Abstract

The refractive and absorptive response of a nonlinear material can be assessed by the use of a spatial scanning technique to characterize the material. This technique generally involves monitoring the normalized trans-mittance of an optical beam focused into a sample of the material. In this paper an analytic solution of the beam propagation equation correct to the first nonlinear order has been obtained for the situation in which the sample has a thickness greater than the depth of focus. In this context simple exact formulas have been obtained for the cumulative phase and the normalized transmittance of a focused beam at any position along the optic axis, and the expressions have been used to investigate the various possible scanning techniques.

© 1993 Optical Society of America

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  1. A. E. Kaplan, “External self focusing of light by a nonlinear layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 869 (1969).
  2. P. P. Banerjee, R. M. Misra, M. Maghraoui, “Theoretical and experimental studies on propagation of beams through a finite sample of a cubically nonlinear material,” J. Opt. Soc. Am. B 8, 1072 (1991).
    [CrossRef]
  3. M. Le Berre, E. Ressayre, A. Tallet, “Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium,” Phys. Rev. A 25, 1604 (1982).
    [CrossRef]
  4. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35 (1975).
    [CrossRef]
  5. S. A. Akhmanov, A. D. Sukhorokov, R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609 (1968).
    [CrossRef]
  6. W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256 (1968).
    [CrossRef]
  7. J. A. Hermann, “Strong self-focusing: an aberrational approach,” J. Mod. Opt. 35, 1777 (1988).
    [CrossRef]
  8. D. J. Hagan, E. W. Van Stryland, M. J. Soileau, Y. Y. Wu, “Self-protecting semiconductor optical limiters,” Opt. Lett. 13, 315 (1988).
    [CrossRef] [PubMed]
  9. E. W. Van Stryland, Y. Y. Wu, D. J. Hagan, M. J. Soileau, K. Mansour, “Optical limiting with semiconductors,” J. Opt. Soc. Am. B 5, 1980 (1988).
    [CrossRef]
  10. D. J. Hagan, E. W. Van Stryland, Y. Y. Wu, T. H. Wei, M. Sheik-Bahae, A. Said, K. Mansour, J. Young, M. J. Soileau, “Passive broadband high dynamic range semiconductor limiters,” in Materials for Optical Switches, Isolators, and Limiters, M. J. Soileau, ed., Proc. Soc. Photo-Opt. In-strum. Eng.1105, 103 (1989).
    [CrossRef]
  11. M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlineari-ties using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
    [CrossRef]
  12. A. A. Said, M. Sheik-bahae, D. J. Hagan, E. J. Canto-Said, Y. Y. Wu, J. Young, T. H. Wei, E. W. Van Stryland, “Non-linearities in semiconductors for optical limiting,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 294 (1990).
    [CrossRef]
  13. In Ref. 11 and elsewhere nsn0is designated γ, and αs is designated β.
  14. V. N. Lugovoi, “Propagation of wave beams in nonlinear media,” Sov. Phys. Dokl. 12, 866 (1968).
  15. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 227.
  16. R. McDuff, A. E. Smith, N. R. Heckenberg, J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” J. Nonlinear Opt. Phys. 1, 265 (1992).
    [CrossRef]
  17. V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199 (1970).
  18. One obtains the virtual solution by interpreting the field at face b as having arisen by propagation in vacuo. By projecting backward, one can locate the position of the virtual waist.
  19. J. A. Hermann, “Self-focusing effects and applications using thin nonlinear media,” J. Nonlinear Opt. Phys. 1, 541 (1992).
    [CrossRef]
  20. J. A. Hermann, P. B. Chappie, “External self-focusing in a two-lens system: shift and compression of the focal profile,” J. Mod. Opt. 38, 1035 (1991).
    [CrossRef]
  21. Equations (37) and (38) imply a maximum effective interaction length, as defined by Eq. (13) of Ref. 22, of leff= 2.71 (90% achievable in a sample length of six Rayleigh lengths). Experimental confirmation of this result was obtained recently by P. B. Chappie, J. Staromlynska, R. G. McDuff, “Z-scan studies in the thin and thick sample limits,” submitted to J. Opt. Soc. Am. B.
  22. M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228 (1991).
    [CrossRef]
  23. In this limit λˆ1/2→ 3 if |ζ0| ≪ ζm, and λˆ1/2 → 1/3 if |ζ0| ~ ζm

