Abstract

Omnidirectional reflection from periodic and Fibonacci quasi-periodic multilayers that are embedded in an isotropic medium is further analyzed. Besides the isotropic structures, birefringent structures are considered that comprise uniaxial layers in the principal-axis system, alternating with isotropic layers so that the refractive index of isotropic layers is equal to the principal extraordinary refractive index of the uniaxial layers. The transfer-matrix method is applied, and the same formalism is used for both the isotropic and the uniaxial media in the principal-axis system. Simple and original relations are obtained for the invariant of the one-dimensional Fibonacci sequences at oblique incidence. Numerical examples are given comparatively for the isotropic and the birefringent structures in the case of periodic and Fibonacci quasi-periodic sequences at different values of the refractive indices.

© 2002 Optical Society of America

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  1. C. M. Bowden, J. P. Dowling, H. O. Everitt, eds., feature on Development and Applications of Materials Exhibiting Photonic Band Gaps, J. Opt. Soc. Am. B 10, 279–413 (1993).
  2. G. Kurizki, J. W. Haus, eds., feature on Photonic Band Structures, J. Mod. Opt. 41, 171–404 (1994).
  3. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
    [CrossRef] [PubMed]
  4. P. St. J. Russell, S. Tredwell, P. J. Roberts, “Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures,” Opt. Commun. 160, 66–71 (1999).
    [CrossRef]
  5. 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]
  6. J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A 11, 2892–2899 (1994).
    [CrossRef]
  7. J. N. Winn, Y. Fink, S. Fan, J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
    [CrossRef]
  8. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
    [CrossRef] [PubMed]
  9. E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filter,” Opt. Lett. 23, 1648–1649 (1998).
    [CrossRef]
  10. W. H. Southwell, “Omnidirectional mirror design with quarter-wave dielectric stacks,” Appl. Opt. 38, 5464–5467 (1999).
    [CrossRef]
  11. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, S. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
    [CrossRef]
  12. I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
    [CrossRef]
  13. I. Abdulhalim, “Reflective phase-only modulation using one-dimensional photonic crystals,” J. Opt. A: Pure Appl. Opt. 2, L9–L11 (2000).
    [CrossRef]
  14. I. Abdulhalim, “Analytic propagation matrix method for anisotropic magneto-optic layered media,” J. Opt. A: Pure Appl. Opt. 2, 557–564 (2000).
    [CrossRef]
  15. E. Cojocaru, “Omnidirectional reflection from Šolc-type anisotropic periodic dielectric structures,” Appl. Opt. 39, 6441–6447 (2000).
    [CrossRef]
  16. M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451–2456 (2000).
    [CrossRef] [PubMed]
  17. P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  18. C. Gu, P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10, 966–973 (1993).
    [CrossRef]
  19. G. D. Landry, T. A. Maldonado, “Complete method to determine transmission and reflection characteristics at a planar interface between arbitrarily oriented biaxial media,” J. Opt. Soc. Am. A 12, 2048–2063 (1995).
    [CrossRef]
  20. C. Sibilia, I. S. Nefedov, M. Scalora, M. Bertolotti, “Electromagnetic mode density for finite quasi-periodic structures,” J. Opt. Soc. Am. B 15, 1947–1952 (1998).
    [CrossRef]
  21. J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
    [CrossRef]
  22. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1987), Chap. 4.
  23. J. He, M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase-matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
    [CrossRef]
  24. A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
    [CrossRef] [PubMed]
  25. A. M. Steinberg, R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
    [CrossRef] [PubMed]
  26. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14.
  27. A. Yariv, P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. 67, 438–448 (1977).
    [CrossRef]
  28. D. Levine, P. J. Steinhardt, “Quasi-crystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
    [CrossRef]
  29. F. Laruelle, B. Etienne, “Fibonacci invariant and electronic properties of GaAs/Ga1-xAlxAs quasi-periodic superlattices,” Phys. Rev. B 37, 4816–4819 (1988).
    [CrossRef]
  30. M. Kohmoto, B. Sutherland, C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasi-crystal model,” Phys. Rev. B 35, 1020–1033 (1987).
    [CrossRef]
  31. W. Gellermann, M. Kohmoto, B. Sutherland, P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
    [CrossRef] [PubMed]

