Abstract

Tapered optical waveguides are of increasing importance for use as mode-size transformers in integrated optics. We present a new Fourier integral interpretation of the first-order coupled-local-mode theory of the modeconversion loss in adiabatically tapered waveguides that provides a new degree of perspective on taper behavior. On the basis of this interpretation, we introduce a technique for synthesizing tapers with desired length-versusloss characteristics. Since this technique involves a Fourier integral, we are able to take advantage of the existing substantial body of literature on filter design. We demonstrate this new technique by analyzing and synthesizing two examples of taper structures, a tapered directional coupler and a mode-size controller, and introduce a class of (near-) optimal tapers from this synthetic approach. We also prove the adiabaticity theorem in tapered waveguides, starting from Maxwell’s equations, on which our first-order perturbation approach is premised.

© 1992 Optical Society of America

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References

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  1. A. W. Snyder, “Coupling of modes in a tapered dielectric cylinder,” IEEE Trans. Microwave Theory Tech. MTT-18, 383–392 (1970).
    [CrossRef]
  2. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 106–111.
  3. A. F. Milton, W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
    [CrossRef]
  4. D. Marcuse, “Radiation losses of step-tapered channel waveguides,” Appl. Opt. 19, 3676–3681 (1980).
    [CrossRef] [PubMed]
  5. B. Hermansson, D. Yevick, P. Danielsen, “Propagating beam analysis of multimode waveguide tapers,” IEEE J. Quantum Electron. QE-19, 1246–1251 (1983).
    [CrossRef]
  6. J. D. Love, W. M. Henry, “Quantifying loss minimization in single-mode fiber tapers,” Electron. Lett. 22, 912–914 (1986).
    [CrossRef]
  7. D. R. Rowland, Y. Chen, A. W. Snyder, “Tapered mismatched couplers,” IEEE J. Lightwave Technol. 9, 567–570 (1991).
    [CrossRef]
  8. A. W. Snyder, “Mode propagation in a nonuniform cylindrical medium,” IEEE Trans. Microwave Theory Tech. MTT-19, 402–403 (1971).
    [CrossRef]
  9. A similar Fourier-transform relationship between the reflection and the local impedance in a transmission-line taper, for which two TEM modes propagating to the opposite directions are the only modes to consider, was found by Bolinder [F. Bolinder, “Fourier transforms in the theory of inhomogeneous transmission lines,” Proc. IRE 38, 1354 (1950)]. By controlling the reflection characteristic, it became possible to design various transmission-line tapers [R. E. Collin, “The optimum transmission line matching section,” Proc. IRE 44, 539–548 (1956)]. Unger used this design principle of controlled reflection for hollow metallic waveguide tapers, which support TE and TM modes and in which the forward coupling is dominant over the backward coupling (reflection) [H.-G. Unger, “Circular waveguide taper of improved design,” Bell Syst. Tech. J. 37, 899–912 (1958)]. It seems proper to consider the forward coupling rather than the backward coupling.
    [CrossRef]
  10. E. A. J. Marcatili, “Dielectric tapers with curved axes and no loss,” IEEE J. Quantum Electron. QE-21, 307–314 (1985).
    [CrossRef]
  11. R. Weder, “Dielectric three-dimensional electromagnetic tapers with no loss,” IEEE J. Quantum Electron. 24, 775–779 (1988).
    [CrossRef]
  12. D. Bertilone, J. Love, C. Pask, “Splicing of optical waveguides with lossless graded-index tapers,” Opt. Quantum Electron. 20, 501–514 (1988).
    [CrossRef]
  13. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).
  14. See Ref. 2, p. 110. The general expression for βj(s) imaginary is given in Ref. 2, p. 108.
  15. When a properly designed taper is already long enough that the sum of the first-order coupling losses is less than, say, 5%, we can safely ignore all those higher-order processes whose overall probability is likely to be less than 2% even when coupled amplitudes build up constructively over the length of the taper for these processes.
  16. W. H. Louisell, “Analysis of the single tapered mode coupler,” Bell Syst. Tech. J. 34, 853–870 (1955).
  17. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, London, 1975).
  18. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978), pp. 143–148.
  19. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  20. J. D. Love, “Spot size, adiabaticity and diffraction in tapered fibers,” Electron. Lett. 23, 993–994 (1987).
    [CrossRef]
  21. J. S. Cook, “Tapered velocity couplers,” Bell Syst. Tech. J. 34, 807–822 (1955).
  22. A. G. Fox, “Wave coupling by warped normal modes,” Bell Syst. Tech. J. 34, 823–852 (1955).
  23. Y. Shani, C. H. Henry, R. C. Kistler, R. F. Kazarinov, K. J. Orlowsky, “Integrated optic adiabatic devices on silicon,” IEEE J. Quantum Electron. 27, 556–566 (1991).
    [CrossRef]
  24. Y. Silberberg, P. Perlmutter, J. E. Baran, “Digital optical switch,” Appl. Phys. Lett. 51, 1230–1232 (1987).
    [CrossRef]
  25. D. Marcuse, “Mode conversion in optical fibers with monotonically increasing core radius,” IEEE J. Lightwave Technol. LT-5, 125–133 (1987).
    [CrossRef]
  26. A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1958), Vol. II.
  27. K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, New York, 1977).
    [CrossRef]
  28. G. H. Song, S.-Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).
    [CrossRef]

