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Optics Express

  • Editor: J. H. Eberly
  • Vol. 3, Iss. 2 — Jul. 20, 1998
  • pp: 63–70
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Growth dynamics of noise-sustained structures in nonlinear optical resonators

Marco Santagiustina, Pere Colet, Maxi San Miguel, and Daniel Walgraef  »View Author Affiliations


Optics Express, Vol. 3, Issue 2, pp. 63-70 (1998)
http://dx.doi.org/10.1364/OE.3.000063


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Abstract

The existence of macroscopic noise-sustained structures in nonlinear optics is theoretically predicted and numerically observed, in the regime of convective instability. The advection-like term, necessary to turn the instability to convective for the parameter region where advection overwhelms the growth, can stem from pump beam tilting or birefringence induced walk-off. The growth dynamics of both noise-sustained and deterministic patterns is exemplified by means of movies. This allows to observe the process of formation of these structures and to confirm the analytical predictions. The amplification of quantum noise by several orders of magnitude is predicted. The qualitative analysis of the near- and far-field is given. It suffices to distinguish noise-sustained from deterministic structures; quantitative informations can be obtained in terms of the statistical properties of the spectra.

© Optical Society of America

1. Introduction

Optical patterns offer the very attractive possibility of studying the interface between classical and quantum patterns. Macroscopic and spatially structured manifestations of quantum correlations are foreseen to occur in these patterns [1

1. L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, and H. Wiedemann, “Quantum aspects of nonlinear optical patterns,” Coherence and Quantum Optics VII (Plenum Press, New York, 1996), p 5.

]. Such correlations are expected since, at a microscopic level, the physical mechanism behind the pattern formation process is often the simultaneous emission of twin photons, four wave mixing processes or other processes involving highly correlated photons. Correlations are easily observed in the far field and should encode specific features of quantum statistics.

A more dramatic manifestation of noise occurs above a convective instability threshold [6

6. R. J. Deissler,“Noise-sustained structure, intermittency, and the Ginzburg-Landau equation,” J. Stat. Phys. 40376 (1985). [CrossRef]

]. Here fluctuations are amplified (instead of being weakly damped) while being advected away from the system. This gives rise to macroscopic structures that are continuously regenerated by noise and hence the name of noise sustained patterns. This phenomenon acts as a microscope, with amplification factors of several orders of magnitude, to observe noise and its spatially dependent manifestations [7

7. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997). [CrossRef]

]. In this paper we give two examples of noise sustained optical patterns. Our emphasis here is in showing how these patterns grow dynamically from noise, invading part of the system and being there maintained by noise.

The two examples to be considered are paradigmatic in the field of optical pattern formation and quantum noise properties. The first is a cavity filled by a Kerr type nonlinear medium and pumped by an external laser beam [7

7. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997). [CrossRef]

]. This system was a prototype model for pattern formation in optics [8

8. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. A 58, 2209 (1987).

,9

9. W. J. Firth, A. G. Scroggie, G. S. McDonald, and L. A. Lugiato, “Hexagonal patterns in optical bistability,” Phys. Rev. A 46, R3609 (1992). [CrossRef] [PubMed]

] and it has also been where the question of quantum fluctuations in patterns was first addressed [10

10. L. A. Lugiato and F. Castelli, “Quantum noise reduction in a spatial dissipative structure,” Phys. Rev. Lett. 68, 3284 (1992). [CrossRef] [PubMed]

]. The second example is an optical parametric oscillator (OPO), also a paradigm for studies of pattern formation [11

11. K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. 91, 82 (1992). [CrossRef]

,12

12. 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]

] and generation of squeezed and non-classical light [13

13. L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520 (1986). [CrossRef] [PubMed]

]. A necessary condition for the existence of a convective instability is the presence of an advection-like term in the governing equations; this term can have different origins. In our first example this originates in any pump misalignment; we will study this example in a simple transverse one-dimensional geometry to clarify the main concepts. For the OPO we consider a type-I phase matching in a uniaxial crystal. Here, the advection term originates in the walk-off between the ordinary and extraordinary rays, due to birefringence.

