Abstract

An analytical method of spatio-temporal design for a Kerr lens mode-locked Ti:sapphire laser is presented. Various spatial and temporal effects are considered to derive simple quadratic equations of the pulse width for both standing- and traveling-wave cavities. By solving the quadratic equation, we obtain the relations between pulse width and other cavity parameters for a given system for optimal design of a Kerr lens mode-locked cavity and to generate the shortest laser pulses. Our theoretical results agree well with the experimental results, and the computation time of this analytic approach for cavity parameters is reduced drastically as compared with an iteratively numerical approach.

© 1996 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. 1642 (1991).
    [Crossref] [PubMed]
  2. G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94221 (1992).
    [Crossref]
  3. T. Brabec, Ch. Spielmann, P. E. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 171292 (1992).
    [Crossref] [PubMed]
  4. G. Cerullo, S. De Silvestri, V. Magni, and L. Pallaro, “Resonators for Kerr-lens mode-locked femtosecond Ti:sapphire lasers,” Opt. Lett. 19807 (1994).
    [Crossref] [PubMed]
  5. J. L. A. Chilla and O. E. Martinez, “Spatial-temporal analysis of the self-mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 10638 (1993).
    [Crossref]
  6. S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 261158 (1990).
    [Crossref]
  7. O. O. Silichev, “Matrix method for calculations on the propagation of coherent laser pulses,” Quantum Electron. 23855 (1993).
    [Crossref]
  8. 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 Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1991), paper CPDP7.
  9. K.-H. Lin and W.-F. Hsieh, “An analytical design of symmetrical Kerr lens mode-locking laser cavities,” J. Opt. Soc. Am. B 11737 (1994).
    [Crossref]
  10. K.-H. Lin, Y. Lai, and W.-F. Hsieh, “A simple analytical method of cavity design for astigmatism compensated Kerr lens mode-locked ring lasers and its applications,” J. Opt. Soc. Am. B 12468 (1995).
    [Crossref]
  11. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 282086 (1992).
    [Crossref]
  12. R. L. Fork, O. E. Martínez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9150 (1984).
    [Crossref] [PubMed]
  13. W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1981), p. 128.
  14. D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode-locking,” Opt. Lett. 17511 (1992).
    [Crossref] [PubMed]
  15. A. J. Alfrey, “Modeling of longitudinally pumped cw Ti:sapphire laser oscillators,” IEEE J. Quantum Electron. 25760 (1989).
    [Crossref]
  16. 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. 161756 (1991).
    [Crossref] [PubMed]
  17. C.-L. Pan, C.-D. Hwang, J.-C. Kuo, J.-M. Shieh, and K.-H. Wu, “Effect of dye concentration on picosecond and femtosecond cw passively mode-locked Ti:sapphire/DDI laser,” Opt. Lett. 171444 (1992).
    [Crossref]
  18. 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. 161022 (1991).
    [Crossref] [PubMed]
  19. 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. 17279 (1992).
    [Crossref] [PubMed]
  20. J.-M. Shieh, F. Ganikhanov, K.-H. Lin, W.-F. Hsieh, and C.-L. Pan, “Completely self-starting picosecond and femtosecond Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 12945 (1995).
    [Crossref]
  21. Ch. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 301100 (1994).
    [Crossref]

1995 (2)

1994 (3)

1993 (2)

J. L. A. Chilla and O. E. Martinez, “Spatial-temporal analysis of the self-mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 10638 (1993).
[Crossref]

O. O. Silichev, “Matrix method for calculations on the propagation of coherent laser pulses,” Quantum Electron. 23855 (1993).
[Crossref]

1992 (6)

1991 (3)

1990 (1)

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 261158 (1990).
[Crossref]

1989 (1)

A. J. Alfrey, “Modeling of longitudinally pumped cw Ti:sapphire laser oscillators,” IEEE J. Quantum Electron. 25760 (1989).
[Crossref]

1984 (1)

’tHooft, G. W.

Acioli, L. H.

Alfrey, A. J.

A. J. Alfrey, “Modeling of longitudinally pumped cw Ti:sapphire laser oscillators,” IEEE J. Quantum Electron. 25760 (1989).
[Crossref]

Brabec, T.

Ch. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 301100 (1994).
[Crossref]

T. Brabec, Ch. Spielmann, P. E. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 171292 (1992).
[Crossref] [PubMed]

Cerullo, G.

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 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. E.

Curley, P. F.

Ch. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 301100 (1994).
[Crossref]

De Silvestri, S.

Demtröder, W.

W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1981), p. 128.

