Abstract

We consider numerical modeling of the optical properties of devices typical of beam-steering devices based on liquid-crystal materials: two-dimensional, anisotropic and inhomogeneous dielectric properties, periodic in one dimension. A mathematical formulation of the system of second-order partial differential equations for the components of the time-harmonic electric field is discretized by using a finite-element method based on curl-conforming edge elements. The discrete equations are also interpreted as equivalent finite-difference equations. It is shown how the resulting large sparse complex system of linear algebraic equations can be solved by an iterative method with convergence accelerated by a preconditioner based on fast Fourier transforms. Benchmarking results and the application to a realistic problem are reported. The practical limitations of the approach and its advantages and disadvantages compared with other approaches are discussed.

© 2004 Optical Society of America

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  1. K. Rokushima, J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. 73, 901–908 (1983).
    [CrossRef]
  2. E. N. Glytsis, 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]
  3. E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1394–1420 (1990).
    [CrossRef]
  4. L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
    [CrossRef]
  5. Em. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for the analysis of tilted nematic liquid crystal structures,” J. Mod. Opt. 46, 1201–1212 (1999).
    [CrossRef]
  6. Em. E. Kriezis, S. J. Elston, “Light wave propagation in periodic tilted liquid crystal structures: a periodic beam propagation method,” Liq. Cryst. 26, 1663–1669 (1999).
    [CrossRef]
  7. Em. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for liquid crystal device calculations,” Appl. Opt. 39, 5707–5714 (2000).
    [CrossRef]
  8. A. A. Fuki, Yu. A. Kravtsov, O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, Amsterdam, 1998).
  9. J. A. Reyes, “Ray propagation in nematic droplets,” Phys. Rev. E 57, 6700–6705 (1998).
    [CrossRef]
  10. J. A. Reyes, “Ray propagation in anisotropic inhomogeneous media,” J. Phys. A 32, 3409–3418 (1999).
    [CrossRef]
  11. G. Panasyuk, J. Kelly, D. W. Allender, E. C. Gartland, “Geometric optics approach in liquid crystal films with three dimensional director variations,” Phys. Rev. E 67, 041702 (10 pages) ( 2003).
    [CrossRef]
  12. B. Witzigmann, P. Regli, W. Fichtner, “Rigorous electromagnetic simulation of liquid crystal displays,” J. Opt. Soc. Am. A 15, 753–757 (1998).
    [CrossRef]
  13. Em. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
    [CrossRef]
  14. Em. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
    [CrossRef]
  15. Em. E. Kriezis, S. J. Elston, “Numerical modelling of multi-dimensional liquid crystal optics: finite-difference time-domain method,” Mol. Cryst. Liq. Cryst. 359, 289–299 (2001).
    [CrossRef]
  16. C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” J. Appl. Phys. 38, 1488–1494 (1999).
    [CrossRef]
  17. C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
    [CrossRef]
  18. G. Bao, D. C. Dobson, “Variational methods for diffractive optics modeling,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, W. Masters, eds. (SIAM, Philadelphia, Pa., 2001).
  19. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1991).
  20. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  21. A. F. Peterson, S. L. Ray, R. Mittra, Computational Methods for Electromagnetics (IEEE Press, Piscataway, N.J., 1998).
  22. M. Cessenat, Mathematical Methods in Electromagnetism (World Scientific, Singapore, 1996).
  23. J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems (Springer, Berlin, 2001).
  24. W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “Solving electromagnetic scattering problems at resonance frequencies,” J. Appl. Phys. 67, 6061–6065 (1990).
    [CrossRef]
  25. X. Chen, A. Friedman, “Maxwell’s equations in a periodic structure,” Trans. Amer. Math. Soc. 323, 465–507 (1991).
  26. J.-C. Nédélec, F. Starling, “Integral-equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
    [CrossRef]
  27. H. Ammari, N. Bereux, J.-C. Nédélec, “Resonances for Maxwell’s equations in a periodic structure,” C. R. Acad. Sci. Paris Sér. I Math. 325, 211–215 (1997).
    [CrossRef]
  28. D. Givoli, Numerical Methods for Problems in Infinite Domains (Elsevier, Amsterdam, 1992).
  29. A. Taflove, Computational Electrodynamics, The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).
  30. R. Hiptmair, “Finite elements in computational electromagnetism,” Acta Numer. 11, 237–339 (2002).
    [CrossRef]
  31. K. W. Morton, D. F. Mayers, Numerical Solution of Partial Differential Equations (Cambridge U. Press, Cambridge, UK, 1994).
  32. 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).
    [CrossRef]
  33. R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).
  34. C. T. Kelly, Iterative Methods for Linear and Nonlinear Equations (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1995).
  35. Y. Saad, Iterative Methods for Sparse Linear Systems (PWS, Boston, Mass., 1996).
  36. P. Monk, “A finite element method for approximating the time-harmonic Maxwell equations,” Numer. Math. 63, 243–261 (1992).
    [CrossRef]
  37. P. Monk, “An analysis of Nédélec’s method for the spatial discretization of Maxwell’s equations,” J. Comput. Appl. Math. 47, 101–121 (1993).
    [CrossRef]
  38. K. H. Kim, S. B. Park, J.-U. Shim, J. Chen, J. H. Souk, “A novel wide viewing angle technology for AM-LCDs,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 175–178.
  39. S. H. Lee, H. Y. Kim, T. K. Jung, I. C. Park, Y. H. Lee, B. G. Rho, J. S. Park, H. S. Park, “Wide viewing-angle homeotropic nematic liquid crystal display controlled by in-plane switching,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 97–100.
  40. R. Kiefer, B. Weber, F. Windscheid, G. Baur, “In plane switching of nematic liquid crystals,” in Proceedings of Japan Display ’92 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1992), pp. 547–550.
  41. W. Liu, J. Kelly, “Optical properties of a switchable diffraction grating,” Mol. Cryst. Liq. Cryst. 358, 199–208 (2001).
    [CrossRef]
  42. W. Liu, J. Kelly, J. Chen, “Electro-optical performance of a self-compensating vertically-aligned liquid crystal display mode,” Jpn. J. Appl. Phys. 38, 2779–2784 (1999).
    [CrossRef]
  43. N. D. Amarasinghe, “2-D liquid crystal optics via numerical solution of time-harmonic Maxwell equations,” M.S. thesis (Kent State University, Kent, Ohio, 2001).

