Abstract

The ABCD-matrix formalism is generalized for the description of self-focusing and radially varying pump power in a nonparabolic approximation. This extended formalism is used for the investigation of the operation and for the optimization of the saturable aperture loss, the power-dependent average gain, and the resonator magnification of Kerr-lens mode-locked lasers with as well as without an internal aperture. The calculated amplitude modulation parameters are compared with the necessary condition for self-starting and for the shortest possible pulse duration limit.

© 1994 Optical Society of America

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  1. D. E. Spence, P. N. Kean, and W. Sibbett, "60-fs pulse generation from a self-mode-locked Ti:sapphire laser," Opt. Lett. 16, 42–44 (1991).
    [CrossRef] [PubMed]
  2. D. K. Negus, L. Spinelli, N. Goldblatt, and G. Feuget, "Sub-100 fs pulse generation by Kerr lens mode locking in Ti:Al2O3," in Advanced Solid State Lasers, Vol. 10 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 120–124, postdeadline paper.
  3. G. Gabetta, D. Huang, R. Ramaswamy, E. P. Ippen, and I. G. Fujimoto, "Femtosecond pulse generation in Ti:Al2O3 using a microdot mirror mode locker," Opt. Lett. 16, 1756–1758 (1991).
    [CrossRef] [PubMed]
  4. Sarukura, Y. Ishida, and N. Nakano, "Generation of 50-fs pulses from a pulse-compressed, cw, passively mode-locked Ti:sapphire laser," Opt. Lett. 16, 153–155 (1991).
    [PubMed]
  5. F. Krausz, C. Spielmann, T. Brabec, E. Wintner, and A. J. Schmidt, "Generation of 33-fs optical pulses from a solid-state laser," Opt. Lett. 17, 204–206 (1992).
    [CrossRef] [PubMed]
  6. C.-P. Huang, M. T. Asaki, S. Backus, M. M. Murnane, H. C. Kapteyn, and H. Nathel, "17-fs pulses from a self-mode-locked Ti:sapphire laser," Opt. Lett. 17, 1289–1291 (1992).
    [CrossRef] [PubMed]
  7. I. M. Jacobson, N. Naganuma, H. A. Haus, and J. G. Fujimoto, "Femtosecond pulse generation in a Ti:Al2O3 laser by using second- and third-order intracavity dispersion," Opt. Lett. 17, 1608–1610 (1992).
    [CrossRef] [PubMed]
  8. F. Kraus, M. E. Ferman, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Winter, and A. J. Schmidt, "Femtosecond solid-state lasers," IEEE J. Quantum Electron. 28, 2097–2121 (1992).
    [CrossRef]
  9. D. E. Spence, J. M. Evans, W. E. Sleat, and W. Sibbett, "Regenerative initiated self-mode-locked Ti:sapphire laser," Opt. Lett. 16, 1762–1764 (1991).
    [CrossRef] [PubMed]
  10. U. Keller, G. W. 'tHooft, W. H. Knox, and J. E. Cunningham, "Femtosecond pulses from a continuously self-starting passively mode-locked Ti:sapphire laser," Opt. Lett. 16, 1022–1024 (1991).
    [CrossRef] [PubMed]
  11. C. Spielmann, F. Krausz, T. Brabec, E. Wintner, and A. J. Schmidt, "Femtosecond pulse generation from a synchronously pumped Ti:sapphire laser," Opt. Lett. 16, 1180–1182 (1991).
    [CrossRef] [PubMed]
  12. N. H. Rizvi, P. M. W. French, and J. R. Taylor, "Continuously self-mode-locked Ti:sapphire laser that produces sub-50-fs pulses," Opt. Lett. 17, 279–281 (1992).
    [CrossRef] [PubMed]
  13. W. S. Delouch, P. E. Powers, C. L. Tang, "Self-starting ring-cavity Ti:sapphire laser," Opt. Lett. 17, 1581–1583 (1992).
    [CrossRef]
  14. G. P. A. Malcolm and A. I. Ferguson, "Self-mode locking of a diode-pumped Nd:YLF laser," Opt. Lett. 16, 1967–1969 (1991).
    [CrossRef] [PubMed]
  15. K. X. Liu, C. J. Flood, D. B. Walker, and H. M. van Driel, "Kerr lens mode locking of a diode-pumped Nd:YAG laser," Opt. Lett. 17, 1361–1363 (1992).
    [CrossRef]
  16. N. V. Rizvi, P. M. French, and J. R. Taylor, "Generation of 33-fs pulses from a passively mode-locked Cr3+:LiSrAlF6 laser," Opt. Lett. 17, 1605–1607 (1992).
    [CrossRef] [PubMed]
  17. M. Piche, "Beam reshaping and self-mode-locking in nonlinear resonators," Opt. Commun. 86, 156–160 (1991).
    [CrossRef]
  18. F. Salin, J. Squier, and M. Piche, "Mode locking of Ti:Al2O3 lasers and self-focusing: a Guassian approximation," Opt. Lett. 16, 1674–1776 (1991).
    [CrossRef] [PubMed]
  19. S. Chen and J. Wang, "Self-starting issues of passive self-focusing mode locking," Opt. Lett. 16, 1689–1691 (1991).
    [CrossRef] [PubMed]
  20. D. Georgiev, J. Herrmann, and U. Stamm, "Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid state lasers," Opt. Commun. 92, 368 (1992); in Ultrafast Processes in Spectroscopy, Inst. Phys. Conf. Ser. 126, 5 (1991).
    [CrossRef]
  21. D. Huang, M. Ulmann, L. H. Acide, H. A. Haus, and J. G. Fujimoto, "Self-focusing induced saturable loss for laser mode locking," Opt. Lett. 17, 511–513 (1992).
    [CrossRef] [PubMed]
  22. J. Herrmann, "The generation conditions and the dynamics of Kerr lens mode locked lasers," presented at the XVII International Quantum Electronics Conference, Vienna, 1992.
  23. H. A. Haus, J. G. Fujimoto, E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086–2096 (1992).
    [CrossRef]
  24. T. Brabec, Ch. Spielmann, P. E. Curley, and F. Krausz, "Ken-lens mode locking," Opt. Lett. 17, 1292–1294 (1992).
    [CrossRef] [PubMed]
  25. P. G. Kryukov and V. S. Lethokov, "Fluctuation mechanism of ultrashort pulse generation by laser with saturable absorber," IEEE J. Quantum Electron. QE-8, 766–782 (1972).
    [CrossRef]
  26. J. A. Fleck, "Mode-locked pulse generation by Q-switched lasers," Phys. Rev. B 1, 84–100 (1970).
    [CrossRef]
  27. J. Herrmann and B. Wilhelmi, Lasers for Ultrashort Light pulses (North-Holland, Amsterdam, 1987), pp. 191ff.
  28. S. V. Chekalin, P. G. Kryukov, Y. A. Matveetz, and O. B. Shatberashvili, "The process of formation of ultrashort laser pulses," Opto-Electronics 6, 249–261 (1974).
    [CrossRef]
  29. J. Herrmann, F. Weidner, and B. Wilhelmi, "Influence of the inversion depletion in the active medium on the evolution of ultrashort pulses in passively modelocked solid-state lasers," Appl. Phys. 20, 237–245 (1979).
    [CrossRef]
  30. J. Herrmann, "Starting dynamic, self-starting condition, and modelocking threshold in passive, coupled-cavity or Kerr-lens modelocked solid-state lasers," Opt. Commun. 98, 111–116 (1993).
    [CrossRef]
  31. H. A. Haus, "Theory of mode locking with a fast saturable absorber," J. Appl. Phys. 46, 3049–3056 (1975).
    [CrossRef]
  32. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Structures for additive-pulse mode locking," J. Opt. Soc. Am. B 8, 2068–2076 (1991).
    [CrossRef]
  33. U. Petrov, D. Georgiev, J. Herrmann, and U. Stamm, "Theory of cw passive mode-locking of solid state lasers with addition of nonlinear index and group velocity dispersion," Opt. Commun. 91, 123–129 (1992).
    [CrossRef]
  34. H. Kogelnik, "On the propagation of Gaussian beams of light through lenslike media including those with loss or gain variation," Appl. Opt. 4, 1562–1569 (1965).
    [CrossRef]
  35. L. W. Casperson and A. Yariv, "The Gaussian mode in optical resonators with a radial gain profile," Appl. Phys. Lett. 12, 355–357 (1968).
    [CrossRef]
  36. L. W. Casperson, "Beam modes in complex lenslike media and resonators," J. Opt. Soc. Am. 66, 1373–1379 (1976).
    [CrossRef]
  37. F. Salin and J. Squier, "Gain guiding in solid-state lasers," Opt. Lett. 17, 1352–1354 (1992).
    [CrossRef] [PubMed]
  38. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  39. L. W. Casperson and S. D. Lunnam, "Gaussian modes in high loss laser resonators," Appl. Opt. 14, 1193–1199 (1975).
    [CrossRef] [PubMed]
  40. A. Yariv and P. Yeh, "Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers," Opt. Commun. 13, 370–374 (1975).
    [CrossRef]
  41. A. L. Alfrey, "Modeling of longitudinal pumped cw Ti:sapphire laser oscillators," IEEE J. Quantum Electron. 25, 760 (1989).
    [CrossRef]
  42. M. Karlsson, D. Anderson, M. Desaix, M. Lisak, "Dynamic effects of Kerr nonlinearity and spatial diffraction on self-phase modulation of optical pulses," Opt. Lett. 16, 1373–1375 (1991).
    [CrossRef] [PubMed]
  43. S. Gatz and J. Herrmann, "Resonator theory for lasers with radially varying gain," to be submitted to J. Opt. Soc. Am. B.
  44. H. A. Haus and E. P. Ippen, "Self-starting of passively mode locked lasers," Opt. Lett. 16, 1331–1333 (1991).
    [CrossRef] [PubMed]
  45. F. Krausz, T. Brabec, and C. Spielmann, "Self-starting passive mode-locking," Opt. Lett. 16, 235–237 (1991).
    [CrossRef] [PubMed]
  46. J. Herrmann and M. Müller, "Theory of mode locking in coupled cavities with a Kerr nonlinearity," to be submitted to J. Opt. Soc. Am. B.
  47. J. Herrmann, "Theory of Mode-locking in femtosecond solid-state lasers with higher order nonlinear effects," submitted to J. Opt. Soc. Am. B.
  48. T. Brabec, Ch. Spielmann, and F. Krausz, "Limits of pulse shortening in solitary lasers," Opt. Lett. 17, 748–750 (1992).
    [CrossRef] [PubMed]

