Abstract

Electromagnetic wave propagation in anisotropic inhomogeneous media is computed by a novel reduced-order model technique, which is based on the restriction of the Marcuvitz–Schwinger equations on Krylov subspaces and on the application of the singular-value decomposition. The model is derived from the standard coupled-wave method and includes both wide-angle diffraction and light scattering at dielectric interfaces. The method, currently implemented for two-dimensional problems, was applied to the analysis of different liquid-crystal test cells. Numerical results are compared with those obtained through the application of the coupled-wave method and the Jones method and with experimental microscopic measurements.

© 2002 Optical Society of America

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References

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  1. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982).
    [CrossRef]
  2. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  3. W. Liu, J. Kelly, “Multidimensional modeling of liquid crystal optics using a ray-tracing technique,” in Proceedings of the Society for Information Display International Symposium 2000 (Society for Information Display, San Jose, Calif., 2000), pp. 847–849.
  4. K. Rokushima, J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. 73, 901–908 (1983).
    [CrossRef]
  5. M. G. Moharam, D. A. Pommet, E. B. Grann, “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]
  6. P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
    [CrossRef]
  7. E. 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]
  8. B. Witzigmann, P. Regli, W. Fichtner, “Rigorous electromagnetic simulation of liquid crystal displays,” J. Opt. Soc. Am. A 15, 753–757 (1998).
    [CrossRef]
  9. J. Van Roey, J. van der Donk, P. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
    [CrossRef]
  10. D. Yevick, M. Glasner, “Forward wide-angle light propagation in semiconductor rib waveguides,” Opt. Lett. 15, 174–176 (1990).
    [CrossRef] [PubMed]
  11. W. P. Huang, C. L. Xu, S. K. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photon. Technol. Lett. 4, 148–151 (1992).
    [CrossRef]
  12. B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
    [CrossRef]
  13. R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction,” J. Opt. Soc. Am. A 9, 265–273 (1992).
    [CrossRef]
  14. P. Galatola, C. Oldano, P. B. Sunil Kumar, “Symmetry properties of anisotropic dielectric gratings,” J. Opt. Soc. Am. A 11, 1332–1341 (1994).
    [CrossRef]
  15. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chaps. 2 and 3.
  16. M. E. Becker, H. Wohler, M. Kamm, J. Kreis, “Numerical modeling of IPS effects: a new approach and more results,” in Proceedings of the Society for Information Display International Symposium 1996 (Society for Information Display, San Jose, Calif., 1996), pp. 596–599.
  17. D. K. G. de Boer, M. T. Johnson, J. A. M. M. van Haaren, M. Fukumoto, M. Yoshiga, Y. Hamawaki, T. Unate, F. A. Fernández, S. E. Day, “Optical response of structured vertically aligned and in-plane switching LCDs,” in Proceedings of the 20th International Display Research Conference (Society for Information Display, San Jose, Calif.,2000), pp. 22–25.
  18. D. K. G. de Boer, R. Cortie, A. D. Pearson, M. E. Becker, H. Whler, D. Olivero, O. A. Peverini, K. Neyts, E. E. Kriezis, S. J. Elston, “Optical simulations and measurements of in-plane switching structures with rapid refractive-index variations,” in Proceedings of the Society for Information Display International Symposium 2001 (Society for Information Display, San Jose, Calif., 2001), pp. 818–821.

1999

E. 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

1997

1995

1994

1992

W. P. Huang, C. L. Xu, S. K. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photon. Technol. Lett. 4, 148–151 (1992).
[CrossRef]

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
[CrossRef]

R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction,” J. Opt. Soc. Am. A 9, 265–273 (1992).
[CrossRef]

1990

1983

1982

1981

1972

Bardyszewski, W.

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
[CrossRef]

Becker, M. E.

D. K. G. de Boer, R. Cortie, A. D. Pearson, M. E. Becker, H. Whler, D. Olivero, O. A. Peverini, K. Neyts, E. E. Kriezis, S. J. Elston, “Optical simulations and measurements of in-plane switching structures with rapid refractive-index variations,” in Proceedings of the Society for Information Display International Symposium 2001 (Society for Information Display, San Jose, Calif., 2001), pp. 818–821.

M. E. Becker, H. Wohler, M. Kamm, J. Kreis, “Numerical modeling of IPS effects: a new approach and more results,” in Proceedings of the Society for Information Display International Symposium 1996 (Society for Information Display, San Jose, Calif., 1996), pp. 596–599.

