6, 7, and 5, respectively.

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Fig. 4. Example of assembly process of three elementary matrices for the derivationn of the

${E_x}$

electric field component given by Eq. (32).

${H_y}$

has 6, 7, and 5 spectral coefficient in elements

${\Omega ^1}$

${\Omega ^2}$

and

${\Omega ^3}$

, respectively, whereas

${E_x}$

has one less.

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Fig. 5. Zeroth transmitted order of grating 1 against the inverse of the truncation order computed with SEM and with PSFM. It is observed that both methods converge to the same value. The inset gives the error of

${T_0}$

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Fig. 6. Specular reflected order of grating 2 against the inverse of the truncation order computed with SEM and with PSFM. It is observed that both methods converge to the same value. The inset gives the error of

${R_0}$

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Fig. 7. Magnitude of

${E_x}$

at the upper surface of grating 1. The straight line fits the data between −4.5 and −3, which is close to the wedge. Its equation is

$y = (.3291 \times x - .48)$

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Tables (3)

Table 1. Parameters of Investigated Dielectric Metallic Gratings

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Table 2. Specular Reflected Efficiency for Lossless Grating 2 Computed with Different Modal Methods^a

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Table 3. Specular Reflected Efficiency for Lossless Grating 2^a

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Equations (50)

Equations on this page are rendered with MathJax. Learn more.

H_{yl} (x, z) = \sum_{n = 1}^{N} A_{ln}^{\pm} \exp (\mp i γ_{ln} z) ψ_{ln} (x),

E_{xl} (x, z) = \frac{1}{ω ε_{l} (x)} \sum_{n = 1}^{N} \pm γ_{l} A_{ln}^{\pm} \exp (\mp i γ_{ln} z) ψ_{ln} (x),

(\frac{d}{d x} \frac{1}{ε (x)} \frac{d}{d x} + k^{2}) ψ_{ln} (x) = γ_{ln}^{2} \frac{1}{ε (x)} ψ_{ln} (x),

\begin{aligned} ψ_{ln} (c^{-}) & = ψ_{ln} (c^{+}), \\ ψ_{ln} (x + d) & = τ ψ_{ln} (x), \\ p_{l}^{-} \frac{d ψ_{ln}}{d x} (c^{-}) & = p_{l}^{+} \frac{d ψ_{ln}}{d x} (c^{+}), \\ p_{l}^{+} \frac{d ψ_{ln}}{d x} (d) & = τ p_{l}^{-} \frac{d ψ_{ln}}{d x} (0), \end{aligned}

γ_{ln} = {\begin{array}{l} \sqrt{γ_{lq}^{2}} & i f γ_{lq}^{2} \in R^{+} \\ - i \sqrt{- γ_{lq}^{2}} & i f γ_{lq}^{2} \in R^{-} \\ \sqrt{γ_{lq}^{2}} & with negative imaginary part if γ_{lq}^{2} \in C \end{array} .

\begin{aligned} R_{q} & = \frac{γ_{1 q}}{γ_{10}} | A_{1 q}^{+} |^{2}, \\ T_{q} & = \frac{n_{1}^{2}}{n_{3}^{2}} \frac{γ_{3 q}}{γ_{10}} | A_{3 q}^{-} |^{2} \end{aligned} .

\begin{aligned} \int_{Ω} ϕ (x) \frac{d}{d x} \frac{1}{ε (x)} \frac{d}{d x} ψ (x) d x + k^{2} \int_{Ω} ϕ (x) ψ (x) d x \\ = γ^{2} \int_{Ω} ϕ (x) \frac{1}{ε (x)} ψ (x) d x . \end{aligned}

\begin{aligned} \int_{Ω^{j}} ϕ (x) \frac{d}{d x} \frac{1}{ε (x)} \frac{d}{d x} ψ (x) d x = I_{j + 1} - I_{j} \\ - \int_{Ω^{j}} \frac{d}{d x} ϕ (x) \frac{1}{ε (x)} \frac{d}{d x} ψ (x) d x, \end{aligned}

\begin{aligned} I_{j + 1} & = ϕ (x_{j + 1}) \frac{1}{ε (x_{j + 1})} {\frac{d}{d x} ψ (x) |}_{x = x_{j + 1}}, \\ I_{j} & = ϕ (x_{j}) \frac{1}{ε (x_{j})} {\frac{d}{d x} ψ (x) |}_{x = x_{j}} . \end{aligned}

