Abstract

Analytical expressions for the pulse variables (duration, energy, chirp, beam size, and curvature) of a self-mode-locked laser are obtained in the form of a five-dimensional iterative map from the full time-and-space (4 × 4) matrix formalism and the gain-equal-to-loss condition. In spite of the great simplifications involved, the model shows good agreement with other authors’ reported experimental data and detailed numerical simulations. The aim is that map theory be used to yield insight into the dynamics of self-mode-locked lasers. As an example of a practical application regions of dynamically stable laser operation in the parameter space are computed.

© 1995 Optical Society of America

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References

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  1. D. Spence, P. Kean, and W. Sibbett, “60-fs pulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 16, 42 (1991).
    [CrossRef] [PubMed]
  2. C. Huang, C. Kapteyn, J. W. McIntosh, and M. M. Murnane, “Generation of transform-limited 32-fs pulses from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 17, 139 (1992).
    [CrossRef] [PubMed]
  3. Y. Silberberg, “Space–time collapse of optical pulses,” in International Quantum Electronics Conference, Vol. 8 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QWB4.
  4. J. P. Likforman, G. Grillon, M. Joffre, C. Le Blanc, A. Migus, and A. Antonetti, “Generation of 27 fs pulses of 70 kW peak power at 80 MHz repetition rate using a cw self-pulsing Ti:sapphire laser,” Appl. Phys. Lett. 58, 2061 (1991).
    [CrossRef]
  5. T. Brabec, Ch. Spielmann, and F. Krausz, “Mode locking in solitary lasers,” Opt. Lett. 16, 1961 (1991).
    [CrossRef] [PubMed]
  6. O. E. Martínez and J. L. A. Chilla, “Self-mode-locking of Ti:sapphire lasers. a matrix formalism,” Opt. Lett. 17, 1210 (1992).
    [CrossRef]
  7. J. L. A. Chilla and O. E. Martínez, “Spatio-temporal analysis of the self-mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 10, 638 (1993).
    [CrossRef]
  8. A. G. Kostenbauder, “Ray pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148 (1990).
    [CrossRef]
  9. See, for example, S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, New York, 1990).
    [CrossRef]
  10. V. Magni, G. Cerullo, and S. De Silvestri, “Closed form Gaussian beam analysis of resonators containing a Kerr medium for fs lasers,” Opt. Commun. 101, 365 (1993).
    [CrossRef]
  11. D. Georgiev, J. Herrmann, and U. Stamm, “Cavity design for optimum nonlinear absorption in Kerr lens mode locked solid state lasers,” Opt. Commun. 92, 368 (1992).
    [CrossRef]
  12. F. Salin, J. Squier, and M. Piché, “Mode locking of Ti:Al2O3lasers and self-focusing: a Gaussian approximation,” Opt. Lett. 16, 1674 (1991).
    [CrossRef] [PubMed]
  13. S. Chen and J. Wang, “Self-starting issues of passive self-focusing mode locking,” Opt. Lett. 16, 1689 (1991).
    [CrossRef] [PubMed]
  14. L. Spinelli, B. Couillaud, N. Goldblat, and D. K. Negus, “Starting and generation of sub-100 fs pulses in Ti:Al2O3by 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.
  15. G. Cerullo, S. De Silvestri, V. Magni, and O. Svelto, “Self-starting fs Kerr-lens mode-locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CWA5.
  16. F. Krausz, Ch. Spielmann, T. Brabec, E. Wintner, and J. A. Schmidt, “Generation of 33-fs optical pulses from a solid-state laser,” Opt. Lett. 17, 204 (1992).
    [CrossRef] [PubMed]

1993 (2)

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

J. L. A. Chilla and O. E. Martínez, “Spatio-temporal analysis of the self-mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 10, 638 (1993).
[CrossRef]

1992 (4)

1991 (5)

1990 (1)

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

Antonetti, A.

J. P. Likforman, G. Grillon, M. Joffre, C. Le Blanc, A. Migus, and A. Antonetti, “Generation of 27 fs pulses of 70 kW peak power at 80 MHz repetition rate using a cw self-pulsing Ti:sapphire laser,” Appl. Phys. Lett. 58, 2061 (1991).
[CrossRef]

Brabec, T.

Cerullo, G.

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

G. Cerullo, S. De Silvestri, V. Magni, and O. Svelto, “Self-starting fs Kerr-lens mode-locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CWA5.

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:Al2O3by 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.

De Silvestri, S.

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

G. Cerullo, S. De Silvestri, V. Magni, and O. Svelto, “Self-starting fs Kerr-lens mode-locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CWA5.