1991 (2)

D. R. Rowland, Y. Chen, A. W. Snyder, “Tapered mismatched couplers,” IEEE J. Lightwave Technol. 9, 567–570 (1991).
[CrossRef]

Y. Shani, C. H. Henry, R. C. Kistler, R. F. Kazarinov, K. J. Orlowsky, “Integrated optic adiabatic devices on silicon,” IEEE J. Quantum Electron. 27, 556–566 (1991).
[CrossRef]

1988 (2)

R. Weder, “Dielectric three-dimensional electromagnetic tapers with no loss,” IEEE J. Quantum Electron. 24, 775–779 (1988).
[CrossRef]

D. Bertilone, J. Love, C. Pask, “Splicing of optical waveguides with lossless graded-index tapers,” Opt. Quantum Electron. 20, 501–514 (1988).
[CrossRef]

1987 (3)

J. D. Love, “Spot size, adiabaticity and diffraction in tapered fibers,” Electron. Lett. 23, 993–994 (1987).
[CrossRef]

Y. Silberberg, P. Perlmutter, J. E. Baran, “Digital optical switch,” Appl. Phys. Lett. 51, 1230–1232 (1987).
[CrossRef]

D. Marcuse, “Mode conversion in optical fibers with monotonically increasing core radius,” IEEE J. Lightwave Technol. LT-5, 125–133 (1987).
[CrossRef]

1986 (1)

J. D. Love, W. M. Henry, “Quantifying loss minimization in single-mode fiber tapers,” Electron. Lett. 22, 912–914 (1986).
[CrossRef]

1985 (2)

1983 (1)

B. Hermansson, D. Yevick, P. Danielsen, “Propagating beam analysis of multimode waveguide tapers,” IEEE J. Quantum Electron. QE-19, 1246–1251 (1983).
[CrossRef]

1980 (1)

1978 (1)

1977 (1)

A. F. Milton, W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
[CrossRef]

1971 (1)

A. W. Snyder, “Mode propagation in a nonuniform cylindrical medium,” IEEE Trans. Microwave Theory Tech. MTT-19, 402–403 (1971).
[CrossRef]

1970 (1)

A. W. Snyder, “Coupling of modes in a tapered dielectric cylinder,” IEEE Trans. Microwave Theory Tech. MTT-18, 383–392 (1970).
[CrossRef]

1955 (3)

J. S. Cook, “Tapered velocity couplers,” Bell Syst. Tech. J. 34, 807–822 (1955).

A. G. Fox, “Wave coupling by warped normal modes,” Bell Syst. Tech. J. 34, 823–852 (1955).

W. H. Louisell, “Analysis of the single tapered mode coupler,” Bell Syst. Tech. J. 34, 853–870 (1955).

1950 (1)

