Pattern formation and competition in nonlinear optics

Abstract

Pattern formation and competition occur in a nonlinear extended medium if dissipation allows for attracting sets, independently of initial and boundary conditions. This intrinsic patterning emerges from a reaction diffusion dynamics (Turing chemical patterns). In optics, the coupling of an electromagnetic field to a polarizable medium and the presence of losses induce a more general (diffraction-diffusion) mechanism of pattern formation. The presence of a coherent phase propagation may lead to a large set of unstable bands and hence to a richer variety with respect to the chemical case. A review of different experimental situations is presented, including a discussion on suitable indicators which characterize the different regimes. Vistas on perspective new phenomena and applications include an extension to atom optics.

Introduction

Pattern formation in extended media is the result of the interaction between a local nonlinear dynamics and space gradient terms which couple neighboring spatial regions. Furthermore, nonlocal terms may bring interactions from far away either in space or time. Some previous review papers are available on the matter [1], [2].

A recent comprehensive review [3] is mainly devoted to fluid dynamic or chemical patterns, where the gradient terms result either from momentum transport of from diffusion processes. Optical patterns, on the other hand, are characterized by a wave transport, mainly pointing in one direction. Furthermore, use of laser sources and resonant media to enhance the nonlinearities restricts the time dependence to a quasi-monochromatic behavior. Therefore, the functional terms we have to deal with in optics are of the type

$$\text{f}\text{(x,}\text{y,}\text{z,}\text{t)}\text{e}{}^{\mathrm{<mtext>i</mtext><mtext>(kz-\omega t)</mtext>}}\text{,}$$

where the exponential term accounts for a plane wave moving along a direction z ($\text{(x,}\text{y)}$ being the plane orthogonal to z) and the residual z and t dependence is slow, i.e.

$$\text{∂}\text{f}\text{∂}\text{z}\text{⪡k|}\text{f}\text{|,}\text{∂}\text{f}\text{∂}\text{t}\text{⪡ω|}\text{f}\text{|}\text{.}$$

This is currently called SVEA (slowly varying envelope approximation) and it is a sensible approximation even when the slow and fast scales differ by a factor less than 10, as it occurs e.g. for femto-second pulses.

Based upon this wave transport feature, optical patterns present all classes of relevant phenomena reported elsewhere, plus some ones which are specific of optics. It seems then appropriate to introduce a general classification of optical patterns, within which it is easy to include also classes of phenomena observed in fluid and chemistry. For convenience, we have collected in Appendix A some introductory facts on nonlinear optics, together with the corresponding jargon.

In the classification we distinguish between the longitudinal space direction z and the transverse plane $\text{(x,}\text{y)}$, since the boundary conditions are usually drastically different for the two cases. We classify patterns as 0, 1, 2, or 3-dimensional depending on the functional space dependence of the envelope f. Notice that, at variance with condensed matter instabilities, a 0-dimensional dynamics (i.e. ruled by an ordinary differential equation for f ) still refers to a wave pattern which is mono-directional and mono-chromatic. Crucial parameters to estimate the dimensionality are the aspect ratios, which will be defined as follows.

Let us confine the optical dynamics within a rectangular box of sides $\text{L}{}_{\mathrm{x}}\text{,}\text{L}{}_{\mathrm{y}}\text{,}\text{L}{}_{\mathrm{z}}$ and take $\text{L}{}_{\mathrm{z}}\text{⪢L}{}_{\mathrm{x}}\text{,}\text{L}{}_{\mathrm{y}}$.

Optical patterns emerge from coupling Maxwell equation to the constitutive matter equations. The Maxwell field $\text{e(}\text{r}\text{,}\text{t)}$ induces a polarization $\text{p(}\text{r}\text{,}\text{t)}$ in the medium. This polarization acts as a source for the field which then obeys the wave equation (for simplicity, we refer to scalar fields, neglecting for the moment the polarizations)

$$\text{□}{}^2\text{e=−μ}\text{∂}{}_{\mathrm{t}}{}^2\text{p}\text{,}$$

where $\text{□}{}^2\text{≡}\text{∂}{}_{\mathrm{x}}{}^2\text{+}\text{∂}{}_{\mathrm{y}}{}^2\text{+}\text{∂}{}_{\mathrm{z}}{}^2\text{,}\text{μ}$ is a suitable parameter, and we are denoting the partial derivations as $\text{∂}\text{∂}\text{x}\text{=}\text{∂}{}_{\mathrm{x}}$, etc. We expand the field as

$$\text{e(}\text{r}\text{,}\text{t)=E(x,}\text{y,}\text{z,}\text{t)}\text{e}{}^{\mathrm{<mtext>i</mtext><mtext>(kz-\omega t)</mtext>}}\text{.}$$

