Abstract

Z-scan studies have been carried out on CS2 with a 6.5-ns, doubled Nd:YAG source. Sample thicknesses ranged from the nearly thin- through the thick-sample regime. The data were analyzed with both an analytic theory, correct to first order in irradiance, and a Gaussian Laguerre modal-decomposition modeling approach, correct to all orders. It was found that the n2 values obtained through both methods of analysis were in agreement to within 10%. The results also indicate that a suitable thickness for a self-focusing optical power limiter is six times the Rayleigh length in the medium.

© 1994 Optical Society of America

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References

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  1. M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity single beam n2measurement,” Opt. Lett. 14, 955–957 (1989).
    [CrossRef] [PubMed]
  2. M. Sheik-Bahai, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
    [CrossRef]
  3. J. A. Hermann, “Beam propagation and optical power limiting with nonlinear media,” J. Opt. Soc. Am. B 1, 729–736 (1984).
    [CrossRef]
  4. J. A. Hermann, “Propagation effects due to nonlinear absorption and refraction within a thin medium,” J. Mod. Opt. 36, 445–470 (1989).
    [CrossRef]
  5. G. L. Wood, W. W. Clark, and E. J. Sharp, “Evaluation of thermal defocusing, nonlinear scattering, and nonlinear quarter-wave stack switches,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartman, M. J. Soileau, and V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 376–394 (1990).
    [CrossRef]
  6. M. Le Berre, E. Ressayre, and A. Tallet, “Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium,” Phys. Rev. A. 25, 1604 (1982).
    [CrossRef]
  7. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  8. W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256 (1968).
    [CrossRef]
  9. M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
    [CrossRef]
  10. J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993).
    [CrossRef]
  11. R. G. McDuff, Department of Physics, University of Queensland, St. Lucia, Queensland 4072, Australia (personal communication).
  12. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1224, 2–14 (1990).
    [CrossRef]
  13. P. B. Chapple, “Beam waist and M2 measurement using a finite slit,” Opt. Eng. (to be published).

1993 (1)

1991 (1)

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

1990 (1)

M. Sheik-Bahai, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

1989 (2)

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity single beam n2measurement,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

J. A. Hermann, “Propagation effects due to nonlinear absorption and refraction within a thin medium,” J. Mod. Opt. 36, 445–470 (1989).
[CrossRef]

1984 (1)

1982 (1)

M. Le Berre, E. Ressayre, and 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 (1)

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

1968 (1)

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

Chapple, P. B.

P. B. Chapple, “Beam waist and M2 measurement using a finite slit,” Opt. Eng. (to be published).

Clark, W. W.

G. L. Wood, W. W. Clark, and E. J. Sharp, “Evaluation of thermal defocusing, nonlinear scattering, and nonlinear quarter-wave stack switches,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartman, M. J. Soileau, and V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 376–394 (1990).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

M. Sheik-Bahai, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Haus, H. A.

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

Hermann, J. A.

Le Berre, M.

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

Marburger, J. H.

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

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

McDuff, R. G.

J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993).
[CrossRef]

R. G. McDuff, Department of Physics, University of Queensland, St. Lucia, Queensland 4072, Australia (personal communication).

Ressayre, E.

M. Le Berre, E. Ressayre, and 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. A.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

M. Sheik-Bahai, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity single beam n2measurement,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Sharp, E. J.

G. L. Wood, W. W. Clark, and E. J. Sharp, “Evaluation of thermal defocusing, nonlinear scattering, and nonlinear quarter-wave stack switches,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartman, M. J. Soileau, and V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 376–394 (1990).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity single beam n2measurement,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Sheik-Bahai, M.

M. Sheik-Bahai, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1224, 2–14 (1990).
[CrossRef]

Soileau, M. J.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Tallet, A.

M. Le Berre, E. Ressayre, and 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, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

M. Sheik-Bahai, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity single beam n2measurement,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Wagner, W. G.

W. G. Wagner, H. A. Haus, and 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-Bahai, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Wood, G. L.