1992

R. McDuff, A. E. Smith, N. R. Heckenberg, J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” J. Nonlinear Opt. Phys. 1, 265 (1992).
[CrossRef]

J. A. Hermann, “Self-focusing effects and applications using thin nonlinear media,” J. Nonlinear Opt. Phys. 1, 541 (1992).
[CrossRef]

1991

J. A. Hermann, P. B. Chappie, “External self-focusing in a two-lens system: shift and compression of the focal profile,” J. Mod. Opt. 38, 1035 (1991).
[CrossRef]

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228 (1991).
[CrossRef]

P. P. Banerjee, R. M. Misra, M. Maghraoui, “Theoretical and experimental studies on propagation of beams through a finite sample of a cubically nonlinear material,” J. Opt. Soc. Am. B 8, 1072 (1991).
[CrossRef]

1990

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlineari-ties using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

1988

1982

M. Le Berre, E. Ressayre, A. Tallet, “Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium,” Phys. Rev. A 25, 1604 (1982).
[CrossRef]

1975

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

1970

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199 (1970).

1969

A. E. Kaplan, “External self focusing of light by a nonlinear layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 869 (1969).

1968

S. A. Akhmanov, A. D. Sukhorokov, R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256 (1968).
[CrossRef]

V. N. Lugovoi, “Propagation of wave beams in nonlinear media,” Sov. Phys. Dokl. 12, 866 (1968).

Abramowitz, M.

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 227.

Akhmanov, S. A.

S. A. Akhmanov, A. D. Sukhorokov, R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Banerjee, P. P.

Canto-Said, E. J.

A. A. Said, M. Sheik-bahae, D. J. Hagan, E. J. Canto-Said, Y. Y. Wu, J. Young, T. H. Wei, E. W. Van Stryland, “Non-linearities in semiconductors for optical limiting,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 294 (1990).
[CrossRef]

Chappie, P. B.

J. A. Hermann, P. B. Chappie, “External self-focusing in a two-lens system: shift and compression of the focal profile,” J. Mod. Opt. 38, 1035 (1991).
[CrossRef]

Equations (37) and (38) imply a maximum effective interaction length, as defined by Eq. (13) of Ref. 22, of leff= 2.71 (90% achievable in a sample length of six Rayleigh lengths). Experimental confirmation of this result was obtained recently by P. B. Chappie, J. Staromlynska, R. G. McDuff, “Z-scan studies in the thin and thick sample limits,” submitted to J. Opt. Soc. Am. B.

Hagan, D. J.

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228 (1991).
[CrossRef]

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlineari-ties using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

D. J. Hagan, E. W. Van Stryland, M. J. Soileau, Y. Y. Wu, “Self-protecting semiconductor optical limiters,” Opt. Lett. 13, 315 (1988).
[CrossRef] [PubMed]

E. W. Van Stryland, Y. Y. Wu, D. J. Hagan, M. J. Soileau, K. Mansour, “Optical limiting with semiconductors,” J. Opt. Soc. Am. B 5, 1980 (1988).
[CrossRef]

D. J. Hagan, E. W. Van Stryland, Y. Y. Wu, T. H. Wei, M. Sheik-Bahae, A. Said, K. Mansour, J. Young, M. J. Soileau, “Passive broadband high dynamic range semiconductor limiters,” in Materials for Optical Switches, Isolators, and Limiters, M. J. Soileau, ed., Proc. Soc. Photo-Opt. In-strum. Eng.1105, 103 (1989).
[CrossRef]

A. A. Said, M. Sheik-bahae, D. J. Hagan, E. J. Canto-Said, Y. Y. Wu, J. Young, T. H. Wei, E. W. Van Stryland, “Non-linearities in semiconductors for optical limiting,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 294 (1990).
[CrossRef]

Haus, H. A.

