Abstract

We show that two linearly polarized counterpropagating waves in a Kerr nonlinear medium with linear dispersion can exhibit multimode temporal instability. We find the boundary of the unstable regime and demonstrate that the fully developed instability results in self-sustained oscillations and the onset of chaos.

© 1989 Optical Society of America

Full Article  |  PDF Article

Errata

C. T. Law and A. E. Kaplan, "Dispersion-related multimode instabilities and self-sustained oscillations in nonlinear counterpropagating waves: erratum," Opt. Lett. 14, 1290-1290 (1989)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-14-22-1290

References

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  1. H. G. Winful, J. H. Marburger, Appl. Phys. Lett. 36, 613 (1980).
    [CrossRef]
  2. A. E. Kaplan, Opt. Lett. 8, 560 (1983).
    [CrossRef] [PubMed]
  3. A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
    [CrossRef]
  4. J. Yumoto, K. Otsuka, Phys. Rev. Lett. 54, 1806 (1985);M. V. Tratnik, J. E. Sipe, Phys. Rev. A 35, 2965 (1987).
    [CrossRef] [PubMed]
  5. I. Bar-Joseph, Y. Silberberg, Phys. Rev. A 36, 1731 (1987);Y. Silberberg, I. Bar-Joseph, Phys. Rev. Lett 48, 1541 (1982);J. Opt. Soc. Am. B 1, 662 (1984).
    [CrossRef] [PubMed]
  6. A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, Phys. Rev. Lett. 58, 2432 (1987);G. Khitrova, J. F. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988);D. J. Gauthier, M. S. Malcuit, R. W. Boyd, Phys. Rev. Lett. 61, 1827 (1988).
    [CrossRef] [PubMed]
  7. S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972).
  8. V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972);A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
    [CrossRef]
  9. G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
    [CrossRef] [PubMed]
  10. Our results were first reported in C. T. Law, A. E. Kaplan, in Digest of International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1988), paper MP 67, and inDigest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988), paper WM4.
  11. C. Paré, M. Piché, P.-A. Bélanger, J. Opt. Soc. Am. B 5, 679 (1988).
    [CrossRef]
  12. A. E. Kaplan, P. Meystre, Opt. Lett 6, 590 (1981);Opt. Commun. 40, 229 (1982);R. A. Bergh, H. C. Le-fevre, H. J. Shaw, Opt. Lett. 7, 282 (1982);S. Eze-kiel, J. L. Davis, R. W. Hellwarth, Opt. Lett. 7, 457 (1982).
    [CrossRef] [PubMed]
  13. W. J. Firth, C. Paré, Opt. Lett. 13, 1096 (1988).
    [CrossRef] [PubMed]
  14. J. E. Bjorkholm, P. W. Smith, W. J. Tomlinson, A. E. Kaplan, Opt. Lett. 6, 345 (1981);A. E. Kaplan, Opt. Lett. 6, 360 (1981);I. C. Khoo, Appl. Phys. Lett. 41, 909 (1982);K. Tai, H. M. Gibbs, N. Peyghambarian, A. Mysyrowicz, Opt. Lett. 10, 220 (1985).
    [CrossRef] [PubMed]
  15. J. W. Fleming, J. Am. Ceram. Soc. 59, 503 (1976);T. Li, IEEE Trans. Comm. COM-26, 946 (1978).
    [CrossRef]
  16. R. Hellwarth, J. Cherlow, T. T. Yang, Phys Rev. B 11, 964 (1975);R. H. Stolen, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979), p. 125.
    [CrossRef]
  17. S. R. Friberg, P. W. Smith, IEEE J. Quantum Electron. QE-23, 2089 (1987).
    [CrossRef]

1988

1987

S. R. Friberg, P. W. Smith, IEEE J. Quantum Electron. QE-23, 2089 (1987).
[CrossRef]

I. Bar-Joseph, Y. Silberberg, Phys. Rev. A 36, 1731 (1987);Y. Silberberg, I. Bar-Joseph, Phys. Rev. Lett 48, 1541 (1982);J. Opt. Soc. Am. B 1, 662 (1984).
[CrossRef] [PubMed]

