Abstract

We investigate the formation of transverse patterns in a doubly resonant degenerate optical parametric oscillator. Extending previous work, we treat the more realistic case of a spherical mirror cavity with a finite–sized input pump field. Using numerical simulations in real space, we determine the conditions on the cavity geometry, pump size and detunings for which pattern formation occurs; we find multistability of different types of optical patterns. Below threshold, we analyze the dependence of the quantum image on the width of the input field, in the near and in the far field.

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Errata

M. Marte, H. Ritsch, K. Petsas, A. Gatti, L. Lugiato, C. Fabre, and D. Leduc, "Spatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effects: errata," Opt. Express 3, 476-476 (1998)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-3-11-476

References

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  1. Special issue on x2 second order nonlinear optics, from fundamentals to applications, edited by C. Fabre and J.-P. Pocholle, in JEOS B: J. Quantum and Semiclassical Opt. 9, 2 (1997).
  2. M. A. M. Marte, H. Ritsch, L. A. Lugiato and C. Fabre, "Simultaneous multimode optical parametric oscillations in a triply resonant cavity," Acta Physica Slovaca 47, 233 (1997).
  3. G. S. Agarwal and S. D. Das Gupta, "Model for mode hopping in optical parametric oscillators," J. Opt. Soc. Am. B 14, 2174 (1997).
    [CrossRef]
  4. C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti and L. A. Lugiato, "Transverse effects and mode couplings in optical parametric oscillators," submitted to Appl. Phys. B, Special Issue on Optical Parametric Oscillators, edited by. J. Mlynek and S. Schiller.
  5. G.-L. Oppo, M. Brambilla and L. A. Lugiato, "Formation and evolution of roll patterns in optical parametric oscillators," Phys. Rev. A 49, 2028 (1994).
    [CrossRef] [PubMed]
  6. A. Gatti and L.A. Lugiato, "Quantum images and critical fluctuations in the optical parametric oscillator below threshold," Phys. Rev. A 52, 1675 (1995).
    [CrossRef] [PubMed]
  7. A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo and S. M. Barnett, "Langevin treatment of quantum uctuations and optical patterns in optical parametric oscillators below threshold," Phys. Rev. A 56, 877 (1997).
    [CrossRef]
  8. A. Gatti, L. A. Lugiato, G.-L. Oppo, R. Martin, P. Di Trapani and A. Berzanskis, "From quantum to classical images," Opt. Express 1, 21 (1997), http://epubs.osa.org/oearchive/source/1968.htm.
    [CrossRef] [PubMed]
  9. C. Schwob and C. Fabre, "Squeezing and quantum correlations in multimode optical parametric oscillators," preprint, to be submitted to JEOS B: J. Quantum and Semiclassical Opt.
  10. E. Lantz and F. Devaux, "Parametric amplification of images," JEOS B: J. Quantum and Semi-classical Opt. 9, 279 (1997).
  11. M. I. Kolobov and L. A. Lugiato, "Noiseless amplification of optical images," Phys. Rev. A 52, 4930 (1995).
    [CrossRef] [PubMed]
  12. L. A. Lugiato and I. Marzoli, "Quantum spatial correlations in the optical parametric oscillator with spherical mirrors," Phys. Rev. A 52, 4886 (1995).
    [CrossRef] [PubMed]
  13. L. A. Lugiato, S. M. Barnett, A. Gatti, I. Marzoli, G.-L. Oppo and H. Wiedemann, "Quantum Images," J. of Nonlinear Optical Phys. and Materials 5, 809 (1996).
    [CrossRef]
  14. L. A. Lugiato and P. Grangier, "Improving quantum-noise reduction with spatially multimode squeezed light," J. Opt. Soc. Am. B 14, 225 (1997).
    [CrossRef]
  15. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes; the art of scientific computing, Cambridge University Press, Cambridge (1986).
  16. M. SanMiguel and R. Toral, Instabilities and Nonequlibrium structures, VI, Kluwer Academic Pub. (1997).
  17. L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino and R. J. Horowicz, "Bistability, self-pulsing and chaos in optical parametric oscillators," Il Nuovo Cimento 10 D, 959 (1988).
  18. K. I. Petsas, A. Gatti and L. A. Lugiato, "Quantum images in optical parametric oscillators with spherical mirrors and gaussian pump," submitted to JEOS B: J. Quantum and Semiclassical Opt.
  19. L. A. Lugiato, A. Gatti, H. Ritsch, I. Marzoli and G.-L. Oppo, "Quantum images in nonlinear optics," J. Mod. Opt. 44, 1899 (1997).
    [CrossRef]
  20. I. Marzoli, A. Gatti and L. A. Lugiato, "Spatial quantum signatures in parametric down-conversion," Phys. Rev. Lett. 78, 2092 (1997).
    [CrossRef]

Other

Special issue on x2 second order nonlinear optics, from fundamentals to applications, edited by C. Fabre and J.-P. Pocholle, in JEOS B: J. Quantum and Semiclassical Opt. 9, 2 (1997).

M. A. M. Marte, H. Ritsch, L. A. Lugiato and C. Fabre, "Simultaneous multimode optical parametric oscillations in a triply resonant cavity," Acta Physica Slovaca 47, 233 (1997).

G. S. Agarwal and S. D. Das Gupta, "Model for mode hopping in optical parametric oscillators," J. Opt. Soc. Am. B 14, 2174 (1997).
[CrossRef]

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti and L. A. Lugiato, "Transverse effects and mode couplings in optical parametric oscillators," submitted to Appl. Phys. B, Special Issue on Optical Parametric Oscillators, edited by. J. Mlynek and S. Schiller.