Georgiev, D.

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

Goldblat, N.

L. Spinelli, B. Couillaud, N. Goldblat, and D. K. Negus, “Starting and generation of sub-100 fs pulses in Ti:Al2O3by 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.

Grillon, G.

J. P. Likforman, G. Grillon, M. Joffre, C. Le Blanc, A. Migus, and A. Antonetti, “Generation of 27 fs pulses of 70 kW peak power at 80 MHz repetition rate using a cw self-pulsing Ti:sapphire laser,” Appl. Phys. Lett. 58, 2061 (1991).
[CrossRef]

Herrmann, J.

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

Huang, C.

Joffre, M.

J. P. Likforman, G. Grillon, M. Joffre, C. Le Blanc, A. Migus, and A. Antonetti, “Generation of 27 fs pulses of 70 kW peak power at 80 MHz repetition rate using a cw self-pulsing Ti:sapphire laser,” Appl. Phys. Lett. 58, 2061 (1991).
[CrossRef]

Kapteyn, C.

Kean, P.

Kostenbauder, A. G.

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

Krausz, F.

Le Blanc, C.

J. P. Likforman, G. Grillon, M. Joffre, C. Le Blanc, A. Migus, and A. Antonetti, “Generation of 27 fs pulses of 70 kW peak power at 80 MHz repetition rate using a cw self-pulsing Ti:sapphire laser,” Appl. Phys. Lett. 58, 2061 (1991).
[CrossRef]

Likforman, J. P.

J. P. Likforman, G. Grillon, M. Joffre, C. Le Blanc, A. Migus, and A. Antonetti, “Generation of 27 fs pulses of 70 kW peak power at 80 MHz repetition rate using a cw self-pulsing Ti:sapphire laser,” Appl. Phys. Lett. 58, 2061 (1991).
[CrossRef]

Magni, V.

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

G. Cerullo, S. De Silvestri, V. Magni, and O. Svelto, “Self-starting fs Kerr-lens mode-locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CWA5.

Martínez, O. E.

McIntosh, J. W.

Migus, A.

J. P. Likforman, G. Grillon, M. Joffre, C. Le Blanc, A. Migus, and A. Antonetti, “Generation of 27 fs pulses of 70 kW peak power at 80 MHz repetition rate using a cw self-pulsing Ti:sapphire laser,” Appl. Phys. Lett. 58, 2061 (1991).
[CrossRef]

Murnane, M. M.

Negus, D. K.

L. Spinelli, B. Couillaud, N. Goldblat, and D. K. Negus, “Starting and generation of sub-100 fs pulses in Ti:Al2O3by 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.

Salin, F.

Schmidt, J. A.

Sibbett, W.

Silberberg, Y.

Y. Silberberg, “Space–time collapse of optical pulses,” in International Quantum Electronics Conference, Vol. 8 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QWB4.

Spence, D.

Spielmann, Ch.

Spinelli, L.

L. Spinelli, B. Couillaud, N. Goldblat, and D. K. Negus, “Starting and generation of sub-100 fs pulses in Ti:Al2O3by 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.

Stamm, U.

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

Svelto, O.

G. Cerullo, S. De Silvestri, V. Magni, and O. Svelto, “Self-starting fs Kerr-lens mode-locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CWA5.

Wang, J.

Wiggins, S.

See, for example, S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, New York, 1990).
[CrossRef]

Wintner, E.

Appl. Phys. Lett. (1)

J. P. Likforman, G. Grillon, M. Joffre, C. Le Blanc, A. Migus, and A. Antonetti, “Generation of 27 fs pulses of 70 kW peak power at 80 MHz repetition rate using a cw self-pulsing Ti:sapphire laser,” Appl. Phys. Lett. 58, 2061 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

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

Opt. Commun. (2)

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

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

Opt. Lett. (7)

Other (4)

See, for example, S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, New York, 1990).
[CrossRef]

Y. Silberberg, “Space–time collapse of optical pulses,” in International Quantum Electronics Conference, Vol. 8 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QWB4.

L. Spinelli, B. Couillaud, N. Goldblat, and D. K. Negus, “Starting and generation of sub-100 fs pulses in Ti:Al2O3by 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.

G. Cerullo, S. De Silvestri, V. Magni, and O. Svelto, “Self-starting fs Kerr-lens mode-locked Ti:sapphire laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CWA5.

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Figures (6)

Fig. 1
Fig. 1

Sketch of the laser cavity: M1, M2, M3, M4, mirrors. The black dots indicate the focal points of M2 and M3. The arrows mark the position and the direction of propagation of the pulse, corresponding to the sets of variables used in the model.

