Abstract

The guided modes supported by two parallel, identical planar dielectric waveguides are investigated exactly by a quasi-optical technique and approximately by using the improved coupled-mode theory. The wave numbers obtained by the two methods are expanded into asymptotic series in terms of a suitable small parameter. By comparing the asymptotic series obtained by the two methods term by term, it is shown that the improved coupled-mode theory is asymptotically exact in the limit of weak coupling and not so in the strong-coupling regime. The merits and the defects of the improved coupled-mode theory are discussed.

© 1988 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).
  2. D. Marcuse, “The coupling of degenerate modes in two parallel, dielectric waveguides,” Bell Syst. Tech. J. 50, 1791–1816 (1971).
  3. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  4. A. Yariv, “Coupled mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [Crossref]
  5. H. F. Taylor, A. Yariv, “Guided wave optics,” Proc. IEEE 62, 1044–1060 (1974).
    [Crossref]
  6. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1979), Chap. 2.
  7. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).
  8. Y. Suematsu, K. Kishino, “Coupling coefficients in strongly coupled dielectric waveguides,” Radio Sci. 12, 587–592 (1977).
    [Crossref]
  9. E. Marom, O. G. Ramer, S. Ruschin, “Relation between normal mode and coupled mode analyses of parallel waveguides,” IEEE J. Quantum Electron. QE-20, 1311–1319 (1984).
    [Crossref]
  10. A. Hardy, W. Streifer, “Analysis of phased-array diode lasers,” Opt. Lett. 10, 335–337 (1985).
    [Crossref] [PubMed]
  11. A. Hardy, W. Streifer, “Coupled mode theory of parallel waveguides,” IEEE J. Lightwave Technol. LT-3, 1135–1146 (1985).
    [Crossref]
  12. A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90–99 (1986).
    [Crossref]
  13. A. Hardy, W. Streifer, “Coupled mode solutions of multi-waveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
    [Crossref]
  14. W. Streifer, M. Osiński, A. Hardy, “Reformulation of the coupled-mode theory of multiwaveguide systems,” IEEE J. Lightwave Technol. LT-5, 1–4 (1987).
    [Crossref]
  15. A. Hardy, S. Shakir, W. Streifer, “Coupled-mode equations for two weakly guiding single-mode fibers,” Opt. Lett. 11, 324–326 (1986).
    [Crossref] [PubMed]
  16. A. Hardy, W. Streifer, M. Osiński, “Coupled-mode equations for multimode waveguide systems in isotropic or anisotropic media,” Opt. Lett. 11, 742–744 (1986).
    [Crossref] [PubMed]
  17. H. A. Haus, W. P. Huang, S. Kawakami, N. A. Whitaker, “Coupled-mode theory of optical waveguides,” IEEE J. Lightwave Technol. LT-5, 16–23 (1987).
    [Crossref]
  18. E. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
    [Crossref]
  19. E. A. J. Marcatili, L. L. Buhl, R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” Appl. Phys. Lett. 49, 1692–1693 (1986).
    [Crossref]
  20. A. W. Snyder, A. Ankiewicz, “Fibre couplers composed of unequal cores,” Electron. Lett. 22, 1237–1238 (1986);erratum, Electron. Lett. 23, 251 (1987).
    [Crossref]
  21. S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” IEEE J. Lightwave Technol. LT-5, 5–15 (1987).
    [Crossref]
  22. S. L. Chuang, “A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” IEEE J. Lightwave Technol. LT-5, 174–183 (1987).
    [Crossref]
  23. S. L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. QE-23, 499–509 (1987).
    [Crossref]
  24. L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguides,” IEEE J. Lightwave Technol. LT-6, 304–311 (1988).
    [Crossref]
  25. H. K. V. Lotsch, “Reflection and refraction of a beam of light at a plane interface,” J. Opt. Soc. Am. 58, 551–561 (1968).
    [Crossref]
  26. N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 73–87.
  27. H. Kogelnik, H. P. Weber, “Rays, stored energy, and power flow in dielectric waveguides,” J. Opt. Soc. Am. 64, 174–185 (1974).
    [Crossref]
  28. S. R. Seshadri, “Quasi-optics of the coupling of guided modes in two parallel, identical dielectric waveguides,” J. Opt. Soc. Am. A 4, 1030–1036 (1987).
    [Crossref]
  29. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 163–171.
  30. N. Agrawal, L. McCaughan, S. R. Seshadri, “A multiple scattering interaction analysis of intersecting waveguides,” J. Appl. Phys. 62, 2187–2193 (1987).
    [Crossref]

