Abstract

We implement numerical modeling of high-energy laser-pulse propagation through bulk nonlinear optical materials using focused beams. An executable program with a graphical user interface is made available to researchers for modeling the propagation of beams through materials much thicker than the diffraction length (up to 103 times longer). Ultrafast nonlinearities of the bound-electronic Kerr effect and two-photon absorption as well as time-dependent excited-state and thermal nonlinearities are taken into account. The hydrodynamic equations describing the rarefaction of the medium that is due to heating are solved to determine thermal index changes for nanosecond laser pulses. We also show how this effect can be simplified in some cases by an approximation that assumes instantaneous expansion (so-called thermal lensing approximation). Comparisons of numerical results with several Z-scan, optical limiting and beam distortion experiments are presented. Possible application to optimization of a passive optical limiter design is discussed.

© 1999 Optical Society of America

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    [CrossRef]
  43. A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, M. A. Decker, S. Khodja, E. W. Van Stryland, “Liquid-based multicell optical limiter,” in Nonlinear Optical Liquids, C. M. Lawson, ed., Proc. SPIE2853, 158–169 (1996).
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1999 (1)

D. I. Kovsh, D. J. Hagan, E. W. Van Stryland, “Numerical modeling of thermal refraction in liquids in the transient regime,” Opt. Exp. 4, 315–327 (1999).
[CrossRef]

1998 (1)

1997 (2)

1995 (3)

1994 (4)

1993 (4)

B. L. Justus, Z. H. Kafafi, A. L. Huston, “Excited-state absorption-enhanced thermal optical limiting in C60,” Opt. Lett. 18, 1603–1605 (1993).
[CrossRef] [PubMed]

J.-G. Tian, C. Zhang, G. Zhang, J. Li, “Position dispersion and optical limiting resulting from thermally induced nonlinearities in Chinese tea,” Appl. Opt. 32, 6628–6632 (1993).
[CrossRef] [PubMed]

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

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993);L. W. Tutt, T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

1992 (5)

G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17, 1426–1428 (1992).
[CrossRef]

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

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, E. W. Van Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron. 24, 1–30 (1992).
[CrossRef]

K. Mansour, E. W. Van Stryland, M. J. Soileau, “Nonlinear properties of carbon-black suspensions (ink),” J. Opt. Soc. Am. B 9, 1100–1109 (1992).
[CrossRef]

R. R. Michael, C. M. Lawson, “Nonlinear transmission and reflection at a dielectric-carbon microparticle suspension interface,” Opt. Lett. 17, 1055–1057 (1992).
[CrossRef] [PubMed]

1990 (2)

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

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

1989 (2)

1986 (1)

1984 (1)

1983 (1)

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

1982 (2)

1981 (2)

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

W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
[CrossRef]

1980 (1)

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

1977 (1)

A. J. Twarowski, D. S. Kliger, “Multiphoton absorption spectra using thermal blooming. I. Theory,” Chem. Phys. 20, 253–258 (1977).
[CrossRef]

1976 (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 130–160 (1976).
[CrossRef]

1975 (1)

1972 (1)

1969 (1)

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

1968 (1)

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

1965 (1)

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

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

Akhmanov, S. A.

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

Belanger, L. J.

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

Bondar, M. V.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992).

Brochard, P.

Brueck, S. R. J.

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

Burzer, J. M.

Cabanel, R.

Campillo, A. J.

Carter, C. A.

Castillo, J.

Castillo, J. A.

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

Cheung, Y. M.

Coulter, D. R.

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

Da Costa, G.

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

Decker, M. A.

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, M. A. Decker, S. Khodja, E. W. Van Stryland, “Liquid-based multicell optical limiter,” in Nonlinear Optical Liquids, C. M. Lawson, ed., Proc. SPIE2853, 158–169 (1996).

Dogariu, A.

Dovichi, N. J.

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

Feit, M. D.