If the longitudinal variations are mainly accounted for by the plane wave, then we can take the envelope E as slowly varying in t and z with respect to the variation rates ω and k in the plane wave exponential (Eq. (2)). Furthermore, we shall define P to be the projection of p on the plane wave. By neglecting second order envelope derivatives in z and t, it is easy to approximate the operator on E as

$$\text{□}{}^2\text{→2}\text{i}\text{k}\text{∂}{}_{\mathrm{z}}\text{+}\text{1}\text{c}\text{∂}{}_{\mathrm{t}}\text{+}\text{∂}{}_{\mathrm{x}}{}^2\text{+}\text{∂}{}_{\mathrm{y}}{}^2\text{,}$$

where c is the light velocity. Eq. (5) is usually called the eikonal approximation of wave optics.

Eq. (5) suggests that the comparison between transverse variations along x and y are ruled by the aspect ratios

$$\text{F}{}_{\mathrm{x}}\text{=}\text{L}{}_{\mathrm{x}}{}^2\text{λL}{}_{\mathrm{z}}\text{,}\text{F}{}_{\mathrm{y}}\text{=}\text{L}{}_{\mathrm{y}}{}^2\text{λL}{}_{\mathrm{z}}\text{,}$$

where λ≡2π/k is the optical wavelength.

The aspect ratios are called Fresnel numbers, and they have the following heuristic meaning. The geometric angle of view of an object of linear size L_x from a distance L_z is L_x/L_z. Within this angle only details of minimal angular separation λ/L_x can be resolved, due to diffraction. Thus, the number of independent resolution elements along L_x detectable at a distance L_z is given by F_x. The same holds for F_y.

So far we have referred to Eq. (5) in free space, considering p as an external perturbation, thus introducing a bare aspect ratio. In fact, the constitutive matter equations provide a generally nonlinear and nonlocal functional dependence p(E).

To define a dressed aspect ratio, we must consider the linear part of the polarization

$$\text{P=ε}{}_0\text{χ}{}^{\mathrm{(1)}}\text{∗E}\text{,}$$

where χ⁽¹⁾ is the linear susceptibility and the star convolution operator accounts for nonisotropic effects (tensor relations) as well as for nonlocalities in time and space (temporal and spatial dispersion). Furthermore we must consider the appropriate boundary conditions. Let us confine the dynamics within a volume bound by two parallel mirrors in the longitudinal direction and consider free lateral boundary conditions. We have the standard longitudinal and transverse modes well known in laser phenomenology.

In the active case, i.e. when the medium provides energy to the field, the frequency width Δω_act of the medium gain line can be compared with the so-called FSR (free spectral range) that is, with the separation Δω_i,i+1≡c/2L_z of the adjacent ith and (i+1)th longitudinal modes. The ratio

$$\text{Γ}{}_{\mathrm{z}}\text{=}\text{Δ}\text{ω}{}_{\mathrm{<mtext>act</mtext>}}\text{Δ}\text{ω}{}_{\mathrm{i,i+1}}$$

defines a longitudinal aspect ratio. If Γ_z<1 only one longitudinal resonance can be excited, and the cavity field is uniform along z, whereas if Γ_z>1 many excited longitudinal modes give rise to a short pulse whose spatial length is smaller than the cavity length L_z.

In Fig. 1a we report the frequency position of the modes and the medium line for 0-, 1- and 2-dimensional cases, in Fig. 1b we report the corresponding wavenumbers.

In fact, for the 1- and 2-dimensional cases, we do not consider different wavenumbers, but we use a pseudo-spectral method consisting in Fourier expanding around the central wavenumber k₀, and considering the residual spread as a slow time-space dependence in terms of evolution equations including the nonlinearities.

In Table 1 we list optical pattern forming systems of different dimensions. It is understood that along a dimension there is no evolution whenever the corresponding aspect ratio is less than 1.

It is well known that a discrete nonlinear dynamical system can undergo a chaotic motion, that is, at least one of its Liapunov exponents can be positive, only when the number of degrees of freedom (phase space dimension) is at least 3.

We find it convenient to refer to dissipative systems, that is, systems with damping terms for which the phase-space volume is not conserved. In such systems the sum of all Liapunov exponents is negative, and initial conditions tend asymptotically to an attractor [4].

For dimensions N=1, the attractor is a fixed point, for N=2 a fixed point or a limit cycle, for N=3 it can be a fixed point (all the three Liapunov exponents negative), a limit cycle (two Liapunov exponents negative and one zero), or a torus (one Liapunov exponent negative and two zero), or even a chaotic attractor (one Liapunov exponent negative, one zero and one positive).

