Abstract

In contrast with scalar modes, the vector modes of two-dimensional waveguides with separable refractive-index profiles cannot be calculated by a simple product ansatz for the fields. It is shown that a separation of the dominant field component into two factors, with each depending on only one of the two Cartesian coordinates, is accurately possible up to first order in the small relative refractive-index difference Δ for symmetric separable profiles. This allows us to calculate the modal propagation constants by decomposing the problem into two independent TE and TM planar wave equations. Birefringence is thereby included to an accuracy of first order in Δ. The minor field component can be separated to first order in Δ, if one of the two factors constituting the major field component is the fundamental mode of the parabolic profile, but cannot be separated in general. The longitudinal field components are separable to their lowest order Δ1/2 for all separable profiles.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ankiewicz, “Ray theory of graded non-circular optical fibres,” Opt. Quantum Electron. 11, 197–203 (1979).
    [CrossRef]
  2. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  3. A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, “Coupling of two parallel multimode parabolic index waveguides: an exact analysis in the weakly guiding approximation,” Opt. Commun. 30, 166–169 (1979).
    [CrossRef]
  4. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
    [CrossRef]
  5. H. Renner, “Far-from-core field of bound modes on non-circular weakly guiding optical waveguides,” J. Mod. Opt. 39, 1–7 (1992).
    [CrossRef]
  6. H. Renner, “Asymptotic coupling coefficients of well-separated single-mode optical waveguides,” J. Mod. Opt. 39, 907–915 (1992).
    [CrossRef]
  7. P. Yeh, H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. 19, 2848–2855 (1980).
    [CrossRef] [PubMed]
  8. A. Kumar, K. Thyagarajan, A. K. Ghatak, “Analysis of rectangular-core dielectric waveguides: an accurate perturbation analysis,” Opt. Lett. 8, 63–65 (1983).
    [CrossRef] [PubMed]
  9. K. Hayata, K. Miura, M. Koshiba, “Full vectorial finite element formalism for lossy anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 37, 875–883 (1989).
    [CrossRef]
  10. F. A. Fernandez, Y. Lu, “A variational finite element formulation for dielectric waveguides in terms of transverse magnetic fields,” IEEE Trans. Magn. 27, 3864–3867 (1991).
    [CrossRef]
  11. Y. Lu, F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 41, 1215–1223 (1993).
    [CrossRef]
  12. G. D. Maxwell, B. J. Ainslie, “Demonstration of directly written directional coupler using UV-induced photosensitivity in a planar silica waveguide,” Electron. Lett. 31, 95–96 (1995).
    [CrossRef]
  13. P. Baldi, P. Aschieri, S. Nouh, M. De Micheli, D. B. Ostrowski, D. Delacourt, M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
    [CrossRef]
  14. C. F. Kaue, R. R. Krchnayet, “Benzocyclobutene optical waveguides,” IEEE Photonics Technol. Lett. 7, 535–537 (1995).
    [CrossRef]
  15. I. Faderl, P. Labeye, P. Gidon, P. Mottier, “Integration of an electrooptic polymer in an integrated optics circuit on silicon,” J. Lightwave Technol. 13, 2020–2026 (1995).
    [CrossRef]
  16. E. R. Hedin, F. J. Goetz, “Experimental studies of electro-optic polymer modulators and waveguides,” Appl. Opt. 34, 1554–1561 (1995).
    [CrossRef] [PubMed]
  17. A. L. Sala, M. G. Mirkov, B. G. Bagley, R. T. Deick, “Derivation of dimensional and material requirements for propagation and processing of temporal optical solitons in planar geometry channel waveguides,” Appl. Opt. 36, 7846–7852 (1997).
    [CrossRef]
  18. P. K. Sinha, “Coupling characteristics of 4×4 elliptical core optical waveguide coupler,” Fiber Integr. Opt. 15, 125–133 (1996).
    [CrossRef]
  19. C. K. Madsen, J. H. Zhao, “A general planar waveguide autoregressive optical filter,” J. Lightwave Technol. 14, 437–447 (1996).
    [CrossRef]
  20. D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, R. C. Tiberio, “Waveguide-coupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 1.6-nm free spectral range,” Opt. Lett. 22, 1244–1246 (1997).
    [CrossRef] [PubMed]
  21. T. Isoshima, K. Tada, “Local normal-mode analysis of second-harmonic generation in a periodic waveguide,” IEEE J. Quantum Electron. 33, 164–175 (1997).
    [CrossRef]
  22. A. Kumar, R. K. Varshney, K. Thyagarajan, “Birefringence calculations in elliptical-core optical fibres,” Electron. Lett. 20, 112–113 (1984).
    [CrossRef]
  23. R. K. Varshney, A. Kumar, “Effect of depressed inner cladding on the polarization characteristics of elliptical-core fibers,” Opt. Lett. 9, 522–524 (1984).
    [CrossRef] [PubMed]
  24. A. Kumar, A. N. Kaul, A. K. Ghatak, “Prediction of coupling length in a rectangular-core directional coupler: an accurate analysis,” Opt. Lett. 10, 86–88 (1985).
    [CrossRef] [PubMed]
  25. R. K. Varshney, A. Kumar, “Birefringence calculations in side-tunnel optical fibers: a rectangular-core waveguide model,” Opt. Lett. 11, 45–47 (1986).
    [CrossRef] [PubMed]
  26. I. Yokohama, K. Okamoto, J. Noda, “Analysis of fiber-optic polarizing beam splitters consisting of fused-taper couplers,” J. Lightwave Technol. LT-4, 1352–1359 (1986).
    [CrossRef]
  27. A. Kumar, U. K. Das, R. K. Varshney, I. C. Goyal, “Design of a mode filter consisting of two dual-mode highly elliptical core fibers,” J. Lightwave Technol. 8, 34–38 (1990).
    [CrossRef]
  28. K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 692–700 (1996).
    [CrossRef]
  29. A. W. Snyder, W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
    [CrossRef]
  30. Ph. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  31. M. Svalgaard, C. V. Poulsen, A. Bjarklev, O. Poulsen, “Direct UV writing of buried singlemode channel waveguides in Ge-doped silica films,” Electron. Lett. 30, 1401–1403 (1994).
    [CrossRef]
  32. D. Johlen, P. Klose, E. Brinkmeyer, “UV-written directional couplers in silica on silicon,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 279–280.