1993 (1)

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

1992 (14)

U. Petrov, D. Georgiev, J. Herrmann, and U. Stamm, "Theory of cw passive mode-locking of solid state lasers with addition of nonlinear index and group velocity dispersion," Opt. Commun. 91, 123–129 (1992).
[CrossRef]

F. Salin and J. Squier, "Gain guiding in solid-state lasers," Opt. Lett. 17, 1352–1354 (1992).
[CrossRef] [PubMed]

T. Brabec, Ch. Spielmann, and F. Krausz, "Limits of pulse shortening in solitary lasers," Opt. Lett. 17, 748–750 (1992).
[CrossRef] [PubMed]

F. Krausz, C. Spielmann, T. Brabec, E. Wintner, and A. J. Schmidt, "Generation of 33-fs optical pulses from a solid-state laser," Opt. Lett. 17, 204–206 (1992).
[CrossRef] [PubMed]

C.-P. Huang, M. T. Asaki, S. Backus, M. M. Murnane, H. C. Kapteyn, and H. Nathel, "17-fs pulses from a self-mode-locked Ti:sapphire laser," Opt. Lett. 17, 1289–1291 (1992).
[CrossRef] [PubMed]

I. M. Jacobson, N. Naganuma, H. A. Haus, and J. G. Fujimoto, "Femtosecond pulse generation in a Ti:Al2O3 laser by using second- and third-order intracavity dispersion," Opt. Lett. 17, 1608–1610 (1992).
[CrossRef] [PubMed]

F. Kraus, M. E. Ferman, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Winter, and A. J. Schmidt, "Femtosecond solid-state lasers," IEEE J. Quantum Electron. 28, 2097–2121 (1992).
[CrossRef]

N. H. Rizvi, P. M. W. French, and J. R. Taylor, "Continuously self-mode-locked Ti:sapphire laser that produces sub-50-fs pulses," Opt. Lett. 17, 279–281 (1992).
[CrossRef] [PubMed]

W. S. Delouch, P. E. Powers, C. L. Tang, "Self-starting ring-cavity Ti:sapphire laser," Opt. Lett. 17, 1581–1583 (1992).
[CrossRef]

K. X. Liu, C. J. Flood, D. B. Walker, and H. M. van Driel, "Kerr lens mode locking of a diode-pumped Nd:YAG laser," Opt. Lett. 17, 1361–1363 (1992).
[CrossRef]

