## Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects |

Optics Express, Vol. 19, Issue 25, pp. 25799-25811 (2011)

http://dx.doi.org/10.1364/OE.19.025799

Acrobat PDF (1392 KB)

### Abstract

This paper proposes a spectral-domain approach to the electromagnetic scattering problem of lamellar grating with defects. The fields in imperfectly periodic structures have continuous spectra in the wavenumber space, and the main problem of the spectral-domain approach is connected to the discretization scheme on the wavenumber. The present approach introduces the pseudo-periodic Fourier transform to consider the discretization scheme in the Brillouin zone. This transformation also makes it possible to apply the conventional grating formulations to the problems of imperfectly periodic structures. The present formulation is based on the rigorous coupled-wave analysis with the help of pseudo-periodic Fourier transform.

© 2011 OSA

## 1. Introduction

1. O. Manyardo, R. Michaely, F. Schädelin, W. Noell, T. Overstolz, N. De Rooij, and H. P. Herzig, “Miniature lamellar grating interferometer based on silicon technology,” Opt. Lett. **29**, 1437–1439 (2004). [CrossRef]

2. Y. Hongbin, Z. Guangya, Ch. F. Siong, L. Feiwen, W. Shouhua, and Z. Mingsheng, “An electromagnetically driven lamellar grating based Fourier transform microspectrometer,” J. Micromech. Microeng. **18**, 055016 (2008). [CrossRef]

3. A. Sato, “Analysis of finite-sized guided-mode resonant gratings using the fast multipole boundary element method,” J. Opt. Soc. Am. A **27**, 1909–1919 (2010). [CrossRef]

4. F. Marquier, C. Arnold, M. Laroche, J. J. Greffet, and Y. Chen, “Degree of polarization of thermal light emitted by gratings supporting surface waves,” Opt. Express **16**, 5305–5313 (2008). [CrossRef] [PubMed]

5. N. Bonod, G. Tayeb, D. Maystre, S. Enoch, and E. Popov, “Total absorption of light by lamellar metallic gratings,” Opt. Express **16**, 15431–15438 (2008). [CrossRef] [PubMed]

6. W. T. Lu, Y. J. Huang, P. Vodo, R. K. Banyal, C. H. Perry, and S. Sridhar, “A new mechanism for negative refraction and focusing using selective diffraction from surface corrugation,” Opt. Express **15**, 9166–9175 (2007). [CrossRef] [PubMed]

7. L. Pajewski, R. Borghi, G. Schettini, F. Frezza, and M. Santarsiero, “Design of a binary grating with subwavelength features that acts as a polarizing beam splitter,” Appl. Opt. **40**, 5898–5905 (2001). [CrossRef]

8. T. Käpfe and O. Parriaux, “Parameter-tolerant binary gratings,” J. Opt. Soc. Am. A **27**, 2660–2669 (2010). [CrossRef]

9. T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys. **46**, 5435–5440 (2007). [CrossRef]

10. R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys. **97**, 053107 (2005). [CrossRef]

2. Y. Hongbin, Z. Guangya, Ch. F. Siong, L. Feiwen, W. Shouhua, and Z. Mingsheng, “An electromagnetically driven lamellar grating based Fourier transform microspectrometer,” J. Micromech. Microeng. **18**, 055016 (2008). [CrossRef]

11. W. Nakagawa, P.-Ch. Sun, Ch-H. Chen, and Y. Fainman, “Wide-field-of-view narrow-band spectral filters based on photonic crystal nanocavities,” Opt. Lett. **27**, 191–193 (2002). [CrossRef]

13. K. Ren, X. Ren, R. Li, J. Zhou, and D. Liu, “Creating “defects” in photonic crystals by controlling polarizations,” Phys. Lett. **325**, 415–419 (2004). [CrossRef]

15. K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Progress in Electromagnetic Res. **PIER 74**, 241–271 (2007). [CrossRef]

*f*(

*x*) be a function of

*x*, and

*d*be a positive real constant. Then the transform is defined by: which is implicitly assumed to converge. This transform introduces a transform parameter

*ξ*, and the inverse transform is formally given by integrating on

*ξ*as where

*k*= 2

_{d}*π*/

*d*. The transformed function

*f̄*(

*x*;

*ξ*) has a pseudo-periodic property with the pseudo-period

*d*in terms of

*x*:

*f̄*(

*x*–

*md*;

*ξ*) =

*f̄*(

*x*;

*ξ*) exp(−

*imdξ*) for any integer

*m*. Also,

*f̄*(

*x*;

