Abstract

An analytical treatment for the propagation of astigmatic Gaussian beams in Kerr media is presented. This method, which is valid to the first order in the beam power, allows us to calculate in closed form the nonlinear variations of the spot size and the radius of curvature. This formalism is applied to the calculation of the self-consistent Gaussian mode of an astigmatic resonator with a Kerr medium, such as those used for Kerr-lens mode-locked lasers. Theoretical predictions are confirmed by the experimental results obtained with a femtosecond Ti:sapphire laser. In particular, the design of an optimized resonator permits self-starting of the mode-locking mechanism.

© 1995 Optical Society of America

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  1. D. E. Spence, P. N. Kean, and W. Sibbett, "60-fsec pulse generation from a self-mode-locked Ti:sapphire laser," Opt. Lett. 16, 42–44 (1991).
    [CrossRef] [PubMed]
  2. F. Salin, J. Squier, and M. Piché, "Mode locking of Ti:Al2O3 lasers and self-focusing: a Gaussian approximation," Opt. Lett. 16, 1674–1676 (1991).
    [CrossRef] [PubMed]
  3. 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. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp. 1151–1228.
  4. J. H. Marburger, "Self-focusing: theory," in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, Oxford, 1977), Vol. 4, pp. 35–110.
    [CrossRef]
  5. Y. R. Shen, "Self-focusing: experimental," in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, Oxford, 1977), Vol. 4, pp. 1–34.
    [CrossRef]
  6. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 17.
  7. P. A. Belanger and C. Pare, "Self-focusing of Gaussian beams: an alternate derivation," Appl. Opt. 22, 1293–1295 (1983).
    [CrossRef] [PubMed]
  8. D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, "Self-focusing-induced saturable absorber loss for laser mode locking," Opt. Lett. 17, 511–513 (1992).
    [CrossRef] [PubMed]
  9. V. Magni, G. Cerullo, and S. De Silvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348–355 (1993).
    [CrossRef]
  10. M. Piché, "Beam reshaping and self-mode-locking in nonlinear laser resonators," Opt. Commun. 86, 156–160 (1991).
    [CrossRef]
  11. D. Georgiev, J. Hermann, and U. Stamm, "Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers," Opt. Commun. 92, 368–375 (1992).
    [CrossRef]
  12. O. E. Martinez and J. L. A. Chilla, "Self-mode-locking of Ti:sapphire lasers: a matrix formalism," Opt. Lett. 17, 1210–1212 (1992).
    [CrossRef] [PubMed]
  13. T. Brabec, Ch. Spielmann, P. F. Curley, and F. Krausz, "Kerr-lens mode locking," Opt. Lett. 17, 1292–1294 (1992).
    [CrossRef] [PubMed]
  14. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086–2096 (1992).
    [CrossRef]
  15. G. W. Pearson, C. Radzewicz, and J. S. Krasinski, "Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements," Opt. Commun. 94, 221–226 (1992).
    [CrossRef]
  16. M. Piché and F. Salin, "Self-mode locking of solid-state lasers without apertures," Opt. Lett. 18, 1041–1043 (1993).
    [CrossRef] [PubMed]
  17. T. Brabec, P. F. Curley, Ch. Spielmann, E. Witner, and A. J. Schmidt, "Hard-aperture Kerr-lens mode locking," J. Opt. Soc. Am. B 10, 1029–1034 (1993).
    [CrossRef]
  18. V. Magni, G. Cerullo, and S. De Silvestri, "Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers," Opt. Commun. 101, 365–370 (1993).
    [CrossRef]
  19. K. H. Lin and W. F. Hsieh, "Analytical design of symmetrical Kerr-lens mode-locking laser cavities," J. Opt. Soc. Am. B 11, 737–741 (1994).
    [CrossRef]
  20. J. Herrmann, "Theory of Kerr-lens mode locking: role of self-focusing and radially varying gain," J. Opt. Soc. Am. B 11, 498–512 (1994).
    [CrossRef]
  21. R. E. Bridges, R. W. Boyd, and G. P. Agrawal, "Effect of beam ellipticity on self-mode locking in lasers," Opt. Lett. 18, 2026–2028 (1993).
    [CrossRef] [PubMed]
  22. C. R. Giuliano, J. H. Marburger, and A. Yariv, "Enhancement of self-focusing threshold in sapphire with elliptical beams," Appl. Phys. Lett. 21, 58–60 (1972).
    [CrossRef]
  23. A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, P. S. Shapovalov, and S. I. Turovets, "Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams," Phys. Lett. A 160, 138–142 (1991).
    [CrossRef]
  24. A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, and P. S. Shapovalov, "Rotating elliptical Gaussian beams in nonlinear media," Opt. Commun. 81, 225–230 (1991).
    [CrossRef]
  25. F. Cornolti, M. Lucchesi, and B. Zambon, "Elliptic Gaussian beam self-focusing in nonlinear media," Opt. Commun. 75, 129–135 (1990).
    [CrossRef]
  26. F. Krausz, T. Brabec, and C. Spielmann, "Self-starting passive mode locking," Opt. Lett. 16, 235–237 (1991).
    [CrossRef] [PubMed]
  27. J. Herrmann, "Starting dynamic, self-starting condition and mode-locking threshold in passive, coupled cavity or Kerr-lens mode-locked solid-state lasers," Opt. Commun. 98, 111–116 (1993).
    [CrossRef]
  28. Ming Lai, "Self-starting, self-mode-locked Ti:sapphire laser," Opt. Lett. 19, 722–724 (1994).
    [CrossRef] [PubMed]
  29. G. Cerullo, S. De Silvestri, and V. Magni, "Self-starting Kerr-lens mode-locking of a Ti:sapphire laser," Opt. Lett. 19, 1040–1042 (1994).
    [CrossRef] [PubMed]
  30. J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt. 8, 1687–1693 (1969).
    [CrossRef] [PubMed]
  31. 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]
  32. M. Desaix, D. Anderson, and M. Lisak, "Variational approach to collapse of optical pulses," J. Opt. Soc. Am. B 8, 2082–2086 (1991).
    [CrossRef]
  33. H. Kogelnik, "Imaging of optical modes—resonators with internal lenses," Bell Syst. Tech. J. 44, 455–494 (1965).
    [CrossRef]
  34. M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, "Highsensitivity, single-beam n2 measurements," Opt. Lett. 14, 955–957 (1989).
    [CrossRef] [PubMed]
  35. A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 17.
  36. Ref. 35, Chap. 15.
  37. G. Cerullo, S. De Silvestri, V. Magni, and L. Pallaro, "Resonators for Kerr-lens mode-locked femtosecond Ti:sapphire lasers," Opt. Lett. 19, 807–809 (1994).
    [CrossRef] [PubMed]
  38. J. P. Taché, "Ray matrices for tilted interfaces in laser resonators," Appl. Opt. 26, 427–429 (1987).
    [CrossRef] [PubMed]
  39. Ref. 35, Chap. 21.
  40. V. Magni, S. De Silvestri, and A. Cybo-Ottone, "On the stability, mode properties, and misalignment sensitivity of femtosecond dye laser resonators," Opt. Commun. 82, 137–144 (1991).
    [CrossRef]
  41. H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, "Astigmatically compensated cavities for cw dye laser," IEEE J. Quantum Electron. QE-8, 373–379 (1972).
    [CrossRef]
  42. P. Baues, "Huygens' principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators," Opto-Electronics 1, 37–44 (1969).
    [CrossRef]