Berreman, D. W.

Chaudhuri, S. K.

W. P. Huang, C. L. Xu, S. K. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photon. Technol. Lett. 4, 148–151 (1992).
[CrossRef]

Cortie, R.

D. K. G. de Boer, R. Cortie, A. D. Pearson, M. E. Becker, H. Whler, D. Olivero, O. A. Peverini, K. Neyts, E. E. Kriezis, S. J. Elston, “Optical simulations and measurements of in-plane switching structures with rapid refractive-index variations,” in Proceedings of the Society for Information Display International Symposium 2001 (Society for Information Display, San Jose, Calif., 2001), pp. 818–821.

Day, S. E.

D. K. G. de Boer, M. T. Johnson, J. A. M. M. van Haaren, M. Fukumoto, M. Yoshiga, Y. Hamawaki, T. Unate, F. A. Fernández, S. E. Day, “Optical response of structured vertically aligned and in-plane switching LCDs,” in Proceedings of the 20th International Display Research Conference (Society for Information Display, San Jose, Calif.,2000), pp. 22–25.

de Boer, D. K. G.

D. K. G. de Boer, M. T. Johnson, J. A. M. M. van Haaren, M. Fukumoto, M. Yoshiga, Y. Hamawaki, T. Unate, F. A. Fernández, S. E. Day, “Optical response of structured vertically aligned and in-plane switching LCDs,” in Proceedings of the 20th International Display Research Conference (Society for Information Display, San Jose, Calif.,2000), pp. 22–25.

D. K. G. de Boer, R. Cortie, A. D. Pearson, M. E. Becker, H. Whler, D. Olivero, O. A. Peverini, K. Neyts, E. E. Kriezis, S. J. Elston, “Optical simulations and measurements of in-plane switching structures with rapid refractive-index variations,” in Proceedings of the Society for Information Display International Symposium 2001 (Society for Information Display, San Jose, Calif., 2001), pp. 818–821.

Elston, S. J.

E. 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]

D. K. G. de Boer, R. Cortie, A. D. Pearson, M. E. Becker, H. Whler, D. Olivero, O. A. Peverini, K. Neyts, E. E. Kriezis, S. J. Elston, “Optical simulations and measurements of in-plane switching structures with rapid refractive-index variations,” in Proceedings of the Society for Information Display International Symposium 2001 (Society for Information Display, San Jose, Calif., 2001), pp. 818–821.

Feit, M. D.

Fernández, F. A.

D. K. G. de Boer, M. T. Johnson, J. A. M. M. van Haaren, M. Fukumoto, M. Yoshiga, Y. Hamawaki, T. Unate, F. A. Fernández, S. E. Day, “Optical response of structured vertically aligned and in-plane switching LCDs,” in Proceedings of the 20th International Display Research Conference (Society for Information Display, San Jose, Calif.,2000), pp. 22–25.

Fichtner, W.

Fleck, J. A.

Fukumoto, M.

D. K. G. de Boer, M. T. Johnson, J. A. M. M. van Haaren, M. Fukumoto, M. Yoshiga, Y. Hamawaki, T. Unate, F. A. Fernández, S. E. Day, “Optical response of structured vertically aligned and in-plane switching LCDs,” in Proceedings of the 20th International Display Research Conference (Society for Information Display, San Jose, Calif.,2000), pp. 22–25.

Galatola, P.

Glasner, M.

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
[CrossRef]

D. Yevick, M. Glasner, “Forward wide-angle light propagation in semiconductor rib waveguides,” Opt. Lett. 15, 174–176 (1990).
[CrossRef] [PubMed]

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chaps. 2 and 3.

Grann, E. B.

Hamawaki, Y.

D. K. G. de Boer, M. T. Johnson, J. A. M. M. van Haaren, M. Fukumoto, M. Yoshiga, Y. Hamawaki, T. Unate, F. A. Fernández, S. E. Day, “Optical response of structured vertically aligned and in-plane switching LCDs,” in Proceedings of the 20th International Display Research Conference (Society for Information Display, San Jose, Calif.,2000), pp. 22–25.

Hermansson, B.

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
[CrossRef]

Huang, W. P.