M. D. Feit, J. A. Fleck, “Simple method for solving propagation problems in cylindrical geometry with fast Fourier transforms,” Opt. Lett. 14, 662–664 (1989).
[CrossRef] [PubMed]

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 130–160 (1976).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Fleck, J. A.

M. D. Feit, J. A. Fleck, “Simple method for solving propagation problems in cylindrical geometry with fast Fourier transforms,” Opt. Lett. 14, 662–664 (1989).
[CrossRef] [PubMed]

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 130–160 (1976).
[CrossRef]

Gayen, S. K.

Gordon, J. P.

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

Grolier-Mazza, V.

Hadley, G. R.

Hagan, D. J.

D. I. Kovsh, D. J. Hagan, E. W. Van Stryland, “Numerical modeling of thermal refraction in liquids in the transient regime,” Opt. Exp. 4, 315–327 (1999).
[CrossRef]

O. V. Przhonska, J. H. Lim, D. J. Hagan, E. W. Van Stryland, M. V. Bondar, Y. L. Slominsky, “Nonlinear light absorption of polymethine dyes in liquid and solid media,” J. Opt. Soc. Am. B 15, 802–809 (1998).
[CrossRef]

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

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993);L. W. Tutt, T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

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

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, E. W. Van Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron. 24, 1–30 (1992).
[CrossRef]

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

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, M. A. Decker, S. Khodja, E. W. Van Stryland, “Liquid-based multicell optical limiter,” in Nonlinear Optical Liquids, C. M. Lawson, ed., Proc. SPIE2853, 158–169 (1996).

D. Kovsh, S. Yang, D. J. Hagan, E. W. Van Stryland, “Software for computer modeling of laser pulse propagation through the optical system with nonlinear optical elements,” in Nonlinear Optical Liquids for Power Limiting and Imaging, C. M. Lawson, ed., Proc. SPIE3472, 163–177 (1998).
[CrossRef]

Harris, J. M.

Hayes, J. N.

Hendow, S. T.

Heritier, J.-M.

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

Hughes, S.

Huston, A. L.

Hutchings, D. C.

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, E. W. Van Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron. 24, 1–30 (1992).
[CrossRef]

Jurgensen, F.

Justus, B. L.

Kafafi, Z. H.

Khodja, S.

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, M. A. Decker, S. Khodja, E. W. Van Stryland, “Liquid-based multicell optical limiter,” in Nonlinear Optical Liquids, C. M. Lawson, ed., Proc. SPIE2853, 158–169 (1996).

Khokhlov, R. V.

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

Kildal, H.

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

Kliger, D. S.

A. J. Twarowski, D. S. Kliger, “Multiphoton absorption spectra using thermal blooming. I. Theory,” Chem. Phys. 20, 253–258 (1977).
[CrossRef]

Knight, L. V.

Kovsh, D.

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, M. A. Decker, S. Khodja, E. W. Van Stryland, “Liquid-based multicell optical limiter,” in Nonlinear Optical Liquids, C. M. Lawson, ed., Proc. SPIE2853, 158–169 (1996).

D. Kovsh, S. Yang, D. J. Hagan, E. W. Van Stryland, “Software for computer modeling of laser pulse propagation through the optical system with nonlinear optical elements,” in Nonlinear Optical Liquids for Power Limiting and Imaging, C. M. Lawson, ed., Proc. SPIE3472, 163–177 (1998).
[CrossRef]

Kovsh, D. I.

D. I. Kovsh, D. J. Hagan, E. W. Van Stryland, “Numerical modeling of thermal refraction in liquids in the transient regime,” Opt. Exp. 4, 315–327 (1999).
[CrossRef]

Kozich, V. P.

Krindach, D. P.

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

Landau, D.

D. Landau, E. M. Lifshitz, Course of Theoretical Physics: Fluid Mechanics (Pergamon, New York, 1996), Vol. 6.

Law, C. T.

C. T. Law, G. A. Swartzlander, “Implementation of a package for optical limiter modeling,” in Nonlinear Optical Liquids and Power Limiters, C. M. Lawson, ed., Proc. SPIE3146, 95–106 (1997).
[CrossRef]

Lawson, C. M.