1997 (3)

1996 (3)

P. K. Sinha, “Coupling characteristics of 4×4 elliptical core optical waveguide coupler,” Fiber Integr. Opt. 15, 125–133 (1996).
[CrossRef]

C. K. Madsen, J. H. Zhao, “A general planar waveguide autoregressive optical filter,” J. Lightwave Technol. 14, 437–447 (1996).
[CrossRef]

K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 692–700 (1996).
[CrossRef]

1995 (5)

G. D. Maxwell, B. J. Ainslie, “Demonstration of directly written directional coupler using UV-induced photosensitivity in a planar silica waveguide,” Electron. Lett. 31, 95–96 (1995).
[CrossRef]

P. Baldi, P. Aschieri, S. Nouh, M. De Micheli, D. B. Ostrowski, D. Delacourt, M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

C. F. Kaue, R. R. Krchnayet, “Benzocyclobutene optical waveguides,” IEEE Photonics Technol. Lett. 7, 535–537 (1995).
[CrossRef]

I. Faderl, P. Labeye, P. Gidon, P. Mottier, “Integration of an electrooptic polymer in an integrated optics circuit on silicon,” J. Lightwave Technol. 13, 2020–2026 (1995).
[CrossRef]

E. R. Hedin, F. J. Goetz, “Experimental studies of electro-optic polymer modulators and waveguides,” Appl. Opt. 34, 1554–1561 (1995).
[CrossRef] [PubMed]

1994 (1)

M. Svalgaard, C. V. Poulsen, A. Bjarklev, O. Poulsen, “Direct UV writing of buried singlemode channel waveguides in Ge-doped silica films,” Electron. Lett. 30, 1401–1403 (1994).
[CrossRef]

1993 (1)

Y. Lu, F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 41, 1215–1223 (1993).
[CrossRef]

1992 (2)

H. Renner, “Far-from-core field of bound modes on non-circular weakly guiding optical waveguides,” J. Mod. Opt. 39, 1–7 (1992).
[CrossRef]

H. Renner, “Asymptotic coupling coefficients of well-separated single-mode optical waveguides,” J. Mod. Opt. 39, 907–915 (1992).
[CrossRef]

1991 (1)

F. A. Fernandez, Y. Lu, “A variational finite element formulation for dielectric waveguides in terms of transverse magnetic fields,” IEEE Trans. Magn. 27, 3864–3867 (1991).
[CrossRef]

1990 (1)