W. P. Huang, C. L. Xu, S. K. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photon. Technol. Lett. 4, 148–151 (1992).
[CrossRef]

Johnson, M. T.

D. K. G. de Boer, M. T. Johnson, J. A. M. M. van Haaren, M. Fukumoto, M. Yoshiga, Y. Hamawaki, T. Unate, F. A. Fernández, S. E. Day, “Optical response of structured vertically aligned and in-plane switching LCDs,” in Proceedings of the 20th International Display Research Conference (Society for Information Display, San Jose, Calif.,2000), pp. 22–25.

Kamm, M.

M. E. Becker, H. Wohler, M. Kamm, J. Kreis, “Numerical modeling of IPS effects: a new approach and more results,” in Proceedings of the Society for Information Display International Symposium 1996 (Society for Information Display, San Jose, Calif., 1996), pp. 596–599.

Kelly, J.

W. Liu, J. Kelly, “Multidimensional modeling of liquid crystal optics using a ray-tracing technique,” in Proceedings of the Society for Information Display International Symposium 2000 (Society for Information Display, San Jose, Calif., 2000), pp. 847–849.

Kreis, J.

M. E. Becker, H. Wohler, M. Kamm, J. Kreis, “Numerical modeling of IPS effects: a new approach and more results,” in Proceedings of the Society for Information Display International Symposium 1996 (Society for Information Display, San Jose, Calif., 1996), pp. 596–599.

Kriezis, E. E.

E. 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]

D. K. G. de Boer, R. Cortie, A. D. Pearson, M. E. Becker, H. Whler, D. Olivero, O. A. Peverini, K. Neyts, E. E. Kriezis, S. J. Elston, “Optical simulations and measurements of in-plane switching structures with rapid refractive-index variations,” in Proceedings of the Society for Information Display International Symposium 2001 (Society for Information Display, San Jose, Calif., 2001), pp. 818–821.

Kumar, P. B. Sunil

Lagasse, P.

Lalanne, P.

Liu, W.

W. Liu, J. Kelly, “Multidimensional modeling of liquid crystal optics using a ray-tracing technique,” in Proceedings of the Society for Information Display International Symposium 2000 (Society for Information Display, San Jose, Calif., 2000), pp. 847–849.

Moharam, M. G.

Neyts, K.

D. K. G. de Boer, R. Cortie, A. D. Pearson, M. E. Becker, H. Whler, D. Olivero, O. A. Peverini, K. Neyts, E. E. Kriezis, S. J. Elston, “Optical simulations and measurements of in-plane switching structures with rapid refractive-index variations,” in Proceedings of the Society for Information Display International Symposium 2001 (Society for Information Display, San Jose, Calif., 2001), pp. 818–821.

Oldano, C.

Olivero, D.

D. K. G. de Boer, R. Cortie, A. D. Pearson, M. E. Becker, H. Whler, D. Olivero, O. A. Peverini, K. Neyts, E. E. Kriezis, S. J. Elston, “Optical simulations and measurements of in-plane switching structures with rapid refractive-index variations,” in Proceedings of the Society for Information Display International Symposium 2001 (Society for Information Display, San Jose, Calif., 2001), pp. 818–821.

Pearson, A. D.

D. K. G. de Boer, R. Cortie, A. D. Pearson, M. E. Becker, H. Whler, D. Olivero, O. A. Peverini, K. Neyts, E. E. Kriezis, S. J. Elston, “Optical simulations and measurements of in-plane switching structures with rapid refractive-index variations,” in Proceedings of the Society for Information Display International Symposium 2001 (Society for Information Display, San Jose, Calif., 2001), pp. 818–821.

Peverini, O. A.

D. K. G. de Boer, R. Cortie, A. D. Pearson, M. E. Becker, H. Whler, D. Olivero, O. A. Peverini, K. Neyts, E. E. Kriezis, S. J. Elston, “Optical simulations and measurements of in-plane switching structures with rapid refractive-index variations,” in Proceedings of the Society for Information Display International Symposium 2001 (Society for Information Display, San Jose, Calif., 2001), pp. 818–821.

Pommet, D. A.

Ratowsky, R. P.

Regli, P.

Rokushima, K.

Unate, T.

