Abstract

An approach to the analysis of complex modes in lossless isotropic dielectric waveguides is presented. The existing theory of complex modes is extended to the case of open structures by techniques of functional analysis. Mathematical and physical properties of complex modes, including some criteria of existence and mechanisms of occurrence, are formulated. The theory is demonstrated by the numerical example of a waveguide with a large core-cladding refractive-index difference. Computed complex-mode dispersion characteristics are analyzed, and the mode vector field, together with the power distribution in the waveguide cross section, is shown.

© 1994 Optical Society of America

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References

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  1. A. M. Beliantsev, A. V. Gaponov, “Waves with complex propagation constants in coupled transmission lines without energy dissipation,” Radio Eng. Electron. Phys. (USSR) 9, 980–988 (1964).
  2. P. J. B. Clarricoats, K. R. Slinn, “Complex modes of propagation in dielectric-loaded circular waveguide,” Electron. Lett. 1, 145–146 (1965).
    [CrossRef]
  3. J. D. Rhodes, “General constraints on propagation characteristics of electromagnetic waves in uniform inhomogeneous waveguides,” Proc. Inst. Electr. Eng. 118, 849–856 (1971).
    [CrossRef]
  4. V. A. Kalmyk, S. B. Rayevskiy, V. P. Ugryumov, “An experimental verification of existence of complex waves in a two layer circular, shielded waveguide,” Radio Eng. Electron. Phys. (USSR) 23, 16–19 (1978).
  5. J. Strube, F. Arndt, “Rigorous hybrid mode analysis of the transition from waveguide to shielded dielectric image line,” IEEE Trans. Microwave Theory Tech. MTT-33, 391–401 (1985).
    [CrossRef]
  6. A. S. Omar, K. F. Schünemann, “Effect of complex modes at finline discontinuities,” IEEE Trans. Microwave Theory Tech. MTT-34, 1508–1514 (1986).
    [CrossRef]
  7. C. Chen, K. A. Zaki, “Resonant frequences of dielectric resonators containing guided complex modes,” IEEE Trans. Microwave Theory Tech. 36, 1455–1457 (1988).
    [CrossRef]
  8. K. A. Zaki, S. Chen, C. Chen, “Modelling discontinuities in dielectric-loaded waveguides,” IEEE Trans. Microwave Theory Tech. 36, 1804–1809 (1988).
    [CrossRef]
  9. U. Crombach, “Complex waves on shielded lossless rectangular dielectric waveguide,” Electron. Lett. 19, 557–558 (1983).
    [CrossRef]
  10. W. Huang, T. Itoh, “Complex modes in lossless shielded microstrip lines,” IEEE Trans. Microwave Theory Tech. 36, 163–165 (1988).
    [CrossRef]
  11. C. J. Railton, T. Rozzi, “Complex modes in boxed microstrip,” IEEE Trans. Microwave Theory Tech. 36, 865–874 (1988).
    [CrossRef]
  12. M. Mrozowski, “Waves in shielded lossless isotropic wave-guiding structures,” Ph.D. dissertation (Telecommunication Institute, Technical University of Gdańsk, Gdańsk, Poland, 1989).
  13. B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, N.J., 1980).
  14. A. S. Omar, K. F. Schünemann, “Complex and backward-wave modes in inhomogeneously and anisotropically filled waveguides,” IEEE Trans. Microwave Theory Tech. MTT-35, 268–275 (1987).
    [CrossRef]
  15. G. I. Viesielov, S. B. Rayevskiy, Stratified Shielded Dielectric Waveguides (Radio i Sviaz, Moscow, 1988) (in Russian).
  16. A. Bamberger, A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. 21, 1487–1510 (1990)
    [CrossRef]
  17. T. F. Jabłoński, “Iterative eigenfunction expansion method for monomode gradient index fibers with arbitrary cross-section,” in Proceedings of the International Union of Radio Science International Symposium on Electromagnetic Theory, Part B (Akadémiai Kiadó, Budapest, 1986), pp. 415–417.
  18. T. F. Jabłoński, M. J. Sowński, “Analysis of dielectric guiding structures by the iterative eigenfunction expansion method,” IEEE Trans. Microwave Theory Tech. 37, 63–70 (1989).
    [CrossRef]
  19. We recall that a function f∈ Hm(ℝn), where m and n are positive integers, if and only if Dαf∈ L2(ℝn) for all multi-indices α= (α1,…,αn) such that ∑i=1nαi≤m, where Dαf is the distribution derivative of f and L2 is the space of square integrable functions.
  20. The Hilbert space ℋ is also equipped with the norm induced by the scalar product: ‖ ‖ℋ≡ (·, ·)ℋ1/2.
  21. If (a) instead of ℝ2 we take a bounded region Ω ⊂ ℝ2, (b) we substitute the domain of the Dirichlet Laplacian ΔD or the domain of the Neumann Laplacian ΔNfor D(l) in Eqs. (10), and (c) we replace H2(ℝ2) in relations (2) with the local Sobolev space H02(Ω), then the operator Tdefined in Eqs. (9) and (10) will correspond to the propagation problem of a shielded inhomogeneously filled dielectric waveguide. See also Appendix A.
  22. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  23. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  24. L. A. Vainstein, Elektromagnitnyje Volny, 2nd ed. (Radio i Sviaz, Moscow, 1988).
  25. Except for the sporadic case in which the h⊥ field of the complex mode with Im(β2) ≠ 0 is spanned equally on its mutually orthogonal real and imaginary parts. Indeed, we have the equivalence(h⊥,h¯⊥)ℋ=0⇔{‖Re(h⊥)‖ℋ=‖Im(h⊥)‖ℋand(Re(h⊥),Im(h⊥))ℋ=0}.
  26. R. Mittra, S. W. Lee, Analytical Techniques in the Theory of Guided Waves (Macmillan, New York, 1971).
  27. This severe energetical test provides the best verification of the computations performed by the IEEM.
  28. Let L and F be unbounded operators in the Hilbert space ℋ. The operator F is said to be relatively compact with respect to L,or simply L-compact, if D(F) ⊇ D(L) and, ∀ρ∉ σ(L), F(L− ρ)−1 is a compact operator in ℋ.
  29. T. F. Jabłoński, “Iterative eigenfunction expansion method for cylindrical fibers,” Research Rep. No. 3 (Institute of Fundamental Technological Research, Warsaw, 1986) (in Polish).
  30. T. F. Jabłoński, “Iterative spectral decomposition method and its application to the analysis of dielectric guiding structures,” Ph.D. dissertation (Polish Academy of Sciences, Institute of Fundamental Technological Research, Warsaw, 1991).
  31. M. Reed, B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1978), Vol. 4.
  32. T. Kato, Perturbation Theory for Linear Operators, 2nd ed. (Springer-Verlag, Berlin, 1980).
  33. We recall that λ belongs to σdisc(T), the discrete spectrum of T,iff λ is an isolated eigenvalue of T with finite multiplicity. The essential spectrum of T is defined as σess(T) ≡ σ(T) − σdisc(T).

