1. Introduction

Cavity solitons are self trapped optical beams that are supported by various kinds of nonlinear resonators both passive and active [1,2]. They may appear in the form of bright/dark peaks of light intensity on a homogeneous background [3,4], or in the form of dark lines separating spatial domains of opposite phase [5,6] or orthogonal polarization [7,8], among others; see Fig. 1. The former are sometimes called amplitude solitons while the latter are usually called phase solitons or phase domain walls (DWs) for obvious reasons.

 

Fig. 1. Schematic representation of one-dimensional phase (upper row) and amplitude (bottom) cavity solitons. The field amplitude is denoted by u (assumed to be real for simplicity). The transverse spatial coordinates are (x,y). Phase solitons (domain walls) connect two states with equal amplitude but opposite phase: they are heteroclinic connections. Amplitude solitons connect the same state (homoclinic connection) by making an excursion to another state. Right column: Density plot of the solitons intensity u ² on the transverse plane showing that domain walls show up as dark lines.

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Phase solitons can appear spontaneously from noise in which case their location at different places of the cavity cross section is arbitrary and uncontrollable. Alternatively, they can be created in a controlled way by addressing laser pulses injected into the nonlinear cavity at particular desired location, see below. Amplitude solitons need to be externally excited [9–14]. Once created, the cavity soliton remains pinned at a fixed position and then can be either erased with another laser pulse or moved to a new position and stored there. Many cavity solitons can be stored and manipulated in an optical cavity of a large Fresnel number, which makes cavity solitons attractive for potential applications in all-optical storage and parallel processing of information [15]. From the experimental side, optically controllable amplitude cavity solitons have been demonstrated in lasers with saturable absorber [9,10], in single-mirror feedback systems [11], and in optically driven semiconductor microcavities [12–14].

Here we shall concentrate on domain walls (DWs) or dark line solitons, Fig. 1, which are topological defects [5] (i.e., once excited, they cannot be removed in a continuous way but through the borders of the illuminated region). These structures appear in extended nonlinear systems with broken phase invariance displaying phase bistability (or multistability). For example, in a degenerate four-wave mixing oscillator the output field phase can take two given, opposite values (i.e., separated by π), while its intensity is the same in both sates. In spatially extended systems part of the output field can be in one of these phases whilst the rest is in the other one. At the boundary between the regions occupied by the different states a front appears. This front is the DW and can be of two types: An Ising wall if the field intensity is null at the wall core, and a Bloch wall if the intensity is not null [16,17]. Ising walls are the type of cavity soliton whose steering we investigate in this article.

DWs have been experimentally observed in a degenerate four-wave-mixing photorefractive oscillator [17–20]. However so far DWs were created only spontaneously, with the exception of [17] where we introduced a technique for writing a single DW at desired locations. In this article we demonstrate experimentally, for the first time, optical control of DWs: Independent switching on/off, location and steering of both isolated DWs as well as of DWs arranged in a linear array.

2. Experimental setup

Figure 2 shows a scheme of the optical arrangement, which is nearly the same used in [17,20]. The idea of the experiment is that by locally injecting into the nonlinear cavity a coherent light beam with appropriate amplitude (which must be at least two times larger than the amplitude of the intracavity field) and phase (π shifted with respect to the intracavity field), one can induce the switching between the two opposite and bistable phases of the intracavity field, leading to the switching on/off of a DW structure. This beam injection is the only difference with respect to the experimental setup in [20].

Our nonlinear cavity contains a parametric gain medium, a BaTiO3 crystal, pumped by two extraordinary polarized and counterpropagating laser beams from a continuous wave Ar+ laser operating in single frequency regime at 515 nm wavelength. The crystal is placed at the center of a plane-plane optical cavity formed by mirrors PM₁ and M, both with reflectivity of 95%. Two pairs of confocal lenses L1 are arranged inside the cavity in near self imaging configuration in order to get a short (few centimeters) effective length of the cavity, and therefore reach a large Fresnel number as needed for soliton formation. Additionally this cavity arrangement provides access to the Fourier planes, which allows making spatial frequency filtering with diaphragm D, and thereby achieving two important features: First, since the gain line is narrower than the free spectral range of the cavity, the diaphragm can cut the high order resonance rings corresponding to high order longitudinal modes thus selecting only one longitudinal mode family; and second, by replacing the iris diaphragm by a slit diaphragm, the cavity can be transformed from two-dimensional (2D) to one-dimensional (1D) in the transverse dimension. The latter configuration is needed to avoid curvature effects in the DWs behavior [17].