Leite, R. C. C.

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

Li, J.

Lifshitz, E. M.

D. Landau, E. M. Lifshitz, Course of Theoretical Physics: Fluid Mechanics (Pergamon, New York, 1996), Vol. 6.

Lim, J. H.

Litvak, M. M.

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

Liu, G.

Longaker, P. R.

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

Mansour, K.

Marcano O., A.

Marz, R.

H.-P. Nolting, R. Marz, “Results of benchmark tests for different numerical BPM algorithms,” IEEE J. Lightware Technol. 13, 216–224 (1995) and references therein.
[CrossRef]

Michael, R. R.

Migulin, A. V.

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

Miles, P.

Moore, R. S.

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

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 130–160 (1976).
[CrossRef]

Nolting, H.-P.

H.-P. Nolting, R. Marz, “Results of benchmark tests for different numerical BPM algorithms,” IEEE J. Lightware Technol. 13, 216–224 (1995) and references therein.
[CrossRef]

Olaizola, A. M.

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

Patel, C. K. N.

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

Perry, J. W.

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

Porto, S. P. S.

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

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Przhonska, O. V.

Said, A. A.

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

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993);L. W. Tutt, T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

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

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

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, M. A. Decker, S. Khodja, E. W. Van Stryland, “Liquid-based multicell optical limiter,” in Nonlinear Optical Liquids, C. M. Lawson, ed., Proc. SPIE2853, 158–169 (1996).

Schroer, W.

Sence, M. J.

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

Shakir, S. A.

Sheik-Bahae, M.

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, E. W. Van Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron. 24, 1–30 (1992).
[CrossRef]

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

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

Sheldon, S. J.

Siegman, E.

Slominsky, Y. L.

Soileau, M. J.

Southwell, W. H.

Spruce, G.

Sukhorukov, A. P.

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

Swartzlander, G. A.

C. T. Law, G. A. Swartzlander, “Implementation of a package for optical limiter modeling,” in Nonlinear Optical Liquids and Power Limiters, C. M. Lawson, ed., Proc. SPIE3146, 95–106 (1997).
[CrossRef]

Sziklas, E. A.

Tam, A. C.

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

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Thorne, J. M.

Tian, J.-G.

Twarowski, A. J.

A. J. Twarowski, D. S. Kliger, “Multiphoton absorption spectra using thermal blooming. I. Theory,” Chem. Phys. 20, 253–258 (1977).
[CrossRef]

Van Stryland, E. W.

D. I. Kovsh, D. J. Hagan, E. W. Van Stryland, “Numerical modeling of thermal refraction in liquids in the transient regime,” Opt. Exp. 4, 315–327 (1999).
[CrossRef]

O. V. Przhonska, J. H. Lim, D. J. Hagan, E. W. Van Stryland, M. V. Bondar, Y. L. Slominsky, “Nonlinear light absorption of polymethine dyes in liquid and solid media,” J. Opt. Soc. Am. B 15, 802–809 (1998).
[CrossRef]

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

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993);L. W. Tutt, T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

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

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, E. W. Van Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron. 24, 1–30 (1992).
[CrossRef]

K. Mansour, E. W. Van Stryland, M. J. Soileau, “Nonlinear properties of carbon-black suspensions (ink),” J. Opt. Soc. Am. B 9, 1100–1109 (1992).
[CrossRef]

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

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

D. Kovsh, S. Yang, D. J. Hagan, E. W. Van Stryland, “Software for computer modeling of laser pulse propagation through the optical system with nonlinear optical elements,” in Nonlinear Optical Liquids for Power Limiting and Imaging, C. M. Lawson, ed., Proc. SPIE3472, 163–177 (1998).
[CrossRef]

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, M. A. Decker, S. Khodja, E. W. Van Stryland, “Liquid-based multicell optical limiter,” in Nonlinear Optical Liquids, C. M. Lawson, ed., Proc. SPIE2853, 158–169 (1996).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Wajsgrus, A.

