Abstract

A formulation of rigorous coupled-wave theory for diffraction gratings in bianisotropic media that exhibit linear birefringence and/or optical activity is presented. The symmetric constitutive relations for bianisotropic materials are adopted. All of the incident, exiting, and grating materials can be isotropic, uniaxial, or biaxial, with or without optical activity. The principal values of the electric permittivity tensor, the magnetic permeability tensor, and the gyrotropic tensor of the media can take arbitrary values, and the principal axes may be arbitrarily and independently oriented. Procedures for Fourier expansion of Maxwell’s equations are described. Distinctive polarization coupling effects due to optical activity are observed in sample calculations.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Tamir, H. C. Wang, and A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. 12, 323–335 (1964).
    [CrossRef]
  2. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  3. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  4. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392(1982).
    [CrossRef]
  5. W. E. Baird, M. G. Moharam, and T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
    [CrossRef]
  6. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
    [CrossRef]
  7. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
    [CrossRef]
  8. K. Rokushima and J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. 73, 901–908 (1983).
    [CrossRef]
  9. E. N. Glytsis and T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A 4, 2061–2080(1987).
    [CrossRef]
  10. S. Mori, K. Mukai, J. Yamakita, and K. Rokushima, “Analysis of dielectric lamellar gratings coated with anisotropic layers,” J. Opt. Soc. Am. A 7, 1661–1665 (1990).
    [CrossRef]
  11. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  12. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  13. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876(1996).
    [CrossRef]
  14. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  15. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  16. L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
    [CrossRef]
  17. K. Rokushima and J. Yamakita, “Analysis of diffraction in periodic liquid crystals: the optics of the chiral smectic C phase,” J. Opt. Soc. Am. A 4, 27–33 (1987).
    [CrossRef]
  18. A. Lakhtakia, V. V. Varadan, and V. K. Varadan, “Scattering by periodic achiral-chiral interfaces,” J. Opt. Soc. Am. A 6, 1675–1681 (1989).
    [CrossRef]
  19. F. Wang and A. Lakhtakia, “Lateral shifts of optical beam on reflection by slanted chiral sculptured thin films,” Opt. Commun. 235, 107–132 (2004).
    [CrossRef]
  20. F. Wang and A. Lakhtakia, “Response of slanted chiral sculptured thin films to dipolar sources,” Opt. Commun. 235, 133–151 (2004).
    [CrossRef]
  21. L. D. Landau and E. M. Lifshitz, “Electromagnetic waves in anisotropic media,” in Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1987), pp. 54, 331–357.
  22. M. Born, “Molekulare optik,” in Optik (Springer-Verlag, 1965), pp. 403–420.
  23. A. Sommerfeld, “Crystal optics,” in Optics, Lectures in Theoretical Physics (Academic, 1949), Vol.  IV, pp. 154–160.
  24. A. Yariv and P. Yeh, “Electromagnetic propagation in anisotropic media,” in Optical Waves in Crystals (Wiley, 1984), pp. 69–102.
  25. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A: Pure Appl. Opt. 5, 345–355 (2003).
    [CrossRef]
  26. K. Watanabe, R. Petit, and M. NeviŁre, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002).
    [CrossRef]
  27. M. P. Silverman, “Reflection and refraction at the surface of a chiral medium: comparison of gyrotropic constitutive relations invariant or noninvariant under a duality transformation,” J. Opt. Soc. Am. A 3, 830–837 (1986).
    [CrossRef]
  28. M. P. Silverman, “Erratum,” J. Opt. Soc. Am. A 4, 1145 (1987).
    [CrossRef]
  29. R. M. Peterson, “Comparison of two theories of optical activity,” Am. J. Phys. 43, 969–972 (1975).
    [CrossRef]
  30. A.Lakhtakia, ed., Selected Papers on Natural Optical Activity, Vol.  MS15 of SPIE Milestone Series, B.J.Thompson, ed. (SPIE Optical Engineering, 1990). This volume contains reprints of .
  31. E. U. Condon, “Theories of optical rotator power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [CrossRef]
  32. F. I. Fedorov, “On the theory of optical activity in crystals,” Opt. Spektrosk. 6, 49–53 (1959).
  33. E. J. Post, Formal Structure of Electromagnetics (North-Holland, 1962).
  34. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, 1989), pp. 13–18.
  35. D. Y. K. Ko and J. R. Sambles, “Scattering matrix method for proparagtion of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
    [CrossRef]
  36. N. P. K. Cotter, T. W. Preist, and J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [CrossRef]
  37. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  38. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  39. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray racing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A 10, 2371–2382 (1993).
    [CrossRef]
  40. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray racing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993).
    [CrossRef]
  41. J. D. Trolinger, Jr., R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–466(1991).
    [CrossRef]
  42. H. C. Chen, “Determination of wave vectors; the Booker quartic,” in Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, 1983), pp. 205–208.
  43. H. G. Booker, “Oblique propagation of electromagnetic waves in a slowly-varying non-isotropic medium,” Proc. R. Soc. A 155, 235–257 (1936).
    [CrossRef]