 

Fig. 2. Sketch of the experimental setup (the same as in [20] but for the beam injection): Degenerate four-wave-mixing BaTiO₃ photorefractive oscillator. PM₁ and M, are the cavity plane mirrors. There are four intracavity lenses L₁ (focal length f) arranged in near self-imaging configuration. D is an iris (or a slit) diaphragm D. The cavity length is actively stabilized and tuned by means of the piezo-mirror PM₁. The mechanical shutter admits for a while a sharply focused (by lens L2) injection beam into the cavity for local illumination of a small area of the crystal. The phase of the injected beam is controlled by piezo-moveable mirror PM₂.. The active stabilizaton mechanism is not shown, see [20] for more details.

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Active stabilization of the cavity length [17,20] (not shown in Fig. 2, see [20] for details) is used for an accurate control of the cavity detuning θ(defined as the frequency difference between the pump field and the cavity resonance frequencies) that plays a key role in the selection of the intracavity field spatial frequencies [20]: For large negative θ the system selects off-axis (tilted) waves forming periodic structures (labyrinth patterns in 2D [18] and a roll pattern in 1D [20]) whose spatial frequency k scales as k ²~-θ; for small negative θ, aperiodic patterns are formed; for positive θ, homogeneous on axis emission is supported by the oscillator. In this positive detuning range phase bistability allows the presence of DWs. It is this region that we explore.

The injection of laser pulses is arranged in the following way. Part of the Ar⁺ laser light is splitted away from the pump beam to serve as a writing/erasing beam. This coherent beam (extraordinarily polarized, like the pump beams and the intracavity field) is focused by lens L₂ and injected coaxially into the nonlinear cavity through the mirror M in order to illuminate a small area of the crystal. By transversely shifting L2 the writing/erasing beam can be addressed to whatever particular location in the crystal cross section. The injection beam is opened for a while by means of a mechanical shutter, in order to admit the injecting light for writing/erasing the DW structures. The amplitude and phase of the injected beam are controlled by a variable attenuator and a piezo-moveable mirror PM₂, respectively. Finally, the observation of the transverse intensity distribution of the generated light beam is carried out with charge-coupled device (CCD) camera placed at the image plane of the nonlinear crystal, which is the near field of the nonlinear cavity.

3. Steering of DWs

To demonstrate optical writing of DWs we start from the intensity homogeneous state of the intracavity field (Fig. 3(a)), which has a constant phase (corresponding to one of the two opposite phase states supported by the degenerate four-wave mixing interaction). Then the 150 µm focused injection beam, adjusted to be π-out of phase with respect to the intracavity field, is applied onto the central part of the homogeneous cross section. It induces the appearance of a small phase domain limited by two DWs (Fig. 3(b)). After the injection beam is blocked an equilibrium (stable) double DW structure of two straight (without curvature) and parallel walls (Fig. 3(c)) forms. Once written the double DW remains stable and self-pinned at fixed position provided the detuning θ is small enough (later we comment on what happens for larger values of θ).

 

Fig. 3. Experimental demonstration of writing (left column), erasing (central column) and locating (right column) of a double DW structure. Initial stable homogeneous state (a) and double DW structure (d), injection of writing (b) and erasing (e) laser beams, final double DW structure (c) and homogeneous state (f) persistent after the writing/erasing beam is blocked. Location and pinning of double DW at fixed positions of the cross section (g), (h) and (i) by writing laser pulses addressed at different places. The horizontal dimension is 1.6 mm

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Next we show optical erasure of the double DW imposed by the writing laser pulse, Figs. 3(d)–3(e). For that the phase of the injection beam was shifted by π with respect to the central phase domain. The application of this beam onto location of the double DW induces its annihilation (Fig. 3(e)). As a result, the system recovers the stable homogeneous state, Fig. 3(f), the same as in Fig. 3(a). The writing/erasing of the double DW can be done at different locations of the homogeneous background by parallelly shifting the injection beam, Figs. 3(g)–3(i).