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256 (1968).
[CrossRef]

Heckenberg, N. R.

R. McDuff, A. E. Smith, N. R. Heckenberg, J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” J. Nonlinear Opt. Phys. 1, 265 (1992).
[CrossRef]

Hermann, J. A.

R. McDuff, A. E. Smith, N. R. Heckenberg, J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” J. Nonlinear Opt. Phys. 1, 265 (1992).
[CrossRef]

J. A. Hermann, “Self-focusing effects and applications using thin nonlinear media,” J. Nonlinear Opt. Phys. 1, 541 (1992).
[CrossRef]

J. A. Hermann, P. B. Chappie, “External self-focusing in a two-lens system: shift and compression of the focal profile,” J. Mod. Opt. 38, 1035 (1991).
[CrossRef]

J. A. Hermann, “Strong self-focusing: an aberrational approach,” J. Mod. Opt. 35, 1777 (1988).
[CrossRef]

Kaplan, A. E.

A. E. Kaplan, “External self focusing of light by a nonlinear layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 869 (1969).

Khokhlov, R. V.

S. A. Akhmanov, A. D. Sukhorokov, R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Le Berre, M.

M. Le Berre, E. Ressayre, A. Tallet, “Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium,” Phys. Rev. A 25, 1604 (1982).
[CrossRef]

Lugovoi, V. N.

V. N. Lugovoi, “Propagation of wave beams in nonlinear media,” Sov. Phys. Dokl. 12, 866 (1968).

Maghraoui, M.

Mansour, K.

E. W. Van Stryland, Y. Y. Wu, D. J. Hagan, M. J. Soileau, K. Mansour, “Optical limiting with semiconductors,” J. Opt. Soc. Am. B 5, 1980 (1988).
[CrossRef]

D. J. Hagan, E. W. Van Stryland, Y. Y. Wu, T. H. Wei, M. Sheik-Bahae, A. Said, K. Mansour, J. Young, M. J. Soileau, “Passive broadband high dynamic range semiconductor limiters,” in Materials for Optical Switches, Isolators, and Limiters, M. J. Soileau, ed., Proc. Soc. Photo-Opt. In-strum. Eng.1105, 103 (1989).
[CrossRef]

Marburger, J. H.

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

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256 (1968).
[CrossRef]

McDuff, R.

R. McDuff, A. E. Smith, N. R. Heckenberg, J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” J. Nonlinear Opt. Phys. 1, 265 (1992).
[CrossRef]

McDuff, R. G.

Equations (37) and (38) imply a maximum effective interaction length, as defined by Eq. (13) of Ref. 22, of leff= 2.71 (90% achievable in a sample length of six Rayleigh lengths). Experimental confirmation of this result was obtained recently by P. B. Chappie, J. Staromlynska, R. G. McDuff, “Z-scan studies in the thin and thick sample limits,” submitted to J. Opt. Soc. Am. B.

Misra, R. M.

Ressayre, E.

M. Le Berre, E. Ressayre, A. Tallet, “Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium,” Phys. Rev. A 25, 1604 (1982).
[CrossRef]

Said, A.

D. J. Hagan, E. W. Van Stryland, Y. Y. Wu, T. H. Wei, M. Sheik-Bahae, A. Said, K. Mansour, J. Young, M. J. Soileau, “Passive broadband high dynamic range semiconductor limiters,” in Materials for Optical Switches, Isolators, and Limiters, M. J. Soileau, ed., Proc. Soc. Photo-Opt. In-strum. Eng.1105, 103 (1989).
[CrossRef]

Said, A. A.

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228 (1991).
[CrossRef]

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlineari-ties using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

A. A. Said, M. Sheik-bahae, D. J. Hagan, E. J. Canto-Said, Y. Y. Wu, J. Young, T. H. Wei, E. W. Van Stryland, “Non-linearities in semiconductors for optical limiting,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 294 (1990).
[CrossRef]

Sheik-bahae, M.