2004 (2)

F. Wang and A. Lakhtakia, “Lateral shifts of optical beam on reflection by slanted chiral sculptured thin films,” Opt. Commun. 235, 107–132 (2004).
[CrossRef]

F. Wang and A. Lakhtakia, “Response of slanted chiral sculptured thin films to dipolar sources,” Opt. Commun. 235, 133–151 (2004).
[CrossRef]

2003 (1)

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A: Pure Appl. Opt. 5, 345–355 (2003).
[CrossRef]

2002 (1)

1998 (1)

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

1997 (1)

1996 (4)

1995 (3)

1993 (2)

1991 (1)

J. D. Trolinger, Jr., R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–466(1991).
[CrossRef]

1990 (1)

1989 (1)

1988 (1)

1987 (3)

1986 (2)

1983 (3)

1982 (1)

1981 (1)

1978 (1)

1975 (1)

R. M. Peterson, “Comparison of two theories of optical activity,” Am. J. Phys. 43, 969–972 (1975).
[CrossRef]

1964 (1)

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

1959 (1)

F. I. Fedorov, “On the theory of optical activity in crystals,” Opt. Spektrosk. 6, 49–53 (1959).

1937 (1)

E. U. Condon, “Theories of optical rotator power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

1936 (1)

H. G. Booker, “Oblique propagation of electromagnetic waves in a slowly-varying non-isotropic medium,” Proc. R. Soc. A 155, 235–257 (1936).
[CrossRef]

Baird, W. E.

W. E. Baird, M. G. Moharam, and T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
[CrossRef]

Booker, H. G.

H. G. Booker, “Oblique propagation of electromagnetic waves in a slowly-varying non-isotropic medium,” Proc. R. Soc. A 155, 235–257 (1936).
[CrossRef]

Born, M.

M. Born, “Molekulare optik,” in Optik (Springer-Verlag, 1965), pp. 403–420.

Chen, H. C.

H. C. Chen, “Determination of wave vectors; the Booker quartic,” in Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, 1983), pp. 205–208.

Chipman, R. A.

Condon, E. U.

E. U. Condon, “Theories of optical rotator power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Cotter, N. P. K.

Fedorov, F. I.

F. I. Fedorov, “On the theory of optical activity in crystals,” Opt. Spektrosk. 6, 49–53 (1959).

Gaylord, T. K.

Glytsis, E. N.

Granet, G.

Grann, E. B.

Guizal, B.

Hillman, L. W.

Knop, K.

Ko, D. Y. K.

Lakhtakia, A.

F. Wang and A. Lakhtakia, “Response of slanted chiral sculptured thin films to dipolar sources,” Opt. Commun. 235, 133–151 (2004).
[CrossRef]

F. Wang and A. Lakhtakia, “Lateral shifts of optical beam on reflection by slanted chiral sculptured thin films,” Opt. Commun. 235, 107–132 (2004).
[CrossRef]

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

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, 1989), pp. 13–18.

Lalanne, P.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, “Electromagnetic waves in anisotropic media,” in Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1987), pp. 54, 331–357.

Li, L.

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A: Pure Appl. Opt. 5, 345–355 (2003).
[CrossRef]

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

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

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

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

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, “Electromagnetic waves in anisotropic media,” in Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1987), pp. 54, 331–357.

McClain, S. C.

Moharam, M. G.

Mori, S.

Morris, G. M.

Mukai, K.

NeviLre, M.

Oliner, A. A.

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

Peterson, R. M.

R. M. Peterson, “Comparison of two theories of optical activity,” Am. J. Phys. 43, 969–972 (1975).
[CrossRef]

Petit, R.

Pommet, D. A.

Post, E. J.

E. J. Post, Formal Structure of Electromagnetics (North-Holland, 1962).

Preist, T. W.

Rokushima, K.

Sambles, J. R.

Silverman, M. P.

Sommerfeld, A.