1990 (1)

A. Bamberger, A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. 21, 1487–1510 (1990)
[CrossRef]

1989 (1)

T. F. Jabłoński, M. J. Sowński, “Analysis of dielectric guiding structures by the iterative eigenfunction expansion method,” IEEE Trans. Microwave Theory Tech. 37, 63–70 (1989).
[CrossRef]

1988 (4)

W. Huang, T. Itoh, “Complex modes in lossless shielded microstrip lines,” IEEE Trans. Microwave Theory Tech. 36, 163–165 (1988).
[CrossRef]

C. J. Railton, T. Rozzi, “Complex modes in boxed microstrip,” IEEE Trans. Microwave Theory Tech. 36, 865–874 (1988).
[CrossRef]

C. Chen, K. A. Zaki, “Resonant frequences of dielectric resonators containing guided complex modes,” IEEE Trans. Microwave Theory Tech. 36, 1455–1457 (1988).
[CrossRef]

K. A. Zaki, S. Chen, C. Chen, “Modelling discontinuities in dielectric-loaded waveguides,” IEEE Trans. Microwave Theory Tech. 36, 1804–1809 (1988).
[CrossRef]

1987 (1)

A. S. Omar, K. F. Schünemann, “Complex and backward-wave modes in inhomogeneously and anisotropically filled waveguides,” IEEE Trans. Microwave Theory Tech. MTT-35, 268–275 (1987).
[CrossRef]

1986 (1)

A. S. Omar, K. F. Schünemann, “Effect of complex modes at finline discontinuities,” IEEE Trans. Microwave Theory Tech. MTT-34, 1508–1514 (1986).
[CrossRef]

1985 (1)

J. Strube, F. Arndt, “Rigorous hybrid mode analysis of the transition from waveguide to shielded dielectric image line,” IEEE Trans. Microwave Theory Tech. MTT-33, 391–401 (1985).
[CrossRef]

1983 (1)

U. Crombach, “Complex waves on shielded lossless rectangular dielectric waveguide,” Electron. Lett. 19, 557–558 (1983).
[CrossRef]

1978 (1)

V. A. Kalmyk, S. B. Rayevskiy, V. P. Ugryumov, “An experimental verification of existence of complex waves in a two layer circular, shielded waveguide,” Radio Eng. Electron. Phys. (USSR) 23, 16–19 (1978).

