Abstract

We study electromagnetic wave interactions in periodic chiral media through an examination of the band-gap structure derived from a coupled-mode solution. For oblique incidence the singly periodic chiral medium possesses three separate fundamental Bragg conditions, which lead to a richer band-gap structure than that observed for its achiral counterpart. We examine these fundamental Bragg conditions in three characteristic domains defined as the subchiral, chiral, and superchiral regions. The conditions defining each region and the band-gap characteristics are presented. The chiral band-gap structure suggests that periodic chiral media may be useful as filters, polarization mode converters, mode discriminators, and multiplexers for circularly polarized waves. It also demonstrates the increased degrees of freedom for the design of distributed-feedback or Bragg-reflection devices.

© 1996 Optical Society of America

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  1. Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philos. Mag. 24, 145–159 (1887).
  2. D. F. Arago, “Sur une modification remarquable qu’eprouvent les rayons lumineux dan leur passage a travers certains corps diaphanes, et sur quelques autres nouveaux phenomenes d’optique,” Mem. Inst. 1, 93–134 (1811).
  3. J. B. Biot, “Mémoire sur les rotations que certaines substances impriment aux axes de polarisation des rayons lumineux,” Mém. Acad.R. Sci. Inst. Fr. 2, 41 (1817).
  4. L. Pasteur, “Sur les relations qui peuvent exister entre la forme cristalline, la composition chimique et le sens de la polarisation rotatoire,” Ann. Chim. Phys. 24, 442–459 (1848).
  5. D. K. Cheng, J. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
    [CrossRef]
  6. A. Lakhtakia, V. K. Varadan, V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics 335 (Springer-Verlag, Berlin, 1989).
  7. D. L. Jaggard, N. Engheta, “Chirality in electrodynamics: modeling and applications,” in Directions in Electromagnetic Wave Modeling, H. L. Bertoni, L. B. Felsen, eds. (Plenum, New York, 1992).
  8. L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953).
  9. T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
    [CrossRef]
  10. C. Yeh, K. F. Casey, Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-13, 297–302 (1965).
    [CrossRef]
  11. D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
    [CrossRef]
  12. R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 25, 651–652 (1974).
    [CrossRef]
  13. P. S. Cross, H. Kogelnik, “Sidelobe suppression in corrugated-waveguide filters,” Opt. Lett. 1, 43–45 (1977).
    [CrossRef] [PubMed]
  14. H. Kogelnik, C. V. Shank, “Stimulated emission in a periodic structure,” Appl. Phys. Lett. 18, 152–154 (1971).
    [CrossRef]
  15. C. V. Shank, J. E. Bjorkholm, H. Kogelnik, “Tunable distributed-feedback dye laser,” Appl. Phys. Lett. 18, 395–397 (1971).
    [CrossRef]
  16. H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  17. D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,”IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
    [CrossRef]
  18. A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Propagation along the direction of inhomogeneity in an inhomogeneous chiral medium,” Int. J. Eng. Sci. 27, 1267–1273 (1989).
    [CrossRef]
  19. K. M. Flood, D. L. Jaggard, “Distributed feedback lasers in chiral media,” IEEE J. Quantum Electron. 30, 339–345 (1994).
    [CrossRef]
  20. A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Scattering by periodic achiral–chiral interfaces,” J. Opt. Soc. Am. A 6, 1675–1681 (1989); erratum 7, 951 (1990).
    [CrossRef]
  21. S. Bassiri, C. H. Papas, N. Engheta, “Electromagnetic wave propagation through a dielectric–chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
    [CrossRef]
  22. D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
    [CrossRef]
  23. W. H. Bragg, W. L. Bragg, The Crystalline State (Bell, London, 1933).
  24. D. L. Jaggard, X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A 9, 804–813 (1992).
    [CrossRef]
  25. D. L. Jaggard, C. Elachi, “Floquet and coupled-waves analysis of higher-order Bragg coupling in a periodic medium,”J. Opt. Soc. Am. 66, 674–682 (1976).
    [CrossRef]
  26. I. V. Lindell, A. H. Sihvola, “Generalized WKB approximation for stratified isotropic chiral media,”J. Electromagn. Waves Appl. 5, 857–872 (1991).
  27. A. J. Viitanen, I. V. Lindell, A. H. Sihvola, “Generalized WKB approximation for stratified isotropic chiral media with obliquely incident plane waves,”J. Electromagn. Waves Appl. 5, 1105–1121 (1991).
  28. D. L. Jaggard, A. R. Mickelson, “The reflection of electromagnetic waves from almost periodic structures,” Appl. Phys. 18, 405–412 (1979).
    [CrossRef]
  29. D. L. Jaggard, X. Sun, J. C. Liu, “On the chiral Riccati equation,” Microwave Opt. Technol. Lett. 5, 107–112 (1992).
    [CrossRef]
  30. J. E. Bjorkholm, T. P. Sosnowski, C. V. Shank, “Distributed-feedback lasers in optical waveguides deposited on anisotropic substrates,” Appl. Phys. Lett. 22, 132–134 (1973).
    [CrossRef]
  31. H. Kogelnik, C. V. Shank, J. E. Bjorkholm, “Hybrid scattering in periodic waveguides,” Appl. Phys. Lett. 22, 135–137 (1973).
    [CrossRef]
  32. A. H. Sihvola, I. V. Lindell, “Chiral Maxwell–Garnett mixing formula,” Electron. Lett. 26, 118–119 (1990).
    [CrossRef]
  33. C. Elachi, C. Yeh, “Stop bands for optical wave propagation in cholesteric liquid crystals,”J. Opt. Soc. Am. 63, 840–842 (1974).
    [CrossRef]
  34. V. A. Ambarzumian, “Diffuse reflection of light by a foggy medium,” C. R. (Dokl.) Acad. Sci. URSS 38, 229–232 (1943).
  35. Equation (A6) includes corrections to the expressions for the off-diagonal terms in the originally published coupling matrix of Ref. 24 [Eq. (7.3)] and Ref. 29 [Eq. (5)]. Errata are to be published.

