Файл: Staliunas K., Sanchez-Morcillo V.J. (eds.) Transverse Patterns in Nonlinear Optical Resonators(ST.pdf

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12.2.3 Stabilization of Spatial Solitons by Gain Di usion

Two di erent interpretations of spatial solitons were discussed in Chap. 8: a spatial soliton can be considered either as a part of an extended pattern, such as a pattern of rolls or hexagons (Fauve and Thual type [5]), or as a homoclinic connection between two stable homogeneous states, forming a domain of minimum size (Rosanov type [4]).

In the first case, the background amplitude (the solution far away from the soliton) corresponds to the stable solution branch. For bright solitons, the upper branch is usually modulationally unstable, and the radiation corresponding to the stable lower branch serves as the background. For dark solitons, the opposite is valid. In the second case, an interaction between locked fronts results from the nonmonotonic decay of the background field far from a front.

In both cases, a spatial modulation is involved in the soliton formation process. As shown in Chap. 11, solitons are more robust, and their stability range is larger, in the case of strong spatial modulation. Consequently, from the analysis of the previous section, it follows that gain di usion must enhance the stability of solitons, since in all cases an increase in the gain di usion always leads to an increase in the growth exponents of the o -axis perturbation modes, i.e. to the enhancement of the spatial modulation. Three di erent situations can be realized:

1.Positive growth exponents become larger, and the parameter range of the modulational instability, and that for solitons of the Fauve and Thual type, increases.

2.Negative growth exponents decrease in magnitude, and spatial oscillations become less damped, resulting in the stabilization of Rosanov-type solitons.

3.Negative growth exponents may become positive. One can then obtain a transformation of solitons of the Rosanov type into solitons of the Fauve and Thual type (this is actually more a transformation of the interpretation than a qualitative transformation of the soliton itself). In all cases, the stability range of the bright solitons is increased.

In order to check the statements above, a numerical investigation of the full system ((12.8) and (12.19)) was performed. The results are summarized in Fig. 12.2, where the existence ranges of bright and dark solitons are plotted on the plane (D0, d), together with the modulational-instability boundary (squares) and the domain equilibrium boundary (full circles). Typical field profiles corresponding to di erent parameter values are shown in Fig. 12.3.

In region A of Fig. 12.2, amplitude spatial solitons of the Fauve and Thual type exist. Their profile is shown in Fig. 12.3d. In region B, the spatial solitons are of the Rosanov type. A contraction of amplitude domains occurs in this region. In region C, the spatial solitons are still of the Rosanov type (shown in Fig. 12.3c), but amplitude domains expand. In regions C and D, dark spatial

Fig. 12.2. Regions corresponding to di erent regimes of localized solutions in the plane (D0, d), as obtained from numerical integration of (12.8) in the purely di usive case. The parameters are as in Fig. 12.1

solitons exist. The line marked by squares (separating region A from regions B and C in Fig. 12.2) corresponds to the modulational-instability threshold. The line marked by filled circles (separating regions B and C) corresponds to the equilibrium state of the two phases corresponding to the upper and lower solution branches, and thus domains neither contract nor expand. The dashed vertical line separates the monostable and bistable regimes of the homogeneous solutions.

In the case of small domains, the domain boundaries lock and result in stable solitons. As Fig. 12.2 indicates, this locking can occur for contracting domains (in region B) and also for expanding domains (in region C).

We note that the spatial solitons in regions A, B, and C appear visually identical: no abrupt changes of the soliton parameters are observed when the modulational-instability threshold line is crossed. The existence range of bright solitons increases with di usion, and the Rosanov-type solitons transform into Fauve and Thual type solitons at the onset of the modulational instability of the upper solution branch.

Figures 12.3a,b show domains at equilibrium, which occurs between region B (contraction) and region C (expansion). However, the domain boundaries do not decay monotonically, but show spatial oscillations. These oscillations are stronger for larger di usion of the gain, as is evident from comparison between Fig. 12.3a and Fig. 12.3b. We note that spatial oscillations are much more prominent on the upper bistability branch, in correspondence with the predictions above.

Dark solitons have also been found numerically. They exist in regions C and D. Curiously enough, enhancement of the modulation of the upperbranch solution stabilizes the dark solitons too. The upper (modulated) so-

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Fig. 12.3. Stationary solutions obtained by numerical integration of the initial equations in the case of one spatial dimension. The parameters are as in Fig. 12.1. (a) Amplitude domain with weakly nonmonotonic tails, for D0 = 2.99, d = 0. (b) Amplitude domain with strongly nonmonotonic tails (close to the modulationalinstability boundary), for D0 = 2.85, d = 5. (c) Soliton in region B (of Rosanov type), for D0 = 2.95, d = 0. (d) Soliton in region A (of Fauve and Thual type), for D0 = 2.7, d = 5. (e) Dark soliton in region D for zero gain di usion (weak spatial modulation), for D0 = 3.05, d = 0. (f ) Dark soliton in region D for strong gain di usion (strong spatial modulation), for D0 = 2.15, d = 5

lution now serves now as the background for the dark solitons. It is usually presumed that a modulation of the solution branch other than that corresponding to the background stabilizes solitons. What follows generally from this study is that enhancement of the modulation of the background also increases the stability of dark solitons.

Figures 12.3e,f show numerically calculated field profiles corresponding to dark solitons. As in the case of large domains, the enhancement of the spatial modulation with gain di usion is also clearly visible in this case.

