Growth dynamics of noise sustained structures in nonlinear optical resonators

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 o The growth dynamics of both noise sustained and deterministic patterns is exempli ed by means of movies This allows to observe the process of formation of these structures and to con rm the analytical predictions The ampli cation of quantum noise by several orders of magnitude is predicted The qualitative anal ysis of the near and far eld is given It su ces to distinguish noise sustained from deterministic structures quantitative informations can be obtained in terms of the statistical properties of the spectra c Optical Society of America OCIS codes Nonlinear optics transverse e ects in Fluc tuations relaxations and noise References and links 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 A Gatti H Wiedemann L A Lugiato I Marzoli G L Oppo S M Barnett A Langevin approach to quantum uctuations and optical patterns formation Phys Rev A A Gatti L A Lugiato G L Oppo R Martin P Di Trapani A Berzanskis From quantum to classical images Opt Express I Marzoli A Gatti L A Lugiato Spatial quantum signatures in parametric downconversion Phys Rev Lett M Hoyuelos P Colet M San Miguel Fluctuations and correlations in polarization patterns of a Kerr medium submitted R J Deissler Noise sustained structure intermittency and the Ginzburg Landau equation J Stat Phys M Santagiustina P Colet M San Miguel D Walgraef Convective noise sustained structures in nonlinear optics Phys Rev Lett L A Lugiato R Lefever Spatial dissipative structures in passive optical systems Phys Rev A W J Firth A G Scroggie G S McDonald L A Lugiato Hexagonal patterns in optical bista bility Phys Rev A R L A Lugiato F Castelli Quantum noise reduction in a spatial dissipative structure Phys Rev Lett K Staliunas Optical vortices during three wave nonlinear coupling Opt Commun G L Oppo M Brambilla L A Lugiato Formation and evolution of roll patterns in optical parametric oscillators Phys Rev A L A Wu H J Kimble J Hall H Wu Generation of squeezed states by parametric down conversion Phys Rev Lett M Santagiustina P Colet M San Miguel D Walgraef Two dimensional noise sustained struc tures in optical parametric oscillators submitted M Haelterman G Vitrant Drift instability and spatiotemporal dissipative structures in a nonlinear Fabry Perot resonator under oblique incidence J Opt Soc Am B G Grynberg Drift instability and light induced spin waves in an alkali vapor with feedback mirror Opt Commun A Petrossian L Dambly G Grynberg Drift insta bility for a laser beam transmitted through a rubidium cell with feedback mirror Europhys Lett N Bloembergen Nonlinear Optics Benjamin Inc Publ Y R Shen The principles of nonlinear optics Wiley P D Drummond K J Mc Neil D F Walls Non equilibrium transitions in sub second harmonic generation semiclassical theory Opt Acta


Introduction
Optical patterns o er 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].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 eld and should encode speci c features of quantum statistics.In the search for the manifestations of quantum noise in optical patterns it is natural to look for situations in which noise is enhanced or ampli ed.Critical uctuations close to an instability point are one of these situations recently considered 2, 3,4].The noisy precursor, observed just below threshold anticipates the pattern to be formed beyond the instability.This precursor occurs because uctuations with the wavenumberto be selected above threshold become weakly damped as the threshold is approached.A more dramatic manifestation of noise occurs above a convective instability threshold 5].Here uctuations are ampli ed (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 ampli cation factors of several orders of magnitude, to observe noise and its spatially dependent manifestations 6].In this paper we give t wo 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 eld of optical pattern formation and quantum noise properties.The rst is a cavity lled by a K e r r t ype nonlinear medium and pumped by an external laser beam 6].This system was a prototype model for pattern formation in optics 7,8] and it has also been where the question of quantum uctuations in patterns was rst addressed 9].The second example is an optical parametric oscillator (OPO), also a paradigm for studies of pattern formation 10] and generation of squeezed and nonclassical light 11].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 di erent origins.In our rst example this originates in any p u m p misalignement 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-o between the ordinary and extraordinary rays, due to birefringence.
Thus, the outline of the paper is as follows.In Section 2 we brie y recall the de nition of the convectively unstable regime and the linear stability analysis which allows to determine the di erent regimes.In Section 3, we describe the convective instabilities and noise sustained structures in Kerr nonlinear resonators with one transverse dimension.We show i n two m o vies 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 exempli ed.The distinction between these two regimes manifests qualitatively in the time evolution of the far eld of the pattern.Section 4 is devoted to noise-sustained structures in OPO with two transverse dimensions.We s h o w the diagram in parameter space where the zero solution becomes unstable either convectively or absolutely.T w o m o vies, displaying the growth dynamics of the noise-sustained and the deterministic patterns, are also presented, for the near-and the far-eld.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.

