Abstract

The detection and characterization of the thermal-lensing effect is measured for a lead oxide–modified silicate glass material. The self-modulation of the exciting beam is measured with a simple experimental setup on both millisecond and nanosecond time scales. The results of our analysis indicate that to first order the operative mechanism of self-focusing is the same on both time scales, namely, the time-dependent modulation of the refractive index that is due to thermal heating. On a millisecond time scale we recognize the occurrence of multiple focal points within the sample. Although these undulations in the beam waist have been reported previously by others, our experimental methods allow these undulations to manifest themselves in new ways, namely, in Z scans and transmittance versus power profiles. These are indicative of a strong thermo-optic effect and are consequences of Maxwell’s equations. With our model a dn/dT value of 1 × 10−5 K−1 is extracted.

© 1992 Optical Society of America

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  1. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum. Electron.,  4, 35–110 (1975).
    [Crossref]
  2. 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. O. Shulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Chap. E3, pp. 1151–1228.
  3. A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms of self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (1973).
    [Crossref]
  4. A. Feldman, D. Horowitz, and R. M. Waxler, “Relative importance of electrostriction and the Kerr effect to self-focusing in optical glasses,” Appl. Phys. Lett. 21, 260–262 (1972).
    [Crossref]
  5. Yu. P. Raizer “Self-focusing and defocusing, instability and stabilization of light beams in weakly absorbing media,” J. Exp. Theor. Phys. (USSR) 52, 470–482 (1967).
  6. E. L. Kerr, “Filamentary tracks formed in transparent optical glass by laser beam self-focusing. II. Theoretical analysis,” Phys. Rev. A 4, 1195–1218 (1971).
    [Crossref]
  7. F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
    [Crossref]
  8. F. M. Durville and R. C. Powell, “Thermal lensing and permanent refractive index changes in rare-earth-doped glasses,” J. Opt. Soc. Am. B 4, 1934–1937 (1987).
    [Crossref]
  9. J. D. Foster and L. M. Oserink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys. 41, 3656–3663 (1970).
    [Crossref]
  10. G. M. Zverev, E. A. Levchuk, E. K. Maldutis, and V. A. Pashkov, “Thermal self-focusing of laser radiation in media with negative dn/dT,” Sov. Phys. JETP Lett. 11, 108–111 (1970).
  11. W. Koechner, Solid State Laser Engineering (Springer-Verlag, Berlin, 1988), Chap. 7, pp. 350–401.
    [Crossref]
  12. G. M. Zverev and V. A. Pashkov, “Self-focusing of laser radiation in solid dielectrics,” Sov. Phys. JETP 30, 616–621 (1970).
  13. E. P. Riedel and G. D. Baldwin, “Theory of dynamic optical distortion in isotropic laser materials,” J. Appl. Phys. 38, 2720–2725 (1967).
    [Crossref]
  14. G. D. Baldwin and E. P. Riedel, “Measurements of dynamic optical distortion in Nd-doped glass laser rods,” J. Appl. Phys. 38, 2726–2737 (1967).
    [Crossref]
  15. B. Taheri, A. Munoz F., W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. (to be published).
  16. M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Opt. Eng. 30, 1228–1235 (1991).
    [Crossref]
  17. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6, pp. 106–135.
  18. A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Multifocus structure of a light beam in a nonlinear medium,” Sov. Phys. JETP 34, 1235–1241 (1972).
  19. A. P. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense laser beams,” Soviet Phys. JETP Lett. 6, 146–149 (1967).
  20. M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [Crossref]
  21. M. D. Feit and J. A. Fleck, “Self-trapping of laser beams in a cylindrical plasma column,” Appl. Phys. Lett. 28, 121–124 (1976).
    [Crossref]
  22. 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, 256–266 (1968).
    [Crossref]
  23. J. Alda and G. D. Boreman, “On-axis and off-axis propagation of Gaussian beams in gradient index media,” Appl. Opt. 29, 2944–2950 (1990).
    [Crossref] [PubMed]
  24. D. Metcalf, P. de Giovanni, J. Zachorowski, and M. Leduc, “Laser resonators containing self-focusing elements,” Appl. Opt. 26, 4508–4517 (1987).
    [Crossref] [PubMed]
  25. H. W. Wyld, Mathematical Methods for Physics (Benjamin/Cummings, Menlo Park, Calif., 1976), pp. 319–326.
  26. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1568 (1965).
    [Crossref]
  27. O and z are more precisely z and z+ δz, where δz is some small iteration in z.
  28. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
    [Crossref]