1971 (1)

J. D. Rhodes, “General constraints on propagation characteristics of electromagnetic waves in uniform inhomogeneous waveguides,” Proc. Inst. Electr. Eng. 118, 849–856 (1971).
[CrossRef]

1965 (1)

P. J. B. Clarricoats, K. R. Slinn, “Complex modes of propagation in dielectric-loaded circular waveguide,” Electron. Lett. 1, 145–146 (1965).
[CrossRef]

1964 (1)

A. M. Beliantsev, A. V. Gaponov, “Waves with complex propagation constants in coupled transmission lines without energy dissipation,” Radio Eng. Electron. Phys. (USSR) 9, 980–988 (1964).

Arndt, F.

J. Strube, F. Arndt, “Rigorous hybrid mode analysis of the transition from waveguide to shielded dielectric image line,” IEEE Trans. Microwave Theory Tech. MTT-33, 391–401 (1985).
[CrossRef]

Bamberger, A.

A. Bamberger, A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. 21, 1487–1510 (1990)
[CrossRef]

Beliantsev, A. M.

A. M. Beliantsev, A. V. Gaponov, “Waves with complex propagation constants in coupled transmission lines without energy dissipation,” Radio Eng. Electron. Phys. (USSR) 9, 980–988 (1964).

Bonnet, A. S.

A. Bamberger, A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. 21, 1487–1510 (1990)
[CrossRef]

Chen, C.

C. Chen, K. A. Zaki, “Resonant frequences of dielectric resonators containing guided complex modes,” IEEE Trans. Microwave Theory Tech. 36, 1455–1457 (1988).
[CrossRef]

K. A. Zaki, S. Chen, C. Chen, “Modelling discontinuities in dielectric-loaded waveguides,” IEEE Trans. Microwave Theory Tech. 36, 1804–1809 (1988).
[CrossRef]

Chen, S.

K. A. Zaki, S. Chen, C. Chen, “Modelling discontinuities in dielectric-loaded waveguides,” IEEE Trans. Microwave Theory Tech. 36, 1804–1809 (1988).
[CrossRef]

Clarricoats, P. J. B.

P. J. B. Clarricoats, K. R. Slinn, “Complex modes of propagation in dielectric-loaded circular waveguide,” Electron. Lett. 1, 145–146 (1965).
[CrossRef]

Crombach, U.

U. Crombach, “Complex waves on shielded lossless rectangular dielectric waveguide,” Electron. Lett. 19, 557–558 (1983).
[CrossRef]

Gaponov, A. V.

A. M. Beliantsev, A. V. Gaponov, “Waves with complex propagation constants in coupled transmission lines without energy dissipation,” Radio Eng. Electron. Phys. (USSR) 9, 980–988 (1964).

Huang, W.

W. Huang, T. Itoh, “Complex modes in lossless shielded microstrip lines,” IEEE Trans. Microwave Theory Tech. 36, 163–165 (1988).
[CrossRef]

Itoh, T.

W. Huang, T. Itoh, “Complex modes in lossless shielded microstrip lines,” IEEE Trans. Microwave Theory Tech. 36, 163–165 (1988).
[CrossRef]

Jablonski, T. F.

T. F. Jabłoński, M. J. Sowński, “Analysis of dielectric guiding structures by the iterative eigenfunction expansion method,” IEEE Trans. Microwave Theory Tech. 37, 63–70 (1989).
[CrossRef]

T. F. Jabłoński, “Iterative eigenfunction expansion method for cylindrical fibers,” Research Rep. No. 3 (Institute of Fundamental Technological Research, Warsaw, 1986) (in Polish).

T. F. Jabłoński, “Iterative spectral decomposition method and its application to the analysis of dielectric guiding structures,” Ph.D. dissertation (Polish Academy of Sciences, Institute of Fundamental Technological Research, Warsaw, 1991).

