Abstract

A simple model for absorption and refraction of a laser beam by a bounded nonlinear medium and for diffraction during the beam’s subsequent free-space propagation is used to calculate analytically the steady-state transverse-field profile in the far-field limit using a number of different laser input profiles. Good qualitative agreement with features found in experiments with indium antimonide at cryogenic temperatures is obtained when the input amplitude is assumed to possess a parabolic cross section. Power-transfer curves for a simple and entirely passive power-limiting device operating below the saturation level for nonlinear absorption have been calculated numerically, and it is concluded that the mechanism for limiting is largely determined by the ratio of nonlinear refraction to nonlinear absorption coefficients. Applications to the protection of sensitive optical components are envisaged.

© 1984 Optical Society of America

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  1. S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. D. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1151–1228.
  2. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609–636 (1968).
    [Crossref]
  3. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [Crossref]
  4. P. D. McWane, “Variable focal length lenses using materials with intensity dependent refractive indices,” Nature 211, 1081–1082 (1966).
    [Crossref]
  5. A. E. Kaplan, “External self-focusing of light by a nonlinear layer,” Radiophys. Quantum Electron. 12, 692–696 (1969).
    [Crossref]
  6. A. E. Kaplan, “Optical bistability due to mutual self-action of counter-propagating light beams,” in Technical Digest Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1981), p. 118.
  7. A. E. Kaplan, “Optical bistability that is due to mutual self-action of counterpropagating beams of light,” Opt. Lett. 6, 360–362 (1981).
    [Crossref] [PubMed]
  8. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 59 (1975).
    [Crossref]
  9. D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D. Smith, “Effect of low-power nonlinear refraction on laser-beam propagation in InSb,” Opt. Lett. 4, 331–333 (1979).
    [Crossref] [PubMed]
  10. D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith, “Nonlinear optical effects in InSb with a cw CO laser,” Opt. Commun. 27, 133–136 (1978).
    [Crossref]
  11. A. Miller, D. A. B. Miller, and S. D. Smith, “Dynamic nonlinear optical processes in semiconductors,” Adv. Phys. 30, 697–800 (1981).
    [Crossref]
  12. A. K. Kar, J. G. H. Mathew, S. D. Smith, B. Davis, and W. Prettl, “Optical bistability in InSb at room temperature with two-photon excitation,” Appl. Phys. Lett. 42, 334–336 (1983).
    [Crossref]
  13. H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, “Room-temperature excitonic optical bistability in a GaAs–GaAlAs superlattice étalon,” Appl. Phys. Lett. 41, 221–222 (1982).
    [Crossref]
  14. D. A. B. Miller, D. S. Chemla, D. J. Eilenberger, P. W. Smith, A. C. Gossard, and W. T. Tsang, “Large room-temperature optical nonlinearity in GaAs/Ga1−x Alx As multiple quantum well structures,” Appl. Phys. Lett. 41, 679–681 (1982).
    [Crossref]
  15. P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
    [Crossref]
  16. S. S. Mitra, L. M. Narducci, R. A. Shatas, Y. F. Tsay, and A. Vaidyanathan, “Nonlinear absorption in direct-gap semiconductors,” Appl. Opt. 14, 3038–3042 (1975).
    [Crossref] [PubMed]
  17. A. M. Johnston, C. R. Pidgeon, and J. Dempsey, “Frequency dependence of two-photon absorption in InSB and Hg1−x Cdx Te,” Phys. Rev. B 22, 825–831 (1980).
    [Crossref]
  18. C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, “Two-photon absorption in zincblende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
    [Crossref]
  19. C. C. Lee and H. Y. Fan, “Two-photon absorption and photoconductivity in GaAs and InP,” Appl. Phys. Lett. 20, 18–20 (1972).
    [Crossref]
  20. J. M. Doviak, A. F. Gibson, M. F. Kimmit, and A. C. Walker, “Two-photon absorption in indium antimonide at 10.6 μ m,” J. Phys. C 6, 593–600 (1973).
    [Crossref]
  21. A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
    [Crossref]
  22. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1566 (1966).
    [Crossref] [PubMed]
  23. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 503.
  24. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 355, 435.
  25. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, London, 1965), p. 459.
  26. J. M. Aaron, C. L. M. Ireland, and C. Grey Morgan, “Aberration effects in the interaction of focused laser beams with matter,” J. Phys. D 7, 1970–1971 (1974).
    [Crossref]
  27. M. J. Soileau, W. E. Williams, and E. W. Van Stryland, “Optical power limiter with picosecond response time,” IEEE J. Quantum Electron QE-19, 731–735 (1983).
    [Crossref]
  28. M. J. Soileau, J. B. Franck, and T. C. Veatch, “On self-focusing and spot-size dependence of laser-induced breakdown,” in Laser-Induced Damage in Optical Materials, Nat. Bur. Stand. Spec. Publ. 620, 385–393 (1980).
  29. M. J. Soileau, W. E. Williams, E. W. Van Stryland, and S. F. Brown, “The use of self-focusing in the prevention of laser-induced damage,” in Proceedings of the 1981 Symposium on Laser-Induced Damage in Optical Materials (National Bureau Standards, Washington, D. C., to be published).