2003 (1)

G. Panasyuk, J. Kelly, D. W. Allender, E. C. Gartland, “Geometric optics approach in liquid crystal films with three dimensional director variations,” Phys. Rev. E 67, 041702 (10 pages) ( 2003).
[CrossRef]

2002 (1)

R. Hiptmair, “Finite elements in computational electromagnetism,” Acta Numer. 11, 237–339 (2002).
[CrossRef]

2001 (3)

W. Liu, J. Kelly, “Optical properties of a switchable diffraction grating,” Mol. Cryst. Liq. Cryst. 358, 199–208 (2001).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “Numerical modelling of multi-dimensional liquid crystal optics: finite-difference time-domain method,” Mol. Cryst. Liq. Cryst. 359, 289–299 (2001).
[CrossRef]

C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
[CrossRef]

2000 (2)

Em. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for liquid crystal device calculations,” Appl. Opt. 39, 5707–5714 (2000).
[CrossRef]

1999 (6)

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

W. Liu, J. Kelly, J. Chen, “Electro-optical performance of a self-compensating vertically-aligned liquid crystal display mode,” Jpn. J. Appl. Phys. 38, 2779–2784 (1999).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

J. A. Reyes, “Ray propagation in anisotropic inhomogeneous media,” J. Phys. A 32, 3409–3418 (1999).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for the analysis of tilted nematic liquid crystal structures,” J. Mod. Opt. 46, 1201–1212 (1999).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “Light wave propagation in periodic tilted liquid crystal structures: a periodic beam propagation method,” Liq. Cryst. 26, 1663–1669 (1999).
[CrossRef]

1998 (3)

J. A. Reyes, “Ray propagation in nematic droplets,” Phys. Rev. E 57, 6700–6705 (1998).
[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]

B. Witzigmann, P. Regli, W. Fichtner, “Rigorous electromagnetic simulation of liquid crystal displays,” J. Opt. Soc. Am. A 15, 753–757 (1998).
[CrossRef]

1997 (1)

H. Ammari, N. Bereux, J.-C. Nédélec, “Resonances for Maxwell’s equations in a periodic structure,” C. R. Acad. Sci. Paris Sér. I Math. 325, 211–215 (1997).
[CrossRef]

1993 (1)

P. Monk, “An analysis of Nédélec’s method for the spatial discretization of Maxwell’s equations,” J. Comput. Appl. Math. 47, 101–121 (1993).
[CrossRef]

1992 (1)

P. Monk, “A finite element method for approximating the time-harmonic Maxwell equations,” Numer. Math. 63, 243–261 (1992).
[CrossRef]

1991 (2)

X. Chen, A. Friedman, “Maxwell’s equations in a periodic structure,” Trans. Amer. Math. Soc. 323, 465–507 (1991).

J.-C. Nédélec, F. Starling, “Integral-equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
[CrossRef]

1990 (2)

W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “Solving electromagnetic scattering problems at resonance frequencies,” J. Appl. Phys. 67, 6061–6065 (1990).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1394–1420 (1990).
[CrossRef]

1987 (1)

1983 (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).
[CrossRef]

Allender, D. W.

G. Panasyuk, J. Kelly, D. W. Allender, E. C. Gartland, “Geometric optics approach in liquid crystal films with three dimensional director variations,” Phys. Rev. E 67, 041702 (10 pages) ( 2003).
[CrossRef]

Amarasinghe, N. D.

N. D. Amarasinghe, “2-D liquid crystal optics via numerical solution of time-harmonic Maxwell equations,” M.S. thesis (Kent State University, Kent, Ohio, 2001).

Ammari, H.

H. Ammari, N. Bereux, J.-C. Nédélec, “Resonances for Maxwell’s equations in a periodic structure,” C. R. Acad. Sci. Paris Sér. I Math. 325, 211–215 (1997).
[CrossRef]

Anderson, J. A.

Bao, G.

G. Bao, D. C. Dobson, “Variational methods for diffractive optics modeling,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, W. Masters, eds. (SIAM, Philadelphia, Pa., 2001).

Barrett, R.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Baur, G.

R. Kiefer, B. Weber, F. Windscheid, G. Baur, “In plane switching of nematic liquid crystals,” in Proceedings of Japan Display ’92 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1992), pp. 547–550.

Bereux, N.

H. Ammari, N. Bereux, J.-C. Nédélec, “Resonances for Maxwell’s equations in a periodic structure,” C. R. Acad. Sci. Paris Sér. I Math. 325, 211–215 (1997).
[CrossRef]

Berry, M.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1991).

Bos, P. J.

C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
[CrossRef]

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

Cessenat, M.

M. Cessenat, Mathematical Methods in Electromagnetism (World Scientific, Singapore, 1996).

Chan, T. F.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Chen, J.

W. Liu, J. Kelly, J. Chen, “Electro-optical performance of a self-compensating vertically-aligned liquid crystal display mode,” Jpn. J. Appl. Phys. 38, 2779–2784 (1999).
[CrossRef]

K. H. Kim, S. B. Park, J.-U. Shim, J. Chen, J. H. Souk, “A novel wide viewing angle technology for AM-LCDs,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 175–178.

Chen, X.

X. Chen, A. Friedman, “Maxwell’s equations in a periodic structure,” Trans. Amer. Math. Soc. 323, 465–507 (1991).

Demmel, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Dobson, D. C.

G. Bao, D. C. Dobson, “Variational methods for diffractive optics modeling,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, W. Masters, eds. (SIAM, Philadelphia, Pa., 2001).

Donato, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Dongarra, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Eijkhout, V.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Elston, S. J.

