Abstract

The general solution for modes in an asymmetric planar waveguide with a homogeneous and isotropic chiral core is given in terms of a pair of parameters related to the eccentricity of the polarization ellipse for the transverse electric field. This formulation provides insight into the transition, with increasing chirality of the core, from TE/TM modes to right-handed and left-handed circular polarization modes. Mode polarization as a function of waveguide thickness and of frequency is discussed in detail. Beyond a mode-dependent maximum thickness (or frequency), the left-handed elliptical modes consist of a slow-wave component whose cutoff properties are examined. The limiting case of a symmetric waveguide is also discussed.

© 2001 Optical Society of America

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References

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  1. D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
    [CrossRef]
  2. A. Lahtakia, V. K. Varadan, V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics Series 335 (Springer-Verlag, Berlin, 1989).
  3. N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE Antennas Propag. Soc. Newsl. 30, 6–12 (1988).
    [CrossRef]
  4. V. K. Varadan, A. Lakhtakia, V. V. Varadan, “Propagation in a parallel plate waveguide wholly filled with a chiral medium,” J. Wave Mater. Interact. 3, 267–272 (1988).
  5. N. Engheta, P. Pelet, “Modes in chirowaveguides,” Opt. Lett. 14, 593–595 (1989).
    [CrossRef] [PubMed]
  6. P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
    [CrossRef]
  7. L. Zhang, Y. Jiao, C. Liang, “The dominant mode on parallel-plate chirowaveguides,” IEEE Trans. Microwave Theory Tech. 42, 2009–2012 (1994).
    [CrossRef]
  8. See, for example, A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Sec. 4.9.
  9. For a recent review, see M. M. Green, J.-W. Park, T. Sato, S. Lifson, R. L. B. Selinger, J. V. Selinger, Angew. Chem. Int. Ed. Engl. 38, 3138–3154 (1999).
    [CrossRef] [PubMed]
  10. Highly chiral oligomers have been realized, for example, in metallocenes [T. J. Katz, A. Sadhakar, M. F. Teasley, A. M. Gilbert, W. E. Geiger, M. P. Robben, M. Wuensch, M. D. Ward, “Synthesis and properties of optically active helical metallocene oligomers,” J. Am. Chem. Soc. 115, 3182–3198 (1993)], but additional requirements of low optical loss and incorporation into polymers with good film-forming properties must be met.
    [CrossRef]
  11. M. Chien, Y. Kim, H. Grebel, “Mode conversion in optically active and isotropic waveguides,” Opt. Lett. 14, 826–828 (1989).
    [CrossRef] [PubMed]
  12. H. Corey, I. Rosenhouse, “Electromagnetic wave propagation along a chiral slab,” IEE Proc. H, 138, 51–54 (1991).
  13. M. Oksanen, P. K. Loivisto, L. V. Lindell, “Dispersion curves and fields for a chiral slab waveguide,” IEE Proc. H 138, 327–334 (1991).
  14. M. I. Oksanen, P. V. Koivisto, S. A. Tretyakov, “Vector circuit method applied for chiral slab waveguides,” J. Lightwave Technol. 10, 150–155 (1992).
    [CrossRef]
  15. J. Xiao, K. Zhang, L. Gong, “Field analysis of a general chiral planar waveguide,” Int. J. Infrared Millim. Waves 18, 939–948 (1997).
    [CrossRef]
  16. S. F. Mahmoud, “Guided modes on open chirowaveguides,” IEEE Trans. Microwave Theory Tech. 43, 205–209 (1995).
    [CrossRef]
  17. K. M. Flood, D. L. Jaggard, “Single-mode operation in symmetric planar waveguides using isotropic chiral media,” Opt. Lett. 21, 474–476 (1996).
    [CrossRef] [PubMed]
  18. S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film devices for integrated optics,” J. Appl. Phys. 43, 5090–5097 (1972).
    [CrossRef]
  19. P. Pelet, N. Engheta, “Coupled-mode theory for chiral waveguides,” J. Appl. Phys. 67, 2742–2745 (1990).
    [CrossRef]
  20. M. I. Oksanen, S. A. Tretyakov, I. V. Lindell, “Vector circuit theory for isotropic and chiral slabs,” J. Electromagn. Waves Appl. 4, 613–643 (1990).
  21. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Norwood, Mass., 1994), Chap. 4.
  22. S. V. Demidov, K. V. Kushnarev, V. V. Shevchenko, “Dispersion properties of the modes of chiral planar optical waveguides,” J. Commun. Technol. Electron. 44, 827–832 (1999) [translated from Radiotekh. Elektron. (Moscow) 44, 885–890 (1999)].
  23. For a discussion of various forms for chiral constitutive equations, see Ref. 2, Chap. 3 and Ref. 21, Chap. 1.
  24. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974) and“Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62, 1566–1571 (1975). See also the discussion in Ref. 2, Chap. 7.
    [CrossRef]
  25. Here we adopt the convention for circularly polarized light used in most optics texts: For RHC (LHC) polarization, the electric field at a fixed position rotates clockwise (counterclockwise) when viewed from the direction toward which the wave is traveling. The opposite designation is used in much of the engineering literature, including Refs. 12-14 cited here. For a discussion, see B. E. A. Salah, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Sec. 6.1.
  26. C. R. Paiva, A. L. Topa, A. M. Barbosa, “Semileaky waves in dielectric chirowaveguides,” Opt. Lett. 17, 1670–1672 (1992).
    [CrossRef] [PubMed]
  27. See, for example, P. K. Cheo, Fiber Optics and Optoelectron-ics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1990), Chap. 3.
  28. Solutions for all examples in this paper were obtained by using built-in routines in Mathcad 2000 Professional (MathSoft, Inc., Cambridge, Mass.).
  29. The TE- and TM-mode equations are normally written by using arctangent rather than arccotangent. However, the arguments of the arctangent are positive definite, and arctan x=arccot(1/x) for x>0,as is true for the achiral case. There is an advantage to using arccotangent in the chiral case because singularities of the argument cause π discontinuities of arccotangent at only the endpoints of the range |g|<1,while if arctangent is used, there are π discontinuities at g=±r0 inside this range.
  30. This merging of effective index curves was first noted in Ref. 12 without discussion of the polarization effects.