N. V. Rizvi, P. M. French, and J. R. Taylor, "Generation of 33-fs pulses from a passively mode-locked Cr3+:LiSrAlF6 laser," Opt. Lett. 17, 1605–1607 (1992).
[CrossRef] [PubMed]

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

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

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

1991 (14)

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

F. Salin, J. Squier, and M. Piche, "Mode locking of Ti:Al2O3 lasers and self-focusing: a Guassian approximation," Opt. Lett. 16, 1674–1776 (1991).
[CrossRef] [PubMed]

S. Chen and J. Wang, "Self-starting issues of passive self-focusing mode locking," Opt. Lett. 16, 1689–1691 (1991).
[CrossRef] [PubMed]

G. P. A. Malcolm and A. I. Ferguson, "Self-mode locking of a diode-pumped Nd:YLF laser," Opt. Lett. 16, 1967–1969 (1991).
[CrossRef] [PubMed]

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

G. Gabetta, D. Huang, R. Ramaswamy, E. P. Ippen, and I. G. Fujimoto, "Femtosecond pulse generation in Ti:Al2O3 using a microdot mirror mode locker," Opt. Lett. 16, 1756–1758 (1991).
[CrossRef] [PubMed]

Sarukura, Y. Ishida, and N. Nakano, "Generation of 50-fs pulses from a pulse-compressed, cw, passively mode-locked Ti:sapphire laser," Opt. Lett. 16, 153–155 (1991).
[PubMed]

D. E. Spence, J. M. Evans, W. E. Sleat, and W. Sibbett, "Regenerative initiated self-mode-locked Ti:sapphire laser," Opt. Lett. 16, 1762–1764 (1991).
[CrossRef] [PubMed]

U. Keller, G. W. 'tHooft, W. H. Knox, and J. E. Cunningham, "Femtosecond pulses from a continuously self-starting passively mode-locked Ti:sapphire laser," Opt. Lett. 16, 1022–1024 (1991).
[CrossRef] [PubMed]

C. Spielmann, F. Krausz, T. Brabec, E. Wintner, and A. J. Schmidt, "Femtosecond pulse generation from a synchronously pumped Ti:sapphire laser," Opt. Lett. 16, 1180–1182 (1991).
[CrossRef] [PubMed]

M. Karlsson, D. Anderson, M. Desaix, M. Lisak, "Dynamic effects of Kerr nonlinearity and spatial diffraction on self-phase modulation of optical pulses," Opt. Lett. 16, 1373–1375 (1991).
[CrossRef] [PubMed]

H. A. Haus and E. P. Ippen, "Self-starting of passively mode locked lasers," Opt. Lett. 16, 1331–1333 (1991).
[CrossRef] [PubMed]

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

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Structures for additive-pulse mode locking," J. Opt. Soc. Am. B 8, 2068–2076 (1991).
[CrossRef]

1989 (1)

A. L. Alfrey, "Modeling of longitudinal pumped cw Ti:sapphire laser oscillators," IEEE J. Quantum Electron. 25, 760 (1989).
[CrossRef]

1979 (1)

J. Herrmann, F. Weidner, and B. Wilhelmi, "Influence of the inversion depletion in the active medium on the evolution of ultrashort pulses in passively modelocked solid-state lasers," Appl. Phys. 20, 237–245 (1979).
[CrossRef]

1976 (1)

1975 (3)

L. W. Casperson and S. D. Lunnam, "Gaussian modes in high loss laser resonators," Appl. Opt. 14, 1193–1199 (1975).
[CrossRef] [PubMed]

A. Yariv and P. Yeh, "Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers," Opt. Commun. 13, 370–374 (1975).
[CrossRef]

H. A. Haus, "Theory of mode locking with a fast saturable absorber," J. Appl. Phys. 46, 3049–3056 (1975).
[CrossRef]

1974 (1)

S. V. Chekalin, P. G. Kryukov, Y. A. Matveetz, and O. B. Shatberashvili, "The process of formation of ultrashort laser pulses," Opto-Electronics 6, 249–261 (1974).
[CrossRef]

1972 (1)

P. G. Kryukov and V. S. Lethokov, "Fluctuation mechanism of ultrashort pulse generation by laser with saturable absorber," IEEE J. Quantum Electron. QE-8, 766–782 (1972).
[CrossRef]

1970 (1)

J. A. Fleck, "Mode-locked pulse generation by Q-switched lasers," Phys. Rev. B 1, 84–100 (1970).
[CrossRef]

1968 (1)

L. W. Casperson and A. Yariv, "The Gaussian mode in optical resonators with a radial gain profile," Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

1965 (1)

’tHooft, G. W.

Casperson, L. W.

Herrmann, J.

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

Piche, M.

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

Powers, P. E.

Ramaswamy, R.

Tang, C. L.

van Driel, H. M.

Acide, L. H.

Alfrey, A. L.

A. L. Alfrey, "Modeling of longitudinal pumped cw Ti:sapphire laser oscillators," IEEE J. Quantum Electron. 25, 760 (1989).
[CrossRef]

Anderson, D.

Asaki, M. T.

Backus, S.

Brabec, T.

Casperson, L. W.

L. W. Casperson and S. D. Lunnam, "Gaussian modes in high loss laser resonators," Appl. Opt. 14, 1193–1199 (1975).
[CrossRef] [PubMed]

L. W. Casperson and A. Yariv, "The Gaussian mode in optical resonators with a radial gain profile," Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Chekalin, S. V.

S. V. Chekalin, P. G. Kryukov, Y. A. Matveetz, and O. B. Shatberashvili, "The process of formation of ultrashort laser pulses," Opto-Electronics 6, 249–261 (1974).
[CrossRef]

Chen, S.

Cunningham, J. E.

Curley, P. E.

Curley, P. F.

F. Kraus, M. E. Ferman, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Winter, and A. J. Schmidt, "Femtosecond solid-state lasers," IEEE J. Quantum Electron. 28, 2097–2121 (1992).
[CrossRef]

Delouch, W. S.

Desaix, M.

Evans, J. M.

Ferguson, A. I.

Ferman, M. E.

F. Kraus, M. E. Ferman, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Winter, and A. J. Schmidt, "Femtosecond solid-state lasers," IEEE J. Quantum Electron. 28, 2097–2121 (1992).
[CrossRef]

Feuget, G.

D. K. Negus, L. Spinelli, N. Goldblatt, and G. Feuget, "Sub-100 fs pulse generation by Kerr lens mode locking in Ti:Al2O3," in Advanced Solid State Lasers, Vol. 10 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 120–124, postdeadline paper.