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228 (1991).
[CrossRef]

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlineari-ties using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

D. J. Hagan, E. W. Van Stryland, Y. Y. Wu, T. H. Wei, M. Sheik-Bahae, A. Said, K. Mansour, J. Young, M. J. Soileau, “Passive broadband high dynamic range semiconductor limiters,” in Materials for Optical Switches, Isolators, and Limiters, M. J. Soileau, ed., Proc. Soc. Photo-Opt. In-strum. Eng.1105, 103 (1989).
[CrossRef]

A. A. Said, M. Sheik-bahae, D. J. Hagan, E. J. Canto-Said, Y. Y. Wu, J. Young, T. H. Wei, E. W. Van Stryland, “Non-linearities in semiconductors for optical limiting,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 294 (1990).
[CrossRef]

Smith, A. E.

R. McDuff, A. E. Smith, N. R. Heckenberg, J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” J. Nonlinear Opt. Phys. 1, 265 (1992).
[CrossRef]

Soileau, M. J.

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228 (1991).
[CrossRef]

E. W. Van Stryland, Y. Y. Wu, D. J. Hagan, M. J. Soileau, K. Mansour, “Optical limiting with semiconductors,” J. Opt. Soc. Am. B 5, 1980 (1988).
[CrossRef]

D. J. Hagan, E. W. Van Stryland, M. J. Soileau, Y. Y. Wu, “Self-protecting semiconductor optical limiters,” Opt. Lett. 13, 315 (1988).
[CrossRef] [PubMed]

D. J. Hagan, E. W. Van Stryland, Y. Y. Wu, T. H. Wei, M. Sheik-Bahae, A. Said, K. Mansour, J. Young, M. J. Soileau, “Passive broadband high dynamic range semiconductor limiters,” in Materials for Optical Switches, Isolators, and Limiters, M. J. Soileau, ed., Proc. Soc. Photo-Opt. In-strum. Eng.1105, 103 (1989).
[CrossRef]

Staromlynska, J.

Equations (37) and (38) imply a maximum effective interaction length, as defined by Eq. (13) of Ref. 22, of leff= 2.71 (90% achievable in a sample length of six Rayleigh lengths). Experimental confirmation of this result was obtained recently by P. B. Chappie, J. Staromlynska, R. G. McDuff, “Z-scan studies in the thin and thick sample limits,” submitted to J. Opt. Soc. Am. B.

Stegun, I.

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 227.

Sukhorokov, A. D.

S. A. Akhmanov, A. D. Sukhorokov, R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Talanov, V. I.

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199 (1970).

Tallet, A.

M. Le Berre, E. Ressayre, A. Tallet, “Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium,” Phys. Rev. A 25, 1604 (1982).
[CrossRef]

Van Stryland, E. W.

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228 (1991).
[CrossRef]

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlineari-ties using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

D. J. Hagan, E. W. Van Stryland, M. J. Soileau, Y. Y. Wu, “Self-protecting semiconductor optical limiters,” Opt. Lett. 13, 315 (1988).
[CrossRef] [PubMed]

E. W. Van Stryland, Y. Y. Wu, D. J. Hagan, M. J. Soileau, K. Mansour, “Optical limiting with semiconductors,” J. Opt. Soc. Am. B 5, 1980 (1988).
[CrossRef]

D. J. Hagan, E. W. Van Stryland, Y. Y. Wu, T. H. Wei, M. Sheik-Bahae, A. Said, K. Mansour, J. Young, M. J. Soileau, “Passive broadband high dynamic range semiconductor limiters,” in Materials for Optical Switches, Isolators, and Limiters, M. J. Soileau, ed., Proc. Soc. Photo-Opt. In-strum. Eng.1105, 103 (1989).
[CrossRef]

A. A. Said, M. Sheik-bahae, D. J. Hagan, E. J. Canto-Said, Y. Y. Wu, J. Young, T. H. Wei, E. W. Van Stryland, “Non-linearities in semiconductors for optical limiting,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 294 (1990).
[CrossRef]

Wagner, W. G.

