Abstract

Grating-assisted backward coupling between two parallel waveguides is analyzed with a set of four coupled equations. The equations were recently derived by a unified approach that is appropriate for other coupling problems as well. Examples include three different operation regions and provide for the field variation along the guides as well as for the reflectivity and transmission coefficients. It is shown that the reduction of the four-wave coupling problem to a set of two coupled equations is possible only for some specific cases and at the cost of reduced accuracy. The model may be especially useful for the design of optical components for various applications such as optical switching and multiplexing.

© 1999 Optical Society of America

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References

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  1. A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
    [CrossRef]
  2. D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
    [CrossRef]
  3. T. L. Koch, E. G. Burkhardt, F. G. Storz, T. J. Bridjes, T. Sizer, “Vertically grating-coupled ARROW structures for III–V integrated optics,” IEEE J. Quantum Electron. 23, 889–897 (1987).
    [CrossRef]
  4. T. L. Koch, J. Corvini, W. T. Tsang, U. Koren, B. I. Miler, “Wavelength selective interlayer directionally grating-coupled InP/InGaAsP waveguide photodetection,” Appl. Phys. Lett. 51, 1060–1062 (1987).
    [CrossRef]
  5. H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  6. A. Yariv, Quantum Electronics (Wiley, New York, 1989).
  7. S. S. Orlov, A. Yariv, S. V. Essen, “Coupled mode analysis of fiber-optic add-drop filters for dense wavelength-division multiplexing,” Opt. Lett. 22, 688–690 (1997).
    [CrossRef] [PubMed]
  8. A. S. Kewitsch, G. A. Rakuljic, P. A. Willems, A. Yariv, “All fiber zero-insertion-loss add-drop filter for wavelength-division multiplexing,” Opt. Lett. 23, 106–108 (1998).
    [CrossRef]
  9. H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).
  10. D. G. Hall, ed., Selected papers on Coupled-Mode Theory in Guided-Wave Optics (SPIE Milestone Series, MS84, Society of Photo-Optical Instrument Engineers, Bellingham, Wash., 1993).
  11. A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum Electron. 34, 1109–1116 (1998).
    [CrossRef]
  12. A. Hardy, W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
    [CrossRef]
  13. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1991).
  14. T. Makino, J. Glinski, “Transfer matrix analysis of the amplified spontaneous emission of DFB semiconductor laser amplifiers,” IEEE J. Quantum Electron. 24, 1507–1518 (1988).
    [CrossRef]
  15. W. E. Boyce, R. C. Diprima, Elementary Differential Equations and Boundary Value Problems (Wiley, New York, 1986).
  16. A. Hardy, W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).
    [CrossRef]
  17. S. Barnett, Matrices Methods and Applications (Oxford University, New York, 1990).
  18. R. E. Collin, Foundations of Microwave Engineering (McGraw-Hill, New York, 1966).
  19. R. C. Hall, R. Mittra, K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
    [CrossRef]
  20. W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in fortran: The Art of Scientific Computing (Cambridge University, New York, 1986).
  21. P. Yeh, H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. 19, 2848–2855 (1980).
    [CrossRef] [PubMed]
  22. D. Marcuse, “Bandwidth of forward and backward coupling directional couplers,” J. Lightwave Technol. 5, 1773–1777 (1987).
    [CrossRef]
  23. H. A. Haus, Y. Lai, “Narrow-band distributed feedback reflector design,” J. Lightwave Technol. 9, 754–760 (1991).
    [CrossRef]
  24. R. Marz, H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. 19, 273–287 (1987).
    [CrossRef]

1998 (2)

1997 (1)

1991 (2)

D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
[CrossRef]

H. A. Haus, Y. Lai, “Narrow-band distributed feedback reflector design,” J. Lightwave Technol. 9, 754–760 (1991).
[CrossRef]

1988 (2)

T. Makino, J. Glinski, “Transfer matrix analysis of the amplified spontaneous emission of DFB semiconductor laser amplifiers,” IEEE J. Quantum Electron. 24, 1507–1518 (1988).
[CrossRef]

R. C. Hall, R. Mittra, K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

1987 (4)

