Abstract

We present a theoretical study of photonic band structures of multilayer grating systems, in which the periodicity in one plane is provided by a grating and the periodicity perpendicular to the plane is obtained by multilayering repeat units. Our method is based on the Chandezon transformation technique together with a scattering matrix approach. It is numerically stable and computationally efficient. Calculations have been performed for several multilayer systems involving a monograting generating a solid with two-dimensional periodicity. Results are presented that show the effect of the amplitudes and the relative phase of the gratings on the subsequent optical band structure. It is found that when all interfaces have the same profile, there are no appreciable bandgaps in the direction of the grating. In contrast, if adjacent interfaces are out of phase, there are large bandgaps in all the directions in the two-dimensional plane containing the grating vector and the layer periodicity vector.

© 1998 Optical Society of America

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References

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  2. K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  3. K. M. Leung, Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
    [CrossRef] [PubMed]
  4. P. Villeneuve, M. Piche, “Photonic band gaps in two dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992).
    [CrossRef]
  5. C. M. Anderson, K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996).
    [CrossRef] [PubMed]
  6. E. Yablonovitch, T. J. Gmitter, K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
    [CrossRef] [PubMed]
  7. F. Gadot, A. Chelnokov, A. DeLustrac, P. Crozat, J. M. Lourtioz, D. Cassagne, C. Jouanin, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl. Phys. Lett. 71, 1780–1803 (1997).
    [CrossRef]
  8. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photon dispersion relation in two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
    [CrossRef] [PubMed]
  9. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  10. T. W. Preist, N. P. Cotter, J. R. Sambles, “Periodic multilayer gratings of arbitrary shape,” J. Opt. Soc. Am. A 12, 1740–1748 (1995).
    [CrossRef]
  11. D. Y. K. Ko, J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
    [CrossRef]
  12. N. P. K. Cotter, T. W. Preist, J. R. Sambles, “A scattering matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [CrossRef]
  13. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]
  14. J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equation in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 838–842 (1995).
    [CrossRef]
  15. P. Lorrain, D. R. Corson, F. Lorrain, Electromagnetic Fields and Waves (Freeman, New York, 1988).
  16. G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
    [CrossRef]
  17. L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
    [CrossRef]
  18. J. B. Pendry, Low Energy Electron Diffraction (Academic, London, 1974).
  19. J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
    [CrossRef] [PubMed]
  20. For a review the reader may consult R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
  21. G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
    [CrossRef]
  22. 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]

1997 (1)

F. Gadot, A. Chelnokov, A. DeLustrac, P. Crozat, J. M. Lourtioz, D. Cassagne, C. Jouanin, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl. Phys. Lett. 71, 1780–1803 (1997).
[CrossRef]

1996 (3)

C. M. Anderson, K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996).
[CrossRef] [PubMed]

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[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]

1995 (5)

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equation in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 838–842 (1995).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

T. W. Preist, N. P. Cotter, J. R. Sambles, “Periodic multilayer gratings of arbitrary shape,” J. Opt. Soc. Am. A 12, 1740–1748 (1995).
[CrossRef]

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

1994 (1)

1992 (3)

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

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photon dispersion relation in two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

P. Villeneuve, M. Piche, “Photonic band gaps in two dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992).
[CrossRef]

1991 (1)

E. Yablonovitch, T. J. Gmitter, K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

1990 (2)

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

K. M. Leung, Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1982 (1)

Anderson, C. M.

C. M. Anderson, K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996).
[CrossRef] [PubMed]

Arjavalingam, G.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photon dispersion relation in two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Brommer, K. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photon dispersion relation in two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Cassagne, D.

F. Gadot, A. Chelnokov, A. DeLustrac, P. Crozat, J. M. Lourtioz, D. Cassagne, C. Jouanin, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl. Phys. Lett. 71, 1780–1803 (1997).
[CrossRef]

Chan, C. T.

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Chandezon, J.

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equation in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 838–842 (1995).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
[CrossRef]

Chelnokov, A.