Em. E. Kriezis, S. J. Elston, “Numerical modelling of multi-dimensional liquid crystal optics: finite-difference time-domain method,” Mol. Cryst. Liq. Cryst. 359, 289–299 (2001).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for liquid crystal device calculations,” Appl. Opt. 39, 5707–5714 (2000).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “Light wave propagation in periodic tilted liquid crystal structures: a periodic beam propagation method,” Liq. Cryst. 26, 1663–1669 (1999).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for the analysis of tilted nematic liquid crystal structures,” J. Mod. Opt. 46, 1201–1212 (1999).
[CrossRef]

Fichtner, W.

Friedman, A.

X. Chen, A. Friedman, “Maxwell’s equations in a periodic structure,” Trans. Amer. Math. Soc. 323, 465–507 (1991).

Fuki, A. A.

A. A. Fuki, Yu. A. Kravtsov, O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, Amsterdam, 1998).

Gartland, E. C.

G. Panasyuk, J. Kelly, D. W. Allender, E. C. Gartland, “Geometric optics approach in liquid crystal films with three dimensional director variations,” Phys. Rev. E 67, 041702 (10 pages) ( 2003).
[CrossRef]

C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
[CrossRef]

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

Gaylord, T. K.

E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1394–1420 (1990).
[CrossRef]

E. N. Glytsis, 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]

Givoli, D.

D. Givoli, Numerical Methods for Problems in Infinite Domains (Elsevier, Amsterdam, 1992).

Glytsis, E. N.

E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1394–1420 (1990).
[CrossRef]

E. N. Glytsis, 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]

Hiptmair, R.

R. Hiptmair, “Finite elements in computational electromagnetism,” Acta Numer. 11, 237–339 (2002).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Jung, T. K.

S. H. Lee, H. Y. Kim, T. K. Jung, I. C. Park, Y. H. Lee, B. G. Rho, J. S. Park, H. S. Park, “Wide viewing-angle homeotropic nematic liquid crystal display controlled by in-plane switching,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 97–100.

Kelly, C. T.

C. T. Kelly, Iterative Methods for Linear and Nonlinear Equations (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1995).

Kelly, J.

G. Panasyuk, J. Kelly, D. W. Allender, E. C. Gartland, “Geometric optics approach in liquid crystal films with three dimensional director variations,” Phys. Rev. E 67, 041702 (10 pages) ( 2003).
[CrossRef]

W. Liu, J. Kelly, “Optical properties of a switchable diffraction grating,” Mol. Cryst. Liq. Cryst. 358, 199–208 (2001).
[CrossRef]

W. Liu, J. Kelly, J. Chen, “Electro-optical performance of a self-compensating vertically-aligned liquid crystal display mode,” Jpn. J. Appl. Phys. 38, 2779–2784 (1999).
[CrossRef]

Kelly, J. R.

C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
[CrossRef]

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

Kiefer, R.

R. Kiefer, B. Weber, F. Windscheid, G. Baur, “In plane switching of nematic liquid crystals,” in Proceedings of Japan Display ’92 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1992), pp. 547–550.

Kim, H. Y.

S. H. Lee, H. Y. Kim, T. K. Jung, I. C. Park, Y. H. Lee, B. G. Rho, J. S. Park, H. S. Park, “Wide viewing-angle homeotropic nematic liquid crystal display controlled by in-plane switching,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 97–100.

Kim, K. H.

K. H. Kim, S. B. Park, J.-U. Shim, J. Chen, J. H. Souk, “A novel wide viewing angle technology for AM-LCDs,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 175–178.

Kravtsov, Yu. A.

A. A. Fuki, Yu. A. Kravtsov, O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, Amsterdam, 1998).

Kriezis, Em. E.

Em. E. Kriezis, S. J. Elston, “Numerical modelling of multi-dimensional liquid crystal optics: finite-difference time-domain method,” Mol. Cryst. Liq. Cryst. 359, 289–299 (2001).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for liquid crystal device calculations,” Appl. Opt. 39, 5707–5714 (2000).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “Light wave propagation in periodic tilted liquid crystal structures: a periodic beam propagation method,” Liq. Cryst. 26, 1663–1669 (1999).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for the analysis of tilted nematic liquid crystal structures,” J. Mod. Opt. 46, 1201–1212 (1999).
[CrossRef]

Lee, S. H.

S. H. Lee, H. Y. Kim, T. K. Jung, I. C. Park, Y. H. Lee, B. G. Rho, J. S. Park, H. S. Park, “Wide viewing-angle homeotropic nematic liquid crystal display controlled by in-plane switching,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 97–100.

Lee, Y. H.

S. H. Lee, H. Y. Kim, T. K. Jung, I. C. Park, Y. H. Lee, B. G. Rho, J. S. Park, H. S. Park, “Wide viewing-angle homeotropic nematic liquid crystal display controlled by in-plane switching,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 97–100.

Li, L.

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

Liu, W.

W. Liu, J. Kelly, “Optical properties of a switchable diffraction grating,” Mol. Cryst. Liq. Cryst. 358, 199–208 (2001).
[CrossRef]

W. Liu, J. Kelly, J. Chen, “Electro-optical performance of a self-compensating vertically-aligned liquid crystal display mode,” Jpn. J. Appl. Phys. 38, 2779–2784 (1999).
[CrossRef]

Mayers, D. F.

K. W. Morton, D. F. Mayers, Numerical Solution of Partial Differential Equations (Cambridge U. Press, Cambridge, UK, 1994).

Mittra, R.

A. F. Peterson, S. L. Ray, R. Mittra, Computational Methods for Electromagnetics (IEEE Press, Piscataway, N.J., 1998).

Monk, P.

P. Monk, “An analysis of Nédélec’s method for the spatial discretization of Maxwell’s equations,” J. Comput. Appl. Math. 47, 101–121 (1993).
[CrossRef]

P. Monk, “A finite element method for approximating the time-harmonic Maxwell equations,” Numer. Math. 63, 243–261 (1992).
[CrossRef]

Morton, K. W.

K. W. Morton, D. F. Mayers, Numerical Solution of Partial Differential Equations (Cambridge U. Press, Cambridge, UK, 1994).

Murphy, W. D.

W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “Solving electromagnetic scattering problems at resonance frequencies,” J. Appl. Phys. 67, 6061–6065 (1990).
[CrossRef]

Naida, O. N.