We found that the separation between DWs in the double DW structure corresponds to a critical (minimal) equilibrium distance, d_c (d_c≈150 µm in the experiment). Fig. 4 shows that if the distance between DWs is shorter than d_c DWs annihilate and the double DW structure disappears. Fig. 4 (left column) shows the failed attempt to write the double DW structure with 100 µm focused injection beam. In this case the distance between two DWs is shorter than d_c and therefore walls annihilate and disappear. The same effect is observed when we try to place a second double DW structure close to an already existing double DW one at a distance shorter than d_c: the interaction produces the annihilation of the inner DWs, right column in Fig. 4. We note that the remaining enlarged pair of DWs is stable, Fig. 4(h).

In order to clarify the interaction between two DWs, Fig. 5 shows the shift of the double DW structure perturbed by an external focused beam injected into the cavity in such a way that it illuminates permanently a small area of the cross section on the left side of the DW structure, Fig. 5(a). The left DW approaches the right one (Fig. 5(b)) and, as a result, the right DW moves to the right. This effect clearly reveals the existence of a repulsion between the two walls forming the double DW structure. The three last snapshots (Figs. 5(c)–5(e)) show that the shifted DW structure finally recovers its original size and remains static.

 

Fig. 4. Annihilation of closely located DWs giving experimental evidence for the existence of a critical equilibrium distance between two DWs. If the second double DW structure is injected too close to the first one, the interaction produces the annihilation of the inner walls (right column). Nevertheless, if the distance is large enough, the writing of the second DW is allowed and the double DW remains static (left column).

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Fig. 5. Shift of double DW occurred in an amplitude gradient created by the external focused beam coaxially injected into the cavity and illuminate permanently small area of the cross section near left DW (a).

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It is also possible to write more than one double DW structure. When DWs are placed close enough to each other they can form clusters. Once one double DW has been written we can write a second one by injecting it close enough to the first double DW thus creating a 4-DW cluster (Fig. 6(a)). Then we add one more double DW and create a 6-DW structure (Fig. 6(e)).

Although the DW clusters are stable structures, we can manipulate them in order to obtain new DW arrangements. As a result of writing and erasing double DW soliton as well as the writing of more than one, we can build a very simple optical memory as an array of double DW solitons which can be switched on/off individually and independently of each other. Fig 6 shows examples of removing individual double DW structures from 4-DW and 6-DW clusters. The erasing procedure is the same describe above (Fig. 3). Thus the double DWs are switched on/off individually and independently of each other.

The static DW structures observed near resonance (θ≈0) are of Ising type [20], that is the intensity is null and phase jumps abruptly by π in the core of the DW. This means that Ising DWs have no phase gradient. However at positive cavity detuning (θ>0) DW can convert from Ising type to Bloch type which has its own phase gradient. Additional interferometric measurement confirms existence of the phase gradient [20]. This is due to the Ising-Bloch transition [16,20] which results in conversion from static to moving DWs. Fig. 7 shows both static double DW structure (left) and moving outward DWs (right).

 

Fig. 6. Switching off individual domains (double-DW structures) in DW clusters with the erasing laser pulses aimed at different places. Initial (a) 4-DW cluster and (e) 6-DW cluster. DW structures after switching off the central (b, f), right (c, d) and left domain (d, h).

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Fig. 7. Static Ising-type DWs (left) and moving Bloch-type DWs (right). Time runs from top to bottom in 5 s steps.

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4. Conclusions

We have reported the first experimental evidence of optically controlled phase domain walls in a nonlinear optical cavity. In particular, our experimental setup has been a degenerate four-wave mixing photorefractive oscillator working in near self-imaging. The 1D case is especially interesting because it allows writing stable structures with different sizes in a wide cavity detuning range, in contrast with the limitations of the 2D case in which curvature effects limit the stability conditions. A simple optical memory has been demonstrated.