1999 (2)

For a recent review, see M. M. Green, J.-W. Park, T. Sato, S. Lifson, R. L. B. Selinger, J. V. Selinger, Angew. Chem. Int. Ed. Engl. 38, 3138–3154 (1999).
[CrossRef] [PubMed]

S. V. Demidov, K. V. Kushnarev, V. V. Shevchenko, “Dispersion properties of the modes of chiral planar optical waveguides,” J. Commun. Technol. Electron. 44, 827–832 (1999) [translated from Radiotekh. Elektron. (Moscow) 44, 885–890 (1999)].

1997 (1)

J. Xiao, K. Zhang, L. Gong, “Field analysis of a general chiral planar waveguide,” Int. J. Infrared Millim. Waves 18, 939–948 (1997).
[CrossRef]

1996 (1)

1995 (1)

S. F. Mahmoud, “Guided modes on open chirowaveguides,” IEEE Trans. Microwave Theory Tech. 43, 205–209 (1995).
[CrossRef]

1994 (1)

L. Zhang, Y. Jiao, C. Liang, “The dominant mode on parallel-plate chirowaveguides,” IEEE Trans. Microwave Theory Tech. 42, 2009–2012 (1994).
[CrossRef]

1993 (1)

Highly chiral oligomers have been realized, for example, in metallocenes [T. J. Katz, A. Sadhakar, M. F. Teasley, A. M. Gilbert, W. E. Geiger, M. P. Robben, M. Wuensch, M. D. Ward, “Synthesis and properties of optically active helical metallocene oligomers,” J. Am. Chem. Soc. 115, 3182–3198 (1993)], but additional requirements of low optical loss and incorporation into polymers with good film-forming properties must be met.
[CrossRef]

1992 (2)

M. I. Oksanen, P. V. Koivisto, S. A. Tretyakov, “Vector circuit method applied for chiral slab waveguides,” J. Lightwave Technol. 10, 150–155 (1992).
[CrossRef]

C. R. Paiva, A. L. Topa, A. M. Barbosa, “Semileaky waves in dielectric chirowaveguides,” Opt. Lett. 17, 1670–1672 (1992).
[CrossRef] [PubMed]

1991 (2)

H. Corey, I. Rosenhouse, “Electromagnetic wave propagation along a chiral slab,” IEE Proc. H, 138, 51–54 (1991).

M. Oksanen, P. K. Loivisto, L. V. Lindell, “Dispersion curves and fields for a chiral slab waveguide,” IEE Proc. H 138, 327–334 (1991).

1990 (3)

P. Pelet, N. Engheta, “Coupled-mode theory for chiral waveguides,” J. Appl. Phys. 67, 2742–2745 (1990).
[CrossRef]

M. I. Oksanen, S. A. Tretyakov, I. V. Lindell, “Vector circuit theory for isotropic and chiral slabs,” J. Electromagn. Waves Appl. 4, 613–643 (1990).