1988 (1)

L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguides,” IEEE J. Lightwave Technol. LT-6, 304–311 (1988).
[Crossref]

1987 (7)

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

S. L. Chuang, “A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” IEEE J. Lightwave Technol. LT-5, 174–183 (1987).
[Crossref]

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

S. R. Seshadri, “Quasi-optics of the coupling of guided modes in two parallel, identical dielectric waveguides,” J. Opt. Soc. Am. A 4, 1030–1036 (1987).
[Crossref]

N. Agrawal, L. McCaughan, S. R. Seshadri, “A multiple scattering interaction analysis of intersecting waveguides,” J. Appl. Phys. 62, 2187–2193 (1987).
[Crossref]

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

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

1986 (7)

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

E. A. J. Marcatili, L. L. Buhl, R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” Appl. Phys. Lett. 49, 1692–1693 (1986).
[Crossref]

A. W. Snyder, A. Ankiewicz, “Fibre couplers composed of unequal cores,” Electron. Lett. 22, 1237–1238 (1986);erratum, Electron. Lett. 23, 251 (1987).
[Crossref]

A. Hardy, S. Shakir, W. Streifer, “Coupled-mode equations for two weakly guiding single-mode fibers,” Opt. Lett. 11, 324–326 (1986).
[Crossref] [PubMed]

A. Hardy, W. Streifer, M. Osiński, “Coupled-mode equations for multimode waveguide systems in isotropic or anisotropic media,” Opt. Lett. 11, 742–744 (1986).
[Crossref] [PubMed]

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90–99 (1986).
[Crossref]

A. Hardy, W. Streifer, “Coupled mode solutions of multi-waveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
[Crossref]

1985 (2)

A. Hardy, W. Streifer, “Analysis of phased-array diode lasers,” Opt. Lett. 10, 335–337 (1985).
[Crossref] [PubMed]

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

1984 (1)

E. Marom, O. G. Ramer, S. Ruschin, “Relation between normal mode and coupled mode analyses of parallel waveguides,” IEEE J. Quantum Electron. QE-20, 1311–1319 (1984).
[Crossref]

1977 (1)

Y. Suematsu, K. Kishino, “Coupling coefficients in strongly coupled dielectric waveguides,” Radio Sci. 12, 587–592 (1977).
[Crossref]

1974 (2)

1973 (1)

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

1971 (1)

D. Marcuse, “The coupling of degenerate modes in two parallel, dielectric waveguides,” Bell Syst. Tech. J. 50, 1791–1816 (1971).

1969 (1)

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

1968 (1)

Agrawal, N.

N. Agrawal, L. McCaughan, S. R. Seshadri, “A multiple scattering interaction analysis of intersecting waveguides,” J. Appl. Phys. 62, 2187–2193 (1987).
[Crossref]

Alferness, R. C.

E. A. J. Marcatili, L. L. Buhl, R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” Appl. Phys. Lett. 49, 1692–1693 (1986).
[Crossref]

Ankiewicz, A.

A. W. Snyder, A. Ankiewicz, “Fibre couplers composed of unequal cores,” Electron. Lett. 22, 1237–1238 (1986);erratum, Electron. Lett. 23, 251 (1987).
[Crossref]

Buhl, L. L.

E. A. J. Marcatili, L. L. Buhl, R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” Appl. Phys. Lett. 49, 1692–1693 (1986).
[Crossref]

Burke, J. J.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 73–87.

Chuang, S. L.

L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguides,” IEEE J. Lightwave Technol. LT-6, 304–311 (1988).
[Crossref]

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

S. L. Chuang, “A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” IEEE J. Lightwave Technol. LT-5, 174–183 (1987).
[Crossref]

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

Hardy, A.

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

A. Hardy, S. Shakir, W. Streifer, “Coupled-mode equations for two weakly guiding single-mode fibers,” Opt. Lett. 11, 324–326 (1986).
[Crossref] [PubMed]

A. Hardy, W. Streifer, M. Osiński, “Coupled-mode equations for multimode waveguide systems in isotropic or anisotropic media,” Opt. Lett. 11, 742–744 (1986).
[Crossref] [PubMed]

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90–99 (1986).
[Crossref]

A. Hardy, W. Streifer, “Coupled mode solutions of multi-waveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
[Crossref]

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

A. Hardy, W. Streifer, “Analysis of phased-array diode lasers,” Opt. Lett. 10, 335–337 (1985).
[Crossref] [PubMed]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 163–171.

