Abstract

The characteristics of prism coupling of finite-width beams into nonlinear waveguides composed of media with diffusive nonlinearities (thermal, etc.) are calculated by including a one-dimensional diffusion equation for the nonlocal nonlinearity. The resulting longitudinal feedback leads to bistability, and the threshold value for the minimum diffusion length varies inversely with the angular detuning.

© 1989 Optical Society of America

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References

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  1. C. Liao, G. I. Stegeman, Appl. Phys. Lett. 44, 164 (1984); C. Liao, G. I. Stegeman, C. T. Seaton, R. L. Shoemaker, J. D. Valera, H. G. Winful, J. Opt. Soc. Am. A 2, 590 (1985).
    [CrossRef]
  2. P. Vincent, N. Paraire, M. Neviere, A. Koster, R. Reinisch, J. Opt. Soc. Am. B 2, 1106 (1985).
    [CrossRef]
  3. G. Assanto, R. M. Fortenberry, C. T. Seaton, G. I. Stegeman, J. Opt. Soc. Am. B 5, 432 (1988).
    [CrossRef]
  4. V. J. Montemayor, R. T. Deck, J. Opt. Soc. Am. B 2, 1010 (1985); J. Opt. Soc. Am. B 3, 1211 (1986).
    [CrossRef]
  5. R. Reinisch, P. Arlot, G. Vitrant, E. Pic, Appl. Phys. Lett. 47, 1248 (1985).
    [CrossRef]
  6. G. M. Carter, Y. J. Chen, Appl. Phys. Lett. 42, 643 (1983).
    [CrossRef]
  7. G. Vitrant, P. Arlot, J. Appl. Phys. 61, 4744 (1987).
    [CrossRef]
  8. H. Chelli, A. Koster, N. Paraire, F. Pardo, H. Sauer, M. Carton, S. Laval, Rev. Phys. Appl. 22, 1273 (1987).
    [CrossRef]
  9. W. Lukosz, P. Pirani, V. Briguet, in Optical Bistability III, H. M. Gibbs, P. Mandel, N. Peyghambarian, S. D. Smith, eds. (Springer-Verlag, Berlin, 1986), p. 109.
    [CrossRef]
  10. G. Assanto, B. Svensson, D. Kuchibhatla, U. J. Gibson, C. T. Seaton, G. I. Stegeman, Opt. Lett. 11, 644 (1986).
    [CrossRef] [PubMed]
  11. P. Martinot, A. Koster, S. Laval, IEEE J. Quantum Electron. QE-21, 1140 (1985).
    [CrossRef]
  12. G. Vitrant, P. Arlot, R. Reinisch, Proc. Soc. Photo-Opt. Instrum. Eng. 800, 169 (1987).
  13. G. I. Stegeman, G. Assanto, R. Zanoni, C. T. Seaton, E. Garmire, A. A. Maradudin, R. Reinisch, G. Vitrant, Appl. Phys. Lett. 52, 869 (1988).
    [CrossRef]
  14. D. Marcuse, Appl. Opt. 19, 3130 (1980).
    [CrossRef] [PubMed]
  15. P. W. Smith, W. J. Tomlinson, IEEE J. Quantum Electron. QE-20, 30 (1984).
    [CrossRef]
  16. H. B. Keller, in Applications of Bifurcation Theory, P. H. Rabinowitz, ed. (Academic, New York, 1977).
  17. E. J. Beltrani, in Mathematics in Science and Engineering (Academic, New York, 1970), Vol. 63.

1988 (2)

G. I. Stegeman, G. Assanto, R. Zanoni, C. T. Seaton, E. Garmire, A. A. Maradudin, R. Reinisch, G. Vitrant, Appl. Phys. Lett. 52, 869 (1988).
[CrossRef]

G. Assanto, R. M. Fortenberry, C. T. Seaton, G. I. Stegeman, J. Opt. Soc. Am. B 5, 432 (1988).
[CrossRef]

1987 (3)

G. Vitrant, P. Arlot, R. Reinisch, Proc. Soc. Photo-Opt. Instrum. Eng. 800, 169 (1987).

G. Vitrant, P. Arlot, J. Appl. Phys. 61, 4744 (1987).
[CrossRef]

H. Chelli, A. Koster, N. Paraire, F. Pardo, H. Sauer, M. Carton, S. Laval, Rev. Phys. Appl. 22, 1273 (1987).
[CrossRef]

1986 (1)

1985 (4)