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256 (1968).
[CrossRef]

Wei, T. H.

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlineari-ties using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

A. A. Said, M. Sheik-bahae, D. J. Hagan, E. J. Canto-Said, Y. Y. Wu, J. Young, T. H. Wei, E. W. Van Stryland, “Non-linearities in semiconductors for optical limiting,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 294 (1990).
[CrossRef]

D. J. Hagan, E. W. Van Stryland, Y. Y. Wu, T. H. Wei, M. Sheik-Bahae, A. Said, K. Mansour, J. Young, M. J. Soileau, “Passive broadband high dynamic range semiconductor limiters,” in Materials for Optical Switches, Isolators, and Limiters, M. J. Soileau, ed., Proc. Soc. Photo-Opt. In-strum. Eng.1105, 103 (1989).
[CrossRef]

Wu, Y. Y.

E. W. Van Stryland, Y. Y. Wu, D. J. Hagan, M. J. Soileau, K. Mansour, “Optical limiting with semiconductors,” J. Opt. Soc. Am. B 5, 1980 (1988).
[CrossRef]

D. J. Hagan, E. W. Van Stryland, M. J. Soileau, Y. Y. Wu, “Self-protecting semiconductor optical limiters,” Opt. Lett. 13, 315 (1988).
[CrossRef] [PubMed]

D. J. Hagan, E. W. Van Stryland, Y. Y. Wu, T. H. Wei, M. Sheik-Bahae, A. Said, K. Mansour, J. Young, M. J. Soileau, “Passive broadband high dynamic range semiconductor limiters,” in Materials for Optical Switches, Isolators, and Limiters, M. J. Soileau, ed., Proc. Soc. Photo-Opt. In-strum. Eng.1105, 103 (1989).
[CrossRef]

A. A. Said, M. Sheik-bahae, D. J. Hagan, E. J. Canto-Said, Y. Y. Wu, J. Young, T. H. Wei, E. W. Van Stryland, “Non-linearities in semiconductors for optical limiting,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 294 (1990).
[CrossRef]

Young, J.

A. A. Said, M. Sheik-bahae, D. J. Hagan, E. J. Canto-Said, Y. Y. Wu, J. Young, T. H. Wei, E. W. Van Stryland, “Non-linearities in semiconductors for optical limiting,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 294 (1990).
[CrossRef]

D. J. Hagan, E. W. Van Stryland, Y. Y. Wu, T. H. Wei, M. Sheik-Bahae, A. Said, K. Mansour, J. Young, M. J. Soileau, “Passive broadband high dynamic range semiconductor limiters,” in Materials for Optical Switches, Isolators, and Limiters, M. J. Soileau, ed., Proc. Soc. Photo-Opt. In-strum. Eng.1105, 103 (1989).
[CrossRef]

IEEE J. Quantum Electron.

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlineari-ties using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

A. E. Kaplan, “External self focusing of light by a nonlinear layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 869 (1969).

J. Mod. Opt.

J. A. Hermann, “Strong self-focusing: an aberrational approach,” J. Mod. Opt. 35, 1777 (1988).
[CrossRef]

J. A. Hermann, P. B. Chappie, “External self-focusing in a two-lens system: shift and compression of the focal profile,” J. Mod. Opt. 38, 1035 (1991).
[CrossRef]

J. Nonlinear Opt. Phys.

R. McDuff, A. E. Smith, N. R. Heckenberg, J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” J. Nonlinear Opt. Phys. 1, 265 (1992).
[CrossRef]

J. A. Hermann, “Self-focusing effects and applications using thin nonlinear media,” J. Nonlinear Opt. Phys. 1, 541 (1992).
[CrossRef]

J. Opt. Soc. Am. B

JETP Lett.

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199 (1970).

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[CrossRef]

In Ref. 11 and elsewhere nsn0is designated γ, and αs is designated β.

