Abstract

The coupled-mode theory (CMT) is a powerful approach routinely used to calculate the effects of spatial mode interactions in perturbed structures, such as optical waveguides. One of its basic hypotheses requires that perturbations are weak. This is usually not the case for devices fabricated with modern semiconductor-based technologies. In this paper, the CMT is studied in these critical cases to assess its validity. Attention will be focused on the quite common case of parallel coupled waveguides. For these structures, results can in fact be compared to the exact ones, obtained using super-modes. The study will show that not all the possible expressions of the coupling coefficients are equivalent, and which one can be pragmatically used to obtain results with minimum errors with respect to exact solutions.

© 2018 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  7. E. A. J. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. 22, 988–993 (1986).
    [Crossref]
  8. H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
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    [Crossref]
  10. S.-L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. 23, 499–509 (1987).
    [Crossref]
  11. W. Streifer, M. Osisnsky, and A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
    [Crossref]
  12. R. G. Peall and R. R. A. Syms, “Comparison between strong coupling theory and experiment for three-arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
    [Crossref]
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    [Crossref]
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    [Crossref]
  19. N. Kohli, S. Srivastava, and E. K. Sharma, “Scalar coupled mode theory and variational analysis for planar SOI waveguide arrays: a detailed comparison,” Opt. Quantum Electron. 48, 265 (2016).
    [Crossref]
  20. A. Macho, M. Morant, and R. Llorente, “Unified model of linear and nonlinear crosstalk in multi-core fiber,” J. Lightwave Technol. 34, 3035–3046 (2016).
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    [Crossref]
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    [Crossref]
  27. R. G. Hunsperger, Integrated Optics, 5th ed. (Springer, 2013), Chap. 8.
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    [Crossref]
  29. P. Orlandi, F. Morichetti, M. Strain, M. Sorel, P. Bassi, and A. Melloni, “Photonic integrated filter with widely tunable bandwidth,” J. Lightwave Technol. 32, 897–907 (2014).
    [Crossref]

2016 (4)

2015 (2)

2014 (2)

2013 (1)

2009 (1)

2008 (1)

2001 (1)

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[Crossref]

1995 (1)

B. E. Little and W. P. Huang, “Coupled-mode theory for optical waveguides,” Prog. Electromagn. Res. 10, 217–270 (1995).

1994 (1)

1989 (1)

R. G. Peall and R. R. A. Syms, “Comparison between strong coupling theory and experiment for three-arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
[Crossref]

1987 (4)

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).

S.-L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987).
[Crossref]

S.-L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. 23, 499–509 (1987).
[Crossref]

W. Streifer, M. Osisnsky, and A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

1986 (1)

E. A. J. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. 22, 988–993 (1986).
[Crossref]

1985 (1)

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

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).

1972 (1)

1955 (1)

S. A. Schelkunoff, “Conversion of Maxwell’s equations into generalized telegraphist’s equations,” Bell Syst. Tech. J. 34, 995–1043 (1955).
[Crossref]

Aschieri, P.

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[Crossref]

Baldi, P.

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[Crossref]

Bandaru, P. R.

Bassi, P.

Bellanca, G.

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[Crossref]

Chen, Y.

Chuang, S.-L.

S.-L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987).
[Crossref]

S.-L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. 23, 499–509 (1987).
[Crossref]

Cooper, M. L.

Fogli, F.

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[Crossref]

Gallion, P.

Gao, Y.

Greco, N.

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[Crossref]

Hardy, A.

W. Streifer, M. Osisnsky, and A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

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

Haus, H. A.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).

Huang, W.

Huang, W. P.

B. E. Little and W. P. Huang, “Coupled-mode theory for optical waveguides,” Prog. Electromagn. Res. 10, 217–270 (1995).

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).

Hunsperger, R. G.

R. G. Hunsperger, Integrated Optics, 5th ed. (Springer, 2013), Chap. 8.

Jian, S.

Kawakami, S.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).

Kogelnik, H.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed., 2nd ed. (Springer, 1979), Chap. 2.

Kohli, N.

N. Kohli, S. Srivastava, and E. K. Sharma, “Scalar coupled mode theory and variational analysis for planar SOI waveguide arrays: a detailed comparison,” Opt. Quantum Electron. 48, 265 (2016).
[Crossref]

N. Kohli, S. Srivastava, and E. K. Sharma, “Orthogonal solutions for asymmetric strongly coupled waveguide arrays: an elegant, analytical approach,” J. Opt. Soc. Am. B 31, 2871–2878 (2014).
[Crossref]

Little, B. E.