A. Sommerfeld, “Crystal optics,” in Optics, Lectures in Theoretical Physics (Academic, 1949), Vol.  IV, pp. 154–160.

Tamir, R.

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

Trolinger, J. D.

J. D. Trolinger, Jr., R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–466(1991).
[CrossRef]

Varadan, V. K.

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

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, 1989), pp. 13–18.

Varadan, V. V.

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

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, 1989), pp. 13–18.

Wang, F.

F. Wang and A. Lakhtakia, “Lateral shifts of optical beam on reflection by slanted chiral sculptured thin films,” Opt. Commun. 235, 107–132 (2004).
[CrossRef]

F. Wang and A. Lakhtakia, “Response of slanted chiral sculptured thin films to dipolar sources,” Opt. Commun. 235, 133–151 (2004).
[CrossRef]

Wang, H. C.

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

Watanabe, K.

Wilson, D. K.

J. D. Trolinger, Jr., R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–466(1991).
[CrossRef]

Yamakita, J.

Yariv, A.

A. Yariv and P. Yeh, “Electromagnetic propagation in anisotropic media,” in Optical Waves in Crystals (Wiley, 1984), pp. 69–102.

Yeh, P.

A. Yariv and P. Yeh, “Electromagnetic propagation in anisotropic media,” in Optical Waves in Crystals (Wiley, 1984), pp. 69–102.

Am. J. Phys. (1)

R. M. Peterson, “Comparison of two theories of optical activity,” Am. J. Phys. 43, 969–972 (1975).
[CrossRef]

Appl. Phys. B (1)

W. E. Baird, M. G. Moharam, and T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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

J. Mod. Opt. (1)

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A: Pure Appl. Opt. 5, 345–355 (2003).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (19)

K. Watanabe, R. Petit, and M. NeviŁre, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002).
[CrossRef]

M. P. Silverman, “Erratum,” J. Opt. Soc. Am. A 4, 1145 (1987).
[CrossRef]

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

M. P. Silverman, “Reflection and refraction at the surface of a chiral medium: comparison of gyrotropic constitutive relations invariant or noninvariant under a duality transformation,” J. Opt. Soc. Am. A 3, 830–837 (1986).
[CrossRef]

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
[CrossRef]

K. Rokushima and J. Yamakita, “Analysis of diffraction in periodic liquid crystals: the optics of the chiral smectic C phase,” J. Opt. Soc. Am. A 4, 27–33 (1987).
[CrossRef]

E. N. Glytsis and T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A 4, 2061–2080(1987).
[CrossRef]

D. Y. K. Ko and J. R. Sambles, “Scattering matrix method for proparagtion of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
[CrossRef]

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

S. Mori, K. Mukai, J. Yamakita, and K. Rokushima, “Analysis of dielectric lamellar gratings coated with anisotropic layers,” J. Opt. Soc. Am. A 7, 1661–1665 (1990).
[CrossRef]

S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray racing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A 10, 2371–2382 (1993).
[CrossRef]

S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray racing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993).
[CrossRef]

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

G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

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

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

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

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

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

Opt. Commun. (2)

F. Wang and A. Lakhtakia, “Lateral shifts of optical beam on reflection by slanted chiral sculptured thin films,” Opt. Commun. 235, 107–132 (2004).
[CrossRef]

F. Wang and A. Lakhtakia, “Response of slanted chiral sculptured thin films to dipolar sources,” Opt. Commun. 235, 133–151 (2004).
[CrossRef]

Opt. Eng. (1)

J. D. Trolinger, Jr., R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–466(1991).
[CrossRef]

Opt. Spektrosk. (1)

F. I. Fedorov, “On the theory of optical activity in crystals,” Opt. Spektrosk. 6, 49–53 (1959).

Proc. R. Soc. A (1)

H. G. Booker, “Oblique propagation of electromagnetic waves in a slowly-varying non-isotropic medium,” Proc. R. Soc. A 155, 235–257 (1936).
[CrossRef]

Rev. Mod. Phys. (1)

E. U. Condon, “Theories of optical rotator power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Other (8)

E. J. Post, Formal Structure of Electromagnetics (North-Holland, 1962).

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, 1989), pp. 13–18.

L. D. Landau and E. M. Lifshitz, “Electromagnetic waves in anisotropic media,” in Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1987), pp. 54, 331–357.

M. Born, “Molekulare optik,” in Optik (Springer-Verlag, 1965), pp. 403–420.

A. Sommerfeld, “Crystal optics,” in Optics, Lectures in Theoretical Physics (Academic, 1949), Vol.  IV, pp. 154–160.

A. Yariv and P. Yeh, “Electromagnetic propagation in anisotropic media,” in Optical Waves in Crystals (Wiley, 1984), pp. 69–102.