A. Kumar, U. K. Das, R. K. Varshney, I. C. Goyal, “Design of a mode filter consisting of two dual-mode highly elliptical core fibers,” J. Lightwave Technol. 8, 34–38 (1990).
[CrossRef]

1989 (1)

K. Hayata, K. Miura, M. Koshiba, “Full vectorial finite element formalism for lossy anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 37, 875–883 (1989).
[CrossRef]

1986 (2)

I. Yokohama, K. Okamoto, J. Noda, “Analysis of fiber-optic polarizing beam splitters consisting of fused-taper couplers,” J. Lightwave Technol. LT-4, 1352–1359 (1986).
[CrossRef]

R. K. Varshney, A. Kumar, “Birefringence calculations in side-tunnel optical fibers: a rectangular-core waveguide model,” Opt. Lett. 11, 45–47 (1986).
[CrossRef] [PubMed]

1985 (1)

1984 (2)

R. K. Varshney, A. Kumar, “Effect of depressed inner cladding on the polarization characteristics of elliptical-core fibers,” Opt. Lett. 9, 522–524 (1984).
[CrossRef] [PubMed]

A. Kumar, R. K. Varshney, K. Thyagarajan, “Birefringence calculations in elliptical-core optical fibres,” Electron. Lett. 20, 112–113 (1984).
[CrossRef]

1983 (1)

1980 (1)

1979 (2)

A. Ankiewicz, “Ray theory of graded non-circular optical fibres,” Opt. Quantum Electron. 11, 197–203 (1979).
[CrossRef]

A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, “Coupling of two parallel multimode parabolic index waveguides: an exact analysis in the weakly guiding approximation,” Opt. Commun. 30, 166–169 (1979).
[CrossRef]

1978 (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]

Ainslie, B. J.

G. D. Maxwell, B. J. Ainslie, “Demonstration of directly written directional coupler using UV-induced photosensitivity in a planar silica waveguide,” Electron. Lett. 31, 95–96 (1995).
[CrossRef]

Ankiewicz, A.

A. Ankiewicz, “Ray theory of graded non-circular optical fibres,” Opt. Quantum Electron. 11, 197–203 (1979).
[CrossRef]

Aschieri, P.

P. Baldi, P. Aschieri, S. Nouh, M. De Micheli, D. B. Ostrowski, D. Delacourt, M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Bagley, B. G.

Baldi, P.

P. Baldi, P. Aschieri, S. Nouh, M. De Micheli, D. B. Ostrowski, D. Delacourt, M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Bjarklev, A.

M. Svalgaard, C. V. Poulsen, A. Bjarklev, O. Poulsen, “Direct UV writing of buried singlemode channel waveguides in Ge-doped silica films,” Electron. Lett. 30, 1401–1403 (1994).
[CrossRef]

Brinkmeyer, E.

D. Johlen, P. Klose, E. Brinkmeyer, “UV-written directional couplers in silica on silicon,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 279–280.

Chiang, K. S.

K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 692–700 (1996).
[CrossRef]

Das, U. K.

A. Kumar, U. K. Das, R. K. Varshney, I. C. Goyal, “Design of a mode filter consisting of two dual-mode highly elliptical core fibers,” J. Lightwave Technol. 8, 34–38 (1990).
[CrossRef]

De Micheli, M.

P. Baldi, P. Aschieri, S. Nouh, M. De Micheli, D. B. Ostrowski, D. Delacourt, M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Deick, R. T.

Delacourt, D.

P. Baldi, P. Aschieri, S. Nouh, M. De Micheli, D. B. Ostrowski, D. Delacourt, M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Faderl, I.

I. Faderl, P. Labeye, P. Gidon, P. Mottier, “Integration of an electrooptic polymer in an integrated optics circuit on silicon,” J. Lightwave Technol. 13, 2020–2026 (1995).
[CrossRef]

Fernandez, F. A.

Y. Lu, F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 41, 1215–1223 (1993).
[CrossRef]

F. A. Fernandez, Y. Lu, “A variational finite element formulation for dielectric waveguides in terms of transverse magnetic fields,” IEEE Trans. Magn. 27, 3864–3867 (1991).
[CrossRef]

Feshbach, H.

Ph. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Ghatak, A. K.

Gidon, P.

I. Faderl, P. Labeye, P. Gidon, P. Mottier, “Integration of an electrooptic polymer in an integrated optics circuit on silicon,” J. Lightwave Technol. 13, 2020–2026 (1995).
[CrossRef]

Goetz, F. J.

