Abstract

A new principle of mode locking is analyzed: additive pulse mode locking. It is shown to be operative in two-cavity soliton lasers, but it also permits mode locking with fibers in the positive dispersion regime. A simple model is developed that displays the pulse-shortening mechanism. Parameter ranges, within which this principle can be exploited, are given. Comparisons with experiments are made.

© 1989 Optical Society of America

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References

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  1. J. Mark, L. Y. Liu, K. L. Hall, H. A. Haus, and E. P. Ippen, “Femtosecond pulse generation in a laser with a nonlinear external resonator,” Opt. Lett. 14, 48–50 (1989).
    [Crossref] [PubMed]
  2. L. F. Mollenauer and R. H. Stolen, “The soliton laser,” Opt. Lett. 9, 13–15 (1984).
    [Crossref] [PubMed]
  3. P. N. Kean, R. S. Grant, X. Zhu, D. W. Crust, D. Burns, and W. Sibbett, “Enhanced mode locking of colour-centre lasers by coupled-cavity feedback control,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD7.
  4. P. N. Kean, X. Zhu, D. W. Crust, R. S. Grant, N. Langford, and W. Sibbett, “Enhanced mode locking of color-center lasers,” Opt. Lett. 14, 39–41 (1989).
    [Crossref] [PubMed]
  5. K. J. Blow and B. P. Nelson, “Non-soliton mode locking of an F-center laser with nonlinear external resonator,” in Ultrafast Phenomena VI, T. Yajima, K. Yoshihara, C. B. Harris, and S. Shionoya, eds., Vol. 48 of Springer Series in Chemical Physics (Springer-Verlag, Berlin, 1988), pp. 67–99.
    [Crossref]
  6. P. A. Belanger, “Soliton laser. I: A simplified model,” J. Opt. Soc. Am. B 5, 793–798 (1988).
    [Crossref]
  7. K. J. Blow and D. Wood, “Mode-locked lasers with nonlinear external cavities,” J. Opt. Soc. Am. B 5, 629–632 (1988).
    [Crossref]
  8. F. Oullette and M. Piché, “Ultrashort pulse reshaping with a nonlinear Fabry–Perot cavity matched to a train of short pulses,” J. Opt. Soc. Am. B 5, 1228–1236 (1988).
    [Crossref]
  9. L. E. Dahlstrom, “Passive mode locking and Q-switching of high power lasers by beams of the optical Kerr effect,” Opt. Commun. 5, 157–162 (1972).
    [Crossref]
  10. F. Oullette and M. Piché, “Pulse shaping and passive mode-locking with a nonlinear Michelson interferometer,” Opt. Commun. 60, 99–103 (1986).
    [Crossref]
  11. H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. QE-22, 325–331 (1986).
    [Crossref]
  12. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).We use a more convenient phase reference for the scattering matrix of the mirror.
  13. M. S. Stix and E. P. Ippen, “Pulse-shaping in passively mode-locked ring dye lasers,” IEEE J. Quantum Electron. QE-19, 520–525 (1983).
    [Crossref]
  14. H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. QE-11, 323–330 (1975).
    [Crossref]
  15. F. M. Mitschke and L. F. Mollenauer, “Stabilizing the soliton laser,” IEEE J. Quantum Electron. QE-22, 2242–2250 (1986).
    [Crossref]

1989 (2)

1988 (3)

1986 (3)

F. Oullette and M. Piché, “Pulse shaping and passive mode-locking with a nonlinear Michelson interferometer,” Opt. Commun. 60, 99–103 (1986).
[Crossref]

H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. QE-22, 325–331 (1986).
[Crossref]

F. M. Mitschke and L. F. Mollenauer, “Stabilizing the soliton laser,” IEEE J. Quantum Electron. QE-22, 2242–2250 (1986).
[Crossref]

1984 (1)

1983 (1)

M. S. Stix and E. P. Ippen, “Pulse-shaping in passively mode-locked ring dye lasers,” IEEE J. Quantum Electron. QE-19, 520–525 (1983).
[Crossref]

1975 (1)

H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. QE-11, 323–330 (1975).
[Crossref]

1972 (1)

L. E. Dahlstrom, “Passive mode locking and Q-switching of high power lasers by beams of the optical Kerr effect,” Opt. Commun. 5, 157–162 (1972).
[Crossref]

Belanger, P. A.

Blow, K. J.