R. Marz, H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. 19, 273–287 (1987).
[CrossRef]

D. Marcuse, “Bandwidth of forward and backward coupling directional couplers,” J. Lightwave Technol. 5, 1773–1777 (1987).
[CrossRef]

T. L. Koch, E. G. Burkhardt, F. G. Storz, T. J. Bridjes, T. Sizer, “Vertically grating-coupled ARROW structures for III–V integrated optics,” IEEE J. Quantum Electron. 23, 889–897 (1987).
[CrossRef]

T. L. Koch, J. Corvini, W. T. Tsang, U. Koren, B. I. Miler, “Wavelength selective interlayer directionally grating-coupled InP/InGaAsP waveguide photodetection,” Appl. Phys. Lett. 51, 1060–1062 (1987).
[CrossRef]

1986 (1)

A. Hardy, W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).
[CrossRef]

1985 (1)

A. Hardy, W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[CrossRef]

1980 (1)

1977 (1)

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
[CrossRef]

1972 (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Barnett, S.

S. Barnett, Matrices Methods and Applications (Oxford University, New York, 1990).

Boyce, W. E.

W. E. Boyce, R. C. Diprima, Elementary Differential Equations and Boundary Value Problems (Wiley, New York, 1986).

Bridjes, T. J.

T. L. Koch, E. G. Burkhardt, F. G. Storz, T. J. Bridjes, T. Sizer, “Vertically grating-coupled ARROW structures for III–V integrated optics,” IEEE J. Quantum Electron. 23, 889–897 (1987).
[CrossRef]

Burkhardt, E. G.

T. L. Koch, E. G. Burkhardt, F. G. Storz, T. J. Bridjes, T. Sizer, “Vertically grating-coupled ARROW structures for III–V integrated optics,” IEEE J. Quantum Electron. 23, 889–897 (1987).
[CrossRef]

Collin, R. E.

R. E. Collin, Foundations of Microwave Engineering (McGraw-Hill, New York, 1966).

Corvini, J.

T. L. Koch, J. Corvini, W. T. Tsang, U. Koren, B. I. Miler, “Wavelength selective interlayer directionally grating-coupled InP/InGaAsP waveguide photodetection,” Appl. Phys. Lett. 51, 1060–1062 (1987).
[CrossRef]

Diprima, R. C.

W. E. Boyce, R. C. Diprima, Elementary Differential Equations and Boundary Value Problems (Wiley, New York, 1986).

Essen, S. V.

Flannery, B.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in fortran: The Art of Scientific Computing (Cambridge University, New York, 1986).

Glinski, J.

T. Makino, J. Glinski, “Transfer matrix analysis of the amplified spontaneous emission of DFB semiconductor laser amplifiers,” IEEE J. Quantum Electron. 24, 1507–1518 (1988).
[CrossRef]

Hall, D. G.

D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
[CrossRef]

Hall, R. C.

R. C. Hall, R. Mittra, K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

Hardy, A.

A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum Electron. 34, 1109–1116 (1998).
[CrossRef]

A. Hardy, W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[CrossRef]

Haus, H. A.

H. A. Haus, Y. Lai, “Narrow-band distributed feedback reflector design,” J. Lightwave Technol. 9, 754–760 (1991).
[CrossRef]

Kewitsch, A. S.

Koch, T. L.

T. L. Koch, J. Corvini, W. T. Tsang, U. Koren, B. I. Miler, “Wavelength selective interlayer directionally grating-coupled InP/InGaAsP waveguide photodetection,” Appl. Phys. Lett. 51, 1060–1062 (1987).
[CrossRef]

T. L. Koch, E. G. Burkhardt, F. G. Storz, T. J. Bridjes, T. Sizer, “Vertically grating-coupled ARROW structures for III–V integrated optics,” IEEE J. Quantum Electron. 23, 889–897 (1987).
[CrossRef]

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).

Koren, U.

T. L. Koch, J. Corvini, W. T. Tsang, U. Koren, B. I. Miler, “Wavelength selective interlayer directionally grating-coupled InP/InGaAsP waveguide photodetection,” Appl. Phys. Lett. 51, 1060–1062 (1987).
[CrossRef]

Lai, Y.