Dienes, A.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 261158 (1990).
[Crossref]

Dijaili, S. P.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 261158 (1990).
[Crossref]

Fork, R. L.

French, P. M. W.

Fujimoto, J. G.

Gabetta, G.

Ganikhanov, F.

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 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.

Haus, H. A.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 282086 (1992).
[Crossref]

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode-locking,” Opt. Lett. 17511 (1992).
[Crossref] [PubMed]

Hsieh, W.-F.

Huang, D.

Hwang, C.-D.

Ippen, E. P.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 282086 (1992).
[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. 161756 (1991).
[Crossref] [PubMed]

Jacobson, J.

Kean, P. N.

Keller, U.

Knox, W. H.

Krasinski, J. S.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94221 (1992).
[Crossref]

Krausz, F.

Ch. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 301100 (1994).
[Crossref]

T. Brabec, Ch. Spielmann, P. E. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 171292 (1992).
[Crossref] [PubMed]

Kuo, J.-C.

Lai, Y.

Lin, K.-H.

Magni, V.

Martinez, O. E.

Martínez, O. E.

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 Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1991), paper CPDP7.

Pallaro, L.

Pan, C.-L.

Pearson, G. W.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94221 (1992).
[Crossref]

Radzewicz, C.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94221 (1992).
[Crossref]

Ramaswamy, M.

Rizvi, N. H.

Shieh, J.-M.

Sibbett, W.

Silichev, O. O.

O. O. Silichev, “Matrix method for calculations on the propagation of coherent laser pulses,” Quantum Electron. 23855 (1993).
[Crossref]

Smith, J. S.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 261158 (1990).
[Crossref]

Spence, D. E.

Spielmann, Ch.

Ch. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 301100 (1994).
[Crossref]

T. Brabec, Ch. Spielmann, P. E. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 171292 (1992).
[Crossref] [PubMed]

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 Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1991), paper CPDP7.

Taylor, J. R.

Ulman, M.

Wu, K.-H.

IEEE J. Quantum Electron. (4)

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 261158 (1990).
[Crossref]

A. J. Alfrey, “Modeling of longitudinally pumped cw Ti:sapphire laser oscillators,” IEEE J. Quantum Electron. 25760 (1989).
[Crossref]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 282086 (1992).
[Crossref]

Ch. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 301100 (1994).
[Crossref]

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94221 (1992).
[Crossref]

Opt. Lett. (9)

T. Brabec, Ch. Spielmann, P. E. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 171292 (1992).
[Crossref] [PubMed]

G. Cerullo, S. De Silvestri, V. Magni, and L. Pallaro, “Resonators for Kerr-lens mode-locked femtosecond Ti:sapphire lasers,” Opt. Lett. 19807 (1994).
[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. 1642 (1991).
[Crossref] [PubMed]

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode-locking,” Opt. Lett. 17511 (1992).
[Crossref] [PubMed]

R. L. Fork, O. E. Martínez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9150 (1984).
[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. 161756 (1991).
[Crossref] [PubMed]

C.-L. Pan, C.-D. Hwang, J.-C. Kuo, J.-M. Shieh, and K.-H. Wu, “Effect of dye concentration on picosecond and femtosecond cw passively mode-locked Ti:sapphire/DDI laser,” Opt. Lett. 171444 (1992).
[Crossref]

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. 161022 (1991).
[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. 17279 (1992).
[Crossref] [PubMed]

Quantum Electron. (1)

O. O. Silichev, “Matrix method for calculations on the propagation of coherent laser pulses,” Quantum Electron. 23855 (1993).
[Crossref]

Other (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 Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1991), paper CPDP7.

W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1981), p. 128.

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Figures (8)

Fig. 1
Fig. 1

Four-mirror figure-x standing-wave laser with a Kerr medium of length Lc placed between two curved mirrors. A pair of Brewster-angle dispersive prisms P1 and P2 are placed in the cavity to achieve group-velocity compensation of the laser pulses.

Fig. 2
Fig. 2

Block diagram illustrating the six temporal effects in Fig. 1.

Fig. 3
Fig. 3

Four-mirror figure-8 ring cavity laser with a Kerr medium of length Lc placed between two curved mirrors. Four Brewster-angle dispersive prisms P1, P2, P3, and P4 are placed in the cavity to achieve group-velocity compensation of the laser pulses.

Fig. 4
Fig. 4

Block diagram illustrating the six temporal effects in Fig. 3.

Fig. 5
Fig. 5

In a real laser cavity, deeper prism insertions are needed to reduce intracavity loss. Here is the apex angle, and dp1 and dp2 are the insertion widths for P1 and P2, respectively.