K. J. Blow and D. Wood, “Mode-locked lasers with nonlinear external cavities,” J. Opt. Soc. Am. B 5, 629–632 (1988).
[Crossref]

K. J. Blow and B. P. Nelson, “Non-soliton mode locking of an F-center laser with nonlinear external resonator,” in Ultrafast Phenomena VI, T. Yajima, K. Yoshihara, C. B. Harris, and S. Shionoya, eds., Vol. 48 of Springer Series in Chemical Physics (Springer-Verlag, Berlin, 1988), pp. 67–99.
[Crossref]

Burns, D.

P. N. Kean, R. S. Grant, X. Zhu, D. W. Crust, D. Burns, and W. Sibbett, “Enhanced mode locking of colour-centre lasers by coupled-cavity feedback control,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD7.

Crust, D. W.

P. N. Kean, X. Zhu, D. W. Crust, R. S. Grant, N. Langford, and W. Sibbett, “Enhanced mode locking of color-center lasers,” Opt. Lett. 14, 39–41 (1989).
[Crossref] [PubMed]

P. N. Kean, R. S. Grant, X. Zhu, D. W. Crust, D. Burns, and W. Sibbett, “Enhanced mode locking of colour-centre lasers by coupled-cavity feedback control,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD7.

Dahlstrom, L. E.

L. E. Dahlstrom, “Passive mode locking and Q-switching of high power lasers by beams of the optical Kerr effect,” Opt. Commun. 5, 157–162 (1972).
[Crossref]

Grant, R. S.

P. N. Kean, X. Zhu, D. W. Crust, R. S. Grant, N. Langford, and W. Sibbett, “Enhanced mode locking of color-center lasers,” Opt. Lett. 14, 39–41 (1989).
[Crossref] [PubMed]

P. N. Kean, R. S. Grant, X. Zhu, D. W. Crust, D. Burns, and W. Sibbett, “Enhanced mode locking of colour-centre lasers by coupled-cavity feedback control,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD7.

Hall, K. L.

Haus, H. A.

J. Mark, L. Y. Liu, K. L. Hall, H. A. Haus, and E. P. Ippen, “Femtosecond pulse generation in a laser with a nonlinear external resonator,” Opt. Lett. 14, 48–50 (1989).
[Crossref] [PubMed]

H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. QE-22, 325–331 (1986).
[Crossref]

H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. QE-11, 323–330 (1975).
[Crossref]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).We use a more convenient phase reference for the scattering matrix of the mirror.

Ippen, E. P.

J. Mark, L. Y. Liu, K. L. Hall, H. A. Haus, and E. P. Ippen, “Femtosecond pulse generation in a laser with a nonlinear external resonator,” Opt. Lett. 14, 48–50 (1989).
[Crossref] [PubMed]

M. S. Stix and E. P. Ippen, “Pulse-shaping in passively mode-locked ring dye lasers,” IEEE J. Quantum Electron. QE-19, 520–525 (1983).
[Crossref]

Kean, P. N.

P. N. Kean, X. Zhu, D. W. Crust, R. S. Grant, N. Langford, and W. Sibbett, “Enhanced mode locking of color-center lasers,” Opt. Lett. 14, 39–41 (1989).
[Crossref] [PubMed]

P. N. Kean, R. S. Grant, X. Zhu, D. W. Crust, D. Burns, and W. Sibbett, “Enhanced mode locking of colour-centre lasers by coupled-cavity feedback control,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD7.

Langford, N.

Liu, L. Y.

Mark, J.

Mitschke, F. M.

F. M. Mitschke and L. F. Mollenauer, “Stabilizing the soliton laser,” IEEE J. Quantum Electron. QE-22, 2242–2250 (1986).
[Crossref]

Mollenauer, L. F.

F. M. Mitschke and L. F. Mollenauer, “Stabilizing the soliton laser,” IEEE J. Quantum Electron. QE-22, 2242–2250 (1986).
[Crossref]

L. F. Mollenauer and R. H. Stolen, “The soliton laser,” Opt. Lett. 9, 13–15 (1984).
[Crossref] [PubMed]

Nelson, B. P.

K. J. Blow and B. P. Nelson, “Non-soliton mode locking of an F-center laser with nonlinear external resonator,” in Ultrafast Phenomena VI, T. Yajima, K. Yoshihara, C. B. Harris, and S. Shionoya, eds., Vol. 48 of Springer Series in Chemical Physics (Springer-Verlag, Berlin, 1988), pp. 67–99.
[Crossref]

Oullette, F.