T. F. Jabłoński, “Iterative eigenfunction expansion method for monomode gradient index fibers with arbitrary cross-section,” in Proceedings of the International Union of Radio Science International Symposium on Electromagnetic Theory, Part B (Akadémiai Kiadó, Budapest, 1986), pp. 415–417.

Kalmyk, V. A.

V. A. Kalmyk, S. B. Rayevskiy, V. P. Ugryumov, “An experimental verification of existence of complex waves in a two layer circular, shielded waveguide,” Radio Eng. Electron. Phys. (USSR) 23, 16–19 (1978).

Kato, T.

T. Kato, Perturbation Theory for Linear Operators, 2nd ed. (Springer-Verlag, Berlin, 1980).

Lee, S. W.

R. Mittra, S. W. Lee, Analytical Techniques in the Theory of Guided Waves (Macmillan, New York, 1971).

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Mittra, R.

R. Mittra, S. W. Lee, Analytical Techniques in the Theory of Guided Waves (Macmillan, New York, 1971).

Mrozowski, M.

M. Mrozowski, “Waves in shielded lossless isotropic wave-guiding structures,” Ph.D. dissertation (Telecommunication Institute, Technical University of Gdańsk, Gdańsk, Poland, 1989).

Omar, A. S.

A. S. Omar, K. F. Schünemann, “Complex and backward-wave modes in inhomogeneously and anisotropically filled waveguides,” IEEE Trans. Microwave Theory Tech. MTT-35, 268–275 (1987).
[CrossRef]

A. S. Omar, K. F. Schünemann, “Effect of complex modes at finline discontinuities,” IEEE Trans. Microwave Theory Tech. MTT-34, 1508–1514 (1986).
[CrossRef]

Parlett, B. N.

B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, N.J., 1980).

Railton, C. J.

C. J. Railton, T. Rozzi, “Complex modes in boxed microstrip,” IEEE Trans. Microwave Theory Tech. 36, 865–874 (1988).
[CrossRef]

Rayevskiy, S. B.

V. A. Kalmyk, S. B. Rayevskiy, V. P. Ugryumov, “An experimental verification of existence of complex waves in a two layer circular, shielded waveguide,” Radio Eng. Electron. Phys. (USSR) 23, 16–19 (1978).

G. I. Viesielov, S. B. Rayevskiy, Stratified Shielded Dielectric Waveguides (Radio i Sviaz, Moscow, 1988) (in Russian).

Reed, M.

M. Reed, B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1978), Vol. 4.

Rhodes, J. D.

J. D. Rhodes, “General constraints on propagation characteristics of electromagnetic waves in uniform inhomogeneous waveguides,” Proc. Inst. Electr. Eng. 118, 849–856 (1971).
[CrossRef]

Rozzi, T.

C. J. Railton, T. Rozzi, “Complex modes in boxed microstrip,” IEEE Trans. Microwave Theory Tech. 36, 865–874 (1988).
[CrossRef]

Schünemann, K. F.

A. S. Omar, K. F. Schünemann, “Complex and backward-wave modes in inhomogeneously and anisotropically filled waveguides,” IEEE Trans. Microwave Theory Tech. MTT-35, 268–275 (1987).
[CrossRef]

A. S. Omar, K. F. Schünemann, “Effect of complex modes at finline discontinuities,” IEEE Trans. Microwave Theory Tech. MTT-34, 1508–1514 (1986).
[CrossRef]

Simon, B.

M. Reed, B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1978), Vol. 4.

Slinn, K. R.

P. J. B. Clarricoats, K. R. Slinn, “Complex modes of propagation in dielectric-loaded circular waveguide,” Electron. Lett. 1, 145–146 (1965).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Sownski, M. J.

T. F. Jabłoński, M. J. Sowński, “Analysis of dielectric guiding structures by the iterative eigenfunction expansion method,” IEEE Trans. Microwave Theory Tech. 37, 63–70 (1989).
[CrossRef]

Strube, J.

J. Strube, F. Arndt, “Rigorous hybrid mode analysis of the transition from waveguide to shielded dielectric image line,” IEEE Trans. Microwave Theory Tech. MTT-33, 391–401 (1985).
[CrossRef]

Ugryumov, V. P.

V. A. Kalmyk, S. B. Rayevskiy, V. P. Ugryumov, “An experimental verification of existence of complex waves in a two layer circular, shielded waveguide,” Radio Eng. Electron. Phys. (USSR) 23, 16–19 (1978).