Haus, H. A.

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

Huang, W. P.

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

Kapany, N. S.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 73–87.

Kawakami, S.

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

Kishino, K.

Y. Suematsu, K. Kishino, “Coupling coefficients in strongly coupled dielectric waveguides,” Radio Sci. 12, 587–592 (1977).
[Crossref]

Kogelnik, H.

H. Kogelnik, H. P. Weber, “Rays, stored energy, and power flow in dielectric waveguides,” J. Opt. Soc. Am. 64, 174–185 (1974).
[Crossref]

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

Lotsch, H. K. V.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

Marcatili, E.

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

Marcatili, E. A. J.

E. A. J. Marcatili, L. L. Buhl, R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” Appl. Phys. Lett. 49, 1692–1693 (1986).
[Crossref]

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

Marcuse, D.

D. Marcuse, “The coupling of degenerate modes in two parallel, dielectric waveguides,” Bell Syst. Tech. J. 50, 1791–1816 (1971).

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

Marom, E.

E. Marom, O. G. Ramer, S. Ruschin, “Relation between normal mode and coupled mode analyses of parallel waveguides,” IEEE J. Quantum Electron. QE-20, 1311–1319 (1984).
[Crossref]

McCaughan, L.

N. Agrawal, L. McCaughan, S. R. Seshadri, “A multiple scattering interaction analysis of intersecting waveguides,” J. Appl. Phys. 62, 2187–2193 (1987).
[Crossref]

Osinski, M.

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

A. Hardy, W. Streifer, M. Osiński, “Coupled-mode equations for multimode waveguide systems in isotropic or anisotropic media,” Opt. Lett. 11, 742–744 (1986).
[Crossref] [PubMed]

Ramer, O. G.

E. Marom, O. G. Ramer, S. Ruschin, “Relation between normal mode and coupled mode analyses of parallel waveguides,” IEEE J. Quantum Electron. QE-20, 1311–1319 (1984).
[Crossref]

Ruschin, S.

E. Marom, O. G. Ramer, S. Ruschin, “Relation between normal mode and coupled mode analyses of parallel waveguides,” IEEE J. Quantum Electron. QE-20, 1311–1319 (1984).
[Crossref]

Seshadri, S. R.

N. Agrawal, L. McCaughan, S. R. Seshadri, “A multiple scattering interaction analysis of intersecting waveguides,” J. Appl. Phys. 62, 2187–2193 (1987).
[Crossref]

S. R. Seshadri, “Quasi-optics of the coupling of guided modes in two parallel, identical dielectric waveguides,” J. Opt. Soc. Am. A 4, 1030–1036 (1987).
[Crossref]

Shakir, S.

Snyder, A. W.

A. W. Snyder, A. Ankiewicz, “Fibre couplers composed of unequal cores,” Electron. Lett. 22, 1237–1238 (1986);erratum, Electron. Lett. 23, 251 (1987).
[Crossref]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

Streifer, W.

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

A. Hardy, S. Shakir, W. Streifer, “Coupled-mode equations for two weakly guiding single-mode fibers,” Opt. Lett. 11, 324–326 (1986).
[Crossref] [PubMed]

A. Hardy, W. Streifer, M. Osiński, “Coupled-mode equations for multimode waveguide systems in isotropic or anisotropic media,” Opt. Lett. 11, 742–744 (1986).
[Crossref] [PubMed]

A. Hardy, W. Streifer, “Coupled mode solutions of multi-waveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
[Crossref]

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90–99 (1986).
[Crossref]

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

A. Hardy, W. Streifer, “Analysis of phased-array diode lasers,” Opt. Lett. 10, 335–337 (1985).
[Crossref] [PubMed]

Suematsu, Y.

Y. Suematsu, K. Kishino, “Coupling coefficients in strongly coupled dielectric waveguides,” Radio Sci. 12, 587–592 (1977).
[Crossref]

Taylor, H. F.

H. F. Taylor, A. Yariv, “Guided wave optics,” Proc. IEEE 62, 1044–1060 (1974).
[Crossref]

Tsang, L.

L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguides,” IEEE J. Lightwave Technol. LT-6, 304–311 (1988).
[Crossref]

Weber, H. P.

Whitaker, N. A.

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

Yariv, A.