De nitions and linear stability analysis
We start by brie y recalling the notion of convective and absolute instabilities readers may refer to 5,6,12] for more details.The steady-state of a generic system is de ned 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 xed position, it actually decays.In this case the state is called convectively unstable.Note that the de nition is unambiguous only if a xed frame of reference is de ned in the cases we consider here the xed frame corresponds to the pump beam.The usual linear stability analysis of the steady-state can be re-formulated in order to take i n to account the above distinction of convectively and absolutely unstable regimes.In general, the calculation of the pump amplitude thresholds of the instability for the systems we are considering entails the evaluation of the linearized asymptotic behaviour of a generic perturbation of the steady-state 5,12].The convective threshold turns out to correspond to the instability threshold which c a n be calculated as if the advection was not present, i.e. < (q)] > 0 where (q) is the linearized eigenvalue of largest real part and q the wave-vector of the perturbation.This means that at threshold all unstable modes are convectively unstable.As regards the threshold of the absolutely unstable regime, its determination reduces to the calculation of the pump amplitude which satis es the following conditions: where the complex vector ks de ned by r k ( k = ks ) = 0 (2) is a saddle point f o r < ( ks )] in the complex vector space.
A detailed mathematical explication of the procedure to get the above conditions can be found in 5,6,12].Here, for the sake of simplicity w e just mention that wave-vectors q are extended to the complex space k in order to evaluate the integral, in the wave-vector space, which determines the asymptotic linearized evolution of a perturbation.

Kerr resonators
For a nonlinear resonator containing a Kerr medium 7] an advection-like term can stem from the input pump beam tilting 13].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 justi ed from an experimental viewpoint 1 4 ] .
The equation governing the electric eld A(x t) is 6,7,13]: @ t A ; 2 0 @ x A = i@ 2 x A ; 1 + i ( ; j Aj 2 )]A + E 0 + p (x t) where: 0 represents the tilt angle, the sign of the nonlinearity, t h e c a vity detuning and E 0 the pump.Di raction is represented by the rst term on the r.h.s. and mirror losses by the rst in the squared brackets (exact de nitions can be found in 6]).We h a ve introduce a complex additive noise (x t), Gaussian, with zero-mean and correlation h (x t) (x 0 t 0 )i = 2 (x ;x 0 ) (t ;t 0 ), 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].In our case it can also account for thermal and input eld uctuations.Through conditions (1,2) applied to the linearized eigenvalue of eq. ( 3) we h a ve estimated the threshold of the absolute instability for a xed set of parameters for the steady-state A 0 , solution of A 0 1 + i ( ; j A 0 j 2 )] = E 0 .F or the same parameters we h a ve i n tegrated 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-eld (A) and far-eld (the space Fourier transform of A) of eq. ( 3) con rm that di erent, unstable regimes actually exist.In movie 1 the near eld intensity t i m e e v olution (left side) can be observed, for the pump amplitude above the threshold of absolute instability.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 at top region where the pump is above the threshold.The evolution of the far-eld (right) shows that after the transitory, w ell de ned harmonics are generated (due to the multiple wave mixing).Their linewidth is scarcely in uenced by t h e presence of noise as demonstrated in 6] in the equivalent time analysis.Movie 2. Near eld (left) and far eld (right) growth dynamics in the convectively unstable regime.
In the convective regime (movie 2) we applied noise ( ' 10 ;5 ) and a pattern forms again.However, note that the structure, even for long times does not invade all the system but rather gathers its saturated value at a random spatial position.This is due to the fact that noise needs to drift for enough time in order to be ampli ed.When the noise source is turned o the pattern (after the delay due to the drifting) eventually disappears.In the far-eld, the noticable broadening of the spectral lines with respect to the previous case con rms the di erent, noisy, nature of the pattern observed.