Goyal, I. C.

A. Kumar, U. K. Das, R. K. Varshney, I. C. Goyal, “Design of a mode filter consisting of two dual-mode highly elliptical core fibers,” J. Lightwave Technol. 8, 34–38 (1990).
[CrossRef]

Hagness, S. C.

Hayata, K.

K. Hayata, K. Miura, M. Koshiba, “Full vectorial finite element formalism for lossy anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 37, 875–883 (1989).
[CrossRef]

Hedin, E. R.

Ho, S. T.

Isoshima, T.

T. Isoshima, K. Tada, “Local normal-mode analysis of second-harmonic generation in a periodic waveguide,” IEEE J. Quantum Electron. 33, 164–175 (1997).
[CrossRef]

Johlen, D.

D. Johlen, P. Klose, E. Brinkmeyer, “UV-written directional couplers in silica on silicon,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 279–280.

Kaue, C. F.

C. F. Kaue, R. R. Krchnayet, “Benzocyclobutene optical waveguides,” IEEE Photonics Technol. Lett. 7, 535–537 (1995).
[CrossRef]

Kaul, A. N.

Khular, E.

A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, “Coupling of two parallel multimode parabolic index waveguides: an exact analysis in the weakly guiding approximation,” Opt. Commun. 30, 166–169 (1979).
[CrossRef]

Klose, P.

D. Johlen, P. Klose, E. Brinkmeyer, “UV-written directional couplers in silica on silicon,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 279–280.

Koshiba, M.

K. Hayata, K. Miura, M. Koshiba, “Full vectorial finite element formalism for lossy anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 37, 875–883 (1989).
[CrossRef]

Krchnayet, R. R.

C. F. Kaue, R. R. Krchnayet, “Benzocyclobutene optical waveguides,” IEEE Photonics Technol. Lett. 7, 535–537 (1995).
[CrossRef]

Kumar, A.

Labeye, P.

I. Faderl, P. Labeye, P. Gidon, P. Mottier, “Integration of an electrooptic polymer in an integrated optics circuit on silicon,” J. Lightwave Technol. 13, 2020–2026 (1995).
[CrossRef]

Love, J. D.

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

Lu, Y.

Y. Lu, F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 41, 1215–1223 (1993).
[CrossRef]

F. A. Fernandez, Y. Lu, “A variational finite element formulation for dielectric waveguides in terms of transverse magnetic fields,” IEEE Trans. Magn. 27, 3864–3867 (1991).
[CrossRef]

Madsen, C. K.

C. K. Madsen, J. H. Zhao, “A general planar waveguide autoregressive optical filter,” J. Lightwave Technol. 14, 437–447 (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]

Maxwell, G. D.

G. D. Maxwell, B. J. Ainslie, “Demonstration of directly written directional coupler using UV-induced photosensitivity in a planar silica waveguide,” Electron. Lett. 31, 95–96 (1995).
[CrossRef]

Mirkov, M. G.

Miura, K.

K. Hayata, K. Miura, M. Koshiba, “Full vectorial finite element formalism for lossy anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 37, 875–883 (1989).
[CrossRef]

Morse, Ph. M.

Ph. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Mottier, P.

I. Faderl, P. Labeye, P. Gidon, P. Mottier, “Integration of an electrooptic polymer in an integrated optics circuit on silicon,” J. Lightwave Technol. 13, 2020–2026 (1995).
[CrossRef]

Noda, J.

I. Yokohama, K. Okamoto, J. Noda, “Analysis of fiber-optic polarizing beam splitters consisting of fused-taper couplers,” J. Lightwave Technol. LT-4, 1352–1359 (1986).
[CrossRef]

Nouh, S.

P. Baldi, P. Aschieri, S. Nouh, M. De Micheli, D. B. Ostrowski, D. Delacourt, M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Okamoto, K.

I. Yokohama, K. Okamoto, J. Noda, “Analysis of fiber-optic polarizing beam splitters consisting of fused-taper couplers,” J. Lightwave Technol. LT-4, 1352–1359 (1986).
[CrossRef]

Ostrowski, D. B.

P. Baldi, P. Aschieri, S. Nouh, M. De Micheli, D. B. Ostrowski, D. Delacourt, M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Papuchon, M.

P. Baldi, P. Aschieri, S. Nouh, M. De Micheli, D. B. Ostrowski, D. Delacourt, M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Poulsen, C. V.