One obtains the virtual solution by interpreting the field at face b as having arisen by propagation in vacuo. By projecting backward, one can locate the position of the virtual waist.

In this limit λˆ1/2→ 3 if |ζ0| ≪ ζm, and λˆ1/2 → 1/3 if |ζ0| ~ ζm

Equations (37) and (38) imply a maximum effective interaction length, as defined by Eq. (13) of Ref. 22, of leff= 2.71 (90% achievable in a sample length of six Rayleigh lengths). Experimental confirmation of this result was obtained recently by P. B. Chappie, J. Staromlynska, R. G. McDuff, “Z-scan studies in the thin and thick sample limits,” submitted to J. Opt. Soc. Am. B.

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Figures (8)

Fig. 1
Fig. 1

Arrangement of the active nonlinear medium in relation to the optical beam and the detector D. The displacement ζ0 is always positive when the waist (in vacuo) is to the right of the entrance face a.

Fig. 2
Fig. 2

Computed Z scans for β = 0.1 with the following increasing values of the effective thickness ζm: a, 0.1; b, 0.5; c, 1.0; d, 5.0; and e, 10.0.

Fig. 3
Fig. 3

Z-scan full peak width at half-maximum, Δζ as a function of ζm.

Fig. 4
Fig. 4

Ratio of Δζ to the interpeak distance ΔZ as a function of ζm.

Fig. 5
Fig. 5

Computed T scans for β = 0.1 and ζm = 10.0 with the following increasing values of the detector distance ζd: a, 0.0; b, 0.5; c, 1.0; d, 10.0 (for convenience, n0 = 1.0).

Fig. 6
Fig. 6

Near-field T-scan profiles for ζm = 10.0 with the following increasing values of the ratio α/β: a, 0.0; b, 0.25; c, 0.50; d, 1.0 (for convenience, n0 = 1.0).

Fig. 7
Fig. 7

Contour profiles of T(ζ0, ζ1) with β = 0.1 and ζm = 10.0; the solid curves represent values of T greater than or equal to 1.0, while dotted curves represent values of T less than 1.0. The oblique dotted–dashed lines are (a) ζ1 = −ζ0 and (b) ζ1 = n0ζmζ0

Fig. 8
Fig. 8

Three-dimensional surface profile corresponding to Fig. 7.

Equations (71)