An example of chaotic motion is offered by the Lorenz model of hydrodynamic instabilities [5], which corresponds to the following equations, where the parameter values have been chosen so as to yield one positive Liapunov exponent:

$$\text{x}\text{̇}\text{=−10x+10y,}\text{y}\text{̇}\text{=−y+28x−xz,}\text{z}\text{̇}\text{=−}\text{8}\text{3}\text{z+xy}\text{,}$$

where now $\text{x,}\text{y,}\text{z}$ are suitable variables, and dots denote temporal derivatives.

Also the fundamental equations for field matter interaction can exhibit chaotic features. Indeed, if we couple Maxwell equations with Schroedinger equations for N atoms confined in a cavity, and expand the field in cavity modes, keeping only the first mode which goes unstable, its amplitude E is coupled with the collective variables P and Δ describing respectively the atomic polarization and the population inversion. The resulting equations are

$$\text{E}\text{̇}\text{=−kE+gP,}\text{P}\text{̇}\text{=−γ}{}_{\mathrm{\perp }}\text{P+gEΔ}\text{,}\text{Δ}\text{̇}\text{=−γ}{}_{\mathrm{\parallel }}\text{(Δ−Δ}{}_0\text{)−4gPE}\text{.}$$

For simplicity, we consider the cavity frequency at resonance with the atomic one, so that we can take E and P as real variables and we have three real equations. Here $\text{k,}\text{γ}{}_{\mathrm{\perp }}\text{,}\text{γ}{}_{\mathrm{\parallel }}$ are the loss rates for field, polarization and population, respectively, g is a coupling constant and Δ₀ is the population inversion which would be established by a pump mechanism in the atomic medium, in the absence of the coupling. While the first equation comes from the Maxwell equation, the two others imply the reduction of each atom to a two-level atom resonantly coupled with the field, that is, a description of each atom is an isospin space of spin 1/2. The last two equations are like Bloch equations which describe the spin precession in the presence of a magnetic field. For such a reason, Eqs. (10) are called Maxwell–Bloch equations.

The presence of loss rates means that the three relevant degrees of freedom are in contact with a sea of other degrees of freedom. In principle, Eqs. (10) could be deduced from microscopic equations by statistical reduction techniques [6], [365].

The similarity of Eqs. (10) with Eqs. (9) would suggest the easy appearance of chaotic instabilities in single mode, homogeneous-line lasers. However, time-scale considerations rule out the full dynamics of Eqs. (10) for most of the available lasers. Eqs. (9) have damping rates which lie within one order of magnitude of each other. On the contrary, in most lasers the three damping rates are widely different from one another.

The following classification has been introduced [7].

Class A lasers (e.g. He–Ne, Ar, Kr, dye): γ_⊥≃γ_∥⪢k. The two last equations of Eqs. (10) can be solved at equilibrium (adiabatic elimination procedure) and one single nonlinear field equation describes the laser. Since N=1, only a fixed point attractor is supported in this case.

Class B lasers (e.g. ruby, Nd, CO): γ_⊥⪢k≃γ_∥. Only polarization is adiabatically eliminated (middle equation of Eqs. (10)) and the dynamics is ruled by two rate equations for field and population. Fixed points and periodic oscillations are then supported.

Class C lasers (e.g. far infrared lasers): γ_⊥∼γ_∥∼k. The complete set of Eqs. (10) has to be taken into account, hence Lorenz like chaos is feasible.

A series of experiments on the birth of deterministic chaos in CO₂ lasers (Class B) was initially carried out in various configurations, namely: with the introduction of a time dependent parameter to make the system nonautonomous [8]; with the injection of a signal from an external laser detuned with respect to the main one [7]; with the use of a bidirectional ring cavity [9], and with the addition of an overall feedback, besides that provided by the cavity mirrors [10], [387]. All these configurations imply a third degree of freedom, besides the two ones intrinsic of a class B laser, thus making chaos possible.

Lorenz chaos was later extensively investigated in far infrared lasers, corresponding to molecular (rotational) transitions with damping rates comparable with the cavity decay rate [11].

A few comprehensive reviews on the subject are available in [12], [13].

In the forthcoming sections, Maxwell–Bloch equations will be generalized as follows:

• introducing a detuning Δω=ω−ω_a between the laser frequency ω and the center ω_a of the atomic line, thus E and P must be considered as complex quantities, and Eqs. (10) transform to a set of 5 real equations;

• increasing the cavity aspect ratios, transforming the problem from a discrete to an extended one in the real space. One should carefully distinguish between the dimensions N of the phase space (N=number of dynamical degrees of freedom) and the dimension D of the real space within which the physics takes place. As one goes from D=0 to D=1, then the time derivative Ė in the equation for E will be replaced by ∂_t+c∂_z. This extension was first considered in connection with the propagation of pulses in an excited two-level medium in presence of scattering losses compensating for the gain, so that soliton pulses (so called π pulses) resulted propagating at the light speed, the nonlinearity compensating for the dispersion [14]. In large aspect ratio cavities (D=2) the time derivative in the equation for E will be replaced by ∂_t+(ic/2k)∇_⊥², where ∇_⊥²≡∂_x²+∂_y² is the transverse Laplacian operator. This extension was first considered in filamentation problems [15].