A. A. Fuki, Yu. A. Kravtsov, O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, Amsterdam, 1998).

Nédélec, J.-C.

H. Ammari, N. Bereux, J.-C. Nédélec, “Resonances for Maxwell’s equations in a periodic structure,” C. R. Acad. Sci. Paris Sér. I Math. 325, 211–215 (1997).
[CrossRef]

J.-C. Nédélec, F. Starling, “Integral-equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
[CrossRef]

J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems (Springer, Berlin, 2001).

Panasyuk, G.

G. Panasyuk, J. Kelly, D. W. Allender, E. C. Gartland, “Geometric optics approach in liquid crystal films with three dimensional director variations,” Phys. Rev. E 67, 041702 (10 pages) ( 2003).
[CrossRef]

Park, H. S.

S. H. Lee, H. Y. Kim, T. K. Jung, I. C. Park, Y. H. Lee, B. G. Rho, J. S. Park, H. S. Park, “Wide viewing-angle homeotropic nematic liquid crystal display controlled by in-plane switching,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 97–100.

Park, I. C.

S. H. Lee, H. Y. Kim, T. K. Jung, I. C. Park, Y. H. Lee, B. G. Rho, J. S. Park, H. S. Park, “Wide viewing-angle homeotropic nematic liquid crystal display controlled by in-plane switching,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 97–100.

Park, J. S.

S. H. Lee, H. Y. Kim, T. K. Jung, I. C. Park, Y. H. Lee, B. G. Rho, J. S. Park, H. S. Park, “Wide viewing-angle homeotropic nematic liquid crystal display controlled by in-plane switching,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 97–100.

Park, S. B.

K. H. Kim, S. B. Park, J.-U. Shim, J. Chen, J. H. Souk, “A novel wide viewing angle technology for AM-LCDs,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 175–178.

Peterson, A. F.

A. F. Peterson, S. L. Ray, R. Mittra, Computational Methods for Electromagnetics (IEEE Press, Piscataway, N.J., 1998).

Pozo, R.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Ray, S. L.

A. F. Peterson, S. L. Ray, R. Mittra, Computational Methods for Electromagnetics (IEEE Press, Piscataway, N.J., 1998).

Regli, P.

Reyes, J. A.

J. A. Reyes, “Ray propagation in anisotropic inhomogeneous media,” J. Phys. A 32, 3409–3418 (1999).
[CrossRef]

J. A. Reyes, “Ray propagation in nematic droplets,” Phys. Rev. E 57, 6700–6705 (1998).
[CrossRef]

Rho, B. G.

S. H. Lee, H. Y. Kim, T. K. Jung, I. C. Park, Y. H. Lee, B. G. Rho, J. S. Park, H. S. Park, “Wide viewing-angle homeotropic nematic liquid crystal display controlled by in-plane switching,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 97–100.

Rokhlin, V.

W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “Solving electromagnetic scattering problems at resonance frequencies,” J. Appl. Phys. 67, 6061–6065 (1990).
[CrossRef]

Rokushima, K.

Romine, C.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Saad, Y.

Y. Saad, Iterative Methods for Sparse Linear Systems (PWS, Boston, Mass., 1996).

Shim, J.-U.

K. H. Kim, S. B. Park, J.-U. Shim, J. Chen, J. H. Souk, “A novel wide viewing angle technology for AM-LCDs,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 175–178.

Shiyanovskii, S. V.

Souk, J. H.

K. H. Kim, S. B. Park, J.-U. Shim, J. Chen, J. H. Souk, “A novel wide viewing angle technology for AM-LCDs,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 175–178.

Starling, F.

J.-C. Nédélec, F. Starling, “Integral-equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
[CrossRef]

Taflove, A.

A. Taflove, Computational Electrodynamics, The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).

Titus, C. M.

C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
[CrossRef]

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

van der Vorst, H.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Vassiliou, M. S.

W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “Solving electromagnetic scattering problems at resonance frequencies,” J. Appl. Phys. 67, 6061–6065 (1990).
[CrossRef]

Weber, B.

R. Kiefer, B. Weber, F. Windscheid, G. Baur, “In plane switching of nematic liquid crystals,” in Proceedings of Japan Display ’92 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1992), pp. 547–550.

Windscheid, F.

R. Kiefer, B. Weber, F. Windscheid, G. Baur, “In plane switching of nematic liquid crystals,” in Proceedings of Japan Display ’92 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1992), pp. 547–550.

Witzigmann, B.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1991).

Yamakita, 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).
[CrossRef]

Acta Numer. (1)

R. Hiptmair, “Finite elements in computational electromagnetism,” Acta Numer. 11, 237–339 (2002).
[CrossRef]

Appl. Opt. (1)

C. R. Acad. Sci. Paris Sér. I Math. (1)

H. Ammari, N. Bereux, J.-C. Nédélec, “Resonances for Maxwell’s equations in a periodic structure,” C. R. Acad. Sci. Paris Sér. I Math. 325, 211–215 (1997).
[CrossRef]

IEEE Trans. Antennas Propag. (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).
[CrossRef]

J. Appl. Phys. (2)

W. D. Murphy, V. Rokhlin, M. S. Vassiliou, “Solving electromagnetic scattering problems at resonance frequencies,” J. Appl. Phys. 67, 6061–6065 (1990).
[CrossRef]

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

J. Comput. Appl. Math. (1)

P. Monk, “An analysis of Nédélec’s method for the spatial discretization of Maxwell’s equations,” J. Comput. Appl. Math. 47, 101–121 (1993).
[CrossRef]

J. Mod. Opt. (2)

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

Em. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for the analysis of tilted nematic liquid crystal structures,” J. Mod. Opt. 46, 1201–1212 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. A (1)

J. A. Reyes, “Ray propagation in anisotropic inhomogeneous media,” J. Phys. A 32, 3409–3418 (1999).
[CrossRef]

Jpn. J. Appl. Phys. (1)

W. Liu, J. Kelly, J. Chen, “Electro-optical performance of a self-compensating vertically-aligned liquid crystal display mode,” Jpn. J. Appl. Phys. 38, 2779–2784 (1999).
[CrossRef]