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, M. A. Decker, S. Khodja, E. W. Van Stryland, “Liquid-based multicell optical limiter,” in Nonlinear Optical Liquids, C. M. Lawson, ed., Proc. SPIE2853, 158–169 (1996).

Wei, T.

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

Wei, T. H.

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993);L. W. Tutt, T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

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

Wherrett, B. S.

Whinnery, J. R.

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

Wu, S.

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

Xia, T.

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

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993);L. W. Tutt, T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

T. Xia, “Modeling and experimental studies of nonlinear optical self-action,” Ph.D. dissertation (University of Central Florida, Orlando, Fla., 1994).

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, M. A. Decker, S. Khodja, E. W. Van Stryland, “Liquid-based multicell optical limiter,” in Nonlinear Optical Liquids, C. M. Lawson, ed., Proc. SPIE2853, 158–169 (1996).

Yang, S.

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, M. A. Decker, S. Khodja, E. W. Van Stryland, “Liquid-based multicell optical limiter,” in Nonlinear Optical Liquids, C. M. Lawson, ed., Proc. SPIE2853, 158–169 (1996).

D. Kovsh, S. Yang, D. J. Hagan, E. W. Van Stryland, “Software for computer modeling of laser pulse propagation through the optical system with nonlinear optical elements,” in Nonlinear Optical Liquids for Power Limiting and Imaging, C. M. Lawson, ed., Proc. SPIE3472, 163–177 (1998).
[CrossRef]

Zhang, C.

Zhang, G.

Appl. Opt. (10)

E. A. Sziklas, E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
[CrossRef] [PubMed]

S. T. Hendow, S. A. Shakir, “Recursive numerical solution for nonlinear wave propagation in fibers and cylindrically symmetric systems,” Appl. Opt. 25, 1759–1764 (1986).
[CrossRef] [PubMed]

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, E. W. Van Stryland, “Optimization of optical limiting devices based on excited-state absorption,” Appl. Opt. 36, 4110–4122 (1997).
[CrossRef] [PubMed]

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

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

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

G. Liu, “Theory of the photoacoustic effect in condensed matter,” Appl. Opt. 21, 955–960 (1982).
[CrossRef] [PubMed]

C. A. Carter, J. M. Harris, “Comparison of model describing the thermal lens effect,” Appl. Opt. 23, 476–481 (1984).
[CrossRef]

J.-G. Tian, C. Zhang, G. Zhang, J. Li, “Position dispersion and optical limiting resulting from thermally induced nonlinearities in Chinese tea,” Appl. Opt. 32, 6628–6632 (1993).
[CrossRef] [PubMed]

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 130–160 (1976).
[CrossRef]

Appl. Phys. B (1)

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

Chem. Phys. (1)

A. J. Twarowski, D. S. Kliger, “Multiphoton absorption spectra using thermal blooming. I. Theory,” Chem. Phys. 20, 253–258 (1977).
[CrossRef]

IEEE J. Lightware Technol. (1)

H.-P. Nolting, R. Marz, “Results of benchmark tests for different numerical BPM algorithms,” IEEE J. Lightware Technol. 13, 216–224 (1995) and references therein.
[CrossRef]

IEEE J. Quantum Electron. (2)

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

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

Int. J. Nonlinear Opt. Phys. (1)

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993);L. W. Tutt, T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

J. Appl. Phys. (3)

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

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

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (5)

Opt. Commun. (2)

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

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

Opt. Eng. (1)

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

Opt. Exp. (1)

D. I. Kovsh, D. J. Hagan, E. W. Van Stryland, “Numerical modeling of thermal refraction in liquids in the transient regime,” Opt. Exp. 4, 315–327 (1999).
[CrossRef]

Opt. Lett. (7)

Opt. Quantum Electron. (1)

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, E. W. Van Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron. 24, 1–30 (1992).
[CrossRef]

Rev. Mod. Phys. (1)

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

Other (9)

H. S. Nalwa, S. Miyata, eds., Nonlinear Optics of Organic Molecules and Polymers (CRC Press, Boca Raton, Fla., 1997); J. W. Perry, “Organic and metal-containing reverse saturable absorbers for optical limiters,” Chap. 13, pp. 813–840.

