Abstract

A comparative study of theoretical models of different three-dimensional photonic bandgap (3D-PBG) structures has been performed, taking into account instability and convergence problems. Some rules for solving these problems and for reducing the computational time by finding symmetries in the structures are also explained. Finally, some applications produced by defects in 3D structures are shown by studying the creation of a complete bandgap in one of them and the variation of partial bandgaps in several 3D-PBG structures when several parameters of the defects, such as the number of layers stacked at each side of the defect, its thickness, and the real and imaginary parts of its index of refraction, are changed.

© 2003 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  19. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
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    [CrossRef]
  22. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  23. N. Chateau, J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
    [CrossRef]
  24. M. G. Moharam, D. A. Pommet, E. B. Grann, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
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    [CrossRef]
  26. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  27. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity though the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  28. P. Lalanne, “Convergence performance of the coupled wave and the differential methods for thin gratings,” J. Opt. Soc. Am. A 14, 1583–1591 (1997).
    [CrossRef]
  29. P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  30. M. M. Sigalas, R. Biswas, C. T. Chan, K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
    [CrossRef]
  31. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  32. E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
  33. P. Lalanne, D. Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
    [CrossRef]
  34. P. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, “Blazed-binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. 23, 1081–1083 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2001 (1)

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppilhalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffers’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

2000 (1)

1999 (3)

1998 (5)

1997 (5)

1996 (9)

1995 (3)

1994 (4)

1993 (2)

1992 (1)

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relation,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Aalto, T.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppilhalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffers’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Abrams, D. S.

Astilean, S.

Bagieu, M.

Bagnoud, V.

Barkou, S. E.

J. Broeng, D. Mogilevstev, S. E. Barkou, A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330 (1999).
[CrossRef]

Biswas, R.

M. M. Sigalas, R. Biswas, C. T. Chan, K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

Bjarklev, A.

J. Broeng, D. Mogilevstev, S. E. Barkou, A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330 (1999).
[CrossRef]

Broeng, J.

J. Broeng, D. Mogilevstev, S. E. Barkou, A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330 (1999).
[CrossRef]

Cambril, E.

Chan, C. T.

M. M. Sigalas, R. Biswas, C. T. Chan, K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

Chandezon, J.

Chateau, N.

Chavel, P.

Cotter, N. P. K.

Daly, J. T.

J. T. Daly, A. C. Greenwald, E. A. Johnson, W. A. Stevenson, J. A. Wollam, T. George, E. W. Jones, “Nanostructured surfaces for tuned infrared emission for spectroscopic applications ,” in Micro- and Nano-photonic Materials and Devices, J. W. Perry, A. Scherer, eds., Proc. SPIE3937, 80–91 (2000).
[CrossRef]

Dansas, P.

Fan, S.

Gaylord, T. K.

George, T.

J. T. Daly, A. C. Greenwald, E. A. Johnson, W. A. Stevenson, J. A. Wollam, T. George, E. W. Jones, “Nanostructured surfaces for tuned infrared emission for spectroscopic applications ,” in Micro- and Nano-photonic Materials and Devices, J. W. Perry, A. Scherer, eds., Proc. SPIE3937, 80–91 (2000).
[CrossRef]

Grann, E. B.

Greenwald, A. C.

J. T. Daly, A. C. Greenwald, E. A. Johnson, W. A. Stevenson, J. A. Wollam, T. George, E. W. Jones, “Nanostructured surfaces for tuned infrared emission for spectroscopic applications ,” in Micro- and Nano-photonic Materials and Devices, J. W. Perry, A. Scherer, eds., Proc. SPIE3937, 80–91 (2000).
[CrossRef]

Haggans, C. W.

Harris, J. B.

Heimala, P.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppilhalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffers’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Ho, K. M.

M. M. Sigalas, R. Biswas, C. T. Chan, K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

Hugonin, J. P.

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Johnson, E. A.

J. T. Daly, A. C. Greenwald, E. A. Johnson, W. A. Stevenson, J. A. Wollam, T. George, E. W. Jones, “Nanostructured surfaces for tuned infrared emission for spectroscopic applications ,” in Micro- and Nano-photonic Materials and Devices, J. W. Perry, A. Scherer, eds., Proc. SPIE3937, 80–91 (2000).
[CrossRef]

Jonannopoulos, J. D.

Jones, E. W.

J. T. Daly, A. C. Greenwald, E. A. Johnson, W. A. Stevenson, J. A. Wollam, T. George, E. W. Jones, “Nanostructured surfaces for tuned infrared emission for spectroscopic applications ,” in Micro- and Nano-photonic Materials and Devices, J. W. Perry, A. Scherer, eds., Proc. SPIE3937, 80–91 (2000).
[CrossRef]

Kuittinen, M.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppilhalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffers’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Lalanne, D.

