Abstract

We examine passive mode locking, to which not only net-gain saturation but also group-velocity dispersion and self-phase modulation contribute significantly. Different sources of self-phase modulation are considered, and closed-form analytical solutions are obtained. The case of the colliding-pulse mode locking is discussed, and an ideal case is considered, in which the self-phase modulation and group-velocity dispersion are balanced so as to take maximum advantage of this additional shaping mechanism. This mechanism is shown to be similar to that which causes solitons in optical fibers; it provides additional reduction of the pulse width and significantly improves the stability against fluctuations in net gain.

© 1985 Optical Society of America

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  1. O. E. Martinez, R. L. Fork, and J. P. Gordon, Opt. Lett. 9, 156 (1984).
    [CrossRef] [PubMed]
  2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45 (13), 1095 (1980).
    [CrossRef]
  3. W. Dietel, E. Dopel, D. Kuhlke, and B. Wilhelmi, Opt. Commun. 43, 433 (1982).
    [CrossRef]
  4. W. Dietel, J. J. Fontaine, and J. C. Diels, Opt. Lett. 8, 4 (1983).
    [CrossRef] [PubMed]
  5. J. P. Gordon and R. L. Fork, Opt. Lett. 9, 153 (1984).
    [CrossRef] [PubMed]
  6. R. L. Fork, O. E. Martinez, and J. P. Gordon, Opt. Lett. 9, 150 (1984).
    [CrossRef] [PubMed]
  7. O. E. Martinez, J. P. Gordon, and R. L. Fork, J. Opt. Soc. Am. A 1, 1003 (1984).
    [CrossRef]
  8. H. A. Haus, IEEE J. Quantum Electron. QE-11, 736 (1975).
    [CrossRef]
  9. H. A. Haus, J. Appl. Phys. 46, 3049 (1975).
    [CrossRef]
  10. R. Cubeddu and O. Svelto, IEEE J. Quantum Electron. QE-5, 495 (1969).
    [CrossRef]
  11. R. C. Ekardt, C. H. Lee, and J. N. Bradford, Opt. Electron. 6, 67 (1974).
    [CrossRef]
  12. D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 694 (1970).
    [CrossRef]
  13. R. L. Fork, B. I. Greene, and C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
    [CrossRef]
  14. J. D. McMullen, Appl. Opt. 18, 737 (1979).
    [CrossRef] [PubMed]
  15. D. N. Dempster, T. Morrow, R. Rankin, and G. F. Thompson, J. Chem. Soc. 68, 1479 (1972).
  16. G. H. C. New, IEEE J. Quantum Electron. QE-19, 521 (1983).
  17. M. S. Stix and E. P. Ippen, IEEE J. Quantum Electron. QE-19, 521 (1983).
  18. J. A. Valdmanis, R. L. Fork, and J. P. Gordon, Opt. Lett. 10, 131 (1985).
    [CrossRef] [PubMed]

1985 (1)

1984 (4)

1983 (3)

W. Dietel, J. J. Fontaine, and J. C. Diels, Opt. Lett. 8, 4 (1983).
[CrossRef] [PubMed]

G. H. C. New, IEEE J. Quantum Electron. QE-19, 521 (1983).

M. S. Stix and E. P. Ippen, IEEE J. Quantum Electron. QE-19, 521 (1983).

1982 (1)

W. Dietel, E. Dopel, D. Kuhlke, and B. Wilhelmi, Opt. Commun. 43, 433 (1982).
[CrossRef]

1981 (1)

R. L. Fork, B. I. Greene, and C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45 (13), 1095 (1980).
[CrossRef]

1979 (1)

1975 (2)

H. A. Haus, IEEE J. Quantum Electron. QE-11, 736 (1975).
[CrossRef]

H. A. Haus, J. Appl. Phys. 46, 3049 (1975).
[CrossRef]

1974 (1)

R. C. Ekardt, C. H. Lee, and J. N. Bradford, Opt. Electron. 6, 67 (1974).
[CrossRef]

1972 (1)

D. N. Dempster, T. Morrow, R. Rankin, and G. F. Thompson, J. Chem. Soc. 68, 1479 (1972).

1970 (1)

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

1969 (1)

R. Cubeddu and O. Svelto, IEEE J. Quantum Electron. QE-5, 495 (1969).
[CrossRef]

Bradford, J. N.