1994 (5)

1993 (6)

R. E. Bridges, R. W. Boyd, and G. P. Agrawal, "Effect of beam ellipticity on self-mode locking in lasers," Opt. Lett. 18, 2026–2028 (1993).
[CrossRef] [PubMed]

M. Piché and F. Salin, "Self-mode locking of solid-state lasers without apertures," Opt. Lett. 18, 1041–1043 (1993).
[CrossRef] [PubMed]

T. Brabec, P. F. Curley, Ch. Spielmann, E. Witner, and A. J. Schmidt, "Hard-aperture Kerr-lens mode locking," J. Opt. Soc. Am. B 10, 1029–1034 (1993).
[CrossRef]

J. Herrmann, "Starting dynamic, self-starting condition and mode-locking threshold in passive, coupled cavity or Kerr-lens mode-locked solid-state lasers," Opt. Commun. 98, 111–116 (1993).
[CrossRef]

V. Magni, G. Cerullo, and S. De Silvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348–355 (1993).
[CrossRef]

V. Magni, G. Cerullo, and S. De Silvestri, "Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers," Opt. Commun. 101, 365–370 (1993).
[CrossRef]

1992 (6)

D. Georgiev, J. Hermann, and U. Stamm, "Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers," Opt. Commun. 92, 368–375 (1992).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, "Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements," Opt. Commun. 94, 221–226 (1992).
[CrossRef]