A.Lakhtakia, ed., Selected Papers on Natural Optical Activity, Vol.  MS15 of SPIE Milestone Series, B.J.Thompson, ed. (SPIE Optical Engineering, 1990). This volume contains reprints of .

H. C. Chen, “Determination of wave vectors; the Booker quartic,” in Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, 1983), pp. 205–208.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Geometry of the grating diffraction problem analyzed herein. (a)  θ in is the angle of incidence and φ in is the azimuthal angle. (b) Grating vector is parallel to the x direction and the mean grating plane normal is parallel to the z direction. Each diffracted order has two component modes, with different wave vectors and polarization states.

Fig. 2
Fig. 2

Diffraction efficiency profiles for the zeroth- and 1 st-order O and E mode transmitted waves calculated with (a) present RCWT formulation and (b) RCWT with only Laurent’s rule. The horizontal axis is the order of Fourier terms N, where 2 N + 1 terms are retained. The O mode (TE mode) wave is incident.

Fig. 3
Fig. 3

Diffraction efficiency profiles for the zeroth- and first-order O and E mode transmitted waves for gyrotopic constants of (a)  G = 0.02 and (b)  G = 0.1 , respectively. A TE mode wave is normally incident on the grating.

Fig. 4
Fig. 4

Diffraction efficiency profiles for the zeroth- and first-order O and E mode transmitted waves of the isotropic grating (where the gyrotopic constant G is 0). A TE mode wave is normally incident on the grating.

Fig. 5
Fig. 5

Structure of the four-step sawtooth diffraction grating analyzed herein.

Fig. 6
Fig. 6

Diffraction efficiency profiles of the first-order O and E mode transmitted waves of the four-step grating in a gyrotopic media with the grating thickness varying from 0 to 20 λ .

Fig. 7
Fig. 7

Diffraction efficiency profiles for the first-order E mode transmitted waves of the four-step grating in the gyrotopic media with various principal gyrotropic constants as a function of the grating depth. O mode (TE mode) wave is normally incident on the grating. G is the principal gyrotropic tensor used in the previous example, where G x = 0.10 , G y = 0.12 , and G z = 0.14 .

Fig. 8
Fig. 8

Diffraction efficiencies for the zeroth- and 1 st-order TE and TM mode transmitted waves as a function of grating depth. The TE mode wave is incident with the angle of incidence of 32.189 ° . The profiles of the diffraction efficiencies seem to be the same as those plotted in Fig. 3 in [10].

Fig. 9
Fig. 9

Diffraction efficiencies for the zeroth- and 1 st-order TE and TM mode transmitted waves as a function of the angle of incidence. The TE mode wave is incident. The depth of the grating is 50 λ . The profiles of the diffraction efficiencies seem to be the same as those plotted in Fig. 4 in [10].

Tables (1)

Tables Icon

Table 1 Diffraction Efficiency Values of the Zeroth- and First-Order Transmitted Waves of the Four-Step Sawtooth Bianisotropic Grating Example ( N = 20 )

Equations (120)