*ξ*) has a periodic property with the period

*k*in terms of

_{d}*ξ*:

*f̄*(

*x*;

*ξ*–

*mk*) =

_{d}*f̄*(

*x*;

*ξ*) for any integer

*m*. PPFT is an extension of the periodic Green function [16], which is defined by the radiation field from periodic line-source array with phase shift. The transform parameter

*ξ*relates to the wavenumber when

*x*is the spatial parameter. If the constant

*d*is chosen to be equal with the fundamental period in the

*x*-direction,

*k*becomes the inverse lattice constant and the period for the transformed function is given by the Brillouin zone. The Wood anomalies are degenerated to a finite number of points in the Brillouin zone, and the required discretization scheme is comparatively easy to consider. Also, the conventional grating theory based on the Floquet theorem becomes possible to be applied for the scattering problem of imperfectly periodic structures because of the pseudo-periodicity of the transformed fields. The present formulation is based on the rigorous coupled-wave analysis (RCWA) [17

_{d}17. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. **68**, 1206–1210 (1978). [CrossRef]

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

## 2. Settings of the problem

*z*-axis, and the direction of original periodicity is parallel to the

*x*-axis. The

*y*-axis is taken so as to be at the center of a ridge, and the top and the bottom of grooves are denoted by

*y*=

*h*and

_{s}*y*=

*h*, respectively. In the grating layer

_{c}*h*<

_{c}*y*<

*h*, the structure is uniform in the

_{s}*y*- and the

*z*-directions, and the grating profile is specified by the period

*d*, the ridge width

*a*, and the groove depth

*t*=

*h*–

_{s}*h*. The structural periodicity is collapsed by removing some ridges. To indicate the removed ridges, we introduce a notation 𝒟, which is a finite subset of integers. If an integer

_{c}*n*is an element of 𝒟, the ridge whose center is at

*x*=

*nd*is removed. The fields are supposed to be also uniform in the

*z*-direction and have a time-dependence in exp(−

*iωt*). Then the fields are represented by complex vectors depending only on the space variables

*x*and

*y*and two fundamental polarizations are expressed by TE and TM, in which the electric and magnetic fields are respectively perpendicular to the

*xy*-plane. We sometimes denote the

*z*-component of the electric field for TE-polarization and the

*z*-component of the magnetic field for TM-polarization by

*ψ*(

*x,y*) to express both polarizations simultaneously. The substrate is linear isotropic medium with the permittivity

*ɛ*and the permeability

_{c}*μ*, and the surrounding medium is denoted by the permittivity

_{c}*ɛ*and the permeability

_{s}*μ*. For lossy media, we use complex values for the permittivity and/or the permeability, and the terms concerning to the current densities are eliminated in the following formulation. The permittivity distribution

_{s}*ɛ*(

*x,y*) and the permeability distribution

*μ*(

*x,y*) are independent of

*y*inside the grating layer

*h*<

_{c}*y*<

*h*and, here, we express them and their reciprocals as follows: for

_{s}*η*=

*ɛ*,

*μ*, where The functions

*h*<

_{c}*y*<

*h*.

_{s}## 3. Formulation

*h*<

_{c}*y*<

*h*, which are described by Equations (10)–(12) come from the Maxwell curl equations, and the constitutive relations yield Eqs. (13)–(15) though they are arranged so as not to include products of two functions with concurrent discontinuities. We apply PPFT to Eqs. (10)–(15), and we may obtain the following relations: Since the transformed fields are pseudo-periodic in terms of

_{s}*x*, they can be approximately expanded in the truncated generalized Fourier series. For example, the

*z*-component of electric field is written as with where

*N*denotes the truncation order and

*Ē*(

_{z,n}*ξ*,

*y*) are the

*n*th-order coefficients. To treat the coefficients systematically, we introduce (2

*N*+ 1) × 1 column matrices; for example, the coefficients of

*Ē*(

_{z}*x*;

*ξ*,

*y*) are expressed by a column matrix

*ē**(*

_{z}*ξ*,

*y*) in such a way that its

*n*th-components are given by

*Ē*(

_{z,n}*ξ*,

*y*). Also, supposing that

*f*(

*x*) is a periodic function with period

*d*, [[

*f*]] denotes the Toeplitz matrix generated by the Fourier coefficients of

*f*(

*x*), in such a way that their (

*n,m*)-entries are given by Similarly, if

*ḡ*(

*x*;

*ξ*) is a transformed function that is pseudo-periodic in terms of

*x*, [[

*ḡ*]](

*ξ*) denotes the Toeplitz matrix generated by the generalized Fourier coefficients of