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, Phys. Rev. Lett. 58, 2432 (1987);G. Khitrova, J. F. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988);D. J. Gauthier, M. S. Malcuit, R. W. Boyd, Phys. Rev. Lett. 61, 1827 (1988).
[CrossRef] [PubMed]

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

1985

A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[CrossRef]

J. Yumoto, K. Otsuka, Phys. Rev. Lett. 54, 1806 (1985);M. V. Tratnik, J. E. Sipe, Phys. Rev. A 35, 2965 (1987).
[CrossRef] [PubMed]

1983

1981

J. E. Bjorkholm, P. W. Smith, W. J. Tomlinson, A. E. Kaplan, Opt. Lett. 6, 345 (1981);A. E. Kaplan, Opt. Lett. 6, 360 (1981);I. C. Khoo, Appl. Phys. Lett. 41, 909 (1982);K. Tai, H. M. Gibbs, N. Peyghambarian, A. Mysyrowicz, Opt. Lett. 10, 220 (1985).
[CrossRef] [PubMed]

A. E. Kaplan, P. Meystre, Opt. Lett 6, 590 (1981);Opt. Commun. 40, 229 (1982);R. A. Bergh, H. C. Le-fevre, H. J. Shaw, Opt. Lett. 7, 282 (1982);S. Eze-kiel, J. L. Davis, R. W. Hellwarth, Opt. Lett. 7, 457 (1982).
[CrossRef] [PubMed]

1980

H. G. Winful, J. H. Marburger, Appl. Phys. Lett. 36, 613 (1980).
[CrossRef]

1976

J. W. Fleming, J. Am. Ceram. Soc. 59, 503 (1976);T. Li, IEEE Trans. Comm. COM-26, 946 (1978).
[CrossRef]

1975

R. Hellwarth, J. Cherlow, T. T. Yang, Phys Rev. B 11, 964 (1975);R. H. Stolen, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979), p. 125.
[CrossRef]

1972

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972);A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

Ackerhalt, J. R.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, Phys. Rev. Lett. 58, 2432 (1987);G. Khitrova, J. F. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988);D. J. Gauthier, M. S. Malcuit, R. W. Boyd, Phys. Rev. Lett. 61, 1827 (1988).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

Akhmanov, S. A.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972).

Bar-Joseph, I.

I. Bar-Joseph, Y. Silberberg, Phys. Rev. A 36, 1731 (1987);Y. Silberberg, I. Bar-Joseph, Phys. Rev. Lett 48, 1541 (1982);J. Opt. Soc. Am. B 1, 662 (1984).
[CrossRef] [PubMed]

Bélanger, P.-A.

Bjorkholm, J. E.

Boyd, R. W.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, Phys. Rev. Lett. 58, 2432 (1987);G. Khitrova, J. F. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988);D. J. Gauthier, M. S. Malcuit, R. W. Boyd, Phys. Rev. Lett. 61, 1827 (1988).
[CrossRef] [PubMed]

Cherlow, J.

R. Hellwarth, J. Cherlow, T. T. Yang, Phys Rev. B 11, 964 (1975);R. H. Stolen, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979), p. 125.
[CrossRef]

Firth, W. J.

Fleming, J. W.

J. W. Fleming, J. Am. Ceram. Soc. 59, 503 (1976);T. Li, IEEE Trans. Comm. COM-26, 946 (1978).
[CrossRef]

Friberg, S. R.

S. R. Friberg, P. W. Smith, IEEE J. Quantum Electron. QE-23, 2089 (1987).
[CrossRef]

Gaeta, A. L.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, Phys. Rev. Lett. 58, 2432 (1987);G. Khitrova, J. F. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988);D. J. Gauthier, M. S. Malcuit, R. W. Boyd, Phys. Rev. Lett. 61, 1827 (1988).
[CrossRef] [PubMed]

Hellwarth, R.

