Abstract

We predict cavity solitons in frequency divide-by-three (3ω=ω+2ω) optical parametric oscillators in the presence of an additional degenerate parametric process 2ω=ω+ω between signal and idler waves. We calculate the stability range of the solitons and investigate their stability properties.

© 2006 Optical Society of America

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  1. K. Koynov and S. Saltiel, "Nonlinear phase shift via multistep chi(2) cascading," Opt. Commun. 152, 96-100 (1998).
    [CrossRef]
  2. Y. S. Kivshar, T. J. Alexander, and S. Saltiel, "Spatial optical solitons resulting from multistep cascading," Opt. Lett. 24, 759-761 (1999).
    [CrossRef]
  3. A. Douillet and J.-J. Zondy, "Low-threshold, self-frequency-stabilized AgGaS2 continuous-wave subharmonic optical parametric oscillator," Opt. Lett. 23, 1259-1261 (1998).
    [CrossRef]
  4. J.-J. Zondy, D. Kolker, and F. N. C. Wong, "Dynamical signatures of self-phase-locking in a triply resonant optical parametric oscillator," Phys. Rev. Lett. 93, 043902 (2004).
    [CrossRef] [PubMed]
  5. S. Longhi, "Spiral waves in a class of optical parametric oscillators," Phys. Rev. E 63, 055202 (2001).
    [CrossRef]
  6. D. W. McLaughlin, J. V. Moloney, and A. C. Newell, "Solitary waves as fixed points of infinite-dimensional maps in an optical bistable ring cavity," Phys. Rev. Lett. 51, 75-78 (1983).
    [CrossRef]
  7. N. N. Rosanov and G. V. Khodova, "Autosolitons in bistable interferometers," Opt. Spectrosc. 65, 449-450 (1988).
  8. W. J. Firth and C. O. Weiss, "Cavity and feedback solitons," Opt. Photonics News 13, 54-58 (2002).
    [CrossRef]
  9. S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
    [CrossRef] [PubMed]
  10. G. S. McDonald and W. J. Firth, "Spatial solitary-wave optical memory," J. Opt. Soc. Am. B 7, 1328-1335 (1990).
    [CrossRef]
  11. N. N. Rosanov, "Diffractive autosolitons in nonlinear interferometers," J. Opt. Soc. Am. B 7, 1057-1065 (1990).
    [CrossRef]
  12. S. Fauve and O. Thual, "Solitary waves generated by subcritical instabilities in dissipative systems," Phys. Rev. Lett. 64, 282-284 (1990).
    [CrossRef] [PubMed]
  13. J. de Valcarcel, E. Roldan, and K. Staliunas, "Cavity solitons in nondegenerate optical parametric oscillation," Opt. Commun. 181, 207-213 (2000).
    [CrossRef]
  14. S. Longhi, "Hydrodynamic equation model for degenerate optical parametric oscillators," J. Mod. Opt. 43, 1089-1094 (1996).
  15. P. Lodahl and M. Saffman, "Pattern formation in singly resonant second-harmonic generation with competing parametric oscillation," Phys. Rev. A 60, 3251-3261 (1999).
    [CrossRef]
  16. P. Coullet and K. Emilsson, "Strong resonances of spatially distributed oscillators: a laboratory to study patterns and defects," Physica D 61, 119-131 (1992).
    [CrossRef]
  17. R. Gallego, D. Walgraef, M. San Miguel, and R. Toral, "Transition from oscillatory to excitable regime in a system forced at three times its natural frequency," Phys. Rev. E 64, 056218 (2001).
    [CrossRef]
  18. A. L. Lin, M. Bertram, K. Martinez, H. L. Swinney, A. Ardelea, and G. F. Carey, "Resonant phase patterns in a reaction-diffusion system," Phys. Rev. Lett. 84, 4240-4243 (2000).
    [CrossRef] [PubMed]

2004

J.-J. Zondy, D. Kolker, and F. N. C. Wong, "Dynamical signatures of self-phase-locking in a triply resonant optical parametric oscillator," Phys. Rev. Lett. 93, 043902 (2004).
[CrossRef] [PubMed]