Fleck, J. A.

J. A. Fleck, "Mode-locked pulse generation by Q-switched lasers," Phys. Rev. B 1, 84–100 (1970).
[CrossRef]

Flood, C. J.

French, P. M.

French, P. M. W.

Fujimoto, I. G.

Fujimoto, J. G.

Gabetta, G.

Gatz, S.

S. Gatz and J. Herrmann, "Resonator theory for lasers with radially varying gain," to be submitted to J. Opt. Soc. Am. B.

Georgiev, D.

U. Petrov, D. Georgiev, J. Herrmann, and U. Stamm, "Theory of cw passive mode-locking of solid state lasers with addition of nonlinear index and group velocity dispersion," Opt. Commun. 91, 123–129 (1992).
[CrossRef]

D. Georgiev, J. Herrmann, and U. Stamm, "Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid state lasers," Opt. Commun. 92, 368 (1992); in Ultrafast Processes in Spectroscopy, Inst. Phys. Conf. Ser. 126, 5 (1991).
[CrossRef]

Goldblatt, N.

D. K. Negus, L. Spinelli, N. Goldblatt, and G. Feuget, "Sub-100 fs pulse generation by Kerr lens mode locking in Ti:Al2O3," in Advanced Solid State Lasers, Vol. 10 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 120–124, postdeadline paper.

Haus, H. A.

Herrmann, J.

U. Petrov, D. Georgiev, J. Herrmann, and U. Stamm, "Theory of cw passive mode-locking of solid state lasers with addition of nonlinear index and group velocity dispersion," Opt. Commun. 91, 123–129 (1992).
[CrossRef]

J. Herrmann, F. Weidner, and B. Wilhelmi, "Influence of the inversion depletion in the active medium on the evolution of ultrashort pulses in passively modelocked solid-state lasers," Appl. Phys. 20, 237–245 (1979).
[CrossRef]

J. Herrmann and M. Müller, "Theory of mode locking in coupled cavities with a Kerr nonlinearity," to be submitted to J. Opt. Soc. Am. B.

J. Herrmann and B. Wilhelmi, Lasers for Ultrashort Light pulses (North-Holland, Amsterdam, 1987), pp. 191ff.

D. Georgiev, J. Herrmann, and U. Stamm, "Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid state lasers," Opt. Commun. 92, 368 (1992); in Ultrafast Processes in Spectroscopy, Inst. Phys. Conf. Ser. 126, 5 (1991).
[CrossRef]

S. Gatz and J. Herrmann, "Resonator theory for lasers with radially varying gain," to be submitted to J. Opt. Soc. Am. B.

J. Herrmann, "Theory of Mode-locking in femtosecond solid-state lasers with higher order nonlinear effects," submitted to J. Opt. Soc. Am. B.

J. Herrmann, "The generation conditions and the dynamics of Kerr lens mode locked lasers," presented at the XVII International Quantum Electronics Conference, Vienna, 1992.

Hofer, M.

F. Kraus, M. E. Ferman, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Winter, and A. J. Schmidt, "Femtosecond solid-state lasers," IEEE J. Quantum Electron. 28, 2097–2121 (1992).
[CrossRef]

Huang, C.-P.

Huang, D.

Ippen, E. P.

Ishida, Y.

Jacobson, I. M.

Kapteyn, H. C.

Karlsson, M.

Kean, P. N.

Keller, U.

Knox, W. H.

Kogelnik, H.

Kraus, F.

F. Kraus, M. E. Ferman, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Winter, and A. J. Schmidt, "Femtosecond solid-state lasers," IEEE J. Quantum Electron. 28, 2097–2121 (1992).
[CrossRef]

Krausz, F.

Kryukov, P. G.

S. V. Chekalin, P. G. Kryukov, Y. A. Matveetz, and O. B. Shatberashvili, "The process of formation of ultrashort laser pulses," Opto-Electronics 6, 249–261 (1974).
[CrossRef]

P. G. Kryukov and V. S. Lethokov, "Fluctuation mechanism of ultrashort pulse generation by laser with saturable absorber," IEEE J. Quantum Electron. QE-8, 766–782 (1972).
[CrossRef]

Lethokov, V. S.

P. G. Kryukov and V. S. Lethokov, "Fluctuation mechanism of ultrashort pulse generation by laser with saturable absorber," IEEE J. Quantum Electron. QE-8, 766–782 (1972).
[CrossRef]

Lisak, M.

Liu, K. X.

Lunnam, S. D.

Malcolm, G. P. A.

Matveetz, Y. A.

S. V. Chekalin, P. G. Kryukov, Y. A. Matveetz, and O. B. Shatberashvili, "The process of formation of ultrashort laser pulses," Opto-Electronics 6, 249–261 (1974).
[CrossRef]

Müller, M.

J. Herrmann and M. Müller, "Theory of mode locking in coupled cavities with a Kerr nonlinearity," to be submitted to J. Opt. Soc. Am. B.

Murnane, M. M.

Naganuma, N.

Nakano, N.

Nathel, H.

Negus, D. K.

D. K. Negus, L. Spinelli, N. Goldblatt, and G. Feuget, "Sub-100 fs pulse generation by Kerr lens mode locking in Ti:Al2O3," in Advanced Solid State Lasers, Vol. 10 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 120–124, postdeadline paper.

Ober, M. H.

F. Kraus, M. E. Ferman, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Winter, and A. J. Schmidt, "Femtosecond solid-state lasers," IEEE J. Quantum Electron. 28, 2097–2121 (1992).
[CrossRef]

Petrov, U.

U. Petrov, D. Georgiev, J. Herrmann, and U. Stamm, "Theory of cw passive mode-locking of solid state lasers with addition of nonlinear index and group velocity dispersion," Opt. Commun. 91, 123–129 (1992).
[CrossRef]

Piche, M.

Rizvi, N. H.

Rizvi, N. V.

Salin, F.

Schmidt, A. J.

Shatberashvili, O. B.

S. V. Chekalin, P. G. Kryukov, Y. A. Matveetz, and O. B. Shatberashvili, "The process of formation of ultrashort laser pulses," Opto-Electronics 6, 249–261 (1974).
[CrossRef]

Sibbett, W.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Sleat, W. E.

Spence, D. E.

Spielmann, C.

Spielmann, Ch.

Spinelli, L.

D. K. Negus, L. Spinelli, N. Goldblatt, and G. Feuget, "Sub-100 fs pulse generation by Kerr lens mode locking in Ti:Al2O3," in Advanced Solid State Lasers, Vol. 10 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 120–124, postdeadline paper.