F. Oullette and M. Piché, “Ultrashort pulse reshaping with a nonlinear Fabry–Perot cavity matched to a train of short pulses,” J. Opt. Soc. Am. B 5, 1228–1236 (1988).
[Crossref]

F. Oullette and M. Piché, “Pulse shaping and passive mode-locking with a nonlinear Michelson interferometer,” Opt. Commun. 60, 99–103 (1986).
[Crossref]

Piché, M.

F. Oullette and M. Piché, “Ultrashort pulse reshaping with a nonlinear Fabry–Perot cavity matched to a train of short pulses,” J. Opt. Soc. Am. B 5, 1228–1236 (1988).
[Crossref]

F. Oullette and M. Piché, “Pulse shaping and passive mode-locking with a nonlinear Michelson interferometer,” Opt. Commun. 60, 99–103 (1986).
[Crossref]

Sibbett, W.

P. N. Kean, X. Zhu, D. W. Crust, R. S. Grant, N. Langford, and W. Sibbett, “Enhanced mode locking of color-center lasers,” Opt. Lett. 14, 39–41 (1989).
[Crossref] [PubMed]

P. N. Kean, R. S. Grant, X. Zhu, D. W. Crust, D. Burns, and W. Sibbett, “Enhanced mode locking of colour-centre lasers by coupled-cavity feedback control,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD7.

Silberberg, Y.

H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. QE-22, 325–331 (1986).
[Crossref]

Stix, M. S.

M. S. Stix and E. P. Ippen, “Pulse-shaping in passively mode-locked ring dye lasers,” IEEE J. Quantum Electron. QE-19, 520–525 (1983).
[Crossref]

Stolen, R. H.

Wood, D.

Zhu, X.

P. N. Kean, X. Zhu, D. W. Crust, R. S. Grant, N. Langford, and W. Sibbett, “Enhanced mode locking of color-center lasers,” Opt. Lett. 14, 39–41 (1989).
[Crossref] [PubMed]

P. N. Kean, R. S. Grant, X. Zhu, D. W. Crust, D. Burns, and W. Sibbett, “Enhanced mode locking of colour-centre lasers by coupled-cavity feedback control,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD7.

IEEE J. Quantum Electron. (4)

H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. QE-22, 325–331 (1986).
[Crossref]

M. S. Stix and E. P. Ippen, “Pulse-shaping in passively mode-locked ring dye lasers,” IEEE J. Quantum Electron. QE-19, 520–525 (1983).
[Crossref]

H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. QE-11, 323–330 (1975).
[Crossref]

F. M. Mitschke and L. F. Mollenauer, “Stabilizing the soliton laser,” IEEE J. Quantum Electron. QE-22, 2242–2250 (1986).
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Commun. (2)

L. E. Dahlstrom, “Passive mode locking and Q-switching of high power lasers by beams of the optical Kerr effect,” Opt. Commun. 5, 157–162 (1972).
[Crossref]

F. Oullette and M. Piché, “Pulse shaping and passive mode-locking with a nonlinear Michelson interferometer,” Opt. Commun. 60, 99–103 (1986).
[Crossref]

Opt. Lett. (3)

Other (3)

K. J. Blow and B. P. Nelson, “Non-soliton mode locking of an F-center laser with nonlinear external resonator,” in Ultrafast Phenomena VI, T. Yajima, K. Yoshihara, C. B. Harris, and S. Shionoya, eds., Vol. 48 of Springer Series in Chemical Physics (Springer-Verlag, Berlin, 1988), pp. 67–99.
[Crossref]

P. N. Kean, R. S. Grant, X. Zhu, D. W. Crust, D. Burns, and W. Sibbett, “Enhanced mode locking of colour-centre lasers by coupled-cavity feedback control,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), paper PD7.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).We use a more convenient phase reference for the scattering matrix of the mirror.

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Figures (14)

Fig. 1
Fig. 1

Schematic of laser cavity and auxiliary cavity.

Fig. 2
Fig. 2

The pulse shortening: ϕ = 0, Φ0 = π, r = 0.8, L = 0.3, τ0 = 1, no dispersion; (a) all pulses normalized to unity peak amplitude; (b) a1 normalized to unity peak amplitude.

Fig. 3
Fig. 3

The pulse shortening: ϕ = −π/2, Φ0 = π/2, r = 0.8, L = 0.3, τ0 = 1, no dispersion; (a) all pulses normalized to unity peak amplitude; (b) a1 normalized to unity peak amplitude.

Fig. 4
Fig. 4

The pulse shortening: ϕ = 0, Φ0 = π, r = 0.8, L = 0.3, D = 0.2, τ0 = 1.