R. C. Ekardt, C. H. Lee, and J. N. Bradford, Opt. Electron. 6, 67 (1974).
[CrossRef]

Cubeddu, R.

R. Cubeddu and O. Svelto, IEEE J. Quantum Electron. QE-5, 495 (1969).
[CrossRef]

Dempster, D. N.

D. N. Dempster, T. Morrow, R. Rankin, and G. F. Thompson, J. Chem. Soc. 68, 1479 (1972).

Diels, J. C.

Dietel, W.

W. Dietel, J. J. Fontaine, and J. C. Diels, Opt. Lett. 8, 4 (1983).
[CrossRef] [PubMed]

W. Dietel, E. Dopel, D. Kuhlke, and B. Wilhelmi, Opt. Commun. 43, 433 (1982).
[CrossRef]

Dopel, E.

W. Dietel, E. Dopel, D. Kuhlke, and B. Wilhelmi, Opt. Commun. 43, 433 (1982).
[CrossRef]

Ekardt, R. C.

R. C. Ekardt, C. H. Lee, and J. N. Bradford, Opt. Electron. 6, 67 (1974).
[CrossRef]

Fontaine, J. J.

Fork, R. L.

Gordon, J. P.

Greene, B. I.

R. L. Fork, B. I. Greene, and C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

Haus, H. A.

H. A. Haus, IEEE J. Quantum Electron. QE-11, 736 (1975).
[CrossRef]

H. A. Haus, J. Appl. Phys. 46, 3049 (1975).
[CrossRef]

Ippen, E. P.

M. S. Stix and E. P. Ippen, IEEE J. Quantum Electron. QE-19, 521 (1983).

Kuhlke, D.

W. Dietel, E. Dopel, D. Kuhlke, and B. Wilhelmi, Opt. Commun. 43, 433 (1982).
[CrossRef]

Kuizenga, D. J.

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

Lee, C. H.

R. C. Ekardt, C. H. Lee, and J. N. Bradford, Opt. Electron. 6, 67 (1974).
[CrossRef]

Martinez, O. E.

McMullen, J. D.

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45 (13), 1095 (1980).
[CrossRef]

Morrow, T.

D. N. Dempster, T. Morrow, R. Rankin, and G. F. Thompson, J. Chem. Soc. 68, 1479 (1972).

New, G. H. C.

G. H. C. New, IEEE J. Quantum Electron. QE-19, 521 (1983).

Rankin, R.

D. N. Dempster, T. Morrow, R. Rankin, and G. F. Thompson, J. Chem. Soc. 68, 1479 (1972).

Shank, C. V.

R. L. Fork, B. I. Greene, and C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

Siegman, A. E.

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

Stix, M. S.

M. S. Stix and E. P. Ippen, IEEE J. Quantum Electron. QE-19, 521 (1983).

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45 (13), 1095 (1980).
[CrossRef]

Svelto, O.

R. Cubeddu and O. Svelto, IEEE J. Quantum Electron. QE-5, 495 (1969).
[CrossRef]

Thompson, G. F.

D. N. Dempster, T. Morrow, R. Rankin, and G. F. Thompson, J. Chem. Soc. 68, 1479 (1972).

Valdmanis, J. A.

Wilhelmi, B.

W. Dietel, E. Dopel, D. Kuhlke, and B. Wilhelmi, Opt. Commun. 43, 433 (1982).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. L. Fork, B. I. Greene, and C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

IEEE J. Quantum Electron. (5)

G. H. C. New, IEEE J. Quantum Electron. QE-19, 521 (1983).

M. S. Stix and E. P. Ippen, IEEE J. Quantum Electron. QE-19, 521 (1983).

R. Cubeddu and O. Svelto, IEEE J. Quantum Electron. QE-5, 495 (1969).
[CrossRef]

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

H. A. Haus, IEEE J. Quantum Electron. QE-11, 736 (1975).
[CrossRef]

J. Appl. Phys. (1)

H. A. Haus, J. Appl. Phys. 46, 3049 (1975).
[CrossRef]

J. Chem. Soc. (1)

D. N. Dempster, T. Morrow, R. Rankin, and G. F. Thompson, J. Chem. Soc. 68, 1479 (1972).

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

Opt. Commun. (1)