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

T. Xia, “Modeling and experimental studies of nonlinear optical self-action,” Ph.D. dissertation (University of Central Florida, Orlando, Fla., 1994).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

D. Kovsh, S. Yang, D. J. Hagan, E. W. Van Stryland, “Software for computer modeling of laser pulse propagation through the optical system with nonlinear optical elements,” in Nonlinear Optical Liquids for Power Limiting and Imaging, C. M. Lawson, ed., Proc. SPIE3472, 163–177 (1998).
[CrossRef]

C. T. Law, G. A. Swartzlander, “Implementation of a package for optical limiter modeling,” in Nonlinear Optical Liquids and Power Limiters, C. M. Lawson, ed., Proc. SPIE3146, 95–106 (1997).
[CrossRef]

D. Landau, E. M. Lifshitz, Course of Theoretical Physics: Fluid Mechanics (Pergamon, New York, 1996), Vol. 6.

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, M. A. Decker, S. Khodja, E. W. Van Stryland, “Liquid-based multicell optical limiter,” in Nonlinear Optical Liquids, C. M. Lawson, ed., Proc. SPIE2853, 158–169 (1996).

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Figures (13)

Fig. 1
Fig. 1

Typical layout of the system under investigation.

Fig. 2
Fig. 2

Thick-sample Z-scan. Closed-aperture Z-scan of CS2 (n 2 = 3.1 × 10-5 cm2/GW), open-aperture Z-scan of 2PA (β2 = 6 cm/GW), and open-aperture Z-scan of a two-element optical limiter. Thickness of the sample is 1 cm (26.5z 0 in air). The two-element tandem limiter is based on a toluene solution of SiNc (thickness of the cells is L 1 = 2 mm, L 2 = 1 mm; separation S = 7 mm).

Fig. 3
Fig. 3

Five-level system. For SiNc, σ G = 2.8 × 10-18 cm2, σ S = 40 × 10-18 cm2, σ T = 120 × 10-18 cm2, τ0 = 1.6 ns, τisc = 6.5 ns, τ S 2 = 1.3 ps, and τT1 = 0.3 µs.

Fig. 4
Fig. 4

Closed-aperture Z-scan of nigrosine solution in water. Nonlinear refractive-index change was computed as a solution to the acoustic equation and its approximation. Parameters of the laser beam are pulse width τ p = 10 ns (HW1/eM), beam size at the waist w 0 = 6 µm (HW1/e2M), and input energy E IN = 2 µJ. Linear transmittance of the sample T L = 90% and thickness of the sample L = 200 µm (τ p ac = 2.5).

Fig. 5
Fig. 5

Closed-aperture Z-scan of nigrosine solution in water. Nonlinear refractive-index change was computed as a solution to the acoustic equation and its approximation. Parameters of the laser beam are τ p = 10 ns, w 0 = 30 µm, and E IN = 50 µJ. Linear transmittance of the sample T L = 90% and L = 200 µm (τ p ac = 0.5).

Fig. 6
Fig. 6

Sensitivity (ΔT P–V) of the closed-aperture Z scan as a function of ratio between pulse width τ p and acoustic transit time τac = w 0/C S. ΔT P–V is normalized to the value obtained with the thermal lensing approximation.

Fig. 7
Fig. 7

Closed-aperture Z-scan of nigrosine in water [τ p = 7 ns (FWHM), w 0 = 6 µm, T L = 40%, L = 50 µm, E IN = 0.55 µJ]. Comparison between modeling and experimental data.

