1. Introduction

Cavity solitons (CS) are robust, self-sustained localized structures that can be excited across the transverse section of the light beam produced by a nonlinear optical resonator with high Fresnel number [1–4]. CSs are interesting objects from the applicative viewpoint as they can serve as bits of an optical memory, while their manipulation and control can allow the parallel processing of information [5]. Especially versatile systems for the generation of CSs are those in which the nature of the nonlinear interaction is phase-sensitive and the generated field can take two possible phase values, differing by π. Thus, such systems display phase bistability. A paradigm of this behavior is represented by the degenerate optical parametric oscillator (DOPO) in which the generated, subharmonic field is phase-locked to the pump field, modulo π. In these phase bistable systems the light field phase can take different values in different regions of the transverse plane, which are separated among them by dark lines. These lines are known as domain walls (DW) or dark cavity-solitons because of their resemblance with the dark solitons of the nonlinear Schrödinger equation. These DWs are essentially 1D structures; related 2D objects are optical droplets (a circular front connecting the two equivalent, oppositely phased homogeneous solutions) [6] and dark-ring cavity solitons (DRCS), where a circular front separates an inner, high intensity peak, from an outer, homogeneous light distribution, of opposite phase [4].

In nonlinear optical cavities DWs were first described in DOPOs [7, 8] and have been experimentally demonstrated in a photorefractive oscillator [9]. In [7] DWs were shown to be stable structures for both one and two transverse dimensions. While for one transverse dimension this seems to be always true irrespectively of the size of the excited region, this is not the case for two transverse dimensions: It has been shown [10] that DWs are only transient solutions of the system that dissapear through the borders of the illuminated region. This would make these structures of reduced practical interest in real systems unless something is done for stabilizing and controlling them. In this sense, the stabilization of DWs in DOPOs pumped by helical waves has been theoretically shown [10].

In this paper two simple methods for controlling and stabilizing DWs are proposed. The methods are illustrated in the case of DOPO but in principle they can be applied to any phase-sensitive nonlinear optical cavity displaying phase bistability. The first method uses an injected field with spatially modulated phase profile [11]. Here the physical mechanism of control relies in the advection exerted by phase gradients on DWs, which tend to be trapped along regions in which the injected field phase is maximum and escape from regions of phase minima, as we show next. The second method uses an injected field with spatially modulated intensity, in the form of a “hole burned” plane wave input. Here the holes of null intensity are able to capture the ends of DWs, and the physical mechanism of control is of topological nature as will be discussed below. When distributed as a regular array these holes delimit spatial optical bits which constitute an optical memory.

It is worth mentioning that although DWs have not yet been observed in DOPOs, their observation in photorefractive oscillators [9] makes possible the implementation of the stabilizing and controlling techniques we present here.

2. Model

In the usual paraxial and mean-field approximations, the dynamics of a doubly resonant, phase-matched, type-I DOPO with planar mirrors is governed by the following dimensionless equations for the pump A ₀(x,y,t) and signal (subharmonic) A ₁(x,y,t) field envelopes [12]:

Denoting by γ ₁ (γ ₀) the cavity decay rate for the signal (pump) field, and by k ₁ the signal field wavenumber within the crystal, t is time normalized to ${\gamma }_1^{-1}$, γ = γ ₀/γ ₁, (x,y) are transverse coordinates normalized to $\sqrt{\frac{c}{{\gamma }_1}{k}_1}$ (c is the speed of light in vacuum), ∇² = ∂ ²/∂X ²+∂ ²/∂y ², Δ₀ and Δ₁ are the cavity detunings of pump and signal fields normalized to γ ₀ and γ ₁ respectively, and E(x,y) is the complex envelope of the injected pump field.