Vainstein, L. A.

L. A. Vainstein, Elektromagnitnyje Volny, 2nd ed. (Radio i Sviaz, Moscow, 1988).

Viesielov, G. I.

G. I. Viesielov, S. B. Rayevskiy, Stratified Shielded Dielectric Waveguides (Radio i Sviaz, Moscow, 1988) (in Russian).

Zaki, K. A.

C. Chen, K. A. Zaki, “Resonant frequences of dielectric resonators containing guided complex modes,” IEEE Trans. Microwave Theory Tech. 36, 1455–1457 (1988).
[CrossRef]

K. A. Zaki, S. Chen, C. Chen, “Modelling discontinuities in dielectric-loaded waveguides,” IEEE Trans. Microwave Theory Tech. 36, 1804–1809 (1988).
[CrossRef]

Electron. Lett. (2)

U. Crombach, “Complex waves on shielded lossless rectangular dielectric waveguide,” Electron. Lett. 19, 557–558 (1983).
[CrossRef]

P. J. B. Clarricoats, K. R. Slinn, “Complex modes of propagation in dielectric-loaded circular waveguide,” Electron. Lett. 1, 145–146 (1965).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (8)

A. S. Omar, K. F. Schünemann, “Complex and backward-wave modes in inhomogeneously and anisotropically filled waveguides,” IEEE Trans. Microwave Theory Tech. MTT-35, 268–275 (1987).
[CrossRef]

W. Huang, T. Itoh, “Complex modes in lossless shielded microstrip lines,” IEEE Trans. Microwave Theory Tech. 36, 163–165 (1988).
[CrossRef]

C. J. Railton, T. Rozzi, “Complex modes in boxed microstrip,” IEEE Trans. Microwave Theory Tech. 36, 865–874 (1988).
[CrossRef]

J. Strube, F. Arndt, “Rigorous hybrid mode analysis of the transition from waveguide to shielded dielectric image line,” IEEE Trans. Microwave Theory Tech. MTT-33, 391–401 (1985).
[CrossRef]

A. S. Omar, K. F. Schünemann, “Effect of complex modes at finline discontinuities,” IEEE Trans. Microwave Theory Tech. MTT-34, 1508–1514 (1986).
[CrossRef]

C. Chen, K. A. Zaki, “Resonant frequences of dielectric resonators containing guided complex modes,” IEEE Trans. Microwave Theory Tech. 36, 1455–1457 (1988).
[CrossRef]

K. A. Zaki, S. Chen, C. Chen, “Modelling discontinuities in dielectric-loaded waveguides,” IEEE Trans. Microwave Theory Tech. 36, 1804–1809 (1988).
[CrossRef]

T. F. Jabłoński, M. J. Sowński, “Analysis of dielectric guiding structures by the iterative eigenfunction expansion method,” IEEE Trans. Microwave Theory Tech. 37, 63–70 (1989).
[CrossRef]

Proc. Inst. Electr. Eng. (1)

J. D. Rhodes, “General constraints on propagation characteristics of electromagnetic waves in uniform inhomogeneous waveguides,” Proc. Inst. Electr. Eng. 118, 849–856 (1971).
[CrossRef]

Radio Eng. Electron. Phys. (USSR) (2)

V. A. Kalmyk, S. B. Rayevskiy, V. P. Ugryumov, “An experimental verification of existence of complex waves in a two layer circular, shielded waveguide,” Radio Eng. Electron. Phys. (USSR) 23, 16–19 (1978).

A. M. Beliantsev, A. V. Gaponov, “Waves with complex propagation constants in coupled transmission lines without energy dissipation,” Radio Eng. Electron. Phys. (USSR) 9, 980–988 (1964).

SIAM J. Math. Anal. (1)

A. Bamberger, A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. 21, 1487–1510 (1990)
[CrossRef]

Other (19)

T. F. Jabłoński, “Iterative eigenfunction expansion method for monomode gradient index fibers with arbitrary cross-section,” in Proceedings of the International Union of Radio Science International Symposium on Electromagnetic Theory, Part B (Akadémiai Kiadó, Budapest, 1986), pp. 415–417.

