Abstract

We show analytically that for the modal analysis of rectangular waveguides, the recently developed effective-index method with built-in perturbation correction is identical to the first-iteration cycle of the variational method that we had developed earlier [Opt. Quantum Electron. 12, 517 (1989)]. We also show that in a number of cases the accuracy improves considerably through iteration beyond the first cycle.

© 2001 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, 2071–2102 (1969).
    [CrossRef]
  2. G. B. Hocker, W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977).
    [CrossRef] [PubMed]
  3. A. Kumar, K. Thyagarajan, A. K. Ghatak, “Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach,” Opt. Lett. 8, 63–65 (1983).
    [CrossRef] [PubMed]
  4. A. Sharma, P. K. Mishra, A. K. Ghatak, “Analysis of single mode waveguides and directional couplers with rectangular cross section,” in Proceedings of the Second European Conference on Integrated Optics (Institute of Electrical Engineers, London, 1983), pp. 9–12.
  5. P. K. 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, 282–286 (1985).
    [CrossRef]
  6. A. Sharma, P. K. Mishra, A. K. Ghatak, “Single-mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis,” J. Lightwave Technol. 6, 1119–1125 (1988).
    [CrossRef]
  7. A. Sharma, “On approximate theories of single mode rectangular waveguides,” Opt. Quantum Electron. 21, 517–520 (1989).
    [CrossRef]
  8. A. Sharma, P. Bindal, “An accurate variational analysis of single mode diffused channel waveguides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
    [CrossRef]
  9. P. C. Kendall, M. J. Adams, S. Ritchie, M. J. Robertson, “Theory for calculating approximate values of the propagation constant of an optical rib waveguide by weighting the refractive index,” IEE Proc. A 134, 699–702 (1987).
  10. F. P. Payne, “A new theory of rectangular optical waveguides,” Opt. Quantum Electron. 14, 525–537 (1982).
    [CrossRef]
  11. F. P. Payne, “A generalised transverse resonance model for planar optical waveguides,” presented at the Tenth European Conference on Optical Communications, Stuttgart, Germany, September 3–6, 1984.
  12. D. A. Roberts, M. S. Stern, “Accuracy of method of moments and weighted index method,” Electron. Lett. 23, 474–475 (1987).
    [CrossRef]
  13. She Shouxian, “Analysis of rectangular-core waveguide structures and directional couplers by an iterated moments method,” Opt. Quantum Electron. 20, 125–136 (1988).
    [CrossRef]
  14. T. M. Benson, P. C. Kendall, “Variational techniques including effective and weighted index methods,” in Progress in Electromagnetic Research (EMW, Cambridge, Mass., 1995), Vol. 10, pp. 1–40.
  15. K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, S113–S134 (1994).
    [CrossRef]
  16. K. S. Chiang, “Dual effective-index method for the analysis of rectangular dielectric waveguides,” Appl. Opt. 25, 2169–2174 (1986).
    [CrossRef] [PubMed]
  17. K. S. Chiang, “Performance of the effective-index method for the analysis of dielectric waveguides,” Opt. Lett. 16, 714–716 (1991).
    [CrossRef] [PubMed]
  18. K. S. Chiang, “Effective-index method for the analysis of optical waveguide couplers and arrays: an asymptotic theory,” J. Lightwave Technol. 9, 62–72 (1991).
    [CrossRef]
  19. K. S. Chiang, “Analysis of rectangular dielectric waveguides: effective-index method with built-in perturbation correction,” Electron. Lett. 28, 388–389 (1992).
    [CrossRef]
  20. K. S. Chiang, K. M. Lo, K. S. Kwok, “Effective-index method with built-in perturbation correction for integrated optical waveguides,” J. Lightwave Technol. 14, 223–228 (1996).
    [CrossRef]
  21. K. S. Chiang, K. S. Kwok, K. M. Lo, “Effective-index method with built-in perturbation correction for the vector modes rectangular-core optical waveguides,” J. Lightwave Technol. 17, 716–722 (1999).
    [CrossRef]
  22. J. P. Meunier, Laboratoire de Traitement du Signal et Instrumentation, UMR CNRS 5516, Faculte des Sciences et Techniques, Université Jean Monnet, 42023 Saint-Etienne Cedex 2, France (personal communication, May, 2000).
  23. A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, A. B. Sotskii, “Methods for approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” Zh. Prikl. Spektrosk. 45, 7–16 (1986) [published in English in J. Appl. Spectrosc. 54, 663–667 (1987)].
  24. R. J. Black, A. Ankiewicz, “Fiber-optic analogies with mechanics,” Am. J. Phys. 53, 554–563 (1985).
    [CrossRef]
  25. P. A. M. Dirac, Principles of Quantum Mechanics (Oxford U. Press, Oxford, UK, 1958).
  26. L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968).