H. F. Taylor, A. Yariv, “Guided wave optics,” Proc. IEEE 62, 1044–1060 (1974).
[Crossref]

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

Appl. Phys. Lett. (1)

E. A. J. Marcatili, L. L. Buhl, R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” Appl. Phys. Lett. 49, 1692–1693 (1986).
[Crossref]

Bell Syst. Tech. J. (2)

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

D. Marcuse, “The coupling of degenerate modes in two parallel, dielectric waveguides,” Bell Syst. Tech. J. 50, 1791–1816 (1971).

Electron. Lett. (1)

A. W. Snyder, A. Ankiewicz, “Fibre couplers composed of unequal cores,” Electron. Lett. 22, 1237–1238 (1986);erratum, Electron. Lett. 23, 251 (1987).
[Crossref]

IEEE J. Lightwave Technol. (7)

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

S. L. Chuang, “A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” IEEE J. Lightwave Technol. LT-5, 174–183 (1987).
[Crossref]

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

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90–99 (1986).
[Crossref]

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

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

L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguides,” IEEE J. Lightwave Technol. LT-6, 304–311 (1988).
[Crossref]

IEEE J. Quantum Electron. (5)

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

A. Hardy, W. Streifer, “Coupled mode solutions of multi-waveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
[Crossref]

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

E. Marom, O. G. Ramer, S. Ruschin, “Relation between normal mode and coupled mode analyses of parallel waveguides,” IEEE J. Quantum Electron. QE-20, 1311–1319 (1984).
[Crossref]

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

J. Appl. Phys. (1)

N. Agrawal, L. McCaughan, S. R. Seshadri, “A multiple scattering interaction analysis of intersecting waveguides,” J. Appl. Phys. 62, 2187–2193 (1987).
[Crossref]

J. Opt. Soc. Am. (2)

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

Opt. Lett. (3)

Proc. IEEE (1)

H. F. Taylor, A. Yariv, “Guided wave optics,” Proc. IEEE 62, 1044–1060 (1974).
[Crossref]

Radio Sci. (1)

Y. Suematsu, K. Kishino, “Coupling coefficients in strongly coupled dielectric waveguides,” Radio Sci. 12, 587–592 (1977).
[Crossref]

Other (5)

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 163–171.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 73–87.

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

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

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Figures (2)

Fig. 1
Fig. 1

Geometry of the two parallel, identical, planar dielectric waveguides.

Fig. 2
Fig. 2

Zigzagging ray directions used in the synthesis of the guided wave supported by planar dielectric waveguide 1.

Equations (178)