1991 (1)

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Opt. Eng. 30, 1228–1235 (1991).
[Crossref]

1990 (1)

1988 (1)

1987 (2)

1976 (1)

M. D. Feit and J. A. Fleck, “Self-trapping of laser beams in a cylindrical plasma column,” Appl. Phys. Lett. 28, 121–124 (1976).
[Crossref]

1975 (1)

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

1973 (1)

A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms of self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (1973).
[Crossref]

1972 (2)

A. Feldman, D. Horowitz, and R. M. Waxler, “Relative importance of electrostriction and the Kerr effect to self-focusing in optical glasses,” Appl. Phys. Lett. 21, 260–262 (1972).
[Crossref]

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Multifocus structure of a light beam in a nonlinear medium,” Sov. Phys. JETP 34, 1235–1241 (1972).

1971 (1)

E. L. Kerr, “Filamentary tracks formed in transparent optical glass by laser beam self-focusing. II. Theoretical analysis,” Phys. Rev. A 4, 1195–1218 (1971).
[Crossref]

1970 (3)

J. D. Foster and L. M. Oserink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys. 41, 3656–3663 (1970).
[Crossref]

G. M. Zverev, E. A. Levchuk, E. K. Maldutis, and V. A. Pashkov, “Thermal self-focusing of laser radiation in media with negative dn/dT,” Sov. Phys. JETP Lett. 11, 108–111 (1970).

G. M. Zverev and V. A. Pashkov, “Self-focusing of laser radiation in solid dielectrics,” Sov. Phys. JETP 30, 616–621 (1970).

1968 (2)

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
[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, 256–266 (1968).
[Crossref]

1967 (4)

A. P. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense laser beams,” Soviet Phys. JETP Lett. 6, 146–149 (1967).

E. P. Riedel and G. D. Baldwin, “Theory of dynamic optical distortion in isotropic laser materials,” J. Appl. Phys. 38, 2720–2725 (1967).
[Crossref]

G. D. Baldwin and E. P. Riedel, “Measurements of dynamic optical distortion in Nd-doped glass laser rods,” J. Appl. Phys. 38, 2726–2737 (1967).
[Crossref]

Yu. P. Raizer “Self-focusing and defocusing, instability and stabilization of light beams in weakly absorbing media,” J. Exp. Theor. Phys. (USSR) 52, 470–482 (1967).

1965 (1)

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Akhmanov, S. A.

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. O. Shulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Chap. E3, pp. 1151–1228.

Alda, J.

Baldwin, G. D.

E. P. Riedel and G. D. Baldwin, “Theory of dynamic optical distortion in isotropic laser materials,” J. Appl. Phys. 38, 2720–2725 (1967).
[Crossref]

G. D. Baldwin and E. P. Riedel, “Measurements of dynamic optical distortion in Nd-doped glass laser rods,” J. Appl. Phys. 38, 2726–2737 (1967).
[Crossref]

Blackburn, D. H.

B. Taheri, A. Munoz F., W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. (to be published).

Boreman, G. D.

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Cranmer, D. C.

B. Taheri, A. Munoz F., W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. (to be published).

Dabby, F. W.

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
[Crossref]

de Giovanni, P.

Durville, F. M.

Dyshko, A. L.