F. Gadot, A. Chelnokov, A. DeLustrac, P. Crozat, J. M. Lourtioz, D. Cassagne, C. Jouanin, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl. Phys. Lett. 71, 1780–1803 (1997).
[CrossRef]

Cornet, G.

Corson, D. R.

P. Lorrain, D. R. Corson, F. Lorrain, Electromagnetic Fields and Waves (Freeman, New York, 1988).

Cotter, N. P.

Cotter, N. P. K.

Crozat, P.

F. Gadot, A. Chelnokov, A. DeLustrac, P. Crozat, J. M. Lourtioz, D. Cassagne, C. Jouanin, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl. Phys. Lett. 71, 1780–1803 (1997).
[CrossRef]

DeLustrac, A.

F. Gadot, A. Chelnokov, A. DeLustrac, P. Crozat, J. M. Lourtioz, D. Cassagne, C. Jouanin, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl. Phys. Lett. 71, 1780–1803 (1997).
[CrossRef]

Dupuis, M. T.

Gadot, F.

F. Gadot, A. Chelnokov, A. DeLustrac, P. Crozat, J. M. Lourtioz, D. Cassagne, C. Jouanin, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl. Phys. Lett. 71, 1780–1803 (1997).
[CrossRef]

Giapis, K. P.

C. M. Anderson, K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996).
[CrossRef] [PubMed]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

Granet, G.

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equation in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 838–842 (1995).
[CrossRef]

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

Harris, J. B.

Ho, K. M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Joannopoulos, J. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photon dispersion relation in two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Jouanin, C.

F. Gadot, A. Chelnokov, A. DeLustrac, P. Crozat, J. M. Lourtioz, D. Cassagne, C. Jouanin, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl. Phys. Lett. 71, 1780–1803 (1997).
[CrossRef]

Ko, D. Y. K.

Leung, K. M.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

K. M. Leung, Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Li, L.

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
[CrossRef]

Liu, Y. F.

K. M. Leung, Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Lorrain, F.

P. Lorrain, D. R. Corson, F. Lorrain, Electromagnetic Fields and Waves (Freeman, New York, 1988).

Lorrain, P.

P. Lorrain, D. R. Corson, F. Lorrain, Electromagnetic Fields and Waves (Freeman, New York, 1988).

Lourtioz, J. M.

F. Gadot, A. Chelnokov, A. DeLustrac, P. Crozat, J. M. Lourtioz, D. Cassagne, C. Jouanin, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl. Phys. Lett. 71, 1780–1803 (1997).
[CrossRef]

MacKinnon, A.

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

Maystre, D.

Meade, R. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photon dispersion relation in two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Pendry, J. B.

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

J. B. Pendry, Low Energy Electron Diffraction (Academic, London, 1974).

Piche, M.

P. Villeneuve, M. Piche, “Photonic band gaps in two dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992).
[CrossRef]

Plumey, J. P.

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equation in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 838–842 (1995).
[CrossRef]

Preist, T. W.

Rappe, A. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photon dispersion relation in two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Robertson, W. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photon dispersion relation in two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Sambles, J. R.

Soukoulis, C. M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Thorpe, R. N.

Villeneuve, P.

P. Villeneuve, M. Piche, “Photonic band gaps in two dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992).
[CrossRef]

Watts, R. A.

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

F. Gadot, A. Chelnokov, A. DeLustrac, P. Crozat, J. M. Lourtioz, D. Cassagne, C. Jouanin, “Experimental demonstration of complete photonic band gap in graphite structure,” Appl. Phys. Lett. 71, 1780–1803 (1997).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. P. Plumey, G. Granet, J. Chandezon, “Differential covariant formalism for solving Maxwell’s equation in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 838–842 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Phys. Rev. B (1)

P. Villeneuve, M. Piche, “Photonic band gaps in two dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992).
[CrossRef]

Phys. Rev. Lett. (7)

C. M. Anderson, K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

K. M. Leung, Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photon dispersion relation in two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

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

Pure Appl. Opt. (3)

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

G. Granet, “Analysis of diffraction by crossed gratings using a non-orthogonal coordinate system,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

Other (3)

J. B. Pendry, Low Energy Electron Diffraction (Academic, London, 1974).

For a review the reader may consult R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).