For a spatially homogeneous input field E (in which case E can be taken as a positive real without loss of generality), the DOPO oscillation threshold occurs at E ² = 1 + ${\mathrm{\Delta }}_0^2$ for Δ₁ ≥ 0, giving rise to a spatially uniform signal field, or at E = 1 for Δ₁< 0, giving rise to a spatially modulated signal field in the form of a roll pattern [12, 13]. Consider the particularly simple, perfectly resonant case Δ₀ =Δ₁ = 0. Above threshold the DOPO admits two equivalent nontrivial solutions $\left\{A_0=1,A_1=\pm \sqrt{E-1}\right\}$ in which the signal field can take two values (±) differing by π. In this case DWs exist [7,8] and are stable in two transverse dimensions as far as the injected field is transversely unbounded. But a real optical system has always a finite transverse extension which, in most cases, is usually determined by the transverse size of the input beam. Oppo et al. [10] have shown that for a pump with circular cross section DWs disappear after a transient time and a single phase ultimately covers the whole transverse plane. We show next how DWs can be stabilized and controlled.

3. Two control methods

3.1. Method based on phase modulated pump

This method was originally introduced by Firth and Scroggie [11] in the context of the “optical bullet holes” (cavity solitons in which a bright light spot is surrounded by a homogeneous background) formed in an optical resonator filled by a saturable absorber. These solitons are of a nature different from that of DWs, as the former do not depend on any phase bistability: They rely on the bistability between a homogeneous solution and an hexagonal pattern. Firth and Scroggie [11] showed that phase gradients are effective in advecting bright structures and used this fact to pin them at desired locations. Oppo et al. [10] also showed that DWs are advected by phase gradients as they found that DWs created in DOPOs pumped by helical waves (with a linearly varying phase in the azymuthal direction) were rotating permanently. Here we demonstrate that the phase gradient method is able to control, select, and pin DWs at desired locations.

In order to get some insight into the effect of a spatially modulated input phase we write E = ∣E(x,y)∣exp[iψ(x,y)] and rewrite Eqs. (1) and (2) for the variables $F_0=A_0\exp [-\frac{1}{2}\mathit{i\psi }\left(x,y\right)]$ [11],

where

is a convective derivative, κ ₀=1 + ∇² ψ/2γ, κ ₁ = 1 + ∇² ψ/2, δ ₀ = Δ₀ +∣∇ψ∣²/2γ and δ ₁ = Δ₁ +∣∇ψ∣²/4. Eqs. (3,4) are formally equivalent to the original DOPO Eqs. (1,2), but now the parameters are, in general, space dependent. On the one hand linear losses (κ _0,1) can increase (reduce) for ∇² ψ< 0 (∇²< 0). On the other hand the cavity mistunings are always increased as Δ_0,1 ≥ Δ_0,1. But the most important modification is the appearance of the advection term (∇ ψ)∙ ∇ accompanying the time derivative in Eq. (5), which indicates that any feature in the transverse plane will move towards phase maxima with velocity v =∇ ψ. This suggests that the use of a convergent (divergent) input field could stop (increase) the natural tendency of DWs to escape the illuminated region.

In order to test these ideas the DOPO Eqs. (1) and (2) were integrated for a top hat shape input pump: Its intensity ∣E∣² = P ² was uniform inside a circle of radius r ₀, falling off very quickly to zero outside it. As a consequence the intracavity pump and signal fields were off outside the circle. Its phase ψ(x,y) was allowed to be spatially varying. A split-step algorithm with periodic boundary conditions was used on a square grid of side L(>2 r ₀) with 256×256 points. Since both pump and signal fields were excited only within a finite region of the integration window, the chosen boundary conditions had no effect on the DOPO dynamics. We show next results obtained for γ= 1, Δ₀ =Δ₁ = 0, P = 3, L = 100, and r ₀ = 45 (other parameters, whenever Δ₁ ≥ 0, yield similar results), and for a temporal step δt= 0.01 (smaller δt ’s were also used for asessing the convergence of the results). Also other spatially confined pumps with different (e.g., Gaussian) intensity profiles were used, yielding similar results.

If the pump phase is flat (ψ= constant) we observe that a straight DW excited in the initial condition leaves the illuminated region after a transient, in agreement with [10]. This behavior is due to local curvature driven dynamics, which develops as the initially straight DW tends to allign itself perpendicularly to the circular boundary. This dynamics is faster for DWs initially excited far from the pump beam center but also occurs for straight DWs initially crossing that center, whenever noise is considered. After our previous discussion one can envisage a way to stabilize this DW: If the pump wavefront is given a cylindrical shape so that it has a maximum along a diameter of the circular input, the DW should move towards that diameter and then remain there forever. Fig. 1 shows this pinning effect for a cylindrical wavefront, convergent along the x direction. The initial condition was a straight DW located at x = 0.56r ₀. The DW moves to the left until it finally settles at the vertical line x = 0, where ψ is maximum.