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Multifocus structure of a light beam in a nonlinear medium,” Sov. Phys. JETP 34, 1235–1241 (1972).

Dyshko, A. P.

A. P. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense laser beams,” Soviet Phys. JETP Lett. 6, 146–149 (1967).

Feit, M. D.

M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
[Crossref]

M. D. Feit and J. A. Fleck, “Self-trapping of laser beams in a cylindrical plasma column,” Appl. Phys. Lett. 28, 121–124 (1976).
[Crossref]

Feldman, A.

A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms of self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (1973).
[Crossref]

A. Feldman, D. Horowitz, and R. M. Waxler, “Relative importance of electrostriction and the Kerr effect to self-focusing in optical glasses,” Appl. Phys. Lett. 21, 260–262 (1972).
[Crossref]

Fleck, J. A.

M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
[Crossref]

M. D. Feit and J. A. Fleck, “Self-trapping of laser beams in a cylindrical plasma column,” Appl. Phys. Lett. 28, 121–124 (1976).
[Crossref]

Foster, J. D.

J. D. Foster and L. M. Oserink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys. 41, 3656–3663 (1970).
[Crossref]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Hagan, D. J.

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Opt. Eng. 30, 1228–1235 (1991).
[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, 256–266 (1968).
[Crossref]

Horowitz, D.

A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms of self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (1973).
[Crossref]

A. Feldman, D. Horowitz, and R. M. Waxler, “Relative importance of electrostriction and the Kerr effect to self-focusing in optical glasses,” Appl. Phys. Lett. 21, 260–262 (1972).
[Crossref]

John, W. D. St.

B. Taheri, A. Munoz F., W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. (to be published).

Kerr, E. L.

E. L. Kerr, “Filamentary tracks formed in transparent optical glass by laser beam self-focusing. II. Theoretical analysis,” Phys. Rev. A 4, 1195–1218 (1971).
[Crossref]

Khokhlov, R. V.

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. O. Shulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Chap. E3, pp. 1151–1228.

Koechner, W.

W. Koechner, Solid State Laser Engineering (Springer-Verlag, Berlin, 1988), Chap. 7, pp. 350–401.
[Crossref]

Kogelnik, H.

Leduc, M.

Levchuk, E. A.

G. M. Zverev, E. A. Levchuk, E. K. Maldutis, and V. A. Pashkov, “Thermal self-focusing of laser radiation in media with negative dn/dT,” Sov. Phys. JETP Lett. 11, 108–111 (1970).

Lugovoi, V. N.

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Multifocus structure of a light beam in a nonlinear medium,” Sov. Phys. JETP 34, 1235–1241 (1972).

A. P. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense laser beams,” Soviet Phys. JETP Lett. 6, 146–149 (1967).

Maldutis, E. K.

G. M. Zverev, E. A. Levchuk, E. K. Maldutis, and V. A. Pashkov, “Thermal self-focusing of laser radiation in media with negative dn/dT,” Sov. Phys. JETP Lett. 11, 108–111 (1970).

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, 256–266 (1968).
[Crossref]

Metcalf, D.

Munoz F., A.

B. Taheri, A. Munoz F., W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. (to be published).

Oserink, L. M.

J. D. Foster and L. M. Oserink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys. 41, 3656–3663 (1970).
[Crossref]

Pashkov, V. A.

G. M. Zverev, E. A. Levchuk, E. K. Maldutis, and V. A. Pashkov, “Thermal self-focusing of laser radiation in media with negative dn/dT,” Sov. Phys. JETP Lett. 11, 108–111 (1970).

G. M. Zverev and V. A. Pashkov, “Self-focusing of laser radiation in solid dielectrics,” Sov. Phys. JETP 30, 616–621 (1970).

Powell, R. C.

F. M. Durville and R. C. Powell, “Thermal lensing and permanent refractive index changes in rare-earth-doped glasses,” J. Opt. Soc. Am. B 4, 1934–1937 (1987).
[Crossref]

B. Taheri, A. Munoz F., W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. (to be published).

Prokhorov, A. M.