1983 (2)

A. K. Kar, J. G. H. Mathew, S. D. Smith, B. Davis, and W. Prettl, “Optical bistability in InSb at room temperature with two-photon excitation,” Appl. Phys. Lett. 42, 334–336 (1983).
[Crossref]

M. J. Soileau, W. E. Williams, and E. W. Van Stryland, “Optical power limiter with picosecond response time,” IEEE J. Quantum Electron QE-19, 731–735 (1983).
[Crossref]

1982 (2)

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, “Room-temperature excitonic optical bistability in a GaAs–GaAlAs superlattice étalon,” Appl. Phys. Lett. 41, 221–222 (1982).
[Crossref]

D. A. B. Miller, D. S. Chemla, D. J. Eilenberger, P. W. Smith, A. C. Gossard, and W. T. Tsang, “Large room-temperature optical nonlinearity in GaAs/Ga1−x Alx As multiple quantum well structures,” Appl. Phys. Lett. 41, 679–681 (1982).
[Crossref]

1981 (2)

A. Miller, D. A. B. Miller, and S. D. Smith, “Dynamic nonlinear optical processes in semiconductors,” Adv. Phys. 30, 697–800 (1981).
[Crossref]

A. E. Kaplan, “Optical bistability that is due to mutual self-action of counterpropagating beams of light,” Opt. Lett. 6, 360–362 (1981).
[Crossref] [PubMed]

1980 (2)

A. M. Johnston, C. R. Pidgeon, and J. Dempsey, “Frequency dependence of two-photon absorption in InSB and Hg1−x Cdx Te,” Phys. Rev. B 22, 825–831 (1980).
[Crossref]

M. J. Soileau, J. B. Franck, and T. C. Veatch, “On self-focusing and spot-size dependence of laser-induced breakdown,” in Laser-Induced Damage in Optical Materials, Nat. Bur. Stand. Spec. Publ. 620, 385–393 (1980).

1979 (2)

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, “Two-photon absorption in zincblende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D. Smith, “Effect of low-power nonlinear refraction on laser-beam propagation in InSb,” Opt. Lett. 4, 331–333 (1979).
[Crossref] [PubMed]

1978 (1)

D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith, “Nonlinear optical effects in InSb with a cw CO laser,” Opt. Commun. 27, 133–136 (1978).
[Crossref]

1976 (1)

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[Crossref]

1975 (3)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 59 (1975).
[Crossref]

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[Crossref]

S. S. Mitra, L. M. Narducci, R. A. Shatas, Y. F. Tsay, and A. Vaidyanathan, “Nonlinear absorption in direct-gap semiconductors,” Appl. Opt. 14, 3038–3042 (1975).
[Crossref] [PubMed]

1974 (1)

J. M. Aaron, C. L. M. Ireland, and C. Grey Morgan, “Aberration effects in the interaction of focused laser beams with matter,” J. Phys. D 7, 1970–1971 (1974).
[Crossref]

1973 (1)

J. M. Doviak, A. F. Gibson, M. F. Kimmit, and A. C. Walker, “Two-photon absorption in indium antimonide at 10.6 μ m,” J. Phys. C 6, 593–600 (1973).
[Crossref]

1972 (1)

C. C. Lee and H. Y. Fan, “Two-photon absorption and photoconductivity in GaAs and InP,” Appl. Phys. Lett. 20, 18–20 (1972).
[Crossref]

1969 (1)

A. E. Kaplan, “External self-focusing of light by a nonlinear layer,” Radiophys. Quantum Electron. 12, 692–696 (1969).
[Crossref]

1968 (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609–636 (1968).
[Crossref]

1966 (2)

P. D. McWane, “Variable focal length lenses using materials with intensity dependent refractive indices,” Nature 211, 1081–1082 (1966).
[Crossref]

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1566 (1966).
[Crossref] [PubMed]

1965 (1)

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[Crossref]

Aaron, J. M.

J. M. Aaron, C. L. M. Ireland, and C. Grey Morgan, “Aberration effects in the interaction of focused laser beams with matter,” J. Phys. D 7, 1970–1971 (1974).
[Crossref]

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609–636 (1968).
[Crossref]

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. D. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1151–1228.

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, London, 1965), p. 459.

Brown, S. F.

M. J. Soileau, W. E. Williams, E. W. Van Stryland, and S. F. Brown, “The use of self-focusing in the prevention of laser-induced damage,” in Proceedings of the 1981 Symposium on Laser-Induced Damage in Optical Materials (National Bureau Standards, Washington, D. C., to be published).