P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
[CrossRef]

1989 (2)

1988 (2)

N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE Antennas Propag. Soc. Newsl. 30, 6–12 (1988).
[CrossRef]

V. K. Varadan, A. Lakhtakia, V. V. Varadan, “Propagation in a parallel plate waveguide wholly filled with a chiral medium,” J. Wave Mater. Interact. 3, 267–272 (1988).

1979 (1)

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

1974 (1)

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974) and“Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62, 1566–1571 (1975). See also the discussion in Ref. 2, Chap. 7.
[CrossRef]

1972 (1)

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film devices for integrated optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

Barbosa, A. M.

Bohren, C. F.

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974) and“Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62, 1566–1571 (1975). See also the discussion in Ref. 2, Chap. 7.
[CrossRef]

Cheo, P. K.

See, for example, P. K. Cheo, Fiber Optics and Optoelectron-ics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1990), Chap. 3.

Chien, M.

Corey, H.

H. Corey, I. Rosenhouse, “Electromagnetic wave propagation along a chiral slab,” IEE Proc. H, 138, 51–54 (1991).

Demidov, S. V.

S. V. Demidov, K. V. Kushnarev, V. V. Shevchenko, “Dispersion properties of the modes of chiral planar optical waveguides,” J. Commun. Technol. Electron. 44, 827–832 (1999) [translated from Radiotekh. Elektron. (Moscow) 44, 885–890 (1999)].

Engheta, N.

P. Pelet, N. Engheta, “Coupled-mode theory for chiral waveguides,” J. Appl. Phys. 67, 2742–2745 (1990).
[CrossRef]

P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
[CrossRef]

N. Engheta, P. Pelet, “Modes in chirowaveguides,” Opt. Lett. 14, 593–595 (1989).
[CrossRef] [PubMed]

N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE Antennas Propag. Soc. Newsl. 30, 6–12 (1988).
[CrossRef]

Flood, K. M.

Geiger, W. E.

Highly chiral oligomers have been realized, for example, in metallocenes [T. J. Katz, A. Sadhakar, M. F. Teasley, A. M. Gilbert, W. E. Geiger, M. P. Robben, M. Wuensch, M. D. Ward, “Synthesis and properties of optically active helical metallocene oligomers,” J. Am. Chem. Soc. 115, 3182–3198 (1993)], but additional requirements of low optical loss and incorporation into polymers with good film-forming properties must be met.
[CrossRef]

Gilbert, A. M.

Highly chiral oligomers have been realized, for example, in metallocenes [T. J. Katz, A. Sadhakar, M. F. Teasley, A. M. Gilbert, W. E. Geiger, M. P. Robben, M. Wuensch, M. D. Ward, “Synthesis and properties of optically active helical metallocene oligomers,” J. Am. Chem. Soc. 115, 3182–3198 (1993)], but additional requirements of low optical loss and incorporation into polymers with good film-forming properties must be met.
[CrossRef]

Gong, L.

J. Xiao, K. Zhang, L. Gong, “Field analysis of a general chiral planar waveguide,” Int. J. Infrared Millim. Waves 18, 939–948 (1997).
[CrossRef]

Grebel, H.

Green, M. M.

For a recent review, see M. M. Green, J.-W. Park, T. Sato, S. Lifson, R. L. B. Selinger, J. V. Selinger, Angew. Chem. Int. Ed. Engl. 38, 3138–3154 (1999).
[CrossRef] [PubMed]

Jaggard, D. L.

K. M. Flood, D. L. Jaggard, “Single-mode operation in symmetric planar waveguides using isotropic chiral media,” Opt. Lett. 21, 474–476 (1996).
[CrossRef] [PubMed]

N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE Antennas Propag. Soc. Newsl. 30, 6–12 (1988).
[CrossRef]

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

Jiao, Y.

L. Zhang, Y. Jiao, C. Liang, “The dominant mode on parallel-plate chirowaveguides,” IEEE Trans. Microwave Theory Tech. 42, 2009–2012 (1994).
[CrossRef]

Katz, T. J.

Highly chiral oligomers have been realized, for example, in metallocenes [T. J. Katz, A. Sadhakar, M. F. Teasley, A. M. Gilbert, W. E. Geiger, M. P. Robben, M. Wuensch, M. D. Ward, “Synthesis and properties of optically active helical metallocene oligomers,” J. Am. Chem. Soc. 115, 3182–3198 (1993)], but additional requirements of low optical loss and incorporation into polymers with good film-forming properties must be met.
[CrossRef]

Kim, Y.

Koivisto, P. V.

M. I. Oksanen, P. V. Koivisto, S. A. Tretyakov, “Vector circuit method applied for chiral slab waveguides,” J. Lightwave Technol. 10, 150–155 (1992).
[CrossRef]

Koyamada, Y.

