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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 9 — Apr. 25, 2011
  • pp: 8233–8241
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Triple-layer guided-mode resonance Brewster filter consisting of a homogenous layer and coupled gratings with equal refractive index

Xin Liu, Shuqi Chen, Weiping Zang, and Jianguo Tian  »View Author Affiliations


Optics Express, Vol. 19, Issue 9, pp. 8233-8241 (2011)
http://dx.doi.org/10.1364/OE.19.008233


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Abstract

A triple-layer guided-mode resonance Brewster filter consisting of a homogeneous layer and two identical gratings with their refractive indices equal to that of the homogeneous layer is presented. The spectral properties of this filter are analyzed based on the coupling modulation of two identical binary gratings at Brewster angle for a TM-polarized wave. The grating layer between substrate and homogeneous layers can significantly change the linewidth and resonant mode position, which are due to the asymmetric field distribution inside the grating layers. The tunability of the resonance can be altered on different resonant channels and a practical filter can be obtained in TM2 waveguide mode. Variation of filling factor can alter the field localization in the grating structure and significantly adjust the linewidth of the filter.

© 2011 OSA

1. Introduction

2. Structure and theory

A schematic diagram of the triple-layer waveguide grating structure under TM polarization light at oblique incident angle is depicted in Fig. 1
Fig. 1 (Color online) Schematic diagram of the GMR Brewster filter. The high and low refractive indices of the identical gratings are nH = 2.25 and nL = 1.8, respectively. The filling factor of the grating is set to F = 0.5. The refractive indices of the cover, substrate and homogeneous layers are set to nc = 1.0, ns = 1.46 and nu = 1.99, respectively. The thickness of the triple-layer structure is dg 1 = dg 2 = 85.6 nm, du = 30 nm. The lateral alignment shift is denoted by S.
. The resonant part above the substrate layer consists of two identical grating layers and a homogeneous layer between the gratings. According to the effective media theory (EMT) [25

25. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

], the second order of effective refractive index of the grating layer under TM illumination for Λ>λ0 can be written in the following form
εeff,TM(2)=εeff,TM(0)+π23F2(1F)2(1εH1εL)2(εeff,TM(0))3εeff,TE(0)(Λλ0)2,
(1)
where Λ and λ0 are the grating period and the central resonant wavelength of the structure, respectively. F is the grating filling factor and εH and εL are high and low permittivity of the grating materials, respectively. The zero-order permittivities under TE and TM-polarization conditions in Eq. (1) are given by

εeff,TE(0)=FεH+(1F)εLεeff,TM(0)=εHεL/[FεL+(1F)εH].
(2)

For a high-spatial frequency waveguide grating (Λ/λ00), the second term of the polynomial expression on right side of Eq. (1) is neglectable and the expression of the effective index of the grating under TM-polarization can be approximately reduced as

neff,TM={εHεL/[FεL+(1F)εH]}1/2.
(3)

For a fixed filling factor, the triple-layer resonant Brewster filter can be obtained through appropriate choosing of the homogeneous layer’s refractive index, which should be equal to the effective refractive indices of gratings calculated by Eq. (3).

In Fourier space, the basic equations of differential theory of gratings can be expressed as follows [24

24. E. Popov and M. Nevière, “Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence,” Opt. Lett. 25(9), 598–600 (2000). [CrossRef]

]
[Hzy]=1/k21[E˜x][E˜xy]=αk21α[Hz][Hz],
(4)
where E˜x=Ex/(iωμ0) (where ω is the circular frequency and μ0 is the permeability of vacuum). Ex and Hz are the x and z components of electric and magnetic field, respectively. k 2 is a periodic function k2(x,y)=k02ε(x,y), where k 0 is the modulus of the wave vector in vacuum. α is a diagonal matrix with elements αn=α0+2πn/Λ, where α0=k0sin(θi) and n. f denotes the Toeplitz matrix generated by the Fourier coefficients of f such that its (p, q) (p, q) element is fp-q, and −1 denotes the matrix inverse [22

22. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13(9), 1870–1876 (1996). [CrossRef]

]. The numerical calculation based on differential Eq. (4) can be implemented to produce high accurate results with rapid convergence by preserving fewer harmonic waves.