\sum_{q = 1}^{Q} U^{q} + V^{q} + I_{N} - I_{0} = \sum_{q = 1}^{Q} W^{q},

U^{q} = \int_{x_{q - 1}}^{x_{q}} \frac{d}{d x} ϕ (x) \frac{1}{ε (x)} \frac{d}{d x} ψ (x) d x,

V^{q} = - \int_{x_{q - 1}}^{x_{q}} ϕ (x) ψ (x) d x,

W^{q} = \int_{x_{q - 1}}^{x_{q}} ϕ (x) \frac{1}{ε (x)} ψ (x) d x .

\begin{aligned} H_{y}^{j} (x) & = \sum_{n = 1}^{N^{j} + 1} H_{y, n}^{j} B_{n} (u), \\ E_{x}^{j} (x) & = \sum_{n = 1}^{N^{j}} E_{x, n}^{j} B_{n} (u) . \end{aligned}

B_{m} (u) = {\begin{array}{l} 0.5 (L_{0} - L_{1}), & m = 1 \\ 0.5 (L_{0} + L_{1}), & m = 2 \\ L_{m} (u) - L_{m - 2} (u), & m >= 2 \end{array},

u = \frac{2}{Δ_{l}} (x - Δ_{l}), Δ_{l} = x_{l} - x_{l + 1}

P^{q} = [\begin{matrix} .5 & - 1 & 0 & \dots & 0 & .5 \\ - .5 & 0 & - 1 & ⋮ & 0 & .5 \\ 0 & 1 & 0 & ⋱ & 0 & 0 \\ 0 & 0 & 1 & ⋱ & - 1 & 0 \\ 0 & 0 & 0 & ⋱ & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}] .

\begin{aligned} U_{mn}^{q} & = \frac{2}{Δ_{j}} \int_{- 1}^{1} \frac{d B_{m}}{d u} \frac{1}{ε (u)} \frac{d B_{n}}{d u} d u, \\ V_{mn}^{q} & = \frac{Δ_{j}}{2} \int_{- 1}^{1} B_{m} B_{n} d u, \\ W_{mn}^{q} & = \frac{Δ_{j}}{2} \int_{- 1}^{1} B_{m} \frac{1}{ε (u)} B_{n} d u . \end{aligned}

\int_{- 1}^{1} L_{m} L_{n} d u = \frac{2}{2 m + 1} δ_{mn},

V^{q} = \frac{Δ_{q}}{2} (P^{q})^{T} D^{q} P^{q}, W^{q} = \frac{1}{ε^{q}} V^{q},

\frac{d L_{m}}{d u} = \sum_{n = 0}^{N} D_{rmn} L_{n} (u) .

{\dot{P}}^{q} = D r^{q} P^{q} .

U^{q} = \frac{Δ_{q}}{ε^{q}} ({\dot{P}}^{q})^{T} D^{q} {\dot{P}}^{q} .

n = p + \sum_{q = 1}^{Q} (N^{q} + 1) - (Q - 1) .

\begin{aligned} X_{N^{p} + 1, 1}^{p}, X_{N^{p} + 1, 2}^{p}, \dots, \\ (X_{N^{p} + 1, N^{p} + 1}^{p} + X_{11}^{p}), \\ \dots, X_{1, 2}^{p + 1}, X_{1, 3}^{p + 1}, X_{1, N^{p + 1} + 1}^{p + 1}, \end{aligned}

H_{y} (x + d) = τ H_{y} (x), τ = \exp (- i α_{0} d) .

\begin{aligned} F_{11} & = X_{11}^{1} + X_{N^{Q} + 1 N^{Q} + 1}^{Q} \\ (F_{1 N - N^{Q}}, F_{1, N - N^{Q} + 1}, \dots, F_{1, N}) \\ = τ^{*} (X_{N^{Q} 1}^{Q}, X_{N^{Q} 2}^{Q}, \dots, X_{N^{Q} N^{Q}}^{Q}), \\ (F_{N - N^{Q} 1}, F_{N - N^{Q} + 11}, \dots, F_{N 1}) \\ = τ (X_{1 N^{Q}}^{Q}, X_{2 N^{Q}}^{Q}, \dots, X_{N^{Q} N^{Q}}^{Q}) . \end{aligned}

(U + V) H_{y} = γ^{2} W H_{y} .

