Abstract

A spatio-temporal model of a self-mode-locked laser is presented. It uses a full wave-optics approximation of the propagation in the laser resonator and includes dispersion, self-phase modulation, and self-focusing of the intracavity radiation. It is shown that the self-consistent evolution of the pulse toward steady state imposes strong space–time focusing in the crystal, where both the space and time foci are located. This combined focusing is inherent for Kerr-lens mode-locked lasers with linear and ring cavities, and with and without intracavity saturable absorbers. It is considered as a general mechanism for compensation of space–time astigmatism within the nonlinear resonator. The predictions of the model agree very well with the experiment.

[Optical Society of America ]

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References

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Other

M. Piche , Opt. Commun. OPCOB8 86 , 156 ( 1991
[CrossRef]

H. A. Haus , J. Appl. Phys. JAPIAU 46 , 3049 ( 1975
[CrossRef]

H. A. Haus and M. N. Islam , IEEE J. Quantum Electron. IEJQA7 QE-21 , 1172 ( 1985
[CrossRef]

H. A. Haus , J. G. Fujimoto , and E. P. Ippen , IEEE J. Quantum Electron. IEJQA7 28 , 2086 ( 1992
[CrossRef]

G. W. Pearson , C. Radzewicz , and J. S. Krasinski , Opt. Commun. OPCOB8 94 , 221 ( 1992
[CrossRef]

I. P. Christov , Opt. Commun. OPCOB8 53 , 364 ( 1985
[CrossRef]

F. Salin , P. Grangier , G. Roger , and A. Brun , Phys. Rev. Lett. PRLTAO 56 , 1132 ( 1986
[CrossRef] [PubMed]

P. F. Curley , Ch. Spielman , T. Brabec , E. Wintner , and F. Krausz , J. Opt. Soc. Am. B JOBPDE 10 , 1025 ( 1993
[CrossRef]

J. Herrmann , J. Opt. Soc. Am. B JOBPDE 11 , 498 ( 1994
[CrossRef]

O. E. Martinez , R. L. Fork , and J. P. Gordon , Opt. Lett. OPLEDP 9 , 156 ( 1984
[CrossRef] [PubMed]

J. A. Valdmanis , R. L. Fork , and J. P. Gordon , Opt. Lett. OPLEDP 10 , 131 ( 1985
[CrossRef] [PubMed]

K. L. Blow and B. P. Nelson , Opt. Lett. OPLEDP 13 , 1026 ( 1988
[CrossRef] [PubMed]

P. N. Kean , X. Zhu , D. W. Crust , R. S. Grant , N. Langford , and W. Sibbett , Opt. Lett. OPLEDP 14 , 39 ( 1989
[CrossRef] [PubMed]

D. E. Spence , P. N. Kean , and W. Sibbett , Opt. Lett. OPLEDP 16 , 42 ( 1991
[CrossRef] [PubMed]

T. Brabec , Ch. Spielman , and F. Krausz , Opt. Lett. OPLEDP 16 , 1961 ( 1991
[CrossRef] [PubMed]

D. R. Heatley , A. M. Dunlop , and W. J. Firth , Opt. Lett. OPLEDP 18 , 170 ( 1993
[CrossRef] [PubMed]

T. Tsang , Opt. Lett. OPLEDP 18 , 293 ( 1993
[CrossRef]

J. Zhou , G. Taft , C.-P. Huang , M. Murnane , H. Kapteyn , and I. P. Christov , Opt. Lett. OPLEDP 19 , 1149 ( 1994
[CrossRef] [PubMed]

I. P. Christov , M. M. Murnane , H. C. Kapteyn , J. Zhou , and C.-P. Huang , Opt. Lett. OPLEDP 19 , 1465 ( 1994
[CrossRef] [PubMed]

V. Magni , G. Cerullo , S. De Silvestri , and A. Monguzzi , J. Opt. Soc. Am. B JOBPDE 12 , 476 ( 1995
[CrossRef]

F. X. Ka rtner and U. Keller , Opt. Lett. OPLEDP 20 , 16 ( 1995
[CrossRef]

I. P. Christov , H. C. Kapteyn , M. M. Murnane , C.-P. Huang , and J. Zhou , Opt. Lett. OPLEDP 20 , 309 ( 1995
[CrossRef] [PubMed]

