Abstract

We present numerical simulations of laser mode-locking using a spatio-temporal master equation. We look at active mode-locking using an amplitude modulator and compare the results with those found using a phase modulator. We find gaussian pulses and stability conditions consistent with the Kuizenga-Siegman theory of mode-locking. We then add a Kerr medium to the cavity and examine the effect this has on the mode-locking process, the stability, and the shape of the final pulses. We find that the pulses are significantly compressed in both space and time, and the profiles become more sech-like.

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References

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  1. T. Deutsch, Appl. Phys. Lett. 7, 80 (1965).<br>
    [CrossRef]
  2. D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 709 (1970).<br>
    [CrossRef]
  3. D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 694 (1970).<br>
    [CrossRef]
  4. H. A. Haus, IEEE J. Quantum Electron. QE-11, 323 (1975).<br>
    [CrossRef]
  5. T. Brabec, C. Spielmann, P. F. Curley, and F. Krausz, Opt. Lett. 17, 1292 (1992).<br>
    [CrossRef] [PubMed]
  6. V. Magni, G. Cerullo, and S. DeSilvestri, Opt. Commun. 96, 348 (1993).<br>
    [CrossRef]
  7. A. Agnesi, IEEE J. Quantum Electron. 30, 1115 (1994).<br>
    [CrossRef]
  8. G. Cerullo, S. DeSilvestri, V. Magni, and L. Pallaro, Opt. Lett. 19, 807 (1994).<br>
    [CrossRef] [PubMed]
  9. H. A. Haus and Y. Silberberg, IEEE J. Quantum Electron. QE-22, 325 (1986).<br>
    [CrossRef]
  10. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991).<br>
    [CrossRef]
  11. O. E. Martinez, R. L. Fork, and J. P. Gordon, J. Opt. Soc. Am. B 2, 753 (1985).<br>
    [CrossRef]
  12. A. M. Dunlop, W. J. Firth, E. M. Wright, and D. R. Heatley, Opt. Lett. 21, 770 (1996).<br>
    [CrossRef] [PubMed]
  13. P. Baues, Opto-Electron. 1, 37 (1969).<br>
    [CrossRef]
  14. A. M. Dunlop, W. J. Firth, and E. M. Wright, Opt. Commun. 138, 211 (1997).<br>
    [CrossRef]
  15. J. V. Roey, J. van der Donk, and P. Lagasse, J. Opt. Soc. Am. 71, 803 (1981).<br>
    [CrossRef]
  16. N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, Opt. Commun. 102, 324 (1993).<br>
    [CrossRef]

Other (16)

T. Deutsch, Appl. Phys. Lett. 7, 80 (1965).<br>
[CrossRef]

D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 709 (1970).<br>
[CrossRef]

D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 694 (1970).<br>
[CrossRef]

H. A. Haus, IEEE J. Quantum Electron. QE-11, 323 (1975).<br>
[CrossRef]

T. Brabec, C. Spielmann, P. F. Curley, and F. Krausz, Opt. Lett. 17, 1292 (1992).<br>
[CrossRef] [PubMed]

V. Magni, G. Cerullo, and S. DeSilvestri, Opt. Commun. 96, 348 (1993).<br>
[CrossRef]

A. Agnesi, IEEE J. Quantum Electron. 30, 1115 (1994).<br>
[CrossRef]

G. Cerullo, S. DeSilvestri, V. Magni, and L. Pallaro, Opt. Lett. 19, 807 (1994).<br>
[CrossRef] [PubMed]

H. A. Haus and Y. Silberberg, IEEE J. Quantum Electron. QE-22, 325 (1986).<br>
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991).<br>
[CrossRef]