Liq. Cryst. (1)

Em. E. Kriezis, S. J. Elston, “Light wave propagation in periodic tilted liquid crystal structures: a periodic beam propagation method,” Liq. Cryst. 26, 1663–1669 (1999).
[CrossRef]

Mol. Cryst. Liq. Cryst. (2)

W. Liu, J. Kelly, “Optical properties of a switchable diffraction grating,” Mol. Cryst. Liq. Cryst. 358, 199–208 (2001).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “Numerical modelling of multi-dimensional liquid crystal optics: finite-difference time-domain method,” Mol. Cryst. Liq. Cryst. 359, 289–299 (2001).
[CrossRef]

Numer. Math. (1)

P. Monk, “A finite element method for approximating the time-harmonic Maxwell equations,” Numer. Math. 63, 243–261 (1992).
[CrossRef]

Opt. Commun. (2)

Em. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

Em. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. E (2)

J. A. Reyes, “Ray propagation in nematic droplets,” Phys. Rev. E 57, 6700–6705 (1998).
[CrossRef]

G. Panasyuk, J. Kelly, D. W. Allender, E. C. Gartland, “Geometric optics approach in liquid crystal films with three dimensional director variations,” Phys. Rev. E 67, 041702 (10 pages) ( 2003).
[CrossRef]

SIAM J. Math. Anal. (1)

J.-C. Nédélec, F. Starling, “Integral-equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell equations,” SIAM J. Math. Anal. 22, 1679–1701 (1991).
[CrossRef]

Trans. Amer. Math. Soc. (1)

X. Chen, A. Friedman, “Maxwell’s equations in a periodic structure,” Trans. Amer. Math. Soc. 323, 465–507 (1991).

Other (17)

D. Givoli, Numerical Methods for Problems in Infinite Domains (Elsevier, Amsterdam, 1992).

A. Taflove, Computational Electrodynamics, The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).

G. Bao, D. C. Dobson, “Variational methods for diffractive optics modeling,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, W. Masters, eds. (SIAM, Philadelphia, Pa., 2001).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1991).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

A. F. Peterson, S. L. Ray, R. Mittra, Computational Methods for Electromagnetics (IEEE Press, Piscataway, N.J., 1998).

M. Cessenat, Mathematical Methods in Electromagnetism (World Scientific, Singapore, 1996).

J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems (Springer, Berlin, 2001).

K. W. Morton, D. F. Mayers, Numerical Solution of Partial Differential Equations (Cambridge U. Press, Cambridge, UK, 1994).

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

C. T. Kelly, Iterative Methods for Linear and Nonlinear Equations (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1995).

Y. Saad, Iterative Methods for Sparse Linear Systems (PWS, Boston, Mass., 1996).

A. A. Fuki, Yu. A. Kravtsov, O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, Amsterdam, 1998).

N. D. Amarasinghe, “2-D liquid crystal optics via numerical solution of time-harmonic Maxwell equations,” M.S. thesis (Kent State University, Kent, Ohio, 2001).

K. H. Kim, S. B. Park, J.-U. Shim, J. Chen, J. H. Souk, “A novel wide viewing angle technology for AM-LCDs,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 175–178.

S. H. Lee, H. Y. Kim, T. K. Jung, I. C. Park, Y. H. Lee, B. G. Rho, J. S. Park, H. S. Park, “Wide viewing-angle homeotropic nematic liquid crystal display controlled by in-plane switching,” in Proceedings of the 4th International Display Workshops, Nagoya, Japan, 1997 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1997), pp. 97–100.

R. Kiefer, B. Weber, F. Windscheid, G. Baur, “In plane switching of nematic liquid crystals,” in Proceedings of Japan Display ’92 (Society for Information Display, Japan Chapter, Tokyo, Japan, 1992), pp. 547–550.

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Figures (7)

Fig. 1
Fig. 1

Geometry of one periodic cell of the model device: uniform in z, 2L periodic in x, anisotropic and inhomogeneous layer Ω of thickness d, various front and back components, interfaces Γ1 and Γ2, illuminated from below by a monochromatic plane-wave source incident from an arbitrary direction in the xy plane.

Fig. 2
Fig. 2

Individual finite-element mesh cell and associated local degrees of freedom of the discrete approximate solution. The complex electric field E(x, y) is approximated locally by low-order polynomials. Ex is constant in x and linear in y, Ey is linear in x and constant in y, and Ez is bilinear in (x, y).

Fig. 3
Fig. 3

Finite-element mesh. Horizontal edges correspond to Ex values, vertical edges to Ey, and nodal values to Ez. The nodal and edge values along the right boundary (boxes) indicate degrees of freedom removed by quasi-periodicity conditions (43).

Fig. 4
Fig. 4

Director field of the idealized tilted cholesteric structure for benchmarking iterative numerical solver and preconditioner. Lengths are in units of the vacuum wavelength. Cholesteric pitch=average wavelength of light in the layer. Cell thickness=2×pitch.

Fig. 5
Fig. 5

Director field of one periodic cell (8×5 μm) of idealized IPS device in the on state.

Fig. 6
Fig. 6

Calculated intensities of reflected and transmitted radiation at the front (y=0) and back (y=5 μm) interfaces for IPS-mode cell of Fig. 5: (a) no polarizers, (b) crossed polarizers.

Fig. 7
Fig. 7

Calculations of Liu and Kelly41 using the BPM and the GOA of intensity of transmitted radiation at the back (y=5 μm) interface for IPS-mode cell of Fig. 5: (a) no polarizers, (b) crossed polarizers.

