Abstract

An efficient full-vectorial finite element beam propagation method is presented that uses higher order vector elements to calculate the wide angle propagation of an optical field through inhomogeneous, anisotropic optical materials such as liquid crystals. The full dielectric permittivity tensor is considered in solving Maxwell’s equations. The wide applicability of the method is illustrated with different examples: the propagation of a laser beam in a uniaxial medium, the tunability of a directional coupler based on liquid crystals and the near-field diffraction of a plane wave in a structure containing micrometer scale variations in the transverse refractive index, similar to the pixels of a spatial light modulator.

©2009 Optical Society of America

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  1. J. M. López-Doña, J. G. Wangüemert-Pérez, and I. Molina-Fernández, “Fast-fourier-based three-dimensional full-vectorial beam propagation method,” IEEE Photonics Technol. Lett.  17, 2319–2321 (2005).
    [Crossref]
  2. C. Ma and E. Van Keuren, “A three-dimensional wide-angle BPM for optical waveguide structures,” Opt. Express  15, 402–407 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-15-2-402.
    [Crossref] [PubMed]
  3. A. J. Davidson and S. J. Elston, “Three-dimensional beam propagation model for the optical path of light through a nematic liquid crystal,” J. Mod. Opt.  53, 979–989 (2006).
    [Crossref]
  4. Q. Wang, G. Farrell, and Y. Semenova, “Modeling liquid-crystal devices with the three-dimensional full-vector beam propagation method,” J. Opt. Soc. Am.  23, 2014–2019 (2006).
    [Crossref]
  5. K. Saitoh and M. Koshiba, “e,” J. Lightwave Technol. 19, 405–413 (2001).
    [Crossref]
  6. D. Schulz, C. Glingener, M. Bludszuweit, and E. Voges, “Mixed finite element beam propagation method,” J. Lightwave Technol.  16, 1336–1342 (1998).
    [Crossref]
  7. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol.  18, 737–743 (2000).
    [Crossref]
  8. F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guid. Wave Lett.  8, 223–225 (1998).
    [Crossref]
  9. J. Jin, The finite element method in electromagnetics, 2nd edition (Wiley, New York US, 2002).
  10. J. Beeckman, R. James, F. A. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” accepted for publication in J. Lightwave Technol. (2009).
    [Crossref]
  11. M. Koshiba and K. Inoue, “Simple and efficient finite-element analysis of microwave and optical waveguides,” IEEE Trans. Microwave Theory Tech.  40, 371–377 (1992).
    [Crossref]
  12. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett.  17, 1743–1745 (1992).
    [Crossref] [PubMed]
  13. GiD, the personal pre and post processor, http://gid.cimne.upc.es/.
  14. R. James, E. Willman, F. A. Fernandez, and S. E. Day, “Finite-Element Modeling of Liquid-Crystal Hydrodynamics With a Variable Degree of Order,” IEEE Trans. Electron Devices  53, 1575–1582 (2006).
    [Crossref]
  15. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron.  37, 95–106 (2005).
    [Crossref]
  16. J. C. Campbell, F. A. Blum, D. W. Shaw, and K. L. Lawlay, “GaAs Electro-optic directional coupler switch,” Appl. Phys. Lett.  27, 202–205 (1975).
    [Crossref]
  17. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd edition (Clarendon, Oxford UK, 1993).
  18. N. Amarasinghe, E. Gartland, and J. Kelly, “Modeling optical properties of liquid-crystal devices by numerical solution of time-harmonic Maxwell equations,” J. Opt. Soc. Am.  21, 1344–1361 (2004).
    [Crossref]
  19. E. Buckley, “Holographic Laser Projection Technology,” SID Int. Symp. Digest Tech. Papers  39, 1074–1079 (2008).
    [Crossref]

2008 (1)

E. Buckley, “Holographic Laser Projection Technology,” SID Int. Symp. Digest Tech. Papers  39, 1074–1079 (2008).
[Crossref]

2007 (1)

2006 (3)

R. James, E. Willman, F. A. Fernandez, and S. E. Day, “Finite-Element Modeling of Liquid-Crystal Hydrodynamics With a Variable Degree of Order,” IEEE Trans. Electron Devices  53, 1575–1582 (2006).
[Crossref]

A. J. Davidson and S. J. Elston, “Three-dimensional beam propagation model for the optical path of light through a nematic liquid crystal,” J. Mod. Opt.  53, 979–989 (2006).
[Crossref]

Q. Wang, G. Farrell, and Y. Semenova, “Modeling liquid-crystal devices with the three-dimensional full-vector beam propagation method,” J. Opt. Soc. Am.  23, 2014–2019 (2006).
[Crossref]