3. Results and discussion

The coupling strength between the electromagnetic fields in the gratings and homogeneous layer can be appropriately affected by the homogeneous layer thickness and lateral alignment condition between two gratings in this kind of devices [14

14. T. Sang, Z. S. Wang, J. T. Zhu, L. Wang, Y. G. Wu, and L. Y. Chen, “Linewidth properties of double-layer surface-relief resonant Brewster filters with equal refractive index,” Opt. Express 15(15), 9659–9665 (2007). [CrossRef] [PubMed]

]. Meanwhile, the tunable range and ability of the lateral alignment shift along transverse direction, which could be applied to alter the line shape and linewidth of spectral response of the Brewster filter, is also dependent on the characteristic of the field inside the homogeneous layer [18

18. W. Nakagawa and Y. Fainman, “Tunable optical nanocavity based on modulation of near-field coupling between subwavelength periodic nanostructures,” IEEE J. Sel. Top. Quantum Electron. 10(3), 478–483 (2004). [CrossRef]

]. Below, the tunable range of the resonant mode location and the characteristic of spectral response of the Brewster filter due to the lateral alignment shift are studied near the same resonant wavelength of 800 nm for three different thicknesses of the homogeneous layer. The reflectance of the Brewster filter as the function of the lateral alignment shift and wavelength for TM0, TM1 and TM2 waveguide modes are shown in Fig. 5(a), (b) and (c)
Fig. 5 (Color online) Calculated reflectance as a function of the lateral alignment shift S and wavelength at Brewster angle (57.13°) for (a) TM0, (b) TM1 and (c) TM2 guided-modes. (d) Calculated reflectance peak as a function of the lateral alignment shift S for TM0 (squares), TM1 (circles) and TM2 (up triangles) guided-modes. Other parameters are the same as those in Fig. 2.
, respectively. For the three waveguide modes, the lateral alignment condition does not almost cause any shift for the resonant peak but the linewidth and reflectance peak. The resonant peaks of the TM0 and TM1 modes shift toward short wavelength as S increases from 0 to 0.5. For example, the wavelength of resonant peak shifts from 800 to 799.12 nm in Fig. 5(a) and from 800 to 799.57 nm in Fig. 5(b). The reason is that the field inside the Brewster filter is mostly confined in the grating layer due to their relative large thicknesses, and the lateral alignment shift S can affect the coupling strength between the two gratings [18

18. W. Nakagawa and Y. Fainman, “Tunable optical nanocavity based on modulation of near-field coupling between subwavelength periodic nanostructures,” IEEE J. Sel. Top. Quantum Electron. 10(3), 478–483 (2004). [CrossRef]

]. With increasing of S from 0 to 0.5, the linewidth of the TM0 mode increases, however, that of the TM1 mode changes more complicated because of the comparable thicknesses of two grating layers to the homogeneous layer. The linewidth of the TM2 guided-mode decreases sharply as S varies from 0 to 0.5, but the location of the resonant peak is invariable. These characteristics are in good agreement with the coupled-mode theory [20

20. H. Y. Song, S. Kim, and R. Magnusson, “Tunable guided-mode resonances in coupled gratings,” Opt. Express 17(26), 23544–23555 (2009). [CrossRef]

].