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film devices for integrated optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

Kushnarev, K. V.

S. V. Demidov, K. V. Kushnarev, V. V. Shevchenko, “Dispersion properties of the modes of chiral planar optical waveguides,” J. Commun. Technol. Electron. 44, 827–832 (1999) [translated from Radiotekh. Elektron. (Moscow) 44, 885–890 (1999)].

Lahtakia, A.

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

Lakhtakia, A.

V. K. Varadan, A. Lakhtakia, V. V. Varadan, “Propagation in a parallel plate waveguide wholly filled with a chiral medium,” J. Wave Mater. Interact. 3, 267–272 (1988).

Liang, C.

L. Zhang, Y. Jiao, C. Liang, “The dominant mode on parallel-plate chirowaveguides,” IEEE Trans. Microwave Theory Tech. 42, 2009–2012 (1994).
[CrossRef]

Lifson, S.

For a recent review, see M. M. Green, J.-W. Park, T. Sato, S. Lifson, R. L. B. Selinger, J. V. Selinger, Angew. Chem. Int. Ed. Engl. 38, 3138–3154 (1999).
[CrossRef] [PubMed]

Lindell, I. V.

M. I. Oksanen, S. A. Tretyakov, I. V. Lindell, “Vector circuit theory for isotropic and chiral slabs,” J. Electromagn. Waves Appl. 4, 613–643 (1990).

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Norwood, Mass., 1994), Chap. 4.

Lindell, L. V.

M. Oksanen, P. K. Loivisto, L. V. Lindell, “Dispersion curves and fields for a chiral slab waveguide,” IEE Proc. H 138, 327–334 (1991).

Loivisto, P. K.

M. Oksanen, P. K. Loivisto, L. V. Lindell, “Dispersion curves and fields for a chiral slab waveguide,” IEE Proc. H 138, 327–334 (1991).

Mahmoud, S. F.

S. F. Mahmoud, “Guided modes on open chirowaveguides,” IEEE Trans. Microwave Theory Tech. 43, 205–209 (1995).
[CrossRef]

Makimoto, T.

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film devices for integrated optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

Mickelson, A. R.

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

Oksanen, M.

M. Oksanen, P. K. Loivisto, L. V. Lindell, “Dispersion curves and fields for a chiral slab waveguide,” IEE Proc. H 138, 327–334 (1991).

Oksanen, M. I.

M. I. Oksanen, P. V. Koivisto, S. A. Tretyakov, “Vector circuit method applied for chiral slab waveguides,” J. Lightwave Technol. 10, 150–155 (1992).
[CrossRef]

M. I. Oksanen, S. A. Tretyakov, I. V. Lindell, “Vector circuit theory for isotropic and chiral slabs,” J. Electromagn. Waves Appl. 4, 613–643 (1990).

Paiva, C. R.

Papas, C. H.

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

Park, J.-W.

For a recent review, see M. M. Green, J.-W. Park, T. Sato, S. Lifson, R. L. B. Selinger, J. V. Selinger, Angew. Chem. Int. Ed. Engl. 38, 3138–3154 (1999).
[CrossRef] [PubMed]

Pelet, P.

P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
[CrossRef]

P. Pelet, N. Engheta, “Coupled-mode theory for chiral waveguides,” J. Appl. Phys. 67, 2742–2745 (1990).
[CrossRef]

N. Engheta, P. Pelet, “Modes in chirowaveguides,” Opt. Lett. 14, 593–595 (1989).
[CrossRef] [PubMed]

Robben, M. P.

Highly chiral oligomers have been realized, for example, in metallocenes [T. J. Katz, A. Sadhakar, M. F. Teasley, A. M. Gilbert, W. E. Geiger, M. P. Robben, M. Wuensch, M. D. Ward, “Synthesis and properties of optically active helical metallocene oligomers,” J. Am. Chem. Soc. 115, 3182–3198 (1993)], but additional requirements of low optical loss and incorporation into polymers with good film-forming properties must be met.
[CrossRef]

Rosenhouse, I.

H. Corey, I. Rosenhouse, “Electromagnetic wave propagation along a chiral slab,” IEE Proc. H, 138, 51–54 (1991).

Sadhakar, A.

Highly chiral oligomers have been realized, for example, in metallocenes [T. J. Katz, A. Sadhakar, M. F. Teasley, A. M. Gilbert, W. E. Geiger, M. P. Robben, M. Wuensch, M. D. Ward, “Synthesis and properties of optically active helical metallocene oligomers,” J. Am. Chem. Soc. 115, 3182–3198 (1993)], but additional requirements of low optical loss and incorporation into polymers with good film-forming properties must be met.
[CrossRef]

Salah, B. E. A.

