Abstract

Two different approximations for the propagation constant of a guided mode in a rectangular dielectric waveguide can be obtained by applying the conventional effective-index method to the waveguide in two different ways. By combining these two approximations with the other two approximations for a complementary mode in a complementary waveguide so that the errors in the approximations can be largely canceled, a more accurate approximation can be derived. Numerical results from analyzing several basic rectangular structures reveal that in most cases the new technique is considerably more accurate than the conventional effective-index method and does not generate ambiguous results as the conventional effective-index method does in the analysis of square dielectric rods.

© 1986 Optical Society of America

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References

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  1. E. A. J. Marcatili, “Dielectric Rectangular Waveguide and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071 (1969).
  2. J. E. Goell, “A Circular Harmonic Computer Analysis of Rectangular Dielectric Waveguides,” Bell Syst. Tech. J. 48, 2133 (1969).
  3. R. M. Knox, P. P. Toulios, “Integrated Circuits for the Millimeter through Optical Frequency Range,” in Proceedings, Symposium on Submillimeter Waves (Polytechnic Press, Brooklyn, 1970), pp. 497–516.
  4. H. Furuta, H. Noda, A. Ihaya, “Novel Optical Waveguide for Integrated Optics,” Appl. Opt. 13, 322 (1974).
    [Crossref] [PubMed]
  5. W. V. McLevige, T. Itoh, R. Mittra, “New Waveguide Structures for Millimeter-Wave and Optical Integrated Circuits,” IEEE Trans. Microwave Theory Tech. MTT-23, 788 (1975).
    [Crossref]
  6. T. Itoh, “Inverted Strip Dielectric Waveguide for Millimeter-Wave Integrated Circuits,” IEEE Trans. Microwave Theory Tech. MTT-24, 821 (1976).
    [Crossref]
  7. G. B. Hocker, W. K. Burns, “Mode Dispersion in Diffused Channel Waveguides by the Effective Index Method,” Appl. Opt. 16, 113 (1977).
    [Crossref] [PubMed]
  8. K. Ogusu, “Numerical Analysis of the Rectangular Dielectric Waveguide and its Modifications,” IEEE Trans. Microwave Theory Tech. MTT-25, 874 (1977).
    [Crossref]
  9. L. Eyges, P. Wintersteiner, P. Gianino, “Modes of Dielectric Waveguides of Arbitrary Cross Sectional Shape,” J. Opt. Soc. Am. 69, 1226 (1979).
    [Crossref]
  10. C. Yeh, K. Ha, S. B. Dong, W. P. Brown, “Single-mode Optical Waveguides,” Appl. Opt. 18, 1490 (1979).
    [Crossref] [PubMed]
  11. R. Mittra, Y. L. Hou, V. Jamnejad, “Analysis of Open Dielectric Waveguides Using Mode-Matching Technique and Variational Methods,” IEEE Trans. Microwave Theory Tech. MTT-28, 36 (1980).
    [Crossref]
  12. U. Crombach, “Analysis of Single and Coupled Rectangular Dielectric Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 870 (1981).
    [Crossref]
  13. S. T. Peng, A. A. Oliner, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part I—Mathematical Formulations,” IEEE Trans. Microwave Theory Tech. MTT-29, 845 (1981).
  14. A. A. Oliner, S. T. Peng, T. I. Hsu, A. Sanchez, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part II—New Physical Effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855 (1981).
    [Crossref]
  15. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981).
  16. F. P. Payne, “A New Theory of Rectangular Optical Waveguides,” Opt. Quantum Electron. 14, 525 (1982).
    [Crossref]
  17. M. N. Armenise, M. De Sario, “Optical Rectangular Waveguide in Titanium-Diffused Lithium Niobate Having its Optical Axis in the Transverse Plane,” J. Opt. Soc. Am. 72, 1514 (1982).
    [Crossref]
  18. A. Kumar, K. Thyagarajan, A. K. Ghatak, “Analysis of Rectangular-Core Dielectric Waveguides: an Accurate Perturbation Approach,” Opt. Lett. 8, 63 (1983).
    [Crossref] [PubMed]
  19. A. Kumar, M. R. Shenoy, K. Thyagarajan, “Modes in Anisotropic Rectangular Waveguides: an Accurate and Simple Perturbation Approach,” IEEE Trans. Microwave Theory Tech. MTT-32, 1415 (1984).
    [Crossref]
  20. P. Mishra, A. Sharma, S. Labroo, A. K. Ghatak, “Scalar Variational Analysis of Single-Mode Waveguides with Rectangular Cross Section,” IEEE Trans. Microwave Theory Tech. MTT-33, 282 (1985).
    [Crossref]
  21. M. Koshiba, M. Suzuki, “Vectorial Wave Analysis of Optical Waveguides with Rectangular Cross-Section Using Equivalent Network Approach,” Electron. Lett. 21, 1026 (1985).
    [Crossref]
  22. K. S. Chiang, “Analysis of Optical Fibers by the Effective-Index Method,” Appl. Opt. 25, 348 (1986).
    [Crossref] [PubMed]
  23. K. S. Chiang, “Geometrical Birefringence in a Class of Step-Index Fiber,” IEEE/OSA J. Lightwave Technol. (submitted for publication).
  24. K. S. Chiang, “Finite-Element Analysis of Optical Fibres with Iterative Treatment of the Infinite 2-D Space,” Opt. Quantum Electron. 17, 381 (1985).
    [Crossref]