2000 (5)

I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
[CrossRef]

I. Abdulhalim, “Reflective phase-only modulation using one-dimensional photonic crystals,” J. Opt. A: Pure Appl. Opt. 2, L9–L11 (2000).
[CrossRef]

I. Abdulhalim, “Analytic propagation matrix method for anisotropic magneto-optic layered media,” J. Opt. A: Pure Appl. Opt. 2, 557–564 (2000).
[CrossRef]

E. Cojocaru, “Omnidirectional reflection from Šolc-type anisotropic periodic dielectric structures,” Appl. Opt. 39, 6441–6447 (2000).
[CrossRef]

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451–2456 (2000).
[CrossRef] [PubMed]

1999 (3)

W. H. Southwell, “Omnidirectional mirror design with quarter-wave dielectric stacks,” Appl. Opt. 38, 5464–5467 (1999).
[CrossRef]

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, S. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

P. St. J. Russell, S. Tredwell, P. J. Roberts, “Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures,” Opt. Commun. 160, 66–71 (1999).
[CrossRef]

1998 (4)

1996 (1)

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

1995 (1)

1994 (4)

J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A 11, 2892–2899 (1994).
[CrossRef]

G. Kurizki, J. W. Haus, eds., feature on Photonic Band Structures, J. Mod. Opt. 41, 171–404 (1994).

A. M. Steinberg, R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

W. Gellermann, M. Kohmoto, B. Sutherland, P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
[CrossRef] [PubMed]

1993 (3)

A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

C. M. Bowden, J. P. Dowling, H. O. Everitt, eds., feature on Development and Applications of Materials Exhibiting Photonic Band Gaps, J. Opt. Soc. Am. B 10, 279–413 (1993).

C. Gu, P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10, 966–973 (1993).
[CrossRef]

1992 (1)

J. He, M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase-matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
[CrossRef]

1990 (1)

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]

1988 (1)

F. Laruelle, B. Etienne, “Fibonacci invariant and electronic properties of GaAs/Ga1-xAlxAs quasi-periodic superlattices,” Phys. Rev. B 37, 4816–4819 (1988).
[CrossRef]

1987 (2)

M. Kohmoto, B. Sutherland, C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasi-crystal model,” Phys. Rev. B 35, 1020–1033 (1987).
[CrossRef]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
[CrossRef] [PubMed]

1984 (1)

D. Levine, P. J. Steinhardt, “Quasi-crystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
[CrossRef]

1977 (2)

Abdulhalim, I.

I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
[CrossRef]

I. Abdulhalim, “Reflective phase-only modulation using one-dimensional photonic crystals,” J. Opt. A: Pure Appl. Opt. 2, L9–L11 (2000).
[CrossRef]

I. Abdulhalim, “Analytic propagation matrix method for anisotropic magneto-optic layered media,” J. Opt. A: Pure Appl. Opt. 2, 557–564 (2000).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1987), Chap. 4.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1987), Chap. 4.

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Bertolotti, M.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14.

Cada, M.

J. He, M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase-matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
[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]

Chen, C.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Chiao, R. Y.

A. M. Steinberg, R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Chigrin, D. N.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, S. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Cojocaru, E.

Dowling, J. P.

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Etienne, B.

F. Laruelle, B. Etienne, “Fibonacci invariant and electronic properties of GaAs/Ga1-xAlxAs quasi-periodic superlattices,” Phys. Rev. B 37, 4816–4819 (1988).
[CrossRef]

Fan, S.

J. N. Winn, Y. Fink, S. Fan, J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Fink, Y.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

J. N. Winn, Y. Fink, S. Fan, J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Gaponenko, S.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, S. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Gellermann, W.