O. E. Martinez and J. L. A. Chilla, "Self-mode-locking of Ti:sapphire lasers: a matrix formalism," Opt. Lett. 17, 1210–1212 (1992).
[CrossRef] [PubMed]

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, "Self-focusing-induced saturable absorber loss for laser mode locking," Opt. Lett. 17, 511–513 (1992).
[CrossRef] [PubMed]

T. Brabec, Ch. Spielmann, P. F. Curley, and F. Krausz, "Kerr-lens mode locking," Opt. Lett. 17, 1292–1294 (1992).
[CrossRef] [PubMed]

1991 (9)

D. E. Spence, P. N. Kean, and W. Sibbett, "60-fsec pulse generation from a self-mode-locked Ti:sapphire laser," Opt. Lett. 16, 42–44 (1991).
[CrossRef] [PubMed]

F. Krausz, T. Brabec, and C. Spielmann, "Self-starting passive mode locking," Opt. Lett. 16, 235–237 (1991).
[CrossRef] [PubMed]

F. Salin, J. Squier, and M. Piché, "Mode locking of Ti:Al2O3 lasers and self-focusing: a Gaussian approximation," Opt. Lett. 16, 1674–1676 (1991).
[CrossRef] [PubMed]

M. Piché, "Beam reshaping and self-mode-locking in nonlinear laser resonators," Opt. Commun. 86, 156–160 (1991).
[CrossRef]

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, P. S. Shapovalov, and S. I. Turovets, "Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams," Phys. Lett. A 160, 138–142 (1991).
[CrossRef]

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, and P. S. Shapovalov, "Rotating elliptical Gaussian beams in nonlinear media," Opt. Commun. 81, 225–230 (1991).
[CrossRef]

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]

V. Magni, S. De Silvestri, and A. Cybo-Ottone, "On the stability, mode properties, and misalignment sensitivity of femtosecond dye laser resonators," Opt. Commun. 82, 137–144 (1991).
[CrossRef]

M. Desaix, D. Anderson, and M. Lisak, "Variational approach to collapse of optical pulses," J. Opt. Soc. Am. B 8, 2082–2086 (1991).
[CrossRef]

1990 (1)

F. Cornolti, M. Lucchesi, and B. Zambon, "Elliptic Gaussian beam self-focusing in nonlinear media," Opt. Commun. 75, 129–135 (1990).
[CrossRef]

1989 (1)

1987 (1)

1983 (1)

1972 (2)

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, "Astigmatically compensated cavities for cw dye laser," IEEE J. Quantum Electron. QE-8, 373–379 (1972).
[CrossRef]

C. R. Giuliano, J. H. Marburger, and A. Yariv, "Enhancement of self-focusing threshold in sapphire with elliptical beams," Appl. Phys. Lett. 21, 58–60 (1972).
[CrossRef]

1969 (2)

P. Baues, "Huygens' principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators," Opto-Electronics 1, 37–44 (1969).
[CrossRef]

J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt. 8, 1687–1693 (1969).
[CrossRef] [PubMed]

1965 (1)

H. Kogelnik, "Imaging of optical modes—resonators with internal lenses," Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

Acioli, L. H.

Agrawal, G. P.

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. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp. 1151–1228.

Anderson, D.

Arnaud, J. A.

Baues, P.

P. Baues, "Huygens' principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators," Opto-Electronics 1, 37–44 (1969).
[CrossRef]

Belanger, P. A.

Boyd, R. W.

Brabec, T.

Bridges, R. E.

Cerullo, G.

G. Cerullo, S. De Silvestri, and V. Magni, "Self-starting Kerr-lens mode-locking of a Ti:sapphire laser," Opt. Lett. 19, 1040–1042 (1994).
[CrossRef] [PubMed]

G. Cerullo, S. De Silvestri, V. Magni, and L. Pallaro, "Resonators for Kerr-lens mode-locked femtosecond Ti:sapphire lasers," Opt. Lett. 19, 807–809 (1994).
[CrossRef] [PubMed]

V. Magni, G. Cerullo, and S. De Silvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348–355 (1993).
[CrossRef]

V. Magni, G. Cerullo, and S. De Silvestri, "Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers," Opt. Commun. 101, 365–370 (1993).
[CrossRef]

Chilla, J. L. A.

Cornolti, F.

F. Cornolti, M. Lucchesi, and B. Zambon, "Elliptic Gaussian beam self-focusing in nonlinear media," Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Curley, P. F.

Cybo-Ottone, A.