G.-L. Oppo, M. Brambilla and L. A. Lugiato, "Formation and evolution of roll patterns in optical parametric oscillators," Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

A. Gatti and L.A. Lugiato, "Quantum images and critical fluctuations in the optical parametric oscillator below threshold," Phys. Rev. A 52, 1675 (1995).
[CrossRef] [PubMed]

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo and S. M. Barnett, "Langevin treatment of quantum uctuations and optical patterns in optical parametric oscillators below threshold," Phys. Rev. A 56, 877 (1997).
[CrossRef]

A. Gatti, L. A. Lugiato, G.-L. Oppo, R. Martin, P. Di Trapani and A. Berzanskis, "From quantum to classical images," Opt. Express 1, 21 (1997), http://epubs.osa.org/oearchive/source/1968.htm.
[CrossRef] [PubMed]

C. Schwob and C. Fabre, "Squeezing and quantum correlations in multimode optical parametric oscillators," preprint, to be submitted to JEOS B: J. Quantum and Semiclassical Opt.

E. Lantz and F. Devaux, "Parametric amplification of images," JEOS B: J. Quantum and Semi-classical Opt. 9, 279 (1997).

M. I. Kolobov and L. A. Lugiato, "Noiseless amplification of optical images," Phys. Rev. A 52, 4930 (1995).
[CrossRef] [PubMed]

L. A. Lugiato and I. Marzoli, "Quantum spatial correlations in the optical parametric oscillator with spherical mirrors," Phys. Rev. A 52, 4886 (1995).
[CrossRef] [PubMed]

L. A. Lugiato, S. M. Barnett, A. Gatti, I. Marzoli, G.-L. Oppo and H. Wiedemann, "Quantum Images," J. of Nonlinear Optical Phys. and Materials 5, 809 (1996).
[CrossRef]

L. A. Lugiato and P. Grangier, "Improving quantum-noise reduction with spatially multimode squeezed light," J. Opt. Soc. Am. B 14, 225 (1997).
[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes; the art of scientific computing, Cambridge University Press, Cambridge (1986).

M. SanMiguel and R. Toral, Instabilities and Nonequlibrium structures, VI, Kluwer Academic Pub. (1997).

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino and R. J. Horowicz, "Bistability, self-pulsing and chaos in optical parametric oscillators," Il Nuovo Cimento 10 D, 959 (1988).

K. I. Petsas, A. Gatti and L. A. Lugiato, "Quantum images in optical parametric oscillators with spherical mirrors and gaussian pump," submitted to JEOS B: J. Quantum and Semiclassical Opt.

L. A. Lugiato, A. Gatti, H. Ritsch, I. Marzoli and G.-L. Oppo, "Quantum images in nonlinear optics," J. Mod. Opt. 44, 1899 (1997).
[CrossRef]

I. Marzoli, A. Gatti and L. A. Lugiato, "Spatial quantum signatures in parametric down-conversion," Phys. Rev. Lett. 78, 2092 (1997).
[CrossRef]

Supplementary Material (1)

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Figures (7)

Figure 1.
Figure 1.

Modulus of steady-state field amplitude as a function of the transverse position: (a) input field, (b) intracavity pump field, (c) signal far field and (d) signal intracavity field for δp = δs = 0, we = 3wp , ξ = 0.05ks and Ep = 42ks . In the near field plots the length scale is wp ; in the far field diagram it is s /(2πwp ), where z is the distance from the cavity.

Figure 2.
Figure 2.

Same as in Fig. 1 except for we = wp , ξ = 0.2ks , δs = -2.5ks , Ep = 65ks : (a) input field, (b) intracavity pump field, (c) signal far field and (d) signal intracavity field.

Figure 3.
Figure 3.

Same as in Fig. 2 but for we = 4wp , ξ = 0.25ks , δs = -1.5ks : (a) input field, (b) intracavity pump field, (c) signal far field and (d) signal intracavity field.

Figure 4.
Figure 4.

The movie shows the formation of a spiral pattern for the same values of the parameters as in Fig. 3 but for E 0 = 70ks . [Media 1]

Figure 5.
Figure 5.

Various quasi–stationary patterns found for the signal field for we = 4wp , ξ = 0.25ks , δp = 0, δs = -1.5ks and E 0 = 85ks .

Figure 6.
Figure 6.

The correlation function F(r, Δϕ, ω) divided by F(r, Δϕ, ω = 0) is plotted as a function of Δϕ for a quadrature component with the phase ϕL specified in the text. We fix ω = 0 and the value of r as described in the text. We set Ēin = 0.9, ξ = 0.5ks , and δs = -1.5ks . (a) Near field, (b) Far field. Red curve: we → ∞, blue curve: we = 2.8ws , black curve: we = 14ws .

Figure 7.
Figure 7.

Same as in Fig. 6, but for Ēin = 0.6, ξ = 0.1ks , and δs = - 0.3ks .

Equations (7)

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t A p r ϕ t = ( k p + i δ p L p ) A p r ϕ t χ 2 A s 2 r ϕ t + E p r ϕ + W p ( t ) ,
t A s r ϕ t = ( k s + i δ s L s ) A s r ϕ t + χ A p r ϕ t A s * r ϕ t + W s ( t ) ,
L k = w k 2 4 T 2 r 2 w k 2 + 1 .
A p r ϕ = k s χ , A s 2 r ϕ = 2 χ [ E p r ϕ k p k s χ ] .
A p r ϕ = E p r ϕ k p , A s r ϕ = 0 .
F ˜ ( r , Δ ϕ , ω ) = + dt e iωt F ( r , Δ ϕ , t ) .
φ L = r 2 w s 2 [ 1 + ( z z r ) 2 ] z z r ( q c + 1 ) tan 1 ( z z r ) + π 2 ,

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