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

i ω H x ( x , z ) = z E y ( x , z ) ,
i ω H z ( x , z ) = x E y ( x , z ) ,
i ω υ E y ( x , z ) = z H x ( x , z ) x H z ( x , z ) ,
υ = c for > | x | > ( b + a ) and ( b a ) > | x | > 0 = f for ( b + a ) > | x | > ( b a ) .
E y ( x , z ) is continuous ;
H z ( x , z ) is continuous .
( 2 x 2 + 2 z 2 + ω 2 υ ) E y ( x , z ) = 0 .
k = ( ω 2 f β 2 ) 1 / 2 ,
E y c ( x , z ) = D exp [ α ( x b a ) ] exp ( i β z ) for > x > ( b + a ) ,
E y f ( x , z ) = { B exp [ i k ( x b + a ) ] + C exp [ i k ( x b + a ) ] } exp ( i β z ) for ( b + a ) > x > ( b a ) ,
E y s ( x , z ) = A { exp [ α ( x b + a ) ] exp [ α ( x + b a ) ] } exp ( i β z ) for ( b a ) > x > 0 ,
α = ( β 2 ω 2 c ) 1 / 2 ,
C = B exp ( i 2 Φ ) ,
tan Φ = [ ( 1 + δ ) / ( 1 δ ) ] tan ϕ ,
tan Φ = α / k ,
δ = exp [ 2 α ( b a ) ] .
z s = 2 Φ / β
t s = 2 Φ / ω .
z c = 2 ϕ / β
t c = 2 ϕ / ω .
tan θ = β / k .
D r ( ω , β ) = 4 k a 2 ϕ 2 Φ = 2 π ( j 1 ) ,
z B = z c + z s + 4 a tan θ = D r / β ,
t B = t c + t s + 4 a tan θ / υ g z = D r / ω ,
sin 2 ( k a ϕ ) = δ sin 2 k a ( odd modes ) .
sin 2 ( k a ϕ ) = δ sin 2 k a ( even modes ) .
δ 0 = exp [ 2 α 0 ( b a ) ] .
β = β 0 + δ 0 β 1 + δ 0 2 β 2 + δ 0 3 β 3 ( even modes ) ,
D r 0 = 4 ( k 0 a ϕ 0 ) = 2 π ( j i ) ,
β 1 = ( 2 sin 2 ϕ 0 ) / z B 0 ,
z B 0 = D r 0
β ev β od = 2 δ 0 β 1 = ( δ 0 4 sin 2 ϕ 0 ) / z B 0 ,
L opt = π z B 0 / ( δ 0 4 sin 2 ϕ 0 ) ,
α = α 0 + δ 0 α 1 = α 0 + δ 0 β 1 α 0 ,
α 1 = β 1 α 0 = β 1 β 0 / α 0 .
δ = exp [ 2 ( b a ) ( α 0 + δ 0 α 1 ) ] = δ 0 + δ 0 2 δ 2 ,
δ 2 = 2 ( b a ) α 1 = ( b a ) β 1 β 0 / α 0 .
δ 0 2 cos π ( j 1 ) [ δ 2 sin 2 ϕ 0 β 1 ( cos 2 ϕ 0 ) 2 a β 0 / k 0 ] .
δ 0 2 cos π ( j 1 ) ( β 2 z B 0 2 1 4 β 1 2 z B 0 ) .
β 2 = β 1 z B 0 [ 1 2 β 1 z B 0 4 ( b a ) β 0 α 0 sin 2 ϕ 0 ( 4 a β 0 / k 0 ) cos 2 ϕ 0 ] .
α = α 0 + δ 0 α 1 + δ 0 2 α 2
= α 0 + ( δ 0 β 1 + δ 0 2 β 2 ) α 0 + ½ δ 0 2 β 1 2 α 0 ,
α 2 = β 2 α 0 + ½ β 1 2 α 0 = β 2 β 0 / α 0 + ½ β 1 2 ( α 0 2 β 0 2 ) / α 0 3 .
δ = exp [ 2 ( b a ) ( α 0 + δ 0 α 1 + δ 0 2 α 2 ) ] = δ 0 + δ 0 2 δ 2 + δ 0 3 δ 3 ,
δ 3 = 2 ( b a ) α 2 + 2 ( b a ) 2 α 1 2 = 2 ( b a ) [ β 2 β 0 / α 0 + ½ ( β 1 2 / α 0 3 ) ( α 0 2 β 0 2 ) ] + 2 ( b a ) 2 ( β 1 β 0 / α 0 ) 2 .
δ 0 3 cos π ( j 1 ) [ 2 a ( β 0 / k 0 ) β 2 cos 2 ϕ 0 + 2 a 2 ( β 0 2 / k 0 2 ) β 1 2 sin 2 ϕ 0 + a ( β 1 2 / k 0 3 ) ( k 0 2 + β 0 2 ) cos 2 ϕ 0 + δ 2 2 a ( β 0 β 1 / k 0 ) cos 2 ϕ 0 δ 3 sin 2 ϕ 0 ] .