In fact, the self focusing and self defocusing phenomena imply the whole 3D structure of the field, and hence require the use of the operator ∂_t+c∂_z+(ic/2k)∇_⊥², where the time derivative will be dropped in case of stationary phenomena [16].

As it has become customary, we call active optical devices those ones in which the medium is in an excited state and it can transfer energy to the light field, whereas we call passive optical devices those in which the medium is in its ground state. As a result, active optics can start from spontaneous emission processes, whereas passive optics always requires an incident field. Laser are prototypical examples of active optics. In the case of photorefractive oscillators, which are based on two or four wave mixing (see Appendix A) one or more input fields will have the role of pumps.

As we will see in this review, evidence of reliable patterns in lasers has been made difficult by two reasons: (i) laser cavities have usually small transverse aspect ratios, (ii) the time scales of the dynamics are so fast that only averaged patterns can be visualized. Both limitations are not present in a photorefractive oscillator which has then become a very useful testbench to explore active pattern formation.

Let us detail what we mean by the title of this report. “Pattern formation” refers to the fact that in an extended nonlinear medium, above a suitable threshold, any uniform amplitude distribution becomes unstable and the space–time distribution of the amplitude splits into correlated domains. The first symmetry is ruled by the boundary conditions. We call a pure pattern an eigenstate of the propagation problem associated with the linear part of the dynamics, in the absence of nonlinearities. Nonlinearities imply the interaction of many eigenstates, or pure patterns. This may induce many scenarios of “Pattern competition”, which, in order of increasing complexity, are respectively:

• winner-takes-all dynamics: one pure pattern prevails on the others;

• cooperation: many pure patterns coexist (e.g. hexagons as coexistence of three roll sets at 120° with each other, mutually coupled by a quadratic nonlinearity);

• locking of space and time phases of the wavevectors corresponding to different hexagon families;

• time alternation of patterns, which fill the whole available region, inducing time chaotic phenomena and yet keeping a spatial coherence;

• segregation of different domains in each of which a different stationary pattern is present;

• space–time chaos with a limited correlation length and correlation time.

Examples of the six cases are discussed in the following.

This report is organized as follows.

In Section 2 we report the theoretical and experimental results on pattern formation and competition in nonlinear active optics. In Section 2.1 we show how Maxwell–Bloch equations can be generalized to account for spatial dependence, and how the two homogeneous stationary solutions (lasing solution and nonlasing solution) bifurcate to patterned states which can be described by suitable amplitude equations. In Section 2.2 we review the most significant results on patterns in laser systems, in both the low and high dimensional cases. In Section 2.3 we report one of the first evidences of spatial dependent chaotic regimes in a photorefractive oscillator, and we show that the main features of its dynamics can be captured by simple normal forms equations accounting for the symmetry requirements.

In Section 3 we review the case of passive nonlinear optics, mainly referring to Kerr media. We consider various experimental situations, namely, filamentation in single-pass systems (Section 3.1), formation of solitons in single-pass systems (Section 3.2), counterpropagating beams in a nonlinear medium (Section 3.3), patterned states originated by a nonlinear medium confined within an optical cavity (Section 3.4), patterns in a nonlinear slice with optical feedback (Section 3.5) and the effects of nonlocal interactions in the selection of the pattern shape and of the relevant pattern size (Section 3.6).

In Section 4 we define what is a phase singularity, or defect, or optical vortex, and we discuss how the presence of these defects is related to the dynamics of the pattern forming system. In Section 4.1, we summarize the main properties of phase singularities and topological defects in linear waves, and we report their scaling laws. In Section 4.2, we analyze the case of phase singularities in nonlinear waves, and we report the experimental evidence of the dynamical transition from a boundary dominated regime to a bulk dominated regime, in terms of the defect statistical properties.

In Section 5 we discuss the perspectives of this area of investigation, referring to some challenging cases, namely, the formation and evolution of localized structures in nonlinear optics (Section 5.1), the problem of stabilizing unstable patterns within space–time chaotic states (Section 5.2), and the pattern formation in atom optics (Section 5.3), where the diffractive properties are associated with the atom Schroedinger field, and the coherence requirements have only recently been reached by the evidence of atomic BEC (Bose–Einstein condensation) [17], [388], [389], as well as of the atom laser [18].