Abstract

Self-focusing by a Kerr nonlinearity in combination with an intracavity aperture creates a power-dependent loss in lasers, which has been used as a method for passive mode locking of lasers. An analytical treatment is presented that yields closed-form expressions for the modulation efficiency. Comparisons between analytical results and numerical calculations are performed. General features of this mode-locking technique are discussed, and a procedure for the optimization of hard-aperture Kerr-lens mode-locking performance is given.

© 1993 Optical Society of America

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  1. D. E. Spence, P. N. Kean, W. Sibbett, Opt. Lett. 16, 42 (1991).
    [CrossRef] [PubMed]
  2. L. Spinelli, B. Couilland, N. Goldblatt, D. K. Negus, 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.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. Ch. Spielmann, F. Krausz, T. Brabec, E. Wintner, A. J. Schmidt, Opt. Lett. 16, 1180 (1991).
    [CrossRef] [PubMed]
  7. A. Miller, P. Li, Kam Wa, B. H. T. Chai, E. W. Van Stryland, Opt. Lett. 17, 195 (1992).
    [CrossRef] [PubMed]
  8. G. Gabetta, D. Huang, J. Jacobson, M. Ramaswamy, E. P. Ippen, J. G. Fujimoto, Opt. Lett. 16, 1756 (1991).
    [CrossRef] [PubMed]
  9. M. Piché, N. McCarthy, F. Salin, in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper MB8.
  10. S. Chen, J. Wang, Opt. Lett. 16, 1689 (1991).
    [CrossRef] [PubMed]
  11. F. Salin, J. Squier, M. Piché, Opt. Lett. 16, 1674 (1991).
    [CrossRef] [PubMed]
  12. M. Piché, Opt. Commun. 86, 156 (1991).
    [CrossRef]
  13. O. E. Martinez, J. L. A. Chilla, Opt. Lett. 17, 1210 (1992).
    [CrossRef]
  14. D. Georgiev, J. Herrmann, U. Stamm, Opt. Commun. 92, 368 (1992).
    [CrossRef]
  15. T. Brabec, Ch. Spielmann, P. F. Curley, F. Krausz, Opt. Lett. 17, 1292 (1992).
    [CrossRef] [PubMed]
  16. H. A. Haus, J. G. Fujimoto, E. P. Ippen, IEEE J. Quantum Electron. 28, 2086 (1992).
    [CrossRef]
  17. Ch. Spielmann, P. F. Curley, T. Brabec, F. Krausz, “Ultra-broadband femtosecond-lasers,” IEEE J. Quantum Electron. (to be published).
  18. J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
    [CrossRef]
  19. P. A. Belanger, C. Pare, Appl. Opt. 22, 1293 (1983).
    [CrossRef]
  20. G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 493 (1961).
    [CrossRef]
  21. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
    [CrossRef]
  22. Note that the distance of critical self-focusing (collapse of the beam) depends on the beam parameters before the nonlinearity. Therefore, operation is also possible for P Pcrwhen the resonator parameters are chosen so that the laser medium is shorter than the critical self-focusing distance.
  23. D. Huang, M. Ulmann, L. H. Acioli, H. A. Haus, J. G. Fujimoto, Opt. Lett. 17, 511 (1992).
    [CrossRef] [PubMed]
  24. E. L. Dawson, J. H. Marburger, Phys. Rev. 179, 862 (1969).
    [CrossRef]
  25. Note that here zr0is the confocal parameter for δ near the limit of the stability range.
  26. H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, IEEE J. Quantum Electron. QE-8, 373 (1972).
    [CrossRef]
  27. Note that the condition for the compensation of the Brewster astigmatism given here differs from that in Ref. 26 because of the different definitions of the length of the Brewster medium. Whereas the length of the medium is measured perpendicular to the surface in Ref. 26, we determine the length of the Brewster medium l along the optical path.

1992 (6)

1991 (9)

1983 (1)

1975 (1)

J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

1972 (1)

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

1969 (1)

E. L. Dawson, J. H. Marburger, Phys. Rev. 179, 862 (1969).
[CrossRef]

1965 (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
[CrossRef]

1961 (1)

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 493 (1961).
[CrossRef]

Acioli, L. H.

Belanger, P. A.

Boyd, G. D.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 493 (1961).
[CrossRef]

Brabec, T.