Thus, the outline of the paper is as follows. In Section 2 we briefly recall the definition of the convectively unstable regime and the linear stability analysis which allows to determine the different regimes. In Section 3, we describe the convective instabilities and noise sustained structures in Kerr nonlinear resonators with one transverse dimension. We show in two movies the growth dynamics of the pattern in the convectively and absolutely unstable regimes. The role of the noise in sustaining the structure in the convective regime is clearly exemplified. The distinction between these two regimes manifests qualitatively in the time evolution of the far field of the pattern. Section 4 is devoted to noise-sustained structures in OPO with two transverse dimensions. We show the diagram in parameter space where the zero solution becomes unstable either convectively or absolutely. Two movies, displaying the growth dynamics of the noise-sustained and the deterministic patterns, are also presented, for the near- and the far-field. Finally we compare a snapshot of the pattern formed in each of the two cases with the noisy precursor below threshold. Conclusions are presented in section V.

2. Definitions and linear stability analysis

We start by briefly recalling the notion of convective and absolute instabilities; readers may refer to [6

6. R. J. Deissler,“Noise-sustained structure, intermittency, and the Ginzburg-Landau equation,” J. Stat. Phys. 40376 (1985). [CrossRef]

,7

7. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997). [CrossRef]

,14

14. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” to be published Phys. Rev. E (1998).

] for more details. The steady-state of a generic system is defined to be absolutely stable (unstable) when a perturbation decays (grows) with time. However, a third possibility is that the perturbation grows (i.e. is unstable) but at the same time is advected so quickly that, at a fixed position, it actually decays. In this case the state is called convectively unstable. Note that the definition is unambiguous only if a fixed frame of reference is defined; in the cases we consider here the fixed frame corresponds to the pump beam.

[λ(ks,Fa)]=0
[k2λ(k=ks),Fa]0
(1)

where the complex vector k s defined by

kλ(k=ks,Fa)=0
(2)

is a saddle point for ℜ[λ(k s, Fa)] in the complex vector space.

3. Kerr resonators

For a nonlinear resonator containing a Kerr medium [8

8. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. A 58, 2209 (1987).

] an advection-like term can stem from the input pump beam tilting [15

15. M. Haelterman and G. Vitrant, “Drift instability and spatiotemporal dissipative structures in a nonlinear Fabry-Perot resonator under oblique incidence,” J. Opt. Soc. Am. B 9, 1563 (1992). [CrossRef]

]. A one-dimensional (1D), transversal model is used in order to simplify the analysis and to clarify the main concepts. The 1D assumption can be also justified from an experimental viewpoint [16

16. G. Grynberg, “Drift instability and light-induced spin waves in an alkali vapor with feedback mirror,” Opt. Commun. 109, 483 (1994). [CrossRef]

,17

17. A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995). [CrossRef]

].

The equation governing the electric field A(x,t) is [7

7. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997). [CrossRef]

,8

8. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. A 58, 2209 (1987).

,15

15. M. Haelterman and G. Vitrant, “Drift instability and spatiotemporal dissipative structures in a nonlinear Fabry-Perot resonator under oblique incidence,” J. Opt. Soc. Am. B 9, 1563 (1992). [CrossRef]

]:

tA2α0xA=ix2A[1+(ΔA2)]A+E0+ξxt,
(3)

where: α 0 represents the tilt angle, η the sign of the nonlinearity, Δ the cavity detuning and E 0 the pump. Diffraction is represented by the first term on the r.h.s. and mirror losses by the first in the squared brackets (exact definitions can be found in [7

7. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997). [CrossRef]

]). We have introduce a complex additive noise ξ(x,t), Gaussian, with zero-mean and correlation ⟨ξ(x,t),ξ *(x′,t′)⟩ = 2δ(x - x′)δ(t - t′), which is a standard semiclassical model of noise. For the linearized version of the Langevin equations of the optical parametric oscillator a similar term describes quantum noise in the Wigner representation, as considered in [2

2. A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A 56, 877 (1997). [CrossRef]

]. The same is true for eq. (3) when α 0 = 0 and linearizing around a steady-state; in our case it can also account for thermal and input field fluctuations.

Through conditions (1,2) applied to the linearized eigenvalue of eq. (3) we have estimated the threshold of the absolute instability for a fixed set of parameters for the steady-state A 0, solution of A 0[1 + (Δ - |A 0|2)] = E 0. For the same parameters we have integrated the equation for a pump amplitude slightly above this threshold and slightly below, in the regime of convective instability.