1994 (1)

K. M. Flood, D. L. Jaggard, “Distributed feedback lasers in chiral media,” IEEE J. Quantum Electron. 30, 339–345 (1994).
[CrossRef]

1992 (2)

D. L. Jaggard, X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A 9, 804–813 (1992).
[CrossRef]

D. L. Jaggard, X. Sun, J. C. Liu, “On the chiral Riccati equation,” Microwave Opt. Technol. Lett. 5, 107–112 (1992).
[CrossRef]

1991 (2)

I. V. Lindell, A. H. Sihvola, “Generalized WKB approximation for stratified isotropic chiral media,”J. Electromagn. Waves Appl. 5, 857–872 (1991).

A. J. Viitanen, I. V. Lindell, A. H. Sihvola, “Generalized WKB approximation for stratified isotropic chiral media with obliquely incident plane waves,”J. Electromagn. Waves Appl. 5, 1105–1121 (1991).

1990 (1)

A. H. Sihvola, I. V. Lindell, “Chiral Maxwell–Garnett mixing formula,” Electron. Lett. 26, 118–119 (1990).
[CrossRef]

1989 (3)

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Scattering by periodic achiral–chiral interfaces,” J. Opt. Soc. Am. A 6, 1675–1681 (1989); erratum 7, 951 (1990).
[CrossRef]

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,”IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Propagation along the direction of inhomogeneity in an inhomogeneous chiral medium,” Int. J. Eng. Sci. 27, 1267–1273 (1989).
[CrossRef]

1988 (1)

1979 (2)

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

D. L. Jaggard, A. R. Mickelson, “The reflection of electromagnetic waves from almost periodic structures,” Appl. Phys. 18, 405–412 (1979).
[CrossRef]

1977 (1)

1976 (1)

1974 (3)

C. Elachi, C. Yeh, “Stop bands for optical wave propagation in cholesteric liquid crystals,”J. Opt. Soc. Am. 63, 840–842 (1974).
[CrossRef]

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
[CrossRef]

R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 25, 651–652 (1974).
[CrossRef]

1973 (2)

J. E. Bjorkholm, T. P. Sosnowski, C. V. Shank, “Distributed-feedback lasers in optical waveguides deposited on anisotropic substrates,” Appl. Phys. Lett. 22, 132–134 (1973).
[CrossRef]

H. Kogelnik, C. V. Shank, J. E. Bjorkholm, “Hybrid scattering in periodic waveguides,” Appl. Phys. Lett. 22, 135–137 (1973).
[CrossRef]

1972 (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

1971 (2)