T. Brabec, Ch. Spielmann, P. F. Curley, F. Krausz, Opt. Lett. 17, 1292 (1992).
[CrossRef] [PubMed]

Ch. Spielmann, F. Krausz, T. Brabec, E. Wintner, A. J. Schmidt, Opt. Lett. 16, 1180 (1991).
[CrossRef] [PubMed]

Ch. Spielmann, P. F. Curley, T. Brabec, F. Krausz, “Ultra-broadband femtosecond-lasers,” IEEE J. Quantum Electron. (to be published).

Chai, B. H. T.

Chen, S.

Chilla, J. L. A.

Couilland, B.

L. Spinelli, B. Couilland, N. Goldblatt, D. K. Negus, 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.

Cunningham, J. E.

Curley, P. F.

T. Brabec, Ch. Spielmann, P. F. Curley, F. Krausz, Opt. Lett. 17, 1292 (1992).
[CrossRef] [PubMed]

P. F. Curley, A. I. Ferguson, Opt. Lett. 16, 1016 (1991).
[CrossRef] [PubMed]

Ch. Spielmann, P. F. Curley, T. Brabec, F. Krausz, “Ultra-broadband femtosecond-lasers,” IEEE J. Quantum Electron. (to be published).

Dawson, E. L.

E. L. Dawson, J. H. Marburger, Phys. Rev. 179, 862 (1969).
[CrossRef]

Dienes, A.

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Ferguson, A. I.

Fujimoto, J. G.

Gabetta, G.

Georgiev, D.

D. Georgiev, J. Herrmann, U. Stamm, Opt. Commun. 92, 368 (1992).
[CrossRef]

Goldblatt, N.

L. Spinelli, B. Couilland, N. Goldblatt, D. K. Negus, 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.

Gordon, J. P.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 493 (1961).
[CrossRef]

Haus, H. A.

D. Huang, M. Ulmann, L. H. Acioli, H. A. Haus, J. G. Fujimoto, Opt. Lett. 17, 511 (1992).
[CrossRef] [PubMed]

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

Herrmann, J.

D. Georgiev, J. Herrmann, U. Stamm, Opt. Commun. 92, 368 (1992).
[CrossRef]

Huang, D.

Ippen, E. P.

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

G. Gabetta, D. Huang, J. Jacobson, M. Ramaswamy, E. P. Ippen, J. G. Fujimoto, Opt. Lett. 16, 1756 (1991).
[CrossRef] [PubMed]

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Jacobson, J.

Kean, P. N.

Keller, U.

Knox, W. H.

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
[CrossRef]

Kogelnik, H. W.

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Krausz, F.

T. Brabec, Ch. Spielmann, P. F. Curley, F. Krausz, Opt. Lett. 17, 1292 (1992).
[CrossRef] [PubMed]

Ch. Spielmann, F. Krausz, T. Brabec, E. Wintner, A. J. Schmidt, Opt. Lett. 16, 1180 (1991).
[CrossRef] [PubMed]

Ch. Spielmann, P. F. Curley, T. Brabec, F. Krausz, “Ultra-broadband femtosecond-lasers,” IEEE J. Quantum Electron. (to be published).

Li, P.

Malcolm, G. P. A.

Marburger, J. H.

J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

E. L. Dawson, J. H. Marburger, Phys. Rev. 179, 862 (1969).
[CrossRef]

Martinez, O. E.

McCarthy, N.

M. Piché, N. McCarthy, F. Salin, in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper MB8.

Miller, A.

Negus, D. K.

L. Spinelli, B. Couilland, N. Goldblatt, D. K. Negus, 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.

Pare, C.

Piché, M.

F. Salin, J. Squier, M. Piché, Opt. Lett. 16, 1674 (1991).
[CrossRef] [PubMed]

M. Piché, Opt. Commun. 86, 156 (1991).
[CrossRef]

M. Piché, N. McCarthy, F. Salin, in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper MB8.

Ramaswamy, M.

Salin, F.

F. Salin, J. Squier, M. Piché, Opt. Lett. 16, 1674 (1991).
[CrossRef] [PubMed]

M. Piché, N. McCarthy, F. Salin, in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper MB8.

Schmidt, A. J.

Shank, C. V.

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Sibbett, W.

Spence, D. E.

Spielmann, Ch.