\sum_{n = 1}^{N^{q}} E_{xn}^{q} B_{n} (u) = \frac{γ}{ω ε^{q}} \sum_{n = 1}^{N^{q} + 1} H_{yn}^{q} B_{n} (u) .

\begin{aligned} \sum_{n = 1}^{N^{q}} E_{xn}^{q} \int_{- 1}^{1} B_{m} (u) B_{n} (u) d u \\ = \frac{γ}{ω ε^{q}} \sum_{n = 0}^{N^{q}} H_{yn}^{j} \int_{- 1}^{1} B_{m} (u) B_{n} (u) d u . \end{aligned}

{E_{x}}^{q} = Z^{q} {H_{y}}^{q},

{E_{x}}^{q} = {[E_{x 1}^{N^{q}}, E_{x 2}^{N^{q}}, \dots, E_{x N^{q}}^{N^{q}}]}^{T},

Z^{q} = \frac{1}{ω ε^{q}} {({({\tilde{P}}^{q})}^{T} {\tilde{D}}^{q} {\tilde{P}}^{q})}^{- 1} ({({\tilde{P}}^{q})}^{T} D^{q} D^{q}) .

\exp (- i α_{q} x) = \sum_{m = 1}^{N^{p} + 1} {\tilde{ψ}}_{mq}^{p} L_{m} (u (x)) x_{p - 1} \leq x \leq x_{p} .

\int_{- 1}^{1} L_{n} (x) \exp (i α x) d x = i^{n} \sqrt{\frac{2 π}{α}} J_{n + 1 / 2} (α)

{\tilde{ψ}}_{mq}^{p} = (- i)^{m} \frac{2 m + 1}{2} \sqrt{\frac{2 π}{ζ_{q}}} J_{m + 1 / 2} (ζ_{q}^{p}) \exp (- i ζ_{q}^{p}),

ζ_{q}^{p} = \frac{α_{q} Δ_{p}}{2} .

\exp (- i α_{q} x) = \sum_{m = 1}^{N^{p} + 1} {\tilde{ψ}}_{mq}^{p} B_{m} (u (x)) x_{p - 1} \leq x \leq x_{p}

ψ_{q}^{p} = (P^{p})^{- 1} {\tilde{ψ}}_{q}^{p} ψ_{q}^{p} = [\begin{matrix} ψ_{1 q}^{p} \\ ⋮ \\ ψ_{N^{q}}^{p} \end{matrix}] {\tilde{ψ}}_{q}^{p} = [\begin{matrix} {\tilde{ψ}}_{1 q}^{p} \\ ⋮ \\ {\tilde{ψ}}_{N^{q}}^{p} \end{matrix}] .

E r r = \log (| T_{0} - T_{0 e x a c t} |) .

E_{x} = A \times (x - c)^{τ - 1},

τ = \frac{2}{π} a r c s i n {(1 - \frac{1}{4} \frac{n_{22}^{2} - n_{1}^{2}}{n_{22}^{2} + n_{1}^{2}})}^{1 / 2} .

\begin{aligned} F (u) & = \sum_{n = 0}^{N} F_{n} L_{n} (u), \\ \frac{d}{d u} F (u) & = G (u) = \sum_{m = 0}^{N} G_{m} L_{m} (u) . \end{aligned}

G_{m} = \sum_{n = 0}^{N} D_{rmn} F_{n} .