D. K. G. de Boer, M. T. Johnson, J. A. M. M. van Haaren, M. Fukumoto, M. Yoshiga, Y. Hamawaki, T. Unate, F. A. Fernández, S. E. Day, “Optical response of structured vertically aligned and in-plane switching LCDs,” in Proceedings of the 20th International Display Research Conference (Society for Information Display, San Jose, Calif.,2000), pp. 22–25.

van der Donk, J.

van Haaren, J. A. M. M.

D. K. G. de Boer, M. T. Johnson, J. A. M. M. van Haaren, M. Fukumoto, M. Yoshiga, Y. Hamawaki, T. Unate, F. A. Fernández, S. E. Day, “Optical response of structured vertically aligned and in-plane switching LCDs,” in Proceedings of the 20th International Display Research Conference (Society for Information Display, San Jose, Calif.,2000), pp. 22–25.

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chaps. 2 and 3.

Van Roey, J.

Whler, H.

D. K. G. de Boer, R. Cortie, A. D. Pearson, M. E. Becker, H. Whler, D. Olivero, O. A. Peverini, K. Neyts, E. E. Kriezis, S. J. Elston, “Optical simulations and measurements of in-plane switching structures with rapid refractive-index variations,” in Proceedings of the Society for Information Display International Symposium 2001 (Society for Information Display, San Jose, Calif., 2001), pp. 818–821.

Witzigmann, B.

Wohler, H.

M. E. Becker, H. Wohler, M. Kamm, J. Kreis, “Numerical modeling of IPS effects: a new approach and more results,” in Proceedings of the Society for Information Display International Symposium 1996 (Society for Information Display, San Jose, Calif., 1996), pp. 596–599.

Xu, C. L.

W. P. Huang, C. L. Xu, S. K. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photon. Technol. Lett. 4, 148–151 (1992).
[CrossRef]

Yamakita, J.

Yeh, P.

Yevick, D.

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
[CrossRef]

D. Yevick, M. Glasner, “Forward wide-angle light propagation in semiconductor rib waveguides,” Opt. Lett. 15, 174–176 (1990).
[CrossRef] [PubMed]

Yoshiga, M.

D. K. G. de Boer, M. T. Johnson, J. A. M. M. van Haaren, M. Fukumoto, M. Yoshiga, Y. Hamawaki, T. Unate, F. A. Fernández, S. E. Day, “Optical response of structured vertically aligned and in-plane switching LCDs,” in Proceedings of the 20th International Display Research Conference (Society for Information Display, San Jose, Calif.,2000), pp. 22–25.

IEEE Photon. Technol. Lett.

W. P. Huang, C. L. Xu, S. K. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photon. Technol. Lett. 4, 148–151 (1992).
[CrossRef]

J. Lightwave Technol.

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Liq. Cryst.

E. 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]

Opt. Lett.

Other

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chaps. 2 and 3.

M. E. Becker, H. Wohler, M. Kamm, J. Kreis, “Numerical modeling of IPS effects: a new approach and more results,” in Proceedings of the Society for Information Display International Symposium 1996 (Society for Information Display, San Jose, Calif., 1996), pp. 596–599.

D. K. G. de Boer, M. T. Johnson, J. A. M. M. van Haaren, M. Fukumoto, M. Yoshiga, Y. Hamawaki, T. Unate, F. A. Fernández, S. E. Day, “Optical response of structured vertically aligned and in-plane switching LCDs,” in Proceedings of the 20th International Display Research Conference (Society for Information Display, San Jose, Calif.,2000), pp. 22–25.

D. K. G. de Boer, R. Cortie, A. D. Pearson, M. E. Becker, H. Whler, D. Olivero, O. A. Peverini, K. Neyts, E. E. Kriezis, S. J. Elston, “Optical simulations and measurements of in-plane switching structures with rapid refractive-index variations,” in Proceedings of the Society for Information Display International Symposium 2001 (Society for Information Display, San Jose, Calif., 2001), pp. 818–821.

W. Liu, J. Kelly, “Multidimensional modeling of liquid crystal optics using a ray-tracing technique,” in Proceedings of the Society for Information Display International Symposium 2000 (Society for Information Display, San Jose, Calif., 2000), pp. 847–849.

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of a liquid-crystal cell in the xz plane.

Fig. 2
Fig. 2

Director profile versus lateral position computed by the 2dimMOS program16 for the data reported in Table 1. The electrode locations are reported in each plot as well. (a) Structured vertical aligned (SVA) cell, (b) in-plane switching (IPS) cell, (c) thin-film transistor (TFT) cell.