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E ( ζ , σ ) = E / E 0 = exp ( i φ ψ ) ,
i E ζ + σ T 2 E + 1 2 B | E | 2 E + 1 2 i α 0 E = 0 ,
χ ζ = 1 2 i α 0 + 1 2 B exp ( 2 ψ ) + σ ( i T 2 χ χ σ 2 ) ,
β = k n 2 | E 0 | 2 z r = 2 k n s I 0 z r n 0 ,
α = α 2 | E 0 | 2 z r = α s I 0 z r .
χ ( ζ , σ ) = k = 0 [ υ k ( ζ ) + i u k ( ζ ) ] σ k
χ ( ζ , σ ) = i [ ( 1 + i ζ ) 1 σ + ln ( 1 + i ζ ) ] + 1 2 B exp ( 2 σ ) F ( ζ , σ ) ,
F ( ζ , σ ) = ζ + i ( 1 4 σ ) ζ 2 + 1 3 ( 7 46 σ + 32 σ 2 ) ζ 3 1 3 i ( 15 150 σ + 216 σ 2 64 σ 3 ) ζ 4 + .
y = tan 1 ζ , x = 4 σ / ( 1 + ζ 2 ) ,
χ y = 1 2 i α 0 sec 2 y + 1 2 B exp ( 1 2 x α 0 tan y ) + 4 x [ i ( χ x x + x 1 χ x ) χ x 2 + 1 2 ( tan y ) χ x ] .
χ ( x , y ) = χ 0 ( x , y ) + 1 2 B g ( x , y ) + O ( B 2 ) .
g y = exp ( 1 2 x α 0 tan y ) + 4 i [ x g x x + ( 1 1 2 x ) g x ] .
g ( x , y ) = k = 0 A k ( y ) Ψ k ( x ) ,
Ψ k ( x ) = exp ( 1 2 x ) L k ( x ) ,
A k ( y ) = m = 0 k ( 1 ) m + k ( k m ) [ 1 exp ( 2 i m y ) 2 i m ] .
g ( x , y ) = 1 4 i 1 u exp ( 1 2 x u ) u d u , u = [ 2 exp ( 2 i y ) 1 ] 1 .
g ( x , y ) = 1 4 [ E 1 ( 1 2 x ) E 1 ( 1 2 x u ) ] ,
g ( 0 , y ) = k = 0 A k ( y ) = 1 4 i ln [ 2 exp ( 2 i y ) 1 ] ,
g ( 0 , y ) = g 0 r + i g 0 i ,
g 0 r = 1 4 tan 1 ( 4 ζ 1 3 ζ 2 ) , g 0 i = 1 8 ln ( 1 + 9 ζ 2 1 + ζ 2 ) .
ψ ( 0 , y ) = 1 2 ln ( 1 + ζ 2 ) + 1 2 ( α g 0 r + β g 0 i ) ,
φ ( 0 , y ) = tan 1 ζ + 1 2 ( β g 0 r α g 0 i ) ,
ζ = Λ ζ , σ = Λ 2 σ , E 0 = Λ 1 E 0 , φ = φ + Λ σ / R ,
E t ( x m , y m ) = E lin ( x m , y m ) [ 1 + 1 2 i B g ( x m , y m ) + O ( B 2 ) ] ,
E lin ( x m , y m ) = ( γ μ ) 1 / 2 exp ( i y m ) × exp { 1 4 x m [ 1 i ( 1 + u 2 ) ξ m + i u 1 ] } ,
x m = 4 σ / γ μ ,
y m = tan 1 ( Λ ξ m ) = tan 1 ζ 0 + tan 1 ( ζ m ζ 0 ) ,
μ = Λ 2 + ξ m 2 ,
γ = 1 + ζ 0 2 ,
Λ = ( 1 ξ m / u ) 1 ,
u = R / γ = ζ 0 1 ,
ξ m = ζ m / γ .
E ( x d , y d ) = i exp ( 1 4 i x d / ξ d ) 4 ξ d 0 J 0 [ 2 ( x m x d ) 1 / 2 / ξ d ] × exp ( 1 4 i x m / ξ d ) ) E ( x m , y m ) d x m ,
E = 1 + 1 2 i B ( 1 + i υ b 3 + i υ b ) k = 0 A k ( y m ) w k ,
E = 1 + 1 8 B In [ ( 1 + w ) exp ( 2 i y m ) w ] ,
υ b = ξ d + u 1 ( 1 + u 2 ) ξ m ,
w = ( 1 i υ b ) / ( 3 + i υ b ) .