\sum_{n = 0}^{N} F_{n} \frac{d}{d u} L_{n} (u) = \sum_{n = 0}^{N} G_{n} L_{n} (u) .

\begin{aligned} \sum_{n = 0}^{N} F_{n} \int_{- 1}^{1} L_{m} (u) \frac{d}{d u} L_{n} (u) d u \\ = \sum_{n = 0}^{N} G_{n} \int_{- 1}^{1} L_{m} (u) L_{n} (u) d u m, n = 0, \dots N . \end{aligned}

\int_{- 1}^{1} L_{m} (u) \frac{d}{d u} L_{n} (u) d u = {\begin{array}{l} 2 & i f n - m > 0 a n d m - n o d d \\ 0 & o t h e r w i s e \end{array}

\int_{- 1}^{1} L_{m} (u) L_{n} (u) d u = \frac{2}{2 m + 1} δ_{mn},

D_{rmn} = \frac{2 m + 1}{2} d_{mn},

d_{mn} = {\begin{array}{l} (1 - {(- 1)}^{n - m}), & m, n \in {0, 1, \dots N}, \\ n - m o d d, n > m \\ 0 & o t h e r w i s e \end{array} .

Parameter	Grating 1	Grating 2
$θ$	0	35
$λ$	0.55	0.45
$c$	$0.3 \times 0.25$	0.5
$d$	0.25	0.2
$h$	0.2	0.521
$n_{3}$	1.5	1.5
$n_{21}$	1	1
$n_{22}$	3.18−4.41j	$- \sqrt{25} j$

TN	SEM	PSMM [14]	AMM [16]
33	1.2955593e-01	1.3059319 (−1)	1.2544877 (−1)
65	1.2812373e-01	1.2835611 (−1)	1.2694599 (−1)
129	1.2779410e-01	1.2787523 (−1)	1.2745205 (−1)
257	1.2771968e-01	1.2771968 (−1)	1.2761832 (−1)
513	1.2770294e-01	1.2769345 (−1)	1.2767238 (−1)
1025	1.2769926e-01	1.2768770 (−1)	1.2768988 (−1)

TN	SEM	PSMM	ASR-FMM
33	1.2955593 (−1)	1.30801852 (−1)	1.2357814 (−1)
65	1.2812373 (−1)	1.2844758 (−1)	1.2776491 (−1)
129	1.2779410 (−1)	1.2787316 (−1)	1.2772640 (−1)
257	1.2771968 (−1)	1.2773789 (−1)	1.2769891 (−1)
513	1.277029 (−1)	1.2770631 (−1)	1.2769842 (−1)
1025	1.2769926 (−1)	1.2770043 (−1)	1.2769835 (−1)
1200	1.27698951 (−1)	1.2769961 (−1)	1.2769824 (−1)
1500	1.2769867 (−1)	1.2769905 (−1)	1.2769812 (−1)
1800	1.2769851 (−1)	1.2769879 (−1)	1.2769809 (−1)
2049	1.2769844 (−1)	1.2769864 (−1)	1.2769814 (−1)
2401	1.2769838 (−1)	1.2769854 (−1)	1.2769821 (−1)
3001	1.2769831 (−1)	1.2769846 (−1)	1.2769821 (−1)
4097	1.2769826 (−1)	1.2769839 (−1)	1.2769817 (−1)

Parameter	Grating 1	Grating 2
$θ$	0	35
$λ$	0.55	0.45
$c$	$0.3 \times 0.25$	0.5
$d$	0.25	0.2
$h$	0.2	0.521
$n_{3}$	1.5	1.5
$n_{21}$	1	1
$n_{22}$	3.18−4.41j	$- \sqrt{25} j$

TN	SEM	PSMM [14]	AMM [16]
33	1.2955593e-01	1.3059319 (−1)	1.2544877 (−1)
65	1.2812373e-01	1.2835611 (−1)	1.2694599 (−1)
129	1.2779410e-01	1.2787523 (−1)	1.2745205 (−1)
257	1.2771968e-01	1.2771968 (−1)	1.2761832 (−1)
513	1.2770294e-01	1.2769345 (−1)	1.2767238 (−1)
1025	1.2769926e-01	1.2768770 (−1)	1.2768988 (−1)

Abstract

References

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