 

Fig. 1. A DOPO DW is advected towards the pump beam centre. A cylindrical pump wavefront, convergent along the horizontal direction, x, is used. Parameters: γ= 1, Δ₀ = Δ₁ = 0, P = 3, r ₀ = 45, L = 100. At t = 0 a straight DW is excited at x = 0.56r ₀. Plots are given at t = 50 (left), 600 (middle), and 2000 (right). Here, and in the rest of figures, the intensity of the signal field is plotted, except when noted.

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One can wonder whether this control mechanism will be effective in the presence of noise. We have integrated the DOPO equations with additive noise terms (δ-correlated Gaussian white noise of zero mean) for different values of the noise variance. The results show that the mechanism is robust and works perfectly well for any noise value that allows the formation of DWs. As an example, in a numerical experiment as that of Fig. 1 but with added noise, we arrive at the same final state (left panel in Fig. 2). The intensity trace along a horizontal cut shown in the right panel of Fig. 2 shows that, even for very large noise levels, the DW is effectively stabilized and kept attached to the prescribed position. Let us remark that noise does not introduce any noticeable delay in reaching the final state.

 

Fig. 2. Final state, for the same parameters as in Fig. 1, reached at t = 2000 in the presence of noise. The right panel shows the intensity trace of the signal field, ∣A ₁∣² , along the horizontal diameter. The noise variance is 3×10^-3 in the dimensionless units of Eqs. (1) and (2).

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We have also investigated the formation and evolution of DWs starting from a random seed. In this case, if the pump phase is flat, we observe the formation of several phase domains of irregular shapes which in the course of time disappear in a way similar to Fig. 9 in [10]. One could ask whether in this so disordered situation a single DW can be picked and stabilized. The positive answer is given in Fig. 3, where the same pump phase profile as in Fig. 1 was used, but now the simulation started from a random seed. After a short transient a labyrinthic pattern is formed. As time goes on all DWs tend to dissapear except one that tends to align itself with the pump phase maximum. We observe that the dynamics can be very slow especially if DRCSs bond to the DW as it occurs in Fig. 3. But one can remove these defects as well by giving an additional curvature to the pump wavefront in the yet neutral direction y, as evidenced in the bottom row of Fig. 3, which forces the defects to exit the illuminated domain, and the final state is a single vertical DW [14]. The previous situation can be generalized to several phase maxima as shown in Fig. 4.

 

Fig. 3. Selection and control of a single DW starting from noise. Rest of conditions as in Fig. 1. From left to right, plots are given at t = 200, 400, 800 (top row) and t = 1250, 2750, 3350 (bottom row). In order to remove the two dark ring cavity solitons bond to the DW, at t = 2750 the pump phase was made divergent along the vertical direction (see text).

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Fig. 4. Selection and control of two DWs starting from noise. Parameters as in Fig. 1, except for the pump wavefront wich has two local maxima at the vertical lines x = ±0.28r ₀ and is divergent along the vertical direction (see text). From left to right, plots are given at t = 10, 50, 200, (top row) and t = 400, 800, 1000 (bottom row).

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We have checked also the effect of noise when the initial seed is random. As in the previous case, we have found that the control mechanism is effective in selecting and pinning a particular DW even in the presence of large noise.

This method can be used as well for stabilizing DRCSs in much the same way as it was used in [11] for stabilizing “optical bullet holes”. We illustrate this in Fig. 5. The left panel shows the initial condition, where nine DRCSs were written in a somewhat disordered manner. Then an egg-box like phase profile with 3×3 maxima was given to the injected field. The result after a transient is shown in the middle panel where the nine DRCSs are finally fixed at the positions of the input field phase maxima. The fact that this constitutes an optical memory is demonstrated in the right panel where the central “bit” has been removed by injecting a narrow pulse of appropriate phase [15], without disturbing the remaining bits.