The time evolution of the near-field (A) and far-field (the space Fourier transform of A) of eq. (3) confirm that different, unstable regimes actually exist. In movie 1 the near field intensity time evolution (left side) can be observed, for the pump amplitude above the threshold of absolute instability.

Movie 1. Near field (left) and far field (right) growth dynamics in the absolutely unstable regime. [Media 1]

The initial condition is the steady-state plus a weak perturbation: noise is not applied ( = 0) because it is not necessary to generate the pattern. After a certain transitory a drifting structure is generated. In spite of the drift, the pattern tends to invade all the flat top region where the pump is above the threshold. The evolution of the far-field (right) shows that after the transitory, well defined harmonics are generated (due to the multiple wave mixing). Their line-width is scarcely influenced by the presence of noise as demonstrated in [7

7. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997). [CrossRef]

] in the equivalent time analysis.

Movie 2. Near field (left) and far field (right) growth dynamics in the convectively unstable regime. [Media 2]

4. Parametric oscillators

In the optical parametric oscillator, i.e. when the nonlinear medium inside the resonator has a quadratic response, the advection-like term stems naturally from the birefringence of the nonlinear crystal, which is exploited to phase-match the nonlinear interaction. In fact, in a birefringent medium the ordinary and extraordinary polarizations can be subject to a transversal walk-off [18

18. N. Bloembergen, Nonlinear Optics, (Benjamin Inc. Publ., Reading, 1965), Chapter 4.2. item Y. R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), Chapter 6.9.

]. In particular, we consider, a degenerate, type I OPO (scattered photons are thus frequency and polarization degenerate). The pump (A 0(x, y, t) at frequency 2ω 0) and the signal (A 1(x, y, t) at frequency ω 0) evolution is described by the following set of coupled equations [11

11. K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. 91, 82 (1992). [CrossRef]

,12

12. 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]

,19

19. P. D. Drummond, K. J. Mc Neil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation: semiclassical theory,” Opt. Acta 27, 321 (1980). [CrossRef]

]:

tA0=γ0[(1+iΔ0)A0+E0+ia02A0+2iK0A12]+0ξ0xyt
tA1=γ1[(1+iΔ1)A1+ρ1yA1+ia12A1+iK0A1*A0]+1ξ1xyt
(4)

where: γ 0,1 represent the losses, Δ0,1 the detunings, a 0,1 the diffraction, K 0 the nonlinearity, ρ 1 the walk-off, E 0 the pump (see [12

12. 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]

,14

14. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” to be published Phys. Rev. E (1998).

] for details). Noise terms ξ 0,1 have the same characteristics of the Kerr case and are uncorrelated.

The uniform steady-state, whose stability we are interested in, is: A 0 = E 0/(1 + iΔ0) , A 1 = 0. It turns out that it can become unstable along the signal component of the eigenvector, A 1, and thus it is necessary to consider only one linearized equation. We can calculate the predicted absolute instability thresholds through conditions (1,2) with λ determined from eqs. (4). In summary, the stability diagram for the OPO is presented in figure 1 as a function of the signal detuning Δ1.

Fig. 1. Stability diagram for the OPO. Shadowed regions are: stable (green), convectively unstable (red). The white region indicates the absolute instability. Absolute threshold shifts upwards by increasing α 0 (red dashed curves).

The linear analysis also reveals that the first mode to become unstable satisfies qx = 0, i.e. is parallel to the x axis. This stems from the breaking of the rotational system due to the walk-off. The walk-off does not affect the growth rate but rather the spreading velocity of the perturbation. The first mode to become absolutely unstable is that which balances the advection with spreading [14

14. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” to be published Phys. Rev. E (1998).

].

The growth dynamics for A 1 is shown in movie 3, for the absolutely unstable, and in movie 4 for the convectively unstable regime. The pump A 0 was a super-Gaussian beam and we show in the movie the central region of space where the pump was flat.

Movie 3. Near field (left) and far field (right) growth dynamics in the absolutely unstable regime. [Media 3]

In the initial stage, noise generates a randomly oriented pattern in both cases; later the two evolutions start to differ. In the former case the stripes generated are parallel to the x-axis, as predicted. The deterministic pattern invades the whole region of pumping, stripes are well defined and no defects of the horizontal orientation can be observed waiting a long enough time (see also figure 2, top, which corresponds to the snapshot of the final time of the evolution).