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

E inc = u ^ exp ( j k inc · r ) ,
E I = E inc + i [ r O , i exp ( j k R O , i · r ) + r E , i exp ( j k R E , i · r ) ] ,
E II = i { t O , i exp [ j k T O , i · ( r h z ^ ) ] + t E , i exp [ j k T E , i · ( r h z ^ ) ] } ,
k q O , i = k x i x ^ + k y i y ^ + k q O z , i z ^ ; ( q = R , T ) ,
k q E , i = k x i x ^ + k y i y ^ + k q E z , i z ^ ; ( q = R , T ) ,
k x i = k inc , x i K g ,
k y i = k inc , y ,
E g = i S i ( z ) exp [ j ( k x i x + k y i y ) ] = i { S x i ( z ) x ^ + S y i ( z ) y ^ + S z i ( z ) z ^ } exp [ j ( k x i x + k y i y ) ] ,
H g = ( ε o μ o ) 1 / 2 i U i ( z ) exp [ j ( k x i x + k y i y ) ] = ( ε o μ o ) 1 / 2 i { U x i ( z ) x ^ + U y i ( z ) y ^ + U z i ( z ) z ^ } exp [ j ( k x i x + k y i y ) ] ,
D = ε o ε · E j ε o μ o G · H ,
B = μ o μ · H + j ε o μ o G · E ,
ε = R ε · ε c · R ε 1 ,
μ = R μ · μ c · R μ 1 ,
G = R G · G c · R G 1 .
D = ε o ε · E ,
B = μ o μ · H ,
ε i k = ε i k ( 0 ) j l ε i k l g l
D = ε o ε ( 0 ) · E j ε o E × g .
[ ε + ( K j G ) · μ 1 · ( K j G ) ] · E = 0 ,
K = 1 k o [ 0 k z k y k z 0 k x k y k x 0 ] ,
det [ ε + ( K j G ) · μ 1 · ( K j G ) ] = 0 .
1 k o × E g = j μ · H g + G · E g ,
1 k o × H g = j ε · E g + G · H g ,
H g = μ o / ε o H g .
1 k 0 [ E z y E y z ] = j ( μ x x H x + μ x y H y + μ x z H z ) + ( G x x E x + G x y E y + G x z E z ) ,
1 k 0 [ E x z E z x ] = j ( μ y x H x + μ y y H y + μ y z H z ) + ( G y x E x + G y y E y + G y z E z ) ,
1 k 0 [ E y x E x y ] = j ( μ z x H x + μ z y H y + μ z z H z ) + ( G z x E x + G z y E y + G z z E z ) ,
1 k 0 [ H z y H y z ] = j ( ε x x E x + ε x y E y + ε x z E z ) + ( G x x H x + G x y H y + G x z H z ) ,
1 k 0 [ H x z H z x ] = j ( ε y x E x + ε y y E y + ε y z E z ) + ( G y x H x + G y y H y + G y z H z ) ,
1 k 0 [ H y x H x y ] = j ( ε z x E x + ε z y E y + ε z z E z ) + ( G z x H x + G z y H y + G z z H z ) ,
R E p = j q μ ˜ p q · U q + q G ˜ S , p q · S q ; ( p , q = x , y , z ) ,
R H p = j q ε ˜ p q · S q + q G ˜ U , p q · U q ; ( p , q = x , y , z ) ,
1 k o z [ S x S y U x U y ] = j [ Γ 11 Γ 12 Γ 13 Γ 14 Γ 21 Γ 22 Γ 23 Γ 24 Γ 31 Γ 32 Γ 33 Γ 34 Γ 41 Γ 42 Γ 43 Γ 44 ] · [ S x S y U x U y ] = j Γ · [ S x S y U x U y ] .
1 k o z V = j Γ · V .
V ( z ) = m c m w m exp [ j k o λ m z ] ,
E g , l = i { S l , x i ( z ) x ^ + S l , y i ( z ) y ^ + S l , z i ( z ) z ^ } exp [ j ( k x i x + k y i y ) ] ,
H g , l = ( ε o μ o ) 1 / 2 i { U l , x i ( z ) x ^ + U l , y i ( z ) y ^ + U l , z i ( z ) z ^ } exp [ j ( k x i x + k y i y ) ] .
1 k o z V l = j Γ l · V l ,
V l ( z ) = m c l , m w l , m exp [ j k o λ l , m ( z z l 1 ) ] ,