A similar Fourier-transform relationship between the reflection and the local impedance in a transmission-line taper, for which two TEM modes propagating to the opposite directions are the only modes to consider, was found by Bolinder [F. Bolinder, “Fourier transforms in the theory of inhomogeneous transmission lines,” Proc. IRE 38, 1354 (1950)]. By controlling the reflection characteristic, it became possible to design various transmission-line tapers [R. E. Collin, “The optimum transmission line matching section,” Proc. IRE 44, 539–548 (1956)]. Unger used this design principle of controlled reflection for hollow metallic waveguide tapers, which support TE and TM modes and in which the forward coupling is dominant over the backward coupling (reflection) [H.-G. Unger, “Circular waveguide taper of improved design,” Bell Syst. Tech. J. 37, 899–912 (1958)]. It seems proper to consider the forward coupling rather than the backward coupling.
[CrossRef]

Baran, J. E.

Y. Silberberg, P. Perlmutter, J. E. Baran, “Digital optical switch,” Appl. Phys. Lett. 51, 1230–1232 (1987).
[CrossRef]

Bertilone, D.

D. Bertilone, J. Love, C. Pask, “Splicing of optical waveguides with lossless graded-index tapers,” Opt. Quantum Electron. 20, 501–514 (1988).
[CrossRef]

Bolinder, F.

A similar Fourier-transform relationship between the reflection and the local impedance in a transmission-line taper, for which two TEM modes propagating to the opposite directions are the only modes to consider, was found by Bolinder [F. Bolinder, “Fourier transforms in the theory of inhomogeneous transmission lines,” Proc. IRE 38, 1354 (1950)]. By controlling the reflection characteristic, it became possible to design various transmission-line tapers [R. E. Collin, “The optimum transmission line matching section,” Proc. IRE 44, 539–548 (1956)]. Unger used this design principle of controlled reflection for hollow metallic waveguide tapers, which support TE and TM modes and in which the forward coupling is dominant over the backward coupling (reflection) [H.-G. Unger, “Circular waveguide taper of improved design,” Bell Syst. Tech. J. 37, 899–912 (1958)]. It seems proper to consider the forward coupling rather than the backward coupling.
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978), pp. 143–148.

Burns, W. K.

A. F. Milton, W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
[CrossRef]

Chadan, K.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, New York, 1977).
[CrossRef]

Chen, Y.

D. R. Rowland, Y. Chen, A. W. Snyder, “Tapered mismatched couplers,” IEEE J. Lightwave Technol. 9, 567–570 (1991).
[CrossRef]

Cook, J. S.

J. S. Cook, “Tapered velocity couplers,” Bell Syst. Tech. J. 34, 807–822 (1955).

Danielsen, P.

B. Hermansson, D. Yevick, P. Danielsen, “Propagating beam analysis of multimode waveguide tapers,” IEEE J. Quantum Electron. QE-19, 1246–1251 (1983).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Fox, A. G.

A. G. Fox, “Wave coupling by warped normal modes,” Bell Syst. Tech. J. 34, 823–852 (1955).

Henry, C. H.

Y. Shani, C. H. Henry, R. C. Kistler, R. F. Kazarinov, K. J. Orlowsky, “Integrated optic adiabatic devices on silicon,” IEEE J. Quantum Electron. 27, 556–566 (1991).
[CrossRef]

Henry, W. M.

J. D. Love, W. M. Henry, “Quantifying loss minimization in single-mode fiber tapers,” Electron. Lett. 22, 912–914 (1986).
[CrossRef]

Hermansson, B.

B. Hermansson, D. Yevick, P. Danielsen, “Propagating beam analysis of multimode waveguide tapers,” IEEE J. Quantum Electron. QE-19, 1246–1251 (1983).
[CrossRef]

Kazarinov, R. F.

Y. Shani, C. H. Henry, R. C. Kistler, R. F. Kazarinov, K. J. Orlowsky, “Integrated optic adiabatic devices on silicon,” IEEE J. Quantum Electron. 27, 556–566 (1991).
[CrossRef]

Kistler, R. C.

Y. Shani, C. H. Henry, R. C. Kistler, R. F. Kazarinov, K. J. Orlowsky, “Integrated optic adiabatic devices on silicon,” IEEE J. Quantum Electron. 27, 556–566 (1991).
[CrossRef]

Louisell, W. H.