Fig. 5
Fig. 5

The pulse shortening: ϕ = 0, Φ0 = π, r = 0.8, L = 0.3, D = −0.2, τ0 = 1.

Fig. 6
Fig. 6

The pulse shortening: ϕ = −π/2, Φ0 = π/2, r = 0.8, L = 0.3, D = 0.2, τ0 = 1.

Fig. 7
Fig. 7

The pulse shortening: ϕ = −π/2, Φ0 = π/2, r = 0.8, L = 0.3, D = −0.2, τ0 = 1.

Fig. 8
Fig. 8

The relative timing of modulation and pulse.

Fig. 9
Fig. 9

Phase of Cb/Ca as function of ϕ and construction of equilibrium point: r = 0.8, L = 0.3. The phase shifts ϕ and ψ are assumed proportional to frequency with equal proportionality constants.

Fig. 10
Fig. 10

Plot of ΔG and ΔB as functions of ϕ: r = 0.9, L = 0.3.

Fig. 11
Fig. 11

(Top) Power in laser cavity, (second trace) power in the fiber, (third trace) output power, (bottom trace) second harmonic.

Fig. 12
Fig. 12

Schematic of measurement. The length of the fiber is varied with a piezoelectric transducer (PZT).

Fig. 13
Fig. 13

Solid curve, pulse shortening per pass. Dashed curve, pulse lengthening per pass. Both in arbitrary units. See text and Eq. (6.2).

Fig. 14
Fig. 14

Spectrum of the APM output for l0 + 10λ (TA = −50 fsec) and l0 − 10λ (TA = +50 fsec).

Equations (54)