Here we adopt the convention for circularly polarized light used in most optics texts: For RHC (LHC) polarization, the electric field at a fixed position rotates clockwise (counterclockwise) when viewed from the direction toward which the wave is traveling. The opposite designation is used in much of the engineering literature, including Refs. 12-14 cited here. For a discussion, see B. E. A. Salah, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Sec. 6.1.

Sato, T.

For a recent review, see M. M. Green, J.-W. Park, T. Sato, S. Lifson, R. L. B. Selinger, J. V. Selinger, Angew. Chem. Int. Ed. Engl. 38, 3138–3154 (1999).
[CrossRef] [PubMed]

Selinger, J. V.

For a recent review, see M. M. Green, J.-W. Park, T. Sato, S. Lifson, R. L. B. Selinger, J. V. Selinger, Angew. Chem. Int. Ed. Engl. 38, 3138–3154 (1999).
[CrossRef] [PubMed]

Selinger, R. L. B.

For a recent review, see M. M. Green, J.-W. Park, T. Sato, S. Lifson, R. L. B. Selinger, J. V. Selinger, Angew. Chem. Int. Ed. Engl. 38, 3138–3154 (1999).
[CrossRef] [PubMed]

Shevchenko, V. V.

S. V. Demidov, K. V. Kushnarev, V. V. Shevchenko, “Dispersion properties of the modes of chiral planar optical waveguides,” J. Commun. Technol. Electron. 44, 827–832 (1999) [translated from Radiotekh. Elektron. (Moscow) 44, 885–890 (1999)].

Sihvola, A. H.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Norwood, Mass., 1994), Chap. 4.

Teasley, M. F.

Highly chiral oligomers have been realized, for example, in metallocenes [T. J. Katz, A. Sadhakar, M. F. Teasley, A. M. Gilbert, W. E. Geiger, M. P. Robben, M. Wuensch, M. D. Ward, “Synthesis and properties of optically active helical metallocene oligomers,” J. Am. Chem. Soc. 115, 3182–3198 (1993)], but additional requirements of low optical loss and incorporation into polymers with good film-forming properties must be met.
[CrossRef]

Teich, M. C.

Here we adopt the convention for circularly polarized light used in most optics texts: For RHC (LHC) polarization, the electric field at a fixed position rotates clockwise (counterclockwise) when viewed from the direction toward which the wave is traveling. The opposite designation is used in much of the engineering literature, including Refs. 12-14 cited here. For a discussion, see B. E. A. Salah, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Sec. 6.1.

Topa, A. L.

Tretyakov, S. A.

M. I. Oksanen, P. V. Koivisto, S. A. Tretyakov, “Vector circuit method applied for chiral slab waveguides,” J. Lightwave Technol. 10, 150–155 (1992).
[CrossRef]

M. I. Oksanen, S. A. Tretyakov, I. V. Lindell, “Vector circuit theory for isotropic and chiral slabs,” J. Electromagn. Waves Appl. 4, 613–643 (1990).

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Norwood, Mass., 1994), Chap. 4.

Varadan, V. K.

V. K. Varadan, A. Lakhtakia, V. V. Varadan, “Propagation in a parallel plate waveguide wholly filled with a chiral medium,” J. Wave Mater. Interact. 3, 267–272 (1988).

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

Varadan, V. V.

V. K. Varadan, A. Lakhtakia, V. V. Varadan, “Propagation in a parallel plate waveguide wholly filled with a chiral medium,” J. Wave Mater. Interact. 3, 267–272 (1988).

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

Viitanen, A. J.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Norwood, Mass., 1994), Chap. 4.

Ward, M. D.

Highly chiral oligomers have been realized, for example, in metallocenes [T. J. Katz, A. Sadhakar, M. F. Teasley, A. M. Gilbert, W. E. Geiger, M. P. Robben, M. Wuensch, M. D. Ward, “Synthesis and properties of optically active helical metallocene oligomers,” J. Am. Chem. Soc. 115, 3182–3198 (1993)], but additional requirements of low optical loss and incorporation into polymers with good film-forming properties must be met.
[CrossRef]

Wuensch, M.

Highly chiral oligomers have been realized, for example, in metallocenes [T. J. Katz, A. Sadhakar, M. F. Teasley, A. M. Gilbert, W. E. Geiger, M. P. Robben, M. Wuensch, M. D. Ward, “Synthesis and properties of optically active helical metallocene oligomers,” J. Am. Chem. Soc. 115, 3182–3198 (1993)], but additional requirements of low optical loss and incorporation into polymers with good film-forming properties must be met.
[CrossRef]

Xiao, J.