T. Brabec, Ch. Spielmann, P. F. Curley, F. Krausz, Opt. Lett. 17, 1292 (1992).
[CrossRef] [PubMed]

Ch. Spielmann, F. Krausz, T. Brabec, E. Wintner, A. J. Schmidt, Opt. Lett. 16, 1180 (1991).
[CrossRef] [PubMed]

Ch. Spielmann, P. F. Curley, T. Brabec, F. Krausz, “Ultra-broadband femtosecond-lasers,” IEEE J. Quantum Electron. (to be published).

Spinelli, L.

L. Spinelli, B. Couilland, N. Goldblatt, D. K. Negus, 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.

Stamm, U.

D. Georgiev, J. Herrmann, U. Stamm, Opt. Commun. 92, 368 (1992).
[CrossRef]

t’Hooft, G. W.

Ulmann, M.

Van Stryland, E. W.

Wa, Kam

Wang, J.

Wintner, E.

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 493 (1961).
[CrossRef]

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
[CrossRef]

IEEE J. Quantum Electron. (2)

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

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Opt. Commun. (2)

M. Piché, Opt. Commun. 86, 156 (1991).
[CrossRef]

D. Georgiev, J. Herrmann, U. Stamm, Opt. Commun. 92, 368 (1992).
[CrossRef]

Opt. Lett. (12)

Phys. Rev. (1)

E. L. Dawson, J. H. Marburger, Phys. Rev. 179, 862 (1969).
[CrossRef]

Prog. Quantum Electron. (1)

J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

Other (6)

Note that here zr0is the confocal parameter for δ near the limit of the stability range.

Note that the condition for the compensation of the Brewster astigmatism given here differs from that in Ref. 26 because of the different definitions of the length of the Brewster medium. Whereas the length of the medium is measured perpendicular to the surface in Ref. 26, we determine the length of the Brewster medium l along the optical path.

L. Spinelli, B. Couilland, N. Goldblatt, D. K. Negus, 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.

M. Piché, N. McCarthy, F. Salin, in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper MB8.

Ch. Spielmann, P. F. Curley, T. Brabec, F. Krausz, “Ultra-broadband femtosecond-lasers,” IEEE J. Quantum Electron. (to be published).

Note that the distance of critical self-focusing (collapse of the beam) depends on the beam parameters before the nonlinearity. Therefore, operation is also possible for P Pcrwhen the resonator parameters are chosen so that the laser medium is shorter than the critical self-focusing distance.

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Figures (5)

Fig. 1
Fig. 1

Schematic of the cavity used in the calculations. The parameters d1 and d2 are the lengths of the short and the long resonator arms, respectively. The focal lengths f1 = f2 = 6 cm and the length of the Kerr medium l = 1.6 cm are fixed throughout this paper. The refractive index of the Kerr medium is assumed to have the value of Ti:sapphire, n = 1.76.

Fig. 2
Fig. 2

Variation of the beam waists at the output couplers of the short- (wc1) and long- (wc2) cavity arms versus the normalized stability parameter δ/δmax [δmax = −(R1 + R2)]. The parameters are d1 = 80 cm, d2 = 120 cm.

Fig. 3
Fig. 3

Absolute change in gain versus the normalized stability parameter [δmax = −(R1 + R2)]. Mode-locked operation is possible for Δg > 0. The parameters are L = 0.1, P ¯ = 0.25, d1 = 80 cm, and d2 = 120 cm. The vertical lines denote the limits of the stability range. Curves 1 and 2 show the results when the aperture is inserted at the output coupler of the short- and the long-cavity arms, respectively. The solid curves are for = 0, and the dashed curves are for a shifted Kerr medium such that = 0.25. (a) Analytical solution of Eqs. (2)(4). (b) Numerical solution of Eqs. (2)(4).

Fig. 4
Fig. 4

Absolute change in gain versus stability parameter δ. The parameters are L = 0.1, P ¯ = 0.25, d2 = 120 cm, and d1 = 60, 90, and 115 cm for curves 1, 2, and 3, respectively. The vertical line denotes the limit of the stability range.

Fig. 5
Fig. 5

Shift of the beam waist from the center of the Kerr medium in the tangential plane y and the sagittal plane x versus the incidence half-angle θ1. Positive x,y means a shift of the Kerr medium towards the shorter resonator arm and hence an enhancement of mode-locked operation.