H. Kogelnik, C. V. Shank, “Stimulated emission in a periodic structure,” Appl. Phys. Lett. 18, 152–154 (1971).
[CrossRef]

C. V. Shank, J. E. Bjorkholm, H. Kogelnik, “Tunable distributed-feedback dye laser,” Appl. Phys. Lett. 18, 395–397 (1971).
[CrossRef]

1968 (1)

D. K. Cheng, J. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
[CrossRef]

1965 (1)

C. Yeh, K. F. Casey, Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-13, 297–302 (1965).
[CrossRef]

1964 (1)

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

1943 (1)

V. A. Ambarzumian, “Diffuse reflection of light by a foggy medium,” C. R. (Dokl.) Acad. Sci. URSS 38, 229–232 (1943).

1887 (1)

Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philos. Mag. 24, 145–159 (1887).

1848 (1)

L. Pasteur, “Sur les relations qui peuvent exister entre la forme cristalline, la composition chimique et le sens de la polarisation rotatoire,” Ann. Chim. Phys. 24, 442–459 (1848).

1817 (1)

J. B. Biot, “Mémoire sur les rotations que certaines substances impriment aux axes de polarisation des rayons lumineux,” Mém. Acad.R. Sci. Inst. Fr. 2, 41 (1817).

1811 (1)

D. F. Arago, “Sur une modification remarquable qu’eprouvent les rayons lumineux dan leur passage a travers certains corps diaphanes, et sur quelques autres nouveaux phenomenes d’optique,” Mem. Inst. 1, 93–134 (1811).

Ambarzumian, V. A.

V. A. Ambarzumian, “Diffuse reflection of light by a foggy medium,” C. R. (Dokl.) Acad. Sci. URSS 38, 229–232 (1943).

Arago, D. F.

D. F. Arago, “Sur une modification remarquable qu’eprouvent les rayons lumineux dan leur passage a travers certains corps diaphanes, et sur quelques autres nouveaux phenomenes d’optique,” Mem. Inst. 1, 93–134 (1811).

Bassiri, S.

Biot, J. B.

J. B. Biot, “Mémoire sur les rotations que certaines substances impriment aux axes de polarisation des rayons lumineux,” Mém. Acad.R. Sci. Inst. Fr. 2, 41 (1817).

Bjorkholm, J. E.

H. Kogelnik, C. V. Shank, J. E. Bjorkholm, “Hybrid scattering in periodic waveguides,” Appl. Phys. Lett. 22, 135–137 (1973).
[CrossRef]

J. E. Bjorkholm, T. P. Sosnowski, C. V. Shank, “Distributed-feedback lasers in optical waveguides deposited on anisotropic substrates,” Appl. Phys. Lett. 22, 132–134 (1973).
[CrossRef]

C. V. Shank, J. E. Bjorkholm, H. Kogelnik, “Tunable distributed-feedback dye laser,” Appl. Phys. Lett. 18, 395–397 (1971).
[CrossRef]

Bragg, W. H.

W. H. Bragg, W. L. Bragg, The Crystalline State (Bell, London, 1933).

Bragg, W. L.

W. H. Bragg, W. L. Bragg, The Crystalline State (Bell, London, 1933).

Brillouin, L.

L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953).

Casey, K. F.

C. Yeh, K. F. Casey, Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-13, 297–302 (1965).
[CrossRef]

Cheng, D. K.

D. K. Cheng, J. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
[CrossRef]

Cross, P. S.

Elachi, C.

Engheta, N.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,”IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

S. Bassiri, C. H. Papas, N. Engheta, “Electromagnetic wave propagation through a dielectric–chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
[CrossRef]

D. L. Jaggard, N. Engheta, “Chirality in electrodynamics: modeling and applications,” in Directions in Electromagnetic Wave Modeling, H. L. Bertoni, L. B. Felsen, eds. (Plenum, New York, 1992).

Flanders, D. C.

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
[CrossRef]

R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 25, 651–652 (1974).
[CrossRef]

Flood, K. M.

K. M. Flood, D. L. Jaggard, “Distributed feedback lasers in chiral media,” IEEE J. Quantum Electron. 30, 339–345 (1994).
[CrossRef]

Jaggard, D. L.