1986 (1)

1985 (3)

K. S. Chiang, “Finite-Element Analysis of Optical Fibres with Iterative Treatment of the Infinite 2-D Space,” Opt. Quantum Electron. 17, 381 (1985).
[Crossref]

P. Mishra, A. Sharma, S. Labroo, A. K. Ghatak, “Scalar Variational Analysis of Single-Mode Waveguides with Rectangular Cross Section,” IEEE Trans. Microwave Theory Tech. MTT-33, 282 (1985).
[Crossref]

M. Koshiba, M. Suzuki, “Vectorial Wave Analysis of Optical Waveguides with Rectangular Cross-Section Using Equivalent Network Approach,” Electron. Lett. 21, 1026 (1985).
[Crossref]

1984 (1)

A. Kumar, M. R. Shenoy, K. Thyagarajan, “Modes in Anisotropic Rectangular Waveguides: an Accurate and Simple Perturbation Approach,” IEEE Trans. Microwave Theory Tech. MTT-32, 1415 (1984).
[Crossref]

1983 (1)

1982 (2)

1981 (3)

U. Crombach, “Analysis of Single and Coupled Rectangular Dielectric Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 870 (1981).
[Crossref]

S. T. Peng, A. A. Oliner, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part I—Mathematical Formulations,” IEEE Trans. Microwave Theory Tech. MTT-29, 845 (1981).

A. A. Oliner, S. T. Peng, T. I. Hsu, A. Sanchez, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part II—New Physical Effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855 (1981).
[Crossref]

1980 (1)

R. Mittra, Y. L. Hou, V. Jamnejad, “Analysis of Open Dielectric Waveguides Using Mode-Matching Technique and Variational Methods,” IEEE Trans. Microwave Theory Tech. MTT-28, 36 (1980).
[Crossref]

1979 (2)

1977 (2)

G. B. Hocker, W. K. Burns, “Mode Dispersion in Diffused Channel Waveguides by the Effective Index Method,” Appl. Opt. 16, 113 (1977).
[Crossref] [PubMed]

K. Ogusu, “Numerical Analysis of the Rectangular Dielectric Waveguide and its Modifications,” IEEE Trans. Microwave Theory Tech. MTT-25, 874 (1977).
[Crossref]

1976 (1)

T. Itoh, “Inverted Strip Dielectric Waveguide for Millimeter-Wave Integrated Circuits,” IEEE Trans. Microwave Theory Tech. MTT-24, 821 (1976).
[Crossref]

1975 (1)

W. V. McLevige, T. Itoh, R. Mittra, “New Waveguide Structures for Millimeter-Wave and Optical Integrated Circuits,” IEEE Trans. Microwave Theory Tech. MTT-23, 788 (1975).
[Crossref]

1974 (1)

1969 (2)

E. A. J. Marcatili, “Dielectric Rectangular Waveguide and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071 (1969).