P. Lorrain, D. R. Corson, F. Lorrain, Electromagnetic Fields and Waves (Freeman, New York, 1988).

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic diagram of a periodic multilayer grating system, (b) irreducible part of its Brillouin zone in the xy plane.

Fig. 2
Fig. 2

(a) Calculated photonic band structure in the kz=0 plane of the PMGS shown in (b): solid curves, TE modes; dashed curves, TM modes. (b) Schematic illustration of the PMGS, in which all the interfaces have the same sinusoidal profile f(x)=½h sin(2πx/lx), where h=40 nm, lx=200 nm, lA=lB=100 nm, A=4, and B=1.

Fig. 3
Fig. 3

(a) Calculated photonic band structure in the kz=0 plane of the PMGS shown in (b): solid curves, TE modes; dashed curves, TM modes. (b) Schematic illustration of the PMGS, in which all the interfaces have the same sinusoidal profile f(x)=½h sin(2πx/lx), where h=200 nm, lx=200 nm, lA=lB=100 nm, A=4, and B=1.

Fig. 4
Fig. 4

(a) Calculated photonic band structure in the kz=0 plane of the PMGS shown in (b): solid curves, TE modes; dashed curves, TM modes. (b) Schematic illustration of the PMGS, in which the profiles of adjacent interfaces are out of phase, i.e., fAB(x)=½h sin(2πx/lx) and fBA(x)=½h sin(2πx/lx+π), where h=54 nm, lx=200 nm, lA=60 nm, lB=140 nm, A=4, and B=1.

Fig. 5
Fig. 5

(a) Calculated photonic band structure in the kz=0 plane of the PMGS shown in (b): solid curves, TE modes; dashed curves, TM modes. (b) Schematic illustration of the PMGS, in which the profiles of adjacent interfaces are out of phase, i.e., fAB(x)=½h sin(2πx/lx) and fBA(x)=½h sin(2πx/lx+π), where h=80 nm, lx=200 nm, lA=lB=100 nm, A=4, and B=1.

Fig. 6
Fig. 6

Calculated photonic band structure in the kz=0.5π/lx plane of the PMGS shown in Fig. 2(b).

Equations (33)

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

yjBA(x)=j(lA+lB)+fBA(x),
yjAB(x)=j(lA+lB)+lA+fAB(x),
u=x,v=y-a(x),w=z,
a(x)=fBA(x)inmediumAfAB(x)inmediumB.
  E=iγ1H,  H=-iγ2E,
γ1=ωZ0cμr,γ2=ωZ0cr,
Fv=DFu+iCγ1-kz2γ2G,
Gv=uDG+iCγ1-kz2/γ2Fu+iγ2F,
F=E3,G=H1forTE-likemode,
F=μrZ0H3,G=-rZ0E1forTM-likemode,
C=11+a2,D=a1+a2,a=da(x)dx.
F(u, v)=mFm(v)exp(iαmu),
G(u, v)=mGm(v)exp(iαmu),
αm=kx+mK,
-iddvξ(v)=Tξ(v),
T=T11T12T21T22.
T11 nm=αmDn-m,
T12 nm=γ1-kz2γ2Cn-m,
T21 nm=-αnαmγ1-kz2/γ2Cn-m+γ2δnm,
T22 nm=αnDn-m.
ξq(v)=Vq exp(irqv),
Vq=V1qV2q,
ζq=αkz(kz2-γ1γ2)-1V1qV1qV2q0,
ζq+2(2N+1)=-r-1Z0V2q0(μrZ0)-1αkz(kz2-γ1γ2)-1V1q(μrZ0)-1V1q.
ψ(v)=q=14(2N+1)bqζq exp(irqv),
rq=rqifq2(2N+1)r[q-2(2N+1)]ifq>2(2N+1).
ψ(v)=Mϕb,
ψj(v)=Mj(v)ϕjbj.
bj=I(j, j-1)bj-1.
bp=I(p, 0)b0.
ψ(v+ly)=exp(ikyly)ψ(v),
bp=exp(ikyly)b0.
I(p, 0)b0=exp(ikyly)b0.

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