1. Introduction

Spatially extended nonlinear systems, far from the thermodynamic equilibrium, can display a rich variety of dissipative structures or patterns [1,2]. Nonlinear optical cavities, also lasers, belong to this class of pattern forming systems [3,4], when the cavity Fresnel number is large enough (wide aperture cavity) for sustaining a large number of transverse modes. In optical systems, dissipative structures appear in the plane orthogonal to the cavity axis, i.e., they are one- or two dimensional structures, hence the name transverse patterns, although three-dimensional patterns have also been predicted [3,4]. Among the variety of possible transverse patterns, localized structures (LSs) are of particular interest because of their potential for information processing purposes [5].

Recently, LSs in bidirectional lasers have been predicted [6] and attracted interest. A bidirectional laser is a ring cavity laser in which emission is possible in two directions, say clockwise and counterclockwise. LSs in this laser were predicted in the model of two coupled Ginzburg-Landau equations [6], see Eqs. (1) below, for slightly anisotropic laser cavities (anisotropic in the sense that the losses for the two counter propagating fields are different). This is the first prediction of LS in “pure” lasers, whereas usually LSs occur in lasers with some modifications of nonlinearities, and/or with the external actions. Such modified systems are lasers with intracavity saturable absorbers [7] or intracavity (Kerr) nonlinearity [8], lasers with a cw [9] or a periodic injection [10,11] (“rocked” lasers), or lasers with frequency-selective optical feedback [12].

The LSs reported in [6] are based on the fact that a plane-wave laser cannot sustain stable bidirectional-, but rather unidirectional cw emission in either of two possible emission directions, as both field components compete for a common population inversion [13]. However, when a plane-mirrors large aspect ratio cavity is considered (i.e., a cavity that allows the oscillation of, in principle, any transverse mode), the above mentioned restriction on bidirectionality holds only locally. This means that the emission in one direction occurs in a given region of the plane transverse to the cavity axis, which we take to be the z-axis, whereas the emission in the other direction occurs in an adjacent transverse region. The continuity of the solutions between these two regions (domains) in the transverse plane implies the existence of a front connecting the two solutions. Fronts in general tend to move, which usually leads to eventual unidirectional motion in the entire transverse plane. However, as shown in [6], if the two intracavity fields have slightly different losses, two of the fronts approach each other and can lock forming stable LSs. These LSs look like bright LSs in the weak field (the field with more loses) while in the strong field they resemble dark LSs, as a dip appears in the otherwise homogeneous (bright) solution.

In [6] these LSs were considered to be cavity solitons (CSs). But as remarked in [14] the LSs must verify a number of requirements in order to be considered as CSs, and not all of them were verified in the bidirectional laser model of [6]. More specifically, the above described LSs [6] cannot be written and/or erased without disturbing other neighboring LSs, which violates the basic requirement for the information storage and manipulation. In [14] a solution for this drawback was proposed using an injection of a weak signal in one or the two counter-propagating fields. The weak injection fixes the intracavity fields phases what prevents the appearance of traveling waves in the transverse space (equivalently tilted waves in a three dimensional representation). This eventually suppresses the long-scale interaction between the LSs. In this way the LSs are no more affected when neighboring LSs are written or erased, and then they belong to the narrower class of CSs [14]. In the present letter we argue that transverse diffusion can also stabilize the LSs of the bidirectional laser in such a way that they behave as true CSs. Moreover, we show that diffusion stabilizes individual LSs in the sense that their stability area in the parameter space becomes larger in the presence of diffusion. We also show that with diffusion the LSs also become stable for the isotropic case (when both unidirectional waves have the same losses), which is also a serious advantage, from the practical point of view.