2005 (2)

J. M. López-Doña, J. G. Wangüemert-Pérez, and I. Molina-Fernández, “Fast-fourier-based three-dimensional full-vectorial beam propagation method,” IEEE Photonics Technol. Lett.  17, 2319–2321 (2005).
[Crossref]

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron.  37, 95–106 (2005).
[Crossref]

2004 (1)

N. Amarasinghe, E. Gartland, and J. Kelly, “Modeling optical properties of liquid-crystal devices by numerical solution of time-harmonic Maxwell equations,” J. Opt. Soc. Am.  21, 1344–1361 (2004).
[Crossref]

2001 (1)

2000 (1)

1998 (2)

D. Schulz, C. Glingener, M. Bludszuweit, and E. Voges, “Mixed finite element beam propagation method,” J. Lightwave Technol.  16, 1336–1342 (1998).
[Crossref]

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guid. Wave Lett.  8, 223–225 (1998).
[Crossref]

1992 (2)

M. Koshiba and K. Inoue, “Simple and efficient finite-element analysis of microwave and optical waveguides,” IEEE Trans. Microwave Theory Tech.  40, 371–377 (1992).
[Crossref]

G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett.  17, 1743–1745 (1992).
[Crossref] [PubMed]

1975 (1)

J. C. Campbell, F. A. Blum, D. W. Shaw, and K. L. Lawlay, “GaAs Electro-optic directional coupler switch,” Appl. Phys. Lett.  27, 202–205 (1975).
[Crossref]

Amarasinghe, N.

N. Amarasinghe, E. Gartland, and J. Kelly, “Modeling optical properties of liquid-crystal devices by numerical solution of time-harmonic Maxwell equations,” J. Opt. Soc. Am.  21, 1344–1361 (2004).
[Crossref]

Beeckman, J.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron.  37, 95–106 (2005).
[Crossref]

J. Beeckman, R. James, F. A. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” accepted for publication in J. Lightwave Technol. (2009).
[Crossref]

Bludszuweit, M.

Blum, F. A.

J. C. Campbell, F. A. Blum, D. W. Shaw, and K. L. Lawlay, “GaAs Electro-optic directional coupler switch,” Appl. Phys. Lett.  27, 202–205 (1975).
[Crossref]

Buckley, E.

E. Buckley, “Holographic Laser Projection Technology,” SID Int. Symp. Digest Tech. Papers  39, 1074–1079 (2008).
[Crossref]

Cambournac, C.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron.  37, 95–106 (2005).
[Crossref]

Campbell, J. C.

J. C. Campbell, F. A. Blum, D. W. Shaw, and K. L. Lawlay, “GaAs Electro-optic directional coupler switch,” Appl. Phys. Lett.  27, 202–205 (1975).
[Crossref]

Chew, W. C.

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guid. Wave Lett.  8, 223–225 (1998).
[Crossref]

Davidson, A. J.

A. J. Davidson and S. J. Elston, “Three-dimensional beam propagation model for the optical path of light through a nematic liquid crystal,” J. Mod. Opt.  53, 979–989 (2006).
[Crossref]

Day, S. E.

R. James, E. Willman, F. A. Fernandez, and S. E. Day, “Finite-Element Modeling of Liquid-Crystal Hydrodynamics With a Variable Degree of Order,” IEEE Trans. Electron Devices  53, 1575–1582 (2006).
[Crossref]

De Cort, W.

J. Beeckman, R. James, F. A. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” accepted for publication in J. Lightwave Technol. (2009).
[Crossref]

de Gennes, P. G.

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd edition (Clarendon, Oxford UK, 1993).

Elston, S. J.

A. J. Davidson and S. J. Elston, “Three-dimensional beam propagation model for the optical path of light through a nematic liquid crystal,” J. Mod. Opt.  53, 979–989 (2006).
[Crossref]

Farrell, G.

Q. Wang, G. Farrell, and Y. Semenova, “Modeling liquid-crystal devices with the three-dimensional full-vector beam propagation method,” J. Opt. Soc. Am.  23, 2014–2019 (2006).
[Crossref]

Fernandez, F. A.

R. James, E. Willman, F. A. Fernandez, and S. E. Day, “Finite-Element Modeling of Liquid-Crystal Hydrodynamics With a Variable Degree of Order,” IEEE Trans. Electron Devices  53, 1575–1582 (2006).
[Crossref]

J. Beeckman, R. James, F. A. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” accepted for publication in J. Lightwave Technol. (2009).
[Crossref]

Gartland, E.