V. Magni, S. De Silvestri, and A. Cybo-Ottone, "On the stability, mode properties, and misalignment sensitivity of femtosecond dye laser resonators," Opt. Commun. 82, 137–144 (1991).
[CrossRef]

Desaix, M.

Dienes, A.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, "Astigmatically compensated cavities for cw dye laser," IEEE J. Quantum Electron. QE-8, 373–379 (1972).
[CrossRef]

Fujimoto, J. G.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, "Self-focusing-induced saturable absorber loss for laser mode locking," Opt. Lett. 17, 511–513 (1992).
[CrossRef] [PubMed]

Georgiev, D.

D. Georgiev, J. Hermann, and U. Stamm, "Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers," Opt. Commun. 92, 368–375 (1992).
[CrossRef]

Giuliano, C. R.

C. R. Giuliano, J. H. Marburger, and A. Yariv, "Enhancement of self-focusing threshold in sapphire with elliptical beams," Appl. Phys. Lett. 21, 58–60 (1972).
[CrossRef]

Goncharenko, A. M.

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, and P. S. Shapovalov, "Rotating elliptical Gaussian beams in nonlinear media," Opt. Commun. 81, 225–230 (1991).
[CrossRef]

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, P. S. Shapovalov, and S. I. Turovets, "Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams," Phys. Lett. A 160, 138–142 (1991).
[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]

Haus, H. A.

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, "Self-focusing-induced saturable absorber loss for laser mode locking," Opt. Lett. 17, 511–513 (1992).
[CrossRef] [PubMed]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

Hermann, J.

D. Georgiev, J. Hermann, and U. Stamm, "Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers," Opt. Commun. 92, 368–375 (1992).
[CrossRef]

Herrmann, J.

J. Herrmann, "Theory of Kerr-lens mode locking: role of self-focusing and radially varying gain," J. Opt. Soc. Am. B 11, 498–512 (1994).
[CrossRef]

J. Herrmann, "Starting dynamic, self-starting condition and mode-locking threshold in passive, coupled cavity or Kerr-lens mode-locked solid-state lasers," Opt. Commun. 98, 111–116 (1993).
[CrossRef]

Hsieh, W. F.

Huang, D.

Ippen, E. P.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, "Astigmatically compensated cavities for cw dye laser," IEEE J. Quantum Electron. QE-8, 373–379 (1972).
[CrossRef]

Kean, P. N.

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. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp. 1151–1228.

Kogelnik, H.

J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt. 8, 1687–1693 (1969).
[CrossRef] [PubMed]

H. Kogelnik, "Imaging of optical modes—resonators with internal lenses," Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

Kogelnik, H. W.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, "Astigmatically compensated cavities for cw dye laser," IEEE J. Quantum Electron. QE-8, 373–379 (1972).
[CrossRef]

Krasinski, J. S.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, "Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements," Opt. Commun. 94, 221–226 (1992).
[CrossRef]

Krausz, F.

Lai, Ming

Lin, K. H.

Lisak, M.

Logvin, Yu. A.

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, P. S. Shapovalov, and S. I. Turovets, "Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams," Phys. Lett. A 160, 138–142 (1991).
[CrossRef]

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, and P. S. Shapovalov, "Rotating elliptical Gaussian beams in nonlinear media," Opt. Commun. 81, 225–230 (1991).
[CrossRef]

Lucchesi, M.

F. Cornolti, M. Lucchesi, and B. Zambon, "Elliptic Gaussian beam self-focusing in nonlinear media," Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Magni, V.

G. Cerullo, S. De Silvestri, and V. Magni, "Self-starting Kerr-lens mode-locking of a Ti:sapphire laser," Opt. Lett. 19, 1040–1042 (1994).
[CrossRef] [PubMed]

G. Cerullo, S. De Silvestri, V. Magni, and L. Pallaro, "Resonators for Kerr-lens mode-locked femtosecond Ti:sapphire lasers," Opt. Lett. 19, 807–809 (1994).
[CrossRef] [PubMed]

V. Magni, G. Cerullo, and S. De Silvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348–355 (1993).
[CrossRef]

V. Magni, G. Cerullo, and S. De Silvestri, "Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers," Opt. Commun. 101, 365–370 (1993).
[CrossRef]

V. Magni, S. De Silvestri, and A. Cybo-Ottone, "On the stability, mode properties, and misalignment sensitivity of femtosecond dye laser resonators," Opt. Commun. 82, 137–144 (1991).
[CrossRef]

Marburger, J. H.