G. I. Viesielov, S. B. Rayevskiy, Stratified Shielded Dielectric Waveguides (Radio i Sviaz, Moscow, 1988) (in Russian).

M. Mrozowski, “Waves in shielded lossless isotropic wave-guiding structures,” Ph.D. dissertation (Telecommunication Institute, Technical University of Gdańsk, Gdańsk, Poland, 1989).

B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, N.J., 1980).

We recall that a function f∈ Hm(ℝn), where m and n are positive integers, if and only if Dαf∈ L2(ℝn) for all multi-indices α= (α1,…,αn) such that ∑i=1nαi≤m, where Dαf is the distribution derivative of f and L2 is the space of square integrable functions.

The Hilbert space ℋ is also equipped with the norm induced by the scalar product: ‖ ‖ℋ≡ (·, ·)ℋ1/2.

If (a) instead of ℝ2 we take a bounded region Ω ⊂ ℝ2, (b) we substitute the domain of the Dirichlet Laplacian ΔD or the domain of the Neumann Laplacian ΔNfor D(l) in Eqs. (10), and (c) we replace H2(ℝ2) in relations (2) with the local Sobolev space H02(Ω), then the operator Tdefined in Eqs. (9) and (10) will correspond to the propagation problem of a shielded inhomogeneously filled dielectric waveguide. See also Appendix A.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

L. A. Vainstein, Elektromagnitnyje Volny, 2nd ed. (Radio i Sviaz, Moscow, 1988).

Except for the sporadic case in which the h⊥ field of the complex mode with Im(β2) ≠ 0 is spanned equally on its mutually orthogonal real and imaginary parts. Indeed, we have the equivalence(h⊥,h¯⊥)ℋ=0⇔{‖Re(h⊥)‖ℋ=‖Im(h⊥)‖ℋand(Re(h⊥),Im(h⊥))ℋ=0}.

R. Mittra, S. W. Lee, Analytical Techniques in the Theory of Guided Waves (Macmillan, New York, 1971).

This severe energetical test provides the best verification of the computations performed by the IEEM.

Let L and F be unbounded operators in the Hilbert space ℋ. The operator F is said to be relatively compact with respect to L,or simply L-compact, if D(F) ⊇ D(L) and, ∀ρ∉ σ(L), F(L− ρ)−1 is a compact operator in ℋ.

T. F. Jabłoński, “Iterative eigenfunction expansion method for cylindrical fibers,” Research Rep. No. 3 (Institute of Fundamental Technological Research, Warsaw, 1986) (in Polish).

T. F. Jabłoński, “Iterative spectral decomposition method and its application to the analysis of dielectric guiding structures,” Ph.D. dissertation (Polish Academy of Sciences, Institute of Fundamental Technological Research, Warsaw, 1991).

M. Reed, B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1978), Vol. 4.

T. Kato, Perturbation Theory for Linear Operators, 2nd ed. (Springer-Verlag, Berlin, 1980).

We recall that λ belongs to σdisc(T), the discrete spectrum of T,iff λ is an isolated eigenvalue of T with finite multiplicity. The essential spectrum of T is defined as σess(T) ≡ σ(T) − σdisc(T).

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Figures (8)

Fig. 1
Fig. 1

Branch cuts of g(β) = (k2coβ2)1/2 for the top Riemann sheet (Re g > 0).

Fig. 2
Fig. 2

Branch cuts of p(β) = (β2k2cl)1/2 for the top Riemann sheet (Im p < 0).

Fig. 3
Fig. 3

Step-index profile; β ( V ) β ( V ) / k cl of the HE11 and EH11 modes. The solid curves are evaluated from Eq. (19). The squares and the circles denote values computed by the IEEM.

Fig. 4
Fig. 4

Step-index profile; complex plane with β2(V) of the EH11 mode. Arrows point in the direction of decreasing V.

Fig. 5
Fig. 5

Step-index profile; complex plane with p[β(V)] of the EH11 mode. Arrows point in the direction of decreasing V.

Fig. 6
Fig. 6

Parabolic-index profile; β ( V ) β ( V ) / k cl of the HE11 and EH11 modes computed by the IEEM.

Fig. 7
Fig. 7

Real HE11 mode: (a) H, (b) E, and (c) Re [ ( e × h ¯ ) ] x 3 ] in the cross-sectional plane of the waveguide; V = 3.5.

Fig. 8
Fig. 8

Complex EH11 mode: (a) H, (b) E, and (c) Re [ ( e × h ¯ ) x 3 ] in the cross-sectional plane of the waveguide; V = 3.0.