N. Amarasinghe, E. Gartland, and J. Kelly, “Modeling optical properties of liquid-crystal devices by numerical solution of time-harmonic Maxwell equations,” J. Opt. Soc. Am.  21, 1344–1361 (2004).
[Crossref]

Glingener, C.

Hadley, G. R.

Haelterman, M.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron.  37, 95–106 (2005).
[Crossref]

Hutsebaut, X.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron.  37, 95–106 (2005).
[Crossref]

Inoue, K.

M. Koshiba and K. Inoue, “Simple and efficient finite-element analysis of microwave and optical waveguides,” IEEE Trans. Microwave Theory Tech.  40, 371–377 (1992).
[Crossref]

James, R.

R. James, E. Willman, F. A. Fernandez, and S. E. Day, “Finite-Element Modeling of Liquid-Crystal Hydrodynamics With a Variable Degree of Order,” IEEE Trans. Electron Devices  53, 1575–1582 (2006).
[Crossref]

J. Beeckman, R. James, F. A. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” accepted for publication in J. Lightwave Technol. (2009).
[Crossref]

Jin, J.

J. Jin, The finite element method in electromagnetics, 2nd edition (Wiley, New York US, 2002).

Kelly, J.

N. Amarasinghe, E. Gartland, and J. Kelly, “Modeling optical properties of liquid-crystal devices by numerical solution of time-harmonic Maxwell equations,” J. Opt. Soc. Am.  21, 1344–1361 (2004).
[Crossref]

Koshiba, M.

Lawlay, K. L.

J. C. Campbell, F. A. Blum, D. W. Shaw, and K. L. Lawlay, “GaAs Electro-optic directional coupler switch,” Appl. Phys. Lett.  27, 202–205 (1975).
[Crossref]

López-Doña, J. M.

J. M. López-Doña, J. G. Wangüemert-Pérez, and I. Molina-Fernández, “Fast-fourier-based three-dimensional full-vectorial beam propagation method,” IEEE Photonics Technol. Lett.  17, 2319–2321 (2005).
[Crossref]

Ma, C.

Molina-Fernández, I.

J. M. López-Doña, J. G. Wangüemert-Pérez, and I. Molina-Fernández, “Fast-fourier-based three-dimensional full-vectorial beam propagation method,” IEEE Photonics Technol. Lett.  17, 2319–2321 (2005).
[Crossref]

Neyts, K.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron.  37, 95–106 (2005).
[Crossref]

J. Beeckman, R. James, F. A. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” accepted for publication in J. Lightwave Technol. (2009).
[Crossref]

Prost, J.

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd edition (Clarendon, Oxford UK, 1993).

Saitoh, K.

Schulz, D.

Semenova, Y.

Q. Wang, G. Farrell, and Y. Semenova, “Modeling liquid-crystal devices with the three-dimensional full-vector beam propagation method,” J. Opt. Soc. Am.  23, 2014–2019 (2006).
[Crossref]

Shaw, D. W.

J. C. Campbell, F. A. Blum, D. W. Shaw, and K. L. Lawlay, “GaAs Electro-optic directional coupler switch,” Appl. Phys. Lett.  27, 202–205 (1975).
[Crossref]

Teixeira, F. L.

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guid. Wave Lett.  8, 223–225 (1998).
[Crossref]

Tsuji, Y.

Van Keuren, E.

Vanbrabant, P. J. M.

J. Beeckman, R. James, F. A. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” accepted for publication in J. Lightwave Technol. (2009).
[Crossref]

Voges, E.

Wang, Q.

Q. Wang, G. Farrell, and Y. Semenova, “Modeling liquid-crystal devices with the three-dimensional full-vector beam propagation method,” J. Opt. Soc. Am.  23, 2014–2019 (2006).
[Crossref]

Wangüemert-Pérez, J. G.

J. M. López-Doña, J. G. Wangüemert-Pérez, and I. Molina-Fernández, “Fast-fourier-based three-dimensional full-vectorial beam propagation method,” IEEE Photonics Technol. Lett.  17, 2319–2321 (2005).
[Crossref]

Willman, E.