J. E. Goell, “A Circular Harmonic Computer Analysis of Rectangular Dielectric Waveguides,” Bell Syst. Tech. J. 48, 2133 (1969).

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981).

Armenise, M. N.

Brown, W. P.

Burns, W. K.

Chiang, K. S.

K. S. Chiang, “Analysis of Optical Fibers by the Effective-Index Method,” Appl. Opt. 25, 348 (1986).
[Crossref] [PubMed]

K. S. Chiang, “Finite-Element Analysis of Optical Fibres with Iterative Treatment of the Infinite 2-D Space,” Opt. Quantum Electron. 17, 381 (1985).
[Crossref]

K. S. Chiang, “Geometrical Birefringence in a Class of Step-Index Fiber,” IEEE/OSA J. Lightwave Technol. (submitted for publication).

Crombach, U.

U. Crombach, “Analysis of Single and Coupled Rectangular Dielectric Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 870 (1981).
[Crossref]

De Sario, M.

Dong, S. B.

Eyges, L.

Furuta, H.

Ghatak, A. K.

P. Mishra, A. Sharma, S. Labroo, A. K. Ghatak, “Scalar Variational Analysis of Single-Mode Waveguides with Rectangular Cross Section,” IEEE Trans. Microwave Theory Tech. MTT-33, 282 (1985).
[Crossref]

A. Kumar, K. Thyagarajan, A. K. Ghatak, “Analysis of Rectangular-Core Dielectric Waveguides: an Accurate Perturbation Approach,” Opt. Lett. 8, 63 (1983).
[Crossref] [PubMed]

Gianino, P.

Goell, J. E.

J. E. Goell, “A Circular Harmonic Computer Analysis of Rectangular Dielectric Waveguides,” Bell Syst. Tech. J. 48, 2133 (1969).

Ha, K.

Hocker, G. B.

Hou, Y. L.

R. Mittra, Y. L. Hou, V. Jamnejad, “Analysis of Open Dielectric Waveguides Using Mode-Matching Technique and Variational Methods,” IEEE Trans. Microwave Theory Tech. MTT-28, 36 (1980).
[Crossref]

Hsu, T. I.

A. A. Oliner, S. T. Peng, T. I. Hsu, A. Sanchez, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part II—New Physical Effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855 (1981).
[Crossref]

Ihaya, A.

Itoh, T.

T. Itoh, “Inverted Strip Dielectric Waveguide for Millimeter-Wave Integrated Circuits,” IEEE Trans. Microwave Theory Tech. MTT-24, 821 (1976).
[Crossref]

W. V. McLevige, T. Itoh, R. Mittra, “New Waveguide Structures for Millimeter-Wave and Optical Integrated Circuits,” IEEE Trans. Microwave Theory Tech. MTT-23, 788 (1975).
[Crossref]

Jamnejad, V.

R. Mittra, Y. L. Hou, V. Jamnejad, “Analysis of Open Dielectric Waveguides Using Mode-Matching Technique and Variational Methods,” IEEE Trans. Microwave Theory Tech. MTT-28, 36 (1980).
[Crossref]

Knox, R. M.

R. M. Knox, P. P. Toulios, “Integrated Circuits for the Millimeter through Optical Frequency Range,” in Proceedings, Symposium on Submillimeter Waves (Polytechnic Press, Brooklyn, 1970), pp. 497–516.

Koshiba, M.

M. Koshiba, M. Suzuki, “Vectorial Wave Analysis of Optical Waveguides with Rectangular Cross-Section Using Equivalent Network Approach,” Electron. Lett. 21, 1026 (1985).
[Crossref]

Kumar, A.

A. Kumar, M. R. Shenoy, K. Thyagarajan, “Modes in Anisotropic Rectangular Waveguides: an Accurate and Simple Perturbation Approach,” IEEE Trans. Microwave Theory Tech. MTT-32, 1415 (1984).
[Crossref]

A. Kumar, K. Thyagarajan, A. K. Ghatak, “Analysis of Rectangular-Core Dielectric Waveguides: an Accurate Perturbation Approach,” Opt. Lett. 8, 63 (1983).
[Crossref] [PubMed]

Labroo, S.