Tables (2)

Tables Icon

Table 1 Error versus Number of Grid Cells for Test Problem a

Tables Icon

Table 2 Time to Solve Test Problem: Preconditioned versus Not a

Equations (183)

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

μ=μ0I,(r)=0r(x, y),
r(x+2L, y)=r(x, y),-<x<,
0<y<d.
×E=iωμ0H,
×H=-iωE.
×(×E)-k2r(x, y)E=0.
k=k0(1+iκ),k0=ωμ00,
Einc(r)=exp(ikr)E0inc,
kk=k2,kE0inc=0.
ET(L, y)=exp(i2kxL)ET(-L, y),
(×E)T(L, y)=exp(i2kxL)(×E)T(-L, y).
Ω{(×E)(×F*)-k2[r(x, y)E]F*}
+Ω[νˆ×(×E)]F*=0.
FT(L, y)=exp(i2kx*L)FT(-L, y),
Ω{(×E)(×F*)-k2[r(x, y)E]F*}
+Γ1+Γ2[νˆ×(×E)]TFT*=0.
C1:ET(, 0)[-yˆ×(×E1)]T(, 0-),
C2:ET(, d)[+yˆ×(×E2)]T(, d+).
Ω{(×E)(×F*)-k2[r(x, y)E]F*}
+Γ1C1(ET)FT*+Γ2C2(ET)FT*=0.
Ω{(×E)(×F*)-k2[r(x, y)E]F*}
+Γ1C1(ET)FT*+Γ1[νˆ×(×E3)]TFT*
+Γ2C2(ET)FT*=0,
Ω{(×E)(×F*)-k2[r(x, y)E]F*}
+Γ1C1(ET)FT*+Γ2C2(ET)FT*
=Γ1g1T(x)FT*,
g1(x)=yˆ×(×E3)(x, 0-).
a(E, F)=l(F),
a(E, F)=Ω{(×E)(×F*)-k2[r(x, y)E]F*}+Γ1C1(ET)FT*+Γ2C2(ET)FT*,
l(F)=Γ1g1T(x)FT*.
E(x, y)=n=-exp[iλnx±iμn(y-y0)]En,
λn=kx+nπL,μn=(k2-λn2)1/2,
n=-,, ,
En=12L-LLexp(-iλnx)E(x, y0)dx.
λnEnx±μnEny=0,
C(ET)=in=-exp(iλnx)μ˜nαnnr2k2Enxβnμ˜n2Enz,
αn
=(nr2μn-μ˜n)exp(iμ˜na)-(nr2μn+μ˜n)exp(-iμ˜na)(nr2μn-μ˜n)exp(iμ˜na)+(nr2μn+μ˜n)exp(-iμ˜na),
βn
=(μ˜n-μn)exp(iμ˜na)-(μ˜n+μn)exp(-iμ˜na)(μ˜n-μn)exp(iμ˜na)+(μ˜n+μn)exp(-iμ˜na).
μ˜n=(nr2k2-λn2)1/2,
EnT=12L-LLexp(-iλnx)ET(x, 0)dx
EnT=12L-LLexp(-iλnx)ET(x, d)dx
C(ET)=-in=-exp(iλnx)μnk2Enxμn2Enz.
Cα[exp(iλnx)ET]=exp(iλnx)MαnET,α=1,2,
C[exp(iλnx)ET]=i exp(iλnx)μ˜nαnnr2k2Exβnμ˜n2Ez=exp(iλnx)iαnnr2k2/μ˜n00iβnμ˜nET,
M1n=M2n=iαnnr2k2/μ˜n00iβnμ˜n
M1n=M2n=-ik2/μn00-iμn.
g1T(x)=-4i exp(ikxx)nr2k2exp(-ikya)E0xinc(nr2ky-k˜y)exp(ik˜ya)+(nr2ky+k˜y)exp(-ik˜ya)kyk˜yexp(-ikya)E0zinc(k˜y-ky)exp(ik˜ya)+(k˜y+ky)exp(-ik˜ya),
g1T(x)=-2ikyexp(ikxx)k2E0xincky2E0zinc.
Eh(x, y)=nexp(iλnx)En(y).
xi=iΔx,i=-I,, I,Δx=L/I,
yj=jΔy,j=0,, J,Δy=d/J.
Exh(x, y)=Ei+1/2,jxyj+1-yΔy+Ei+1/2,j+1xy-yjΔy,
Eyh(x, y)=Ei,j+1/2yxi+1-xΔx+Ei+1,j+1/2yx-xiΔx,
Ezh(x, y)=Ei,jzxi+1-xΔxyj+1-yΔy+Ei+1,jzx-xiΔxyj+1-yΔy+Ei,j+1zxi+1-xΔxy-yjΔy+Ei+1,j+1zx-xiΔxy-yjΔy.
Ei+1/2,jx,j=0,, J,
Ei,j+1/2y,j=0,, J-1,
Ei,jz,j=0,, J,
EI,j+1/2y=exp(i2kxL)E-I,j+1/2y,j=0,, J-1,
EI,jz=exp(i2kxL)E-I,jz,j=0,, J.
Ae=b,
l(F)=Γ1g1T(x)  FT*+Γ2g2T(x)  FT*+ΩG(x, y)  F*.
×(×E)-k2D=G(x, y),inΩ,
y(xEy-yEx)-k2Dx=Gx(x, y),
-x(xEy-yEx)-k2Dy=Gy(x, y),
-x2Ez-y2Ez-k2Dz=Gz(x, y),
[yˆ×(×E)]T+C1(ET)=g1T(x),onΓ1,
-[yˆ×(×E)]T+C2(ET)=g2T(x),onΓ2,
(xEy-yEx)(x, 0)+C1x(x)=g1x(x),
-yEz(x, 0)+C1z(x)=g1z(x),
-(xEy-yEx)(x, d)+C2x(x)=g2x(x),
yEz(x, d)+C2z(x)=g2z(x),
r(x, y)r(xi+1/2, yj+1/2),
xi<x<xi+1,yj<y<yj+1,
yjyj+1xixi+1f(x, y)dxdy