C. R. Giuliano, J. H. Marburger, and A. Yariv, "Enhancement of self-focusing threshold in sapphire with elliptical beams," Appl. Phys. Lett. 21, 58–60 (1972).
[CrossRef]

J. H. Marburger, "Self-focusing: theory," in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, Oxford, 1977), Vol. 4, pp. 35–110.
[CrossRef]

Martinez, O. E.

Pallaro, L.

Pare, C.

Pearson, G. W.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, "Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements," Opt. Commun. 94, 221–226 (1992).
[CrossRef]

Piché, M.

Radzewicz, C.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, "Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements," Opt. Commun. 94, 221–226 (1992).
[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-Bahae, A. A. Said, and E. W. Van Stryland, "Highsensitivity, single-beam n2 measurements," Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Salin, F.

Sanson, A. M.

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, P. S. Shapovalov, and S. I. Turovets, "Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams," Phys. Lett. A 160, 138–142 (1991).
[CrossRef]

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, and P. S. Shapovalov, "Rotating elliptical Gaussian beams in nonlinear media," Opt. Commun. 81, 225–230 (1991).
[CrossRef]

Schmidt, A. J.

Shank, C. V.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, "Astigmatically compensated cavities for cw dye laser," IEEE J. Quantum Electron. QE-8, 373–379 (1972).
[CrossRef]

Shapovalov, P. S.

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, and P. S. Shapovalov, "Rotating elliptical Gaussian beams in nonlinear media," Opt. Commun. 81, 225–230 (1991).
[CrossRef]

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, P. S. Shapovalov, and S. I. Turovets, "Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams," Phys. Lett. A 160, 138–142 (1991).
[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, "Highsensitivity, single-beam n2 measurements," Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Shen, Y. R.

Y. R. Shen, "Self-focusing: experimental," in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, Oxford, 1977), Vol. 4, pp. 1–34.
[CrossRef]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 17.

Sibbett, W.

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 17.

Silvestri, S. De

G. Cerullo, S. De Silvestri, and V. Magni, "Self-starting Kerr-lens mode-locking of a Ti:sapphire laser," Opt. Lett. 19, 1040–1042 (1994).
[CrossRef] [PubMed]

G. Cerullo, S. De Silvestri, V. Magni, and L. Pallaro, "Resonators for Kerr-lens mode-locked femtosecond Ti:sapphire lasers," Opt. Lett. 19, 807–809 (1994).
[CrossRef] [PubMed]

V. Magni, G. Cerullo, and S. De Silvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348–355 (1993).
[CrossRef]

V. Magni, G. Cerullo, and S. De Silvestri, "Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers," Opt. Commun. 101, 365–370 (1993).
[CrossRef]

V. Magni, S. De Silvestri, and A. Cybo-Ottone, "On the stability, mode properties, and misalignment sensitivity of femtosecond dye laser resonators," Opt. Commun. 82, 137–144 (1991).
[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]

Spence, D. E.

Spielmann, C.

Spielmann, Ch.

Squier, J.

Stamm, U.

D. Georgiev, J. Hermann, and U. Stamm, "Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers," Opt. Commun. 92, 368–375 (1992).
[CrossRef]

Stryland, E. W. Van

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, "Highsensitivity, single-beam n2 measurements," Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

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. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp. 1151–1228.

Taché, J. P.

Turovets, S. I.

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, P. S. Shapovalov, and S. I. Turovets, "Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams," Phys. Lett. A 160, 138–142 (1991).
[CrossRef]

Ulman, M.

Witner, E.

Yariv, A.

C. R. Giuliano, J. H. Marburger, and A. Yariv, "Enhancement of self-focusing threshold in sapphire with elliptical beams," Appl. Phys. Lett. 21, 58–60 (1972).
[CrossRef]

Zambon, B.

F. Cornolti, M. Lucchesi, and B. Zambon, "Elliptic Gaussian beam self-focusing in nonlinear media," Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

C. R. Giuliano, J. H. Marburger, and A. Yariv, "Enhancement of self-focusing threshold in sapphire with elliptical beams," Appl. Phys. Lett. 21, 58–60 (1972).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, "Imaging of optical modes—resonators with internal lenses," Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

IEEE J. Quantum Electron. (2)

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, "Astigmatically compensated cavities for cw dye laser," IEEE J. Quantum Electron. QE-8, 373–379 (1972).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (9)

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, and P. S. Shapovalov, "Rotating elliptical Gaussian beams in nonlinear media," Opt. Commun. 81, 225–230 (1991).
[CrossRef]