*ḡ*(

*x*;

*ξ*), in such a way that their (

*n,m*)-entries are Then Eqs. (16)–(21) yield the following relations: where

**(**

*X̄**ξ*) denotes a diagonal matrix whose

*n*th-diagonal components are

*α*(

_{n}*ξ*). Since the right-hand sides of Eqs. (19)–(21) consist of the products of periodic or pseudo-periodic functions with no concurrent jump discontinuities, the Laurent rule [19

19. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A **13**, 1870–1876 (1996). [CrossRef]

*ξ*∈ (−

*k*/2,

_{d}*k*/2]. However, we introduce here a discretization in the transform parameter

_{d}*ξ*for numerical purposes. We take

*L*sample points, and suppose that the coefficients satisfy Eqs. (26)–(31) at these sample points. Also the integrations in Eqs. (29)–(31) are approximated by an appropriate numerical integration scheme using the same sample points. Let

*ẽ**(*

_{z}*y*) denotes a column matrix generated by the generalized Fourier coefficients of

*Ē*(

_{z}*x*;

*ξ*,

*y*) and the coefficients of the other field components are expressed in the same way. The matrices in the equations are defined as follows: for

*υ*=

*ɛ*,

*μ*, where

*δ*

_{l,l}_{′}denotes the Kronecker delta.

*n*th-eigenvalue and the associated eigenvector of

*ẽ**(*

_{z}*y*) and

*h̃**(*

_{x}*y*) are written in the following form: with Two

*L*(2

*N*+ 1) × 1 column matrices

*y*-directions, respectively. We include the

*y*-dependence of each amplitude, and their

*y*-dependences are expressed by with for

*h*<

_{c}*y*,

*y*′ <

*h*.

_{s}*y*>

*h*and

_{s}*y*<

*h*), the permittivity and the permeability are constant, and the electromagnetic fields can be approximately expressed by the truncated Rayleigh expansions [16]. Each term of this expansion represents a plane wave, and we include the exponential

_{c}*y*-dependence of each term in the Rayleigh coefficient. Here, the regions

*y*>

*h*and

_{s}*y*<

*h*are specified by

_{c}*s*and

*c*, respectively, and the wavenumber in each region is denoted by

*k*=

_{r}*ω*(

*ɛ*

_{r}*μ*)

_{r}^{1/2}for

*r*=

*s,c*. The relations between the generalized Fourier coefficients of the field components and the amplitudes of the plane-waves can be written in the following forms: with for

*r*=

*s,c*, where

**denotes the identity matrix.**

*I**H̄*(

_{z}*x*;

*ξy*) and

*Ē*(

_{x}*x*;

*ξy*) are expressed in the same forms with Eqs. (46) and (52), though we specify below the matrices by the superscript

*h*for the TM-polarization instead of those with superscript

*e*for the TE-polarization. The matrices

## 4. Numerical experiments

*z*-axis at (

*x,y*) = (

*x*

_{0},

*y*

_{0}) where

*y*

_{0}>

*h*. The incident field is then expressed as where

_{s}*ρ*(

*x,y*) = (

*x*

^{2}+

*y*

^{2})

^{1/2}. The grating parameters are chosen as follows:

*d*= 0.6

*λ*

_{0},

*a*= 0.6

*d*,

*h*= 0,

_{s}*h*= −0.5

_{c}*d*,

*ɛ*=

_{s}*ɛ*

_{0},

*ɛ*= (1.3 +

_{c}*i*7.6)

^{2}

*ɛ*

_{0}, and

*μ*=

_{s}*μ*=

_{c}*μ*

_{0}. We also set the position of line-source at (

*x, y*) = (0,2

*d*) and the observation point for convergence tests at (

*x,y*) = (0

*,d*).

*x*= 0 is removed from the perfectly periodic grating. Figure 2(a) shows the obtained results of the field intensity at the observation point as a function of the number of sample points

*L*. The truncation order for each generalized Fourier series expansion is set to

*N*= 5 for this computation. The dotted curves are the results of the trapezoidal scheme that uses equidistant sample points and equal weights. The trapezoidal scheme is known to usually provide a fast convergence for the integration of smooth periodic function over one period, but they converge very slowly. This implies that the spectra under consideration may be non-smooth. The fields in perfectly periodic structures are known to be non-smooth at the Wood-Rayleigh anomalies in the wavenumber space. We expect the spectra of fields in imperfectly periodic structure have a similar singularity to those for the periodic structure without defect, and apply here the same discretization scheme proposed for the perfectly periodic grating with the line-source excitation in Ref. [15