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-o 15].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 2! 0 ) and the signal (A 1 (x y t) at frequency !0 ) evolution is described by the following set of coupled equations 10,16]: @ t A 0 = 0 ;(1 + i 0 )A 0 + E 0 + ia 0 r 2 A 0 + 2 iK 0 A 2  1 ] + p 0 0 (x y t) @ t A 1 = 1 ;(1 + i 1 )A 1 + 1 @ y A 1 + ia 1 r 2 A 1 + iK 0 A 1 A 0 ] + p 1 1 (x y t) (4) where: 0 1 represent the losses, 0 1 the detunings, a 0 1 the di raction, K 0 the nonlinearity, 1 the walk-o , E 0 the pump (see 10,12] for details).Noise terms 0 1 have t h e 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 + i 0 ) A 1 = 0.It turns out that it can become unstable along the signal component o f the eigenvector, A 1 , a n d t h us it is necessary to consider only one linearized equation.
We can calculate the predicted absolute instability thresholds through conditions (1,2) with determined from eqs. (4).In summary, the stability diagram for the OPO is presented in gure 1 as a function of the signal detuning 1 .The linear analysis also reveals that the rst mode to become unstable satis es q x = 0 , i.e. is parallel to the x axis.This stems from the breaking of the rotational system due to the walk-o .The walk-o does not a ect the growth rate but rather the spreading velocity of the perturbation.The rst mode to become absolutely unstable is that which balances the advection with spreading 12].
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 supergaussian beam and we show i n t h e m o vie the central region of space where the pump was at.
Movie 3. Near eld (left) and far eld (right) growth dynamics in the absolutely unstable regime.
In the initial stage, noise generates a randomly oriented pattern in both cases later the two e v olutions start to di er.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 de ned and no defects of the horizontal orientation can be observed waiting a long enough time (see also gure 2, top, which corresponds to the snapshot of the nal time of the evolution).In the convective regime (movie 4) the pattern is continuously generated by noise with random orientation at the bottom of the window and ampli ed while drifting.During this process stripes get parallel to the x-axis.Note that the dynamical orientation is less marked than in the previous case and defects can be still seen.The location where the pattern gathers its saturated value is random as in the Kerr case (see gure 2, middle).The average space delay a n d i t s v ariance depends on the noise intensity.The far eld observation su ces to distinguish the two di erent regimes.At the rst stage all modes on a r i n g o f radius q c = p ; 1 =a 1 are excited later, in the absolute regime two narrow spots form, in correspondance with the rst mode that become absolutely unstable (q x = 0), in the convective regime two broadenend arcs of the ring remain visible even for very large times (see gure 2b).
Quantitative results which help to sharply distinguish the two regimes can be obtained by means of a time spectral analysis 12].To summarize we p r e s e n t three situations in gure 2, i.e. from the top to the bottom: absolutely unstable, convectively unstable and absolutely stable (close to threshold).The rst is a deterministic pattern, the second a noise-sustained one and the last is a noisy-precursor we h a ve 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 eld.Noise-sustained structures show: high intensity due to large noise magni cation factors, preferential selection of the stripe orientation (in 2D), although defects are clearly observable, only partial and random occupancy of the system, broadened far elds.Noisy precursors below t h e instability threshold are characterized by: low intensities and random orientation (in 2D), because noise is only selectively enhanced by the ltering e ect of the nonlinearity, and very broaden far eld.

Conclusions
We h a ve theoretically predicted the existence of macroscopic, noise-sustained transversal structures in nonlinear optical resonators.Numerical solutions con rm 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-o 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.

Movie 1 .
Near eld (left) and far eld (right) growth dynamics in the absolutely unstable regime.

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

Movie 4 .
Near eld (left) and far eld (right) growth dynamics in the convectively unstable regime.