Abstract

We present the results of modeling of nanosecond pulse propagation in optically absorbing liquid media. Acoustic and electromagnetic wave equations must be solved simultaneously to model refractive index changes due to thermal expansion and/or electrostriction, which are highly transient phenomena on a nanosecond time scale. Although we consider situations with cylindrical symmetry and where the paraxial approximation is valid, this is still a computation-intensive problem, as beam propagation through optically thick media must be modeled. We compare the full solution of the acoustic wave equation with the approximation of instantaneous expansion (steady-state solution) and hence determine the regimes of validity of this approximation. We also find that the refractive index change obtained from the photo-acoustic equation overshoots its steady-state value once the ratio between the pulsewidth and the acoustic transit time exceeds a factor of unity.

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References

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  1. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto and J. R. Whinnery, "Long-transient effects in lasers with inserted liquid samples," J. Appl. Phys. 36, 3-8 (1965).
    [CrossRef]
  2. S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov and R. V. Khokhlov, "Thermal self-action of laser beams," IEEE J. Quantum Electron. QE-4, 568-575 (1968).
    [CrossRef]
  3. C. K. N. Patel and A. C. Tam, "Pulsed optoacoustic spectroscopy of condensed matter," Rev. Mod. Phys. 53, 517-550 (1981).
    [CrossRef]
  4. J. N. Hayes, "Thermal blooming of laser beams in fluids," Appl. Opt. 11, 455-461 (1972).
    [CrossRef] [PubMed]
  5. A. J. Twarowski and D. S. Kliger, "Multiphoton absorption spectra using thermal blooming. I. Theory," Chem. Phys. 20, 251-258 (1977).
  6. S. J. Sheldon, L. V. Knight and J. M. Thorne, "Laser-induced thermal lens effect: a new theoretical model," Appl. Opt. 21, 1663-1669 (1982).
    [CrossRef] [PubMed]
  7. P. R. Longaker and M. M. Litvak, "Perturbation of the refractive index of absorbing media by a pulsed laser beam," J. Appl. Phys. 40, 4033-4041 (1969).
    [CrossRef]
  8. Gu Liu, "Theory of the photoacoustic effect in condensed matter," Appl. Opt. 21, 955-960 (1982).
    [CrossRef] [PubMed]
  9. C. A. Carter and J. M. Harris, "Comparison of models describing the thermal lens effect," Appl. Opt. 23, 476-481 (1984).
    [CrossRef] [PubMed]
  10. A. M. Olaizola, G. Da Costa and J. A. Castillo, "Geometrical interpretation of a laser-induced thermal lens," Opt. Eng. 32, 1125-1130 (1993).
    [CrossRef]
  11. F. Jurgensen and W. Schroer, "Studies on the diffraction image of a thermal lens," Appl. Opt. 34, 41- 50 (1995).
    [CrossRef] [PubMed]
  12. S. Wu and N. J. Dovichi, "Fresnel diffraction theory for steady-state thermal lens measurements in thin films," J. Appl. Phys. 67, 1170-1182 (1990).
    [CrossRef]
  13. S. R. J. Brueck, H. Kildal and L. J. Belanger, "Photo-acoustic and photo-refractive detection of small absorptions in liquids," Opt. Comm. 34, 199-204 (1980).
    [CrossRef]
  14. J. -M. Heritier, "Electrostrictive limit and focusing effects in pulsed photoacoustic detection," Opt. Comm. 44, 267-272 (1983).
    [CrossRef]
  15. P. Brochard, V. Grolier-Mazza and R. Cabanel, "Thermal nonlinear refraction in dye solutions: a study of the transient regime," J. Opt. Soc. Am. B 14, 405-414 (1997)
    [CrossRef]
  16. D. J. Hagan, T. Xia, A. A. Said, T. H. Wei and E. W. Van Stryland, "High Dynamic Range Passive Optical Limiters," Int. J. Nonlinear Opt. Phys. 2, 483-501 (1993).
    [CrossRef]
  17. P. Miles, "Bottleneck optical limiters: the optimal use of excited-state absorbers," Appl. Opt. 33, 6965-6979 (1994).
    [CrossRef] [PubMed]
  18. T. Xia, D. J. Hagan, A. Dogariu, A. A. Said and E. W. Van Stryland, "Optimization of optical limiting devices based on excited-state absorption," Appl. Opt. 36, 4110-4122 (1997).
    [CrossRef] [PubMed]
  19. T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry and D. R. Coulter, "Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines," Appl. Phys. B 54, 46-51 (1992).
    [CrossRef]
  20. Jian-Gio Tian et al, "Position dispersion and optical limiting resulting from thermally induced nonlinearities in Chinese tea," Appl. Opt. 32, (1993).
    [CrossRef] [PubMed]
  21. Y. M. Cheung and S. K. Gayen, "Optical nonlinearities of tea studied by Z-scan and four-wave mixing techniques," J. Opt. Soc. Am. B 11, 636-643 (1994).
    [CrossRef]
  22. J. Castillo, V. P. Kozich et al, "Thermal lensing resulting from one- and two-photon absorption studied with a two-color time-resolved Z-scan," Opt. Lett. 19, 171-173 (1994).
    [CrossRef] [PubMed]
  23. D. Landau and E. M. Lifshitz, Course of theoretical physics. Volume 6. Fluid mechanics, (Pergamon Press).
  24. T. Xia, "Modeling and experimental studies of nonlinear optical self-action," Ph.D. thesis, Univ. of Central Florida (1994).
  25. R. W. Boyd, Nonlinear optics, (Academic Press, Inc. 1992).
  26. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical recipes. The art of scientific computing, (Cambridge University Press, 1986).
  27. D. Kovsh, S Yang, D. J. Hagan and E. W. Van Stryland; "Software for computer modeling of laser pulse propagation through the optical system with nonlinear optical elements," Proc. SPIE 3472, 163- 177 (1998).
    [CrossRef]
  28. D. Kovsh, S. Yang, D. Hagan and E. Van Stryland, "Nonlinear optical beam propagation for optical limiting," submitted to Applied Optics.
  29. M. Sheik-Bahae, A. A. Said and E. W. Van Stryland, "High-sensitivity, single-beam n2 measurements," Opt. Lett. 14, 955-957 (1989).
    [CrossRef] [PubMed]