J. Xiao, K. Zhang, L. Gong, “Field analysis of a general chiral planar waveguide,” Int. J. Infrared Millim. Waves 18, 939–948 (1997).
[CrossRef]

Yamamoto, S.

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-mode analysis of anisotropic and gyrotropic thin-film devices for integrated optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

Yariv, A.

See, for example, A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Sec. 4.9.

Yeh, P.

See, for example, A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Sec. 4.9.

Zhang, K.

J. Xiao, K. Zhang, L. Gong, “Field analysis of a general chiral planar waveguide,” Int. J. Infrared Millim. Waves 18, 939–948 (1997).
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[CrossRef]

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[CrossRef]

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[CrossRef]

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Opt. Lett. (4)

Other (9)

See, for example, P. K. Cheo, Fiber Optics and Optoelectron-ics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1990), Chap. 3.

Solutions for all examples in this paper were obtained by using built-in routines in Mathcad 2000 Professional (MathSoft, Inc., Cambridge, Mass.).

The TE- and TM-mode equations are normally written by using arctangent rather than arccotangent. However, the arguments of the arctangent are positive definite, and arctan x=arccot(1/x) for x>0,as is true for the achiral case. There is an advantage to using arccotangent in the chiral case because singularities of the argument cause π discontinuities of arccotangent at only the endpoints of the range |g|<1,while if arctangent is used, there are π discontinuities at g=±r0 inside this range.

This merging of effective index curves was first noted in Ref. 12 without discussion of the polarization effects.

For a discussion of various forms for chiral constitutive equations, see Ref. 2, Chap. 3 and Ref. 21, Chap. 1.

Here we adopt the convention for circularly polarized light used in most optics texts: For RHC (LHC) polarization, the electric field at a fixed position rotates clockwise (counterclockwise) when viewed from the direction toward which the wave is traveling. The opposite designation is used in much of the engineering literature, including Refs. 12-14 cited here. For a discussion, see B. E. A. Salah, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Sec. 6.1.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Norwood, Mass., 1994), Chap. 4.

See, for example, A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Sec. 4.9.

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

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Figures (9)

Fig. 1
Fig. 1

Coordinate system for an asymmetric planar waveguide with an isotropic chiral core.

Fig. 2
Fig. 2

Plots of eccentricity parameter g versus chirality for the first six elliptical modes in a planar waveguide with a 3-µm-thick core and ng=1.59, n0=1.00, and ns=1.51: (top) modes that are RHE for γ>0 (and LHE for γ<0) evolving from TM modes at γ=0 and (bottom) modes that are LHE for γ>0 (and RHE for γ<0) evolving from TE modes at γ=0.

Fig. 3
Fig. 3

Curves of effective index neff versus thickness d for TE and TM modes in an achiral (γ=0) asymmetric planar waveguide with ng=1.59, n0=1.00, and ns=1.51.

Fig. 4
Fig. 4

Curves of effective index neff versus thickness d for RHE and LHE modes in a chiral (γ=45 pm) asymmetric planar waveguide with ng=1.59, n0=1.00, and ns=1.51. The dotted portions of the LHE-mode curves indicate the regime where the RHC component is a slow wave. The inset is a blowup emphasizing the mode crossover points.

Fig. 5
Fig. 5

Plot of effective index (left axis) and eccentricity parameter g (right axis) for the RHE0 mode versus thickness, illustrating the variation in eccentricity in the vicinity of the intersections of the RHE0 mode effect index (left axis) with the LHE modes (thin curves) for the same waveguide parameters as those in Fig. 4. The icons included to help visualize the eccentricity are vertical double arrow (TM), horizontal double arrow (TE), and ellipses (elliptical polarization with x or y major axis).

Fig. 6
Fig. 6

Relationship of the substrate eccentricity parameter h (short-dashed curves) to the upper cladding eccentricity parameter g (solid curves) as a function of thickness for the RHE0 mode of Figs. 4 and 5. The inset shows hg when m--m+ is even and h1/g when m--m+ is odd. Also shown for reference is core amplitude ratio B+/B- versus thickness.

Fig. 7
Fig. 7

Detail of the crossover of mode RHE0 (dotted curves) with LHE1 (solid curves) in Figs. 46:(top) effective indices versus thickness and (bottom) eccentricity parameter g for RH polarization (1/g for LH polarization) versus thickness. In the narrow transition region, indicated by vertical dotted lines, only LH polarization exists.