W. H. Louisell, “Analysis of the single tapered mode coupler,” Bell Syst. Tech. J. 34, 853–870 (1955).

Love, J.

D. Bertilone, J. Love, C. Pask, “Splicing of optical waveguides with lossless graded-index tapers,” Opt. Quantum Electron. 20, 501–514 (1988).
[CrossRef]

Love, J. D.

J. D. Love, “Spot size, adiabaticity and diffraction in tapered fibers,” Electron. Lett. 23, 993–994 (1987).
[CrossRef]

J. D. Love, W. M. Henry, “Quantifying loss minimization in single-mode fiber tapers,” Electron. Lett. 22, 912–914 (1986).
[CrossRef]

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

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric tapers with curved axes and no loss,” IEEE J. Quantum Electron. QE-21, 307–314 (1985).
[CrossRef]

Marcuse, D.

D. Marcuse, “Mode conversion in optical fibers with monotonically increasing core radius,” IEEE J. Lightwave Technol. LT-5, 125–133 (1987).
[CrossRef]

D. Marcuse, “Radiation losses of step-tapered channel waveguides,” Appl. Opt. 19, 3676–3681 (1980).
[CrossRef] [PubMed]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 106–111.

Messiah, A.

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1958), Vol. II.

Milton, A. F.

A. F. Milton, W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, London, 1975).

Orlowsky, K. J.

Y. Shani, C. H. Henry, R. C. Kistler, R. F. Kazarinov, K. J. Orlowsky, “Integrated optic adiabatic devices on silicon,” IEEE J. Quantum Electron. 27, 556–566 (1991).
[CrossRef]

Pask, C.

D. Bertilone, J. Love, C. Pask, “Splicing of optical waveguides with lossless graded-index tapers,” Opt. Quantum Electron. 20, 501–514 (1988).
[CrossRef]

Perlmutter, P.

Y. Silberberg, P. Perlmutter, J. E. Baran, “Digital optical switch,” Appl. Phys. Lett. 51, 1230–1232 (1987).
[CrossRef]

Rowland, D. R.

D. R. Rowland, Y. Chen, A. W. Snyder, “Tapered mismatched couplers,” IEEE J. Lightwave Technol. 9, 567–570 (1991).
[CrossRef]

Sabatier, P. C.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, New York, 1977).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, London, 1975).

Shani, Y.

Y. Shani, C. H. Henry, R. C. Kistler, R. F. Kazarinov, K. J. Orlowsky, “Integrated optic adiabatic devices on silicon,” IEEE J. Quantum Electron. 27, 556–566 (1991).
[CrossRef]

Shin, S.-Y.

Silberberg, Y.

Y. Silberberg, P. Perlmutter, J. E. Baran, “Digital optical switch,” Appl. Phys. Lett. 51, 1230–1232 (1987).
[CrossRef]

Snyder, A. W.

D. R. Rowland, Y. Chen, A. W. Snyder, “Tapered mismatched couplers,” IEEE J. Lightwave Technol. 9, 567–570 (1991).
[CrossRef]

A. W. Snyder, “Mode propagation in a nonuniform cylindrical medium,” IEEE Trans. Microwave Theory Tech. MTT-19, 402–403 (1971).
[CrossRef]

A. W. Snyder, “Coupling of modes in a tapered dielectric cylinder,” IEEE Trans. Microwave Theory Tech. MTT-18, 383–392 (1970).
[CrossRef]

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

Song, G. H.

Weder, R.

R. Weder, “Dielectric three-dimensional electromagnetic tapers with no loss,” IEEE J. Quantum Electron. 24, 775–779 (1988).
[CrossRef]

Yevick, D.