Fig. 2
Fig. 2

Variation of pulse duration τ and energy U at the fixed point P1 as a function of the negative GVD introduced by the prisms, D (in thousands of femtoseconds squared). The dashed lines indicate the results of the model; the solid curves are the results of the detailed numerical simulations6 (Ω = 7 cm−1, d = 0.4 cm, k = 0.3, = 0.03, Γ = 20).

Fig. 3
Fig. 3

Logarithm of the absolute value of Φs as a function of υ (Ω = 7 cm−1, k = 0.5, = 0.025, Γ = 15, D = 5100 fs2). The positive values correspond to unstable regions.

Fig. 4
Fig. 4

Regions of instability (in gray) of the eigenvalues, showing instability as a function of υ (0 < υ < 4) and D (in thousands of femtoseconds squared): (a) Φs; (b) Φr and Φq (the unstable region is to the left of the curve). The value of D that exactly compensates for the positive GVD introduced by the rod in one round trip (2α = 2200 fs2) is at the origin. Parameter values are as in Fig. 3.

Fig. 5
Fig. 5

Interruption and disappearance of bands of Φr at low gain (Γ = 5; other parameters as in Fig. 3). (b) Fingers in Φs at high gain (Γ = 40; other parameters as in Fig. 2). (c) Island in Φr [same parameters as in (b)]. Islands often also appear in Φs.

Fig. 6
Fig. 6

Regions of stability (not shadowed) for several values of gain: (a) Γ = 10, (b) Γ = 20, (c) Γ = 30. Other parameters are as in Fig. 2. The dotted curves are isobars joining points of equal pulse duration (indicated in femtoseconds).

Equations (32)

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Ω = 8 L 1 / r 2 , Ω = 4 m / r 2 .
Ω Ω , 2 L 1 / r 1 , z L 1 , L 2 ,
[ 1 z g 1 ] ,
g = C z U / τ σ 4 ,
[ 1 α β 1 ] ,
β = C z U / l σ 2 τ 3 ,
S = 1 / σ 2 , T = 1 / τ 2 .
β / β = g / g q .
[ 1 υ d ( 2 υ ) h Ω ( 1 + hd ) hd + ( 1 υ ) ( hd + 1 ) ] ,
[ 1 β ( δ + α ) δ β + β 1 β ( δ + α ) ] ,
Q n + 1 = ( β + β + Q n ) ( 1 δ Q n ) δ ( T n / π ) 2 ( 1 δ Q n ) 2 + ( δ T n / π ) 2 ,
R n + 1 = Y { h Ω ( 1 + h d ) + X [ R n + ( l S n / π ) 2 d ( 2 υ ) / Y ] } Y 2 + ( l S n / π ) 2 ( 2 υ ) 2 d 2 ,
S n + 1 = S n [ X R n + 1 d ( 2 υ ) ] / Y ,
T n + 1 = T n ( 1 + δ Q n + 1 ) / ( 1 δ Q n ) ,
G = Γ / ( 1 + I / I sat ) ( G 2 k = 1 ) ,
U n + 1 = U n S n / S n + 1 F / S n + 1 = U n Y / [ X R n + 1 d ( 2 υ ) ] ,
τ = l q δ π 2 C z F ( q + 1 ) f β A δ ,
τ = A δ + p .
σ = σ 0 ( 1 + 2 BF σ 0 2 / τ ) 1 / 2 ,
σ 0 4 = ( 2 l d / π ) 2 υ ( 4 υ ) ,
B = ( C z / q ) ( π / l ) 2 ( q + 1 ) f g ( 1 ) ( G 1 ) [ 2 ( 2 υ ) 2 ] 8 d ln ( G ) ( 2 υ ) .
R = h / 2 = ½ [ g ( q + 1 ) / q Ω ] ,
R = Ω / 2 , T , Q = 0 , U = F σ 0 2 , σ = σ 0 .
| ( 1 υ ) 2 ½ υ 2 y d ( 1 ½ υ ) | | x | 1 ,
det ( J Φ I ) = 0 ,
S n + 1 / S n = R n + 1 / R n = 2 x 2 1 ,
U = U n ( 1 ) ( G 1 ) / ln ( G ) ,
β = f β ( T n / T n ) U n S n T n 3 / 2 C z / l ,
g = f g ( T n / T n ) U n S n T n 1 / 2 C z ,
β = f β ( T n + 1 / T n ) U n S n T n + 1 3 / 2 C z / l ,
g = f g ( T n + 1 / T n ) U n S n T n + 1 1 / 2 C z ,
U n = U n G k ( G 1 ) / ln ( G ) .

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