2. Bidirectional ring laser model.

In [6] it was shown that a wide-aperture class-A bidirectional laser (for which all material variables relax much faster than the intracavity field), close to the emission threshold; can be modeled by the following coupled Ginzburg-Landau equations:

where F_i(ξ,τ) are proportional to the intracavity field amplitudes of the two counter-propagating fields, A accounts for the pump strength, σ = κ₂/κ₁ with κ_i, the decay rate of field F_i (we take σ ≥ 1 without loss of generality, hence F ₁(F ₂) is the strong (weak) field), τ = κ₁ t, and ξ is the transverse coordinate normalized to the square of the diffraction coefficient. We must mention that σ = 1 in standard ring cavities [15], and a use of intracavity nonreciprocal devices, such as Faraday rotators, would be necessary in order to have σ ≠ 1.

In [6] only diffraction was considered: d = 0. Here we also account for the diffusion term d∂_ξ ² F ₂ that we introduce phenomenologically (d is the diffusion coefficient normalized to diffraction) and study its influence on the stability and dynamics of LSs. There are several origins of the transverse diffusion: On one hand, the diffusion is always present in any nonlinear optical cavity because of diffraction losses in finite aperture cavities, which are properly modeled with the diffusion term [16]; on the other hand, the appropriate model for large aperture class-C lasers includes a Swift-Hohenberg term of the form (δ + ∂_ξ ²)² F _i (δ is the cavity detuning) that works as a “superdiffusion” [3,4]. One must keep in mind however that for these more general conditions (including nonzero detuning and arbitrary decay rates) the cross-coupling coefficients can become complex and cross-phase modulation appears besides the cross-gain modulation. As we want to keep our study as general as possible, we study the simplest Eqs. (1).

It is worth mentioning that there are other types of lasers that under appropriate conditions are also described by two coupled Ginzburg-Landau equations similar to Eqs. (1). These lasers include, e.g., lasers with two orthogonal polarizations (Zeeman lasers) [17] and cascade lasers [18]. In Zeeman lasers, the two fields correspond to the two orthogonal polarization components of the field (see [19] for recent experimental research on CSs in VCSELs involving the two polarization modes). In cascade lasers, the two fields of different frequencies, are coupled to two adjacent atomic transitions. Hence, our results on the positive role of diffusion could be extended to such laser systems.

3. Numerical results.

We have numerically integrated Eqs. (1) by means of a split-step method on a grid of 512 spatial points and with total integration length L = 316 (also L = 707 when needed). We have considered a flat-top profile for the pumping A (modeled with a super-Gaussian function) in order to take into account the finite transverse extension of the laser excitation.

Figure 1 contains the first main result of this letter: LSs exist in the isotropic cavity provided there is some diffusion. Figure 1(a) shows the intensity profile of a stationary LS obtained for σ = 1, A = 1.4 and d = 0.3. The two counter-propagating fields are represented by continuous and dashed lines, respectively. Figure 1(b) shows the evolution of the corresponding phase profile. The tilted equiphase contours indicate the emission of traveling waves from the LSs, whose relevance is considered below. Note that the existence of the stationary LSs for σ = 1 is a new result made possible by the introduction of diffusion. The result is important as it implies that there is no need to introduce any nonreciprocal device within the laser cavity in order to achieve asymmetric losses.

 

Fig. 1. Bidirectional laser localized structure as obtained from Eqs. (1) for σ= 1 (isotropic case), d = 0.3 and A = 1.4. In (a) the dashed (full) lines correspond to the transverse intensity distribution for the strong (weak) field. In (b) the phase is shown as a function of time.

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In Fig. 2 we represent the domain of existence, in the plane <σ,d>, of the LSs for a fixed value of the incoherent pump A = 1.4. The case d = 0 was already treated in [6] where for σ→1 no proper LSs exist. Figure 2 shows that the influence of diffusion is twofold: On one hand, it reduces the domain of the non stationary (i.e., self-pulsing) LSs till it disappears for d ≈ 0.1, allowing for the existence of steady LSs even in the isotropic case σ = 1; and on the other hand, the diffusion first enhances the domain of existence of LSs, and then reduces it, but quite weakly. In conclusion, from the view point of the domain of existence of LSs, a reasonable diffusion (say 0.1 < d < 0.5) plays a positive role for the case shown in Fig. 2. Thus diffusion is advantageous for the individual LSs as they become more stable (the pulsing domain shrinks and then disappears) and can exist for σ = 1.

