Abstract

It is shown theoretically that a degenerate optical parametric oscillator with a frequency limiter and continuous-wave pumping can generate stable dark solitons when it operates near antiresonance. These solitons are supported by the combined effect of pump depletion and spectral filtering, and their stability is ensured by the phase-dependent nature of the parametric gain.

© 1996 Optical Society of America

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References

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  1. See, for instance, the feature on optical parametric oscillation and amplification, J. Opt. Soc. Am. B 10, 2147–2243 (1993), and the feature on optical parametric devices, J. Opt. Soc. Am. B 12, 2083–2320 (1995).
  2. L. Wu, H. J. Kimble, J. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986); A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
    [CrossRef] [PubMed]
  3. J. Falk, IEEE J. Quantum Electron. QE-7, 230 (1971).
    [CrossRef]
  4. L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, R. J. Horowicz, Nuovo Cimento D 10, 959 (1988).
    [CrossRef]
  5. R. C. Eckardt, C. D. Nabors, W. J. Kozlovsky, R. L. Byer, J. Opt. Soc. Am. B 8, 646 (1991); T. Debuisschert, A. Sizmann, E. Giacobino, C. Fabre, J. Opt. Soc. Am. B 10, 1668 (1993); S. T. Yang, R. C. Eckardt, R. L. Byer, J. Opt. Soc. Am. B 10, 1684 (1993); D. Lee, N. C. Wang, J. Opt. Soc. Am. B 10, 1659 (1993).
    [CrossRef]
  6. C. Richy, K. I. Petsas, E. Giacobino, C. Fabre, L. Lugiato, J. Opt. Soc. Am. B 12, 456 (1995).
    [CrossRef]
  7. G-L. Oppo, M. Brambilla, L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); K. Staliunas, J. Mod. Opt. 42, 1261 (1995).
    [CrossRef] [PubMed]
  8. S. Longhi, Opt. Lett. 20, 695 (1995); S. Longhi, A. Geraci, Appl. Phys. Lett. 67, 3060 (1995).
    [CrossRef] [PubMed]
  9. A. Mecozzi, W. L. Kath, P. Kumar, C. G. Goedde, Opt. Lett. 19, 2050 (1994).
    [CrossRef] [PubMed]
  10. For a review of dark solitons in nonlinear optics, see, for instance, Y. S. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993); for recent developments on dark solitons in dispersive quadratic media, seeA. V. Buryak, Y. S. Kivshar, Opt. Lett. 20, 834 (1995), and references therein.
    [CrossRef] [PubMed]
  11. H. A. Haus, J. Appl. Phys. 46, 3049 (1975); H. A. Haus, J. G. Fujimoto, E. P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991).
    [CrossRef]
  12. S. Longhi, “Hydrodynamic equation model for degenerate optical parametric oscillators,”J. Mod. Opt. (to be published).
  13. M. C. Cross, P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
    [CrossRef]
  14. L. N. Bulaevskii, V. L. Ginzburg, Sov. Phys. JETP 18, 530 (1964); S. Sarker, S. E. Trullinger, A. R. Bishop, Phys. Lett. A 59, 255 (1976).
    [CrossRef]
  15. G. Pöschl, E. Teller, Z. Phys. 83, 143 (1933); P. M. Morse, H. Feshbach, in Methods of Theoretical Physics, P. M. Morse, H. Feshbach, eds. (McGraw-Hill, New York, 1953), Vol. 2, p. 1650.
    [CrossRef]
  16. Strictly speaking, in the nonvariational case, i.e., for ν ≠ 0, the eigenvalue Λ1 should belong to the continuous spectrum because, although ξ2 vanishes at infinity, when Eq. (10a) is solved its solution ξ1 is not in general a bound state.

1995

1994

G-L. Oppo, M. Brambilla, L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); K. Staliunas, J. Mod. Opt. 42, 1261 (1995).
[CrossRef] [PubMed]

A. Mecozzi, W. L. Kath, P. Kumar, C. G. Goedde, Opt. Lett. 19, 2050 (1994).
[CrossRef] [PubMed]

1993

For a review of dark solitons in nonlinear optics, see, for instance, Y. S. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993); for recent developments on dark solitons in dispersive quadratic media, seeA. V. Buryak, Y. S. Kivshar, Opt. Lett. 20, 834 (1995), and references therein.
[CrossRef] [PubMed]

See, for instance, the feature on optical parametric oscillation and amplification, J. Opt. Soc. Am. B 10, 2147–2243 (1993), and the feature on optical parametric devices, J. Opt. Soc. Am. B 12, 2083–2320 (1995).