Squier, J.

Stamm, U.

U. Petrov, D. Georgiev, J. Herrmann, and U. Stamm, "Theory of cw passive mode-locking of solid state lasers with addition of nonlinear index and group velocity dispersion," Opt. Commun. 91, 123–129 (1992).
[CrossRef]

D. Georgiev, J. Herrmann, and U. Stamm, "Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid state lasers," Opt. Commun. 92, 368 (1992); in Ultrafast Processes in Spectroscopy, Inst. Phys. Conf. Ser. 126, 5 (1991).
[CrossRef]

Taylor, J. R.

Ulmann, M.

Walker, D. B.

Wang, J.

Weidner, F.

J. Herrmann, F. Weidner, and B. Wilhelmi, "Influence of the inversion depletion in the active medium on the evolution of ultrashort pulses in passively modelocked solid-state lasers," Appl. Phys. 20, 237–245 (1979).
[CrossRef]

Wilhelmi, B.

J. Herrmann, F. Weidner, and B. Wilhelmi, "Influence of the inversion depletion in the active medium on the evolution of ultrashort pulses in passively modelocked solid-state lasers," Appl. Phys. 20, 237–245 (1979).
[CrossRef]

J. Herrmann and B. Wilhelmi, Lasers for Ultrashort Light pulses (North-Holland, Amsterdam, 1987), pp. 191ff.

Winter, E.

F. Kraus, M. E. Ferman, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Winter, and A. J. Schmidt, "Femtosecond solid-state lasers," IEEE J. Quantum Electron. 28, 2097–2121 (1992).
[CrossRef]

Wintner, E.

Yariv, A.

A. Yariv and P. Yeh, "Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers," Opt. Commun. 13, 370–374 (1975).
[CrossRef]

L. W. Casperson and A. Yariv, "The Gaussian mode in optical resonators with a radial gain profile," Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Yeh, P.

A. Yariv and P. Yeh, "Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers," Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. (1)

J. Herrmann, F. Weidner, and B. Wilhelmi, "Influence of the inversion depletion in the active medium on the evolution of ultrashort pulses in passively modelocked solid-state lasers," Appl. Phys. 20, 237–245 (1979).
[CrossRef]

Appl. Phys. Lett. (1)

L. W. Casperson and A. Yariv, "The Gaussian mode in optical resonators with a radial gain profile," Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

IEEE J. Quantum Electron. (4)

F. Kraus, M. E. Ferman, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Winter, and A. J. Schmidt, "Femtosecond solid-state lasers," IEEE J. Quantum Electron. 28, 2097–2121 (1992).
[CrossRef]

P. G. Kryukov and V. S. Lethokov, "Fluctuation mechanism of ultrashort pulse generation by laser with saturable absorber," IEEE J. Quantum Electron. QE-8, 766–782 (1972).
[CrossRef]

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

A. L. Alfrey, "Modeling of longitudinal pumped cw Ti:sapphire laser oscillators," IEEE J. Quantum Electron. 25, 760 (1989).
[CrossRef]

J. Appl. Phys. (1)

H. A. Haus, "Theory of mode locking with a fast saturable absorber," J. Appl. Phys. 46, 3049–3056 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Opt. Commun. (4)

U. Petrov, D. Georgiev, J. Herrmann, and U. Stamm, "Theory of cw passive mode-locking of solid state lasers with addition of nonlinear index and group velocity dispersion," Opt. Commun. 91, 123–129 (1992).
[CrossRef]

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

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

A. Yariv and P. Yeh, "Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers," Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Opt. Lett. (23)

F. Salin, J. Squier, and M. Piche, "Mode locking of Ti:Al2O3 lasers and self-focusing: a Guassian approximation," Opt. Lett. 16, 1674–1776 (1991).
[CrossRef] [PubMed]

F. Salin and J. Squier, "Gain guiding in solid-state lasers," Opt. Lett. 17, 1352–1354 (1992).
[CrossRef] [PubMed]

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

Sarukura, Y. Ishida, and N. Nakano, "Generation of 50-fs pulses from a pulse-compressed, cw, passively mode-locked Ti:sapphire laser," Opt. Lett. 16, 153–155 (1991).
[PubMed]

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

U. Keller, G. W. 'tHooft, W. H. Knox, and J. E. Cunningham, "Femtosecond pulses from a continuously self-starting passively mode-locked Ti:sapphire laser," Opt. Lett. 16, 1022–1024 (1991).
[CrossRef] [PubMed]

C. Spielmann, F. Krausz, T. Brabec, E. Wintner, and A. J. Schmidt, "Femtosecond pulse generation from a synchronously pumped Ti:sapphire laser," Opt. Lett. 16, 1180–1182 (1991).
[CrossRef] [PubMed]

H. A. Haus and E. P. Ippen, "Self-starting of passively mode locked lasers," Opt. Lett. 16, 1331–1333 (1991).
[CrossRef] [PubMed]

M. Karlsson, D. Anderson, M. Desaix, M. Lisak, "Dynamic effects of Kerr nonlinearity and spatial diffraction on self-phase modulation of optical pulses," Opt. Lett. 16, 1373–1375 (1991).
[CrossRef] [PubMed]

S. Chen and J. Wang, "Self-starting issues of passive self-focusing mode locking," Opt. Lett. 16, 1689–1691 (1991).
[CrossRef] [PubMed]

G. Gabetta, D. Huang, R. Ramaswamy, E. P. Ippen, and I. G. Fujimoto, "Femtosecond pulse generation in Ti:Al2O3 using a microdot mirror mode locker," Opt. Lett. 16, 1756–1758 (1991).
[CrossRef] [PubMed]

D. E. Spence, J. M. Evans, W. E. Sleat, and W. Sibbett, "Regenerative initiated self-mode-locked Ti:sapphire laser," Opt. Lett. 16, 1762–1764 (1991).
[CrossRef] [PubMed]

G. P. A. Malcolm and A. I. Ferguson, "Self-mode locking of a diode-pumped Nd:YLF laser," Opt. Lett. 16, 1967–1969 (1991).
[CrossRef] [PubMed]

F. Krausz, C. Spielmann, T. Brabec, E. Wintner, and A. J. Schmidt, "Generation of 33-fs optical pulses from a solid-state laser," Opt. Lett. 17, 204–206 (1992).
[CrossRef] [PubMed]

N. H. Rizvi, P. M. W. French, and J. R. Taylor, "Continuously self-mode-locked Ti:sapphire laser that produces sub-50-fs pulses," Opt. Lett. 17, 279–281 (1992).
[CrossRef] [PubMed]