H. A. Haus, Y. Lai, “Narrow-band distributed feedback reflector design,” J. Lightwave Technol. 9, 754–760 (1991).
[CrossRef]

Makino, T.

T. Makino, J. Glinski, “Transfer matrix analysis of the amplified spontaneous emission of DFB semiconductor laser amplifiers,” IEEE J. Quantum Electron. 24, 1507–1518 (1988).
[CrossRef]

Marcuse, D.

D. Marcuse, “Bandwidth of forward and backward coupling directional couplers,” J. Lightwave Technol. 5, 1773–1777 (1987).
[CrossRef]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1991).

Marz, R.

R. Marz, H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. 19, 273–287 (1987).
[CrossRef]

Miler, B. I.

T. L. Koch, J. Corvini, W. T. Tsang, U. Koren, B. I. Miler, “Wavelength selective interlayer directionally grating-coupled InP/InGaAsP waveguide photodetection,” Appl. Phys. Lett. 51, 1060–1062 (1987).
[CrossRef]

Mittra, R.

R. C. Hall, R. Mittra, K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

Mitzner, K. M.

R. C. Hall, R. Mittra, K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

Nakamura, M.

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
[CrossRef]

Nolting, H. P.

R. Marz, H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. 19, 273–287 (1987).
[CrossRef]

Orlov, S. S.

Press, W.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in fortran: The Art of Scientific Computing (Cambridge University, New York, 1986).

Rakuljic, G. A.

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Sizer, T.

T. L. Koch, E. G. Burkhardt, F. G. Storz, T. J. Bridjes, T. Sizer, “Vertically grating-coupled ARROW structures for III–V integrated optics,” IEEE J. Quantum Electron. 23, 889–897 (1987).
[CrossRef]

Storz, F. G.

T. L. Koch, E. G. Burkhardt, F. G. Storz, T. J. Bridjes, T. Sizer, “Vertically grating-coupled ARROW structures for III–V integrated optics,” IEEE J. Quantum Electron. 23, 889–897 (1987).
[CrossRef]

Streifer, W.

A. Hardy, W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[CrossRef]

Taylor, H. F.

Teudolsky, S.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in fortran: The Art of Scientific Computing (Cambridge University, New York, 1986).

Tsang, W. T.

T. L. Koch, J. Corvini, W. T. Tsang, U. Koren, B. I. Miler, “Wavelength selective interlayer directionally grating-coupled InP/InGaAsP waveguide photodetection,” Appl. Phys. Lett. 51, 1060–1062 (1987).
[CrossRef]

Vetterling, W.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in fortran: The Art of Scientific Computing (Cambridge University, New York, 1986).

Willems, P. A.

Yariv, A.

Yeh, P.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

T. L. Koch, J. Corvini, W. T. Tsang, U. Koren, B. I. Miler, “Wavelength selective interlayer directionally grating-coupled InP/InGaAsP waveguide photodetection,” Appl. Phys. Lett. 51, 1060–1062 (1987).
[CrossRef]

IEEE J. Quantum Electron. (5)

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
[CrossRef]

A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum Electron. 34, 1109–1116 (1998).
[CrossRef]

T. Makino, J. Glinski, “Transfer matrix analysis of the amplified spontaneous emission of DFB semiconductor laser amplifiers,” IEEE J. Quantum Electron. 24, 1507–1518 (1988).
[CrossRef]

A. Hardy, W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).
[CrossRef]

T. L. Koch, E. G. Burkhardt, F. G. Storz, T. J. Bridjes, T. Sizer, “Vertically grating-coupled ARROW structures for III–V integrated optics,” IEEE J. Quantum Electron. 23, 889–897 (1987).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

R. C. Hall, R. Mittra, K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. Lightwave Technol. (3)

A. Hardy, W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[CrossRef]

D. Marcuse, “Bandwidth of forward and backward coupling directional couplers,” J. Lightwave Technol. 5, 1773–1777 (1987).
[CrossRef]

H. A. Haus, Y. Lai, “Narrow-band distributed feedback reflector design,” J. Lightwave Technol. 9, 754–760 (1991).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

R. Marz, H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. 19, 273–287 (1987).
[CrossRef]

Prog. Opt. (1)

D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
[CrossRef]

Other (8)

A. Yariv, Quantum Electronics (Wiley, New York, 1989).

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).

