We study the influence of field diffusion on the spatial localized structures (cavity solitons) in bidirectional lasers. We find threefold positive role of the diffusion: 1) it allows for the existence of solitons in cavities with equal losses for the two fields; 2) it increases the stability range of the individual (isolated) solitons; and 3) it reduces the long-range interaction between cavity solitons, allowing for the independent manipulation (writing and erasing) of individual structures.
©2009 Optical Society of America
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 .
Recently, LSs in bidirectional lasers have been predicted  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 , 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  or intracavity (Kerr) nonlinearity , lasers with a cw  or a periodic injection [10,11] (“rocked” lasers), or lasers with frequency-selective optical feedback .
The LSs reported in  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 . 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 , 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  these LSs were considered to be cavity solitons (CSs). But as remarked in  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 . More specifically, the above described LSs  cannot be written and/or erased without disturbing other neighboring LSs, which violates the basic requirement for the information storage and manipulation. In  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 . 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  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 Fi(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 κi, the decay rate of field Fi (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 , and a use of intracavity nonreciprocal devices, such as Faraday rotators, would be necessary in order to have σ ≠ 1.
In  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 ; 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)  and cascade lasers . In Zeeman lasers, the two fields correspond to the two orthogonal polarization components of the field (see  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.
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  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.
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.
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.
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  where, as shown analytically , the velocity of the traveling waves decreases with increasing diffusion.
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.
We have numerically demonstrated that transverse diffusion stabilizes localized structures (LSs) in the wide aperture bidirectional class-A laser model of . 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.
This work has been supported by the Spanish MEC, MICINN and the European Union FEDER through Projects FIS2005-07931-C03 and FIS2008-06024-C03. K.S. acknowledges financial support from Generalitat Valenciana through the program Ajudes per a estades d’investigadors invitats en centres de la Comunitat Valenciana.
References and links
1. M. C. Cross and P. C. Hohenberg, Pattern Formation Out of Equilibrium, Rev. Mod. Phys. 65, 851–1112 (1993). [CrossRef]
2. M. Tlidi, M. Taki, and Th. Kolokolnikov eds., Focus Issue: Dissipative Localized structures in extended systems, Chaos 17, Issue 3 (2007).
3. P. Mandel and M. Tlidi, Transverse dynamics in cavity nonlinear optics (2000-2003), J. Opt. B: Quantum and Semiclass. Opt. 6, R60–R75 (2004). [CrossRef]
4. K. Staliunas and V.J. Sánchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators, (Springer, Berlín, 2003)
5. W.J. Firth and C.O. Weiss, Cavity and feedback solitons, Opt. Photonic News 13, Issue 2, 54–58 (2002). [CrossRef]
7. G. Slekys, K. Staliunas, and C. O. Weiss, Spatial localized structures in resonators with saturable absorber, Opt. Commun. 149, 113–116 (1998). [CrossRef]
8. A. M. Dunlop, E. M. Wright, and W. J. Firth, Spatial solitons laser, Opt. Commun. 147, 393–401 (1998). [CrossRef]
9. X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, Cavity solitons in a driven VCSEL above threshold, IEEE J. Sel. Top. Quantum Electron. 12, 339–351 (2006). [CrossRef]
10. G. J. de Valcárcel and K. Staliunas, Excitation of phase patterns and spatial solitons via two-frequency forcing of a 1:1 resonance, Phys. Rev. E 67, 026604 (2003). [CrossRef]
11. A. Esteban-Martín, M. Martínez-Quesada, V. B. Taranenko, E. Roldán, and G. J. de Valcárcel, Bistable Phase Locking of a Nonlinear Optical Cavity via Rocking: Transmuting Vortices into Phase Patterns, Phys. Rev. Lett. 97, 093903 (2006). [CrossRef] [PubMed]
12. P. V. Paulau, A. J. Scroggie, A. Naumenko, T. Ackemann, N. A. Loiko, and W. J. Firth, Localized traveling waves in vertical-cavity surface-emitting lasers with frequency-selective optical feedback, Phys. Rev. E 75, 056208 (2007). [CrossRef]
13. H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, Bidirectional ring laser: Stability analysis and time-dependent solutions, Phys. Rev. A 37, 470–497 (1988). [CrossRef] [PubMed]
15. M. Piché and P. Belánger, On the loses of the counter-propagating modes of ring cavities, IEEE Journ. Quantm Electron. QE-20, 1303–1307 (2008).
16. A. M. Dunlop, W. J. Firth, and E. M. Wright, Opt. Commun. 138, 211 (1997). [CrossRef]
18. J. García-Ojalvo, R. Vilaseca, and M. C. Torrent, Coupled pattern formation near threshold in a broad-area cascade laser, Phys. Rev. A 56, 5111–5120 (1997). [CrossRef]
19. X. Hachair, G. Tissoni, H. Thienpont, and K. Panajotov, Linearly polarized bistable localized structure in medium-size vertical-cavity surface-emitting lasers, Phys. Rev. A 79, 011801(R) (2009). [CrossRef]
20. W. van Saarloos and P. C. Hohenberg, Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Physica D 56, 306–367 (1992). [CrossRef]
21. P. Coullet, T. Frisch, and F. Plaza, Sources and sinks of wave patterns, Physica D 62, 75–79 (1993). [CrossRef]