Abstract

We used the ABCD law to derive the transfer matrix of beam propagation in a Kerr-lens mode-locked laser. On the basis of the transfer matrix and the formulas of the cavity stability G parameters that differ from the simple cavity g parameters, we give the expressions for Kerr-lens mode-locking (KLM) strength and some equations for cavity geometric parameters. The expressions and the equations can be used to construct a laser resonator to achieve self-starting oscillation of KLM. In addition we made a numerical simulation of the KLM strength at different positions in a mode-locked Ti:sapphire laser. The numerical results provide a good explanation of some of the physical phenomena observed in our experiments.

© 2002 Optical Society of America

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References

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  1. B. E. Lemoff, C. P. J. Barty, “Generation of high-peak-power 20-fs pulses from a regeneratively initiated, self-mode-locked Ti:sapphire laser,” Opt. Lett. 17, 1367–1369 (1992).
    [CrossRef] [PubMed]
  2. D. Georgiev, J. Herrmann, U. Stamm, “Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers,” Opt. Commun. 92, 368–375 (1992).
    [CrossRef]
  3. M. T. Asaki, C. P. Huang, D. Garvey, “Generation of 11-fs pulses from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 18, 977–979 (1993).
    [CrossRef] [PubMed]
  4. K. Tamura, J. Jacobson, E. P. Ippen, H. A. Haus, J. G. Fujimoto, “Unidirectional ring resonators for self-starting passively mode-locked lasers,” Opt. Lett. 18, 220–222 (1993).
    [CrossRef] [PubMed]
  5. V. Magni, G. Cerullo, S. D. Silvestri, “ABCD matrix analysis of propagation of Gaussian beams through Kerr media,” Opt. Commun. 96, 348–355 (1993).
    [CrossRef]
  6. V. Magni, G. Cerullo, S. D. Silvestri, “Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
    [CrossRef]
  7. G. Cerullo, S. D. Silvestri, V. Magni, “Self-starting Kerr-lens mode locking of a Ti:sapphire laser,” Opt. Lett. 19, 1040–1042 (1994).
    [CrossRef] [PubMed]
  8. J. Solis, J. Siegel, C. N. Afonso, N. P. Barry, R. Mellish, P. M. W. French, “Experimental study of a self-starting Kerr-lens mode-locked titanium-doped sapphire laser,” Opt. Commun. 123, 547–552 (1996).
    [CrossRef]
  9. G. H. Xu, H. Feiran, W. K. Lee, M. R. Wang, “Cavity design of a compact Kerr-lens mode-locking laser,” Opt. Commun. 142, 249–252 (1997).
    [CrossRef]
  10. U. Morgner, F. X. Kartner, S. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411–413 (1999).
    [CrossRef]
  11. M. A. Marioni, A. A. Hnilo, “Self-starting of self mode-locking Ti:sapphire lasers: description with a Poincaré map,” Opt. Commun. 147, 89–94 (1998).
    [CrossRef]
  12. W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, New York, 1999), p. 538.

1999

1998

M. A. Marioni, A. A. Hnilo, “Self-starting of self mode-locking Ti:sapphire lasers: description with a Poincaré map,” Opt. Commun. 147, 89–94 (1998).
[CrossRef]

1997

G. H. Xu, H. Feiran, W. K. Lee, M. R. Wang, “Cavity design of a compact Kerr-lens mode-locking laser,” Opt. Commun. 142, 249–252 (1997).
[CrossRef]

1996

J. Solis, J. Siegel, C. N. Afonso, N. P. Barry, R. Mellish, P. M. W. French, “Experimental study of a self-starting Kerr-lens mode-locked titanium-doped sapphire laser,” Opt. Commun. 123, 547–552 (1996).
[CrossRef]

1994

1993

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

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

K. Tamura, J. Jacobson, E. P. Ippen, H. A. Haus, J. G. Fujimoto, “Unidirectional ring resonators for self-starting passively mode-locked lasers,” Opt. Lett. 18, 220–222 (1993).
[CrossRef] [PubMed]

M. T. Asaki, C. P. Huang, D. Garvey, “Generation of 11-fs pulses from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 18, 977–979 (1993).
[CrossRef] [PubMed]

1992

B. E. Lemoff, C. P. J. Barty, “Generation of high-peak-power 20-fs pulses from a regeneratively initiated, self-mode-locked Ti:sapphire laser,” Opt. Lett. 17, 1367–1369 (1992).
[CrossRef] [PubMed]

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

Afonso, C. N.