M. Svalgaard, C. V. Poulsen, A. Bjarklev, O. Poulsen, “Direct UV writing of buried singlemode channel waveguides in Ge-doped silica films,” Electron. Lett. 30, 1401–1403 (1994).
[CrossRef]

Poulsen, O.

M. Svalgaard, C. V. Poulsen, A. Bjarklev, O. Poulsen, “Direct UV writing of buried singlemode channel waveguides in Ge-doped silica films,” Electron. Lett. 30, 1401–1403 (1994).
[CrossRef]

Rafizadeh, D.

Renner, H.

H. Renner, “Far-from-core field of bound modes on non-circular weakly guiding optical waveguides,” J. Mod. Opt. 39, 1–7 (1992).
[CrossRef]

H. Renner, “Asymptotic coupling coefficients of well-separated single-mode optical waveguides,” J. Mod. Opt. 39, 907–915 (1992).
[CrossRef]

Sala, A. L.

Sharma, A.

A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, “Coupling of two parallel multimode parabolic index waveguides: an exact analysis in the weakly guiding approximation,” Opt. Commun. 30, 166–169 (1979).
[CrossRef]

Sinha, P. K.

P. K. Sinha, “Coupling characteristics of 4×4 elliptical core optical waveguide coupler,” Fiber Integr. Opt. 15, 125–133 (1996).
[CrossRef]

Snyder, A. W.

A. W. Snyder, W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
[CrossRef]

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

Stair, K. A.

Svalgaard, M.

M. Svalgaard, C. V. Poulsen, A. Bjarklev, O. Poulsen, “Direct UV writing of buried singlemode channel waveguides in Ge-doped silica films,” Electron. Lett. 30, 1401–1403 (1994).
[CrossRef]

Tada, K.

T. Isoshima, K. Tada, “Local normal-mode analysis of second-harmonic generation in a periodic waveguide,” IEEE J. Quantum Electron. 33, 164–175 (1997).
[CrossRef]

Taflove, A.

Taylor, H. F.

Thyagarajan, K.

A. Kumar, R. K. Varshney, K. Thyagarajan, “Birefringence calculations in elliptical-core optical fibres,” Electron. Lett. 20, 112–113 (1984).
[CrossRef]

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

A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, “Coupling of two parallel multimode parabolic index waveguides: an exact analysis in the weakly guiding approximation,” Opt. Commun. 30, 166–169 (1979).
[CrossRef]

Tiberio, R. C.

Varshney, R. K.

A. Kumar, U. K. Das, R. K. Varshney, I. C. Goyal, “Design of a mode filter consisting of two dual-mode highly elliptical core fibers,” J. Lightwave Technol. 8, 34–38 (1990).
[CrossRef]

R. K. Varshney, A. Kumar, “Birefringence calculations in side-tunnel optical fibers: a rectangular-core waveguide model,” Opt. Lett. 11, 45–47 (1986).
[CrossRef] [PubMed]

A. Kumar, R. K. Varshney, K. Thyagarajan, “Birefringence calculations in elliptical-core optical fibres,” Electron. Lett. 20, 112–113 (1984).
[CrossRef]

R. K. Varshney, A. Kumar, “Effect of depressed inner cladding on the polarization characteristics of elliptical-core fibers,” Opt. Lett. 9, 522–524 (1984).
[CrossRef] [PubMed]

Yeh, P.

Yokohama, I.

I. Yokohama, K. Okamoto, J. Noda, “Analysis of fiber-optic polarizing beam splitters consisting of fused-taper couplers,” J. Lightwave Technol. LT-4, 1352–1359 (1986).
[CrossRef]

Young, W. R.

Zhang, J. P.

Zhao, J. H.

C. K. Madsen, J. H. Zhao, “A general planar waveguide autoregressive optical filter,” J. Lightwave Technol. 14, 437–447 (1996).
[CrossRef]

Appl. Opt. (3)

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. (3)

A. Kumar, R. K. Varshney, K. Thyagarajan, “Birefringence calculations in elliptical-core optical fibres,” Electron. Lett. 20, 112–113 (1984).
[CrossRef]

G. D. Maxwell, B. J. Ainslie, “Demonstration of directly written directional coupler using UV-induced photosensitivity in a planar silica waveguide,” Electron. Lett. 31, 95–96 (1995).
[CrossRef]

