Abstract

We describe an efficient implementation of the finite-difference time-domain (FDTD) method as applied to lightwave propagation through periodic media with arbitrary anisotropy (birefringence). A permittivity tensor with non-diagonal elements is successfully integrated here with periodic boundary conditions, bounded computation space, and the split-field update technique. This enables modeling of periodic structures using only one period even with obliquely incident light in combination with both monochromatic (sinusoidal) and wideband (time-domain pulse) sources. Comparisons with results from other techniques in four validation cases are presented and excellent agreement is obtained. Our implementation is freely available on the Web.

© 2006 Optical Society of America

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  1. R. C. Jones, "A new calculus for the treatment of optical systems. I. Description and discussion of the calculus," J. Opt. Soc. Am. 31, 488-493 (1941).
  2. A. Lien, "Extended Jones matrix presentation for the twisted nematic liquid-crystal display at oblique incidence," Appl. Phys. Lett. 57, 2767-2769 (1990).
    [CrossRef]
  3. D. W. Berreman, "Optics in stratified and anisotropic media: 4×4-matrix formulation," J. Opt. Soc. Am. 62, 502-510 (1972).
  4. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
  5. E. K. Miller, L. Medgyesi-Mitschang, and E. H. Newman, Computational electrodynamics - frequency-domain method of moments (IEEE Press, 1992).
  6. S. G. Garcia, T. M. HungBao, R. G. Martin, and B. G. Olmedo, "On application of finite methods in time domain to anisotropic dielectric waveguides," IEEE Trans. Microwave Theory Tech. 44, 2195-2206 (1996).
  7. Y. A. Kao, "Finite-difference time-domian modeling of oblique incidence scattering from periodic surfaces," Master’s thesis, Massachusetts Institute of Technology, Cambridge, MA (1997).
  8. A. Taflove and S. C. Hagness, Computational electrodynamics: finite-difference time-domain method, 2nd ed. (Artech House, Norwood, MA, 2000), Chap. 13.
  9. J. A. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, "Time-domain analysis of periodic structures at oblique incidence: orthorgonal and nonorthogonal FDTD implementation," IEEE Trans. Microwave Theory Tech. 46, 420-427 (1998).
  10. S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996).
    [CrossRef]
  11. J. A. Kong, Theory of electromagnetic waves (Wiley, New York, 1975).
  12. G. B. Arfken and H. J. Weber, Mathematical methods for physicists, 4th ed. (Academic Press, San Diego, 1995).
  13. J. Schneider and S. Hudson, "The finite-difference time-domain method applied to anisotropic material," IEEE Trans. Antennas Propag. 41, 994-999 (1993).
    [CrossRef]
  14. C. M. Titus, P. J. Bos, J. R. Kelly, and E. C. Gartland, "Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures," Jpn. J. Appl. Phys. 38, 1488-1494 (1999).
    [CrossRef]
  15. X. Zhang, J. Fang, K. K. Mei, and Y. Liu, "Calculations of the dispersive characteristics of microstrips by the time-domain finite difference method," IEEE Trans. Microwave Theory Tech. 36, 263-267 (1988).
  16. C. M. Furse, S. P. Mathur, and O. P. Gandhi, "Improvements to the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target," IEEE Trans. Microwave Theory Tech. 38, 919-927 (1990).
  17. P. Yeh and C. Gu, Optics of liquid crystal displays (Wiley, New York, 1999).
  18. C. H. Gooch and H. A. Tarry, "The optical properties of twisted nematic liquid crystal structures with twist angles ≤ 90 degrees," J. Phys. D: Appl. Phys. 8, 1575-1584 (1975).
    [CrossRef]
  19. W. H. Southwell, "Gradient-index antireflection coatings," Opt. Lett. 8, 584-586 (1983).
  20. G. R. Fowles, Introduction to modern optics (Holt, Rinehart and Winston, New York, 1968).
  21. I. Richter, Z. Ryzi, and P. Fiala, "Analysis of binary diffraction gratings: comparison of different approaches," J. Mod. Opt. 16, 1915-1917 (1991).
  22. M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am. 71, 811-818 (1981).
  23. 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," Proc. IEEE 73, 894-937 (1985).
  24. S. D. Kakichashvili, "Method of recording phase polarization holograms," Sov. J. Quantum. Electron. 4, 795-798 (1974).
    [CrossRef]
  25. L. Nikolova and T. Todorov, "Diffraction efficiency and selectivity of polarization holographic recording," Opt. Acta 31, 579-588 (1984).
  26. I. Naydenova, L. Nikolova, T. Todorov, N. Holme, P. Ramanujam, and S. Hvilsted, "Diffraction from polarization holographic gratings with surface relief in side-chain azobenzene polyesters," J. Opt. Soc. Am. B 15, 1257-1265 (1998).
  27. J. Tervo and J. Turunen, "Paraxial-domain diffractive elements with 100% efficieincy based on polarization gratings," Opt. Lett. 25, 785-786 (2000).
  28. G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. Callan-Jones, and R. A. Pelcovits, "Liquid-crystal diffraction gratings using polarization holography alignment techniques," J. Appl. Phys. 98 and 123,102 (2005).
  29. M. J. Escuti and W. M. Jones, "Polarization independent switching with high contrast from a liquid crystal polarization grating," SID Int. Symp. Digest Tech. Papers  37, 1443-1446 (2006).
  30. C. Oh, R. Komanduri, and M. J. Escuti, "FDTD and elastic continuum analysis of the liquid crystal polarization grating," SID Int. Symp. Digest Tech. Papers  37, 844-847 (2006).