R. Hellwarth, J. Cherlow, T. T. Yang, Phys Rev. B 11, 964 (1975);R. H. Stolen, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979), p. 125.
[CrossRef]

Kaplan, A. E.

A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[CrossRef]

A. E. Kaplan, Opt. Lett. 8, 560 (1983).
[CrossRef] [PubMed]

A. E. Kaplan, P. Meystre, Opt. Lett 6, 590 (1981);Opt. Commun. 40, 229 (1982);R. A. Bergh, H. C. Le-fevre, H. J. Shaw, Opt. Lett. 7, 282 (1982);S. Eze-kiel, J. L. Davis, R. W. Hellwarth, Opt. Lett. 7, 457 (1982).
[CrossRef] [PubMed]

J. E. Bjorkholm, P. W. Smith, W. J. Tomlinson, A. E. Kaplan, Opt. Lett. 6, 345 (1981);A. E. Kaplan, Opt. Lett. 6, 360 (1981);I. C. Khoo, Appl. Phys. Lett. 41, 909 (1982);K. Tai, H. M. Gibbs, N. Peyghambarian, A. Mysyrowicz, Opt. Lett. 10, 220 (1985).
[CrossRef] [PubMed]

Our results were first reported in C. T. Law, A. E. Kaplan, in Digest of International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1988), paper MP 67, and inDigest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988), paper WM4.

Khokhlov, R. V.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972).

Law, C. T.

A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[CrossRef]

Our results were first reported in C. T. Law, A. E. Kaplan, in Digest of International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1988), paper MP 67, and inDigest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988), paper WM4.

Marburger, J. H.

H. G. Winful, J. H. Marburger, Appl. Phys. Lett. 36, 613 (1980).
[CrossRef]

Meystre, P.

A. E. Kaplan, P. Meystre, Opt. Lett 6, 590 (1981);Opt. Commun. 40, 229 (1982);R. A. Bergh, H. C. Le-fevre, H. J. Shaw, Opt. Lett. 7, 282 (1982);S. Eze-kiel, J. L. Davis, R. W. Hellwarth, Opt. Lett. 7, 457 (1982).
[CrossRef] [PubMed]

Milonni, P. W.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, Phys. Rev. Lett. 58, 2432 (1987);G. Khitrova, J. F. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988);D. J. Gauthier, M. S. Malcuit, R. W. Boyd, Phys. Rev. Lett. 61, 1827 (1988).
[CrossRef] [PubMed]

Otsuka, K.

J. Yumoto, K. Otsuka, Phys. Rev. Lett. 54, 1806 (1985);M. V. Tratnik, J. E. Sipe, Phys. Rev. A 35, 2965 (1987).
[CrossRef] [PubMed]

Paré, C.

Piché, M.

Shabat, A. B.

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972);A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

Silberberg, Y.

I. Bar-Joseph, Y. Silberberg, Phys. Rev. A 36, 1731 (1987);Y. Silberberg, I. Bar-Joseph, Phys. Rev. Lett 48, 1541 (1982);J. Opt. Soc. Am. B 1, 662 (1984).
[CrossRef] [PubMed]

Smith, P. W.

Sukhorukov, A. P.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972).

Tomlinson, W. J.

Winful, H. G.

H. G. Winful, J. H. Marburger, Appl. Phys. Lett. 36, 613 (1980).
[CrossRef]

Yang, T. T.

R. Hellwarth, J. Cherlow, T. T. Yang, Phys Rev. B 11, 964 (1975);R. H. Stolen, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979), p. 125.
[CrossRef]

Yumoto, J.

J. Yumoto, K. Otsuka, Phys. Rev. Lett. 54, 1806 (1985);M. V. Tratnik, J. E. Sipe, Phys. Rev. A 35, 2965 (1987).
[CrossRef] [PubMed]

Zakharov, V. E.