K. M. Flood, D. L. Jaggard, “Distributed feedback lasers in chiral media,” IEEE J. Quantum Electron. 30, 339–345 (1994).
[CrossRef]

D. L. Jaggard, X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A 9, 804–813 (1992).
[CrossRef]

D. L. Jaggard, X. Sun, J. C. Liu, “On the chiral Riccati equation,” Microwave Opt. Technol. Lett. 5, 107–112 (1992).
[CrossRef]

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,”IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

D. L. Jaggard, A. R. Mickelson, “The reflection of electromagnetic waves from almost periodic structures,” Appl. Phys. 18, 405–412 (1979).
[CrossRef]

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

D. L. Jaggard, C. Elachi, “Floquet and coupled-waves analysis of higher-order Bragg coupling in a periodic medium,”J. Opt. Soc. Am. 66, 674–682 (1976).
[CrossRef]

D. L. Jaggard, N. Engheta, “Chirality in electrodynamics: modeling and applications,” in Directions in Electromagnetic Wave Modeling, H. L. Bertoni, L. B. Felsen, eds. (Plenum, New York, 1992).

Kaprielian, Z. A.

C. Yeh, K. F. Casey, Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-13, 297–302 (1965).
[CrossRef]

Kim, Y.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,”IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

Kogelnik, H.

P. S. Cross, H. Kogelnik, “Sidelobe suppression in corrugated-waveguide filters,” Opt. Lett. 1, 43–45 (1977).
[CrossRef] [PubMed]

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
[CrossRef]

H. Kogelnik, C. V. Shank, J. E. Bjorkholm, “Hybrid scattering in periodic waveguides,” Appl. Phys. Lett. 22, 135–137 (1973).
[CrossRef]

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

C. V. Shank, J. E. Bjorkholm, H. Kogelnik, “Tunable distributed-feedback dye laser,” Appl. Phys. Lett. 18, 395–397 (1971).
[CrossRef]

H. Kogelnik, C. V. Shank, “Stimulated emission in a periodic structure,” Appl. Phys. Lett. 18, 152–154 (1971).
[CrossRef]

Kong, J.

D. K. Cheng, J. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
[CrossRef]

Kowarz, M. W.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,”IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Scattering by periodic achiral–chiral interfaces,” J. Opt. Soc. Am. A 6, 1675–1681 (1989); erratum 7, 951 (1990).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Propagation along the direction of inhomogeneity in an inhomogeneous chiral medium,” Int. J. Eng. Sci. 27, 1267–1273 (1989).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics 335 (Springer-Verlag, Berlin, 1989).

Lindell, I. V.

A. J. Viitanen, I. V. Lindell, A. H. Sihvola, “Generalized WKB approximation for stratified isotropic chiral media with obliquely incident plane waves,”J. Electromagn. Waves Appl. 5, 1105–1121 (1991).

I. V. Lindell, A. H. Sihvola, “Generalized WKB approximation for stratified isotropic chiral media,”J. Electromagn. Waves Appl. 5, 857–872 (1991).

A. H. Sihvola, I. V. Lindell, “Chiral Maxwell–Garnett mixing formula,” Electron. Lett. 26, 118–119 (1990).
[CrossRef]

Liu, J. C.

D. L. Jaggard, X. Sun, J. C. Liu, “On the chiral Riccati equation,” Microwave Opt. Technol. Lett. 5, 107–112 (1992).
[CrossRef]

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,”IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

Mickelson, A. R.

D. L. Jaggard, A. R. Mickelson, “The reflection of electromagnetic waves from almost periodic structures,” Appl. Phys. 18, 405–412 (1979).
[CrossRef]

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

Oliner, A. A.

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Papas, C. H.

Pasteur, L.

L. Pasteur, “Sur les relations qui peuvent exister entre la forme cristalline, la composition chimique et le sens de la polarisation rotatoire,” Ann. Chim. Phys. 24, 442–459 (1848).

Pelet, P.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,”IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

Rayleigh,

Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philos. Mag. 24, 145–159 (1887).

Schmidt, R. V.

R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 25, 651–652 (1974).
[CrossRef]

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
[CrossRef]

Shank, C. V.