 

Fig. 2. Domain of existence of the LSs in the plane <σ,d> for a pump value A = 1.4. Shaded areas correspond to the regions the stable LSs are found.

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Next we show the second main result of this letter namely that the LSs become true CSs for appropriate diffusion values. With that aim, we explore the dynamics of a cluster of LSs in the presence of diffusion. Figure 3 shows the evolution of such a cluster with eight LSs which are injected at different times: The central LS is injected at τ= 0, and the rest are injected later. The panels correspond to three different (increasing) values of diffusion for a fixed σ = 1 (however are representative of what happens for other values of σ). The bright LS corresponding to the weak field is depicted in the figure.

 

Fig. 3. Evolution of a cluster of eight LSs for σ = 1 for different values of the diffusion coefficient. From left to right: d = 0.1 (a), d = 0.2 (b) and d = 0.3 (c). Time span is 5000 and the transverse dimension L=707.

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It is evident that for the lower values of d the injected LSs do not survive, see cases (a) and (b) in Fig. 3. This is because they are advected by the traveling waves emitted by the sources located at the central LS, see Fig. 1(b). As the diffusion coefficient is increased, the injected LSs survive longer and for large enough d, see case (c) in Fig. 3, they remain. Note also that, in this parameter region, some of the LSs can be erased, without affecting significantly the other ones in the array. In this sense, as stated in the introduction, they can be considered as true CSs.

 

Fig. 4. Evolution of a cluster of two LSs for σ = 1 and d = 0.1 (upper row) and d = 0.3 (lower row). Left column shows the full picture, with L = 316 and time span 5000. Central column shows a magnification of the red box in the left column, with L = 120 and time running from 750 to 2000. Right column shows the spatio-temporal distribution of the phase for the field distribution in the inset

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The stabilization by diffusion can be traced back to the dynamics of the traveling waves emitted by the sources. Each LS is a source of two traveling waves propagating outwards from the structure. (More precisely the LS consists of two sources and one sink, see Fig. 1(b), but in overall it behaves as a source from a long distance perspective.) In the absence of diffusion, the traveling waves emitted by the LSs interact with the neighboring ones, pushing them apart, and eventually destroying them, see Fig. 4. The increasing diffusion weakens the effect of these traveling waves as their velocity (equivalently the tilt angle of the tilted wave) reduces with increasing d. The effect resembles the dependence of the tilted waves emitted by sources in the usual complex Ginzburg-Landau Equation [20] where, as shown analytically [21], the velocity of the traveling waves decreases with increasing diffusion.

 

Fig. 5. Phase velocity as a function of diffusion d for σ = 1.020 (a), 1.010 (b), 1.005 (c), 1 (d). The velocity is calculated by evaluating the derivative of the phase with respect the time, at a point sufficiently far from the center of a single LS.

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The situation here is more involved, and the analytical evaluation is more complicated. Therefore we numerically calculated the phase velocity of the traveling wave as emitted by an isolated LS, which is shown in Fig. 5. Notice the rapid decrease of this phase velocity with increasing diffusion for different values of σ, which qualitatively explains the numerically observed effect.

4. Conclusions

We have numerically demonstrated that transverse diffusion stabilizes localized structures (LSs) in the wide aperture bidirectional class-A laser model of [6]. The stabilization is threefold as diffusion: (i), allows for the existence of LSs in isotropic cavities; (ii), reduces the parameter region of self-pulsing LSs; and (iii), converts the LSs in true CSs, as the LSs in an ensemble can be written/erased independently.

We have traced back the reason why diffusion affects in such an important way the stability and dynamics of LSs to the dynamics of the traveling-waves emitted by the LSs. We have shown numerically that the phase velocity of these traveling waves is reduced by diffusion, which suggests a weakening of the interaction between neighboring LSs. However this is a point that would require further attention as the interaction between LSs is not well understood yet.