ΔxΔy4 (fi,j+fi+1,j+fi,j+1+fi+1,j+1),
δxui=ui+1/2-ui-1/2Δx,
δx2ui=δx(δxui)=ui+1-2ui+ui-1Δx2.
δy(δxEi+1/2,jy-δyEi+1/2,jx)-k2D¯i+1/2,jx=Gi+1/2,jx,
j=1,, J-1,
-δx(δxEi,j+1/2y-δyEi,j+1/2x)-k2D¯i,j+1/2y=Gi,j+1/2y,
j=0,, J-1,
-δx2Ei,jz-δy2Ei,jz-k2D¯i,jz=Gi,jz,
j=1,, J-1,
D¯i+1/2,jx=i+1/2,j-1/2xx+i+1/2,j+1/2xx2 Ei+1/2,jx+12i+1/2,j-1/2xyEi,j-1/2y+Ei+1,j-1/2y2+i+1/2,j+1/2xyEi,j+1/2y+Ei+1,j+1/2y2+i+1/2,j-1/2xz+i+1/2,j+1/2xz2Ei,jz+Ei+1,jz2,
D¯i,j+1/2y=12i-1/2,j+1/2yxEi-1/2,jx+Ei-1/2,j+1x2+i+1/2,j+1/2yxEi+1/2,jx+Ei+1/2,j+1x2+i-1/2,j+1/2yy+i+1/2,j+1/2yy2 Ei,j+1/2y+i-1/2,j+1/2yz+i+1/2,j+1/2yz2Ei,jz+Ei,j+1z2,
D¯i,jz=12i-1/2,j-1/2zx+i-1/2,j+1/2zx2 Ei-1/2,jx+i+1/2,j-1/2zx+i+1/2,j+1/2zx2 Ei+1/2,jx+12i-1/2,j-1/2zy+i+1/2,j-1/2zy2 Ei,j-1/2y+i-1/2,j+1/2zy+i+1/2,j+1/2zy2 Ei,j+1/2y+i-1/2,j-1/2zz+i+1/2,j-1/2zz+i-1/2,j+1/2zz+i+1/2,j+1/2zz4 Ei,jz.
(δxEi+1/2,1/2y-δyEi+1/2,1/2x)+Ci+1/21x
=gi+1/21x+Δy2 (k2D¯i+1/2,0x+Gi+1/2,0x),
-δyEi,1/2z+Ci1z
=gi1z+Δy2 (δx2Ei,0z+k2D¯i,0z+Gi,0z),
-(δxEi+1/2,J-1/2y-δyEi+1/2,J-1/2x)+Ci+1/22x
=gi+1/22x+Δy2 (k2D¯i+1/2,Jx+Gi+1/2,Jx),
δyEi,J-1/2z+Ci2z
=gi2z+Δy2 (δx2Ei,Jz+k2D¯i,Jz+Gi,Jz),
D¯i+1/2,0x=i+1/2,1/2xxEi+1/2,0x+i+1/2,1/2xyEi,1/2y+Ei+1,1/2y2+i+1/2,1/2xzEi,0z+Ei+1,0z2,
D¯i,0z=12 (i-1/2,1/2zxEi-1/2,0x+i+1/2,1/2zxEi+1/2,0x)+i-1/2,1/2zy+i+1/2,1/2zy2 Ei,1/2y+i-1/2,1/2zz+i+1/2,1/2zz2 Ei,0z,
D¯i+1/2,Jx=i+1/2,J-1/2xxEi+1/2,Jx+i+1/2,J-1/2xyEi,J-1/2y+Ei+1,J-1/2y2+i+1/2,J-1/2xzEi,Jz+Ei+1,Jz2,
D¯i,Jz=12 [i-1/2,J-1/2zxEi-1/2,Jx+i+1/2,J-1/2zxEi+1/2,Jx]+i-1/2,J-1/2zy+i+1/2,J-1/2zy2 Ei,J-1/2y+i-1/2,J-1/2zz+i+1/2,J-1/2zz2 Ei,Jz,
El+1/2,mx=n=-II-1E^n,mxexp(iλnxl+1/2),m=0,, J,
El,m+1/2y=n=-II-1E^n,m+1/2yexp(iλnxl),m=0,, J-1,
El,mz=n=-II-1E^n,mzexp(iλnxl),m=0,, J,
E^n,mx=12Il=-II-1El+1/2,mxexp(-iλnxl+1/2),
m=0,, J,
E^n,m+1/2y=12Il=-II-1El,m+1/2yexp(-iλnxl),
m=0,, J-1,
E^n,mz=12Il=-II-1El,mzexp(-iλnxl),m=0,, J.
Ω{(×E)  (×F*)-k2[r(x, y)E]  F*},
ΓC(ET)  FT*=ΓET C(FT)*,
C(ET)=-in=-exp(iλnx)μnk2Enxμn2Enz,
C(FT)=in=-exp(iλn*x)μn*(k*)2Fnx(μn*)2Fnz.
M-1Ae=M-1b.
×(×E)-k2r(x, y)E=G(x, y),inΩ,
[yˆ×(×E)]T+C1(ET)=0,onΓ1,
-[yˆ×(×E)]T+C2(ET)=0,onΓ2,
r(x, y)p(y).
p(y)=np2I,0<y<d,
p(y)=12L-LLr(x, y)dx,0<y<d,
2Δy (δxEl+1/2,1/2y-δyEl+1/2,1/2x)-k2np2El+1/2,0x+2Δy Cl+1/21x=Gl+1/2,0x,
δy(δxEl+1/2,my-δyEl+1/2,mx)-k2np2El+1/2,mx=Gl+1/2,mx,m=1,, J-1,
-2Δy (δxEl+1/2,J-1/2y-δyEl+1/2,J-1/2x)-k2np2El+1/2,Jx+2Δy Cl+1/22x=Gl+1/2,Jx,
-δx(δxEl,m+1/2y-δyEl,m+1/2x)-k2np2El,m+1/2y=Gl,m+1/2y,m=0,, J-1,
-δx2El,0z-2Δy δyEl,1/2z-k2np2El,0z+2Δy Cl1z=Gl,0z,
-δx2El,mz-δy2El,mz-k2np2El,mz=Gl,mz,m=1,, J-1,
-δx2El,Jz+2Δy δyEl,J-1/2z-k2np2El,Jz+2Δy Cl2z=Gl,Jz.