15. K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Progress in Electromagnetic Res. **PIER 74**, 241–271 (2007). [CrossRef]

*ξ*has to satisfy

*α*(

_{n}*ξ*) = ±

*k*,

_{s}*k*at the anomalies. Since we consider here a lossy material for the substrate, the anomalies degenerate at two points

_{c}*ξ*= ±0.4

*k*in the Brillouin zone. We split the Brillouin zone at these points and apply the commonly used numerical integration schemes for each subinterval. This discretization scheme is also valid for another imperfectly periodic structure, in which some circular cylinders are located near the periodic cylinder array [20]. The solid curves in Fig. 2(a) are the result of the double exponential scheme [21

_{d}21. H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. **9**, 721–741 (1974). [CrossRef]

*N*are shown in Fig. 2(b). The values are computed for

*L*= 80 and the sample points and weights are determined by the double exponential scheme applying for the subintervals. It is observed that the convergences are fast. Figure 3 shows the field intensities computed with

*L*= 80 and

*N*= 5 by changing the observation point. The position of grating surface is indicated by the dashed line. The fields decay rapidly inside the substrate because the substrate is assumed to be a conducting material. The field for TE-polarization does not propagate into the narrow grooves because of the cutoff. Therefore, the horizontal interference fringes observed in Fig. 3(a) are similar to those for the plane conducting plate though the phase shift is also observed near the defect. For the TM-polarization, the fields can propagate into the narrow grooves and the vertical fringes are stronger than these for the TE-polarization.

*L*= 80 and

*N*= 5 are shown in Fig. 5. The obtained results seems to be proper. We examine the reciprocal property to verify the present formulation. Here, we define the reciprocity error for two points (

*x*,

_{p}*y*) and (

_{p}*x*,

_{q}*y*) by where

_{q}*ψ*(

*x*,

_{p}*y*;

_{p}*x*,

_{q}*y*) denotes the field observed at (

_{q}*x*,

_{p}*y*) for a line-source located at (

_{p}*x*,

_{q}*y*). The reciprocity theorem requires that this function is zero when both (

_{q}*x*,

_{p}*y*) and (

_{p}*x*,

_{q}*y*) are located in the surrounding medium. We fix one point at (

_{q}*x*,

_{p}*y*) = (0, 2

_{p}*d*) and the other point (

*x*,

_{q}*y*) is moved on the line

_{q}*y*=

*d*. The reciprocity errors at 101 equidistant points in −7

*d*≤

*x*≤ 7

*d*are calculated with

*L*= 80 and

*N*= 5 in the standard double-precision arithmetic, and Fig. 6 shows the obtained results. The largest values are about 2.1 × 10

^{−8}for the TE-polarization and 3.7 × 10

^{−8}for the TM-polarization, and the reciprocity relation is well satisfied.

## 5. Concluding remarks

## Acknowledgments

## References and links

1. | O. Manyardo, R. Michaely, F. Schädelin, W. Noell, T. Overstolz, N. De Rooij, and H. P. Herzig, “Miniature lamellar grating interferometer based on silicon technology,” Opt. Lett. |

2. | Y. Hongbin, Z. Guangya, Ch. F. Siong, L. Feiwen, W. Shouhua, and Z. Mingsheng, “An electromagnetically driven lamellar grating based Fourier transform microspectrometer,” J. Micromech. Microeng. |

3. | A. Sato, “Analysis of finite-sized guided-mode resonant gratings using the fast multipole boundary element method,” J. Opt. Soc. Am. A |

4. | F. Marquier, C. Arnold, M. Laroche, J. J. Greffet, and Y. Chen, “Degree of polarization of thermal light emitted by gratings supporting surface waves,” Opt. Express |

5. | N. Bonod, G. Tayeb, D. Maystre, S. Enoch, and E. Popov, “Total absorption of light by lamellar metallic gratings,” Opt. Express |

6. | W. T. Lu, Y. J. Huang, P. Vodo, R. K. Banyal, C. H. Perry, and S. Sridhar, “A new mechanism for negative refraction and focusing using selective diffraction from surface corrugation,” Opt. Express |

7. | L. Pajewski, R. Borghi, G. Schettini, F. Frezza, and M. Santarsiero, “Design of a binary grating with subwavelength features that acts as a polarizing beam splitter,” Appl. Opt. |

8. | T. Käpfe and O. Parriaux, “Parameter-tolerant binary gratings,” J. Opt. Soc. Am. A |

