9.3 Numerical Simulations

129

phenomena balance this double narrowing e ect. The balance in the present model comes from the di usion, as (9.12b) shows.

The counterbalance between nonlinearities on the one hand, and di usion on the other hand, can be seen in a plot of the vector field generated by (9.12), as shown in Fig. 9.1. When the di usion is large enough to compensate the narrowing e ect of the nonlinearities, a node point appears, signaling the possibility of stable solitons.

		node
		node
		saddle
a)	b)	c)

Fig. 9.1. The vector field of the evolution of the parameters of the spatial soliton (amplitude a(t) in the vertical direction, and inverse width c(t) in the horizontal direction) as given by the parabolic expansion (9.12). (a) Below the threshold, p = 0.75, aRe = 0.127; (b) situation corresponding to bistability, p = 0.79, aRe = 0.13; (c) in the monostable regime, p = 1.6, aRe = 0.29. Other parameters are Ip = 1, Is = 1; the amplitude a(t) varies from 0 to 8.4; and the inverse width c(t) from 0 to 1.4 in all three cases

9.3 Numerical Simulations

In order to check the analytical results described above, the models (9.2) and (9.6) were integrated numerically by using the split-step method with periodic boundary conditions. Two stages can be distinguished: (1) the process of soliton formation starting from noisy initial conditions, and (2) the manipulation of a single soliton once it has been formed. Experiments corresponding to these results are described in Sect. 9.4.

9.3.1 Soliton Formation

We start by considering simulations of a laser in the first configuration (with both nonlinear elements located in the near field of the resonator), described by (9.2). In this case no competition between localized structures is expected.

The emission was initiated with a large pump value (p = 2.5), at which the absorber was completely saturated. The field developed several optical

130 9 Subcritical Solitons I: Saturable Absorber

(a)

(b)

(c)

(d)

(e)

Fig. 9.2. Evolution of an initial randomly distributed field as obtained by numerical integration of (9.2). The parameters are aRe = 0.5, aIm = 2, Ip = 1, Is = 0.06, α = 3. The first plot was calculated for a relatively large value of the pump intensity (p = 2.5), corresponding to an almost saturated absorber. Later the pump intensity was lowered to p = 2.15. The time between plots is t = 30

vortices at arbitrary locations, separated by shocks, as Fig. 9.2a shows. Next, the pump value was decreased to p = 2.15, at which value the absorber was unsaturated, and field discrimination occurred. The vortices converted into dark domains, and the shocks into bright domains, the precursors of the solitons (Figs. 9.2b,c). The bright domains take the form of arbitrarily oriented stripes, which correspond to one-dimensional solitons (Fig. 9.2d,e). If the pump is left unchanged, the stripes shorten during the evolution and finally disappear, leaving the system in a homogeneous state. The excitation of two-dimensional solitons requires a slight increase of the pump in the final stage of contraction. The shortening of the stripes then stops, allowing the formation of stationary solitons.

The need for this complicated procedure, based on variations of the pump intensity, to control the patterns and obtain stable solitons, can be under-

Fig. 9.3. The peak amplitude of a stable soliton versus the pump parameter in the case of one spatial dimension (circles) and two spatial dimensions (triangles), as obtained by numerical integration of (9.2). The parameters are as in Fig. 9.2

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131

stood from an analysis of single solitons in 1D and 2D. In Fig. 9.3 the peak amplitudes of 1D and 2D solitons, evaluated numerically, are plotted together with the amplitude of the homogeneous solution given by (9.8), as a variation of the pump parameter. We clearly see that localized stripes (1D solitons) are stable for pump values smaller than those required for localized spots (2D solitons), and that their existence ranges do not coincide. The explanation lies in the fact that the solitons have larger losses (di ractive and di usive) in two dimensions than in one, and therefore require a larger pump value to compensate such losses. It is also important to note that the existence range of the solitons corresponds only to a small portion of the full bistability range (9.10).

Let us now consider the other laser configuration (with a near-field–far- field separation), where, as discussed in the previous section, competition among solitons is expected. The numerical integration of (9.6) in this case shows the formation of a single soliton from a noisy spatial distribution (Fig. 9.4).

(a)

(b)

(c)

Fig. 9.4. Formation of a symmetric, stable, localized structure from an initial randomly distributed field, as obtained by numerical in-

(d) tegration of (9.6) in the case of two spatial dimensions: the left column shows the spatial distributions (near field), and the right column the spatial spectra (far field)

The temporal series shown in Fig. 9.4 shows the near field (left) and the far field (right), where the absorber and the gain medium, respectively, are placed. Figure 9.4a shows the initial random distribution. During the linear evolution, several bright spots emerge, and a filtering in the Fourier plane is observed, which corresponds to a broadening of the spots in the spatial domain (Fig. 9.4b). In the nonlinear stage, the saturation of both the gain and the absorption contributes to a broadening of the spatial spectrum (or, equivalently, to a narrowing of the spots), forming an ensemble of round spots

132 9 Subcritical Solitons I: Saturable Absorber

or solitons (Fig. 9.4c). Finally, owing to the nonlinear absorption, only the strongest soliton survives the competition (Fig. 9.4d).

We remark that this scenario of the developement of a single soliton in the transverse plane is analogous to the temporal pulse-formation process in lasers with passive mode locking. In both cases one starts with a random field distribution (a random ensemble of pulses or solitons). A spectral filtering in the frequency spectrum or in the spatial Fourier spectrum occurs in the linear stage of the evolution. In both cases, only one pulse or soliton survives in the nonlinear stage of the evolution. Competition occurs because several pulses or solitons share a common population inversion. For a mode-locked laser the amplifying medium is relatively slow, and the amplification depends on the integral energy. In our pattern-forming laser, the spatial spectra of individual pulses overlap in the focal plane, and the amplification again depends on the integral characteristics (in space) of the radiation.

9.3.2 Soliton Manipulation: Positioning, Propagation, Trapping

and Switching

One of the most promising applications of transverse spatial solitons is expected in the field of information processing and storage, where the solitons would be used as bits or basic information units [5]. To realize such practical applications, one needs to be able to manipulate solitons by external actions. In this section we show how such external actions can be incorporated into the numerical simulations, and how the solitons can be influenced in the desired way.

An important property of a spatial soliton is its position: in the absence of spatial inhomogeneities, a soliton has spatial multistability, i.e. it is stable independently of its location in the laser cross section. Consequently, it is expected that a soliton will move under the action of gradient forces, in particular those produced by phase gradients. We have tested the drift of a soliton under the influence of two di erent phase gradients: a linear (ramped) and a parabolic gradient.

The easiest way to physically introduce a linear phase gradient is by tilting one of the resonator mirrors. This e ect can be taken into account in the model by adding phenomenologically a symmetry-breaking term to (9.1), which now reads

∂A	= −kt A − αA + TNL + (aRe + iaIm) 2A ,	(9.13)
∂t

where TNL represents the same nonlinear terms as in (9.1) and kt describes the tilt of the mirror, which has the dimensions of a velocity. The first term in (9.13) results in the drift of a soliton in a direction determined by the mirror tilt, with a constant velocity proportional to kt.

Similarly, a phase trough can be modeled by placing one of the intracavity lenses closer to the resonator center or, alternatively, by a spatially dependent