Fig. 2. Snapshots of the near(far)-field at time t = 2000 on the left (right) hand side. Parameters of the top, middle and bottom images correspond respectively to (*, +, X) of Fig. 1.

Movie 4. Near field (left) and far field (right) growth dynamics in the convectively unstable regime. [Media 4]

The far field observation suffices to distinguish the two different regimes. At the first stage all modes on a ring of radius qc=Δ1a1 are excited; later, in the absolute regime two narrow spots form, in correspondence with the first mode that become absolutely unstable (qx = 0), in the convective regime two broadened arcs of the ring remain visible even for very large times (see figure 2b).

Quantitative results which help to sharply distinguish the two regimes can be obtained by means of a time spectral analysis [14

14. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” to be published Phys. Rev. E (1998).

]. To summarize we present three situations in figure 2, i.e. from the top to the bottom: absolutely unstable, convectively unstable and absolutely stable (close to threshold). The first is a deterministic pattern, the second a noise-sustained one and the last is a noisy-precursor we have referred to in the introduction. Signatures of a deterministic pattern are: the high intensity, the pattern orientation (if 2D), orthogonal to the drift direction due to the symmetry breaking, the fact that it invades all the system, narrow spatial dispersion in the far field. Noise-sustained structures show: high intensity due to large noise magnification factors, preferential selection of the stripe orientation (in 2D), although defects are clearly observable, only partial and random occupancy of the system, broadened far fields. Noisy precursors below the instability threshold are characterized by: low intensities and random orientation (in 2D), because noise is only selectively enhanced by the filtering effect of the nonlinearity, and very broaden far field.

5. Conclusions

We have theoretically predicted the existence of macroscopic, noise-sustained transversal structures in nonlinear optical resonators. Numerical solutions confirm the qualitative and quantitative predictions. Noise-sustained structures can be found in the regime of convective instability which can be induced either by a tilt in the input pump beam or by the walk-off due to birefringence. The growth dynamics of noise-sustained as well as deterministic patterns is presented and helps to distinguish the nature of the structures. This work is supported by QSTRUCT (Project ERB FMRX-CT96-0077). Financial support from DGICYT (Spain) Project PB94-1167 is also acknowledged.

References and links

1.

L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, and H. Wiedemann, “Quantum aspects of nonlinear optical patterns,” Coherence and Quantum Optics VII (Plenum Press, New York, 1996), p 5.

2.

A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A 56, 877 (1997). [CrossRef]

3.

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). [CrossRef] [PubMed]

4.

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]

5.

M. Hoyuelos, P. Colet, and M. San Miguel, “Fluctuations and correlations in polarization patterns of a Kerr medium,” to be published, Phys. Rev. E (1998). [CrossRef]

6.

R. J. Deissler,“Noise-sustained structure, intermittency, and the Ginzburg-Landau equation,” J. Stat. Phys. 40376 (1985). [CrossRef]

7.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997). [CrossRef]

8.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. A 58, 2209 (1987).

9.

W. J. Firth, A. G. Scroggie, G. S. McDonald, and L. A. Lugiato, “Hexagonal patterns in optical bistability,” Phys. Rev. A 46, R3609 (1992). [CrossRef] [PubMed]

10.

L. A. Lugiato and F. Castelli, “Quantum noise reduction in a spatial dissipative structure,” Phys. Rev. Lett. 68, 3284 (1992). [CrossRef] [PubMed]

11.

K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. 91, 82 (1992). [CrossRef]

12.

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]

13.

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520 (1986). [CrossRef] [PubMed]

14.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” to be published Phys. Rev. E (1998).

15.

M. Haelterman and G. Vitrant, “Drift instability and spatiotemporal dissipative structures in a nonlinear Fabry-Perot resonator under oblique incidence,” J. Opt. Soc. Am. B 9, 1563 (1992). [CrossRef]

16.

G. Grynberg, “Drift instability and light-induced spin waves in an alkali vapor with feedback mirror,” Opt. Commun. 109, 483 (1994). [CrossRef]

17.

A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995). [CrossRef]

18.

N. Bloembergen, Nonlinear Optics, (Benjamin Inc. Publ., Reading, 1965), Chapter 4.2. item Y. R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), Chapter 6.9.