Chemla, D. S.

D. A. B. Miller, D. S. Chemla, D. J. Eilenberger, P. W. Smith, A. C. Gossard, and W. T. Tsang, “Large room-temperature optical nonlinearity in GaAs/Ga1−x Alx As multiple quantum well structures,” Appl. Phys. Lett. 41, 679–681 (1982).
[Crossref]

Davis, B.

A. K. Kar, J. G. H. Mathew, S. D. Smith, B. Davis, and W. Prettl, “Optical bistability in InSb at room temperature with two-photon excitation,” Appl. Phys. Lett. 42, 334–336 (1983).
[Crossref]

Dempsey, J.

A. M. Johnston, C. R. Pidgeon, and J. Dempsey, “Frequency dependence of two-photon absorption in InSB and Hg1−x Cdx Te,” Phys. Rev. B 22, 825–831 (1980).
[Crossref]

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, “Two-photon absorption in zincblende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

Doviak, J. M.

J. M. Doviak, A. F. Gibson, M. F. Kimmit, and A. C. Walker, “Two-photon absorption in indium antimonide at 10.6 μ m,” J. Phys. C 6, 593–600 (1973).
[Crossref]

Eilenberger, D. J.

D. A. B. Miller, D. S. Chemla, D. J. Eilenberger, P. W. Smith, A. C. Gossard, and W. T. Tsang, “Large room-temperature optical nonlinearity in GaAs/Ga1−x Alx As multiple quantum well structures,” Appl. Phys. Lett. 41, 679–681 (1982).
[Crossref]

Fan, H. Y.

C. C. Lee and H. Y. Fan, “Two-photon absorption and photoconductivity in GaAs and InP,” Appl. Phys. Lett. 20, 18–20 (1972).
[Crossref]

Franck, J. B.

M. J. Soileau, J. B. Franck, and T. C. Veatch, “On self-focusing and spot-size dependence of laser-induced breakdown,” in Laser-Induced Damage in Optical Materials, Nat. Bur. Stand. Spec. Publ. 620, 385–393 (1980).

Gibbs, H. M.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, “Room-temperature excitonic optical bistability in a GaAs–GaAlAs superlattice étalon,” Appl. Phys. Lett. 41, 221–222 (1982).
[Crossref]

Gibson, A. F.

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[Crossref]

J. M. Doviak, A. F. Gibson, M. F. Kimmit, and A. C. Walker, “Two-photon absorption in indium antimonide at 10.6 μ m,” J. Phys. C 6, 593–600 (1973).
[Crossref]

Gossard, A. C.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, “Room-temperature excitonic optical bistability in a GaAs–GaAlAs superlattice étalon,” Appl. Phys. Lett. 41, 221–222 (1982).
[Crossref]

D. A. B. Miller, D. S. Chemla, D. J. Eilenberger, P. W. Smith, A. C. Gossard, and W. T. Tsang, “Large room-temperature optical nonlinearity in GaAs/Ga1−x Alx As multiple quantum well structures,” Appl. Phys. Lett. 41, 679–681 (1982).
[Crossref]

Grey Morgan, C.

J. M. Aaron, C. L. M. Ireland, and C. Grey Morgan, “Aberration effects in the interaction of focused laser beams with matter,” J. Phys. D 7, 1970–1971 (1974).
[Crossref]

Hatch, C. B.

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[Crossref]

Ireland, C. L. M.

J. M. Aaron, C. L. M. Ireland, and C. Grey Morgan, “Aberration effects in the interaction of focused laser beams with matter,” J. Phys. D 7, 1970–1971 (1974).
[Crossref]

Jewell, J. L.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, “Room-temperature excitonic optical bistability in a GaAs–GaAlAs superlattice étalon,” Appl. Phys. Lett. 41, 221–222 (1982).
[Crossref]

Johnston, A. M.

A. M. Johnston, C. R. Pidgeon, and J. Dempsey, “Frequency dependence of two-photon absorption in InSB and Hg1−x Cdx Te,” Phys. Rev. B 22, 825–831 (1980).
[Crossref]

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, “Two-photon absorption in zincblende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

Kaplan, A. E.

A. E. Kaplan, “Optical bistability that is due to mutual self-action of counterpropagating beams of light,” Opt. Lett. 6, 360–362 (1981).
[Crossref] [PubMed]

A. E. Kaplan, “External self-focusing of light by a nonlinear layer,” Radiophys. Quantum Electron. 12, 692–696 (1969).
[Crossref]

A. E. Kaplan, “Optical bistability due to mutual self-action of counter-propagating light beams,” in Technical Digest Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1981), p. 118.

Kar, A. K.