9. | T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys. |

10. | R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys. |

11. | W. Nakagawa, P.-Ch. Sun, Ch-H. Chen, and Y. Fainman, “Wide-field-of-view narrow-band spectral filters based on photonic crystal nanocavities,” Opt. Lett. |

12. | B. G. Zhai, Y. G. Cai, and Y. M. Huang, “Transmission spectra of one-dimensional photonic crystal with a centered defect,” Mat. Sci. For. |

13. | K. Ren, X. Ren, R. Li, J. Zhou, and D. Liu, “Creating “defects” in photonic crystals by controlling polarizations,” Phys. Lett. |

14. | E. G. Loewen and E. Popov, |

15. | K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Progress in Electromagnetic Res. |

16. | R. Petit, ed., |

17. | K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. |

18. | M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. |

19. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A |

20. | K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B |

21. | H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. |

22. | P. J. Davis and P. Rabinowitz, |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: September 2, 2011

Revised Manuscript: October 7, 2011

Manuscript Accepted: October 10, 2011

Published: December 2, 2011

**Citation**

Koki Watanabe, Jaromír Pištora, and Yoshimasa Nakatake, "Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects," Opt. Express **19**, 25799-25811 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25799

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### References

- O. Manyardo, R. Michaely, F. Schädelin, W. Noell, T. Overstolz, N. De Rooij, and H. P. Herzig, “Miniature lamellar grating interferometer based on silicon technology,” Opt. Lett.29, 1437–1439 (2004). [CrossRef]
- Y. Hongbin, Z. Guangya, Ch. F. Siong, L. Feiwen, W. Shouhua, and Z. Mingsheng, “An electromagnetically driven lamellar grating based Fourier transform microspectrometer,” J. Micromech. Microeng.18, 055016 (2008). [CrossRef]
- A. Sato, “Analysis of finite-sized guided-mode resonant gratings using the fast multipole boundary element method,” J. Opt. Soc. Am. A27, 1909–1919 (2010). [CrossRef]
- F. Marquier, C. Arnold, M. Laroche, J. J. Greffet, and Y. Chen, “Degree of polarization of thermal light emitted by gratings supporting surface waves,” Opt. Express16, 5305–5313 (2008). [CrossRef] [PubMed]
- N. Bonod, G. Tayeb, D. Maystre, S. Enoch, and E. Popov, “Total absorption of light by lamellar metallic gratings,” Opt. Express16, 15431–15438 (2008). [CrossRef] [PubMed]
- W. T. Lu, Y. J. Huang, P. Vodo, R. K. Banyal, C. H. Perry, and S. Sridhar, “A new mechanism for negative refraction and focusing using selective diffraction from surface corrugation,” Opt. Express15, 9166–9175 (2007). [CrossRef] [PubMed]
- L. Pajewski, R. Borghi, G. Schettini, F. Frezza, and M. Santarsiero, “Design of a binary grating with subwavelength features that acts as a polarizing beam splitter,” Appl. Opt.40, 5898–5905 (2001). [CrossRef]
- T. Käpfe and O. Parriaux, “Parameter-tolerant binary gratings,” J. Opt. Soc. Am. A27, 2660–2669 (2010). [CrossRef]
- T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys.46, 5435–5440 (2007). [CrossRef]
- R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys.97, 053107 (2005). [CrossRef]
- W. Nakagawa, P.-Ch. Sun, Ch-H. Chen, and Y. Fainman, “Wide-field-of-view narrow-band spectral filters based on photonic crystal nanocavities,” Opt. Lett.27, 191–193 (2002). [CrossRef]
- B. G. Zhai, Y. G. Cai, and Y. M. Huang, “Transmission spectra of one-dimensional photonic crystal with a centered defect,” Mat. Sci. For.663–665, 733–736 (2010).
- K. Ren, X. Ren, R. Li, J. Zhou, and D. Liu, “Creating “defects” in photonic crystals by controlling polarizations,” Phys. Lett.325, 415–419 (2004). [CrossRef]
- E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).
- K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Progress in Electromagnetic Res.PIER 74, 241–271 (2007). [CrossRef]
- R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
- K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am.68, 1206–1210 (1978). [CrossRef]
- M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am.72, 1385–1392 (1982). [CrossRef]
- L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A13, 1870–1876 (1996). [CrossRef]
- K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B31, 219–237 (2011).
- H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ.9, 721–741 (1974). [CrossRef]
- P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic Press, New York, 1984).

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