1999 (1)

1996 (1)

K. S. Chiang, K. M. Lo, K. S. Kwok, “Effective-index method with built-in perturbation correction for integrated optical waveguides,” J. Lightwave Technol. 14, 223–228 (1996).
[CrossRef]

1994 (1)

K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, S113–S134 (1994).
[CrossRef]

1992 (2)

K. S. Chiang, “Analysis of rectangular dielectric waveguides: effective-index method with built-in perturbation correction,” Electron. Lett. 28, 388–389 (1992).
[CrossRef]

A. Sharma, P. Bindal, “An accurate variational analysis of single mode diffused channel waveguides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
[CrossRef]

1991 (2)

K. S. Chiang, “Performance of the effective-index method for the analysis of dielectric waveguides,” Opt. Lett. 16, 714–716 (1991).
[CrossRef] [PubMed]

K. S. Chiang, “Effective-index method for the analysis of optical waveguide couplers and arrays: an asymptotic theory,” J. Lightwave Technol. 9, 62–72 (1991).
[CrossRef]

1989 (1)

A. Sharma, “On approximate theories of single mode rectangular waveguides,” Opt. Quantum Electron. 21, 517–520 (1989).
[CrossRef]

1988 (2)

A. Sharma, P. K. Mishra, A. K. Ghatak, “Single-mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis,” J. Lightwave Technol. 6, 1119–1125 (1988).
[CrossRef]

She Shouxian, “Analysis of rectangular-core waveguide structures and directional couplers by an iterated moments method,” Opt. Quantum Electron. 20, 125–136 (1988).
[CrossRef]

1987 (2)

D. A. Roberts, M. S. Stern, “Accuracy of method of moments and weighted index method,” Electron. Lett. 23, 474–475 (1987).
[CrossRef]

P. C. Kendall, M. J. Adams, S. Ritchie, M. J. Robertson, “Theory for calculating approximate values of the propagation constant of an optical rib waveguide by weighting the refractive index,” IEE Proc. A 134, 699–702 (1987).

1986 (2)

K. S. Chiang, “Dual effective-index method for the analysis of rectangular dielectric waveguides,” Appl. Opt. 25, 2169–2174 (1986).
[CrossRef] [PubMed]

A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, A. B. Sotskii, “Methods for approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” Zh. Prikl. Spektrosk. 45, 7–16 (1986) [published in English in J. Appl. Spectrosc. 54, 663–667 (1987)].

1985 (2)

R. J. Black, A. Ankiewicz, “Fiber-optic analogies with mechanics,” Am. J. Phys. 53, 554–563 (1985).
[CrossRef]

P. K. 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, 282–286 (1985).
[CrossRef]

1983 (1)

1982 (1)

F. P. Payne, “A new theory of rectangular optical waveguides,” Opt. Quantum Electron. 14, 525–537 (1982).
[CrossRef]

1977 (1)

1969 (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[CrossRef]

Adams, M. J.

P. C. Kendall, M. J. Adams, S. Ritchie, M. J. Robertson, “Theory for calculating approximate values of the propagation constant of an optical rib waveguide by weighting the refractive index,” IEE Proc. A 134, 699–702 (1987).

Ankiewicz, A.

R. J. Black, A. Ankiewicz, “Fiber-optic analogies with mechanics,” Am. J. Phys. 53, 554–563 (1985).
[CrossRef]

Benson, T. M.

T. M. Benson, P. C. Kendall, “Variational techniques including effective and weighted index methods,” in Progress in Electromagnetic Research (EMW, Cambridge, Mass., 1995), Vol. 10, pp. 1–40.

Bindal, P.

A. Sharma, P. Bindal, “An accurate variational analysis of single mode diffused channel waveguides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
[CrossRef]

Black, R. J.

R. J. Black, A. Ankiewicz, “Fiber-optic analogies with mechanics,” Am. J. Phys. 53, 554–563 (1985).
[CrossRef]

Burns, W. K.

Chiang, K. S.