B. E. Little and W. P. Huang, “Coupled-mode theory for optical waveguides,” Prog. Electromagn. Res. 10, 217–270 (1995).

Llorente, R.

Macho, A.

Marcatili, E. A. J.

E. A. J. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. 22, 988–993 (1986).
[Crossref]

Marcuse, D.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, 1982).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

Melloni, A.

Mookherjea, S.

Morant, M.

Morichetti, F.

Napierala, M.

Nasilowski, T.

Orlandi, P.

Osisnsky, M.

W. Streifer, M. Osisnsky, and A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

Peall, R. G.

R. G. Peall and R. R. A. Syms, “Comparison between strong coupling theory and experiment for three-arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
[Crossref]

Pytel, A.

Ren, G.

Schelkunoff, S. A.

S. A. Schelkunoff, “Conversion of Maxwell’s equations into generalized telegraphist’s equations,” Bell Syst. Tech. J. 34, 995–1043 (1955).
[Crossref]

Sharma, E. K.

N. Kohli, S. Srivastava, and E. K. Sharma, “Scalar coupled mode theory and variational analysis for planar SOI waveguide arrays: a detailed comparison,” Opt. Quantum Electron. 48, 265 (2016).
[Crossref]

N. Kohli, S. Srivastava, and E. K. Sharma, “Orthogonal solutions for asymmetric strongly coupled waveguide arrays: an elegant, analytical approach,” J. Opt. Soc. Am. B 31, 2871–2878 (2014).
[Crossref]

Snyder, A. W.

Sorel, M.

Srivastava, S.

N. Kohli, S. Srivastava, and E. K. Sharma, “Scalar coupled mode theory and variational analysis for planar SOI waveguide arrays: a detailed comparison,” Opt. Quantum Electron. 48, 265 (2016).
[Crossref]

N. Kohli, S. Srivastava, and E. K. Sharma, “Orthogonal solutions for asymmetric strongly coupled waveguide arrays: an elegant, analytical approach,” J. Opt. Soc. Am. B 31, 2871–2878 (2014).
[Crossref]

Strain, M.

Streifer, W.

W. Streifer, M. Osisnsky, and A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

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

Syms, R. R. A.

R. G. Peall and R. R. A. Syms, “Comparison between strong coupling theory and experiment for three-arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
[Crossref]

Szostkiewicz, L.

Tenderenda, T.

Wan, C.

Wang, Q.

Whitaker, N. A.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).

Wu, B.

Xu, J.

Yang, S.

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).

Zhou, J.

Zhu, B.

Ziolowicz, A.

Bell Syst. Tech. J. (1)

S. A. Schelkunoff, “Conversion of Maxwell’s equations into generalized telegraphist’s equations,” Bell Syst. Tech. J. 34, 995–1043 (1955).
[Crossref]

IEEE J. Quantum Electron. (3)

E. A. J. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. 22, 988–993 (1986).
[Crossref]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).

S.-L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. 23, 499–509 (1987).
[Crossref]

J. Lightwave Technol. (7)

W. Streifer, M. Osisnsky, and A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

R. G. Peall and R. R. A. Syms, “Comparison between strong coupling theory and experiment for three-arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
[Crossref]

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

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).

S.-L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987).
[Crossref]

P. Orlandi, F. Morichetti, M. Strain, M. Sorel, P. Bassi, and A. Melloni, “Photonic integrated filter with widely tunable bandwidth,” J. Lightwave Technol. 32, 897–907 (2014).
[Crossref]

A. Macho, M. Morant, and R. Llorente, “Unified model of linear and nonlinear crosstalk in multi-core fiber,” J. Lightwave Technol. 34, 3035–3046 (2016).
[Crossref]

J. Opt. Soc. Am. (1)

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

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

Opt. Express (5)

Opt. Lett. (2)

Opt. Quantum Electron. (2)

N. Kohli, S. Srivastava, and E. K. Sharma, “Scalar coupled mode theory and variational analysis for planar SOI waveguide arrays: a detailed comparison,” Opt. Quantum Electron. 48, 265 (2016).
[Crossref]

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[Crossref]

Prog. Electromagn. Res. (1)

B. E. Little and W. P. Huang, “Coupled-mode theory for optical waveguides,” Prog. Electromagn. Res. 10, 217–270 (1995).

Other (5)

“COMSOL Multiphysics software,” http://www.comsol.com .