Figure 5(d) shows the variety of reflectance peak around 800 nm as a function of the lateral alignment shift S for TM0 (squares), TM1 (circles) and TM2 (up triangles) waveguide modes. For the TM0 and TM1 guided-modes, the electromagnetic field of the excited guided-mode inside the filter can be greatly affected due to oblique incident angle when the two gratings are not perfectly aligned. Thus, as the lateral alignment shift S changes, the reflectance peak varies in the range of 0.2-1 for TM0 mode and 0.75-1 for TM1 mode, respectively. For the TM2 waveguide mode, the field distribution in the homogeneous layer reduced the difference between the diffractive characteristics of the two gratings when the lateral alignment S varies and the condition of phase matching is satisfied. Thus, the spectral reflectance peak of the TM2 waveguide mode in Fig. 5(d) is almost unchanged. This configuration can be considered as a perfect GMR filter at Brewster incident angle for practical applications of narrowing the linewidth of reflective spectrum. The spectral responses under particular alignment conditions (S = 0, 0.25 and 0.5) TM0, TM1 and TM2 waveguide modes are shown in Fig. 6(a), (b) and (c)
Fig. 6 (Color online) Reflection spectral response of the triple-layer filter for (a) TM0, (b) TM1 and (c) TM2 guided-modes under different alignment conditions: (solid curves) perfect alignment (S = 0), (dashed curves) quarter-period shifted (S = 0.25), and (dotted curves) half-period shifted (S = 0.5). Other parameters are the same as those in Fig. 2.
, respectively. As can be seen, by adjusting the lateral alignment condition, TM0 and TM2 guided-modes can be respectively used for expanding or narrowing the linewidth of the spectral response of the Brewster filter with symmetric line shape and low sideband features maintained.

Since the field inside the GMR filter is mostly confined in the medium with high refractive index [19

19. R. Magnusson and Y. Ding, “MENS tunable resonant leaky mode filters,” IEEE Photon. Technol. Lett. 18(14), 1479–1481 (2006). [CrossRef]

], therefore, the tunable range of the resonant mode location and the characteristic of spectral response of the Brewster filter due to the lateral alignment shift could be affected by the filling factors of the gratings. Figures 7(a) and (b)
Fig. 7 (Color online) Reflection spectral response of the triple-layer Brewster filter for (a) F = 0.1 and (b) F = 0.9 under different alignment conditions: (solid curves) perfect alignment (S = 0), (dashed curves) quarter-period shifted (S = 0.25), and (dotted curves) half-period shifted (S = 0.5). The other parameters are the same as those in Fig. 2 except the period of the grating: (a) Λ=342.10 nmand (b) Λ=321.71 nm.
show the spectrums of the Brewster filter with a grating filling factor of F = 0.1 and F = 0.9, respectively, which correspond to two modulated grating structures. For locating the resonant wavelength at 800 nm, the periods of the gratings are adjusted to 342.10 nm and 321.71 nm in Fig. 7(a) and (b), respectively. The other parameters are the same as those in Fig. 5(a). In Fig. 7(a), results show that the GMR effect does not vanish under the perfect alignment condition (S = 0). The range of the spectral linewidth is still tunable as S varies, since the field in the gratings is extremely confined in the narrow high refractive index region. In Fig. 7(a), the effective refractive indices of the identical gratings are close to that of the substrate. Therefore, the location of the resonant peak cannot reach the minimum value of the tunable range when S = 0.5. When the filling factor is set to 0.9, a strong coupling strength between the fields in the gratings is kept. Therefore, the linewidth of the spectral response is almost unchanged as varying of S from 0 to 0.5, which is shown in Fig. 7(b).

4. Conclusion

In summary, a triple-layer GMR Brewster filter can be fabricated by selecting a homogeneous layer with refractive index equal to two identical gratings with filling factors of 0.5. It is shown that the linewidth of the spectrum can be significantly changed by altering the thickness of the lower grating layer, but the upper grating layer mainly affects the resonant mode location and sideband levels. Higher order of guided-mode can be excited when the thickness of homogeneous layer increase and different line shape of spectral response can be obtained by selecting different homogeneous layer thicknesses. The tunable range and ability of the lateral alignment shift along the transverse direction is also dependent on the thickness of the homogeneous layer. Different dependence of the linewidth and reflectance peak on the lateral alignment shift S can be obtained at TM0 and TM2 waveguide modes, the later case can be used as a perfect GMR filter with the reflectance peak unchanged as S varies. For practical applications, the triple-layer Brewster filter can still be used to obtain extremely narrow linewidth of spectral response when the filling factor is set to 0.1. Meanwhile, the resonant mode location has a certain tunable range when the filling factor is set to 0.9.