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972);A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

Appl. Phys. Lett.

H. G. Winful, J. H. Marburger, Appl. Phys. Lett. 36, 613 (1980).
[CrossRef]

IEEE J. Quantum Electron.

A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[CrossRef]

S. R. Friberg, P. W. Smith, IEEE J. Quantum Electron. QE-23, 2089 (1987).
[CrossRef]

J. Am. Ceram. Soc.

J. W. Fleming, J. Am. Ceram. Soc. 59, 503 (1976);T. Li, IEEE Trans. Comm. COM-26, 946 (1978).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett

A. E. Kaplan, P. Meystre, Opt. Lett 6, 590 (1981);Opt. Commun. 40, 229 (1982);R. A. Bergh, H. C. Le-fevre, H. J. Shaw, Opt. Lett. 7, 282 (1982);S. Eze-kiel, J. L. Davis, R. W. Hellwarth, Opt. Lett. 7, 457 (1982).
[CrossRef] [PubMed]

Opt. Lett.

Phys Rev. B

R. Hellwarth, J. Cherlow, T. T. Yang, Phys Rev. B 11, 964 (1975);R. H. Stolen, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979), p. 125.
[CrossRef]

Phys. Rev. A

I. Bar-Joseph, Y. Silberberg, Phys. Rev. A 36, 1731 (1987);Y. Silberberg, I. Bar-Joseph, Phys. Rev. Lett 48, 1541 (1982);J. Opt. Soc. Am. B 1, 662 (1984).
[CrossRef] [PubMed]

Phys. Rev. Lett.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, Phys. Rev. Lett. 58, 2432 (1987);G. Khitrova, J. F. Valley, H. M. Gibbs, Phys. Rev. Lett. 60, 1126 (1988);D. J. Gauthier, M. S. Malcuit, R. W. Boyd, Phys. Rev. Lett. 61, 1827 (1988).
[CrossRef] [PubMed]

J. Yumoto, K. Otsuka, Phys. Rev. Lett. 54, 1806 (1985);M. V. Tratnik, J. E. Sipe, Phys. Rev. A 35, 2965 (1987).
[CrossRef] [PubMed]

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

Sov. Phys. JETP

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972);A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

Other

Our results were first reported in C. T. Law, A. E. Kaplan, in Digest of International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1988), paper MP 67, and inDigest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988), paper WM4.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972).

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Figures (3)

Fig. 1
Fig. 1

Normalized threshold intensity pcr versus the normalized dispersion d. The solid curve is the boundary of instability above which the system is unstable. The dashed curves indicate the boundaries of the instability of individual modes. The inset shows the number of unstable modes versus |p|/pcr, where p is the normalized driving intensity.

Fig. 2
Fig. 2

Normalized intensity pos of oscillations versus the normalized time T for a driving intensity |p| = 2.02 (above-threshold pumping, |p|/pcr = 1.02, with single-mode excitation) when normalized dispersion d is fixed at 0.01. The inset is an enlargement of the region indicated by the arrow and demonstrates the single-mode nature of excitation. The frequency of oscillations is ≃17.2υgL−1.

Fig. 3
Fig. 3

The same conditions as in Fig. 2 except with a driving intensity |p| = 2.5 (|p|/pcr = 1.25 with three unstable modes involved). The excitation shows the formation of a subharmonic and the onset of chaos. The initial frequency of oscillations is slightly less than that shown in Fig. 2.

Equations (3)

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

i [ ( 1 ) j + 1 E j z + 1 υ g E j t ] μ 2 2 E j t 2 = β ( 2 I 3 j + I j ) E j , j = 1 , 2 ,
E j = E j s ( z ) [ 1 + δ A j ( z ) exp ( λ t ) + δ B j * ( z ) exp ( λ * t ) ] , j = 1 , 2 ,
P cr = sgn ( β d ) | log η | d | | ,

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