P. Mishra, A. Sharma, S. Labroo, A. K. Ghatak, “Scalar Variational Analysis of Single-Mode Waveguides with Rectangular Cross Section,” IEEE Trans. Microwave Theory Tech. MTT-33, 282 (1985).
[Crossref]

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric Rectangular Waveguide and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071 (1969).

McLevige, W. V.

W. V. McLevige, T. Itoh, R. Mittra, “New Waveguide Structures for Millimeter-Wave and Optical Integrated Circuits,” IEEE Trans. Microwave Theory Tech. MTT-23, 788 (1975).
[Crossref]

Mishra, P.

P. Mishra, A. Sharma, S. Labroo, A. K. Ghatak, “Scalar Variational Analysis of Single-Mode Waveguides with Rectangular Cross Section,” IEEE Trans. Microwave Theory Tech. MTT-33, 282 (1985).
[Crossref]

Mittra, R.

R. Mittra, Y. L. Hou, V. Jamnejad, “Analysis of Open Dielectric Waveguides Using Mode-Matching Technique and Variational Methods,” IEEE Trans. Microwave Theory Tech. MTT-28, 36 (1980).
[Crossref]

W. V. McLevige, T. Itoh, R. Mittra, “New Waveguide Structures for Millimeter-Wave and Optical Integrated Circuits,” IEEE Trans. Microwave Theory Tech. MTT-23, 788 (1975).
[Crossref]

Noda, H.

Ogusu, K.

K. Ogusu, “Numerical Analysis of the Rectangular Dielectric Waveguide and its Modifications,” IEEE Trans. Microwave Theory Tech. MTT-25, 874 (1977).
[Crossref]

Oliner, A. A.

S. T. Peng, A. A. Oliner, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part I—Mathematical Formulations,” IEEE Trans. Microwave Theory Tech. MTT-29, 845 (1981).

A. A. Oliner, S. T. Peng, T. I. Hsu, A. Sanchez, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part II—New Physical Effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855 (1981).
[Crossref]

Payne, F. P.

F. P. Payne, “A New Theory of Rectangular Optical Waveguides,” Opt. Quantum Electron. 14, 525 (1982).
[Crossref]

Peng, S. T.

A. A. Oliner, S. T. Peng, T. I. Hsu, A. Sanchez, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part II—New Physical Effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855 (1981).
[Crossref]

S. T. Peng, A. A. Oliner, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part I—Mathematical Formulations,” IEEE Trans. Microwave Theory Tech. MTT-29, 845 (1981).

Sanchez, A.

A. A. Oliner, S. T. Peng, T. I. Hsu, A. Sanchez, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part II—New Physical Effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855 (1981).
[Crossref]

Sharma, A.

P. Mishra, A. Sharma, S. Labroo, A. K. Ghatak, “Scalar Variational Analysis of Single-Mode Waveguides with Rectangular Cross Section,” IEEE Trans. Microwave Theory Tech. MTT-33, 282 (1985).
[Crossref]

Shenoy, M. R.

A. Kumar, M. R. Shenoy, K. Thyagarajan, “Modes in Anisotropic Rectangular Waveguides: an Accurate and Simple Perturbation Approach,” IEEE Trans. Microwave Theory Tech. MTT-32, 1415 (1984).
[Crossref]

Suzuki, M.

M. Koshiba, M. Suzuki, “Vectorial Wave Analysis of Optical Waveguides with Rectangular Cross-Section Using Equivalent Network Approach,” Electron. Lett. 21, 1026 (1985).
[Crossref]

Thyagarajan, K.

A. Kumar, M. R. Shenoy, K. Thyagarajan, “Modes in Anisotropic Rectangular Waveguides: an Accurate and Simple Perturbation Approach,” IEEE Trans. Microwave Theory Tech. MTT-32, 1415 (1984).
[Crossref]

A. Kumar, K. Thyagarajan, A. K. Ghatak, “Analysis of Rectangular-Core Dielectric Waveguides: an Accurate Perturbation Approach,” Opt. Lett. 8, 63 (1983).
[Crossref] [PubMed]

Toulios, P. P.