δ 0 3 cos π ( j 1 ) ( β 3 z B 0 2 1 2 β 2 β 1 z B 0 + 1 48 β 1 3 z B 0 3 1 12 β 1 3 z B 0 ) .
β 3 = 1 z B 0 [ β 1 β 2 z B 0 + 1 24 β 1 3 z B 0 3 1 6 β 1 3 z B 0 4 a β 0 β 2 k 0 cos 2 ϕ 0 4 a 2 β 0 2 β 1 2 k 0 2 sin 2 ϕ 0 2 a ( k 0 2 + β 0 2 ) k 0 3 β 1 2 cos 2 ϕ 0 δ 2 4 a β 0 β 1 k 0 cos 2 ϕ 0 + 2 δ 3 sin 2 ϕ 0 ] .
L opt = π z B 0 / ( δ 0 4 sin 2 ϕ 0 ) [ 1 + ( β 3 / β 1 ) δ 0 2 ] .
i ω υ E x ( x , z ) = z H y ( x , z ) ,
i ω υ E z ( x , z ) = x H y ( x , z ) ,
i ω H y ( x , z ) = z E x ( x , z ) x E z ( x , z ) ,
H y ( x , z ) is continuous ,
E z ( x , z ) is continuous .
( 2 x 2 + 2 z 2 + ω 2 υ ) H y ( x , z ) = 0 .
tan ϕ = f α / c k .
w z [ E ( 1 ) × H ( 2 ) E ( 2 ) × H ( 1 ) ] · z ̂ d x = w i ω [ ( 2 ) ( 1 ) ] E ( 1 ) · E ( 2 ) d x .
( a ) = c for > x > ( b + a ) and for ( b a ) > x > ,
( a ) = f for ( b + a ) > x > ( b a ) .
( b ) = c for > x > ( b a ) and for ( b + a ) > x > ,
( b ) = f for ( b a ) > x > ( b + a ) .
= c for > | x | > ( b + a ) and for ( b a ) > | x | ,
= f for ( b + a ) > | x | ( b a ) .
Δ ( a ) = ( a ) = ( f c ) for ( b a ) > x > ( b + a )
= 0 elsewhere
Δ ( b ) = ( b ) = ( f c ) for ( b + a ) > x > ( b a )
= 0 elsewhere .
K ¯ a b = w ω 4 Δ ( b ) E y ( a ) E y ( b ) d x ,
C a b = w 2 E y ( b ) H x ( a ) d x .
( 1 ) = ( a ) ,
E y ( 1 ) = E y ( a ) ( x ) exp ( i β 0 z ) ,
H ( 1 ) = [ x ̂ H x ( a ) ( x ) + z ̂ H z ( a ) ( x ) ] exp ( i β 0 z ) ,
( 2 ) = ( b ) ,
E y ( 2 ) = E y ( b ) ( x ) exp ( i β 0 z ) ,
H ( 2 ) = [ x ̂ H x ( b ) ( x ) + z ̂ H z ( b ) ( x ) ] exp ( i β 0 z ) .
K ¯ a b = K ¯ b a .
( 2 ) = ( b ) ,
E y ( 2 ) = E y ( b ) ( x ) exp ( i β 0 z ) ,
H ( 2 ) = [ x ̂ H x ( b ) ( x ) + z ̂ H z ( b ) ( x ) ] exp ( i β 0 z ) .
C b a = C a b .
( 1 ) = ,
E y ( 1 ) = A 1 ( z ) E y ( a ) ( x ) + A 2 z E y ( b ) ( x ) ,
H x ( 1 ) = A 1 ( z ) H x ( a ) ( x ) + A 2 ( z ) H x ( b ) ( x ) ,
H z ( 1 ) = A 1 ( z ) H z ( a ) ( x ) + A 2 ( z ) H z ( b ) ( x ) .
( 2 ) = ( a ) ,
E y ( 2 ) = E y ( a ) ( x ) exp ( i β 0 z ) ,
H ( 2 ) = [ x ̂ H x ( a ) ( x ) + z ̂ H z ( a ) ( x ) ] exp ( i β 0 z ) ,
C a a d A 1 d z + C a b d A 2 d z = i ( β 0 C a a + K ¯ a a ) A 1 + i ( β 0 C a b + K ¯ b a ) A 2 .
( 2 ) = ( b ) ,
E y ( 2 ) = E y ( b ) ( x ) exp ( i β 0 z ) ,
H ( 2 ) = [ x ̂ H x ( b ) ( x ) + z ̂ H z ( b ) ( x ) ] exp ( i β 0 z ) ,
C a b d A 1 d z + C b b d A 2 d z = i ( β 0 C a b + K ¯ b a ) A 1 + i ( β 0 C b b + K ¯ b b ) A 2 .