Acknowledgments

This work is supported by the Chinese National Key Basic Research Special Fund (grant 2011CB922003), the Natural Science Foundation of China (grant 60678025 and 61008002), 111 Project (grant B07013), the Specialized Research Fund for the Doctoral Program of Higher Education (grant 20100031120005), and the Fundamental Research Funds for the Central Universities.

References and links

1.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Proc. Phys. Soc. Lond. 18(1), 269–275 (1901).

2.

S. S. Wang and R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19(12), 919–921 (1994). [CrossRef] [PubMed]

3.

S. Tibuleac and R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A 14(7), 1617–1626 (1997). [CrossRef]

4.

R. Magnusson, S. S. Wang, T. D. Black, and A. Sohn, “Resonance properties of dielectric waveguide gratings: theory and experiments at 4–18 GHz,” IEEE Trans. Antenn. Propag. 42(4), 567–569 (1994). [CrossRef]

5.

A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating–waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14(11), 2985–2993 (1997). [CrossRef]

6.

P. S. Priambodo, T. A. Maldonado, and R. Magnusson, “Fabrication and characterization of high-quality waveguide-mode resonant optical filters,” Appl. Phys. Lett. 83(16), 3248–3250 (2003). [CrossRef]

7.

K. J. Lee, R. LaComb, B. Britton, M. Shokooh-Saremi, H. Silva, E. Donkor, Y. Ding, and R. Magnusson, “Silicon-layer guided-mode resonance polarizer with 40-nm bandwidth,” IEEE Photon. Technol. Lett. 20(22), 1857–1859 (2008). [CrossRef]

8.

A. K. Kodali, M. Schulmerich, J. Ip, G. Yen, B. T. Cunningham, and R. Bhargava, “Narrowband midinfrared reflectance filters using guided mode resonance,” Anal. Chem. 82(13), 5697–5706 (2010). [CrossRef] [PubMed]

9.

T. Kobayashi, Y. Kanamori, and K. Hane, “Surface laser emission from solid polymer dye in a guided mode resonant grating filter structure,” Appl. Phys. Lett. 87(15), 151106 (2005). [CrossRef]

10.

N. Kaiser, T. Feigl, O. Stenzel, U. Schulz, and M. Yang, “Optical coatings: trends and challenges,” Opt. Precis. Eng. 13(4), 389–396 (2005).

11.

Q. M. Ngo, S. Kim, S. H. Song, and R. Magnusson, “Optical bistable devices based on guided-mode resonance in slab waveguide gratings,” Opt. Express 17(26), 23459–23467 (2009). [CrossRef]

12.

Z. S. Wang, T. Sang, L. Wang, J. T. Zhu, Y. G. Wu, and L. Y. Chen, “Guided-mode resonance Brewster filters with multiple channels,” Appl. Phys. Lett. 88(25), 251115 (2006). [CrossRef]

13.

Z. S. Wang, T. Sang, J. T. Zhu, L. Wang, Y. G. Wu, and L. Y. Chen, “Double-layer resonant Brewster filters consisting of a homogeneous layer and a grating with equal refractive index,” Appl. Phys. Lett. 89(24), 241119 (2006). [CrossRef]

14.

T. Sang, Z. S. Wang, J. T. Zhu, L. Wang, Y. G. Wu, and L. Y. Chen, “Linewidth properties of double-layer surface-relief resonant Brewster filters with equal refractive index,” Opt. Express 15(15), 9659–9665 (2007). [CrossRef] [PubMed]

15.

Q. Wang, D. W. Zhang, Y. S. Huang, Z. J. Ni, J. B. Chen, Y. W. Zhong, and S. L. Zhuang, “Type of tunable guided-mode resonance filter based on electro-optic characteristic of polymer-dispersed liquid crystal,” Opt. Lett. 35(8), 1236–1238 (2010). [CrossRef] [PubMed]

16.