Fig. 8
Fig. 8

(top) Mode curves of effective index neff versus wave number k0 for a 2-µm-thick core with ng=1.59, n0=1.00, and ns=1.57 and (bottom) eccentricity parameters g and h for the LHE0 and RHE0 modes. Dramatic variation in polarization occurs near mode crossover points.

Fig. 9
Fig. 9

Eccentricity parameter g for the first three RHE modes (top) and 1/g for the first three LHE modes (bottom) versus thickness for chirality γ=0.45 pm (solid curves) and γ=4.5 pm (dashed curves) in a symmetric waveguide with ng=1.59 and n0=ns=1.57. The values at cutoff [see Eq. (52)] for all but the zero-order modes are gc=±rs, where, in this example, rs=0.975; note the initial rapid falloff and subsequent rise for d>dc.

Equations (89)

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×E+iωB=0,×H-iωD=0,
D=(E+γ×E),B=μ(H+γ×H),
E=12(F++F-),H=12i μ1/2(F+-F-),
F±E±iμ/H.
×F±=k0n±F±,
n±=ng1±δ,ng=01/2, δ=ωμγ=k0ngγ,k0=2πλ.
2F±+(k0n±)2F±=0.
F±(y, z)=Ψ±(y)exp(-ik0neffz),
Ψx±(y)ψ±(y),Ψy±(y)=±i neffnqψ±(y), Ψz±(y)=±1k0nq dψ±(y)dy,
d2ψ±(y)dy2+k02(nq2-neff2)ψ±(y)=0.
ψ±(y)=A± exp(-vy),y0B± cos(u±y+ϕ±),0<y<-d,C± exp[w(y+d)],y-d
u±k0(n±2-neff2)1/2,vk0(neff2-n02)1/2, wk0(neff2-ns2)1/2.
sin θ±=neff/n±,cos θ±=u±/(k0n±).
Φ1±=B± exp(-iϕ±)(sˆ±ipˆ1±)exp(-iK1±·r)
Φ2±=B± exp(iϕ±)(sˆ±ipˆ2±)exp(-iK2±·r)
F±=12(Φ1±+Φ2±).
Mγ(neff)B+ sin ϕ+B+ cos ϕ+B- sin ϕ-B- cos ϕ-=0,
Mγ(neff)=σ0+-1σ0--1r0σ0+-1-r0σ0-1Cs+-Ss+Cs--Ss-Crs+-Srs+-Crs-Srs-,
Cp±=σp± cos(u±d)+sin(u±d),
Crp±=rpσp± cos(u±d)+sin(u±d),
Sp±=σp± sin(u±d)-cos(u±d),
Srp±=rpσp± sin(u±d)-cos(u±d),
σ0±=(1±δ) u±v,σs±=(1±δ) u±w,
rp=np2ng2,p=0, s.
Dγ(neff)=|Mγ(neff)|=0,
D0(neff)=4|MTE||MTM|,
MTE=σ0-1Cs-Ss,MTM=r0σ0-1Crs-Srs.
11-cot ϕ+σ0++11-cot ϕ-σ0-=1Δ0,
11-cot(u+d-ϕ+)σs++11-cot(u-d-ϕ-)σs-=1Δs,
cot ϕ±=σ0±r0±g1±g,
cot(u±d-ϕ±)=σs±rs±h1±h.
B+ σ0+ sin ϕ+1+g+B- σ0- sin ϕ-1-g=0,
B+ σs+ sin(u+d-ϕ+)1+h+B- σs- sin(u-d-ϕ-)1-h=0.
1+h1-h=Sr0++S0+gSr0--S0-g,
u±d=cot-1σ0± r0±g1±g+cot-1σs± rs±h1±h+m±π,
h(g, neff)=(Sr0+-Sr0-)+(S0++S0-)g(Sr0++Sr0-)+(S0+-S0-)g,
g¯=1/g,h¯=1/h
 u±d=cot-1σ0± r0g¯±1g¯±1+cot-1σs± rsh¯±1h¯±1+m±π,
h¯(g¯, neff)=(Sr0++Sr0-)g¯+(S0+-S0-)(Sr0+-Sr0-)g¯+(S0++S0-),
B+=B-g+r0g-r0 cos ϕ-cos ϕ+,