K. S. Chiang, K. S. Kwok, K. M. Lo, “Effective-index method with built-in perturbation correction for the vector modes rectangular-core optical waveguides,” J. Lightwave Technol. 17, 716–722 (1999).
[CrossRef]

K. S. Chiang, K. M. Lo, K. S. Kwok, “Effective-index method with built-in perturbation correction for integrated optical waveguides,” J. Lightwave Technol. 14, 223–228 (1996).
[CrossRef]

K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, S113–S134 (1994).
[CrossRef]

K. S. Chiang, “Analysis of rectangular dielectric waveguides: effective-index method with built-in perturbation correction,” Electron. Lett. 28, 388–389 (1992).
[CrossRef]

K. S. Chiang, “Performance of the effective-index method for the analysis of dielectric waveguides,” Opt. Lett. 16, 714–716 (1991).
[CrossRef] [PubMed]

K. S. Chiang, “Effective-index method for the analysis of optical waveguide couplers and arrays: an asymptotic theory,” J. Lightwave Technol. 9, 62–72 (1991).
[CrossRef]

K. S. Chiang, “Dual effective-index method for the analysis of rectangular dielectric waveguides,” Appl. Opt. 25, 2169–2174 (1986).
[CrossRef] [PubMed]

Dirac, P. A. M.

P. A. M. Dirac, Principles of Quantum Mechanics (Oxford U. Press, Oxford, UK, 1958).

Ghatak, A. K.

A. Sharma, P. K. Mishra, A. K. Ghatak, “Single-mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis,” J. Lightwave Technol. 6, 1119–1125 (1988).
[CrossRef]

P. K. 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, 282–286 (1985).
[CrossRef]

A. Kumar, K. Thyagarajan, A. K. Ghatak, “Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach,” Opt. Lett. 8, 63–65 (1983).
[CrossRef] [PubMed]

A. Sharma, P. K. Mishra, A. K. Ghatak, “Analysis of single mode waveguides and directional couplers with rectangular cross section,” in Proceedings of the Second European Conference on Integrated Optics (Institute of Electrical Engineers, London, 1983), pp. 9–12.

Goncharenko, A. M.

A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, A. B. Sotskii, “Methods for approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” Zh. Prikl. Spektrosk. 45, 7–16 (1986) [published in English in J. Appl. Spectrosc. 54, 663–667 (1987)].

Hocker, G. B.

Karpenko, V. A.

A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, A. B. Sotskii, “Methods for approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” Zh. Prikl. Spektrosk. 45, 7–16 (1986) [published in English in J. Appl. Spectrosc. 54, 663–667 (1987)].

Kendall, P. C.

P. C. Kendall, M. J. Adams, S. Ritchie, M. J. Robertson, “Theory for calculating approximate values of the propagation constant of an optical rib waveguide by weighting the refractive index,” IEE Proc. A 134, 699–702 (1987).

T. M. Benson, P. C. Kendall, “Variational techniques including effective and weighted index methods,” in Progress in Electromagnetic Research (EMW, Cambridge, Mass., 1995), Vol. 10, pp. 1–40.

Kumar, A.

Kwok, K. S.

K. S. Chiang, K. S. Kwok, K. M. Lo, “Effective-index method with built-in perturbation correction for the vector modes rectangular-core optical waveguides,” J. Lightwave Technol. 17, 716–722 (1999).
[CrossRef]

K. S. Chiang, K. M. Lo, K. S. Kwok, “Effective-index method with built-in perturbation correction for integrated optical waveguides,” J. Lightwave Technol. 14, 223–228 (1996).
[CrossRef]

Labroo, S.

P. K. 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, 282–286 (1985).
[CrossRef]

Lo, K. M.

K. S. Chiang, K. S. Kwok, K. M. Lo, “Effective-index method with built-in perturbation correction for the vector modes rectangular-core optical waveguides,” J. Lightwave Technol. 17, 716–722 (1999).
[CrossRef]

K. S. Chiang, K. M. Lo, K. S. Kwok, “Effective-index method with built-in perturbation correction for integrated optical waveguides,” J. Lightwave Technol. 14, 223–228 (1996).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[CrossRef]

Meunier, J. P.

J. P. Meunier, Laboratoire de Traitement du Signal et Instrumentation, UMR CNRS 5516, Faculte des Sciences et Techniques, Université Jean Monnet, 42023 Saint-Etienne Cedex 2, France (personal communication, May, 2000).

Mishra, P. K.

A. Sharma, P. K. Mishra, A. K. Ghatak, “Single-mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis,” J. Lightwave Technol. 6, 1119–1125 (1988).
[CrossRef]

P. K. 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, 282–286 (1985).
[CrossRef]

A. Sharma, P. K. Mishra, A. K. Ghatak, “Analysis of single mode waveguides and directional couplers with rectangular cross section,” in Proceedings of the Second European Conference on Integrated Optics (Institute of Electrical Engineers, London, 1983), pp. 9–12.