Fig. 3
Fig. 3

Dynamic range of the singular values of matrix M built as in Eq. (22) in the case of the SVA cell. Computations were performed with 59 beams and by dividing the cell thickness into Nz=20 layers. The reported singular values refer to the last layer.

Fig. 4
Fig. 4

Computed transmittance of the SVA cell versus lateral position. Computations were performed by means of the reduced-order grating method (RGM) for different values of the number Q of Krylov vectors. Simulation parameters: 59 beams, Nz=20 layers, and two forward iterations.

Fig. 5
Fig. 5

Computed transmittance of the SVA cell versus lateral position. Computations were performed by means of the grating method (GM), the RGM, and the Jones method. In the GM and the RGM, 59 beams were considered. Other simulation parameters for the RGM: Q=6 Krylov vectors, Nz=20 layers, and two forward iterations.

Fig. 6
Fig. 6

Computed transmittance of the SVA cell versus lateral position. Computations were performed by means of the GM, the RGM, and the fourth-order Runge–Kutta (R.K.) algorithm for the field propagation in the longitudinal direction. In all the simulations 21 beams were considered. In both the GM and the RGM, Nz=20 longitudinal steps were used, corresponding to Δzλ/2. In the fourth-order Runge-Kutta algorithm, values of Δz=λ/10 and Δz=λ/20 were adopted. Other simulation parameters for the RGM: Q=6 Krylov vectors and two forward iterations.

Fig. 7
Fig. 7

Transmittance of the IPS cell versus lateral position. Comparison between microscopic measurements (see the text) and simulations was performed by means of the RGM and the Jones method. Simulation parameters for the RGM: 51 beams, Q=5 Krylov vectors, Nz=20 layers, and two forward iterations.

Fig. 8
Fig. 8

Computed transmittance of the TFT cell versus lateral position. Computations were performed by means of the GM and the RGM with 81 beams. (a) Normal incidence, (b) oblique incidence at an angle of θinc=60° in air. The RGM simulations were carried out with Q=7 Krylov vectors, Nz=20 longitudinal layers, and two forward iterations.

Fig. 9
Fig. 9

Ratio between the CPU time required by the RGM (TRGM) and that of the standard GM (TGM) as a function of the number of beams considered in the simulations of the TFT cell for normal incidence. The RGM simulations were carried out with Q=7 Krylov vectors, Nz=20 longitudinal layers, and two forward iterations.

Tables (1)

Tables Icon

Table 1 Cell Parameters Used in the Simulations

Equations (52)