D. G. Hall, ed., Selected papers on Coupled-Mode Theory in Guided-Wave Optics (SPIE Milestone Series, MS84, Society of Photo-Optical Instrument Engineers, Bellingham, Wash., 1993).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1991).

W. E. Boyce, R. C. Diprima, Elementary Differential Equations and Boundary Value Problems (Wiley, New York, 1986).

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in fortran: The Art of Scientific Computing (Cambridge University, New York, 1986).

S. Barnett, Matrices Methods and Applications (Oxford University, New York, 1990).

R. E. Collin, Foundations of Microwave Engineering (McGraw-Hill, New York, 1966).

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Figures (9)

Fig. 1
Fig. 1

Schematic illustration of the structure for grating-assisted backward coupling: two parallel single-mode waveguides, a rectangular grating between them, and the four involved waves.

Fig. 2
Fig. 2

Spectral values of the squared amplitudes in example 1. Solid curves, full unified four-wave model; dashed curves, its degenerate two-wave approximation; (a) |ua+(L)/ua+(0)|2 and |ua-(0)/ua+(0)|2 as functions of wavelength, (b) |ub-(0)/ua+(0)|2 and |ub+(L)/ua+(0)|2 as functions of wavelength.

Fig. 3
Fig. 3

Power spectrum response of the whole structure in example 1 from the unified four-wave model. T(λ) denotes the power transmission coefficient, and R(λ) denotes the power reflection coefficient.

Fig. 4
Fig. 4

Variation of the squared normalized amplitudes along the structure of example 2, calculated by means of the unified four-wave model, at λ0. (a) |ua+(z)/ua+(0)|2 and |ub-(z)/ua+(0)|2 along the structure, (b) |ua-(z)/ua+(0)|2 and |ub+(z)/ua+(0)|2 along the structure.

Fig. 5
Fig. 5

Normalized forward-propagating power p+(z)/p+(0) and normalized backward-propagating power p-(z)/p+(0) in the whole structure for example 2, employing the unified four-wave model, at λ0.

Fig. 6
Fig. 6

Spectral values of the squared amplitudes in example 2 from the unified four-wave model. (a) |ua+(L)/ua+(0)|2 and |ua-(0)/ua+(0)|2 as functions of wavelength, (b) |ub-(0)/ua+(0)|2 and |ub+(L)/ua+(0)|2 as functions of wavelength.

Fig. 7
Fig. 7

Power spectrum response of the whole structure in example 2 from the unified four-wave model. T(λ) denotes the power transmission coefficient, and R(λ) denotes the power reflection coefficient.

Fig. 8
Fig. 8

Power spectrum response of the whole structure in example 3. Solid curves, T(λ) and R(λ) derived by the unified four-wave model. Dashed curves, T2(λ) and R2(λ) calculated by the degenerate two-wave approximation to the unified four-wave model.

Fig. 9
Fig. 9

Normalized transmitted and reflected power spectra for two different coupling schemes. (a) Forward power transmission Ta(λ) through waveguide a and backward power reflection Rb(λ) through waveguide b for the geometry and boundary conditions shown by the inset. (b) Forward power transmission Tb(λ) through waveguide b and backward power reflection Ra(λ) through waveguide a for the geometry and boundary conditions shown by the inset.

Tables (1)

Tables Icon

Table 1 Structure Parameters for Three Examples

Equations (47)