A. K. Kar, J. G. H. Mathew, S. D. Smith, B. Davis, and W. Prettl, “Optical bistability in InSb at room temperature with two-photon excitation,” Appl. Phys. Lett. 42, 334–336 (1983).
[Crossref]

Kelley, P. L.

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[Crossref]

Khokhlov, R. V.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609–636 (1968).
[Crossref]

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. D. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1151–1228.

Kimmit, M. F.

J. M. Doviak, A. F. Gibson, M. F. Kimmit, and A. C. Walker, “Two-photon absorption in indium antimonide at 10.6 μ m,” J. Phys. C 6, 593–600 (1973).
[Crossref]

Kogelnik, H.

Lee, C. C.

C. C. Lee and H. Y. Fan, “Two-photon absorption and photoconductivity in GaAs and InP,” Appl. Phys. Lett. 20, 18–20 (1972).
[Crossref]

Li, T.

Maggs, P. N. D.

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[Crossref]

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[Crossref]

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 59 (1975).
[Crossref]

Mathew, J. G. H.

A. K. Kar, J. G. H. Mathew, S. D. Smith, B. Davis, and W. Prettl, “Optical bistability in InSb at room temperature with two-photon excitation,” Appl. Phys. Lett. 42, 334–336 (1983).
[Crossref]

McCall, S. L.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, “Room-temperature excitonic optical bistability in a GaAs–GaAlAs superlattice étalon,” Appl. Phys. Lett. 41, 221–222 (1982).
[Crossref]

McWane, P. D.

P. D. McWane, “Variable focal length lenses using materials with intensity dependent refractive indices,” Nature 211, 1081–1082 (1966).
[Crossref]

Miller, A.

A. Miller, D. A. B. Miller, and S. D. Smith, “Dynamic nonlinear optical processes in semiconductors,” Adv. Phys. 30, 697–800 (1981).
[Crossref]

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, “Two-photon absorption in zincblende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith, “Nonlinear optical effects in InSb with a cw CO laser,” Opt. Commun. 27, 133–136 (1978).
[Crossref]

Miller, D. A. B.

D. A. B. Miller, D. S. Chemla, D. J. Eilenberger, P. W. Smith, A. C. Gossard, and W. T. Tsang, “Large room-temperature optical nonlinearity in GaAs/Ga1−x Alx As multiple quantum well structures,” Appl. Phys. Lett. 41, 679–681 (1982).
[Crossref]

A. Miller, D. A. B. Miller, and S. D. Smith, “Dynamic nonlinear optical processes in semiconductors,” Adv. Phys. 30, 697–800 (1981).
[Crossref]

D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D. Smith, “Effect of low-power nonlinear refraction on laser-beam propagation in InSb,” Opt. Lett. 4, 331–333 (1979).
[Crossref] [PubMed]

D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith, “Nonlinear optical effects in InSb with a cw CO laser,” Opt. Commun. 27, 133–136 (1978).
[Crossref]

Mitra, S. S.

Mozolowski, M. H.

D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith, “Nonlinear optical effects in InSb with a cw CO laser,” Opt. Commun. 27, 133–136 (1978).
[Crossref]

Narducci, L. M.

Passner, A.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, “Room-temperature excitonic optical bistability in a GaAs–GaAlAs superlattice étalon,” Appl. Phys. Lett. 41, 221–222 (1982).
[Crossref]

Pidgeon, C. R.

A. M. Johnston, C. R. Pidgeon, and J. Dempsey, “Frequency dependence of two-photon absorption in InSB and Hg1−x Cdx Te,” Phys. Rev. B 22, 825–831 (1980).
[Crossref]

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, “Two-photon absorption in zincblende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

Prettl, W.

A. K. Kar, J. G. H. Mathew, S. D. Smith, B. Davis, and W. Prettl, “Optical bistability in InSb at room temperature with two-photon excitation,” Appl. Phys. Lett. 42, 334–336 (1983).
[Crossref]

Shatas, R. A.

Smith, P. W.

D. A. B. Miller, D. S. Chemla, D. J. Eilenberger, P. W. Smith, A. C. Gossard, and W. T. Tsang, “Large room-temperature optical nonlinearity in GaAs/Ga1−x Alx As multiple quantum well structures,” Appl. Phys. Lett. 41, 679–681 (1982).
[Crossref]

Smith, S. D.

A. K. Kar, J. G. H. Mathew, S. D. Smith, B. Davis, and W. Prettl, “Optical bistability in InSb at room temperature with two-photon excitation,” Appl. Phys. Lett. 42, 334–336 (1983).
[Crossref]

A. Miller, D. A. B. Miller, and S. D. Smith, “Dynamic nonlinear optical processes in semiconductors,” Adv. Phys. 30, 697–800 (1981).
[Crossref]

D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D. Smith, “Effect of low-power nonlinear refraction on laser-beam propagation in InSb,” Opt. Lett. 4, 331–333 (1979).
[Crossref] [PubMed]

D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith, “Nonlinear optical effects in InSb with a cw CO laser,” Opt. Commun. 27, 133–136 (1978).
[Crossref]

Soileau, M. J.