R. Magnusson, D. Shin, and Z. S. Liu, “Guided-mode resonance Brewster filter,” Opt. Lett. 23(8), 612–614 (1998). [CrossRef]

17.

D. Shin, Z. S. Liu, and R. Magnusson, “Resonant Brewster filters with absentee layers,” Opt. Lett. 27(15), 1288–1290 (2002). [CrossRef]

18.

W. Nakagawa and Y. Fainman, “Tunable optical nanocavity based on modulation of near-field coupling between subwavelength periodic nanostructures,” IEEE J. Sel. Top. Quantum Electron. 10(3), 478–483 (2004). [CrossRef]

19.

R. Magnusson and Y. Ding, “MENS tunable resonant leaky mode filters,” IEEE Photon. Technol. Lett. 18(14), 1479–1481 (2006). [CrossRef]

20.

H. Y. Song, S. Kim, and R. Magnusson, “Tunable guided-mode resonances in coupled gratings,” Opt. Express 17(26), 23544–23555 (2009). [CrossRef]

21.

C. Kappel, A. Selle, M. A. Bader, and G. Marowsky, “Resonant double-grating waveguide structure as inverted Fabry-Perot interferometers,” J. Opt. Soc. Am. B 21(6), 1127–1136 (2004). [CrossRef]

22.

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13(9), 1870–1876 (1996). [CrossRef]

23.

N. Chateau and J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11(4), 1321–1331 (1994). [CrossRef]

24.

E. Popov and M. Nevière, “Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence,” Opt. Lett. 25(9), 598–600 (2000). [CrossRef]

25.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

26.

K. Kawano, and T. Kitoh, Introduction to optical waveguide analysis: Solving Maxwell’s equations and the Schrödinger Equation (John Wiley & Sons, Inc., 2001).

27.

D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, and J. M. Bendickson, “Effects of modulation strength in guided-mode resonant subwavelength gratings at normal incidence,” J. Opt. Soc. Am. A 17(7), 1221–1230 (2000). [CrossRef]

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(120.2440) Instrumentation, measurement, and metrology : Filters
(310.2790) Thin films : Guided waves

ToC Category:
Diffraction and Gratings

History
Original Manuscript: February 10, 2011
Revised Manuscript: March 27, 2011
Manuscript Accepted: March 28, 2011
Published: April 14, 2011

Citation
Xin Liu, Shuqi Chen, Weiping Zang, and Jianguo Tian, "Triple-layer guided-mode resonance Brewster filter consisting of a homogenous layer and coupled gratings with equal refractive index," Opt. Express 19, 8233-8241 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8233