Equations (55)

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: 2 x = ( x 1 , x 2 ) , ( x ) = cl + ( co cl ) s ( x ) ,
x 2 , s ( x ) 0 , sup x 2 s ( x ) = 1 , 2 supp ( s ) is compact , s H 2 ( 2 ) ,
H = h exp [ i ( β x 3 ω t ) ] , h ( x ) = [ h x 1 ( x ) , h x 2 ( x ) ]
E = e exp [ i ( β x 3 ω t ) ] , e ( x ) = [ e x 2 ( x ) , e x 2 ( x ) ] ,
2 h + k 2 h + 1 [ × ( × h ) ] β 2 h = 0 ,
2 e + k 2 e + ( 1 e ) β 2 e = 0 ,
h = i β 1 ( h ) , e = i ν ( k ) 1 ( × h ) , e = i ν ( k ) 1 ( × h + × h ) ,
( u , w ) ( u x 1 , w x 1 ) 2 + ( u x 2 , w x 2 ) 2 , u = [ u x 1 , u x 2 ] , w = [ w x 1 , w x 2 ] , ( u x j , w x j ) 2 = 2 u x j w ¯ x j d x ,
T u 2 u + k 2 u + 1 [ × ( × u ) ] ,
D ( l ) = { υ L 2 ( 2 ) : Δ υ L 2 ( 2 ) in the distribution sense }
T u = ( 2 + k 2 cl ) u + k 2 ( co cl ) s u + 1 [ × ( × u ) ] = T 1 u + T 2 u + T 3 u .
( T λ ) u = 0 , u D ( T ) , u = 1 .
u D ( T ) , u = 1 , ( ( T 1 + T 2 ) u , u ) k 2 co .
Re [ ( T 3 h , h ) ] Re ( β 2 ) k 2 co .
Im ( β 2 ) = Im [ ( T 3 h , h ) ] .
( e , h ) = 2 ( e × h ¯ ) x 3 d x = 0 .
g ( h , β ) β 1 h 2 + 1 β 1 h 2 2 1 β ( h , 1 2 h ) 2 = β 1 h 2 + β ¯ 1 h 2 2 + i ( h , 1 2 h ) 2 .
( T β 2 ¯ ) h ¯ = 0
υ g ω β = c β k 2 ( e × h ¯ ) x 3 d x / 2 ( e × h ¯ ) x 3 d x ,
υ g = c k g ( h , β ) h 2 + h 2 .
β 3 = β , β 1 = β , H 3 = h exp ( i β x 3 ) , H 1 = h exp ( i β x 3 ) , E 3 = e exp ( i β x 3 ) , E 1 = e exp ( i β x 3 ) , x 3 ( h , β 2 ) x 3 β 4 = β ¯ , β 2 = β ¯ , H 4 = h ¯ exp ( i β ¯ x 3 ) = H ¯ 3 , H 2 = h ¯ exp ( i β ¯ x 3 ) = H ¯ 1 , E 4 = e ¯ exp ( i β ¯ x 3 ) = E ¯ 3 , E 2 = e ¯ exp ( i β ¯ x 3 ) = E ¯ 1 ,
2 [ ( E 1 + E 2 ) × ( H 1 + H 2 ) ¯ ] x 3 d x = 2 [ ( E 1 × H ¯ 2 ) + ( E 2 × H ¯ 1 ) ] x 3 d x = exp ( 2 i β x 3 ) ( e , h ¯ ) exp ( 2 i β ¯ x 3 ) ( e , h ¯ ) ¯ = 2 Im [ exp ( 2 i β x 3 ) ( e , h ¯ ) ] .
H = α H 1 + α ¯ H ¯ 1 = | α | { h exp [ i ( β x 3 + ϕ ) ] + h ¯ exp [ i ( β ¯ x 3 + ϕ ) ] } = 2 | α | | h | exp ( Im β ) cos [ ( Re β ) x 3 + ψ ]
E = 2 | α | | e | exp ( Im β ) sin [ ( Re β ) x 3 + ψ ] .
( co f F ) ( f F ) = m 2 [ 1 ( g r ) 2 + 1 ( p r ) 2 ] × [ co ( g r ) 2 + 1 ( p r ) 2 ] , f = J m ( g r ) g r J m ( g r ) , F = K m ( p r ) p r K m ( p r ) ,