E y = ( 2 ω / β 0 ) 1 / 2 A ( z ) u ( x ) ,
H x = ( 2 β 0 / ω ) 1 / 2 A ( z ) u ( x ) ,
H z = ( i α 0 / ω ) ( 2 ω / β 0 ) 1 / 2 A ( z ) p ( x ) ,
u ( x ) = p ( x ) = exp [ α 0 ( x a ) ] / N ;
u ( x ) = cos k 0 x / N cos k 0 a ,
p ( x ) = sin k 0 x / N sin k 0 a ;
u ( x ) = p ( x ) = exp [ α 0 ( x + a ) ] / N ,
k 0 = ( ω 2 f β 0 2 ) 1 / 2 ,
α 0 = ( β 0 2 ω 2 c ) 1 / 2 ,
N = [ w ( k 0 2 + α 0 2 ) ( 1 + α 0 a ) / k 0 2 α 0 ] 1 / 2 .
k 0 tan k 0 a = α 0 .
A n ( z ) = A n exp ( i β 0 z ) for n = 1 , 2 .
C a a = w 2 E y ( a ) H x ( a ) d x = w u ( a ) ( x ) u ( a ) ( x ) d x = 1
C b b = w 2 E y ( b ) H x ( b ) d x = w u ( b ) ( x ) u ( b ) ( x ) d x = 1 .
K ¯ a a = K ¯ b b = k 0 2 δ 0 2 [ 1 exp ( 4 α 0 a ) ] / 4 β 0 ( 1 + α 0 a ) .
A ev = A 1 + A 2 , A od = A 1 A 2 ,
d A ev d z = i ( β 0 + K ¯ a a + K ¯ b a 1 + C a b ) A ev
d A od d z = i ( β 0 + K ¯ a a K ¯ b a 1 + C a b ) A od .
β ev = β 0 + ( K ¯ a a + K ¯ b a ) / ( 1 + C a b )
β od = β 0 + ( K ¯ a a K ¯ b a ) / ( 1 C a b ) .
K ¯ a b = K ¯ b a = k 0 2 α 0 2 δ 0 / β 0 ( k 0 2 + α 0 2 ) ( 1 + a 0 a ) .
C a b = δ 0 P ,
P = k 0 2 α 0 [ k 0 2 + α 0 2 ) ( 1 + α 0 a ) ] 1 [ 2 ( b a ) + 4 α 0 / ( k 0 2 + α 0 2 ) + exp ( 2 α 0 a ) / α 0 ] .
β ev = β 0 + δ 0 β 1 + δ 0 2 β 2 + δ 0 3 β 3 .
β 0 = β 0 ,
β 1 = k 0 2 α 0 2 / β 0 ( k 0 2 + α 0 2 ) ( 1 + α 0 a ) = ( 2 sin 2 ϕ 0 ) / z B 0 ,
β 2 = ( 1 / z B 0 ) { ( k 0 / α 0 ) [ 1 exp ( 4 α 0 a ) ] 4 k 0 α 0 P / ( k 0 2 + α 0 2 ) } ,
β 3 = ( P / z B 0 ) { ( k 0 / α 0 ) [ 1 exp ( 4 α 0 a ) ] 4 k 0 α 0 P / ( k 0 2 + α 0 2 ) } .
( a ) / = c / f for ( b a ) > x > ( b + a )
= 1 elsewhere
( b ) / = c / f for ( b + a ) > x > ( b a )
= 1 elsewhere .
K ¯ a b t = w ω 4 Δ ( b ) E x ( a ) E x ( b ) d x ,
K ¯ a b z = w ω 4 Δ ( b ) E z ( a ) E z ( b ) d x ,
K ¯ a b = K ¯ a b t K ¯ a b z ,
K ¯ i i = K ¯ i i t ( c / f ) K ¯ i i z , i = a or i = b ,
K i j = K ¯ i j , i = a , j = b and i = b , j = b ,
C a b = 1 2 w E x ( b ) H y ( a ) d x .
( 1 ) = ( a ) ,
H y ( 1 ) = H y ( a ) ( x ) exp ( i β 0 z ) ,
E ( 1 ) = [ x ̂ E x ( a ) ( x ) + z ̂ E z ( a ) ( x ) ] exp ( i β 0 z ) ,
( 2 ) = ( b ) ,
H y ( 2 ) = H y ( b ) ( x ) exp ( i β 0 z ) ,
E ( 2 ) = [ x ̂ E x ( b ) ( x ) z ̂ E z ( b ) ( x ) ] exp ( i β 0 z ) .
K ¯ b a = K ¯ a b .
K b a = K a b .
( 2 ) = ( b ) ,
H y ( 2 ) = H y ( b ) ( x ) exp ( i β 0 z ) ,
E 2 = [ x ̂ E x ( b ) ( x ) + z ̂ E z ( b ) ( x ) ] exp ( i β 0 z ) .