J. Solis, J. Siegel, C. N. Afonso, N. P. Barry, R. Mellish, P. M. W. French, “Experimental study of a self-starting Kerr-lens mode-locked titanium-doped sapphire laser,” Opt. Commun. 123, 547–552 (1996).
[CrossRef]

Angelow, G.

Asaki, M. T.

Barry, N. P.

J. Solis, J. Siegel, C. N. Afonso, N. P. Barry, R. Mellish, P. M. W. French, “Experimental study of a self-starting Kerr-lens mode-locked titanium-doped sapphire laser,” Opt. Commun. 123, 547–552 (1996).
[CrossRef]

Barty, C. P. J.

Cerullo, G.

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

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

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

Chen, Y.

Cho, S.

Feiran, H.

G. H. Xu, H. Feiran, W. K. Lee, M. R. Wang, “Cavity design of a compact Kerr-lens mode-locking laser,” Opt. Commun. 142, 249–252 (1997).
[CrossRef]

French, P. M. W.

J. Solis, J. Siegel, C. N. Afonso, N. P. Barry, R. Mellish, P. M. W. French, “Experimental study of a self-starting Kerr-lens mode-locked titanium-doped sapphire laser,” Opt. Commun. 123, 547–552 (1996).
[CrossRef]

Fujimoto, J. G.

Garvey, D.

Georgiev, D.

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

Haus, H. A.

Herrmann, J.

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

Hnilo, A. A.

M. A. Marioni, A. A. Hnilo, “Self-starting of self mode-locking Ti:sapphire lasers: description with a Poincaré map,” Opt. Commun. 147, 89–94 (1998).
[CrossRef]

Huang, C. P.

Ippen, E. P.

Jacobson, J.

Kartner, F. X.

Koechner, W.

W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, New York, 1999), p. 538.

Lee, W. K.

G. H. Xu, H. Feiran, W. K. Lee, M. R. Wang, “Cavity design of a compact Kerr-lens mode-locking laser,” Opt. Commun. 142, 249–252 (1997).
[CrossRef]

Lemoff, B. E.

Magni, V.

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

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

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

Marioni, M. A.

M. A. Marioni, A. A. Hnilo, “Self-starting of self mode-locking Ti:sapphire lasers: description with a Poincaré map,” Opt. Commun. 147, 89–94 (1998).
[CrossRef]

Mellish, R.

J. Solis, J. Siegel, C. N. Afonso, N. P. Barry, R. Mellish, P. M. W. French, “Experimental study of a self-starting Kerr-lens mode-locked titanium-doped sapphire laser,” Opt. Commun. 123, 547–552 (1996).
[CrossRef]

Morgner, U.

Scheuer, V.

Siegel, J.

J. Solis, J. Siegel, C. N. Afonso, N. P. Barry, R. Mellish, P. M. W. French, “Experimental study of a self-starting Kerr-lens mode-locked titanium-doped sapphire laser,” Opt. Commun. 123, 547–552 (1996).
[CrossRef]

Silvestri, S. D.

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

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

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

Solis, J.

J. Solis, J. Siegel, C. N. Afonso, N. P. Barry, R. Mellish, P. M. W. French, “Experimental study of a self-starting Kerr-lens mode-locked titanium-doped sapphire laser,” Opt. Commun. 123, 547–552 (1996).
[CrossRef]

Stamm, U.

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

Tamura, K.

Tschudi, T.

Wang, M. R.

G. H. Xu, H. Feiran, W. K. Lee, M. R. Wang, “Cavity design of a compact Kerr-lens mode-locking laser,” Opt. Commun. 142, 249–252 (1997).
[CrossRef]

Xu, G. H.