M. J. Soileau, W. E. Williams, and E. W. Van Stryland, “Optical power limiter with picosecond response time,” IEEE J. Quantum Electron QE-19, 731–735 (1983).
[Crossref]

M. J. Soileau, J. B. Franck, and T. C. Veatch, “On self-focusing and spot-size dependence of laser-induced breakdown,” in Laser-Induced Damage in Optical Materials, Nat. Bur. Stand. Spec. Publ. 620, 385–393 (1980).

M. J. Soileau, W. E. Williams, E. W. Van Stryland, and S. F. Brown, “The use of self-focusing in the prevention of laser-induced damage,” in Proceedings of the 1981 Symposium on Laser-Induced Damage in Optical Materials (National Bureau Standards, Washington, D. C., to be published).

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609–636 (1968).
[Crossref]

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. D. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1151–1228.

Tai, K.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, “Room-temperature excitonic optical bistability in a GaAs–GaAlAs superlattice étalon,” Appl. Phys. Lett. 41, 221–222 (1982).
[Crossref]

Tarng, S. S.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, “Room-temperature excitonic optical bistability in a GaAs–GaAlAs superlattice étalon,” Appl. Phys. Lett. 41, 221–222 (1982).
[Crossref]

Tilley, D. R.

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[Crossref]

Tsang, W. T.

D. A. B. Miller, D. S. Chemla, D. J. Eilenberger, P. W. Smith, A. C. Gossard, and W. T. Tsang, “Large room-temperature optical nonlinearity in GaAs/Ga1−x Alx As multiple quantum well structures,” Appl. Phys. Lett. 41, 679–681 (1982).
[Crossref]

Tsay, Y. F.

Vaidyanathan, A.

Van Stryland, E. W.

M. J. Soileau, W. E. Williams, and E. W. Van Stryland, “Optical power limiter with picosecond response time,” IEEE J. Quantum Electron QE-19, 731–735 (1983).
[Crossref]

M. J. Soileau, W. E. Williams, E. W. Van Stryland, and S. F. Brown, “The use of self-focusing in the prevention of laser-induced damage,” in Proceedings of the 1981 Symposium on Laser-Induced Damage in Optical Materials (National Bureau Standards, Washington, D. C., to be published).

Veatch, T. C.

M. J. Soileau, J. B. Franck, and T. C. Veatch, “On self-focusing and spot-size dependence of laser-induced breakdown,” in Laser-Induced Damage in Optical Materials, Nat. Bur. Stand. Spec. Publ. 620, 385–393 (1980).

Walker, A. C.

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[Crossref]

J. M. Doviak, A. F. Gibson, M. F. Kimmit, and A. C. Walker, “Two-photon absorption in indium antimonide at 10.6 μ m,” J. Phys. C 6, 593–600 (1973).
[Crossref]

Weaire, D.

Weigmann, W.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, “Room-temperature excitonic optical bistability in a GaAs–GaAlAs superlattice étalon,” Appl. Phys. Lett. 41, 221–222 (1982).
[Crossref]

Weinberger, D. A.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, “Room-temperature excitonic optical bistability in a GaAs–GaAlAs superlattice étalon,” Appl. Phys. Lett. 41, 221–222 (1982).
[Crossref]

Wherrett, B. S.

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, “Two-photon absorption in zincblende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D. Smith, “Effect of low-power nonlinear refraction on laser-beam propagation in InSb,” Opt. Lett. 4, 331–333 (1979).
[Crossref] [PubMed]

Williams, W. E.

M. J. Soileau, W. E. Williams, and E. W. Van Stryland, “Optical power limiter with picosecond response time,” IEEE J. Quantum Electron QE-19, 731–735 (1983).
[Crossref]

M. J. Soileau, W. E. Williams, E. W. Van Stryland, and S. F. Brown, “The use of self-focusing in the prevention of laser-induced damage,” in Proceedings of the 1981 Symposium on Laser-Induced Damage in Optical Materials (National Bureau Standards, Washington, D. C., to be published).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, London, 1965), p. 459.