T = 1 1 8 β F 1 ( υ a , υ b ) 1 4 α F 2 ( υ a , υ b ) ,
F 1 ( υ a , υ b ) = ln ( 9 + υ b 2 1 + υ b 2 × 1 + υ a 2 9 + υ a 2 ) ,
F 2 ( υ a , υ b ) = tan 1 ( 4 υ a 3 υ a 2 ) tan 1 ( 4 υ b 3 υ b 2 ) + ,
Δ ϕ NL = 1 16 α F 1 ( υ a , υ b ) + 1 8 β F 2 ( υ a , υ b ) ,
υ a = υ ( ζ 0 , ζ 1 , ζ m ) = ζ 0 ( 1 + ζ 0 2 ) / ( ζ 1 + ξ 0 ) , ζ 1 = ζ 1 Δ ζ ,
υ b = υ ( ζ 0 ζ m , ζ 1 , ζ m ) ,
T ζ 0 = 2 β υ b + α ( 3 + υ b 2 ) [ 1 + ( ζ m ζ 0 ) 2 ] ( 9 + υ b 2 ) 2 β υ a + α ( 3 + υ a 2 ) ( 1 + ζ 0 2 ) ( 9 + υ a 2 ) ,
Δ ϕ NL ζ 0 = α υ b 1 2 β ( 3 + υ b 2 ) [ 1 + ( ζ m ζ 0 ) 2 ] ( 9 + υ b 2 ) α υ a 1 2 β ( 3 + υ a 2 ) ( 1 + ζ 0 2 ) ( 9 + υ a 2 ) .
3 x 2 + ( 10 + ζ m 2 ) x 9 = 0 , x = ( ζ 0 ζ m ) ζ 0 .
Δ T = 1 4 β ln [ h ( ζ m ) ] ,
h ( ζ m ) = 9 + Ω Γ 1 + Ω Γ × 1 + Ω + Γ 9 + Ω + Γ ,
Ω = η + 1 3 ζ m 2 ,
Γ = ζ m ( η + 1 12 ζ m 2 ) 1 / 2 ,
η = 5 3 + [ 3 + 1 9 ( 5 + 1 2 ζ m 2 ) 2 ] 1 / 2 .
lim ζ m Δ T = 1 2 β ln 3 ,
lim ζ m Δ T = 0.406 κ ¯ , κ ¯ = 1 2 β ζ m .
Δ Z thick 2 = Δ Z thin 2 + U ( ζ m ) ,
U ( ζ m ) = 1 3 ζ m 2 + 4 { [ 3 + 1 9 ( 5 + 1 2 ζ m 2 ) 2 ] 1 / 2 ( 3 + 25 / 9 ) 1 / 2 } ,
Δ Z thin = 1.717.
λ 1 / 2 = exp ( 2 Δ T / β ) = [ h ( ζ m ) ] 1 / 2 ,
( 9 + ζ 0 2 ) [ 1 + ( ζ 0 ζ m ) 2 ] = λ ˆ 1 / 2 ( 1 + ζ 0 2 ) [ 9 + ( ζ 0 ζ m ) 2 ] ,
ζ 0 2 ( ζ m + ζ d ) 2 ( ζ m + 2 ζ d ) ( 2 ζ 0 ζ m ) + 1 = 0.
ζ d 1 1 2 ζ m + ( 1 + 1 4 ζ m 2 ) 1 / 2 .
A ( ρ , ζ 0 , ζ ) = ρ w 1 exp ( ρ 2 / ρ w 2 ) , ρ w 2 = γ μ = 1 + ( ζ 0 ζ ) 2 ,
φ 0 ζ = ρ w 2 , A 0 ζ = A 0 R ,
R 1 = z r R ( ζ 0 , ζ ) 1 = ( ζ 0 ζ ) / ρ w 2 .
y m = tan 1 ( γ 1 Λ ζ m ) = tan 1 [ ζ m / ( 1 + ζ 0 2 ζ 0 ζ m ) ] .
υ ( ζ 0 , η ) = ζ 0 ( 1 + ζ 0 2 ) / ( ζ 0 + η ) = tan ( π / 2 + tan 1 ζ 0 + tan 1 η )
tan 1 υ a tan 1 υ b = tan 1 ζ 0 tan 1 ( ζ 0 ζ m ) = y m .
E = 1 + 1 8 B ln ψ ,
ψ = ( 1 + w ) exp ( 2 i y m ) w , w = ( 1 i υ b ) / ( 3 + i υ b ) .
ψ = 1 + i υ b 3 + i υ b [ 2 + exp ( 2 i λ ) ] exp ( 2 i y m ) ,
ψ = ( 1 + i υ a 3 i υ a ) / ( 1 + i υ b 3 i υ b ) .
T = | E | = 1 + 1 4 Re ( B ln ψ ) + O ( B 2 ) = 1 1 8 β ln ( 9 + υ b 2 1 + υ b 2 1 + υ a 2 9 + υ a 2 ) 1 4 α [ tan 1 ( 4 υ a 3 υ a 2 ) tan 1 ( 4 υ b 3 υ b 2 ) ] + .

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