β 0 ( C b a C a b ) = ( K ¯ b a t ) + ( K ¯ b a z ) ( K ¯ a b t + K ¯ a b z ) .
K ¯ b a t = K ¯ a b t ,
K ¯ b a z = K ¯ a b z .
C b a = C a b .
( 1 ) = ,
H y ( 1 ) = A 1 ( z ) H y ( a ) ( x ) + A 2 ( z ) H y ( b ) ( x ) ,
E x ( 1 ) = A 1 ( z ) E x ( a ) ( x ) + A 2 ( z ) E x ( b ) ( x ) ,
E z ( 1 ) = A 1 ( z ) [ ( a ) / ] E z ( a ) ( x ) + A 2 ( z ) [ ( b ) / ] E z ( b ) ( x ) .
( 2 ) = ( a ) ,
H y ( 2 ) = H y ( a ) ( x ) exp ( i β 0 z ) ,
E ( 2 ) = [ x ̂ E x ( a ) ( x ) z ̂ E z ( a ) ( x ) ] exp ( i β 0 z ) .
( 2 ) = ( b ) ,
H y ( 2 ) = H y ( b ) ( x ) exp ( i β 0 z ) ,
E 2 = [ x ̂ E x ( b ) ( x ) z ̂ E z ( b ) ( x ) ] exp ( i β 0 z ) .
H y = ( 2 ω / β 0 ) 1 / 2 A ( z ) u ( x ) ,
E x = ( 2 β 0 / ω ) 1 / 2 A ( z ) u ( x ) / υ ,
E z = ( i α 0 / ω c ) ( 2 ω / β 0 ) 1 / 2 A ( z ) p ( x ) ,
υ = c for > x > a and a > x >
= f for a > x > a .
N = [ w z B 0 ( c 2 k 0 2 + f 2 α 0 2 ) / 4 β 0 k 0 f c 2 ] 1 / 2 .
0 k 0 tan k 0 a = f α 0 .
C a a = w 2 E x ( a ) H y ( a ) d x = w 1 υ ( a ) u ( a ) ( x ) u ( a ) ( x ) d x = 1
C b b = w 2 E x ( b ) H y ( b ) d x = w 1 υ ( b ) u ( b ) ( x ) u ( b ) ( x ) d x = 1 .
K ¯ a a t = K ¯ b b t = [ k 0 ( f c ) f β 0 2 δ 0 2 / α 0 z B 0 ( c 2 k 0 2 + f 2 α 0 2 ) ] × [ 1 exp ( 4 α 0 a ) ] ,
K ¯ a a z = K ¯ b b z = [ k 0 ( f c ) f α 0 2 δ 0 2 / α 0 z B 0 ( c 2 k 0 2 + f 2 α 0 2 ) ] × [ 1 exp ( 4 α 0 a ) ] ,
K ¯ a a = K b b = K ¯ a a t c f K ¯ a a z = k 0 ( f c ) ( f β 0 2 + c α 0 2 ) α 0 z B 0 ( c 2 k 0 2 + f 2 α 0 2 ) × δ 0 2 [ 1 exp ( 4 α 0 a ) ] .
β ¯ ev = β 0 + ( K a a + K b b ) / ( 1 + C a b )
β od = β 0 + ( K a a K b a ) / ( 1 C a b ) .
K ¯ a b t = K ¯ b a t = w ( f c ) β 0 α 0 2 ( k 0 2 + α 0 2 ) f c 2 N 2 × [ exp ( 2 α 0 b ) ( f c ) + δ 0 ( f + c ) ] ,
K ¯ a b z = K ¯ b a z = w ( f c ) α 0 2 ( k 0 2 + α 0 2 ) f c 2 N 2 β 0 × [ exp ( 2 α 0 b ) ( f α 0 2 + c k 0 2 ) + δ 0 ( f α 0 2 + c k 0 2 ) ] ,
K b a = K a b = K ¯ a b = K ¯ a b t K ¯ a b z = 4 α 0 k 0 f c δ 0 / z B 0 ( c 2 k c 2 + f 2 α 0 2 ) .
C b a = C a b = δ 0 P ,
P = 4 β 0 k 0 f c z B 0 ( c 2 k 0 2 + f 2 α 0 2 ) { 2 ( b a ) + 1 α 0 exp ( 2 α 0 a ) × [ 1 + ( f 2 c 2 ) α 0 2 f c ( k 0 2 + α 0 2 ) ] + ( f + c ) 2 α 0 f c ( k 0 2 + α 0 2 ) } .
β 0 = β 0 ,
β 1 = 4 f α 0 c k 0 z B 0 ( c 2 k 0 2 + f 2 α 0 2 ) = 2 sin 2 ϕ 0 z B 0 ,
β 2 = k 0 z B 0 ( c 2 k 0 2 + f 2 α 0 2 ) { ( f c ) ( f β 0 2 + c α 0 2 ) α 0 × [ 1 exp ( 4 α 0 a ) ] 4 f c α 0 P } ,
β 3 = P β 2 .

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