Mogilevich, V. N.

A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, A. B. Sotskii, “Methods for approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” Zh. Prikl. Spektrosk. 45, 7–16 (1986) [published in English in J. Appl. Spectrosc. 54, 663–667 (1987)].

Payne, F. P.

F. P. Payne, “A new theory of rectangular optical waveguides,” Opt. Quantum Electron. 14, 525–537 (1982).
[CrossRef]

F. P. Payne, “A generalised transverse resonance model for planar optical waveguides,” presented at the Tenth European Conference on Optical Communications, Stuttgart, Germany, September 3–6, 1984.

Ritchie, S.

P. C. Kendall, M. J. Adams, S. Ritchie, M. J. Robertson, “Theory for calculating approximate values of the propagation constant of an optical rib waveguide by weighting the refractive index,” IEE Proc. A 134, 699–702 (1987).

Roberts, D. A.

D. A. Roberts, M. S. Stern, “Accuracy of method of moments and weighted index method,” Electron. Lett. 23, 474–475 (1987).
[CrossRef]

Robertson, M. J.

P. C. Kendall, M. J. Adams, S. Ritchie, M. J. Robertson, “Theory for calculating approximate values of the propagation constant of an optical rib waveguide by weighting the refractive index,” IEE Proc. A 134, 699–702 (1987).

Schiff, L. I.

L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968).

Sharma, A.

A. Sharma, P. Bindal, “An accurate variational analysis of single mode diffused channel waveguides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
[CrossRef]

A. Sharma, “On approximate theories of single mode rectangular waveguides,” Opt. Quantum Electron. 21, 517–520 (1989).
[CrossRef]

A. Sharma, P. K. Mishra, A. K. Ghatak, “Single-mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis,” J. Lightwave Technol. 6, 1119–1125 (1988).
[CrossRef]

P. K. 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, 282–286 (1985).
[CrossRef]

A. Sharma, P. K. Mishra, A. K. Ghatak, “Analysis of single mode waveguides and directional couplers with rectangular cross section,” in Proceedings of the Second European Conference on Integrated Optics (Institute of Electrical Engineers, London, 1983), pp. 9–12.

Shouxian, She

She Shouxian, “Analysis of rectangular-core waveguide structures and directional couplers by an iterated moments method,” Opt. Quantum Electron. 20, 125–136 (1988).
[CrossRef]

Sotskii, A. B.

A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, A. B. Sotskii, “Methods for approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” Zh. Prikl. Spektrosk. 45, 7–16 (1986) [published in English in J. Appl. Spectrosc. 54, 663–667 (1987)].

Stern, M. S.

D. A. Roberts, M. S. Stern, “Accuracy of method of moments and weighted index method,” Electron. Lett. 23, 474–475 (1987).
[CrossRef]

Thyagarajan, K.

Am. J. Phys. (1)

R. J. Black, A. Ankiewicz, “Fiber-optic analogies with mechanics,” Am. J. Phys. 53, 554–563 (1985).
[CrossRef]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[CrossRef]

Electron. Lett. (2)

D. A. Roberts, M. S. Stern, “Accuracy of method of moments and weighted index method,” Electron. Lett. 23, 474–475 (1987).
[CrossRef]

K. S. Chiang, “Analysis of rectangular dielectric waveguides: effective-index method with built-in perturbation correction,” Electron. Lett. 28, 388–389 (1992).
[CrossRef]

IEE Proc. A (1)

P. C. Kendall, M. J. Adams, S. Ritchie, M. J. Robertson, “Theory for calculating approximate values of the propagation constant of an optical rib waveguide by weighting the refractive index,” IEE Proc. A 134, 699–702 (1987).

