## Spatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effects

Optics Express, Vol. 3, Issue 2, pp. 71-80 (1998)

http://dx.doi.org/10.1364/OE.3.000071

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### 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.

© Optical Society of America

## 1. Introduction

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

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]

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]

## 2. Model

*thin*nonlinear

*χ*

^{(2)}-crystal with an effective nonlinear coupling strength

*χ*. The cavity is coherently pumped by an input field

*E*

_{p}at a frequency

*ω*

_{p}from the outside and the single input/output mirror is assumed highly reflective at the pump frequency as well as at the signal frequency

*ω*

_{s}=

*ω*

_{p}/2. We consider the doubly resonant case of an OPO with a common cavity for the two fields, which have a common Rayleigh length

*z*

_{r}. For the overall geometry we restrict ourselves to a quasi–planar or a quasi–confocal geometry. In this case, the effective transverse mode spacing

*ξ*is on the order of, or even less than, the cavity linewidths

*k*

_{s}and

*k*

_{p}for the signal and pump field, respectively, and many transverse modes can contribute to the dynamics [4]. We will concentrate on the classical aspects of the field dynamics first. Eliminating the longitudinal dependence by using the paraxial and the mean field approximations, we find the following equations for the transverse dynamics of the slowly varying pump field amplitude

*A*

_{p}(

*r, ϕ, t*) and the signal field

*A*

_{s}(

*r, ϕ, t*) [7

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

*r*and

*ϕ*denote the distance from the axis of the system and the angular variable, respectively. The effect of diffraction and spherical mirrors is contained in the differential operator [12

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]

*δ*

_{k}=

*ω*

_{k}(

*k*∈ {

*p, s*}) is the detuning between the chosen carrier frequency of the fields and the eigenfrequency of the

*TEM*

_{00}-mode closest to resonance.

*E*

_{p}(

*r, ϕ*) represents an externally applied pump amplitude and

*W*

_{k}have been introduced to model fluctuations of the pump field or other noise sources. On one hand such noise are helpful to speed up the convergence of the numerical solutions; on the other hand, for a proper choice of the time-correlation functions of

*W*

_{p}and

*W*

_{s}Eqs. (1,2) are Langevin equations which govern the dynamics of the system on a quantum level, in the Wigner representation [7

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

*A*

_{s}= 0,

*A*

_{p}=

*E*

_{p}/ (

*k*

_{p}+

*iδ*

_{p}). This solution is stable only below threshold.

*L*

_{k}are just the usual transverse Gauss-Laguerre functions. Inserting the corresponding mode expansions for all fields leads to the standard coupled mode equations [4]. Unfortunately, any analytical treatment of these equations seems impossible at present. Let us, however, emphasize here that for the ideal degenerate confocal cavity,

*i.e*. for

*ξ*= 0, the spatial differential operator disappears from the above equations and we get independent OPO’s at each spatial point. Such a system has quite intriguing physical properties, as e.g. localized squeezing [14

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]

*ξ*→ 0 but, simultaneously,

*w*

_{k}→ ∞ in such a way that

*L*

_{k}to the mirror surface, and by avoiding periodic boundary conditions. Although in practice one usually works in a regime where the mirror boundaries seem to play no essential role, the spontaneous formation of optical patterns with broken rotational symmetry could be very sensitive to even small boundary effects.

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

## 3. Spatial patterns in a doubly resonant degenerate quasi–confocal OPO

*w*

_{e},

*i.e*.

*E*

_{p}(

*r, ϕ*) =

*E*

_{0}exp (-

*r*

^{2}/

*A*

_{k}(

*r, ϕ*) =

*A*

_{k}(

*r, ϕ*+

*π*) [or alternatively

*A*

_{k}(

*r, ϕ*) = -

*A*

_{k}(

*r, ϕ*+

*π*)] of all intracavity fields; this cuts the number of points needed for the numerical solution in half.

*ξ*≫

*k*

_{s}to a large number of contributing modes for

*ξ*≪

*k*

_{s}. By varying the detunings we can choose the modes out of the transverse manifold which are dominantly excited. Similarly, by changing the shape (e.g. the size) of the pump field we can select which modes are effectively excited. In addition, via the pump–

*size*we can also control the nonlinear intermode coupling [2,4] from independent excitation (large pump

*w*

_{e}≫

*w*

_{p}) to a strong mode coupling (

*w*

_{e}≈

*w*

_{s}). Finally, we can influence the dynamics by changing the pump

*strength*.