W. Dietel, E. Dopel, D. Kuhlke, and B. Wilhelmi, Opt. Commun. 43, 433 (1982).
[CrossRef]

Opt. Electron. (1)

R. C. Ekardt, C. H. Lee, and J. N. Bradford, Opt. Electron. 6, 67 (1974).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. Lett. (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45 (13), 1095 (1980).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

(a) Absorption cross section for the ground state of the DODCI (peak at λ = 590 nm) and the photoisomer (peak at λ = 620 nm) as from Ref. 14. (b) Dispersion constant (per molecule/cubic centimeter) for the ground state of the DODCI. Obtained by a Kramers–Kronig transformation of (a). The dispersion of the photoisomer is shown in the text not to have a significant contribution.

Fig. 2
Fig. 2

Parameters for the pulse inside the cavity for different values of the net gain g′i: g′i = 0.03; Δgi = 0.031; gi = 0.032; + gi = 0.033. The other parameters are α0 = 0.25; α1 = 0.025; γ = 0.015; a1 = 25 fsec2; b2 = 0; s0 = 100; s1 = 5.55; and d = −2. (a) Pulse energy as a function of the group-velocity dispersion (−a2). The energy is given in units of the saturation energy for the photoisomer. Note that the shaping is done by the ground state, with a saturation energy 18 times larger. (b) Pulse width (τ). (c) Chirp parameter X (indicates how much the pulse may be shortened with a linear compressor).

Fig. 3
Fig. 3

Pulse width obtained after including the fast self-phase modulation in the case shown in Fig. 2, for gi = 0.031. △, b2 = 0; ×, b2 = 0.01 fsec; ○, b2 = 0.025 fsec. For the example given, b2 = 3 × 10−4 fsec per micrometer thickness of the jet.

Fig. 4
Fig. 4

Ideal case with d = 1, i.e., opposite sign for the slow self-phase modulation resulting from the absorber with respect to that shown in previous figures, (a) Pulse width for different values for the fast self-phase modulation parameter b2 ○, b2 = 0; ×, b2 = 0.2 fsec; △, b2 = 0.5 fsec; □, b2 = 1.0 fsec. In this case b = 4 × 10−3 fsec per micrometer thickness of jet (the difference is due to the difference in the saturation energy of the absorber). Other parameters: α, 0.01; γ, 0.06; g1, 0.13; a1, 25 fsec2; s, 5. (b) Chirp parameter X for the same case. No difference was observed for the different values of b2.

Tables (1)

Tables Icon

Table 1 Typical Parameters for a Colliding-Pulse Mode-Locked Lasera

Equations (54)