R. M. Knox, P. P. Toulios, “Integrated Circuits for the Millimeter through Optical Frequency Range,” in Proceedings, Symposium on Submillimeter Waves (Polytechnic Press, Brooklyn, 1970), pp. 497–516.

Wintersteiner, P.

Yeh, C.

Appl. Opt. (4)

Bell Syst. Tech. J. (2)

E. A. J. Marcatili, “Dielectric Rectangular Waveguide and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071 (1969).

J. E. Goell, “A Circular Harmonic Computer Analysis of Rectangular Dielectric Waveguides,” Bell Syst. Tech. J. 48, 2133 (1969).

Electron. Lett. (1)

M. Koshiba, M. Suzuki, “Vectorial Wave Analysis of Optical Waveguides with Rectangular Cross-Section Using Equivalent Network Approach,” Electron. Lett. 21, 1026 (1985).
[Crossref]

IEEE Trans. Microwave Theory Tech. (9)

A. Kumar, M. R. Shenoy, K. Thyagarajan, “Modes in Anisotropic Rectangular Waveguides: an Accurate and Simple Perturbation Approach,” IEEE Trans. Microwave Theory Tech. MTT-32, 1415 (1984).
[Crossref]

P. Mishra, A. Sharma, S. Labroo, A. K. Ghatak, “Scalar Variational Analysis of Single-Mode Waveguides with Rectangular Cross Section,” IEEE Trans. Microwave Theory Tech. MTT-33, 282 (1985).
[Crossref]

K. Ogusu, “Numerical Analysis of the Rectangular Dielectric Waveguide and its Modifications,” IEEE Trans. Microwave Theory Tech. MTT-25, 874 (1977).
[Crossref]

W. V. McLevige, T. Itoh, R. Mittra, “New Waveguide Structures for Millimeter-Wave and Optical Integrated Circuits,” IEEE Trans. Microwave Theory Tech. MTT-23, 788 (1975).
[Crossref]

T. Itoh, “Inverted Strip Dielectric Waveguide for Millimeter-Wave Integrated Circuits,” IEEE Trans. Microwave Theory Tech. MTT-24, 821 (1976).
[Crossref]

R. Mittra, Y. L. Hou, V. Jamnejad, “Analysis of Open Dielectric Waveguides Using Mode-Matching Technique and Variational Methods,” IEEE Trans. Microwave Theory Tech. MTT-28, 36 (1980).
[Crossref]

U. Crombach, “Analysis of Single and Coupled Rectangular Dielectric Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 870 (1981).
[Crossref]

S. T. Peng, A. A. Oliner, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part I—Mathematical Formulations,” IEEE Trans. Microwave Theory Tech. MTT-29, 845 (1981).

A. A. Oliner, S. T. Peng, T. I. Hsu, A. Sanchez, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part II—New Physical Effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855 (1981).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Lett. (1)

Opt. Quantum Electron. (2)

F. P. Payne, “A New Theory of Rectangular Optical Waveguides,” Opt. Quantum Electron. 14, 525 (1982).
[Crossref]

K. S. Chiang, “Finite-Element Analysis of Optical Fibres with Iterative Treatment of the Infinite 2-D Space,” Opt. Quantum Electron. 17, 381 (1985).
[Crossref]

Other (3)

K. S. Chiang, “Geometrical Birefringence in a Class of Step-Index Fiber,” IEEE/OSA J. Lightwave Technol. (submitted for publication).

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981).

R. M. Knox, P. P. Toulios, “Integrated Circuits for the Millimeter through Optical Frequency Range,” in Proceedings, Symposium on Submillimeter Waves (Polytechnic Press, Brooklyn, 1970), pp. 497–516.

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Figures (8)

Fig. 1
Fig. 1

(a) Rectangular dielectric rod with long dimension a along the x axis and short dimension b along the y axis can be analyzed by the simple effective-index method in two different ways: (b) using the slab of thickness b to define the effective index for the slab of thickness a; or (c) using the slab of thickness a to define the effective index for the slab of thickness b.