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

T. Brabec, Ch. Spielmann, and F. Krausz, "Limits of pulse shortening in solitary lasers," Opt. Lett. 17, 748–750 (1992).
[CrossRef] [PubMed]

C.-P. Huang, M. T. Asaki, S. Backus, M. M. Murnane, H. C. Kapteyn, and H. Nathel, "17-fs pulses from a self-mode-locked Ti:sapphire laser," Opt. Lett. 17, 1289–1291 (1992).
[CrossRef] [PubMed]

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

K. X. Liu, C. J. Flood, D. B. Walker, and H. M. van Driel, "Kerr lens mode locking of a diode-pumped Nd:YAG laser," Opt. Lett. 17, 1361–1363 (1992).
[CrossRef]

W. S. Delouch, P. E. Powers, C. L. Tang, "Self-starting ring-cavity Ti:sapphire laser," Opt. Lett. 17, 1581–1583 (1992).
[CrossRef]

N. V. Rizvi, P. M. French, and J. R. Taylor, "Generation of 33-fs pulses from a passively mode-locked Cr3+:LiSrAlF6 laser," Opt. Lett. 17, 1605–1607 (1992).
[CrossRef] [PubMed]

I. M. Jacobson, N. Naganuma, H. A. Haus, and J. G. Fujimoto, "Femtosecond pulse generation in a Ti:Al2O3 laser by using second- and third-order intracavity dispersion," Opt. Lett. 17, 1608–1610 (1992).
[CrossRef] [PubMed]

Opto-Electronics (1)

S. V. Chekalin, P. G. Kryukov, Y. A. Matveetz, and O. B. Shatberashvili, "The process of formation of ultrashort laser pulses," Opto-Electronics 6, 249–261 (1974).
[CrossRef]

Phys. Rev. B (1)

J. A. Fleck, "Mode-locked pulse generation by Q-switched lasers," Phys. Rev. B 1, 84–100 (1970).
[CrossRef]

Other (8)

J. Herrmann and B. Wilhelmi, Lasers for Ultrashort Light pulses (North-Holland, Amsterdam, 1987), pp. 191ff.

D. K. Negus, L. Spinelli, N. Goldblatt, and G. Feuget, "Sub-100 fs pulse generation by Kerr lens mode locking in Ti:Al2O3," in Advanced Solid State Lasers, Vol. 10 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 120–124, postdeadline paper.

D. Georgiev, J. Herrmann, and U. Stamm, "Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid state lasers," Opt. Commun. 92, 368 (1992); in Ultrafast Processes in Spectroscopy, Inst. Phys. Conf. Ser. 126, 5 (1991).
[CrossRef]

J. Herrmann, "The generation conditions and the dynamics of Kerr lens mode locked lasers," presented at the XVII International Quantum Electronics Conference, Vienna, 1992.

S. Gatz and J. Herrmann, "Resonator theory for lasers with radially varying gain," to be submitted to J. Opt. Soc. Am. B.

J. Herrmann and M. Müller, "Theory of mode locking in coupled cavities with a Kerr nonlinearity," to be submitted to J. Opt. Soc. Am. B.

J. Herrmann, "Theory of Mode-locking in femtosecond solid-state lasers with higher order nonlinear effects," submitted to J. Opt. Soc. Am. B.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

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Figures (10)

Fig. 1
Fig. 1

Equivalent cavity configuration for a four-mirror cavity. M1 and M4, plane mirrors; M2 and M3, curved focusing mirrors with radii 2f; LC, laser crystal with length L.

Fig. 2
Fig. 2

Beam spot size at mirror M2 as a function of the laser power. Solid curves, wp = 150 μm, g0 = 0.1 cm−1; dotted curves, wp = 75 μm, g = 0.1 cm−1; dashed curve, g0 = 0. (a) Stable cavity: a′ = 50 μm, b′ = 0.73 cm. (b) Unstable cavity: a′ = −63 μm, b′ = 0.73 cm. (c) Parabolic approximation for a′ = −67 μm, b′ = 0.73 cm. Other parameters: c = 90 cm, d = 93 cm, f = 5 cm, k0 = 8 × 104 cm−1, L = 2 cm, n = 1.76, n2 = 2 × 10−16 cm2 W−1.

Fig. 3
Fig. 3

Beam overlap in the nonlinear laser rod. Solid lines, pump spot wp = 75 μm); dotted curves, laser spot size for P = 0; dashed curves, laser spot size w2 for P = 2 × 105 W. (a) Forward direction, (b) backward direction, (c) forward direction in parabolic approximation, (d) backward direction in parabolic approximation, (e) without gain guiding (g = 0) for a′ = 50 μm, b′ = 0.68 cm; other parameters are as in Fig. 2(b).

Fig. 4
Fig. 4

Nonlinear loss coefficient γa at an aperture for a uniform pump beam (wp → ∞). (a) Change in distance a(a′ = a + b − 2ff2/cf) for b′ = bf = 0.68 cm, (b) change in distance b(b′ = bf) for a′ = 50 μm; other parameters as in Fig. 2; qNL = 0.5.

Fig. 5
Fig. 5

Nonlinear loss coefficient γa, taking into account the radially varying gain. (a) Change in the variable a′ for b′ = bopt′ = 0.73 cm, (b) change in variable b′ for a′ = aopt′. Solid curves, wp = 150 μm, g = 0.1 cm−1, aopt′ = −67 μm; dotted curves, wp = 75 μm, g = 0.1 cm−1, aopt′ = −63 μm; other parameters as in Fig. 2.

Fig. 6
Fig. 6

Nonlinear loss coefficient γa for an asymmetric cavity. Parameters: c = 80 cm, d = 120 cm, f = 5 cm, L = 2 cm, g = 0.1 cm−1, wp = 40 μm; (a) b′ = 0.64 cm, (b) a′ = 50 μm.

Fig. 7
Fig. 7

Characteristic mode-locking parameters γ0+, γ0, and γ0 caused by a power-dependent beam overlap. Solid curves, γ0 for a complete resonator trip; dashed curves, γ0+ for a forward transition; dotted curves, γ0 for a backward transition. (a), (b) wp = 75 μm, a′ = −63 μm, b′ = 0.73 cm, g = 0.1 cm−1; (c), (d) wp = 150 μm, a′ = −67 μm, b′ = 0.73 cm, g = 0.1 cm−1.