V. J. Montemayor, R. T. Deck, J. Opt. Soc. Am. B 2, 1010 (1985); J. Opt. Soc. Am. B 3, 1211 (1986).
[CrossRef]

P. Vincent, N. Paraire, M. Neviere, A. Koster, R. Reinisch, J. Opt. Soc. Am. B 2, 1106 (1985).
[CrossRef]

P. Martinot, A. Koster, S. Laval, IEEE J. Quantum Electron. QE-21, 1140 (1985).
[CrossRef]

R. Reinisch, P. Arlot, G. Vitrant, E. Pic, Appl. Phys. Lett. 47, 1248 (1985).
[CrossRef]

1984 (2)

C. Liao, G. I. Stegeman, Appl. Phys. Lett. 44, 164 (1984); C. Liao, G. I. Stegeman, C. T. Seaton, R. L. Shoemaker, J. D. Valera, H. G. Winful, J. Opt. Soc. Am. A 2, 590 (1985).
[CrossRef]

P. W. Smith, W. J. Tomlinson, IEEE J. Quantum Electron. QE-20, 30 (1984).
[CrossRef]

1983 (1)

G. M. Carter, Y. J. Chen, Appl. Phys. Lett. 42, 643 (1983).
[CrossRef]

1980 (1)

Arlot, P.

G. Vitrant, P. Arlot, R. Reinisch, Proc. Soc. Photo-Opt. Instrum. Eng. 800, 169 (1987).

G. Vitrant, P. Arlot, J. Appl. Phys. 61, 4744 (1987).
[CrossRef]

R. Reinisch, P. Arlot, G. Vitrant, E. Pic, Appl. Phys. Lett. 47, 1248 (1985).
[CrossRef]

Assanto, G.

Beltrani, E. J.

E. J. Beltrani, in Mathematics in Science and Engineering (Academic, New York, 1970), Vol. 63.

Briguet, V.

W. Lukosz, P. Pirani, V. Briguet, in Optical Bistability III, H. M. Gibbs, P. Mandel, N. Peyghambarian, S. D. Smith, eds. (Springer-Verlag, Berlin, 1986), p. 109.
[CrossRef]

Carter, G. M.

G. M. Carter, Y. J. Chen, Appl. Phys. Lett. 42, 643 (1983).
[CrossRef]

Carton, M.

H. Chelli, A. Koster, N. Paraire, F. Pardo, H. Sauer, M. Carton, S. Laval, Rev. Phys. Appl. 22, 1273 (1987).
[CrossRef]

Chelli, H.

H. Chelli, A. Koster, N. Paraire, F. Pardo, H. Sauer, M. Carton, S. Laval, Rev. Phys. Appl. 22, 1273 (1987).
[CrossRef]

Chen, Y. J.

G. M. Carter, Y. J. Chen, Appl. Phys. Lett. 42, 643 (1983).
[CrossRef]

Deck, R. T.

Fortenberry, R. M.

Garmire, E.

G. I. Stegeman, G. Assanto, R. Zanoni, C. T. Seaton, E. Garmire, A. A. Maradudin, R. Reinisch, G. Vitrant, Appl. Phys. Lett. 52, 869 (1988).
[CrossRef]

Gibson, U. J.

Keller, H. B.

H. B. Keller, in Applications of Bifurcation Theory, P. H. Rabinowitz, ed. (Academic, New York, 1977).

Koster, A.

H. Chelli, A. Koster, N. Paraire, F. Pardo, H. Sauer, M. Carton, S. Laval, Rev. Phys. Appl. 22, 1273 (1987).
[CrossRef]

P. Martinot, A. Koster, S. Laval, IEEE J. Quantum Electron. QE-21, 1140 (1985).
[CrossRef]

P. Vincent, N. Paraire, M. Neviere, A. Koster, R. Reinisch, J. Opt. Soc. Am. B 2, 1106 (1985).
[CrossRef]

Kuchibhatla, D.

Laval, S.

H. Chelli, A. Koster, N. Paraire, F. Pardo, H. Sauer, M. Carton, S. Laval, Rev. Phys. Appl. 22, 1273 (1987).
[CrossRef]

P. Martinot, A. Koster, S. Laval, IEEE J. Quantum Electron. QE-21, 1140 (1985).
[CrossRef]

Liao, C.

C. Liao, G. I. Stegeman, Appl. Phys. Lett. 44, 164 (1984); C. Liao, G. I. Stegeman, C. T. Seaton, R. L. Shoemaker, J. D. Valera, H. G. Winful, J. Opt. Soc. Am. A 2, 590 (1985).
[CrossRef]

Lukosz, W.