2002

W. J. Firth and C. O. Weiss, "Cavity and feedback solitons," Opt. Photonics News 13, 54-58 (2002).
[CrossRef]

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

2001

S. Longhi, "Spiral waves in a class of optical parametric oscillators," Phys. Rev. E 63, 055202 (2001).
[CrossRef]

R. Gallego, D. Walgraef, M. San Miguel, and R. Toral, "Transition from oscillatory to excitable regime in a system forced at three times its natural frequency," Phys. Rev. E 64, 056218 (2001).
[CrossRef]

2000

A. L. Lin, M. Bertram, K. Martinez, H. L. Swinney, A. Ardelea, and G. F. Carey, "Resonant phase patterns in a reaction-diffusion system," Phys. Rev. Lett. 84, 4240-4243 (2000).
[CrossRef] [PubMed]

J. de Valcarcel, E. Roldan, and K. Staliunas, "Cavity solitons in nondegenerate optical parametric oscillation," Opt. Commun. 181, 207-213 (2000).
[CrossRef]

1999

P. Lodahl and M. Saffman, "Pattern formation in singly resonant second-harmonic generation with competing parametric oscillation," Phys. Rev. A 60, 3251-3261 (1999).
[CrossRef]

Y. S. Kivshar, T. J. Alexander, and S. Saltiel, "Spatial optical solitons resulting from multistep cascading," Opt. Lett. 24, 759-761 (1999).
[CrossRef]

1998

1996

S. Longhi, "Hydrodynamic equation model for degenerate optical parametric oscillators," J. Mod. Opt. 43, 1089-1094 (1996).

1992

P. Coullet and K. Emilsson, "Strong resonances of spatially distributed oscillators: a laboratory to study patterns and defects," Physica D 61, 119-131 (1992).
[CrossRef]

1990

1988

N. N. Rosanov and G. V. Khodova, "Autosolitons in bistable interferometers," Opt. Spectrosc. 65, 449-450 (1988).

1983

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, "Solitary waves as fixed points of infinite-dimensional maps in an optical bistable ring cavity," Phys. Rev. Lett. 51, 75-78 (1983).
[CrossRef]

Alexander, T. J.

Ardelea, A.

A. L. Lin, M. Bertram, K. Martinez, H. L. Swinney, A. Ardelea, and G. F. Carey, "Resonant phase patterns in a reaction-diffusion system," Phys. Rev. Lett. 84, 4240-4243 (2000).
[CrossRef] [PubMed]

Balle, S.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Barland, S.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Bertram, M.

A. L. Lin, M. Bertram, K. Martinez, H. L. Swinney, A. Ardelea, and G. F. Carey, "Resonant phase patterns in a reaction-diffusion system," Phys. Rev. Lett. 84, 4240-4243 (2000).
[CrossRef] [PubMed]

Brambilla, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Carey, G. F.

A. L. Lin, M. Bertram, K. Martinez, H. L. Swinney, A. Ardelea, and G. F. Carey, "Resonant phase patterns in a reaction-diffusion system," Phys. Rev. Lett. 84, 4240-4243 (2000).
[CrossRef] [PubMed]

Coullet, P.

P. Coullet and K. Emilsson, "Strong resonances of spatially distributed oscillators: a laboratory to study patterns and defects," Physica D 61, 119-131 (1992).
[CrossRef]

de Valcarcel, J.

J. de Valcarcel, E. Roldan, and K. Staliunas, "Cavity solitons in nondegenerate optical parametric oscillation," Opt. Commun. 181, 207-213 (2000).
[CrossRef]

Douillet, A.

Emilsson, K.

P. Coullet and K. Emilsson, "Strong resonances of spatially distributed oscillators: a laboratory to study patterns and defects," Physica D 61, 119-131 (1992).
[CrossRef]

Fauve, S.

S. Fauve and O. Thual, "Solitary waves generated by subcritical instabilities in dissipative systems," Phys. Rev. Lett. 64, 282-284 (1990).
[CrossRef] [PubMed]

Firth, W. J.

W. J. Firth and C. O. Weiss, "Cavity and feedback solitons," Opt. Photonics News 13, 54-58 (2002).
[CrossRef]

G. S. McDonald and W. J. Firth, "Spatial solitary-wave optical memory," J. Opt. Soc. Am. B 7, 1328-1335 (1990).
[CrossRef]

Gallego, R.