A±=B- cos ϕ-g±r0g-r0,
C±=B- cos(u-d-ϕ-)h±rsh-rs.
ET=12 [ψ+(y)+ψ-(y)]xˆ+ineffψ+(y)n+-ψ-(y)n-yˆ×exp(-ik0neffz),
EyExy0=i neffn0 r0g=i neffng 1g.
 TM:g0, TE:g±(org¯0),RHC:g+1(org¯+1),LHC:g-1(org¯-1).
EyExy-d=i neffns rsh=i neffng 1h.
B+B-=sgng+1g-1g+1σ0+2+(g+r0)2g-1σ0-2+(g-r0)21/2,
uc+[ϕc-(k0, gc)+Mπ]-uc-[ϕc+(k0, gc)+Mπ]=0,
uc±=k0[(n±)2-ns2]1/2,
ϕc±(k0, gc)=cot-1(1±δ)×(n±)2-ns2ns2-n021/2 r0±gc1±gc,
k0dc=ϕc±(k0, gc)+M±π[(n±)2-ns2]1/2.
hc=rs1-2cos(uc-dc)(r0+gc)cos(uc+dc)(r0-gc)+1-1.
γ>0:RHE(M=M+)  0gc1,LHE(M=M-) -<gc-r0,γ<0:RHE(M=M+)  r0gc<,LHE(M=M-)-1gc0.
dc2(M-)=cot-1(1-δ)(n-)2-(n+)2(n+)2-n021/2 r0+12+cot-1(1-δ)(n-)2-(n+)2(n+)2-ns21/2 rs+12+M-πk0[(n-)2-(n+)2]1/2.
dc2(M-)12k0ng M-+1δπ-1+r01-r0+1+rs1-rs.
M--M+=oddinteger:low-dside:1/g=0, h=0,high-dside:g=0, 1/h=0;
M--M+=eveninteger:low-dside:1/g=0, 1/h=0,high-dside:g=0, h=0.
uc±d=ϕc±(k0, gc)+m±π,
ngk0c2=(M+1)π4dγ2/3[(1+α3+1)1/3-α(1+α3+1)-1/3]2,
α=23 1-Δ0Δ0+1-ΔsΔs2γ/d[(M+1)π]21/3.
u±d=2ϕ±+m±π,
ψcore±(y)=B± cos[u±(y+d/2)+m±π/2], 0<y<-d,
gc±=-G±G2+rs,M±=0±rs,M±0,
G=Δs2δ(1-Δs).
Ex aty=0:B+ cos ϕ++B- cos ϕ-=A++A-,
Hx aty=0:B+ cos ϕ+-B- cos ϕ-=r0(A+-A-),
Ez aty=0:B+r0σ0+ sin ϕ++B-r0σ0- sin ϕ-=A+-A-,
Hz aty=0:B+σ0+ sin ϕ++B-σ0- sin ϕ-=A++A-,
Ex aty=-d:B+ cos(u+d-ϕ+)+B- cos(u-d-ϕ-)=C++C-,
Hx aty=-d:B+ cos(u+d-ϕ+)-B- cos(u-d-ϕ-)=rs(C+-C-),
Ez aty=-d:B+rsσs+ sin(u+d-ϕ+)-B-rsσs- sin(u-d-ϕ-=C+-C-,
Hz aty=-d:B+σs+ sin(u+d-ϕ+)+B-σs- sin(u-d-ϕ-)=C++C-,
B+(σ0+ sin ϕ+-cos ϕ+)+B-(σ0- sin ϕ--cos ϕ-)=0,
B+(r0σ0+ sin ϕ+-cos ϕ+)-B-(r0σ0- sin ϕ--cos ϕ-)=0,
B+[σs+ sin(u+d-ϕ+)-cos(u+d-ϕ+)]+B-[σs- sin(u-d-ϕ-)-cos(u-d-ϕ-)]=0,
B+[rsσs+ sin(u+d-ϕ+)-cos(u+d-ϕ+)]-B-[rsσs- sin(u-d-ϕ-)-cos(u-d-ϕ-)]=0.
cot ϕ+(1+δ)u+-1-Δ0vcot ϕ-(1-δ)u--1-Δ0v=Δ02v2,
cot(u+d-ϕ+)σs+-(1-Δs)×cot(u-d-ϕ-)σs--(1-Δs)=Δs2.
Δs21,Δ02/v21,
u±d=cot-1σs±(1-Δs)+cot-1σ0±(1-Δ0)+m±π,
cot ϕ(1-δ)u±-1-Δ0v=Δ0v g1g±1,
cot(u±d-ϕ±)σs±-(1-Δs)=Δs h1h±1.
g(neff, k0, d)=Cr0±-(1-Δs)σs±Sr0±+Δsσs±Sr0C0±-(1-Δs)σs±S0±-Δsσs±Sr0,
gc±=Sr0c-Sr0c+S0c-±S0c+,
hc=rsS0c--S0c+S0c-+S0c+,
gslow(neff)-1-iσ0+r01-iσ0+.
(σ0+)asymptotic=i2δΔ0+δ1/2.
g=Srs12±Ss12±(evenmodes),
g=Crs12±Cs12±(oddmodes),

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