P. Lalanne, D. Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am. A 14, 450–458 (1997).
[CrossRef]

P. Lalanne, D. Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Lalanne, P.

Launois, H.

Leppilhalme, M.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppilhalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffers’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Li, L.

Liang, T.

MacKinnon, A.

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relation,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Mainguy, S.

Maystre, D.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Mogilevstev, D.

J. Broeng, D. Mogilevstev, S. E. Barkou, A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330 (1999).
[CrossRef]

Moharam, M. G.

Montiel, F.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity though the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

Morris, G. M.

Nevière, M.

Noponen, E.

Paraire, N.

Pendry, J. B.

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relation,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Peyrot, P.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

Pommet, D. A.

Popov, E.

Preist, T. W.

Sambles, J. R.

Sigalas, M. M.

M. M. Sigalas, R. Biswas, C. T. Chan, K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

Sobnack, M. B.

Stevenson, W. A.

J. T. Daly, A. C. Greenwald, E. A. Johnson, W. A. Stevenson, J. A. Wollam, T. George, E. W. Jones, “Nanostructured surfaces for tuned infrared emission for spectroscopic applications ,” in Micro- and Nano-photonic Materials and Devices, J. W. Perry, A. Scherer, eds., Proc. SPIE3937, 80–91 (2000).
[CrossRef]

Tan, W. C.

Tervo, J.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppilhalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffers’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Thorpe, R. N.

Tran, P.

Turunen, J.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppilhalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffers’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

Vahimaa, P.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppilhalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffers’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Villeneuve, P. R.

Wanstall, N. P.

Watts, R. A.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Wollam, J. A.

J. T. Daly, A. C. Greenwald, E. A. Johnson, W. A. Stevenson, J. A. Wollam, T. George, E. W. Jones, “Nanostructured surfaces for tuned infrared emission for spectroscopic applications ,” in Micro- and Nano-photonic Materials and Devices, J. W. Perry, A. Scherer, eds., Proc. SPIE3937, 80–91 (2000).
[CrossRef]

Yablonovitch, E.

Yang, H. Y. D.

H. Y. D. Yang, “Finite difference analysis of 2-D photonic crystals,” IEEE Trans. Microwave Theory Tech. 44, 2688–2695 (1996).
[CrossRef]

Ziolkowski, R. W.

IEEE Trans. Microwave Theory Tech. (1)

H. Y. D. Yang, “Finite difference analysis of 2-D photonic crystals,” IEEE Trans. Microwave Theory Tech. 44, 2688–2695 (1996).
[CrossRef]

J. Mod. Opt. (2)

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

P. Lalanne, D. Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

J. Opt. Soc. Am. A (22)

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

E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

P. Lalanne, D. Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am. A 14, 450–458 (1997).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity though the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

P. Lalanne, “Convergence performance of the coupled wave and the differential methods for thin gratings,” J. Opt. Soc. Am. A 14, 1583–1591 (1997).
[CrossRef]

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

N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
[CrossRef]

W. C. Tan, T. W. Preist, J. R. Sambles, M. B. Sobnack, N. P. Wanstall, “Calculation of photonic band structures of periodic multilayer grating systems by use of curvilinear coordinate transformation,” J. Opt. Soc. Am. A 15, 2365–2372 (1998).
[CrossRef]

J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
[CrossRef]

L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
[CrossRef]

L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
[CrossRef]

V. Bagnoud, S. Mainguy, “Diffraction of electromagnetic waves by dielectric crossed gratings: a three-dimensional Rayleigh–Fourier solution,” J. Opt. Soc. Am. A 16, 1277–1285 (1999).
[CrossRef]

M. Bagieu, D. Maystre, “Waterman and Rayleigh methods for diffraction grating problems: extension of the convergence domain,” J. Opt. Soc. Am. A 15, 1566–1576 (1998).
[CrossRef]

M. Bagieu, D. Maystre, “Regularized Waterman and Rayleigh methods: extension to two-dimensional gratings,” J. Opt. Soc. Am. A 16, 284–292 (1999).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
[CrossRef]

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

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

M. G. Moharam, D. A. Pommet, E. B. Grann, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

P. Dansas, N. Paraire, “Fast modeling of photonic bandgap structures by use of a diffraction-grating approach,” J. Opt. Soc. Am. A 15, 1586–1598 (1998).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppilhalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffers’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Opt. Fiber Technol. (1)

J. Broeng, D. Mogilevstev, S. E. Barkou, A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. 5, 305–330 (1999).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. B (1)

M. M. Sigalas, R. Biswas, C. T. Chan, K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relation,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Other (2)

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

J. T. Daly, A. C. Greenwald, E. A. Johnson, W. A. Stevenson, J. A. Wollam, T. George, E. W. Jones, “Nanostructured surfaces for tuned infrared emission for spectroscopic applications ,” in Micro- and Nano-photonic Materials and Devices, J. W. Perry, A. Scherer, eds., Proc. SPIE3937, 80–91 (2000).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

3D-PBG structure with cubic holes in a cubic lattice composed of six layers.        