R. Gallego, D. Walgraef, M. San Miguel, and R. Toral, "Transition from oscillatory to excitable regime in a system forced at three times its natural frequency," Phys. Rev. E 64, 056218 (2001).
[CrossRef]

Giudici, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Jäger, R.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Khodova, G. V.

N. N. Rosanov and G. V. Khodova, "Autosolitons in bistable interferometers," Opt. Spectrosc. 65, 449-450 (1988).

Kivshar, Y. S.

Knödl, T.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Kolker, D.

J.-J. Zondy, D. Kolker, and F. N. C. Wong, "Dynamical signatures of self-phase-locking in a triply resonant optical parametric oscillator," Phys. Rev. Lett. 93, 043902 (2004).
[CrossRef] [PubMed]

Koynov, K.

K. Koynov and S. Saltiel, "Nonlinear phase shift via multistep chi(2) cascading," Opt. Commun. 152, 96-100 (1998).
[CrossRef]

Lin, A. L.

A. L. Lin, M. Bertram, K. Martinez, H. L. Swinney, A. Ardelea, and G. F. Carey, "Resonant phase patterns in a reaction-diffusion system," Phys. Rev. Lett. 84, 4240-4243 (2000).
[CrossRef] [PubMed]

Lodahl, P.

P. Lodahl and M. Saffman, "Pattern formation in singly resonant second-harmonic generation with competing parametric oscillation," Phys. Rev. A 60, 3251-3261 (1999).
[CrossRef]

Longhi, S.

S. Longhi, "Spiral waves in a class of optical parametric oscillators," Phys. Rev. E 63, 055202 (2001).
[CrossRef]

S. Longhi, "Hydrodynamic equation model for degenerate optical parametric oscillators," J. Mod. Opt. 43, 1089-1094 (1996).

Lugiato, L. A.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Maggipinto, T.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Martinez, K.

A. L. Lin, M. Bertram, K. Martinez, H. L. Swinney, A. Ardelea, and G. F. Carey, "Resonant phase patterns in a reaction-diffusion system," Phys. Rev. Lett. 84, 4240-4243 (2000).
[CrossRef] [PubMed]

McDonald, G. S.

McLaughlin, D. W.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, "Solitary waves as fixed points of infinite-dimensional maps in an optical bistable ring cavity," Phys. Rev. Lett. 51, 75-78 (1983).
[CrossRef]

Miller, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Moloney, J. V.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, "Solitary waves as fixed points of infinite-dimensional maps in an optical bistable ring cavity," Phys. Rev. Lett. 51, 75-78 (1983).
[CrossRef]

Newell, A. C.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, "Solitary waves as fixed points of infinite-dimensional maps in an optical bistable ring cavity," Phys. Rev. Lett. 51, 75-78 (1983).
[CrossRef]

Roldan, E.

J. de Valcarcel, E. Roldan, and K. Staliunas, "Cavity solitons in nondegenerate optical parametric oscillation," Opt. Commun. 181, 207-213 (2000).
[CrossRef]

Rosanov, N. N.

N. N. Rosanov, "Diffractive autosolitons in nonlinear interferometers," J. Opt. Soc. Am. B 7, 1057-1065 (1990).
[CrossRef]

N. N. Rosanov and G. V. Khodova, "Autosolitons in bistable interferometers," Opt. Spectrosc. 65, 449-450 (1988).

Saffman, M.

P. Lodahl and M. Saffman, "Pattern formation in singly resonant second-harmonic generation with competing parametric oscillation," Phys. Rev. A 60, 3251-3261 (1999).
[CrossRef]

Saltiel, S.

Y. S. Kivshar, T. J. Alexander, and S. Saltiel, "Spatial optical solitons resulting from multistep cascading," Opt. Lett. 24, 759-761 (1999).
[CrossRef]

K. Koynov and S. Saltiel, "Nonlinear phase shift via multistep chi(2) cascading," Opt. Commun. 152, 96-100 (1998).
[CrossRef]

San Miguel, M.

R. Gallego, D. Walgraef, M. San Miguel, and R. Toral, "Transition from oscillatory to excitable regime in a system forced at three times its natural frequency," Phys. Rev. E 64, 056218 (2001).
[CrossRef]

Spinelli, L.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Staliunas, K.