Other (29)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto and J. R. Whinnery, "Long-transient effects in lasers with inserted liquid samples," J. Appl. Phys. 36, 3-8 (1965).
[CrossRef]

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov and R. V. Khokhlov, "Thermal self-action of laser beams," IEEE J. Quantum Electron. QE-4, 568-575 (1968).
[CrossRef]

C. K. N. Patel and A. C. Tam, "Pulsed optoacoustic spectroscopy of condensed matter," Rev. Mod. Phys. 53, 517-550 (1981).
[CrossRef]

J. N. Hayes, "Thermal blooming of laser beams in fluids," Appl. Opt. 11, 455-461 (1972).
[CrossRef] [PubMed]

A. J. Twarowski and D. S. Kliger, "Multiphoton absorption spectra using thermal blooming. I. Theory," Chem. Phys. 20, 251-258 (1977).

S. J. Sheldon, L. V. Knight and J. M. Thorne, "Laser-induced thermal lens effect: a new theoretical model," Appl. Opt. 21, 1663-1669 (1982).
[CrossRef] [PubMed]

P. R. Longaker and M. M. Litvak, "Perturbation of the refractive index of absorbing media by a pulsed laser beam," J. Appl. Phys. 40, 4033-4041 (1969).
[CrossRef]

Gu Liu, "Theory of the photoacoustic effect in condensed matter," Appl. Opt. 21, 955-960 (1982).
[CrossRef] [PubMed]

C. A. Carter and J. M. Harris, "Comparison of models describing the thermal lens effect," Appl. Opt. 23, 476-481 (1984).
[CrossRef] [PubMed]

A. M. Olaizola, G. Da Costa and J. A. Castillo, "Geometrical interpretation of a laser-induced thermal lens," Opt. Eng. 32, 1125-1130 (1993).
[CrossRef]

F. Jurgensen and W. Schroer, "Studies on the diffraction image of a thermal lens," Appl. Opt. 34, 41- 50 (1995).
[CrossRef] [PubMed]

S. Wu and N. J. Dovichi, "Fresnel diffraction theory for steady-state thermal lens measurements in thin films," J. Appl. Phys. 67, 1170-1182 (1990).
[CrossRef]

S. R. J. Brueck, H. Kildal and L. J. Belanger, "Photo-acoustic and photo-refractive detection of small absorptions in liquids," Opt. Comm. 34, 199-204 (1980).
[CrossRef]

J. -M. Heritier, "Electrostrictive limit and focusing effects in pulsed photoacoustic detection," Opt. Comm. 44, 267-272 (1983).
[CrossRef]

P. Brochard, V. Grolier-Mazza and R. Cabanel, "Thermal nonlinear refraction in dye solutions: a study of the transient regime," J. Opt. Soc. Am. B 14, 405-414 (1997)
[CrossRef]

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei and E. W. Van Stryland, "High Dynamic Range Passive Optical Limiters," Int. J. Nonlinear Opt. Phys. 2, 483-501 (1993).
[CrossRef]

P. Miles, "Bottleneck optical limiters: the optimal use of excited-state absorbers," Appl. Opt. 33, 6965-6979 (1994).
[CrossRef] [PubMed]

T. Xia, D. J. Hagan, A. Dogariu, A. A. Said and E. W. Van Stryland, "Optimization of optical limiting devices based on excited-state absorption," Appl. Opt. 36, 4110-4122 (1997).
[CrossRef] [PubMed]

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry and D. R. Coulter, "Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines," Appl. Phys. B 54, 46-51 (1992).
[CrossRef]

Jian-Gio Tian et al, "Position dispersion and optical limiting resulting from thermally induced nonlinearities in Chinese tea," Appl. Opt. 32, (1993).
[CrossRef] [PubMed]

Y. M. Cheung and S. K. Gayen, "Optical nonlinearities of tea studied by Z-scan and four-wave mixing techniques," J. Opt. Soc. Am. B 11, 636-643 (1994).
[CrossRef]

J. Castillo, V. P. Kozich et al, "Thermal lensing resulting from one- and two-photon absorption studied with a two-color time-resolved Z-scan," Opt. Lett. 19, 171-173 (1994).
[CrossRef] [PubMed]

D. Landau and E. M. Lifshitz, Course of theoretical physics. Volume 6. Fluid mechanics, (Pergamon Press).

T. Xia, "Modeling and experimental studies of nonlinear optical self-action," Ph.D. thesis, Univ. of Central Florida (1994).