F. Cornolti, M. Lucchesi, and B. Zambon, "Elliptic Gaussian beam self-focusing in nonlinear media," Opt. Commun. 75, 129–135 (1990).
[CrossRef]

J. Herrmann, "Starting dynamic, self-starting condition and mode-locking threshold in passive, coupled cavity or Kerr-lens mode-locked solid-state lasers," Opt. Commun. 98, 111–116 (1993).
[CrossRef]

V. Magni, G. Cerullo, and S. De Silvestri, "Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers," Opt. Commun. 101, 365–370 (1993).
[CrossRef]

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, "Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements," Opt. Commun. 94, 221–226 (1992).
[CrossRef]

V. Magni, G. Cerullo, and S. De Silvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348–355 (1993).
[CrossRef]

M. Piché, "Beam reshaping and self-mode-locking in nonlinear laser resonators," Opt. Commun. 86, 156–160 (1991).
[CrossRef]

D. Georgiev, J. Hermann, and U. Stamm, "Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers," Opt. Commun. 92, 368–375 (1992).
[CrossRef]

V. Magni, S. De Silvestri, and A. Cybo-Ottone, "On the stability, mode properties, and misalignment sensitivity of femtosecond dye laser resonators," Opt. Commun. 82, 137–144 (1991).
[CrossRef]

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. (12)

Ming Lai, "Self-starting, self-mode-locked Ti:sapphire laser," Opt. Lett. 19, 722–724 (1994).
[CrossRef] [PubMed]

G. Cerullo, S. De Silvestri, and V. Magni, "Self-starting Kerr-lens mode-locking of a Ti:sapphire laser," Opt. Lett. 19, 1040–1042 (1994).
[CrossRef] [PubMed]

F. Krausz, T. Brabec, and C. Spielmann, "Self-starting passive mode locking," Opt. Lett. 16, 235–237 (1991).
[CrossRef] [PubMed]

G. Cerullo, S. De Silvestri, V. Magni, and L. Pallaro, "Resonators for Kerr-lens mode-locked femtosecond Ti:sapphire lasers," Opt. Lett. 19, 807–809 (1994).
[CrossRef] [PubMed]

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

O. E. Martinez and J. L. A. Chilla, "Self-mode-locking of Ti:sapphire lasers: a matrix formalism," Opt. Lett. 17, 1210–1212 (1992).
[CrossRef] [PubMed]

T. Brabec, Ch. Spielmann, P. F. Curley, and F. Krausz, "Kerr-lens mode locking," Opt. Lett. 17, 1292–1294 (1992).
[CrossRef] [PubMed]

D. E. Spence, P. N. Kean, and W. Sibbett, "60-fsec pulse generation from a self-mode-locked Ti:sapphire laser," Opt. Lett. 16, 42–44 (1991).
[CrossRef] [PubMed]

F. Salin, J. Squier, and M. Piché, "Mode locking of Ti:Al2O3 lasers and self-focusing: a Gaussian approximation," Opt. Lett. 16, 1674–1676 (1991).
[CrossRef] [PubMed]

M. Piché and F. Salin, "Self-mode locking of solid-state lasers without apertures," Opt. Lett. 18, 1041–1043 (1993).
[CrossRef] [PubMed]

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, "Self-focusing-induced saturable absorber loss for laser mode locking," Opt. Lett. 17, 511–513 (1992).
[CrossRef] [PubMed]

R. E. Bridges, R. W. Boyd, and G. P. Agrawal, "Effect of beam ellipticity on self-mode locking in lasers," Opt. Lett. 18, 2026–2028 (1993).
[CrossRef] [PubMed]

Opto-Electronics (1)

P. Baues, "Huygens' principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators," Opto-Electronics 1, 37–44 (1969).
[CrossRef]

Phys. Lett. A (1)

A. M. Goncharenko, Yu. A. Logvin, A. M. Sanson, P. S. Shapovalov, and S. I. Turovets, "Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams," Phys. Lett. A 160, 138–142 (1991).
[CrossRef]

Other (7)

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 17.

Ref. 35, Chap. 15.

Ref. 35, Chap. 21.

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. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp. 1151–1228.

J. H. Marburger, "Self-focusing: theory," in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, Oxford, 1977), Vol. 4, pp. 35–110.
[CrossRef]

Y. R. Shen, "Self-focusing: experimental," in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, Oxford, 1977), Vol. 4, pp. 1–34.
[CrossRef]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 17.