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b 1 = r a 1 + 1 r 2 a 2 ,
b 2 = 1 r 2 a 1 r a 2 .
a 2 ( t ) = L exp j { ϕ + κ [ | a 2 ( t ) | 2 | a 2 ( 0 ) | 2 ] } b 2 ( t ) = L exp [ j ( ϕ + Φ ) ] b 2 ( t ) ,
Φ κ [ | a 2 ( t ) | 2 | a 2 ( 0 ) | 2 ] .
b 1 = 1 1 r 2 { 1 + r L exp [ j ( ϕ + Φ ) ] } a 2 ,
a 1 = 1 1 r 2 { r + 1 L exp [ j ( ϕ + Φ ) ] } a 2 .
Γ = b 1 a 1 = 1 + r L exp [ j ( ϕ + Φ ) ] r + 1 L exp [ j ( ϕ + Φ ) ] .
Γ = { r + L ( 1 r 2 ) exp [ j ( ϕ + Φ ) ] } .
Γ = r + L ( 1 r 2 ) e j ϕ ( 1 j Φ ) .
| Γ | r + L ( 1 r 2 ) Φ .
D ̂ ( 1 + j D d 2 d t 2 ) ,
b 2 ( t ) = e j ϕ L Ô 1 a 2 ( t ) ,
Ô 1 = e j Φ ( 1 j D d 2 d t 2 ) .
g = g 0 1 + ( E / E s ) ,
g 1 + ( ω ω 0 + Δ ω ) 2 ω g 2 g ( 1 ( ω ω 0 + Δ ω ) 2 ω g 2 ) ,
j ( ω ω 0 ) d d t ,
g ( 1 + 1 ω g 2 d 2 d t 2 Δ ω 2 ω g 2 + 2 j Δ ω ω g 2 d d t ) .
loss l + M ω M 2 ( t Δ T ) 2 = l + M ω M 2 t 2 2 M ω M 2 Δ T t + M ω M 2 Δ T 2 ,
a 1 = M ̂ b 1 ,
M ̂ = e j ψ ( 1 + g l + m ̂ ) , = e j ψ [ 1 + g l + g ( 1 ω g 2 d 2 d t 2 Δ ω 2 ω g 2 + 2 j Δ ω ω g 2 d d t ) M ω M 2 ( t 2 2 Δ T t + Δ T 2 ) + T L d d t ] .
b 1 = Ĉ b a 2 ,
a 1 = Ĉ a a 2 ,
Ĉ b = 1 1 r 2 ( 1 + r L e j ϕ Ô 1 ) ,
Ĉ a = 1 1 r 2 ( r + 1 L e j ϕ Ô 1 ) .
Ô = ( 1 + j D d 2 d t 2 + j Φ 0 t 2 τ 2 + T A d d t ) 1 + ô ,
Ĉ b = C b ( 1 + d ̂ b ) ,
C b = 1 1 r 2 ( 1 + r L e j ϕ ) ,
d ̂ b = r L e j ϕ 1 + r L e j ϕ ô = ( G b + j B b ) ô .
Ĉ a = C a ( 1 + d ̂ a ) ,
C a 1 1 r 2 ( r + 1 L e j ϕ ) ,
d ̂ a = 1 L e j ϕ r + 1 L e j ϕ ô = ( G a + j B a ) ô .
b 1 + Δ b 1 = Ĉ b Ĉ a 1 M ̂ b 1 .
Δ b 1 = ( Ĉ b Ĉ a 1 M ̂ 1 ) b 1 = 0 .
d d t = t / τ 2 = ( t / τ 0 2 ) ( 1 + j γ )
d 2 d t 2 = t 2 / τ 4 1 / τ 2 = ( t 2 / τ 0 4 ) ( 1 + 2 j γ ) ( 1 + j γ ) / τ 0 2 .
C b C a e j ψ { 1 + g [ 1 1 ω g 2 τ 0 2 ( 1 + j γ ) ] l g Δ ω 2 ω g 2 M ω M 2 Δ T 2 } 1 = 0 .
2 j Δ ω g ω g 2 1 τ 2 + 2 M ω M 2 Δ T T L 1 τ 2 + ( Δ G + j Δ B ) T A 1 τ 2 = 0 ,
g ω g 2 1 τ 4 M ω M 2 ( Δ G + j Δ B ) j ( D 1 τ 4 + Φ 0 1 τ 0 2 ) = 0 .
arg ( C b / C a ) = ψ
| C b C a | [ 1 + g ( 1 Δ ω 2 ω g 2 1 ω g 2 τ 0 2 ) l M ω M 2 Δ T 2 ] 1 = 0 .
2 M ω M 2 Δ T T L τ 0 2 + Δ G T A τ 0 2 = 0 ,
2 Δ ω g ω g 2 1 τ 0 2 + Δ B T A τ 0 2 = 0 .
g ω g 2 1 τ 0 4 M ω M 2 + Δ B ( D 1 τ 0 4 + Φ 0 τ 0 2 ) + Δ G D 1 τ 0 4 2 γ = 0 ,
g ω g 2 1 τ 0 4 2 γ Δ G ( D 1 τ 0 4 + Φ 0 τ 0 2 ) + Δ B D 1 τ 0 4 2 γ = 0 .
2 M ω M 2 Δ T + Δ G T A τ 0 2 = 0 .
g ω g 2 1 τ 0 4 M ω M 2 + Δ B Φ 0 τ 0 2 = 0 .
M ω M 2 Δ B K τ 0 3
g ω g 2 τ 0 2 = Δ B ( D τ 0 2 + Φ 0 ) .
Δ ω τ 0 = T A / τ 0 2 ( D τ 0 2 + Φ 0 ) .
Δ b 1 = ( B 1 + δ B 1 ) exp [ j ( ψ + δ ψ ) ] exp [ j ( ω 0 + δ ω ) t ] × exp ( { 1 2 ( t + δ T ) 2 [ 1 τ 0 2 + δ ( 1 τ 0 2 ) ] [ 1 + j ( γ + δ γ ) ] } ) B 1 e j ψ exp ( j ω 0 t ) exp [ ( t 2 / 2 τ 0 2 ) ( 1 + j γ ) ] { δ B 1 j δ ψ B 1 + [ j δ ω t t δ T τ 0 2 ( 1 + j γ ) t 2 2 δ ( 1 τ 0 2 ) ( 1 + j γ ) t 2 2 τ 0 2 j δ γ ] B 1 } × exp ( j ψ ) exp ( j ω 0 t ) exp [ t 2 2 τ 0 2 ( 1 + j γ ) ] ,
T R d d t ( δ B 1 + j δ ψ B 1 ) = { C B C a e j ψ ( 1 1 ω g 2 τ 2 ) M ω M 2 Δ T 2 g Δ ω 2 ω g 2 1 } B 1 .
T R d d t [ δ T τ 0 2 ( 1 + j γ ) + j δ ω ] = 2 j Δ ω g ω g 2 1 τ 2 + 2 M ω M 2 Δ T T L τ 2 + ( Δ G + j Δ B ) T A τ 2 .
T R d d T [ j δ γ 2 τ 0 2 1 2 δ ( 1 τ 0 2 ) ( 1 + j γ ) ] = g ω g 2 1 τ 4 M ω M 2 ( Δ G + j Δ B ) j ( D τ 4 + Φ 0 τ 0 2 ) .
g ω g 2 + Δ B D > 0 .

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