W. Gellermann, M. Kohmoto, B. Sutherland, P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
[CrossRef] [PubMed]

Gilbert, L. R.

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451–2456 (2000).
[CrossRef] [PubMed]

Gu, C.

He, J.

J. He, M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase-matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
[CrossRef]

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]

Hong, C. S.

Joannopoulos, J. D.

J. N. Winn, Y. Fink, S. Fan, J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Kohmoto, M.

W. Gellermann, M. Kohmoto, B. Sutherland, P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
[CrossRef] [PubMed]

M. Kohmoto, B. Sutherland, C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasi-crystal model,” Phys. Rev. B 35, 1020–1033 (1987).
[CrossRef]

Kwiat, P. G.

A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Landry, G. D.

Laruelle, F.

F. Laruelle, B. Etienne, “Fibonacci invariant and electronic properties of GaAs/Ga1-xAlxAs quasi-periodic superlattices,” Phys. Rev. B 37, 4816–4819 (1988).
[CrossRef]

Lavrinenko, A. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, S. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Lekner, J.

Levine, D.

D. Levine, P. J. Steinhardt, “Quasi-crystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
[CrossRef]

Maldonado, T. A.

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Nefedov, I. S.

Nevitt, T. J.

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451–2456 (2000).
[CrossRef] [PubMed]

Ouderkirk, A. J.

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451–2456 (2000).
[CrossRef] [PubMed]

Roberts, P. J.

P. St. J. Russell, S. Tredwell, P. J. Roberts, “Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures,” Opt. Commun. 160, 66–71 (1999).
[CrossRef]

Russell, P. St. J.

P. St. J. Russell, S. Tredwell, P. J. Roberts, “Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures,” Opt. Commun. 160, 66–71 (1999).
[CrossRef]

Scalora, M.

C. Sibilia, I. S. Nefedov, M. Scalora, M. Bertolotti, “Electromagnetic mode density for finite quasi-periodic structures,” J. Opt. Soc. Am. B 15, 1947–1952 (1998).
[CrossRef]

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Sibilia, C.

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]

Southwell, W. H.

Steinberg, A. M.

A. M. Steinberg, R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Steinhardt, P. J.

D. Levine, P. J. Steinhardt, “Quasi-crystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
[CrossRef]

Stover, C. A.

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451–2456 (2000).
[CrossRef] [PubMed]

Sutherland, B.

W. Gellermann, M. Kohmoto, B. Sutherland, P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
[CrossRef] [PubMed]

M. Kohmoto, B. Sutherland, C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasi-crystal model,” Phys. Rev. B 35, 1020–1033 (1987).
[CrossRef]

Tang, C.

M. Kohmoto, B. Sutherland, C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasi-crystal model,” Phys. Rev. B 35, 1020–1033 (1987).
[CrossRef]

Taylor, P. C.

W. Gellermann, M. Kohmoto, B. Sutherland, P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
[CrossRef] [PubMed]

Thomas, E. L.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Tredwell, S.

P. St. J. Russell, S. Tredwell, P. J. Roberts, “Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures,” Opt. Commun. 160, 66–71 (1999).
[CrossRef]

Weber, M. F.

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451–2456 (2000).
[CrossRef] [PubMed]

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

J. N. Winn, Y. Fink, S. Fan, J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14.

Yablonovitch, E.

E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filter,” Opt. Lett. 23, 1648–1649 (1998).
[CrossRef]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
[CrossRef] [PubMed]

Yariv, A.

Yarotsky, D. A.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, S. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Yeh, P.