R. G. Hunsperger, Integrated Optics, 5th ed. (Springer, 2013), Chap. 8.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed., 2nd ed. (Springer, 1979), Chap. 2.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, 1982).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

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Figures (12)

Fig. 1.
Fig. 1. Schematic of the transversal section of the studied parallel waveguide coupler. The z axis is orthogonal to the design plane; ϵ s is the dielectric constant of the structure substrate; Δ ϵ l and Δ ϵ r denote the increase of the substrate dielectric constant which constitute the two, left and right, waveguide cores.
Fig. 2.
Fig. 2. Imaginary part of E x (left) and real part of E z (right) of the quasi-TE mode of the left high-index-contrast waveguide ( 480 × 220    nm size) alone. The right waveguide, separated by a 200 nm gap, is not present in the studied structure but is sketched anyway for visual aid as explained in the text. The ratio between the maximum value of the moduli of E z and E x is about 0.5.
Fig. 3.
Fig. 3. Imaginary part of E y (left) and real part of E z (right) of the quasi-TM mode of left high-index-contrast waveguide ( 480 × 220    nm size) alone. The right waveguide is not present in the studied structure but is sketched anyway for visual aid as explained in the text. The ratio between the maximum value of the moduli of E z and E y is about 0.6.
Fig. 4.
Fig. 4. Imaginary part of E x (left) and real part of E z (right) of the two quasi-TE super-modes of the high-index-contrast coupler with even (upper line) and odd (lower line) symmetries. Waveguides have 480 × 220    nm size with a 200 nm gap. In both cases, the ratio between the maximum value of the moduli of E z and E x is about 0.5.
Fig. 5.
Fig. 5. Imaginary part of E y (left) and real part of E z (right) of the two quasi-TM super-modes of the high-index-contrast coupler with even (upper line) and odd (lower line) symmetries. Waveguides have 480 × 220    nm size with a 200 nm gap. In both cases, the ratio between the maximum value of the moduli of E z and E x is about 0.6.
Fig. 6.
Fig. 6. Coupling coefficient κ versus waveguide gap for the quasi-TE mode of a Si waveguide obtained with the O-CMT. The black long dashed line with open circles comes using Eq. (38). The red solid line with crosses shows κ t . The blue dotted line with triangles shows κ t + κ z a . The green dashed-dotted line with squares (almost superimposed to the black long dashed line) shows κ t + κ z b .
Fig. 7.
Fig. 7. Coupling coefficient κ versus waveguide gap for the quasi-TM mode of a Si waveguide with the O-CMT. Line types as in Fig. 6.
Fig. 8.
Fig. 8. Coupling coefficient κ versus waveguide gap for the quasi-TE mode of the average-index-contrast waveguide obtained using O-CMT. Line types as in Fig. 6.
Fig. 9.
Fig. 9. Coupling coefficient κ versus waveguide gap for the quasi-TM mode of the average-index-contrast waveguide obtained using O-CMT. Line types as in Fig. 6.
Fig. 10.
Fig. 10. Effective indices of the quasi-TE and quasi-TM modes versus gap. Red solid lines refer to the two quasi-TE polarized super-modes; blue dashed lines refer to the quasi-TM polarized super-modes.
Fig. 11.
Fig. 11. Coupling coefficient κ versus waveguide gap for the quasi-TE mode of the low-index-contrast waveguide. Top: O-CMT; bottom: NO-CMT. Line types as in Fig. 6.
Fig. 12.
Fig. 12. Coupling coefficient κ versus waveguide gap for the quasi-TM mode of the low-index-contrast waveguide. Top: O-CMT; bottom: NO-CMT. Line types as in Fig. 6.

Equations (40)