B. Hermansson, D. Yevick, P. Danielsen, “Propagating beam analysis of multimode waveguide tapers,” IEEE J. Quantum Electron. QE-19, 1246–1251 (1983).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

Y. Silberberg, P. Perlmutter, J. E. Baran, “Digital optical switch,” Appl. Phys. Lett. 51, 1230–1232 (1987).
[CrossRef]

Bell Syst. Tech. J. (3)

W. H. Louisell, “Analysis of the single tapered mode coupler,” Bell Syst. Tech. J. 34, 853–870 (1955).

J. S. Cook, “Tapered velocity couplers,” Bell Syst. Tech. J. 34, 807–822 (1955).

A. G. Fox, “Wave coupling by warped normal modes,” Bell Syst. Tech. J. 34, 823–852 (1955).

Electron. Lett. (2)

J. D. Love, “Spot size, adiabaticity and diffraction in tapered fibers,” Electron. Lett. 23, 993–994 (1987).
[CrossRef]

J. D. Love, W. M. Henry, “Quantifying loss minimization in single-mode fiber tapers,” Electron. Lett. 22, 912–914 (1986).
[CrossRef]

IEEE J. Lightwave Technol. (2)

D. R. Rowland, Y. Chen, A. W. Snyder, “Tapered mismatched couplers,” IEEE J. Lightwave Technol. 9, 567–570 (1991).
[CrossRef]

D. Marcuse, “Mode conversion in optical fibers with monotonically increasing core radius,” IEEE J. Lightwave Technol. LT-5, 125–133 (1987).
[CrossRef]

IEEE J. Quantum Electron. (5)

B. Hermansson, D. Yevick, P. Danielsen, “Propagating beam analysis of multimode waveguide tapers,” IEEE J. Quantum Electron. QE-19, 1246–1251 (1983).
[CrossRef]

E. A. J. Marcatili, “Dielectric tapers with curved axes and no loss,” IEEE J. Quantum Electron. QE-21, 307–314 (1985).
[CrossRef]

R. Weder, “Dielectric three-dimensional electromagnetic tapers with no loss,” IEEE J. Quantum Electron. 24, 775–779 (1988).
[CrossRef]

Y. Shani, C. H. Henry, R. C. Kistler, R. F. Kazarinov, K. J. Orlowsky, “Integrated optic adiabatic devices on silicon,” IEEE J. Quantum Electron. 27, 556–566 (1991).
[CrossRef]

A. F. Milton, W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

A. W. Snyder, “Coupling of modes in a tapered dielectric cylinder,” IEEE Trans. Microwave Theory Tech. MTT-18, 383–392 (1970).
[CrossRef]

A. W. Snyder, “Mode propagation in a nonuniform cylindrical medium,” IEEE Trans. Microwave Theory Tech. MTT-19, 402–403 (1971).
[CrossRef]

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

Opt. Quantum Electron. (1)

D. Bertilone, J. Love, C. Pask, “Splicing of optical waveguides with lossless graded-index tapers,” Opt. Quantum Electron. 20, 501–514 (1988).
[CrossRef]

Proc. IRE (1)

A similar Fourier-transform relationship between the reflection and the local impedance in a transmission-line taper, for which two TEM modes propagating to the opposite directions are the only modes to consider, was found by Bolinder [F. Bolinder, “Fourier transforms in the theory of inhomogeneous transmission lines,” Proc. IRE 38, 1354 (1950)]. By controlling the reflection characteristic, it became possible to design various transmission-line tapers [R. E. Collin, “The optimum transmission line matching section,” Proc. IRE 44, 539–548 (1956)]. Unger used this design principle of controlled reflection for hollow metallic waveguide tapers, which support TE and TM modes and in which the forward coupling is dominant over the backward coupling (reflection) [H.-G. Unger, “Circular waveguide taper of improved design,” Bell Syst. Tech. J. 37, 899–912 (1958)]. It seems proper to consider the forward coupling rather than the backward coupling.
[CrossRef]

Other (8)

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

See Ref. 2, p. 110. The general expression for βj(s) imaginary is given in Ref. 2, p. 108.

When a properly designed taper is already long enough that the sum of the first-order coupling losses is less than, say, 5%, we can safely ignore all those higher-order processes whose overall probability is likely to be less than 2% even when coupled amplitudes build up constructively over the length of the taper for these processes.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, London, 1975).

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978), pp. 143–148.

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1958), Vol. II.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, New York, 1977).
[CrossRef]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 106–111.

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Figures (16)

Fig. 1
Fig. 1

Directional mismatch coupler of a linear-shape design.

Fig. 2
Fig. 2

Normalized propagation constants of the several local eigenmodes for the directional mismatch coupler of Fig. 1. The curve at the top represents that of the fundamental mode. The horizontal axis represents steps used in the calculation that correspond to uniform increments of the taper variable.