Adv. Phys. (1)

A. Miller, D. A. B. Miller, and S. D. Smith, “Dynamic nonlinear optical processes in semiconductors,” Adv. Phys. 30, 697–800 (1981).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (4)

A. K. Kar, J. G. H. Mathew, S. D. Smith, B. Davis, and W. Prettl, “Optical bistability in InSb at room temperature with two-photon excitation,” Appl. Phys. Lett. 42, 334–336 (1983).
[Crossref]

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, “Room-temperature excitonic optical bistability in a GaAs–GaAlAs superlattice étalon,” Appl. Phys. Lett. 41, 221–222 (1982).
[Crossref]

D. A. B. Miller, D. S. Chemla, D. J. Eilenberger, P. W. Smith, A. C. Gossard, and W. T. Tsang, “Large room-temperature optical nonlinearity in GaAs/Ga1−x Alx As multiple quantum well structures,” Appl. Phys. Lett. 41, 679–681 (1982).
[Crossref]

C. C. Lee and H. Y. Fan, “Two-photon absorption and photoconductivity in GaAs and InP,” Appl. Phys. Lett. 20, 18–20 (1972).
[Crossref]

IEEE J. Quantum Electron (1)

M. J. Soileau, W. E. Williams, and E. W. Van Stryland, “Optical power limiter with picosecond response time,” IEEE J. Quantum Electron QE-19, 731–735 (1983).
[Crossref]

J. Phys. C (2)

J. M. Doviak, A. F. Gibson, M. F. Kimmit, and A. C. Walker, “Two-photon absorption in indium antimonide at 10.6 μ m,” J. Phys. C 6, 593–600 (1973).
[Crossref]

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[Crossref]

J. Phys. D (1)

J. M. Aaron, C. L. M. Ireland, and C. Grey Morgan, “Aberration effects in the interaction of focused laser beams with matter,” J. Phys. D 7, 1970–1971 (1974).
[Crossref]

Laser-Induced Damage in Optical Materials (1)

M. J. Soileau, J. B. Franck, and T. C. Veatch, “On self-focusing and spot-size dependence of laser-induced breakdown,” in Laser-Induced Damage in Optical Materials, Nat. Bur. Stand. Spec. Publ. 620, 385–393 (1980).

Nature (1)

P. D. McWane, “Variable focal length lenses using materials with intensity dependent refractive indices,” Nature 211, 1081–1082 (1966).
[Crossref]

Opt. Commun. (1)

D. A. B. Miller, M. H. Mozolowski, A. Miller, and S. D. Smith, “Nonlinear optical effects in InSb with a cw CO laser,” Opt. Commun. 27, 133–136 (1978).
[Crossref]

Opt. Lett. (2)

Phys. Rev. B (1)

A. M. Johnston, C. R. Pidgeon, and J. Dempsey, “Frequency dependence of two-photon absorption in InSB and Hg1−x Cdx Te,” Phys. Rev. B 22, 825–831 (1980).
[Crossref]

Phys. Rev. Lett. (2)

C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, “Two-photon absorption in zincblende semiconductors,” Phys. Rev. Lett. 42, 1785–1788 (1979).
[Crossref]

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[Crossref]

Prog. Quantum Electron. (2)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[Crossref]

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 59 (1975).
[Crossref]

Radiophys. Quantum Electron. (1)

A. E. Kaplan, “External self-focusing of light by a nonlinear layer,” Radiophys. Quantum Electron. 12, 692–696 (1969).
[Crossref]

Sov. Phys. Usp. (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609–636 (1968).
[Crossref]

Other (6)

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. D. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1151–1228.

A. E. Kaplan, “Optical bistability due to mutual self-action of counter-propagating light beams,” in Technical Digest Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1981), p. 118.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 503.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 355, 435.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, London, 1965), p. 459.

M. J. Soileau, W. E. Williams, E. W. Van Stryland, and S. F. Brown, “The use of self-focusing in the prevention of laser-induced damage,” in Proceedings of the 1981 Symposium on Laser-Induced Damage in Optical Materials (National Bureau Standards, Washington, D. C., to be published).

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Figures (6)

Fig. 1
Fig. 1

Far-field radial profiles given by the (dimensionless) intensity function J(ψ, Y) with ψ = 2θ/ζ and Y = |ρ| for an input laser amplitude with parabolic cross section: a, Y = 0; b, Y = 4; c, Y = 12; d, Y = 20.

Fig. 2
Fig. 2

Radial profiles as in Fig. 1 but with Gaussian-input cross section: a, Y = 0; b, Y = 4; c, Y = 10; d, Y = 20.

Fig. 3
Fig. 3

Power-transfer curve X(Y) for an optical power limiter with input amplitude possessing a parabolic cross section and a far-field aperture allowing 99% transmission at low laser powers.

Fig. 4
Fig. 4

Power-transfer curve as in Fig. 3 but with a Gaussian-input cross section. The input function X and transmission function Y are defined differently from in Fig. 3 because ν is different (see text). The dashed line shows the effect of increasing the radius of the aperture by a factor of 2.

Fig. 5
Fig. 5

Power-transfer curves X ¯ ( Y ¯ ) for parabolic cross-section input amplitude, 99% transmission at low powers, with increasing values of σ. The dashed line is the net transmission curve Xt( Y ¯) for the nonlinear medium.