Appl. Opt. (2)

Appl. Phys. A (1)

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, S. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Appl. Phys. Lett. (1)

J. He, M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase-matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
[CrossRef]

J. Mod. Opt. (1)

G. Kurizki, J. W. Haus, eds., feature on Photonic Band Structures, J. Mod. Opt. 41, 171–404 (1994).

J. Opt. A: Pure Appl. Opt. (2)

I. Abdulhalim, “Reflective phase-only modulation using one-dimensional photonic crystals,” J. Opt. A: Pure Appl. Opt. 2, L9–L11 (2000).
[CrossRef]

I. Abdulhalim, “Analytic propagation matrix method for anisotropic magneto-optic layered media,” J. Opt. A: Pure Appl. Opt. 2, 557–564 (2000).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (2)

C. M. Bowden, J. P. Dowling, H. O. Everitt, eds., feature on Development and Applications of Materials Exhibiting Photonic Band Gaps, J. Opt. Soc. Am. B 10, 279–413 (1993).

C. Sibilia, I. S. Nefedov, M. Scalora, M. Bertolotti, “Electromagnetic mode density for finite quasi-periodic structures,” J. Opt. Soc. Am. B 15, 1947–1952 (1998).
[CrossRef]

Opt. Commun. (2)

I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
[CrossRef]

P. St. J. Russell, S. Tredwell, P. J. Roberts, “Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures,” Opt. Commun. 160, 66–71 (1999).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

A. M. Steinberg, R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

Phys. Rev. B (2)

F. Laruelle, B. Etienne, “Fibonacci invariant and electronic properties of GaAs/Ga1-xAlxAs quasi-periodic superlattices,” Phys. Rev. B 37, 4816–4819 (1988).
[CrossRef]

M. Kohmoto, B. Sutherland, C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasi-crystal model,” Phys. Rev. B 35, 1020–1033 (1987).
[CrossRef]

Phys. Rev. E (1)

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic bandgap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Phys. Rev. Lett. (5)

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]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
[CrossRef] [PubMed]

W. Gellermann, M. Kohmoto, B. Sutherland, P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72, 633–636 (1994).
[CrossRef] [PubMed]

D. Levine, P. J. Steinhardt, “Quasi-crystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
[CrossRef]

A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Science (2)

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451–2456 (2000).
[CrossRef] [PubMed]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Other (2)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1987), Chap. 4.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14.

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Figures (14)

Fig. 1
Fig. 1

Variation in (a) Rp(N) and (b) Rs(N) by the normalized, dimensionless frequency Ω and the incidence angle θ0 for a periodic isotropic structure that comprises N = 45 periods with alternating layers of equal thicknesses (τ = d1/d2 = 1) and refractive indices n1 = 2.4 and n2 = 1.46 in air (n0 = 1). Ω is varied in increments of 0.0075 and θ0 in increments of 2.75°.

Fig. 2
Fig. 2

Variations in the Ω interval in the gap by the incidence angle θ0 for the s mode (hatched region) and the p mode (crosshatched region) for the periodic isotropic structure with τ = d1/d2 = 1 for different pairs of layer refractive indices: (a), (b) n1 = 2.4, n2 = 1.46 when n0 = 1 and 1.4, respectively; (c), (d) n1 = 2.3, n2 = 1.7 when n0 = 1 and 1.4, respectively; (e), (f) n1 = 1.9, n2 = 1.5 when n0 = 1 and 1.4, respectively. The curves at the center of the gaps are determined by Eq. (10) with τ = 1. Ω is varied in increments of 0.001 and θ0 in increments of 1°.

Fig. 3
Fig. 3

Variation in the ODR bandwidth regardless of polarization ΔΩODR with the ratio τ of layer thicknesses, τ = d1/d2, when n1 = 2.4, n2 = 1.46, and n0 = 1. Ω is varied in increments of 0.0001 and τ in increments of 0.05.

Fig. 4
Fig. 4

Variation in the normalized group velocity Vg/c by Ω for normally incident plane waves on a multilayer with N = 45 periods of alternating isotropic layers of equal thicknesses (τ = 1) and refractive indices, curve 1, n1 = 2.3, n2 = 1.7 and, curve 2, n1 = 2.4, n2 = 1.46 in air (n0 = 1). Ω is varied in increments of 0.001.