Fig. 8
Fig. 8

Modeling of spatial irradiance distribution in the center of the pulse passing through the cell filled with the toluene solution of SiNc [τ p = 4.6 ns (FWHM), w 0 = 8 µm, T L = 60%, L = 1 mm, E IN = 2 µJ]: (a) with RSA only and (b) with RSA and thermal refraction.

Fig. 9
Fig. 9

Beam size (computed as a second moment of irradiance) inside the sample as a function of z and time during the pulse. Parameters of the system are the same as for Fig. 8: (a) with RSA only and (b) with RSA and thermal refraction.

Fig. 10
Fig. 10

Limiting curves showing the performance of the 1-mm SiNc–toluene limiter for two positions of the limiting element with respect to the focus as a function of input fluence [w 0 = 6 µm, τ p = 7 ns (FWHM), T L = 48%]. Solid curves show the results of modeling if the aperture (S = 99%) is placed in front of the detector. Arrows show the values of input fluence at which the nonlinear scattering becomes noticeable as observed by a detector positioned off to the side of the sample.

Fig. 11
Fig. 11

Limiting performance of the SiNc–toluene limiter for various positions of the sample. Parameters are the same as for Fig. 10.

Fig. 12
Fig. 12

Normalized radial fluence distribution at the image plane of the system with the flat-top input beam spatial profile [input radius w IN = 3 mm, τ p = 7 ns (FWHM), E IN = 0.417 µJ, T L = 47.5%, L = 50 µm]. A sample filled with nigrosine dissolved in water is placed at the position corresponding to the minimum of the closed-aperture Z-scan curve.

Fig. 13
Fig. 13

Closed-aperture Z-scan of nigrosine in water. Flat-top input beam geometry was used. Parameters of the system are the same as for Fig. 12.

Equations (36)

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

××Er, t+1c22Er, tt2=-μ02Pr, tt2,
-2Er, t+1+χLc22jω Er, tt-ω2Er, t=-μ0ω2PNLr, t,
2Et2  2ωEt.
2jk Ψr, z, tz=2Ψr, z, t+k02χNLr, z, t-jkαLΨr, z, t,
2Ψz2  2kΨz.
χNLr, z, t=χNLinsr, z+χNLcumr, z, t.
Ψr, z+Δz=exp-jSˆr, zΔzΨr, z,
Sˆr, z=12k2r2+1rr+k02χNLr, z-jkαL.
χNLr, z=χNLr, z-j n0k0 αL.
ReχNLr, z=2n0Δnr, z,
ImχNLr, z=-n0k0 αr, z=-n0k0αL+αNLr, z.
Δnr, z=n2Ir, z,
αNLr, z=β2Ir, z.
α=σGNG+σSNS1+σTNT1,
dNS1dt=σGNGIω-NS1τS1, dNT1dt=NS1τisc, NG+NS1+NT1=N0.
Δn=σS1,rNS1+σT1,rNT1k,
ρcpTt-κ2T=αLI,
ΔTt=1ρcp0t αLItdt.
Δn=nρTΔρ+nTρΔT.
2Δn-1CS22Δnt2=-γeβ2n 2ΔT,
ΔndndT0ΔT,
τpτac1.5.
ρt+ρu=0,
ρut+uu+p+η2u+ζ+η3u=F,
ρTst+us=κT+σikuixk+Q,
Fr, t=ρnρTIr, tc.
Qr, t=αLIr, t
-2t2+CS2γ 2+ηρt 2ρ+CS2βργ 2T=γe2nc 2I,
ρcVt-κ2T-cp-cVβρt=αLI.
t2ρt2-CS22ρ=CS2βαLcp 2I-γe2nct 2I,
Δn=nρTρ+nTρT=γe2nρ ρ+nTρT.
2Δnt2-CS22Δn=γe2nρβCS2cp- 2αLIr, tdt.
ρcpTt-κ2T=αLI
Tr, t=1ρcp- αLIr, tdt.
2Δnt2-CS22Δn=CS2γeβ2n 2Tr, t.
Δn=dndTT,

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