2006 (2)

M. J. Escuti and W. M. Jones, "Polarization independent switching with high contrast from a liquid crystal polarization grating," SID Int. Symp. Digest Tech. Papers  37, 1443-1446 (2006).

C. Oh, R. Komanduri, and M. J. Escuti, "FDTD and elastic continuum analysis of the liquid crystal polarization grating," SID Int. Symp. Digest Tech. Papers  37, 844-847 (2006).

2000 (1)

1999 (1)

C. M. Titus, P. J. Bos, J. R. Kelly, and E. C. Gartland, "Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures," Jpn. J. Appl. Phys. 38, 1488-1494 (1999).
[CrossRef]

1998 (2)

J. A. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, "Time-domain analysis of periodic structures at oblique incidence: orthorgonal and nonorthogonal FDTD implementation," IEEE Trans. Microwave Theory Tech. 46, 420-427 (1998).

I. Naydenova, L. Nikolova, T. Todorov, N. Holme, P. Ramanujam, and S. Hvilsted, "Diffraction from polarization holographic gratings with surface relief in side-chain azobenzene polyesters," J. Opt. Soc. Am. B 15, 1257-1265 (1998).

1996 (2)

S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996).
[CrossRef]

S. G. Garcia, T. M. HungBao, R. G. Martin, and B. G. Olmedo, "On application of finite methods in time domain to anisotropic dielectric waveguides," IEEE Trans. Microwave Theory Tech. 44, 2195-2206 (1996).

1993 (1)

J. Schneider and S. Hudson, "The finite-difference time-domain method applied to anisotropic material," IEEE Trans. Antennas Propag. 41, 994-999 (1993).
[CrossRef]

1991 (1)

I. Richter, Z. Ryzi, and P. Fiala, "Analysis of binary diffraction gratings: comparison of different approaches," J. Mod. Opt. 16, 1915-1917 (1991).

1990 (2)

A. Lien, "Extended Jones matrix presentation for the twisted nematic liquid-crystal display at oblique incidence," Appl. Phys. Lett. 57, 2767-2769 (1990).
[CrossRef]

C. M. Furse, S. P. Mathur, and O. P. Gandhi, "Improvements to the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target," IEEE Trans. Microwave Theory Tech. 38, 919-927 (1990).

1988 (1)

X. Zhang, J. Fang, K. K. Mei, and Y. Liu, "Calculations of the dispersive characteristics of microstrips by the time-domain finite difference method," IEEE Trans. Microwave Theory Tech. 36, 263-267 (1988).

1985 (1)

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," Proc. IEEE 73, 894-937 (1985).

1984 (1)

L. Nikolova and T. Todorov, "Diffraction efficiency and selectivity of polarization holographic recording," Opt. Acta 31, 579-588 (1984).