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E ( t ) = υ ( t ) exp [ i ϕ ( t ) ] exp ( i w 0 t ) ,
υ ( t + T cav ) exp [ i ϕ ( t + T cav ) ] = exp [ ( α + i β ) ] υ ( t ) × exp [ i ϕ ( t ) ] .
U ( t ) = t υ 2 ( t ) d t .
υ ( t + T cav ) exp [ i ϕ ( t + T cav ) ] = ( 1 α i β ) υ ( t ) exp [ i ϕ ( t ) ] .
U = U U 0 ,
α = g 0 + g U + g U 2 + c 2 ( w w 0 ) + c 2 U ( w w 0 ) a 1 ( w w 0 ) 2 + b 1 υ 2 ,
β = ψ 0 + ψ U + ψ U 2 c 1 ( w w 0 ) c 1 U ( w w 0 ) a 2 ( w w 0 ) 2 + b 2 υ 2 .
( g 0 + g U + g U 2 + c 1 d d t + c 1 U d d t + a 1 d 2 d t 2 + b 1 υ 2 ) i ( ψ 0 + ψ U + ψ U 2 + c 2 d d t + c 2 U d d t + a 2 d 2 d t 2 + b 2 υ 2 ) υ ( t ) exp [ i ϕ ( t ) ] = 0 .
d ϕ / d t = ξ U ,
[ g 0 + g U + g U 2 + c 1 d d t + c 1 U d d t + c 2 ξ U a 1 ξ 2 U 2 + ( b 1 + a 2 ξ ) υ 2 + c 2 ξ U 2 + 2 a 2 ξ U d d t + a 1 d 2 d t 2 ] υ ( t ) = 0 ,
[ ψ 0 + ψ U + ψ U 2 c 1 ξ U c 1 ξ U 2 + c 2 d d t + c 2 U d d t a 2 ξ 2 U 2 + ( b 2 a 1 ξ ) υ 2 2 a 1 ξ U d d t + a 2 d 2 d t 2 ] υ ( t ) = 0 .
υ = ( U 0 τ ) 1 / 2 sech ( t / τ ) ,
ϕ ( t ) = X ln ( υ ) ,
U = U 0 tanh ( t / τ ) ,
d υ d t = υ U Y ,
d 2 υ d t 2 = υ ( 2 U 2 U 0 2 ) / Y 2 ,
U 2 = U 0 2 Y υ 2 ,
Y = U 0 τ , X = ξ Y .
g 0 τ 2 + b 1 Y + a 2 X a 1 = 0 ,
g Y 2 c 2 X Y + a 1 X 2 + ( c 1 + b 1 ) Y + 3 a 2 X 2 a 1 = 0 ,
ψ Y 2 c 1 X Y + a 2 X 2 + ( c 2 + b 2 ) Y 3 a 1 X 2 a 2 = 0 ,
c 1 = g Y + c 2 X ,
c 2 = ψ Y c 1 X ,
ψ 0 τ 2 = b 2 Y + a 1 X + a 2 .
τ 2 = a 1 a 2 X b 1 Y / g 0 .
X = [ a 1 ( c 2 + b 2 ) a 2 ( c 1 + b 1 ) ] Y + [ a 2 g a 1 ψ ] Y 2 3 ( a 1 2 + a 2 2 ) ( a 1 c 1 + a 2 c 2 ) Y .
u = U / U s ,
η = ( w 0 w B ) / Δ w B .
g U 2 = α 0 2 ( 1 + η 2 ) e u u 2 ,
ψ U 0 2 = g U 0 2 η ,
c 1 U 0 = ( 1 η 2 ) Δ w B ( 1 + η 2 ) 2 e u u ,
c 2 U 0 = 2 α 0 η Δ w B ( 1 + η 2 ) 2 e u u .
| c 1 | | g Y | ,
u τ Δ w b 1 .
| c 1 Y | | 3 a 1 | .
Δ w B w τ α 0 3 e g 0 ,
| c 2 | | g Y |
| a 2 c 2 Y | 3 | a 1 a 2 | < 3 ( a 1 2 + a 2 2 ) ,
d 2 ϕ d t 2 = ξ U 0 τ = X τ 2 ,
υ ( t ) = υ 0 exp ( 2 t 2 T 2 + i δ w m t 2 2 T ) ,
T x = 4 δ w m = T / X ,
s 0 = σ 0 σ e ρ = 80 120 ,
s 1 = σ 1 σ 3 ρ = 4.4 6.6 ,
U s = h ν σ 0 x area = 0.1 nJ 0.2 nJ
b 2 = 2 π n 2 l ρ U s ,
a 2 X = a 1 ( 2 + Z ) ( 3 + 2 Z ) ,
Z = g U 0 2 / s 0 .
b 2 Y 2 a 2 .
τ 2 = a 1 g 0 ( 1 + Z ) ( 3 + 2 Z ) .
g 0 + g U 0 + g U 0 2 < 0 ,
g 0 g U 0 + g U 0 2 < 0 ,
Z > 1 ,
τ 2 > 2 5 a 1 g 0 .
g 0 = g i exp ( u / s 0 ) α 0 e M α 1 exp ( u s 1 / s 0 ) α ; g = s 0 2 s 1 exp ( u s 1 / s 0 ) ; U = U 0 / U s .

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