Fig. 8
Fig. 8

Averaged power amplification factor Γgg as a function of the laser power. (a) Stable cavity: a′ = 50 am, b′ = 0.73 cm. (b) Unstable cavity: a′ = −63 μm, b′ = 0.73 cm. (c) Parabolic approximation for a′ = −67 μm, b′ = 0.73 cm; other parameters as in Fig. 2.

Fig. 9
Fig. 9

Characteristic mode-locking parameter γgg for KLM without an aperture, caused by a power-dependent radiation redistribution and beam overlap. (a) Change of the distance a′ for b′ = 0.73 cm, (b) change of the distance b′ for a′ = aopt′; parameters as in Fig. 5.

Fig. 10
Fig. 10

Characteristic mode-locking parameter γgg for an asymmetric cavity. Parameters as in Fig. 6.

Equations (56)

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A z + i 2 k ( 2 A x 2 + 2 A y 2 ) = g exp ( - 2 r 2 w p 2 ) A - i κ A 2 A ,
A = α 3 ( z ) B 0 exp [ G ˜ ( z ) + i ϕ ( z ) + i Q 3 ( z ) r 2 ] ,
d G ˜ d z + 1 α 3 d α 3 d z + i d ϕ d z - 2 k Q 3 + i ( d Q 3 d z - 2 k Q 3 2 ) 1 2 Q 3 - g Q 3 w p 2 1 + Q 3 w p 2 + 1 2 i κ α 3 B 0 2 exp ( 2 G ˜ ) = 0 ,
d G ˜ d z + 1 α 3 d α 3 d z + i d ϕ d z - 2 k Q 3 + i ( d Q 3 d z - 2 k Q 3 2 ) 1 Q 3 - g ( Q 3 w p 2 1 + Q 3 w p 2 ) 2 + 1 4 i κ α 3 B 0 2 exp ( 2 G ˜ ) = 0.
1 α 3 d α 3 d z = Q 3 2 k ,
Q ( z ) = - k 2 1 q ( z ) ,             1 q = 1 R + i λ π w 2 ,
d Q 3 d z - 2 k Q 3 2 = h ( z ) = κ π Q 3 ( z ) Q 30 α 3 2 exp [ 2 G ˜ ( z ) ] P + 2 i g w p 2 ( Q 3 w p 2 1 + Q 3 w p 2 ) 2 ,
d G ˜ d z = g Q 2 w p 2 2 + Q 3 w p 2 ( 1 + Q 3 w p 2 ) 2 ,
d ϕ d z = - 3 4 κ α 3 B 0 2 exp ( 2 G ˜ ) ,
Q 3 + ( z ) = D 3 + ( z ) Q 30 + - k C 3 + / 2 A 3 + ( z ) - 2 B 3 + ( z ) Q 30 + / k .
d A 3 + d z = 1 n 0 C 3 + , d B 3 + d z = 1 n 0 D 3 + , d C 3 + d z = - 2 n 0 h + ( z ) A 3 + , d D 3 + d z = - 2 k h + ( z ) B 3 + ,
α 3 + ( z ) = 1 A 3 + ( z ) - 2 B 3 + ( z ) Q 30 + / k .
α 3 + ( z ) = w 3 + ( 0 ) w 3 + ( z ) .
M + = [ A + B + C + D + ] = [ a 2 b 2 c 2 d 2 ] [ A 3 + B 3 + C 3 + D 3 + ] [ d 1 b 1 c 1 a 1 ] = [ a 2 d 1 A 3 + + c 1 b 2 D 3 + + b 2 d 1 C 3 + + a 2 c 1 B 3 a 2 d 1 A 3 + + a 1 b 2 D 3 + + b 1 b 2 C 3 + + a 1 a 2 B 3 + d 1 c 2 A 3 + + c 1 d 2 D 3 + + d 1 d 2 C 3 + + c 1 c 2 B 3 + c 2 b 1 A 3 + + d 2 a 1 D 3 + + b 1 d 2 C 3 + + c 2 a 1 B 3 + ] ,
a 1 = 1 - c f , b 1 = ( a - L 2 n ) ( 1 - c f ) + c , c 1 = - c f , d 1 = 1 - ( a - L 2 n ) 1 f , a 2 = 1 - d f , b 2 = ( b - L 2 n ) ( 1 - d f ) + d , c 2 = - 1 f , d 2 = 1 - ( b - L 2 n ) 1 f .
Q 3 - ( z ) = D ˜ 3 - ( z , L ) Q 30 - - k C ˜ 3 - ( z , L ) / 2 A ˜ 3 - ( z , L ) - 2 B ˜ 3 - ( z , L ) Q 30 - / k , α 3 - ( z ) = [ A ˜ 3 - ( z , L ) - 2 B ˜ 3 - ( z , L ) Q 30 - / k ] - 1 ,
A ˜ 3 - ( z , L ) = D 3 - ( L ) A 3 - ( z ) - C 3 - ( L ) B 3 - ( z ) , B ˜ 3 - ( z , L ) = B 3 - ( L ) A 3 - ( z ) - A 3 - ( L ) B 3 - ( z ) , C ˜ 3 - ( z , L ) = C 3 - ( L ) D 3 - ( z ) - D 3 - ( L ) C 3 - ( z ) , D ˜ 3 - ( z , L ) = A 3 - ( L ) D 3 - ( z ) - B 3 - ( L ) C 3 - ( z ) .
h - ( z ) = κ π Q 3 - ( z ) Q 30 - α 3 - ( z ) 2 exp [ 2 G ˜ - ( z ) ] P + 2 i g w p 2 ( Q 3 - w p 2 1 + Q 3 - w p 2 ) 2 .
M - = [ D - B - C - A - ] = [ a 1 b 1 c 1 d 1 ] [ D 3 - B 3 - C 3 - A 3 - ] [ d 2 b 2 c 2 a 2 ] = [ c 2 b 1 A 3 - + a 1 c 2 B 3 - + d 2 b 1 C 3 - + d 2 a 1 D 3 - , a 2 b 1 A 3 - + a 1 a 2 B 3 - + b 1 b 2 C 3 - + a 1 b 2 D 3 - d 1 c 2 A 3 - + c 1 c 2 B 3 - + d 1 d 2 C 3 - + c 1 d 2 D 3 - , a 2 d 1 A 3 - + a 2 c 1 B 3 - + d 1 b 2 C 3 - + b 2 c 1 D 3 - ] .