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
[CrossRef]

R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 25, 651–652 (1974).
[CrossRef]

H. Kogelnik, C. V. Shank, J. E. Bjorkholm, “Hybrid scattering in periodic waveguides,” Appl. Phys. Lett. 22, 135–137 (1973).
[CrossRef]

J. E. Bjorkholm, T. P. Sosnowski, C. V. Shank, “Distributed-feedback lasers in optical waveguides deposited on anisotropic substrates,” Appl. Phys. Lett. 22, 132–134 (1973).
[CrossRef]

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

C. V. Shank, J. E. Bjorkholm, H. Kogelnik, “Tunable distributed-feedback dye laser,” Appl. Phys. Lett. 18, 395–397 (1971).
[CrossRef]

H. Kogelnik, C. V. Shank, “Stimulated emission in a periodic structure,” Appl. Phys. Lett. 18, 152–154 (1971).
[CrossRef]

Sihvola, A. H.

I. V. Lindell, A. H. Sihvola, “Generalized WKB approximation for stratified isotropic chiral media,”J. Electromagn. Waves Appl. 5, 857–872 (1991).

A. J. Viitanen, I. V. Lindell, A. H. Sihvola, “Generalized WKB approximation for stratified isotropic chiral media with obliquely incident plane waves,”J. Electromagn. Waves Appl. 5, 1105–1121 (1991).

A. H. Sihvola, I. V. Lindell, “Chiral Maxwell–Garnett mixing formula,” Electron. Lett. 26, 118–119 (1990).
[CrossRef]

Sosnowski, T. P.

J. E. Bjorkholm, T. P. Sosnowski, C. V. Shank, “Distributed-feedback lasers in optical waveguides deposited on anisotropic substrates,” Appl. Phys. Lett. 22, 132–134 (1973).
[CrossRef]

Standley, R. D.

R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 25, 651–652 (1974).
[CrossRef]

Sun, X.

D. L. Jaggard, X. Sun, J. C. Liu, “On the chiral Riccati equation,” Microwave Opt. Technol. Lett. 5, 107–112 (1992).
[CrossRef]

D. L. Jaggard, X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A 9, 804–813 (1992).
[CrossRef]

Tamir, T.

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Varadan, V. K.

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Scattering by periodic achiral–chiral interfaces,” J. Opt. Soc. Am. A 6, 1675–1681 (1989); erratum 7, 951 (1990).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Propagation along the direction of inhomogeneity in an inhomogeneous chiral medium,” Int. J. Eng. Sci. 27, 1267–1273 (1989).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics 335 (Springer-Verlag, Berlin, 1989).

Varadan, V. V.

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Scattering by periodic achiral–chiral interfaces,” J. Opt. Soc. Am. A 6, 1675–1681 (1989); erratum 7, 951 (1990).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Propagation along the direction of inhomogeneity in an inhomogeneous chiral medium,” Int. J. Eng. Sci. 27, 1267–1273 (1989).
[CrossRef]

A. Lakhtakia, V. K. Varadan, V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics 335 (Springer-Verlag, Berlin, 1989).

Viitanen, A. J.

A. J. Viitanen, I. V. Lindell, A. H. Sihvola, “Generalized WKB approximation for stratified isotropic chiral media with obliquely incident plane waves,”J. Electromagn. Waves Appl. 5, 1105–1121 (1991).

Wang, H. C.

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Equation (A6) includes corrections to the expressions for the off-diagonal terms in the originally published coupling matrix of Ref. 24 [Eq. (7.3)] and Ref. 29 [Eq. (5)]. Errata are to be published.

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Figures (7)

Fig. 1
Fig. 1

Geometry for the periodic chiral media analysis. The four modes corresponding to the mode amplitudes F+(z), F(z), B+(z), and B(z) are shown schematically along with the coupling coefficients that connect them.

Fig. 2
Fig. 2

Comparison of coupling coefficients for achiral and chiral periodic media. For all cases = 40, μ = μ0, and ψL/λ0 = 1. Plot (a) shows the cross-polarization and copolarization coupling coefficients for the achiral case. In the remaining plots ξ ¯ c = 0.05, ψc = 0 in (b) and (c), and ψcL/λ0 = 10 in (d). For the chiral case the LCP critical angle occurs at a RCP incident angle of 64.8°.

Fig. 3
Fig. 3

Stop-band regions for a periodic chiral medium with /0 = 4, μ/μ0 = 1, ξ ¯ c = 0.001, ψ = ψc = 0.001, ψμ = 0, and L/λ0 = 1000. The cross-hatched parts of (a) and (b) show the regions in which the first and the second eigenvalues, respectively, are complex.