 

Fig. 5. Control of DRCSs. A signal seed in a quasi-ordered shape (left) is controlled by an egg-box phase profile consisting of 3×3 phase maxima. At t = 100 (middle panel) the marks have developed into DRCSs that are pinned by the phase maxima. The right panel shows the erasure of the central DRCS obtained 10 time units after a narrow pulse superimposed to it has been injected. Other parameters as in Fig. 1.

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The calculation was repeated in the presence of additive noise (same variance as in Fig. 2) obtaining the same results. As an example we just show in Fig. 6 the state analogous to the middle panel in Fig. 5. The right panel in Fig. 6 evidences the high level of noise present in the simulation.

 

Fig. 6. Control of DRCSs in the presence of noise. The right panel shows the intensity trace of the signal field along the horizontal diameter at t = 100 for the same initial condition and parameters as in Fig. 5 (the noise variance is 3×10^-3).

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3.2. Method based on intensity modulated pump

As DWs are topological solitons [7] a different control method can be envisaged that makes use of input pumps whose transverse profile contains zeroes (or regions below DOPO oscillation). In such case there are regions in the transverse plane where the signal field is off, to which a DW can be attached. The point is that a DW cannot be detached from a “hole” as, in that case, a DW with a free end would exist. But this cannot occur in phase-locked systems as a DW separates regions with opposite phases and the only possibility for that would be that a continuous 2π variation of the signal phase would develop around that free end, similarly to a vortex. An example of DW stabilization using this method is given in Fig. 7. In the left panel the initial condition is shown: A pair of DWs (note that an even number of DWs ending in the hole must be excited as the signal phase must be continuous after a circulation around the hole) were written in such a way that they were not perpendicular to the pumped region boundaries. Curvature forces the two DW to move (central panel) until they become perpendicular to the boundaries (right panel) where the DW remain at rest. If the initial condition is such that the DW are perpendicular to the boundaries, they do not move. This method can be combined with the phase gradient method in order to move the DWs at will.

 

Fig. 7. Stabilization of a DW by a doughnut shape pump. Same parameters as in Fig. 1. The time is t = 5, 50, 1000 from left to right.

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The method can be exploited further to constuct an optical memory as shown in Fig. 8. In this case the input pump contains an array of 4×4 holes. In the left panel the signal field phase is spatially uniform, and bits of information can be addressed by applying gaussian pulses whose phases are opposite (with a large tollerance) to the existing one, middle panel. In this way square patches (phase domains) of a given phase can be written. This can be done independently in each of the square regions delimited by holes, as well as use of oppositely phase pulses allows to erase existing domains, right panel.

 

Fig. 8. Codification in a 3×3 matrix. White corresponds to a positive value, dark gray corresponds to a negative value, and light gray (like in the circular holes) indicates a null value of the real part of the signal field. The left panel shows the field without any codification. After injecting appropriate signals, the system exhibits the distribution shown in the central panel where cells (3,1), (2,2), (3,2) have been marked (the phase value has been flipped from negative to positive). These marks can be changed after injecting new signals: The right panel shows new marks in cells (1,3), (2,2), (3,1). Time difference between the different panels is t = 10. Other parameters as in Fig. 1, except Δ₁ = Δ₀/2 = 1.2.

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4. Conclusions

We have demonstrated two robust methods for stabilizing and controlling DWs. The first one is based on the advection introduced by pump phase gradients, so that DWs tend to be trapped along the lines of pump phase local maxima. The use of divergent or tilted wavefronts in order to get rid of defects has been demonstrated as well. The method is also useful for pinning DRCSs at desired locations. The robustness of the method has been demonstrated in the presence of (high levels of) noise. The second method is based on the topological nature of DWs and uses input pumps with zeroes in its transverse spatial distribution which trap DWs. Both methods have been shown to permit the development of optical memories. The techniques have been applied to DOPOs, but they should allow the stabilization and control of DWs and DRCSs in any nonlinear optical resonator exhibiting phase bistability like, e.g., degenerate four-wave mixing oscillators [9] and vectorial Kerr cavities [16,17].