Abstract

We employ our extended spatial–temporal matrices to analyze the Ti: sapphire self-mode-locked laser. Our results agree with previously reported experimental work and give us a deeper understanding of the way this laser functions. We found the pulse-shaping mechanism to be essentially solitonlike; the role of the aperture is to discriminate against cw operation. We also study the buildup of the pulse starting from a fluctuation.

© 1993 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. L. Spinelli, B. Couillaud, N. Goldblat, and D. K. Negus, “Starting and generation of sub-100 fs pulses in Ti:Al2O3 by self focusing,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.
  3. G. Gabetta, D. Huang, J. Jacobson, M. Ramaswamy, E. P. Ippen, and J. G. Fujimoto, “Femtosecond pulse generation in Ti:Al2O3 using a microdot mirror mode locker,” Opt. Lett. 16, 1756–1758 (1991).
    [CrossRef] [PubMed]
  4. F. Salin, J. A. Squier, and M. Piché, “Mode locking of Ti:Al2O3 lasers and self-focusing: a Gaussian approximation,” Opt. Lett. 16, 1674–1676 (1991).
    [CrossRef] [PubMed]
  5. T. Bravec, Ch. Spielmann, and F. Krausz, “Mode locking in solitary lasers,” Opt. Lett. 16, 1961–1963 (1991).
    [CrossRef]
  6. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068 (1991).
    [CrossRef]
  7. S. Chen and J. Wang, “Self-starting issues of passive self-focusing mode locking,” Opt. Lett. 16, 1689–1691 (1991).
    [CrossRef] [PubMed]
  8. A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
    [CrossRef]
  9. 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]
  10. F. Krausz, Ch. Spielmann, T. Bravec, 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]
  11. J. M. Jacobson, A. G. Jacobson, K. Naganuma, H. A. Haus, and J. G. Fujimoto, “Generation of 28 fs pulses from a Ti:Al2O3 laser using second and third order intracavity dispersion compensation,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU2.
  12. C. P. Huang, M. Asaki, J. W. McIntosh, M. M. Murnane, and H. C. Kapteyn, “Short pulse optimization of a mode locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU5.
  13. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 15, 16, and 27.
  14. S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
    [CrossRef]
  15. Y. M. Liu, K. W. Sun, P. R. Prucnal, and S. A. Lyon, “Simple method to start and maintain self-mode-locking of a Ti: sapphire laser,” Opt. Lett. 17, 1219–1221 (1992).
    [CrossRef] [PubMed]
  16. U. Keller, G. W. ‘tHooft, and W. H. Knox, “Femtosecond pulses from a continuously self-starting passively mode-locked Ti:sapphire laser,” Opt. Lett. 16, 1022–1024 (1991).
    [CrossRef] [PubMed]

1992 (3)

1991 (7)

1990 (2)

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

‘tHooft, G. W.

Asaki, M.

C. P. Huang, M. Asaki, J. W. McIntosh, M. M. Murnane, and H. C. Kapteyn, “Short pulse optimization of a mode locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU5.

Bravec, T.

Chen, S.

Chilla, J. L. A.

Couillaud, B.

L. Spinelli, B. Couillaud, N. Goldblat, and D. K. Negus, “Starting and generation of sub-100 fs pulses in Ti:Al2O3 by self focusing,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

Dienes, A.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

Dijaili, S. P.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

Fujimoto, J. G.

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

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

J. M. Jacobson, A. G. Jacobson, K. Naganuma, H. A. Haus, and J. G. Fujimoto, “Generation of 28 fs pulses from a Ti:Al2O3 laser using second and third order intracavity dispersion compensation,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU2.

Gabetta, G.

Goldblat, N.

L. Spinelli, B. Couillaud, N. Goldblat, and D. K. Negus, “Starting and generation of sub-100 fs pulses in Ti:Al2O3 by self focusing,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

Haus, H. A.

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

J. M. Jacobson, A. G. Jacobson, K. Naganuma, H. A. Haus, and J. G. Fujimoto, “Generation of 28 fs pulses from a Ti:Al2O3 laser using second and third order intracavity dispersion compensation,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU2.

Huang, C. P.

C. P. Huang, M. Asaki, J. W. McIntosh, M. M. Murnane, and H. C. Kapteyn, “Short pulse optimization of a mode locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU5.

Huang, D.

Ippen, E. P.

Jacobson, A. G.

J. M. Jacobson, A. G. Jacobson, K. Naganuma, H. A. Haus, and J. G. Fujimoto, “Generation of 28 fs pulses from a Ti:Al2O3 laser using second and third order intracavity dispersion compensation,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU2.

Jacobson, J.

Jacobson, J. M.

J. M. Jacobson, A. G. Jacobson, K. Naganuma, H. A. Haus, and J. G. Fujimoto, “Generation of 28 fs pulses from a Ti:Al2O3 laser using second and third order intracavity dispersion compensation,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU2.

Kapteyn, H. C.

C. P. Huang, M. Asaki, J. W. McIntosh, M. M. Murnane, and H. C. Kapteyn, “Short pulse optimization of a mode locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU5.

Kean, P. N.

Keller, U.

Knox, W. H.

Kostenbauder, A. G.

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

Krausz, F.

Liu, Y. M.