R. James, E. Willman, F. A. Fernandez, and S. E. Day, “Finite-Element Modeling of Liquid-Crystal Hydrodynamics With a Variable Degree of Order,” IEEE Trans. Electron Devices  53, 1575–1582 (2006).
[Crossref]

Appl. Phys. Lett. (1)

J. C. Campbell, F. A. Blum, D. W. Shaw, and K. L. Lawlay, “GaAs Electro-optic directional coupler switch,” Appl. Phys. Lett.  27, 202–205 (1975).
[Crossref]

IEEE Microw. Guid. Wave Lett. (1)

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guid. Wave Lett.  8, 223–225 (1998).
[Crossref]

IEEE Photonics Technol. Lett. (1)

J. M. López-Doña, J. G. Wangüemert-Pérez, and I. Molina-Fernández, “Fast-fourier-based three-dimensional full-vectorial beam propagation method,” IEEE Photonics Technol. Lett.  17, 2319–2321 (2005).
[Crossref]

IEEE Trans. Electron Devices (1)

R. James, E. Willman, F. A. Fernandez, and S. E. Day, “Finite-Element Modeling of Liquid-Crystal Hydrodynamics With a Variable Degree of Order,” IEEE Trans. Electron Devices  53, 1575–1582 (2006).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

M. Koshiba and K. Inoue, “Simple and efficient finite-element analysis of microwave and optical waveguides,” IEEE Trans. Microwave Theory Tech.  40, 371–377 (1992).
[Crossref]

J. Lightwave Technol. (3)

J. Mod. Opt. (1)

A. J. Davidson and S. J. Elston, “Three-dimensional beam propagation model for the optical path of light through a nematic liquid crystal,” J. Mod. Opt.  53, 979–989 (2006).
[Crossref]

J. Opt. Soc. Am. (2)

Q. Wang, G. Farrell, and Y. Semenova, “Modeling liquid-crystal devices with the three-dimensional full-vector beam propagation method,” J. Opt. Soc. Am.  23, 2014–2019 (2006).
[Crossref]

N. Amarasinghe, E. Gartland, and J. Kelly, “Modeling optical properties of liquid-crystal devices by numerical solution of time-harmonic Maxwell equations,” J. Opt. Soc. Am.  21, 1344–1361 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect,” Opt. Quantum Electron.  37, 95–106 (2005).
[Crossref]

SID Int. Symp. Digest Tech. Papers (1)

E. Buckley, “Holographic Laser Projection Technology,” SID Int. Symp. Digest Tech. Papers  39, 1074–1079 (2008).
[Crossref]

Other (4)

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd edition (Clarendon, Oxford UK, 1993).

GiD, the personal pre and post processor, http://gid.cimne.upc.es/.

J. Jin, The finite element method in electromagnetics, 2nd edition (Wiley, New York US, 2002).

J. Beeckman, R. James, F. A. Fernandez, W. De Cort, P. J. M. Vanbrabant, and K. Neyts, “Calculation of fully anisotropic liquid crystal waveguide modes,” accepted for publication in J. Lightwave Technol. (2009).
[Crossref]

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Figures (12)

Fig. 1.
Fig. 1. (a) Transverse cross section of a directional coupler, indicating the different PML regions 1–8 at the edges of the computational window, (b) Sketch of a hybrid linear tangential/ quadratic normal (LT/QN) vector element indicating the transverse and nodal unknowns Φ ti (i=1 to 8) and Φ zj (j=1 to 6), respectively.

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2. Input optical field at z=0µm: (a) transverse electric field, (b) Ex component, (c) Ey component, (d) Ez component.

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Fig. 3.
Fig. 3. Evolution of Ez upon propagation through the uniaxial layer: (a) d=1.25µm, (b) d=2.5µm, (c) d=3.75µm, (d) d=5mm.

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Fig. 4.
Fig. 4. (a) Wave vectors of incident and transmitted plane waves at an interface between air and an anisotropic material ε̿. (b) Definition of inclination θ and azimuth ϕ angles.

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Fig. 5.
Fig. 5. Intensity profile of the gaussian optical field (a) at z=0µm and (b) at z=10µm.

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Fig. 6.
Fig. 6. The relative error on the beam displacement after propagation over 10µm as a function of the number of elements.

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Fig. 7.
Fig. 7. Director profile of the liquid crystal molecules when 0V is applied. The tilt angle θ of the director in the liquid crystal layer is indicated in the color bar.

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Fig. 8.
Fig. 8. Director profile of the liquid crystal molecules when 7V is applied. The tilt angle θ of the director in the liquid crystal layer is indicated in the color bar.

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Fig. 9.
Fig. 9. Evolution of the electric field components Ey (top) and Ez (bottom) upon propagation through the directional coupler when no voltage is applied. The contour plots are normalized to have equal maximum values for each frame.

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Fig. 10.
Fig. 10. Evolution of the electric field components Ey (top) and Ez (bottom) upon propagation through the directional coupler when 7V is applied over the electrodes. The contour plots are normalized to have equal maximum values for each frame.