W. Lukosz, P. Pirani, V. Briguet, in Optical Bistability III, H. M. Gibbs, P. Mandel, N. Peyghambarian, S. D. Smith, eds. (Springer-Verlag, Berlin, 1986), p. 109.
[CrossRef]

Maradudin, A. A.

G. I. Stegeman, G. Assanto, R. Zanoni, C. T. Seaton, E. Garmire, A. A. Maradudin, R. Reinisch, G. Vitrant, Appl. Phys. Lett. 52, 869 (1988).
[CrossRef]

Marcuse, D.

Martinot, P.

P. Martinot, A. Koster, S. Laval, IEEE J. Quantum Electron. QE-21, 1140 (1985).
[CrossRef]

Montemayor, V. J.

Neviere, M.

Paraire, N.

H. Chelli, A. Koster, N. Paraire, F. Pardo, H. Sauer, M. Carton, S. Laval, Rev. Phys. Appl. 22, 1273 (1987).
[CrossRef]

P. Vincent, N. Paraire, M. Neviere, A. Koster, R. Reinisch, J. Opt. Soc. Am. B 2, 1106 (1985).
[CrossRef]

Pardo, F.

H. Chelli, A. Koster, N. Paraire, F. Pardo, H. Sauer, M. Carton, S. Laval, Rev. Phys. Appl. 22, 1273 (1987).
[CrossRef]

Pic, E.

R. Reinisch, P. Arlot, G. Vitrant, E. Pic, Appl. Phys. Lett. 47, 1248 (1985).
[CrossRef]

Pirani, P.

W. Lukosz, P. Pirani, V. Briguet, in Optical Bistability III, H. M. Gibbs, P. Mandel, N. Peyghambarian, S. D. Smith, eds. (Springer-Verlag, Berlin, 1986), p. 109.
[CrossRef]

Reinisch, R.

G. I. Stegeman, G. Assanto, R. Zanoni, C. T. Seaton, E. Garmire, A. A. Maradudin, R. Reinisch, G. Vitrant, Appl. Phys. Lett. 52, 869 (1988).
[CrossRef]

G. Vitrant, P. Arlot, R. Reinisch, Proc. Soc. Photo-Opt. Instrum. Eng. 800, 169 (1987).

R. Reinisch, P. Arlot, G. Vitrant, E. Pic, Appl. Phys. Lett. 47, 1248 (1985).
[CrossRef]

P. Vincent, N. Paraire, M. Neviere, A. Koster, R. Reinisch, J. Opt. Soc. Am. B 2, 1106 (1985).
[CrossRef]

Sauer, H.

H. Chelli, A. Koster, N. Paraire, F. Pardo, H. Sauer, M. Carton, S. Laval, Rev. Phys. Appl. 22, 1273 (1987).
[CrossRef]

Seaton, C. T.

Smith, P. W.

P. W. Smith, W. J. Tomlinson, IEEE J. Quantum Electron. QE-20, 30 (1984).
[CrossRef]

Stegeman, G. I.

G. Assanto, R. M. Fortenberry, C. T. Seaton, G. I. Stegeman, J. Opt. Soc. Am. B 5, 432 (1988).
[CrossRef]

G. I. Stegeman, G. Assanto, R. Zanoni, C. T. Seaton, E. Garmire, A. A. Maradudin, R. Reinisch, G. Vitrant, Appl. Phys. Lett. 52, 869 (1988).
[CrossRef]

G. Assanto, B. Svensson, D. Kuchibhatla, U. J. Gibson, C. T. Seaton, G. I. Stegeman, Opt. Lett. 11, 644 (1986).
[CrossRef] [PubMed]

C. Liao, G. I. Stegeman, Appl. Phys. Lett. 44, 164 (1984); C. Liao, G. I. Stegeman, C. T. Seaton, R. L. Shoemaker, J. D. Valera, H. G. Winful, J. Opt. Soc. Am. A 2, 590 (1985).
[CrossRef]

Svensson, B.

Tomlinson, W. J.

P. W. Smith, W. J. Tomlinson, IEEE J. Quantum Electron. QE-20, 30 (1984).
[CrossRef]

Vincent, P.

Vitrant, G.

G. I. Stegeman, G. Assanto, R. Zanoni, C. T. Seaton, E. Garmire, A. A. Maradudin, R. Reinisch, G. Vitrant, Appl. Phys. Lett. 52, 869 (1988).
[CrossRef]

G. Vitrant, P. Arlot, J. Appl. Phys. 61, 4744 (1987).
[CrossRef]

G. Vitrant, P. Arlot, R. Reinisch, Proc. Soc. Photo-Opt. Instrum. Eng. 800, 169 (1987).

R. Reinisch, P. Arlot, G. Vitrant, E. Pic, Appl. Phys. Lett. 47, 1248 (1985).
[CrossRef]

Zanoni, R.