A. Stingl , M. Lenzner , Ch. Spielman , F. Krausz , and R. Szipo cs , Opt. Lett. OPLEDP 20 , 602 ( 1995
[CrossRef] [PubMed]

I. P. Christov , V. D. Stoev , M. M. Murnane , and H. C. Kapteyn , Opt. Lett. OPLEDP 20 , 2111 ( 1995
[CrossRef] [PubMed]

I. D. Jung , F. X. Kartner , N. Matuschek , D. H. Sutter , F. Morier-Genoud , G. Zhang , U. Keller , V. Scheuer , M. Tilsch , and T. Tschudi , Opt. Lett. OPLEDP 22 , 1009 ( 1997
[CrossRef] [PubMed]

S. T. Cundiff , W. H. Knox , E. P. Ippen , and H. A. Haus , Opt. Lett. OPLEDP 21 , 662 ( 1996
[CrossRef] [PubMed]

I. P. Christov , V. D. Stoev , M. M. Murnane , and H. C. Kapteyn , Opt. Lett. OPLEDP 21 , 1493 ( 1996
[CrossRef] [PubMed]

F. Salin , J. Squier , and M. Piche , Opt. Lett. OPLEDP 16 , 1674 ( 1991
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

Schematic of the linear and ring resonators used in our model calculations. The focal length of the curved mirrors is 3.5 cm, the crystal length is 0.23 cm, and D is the dispersion control. The distances for the linear resonator are M1–M2=58.8 cm, M2–C=3.443 cm, C–M3=3.438 cm, and M3–M4=101 cm. For the ring resonator, M1–M2=50 cm, M2–C=3.438 cm, C–M3=3.449 cm, M3–M4=50 cm, M4–M1=64 cm.

Fig. 2
Fig. 2

Transmission of the resonator for an optimally chirped pulse (solid curve) and for an unchirped pulse (dashed curve) for a different peak intensity in the laser crystal.

Fig. 3
Fig. 3

Evolution of the pulse spectrum, pulse duration, and mode size at the output coupler M1 versus the number of round trips in a linear resonator with no dispersion for 50 fs [(a), (c), and (e)] and for 5 fs [(b), (d), and (f)].

Fig. 4
Fig. 4

Initial (dashed curve) and steady-state (solid curve) spectrum in a linear resonator with no dispersion for (a) 50-fs pulse, and (b) 5-fs pulse.

Fig. 5
Fig. 5

Evolution of the pulse spectrum, pulse duration, and pulse energy at the output coupler M1 versus the number of round trips in a linear resonator with no dispersion for a linear resonator [(a), (c), and (e)] and for a ring resonator [(b), (d), and (f)].

Fig. 6
Fig. 6

Space–time focusing in the laser crystal in steady state (a) for a linear resonator, and (c) for a ring resonator. Curves (b) and (d) show the peak intensity of the pulses versus distance in the crystal. The solid curve denotes pulse duration, and the dashed curve denotes spot size.

Fig. 7
Fig. 7

Same as in Fig. 5 but for nonoptimum offset of the laser mirrors.

Fig. 8
Fig. 8

(a) Pulse width and (b) beam size versus the position in the laser crystal for crystal length 1.15 mm and 2.3 mm.

Fig. 9
Fig. 9

(a) Laser spectra in steady state for crystal length 1.15 mm (solid curve) and 2.3 mm (dashed curve). (b) Spectral dependence of the beam size close to the face of the crystal: the solid curve denotes a 1.15-mm crystal; the dashed curve denotes a 2.3-mm crystal.

Equations (8)

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

Ei+1(r, t)=RTlTrEi(r, t),
Tr=FDFfCFfF.
Ei+1(r, t)=RTcEi(r, t),
Tc=FD1FFfCFfD2.
E(r, z, t)=iλzE(r0, 0, ω)expir02ω2c1f-1z×exp-ir2ω2czexpirr0ωcz-iωtdr0dω,
Ei+2(r, z, t)=Ni+1(r, z, t)Ei(k, ω)P(k, ω)×exp[i(kr-ωt)]dkdω,
P(k, ω)=exp{i[(ωn0(ω)/c)2-k2]1/2z},
Ni+1(r, z, t)=exp[ik0n2I(r, z, t)z],

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