[ δ i 0 u ^ + r O , i + r E , i ] q = S 1 , q , i ( 0 ) ; ( q = x , y ) ,
[ δ i , 0 K ˜ inc · u ^ + K ˜ R O , i · r O , i + K ˜ R E , i · r E , i ] q = U 1 , q , i ( 0 ) ; ( q = x , y ) ,
[ t O , i + t E , i ] q = S L , q , i ( z n ) ; ( q = x , y ) ,
[ K ˜ T O , i · t O , i + K ˜ T E , i · t E , i ] q = U L , q , i ( z n ) ; ( q = x , y ) .
K ˜ = μ 1 · [ K j G ] .
r O , i = [ r O x , i x ^ + r O y , i y ^ + r O z , i z ^ ] r O , i ,
r E , i = [ r E x , i x ^ + r E y , i y ^ + r E z , i z ^ ] r E , i ,
t O , i = [ t O x , i x ^ + t O y , i y ^ + t O z , i z ^ ] t O , i ,
t E , i = [ t E x , i x ^ + t E y , i y ^ + t E z , i z ^ ] t E , i .
[ u x δ u y δ μ x δ μ y δ ] + [ r O x r O y ρ O x ρ O y r E x r E y ρ E x ρ E y ] · [ r O r E ] = [ S 1 , x ( 0 ) S 1 , y ( 0 ) U 1 , x ( 0 ) U 1 , y ( 0 ) ] = W 1 · C 1 ,
[ t O x t O y τ O x τ O y t E x t E y τ E x τ E y ] · [ t O t E ] = [ S L , x ( z L ) S L , y ( z L ) U L , x ( z L ) U L , y ( z L ) ] = W L · Q L ( h L ) · C L ,
ρ O p , i = q K ˜ R O , p q , i r O q , i , ρ E p , i = q K ˜ R E , p q , i r E q , i ,
τ O p , i = q K ˜ T O , p q , i t O q , i , τ E p , i = q K ˜ T E , p q , i t E q , i ,
μ p = q K ˜ inc , p q u q ; ( p = x , y , q = x , y , z ) .
W l · Q l ( h l ) · C l = W l + 1 · C l + 1 ; ( l = 1 , 2 , , L 1 ) .
D E r q , i = | r q , i | 2 Re [ r q , i × ( K ˜ R q , i · r q , i ) * ] · z ^ Re [ u ^ × ( K ˜ inc · u ^ ) * ] · z ^ ; ( q = O , E ) ,
D E t q , i = | t q , i | 2 Re [ t q , i × ( K ˜ T q , i · t q , i ) * ] · z ^ Re [ u ^ × ( K ˜ inc · u ^ ) * ] · z ^ ; ( q = O , E ) .
k 2 = k o 2 γ 2 ± [ γ 2 2 4 γ 0 ( γ 4 a + γ 4 b ) ] 1 / 2 2 ( γ 4 a + γ 4 b ) ,
γ 4 a = α x s x 4 + α y s y 4 + α z s z 4 ,
γ 4 b = β x y s x 2 s y 2 + β y z s y 2 s z 2 + β z x s z 2 s x 2 ,
γ 2 = α x β y z + s x 2 + α y β z x + s y 2 + α z β x y + s z 2 ,
γ 0 = α x α y α z ,
α x = ε x G x 2 , β x y ± = ε x + ε y ± 2 G x G y ,
α y = ε y G y 2 , β y z ± = ε y + ε z ± 2 G y G z ,
α z = ε z G z 2 , β z x ± = ε z + ε x ± 2 G z G x ,
R H x = j α x x μ x x [ ( E x + β x y α x x E y + β x z α x x E z ) j ( σ x y α x x H y + σ x z α x x H z ) + G x x α x x R E x ] ,
R E x = j α x x ε x x [ ( H x + γ x y α x x H y + γ x z α x x H z ) + j ( ρ x y α x x E y + ρ x z α x x E z ) + G x x α x x R H x ] ,
R E x = 1 k o [ E z y E y z ] , R H x = 1 k o [ H z y H y z ] ,
α x x = ε x x μ x x G x x G x x ,
β p q = ε p q μ x x G x x G p q , γ p q = ε x x μ p q G x x G p q ,
ρ p q = ε x x G p q ε p q G x x , σ p q = μ x x G p q μ p q G x x .