IEEE Trans. Microwave Theory Tech. (1)

P. K. 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, 282–286 (1985).
[CrossRef]

J. Lightwave Technol. (4)

A. Sharma, P. K. Mishra, A. K. Ghatak, “Single-mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis,” J. Lightwave Technol. 6, 1119–1125 (1988).
[CrossRef]

K. S. Chiang, K. M. Lo, K. S. Kwok, “Effective-index method with built-in perturbation correction for integrated optical waveguides,” J. Lightwave Technol. 14, 223–228 (1996).
[CrossRef]

K. S. Chiang, K. S. Kwok, K. M. Lo, “Effective-index method with built-in perturbation correction for the vector modes rectangular-core optical waveguides,” J. Lightwave Technol. 17, 716–722 (1999).
[CrossRef]

K. S. Chiang, “Effective-index method for the analysis of optical waveguide couplers and arrays: an asymptotic theory,” J. Lightwave Technol. 9, 62–72 (1991).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (5)

F. P. Payne, “A new theory of rectangular optical waveguides,” Opt. Quantum Electron. 14, 525–537 (1982).
[CrossRef]

A. Sharma, “On approximate theories of single mode rectangular waveguides,” Opt. Quantum Electron. 21, 517–520 (1989).
[CrossRef]

A. Sharma, P. Bindal, “An accurate variational analysis of single mode diffused channel waveguides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
[CrossRef]

She Shouxian, “Analysis of rectangular-core waveguide structures and directional couplers by an iterated moments method,” Opt. Quantum Electron. 20, 125–136 (1988).
[CrossRef]

K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, S113–S134 (1994).
[CrossRef]

Zh. Prikl. Spektrosk. (1)

A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, A. B. Sotskii, “Methods for approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” Zh. Prikl. Spektrosk. 45, 7–16 (1986) [published in English in J. Appl. Spectrosc. 54, 663–667 (1987)].

Other (6)

P. A. M. Dirac, Principles of Quantum Mechanics (Oxford U. Press, Oxford, UK, 1958).

L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968).

T. M. Benson, P. C. Kendall, “Variational techniques including effective and weighted index methods,” in Progress in Electromagnetic Research (EMW, Cambridge, Mass., 1995), Vol. 10, pp. 1–40.

J. P. Meunier, Laboratoire de Traitement du Signal et Instrumentation, UMR CNRS 5516, Faculte des Sciences et Techniques, Université Jean Monnet, 42023 Saint-Etienne Cedex 2, France (personal communication, May, 2000).

F. P. Payne, “A generalised transverse resonance model for planar optical waveguides,” presented at the Tenth European Conference on Optical Communications, Stuttgart, Germany, September 3–6, 1984.

A. Sharma, P. K. Mishra, A. K. Ghatak, “Analysis of single mode waveguides and directional couplers with rectangular cross section,” in Proceedings of the Second European Conference on Integrated Optics (Institute of Electrical Engineers, London, 1983), pp. 9–12.

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Figures (4)

Fig. 1
Fig. 1

Rectangular waveguide whose modes are to be obtained.

Fig. 2
Fig. 2

Planar waveguide varying in the y direction.

Fig. 3
Fig. 3

Equivalent planar x waveguide obtained in (a) the EIMPC-X method and (b) the CEVAR* method.

Fig. 4
Fig. 4

Strip-loaded waveguide with n1=1.55,n2=1.50,n3=1.00,w=3 μm,h=t=2 μm,λ=1.55 μm.

Tables (1)

Tables Icon

Table 1 Convergence of Results for B=[(β/k0)2-n22]/[k02(n12-n22)]

Equations (23)

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

ny2(y)=n12core
=n22substrate
=n32cover.
nx2(x)=[n2(x, y)-ny2(y)]|ψy(y)|2dy+βy2/k02,
ny2(y)|improved=[n2(x, y)-nx2(x)]|ψx(x)|2dx+βx2/k02,
n˜x2(x)[core]=nex2
n˜x2(x)[clad]=n42-γ(n12-nex2),
γ=1-n12-n42n12-nex2Pcly=1-(1-Pcoy)n12-n42n12-nex2,
nx2(x)[core]=βy2/k02nex2,
nx2(x)[clad]=(n42-n12)coreψy2(y)dy+βy2/k02,
=(n42-n12)Pcoy+nex2.
n˜x2(x)[clad]EIMPC=n42-1-(1-Pcoy)n12-n42n12-nex2×(n12-nex2),
=nex2-Pcoy(n12-n42)nx2(x)[clad]CEVAR*,
H|ϕ=E|ϕ,
ϕ|ϕ=1.
Et(p)=ϕt(p)|H|ϕt(p).
Etp=0,
H0|ϕ0=E0|ϕ0,orE0=ϕ0|H0|ϕ0,
E1=E0+ϕ0|H-H0|ϕ0.
E1=ϕ0|H0|ϕ0+ϕ0|H-H0|ϕ0=ϕ0|H|ϕ0,
Et(p)=ϕt(p)|H|ϕt(p).
Et(p=p0+Δp)=E(p0)+Δp Etpp=p0+ ,
Etpp=p0=0,

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