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Multifocus structure of a light beam in a nonlinear medium,” Sov. Phys. JETP 34, 1235–1241 (1972).

A. P. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense laser beams,” Soviet Phys. JETP Lett. 6, 146–149 (1967).

Raizer, Yu. P.

Yu. P. Raizer “Self-focusing and defocusing, instability and stabilization of light beams in weakly absorbing media,” J. Exp. Theor. Phys. (USSR) 52, 470–482 (1967).

Riedel, E. P.

G. D. Baldwin and E. P. Riedel, “Measurements of dynamic optical distortion in Nd-doped glass laser rods,” J. Appl. Phys. 38, 2726–2737 (1967).
[Crossref]

E. P. Riedel and G. D. Baldwin, “Theory of dynamic optical distortion in isotropic laser materials,” J. Appl. Phys. 38, 2720–2725 (1967).
[Crossref]

Said, A. A.

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Opt. Eng. 30, 1228–1235 (1991).
[Crossref]

Sheik-bahae, M.

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Opt. Eng. 30, 1228–1235 (1991).
[Crossref]

Soileau, M. J.

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Opt. Eng. 30, 1228–1235 (1991).
[Crossref]

Sukhorukov, A. P.

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. O. Shulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Chap. E3, pp. 1151–1228.

Taheri, B.

B. Taheri, A. Munoz F., W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. (to be published).

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Van Stryland, E. W.

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Opt. Eng. 30, 1228–1235 (1991).
[Crossref]

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, 256–266 (1968).
[Crossref]

Waxler, R. M.

A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms of self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (1973).
[Crossref]

A. Feldman, D. Horowitz, and R. M. Waxler, “Relative importance of electrostriction and the Kerr effect to self-focusing in optical glasses,” Appl. Phys. Lett. 21, 260–262 (1972).
[Crossref]

Whinnery, J. R.

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
[Crossref]

Wicksted, J. P.

B. Taheri, A. Munoz F., W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. (to be published).

Wyld, H. W.

H. W. Wyld, Mathematical Methods for Physics (Benjamin/Cummings, Menlo Park, Calif., 1976), pp. 319–326.

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6, pp. 106–135.

Zachorowski, J.

Zverev, G. M.

G. M. Zverev and V. A. Pashkov, “Self-focusing of laser radiation in solid dielectrics,” Sov. Phys. JETP 30, 616–621 (1970).

G. M. Zverev, E. A. Levchuk, E. K. Maldutis, and V. A. Pashkov, “Thermal self-focusing of laser radiation in media with negative dn/dT,” Sov. Phys. JETP Lett. 11, 108–111 (1970).

Appl. Opt. (3)

Appl. Phys. Lett. (3)

M. D. Feit and J. A. Fleck, “Self-trapping of laser beams in a cylindrical plasma column,” Appl. Phys. Lett. 28, 121–124 (1976).
[Crossref]

A. Feldman, D. Horowitz, and R. M. Waxler, “Relative importance of electrostriction and the Kerr effect to self-focusing in optical glasses,” Appl. Phys. Lett. 21, 260–262 (1972).
[Crossref]

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms of self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (1973).
[Crossref]

J. Appl. Phys. (3)

E. P. Riedel and G. D. Baldwin, “Theory of dynamic optical distortion in isotropic laser materials,” J. Appl. Phys. 38, 2720–2725 (1967).
[Crossref]

G. D. Baldwin and E. P. Riedel, “Measurements of dynamic optical distortion in Nd-doped glass laser rods,” J. Appl. Phys. 38, 2726–2737 (1967).
[Crossref]

J. D. Foster and L. M. Oserink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys. 41, 3656–3663 (1970).
[Crossref]

J. Exp. Theor. Phys. (USSR) (1)

Yu. P. Raizer “Self-focusing and defocusing, instability and stabilization of light beams in weakly absorbing media,” J. Exp. Theor. Phys. (USSR) 52, 470–482 (1967).