Fig. 2
Fig. 2

Dispersion curves for rectangular dielectric rods with n1 = 1.5 and n2 = 1.0: (a) R = 1 (square); and (b) R = 2.

Fig. 3
Fig. 3

Dispersion curves for rectangular dielectric rods with n1n2: (a) R = 1 (square); (b) R = 2.

Fig. 4
Fig. 4

Dispersion curves of the E 21 x , y mode in rectangular rods with n1n2 for various aspect ratios R.

Fig. 5
Fig. 5

(a) Rectangular channel waveguide with width a and depth b. (b) Rectangular channel waveguide with width b and depth a, which is complementary to the one as shown in (a).

Fig. 6
Fig. 6

Dispersion curves for the R = 2 channel waveguide with n1 = 1.5, n2 = 1.45, and n3 = 1.0.

Fig. 7
Fig. 7

Rectangular embossed waveguides which are complementary to each other.

Fig. 8
Fig. 8

Dispersion curves of the E 11 y mode in embossed waveguides with n1 = 1.5, n2 = 1.45, and n3 = 1.0 for various aspect ratios R.

Equations (24)

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

P 2 = ( β / k ) 2 - n 2 2 n 1 2 - n 2 2
B = b π k ( n 1 2 - n 2 2 ) 1 / 2 ,
P x 1 2 = [ TE n - 1 b ( n 2 n 1 n 2 ) , TM m - 1 a ( n 2 n eff n 2 ) ] , for E m n x ,
P x 2 2 = [ TM m - 1 a ( n 2 n 1 n 2 ) , TE n - 1 b ( n 2 n eff n 2 ) ] , for E m n x ,
P y 1 2 = [ TM m - 1 b ( n 2 n 1 n 2 ) , TE n - 1 a ( n 2 n eff n 2 ) ] , for E n m y ,
P y 2 2 = [ TE n - 1 a ( n 2 n 1 n 2 ) , TM m - 1 b ( n 2 n eff n 2 ) ] , for E n m y ,
P x 1 2 = P x 2 + x ( R , P x 2 ) ,
P y 1 2 = P y 2 + y ( R , P y 2 ) ,
P x 2 2 = P x 2 + G x ( R , P x 2 ) y ( R , P x 2 ) ,
P y 2 2 = p y 2 + G y ( R , P y 2 ) x ( R , P y 2 ) ,
G x ( R , P 2 ) = G y ( R , P 2 ) = G ( R , P 2 ) .
P x 2 = P y 2 = P 2
P x 2 = G P x 1 2 - P y 2 2 G - 1 ,
P y 2 = G P y 1 2 - P x 2 2 G - 1 .
G P x 1 2 + P x 2 2 = G P y 1 2 + P y 2 2 .
P y 2 2 = P y 2 + x ( 1 / R , P y 2 ) ,
P x 1 2 = P x 2 + G ( 1 / R , P x 2 ) x ( 1 / R , P x 2 ) .
G ( R , P 2 ) G ( 1 / R , P 2 ) = 1
G ( R , P 2 ) = R α ,             α = α ( P 2 ) .
α ( P 2 ) = 0.75 exp [ - 0.48 ( P 2 - 1 ) 2 ] + 0.25 1 - 0.36 ( P 2 - 1 ) 2 .
P x 1 2 = [ TE n - 1 b ( n 3 n 1 n 2 ) , TM m - 1 a ( n 2 n eff n 2 ) ] , for E m n x ,
P x 2 2 = [ TM m - 1 a ( n 2 n 1 n 2 ) , TE n - 1 b ( n 3 n eff n 2 ) ] , for E m n x ,
P y 1 2 = [ TM m - 1 b ( n 2 n 1 n 2 ) , TE n - 1 a ( n 3 n eff n 2 ) ] , for E n m y ,
P y 2 2 = [ TE n - 1 a ( n 3 n 1 n 2 ) , TM m - 1 b ( n 2 n eff n 2 ) ] , for E n m y .

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