1983 (1)

1981 (1)

1975 (1)

C. H. Gooch and H. A. Tarry, "The optical properties of twisted nematic liquid crystal structures with twist angles ≤ 90 degrees," J. Phys. D: Appl. Phys. 8, 1575-1584 (1975).
[CrossRef]

1974 (1)

S. D. Kakichashvili, "Method of recording phase polarization holograms," Sov. J. Quantum. Electron. 4, 795-798 (1974).
[CrossRef]

1972 (1)

1966 (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).

1941 (1)

Berreman, D. W.

Bos, P. J.

C. M. Titus, P. J. Bos, J. R. Kelly, and E. C. Gartland, "Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures," Jpn. J. Appl. Phys. 38, 1488-1494 (1999).
[CrossRef]

Escuti, M. J.

M. J. Escuti and W. M. Jones, "Polarization independent switching with high contrast from a liquid crystal polarization grating," SID Int. Symp. Digest Tech. Papers  37, 1443-1446 (2006).

C. Oh, R. Komanduri, and M. J. Escuti, "FDTD and elastic continuum analysis of the liquid crystal polarization grating," SID Int. Symp. Digest Tech. Papers  37, 844-847 (2006).

Fang, J.

X. Zhang, J. Fang, K. K. Mei, and Y. Liu, "Calculations of the dispersive characteristics of microstrips by the time-domain finite difference method," IEEE Trans. Microwave Theory Tech. 36, 263-267 (1988).

Fiala, P.

I. Richter, Z. Ryzi, and P. Fiala, "Analysis of binary diffraction gratings: comparison of different approaches," J. Mod. Opt. 16, 1915-1917 (1991).

Furse, C. M.

C. M. Furse, S. P. Mathur, and O. P. Gandhi, "Improvements to the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target," IEEE Trans. Microwave Theory Tech. 38, 919-927 (1990).

Gandhi, O. P.

C. M. Furse, S. P. Mathur, and O. P. Gandhi, "Improvements to the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target," IEEE Trans. Microwave Theory Tech. 38, 919-927 (1990).

Garcia, S. G.

S. G. Garcia, T. M. HungBao, R. G. Martin, and B. G. Olmedo, "On application of finite methods in time domain to anisotropic dielectric waveguides," IEEE Trans. Microwave Theory Tech. 44, 2195-2206 (1996).

Gartland, E. C.

C. M. Titus, P. J. Bos, J. R. Kelly, and E. C. Gartland, "Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures," Jpn. J. Appl. Phys. 38, 1488-1494 (1999).
[CrossRef]

Gaylord, T. K.

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," Proc. IEEE 73, 894-937 (1985).

M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am. 71, 811-818 (1981).

Gedney, S. D.

J. A. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, "Time-domain analysis of periodic structures at oblique incidence: orthorgonal and nonorthogonal FDTD implementation," IEEE Trans. Microwave Theory Tech. 46, 420-427 (1998).

S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996).
[CrossRef]

Gooch, C. H.

C. H. Gooch and H. A. Tarry, "The optical properties of twisted nematic liquid crystal structures with twist angles ≤ 90 degrees," J. Phys. D: Appl. Phys. 8, 1575-1584 (1975).
[CrossRef]

Grann, E. B.

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," Proc. IEEE 73, 894-937 (1985).

Harms, P. H.

J. A. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, "Time-domain analysis of periodic structures at oblique incidence: orthorgonal and nonorthogonal FDTD implementation," IEEE Trans. Microwave Theory Tech. 46, 420-427 (1998).

Holme, N.

Hudson, S.

J. Schneider and S. Hudson, "The finite-difference time-domain method applied to anisotropic material," IEEE Trans. Antennas Propag. 41, 994-999 (1993).
[CrossRef]

Hvilsted, S.

Jones, R. C.

Jones, W. M.

M. J. Escuti and W. M. Jones, "Polarization independent switching with high contrast from a liquid crystal polarization grating," SID Int. Symp. Digest Tech. Papers  37, 1443-1446 (2006).

Kakichashvili, S. D.