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, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Opt. Eng. 30, 1228–1235 (1991).
[Crossref]

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, 256–266 (1968).
[Crossref]

Phys. Rev. A (1)

E. L. Kerr, “Filamentary tracks formed in transparent optical glass by laser beam self-focusing. II. Theoretical analysis,” Phys. Rev. A 4, 1195–1218 (1971).
[Crossref]

Phys. Rev. Lett. (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Prog. Quantum. Electron. (1)

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

Sov. Phys. JETP (2)

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Multifocus structure of a light beam in a nonlinear medium,” Sov. Phys. JETP 34, 1235–1241 (1972).

G. M. Zverev and V. A. Pashkov, “Self-focusing of laser radiation in solid dielectrics,” Sov. Phys. JETP 30, 616–621 (1970).

Sov. Phys. JETP Lett. (1)

G. M. Zverev, E. A. Levchuk, E. K. Maldutis, and V. A. Pashkov, “Thermal self-focusing of laser radiation in media with negative dn/dT,” Sov. Phys. JETP Lett. 11, 108–111 (1970).

Soviet Phys. JETP Lett. (1)

A. P. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense laser beams,” Soviet Phys. JETP Lett. 6, 146–149 (1967).

Other (6)

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6, pp. 106–135.

B. Taheri, A. Munoz F., W. D. St. John, J. P. Wicksted, R. C. Powell, D. H. Blackburn, and D. C. Cranmer, “Effects of the structure and composition of lead glasses on the thermal lensing of pulsed laser radiation,” J. Appl. Phys. (to be published).

W. Koechner, Solid State Laser Engineering (Springer-Verlag, Berlin, 1988), Chap. 7, pp. 350–401.
[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. O. Shulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Chap. E3, pp. 1151–1228.

O and z are more precisely z and z+ δz, where δz is some small iteration in z.

H. W. Wyld, Mathematical Methods for Physics (Benjamin/Cummings, Menlo Park, Calif., 1976), pp. 319–326.

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Figures (6)

Fig. 1
Fig. 1

Shape of a nanosecond YAG pulse. Dotted lines show the triangular approximation.

Fig. 2
Fig. 2

Experimental setup for Z scans and transmission profiles: M’s mirrors; BS, beam splitter; AOM, acousto-optic modulator; Z1, Z1, the distance from focusing lens to front face of sample.

Fig. 3
Fig. 3

Millisecond temporal response at four values of Z1 with laser power at 275 mW. (a) Z1 = 4.77 cm, (b) Z1 = 4.79 cm, (c) Z1 = 4.84 cm, (d) Z1 = 4.88 cm. The solid curves through the circled data points are theoretical fits described in the text.

Fig. 4
Fig. 4

(a) Z scans and (b) normalized transmittance versus input power at two different values of Z1. The solid curves through the circled data points are theoretical fits described in the text.

Fig. 5
Fig. 5

(a) Theoretical Z scans at three different powers; (b) theoretical change in waist size within the sample for each adjacent Z scan for a fixed sample position Z1 = 4.9 cm.

Fig. 6
Fig. 6

Results of nanosecond experiments: (a) Z scans, (b) output fluence versus input fluence (Z1 fixed). Solid curves are theoretical fits.