M. C. Cross, P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

1991

1988

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, R. J. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

1986

L. Wu, H. J. Kimble, J. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986); A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

1975

H. A. Haus, J. Appl. Phys. 46, 3049 (1975); H. A. Haus, J. G. Fujimoto, E. P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991).
[CrossRef]

1971

J. Falk, IEEE J. Quantum Electron. QE-7, 230 (1971).
[CrossRef]

1964

L. N. Bulaevskii, V. L. Ginzburg, Sov. Phys. JETP 18, 530 (1964); S. Sarker, S. E. Trullinger, A. R. Bishop, Phys. Lett. A 59, 255 (1976).
[CrossRef]

1933

G. Pöschl, E. Teller, Z. Phys. 83, 143 (1933); P. M. Morse, H. Feshbach, in Methods of Theoretical Physics, P. M. Morse, H. Feshbach, eds. (McGraw-Hill, New York, 1953), Vol. 2, p. 1650.
[CrossRef]

Brambilla, M.

G-L. Oppo, M. Brambilla, L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); K. Staliunas, J. Mod. Opt. 42, 1261 (1995).
[CrossRef] [PubMed]

Bulaevskii, L. N.

L. N. Bulaevskii, V. L. Ginzburg, Sov. Phys. JETP 18, 530 (1964); S. Sarker, S. E. Trullinger, A. R. Bishop, Phys. Lett. A 59, 255 (1976).
[CrossRef]

Byer, R. L.

Cross, M. C.

M. C. Cross, P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

Eckardt, R. C.

Fabre, C.

C. Richy, K. I. Petsas, E. Giacobino, C. Fabre, L. Lugiato, J. Opt. Soc. Am. B 12, 456 (1995).
[CrossRef]

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, R. J. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

Falk, J.

J. Falk, IEEE J. Quantum Electron. QE-7, 230 (1971).
[CrossRef]

Giacobino, E.

C. Richy, K. I. Petsas, E. Giacobino, C. Fabre, L. Lugiato, J. Opt. Soc. Am. B 12, 456 (1995).
[CrossRef]

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, R. J. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

Ginzburg, V. L.

L. N. Bulaevskii, V. L. Ginzburg, Sov. Phys. JETP 18, 530 (1964); S. Sarker, S. E. Trullinger, A. R. Bishop, Phys. Lett. A 59, 255 (1976).
[CrossRef]

Goedde, C. G.

Hall, J.

L. Wu, H. J. Kimble, J. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986); A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Haus, H. A.

H. A. Haus, J. Appl. Phys. 46, 3049 (1975); H. A. Haus, J. G. Fujimoto, E. P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991).
[CrossRef]

Hohenberg, P. C.

M. C. Cross, P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

Horowicz, R. J.

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, R. J. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

Kath, W. L.

Kimble, H. J.

L. Wu, H. J. Kimble, J. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986); A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Kivshar, Y. S.

For a review of dark solitons in nonlinear optics, see, for instance, Y. S. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993); for recent developments on dark solitons in dispersive quadratic media, seeA. V. Buryak, Y. S. Kivshar, Opt. Lett. 20, 834 (1995), and references therein.
[CrossRef] [PubMed]

Kozlovsky, W. J.

Kumar, P.

Longhi, S.

S. Longhi, Opt. Lett. 20, 695 (1995); S. Longhi, A. Geraci, Appl. Phys. Lett. 67, 3060 (1995).
[CrossRef] [PubMed]

S. Longhi, “Hydrodynamic equation model for degenerate optical parametric oscillators,”J. Mod. Opt. (to be published).

Lugiato, L.

Lugiato, L. A.

G-L. Oppo, M. Brambilla, L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); K. Staliunas, J. Mod. Opt. 42, 1261 (1995).
[CrossRef] [PubMed]

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, R. J. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

Mecozzi, A.

Nabors, C. D.

Oldano, C.

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, R. J. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

Oppo, G-L.

G-L. Oppo, M. Brambilla, L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); K. Staliunas, J. Mod. Opt. 42, 1261 (1995).
[CrossRef] [PubMed]

Petsas, K. I.

Pöschl, G.

G. Pöschl, E. Teller, Z. Phys. 83, 143 (1933); P. M. Morse, H. Feshbach, in Methods of Theoretical Physics, P. M. Morse, H. Feshbach, eds. (McGraw-Hill, New York, 1953), Vol. 2, p. 1650.
[CrossRef]

Richy, C.