The above results correspond to the purely di usive case. In the presence of di raction, the mathematical expressions obtained from the linear stability analysis are not so transparent. The corresponding plots are given in Fig. 12.4. In general, a relatively small ammount of di raction of the field does not bring about qualitative changes: an enhancement of the modulational instability

Fig. 12.4. Growth rate of perturbations in the di ractive case. Parameters as in Fig. 12.1, except for the di raction coe cient. (a) Weak di raction, a = 0.25; for nonzero gain di usion, a small region of locking between amplitude and phase perturbations appears (indicated by vertical dashed lines). (b) Strong di raction, a = 2.5; the amplitude and phase perturbations are locked everywhere except for relatively small transverse wavenumbers k

(Fig. 12.4a) is observed because of gain di usion, as in the purely di usive case studied above. The additional feature compared with the purely di usive case is a locking between amplitude and phase instabilities in a certain band of transverse wavenumbers (where the amplitude and phase λ-branches are su ciently close). As the numerical calculations show, the enhancement of the soliton stability range due to gain di usion is similar to that in the purely di usive case.

For larger di raction (Fig. 12.4b), the locking between amplitude and phase perturbations is stronger. As a consequence, a nonstationary modulational instability is predicted (an instability of Hopf type). One may, therefore, expect oscillatory solitons in the case of strong di raction.

Some general conclusions following from the above analytical and numerical study are:

1.Di usion of the saturable gain enhances the growth of the o -axis field components. Overall, gain di usion results in “antidi usion” of the order parameter. As a result, gain di usion can enhance and/or initiate a modulational instability. In the monostable case, a modulational instability is never achieved; however, the maximum growth rate can approach very close to the zero axis from below, thus causing weakly decaying spatial oscillations. In the bistable case, a modulational instability can appear in some band of transverse wavenumbers.

2.The solution corresponding to the upper bistability branch is predominantly a ected by gain di usion. The solution corresponding to the lower branch is less a ected, or almost una ected, since its amplitude is significantly smaller than that of the upper branch.

3.As a result, gain di usion increases the stability range of solitons of both types: in the case of a Fauve and Thual type soliton, which a priori requires modulationally unstable solutions, the enhancement of the modulational instability obviously increases its existence range. In the case of

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solitons of the Rosanov type, where the stabilization is due to the nonmonotonic decay of the domain fronts, an increase of the gain di usion results in an increase of the spatial oscillations, and the existence range of solitons increases correspondingly.

4.The transition between the solitons of the two types is smooth (no singular behavior appears at the boundary between solitons of Rosanov type and solitons of Fauve and Thual type). This suggests that distinguishing the two types of solitons is only a matter of interpretation. In essence, the solitons of the two types are similar, as they convert one into another smoothly.

5.As the analysis of bright solitons shows, an enhancement of the modulation of the upper state (unlike a enhancement of the modulaton of the background solution) increases the stability of a soliton. However, the analysis of dark solitons shows that an enhancement of the background modulation can also stabilize those solitons.

6.In the di ractive case, pump di usion can enhance or initiate not only stationary modulational instabilities, but also nonstationary ones (instabilities of Hopf type).

12.3 Optical Parametric Oscillator with Di racting Pump

Consider now a system where the two competing fields are di racting. One example is given by a DOPO, whose mean-field dynamical equations for the signal (subharmonic) A1(r, t) and the pump wave A0(r, t) are

where the parameters are defined in Chap. 3.

Throughout this section, the case of a resonant pump ω0 = 0 and equal decay rates γ1 = γ0 is considered to simplify the analysis. Also, we normalize the spatial coordinates to √a1, which is equivalent to setting a1 = 1 and a0 = a in (12.22), where a = a0/a1 is the relative di raction parameter.

We note here that, as di raction in a laser depends on the resonator length in the case of a self-imaging cavity, in a DOPO the use of such a cavity allows one to choose freely the value of the di raction parameter a. In fact, when the optical cavity is formed by plane mirrors, the di raction coe cients of the signal and pump fields are related by a1 = 2a0, as a result of the phase-matching condition [7]. In the present case, in order to study the influence of di raction, we assume that each field resonates in a near- self-imaging cavity, with di erent lengths for the two fields, and consider a = a0/a1 a free parameter.

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At the threshold of the pattern-forming instability, the real part of the eigenvalue of the wavenumber with maximum growth is zero. In Fig. 12.6, the o -resonance and LALI instability regions are plotted in the parameter space (ω1, E) for a specific value of the di raction parameter. The regions are well separated in the parameter space, and therefore can be associated with di erent mechanisms. The o -resonance instability exists for all values of the pump intensity above threshold, whereas the LALI instability appears only at some critical pump value that depends on the di raction parameter a.

Fig. 12.6. Instability regions in the parameter space (ω1, E) for nonzero di raction parameter a = 5, evaluated from a linear stability analysis. There are two instability regions: for negative detuning, the traditional o -resonance instability; for positive detuning, the Turing instability

We note that pump di raction not only creates the LALI instability, but also modifies the o -resonance instability range, as can be seen from Fig. 12.6. For zero pump di raction the o -resonance instability occurs between the dashed curve and the left part of the solid curve corresponding to the neutral-stability line, as follows from the standard analysis. Pump di raction increases significantly the o -resonance instability region. However, the spatial scale of the o -resonance pattern is not modified by the presence of pump di raction.

Another important feature that reveals the di erent nature of the patterns on both sides of the resonance is the corresponding wavelength. In the case of o -resonance patterns, this wavelength depends mainly on the resonator detuning and the di raction coe cient of the signal wave. In contrast, the wavelength of the pattern in the LALI region depends essentially on the pump and on the ratio of the di raction coe cients a, and very weakly on the resonator detuning. This behavior is shown in Fig. 12.7, where the squared wavenumber of the maximally growing mode is plotted against the detuning (full line). The broken part of the curve, in the neighborhood of the resonance, corresponds to negative eigenvalues.

Some analytical expressions can be found in di erent limits. For negative detuning, the wavenumber is given by