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

σijkjEk=-Jωμ0Hi,
σijkjHk=Jωϵ0ϵijEj,i, j, k=x, y, z,
σzνσνeσ=-Jk0hz,
σzνσνhσ=Jk0ϵzσeσ+Jk0ϵzzez,
σανzνez+σαzσzeσ=-Jk0hα,
σανzνhz+σαzσzhσ=Jk0ϵασeσ+Jk0ϵαzez,
σανzν1Jk0ϵzzσzνσνhσ-ϵzσϵzzeσ+σαzσzeσ=-Jk0hα,
σανzν-1Jk0σzνσνeσ+σαzσzhσ=Jk0ϵασeσ+Jk0ϵαz1jk0ϵzzσzνσνhσ-ϵzσϵzzeσ.
zΨ=Jk0AΨ,
ϵi,j(x, z)=m=-NNϵ˜i,j(m)(z)exp(-Jkx,mx),
kx,inc=k0 sin(θ)cos(ϕ),ky,inc=k0 sin(θ)sin(ϕ).
Ψ(x, y, z)=n=-Nn=NΨ˜(n)(z)exp[-J(kx,n+kx,inc)x]exp(-Jky,inc y).
e˜x=e˜x(-N)e˜x(N),e˜y=e˜y(-N)e˜y(N),
h˜x=h˜x(-N)h˜x(N),h˜y=h˜y(-N)h˜y(N)
zΨ˜=Jk0A˜Ψ˜,
A˜=A˜ex,exA˜ex,eyA˜ex,hxA˜ex,hyA˜ey,exA˜ey,eyA˜ey,hxA˜ey,hyA˜hx,exA˜hx,eyA˜hx,hxA˜hx,hyA˜hy,exA˜hy,eyA˜hy,hxA˜hy,hy.
[A˜ex,ex]p,m=kx,p+kx,inck0γ˜zx(p-m),
T(l)=exp[Jk0A˜(l)h(l)],
T=l=1NzT(l).
K(Q)(A˜;Ψ˜A)=span(Ψ˜A, A˜·Ψ˜A, , A˜Q-1·Ψ˜A)
exp(Jk0A˜h)I+Jk0A˜h+12!(Jk0A˜h)2++1(Q-1)!(Jk0A˜h)Q-1,
Ψ˜(z=h)Ψ˜A+Jk0hA˜·Ψ˜A++1(Q-1)!(Jk0hA˜)Q-1·Ψ˜A.
M=[r1r2rQ-1].
Ψ˜=UPΥ˜.
dΥ˜dz=Jk0UPHA˜·UP·Υ˜.
Υ˜(z=h)=exp(Jk0A˜Ph)·Υ˜A.
Ψ˜(z=h)=UP exp(Jk0A˜Ph)·UPHΨ˜A.
e˜x,k=Aex,ex·e˜x,k-1+Aex,ey·e˜y,k-1+Aex,hx·h˜x,k-1+Aex,hy·h˜y,k-1.
e˜x,k(p)=m=-NNkx,p+kx,inck0γ˜zx(p-m)e˜x,k-1(m)p=-N, , N.
e˜x,k=kx+kx,inck0(gzx*e˜x,k-1),
Ψ˜B(0)=Pglass+cB++(0)+Pglass-cB+-(0).
Ψ˜A(1)=Pglass+cA-+(1)+Pglass-cA--(1),
Ψ˜A(2)=Pglass+cˆA-++Pglass-cA--(1).
A=1k02 Jk0(xγzx+γzxx)Jk0(xγzy+γzyx)(xγzzy+γzzxy)-(xγzzx+γzzx2+k02)Jk0(yγzx+γzxy)Jk0(yγzy+γzyy)(yγzzy+γzzy2+k02)-(yγzzx+γzzxy)-(xy-k02ϵyx+k02γyx)(x2+k02ϵyy-k02γyy)-Jk0(γzyy)Jk0(γzyx)-(k02ϵxx-k02γxx+y2)(k02γyx-k02ϵyx+yx)-Jk0(γzxy)Jk0(γzxx),
γzx=ϵzxϵzz,γzy=ϵzyϵzz,γxx=ϵzx2ϵzz,
γyx=ϵzyϵzxϵzz,γzz=1ϵzz,γyy=ϵyz2ϵzz.
[A˜ex,ex]p,m=kx,p+kx,inck0γ˜zx(p-m),
[A˜ex,ey]p,m=kx,p+kx,inck0γ˜zy(p-m),
[A˜ex,hx]p,m=-kx,p-mky,inck02γ˜zz(p-m)-(kx,m+kx,inc)ky,inck02γ˜zz(p-m),
[A˜ex,hy]p,m=kx,p-m(kx,m+kx,inc)k02γ˜zz(p-m)+(kx,m+kx,inc)2k02γ˜zz(p-m)-δpm,
[A˜ey,ex]p,m=ky,inck0γ˜zx(p-m),
[A˜ey,ey]p,m=ky,inck0γ˜zy(p-m),
[A˜ey,hx]p,m=-ky,inc2k02γ˜zz(p-m)+δpm,
[A˜ey,hy]p,m=(kx,m+kx,inc)ky,inck02γ˜zz(p-m),
[A˜hx,ex]p,m=ϵ˜yx(p-m)-γ˜yx(p-m)+(kx,p+kx,inc)ky,inck02δpm,
[A˜hx,ey]p,m=ϵ˜yy(p-m)-γ˜yy(p-m)-(kx,p+kx,inc)2k02δpm,
[A˜hx,hx]p,m=ky,inck0γ˜zy(p-m),
[A˜hx,hy]p,m=-kx,m+kx,inck0γ˜zy(p-m),
[A˜hy,ex]p,m=-ϵ˜xx(p-m)+γ˜xx(p-m)+ky,inc2k02δpm,
[A˜hy,ey]p,m=-ϵ˜yx(p-m)+γ˜yx(p-m)-(kx,p+kx,inc)ky,inck02δpm,
[A˜hy,hx]p,m=-ky,inck0γ˜zx(p-m),
[A˜hy,hy]p,m=kx,m+kx,inck0γ˜zx(p-m).

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