R. W. Boyd, Nonlinear optics, (Academic Press, Inc. 1992).

W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical recipes. The art of scientific computing, (Cambridge University Press, 1986).

D. Kovsh, S Yang, D. J. Hagan and E. W. Van Stryland; "Software for computer modeling of laser pulse propagation through the optical system with nonlinear optical elements," Proc. SPIE 3472, 163- 177 (1998).
[CrossRef]

D. Kovsh, S. Yang, D. Hagan and E. Van Stryland, "Nonlinear optical beam propagation for optical limiting," submitted to Applied Optics.

M. Sheik-Bahae, A. A. Said and E. W. Van Stryland, "High-sensitivity, single-beam n2 measurements," Opt. Lett. 14, 955-957 (1989).
[CrossRef] [PubMed]

Supplementary Material (2)

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Figures (10)

Fig. 1.
Fig. 1.

Spatial distribution of the thermally induced refractive index change (τp = 4 ns, w 0 = 6 μm, TL = 80%, L = 1 mm, EIN = 5μJ). [Media 1]

Fig. 2(a).
Fig. 2(a).

Radial fluence distribution on the back surface of the sample (near field).

Fig. 2(b).
Fig. 2(b).

Radial fluence distribution on the detector (far field).

Fig. 3.
Fig. 3.

Spatial distribution of the thermally induced refractive index change (τp = 4 ns, w0 = 30 μm, TL = 80%, L = 1 mm, EIN = 125 μJ). [Media 2]

Fig. 4(a).
Fig. 4(a).

Radial fluence distribution on the back surface of the sample (near field).

Fig. 4(b).
Fig. 4(b).

Radial fluence distribution on the detector (far field).

Fig. 5.
Fig. 5.

Closed-aperture Z-scan of nigrosine solution in water (τp = 10 ns, w0 = 6 μm, TL = 90%, L = 200 μm, EIN = 2 μJ).

Fig. 6.
Fig. 6.

Closed-aperture Z-scan of nigrosine solution in water (τp = 10 ns, w0 = 30 μm, TL = 90%, L = 200 μm, EIN = 50 μJ).

Figure 7.
Figure 7.

Sensitivity (ΔTP-V ) of the closed-aperture Z-scan as a function of ratio between pulse width, τp , and acoustic transit time τac = w0 /CS . ΔTP-V is normalized to the value obtained for the steady state solution (s.s.).

Figure 8.
Figure 8.

On-axis refractive index change computed as a solution to the acoustic wave equation, Δnac(t), normalized to the steady state index distribution, Δnss(t). Values of the parameter τpac were chosen to be 0.5 (solid red), 1.5 (solid blue), 2.5 (solid green), 5.0 (dash red), 10 (dashed blue) and 15 (dashed green). The normalized Δnss(t) (with negative sign) is shown with dashed black line. The normalized intensity distribution (solid black) is plotted to show the time scale of the index changes.

Equations (12)

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ρ c p T t κ 2 T = Q ,
Δ n = ( n ρ ) T Δ ρ + ( n T ) ρ Δ T .
t [ 2 ( Δ ρ ) t 2 C S 2 2 ( Δ ρ ) ] = C S 2 β c p 2 ( α L I ) γ e 2 nc t 2 I .
2 ( Δ n ) t 2 C S 2 2 ( Δ n ) = γ e 2 β C S 2 c p t 2 ( α L I ( r , t′ ) ) dt .
Δ T ( r , t ) = 1 ρ c p t α L I ( r , t′ ) dt .
2 ( Δ n ) t 2 C S 2 2 ( Δ n ) = γ e β C S 2 2 n 2 ( Δ T ) .
Δ n ( d n d T ) Δ T ,
× × E ( r , t ) + 1 c 2 2 E ( r , t ) t 2 = μ 0 2 P ( r , t ) t 2 ,
2 jk ψ ( r , z , t ) z = 2 ψ ( r , z , t ) + ( k 0 2 χ NL ( r , z , t ) jk α L ) ψ ( r , z , t ) ,
χ NL ( r , z , t ) = χ NL ins ( r , z ) + χ NL cum ( r , z . t ) .
Re { χ NL } = 2 n 0 Δ n
Im { χ NL } = n 0 k 0 α = n 0 k 0 ( α L + α NL ) .

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