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References

  1. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Proc. Phys. Soc. Lond. 18(1), 269–275 (1901).
  2. S. S. Wang and R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19(12), 919–921 (1994). [CrossRef] [PubMed]
  3. S. Tibuleac and R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A 14(7), 1617–1626 (1997). [CrossRef]
  4. R. Magnusson, S. S. Wang, T. D. Black, and A. Sohn, “Resonance properties of dielectric waveguide gratings: theory and experiments at 4–18 GHz,” IEEE Trans. Antenn. Propag. 42(4), 567–569 (1994). [CrossRef]
  5. A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating–waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14(11), 2985–2993 (1997). [CrossRef]
  6. P. S. Priambodo, T. A. Maldonado, and R. Magnusson, “Fabrication and characterization of high-quality waveguide-mode resonant optical filters,” Appl. Phys. Lett. 83(16), 3248–3250 (2003). [CrossRef]
  7. K. J. Lee, R. LaComb, B. Britton, M. Shokooh-Saremi, H. Silva, E. Donkor, Y. Ding, and R. Magnusson, “Silicon-layer guided-mode resonance polarizer with 40-nm bandwidth,” IEEE Photon. Technol. Lett. 20(22), 1857–1859 (2008). [CrossRef]
  8. A. K. Kodali, M. Schulmerich, J. Ip, G. Yen, B. T. Cunningham, and R. Bhargava, “Narrowband midinfrared reflectance filters using guided mode resonance,” Anal. Chem. 82(13), 5697–5706 (2010). [CrossRef] [PubMed]
  9. T. Kobayashi, Y. Kanamori, and K. Hane, “Surface laser emission from solid polymer dye in a guided mode resonant grating filter structure,” Appl. Phys. Lett. 87(15), 151106 (2005). [CrossRef]
  10. N. Kaiser, T. Feigl, O. Stenzel, U. Schulz, and M. Yang, “Optical coatings: trends and challenges,” Opt. Precis. Eng. 13(4), 389–396 (2005).
  11. Q. M. Ngo, S. Kim, S. H. Song, and R. Magnusson, “Optical bistable devices based on guided-mode resonance in slab waveguide gratings,” Opt. Express 17(26), 23459–23467 (2009). [CrossRef]
  12. Z. S. Wang, T. Sang, L. Wang, J. T. Zhu, Y. G. Wu, and L. Y. Chen, “Guided-mode resonance Brewster filters with multiple channels,” Appl. Phys. Lett. 88(25), 251115 (2006). [CrossRef]
  13. Z. S. Wang, T. Sang, J. T. Zhu, L. Wang, Y. G. Wu, and L. Y. Chen, “Double-layer resonant Brewster filters consisting of a homogeneous layer and a grating with equal refractive index,” Appl. Phys. Lett. 89(24), 241119 (2006). [CrossRef]
  14. T. Sang, Z. S. Wang, J. T. Zhu, L. Wang, Y. G. Wu, and L. Y. Chen, “Linewidth properties of double-layer surface-relief resonant Brewster filters with equal refractive index,” Opt. Express 15(15), 9659–9665 (2007). [CrossRef] [PubMed]
  15. Q. Wang, D. W. Zhang, Y. S. Huang, Z. J. Ni, J. B. Chen, Y. W. Zhong, and S. L. Zhuang, “Type of tunable guided-mode resonance filter based on electro-optic characteristic of polymer-dispersed liquid crystal,” Opt. Lett. 35(8), 1236–1238 (2010). [CrossRef] [PubMed]
  16. R. Magnusson, D. Shin, and Z. S. Liu, “Guided-mode resonance Brewster filter,” Opt. Lett. 23(8), 612–614 (1998). [CrossRef]
  17. D. Shin, Z. S. Liu, and R. Magnusson, “Resonant Brewster filters with absentee layers,” Opt. Lett. 27(15), 1288–1290 (2002). [CrossRef]
  18. W. Nakagawa and Y. Fainman, “Tunable optical nanocavity based on modulation of near-field coupling between subwavelength periodic nanostructures,” IEEE J. Sel. Top. Quantum Electron. 10(3), 478–483 (2004). [CrossRef]
  19. R. Magnusson and Y. Ding, “MENS tunable resonant leaky mode filters,” IEEE Photon. Technol. Lett. 18(14), 1479–1481 (2006). [CrossRef]
  20. H. Y. Song, S. Kim, and R. Magnusson, “Tunable guided-mode resonances in coupled gratings,” Opt. Express 17(26), 23544–23555 (2009). [CrossRef]
  21. C. Kappel, A. Selle, M. A. Bader, and G. Marowsky, “Resonant double-grating waveguide structure as inverted Fabry-Perot interferometers,” J. Opt. Soc. Am. B 21(6), 1127–1136 (2004). [CrossRef]
  22. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13(9), 1870–1876 (1996). [CrossRef]
  23. N. Chateau and J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11(4), 1321–1331 (1994). [CrossRef]
  24. E. Popov and M. Nevière, “Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence,” Opt. Lett. 25(9), 598–600 (2000). [CrossRef]
  25. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).
  26. K. Kawano, and T. Kitoh, Introduction to optical waveguide analysis: Solving Maxwell’s equations and the Schrödinger Equation (John Wiley & Sons, Inc., 2001).
  27. D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, and J. M. Bendickson, “Effects of modulation strength in guided-mode resonant subwavelength gratings at normal incidence,” J. Opt. Soc. Am. A 17(7), 1221–1230 (2000). [CrossRef]

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