J. de Valcarcel, E. Roldan, and K. Staliunas, "Cavity solitons in nondegenerate optical parametric oscillation," Opt. Commun. 181, 207-213 (2000).
[CrossRef]

Swinney, H. L.

A. L. Lin, M. Bertram, K. Martinez, H. L. Swinney, A. Ardelea, and G. F. Carey, "Resonant phase patterns in a reaction-diffusion system," Phys. Rev. Lett. 84, 4240-4243 (2000).
[CrossRef] [PubMed]

Thual, O.

S. Fauve and O. Thual, "Solitary waves generated by subcritical instabilities in dissipative systems," Phys. Rev. Lett. 64, 282-284 (1990).
[CrossRef] [PubMed]

Tissoni, G.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Toral, R.

R. Gallego, D. Walgraef, M. San Miguel, and R. Toral, "Transition from oscillatory to excitable regime in a system forced at three times its natural frequency," Phys. Rev. E 64, 056218 (2001).
[CrossRef]

Tredicce, J. R.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Walgraef, D.

R. Gallego, D. Walgraef, M. San Miguel, and R. Toral, "Transition from oscillatory to excitable regime in a system forced at three times its natural frequency," Phys. Rev. E 64, 056218 (2001).
[CrossRef]

Weiss, C. O.

W. J. Firth and C. O. Weiss, "Cavity and feedback solitons," Opt. Photonics News 13, 54-58 (2002).
[CrossRef]

Wong, F. N. C.

J.-J. Zondy, D. Kolker, and F. N. C. Wong, "Dynamical signatures of self-phase-locking in a triply resonant optical parametric oscillator," Phys. Rev. Lett. 93, 043902 (2004).
[CrossRef] [PubMed]

Zondy, J.-J.

J.-J. Zondy, D. Kolker, and F. N. C. Wong, "Dynamical signatures of self-phase-locking in a triply resonant optical parametric oscillator," Phys. Rev. Lett. 93, 043902 (2004).
[CrossRef] [PubMed]

A. Douillet and J.-J. Zondy, "Low-threshold, self-frequency-stabilized AgGaS2 continuous-wave subharmonic optical parametric oscillator," Opt. Lett. 23, 1259-1261 (1998).
[CrossRef]

J. Mod. Opt.

S. Longhi, "Hydrodynamic equation model for degenerate optical parametric oscillators," J. Mod. Opt. 43, 1089-1094 (1996).

J. Opt. Soc. Am. B

Nature

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

Opt. Commun.

J. de Valcarcel, E. Roldan, and K. Staliunas, "Cavity solitons in nondegenerate optical parametric oscillation," Opt. Commun. 181, 207-213 (2000).
[CrossRef]

K. Koynov and S. Saltiel, "Nonlinear phase shift via multistep chi(2) cascading," Opt. Commun. 152, 96-100 (1998).
[CrossRef]

Opt. Lett.

Opt. Photonics News

W. J. Firth and C. O. Weiss, "Cavity and feedback solitons," Opt. Photonics News 13, 54-58 (2002).
[CrossRef]

Opt. Spectrosc.

N. N. Rosanov and G. V. Khodova, "Autosolitons in bistable interferometers," Opt. Spectrosc. 65, 449-450 (1988).

Phys. Rev. A

P. Lodahl and M. Saffman, "Pattern formation in singly resonant second-harmonic generation with competing parametric oscillation," Phys. Rev. A 60, 3251-3261 (1999).
[CrossRef]

Phys. Rev. E

R. Gallego, D. Walgraef, M. San Miguel, and R. Toral, "Transition from oscillatory to excitable regime in a system forced at three times its natural frequency," Phys. Rev. E 64, 056218 (2001).
[CrossRef]

S. Longhi, "Spiral waves in a class of optical parametric oscillators," Phys. Rev. E 63, 055202 (2001).
[CrossRef]

Phys. Rev. Lett.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, "Solitary waves as fixed points of infinite-dimensional maps in an optical bistable ring cavity," Phys. Rev. Lett. 51, 75-78 (1983).
[CrossRef]