G. H. Xu, H. Feiran, W. K. Lee, M. R. Wang, “Cavity design of a compact Kerr-lens mode-locking laser,” Opt. Commun. 142, 249–252 (1997).
[CrossRef]

Opt. Commun.

D. Georgiev, J. Herrmann, 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, G. Cerullo, S. D. Silvestri, “ABCD matrix analysis of propagation of Gaussian beams through Kerr media,” Opt. Commun. 96, 348–355 (1993).
[CrossRef]

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

J. Solis, J. Siegel, C. N. Afonso, N. P. Barry, R. Mellish, P. M. W. French, “Experimental study of a self-starting Kerr-lens mode-locked titanium-doped sapphire laser,” Opt. Commun. 123, 547–552 (1996).
[CrossRef]

G. H. Xu, H. Feiran, W. K. Lee, M. R. Wang, “Cavity design of a compact Kerr-lens mode-locking laser,” Opt. Commun. 142, 249–252 (1997).
[CrossRef]

M. A. Marioni, A. A. Hnilo, “Self-starting of self mode-locking Ti:sapphire lasers: description with a Poincaré map,” Opt. Commun. 147, 89–94 (1998).
[CrossRef]

Opt. Lett.

Other

W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, New York, 1999), p. 538.

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Figures (6)

Fig. 1
Fig. 1

Schematic of an equivalent four-mirror cavity configuration.

Fig. 2
Fig. 2

KLM strength at the center of the Kerr medium versus the defocusing distance of curved mirrors.

Fig. 3
Fig. 3

KLM strength at the end mirrors versus the defocusing distance of curved mirrors.

Fig. 4
Fig. 4

KLM strength at the center of the Kerr medium versus the length of the laser arm.

Fig. 5
Fig. 5

Beam spot size at the center of the Ti:sapphire rod as a function of the intracavity peak power.

Fig. 6
Fig. 6

Beam spot size on the end mirror as a function of the intracavity peak power.

Equations (24)

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Mc=1L2n0110-D11L2n01,D=4n2I0Lw02=4n2pLπw04.
M=abcd=1L20110- 1f211r2011L2n0110-D1×1L2n011r10110- 1f111L101=a1a2+b2c1-a1b2Da2b1+b2d1-b1b2Da1c2+c1d2-a1d2Db1c2+d1d2-b1d2D,
a1=1-r1f1-L2nf1,b1=L1+r1+L2n-L1r1f1-L1L2nf1,c1=-1f1, d1=1-L1f1,a2=1-L2f2, b2=L2+r2+L2n-L2r2f2-L2L2nf2,c2=- 1f2, d2=1-r2f2-L2nf2.
G1=a-bρ1, G2=d-bρ2.
0<G1G2<1.
w14=λbπ2G2G11-G1G2,w24=λbπ2G1G21-G1G2.
w014=λbπ2G1G21-G1G2G1+a2G2-2aG1G22,w024=λbπ2G1G21-G1G2G1+d2G2-2dG1G22.
L01=bG2a-G1G1+a2G2-2aG1G2,L02=bG1d-G2G2+d2G1-2dG1G2.
wc2=f12w012L1-L01-f12+z012, z01=πw012λ,
δ=-1wdwdpp=0,
δ1=-1w1dw1dpp=0=-12bdbdp-141G1-G21-G1G2dG1dp+141G2+G11-G1G2dG2dpp=0,
δc=-1wcdwcdpp=0=δ011-2z012L1-L01-f12+z012p=0,δ01=-1w01dw01dpp=0.
M=abcd=1L20110-1f211r2011Ln01×1r10110-1f111L101.
f1+f2+L1-1n-r1+r2+L=Δ,
r1+r2+Ln=f1+f2-Δ.
G1=-f2f1, G2=-f1f2, G1G2=1.
G1=-1+L1+L2-2fρ1, G2=-1+L1+L2-2fρ2.
ρ1=L1+L2-2f,
ρ2=L1+L2-2f.
L1+L2=2f,
ρ1+ρ2=L1+L2-2f,
ρ1=ρ2=.
L1=L2=f,
ρ1=ρ2=L1-f=L2-f.

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