δxexp(iλx)=exp[iλ(x+Δx/2)]-exp[iλ(x-Δx/2)]Δx=2iΔxsinλ Δx2exp(iλx).
C^n1xC^n1z=Mxx1nMxz1nMzx1nMzz1n E^n,0xE^n,0z,
C^n2xC^n2z=Mxx2nMxz2nMzx2nMzz2n E^n,JxE^n,Jz,
2Δy2iΔxsinλnΔx2E^n,1/2y-δyE^n,1/2x-k2np2E^n,0x
+2Δy (Mxx1nE^n,0x+Mxz1nE^n,0z)
=G^n,0x,
2iΔxsinλnΔx2δyE^n,my-δy2E^n,mx-k2np2E^n,mx
=G^n,mx,m=1,, J-1,
-2Δy2iΔxsinλnΔx2E^n,j-1/2y-δyE^n,j-1/2x-k2np2E^n,Jx
+2Δy (Mxx2nE^n,Jx+Mxz2nE^n,Jz)
=G^n,Jx,
4Δx2sin2λnΔx2E^n,m+1/2y+2iΔxsinλnΔx2δyE^n,m+1/2x
-k2np2E^n,m+1/2y
=G^n,m+1/2y,m=0,, J-1,
4Δx2sin2λnΔx2E^n,0z-2Δy δyE^n,1/2z-k2np2E^n,0z
+2Δy (Mzx1nE^n,0x+Mzz1nE^n,0z)
=G^n,0z,
4Δx2sin2λnΔx2E^n,mz-δy2E^n,mz-k2np2E^n,mz=G^n,mz,m=1,, J-1,
4Δx2sin2λnΔx2E^n,Jz+2Δy δyE^n,j-1/2z-k2np2E^n,Jz+2Δy (Mzx2nE^n,Jx+Mzz2nE^n,Jz)=G^n,Jz.
r(x, y)=no2I+(ne2-no2)nˆ(x, y)nˆ(x, y).
E(x, y, z)=exp(ikzz)E˜(x, y).
2E1+nr2k2E1=0,E1=0,
-a<y<0,
2E1+k2E1=0,E1=0,
-<y<-a,
E1T(x, 0)=ET(x, 0),-L<x<L.
E1(x, y)=n=-exp(iλnx)[exp(iμ˜ny)An+exp(-iμ˜ny)Bn],-a<y<0n=-exp(iλnx)exp[-iμn(y+a)]Cn,-<y<-a,
μ˜n=(nr2k2-λn2)1/2,
E1T(x, 0)=ET(x, 0)=n=-exp(iλnx)EnT,
EnT=12L-LLexp(-iλnx)ET(x, 0)dx.
λnAnx+μ˜nAny=0,
λnBnx-μ˜nBny=0,
λnCnx-μnCny=0,
Anx
=(nr2μn-μ˜n)exp(iμ˜na)(nr2μn-μ˜n)exp(iμ˜na)+(nr2μn+μ˜n)exp(-iμ˜na)× Enx,
Bnx
=(nr2μn+μ˜n)exp(-iμ˜na)(nr2μn-μ˜n)exp(iμ˜na)+(nr2μn+μ˜n)exp(-iμ˜na)× Enx,
Cnx
=2nr2μn(nr2μn-μ˜n)exp(iμ˜na)+(nr2μn+μ˜n)exp(-iμ˜na)× Enx,
Anz=(μ˜n-μn)exp(iμ˜na)(μ˜n-μn)exp(iμ˜na)+(μ˜n+μn)exp(-iμ˜na)× Enz,
Bnz=(μ˜n+μn)exp(-iμ˜na)(μ˜n-μn)exp(iμ˜na)+(μ˜n+μn)exp(-iμ˜na)× Enz,
Cnz=2μ˜n(μ˜n-μn)exp(iμ˜na)+(μ˜n+μn)exp(-iμ˜na)× Enz.
C1(ET)=limy0-[-yˆ×(×E1)]T(x, y)=in=-exp(iλnx)μ˜nαnnr2k2Enxβnμ˜n2Enz,
EnT=12L-LLexp(-iλnx)ET(x, d)dx.
E2(x, y)=E1x(x, d-y)-E1y(x, d-y)E1z(x, d-y)
[yˆ×(×E2)]T(x, d+)=[-yˆ×(×E1)]T(x, 0-).
E3T(x, 0)=0,-L<x<L,
E3(x, y)=exp(ikxx)exp(ik˜yy)A+exp(-ik˜yy)B,-a<y<0,exp(ikyy)E0inc+exp[-iky(y+a)]C,-<y<-a,
Ax=2k˜yexp(-ikya)E0xinc(nr2ky-k˜y)exp(ik˜ya)+(nr2ky+k˜y)exp(-ik˜ya),
Bx=-2k˜yexp(-ikya)E0xinc(nr2ky-k˜y)exp(ik˜ya)+(nr2ky+k˜y)exp(-ik˜ya),
Cx=-(nr2ky+k˜y)exp(ik˜ya)+(nr2ky-k˜y)exp(-ik˜ya)(nr2ky-k˜y)exp(ik˜ya)+(nr2ky+k˜y)exp(-ik˜ya)×exp(-ikya)E0xinc,
Az=2kyexp(-ikya)E0zinc(k˜y-ky)exp(ik˜ya)+(k˜y+ky)exp(-ik˜ya),
Bz=-2kyexp(-ikya)E0zinc(k˜y-ky)exp(ik˜ya)+(k˜y+ky)exp(-ik˜ya),
Cz=-(k˜y+ky)exp(ik˜ya)+(k˜y-ky)exp(-ik˜ya)(k˜y-ky)exp(ik˜ya)+(k˜y+ky)exp(-ik˜ya)×exp(-ikya)E0zinc,
Ay=-kxk˜y Ax,By=kxk˜y Bx,Cy=kxk˜y Cx.
g1T(x)=limy0-[yˆ×(×E3)]T(x, y)=-4i exp(ikxx)
×nr2k2exp(-ikya)E0xinc(nr2ky-k˜y)exp(ik˜ya)+(nr2ky+k˜y)exp(-ik˜ya)kyk˜yexp(-ikya)E0zinc(k˜y-ky)exp(ik˜ya)+(k˜y+ky)exp(-ik˜ya),

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