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Figures (11)

Fig. 1
Fig. 1

Coordinate system used for study of nonlinear propagation in a Kerr medium.

Fig. 2
Fig. 2

(a) Z-scan experiment simulation for an astigmatic Gaussian beam with wavelength λ = 0.8 μm, normalized power p = 0.1, and coincident waists with spot sizes wx0 = 60 μm and wy0 = 20 μm, through a 20-mm-long Kerr medium with n = 1.5. The Kerr medium position is the distance between its center and the beam waist without the Kerr medium. Spot-size variations on the output plane (normalized to the linear spot size) are observed in the beam far field at a distance of 300 mm from the waist. Dots are obtained with the numerical solution of Eqs. (1). (b) Same as in (a), but p = 0.5.

Fig. 3
Fig. 3

(a) Same as in Fig. 2(a), with wx0 = 30 μm, wy0 = 10 μm, and p = 0.1. (b) Same as in (a), but p = 0.5.

Fig. 4
Fig. 4

Comparison of the analytical astigmatic beam results (solid curves) with those obtained by use of, for the (x, z) and the (y, z) planes, cylindrically symmetric beam propagation (dashed curves). Parameters for the calculations are as in Fig. 2(b).

Fig. 5
Fig. 5

General configuration of a resonator containing a Kerr medium; the arrows indicate the directions to which the matrices refer. Subscripts x and y of the matrix elements are omitted. L.O.E., linear optical elements; M1, mirror M1; M2, mirror M2.

Fig. 6
Fig. 6

Resonator configuration of a KLM laser used for both the calculations and the experiments. M1, M2 flat mirrors; M3, M4, concave folding mirrors (radius of curvature 100 mm, incidence angle θ = 14.8°); S, slit; the medium is a Brewster-cut Ti:sapphire rod with l = 20 mm and n = 1.76. Folding-mirror distance b is the physical path length between the mirrors.

Fig. 7
Fig. 7

(a) Kerr-lens sensitivity on mirror M1 in the tangential plane for a resonator with L1 = L2 = 850 mm and b = 117 mm as a function of the rod position a. The solid curve is obtained from Eq. (20); points are the results of an iterative numerical calculation performed with p = 0.01; the dashed curve is obtained by application of the cylindrically symmetric beam treatment to the equivalent uncoupled one-dimensional resonator. (b) Same as in (a) but for the sagittal plane.

Fig. 8
Fig. 8

Contour lines of the Kerr-lens sensitivity δ1x (parameter of the curves) as a function of a and b in the tangential plane for an asymmetric resonator with L1 = 500 mm and L2 = 1100 mm. The squares mark the points for which KLM was experimentally achieved with a slit on mirror M1, cutting the beam in the tangential plane. The horizontal lines are the resonator optical stability limits.

Fig. 9
Fig. 9

Same as in Fig. 8 but in the sagittal plane.

Fig. 10
Fig. 10

Same as in Fig. 8 but for a symmetric resonator with L1 = L2 = 850 mm in the tangential plane. The squares mark the points at which KLM was initiated by our tapping on one of the end mirrors; the triangles correspond to the self-starting condition.

Fig. 11
Fig. 11

Same as in Fig. 10 but in the sagittal plane.

Equations (35)