R H x = j [ [ μ x x α x x ] ] 1 { ( S x + [ [ β x y α x x ] ] · S y + [ [ β x z α x x ] ] · S z ) j ( [ [ σ x y α x x ] ] · U y + [ [ σ x z α x x ] ] · U z ) + [ [ G x x α x x ] ] · R E x } ,
R E x = j [ [ ε x x α x x ] ] 1 { ( U x + [ [ γ x y α x x ] ] · U y + [ [ γ x z α x x ] ] · U z ) + j ( [ [ ρ x y α x x ] ] · S y + [ [ ρ x z α x x ] ] · S z ) + [ [ G x y α x x ] ] · R H x } ,
R E κ = j ( p κ y H y + p κ z H z ) ( v κ y E y + v κ z E z ) + γ κ x α x x R E x j σ κ x α x x R H x ; ( κ = y , z ) ,
R H κ = j ( q κ y E y + q κ z E z ) ( w κ y H y + w κ z H z ) + β κ x α x x R H x + j ρ κ x α x x R E x ; ( κ = y , z ) ,
R E κ = 1 k o [ × E ] κ , R H κ = 1 k o [ × H ] κ ,
p l m = ( μ l x γ x m + G l x σ x m α x x μ l m ) / α x x ,
q l m = ( ε l x β x m + G l x ρ x m α x x ε l m ) / α x x ,
v l m = ( μ l x ρ x m + G l x β x m α x x G l m ) / α x x ,
w l m = ( ε l x σ x m + G l x γ x m α x x G l m ) / α x x .
R E κ = j ( [ [ p κ y ] ] · U y + [ [ p κ z ] ] · U z ) ( [ [ v κ y ] ] · S y + [ [ v κ z ] ] · S z ) + [ [ γ κ x α x x ] ] · R E x j [ [ σ κ x α x x ] ] · R H x ,
R H κ = j ( [ [ q κ y ] ] · S y + [ [ q κ z ] ] · S z ) ( [ [ w κ y ] ] · U y + [ [ w κ z ] ] · U z ) + [ [ β κ x α x x ] ] · R H x + j [ [ ρ κ x α x x ] ] · R E x .
ε ˜ p q = { [ μ x x ] 1 · β x q + [ μ x x ] 1 · G x x · A x q ( p = x q = x , y , z ) β p x · D x q + ρ p x · A x q Q p q ( p = y , z q = x , y , z ) ,
μ ˜ p q = { [ ε x x ] 1 · γ x q + [ ε x x ] 1 · G x x · B x q ( p = x q = x , y , z ) γ p x · C x q + σ p x · B x q P p q ( p = y , z q = x , y , z ) ,
G ˜ S , p q = { [ ε x x ] 1 · ρ x q + [ ε x x ] 1 · G x x · D x q ( p = x q = x , y , z ) γ p x · A x q + σ p x · D x q V p q ( p = y , z q = x , y , z ) ,
G ˜ U , p q = { [ μ x x ] 1 · σ x q + [ μ x x ] 1 · G x x · C x q ( p = x q = x , y , z ) β p x · B x q + ρ p x · C x q W p q ( p = y , z q = x , y , z ) ;
A x q = [ X E ] 1 · [ ε x x ] 1 · { ρ x q + G x x · [ μ x x ] 1 · β x q } ,
B x q = [ X H ] 1 · [ μ x x ] 1 · { σ x q + G x x · [ ε x x ] 1 · γ x q } ,
C x q = [ X E ] 1 · [ ε x x ] 1 · { γ x q + G x x · [ μ x x ] 1 · σ x q } ,
D x q = [ X H ] 1 · [ μ x x ] 1 · { β x q + G x x · [ ε x x ] 1 · ρ x q } ,
X E = I [ ε x x ] 1 · G x x · [ μ x x ] 1 · G x x ,
X H = I [ μ x x ] 1 · G x x · [ ε x x ] 1 · G x x .
P y x = P z x = 0 , Q y x = Q z x = 0 ,
V y x = V z x = 0 , W y x = W z x = 0 ,
α x x = β x x = γ x x , ρ x x = σ x x = 0 .
Γ 11 = k x · [ ε ¯ z z ] 1 · ε ¯ z x μ ˜ y z · [ μ ¯ z z ] 1 · k ¯ ε y + j ( G ˜ S , y x G ˜ S , y z · [ ε ¯ z z ] 1 · ε ¯ z x ) ,
Γ 12 = k x · [ ε ¯ z z ] 1 · ε ¯ z y + μ ˜ y z · [ μ ¯ z z ] 1 · k ¯ ε x + j ( G ˜ S , y y G ˜ S , y z · [ ε ¯ z z ] 1 · ε ¯ z y ) ,