Fig. 6
Fig. 6

The intensity function J ( ψ , Y ¯ , σ ) for parabolic cross-section input amplitude and fixed Y ¯. The input is constrained by ρ = βσ = 4.0, with A = (β/σ)1/2α2 taking the values a, A = 0.02; b, A = 0.20; c, A = 2.00.

Tables (1)

Tables Icon

Table 1 Possible Cross Sections of an Input Laser Beam Expressed in Terms of the Amplitude Profile F(θ) = A(θ, 0)/A0a

Equations (57)

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k 0 A 2 / x + T · ( A 2 T ϕ ) = - b A 2 χ ( A 2 ) ,
2 k 0 ϕ / x + ( T ϕ ) 2 - A - 1 T 2 A = + b χ ( A 2 ) ,
I ( r , ω ) = ½ n 0 c 0 A 2 ( r , x ) ,
θ = r / w 0 ,             θ = r / w 0 ,             ζ = x / R d ,
I ζ + 1 2 ( I θ ϕ θ + I T 2 ϕ ) = - α 1 I - α 2 I 2 ,
ϕ ζ + 1 4 [ ( ϕ θ ) 2 - A - 1 T 2 A ] = κ I .
P i = 2 π w 0 2 0 I ( θ , 0 ) θ d θ = ν - 1 π w 0 2 I 0 ,
Δ ϕ R d / L ,
I ( θ , ζ ) = I ( θ , 0 ) exp ( - α 1 ζ ) ,
Δ ϕ ( θ , ζ ) = κ a 1 [ 1 - exp ( - α 1 ζ ) ] I ( θ , 0 ) .
E ( θ , ζ ) = - 2 i ζ exp ( i θ / ζ ) × 0 R J 0 ( 2 θ θ / ζ ) A ( θ , ζ L ) exp [ i Δ ϕ ( θ , ζ L ) ] θ d θ ,
E 1 , 2 ( ψ ) = 0 R J 0 ( θ ψ ) F ( θ ) cos sin [ ( ν P i / P 1 ) F ( θ ) 2 ] θ d θ ,
E ( θ , ζ ) = - 2 i ζ exp ( i ψ / 2 ) [ E 1 ( ψ ) + i E 2 ( ψ ) ] ,
P 1 = 2 π n 0 α 1 { [ 1 - exp ( - α 1 ζ L ) ] k 0 2 n 2 } - 1 .
ρ ( ν ) = ν P i / P 1 .
E 1 + i E 2 = k = 0 ( i ρ ) k k ! 0 R F ( θ ) 2 k + 1 θ d θ ,
E 1 + i E 2 = exp ( i ρ ) 2 ( ν - 1 ) F 1 1 [ 1 ; ½ ( ν + 1 ) , - i ρ ] = 1 4 k = 0 ( i ρ ) k k ! ( k + ½ ( ν - 1 ) )
ν = 2 ( Gaussian ) :             E 1 + i E 2 = 1 2 [ C ( η ) + i S ( η ) ] / η ,             η = 2 ρ / π ,
ν = 3 ( parabolic ) :             E 1 + i E 2 = 1 4 ρ [ sin ρ + i ( 1 - cos ρ ) ] ,
E 1 + i E 2 = ρ - i 4 ρ e i ρ [ 1 + 0 ( ρ - 1 ) ] + ¼ Γ [ ½ ( ν - 1 ) ] ( - i ρ ) 1 / 2 ( ν - 1 ) [ 1 + 0 ( ρ - 1 ) ] .
E 1 ( ψ ) + i E 2 ( ψ ) = k = 0 ( i ρ ) k k ! 0 R J 0 ( θ ψ ) F ( θ ) 2 k + 1 θ d θ .
ν = 1 ( Lorentzian )             E 1 + i E 2 = k = 0 ( ¼ i ρ ψ 2 ) k k ! ( 2 k ) ! K 2 k ( ψ ) ,
ν = 2 ( Gaussian )             E 1 + i E 2 = 1 2 k = 0 ( i ρ ) k k ! ( 2 k + 1 ) × exp [ - ψ 2 / 4 ( 2 k + 1 ) ] ,
ν = 3 ( parabolic )             E 1 + i E 2 = 2 k = 0 ( 4 i ρ ) k k ! ( 2 k + 1 ) ! × J 2 ( k + 1 ) ( ψ ) / ψ 2 ( k + 1 ) ,
ν = 4 ( elliptical )             E 1 + i E 2 = k = 0 ( 2 i ρ ) k k ! Γ ( k + 3 / 2 ) Γ ( 1 / 2 ) × j k + 1 ( ψ / 2 ) / ( ψ / 2 ) k + 1 .