Fig. 3
Fig. 3

Directional taper coupler with the linear taper shape. (a) Taper T functions: a, T21(u); b, T31(u); c, T41(u); d, T51(u); e, T61(u). (b) Total conversion loss and the loss components from individual local-mode couplings. The total loss and the loss from 1 → 2 shown as a solid curve are virtually identical. The loss from the 1 → 3 conversion is shown as a broken curve. Only the main lobe of the loss from 1 → 4 is shown at the bottom left-hand corner.

Fig. 4
Fig. 4

Directional taper coupler with rectangular T21(u). (a) Taper characteristic functions for several orders of mode conversion: a, T21(u) = constant; b, T31(u); c, T41(u); d, T51(u); e, T61(u). (b) Total loss and the loss components from individual local-mode couplings. The total loss is represented by a solid curve immediately above the loss from the 1 → 2 conversion, i.e., |21(L)|2. Losses from 1 → 3 and 1 → 5 appear throughout to the taper length of 4 mm, while that of 1 → 4 appears only in the bottom left-hand corner of the plot. The higher the order of conversion, the shorter the period of fluctuation.

Fig. 5
Fig. 5

Directional taper coupler with the Hamming design. Curves for mode conversions from the fundamental mode (1). (a) Taper T functions: a, T21(u); b, T31(u); c, T41(u); d, T51(u); e, T61(u). (b) Total conversion loss and the loss components from individual local-mode couplings. The higher the order of mode conversion, the shorter the period of fluctuation. The solid curve represents the total loss, and the dotted curve below the solid curve represents the loss from the 1 → 2 conversion. The latter, |21(L)|2, has particularly small first two sidelobes, which are even below the loss curves from higher-order mode conversions such as the 1 → 3 and the 1 → 5 conversions, |31(L)|2 and |51(L)|2.

Fig. 6
Fig. 6

Directional taper coupler with the Hamming design. Curves for mode conversions from the first-higher mode 2. (a) Taper T functions: a, T12(u); b, T32(u); c, T42(u); d, T52(u); e, T62(u). (b) Total conversion loss and the loss components from individual local-mode couplings. The higher the order of mode conversion, the shorter the period of fluctuation. The solid curve represents the total loss, and the dotted curve below the solid curve represents the loss from the 2 → 1 conversion. The latter, |12(L)|2, has particularly small first two sidelobes, which are even below the loss curves from higher-order mode conversions such as the 1 → 3 and the 1 → 5 conversions, |32(L)|2 and |52(L)|2.

Fig. 7
Fig. 7

Directional taper couplers of (a) the rectangular function design and (b) the Hamming function design for T21(u).

Fig. 8
Fig. 8

Wedge-type linear taper of length L. The left-hand graph represents the refractive-index profile at z = 0 of a wedge taper on top of the silver-exchanged glass waveguide.

Fig. 9
Fig. 9

Field amplitude profiles of the local eigenmodes at each section of the wedge taper shown in Fig. 8. The steps represent equal increments in the taper variable, which is the thickness of the high-index wedge.

Fig. 10
Fig. 10

Normalized propagation constants of the several local eigenmodes in the wedge-shaped taper. As in Fig. 2, the curve at the top represents the propagation constant of the fundamental mode, and the horizontal axis represents the steps used in the calculation.

Fig. 11
Fig. 11

Wedge-shaped taper with the linear-shape design. (a) Taper characteristic functions for several orders of mode conversion: a, T21(u); b, T31(u); c, T41(u); d, T51(u); e, T61(u). (b) Total loss and the loss components from individual local-mode couplings: a, total loss; b, loss from the 1 → 2 conversion, |21(L)|2; c, |31(L)|2; d, |41(L)|2; e, |51(L)|2; f, |61(L)|2.

Fig. 12
Fig. 12

Wedge-shaped taper with T21(u) rectangular. (a) Taper characteristic functions for several orders of mode conversion: a, T21(u) = constant; b, T31(u); c, T41(u); d, T51(u); e, T61(u). Total loss and the loss components from individual local-mode couplings. The solid curve represents the total conversion loss. The higher the order of mode conversion, the shorter the period of fluctuation.