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

Et(x, y, z)=[ua+(z)+ua-(z)]Et(a)(x, y)+[ub+(z)+ub-(z)]Et(b)(x, y),
Ht(x, y, z)=[ua+(z)-ua-(z)]Ht(a)(x, y)+[ub+(z)-ub-(z)]Ht(b)(x, y),
dU(z)dz=iM(z)U(z),
M(z)=C-1[BC+K(z)].
βa(λ0)+βb(λ0)=2πmΛ,
P(z)=P+(z)-P-(z)+12c-{Re[ua+(z)ub-*(z)]-Re[ua-(z)ub+*(z)]},
P+(z)=14[ua+(z), ub+(z)]1c+c+1ua+(z)ub+(z),
P-(z)=14[ua-(z), ub-(z)]1c+c+1ua-(z)ub-(z).
R(λ)=P-(0)P+(0)(λ),
T(λ)=P+(L)P+(0)(λ).
Pq±(ze)=14|uq±(ze)+c+uq¯±(ze)|2.
ua+(0)=1,ub+(0)=ua-(L)=ub-(L)=0.
M(z)=m11m12m13 exp(i2β0z)m14 exp(i2β0z)m21m22m23 exp(i2β0z)m24 exp(i2β0z)m31 exp(-i2β0z)m32 exp(-i2β0z)m33m34m41 exp(-i2β0z)m42 exp(-i2β0z)m43m44,
W(z)w1(z)w2(z)w3(z)w4(z)ua+(z)exp(-iβ0z)ub+(z)exp(-iβ0z)ua-(z)exp(iβ0z)ub-(z)exp(iβ0z).
dW(z)dz=iQW(z),
Q=m11-β0m12m13m14m21m22-β0m23m24m31m32-m11+β0m34m41m42m43-m22+β0.
wk(z)=j=14cjvk(σj)exp(iσjz),
U(Λ)=T1U(0),
T1A2R2(Λ2)A2-1A1R1(Λ1)A1-1.
T=T1n,
U˜out=SU˜in,
B=diag(βa, βb,-βa,-βb),
C=1c+0c-c+1-c-00c-1c+-c-0c+1,
2zˆ·-[Et(q)×Ht(q)]dxdy=1(q=a, b),
c±=zˆ·-[Et(b)×Ht(a)±Et(a)×Ht(b)]dxdy.
K(z)=kaa-kab-kaa+kab+kba-kbb-kba+kbb+-kaa+-kab+-kaa--kab--kba+-kbb+-kba--kbb-,
kpq±=kpqt±kpqz(p=a, b;q=a, b),
kpqt=ω-Δp(x, y, z)Et(p)(x, y)Et(q)(x, y)dxdy,
kpqz=ω-q(x, y)Δp(x, y, z)(x, y, z)×Ez(p)(x, y)Ez(q)(x, y)dxdy.
ΔEp(x, y, z)=E(x, y, z)-Ep(x, y)(p=a, b).
m11=11-(c+)2+(c-)2{βa+[(c-)2-(c+)2]βb+[kaa--c+kba--c-kba+](l=0)},
m22=11-(c+)2+(c-)2{βb+[(c-)2-(c+)2]βa+[kbb--c+kab-+c-kab+](l=0)},
m12=11-(c+)2+(c-)2{c+(βa-βb)+[kab--c+kbb--c-kbb+](l=0)},
m21=11-(c+)2+(c-)2{c+(βb-βa)+[kba--c+kaa-+c-kaa+](l=0)},
m33=-m11
m44=-m22
m34=11-(c+)2+(c-)2{c+(βb-βa)+[-kab-+c+kbb-+c-kbb+](l=0)},
m43=11-(c+)2+(c-)2{c+(βa-βb)-[kba--c+kaa-+c-kaa+](l=0)},
m13=11-(c+)2+(c-)2[kaa+-c+kba+-c-kba-](l=m),
m24=11-(c+)2+(c-)2[kbb+-c+kab++c-kab-](l=m),
m23=11-(c+)2+(c-)2[kba+-c+kaa++c-kaa-](l=m),
m14=11-(c+)2+(c-)2[kab+-c+kbb+-c-kbb-](l=m),
m31=11-(c+)2+(c-)2[-kaa++c+kba++c-kba-](l=-m),
m42=11-(c+)2+(c-)2[-kbb++c+kab+-c-kab-](l=-m),
m32=11-(c+)2+(c-)2[-kab++c+kbb++c-kbb-](l=-m),
m41=11-(c+)2+(c-)2[-kba++c+kaa+-c-kaa-](l=-m).
S=t34t41-t31t44Δ34t34t42-t32t34Δ34t44Δ34-t34Δ34t31t43-t33t41Δ34t32t43-t33t42Δ34-t43Δ34t33Δ34t13s11+t14s21+t11t13s12+t14s22+t12t13s13+t14s23t13s14+t14s24t23s11+t24s21+t21t23s12+t24s22+t22t23s13+t24s23t23s14+t24s24,

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