Teller, E.

G. Pöschl, E. Teller, Z. Phys. 83, 143 (1933); P. M. Morse, H. Feshbach, in Methods of Theoretical Physics, P. M. Morse, H. Feshbach, eds. (McGraw-Hill, New York, 1953), Vol. 2, p. 1650.
[CrossRef]

Wu, H.

L. Wu, H. J. Kimble, J. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986); A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Wu, L.

L. Wu, H. J. Kimble, J. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986); A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

IEEE J. Quantum Electron.

J. Falk, IEEE J. Quantum Electron. QE-7, 230 (1971).
[CrossRef]

For a review of dark solitons in nonlinear optics, see, for instance, Y. S. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993); for recent developments on dark solitons in dispersive quadratic media, seeA. V. Buryak, Y. S. Kivshar, Opt. Lett. 20, 834 (1995), and references therein.
[CrossRef] [PubMed]

J. Appl. Phys.

H. A. Haus, J. Appl. Phys. 46, 3049 (1975); H. A. Haus, J. G. Fujimoto, E. P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991).
[CrossRef]

J. Opt. Soc. Am. B

Nuovo Cimento D

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, R. J. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

Opt. Lett.

Phys. Rev. A

G-L. Oppo, M. Brambilla, L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); K. Staliunas, J. Mod. Opt. 42, 1261 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett.

L. Wu, H. J. Kimble, J. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986); A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Rev. Mod. Phys.

M. C. Cross, P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

Sov. Phys. JETP

L. N. Bulaevskii, V. L. Ginzburg, Sov. Phys. JETP 18, 530 (1964); S. Sarker, S. E. Trullinger, A. R. Bishop, Phys. Lett. A 59, 255 (1976).
[CrossRef]

Z. Phys.

G. Pöschl, E. Teller, Z. Phys. 83, 143 (1933); P. M. Morse, H. Feshbach, in Methods of Theoretical Physics, P. M. Morse, H. Feshbach, eds. (McGraw-Hill, New York, 1953), Vol. 2, p. 1650.
[CrossRef]

Other

Strictly speaking, in the nonvariational case, i.e., for ν ≠ 0, the eigenvalue Λ1 should belong to the continuous spectrum because, although ξ2 vanishes at infinity, when Eq. (10a) is solved its solution ξ1 is not in general a bound state.

S. Longhi, “Hydrodynamic equation model for degenerate optical parametric oscillators,”J. Mod. Opt. (to be published).

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Figures (1)

Fig. 1
Fig. 1

Characteristic profile of the dark-pulse solution u of Eq. (7) (solid curve) and corresponding intensity (dashed curve) for parameter values ν = 0 and C = 2. The dotted lines represent the two stable homogeneous solutions u± of Eq. (7), which correspond to the asymptotic tails of the solitary wave.

Equations (14)

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( z + 1 c t ) A 1 = σ A 1 * A 0 + 1 L ω g 2 t 2 A 1 ,
( z + 1 c t ) A 0 = σ 2 A 1 2 ,
A 0 ( 0 , t ) = E p , A 1 ( 0 , t ) = r A 1 ( L , t ) exp ( i k 0 L ) ,
z a ± = ( Λ c + Λ 2 L ω g 2 ) a ± + σ E p a .
r 2 ρ 2 2 r ρ cos ( k 0 L ) cosh γ + 1 = 0 ,
Ω m = m π c / L ,
cosh γ m = 1 + r 2 exp ( 2 Ω m 2 / ω g 2 ) 2 ( 1 ) m cos ( k 0 L ) r exp ( Ω m 2 / ω g 2 ) ,
u ( η + Δ ) = [ u + ( λ + i θ + λ η 2 ) u + γ u * λ | u | 2 u ] ,
T u = ( 1 + i ν ) u + C u * + η 2 u | u | 2 u ,
u ( T , ) = u ( T , ) ,
u ( η ) = 2 β tanh ( β η ) exp ( i φ ) ,
2 β 2 = 1 + ( C 2 ν 2 ) 1 / 2 , sin ( 2 φ ) = ν / C ,
Λ ξ 1 = ( 1 + C cos 2 φ ) ξ 1 2 ν ξ 2 + η 2 ξ 1 6 β 2 tanh 2 ( β η ) ξ 1 ,
Λ ξ 2 = ( 1 C cos 2 φ ) ξ 2 + η 2 ξ 2 2 β 2 tanh 2 ( β η ) ξ 2 ,

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