19.

P. D. Drummond, K. J. Mc Neil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation: semiclassical theory,” Opt. Acta 27, 321 (1980). [CrossRef]

OCIS Codes
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(270.2500) Quantum optics : Fluctuations, relaxations, and noise

ToC Category:
Focus Issue: Quantum structures in nonlinear optics and atomic physics

History
Original Manuscript: March 27, 1998
Published: July 20, 1998

Citation
Marco Santagiustina, Pere Colet, Maxi San Miguel, and Daniel Walgraef, "Growth dynamics of noise-sustained structures in nonlinear optical resonators," Opt. Express 3, 63-70 (1998)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-2-63


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References

  1. L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, H. Wiedemann, "Quantum aspects of nonlinear optical patterns," Coherence and Quantum Optics VII (Plenum Press, New York, 1996), p 5.
  2. A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G- L. Oppo, S.M. Barnett, "A Langevin approach to quantum fluctuations and optical patterns formation," Phys. Rev. A 56, 877 (1997). [CrossRef]
  3. A. Gatti, L. A. Lugiato, G-L. Oppo, R. Martin, P. Di Trapani, A. Berzanskis, "From quantum to classical images," Opt. Express 1, 21 (1997). [CrossRef] [PubMed]
  4. A. Gatti, L. A. Lugiato, "Quantum images and critical uctuations in the optical parametric oscillator below threshold," Phys. Rev. A 52, 1675 (1995). [CrossRef] [PubMed]
  5. M. Hoyuelos, P. Colet, M. San Miguel, "Fluctuations and correlations in polarization patterns of a Kerr medium," to be published, Phys. Rev. E (1998). [CrossRef]
  6. R. J. Deissler,"Noise-sustained structure, intermittency, and the Ginzburg-Landau equation," J. Stat. Phys. 40, 376 (1985). [CrossRef]
  7. M. Santagiustina, P. Colet, M. San Miguel, D. Walgraef, "Convective noise-sustained structures in nonlinear optics," Phys. Rev. Lett. 79, 3363 (1997). [CrossRef]
  8. L. A. Lugiato, R. Lefever, "Spatial dissipative structures in passive optical systems," Phys. Rev. A 58, 2209 (1987).
  9. W. J. Firth, A. G. Scroggie, G. S. McDonald, L. A. Lugiato, "Hexagonal patterns in optical bistability," Phys. Rev. A 46, R3609 (1992). [CrossRef] [PubMed]
  10. L. A. Lugiato, F. Castelli, "Quantum noise reduction in a spatial dissipative structure," Phys. Rev. Lett. 68, 3284 (1992). [CrossRef] [PubMed]
  11. K. Staliunas, "Optical vortices during three-wave nonlinear coupling," Opt. Commun. 91, 82 (1992). [CrossRef]
  12. G-L. Oppo, M. Brambilla, L. A. Lugiato, "Formation and evolution of roll patterns in optical parametric oscillators," Phys. Rev. A 49, 2028 (1994). [CrossRef] [PubMed]
  13. L. A. Wu, H. J. Kimble, J. Hall, H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520 (1986). [CrossRef] [PubMed]
  14. M. Santagiustina, P. Colet, M. San Miguel, D. Walgraef, "Two-dimensional noise-sustained structures in optical parametric oscillators," to be published Phys. Rev. E (1998).
  15. M. Haelterman, G. Vitrant, "Drift instability and spatiotemporal dissipative structures in a nonlinear Fabry-Perot resonator under oblique incidence," J. Opt. Soc. Am. B 9, 1563 (1992). [CrossRef]
  16. G. Grynberg, "Drift instability and light-induced spin waves in an alkali vapor with feedback mirror," Opt. Commun. 109, 483 (1994). [CrossRef]
  17. A. Petrossian, L. Dambly, G. Grynberg, "Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror," Europhys. Lett. 29, 209 (1995). [CrossRef]
  18. N. Bloembergen, Nonlinear Optics, (Benjamin Inc. Publ., Reading, 1965), Chapter 4.2. item Y. R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), Chapter 6.9.
  19. P. D. Drummond, K. J. Mc Neil, D. F. Walls, "Non-equilibrium transitions in sub/second harmonic generation: semiclassical theory," Opt. Acta 27, 321 (1980). [CrossRef]

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