Lyon, S. A.

Martinez, O. E.

McIntosh, J. W.

C. P. Huang, M. Asaki, J. W. McIntosh, M. M. Murnane, and H. C. Kapteyn, “Short pulse optimization of a mode locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU5.

Murnane, M. M.

C. P. Huang, M. Asaki, J. W. McIntosh, M. M. Murnane, and H. C. Kapteyn, “Short pulse optimization of a mode locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU5.

Naganuma, K.

J. M. Jacobson, A. G. Jacobson, K. Naganuma, H. A. Haus, and J. G. Fujimoto, “Generation of 28 fs pulses from a Ti:Al2O3 laser using second and third order intracavity dispersion compensation,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU2.

Negus, D. K.

L. Spinelli, B. Couillaud, N. Goldblat, and D. K. Negus, “Starting and generation of sub-100 fs pulses in Ti:Al2O3 by self focusing,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

Piché, M.

Prucnal, P. R.

Ramaswamy, M.

Salin, F.

Schmidt, A. J.

Sibbett, W.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 15, 16, and 27.

Smith, J. S.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

Spence, D. E.

Spielmann, Ch.

Spinelli, L.

L. Spinelli, B. Couillaud, N. Goldblat, and D. K. Negus, “Starting and generation of sub-100 fs pulses in Ti:Al2O3 by self focusing,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

Squier, J. A.

Sun, K. W.

Wang, J.

Wintner, E.

IEEE J. Quantum Electron. (2)

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

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

Opt. Lett. (9)

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

Y. M. Liu, K. W. Sun, P. R. Prucnal, and S. A. Lyon, “Simple method to start and maintain self-mode-locking of a Ti: sapphire laser,” Opt. Lett. 17, 1219–1221 (1992).
[CrossRef] [PubMed]

U. Keller, G. W. ‘tHooft, and W. H. Knox, “Femtosecond pulses from a continuously self-starting passively mode-locked Ti:sapphire laser,” Opt. Lett. 16, 1022–1024 (1991).
[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]

F. Krausz, Ch. Spielmann, T. Bravec, 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]

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]

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

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

T. Bravec, Ch. Spielmann, and F. Krausz, “Mode locking in solitary lasers,” Opt. Lett. 16, 1961–1963 (1991).
[CrossRef]

Other (4)

L. Spinelli, B. Couillaud, N. Goldblat, and D. K. Negus, “Starting and generation of sub-100 fs pulses in Ti:Al2O3 by self focusing,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

J. M. Jacobson, A. G. Jacobson, K. Naganuma, H. A. Haus, and J. G. Fujimoto, “Generation of 28 fs pulses from a Ti:Al2O3 laser using second and third order intracavity dispersion compensation,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU2.

C. P. Huang, M. Asaki, J. W. McIntosh, M. M. Murnane, and H. C. Kapteyn, “Short pulse optimization of a mode locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CTuU5.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 15, 16, and 27.

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Figures (5)

Fig. 1
Fig. 1

Cavity schematic of the self-mode-locked Ti: sapphire laser. Ml and M4 are flat mirrors (Ml is the output coupler). M2 and M3 are 10-cm mirrors, located 85 cm apart from the flat ones, BWL is the Lyot filter (bandwidth limitation), Ap is aperture, P1 and P2 are prisms for dispersion compensation. Point a is the starting point of our analysis.

Fig. 2
Fig. 2

Evolution of the laser beam radius in the cavity for mode-locked (solid) and cw operation (dashed). The insets show the beam size near the rod. The origin of the abscissa corresponds to point a in Fig. 1.

Fig. 3
Fig. 3

Performance of the laser (a, output power; b, pulse width) for three cavity configurations. Solid curves: displaced rod (53.9 and 49.7 mm to M2 and M3, respectively); there is no aperture. Dashed curves: displaced rod; the aperture is 3 mm. Dotted curves: centered rod (51.5 mm to both M2 and M3); the aperture is 0.6 mm.

Fig. 4
Fig. 4

Performance of the laser (a, output power; b, pulse width) for different pump powers. Configuration: displaced rod (53.9 and 49.7 mm to M2 and M3, respectively); there is no aperture.

Fig. 5
Fig. 5

Competition between a fluctuation and the cw mode for the cavity of Fig. 2. If the fluctuation is in the shaded zone, it grows and quenches cw laser emission, thus starting mode locking.

Tables (1)

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Table 1 Evolution of the Pulse Width in the Cavity

Equations (7)

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

M = [ A B 0 0 C D 0 0 0 0 K I 0 0 J L ] .
E ( t ) = ( U τ σ 2 ) 1 / 2 exp ( - j k r 2 / 2 q ) exp ( j k t 2 / 2 p ) ,
1 q = n R - j n λ π σ 2 ,             1 p = 1 k 2 ϕ t 2 ,             1 p = 1 k 2 ϕ t 2 + j n λ π τ 2 ,
q out = A q in + B C q in + D , p out = K p in + I / λ λ J p in + L .
Re ( C ) = - 4 γ U z τ σ 4 ( 2 π ) 3 / 2
J = 4 γ U z λ τ 3 σ 2 ( 2 π ) 3 / 2
Im ( C ) = - 2 λ π σ p 2 ( g 1 + g ) .

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