Fig. 4
Fig. 4

Band-gap diagrams for a periodic chiral medium for which χ++L = χ−−L = 0.5 and χ+−L = χ−+L = −1 for (a)–(d) and χ++L = 1.5, χ−+ = −0.8, χ+− = −1.2, and χ−− = 0.5 for (d). The degree of chirality, as measured by the relative phase mismatch, was varied for each plot so that (δ++δ−−)L is (a) 0, subchiral regime; (b) 0.5, chiral regime; (c) 1, chiral regime; and (d) and (e) 4, superchiral regime. The eigenvalue roots of Eq. (15) are identified in (a) and (c); (d) shows the association between the eigenmodes and the eigenvalues.

Fig. 5
Fig. 5

Band-gap diagrams [(a) and (b)] and field amplitudes [(c) and (d)], F+(L), F(L), B+(0), and B(0), for a periodic chiral medium with χ++L = χ−−L = 0.5, χ+−L = χ−+L = −1, and (δ++δ−−)L = 8. The field amplitudes and the medium boundaries are given for (c) an incident RCP wave and (d) an incident LCP wave. The stop-band regions are shaded, and the Bragg conditions are identified with arrows.

Fig. 6
Fig. 6

Band-gap diagrams [(a) and (b)] and mode amplitudes [(c) and (d)] for a periodic chiral medium in the LCP total-internal-reflection region. All coupling strength magnitudes were set equal to unity, where χ++L = 1, χ - + L = - ( 1 + i ) / 2, χ + - L = - ( 1 - i ) / 2, and χ−−L = −1. The relative phase angles of the cross-polarization coupling coefficients are of little significance provided that their overall magnitudes are not excessive. For the mode amplitude graphs (c) is the response for an incident RCP wave, and (d) is the response when F(0) = 1.

Fig. 7
Fig. 7

Illustration of the computational technique used to determine the matrix reflection and transmission coefficients for the chiral Riccati equation. The drawing shows an infinitesimally thin section of width Δz in a stratified chiral medium.

Equations (40)