P 0 / P i = g ( ψ m , P i ) = 2 ν exp ( - α 1 ζ L ) × 0 ψ m [ E 1 2 ( ψ ) + E 2 2 ( ψ ) ] ψ d ψ ,
1 - f = 2 ν 0 ψ m E 1 2 ( ψ ) ψ d ψ ,
E 1 ( ψ ) = 0 R J 0 ( θ ψ ) F ( θ ) θ d θ .
ν = 1 :             f = ψ m 2 [ K 1 2 ( ψ m ) - K 0 2 ( ψ m ) ] ,
ν = 2 :             f = exp ( - ψ m 2 / 2 ) ,
ν = 3 :             f = [ J 1 2 ( ψ m ) + J 2 2 ( ψ m ) ] / 4 ψ m 2 ,
ν = 4 :             f = j 0 2 ( ψ m / 2 ) + j 1 2 ( ψ m / 2 ) .
ν = 1 : ψ m = 2.4915 , ν = 2 : ψ m = 3.0349 , ν = 3 : ψ m = 6.6450.
ϕ ( θ , ζ ) = κ I ( θ , 0 ) ζ + 0 ( ζ 2 ) ,
I ζ + 1 2 κ ζ G ( θ ) = - α 1 I - α 2 I 2 ,
I ( θ / ζ ) / I ( θ , 0 ) = e - α 1 ζ { 1 + I ( θ , 0 ) [ α 2 α 1 ( 1 - e - α 1 ζ ) + κ ζ 2 G ( θ ) ] + 0 ( ζ 3 ) } - 1 .
ϕ ( θ , ζ ) = κ α 2 ln { 1 + I ( θ , 0 ) α 2 α 1 [ 1 - e - α 1 ζ ] } .
P t / P i = 2 ν e - α 1 ζ L 0 R F ( θ ) 2 θ d θ 1 + ν P i P 2 F ( θ ) 2 ,
P 2 = π w 0 2 α 1 α 2 [ 1 - exp ( - α 1 ζ L ) ] - 1 .
ν = 1 :             P T = P I 1 / 2 [ π 2 - tan - 1 ( P I - 1 / 2 ) ] ,
ν = 2 :             P T = ½ ln ( 1 + 2 P I ) ,
ν = 3 :             P T = 1 - [ tan - 1 ( 3 P I ) 1 / 2 ] / ( 3 P I ) 1 / 2 ,
ν = 4 :             P T = ½ { 1 - [ ln ( 1 + 4 P I ) ] / 4 P I } ,
ν = 1 :             FWHM = 2 ( 2 + P I - 1 ) 1 / 2 ,
ν = 2 :             FWHM = 2 [ 1 2 ln ( 2 + 2 P I ) ] 1 / 2 ,
ν = 3 :             FWHM = 2 [ 1 - ( 2 + 3 P I ) - 1 / 2 ] 1 / 2 ,
ν = 4 :             FWHM = [ ( 1 + 4 P I ) / ( 1 + 2 P I ) ] 1 / 2 .
E 1 ( ψ ) + i E 2 ( ψ ) = k = 0 ( i σ - ½ k ) β k 0 R × J 0 ( θ ψ ) F ( θ ) 2 k + 1 θ d θ ,
σ = κ / α 2 ,             β ( ν ) = ν P i / ( σ P 1 ) .
( i σ - ½ k ) = Γ ( i σ + ½ ) / [ k ! Γ ( i σ - k + ½ ) ]
E 1 + i E 2 = 1 2 ( 1 - ν ) F 2 1 [ ½ - i σ , ½ ( ν - 1 ) ; ½ ( ν + 1 ) ; - β ] = 1 4 0 1 t ( ν - 3 ) / 2 ( 1 + β t ) i σ - 1 / 2 d t
E 1 ( ψ ) + i E 2 ( ψ ) = exp ( i σ ln β ) 2 β 1 / 2 ( ν - 2 + 2 i σ ) [ F ( ψ , ν ) + 0 ( β - 1 ) ] ,
E 1 , 2 ( ψ ) = 0 1 J 0 ( θ ψ ) ( 1 - θ 2 ) [ 1 + Y ¯ ( 1 - θ 2 ) 2 ] - 1 / 2 × cos sin { σ ln [ 1 + Y ¯ ( 1 - θ 2 ) 2 ] } θ d θ .
X t = 3 [ 1 - tan - 1 ( Y ¯ 1 / 2 ) / Y ¯ 1 / 2 ] .
I ( θ , ζ ) = I ( θ , 0 ) [ 1 + 2 α 3 I ( θ , 0 ) 2 ζ ] - 1 / 2 .
1 ζ 2 | 0 R J 0 ( 2 θ θ / ζ ) A ( θ , ζ L ) exp [ i Ψ ( ρ , θ , ζ L , ζ , y ) ] θ d θ | 2 ,
Ψ ( ρ , θ , ζ L , ζ , y ) = Δ ϕ ( ρ , θ , ζ L ) - y θ 2 / ζ 2 - Φ ( ζ , θ ) ,

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