M. Svalgaard, C. V. Poulsen, A. Bjarklev, O. Poulsen, “Direct UV writing of buried singlemode channel waveguides in Ge-doped silica films,” Electron. Lett. 30, 1401–1403 (1994).
[CrossRef]

Fiber Integr. Opt. (1)

P. K. Sinha, “Coupling characteristics of 4×4 elliptical core optical waveguide coupler,” Fiber Integr. Opt. 15, 125–133 (1996).
[CrossRef]

IEEE J. Quantum Electron. (2)

P. Baldi, P. Aschieri, S. Nouh, M. De Micheli, D. B. Ostrowski, D. Delacourt, M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

T. Isoshima, K. Tada, “Local normal-mode analysis of second-harmonic generation in a periodic waveguide,” IEEE J. Quantum Electron. 33, 164–175 (1997).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

C. F. Kaue, R. R. Krchnayet, “Benzocyclobutene optical waveguides,” IEEE Photonics Technol. Lett. 7, 535–537 (1995).
[CrossRef]

IEEE Trans. Magn. (1)

F. A. Fernandez, Y. Lu, “A variational finite element formulation for dielectric waveguides in terms of transverse magnetic fields,” IEEE Trans. Magn. 27, 3864–3867 (1991).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

Y. Lu, F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 41, 1215–1223 (1993).
[CrossRef]

K. Hayata, K. Miura, M. Koshiba, “Full vectorial finite element formalism for lossy anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 37, 875–883 (1989).
[CrossRef]

K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 692–700 (1996).
[CrossRef]

J. Lightwave Technol. (4)

I. Yokohama, K. Okamoto, J. Noda, “Analysis of fiber-optic polarizing beam splitters consisting of fused-taper couplers,” J. Lightwave Technol. LT-4, 1352–1359 (1986).
[CrossRef]

A. Kumar, U. K. Das, R. K. Varshney, I. C. Goyal, “Design of a mode filter consisting of two dual-mode highly elliptical core fibers,” J. Lightwave Technol. 8, 34–38 (1990).
[CrossRef]

C. K. Madsen, J. H. Zhao, “A general planar waveguide autoregressive optical filter,” J. Lightwave Technol. 14, 437–447 (1996).
[CrossRef]

I. Faderl, P. Labeye, P. Gidon, P. Mottier, “Integration of an electrooptic polymer in an integrated optics circuit on silicon,” J. Lightwave Technol. 13, 2020–2026 (1995).
[CrossRef]

J. Mod. Opt. (2)

H. Renner, “Far-from-core field of bound modes on non-circular weakly guiding optical waveguides,” J. Mod. Opt. 39, 1–7 (1992).
[CrossRef]

H. Renner, “Asymptotic coupling coefficients of well-separated single-mode optical waveguides,” J. Mod. Opt. 39, 907–915 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, “Coupling of two parallel multimode parabolic index waveguides: an exact analysis in the weakly guiding approximation,” Opt. Commun. 30, 166–169 (1979).
[CrossRef]

Opt. Lett. (5)

Opt. Quantum Electron. (1)

A. Ankiewicz, “Ray theory of graded non-circular optical fibres,” Opt. Quantum Electron. 11, 197–203 (1979).
[CrossRef]

Other (3)

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

Ph. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

D. Johlen, P. Klose, E. Brinkmeyer, “UV-written directional couplers in silica on silicon,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 279–280.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Rectangular waveguide profile composed of two planar three-layer waveguide profiles.

Fig. 2
Fig. 2

Refractive-index profile of a channel waveguide fabricated by direct UV-laser beam writing in a germanosilicate film.

Fig. 3
Fig. 3

Relative orientation of the major field component xˆex(x, y) and the refractive-index distribution of Fig. 2 according to the situations in (a) Subsection 3.A and (b) Subsection 3.B.

Fig. 4
Fig. 4

Field functions of Subsection 3.A, with g(x) being a step function and h(y) being a parabola: (a) G(x), (b) Fx(x)/2Δ, (c) dG(x)/x, (d) H(y), (e) Fy(y)/2Δ, and (f) dH(y)/y.

Fig. 5
Fig. 5

Field functions of Subsection 3.B, with g(x) being a parabola and h(y) being a step function: (a) G(x), (b) Fx(x)/2Δ, (c) dG(x)/x, (d) H(y), (e) Fy(y)/2Δ, and (f) dH(y)/y.