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D = E + i ξ c B ,
H = B / μ + i ξ c E ,
k ± = ± ω μ ξ c + [ ( ω μ ξ c ) 2 + ω 2 μ ] 1 / 2 ,
k z = n K 2 ,             n = 1 , 2 , 3 ,
{ k z + + k z - 2 k z + k z - } = n K 2 ,
( ɛ ( z ) μ ( z ) ξ c ( z ) ) = ( ɛ μ ξ c ) [ 1 + ( ψ ψ μ ψ c ) cos ( K z ) ] ,
E ( x , z ) = F + ( z ) exp { i [ k x x + ( K / 2 ) z ] } × [ ( cos θ + ) e ^ x + i e ^ y - ( sin θ + ) e ^ z ] + F - ( z ) exp { i [ k x x + ( K / 2 ) z ] } × [ ( cos θ - ) e ^ x - i e ^ y - ( sin θ - ) e ^ z ] + B + ( z ) exp { i [ k x x - ( K / 2 ) z ] } × [ ( cos θ + ) e ^ x - i e ^ y + ( sin θ + ) e ^ z ] + B - ( z ) exp { i [ k x x - ( K / 2 ) z ] } × [ ( cos θ - ) e ^ x + i e ^ y + ( sin θ - ) e ^ z ] ,
d F + d z - i δ + + F + = i χ + + B + + i χ - + B - ,
d F - d z - i δ - - F - = i χ + - B + + i χ - - B - ,
- d B + d z - i δ + + B + = i χ + + F + + i χ - + F - ,
- d B - d z - i δ - - B - = i χ + - F + + i χ - - F - ,
δ ± ± = k z ± - K 2 ,
χ ± ± = 1 4 ( k x k z ± ) 2 [ k ± ( ψ μ + ψ c ) + k ( ψ - ψ c ) ] ,
χ ± = 1 4 [ k + k - k + + k - ( ψ μ - ψ ) - 1 2 ( k + - k - ) 2 k + + k - ψ c ] × ( 1 + k k ± k k z ± k z ) .
θ - = sin - 1 ( k + k - sin θ + ) .
θ + = sin - 1 ( 1 - 2 ξ ¯ c 1 + ξ ¯ c 2 + 2 ξ ¯ c 2 ) π 2 ( 1 - 2 ξ ¯ c ) ,
| δ + + - γ 0 χ + + χ - + 0 δ - - - γ χ + - χ - - - χ + + - χ - + - δ + + - γ 0 - χ + - - χ - - 0 - δ - - - γ | = 0.
γ 2 1 = ( δ + - 2 + ( δ + + - δ - - 2 ) 2 - χ + + 2 + 2 χ + - χ - + + χ - - 2 2 ± { [ δ + - ( δ + + - δ - - ) - χ + + 2 - χ - - 2 2 ] 2 + [ ( χ + + + χ - - ) 2 - ( δ + + - δ - - ) 2 ] χ + - χ - + } 1 / 2 ) 1 / 2 .
δ + + - δ - - | 2 ( χ + + 2 + 2 χ + - χ - + + χ - - 2 ) | { subchiral chiral superchiral } .
γ 2 1 [ δ + - 2 - χ + + 2 + 2 χ + - χ - + + χ - - 2 2 ± ( χ + + + χ - - ) × ( χ + + - χ - - 2 ) 2 + χ + - + χ - + ] 1 / 2 .
γ 2 1 [ δ + - 2 + ( δ + + - δ - - 2 ) 2 - χ + + 2 + 2 χ + - χ - + + χ - - 2 2 ± ( δ + + + δ - - ) δ + - 2 - δ + - χ + + 2 - χ - - 2 δ + + - δ - - - χ + - χ - + ] 1 / 2 .
γ 1 ( δ + + 2 + δ + + χ + + 2 - 2 χ + - χ - + - χ - - 2 δ + + - δ - - - χ + + 2 ) 1 / 2 ,
γ 2 - δ - - .
γ 1 δ + + ,
γ 2 ( δ - - 2 + δ - - χ + + 2 + 2 χ + - χ - + - χ - - 2 δ + + - δ - - - χ - - 2 ) 1 / 2 ,
E ( x , z ) = F ˜ + ( z ) exp [ i k x ( z ) x ] { cos [ θ + ( z ) ] e ^ x + i e ^ y - sin [ θ + ( z ) ] e ^ z } + F ˜ - ( z ) exp [ i k x ( z ) x ] { cos [ θ - ( z ) ] e ^ x - i e ^ y - sin [ θ - ( z ) ] e ^ z } + B ˜ + ( z ) exp [ i k x ( z ) x ] { cos [ θ + ( z ) ] e ^ x - i e ^ y + sin [ θ + ( z ) ] e ^ z } + B ˜ - ( z ) exp [ i k x ( z ) x ] { cos [ θ - ( z ) ] e ^ x + i e ^ y + sin [ θ - ( z ) ] e ^ z } ,
B ˜ + = r ˜ + + F ˜ + + r ˜ - + F ˜ - ,
B ˜ - = r ˜ + - F ˜ + + r ˜ - - F ˜ - .
d d z R ˜ ̳ = χ ˜ ̳ - R ˜ ̳ χ ˜ ̳ R ˜ ̳ - i ( κ ̳ R ˜ ̳ + R ˜ ̳ κ ̳ ) ,
R ˜ ̳ = [ r ˜ + + r ˜ - + r ˜ + - r ˜ - - ] ,
χ ˜ ̳ = [ 1 2 k + d k + d z ( k x k z + ) 2 1 2 η c d η c d z ( 1 + k + k - k z - k z + ) 1 2 η c d η c d z ( 1 + k - k + k z + k z - ) 1 2 k - d k - d z ( k x k z - ) 2 ] ,
κ ̳ = [ k z + 0 0 k z - ] ,
( ( z ) μ ( z ) ξ c ( z ) ) = ( μ ξ c ) [ 1 + ( ψ ψ μ ψ c ) cos ( K z ) ] ,
F ± ( z ) = F ˜ ± ( z ) exp ( - i K 2 z ) ,
B ± ( z ) = B ˜ ± ( z ) exp ( i K 2 z ) .
d d z R ̳ = - i χ ̳ - i R ̳ χ ̳ R ̳ - i ( δ ̳ R ̳ + R δ ̳ ) ,
χ ̳ = [ χ + + χ - + χ + - χ - - ] ,
δ = [ δ + + 0 0 δ - - ] .
d d z F - i δ F = i χ B ,
- d d z B - i δ B = i χ F .

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