Fig. 2
Fig. 2

3D-PBG structure with crossed circular holes in the x and y axes composed of six layers.

Fig. 3
Fig. 3

Convergence of the (0, 0) transmitted order of a lamellar grating with cylindrical grooves of n g = 2 etched in a substrate of refractive index n out = 2 . The other parameters are n in = 1 , n 1 = 1 , Λ x = Λ y = 0.1 λ , f x = f y = 0.443 , θ = ϕ = 45 , and ψ = 0 .

Fig. 4
Fig. 4

Convergence of the (1, 0) transmitted order of a lamellar grating with square grooves of refractive index n g = 1.5 + 1 i etched in a substrate of refractive index n out = 1.5 . The other parameters are n in = 1 , n 1 = 1 , c z = λ , Λ x = Λ y = 1.2 λ , f x = f y = 0.5 , θ = ϕ = 15 , and ψ = 90 .

Fig. 5
Fig. 5

Transmission efficiencies for a 3D-CUB-CYL structure for three convergence methods. Parameters are n in = 1 , n out = 1 , n g = 3.6056 , n 1 = 1 , Λ x = Λ y = Λ z = 440   nm , f x = f y = 0.43 , c z = 189.2   nm , and θ = 0 (normal incidence). Number of diffraction orders: 441. NL = 9 (the last layer is a grating).

Fig. 6
Fig. 6

Transmission efficiencies for a 3D-CUB-CUB structure for three convergence methods. Parameters are n in = 1 , n out = 1 , n g = 1 , n 1 = 3.6056 , Λ x = Λ y = Λ z = 360   nm , f x = f y = 0.9 , c z = 270   nm , and θ = 0 (normal incidence). Number of diffraction orders: 441. NL = 13 (the last layer is homogeneous).

Fig. 7
Fig. 7

Analysis of the transmission efficiency for the same parameters as in Fig. 5 but with different angles.

Fig. 8
Fig. 8

Analysis of the transmission efficiency for the same parameters as in Fig. 6 but with different angles.

Fig. 9
Fig. 9

Variation of the transmission efficiency for the same parameters as in Fig. 5 with the introduction of a defect of different thicknesses in the middle of the structure. The number of layers is 11 (the last layer is a grating).

Fig. 10
Fig. 10

Variation of the transmission efficiency for the same parameters as in Fig. 6 with the introduction of a defect of double thickness in the middle of the structure and variation of the refractive index of the defect and the number of layers stacked at each side of it.

Fig. 11
Fig. 11

Transmission efficiencies for a 3D-CUB-CRO structure, varying the thickness of the ninth layer. Parameters are n in = 1 , n out = 1 , n g = 1 , n 1 = 3.3764 , Λ x = Λ y = Λ z = 400   nm , c zx = c zx = 192   nm , and θ = 0 (normal). Diffraction orders: 441. NL = 19 (the last layer is homogeneous, and the last grating is oriented in the x axis).

Fig. 12
Fig. 12

Variation of the transmission efficiency for the same parameters as in Fig. 11 for a ninth-layer defect of thickness of 128 nm, by varying the imaginary part of the defect.

Equations (6)

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S y z S x z U y z U x z = k 0 0 K Y E - 1 K X I - K Y E - 1 K Y K X E - 1 K X - I - K X E - 1 K Y K Y K X E - K Y 2 K X 2 - E - K X K Y 0 S y S x U y U x ,
[ C 0 ] I R = i = 1 L W i W i X i V i - V i X i   W i X i W i V i X i - V i - 1 × [ Cd ] T O ,
I n + 1 2 = cos   ψ   cos   θ   sin   ϕ + sin   ψ   cos   ϕ ,
I n + n + 1 2 = cos   ψ   cos   θ   cos   ϕ - sin   ψ   sin   ϕ .
C 0 = I 0 I 0 0 I 0 I j   K X K Y k 0 K I , z j   n I 2 k 0 2 - K Y 2 k 0 K I , z - j   K X K Y k 0 K I , z - j   n I 2 k 2 0 - K Y 2 k 0 K I , z - j   n I 2 k 0 2 - K X 2 k 0 K I , z - 1 j   K X K Y k 0 K I , z j   n I 2 k 0 2 - K X 2 k 0 K I , z j   K X K Y k 0 K I , z ,
Cd = I 0 I 0 0 I 0 I j   K X K Y k 0 K II , z j   n II 2 k 0 2 - K Y 2 k 0 K II , z - j   K X K Y k 0 K II , z - j   n II 2 k 2 0 - K Y 2 k 0 K II , z - j   n II 2 k 0 2 - K X 2 k 0 K II , z - 1 j   K X K Y k 0 K II , z j   n II 2 k 0 2 - K X 2 k 0 K II , z j   K X K Y k 0 K II , z ,

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