Equations (100)

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

n2(x, y)=n12{1-2Δ[g(x)+h(y)]}.
g(x)=1forx>a0for|x|<a1forx<-a,
h(y)=(n12-n32)/(n12-n22)fory>b0for|y|<b1fory<-b.
[t2+k2n2(x, y)-β2]et(x, y)
=-t[et(x, y)t ln n2(x, y)],
et(x, y)eN=xˆG(x)H(y)+yˆF(x, y)+O(Δ2).
xˆ:t2G(x)H(y)+[k2n2(x, y)-β2]G(x)H(y)
=2ΔddxG(x) dg(x)dxH(y)+F(x, y)xdh(y)dy+O(Δ2),
yˆ:t2F(x, y)+[k2n2(x, y)-β2]F(x, y)
=2ΔG(x) dg(x)dxdH(y)dy+yF(x, y) dh(y)dy+O(Δ2)
xˆ:t2G(x)H(y)+[k2n2(x, y)-β2]G(x)H(y)
=2Δ ddxG(x) dg(x)dxH(y)+O(Δ2),
yˆ:t2F(x, y)+[k2n2(x, y)-β2]F(x, y)
=2ΔG(x) dg(x)dxdH(y)dy+O(Δ2)
d2G(x)dx2+{k2n12[1-2Δg(x)]-βx2}G(x)
=2Δ ddxG(x) dg(x)dx+O(Δ2),
d2H(y)dy2+{k2n12[1-2Δh(y)]-βy2}H(y)=O(Δ2),
β2=βx2+βy2-k2n12
2Δ ddxG(x) dg(x)dx
2Δ ddxn12n12[1-2Δg(x)]G(x) dg(x)dx+O(Δ2),
d2G(x)dx2+[k2nx2(x)-βx2]G(x)
=-ddxG(x)nx2(x)dnx2(x)dx+O(Δ2).
1Fx(x)d2Fx(x)dx2+1Fy(y)d2Fy(y)dy2
+k2n12{1-2Δ[g(x)+h(y)]}-β2
=2Δ G(x)Fx(x)dg(x)dx1Fy(y)dH(y)dy+O(Δ2).
Fy(y)=A2Δ dH(y)dy
d2Fx(x)dx2+{k2n12[1-2Δg(x)]-γx2}Fx(x)
=2ΔAG(x) dg(x)dx,
d2Fy(y)dy2+{k2n12[1-2Δh(y)]-γy2}Fy(y)=0,
β2=γx2+γy2-k2n12
(βy2-γy2) Hy(y)dy-k2n122Δ h(y)dyH(y)=0.
H(y)exp-k2n122Δβy2-γy2h(y).
h(y)=y2/b2,
H(y)=exp-Vyy22b2
βy2=k2n12-Vy/b2,
Fy(y)=-A2Δ Vyyb2exp-Vyy22b2
γy2=k2n12-3Vy/b2.
β2=βx2-Vy/b2,
γx2=βx2+2Vy/b2>βx2.
Fx(x)=2ΔA-+G(x, x)G(x) dg(x)dxdx,
d2G(x, x)dx2+{k2n12[1-2Δg(x)]-γx2}G(x, x)
=δ(x-x).
G(x, x)=1W(ϕ1, ϕ2)ϕ1(x)ϕ2(x)forxxϕ1(x)ϕ2(x)forxx.
W(ϕ1, ϕ2)=dϕ1(x)dxϕ2(x)-ϕ1(x) dϕ2(x)dx
g(x)=θ1-Sx̲a=0for|x|<aθ>0for|x|>a,
n2(ζ, ξ)=n221-2Δξ1-Sξξ0+n122ΔζSξξ0exp-ζ2ζ02n221-2Δξ1-Sξξ0+n122ΔζSξξ01-ζ2ζ02,
n2(ζ, ξ)n121-2Δξ1-Sξξ0-2Δζ ζ2ζ02=n121-2Δθ-θSξξ0+ζ2ζ02,
Fx(x)=2ΔθA[-G(-a)G(x,-a)+G(a)G(x, a)].
Fx(x)=-2ΔθG(a)sin(κ¯a)A[κ¯cos(κ¯a)+σ¯sin(κ¯a)]×sin(κ¯x)/sin(κ¯a)for|x|a(x/|x|)exp[-σ¯(|x|-a)]for|x|a,
G(x)=(1-2Δθ)cos(κx)/cos(κa)for|x|aexp[-σ(|x|-a)]for|x|a,