Γ 21 = k y · [ ε ¯ z z ] 1 · ε ¯ z x + μ ˜ x z · [ μ ¯ z z ] 1 · k ¯ ε y j ( G ˜ S , x x G ˜ S , x z · [ ε ¯ z z ] 1 · ε ¯ z x ) ,
Γ 22 = k y · [ ε ¯ z z ] 1 · ε ¯ z y μ ˜ x z · [ μ ¯ z z ] 1 · k ¯ ε x j ( G ˜ S , x y G ˜ S , x z · [ ε ¯ z z ] 1 · ε ¯ z y ) ,
Γ 13 = + k x · [ ε ¯ z z ] 1 · k ¯ μ y + μ ˜ y x μ ˜ y z · [ μ ¯ z z ] 1 · μ ¯ z x + j ( G ˜ S , y z · [ ε ¯ z z ] 1 · k ¯ μ y ) ,
Γ 14 = k x · [ ε ¯ z z ] 1 · k ¯ μ x + μ ˜ y y μ ˜ y z · [ μ ¯ z z ] 1 · μ ¯ z y j ( G ˜ S , y z · [ ε ¯ z z ] 1 · k ¯ μ x ) ,
Γ 23 = + k y · [ ε ¯ z z ] 1 · k ¯ μ y μ ˜ x x + μ ˜ x z · [ μ ¯ z z ] 1 · μ ¯ z x j ( G ˜ S , x z · [ ε ¯ z z ] 1 · k ¯ μ y ) ,
Γ 24 = k y · [ ε ¯ z z ] 1 · k ¯ μ x μ ˜ x y + μ ˜ x z · [ μ ¯ z z ] 1 · μ ¯ z y + j ( G ˜ S , x z · [ ε ¯ z z ] 1 · k ¯ μ x ) ,
Γ 31 = k x · [ μ ¯ z z ] 1 · k ¯ ε y ε ˜ y x + ε ˜ y z · [ ε ¯ z z ] 1 · ε ¯ z x j ( G ˜ U , y z · [ μ ¯ z z ] 1 · k ¯ ε y ) ,
Γ 32 = + k x · [ μ ¯ z z ] 1 · k ¯ ε x ε ˜ y y + ε ˜ y z · [ ε ¯ z z ] 1 · ε ¯ z y + j ( G ˜ U , y z · [ μ ¯ z z ] 1 · k ¯ ε x ) ,
Γ 41 = k y · [ μ ¯ z z ] 1 · k ¯ ε y + ε ˜ x x ε ˜ x z · [ ε ¯ z z ] 1 · ε ¯ z x + j ( G ˜ U , x z · [ μ ¯ z z ] 1 · k ¯ ε y ) ,
Γ 42 = + k y · [ μ ¯ z z ] 1 · k ¯ ε x + ε ˜ x y ε ˜ x z · [ ε ¯ z z ] 1 · ε ¯ z y j ( G ˜ U , x z · [ μ ¯ z z ] 1 · k ¯ ε x ) ,
Γ 33 = k x · [ μ ¯ z z ] 1 · μ ¯ z x ε ˜ y z · [ ε ¯ z z ] 1 · k ¯ μ y + j ( G ˜ U , y x G ˜ U , y z · [ μ ¯ z z ] 1 · μ ¯ z x ) ,
Γ 34 = k x · [ μ ¯ z z ] 1 · μ ¯ z y + ε ˜ y z · [ ε ¯ z z ] 1 · k ¯ μ x + j ( G ˜ U , y y G ˜ U , y z · [ μ ¯ z z ] 1 · μ ¯ z y ) ,
Γ 43 = k y · [ μ ¯ z z ] 1 · μ ¯ z x + ε ˜ x z · [ ε ¯ z z ] 1 · k ¯ μ y j ( G ˜ U , x x G ˜ U , x z · [ μ ¯ z z ] 1 · μ ¯ z x ) ,
Γ 44 = k y · [ μ ¯ z z ] 1 · μ ¯ z y ε ˜ x z · [ ε ¯ z z ] 1 · k ¯ μ x j ( G ˜ U , x y G ˜ U , x z · [ μ ¯ z z ] 1 · μ ¯ z y ) ,
ε ¯ z x = ε ˜ z x G ˜ U , z z · [ μ ˜ z z ] 1 · G ˜ S , z x + j G ˜ U , z z · [ μ ˜ z z ] 1 · k y ,
ε ¯ z y = ε ˜ z y G ˜ U , z z · [ μ ˜ z z ] 1 · G ˜ S , z y j G ˜ U , z z · [ μ ˜ z z ] 1 · k x ,
ε ¯ z z = ε ˜ z z G ˜ U , z z · [ μ ˜ z z ] 1 · G ˜ S , z z ,
μ ¯ z x = μ ˜ z x G ˜ S , z z · [ ε ˜ z z ] 1 · G ˜ U , z x + j G ˜ S , z z · [ ε ˜ z z ] 1 · k y ,
μ ¯ z y = μ ˜ z y G ˜ S , z z · [ ε ˜ z z ] 1 · G ˜ U , z y j G ˜ S , z z · [ ε ˜ z z ] 1 · k x ,
μ ¯ z z = μ ˜ z z G ˜ S , z z · [ ε ˜ z z ] 1 · G ˜ U , z z ,
k ¯ ε x = k x j ( G ˜ S , z y G ˜ S , z z · [ ε ˜ z z ] 1 · ε ˜ z y ) ,
k ¯ ε y = k y + j ( G ˜ S , z x G ˜ S , z z · [ ε ˜ z z ] 1 · ε ˜ z x ) ,
k ¯ μ x = k x j ( G ˜ U , z y G ˜ U , z z · [ μ ˜ z z ] 1 · μ ˜ z y ) ,
k ¯ μ y = k y + j ( G ˜ U , z x G ˜ U , z z · [ μ ˜ z z ] 1 · μ ˜ z x ) ,

Metrics