Fig. 13
Fig. 13

Wedge-shaped taper synthesized with the Hamming function T21(u). (a) Taper T functions for a, T21(u); b, T31(u); c, T41(u); d, T51(u); e, T61(u). (b) Total loss and the loss components from individual local-mode couplings: a, total loss; b, |21(L)|2; c, |31(L)|2; d, |41(L)|2;e, |51(L)|2; f, |61(L)12.

Fig. 14
Fig. 14

Wedge-shaped taper synthesized with the Hamming function T31(u). (a) Taper T functions for a, T21(u); b, T31(u); c, T41(u); d, T51(u); e, T61(u). (b) Total loss and the loss components from individual local-mode couplings: a, total loss; b, |21(L)|2; c, |31(L)|2; d, |41(L)|2; e, |51(L)|2.

Fig. 15
Fig. 15

Local thicknesses of the curved wedge with three different designs. The shape of the solid curve gives a rectangular T21(u), while the shapes of the dotted and the dashed curves give the Hamming function for T21(u) and T31(u), respectively. The local thickness of the wedge is plotted in the normalized scale s.

Fig. 16
Fig. 16

Mode-conversion losses obtained by different methods for the first example of Fig. 1 with the result given in Fig. 5. The solid curve was obtained by the first-order coupled-local-mode theory of relation (1). The dotted curve was obtained by a beam propagation method (BPM) analysis. The dashed curve was obtained by taking only one term in relation (1) for the dominant local-mode coupling.

Equations (16)

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C ^ j i ( s ) ω ɛ 0 4 [ β j ( s ) - β i ( s ) ] × - - d n 2 ( x , y , s L ) d s E j * ( s ) · E i ( s ) d x d y ,
T j i ( u ) [ C ^ j i ( s ) β j ( s ) - β i ( s ) ] s = f j i - 1 ( u )
u = f j i ( s ) 0 s [ β j ( s ) - β 1 ( s ) ] d s .
T 21 ( u ) = ( const . ) Π [ u / f 21 ( 1 ) + 1 / 2 ] , Π ( t ) { 1 t < 1 / 2 0 t > 1 / 2 .
T 21 ( u ) 0.54 - 0.46 cos [ 2 π u / f 21 ( 1 ) ] .
T j 1 ( u ) ~ 1 - cos [ 2 π u / f j 1 ( 1 ) ] .
E ( r , t ) = E ( r ) exp ( j ω t ) ,             H ( r , t ) = H ( r ) exp ( j ω t ) .
[ D z O O D z ] [ E H ] = [ D t j ω μ - j ω ɛ D t ] [ E H ] ,
E = [ E x E y E z ] ,             H [ H x H y H z ] , D z [ 0 z 0 - z 0 0 0 0 0 ] ,             D t [ 0 0 y 0 0 - x - y x 0 ] ;
z ( Q F ) = G F ,
F [ E x E y H x H y ] ,             Q [ 0 1 0 0 - 1 0 0 0 0 0 0 1 0 0 - 1 0 ] , G [ 0 0 j ω μ x x j ω μ x y 0 0 j ω μ y x j ω μ y y - j ω ɛ x x - j ω ɛ x y 0 0 - j ω ɛ y x - j ω ɛ y y 0 0 ] + [ y j ω μ x z - x j ω μ u z - j ω ɛ x z y - j ω ɛ y z - x ] [ [ - j ω ɛ z x - j ω ɛ z y - y x ] j ω ɛ z z [ - y x j ω μ z x j ω μ z y ] - j ω μ z z ] .
j z F = H F ,             H j Q T G ,
F ( x , y , z ) = exp [ - j 0 z H ( x , y , z ) d z ] F ( x , y , 0 ) .
F ( x , y , z ) U ( s ) F ( x , y , 0 ) , s z / L , j L d U ( s ) d s = H ( s ) U ( s ) , H ( s ) H ( x , y , s L ) ,
P j ( s ) = A ( s ) P j ( 0 ) A ( s )
j = d U ( A ) d s = ( L A H A - j A d A d s ) U ( A ) .

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