Equations (27)

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n = n 0 + d n d T Δ T ,
2 T ( r , t ) - 1 η T ( r , t ) t = - 4 π Q ( r , t ) ,
G ( r , t ; r , t ) = - 1 4 π ( t - t ) exp [ - r - r 2 4 η ( t - t ) ] ,
T ( r , t ) = t 0 t 0 0 2 π exp [ - r - r 2 4 η ( t - t ) ] ( t - t ) Q ( r , t ) r d r d ϕ d t + 1 4 π η ( t - t 0 ) × 0 0 2 π exp [ - r - r 2 4 η ( t - t 0 ) ] T ( r , t 0 ) r d r d ϕ ,
Q ( r , t ) = I ( r , t ) α abs κ = 2 α abs π w 2 k exp [ - 2 ( r ) 2 w 2 ] P 0 ( t ) ,
P ( t ) = P 0 t 0 t ,             t t 0 , P ( t ) = P 0 ( 2 - t t 0 ) ,             t 0 t 2 t 0 ,
n ( r , t ) = n 0 + d n d T P 0 α abs κ [ ln ( 1 + t t c ) - 2 r 2 w 2 [ t t c 1 + t t c ] ] ;
n ( r , t ) = n 0 + d n d T P 0 α abs κ { t t 0 + t c t 0 ln [ 1 + t t c ] - 2 r 2 w 2 t c t 0 [ ln ( 1 + t t c ) + 1 1 + t t c - 1 ] } ,
n ( r , t ) = n 0 + d n d T P 0 α abs κ [ 2 - t t 0 + ( 2 + t c t 0 ) ln ( 1 + t t c ) - 4 r 2 w 2 t - t 0 t + t c ] ,
n ( r , t ) = n 3 ( t ) - n 4 ( t ) r 2 .
E + k 2 ( r ) E = 0 ,
k 2 ( r ) = n 2 ( r ) k 0 2 = k 0 2 ( n 3 2 - 2 n 3 n 4 r 2 + n 4 2 r 4 ) k 0 2 ( n 3 2 - 2 n 3 n 4 r 2 ) ,
E = ψ ( x , y , z ) exp ( - i k z ) .
t 2 ψ - 2 i k 0 n 3 ψ - 2 k 0 2 n 3 n 4 r 2 ψ = 0 ,
ψ = exp [ - i ( P + ½ Q r 2 ) ] ,
P = - i Q k 0 n 3 , Q 2 + k 0 n 3 Q + 2 k 0 2 n 3 n 4 = 0 ,
q ( z ) = cos ( φ z ) q ( 0 ) + ( 1 / φ ) sin ( φ z ) - φ sin ( φ z ) q ( 0 ) + cos ( φ z ) ,
φ = 2 n 4 / n 3 .
A B C D [ cos ( φ z ) ( 1 / φ ) sin ( φ z ) - φ sin ( φ z ) cos ( φ z ) ] .
1 q ( z ) = 1 R ( z ) - i λ π w 2 ( z ) ,
w 2 ( z ) = w 2 ( 0 ) { [ cos ( φ z ) + 1 R ( 0 ) sin ( φ z ) φ ] 2 + [ λ π w 2 ( 0 ) sin ( φ z ) φ ] 2 } , 1 R ( z ) = cos 2 ( φ z ) - sin 2 ( φ z ) R ( 0 ) + sin ( φ z ) cos ( φ z ) { 1 φ R 2 ( 0 ) - φ + 1 φ [ λ π w 2 ( 0 ) ] 2 } ,
φ [ w ( z ) ] = [ 4 P 0 α abs n 0 w 2 ( z ) κ d n d T t t c 1 + t t c ] 1 / 2 ,
w 2 ( z ) w 2 ( 0 ) = cos 2 ( φ z ) [ 1 - ( λ φ π w 2 ( 0 ) ) 2 ] + ( λ φ π w 2 ( 0 ) ) 2 .
w 2 ( z ) w 2 ( 0 ) = cos 2 ( φ z ) ( 1 - P c r P 0 ) + P c r P 0 ,
P c r = n 0 κ λ 2 4 π 2 w 2 ( 0 ) α abs d n d T ( 1 + t t c / t t c ) .
P cr ( ss ) = n 0 κ λ 2 4 π 2 w 2 ( 0 ) α abs d n d T .
w 2 ( z ) w 2 ( 0 ) = 1 - φ 2 z 2 ( 1 - P c r ( ss ) P 0 ) .

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