Abstract

Analytic solutions are presented for the mode amplitudes in two-mode couplers which have a constant coupling coefficient and propagation coefficients that vary linearly. The solutions are obtained in terms of parabolic cylinder functions which reduce, asymptotically toward the ends of the coupler, to simpler forms in elementary functions. These lead to a simple analytic expression for the coupler efficiency, and also to quantitative criteria for the coupler length.

© 1976 Optical Society of America

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  1. D. G. Dalgoutte, R. B. Smith, G. Achutaramayya, and J. H. Harris, Appl. Opt. 14, 1860 (1975).
  2. M. G. F. Wilson and G. A. Teh, Electron. Lett. 9, 453 (1973).
  3. M. G. F. Wilson and G. A. Teh, IEEE Trans. M.T.T. MTT-23, 85 (1975). (This paper was published without labels for the axes and lines of the figures. Figure 4, showing the coupling efficiency, may be found correctly labeled in Ref. 2.)
  4. J. S. Cook, Bell Syst. Tech. J. 34, 807 (1955).
  5. A. G. Fox, Bell Syst. Tech. J. 34, 853 (1955).
  6. W. H. Louisell, Bell Syst. Tech. J. 34, 853 (1955).
  7. A. F. Milton and W. K. Burns, Appl. Opt. 14, 1207 (1975).
  8. A. L. Jones, J. Opt. Soc. Am. 55, 261 (1965).
  9. Y. S. Chen and A. Ishimaru, Proc. IEEE (Lett.) 54, 1071 (1966).
  10. E. Schloman and R. L. Joseph, J. Appl. Phys. 35, 2382 (1964); 36, 875 (1965).
  11. E. K. Kirchner, L. F. Donaghey, and F. A. Olson, J. Appl. Phys. 37, 988 (1966).
  12. S. M. Rezende and F. R. Morgenthaler, J. Appl. Phys. 40, 524 (1969).
  13. J. L. Doane, J. Appl. Phys. 45, 2748 (1974).
  14. R. B. Smith, Electron. Lett. 10, 204 (1975).
  15. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, England, 1952), p. 323.
  16. J. C. P. Miller, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (NBS, Washington, D. C., 1964), Chap. 19, p. 686.
  17. A. L. Cullen and O. J. Davies, Electron. Lett. 5, 90 (1969).
  18. J. B. Knorr, Proc. IEEE (Lett.) 59, 1624 (1971).
  19. A. W. Snyder, Proc. IEEE (Lett.) 58, 168 (1970).
  20. P. Horowitz, Appl. Phys. Lett. 23, 306 (1975).
  21. M. Matsuhara, K. O. Hill, and A. Watanabe, J. Opt. Soc. Am. 65, 804 (1975).
  22. G. H. Wannier, Physics 5, 251 (1965).
  23. C. Zener, Proc. R. Soc. Lond. A 137, 696 (1932).
  24. R. B. Smith (unpublished).
  25. H. B. Keller and J. B. Keller, J. Soc. Indust. Appl. Math. 10, 246 (1962).
  26. N. G. Van Kampen, Physica (Utr.) 35, 70 (1967).

Achutaramayya, G.

D. G. Dalgoutte, R. B. Smith, G. Achutaramayya, and J. H. Harris, Appl. Opt. 14, 1860 (1975).

Burns, W. K.

A. F. Milton and W. K. Burns, Appl. Opt. 14, 1207 (1975).

Chen, Y. S.

Y. S. Chen and A. Ishimaru, Proc. IEEE (Lett.) 54, 1071 (1966).

Cook, J. S.

J. S. Cook, Bell Syst. Tech. J. 34, 807 (1955).

Cullen, A. L.

A. L. Cullen and O. J. Davies, Electron. Lett. 5, 90 (1969).

Dalgoutte, D. G.

D. G. Dalgoutte, R. B. Smith, G. Achutaramayya, and J. H. Harris, Appl. Opt. 14, 1860 (1975).

Davies, O. J.

A. L. Cullen and O. J. Davies, Electron. Lett. 5, 90 (1969).

Doane, J. L.

J. L. Doane, J. Appl. Phys. 45, 2748 (1974).

Donaghey, L. F.

E. K. Kirchner, L. F. Donaghey, and F. A. Olson, J. Appl. Phys. 37, 988 (1966).

Fox, A. G.

A. G. Fox, Bell Syst. Tech. J. 34, 853 (1955).

Harris, J. H.

D. G. Dalgoutte, R. B. Smith, G. Achutaramayya, and J. H. Harris, Appl. Opt. 14, 1860 (1975).

Hill, K. O.

M. Matsuhara, K. O. Hill, and A. Watanabe, J. Opt. Soc. Am. 65, 804 (1975).

Horowitz, P.

P. Horowitz, Appl. Phys. Lett. 23, 306 (1975).

Ishimaru, A.

Y. S. Chen and A. Ishimaru, Proc. IEEE (Lett.) 54, 1071 (1966).

Jones, A. L.

A. L. Jones, J. Opt. Soc. Am. 55, 261 (1965).

Joseph, R. L.

E. Schloman and R. L. Joseph, J. Appl. Phys. 35, 2382 (1964); 36, 875 (1965).

Keller, H. B.

H. B. Keller and J. B. Keller, J. Soc. Indust. Appl. Math. 10, 246 (1962).

Keller, J. B.

H. B. Keller and J. B. Keller, J. Soc. Indust. Appl. Math. 10, 246 (1962).

Kirchner, E. K.