Fig. 5
Fig. 5

Values of Ω at which Vg/c is maximum at normal incidence plotted by τ when n1 = 2.4, n2 = 1.46, n0 = 1, and N = 45. Dotted curve, variation in Ωc determined by Eq. (10) with τ. Ω is varied in increments of 0.001 and τ in increments of 0.01.

Fig. 6
Fig. 6

Periodic birefringent structure consisting of N periods of uniaxial layers of thickness d1 and principal refractive indices n and n, alternating with isotropic layers of thickness, d2 = d1/τ, and refractive index, n2 = n. The structure is embedded in the isotropic medium of refractive index n0. Plane waves of unit amplitude are incident at angle θ0; rν(N) and tν(N), with ν = p, s are the overall reflection and transmission coefficients.

Fig. 7
Fig. 7

Variation in (a) Rp(N) and (b) Rs(N) by Ω and θ0 for a periodic birefringent structure that comprises N = 45 periods of alternating uniaxial layers (n = 1.7, n = 2.3) and isotropic layers (n2 = n = 2.3) of equal thickness (τ = d1/d2 = 1) in air (n0 = 1). Ω is varied in increments of 0.0075 and θ0 in increments of 2.75°.

Fig. 8
Fig. 8

Variations in the Ω interval in the gap by the incidence angle θ0 for the s mode (broader hatched region) and the p mode (narrower hatched region) in the case of a periodic birefringent structure with τ = d1/d2 = 1 for different layer refractive indices: (a), (b), n = 1.7, n = 2.3, n2 = n = 2.3 when n0 = 1 and 1.4; (c), (d), positive uniaxial layers, n = 1.5, n = 1.9, n2 = n = 1.9, when n0 = 1 and 1.4; (e), (f), negative uniaxial layers, n = 1.9, n = 1.5, n2 = n = 1.5, when n0 = 1 and 1.4. The curves (for the p mode marked by asterisks) represent the centers of the gaps that are determined by Eqs. (16 ) with τ = 1. Ω is varied in increments of 0.001 and θ0 in increments of 1°.

Fig. 9
Fig. 9

Variation in (a) Rp(N) and (b) Rs(N) with Ω and θ0 for a periodic birefringent structure that comprises N = 45 periods of alternating negative uniaxial layers (n = 1.9, n = 1.5) and isotropic layers (n2 = n = 1.5) of equal thicknesses (τ = 1) in air (n0 = 1). Variation in (c) Rp(N) and (d) Rs(N) by Ω and θ0 for the same periodic birefringent structure in glass (n0 = 1.4). Ω is varied in increments of 0.0075 and θ0 in increments of 2.75°.

Fig. 10
Fig. 10

Variation in the ODR bandwidth regardless of polarization ΔΩODR by the ratio τ of layer thicknesses, τ = d1/d2, for a birefringent periodic structure with layer refractive indices n = 1.7, n = 2.3, n2 = n = 2.3, in air, n0 = 1. Ω is varied in increments of 0.0001 and τ in increments of 0.05.

Fig. 11
Fig. 11

Variation in (a) Rp(k) and (b) Rs(k) with Ω and θ0 for a Fibonacci sequence of the order of k = 9 that comprises 89 isotropic layers of equal thicknesses (τ = d1/d2 = 1) and refractive indices n1 = 2.4 and n2 = 1.46 in air (n0 = 1). Ω is varied in increments of 0.0075 and θ0 in increments of 2.75°.

Fig. 12
Fig. 12

Variation in reflectivity R(k) with Ω at normal incidence for a Fibonacci sequence with k = 9, n1 = 2.4, n2 = 1.46, and n0 = 1 when (a) τ = 1 and (b) τ = gm, where gm is the so-called golden mean. Ω is varied in increments of 0.001.