*ξ*= 0. One easily finds that for points (

*r, ϕ*) where the pump intensity is large enough so that |

*E*

_{p}(

*r, ϕ*)|

^{2}(

*χ*

^{2}, the OPO is above threshold. For instance, for

*δ*

_{s}=

*δ*

_{p}= 0 and by choosing

*E*

_{p}real, one has in steady-state:

*E*

_{p}(

*r, ϕ*)|

^{2}< (

*χ*

^{2}, the OPO is below threshold. For

*δ*

_{s}=

*δ*

_{p}= 0 and

*E*

_{p}real one has, e.g.:

*ξ*≠ 0, the treshold has to be determined numerically. In the following, we will focus on a number of selected examples. In all calculations of this section we have

*χ*= 0.1

*k*

_{s}and

*k*

_{p}= 3

*k*

_{s}. For such values, the plane wave threshold for a planar cavity corresponds to

*E*

_{p}= 30

*k*

_{s}for

*δ*

_{s}=

*δ*

_{p}= 0.

### 3.1 Resonant multimode case

*ξ*≈

*k*

_{s}), or even very small (

*ξ*≪

*k*

_{s}) transverse mode spacing. As we can see from Eqs. (1,2), this decreases the influence of diffraction, which mediates spatial cross-coupling, and leads eventually to individual spatial points oscillating independently. Let us for the moment assume

*δ*

_{p}=

*δ*

_{s}= 0, the corresponding stationary signal field intensity distribution is shown in Fig. 1.

*ξ*= 0.05

*k*

_{s}, we see that inside a central region, where the pump field is locally above threshold, the intracavity pump field is clamped by the dynamics to its threshold value, yielding a flat top. This behaviour is therefore very close to the one encountered in the perfect confocal configuration. In the outside region, where the pump is below threshold, the intracavity pump field is merely proportional to the input. This behaviour gets more pronounced for smaller

*ξ*and larger

*w*

_{e}, where more and more modes contribute to the intracavity fields. The signal field is strongly confined to the above threshold region, where its shape roughly corresponds to the input pump field. Outside this region it is almost zero. The strong directional confinement of the far field shows the transverse phase coherence of the total signal field, which is still present despite the weak cross-coupling.

### 3.2 Detuned multimode operation: transverse patterns

*TEM*

_{00}mode. As it is known this leads to spatial instabilities and optical pattern formation [5

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]

*TEM*

_{00}input field. Many modes are excited and we have a fairly strong cross-coupling between the different modes. This leads to a fixed relative phase operation of several modes, yielding various rather smooth spatial distributions, which can in some sense be interpreted as an effective oscillating mode. This notion is, however, somewhat artificial as the effective mode depends on the pump strength and pump shape. An example is shown in Fig. 2, where we plot the stationary intracavity pump and signal fields for a small transverse mode spacing.

**56**, 877 (1997). [CrossRef]

*ξ*= 0.075

*k*

_{s}, we find a pattern with broken rotational symmetry, similar to the roll patterns discussed just below.

*w*

_{e}≫

*w*

_{p}) again many modes are possibly excited, but the relative phase coupling is less strong and there is more room for dynamical adjustment of the fields. This allows for breaking rotational symmetry and for the spontaneous dynamical formation of various optical patterns. The situation is richer than in the plane wave case. In the following we will show this on some specific examples.

*δ*

_{s}and

*E*

_{p}. In some rare cases a ring shape pattern appears. Once such a pattern has formed, it seems to be stable against the formation of stripes even for a fairly large amount of noise and for slow changes of the system parameters. As one might expect, the stripe pattern leads to a two-peaked intensity distribution in the far field.

*i.e*. the end of the arms grow. Again, these turn out to be quite stable against noise and slow changes of the system parameters. The rotation speed depends on

*E*

_{p}and

*w*

_{p}and is much slower than the other time scales in the system. For these parameter ranges stripes and spirals are stable at the same time. Even seeding a stripe pattern into the input for an existing spiral has no effect, as the modification of the pump field induced by the spiral prevents gain for the seeded stripes, and vice versa. Hence there is multistability of various patterns induced

*via*the backaction of the existing pattern on the pump field.

## 4. Quantum images below threshold

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]

*A*

_{p}in Eq. (2) can be expressed as a given function of

*r, ϕ*equal to the input field

*E*

_{in}. Hence, Eq. (2) becomes self-contained for the signal field

*A*

_{s}(

*r,ϕ,t*), and Eq. (1) can be dropped altogether.

*r, ϕ, t*) is uniform on average over the circle because of the cylindrical symmetry and is constant in time. However, if we consider the spatial correlation function at steady-state

*F*(

*r*, Δ

*ϕ, t*) = ⟨𝓞 (

*r, ϕ, t*) 𝓞 (

*r, ϕ′*, 0)⟩ where

*r*is the radius of circle 𝑐, we obtain a function of Δ

*ϕ*=

*ϕ*-

*ϕ′*which exhibits a spatial modulation. This structure, visible in the spatial correlation function, has been called “

*quantum image*” [6

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]