E. K. Kirchner, L. F. Donaghey, and F. A. Olson, J. Appl. Phys. 37, 988 (1966).

Knorr, J. B.

J. B. Knorr, Proc. IEEE (Lett.) 59, 1624 (1971).

Louisell, W. H.

W. H. Louisell, Bell Syst. Tech. J. 34, 853 (1955).

Matsuhara, M.

M. Matsuhara, K. O. Hill, and A. Watanabe, J. Opt. Soc. Am. 65, 804 (1975).

Miller, J. C. P.

J. C. P. Miller, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (NBS, Washington, D. C., 1964), Chap. 19, p. 686.

Milton, A. F.

A. F. Milton and W. K. Burns, Appl. Opt. 14, 1207 (1975).

Morgenthaler, F. R.

S. M. Rezende and F. R. Morgenthaler, J. Appl. Phys. 40, 524 (1969).

Olson, F. A.

E. K. Kirchner, L. F. Donaghey, and F. A. Olson, J. Appl. Phys. 37, 988 (1966).

Rezende, S. M.

S. M. Rezende and F. R. Morgenthaler, J. Appl. Phys. 40, 524 (1969).

Schloman, E.

E. Schloman and R. L. Joseph, J. Appl. Phys. 35, 2382 (1964); 36, 875 (1965).

Smith, R. B.

D. G. Dalgoutte, R. B. Smith, G. Achutaramayya, and J. H. Harris, Appl. Opt. 14, 1860 (1975).

R. B. Smith, Electron. Lett. 10, 204 (1975).

R. B. Smith (unpublished).

Snyder, A. W.

A. W. Snyder, Proc. IEEE (Lett.) 58, 168 (1970).

Teh, G. A.

M. G. F. Wilson and G. A. Teh, IEEE Trans. M.T.T. MTT-23, 85 (1975). (This paper was published without labels for the axes and lines of the figures. Figure 4, showing the coupling efficiency, may be found correctly labeled in Ref. 2.)

M. G. F. Wilson and G. A. Teh, Electron. Lett. 9, 453 (1973).

Van Kampen, N. G.

N. G. Van Kampen, Physica (Utr.) 35, 70 (1967).

Wannier, G. H.

G. H. Wannier, Physics 5, 251 (1965).

Watanabe, A.

M. Matsuhara, K. O. Hill, and A. Watanabe, J. Opt. Soc. Am. 65, 804 (1975).

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, England, 1952), p. 323.

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, England, 1952), p. 323.

Wilson, M. G. F.

M. G. F. Wilson and G. A. Teh, IEEE Trans. M.T.T. MTT-23, 85 (1975). (This paper was published without labels for the axes and lines of the figures. Figure 4, showing the coupling efficiency, may be found correctly labeled in Ref. 2.)

M. G. F. Wilson and G. A. Teh, Electron. Lett. 9, 453 (1973).

Zener, C.

C. Zener, Proc. R. Soc. Lond. A 137, 696 (1932).

Other (26)

D. G. Dalgoutte, R. B. Smith, G. Achutaramayya, and J. H. Harris, Appl. Opt. 14, 1860 (1975).

M. G. F. Wilson and G. A. Teh, Electron. Lett. 9, 453 (1973).

M. G. F. Wilson and G. A. Teh, IEEE Trans. M.T.T. MTT-23, 85 (1975). (This paper was published without labels for the axes and lines of the figures. Figure 4, showing the coupling efficiency, may be found correctly labeled in Ref. 2.)

J. S. Cook, Bell Syst. Tech. J. 34, 807 (1955).

A. G. Fox, Bell Syst. Tech. J. 34, 853 (1955).

W. H. Louisell, Bell Syst. Tech. J. 34, 853 (1955).

A. F. Milton and W. K. Burns, Appl. Opt. 14, 1207 (1975).

A. L. Jones, J. Opt. Soc. Am. 55, 261 (1965).

Y. S. Chen and A. Ishimaru, Proc. IEEE (Lett.) 54, 1071 (1966).

E. Schloman and R. L. Joseph, J. Appl. Phys. 35, 2382 (1964); 36, 875 (1965).

E. K. Kirchner, L. F. Donaghey, and F. A. Olson, J. Appl. Phys. 37, 988 (1966).

S. M. Rezende and F. R. Morgenthaler, J. Appl. Phys. 40, 524 (1969).

J. L. Doane, J. Appl. Phys. 45, 2748 (1974).

R. B. Smith, Electron. Lett. 10, 204 (1975).

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, England, 1952), p. 323.

J. C. P. Miller, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (NBS, Washington, D. C., 1964), Chap. 19, p. 686.

A. L. Cullen and O. J. Davies, Electron. Lett. 5, 90 (1969).

J. B. Knorr, Proc. IEEE (Lett.) 59, 1624 (1971).

A. W. Snyder, Proc. IEEE (Lett.) 58, 168 (1970).

P. Horowitz, Appl. Phys. Lett. 23, 306 (1975).

M. Matsuhara, K. O. Hill, and A. Watanabe, J. Opt. Soc. Am. 65, 804 (1975).

G. H. Wannier, Physics 5, 251 (1965).

C. Zener, Proc. R. Soc. Lond. A 137, 696 (1932).

R. B. Smith (unpublished).

H. B. Keller and J. B. Keller, J. Soc. Indust. Appl. Math. 10, 246 (1962).

N. G. Van Kampen, Physica (Utr.) 35, 70 (1967).

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