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Fig. 11.
Fig. 11. Sketch of the four pixel structure, indicating the ON and OFF pixels.

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Fig. 12.
Fig. 12. Amplitude of the near-field Ex component obtained after propagation of a plane wave [0,Ey ,0] over 5µm through the 4-pixel structure for (a) 4µm by 4µm and (b) 2µm by 2µm pixels.

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Equations (32)

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

∇×(μ̿−1∇×Eˉ)−k02ε̿.Eˉ=0 ,
Eˉ(x,y,z)=Φˉ(x,y,z)exp(−jk0n0z),
[ϕxϕyϕz]=[[Nx]T[0][Ny]T[0][0]j[L]T][ϕteϕze],
[[Btt][0][0][0]]d2dz2{[ϕtϕz]}−[2jk0n0[Btt]j[Btz]j[Bzt][0]]ddz{[ϕtϕz]}
−{[[Att]−j[Ctz]j[Czt][Bzz]]−jk0n0[[0]j[Btz]j[Bzt][0]]+k02n02[[Btt][0][0][0]]}{[[ϕt][ϕz]]}=0
[Att]=Σe∬epzz(∂{Nx}∂y∂{Nx}T∂y+∂{Ny}∂x∂{Ny}T∂x−∂{Nx}∂y∂{Ny}T∂x−∂{Ny}∂x∂{Nx}T∂y)
−k02(qxx{Nx}{Nx}T+qxy{Nx}{Ny}T+qyx{Ny}{Nx}T+qyy{Ny}{Ny}T)dxdy
[Btt]=Σe∬e[pyy{Nx}{Nx}T+pxx{Ny}{Ny}T]dxdy
[Btz]=Σe∬e[pyy{Nx}∂{L}T∂x+pxx{Ny}∂{L}T∂y]dxdy
[Btz]=Σe∬e[pyy∂{L}∂x{Nx}T+pxx∂{L}∂y{Ny}T]dxdy
[Bzz]=Σe∬e[pyy∂{L}∂x∂{L}T∂x+pxx∂{L}∂y∂{L}T∂y−k02qzz{L}{L}T]dxdy
[Ctz]=Σe∬ek02[qxz{Nx}{L}T+qyz{Ny}{L}T]dxdy
[Czt]=Σe∬ek02[qzx{L}{Nx}T+qzy{L}{Ny}T]dxdy ,
ϕz=−j Bzz−1 Bzt dϕtdz −Bzz−1(jCzt+k0n0Bzt)ϕt.
{[Btt]−[Btz][Bzz]−1[Bzt]}d2ϕtdz2−{2jk0n0[Btt]−j[Btz][Bzz]−1(j[Czt]+k0n0[Bzt])+j(j[Ctz]−k0n0[Bzz])[Bzz]−1[Bzt]dϕtdz}−{[Att]+k02n02[Btt]+(j[Ctz]−k0n0[Btz])[Bzz]−1(j[Czt]+k0n0[Bzt])}ϕt=0,
Addz[ϕtdϕtdz]+B[ϕtdϕtdz]=0,
A=[[A11][A12][1][0]]
B=[[B11][0][0]−[1]],
[A11]=−2 j k0 n0 [Btt]−[Btz][Bzz]−1([Czt]−jk0n0[Bzt])+([Ctz]+jk0n0[Btz])[Bzz]−1[Bzt]
[A12]=[Btt]−[Btz][Bzz]−1[Bzt]
[B11]=−[Att]−k02n02[Btt]−(j[Ctz]−k0n0[Btz])[Bzz]−1(j[Czt]+k0n0[Bzt]).
[Y]i[ϕtdϕtdz]i+1=[Z]i[ϕtdϕtdz]i
[Y]i=[A]i+ϑΔz [B]i
[Z]i=[A]i−(1−ϑ)Δz[B]i,
[Y′]=[A11]+ϑΔz[B11]
[Z′]=[A11]−(1−ϑ)Δz[B11].
dϕtdz∣z=0 ≈ϕt∣z=Δz′−ϕt∣z=0Δz′ ,
0=Dzt
=εzyEyt+εzzEzt .
tanδ=SytSzt=−EztHxtEytHxt=εzyεzz.
ε̿=[ε⊥+Δεsin2θcos2ϕΔεsinθcosθcosϕΔεsin2θcosϕsinϕΔεsinθcosθcosϕε⊥+Δεcos2θΔεsinθcosθsinϕΔεsin2θcosϕsinϕΔεsinθcosθsinϕε⊥+Δεsin2θsin2ϕ],
δ=arctan (εzyεzz) ≈3.688°.

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