Equations (25)

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z r 0 2 ( δ ) = ( π w r 0 2 n λ ) 2 = n 2 t ( R 1 - t ) ( R 2 - t ) ( R 1 + R 2 - t ) ( R 1 + R 2 - 2 t ) 2 .
z r 0 2 ( δ = - R 2 / 2 ) = n 2 [ 1 - ( R 2 2 R 1 ) 2 ] ( R 2 2 ) 2 .
R ¯ 1 ( η ) = l x 1 ( η ) n { 1 + [ z r ( η ) l x 1 ( η ) η ] 2 } ,
R ¯ 2 ( η ) = l x 2 ( η ) n { 1 + [ z r ( η ) l x 2 ( η ) η ] 2 } .
R ¯ 1 2 ( η ) - R ¯ 1 ( η ) [ R ¯ 10 + ( 1 x 10 - 1 x 1 ( η ) η 2 ) n l B 1 A 1 D 1 C 1 ] - 1 - η 2 η 2 B 1 D 1 A 1 C 1 = 0 ,
R ¯ 2 2 ( η ) - R ¯ 2 ( η ) [ R ¯ 20 + ( 1 x 20 - 1 x 2 ( η ) η 2 ) n l B 2 D 2 A 2 C 2 ] - 1 - η 2 η 2 B 2 D 2 A 2 C 2 = 0 ,
T ^ 1 = [ A 1 B 1 C 1 D 1 ] = [ f 1 R 1 f 1 R 1 { R 1 - [ l x 10 n - δ ( δ + R 2 ) 2 δ + R 1 + R 2 ] } - 1 f 1 1 f 1 [ l x 10 n - δ ( δ + R 2 ) 2 δ + R 1 + R 2 ] ] ,
T ^ 2 = [ A 2 B 2 C 2 D 2 ] = [ f 2 R 2 f 2 R 2 { R 2 - [ l x 20 n - δ ( δ + R 1 ) 2 δ + R 1 + R 2 ] } - 1 f 2 1 f 2 [ l x 2 n - δ ( δ + R 1 ) 2 δ + R 1 + R 2 ] ] .
P ¯ F 1 - 4 Δ w r R ¯ 10 - Δ x 1 n l R ¯ 10 z r 0 2 ( 2 l x 10 n + F 1 ) = 0 ,
P ¯ F 2 - 4 Δ w r R ¯ 20 + Δ x 1 n l R ¯ 20 z r 0 2 ( 2 l x 20 n + F 2 ) = 0 ,
F 1 = ( A 1 D 1 + B 1 C 1 ) / ( A 1 C 1 ) ,
F 2 = ( A 2 D 2 + B 2 C 2 ) / ( A 2 C 2 ) .
Δ x 1 = P ¯ z r 0 2 n l [ ( F 1 / R ¯ 10 ) - ( F 2 / R ¯ 20 ) ( 2 l / n ) + F 1 + F 2 ] ,
Δ w r = P ¯ 8 [ R 1 2 - R 2 2 ( 2 δ + R 1 + R 2 ) 2 ( F 2 R ¯ 20 - F 1 R ¯ 10 ) + ( F 2 R ¯ 20 + F 1 R ¯ 10 ) ] .
z c i 2 = - B i D i ( A i + B i / R ¯ i C i + D i / R ¯ i ) .
Δ w c i 2 = - 1 2 R i ( P ¯ R ¯ i 0 - n l z r 0 2 Δ x i ) ,
Δ w c i 2 = P P cr n R i l [ 1 + ( 2 z r 0 / l ) 2 ] ( 1 + ( - 1 ) i { R 1 2 - R 2 2 ( R 1 + R 2 + 2 δ ) 2 - 8 l n [ 1 + ( l / 2 z r 0 ) 2 ] 1 R 1 + R 2 + 2 δ } ) ,
Δ g i = L Δ w c i 2 .
( l / n 3 ) ( n 2 - 1 ) = f [ ( sin θ 1 ) ( tan θ 1 ) + ( sin θ 2 ) ( tan θ 2 ) ] ,
l / n 2 ( n 2 - 1 ) = f [ ( sin θ m ) ( tan θ m ) ] .
d f 1 x = d air + l n ( 1 2 + x ) = f cos θ 1 ,
d f 1 y = d air + l n 3 ( 1 2 + y ) = f cos θ 1 ,
( ξ - ½ ) ( n 2 - 1 ) = n 2 x - y ,
ξ = f n 3 ( sin θ 1 ) ( tan θ 1 ) l ( n 2 - 1 ) .
y = ( ξ - 1 2 ) ( 1 - n 2 ) ( 1 + n 2 )

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