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d 2 w x d z 2 = ( λ n π ) 2 1 w x 3 ( 1 w x w y p ) , d w x d z = w x R x ,
ω x = w x 2 , ρ x = w x 2 / R x .
d ω x d z = 2 ρ x , d ρ x d z = ρ x 2 ω x + ( λ n π ) 2 1 ω x p ( λ n π ) 2 1 ( ω x ω y ) 1 / 2 .
ω L x = ω x 0 [ ( 1 + ρ x 0 ω x 0 z ) 2 + ( λ z π n ω x 0 ) 2 ] , ρ L x = ρ x 0 ( 1 + ρ x 0 ω x 0 z ) + ( λ π n ) 2 z ω x 0 ,
d ω x = 2 ρ x d z , d ρ x = [ ρ x 2 ω x + ( λ n π ) 2 1 ω x ] d z p ( λ n π ) 2 1 ( ω x ω y ) 1 / 2 d z .
ϕ x = p ( λ n π ) 2 ( ω x ω y ) 1 / 2 1 ω x 2 d z .
ω x ( p ) ω L x + p ω K x , ρ x ( p ) ρ L x + p ρ K x .
d ω K x d z = 2 ρ K x , ω L x d ρ K x d z = 2 ρ L x ρ K x d ρ L x d z ω K x ( λ n π ) 2 ( ω L x ω L y ) 1 / 2 .
d ω K x ( z ) = ( d ω x ( z ) d ϕ x ) ϕ x = 0 ( d ϕ x d p ) p = 0 ,
ω x ( z ) = ω L x ( ζ ) { [ 1 ( z ζ ) ϕ x + ( z ζ ) ρ L x ω L x ] 2 + [ λ ( z ζ ) π n ω L x ] 2 } .
d ω K x ( z ) = 2 ( λ π n ) 2 ( ω L x ω L y ) 1 / 2 [ 1 + z ζ ω L x ( ζ ) ρ L x ( ζ ) ] × z ζ ω L x ( ζ ) d ζ .
d ω K x ( z ) = ( λ π n ) 2 [ ω L x ( ζ ) ω L y ( ζ ) ] 1 / 2 d d ζ [ ( z ζ ) 2 ω L x ( ζ ) ] d ζ .
ω K x ( z ) = ( λ π n ) 2 0 z [ ω L x ( ζ ) ω L y ( ζ ) ] 1 / 2 d d ζ [ ( z ζ ) 2 ω L x ( ζ ) ] d ζ .
ρ K x ( z ) = ( λ π n ) 2 0 z [ ω L x ( ζ ) ω L y ( ζ ) ] 1 / 2 d d ζ ( z ζ ω L x ( ζ ) ) d ζ .
ω K x ( z ) = ω K y ( z ) = ( λ π n ) 2 z 2 ω L x ( 0 ) .
δ = ( 1 2 ω d ω d p ) p = 0 .
d δ 1 x = ( 1 2 ω 1 x d ω 1 x d ϕ x ) ϕ x = 0 ( d ϕ x d p ) p = 0 ,
δ 1 x = 1 n ( 1 S y 2 1 S x 2 ) 1 / 4 0 l | B x B y | 1 / 2 × B 2 x D 2 x S x + B 1 x D 1 x B x 2 d ζ .
S x = Ã x + D x 2 = A x D x + B x C x ,
δ 1 x = 1 2 n ( 1 S y 2 1 S x 2 ) 1 / 4 0 l | B x B y | 1 / 2 d d ζ ( B 2 x D 2 x B x ) d ζ .
δ 1 = 1 2 n [ ( B 2 D 2 B ) ζ = l ( B 2 D 2 B ) ζ = 0 ] ,
δ 1 x 1 2 ω 1 L x ω 1 x ω 1 L x p ,
[ A j x B j x C j x D j x ] ,
[ A x ( ϕ x ) B x ( ϕ x ) C x ( ϕ x ) D x ( ϕ x ) ]
[ A x ( ϕ x ) B x ( ϕ x ) C x ( ϕ x ) D x ( ϕ x ) ] = [ A x D 1 x B 2 x ϕ x / n C x D 1 x D 2 x ϕ x / n B x B 1 x B 2 x ϕ x / n D x B 1 x D 2 x ϕ x / n ] ,
ω 1 x 2 = ( λ π ) 2 B x ( ϕ x ) D x ( ϕ x ) A x ( ϕ x ) C x ( ϕ x ) .
( 1 2 ω 1 x d ω 1 x d ϕ x ) = 1 n ( S x 2 1 ) ( B 2 x D 2 x S x + B 1 x D 1 x ) ,
( λ π n ω L x ) 2 = 1 S x 2 B x 2 .
d δ 1 x = 1 n ( 1 S y 2 1 S x 2 ) 1 / 4 | B x B y | 1 / 2 B 2 x D 2 x S x + B 1 x D 1 x B x 2 d ζ .
d B x d ζ = D x à x ,
d B 2 x D 2 x d ζ = ( A 2 x D 2 x + B 2 x C 2 x ) .
d d ζ ( B 2 x D 2 x B x ) = ( A 2 x D 2 x + B 2 x C 2 x ) B x + B 2 x D 2 x ( D x à x ) B x 2 = 2 B 2 x D 2 x S x + B 1 x D 1 x B x 2 ,
[ A x C x B x D x ]
ω x = ω 1 x [ A x 2 + ( λ B x π ω 1 ) 2 ] .
δ x = ( 1 2 ω x d ω x d p ) p = 0 = ( ω 1 x ω x d ω x d ω 1 x ) p = 0 δ 1 x = A x 2 ( λ B x / π ω 1 L x ) 2 A x 2 + ( λ B x / π ω 1 L x ) 2 δ 1 x ,

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