O. E. Martinez, R. L. Fork, and J. P. Gordon, J. Opt. Soc. Am. B 2, 753 (1985).<br>
[CrossRef]

A. M. Dunlop, W. J. Firth, E. M. Wright, and D. R. Heatley, Opt. Lett. 21, 770 (1996).<br>
[CrossRef] [PubMed]

P. Baues, Opto-Electron. 1, 37 (1969).<br>
[CrossRef]

A. M. Dunlop, W. J. Firth, and E. M. Wright, Opt. Commun. 138, 211 (1997).<br>
[CrossRef]

J. V. Roey, J. van der Donk, and P. Lagasse, J. Opt. Soc. Am. 71, 803 (1981).<br>
[CrossRef]

N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, Opt. Commun. 102, 324 (1993).<br>
[CrossRef]

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Figures (4)

Figure 1.
Figure 1.

Temporal profiles. |α| = 0.09, β = (6.0 - 9.0i) × 10-3, κ = 0.0, zero net gain. Red, green and blue lines are for arg(α) = π (amplitude modulation), π/2 - π/2 (phase modulation) respectively.

Figure 2.
Figure 2.

Dynamics of field in amplitude modulated and phase modulated cavities with a Kerr medium. |α| = 0.09, β = (6.0 - 9.0i) × 10-3, κ = 0.1. Red and blue lines are for arg(α) = π(AM) and arg(α) = -π/2 (PM) respectively, with no net gain, green and magenta lines are AM and PM respectively, with a net gain of 0.025.

Figure 3.
Figure 3.

Temporal profiles. Comparison of linear and nonlinear results using amplitude modulation. Profiles have been scaled to equal heights. Red and blue lines are linear and nonlinear results, respectively, with zero gain, green line is nonlinear result with a net gain of 0.025. A similar result is also found for the spatial profiles.

Figure 4.
Figure 4.

Temporal profiles. Comparison of linear and nonlinear results using phase modulation. Profiles have been scaled to equal heights. Red and blue lines are linear and nonlinear results, respectively, with zero gain, green line is nonlinear result with a net gain of 0.025. A similar result is also found for the spatial profiles.

Equations (20)

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E ( x , t , T ) T = π ̂ x E + π ̂ t E + g E + N ( E ) .
π ̂ x = 2 T R sin ψ ( B k 2 x 2 + i ( A D ) ( x x + 1 2 ) + k C x 2 ) .
π ̂ t = i β 2 E t 2 ,
π ̂ t = i β 2 E t 2 + α t 2 .
E ( x , t , T ) T = 2 T R sin ψ ( B k 2 x 2 + i ( A D ) ( x x + 1 2 ) + k C x 2 )
+ i β 2 E t 2 + α t 2 + g E .
exp ( i k x 2 2 q x ) ; 1 q x = 1 R + 2 i k w 2 ,
E ( x , t , T ) = A ( T ) exp ( i k x 2 2 q x ( T ) ) exp ( i ω t 2 2 q t ( T ) ) ,
i k 2 q x 2 d q x d T = i ψ 2 T R sin ψ ( B k q x 2 k ( A D ) q x + k C ) ,
i ω 2 q t 2 d q t d T = i β ω 2 q t 2 + α ,
1 A d A d T = i ψ 2 T R sin ψ ( i B q x + i ( A D ) 2 ) β ω q t .
1 q x Q x = D A 2 B ± i sin ψ B = Q x m ,
1 q t Q t = ± 1 ω i α β = Q t m ,
1 A d A d T = i ψ 2 i α β + g .
1 Δ x d Δ x d T = 2 i ψ .
1 Δ t d Δ t d T = 4 i α β .
E ˜ ( x , t , T ) = U ( x ) E ( x , t , T ) ,
U ( x ) = exp [ i k 2 B ( A D 2 ) x 2 ] .
E ˜ T = i ψ 2 T R sin ψ ( B k 2 x 2 k sin 2 ψ B x 2 ) E ˜ + i β 2 t 2 E ˜ + α t 2 E ˜
+ g E ˜ + i k U ( x ) 1 E ˜ 2 E ˜ .

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