S. D. Kakichashvili, "Method of recording phase polarization holograms," Sov. J. Quantum. Electron. 4, 795-798 (1974).
[CrossRef]

Kelly, J. R.

C. M. Titus, P. J. Bos, J. R. Kelly, and E. C. Gartland, "Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures," Jpn. J. Appl. Phys. 38, 1488-1494 (1999).
[CrossRef]

Kesler, M. P.

J. A. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, "Time-domain analysis of periodic structures at oblique incidence: orthorgonal and nonorthogonal FDTD implementation," IEEE Trans. Microwave Theory Tech. 46, 420-427 (1998).

Komanduri, R.

C. Oh, R. Komanduri, and M. J. Escuti, "FDTD and elastic continuum analysis of the liquid crystal polarization grating," SID Int. Symp. Digest Tech. Papers  37, 844-847 (2006).

Lien, A.

A. Lien, "Extended Jones matrix presentation for the twisted nematic liquid-crystal display at oblique incidence," Appl. Phys. Lett. 57, 2767-2769 (1990).
[CrossRef]

Liu, Y.

X. Zhang, J. Fang, K. K. Mei, and Y. Liu, "Calculations of the dispersive characteristics of microstrips by the time-domain finite difference method," IEEE Trans. Microwave Theory Tech. 36, 263-267 (1988).

Maloney, J. G.

J. A. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, "Time-domain analysis of periodic structures at oblique incidence: orthorgonal and nonorthogonal FDTD implementation," IEEE Trans. Microwave Theory Tech. 46, 420-427 (1998).

Mathur, S. P.

C. M. Furse, S. P. Mathur, and O. P. Gandhi, "Improvements to the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target," IEEE Trans. Microwave Theory Tech. 38, 919-927 (1990).

Mei, K. K.

X. Zhang, J. Fang, K. K. Mei, and Y. Liu, "Calculations of the dispersive characteristics of microstrips by the time-domain finite difference method," IEEE Trans. Microwave Theory Tech. 36, 263-267 (1988).

Moharam, M. G.

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," Proc. IEEE 73, 894-937 (1985).

M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am. 71, 811-818 (1981).

Naydenova, I.

Nikolova, L.

Oh, C.

C. Oh, R. Komanduri, and M. J. Escuti, "FDTD and elastic continuum analysis of the liquid crystal polarization grating," SID Int. Symp. Digest Tech. Papers  37, 844-847 (2006).

Pommet, D. A.

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," Proc. IEEE 73, 894-937 (1985).

Ramanujam, P.

Richter, I.

I. Richter, Z. Ryzi, and P. Fiala, "Analysis of binary diffraction gratings: comparison of different approaches," J. Mod. Opt. 16, 1915-1917 (1991).

Roden, J. A.

J. A. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, "Time-domain analysis of periodic structures at oblique incidence: orthorgonal and nonorthogonal FDTD implementation," IEEE Trans. Microwave Theory Tech. 46, 420-427 (1998).

Ryzi, Z.

I. Richter, Z. Ryzi, and P. Fiala, "Analysis of binary diffraction gratings: comparison of different approaches," J. Mod. Opt. 16, 1915-1917 (1991).

Schneider, J.

J. Schneider and S. Hudson, "The finite-difference time-domain method applied to anisotropic material," IEEE Trans. Antennas Propag. 41, 994-999 (1993).
[CrossRef]

Southwell, W. H.

Tarry, H. A.

C. H. Gooch and H. A. Tarry, "The optical properties of twisted nematic liquid crystal structures with twist angles ≤ 90 degrees," J. Phys. D: Appl. Phys. 8, 1575-1584 (1975).
[CrossRef]

Tervo, J.

Titus, C. M.

C. M. Titus, P. J. Bos, J. R. Kelly, and E. C. Gartland, "Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures," Jpn. J. Appl. Phys. 38, 1488-1494 (1999).
[CrossRef]

Todorov, T.

Turunen, J.

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).

Zhang, X.

X. Zhang, J. Fang, K. K. Mei, and Y. Liu, "Calculations of the dispersive characteristics of microstrips by the time-domain finite difference method," IEEE Trans. Microwave Theory Tech. 36, 263-267 (1988).