G. I. Stegeman, G. Assanto, R. Zanoni, C. T. Seaton, E. Garmire, A. A. Maradudin, R. Reinisch, G. Vitrant, Appl. Phys. Lett. 52, 869 (1988).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

C. Liao, G. I. Stegeman, Appl. Phys. Lett. 44, 164 (1984); C. Liao, G. I. Stegeman, C. T. Seaton, R. L. Shoemaker, J. D. Valera, H. G. Winful, J. Opt. Soc. Am. A 2, 590 (1985).
[CrossRef]

R. Reinisch, P. Arlot, G. Vitrant, E. Pic, Appl. Phys. Lett. 47, 1248 (1985).
[CrossRef]

G. M. Carter, Y. J. Chen, Appl. Phys. Lett. 42, 643 (1983).
[CrossRef]

G. I. Stegeman, G. Assanto, R. Zanoni, C. T. Seaton, E. Garmire, A. A. Maradudin, R. Reinisch, G. Vitrant, Appl. Phys. Lett. 52, 869 (1988).
[CrossRef]

IEEE J. Quantum Electron. (2)

P. W. Smith, W. J. Tomlinson, IEEE J. Quantum Electron. QE-20, 30 (1984).
[CrossRef]

P. Martinot, A. Koster, S. Laval, IEEE J. Quantum Electron. QE-21, 1140 (1985).
[CrossRef]

J. Appl. Phys. (1)

G. Vitrant, P. Arlot, J. Appl. Phys. 61, 4744 (1987).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

G. Vitrant, P. Arlot, R. Reinisch, Proc. Soc. Photo-Opt. Instrum. Eng. 800, 169 (1987).

Rev. Phys. Appl. (1)

H. Chelli, A. Koster, N. Paraire, F. Pardo, H. Sauer, M. Carton, S. Laval, Rev. Phys. Appl. 22, 1273 (1987).
[CrossRef]

Other (3)

W. Lukosz, P. Pirani, V. Briguet, in Optical Bistability III, H. M. Gibbs, P. Mandel, N. Peyghambarian, S. D. Smith, eds. (Springer-Verlag, Berlin, 1986), p. 109.
[CrossRef]

H. B. Keller, in Applications of Bifurcation Theory, P. H. Rabinowitz, ed. (Academic, New York, 1977).

E. J. Beltrani, in Mathematics in Science and Engineering (Academic, New York, 1970), Vol. 63.

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Figures (3)

Fig. 1
Fig. 1

Influence of the diffusion length D on the response of a nonlinear prism coupler for D = 0 (the local case), 1.5, 10, and 100. The D = 100 case corresponds to a complete derealization of the nonlinear effect. The other numerical parameters are Δβ = 1.8, βi = 0.5, w0 = 10, l1 = 25, l2 = 0, δ = 1, and t = 1.

Fig. 2
Fig. 2

Variation in Dth−1 with detuning Δβ for various values of beam spot size w0. The parameters are βi = 1, l1 = 25, l2 = 0, δ = 1, and t = 1.

Fig. 3
Fig. 3

(a) Input and (b) output field amplitudes required at the threshold for bistability as a function of detuning Δβ for various values of w0. The parameters are βi = 1, l1 = 25, l2 = 0, δ = 1, and t = 1.

Equations (9)

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d d x A gw ( x ) = j [ Δ β + j β i + U ( x ) ] A gw ( x ) + t A i G ( x ) ,
E i ( x ) = A i G ( x ) exp ( i β 0 x ) ,
E gw ( x ) = A gw ( x ) Φ ( y ) exp [ j ( β r + j β i ) x ] ,
Δ n ( x , y ) = n 2 ( y ) | Φ ( y ) | 2 | A gw ( x ) | 2 .
U ( x ) = 4 π 2 β r λ 2 d yn ( y ) n 2 ( y ) | Φ ( y ) | 4 × | A gw ( x ) | 2 = Δ β 0 | A gw ( x ) | 2 ,
D 2 d U 2 ( x ) d x 2 = U ( x ) Δ β 0 | A gw ( x ) | 2 ,
U ( l 1 ) D d U ( x ) d x = 0 at x = l 1 ,
U ( l 2 ) + D d U ( x ) d x = Δ β 0 | A gw ( l 2 ) | 2 at x = l 2 .
D th = a / [ Δ β Δ β min ( w 0 ) ] ,

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