Fig. 13
Fig. 13

Variation in (a) Rp(k) and (b) Rs(k) with Ω and θ0 for a birefringent Fibonacci sequence of the order of k = 9 that comprises uniaxial (n = 1.7, n = 2.3) and isotropic (n2 = n = 2.3) layers of equal thicknesses (τ = d1/d2 = 1) in air (n0 = 1). Ω is varied in increments of 0.0075 and θ0 in increments of 2.75°.

Fig. 14
Fig. 14

Variation in the normalized group velocity Vg/c with Ω at normal incidence for a Fibonacci sequence of the order of k = 9 that comprises 89 layers of equal thicknesses (τ = 1) in air (n0 = 1). The lower curve corresponds to the pair of layer refractive indices (n1 = 2.3, n2 = 1.7) and the upper curve to the pair (n1 = 2.4, n2 = 1.46). Ω is varied in increments of 0.001.

Tables (1)

Tables Icon

Table 1 Widths of ODR Bands in Terms of Normalized, Dimensionless Frequency Ω for Periodic Isotropic and Birefringent Structures with Alternating Layers of Equal Thickness (τ = 1)

Equations (40)

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ζ=n cos θ, ρ=cos θ/n,
r0is=ζ0-ζi/ζ0+ζi,
r0ip=ρ0-ρi/ρ0+ρi,
t0is=2ζ0/ζ0+ζi,
t0ip=2n0ρ0/niρ0+ρi.
A0s+A0s-=1/21+ζi/ζ01-ζi/ζ01-ζi/ζ01+ζi/ζ0Ais+Ais-,
A0p+A0p-=ni/2n01+ρi/ρ01-ρi/ρ01-ρi/ρ01+ρi/ρ0Aip+Aip-.
Pi=expjϕi00exp-jϕi,
Miν=1/tiνriν*/tiν*riν/tiν1/tiν*, ν=p, s,
cos βν=real[1/tν(1), ν=p, s,
Mν(1)N=Mν(1) sin Nβν/sin βν-I2 sinN-1βν/sin βν,
1/tN=ξN-jηN,
zN=ηN/ξN,
Vg/c=DN/c1+zN2/zN,
Ωωd1/2πc=d1/λ,
Ωc=1/2ζ1+ζ2/τ.
ζo=n2-n02 sin2 θ01/2,
ζeo=n/nn2-n02 sin2 θ01/2,
neo=n2+1-n2/n2n02 sin2 θ01/2,
ρeo=ζeo/n2.
r01s=ζ0-ζo/ζ0+ζo,
r01p=ρ0-ρeo/ρ0+ρeo,
t01s=2ζ0/ζ0+ζo,
t01p=2n0ρ0/neoρ0+ρeocos δ,
cos δ=n2+n2-n2cos2 θeo/n4+n4-n4cos2 θeo1/2.
A0s+A0s-=1/21+ζo/ζ01-ζo/ζ01-ζo/ζ01+ζo/ζ0Ao+Ao-,
A0p+A0p-=neo cos δ/2n01+ρeo/ρ01-ρeo/ρ01-ρeo/ρ01+ρeo/ρ0×Aeo+Aeo-.
sinϕo+ϕ2/τ=0 for the s mode, sinϕeo+ϕ2/τ=0 for the p mode,
Ωcs=1/2ζo+ζ2/τ,
Ωcp=1/2ζeo+ζ2/τ.
Sk+1=Sk-1Sk.
Fk=Fk-1+Fk-2
Dk+1=Dk-1+Dk
1/tνk=ξνk-jηνk, ν=p, s.
ξνk=2ξνk-1ξνk-2-ξνk-3,
ηνk=2ξνk-1ηνk-2+ηνk-3.
Jν=-1+ξνk2+ξνk-12+ξνk-22-2ξνkξνk-1ξνk-2,
Js=1/2ζ1/ζ2-ζ2/ζ1sin ϕ1 sin ϕ22,
Jp=1/2ρ1/ρ2-ρ2/ρ1sin ϕ1 sin ϕ22,
J=1/2n1/n2-n2/n1sin ϕ1 sin ϕ22.

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