**52**, 4886 (1995). [CrossRef] [PubMed]

**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]

*A*

_{s}exp(-

*iφ*

_{L}) +

*iφ*

_{L}), for an appropriate phase

*φ*

_{L}of the local oscillator field used to detect the quadrature. While in [12

**52**, 4886 (1995). [CrossRef] [PubMed]

**5**, 809 (1996). [CrossRef]

*w*

_{e}, and analyze how the result changes when we decrease gradually

*w*

_{e}from infinity (plane wave case) to values on the order of

*w*

_{s}. In particular, we want to see whether the main phenomena identified in [12

**52**, 4886 (1995). [CrossRef] [PubMed]

**5**, 809 (1996). [CrossRef]

**52**, 4886 (1995). [CrossRef] [PubMed]

**5**, 809 (1996). [CrossRef]

*w*

_{e}couples the modes [2,4]; a complete description of the calculation will be given in a future paper [18]. Figures 6 and 7 show the spectral density

*F̃*(

*r*, Δ

*ϕ*,

*ω*) [normalized to

*F̃*(

*r*, Δ

*ϕ*= 0,

*ω*)] for

*ω*= 0. In the near field [

*i.e*. in Figs. 6(a) and 7(a)] we take

*r*=

*w*

_{s}and

*φ*

_{L}= 0. In the far field [Figs. 6(b) and 7(b)], on the other hand, we evaluate the correlation function at a distance

*z*= 20

*z*

_{r}from cavity center, and we take

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]

*q*

_{c}is the order of a frequency-degenerate family of Gauss-Laguerre modes which is exactly resonant with the signal frequency

*ω*

_{s}[12

**52**, 4886 (1995). [CrossRef] [PubMed]

*q*= 2

*p*+

*l*, where

*p*and

*l*are the radial and angular indices of the Gauss-Laguerre functions, respectively. In both figures 6 and 7 we have

*q*

_{c}= 3. The value of the input field is measured as a dimensioneless parameter

*Ē*

_{in}=

*E*

_{in}

*χ*/

*k*

_{s}; in the plane wave limit

*w*

_{e}→ ∞, the threshold value is

*Ē*

_{in}= 1.

*ϕ*. For the value of

*Ē*

_{in}considered in Fig. 6, the system is 10% below threshold in the plane wave limit

*w*

_{e}→ ∞. As shown in [12

**52**, 4886 (1995). [CrossRef] [PubMed]

*w*

_{e}= ∞, when approaching the threshold the contribution of the resonant family becomes dominant and, because the circle 𝑐 is chosen in such a way that the modes

*p*= 1,

*l*= 1 vanish, the correlation function

*F̃*(

*r*, Δ

*ϕ*,

*ω*= 0)/

*F̃*(

*r*, Δ

*ϕ*= 0,

*ω*= 0) arises only from the contribution of the two modes

*p*= 0,

*l*= 3, so that getting close to threshold the curve approaches the function cos (3Δ

*ϕ*), as shown by the red line in Fig. 6(a). Reducing

*w*

_{e}, the modulation becomes less regular and less pronounced, but it is still quite remarkable for

*w*

_{e}= 14

*w*

_{s}. It must be taken into account, in addition, that decreasing

*w*

_{e}the threshold value for

*Ē*

_{in}increases.

**5**, 809 (1996). [CrossRef]

*ϕ*=

*π*higher than the peak in Δ

*ϕ*= 0. As shown in [19

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]

*ξ*is decreased) is quite robust with respect to the reduction of

*w*

_{e}, up to when

*w*

_{e}becomes on the order of

*w*

_{s}. The correlation as a function of Δ

*ϕ*becomes broader as

*w*

_{e}is decreased.

## Acknowledgements

*Quantum Structures*”) of the TMR program of the EU, and supported by the Austrian Science Foundation FWF Project No. S6506. M. M. was supported by an

*APART*fellowship of the Austrian Academy of Sciences.

## References and links

1. | Special issue on |

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 |

3. | G. S. Agarwal and S. D. Das Gupta, “Model for mode hopping in optical parametric oscillators,” J. Opt. Soc. Am. B |

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. 1Schiller. |

5. | G.-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A |

6. | A. Gatti and L.A. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A |

7. | A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A |

8. | A. Gatti, L. A. Lugiato, G.-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” Opt. Express |

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. |

11. | M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A |

12. | L. A. Lugiato and I. Marzoli, “Quantum spatial correlations in the optical parametric oscillator with spherical mirrors,” Phys. Rev. A |

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 |

14. | L. A. Lugiato and P. Grangier, “Improving quantum-noise reduction with spatially multimode squeezed light,” J. Opt. Soc. Am. B |

15. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, |

16. | M. SanMiguel and R. Toral, |

17. | L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, “Bistability, self-pulsing snd 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. |

20. | I. Marzoli, A. Gatti, and L. A. Lugiato, “Spatial quantum signatures in parametric down-conversion,” Phys. Rev. Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Focus Issue: Quantum structures in nonlinear optics and atomic physics

**History**

Original Manuscript: April 13, 1998

Published: July 20, 1998

**Citation**

Monika A. M. Marte, H Ritsch, K. Petsas, Alessandra Gatti, Luigi Lugiato, Claude Fabre, and D. Leduc, "Spatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effects," Opt. Express **3**, 71-80 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-2-71

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### References

- 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]

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