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× E 1 = j ω μ H 1 ,
× H 1 = j ω ϵ 1 E 1
× E 2 = j ω μ H 2 ,
× H 2 = j ω ϵ 2 E 2 .
S z ( E 1 × H 2 * + E 2 * × H 1 ) · z ^    d S = j ω S Δ ϵ E 1 · E 2 * d S .
E t ( z ) = ν ( a ν ( z ) + b ν ( z ) ) E t ν ( z ) + ( a ρ ( z ) + b ρ ( z ) ) E t ρ ( z ) d ρ ,
H t ( z ) = ν ( a ν ( z ) b ν ( z ) ) H t ν ( z ) + ( a ρ ( z ) b ρ ( z ) ) H t ρ ( z ) d ρ ,
E t ( z ) = ν a ν ( z ) E t ν = ν A ν ( z ) e j β ν z E t ν ,
H t ( z ) = ν a ν ( z ) H t ν = ν A ν ( z ) e j β ν z H t ν .
E 1 t ( z ) = ν a ν E t ν ,
H 1 t ( z ) = ν a ν H t ν .
E 2 t ( z ) = E t μ e j β μ z ,
H 2 t ( z ) = H t μ e j β μ z ,
S z [ ν a ν E t ν × H t μ * e j β μ z + E t μ * e j β μ z × ν a ν H t ν ] · z ^    d S .
j 4 ν κ A ν e j ( β ν β μ ) z ,
κ ν μ t = ω 4 S Δ ϵ Δ ϵ E t ν · E t μ * d S ,
κ ν μ z = ω 4 S Δ ϵ ϵ Δ ϵ ϵ + Δ ϵ E z ν E z μ * d S ,
κ ν μ z = ω 4 S Δ ϵ Δ ϵ E z ν E z μ * d S .
S E t ν × H t μ * · z ^    d S = 2 P δ ν μ ,
S E t μ * × H t ν · z ^    d S = 2 P δ μ ν ,
S z [ a μ E t μ × H t μ * e j β μ z + E t μ * e j β μ z × a μ H t μ ] · z ^    d S = 4 P A μ z .
A μ z = j ν [ A ν ( κ ν μ t + κ ν μ z ) e j ( β ν β μ ) z ] .
A 1 z = j A 1 ( κ 11 t + κ 11 z ) j A 2 ( κ 21 t + κ 21 z ) e j ( β 2 β 1 ) z ,
A 2 z = j A 1 ( κ 12 t + κ 12 z ) e j ( β 1 β 2 ) z j A 2 ( κ 22 t + κ 22 z ) .
S E t ν × H t μ * · z ^    d S = 2 P ν μ
S E t μ * × H t ν · z ^    d S = 2 P μ ν .
4 ν [ ( A ν z j ( β ν β μ ) A ν ) e j ( β ν β μ ) z P ν μ ] .
ν ( A ν z j ( β ν β μ ) A ν ) e j ( β ν β μ ) z P ν μ = j ν A ν e j ( β ν β μ ) z ( κ ν μ t + κ ν μ z ) .
A 1 z P 11 + ( A 2 z j ( β 2 β 1 ) A 2 ) e j ( β 2 β 1 ) z P 21 = j A 1 κ 11 j A 2 e j ( β 2 β 1 ) z κ 21 ,
( A 1 z j ( β 1 β 2 ) A 1 ) e j ( β 1 β 2 ) z P 12 + A 2 z P 22 = j A 1 e j ( β 1 β 2 ) z κ 12 j A 2 κ 22 .
[ P 11 e j β 1 z P 21 e j β 2 z P 12 e j β 1 z P 22 e j β 2 z ] z [ A 1 A 2 ] = [ j κ 11 e j β 1 z j [ P 21 ( β 2 β 1 ) κ 21 ] e j β 2 z j [ P 12 ( β 1 β 2 ) κ 12 ] e j β 1 z j κ 22 e j β 2 z ] [ A 1 A 2 ] ,
z [ A 1 A 2 ] = 1 P 11 P 22 P 12 P 21 [ P 22 e j β 1 z P 21 e j β 1 z P 12 e j β 2 z P 11 e j β 2 z ] [ j κ 11 e j β 1 z j [ P 21 ( β 2 β 1 ) κ 21 ] e j β 2 z j [ P 12 ( β 1 β 2 ) κ 12 ] e j β 1 z j κ 22 e j β 2 z ] [ A 1 A 2 ] .
z [ A 1 A 2 ] = j [ κ ˜ 11 κ ˜ 21 e j ( β 1 β 2 ) z κ ˜ 12 e j ( β 1 β 2 ) z κ ˜ 22 ] [ A 1 A 2 ] ,
κ ˜ 11 = P 22 κ 11 P 21 κ 12 + P 21 P 12 ( β 1 β 2 ) P 11 P 22 P 12 P 21 ,
κ ˜ 21 = P 22 κ 21 P 21 κ 22 P 22 P 21 ( β 2 β 1 ) P 11 P 22 P 12 P 21 ,
κ ˜ 12 = P 11 κ 12 P 12 κ 11 P 11 P 12 ( β 1 β 2 ) P 11 P 22 P 12 P 21 ,
κ ˜ 22 = P 11 κ 22 P 12 κ 21 + P 12 P 21 ( β 2 β 1 ) P 11 P 22 P 12 P 21 .
L c = π Δ β ,
L c = π 2 κ .
κ = Δ β 2 .

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