G. L. Wood, W. W. Clark, and E. J. Sharp, “Evaluation of thermal defocusing, nonlinear scattering, and nonlinear quarter-wave stack switches,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartman, M. J. Soileau, and V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 376–394 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahai, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

J. Mod. Opt. (1)

J. A. Hermann, “Propagation effects due to nonlinear absorption and refraction within a thin medium,” J. Mod. Opt. 36, 445–470 (1989).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Eng. (1)

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

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

Phys. Rev. A. (1)

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

Prog. Quantum Electron. (1)

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

Other (4)

G. L. Wood, W. W. Clark, and E. J. Sharp, “Evaluation of thermal defocusing, nonlinear scattering, and nonlinear quarter-wave stack switches,” in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartman, M. J. Soileau, and V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 376–394 (1990).
[CrossRef]

R. G. McDuff, Department of Physics, University of Queensland, St. Lucia, Queensland 4072, Australia (personal communication).

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1224, 2–14 (1990).
[CrossRef]

P. B. Chapple, “Beam waist and M2 measurement using a finite slit,” Opt. Eng. (to be published).

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Figures (10)

Fig. 1
Fig. 1

Z-scan technique. The far-field on-axis transmission of a nonlinear sample NL is recorded as a function of z, the distance from the beam waist, by use of detector D.

Fig. 2
Fig. 2

Z scan for a thin self-focusing material.

Fig. 3
Fig. 3

Experimental arrangement; D1, D2, detectors; NL, nonlinear sample.

Fig. 4
Fig. 4

Beam radius versus axial position, determined from the second moment 〈y2〉, from scanning slit measurements.

Fig. 5
Fig. 5

Temporal profile of the laser pulse.

Fig. 6
Fig. 6

Raw Z-scan data for the 20-mm sample thickness at high and low powers (peak irradiances of approximately 1 and 0.2 GW/cm2).

Fig. 7
Fig. 7

Corrected Z-scan data for different sample lengths, together with fitted theoretical curve from the numerical calculation. (a) High power (peak irradiance ≃1 GW/cm2); (b) Medium power (peak irradiance ≃0.4 GW/cm2).

Fig. 8
Fig. 8

Experimental data together with fitted theoretical curve from the analytical (first-order) theory for 1- and 20-mm sample lengths.

Fig. 9
Fig. 9

First- and second-order contributions to the normalized transmission T.

Fig. 10
Fig. 10

ΔT versus sample length; theory and experiment. The effective length corresponding to an infinite length is shown.

Tables (1)

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Table 1 Experimental Values of γ

Equations (10)

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Δ T = 0.203 β l n 0 z r ,
β = 2 k γ I 0 z r n 0 2 = k n 2 E 0 2 z r n 0 .
γ ( m 2 / W ) = 40 π c ( m / s ) n 0 n s ( esu ) .
Δ T = ¼ β ln ( 9 + Ω - Γ 1 + Ω - Γ 1 + Ω + Γ 9 + Ω + Γ ) ,
Ω = η + 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 , ζ m = l / ( n 0 z r ) .
T ( β , z ) = 1 + P 1 ( z ) β + P 2 ( z ) β 2 + P 3 ( z ) β 3 .
T * ( β , z ) = 1 + T ( β , z ) - T ( r β , z ) = 1 + P 1 ( z ) ( 1 - r ) β + P 2 ( z ) ( 1 - r 2 ) β 2 + P 3 ( z ) ( 1 - r 3 ) β 3 .
β ( t ) = β 0 exp ( - t 2 / τ 2 ) .
T * ¯ ( β 0 , z ) = T * [ β ( t ) , z ] β ( t ) d t β ( t ) d t = 1 + P 1 ( z ) ( 1 - r ) β 0 2 + P 2 ( z ) ( 1 - r 2 ) ( β 0 2 / 3 ) + P 3 ( z ) ( 1 - r 3 ) ( β 0 3 / 2 ) .
l eff = Δ T thick Δ T thin l .

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