Fig. 6
Fig. 6

Calculated FWHM pulse width at the output coupler of a standing-wave cavity for 10-nm and 20-nm intracavity bandwidths as functions of prism-pair dispersion with (a) intracavity pulse energy of 120 nJ and (b) intracavity pulse energy of 180 nJ. The distance between curved mirrors is 114 mm.

Fig. 7
Fig. 7

Calculated pulse width at the output coupler of a standing-wave cavity as a function of intracavity pulse energy: (a) 10-nm bandwidth, -2780-fs2 prism-pair dispersion; (b) 20-nm bandwidth, -1772-fs2 prism-pair dispersion.

Fig. 8
Fig. 8

(a) Output pulse width of a traveling wave cavity for a 10-nm intracavity bandwidth as a function of prism separation with intracavity pulse energy 240 nJ. The distance between curved mirrors is 112 mm. (b) Output pulse width of a traveling-wave cavity as a function of intracavity pulse energy. The bandwidth is 10 nm, and the prism-pair dispersion is -926 fs2.

Tables (1)

Tables Icon

Table 1 Prism Separation and Temporal Characteristics of Published Ti:Sapphire Laser Systems

Equations (37)

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A(τ)=A0 expicτ22pexp(icτ/η),
1p=2ρc+i2cσ2,
MS=1S01=12zn101;
Ma=102ac1.
pout=At p+BtCt p+Dt,
MNR=1-iNσc zT22c01=1-iGT22c01,
MNA=102ac1=10i4κA02cσ21.
β(ω)=ωcn(ω)β0+β(ω-ω0)+12β(ω-ω0)2.
ωcn(ω)β0+ωc(n0+n1ω),
MDTC=12zn101=1βcz01.
TT0 expi4πn2A02Lcαcλτ2σ2=exp(iaτ2),
MSPM=102ac1=108πn2A02Lcαcλ1cσ21.
MSPM=102ac1=108πn2αcλ1cσ20LcA02dz1.
1q1=1R1-i1y1=1R1-iλnπw12,
1q2=1R2-i1y2=1R2-iλnπw22,
A02=2Pπw22=2nPλy2=2nPλ1y1nR12+1-K2y1z2+2y1R1z+ny1,
0LcA02dz=2nPλ0Lc1y1nR12+1-K2y1z2+2y1R1z+ny1dz.
dzaz2+bz+c=2(4ac-b2)1/2tan-12az+b(4ac-b2)1/2.
T=T0 exp-ω2Δ2exp-i ωn0zc,
T=T0 exp-iω(n0+n1ω)zc.
MBWL=12zn101=1-i 2cΔ201.
βz=-2λ3πc2dndλ2Lp+λ3πc2d2ndλ2+2n-1n3dndλ2dp,
dp=4 sin(/2)[1-n2 sin2(/2)]1/2dndλΔλLp,
βz=-2λ3πc2dndλ2Lp+λ3πc2d2ndλ2+2n-1n3×dndλ2 4 sin/2[1-n2 sin2(/2)]1/2dndλΔλLp+λ3πc2d2ndλ2tan(/2)(dp1+dp2).
MDPP=12zn101=1βzc01,
a4(y02)4+a3(y02)3+a2(y02)2+a1(y02)+a1=0,
1α+iβ01,
10γ+iδ1.
AtBtCtDt=1+4(α+iβ)(γ+iδ)cσ22(α+iβ)2(γ+iδ)cσ21.
1p=Dt-At2Bt±[(Dt+At)2-4]1/22Bt.
2ρc+i 2cσ2=-(γ+iδ)cσ2±16(a+iβ)2(γ+iδ)2c2σ4+16(α+iβ)(γ+iδ)cσ21/24(α+iβ).
4ρ2σ4+4γρσ2-4(δ+1)=c(αγ+βδ)α2+β2σ2,
4γ+4(δ+2)ρσ2=c(αδ-βγ)α2+β2σ2.
c2(αδ-βγ)2σ4+4c(α2+β2)[γδ(αδ-βγ)-(δ+2)2(αγ+βδ)]σ2-16(δ+1)×(α2+β2)2[γ2+(δ+2)2]=0,
ρ=14(δ+2)c(αδ-βγ)α2+β2-4 γσ2.
ρ1=ρ-2αρ2c-2αcσ41-2αρc-2βcσ22+2αcσ2-2βρc2,
σ1=1-2αρc-2βcσ22+2αcσ2-2βρc21σ2-2βρ2c-2βcσ4.

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