(1+2Δθ)σ=κ tan(κa),
G(x)=cos(κx)/cos(κa)for|x|aexp[-σ(|x|-a)]for|x|a,
σ=κ tan(κa).
A2=θG(a)b sin(κ¯a)exp(Vy/2)Vy[κ ¯cos(κ¯a)+σ ¯sin(κ¯a)]
Fx(x)=A2ΔG(x) dg(x)dx,
d2Fx(x)dx2+{k2n12[1-2Δg(x)]-γx2}Fx(x)=0,
d2Fy(y)dy2+{k2n12[1-2Δh(y)]-γy2}Fy(y)
=2ΔAdH(y)dy,
d2G(x)dx2+[k2nx2(x)-βx2]G(x)
=-dg(x)dx-12 d2g(x)dx2G(x)dx+d3g(x)dx3+(βx2-γx2) dg(x)dxG(x).
2 d2g(x)dx2G(x)dx+d3g(x)dx3+(βx2-γx2) dg(x)dxG(x)
=0.
G(x)d2g(x)dx2-1/2
×exp-βx2-γx22xd2g(t)dt2-1 dg(t)dtdt.
G(x)=G0(x)=exp-Vxx22a2
g(x)=x2/a2,
βx2=k2n12-Vx/a2,
Fx(x)=A2Δ 2xa2exp-Vxx22a2
γx2=k2n12-3Vx/a2
G(x)=exp-Vxx22a21-2ΔVx,
βx2=k2n12-Vxa21+2ΔVx.
β2=βy2-Vxa21+2ΔVx,
Fy(y)=2ΔA-+G(y, y) dH(y)dydy,
h(y)=θ1-Syb=0for|y|<bθ>0for|y|>b,
H(y)=cos(κy)/cos(κb)for|y|bexp[-σ(|y|-b)]for|y|b,
Fy(y)=2ΔAb2(γy2-βy2)b2κ sin(κy)cos(κb)-Vy2 sin(κ¯y)κ¯cos(κ¯b)+σ¯sin(κ¯b)
for|y|b,
Fy(y)=y|y|2Δ exp(σb)Ab2(γy2-βy2)b2σ exp(-σ|y|)-Vy2 sin(κ¯b)κ¯cos(κ¯b)+σ¯sin(κ¯b)exp(-σ¯|y|)
for|y|b.
A2=a exp(Vx/2)2b2(γy2-βy2)b2κ sin(κb)cos(κb)-Vy2 sin(κ¯b)κ ¯cos(κ¯b)+σ ¯sin(κ¯b).
g(x)+h(y)=(x/a)2+(y/b)2,
G(x)H(y)=exp-Vxx22a21-2ΔVx+Vyy22b2,
β2=βx2+βy2-k2n12=k2n12-Vxa21+2ΔVx-Vyb2,
Fx(x)Fy(y)=4Δxya(a+b)exp-Vxx22a2+Vyy22b2,
ez=iβn2(x, y)[n2(x, y)t·et+et·tn2(x, y)],
ez(x, y)eNiβdG(x)dxH(y)+O(Δ1).
ezeN=iβn2(x, y)n2(x, y)G(x)dxH(y)+Fx(x) Fy(y)dy-2Δn12G(x) dg(x)dxH(y)+Fx(x)Fy(y) dh(y)dy+O(Δ2),
=iβG(x)dxH(y)+Fx(x) Fy(y)dy-2ΔG0(x) dg(x)dxH(y)+O(Δ2),
dG(x)dx=-κ sin(κx)/cos(κa)for|x|aσx/|x|exp[-σ(|x|-a)]for|x|a,
dG(x)dx=-Vxxa2exp-Vxx22a2,
[t2+k2n2(x, y)-β2]ht(x, y)
=[t×ht(x, y)]×t ln n2(x, y).
xˆ:t2F(x, y)+[k2n2(x, y)-β2]F(x, y)
=2Δ dG(x)dxH(y) dh(y)dy+O(Δ2),
yˆ:t2G(x)H(y)+[k2n2(x, y)-β2]G(x)H(y)
=-2Δ dG(x)dxH(y) dg(x)dx+O(Δ2).
hz=-ik0μ01/2zˆ·t×et,
hz(x, y)hNik0μ01/2G(x) dH(y)dy+O(Δ1)
dH(y)dy=-Vyyb2exp-Vyy22b2,
dH(y)dy=-κ sin(κy)/cos(κb)for|y|bσy/|y|exp[-σ(|y|-b)]for|y|b

Metrics