J.-J. Zondy, D. Kolker, and F. N. C. Wong, "Dynamical signatures of self-phase-locking in a triply resonant optical parametric oscillator," Phys. Rev. Lett. 93, 043902 (2004).
[CrossRef] [PubMed]

A. L. Lin, M. Bertram, K. Martinez, H. L. Swinney, A. Ardelea, and G. F. Carey, "Resonant phase patterns in a reaction-diffusion system," Phys. Rev. Lett. 84, 4240-4243 (2000).
[CrossRef] [PubMed]

S. Fauve and O. Thual, "Solitary waves generated by subcritical instabilities in dissipative systems," Phys. Rev. Lett. 64, 282-284 (1990).
[CrossRef] [PubMed]

Physica D

P. Coullet and K. Emilsson, "Strong resonances of spatially distributed oscillators: a laboratory to study patterns and defects," Physica D 61, 119-131 (1992).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Stability balloon (gray areas) of CSs as found by numerical integration of Eq. (2) in (a) one spatial dimension and in (b) two spatial dimensions with parameters α = 0.5 , β R = 1 , β I = 0 , δ R δ I = 0.1 , and δ I = 0.002 . The loose stability of the CSs by a Hopf bifurcation by crossing the boundaries is indicated by crosses. Squares show the boundary for CS existence. In (a), the solid circles indicate Maxwellian states characterized by the balance between trivial zero and nonzero homogeneous states. The two plots below (a) show the amplitude (left plot) and phase (right plot) profiles of the solitons for parameter values corresponding to point A inside the stability balloons. In the 2D case, CSs are radially symmetric and the figure shows the amplitude and phase profiles on a radial line crossing the center of the soliton. In (a), point A corresponds to ν = 0.19 and p = 0.66 , whereas point B corresponds to ν = 0.19 and p = 0.70 .

Fig. 2
Fig. 2

(a) Temporal 1D soliton evolution (peak intensity) in the Hopf instability region [point B in Fig. 1a]. (b) Diagram of numerically computed eigenvalues λ in the linearized system of CSs for parameter values inside the stability balloon (squares, point A) and outside the stability balloon (circles, point B).

Fig. 3
Fig. 3

Peak intensity of the 1D soliton (solid squares) versus nonlinear gain parameter p for fixed detuning ν = 0.19 ; the other parameters are as in Fig. 1a. The intensity of the plane-wave solution is given by the solid (dashed) curves as corresponding to the stable (unstable) branch. The open circles depict the numerically calculated growth and decay rate of the Hopf oscillations (definition in the text).

Fig. 4
Fig. 4

Two amplitude profiles of the 2D soliton on a radial line crossing the center in the Hopf unstable regime, taken at two times corresponding to the minimal and maximal amplitude of the oscillations. ν = 0.34 , p = 1.94 ; other parameters are as in Fig. 1. The oscillation period is Δ t = 41 in this case.

Fig. 5
Fig. 5

Diagram of (a) amplitude and (b) phase of stable CSs as obtained by numerical integration of the microscopic Eqs. (1) for parameter values μ = 0.96 , σ = 10 , a 1 = 0.001 , a 2 = 0.002 , γ 1 = γ 2 = 1 , Δ 1 = Δ 2 = 0 . Solid curves, profiles of the signal wave; dashed curves, profiles of the idler wave.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

t A 1 = γ 1 [ ( 1 + i Δ 1 ) A 1 + i a 1 2 A 1 + μ A 2 * + σ A 1 * A 2 A 2 2 A 1 ] ,
t A 2 = γ 2 [ ( 1 + i Δ 2 ) A 2 + i a 2 2 A 2 + μ A 1 * σ 2 A 1 2 A 1 2 A 2 ] .
τ A = ( α + i ν ) A + ( δ R + i δ I ) 2 A + p A * 2 ( β R + i β I ) A 2 A .
F [ B 1 B 2 ] = λ [ B 1 B 2 ] ,
F = [ α + i ν + ( δ R + i δ I ) 2 x 2 2 ( β R + i β I ) A s 2 2 p A s * ( β R + i β I ) A s 2 2 p A s ( β R i β I ) A s * 2 α i ν + ( δ R i δ I ) 2 x 2 2 ( β R i β I ) A s 2 ] .

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