M ^ = [ A ^ B ^ C ^ D ^ ] = [ A + D - + C - B + A + B - + A - B + C + D - + D + C - C + B - + D + A - ] .
Q 2 = k 4 B ^ { A ^ - D ^ ± [ ( A ^ + D ^ ) 2 - 4 ] 1 / 2 } ,
α 2 = 1 A ^ - 2 B ^ Q 2 / k = 2 A ^ + D ^ ± [ ( A ^ + D ^ ) 2 - 4 ] 1 / 2 .
Q 1 = A - Q 2 - k C - / 2 D - - 2 B - Q 2 / k .
A + = D - ,             B + = B - ,             C + = C - ,             D + = A - .
Q 3 ( z ) = Q 30 ( 1 - 2 z n k Q 30 ) - 1 , Q 30 = a 1 Q 1 - k c 1 / 2 d 1 - 2 b 1 Q 1 / k .
A 3 ( z ) = 1 - p a P = 1 - 2 k 0 z d z 0 z h k ( z ) d z P , B 3 ( z ) = z n 0 - p b P = z n 0 - 2 k n 0 0 z d z 0 z h k ( z ) d z P , C 3 ( z ) = - p c P = - 2 k 0 z h k ( z ) d z P , D 3 ( z ) = 1 - p d P = 1 - 2 k 0 z h k ( z ) z d z P ,
h k ( z ) = ( κ / π ) [ Q 3 ( z ) ] 2 P .
w 1 4 = - ( λ π ) 2 D + B + C + A + ,             w 2 4 = - ( λ π ) 2 A + B + C + D + .
ξ A = 2 f + f 2 d - f , ξ B = 2 f + f 2 c - f + f 2 d - f , ξ C = 2 f , ξ D = 2 f + f 2 c - f .
ξ = ξ D + a ,             b = f + b ,
h = 4 κ π [ Q 3 ( z ) ] 2 P + 2 i g w p 2 .
A 3 ( z ) = D 3 ( z ) = cos ( ν z ) , B 3 ( z ) = 1 n ν sin ( ν z ) ,             C 3 ( z ) = - n ν sin ( ν z ) ,
A 3 ± ( z ) = 1 + 1 n 0 0 z C 3 ± ( z ) d z , B 3 ± ( z ) = 1 n 0 0 z D 3 ± ( z ) d z , C 3 ± ( z ) = - 2 k 0 z h ± ( z ) A 3 ± ( z ) d z , D 3 ± ( z ) = 1 - 2 k 0 z h ± ( z ) B 3 ± ( z ) d z .
T a = 1 - Γ a = - - d x d y S ( x , y ) 2 π w 2 2 exp ( - 2 x 2 + y 2 w 2 2 ) 1 - q L + q NL 1 w 2 w 2 P P .
q L = exp ( - 2 w 0 2 w 2 2 ) ,             q NL = 4 w 0 2 w 2 2 exp ( - 2 w 0 2 w 2 2 ) ,
q L = 1 - erf ( 2 w 0 w 2 ) , q NL = 2 ( 2 π ) 1 / 2 w 0 w 2 exp ( - 2 w 0 2 w 2 2 ) .
γ a = - q NL 1 w 2 w 2 P = [ ( a 2 d 1 A + a 2 b 1 B - d 1 c 2 C - c 2 b 1 D ) p A + ( a 2 c 1 A + a 1 a 2 B - c 1 c 2 C - a 1 c 2 D ) p B + ( b 2 d 1 A + b 1 b 2 B - d 1 d 2 C - b 1 d 2 D ) p C + ( c 1 b 2 A + a 1 b 2 B - c 1 d 2 C - d 2 a 1 D ) p D ] q NL 4 .
A = [ Q 3 ( z ) ] 1 / 2 B 0 exp [ G ( z ) + i ϕ ( z ) + i Ω 3 ( z ) r 2 ] ,
d G ± d z = g Q 3 ± w p 2 1 + Q 3 ± w p 2 .
γ 0 = γ 0 + + γ 0 - = ( d G + d P + d G - d P ) | P = 0 .
A ( 1 ) ( x , y ) = λ A ( 0 ) ( x , y ) .
λ = 1 A ^ - 2 B ^ Q 2 / k exp ( G ˜ + + G ˜ - + i ϕ ) = 1 M exp ( G ˜ + + G ˜ - + i ϕ ) ,
Γ g g = 4 exp ( 2 G ˜ + + 2 G ˜ - ) A ^ + D ^ ± [ ( A ^ + D ^ ) 2 - 4 ] 1 / 2 2
γ g g = Γ g g P | P = 0 .
Γ g g ( P = 0 ) R 1 R 2 = 1 ,
γ > γ ss = σ ( T c g 0 T 1 P ˜ th ) 1 / 2 13.2 ω A gain R ˜ P ˜ 1 / 2 ,
( c 2 d 2 d η 2 + c 1 d d η + c 0 + c NL Ψ 2 ) Ψ = 0 ,
c 0 = G ( ω L ) 1 + U Re [ L ( ω L ) ] - l ( ω L ) , c 1 = - i d d ω [ G ( ω ) - l ( ω ) ] ω L - h exp ( i ξ ) , c 2 = - d 2 d ω 2 [ G ( ω ) - l ( ω ) ] + i Φ p - 0.5 h 2 exp ( i ξ ) , c NL = - r P gain ( γ a + γ g g ) + i r δ P gain ,
( β 2 - 2 ) c 2 + 3 i β c 2 + v 0 2 τ 2 c NL = 0 , ( 1 - β 2 ) c 2 - 2 i β c 2 + τ 2 c 0 = 0.
β = - 3 2 χ - sgn ( N - D ) [ 2 + ( 3 2 χ ) 2 ] 1 / 2 , τ = 1 K Ω 1 U c 2 , μ = U - Q 2 ( U 3 + U 2 ) ,
Q = Ω 2 K / l Ω 1 , μ = ( G - l ) / l , K = T c c NL / ( 2 Re L , c 2 T 2 ) , Ω 1 = 2 - β 2 + 3 β D , Ω 2 = 1 - β 2 + 2 β D , D = c 2 / c 1 , N = c NL / c NL , χ = ( 1 + N D ) / ( N - D ) .
U < U * ( μ ) 3 μ - 1 4 + [ μ + ( 3 μ - 1 4 ) 2 ] 1 / 2 .
τ L > ( τ L ) min = 1.76 T 2 c 2 Z * l > ˜ 3 T 2 ,
( τ L ) min = 1.76 T 2 c 2 ζ Z * l ,
ζ = α ( D N ) * = - ( 1 3 + 4 9 α 2 ) 1 / 2 + 2 3 α ,             α = 2 Q * K l .
1 4 < D N < ( D N ) * 1 α [ - ( 1 3 + 4 9 α 2 ) 2 + 2 3 α ] .

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