g = g ( β ) = ( k 2 co β 2 ) 1 / 2 , p = p ( β ) = ( β 2 k 2 cl ) 1 / 2 .
K m ( z ) ( π 2 z ) 1 / 2 exp ( z ) ( 1 + 4 m 2 1 4 z + ) , | z | , | arg ( z ) | < 3 π 2 ,
u D ( L ) , F u a L u + b u ;
( T u , u ) γ T γ L + max [ b ( 1 a ) 1 , a | γ L | + b ] ,
q ( u , w ) = u w ¯ d x
C 0 ( Ω ) for Δ D , H 1 ( Ω ) = { f L 2 ( Ω ) : f L 2 ( n ) } for Δ N ,
ϕ C 0 ( n ) , ( u ) ϕ d x = u ( ϕ ) d x .
D D = { f H 0 1 ( Ω ) : f C down to Ω , f | Ω = 0 } , D N = { f H 0 1 ( Ω ) : f C down to Ω , ( / n ) f | Ω = 0 } ,
u , w D ( T ) such that ( T 3 u , w ) ( u , T 3 w ) .
u = [ u 1 , 0 ] , w = [ w 1 , 0 ] , u 1 ( x ) = x 2 χ ( x ) , w 1 ( x ) = x 1 χ ( x ) ,
( T 3 u , w ) 0
( u , T 3 w ) = 0 .
( T 1 h , h ) + ( T 2 h , h ) + ( T 3 h , h ) = β 2 .
Re [ ( T 3 h , h ) ] = Re ( β 2 ) ( T 1 h , h ) ( T 2 h , h ) Re ( β 2 ) k 2 co .
T * w = [ ( 2 + k 2 ¯ ) w 1 x 2 [ 1 ( ¯ x 1 w 2 ¯ x 2 w 1 ) ] , ( 2 + k 2 ¯ ) w 2 + x 1 [ 1 ( ¯ x 1 w 2 ¯ x 2 w 1 ) ] ] .
( T * β 2 ) e = 0
β 2 ( e , h ) = ( T * e , h ) = ( e , T h ) = ( e , β 2 h ) = β 2 ¯ ( e , h ) .
0 = ( e , h ) = ( e 2 , h 1 ) 2 + ( e 1 , h 2 ) 2 = 2 e 1 h ¯ 2 e 2 h ¯ 1 d x = 2 ( e × h ¯ ) x 3 d x .
e ( h , β ) = [ e 1 , e 2 ] = [ β h 2 1 β x 2 ( h 1 x 1 + h 2 x 2 ) , β h 1 + 1 β x 1 ( h 1 x 1 + h 2 x 2 ) ] ν k .
( e , h ) = ν k 1 [ β ( 1 h , h ) β 1 ( 1 G h , h ) ] ,
( G u , u ) = ( x 1 ( u 1 x 1 + u 2 x 2 ) , u 1 ) 2 + ( x 2 ( u 1 x 1 + u 2 x 2 ) , u 2 ) 2 = u 2 2 0 .
( 1 G u , u ) = ( G u , 1 u ) = 1 / 2 u 2 2 + ( u , 2 u ) 2 ,
( e , h ) = ν k 1 [ β 1 / 2 h 2 + β 1 1 / 2 h 2 2 β 1 ( h , 2 h ) 2 ] .
H = h ( x 1 , x 2 ) exp [ i ( Re β ) x 3 ω t ] exp [ ( Im β ) x 3 ] , h = [ h 1 ( x 1 , x 2 ) , h 2 ( x 1 , x 2 ) ] .
h j ( x 1 , x 2 ) = | h j ( x 1 , x 2 ) | exp [ i ϕ j ( x 1 , x 2 ) ] , j = 1 , 2 .
| h | = ( h 1 h ¯ 1 + h 2 h ¯ 2 ) 1 / 2 , Γ = Γ ( | h 1 | sign [ Re ( h 1 ) ] , | h 2 | sign [ Re ( h 2 ) ] ) , Ψ 1 = ϕ 1 ( x 1 , x 2 ) φ , Ψ 2 = ϕ 2 ( x 1 , x 2 ) φ ,
V = k r ( co cl ) 1 / 2 , D = ( co cl ) ( 2 cl ) 1 , Z = ( β 2 k 2 cl ) [ k 2 ( co cl ) ] 1 ,
s ( x ) { 1 ( | x | / r ) α for | x | r 0 for | x | > r .
(h,h¯)=0{Re(h)=Im(h)
(Re(h),Im(h))=0}.

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