Appl. Phys. Lett. (1)

A. Lien, "Extended Jones matrix presentation for the twisted nematic liquid-crystal display at oblique incidence," Appl. Phys. Lett. 57, 2767-2769 (1990).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).

S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996).
[CrossRef]

J. Schneider and S. Hudson, "The finite-difference time-domain method applied to anisotropic material," IEEE Trans. Antennas Propag. 41, 994-999 (1993).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (4)

S. G. Garcia, T. M. HungBao, R. G. Martin, and B. G. Olmedo, "On application of finite methods in time domain to anisotropic dielectric waveguides," IEEE Trans. Microwave Theory Tech. 44, 2195-2206 (1996).

X. Zhang, J. Fang, K. K. Mei, and Y. Liu, "Calculations of the dispersive characteristics of microstrips by the time-domain finite difference method," IEEE Trans. Microwave Theory Tech. 36, 263-267 (1988).

C. M. Furse, S. P. Mathur, and O. P. Gandhi, "Improvements to the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target," IEEE Trans. Microwave Theory Tech. 38, 919-927 (1990).

J. A. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, "Time-domain analysis of periodic structures at oblique incidence: orthorgonal and nonorthogonal FDTD implementation," IEEE Trans. Microwave Theory Tech. 46, 420-427 (1998).

J. Mod. Opt. (1)

I. Richter, Z. Ryzi, and P. Fiala, "Analysis of binary diffraction gratings: comparison of different approaches," J. Mod. Opt. 16, 1915-1917 (1991).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. B (1)

J. Phys. D: Appl. Phys. (1)

C. H. Gooch and H. A. Tarry, "The optical properties of twisted nematic liquid crystal structures with twist angles ≤ 90 degrees," J. Phys. D: Appl. Phys. 8, 1575-1584 (1975).
[CrossRef]

Jpn. J. Appl. Phys. (1)

C. M. Titus, P. J. Bos, J. R. Kelly, and E. C. Gartland, "Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures," Jpn. J. Appl. Phys. 38, 1488-1494 (1999).
[CrossRef]

Opt. Acta (1)

L. Nikolova and T. Todorov, "Diffraction efficiency and selectivity of polarization holographic recording," Opt. Acta 31, 579-588 (1984).

Opt. Lett. (2)

Proc. IEEE (1)

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," Proc. IEEE 73, 894-937 (1985).

SID (1)

M. J. Escuti and W. M. Jones, "Polarization independent switching with high contrast from a liquid crystal polarization grating," SID Int. Symp. Digest Tech. Papers  37, 1443-1446 (2006).

SID Int. Symp. Digest (1)

C. Oh, R. Komanduri, and M. J. Escuti, "FDTD and elastic continuum analysis of the liquid crystal polarization grating," SID Int. Symp. Digest Tech. Papers  37, 844-847 (2006).

Sov. J. Quantum. Electron. (1)

S. D. Kakichashvili, "Method of recording phase polarization holograms," Sov. J. Quantum. Electron. 4, 795-798 (1974).
[CrossRef]

Other (8)

G. R. Fowles, Introduction to modern optics (Holt, Rinehart and Winston, New York, 1968).

P. Yeh and C. Gu, Optics of liquid crystal displays (Wiley, New York, 1999).

J. A. Kong, Theory of electromagnetic waves (Wiley, New York, 1975).

G. B. Arfken and H. J. Weber, Mathematical methods for physicists, 4th ed. (Academic Press, San Diego, 1995).

Y. A. Kao, "Finite-difference time-domian modeling of oblique incidence scattering from periodic surfaces," Master’s thesis, Massachusetts Institute of Technology, Cambridge, MA (1997).

A. Taflove and S. C. Hagness, Computational electrodynamics: finite-difference time-domain method, 2nd ed. (Artech House, Norwood, MA, 2000), Chap. 13.

E. K. Miller, L. Medgyesi-Mitschang, and E. H. Newman, Computational electrodynamics - frequency-domain method of moments (IEEE Press, 1992).

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. Callan-Jones, and R. A. Pelcovits, "Liquid-crystal diffraction gratings using polarization holography alignment techniques," J. Appl. Phys. 98 and 123,102 (2005).

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