Файл: Staliunas K., Sanchez-Morcillo V.J. (eds.) Transverse Patterns in Nonlinear Optical Resonators(ST.pdf
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172 12 Turing Patterns in Nonlinear Optics
semiconductor laser) should weaken the spatial inhomogeneity of the emitted optical field. As a consequence, gain di usion should reduce or suppress a modulational instability, and might destroy spatial solitons that would exist in its absence.
The opposite phenomenon is shown to be true in this section, namely that the di usion of a saturating gain enhances the spatial modulation of the optical field. This enhancement of modulation supports solitons and increases their stability range.
12.2.1 General Case
Consider a general model, where the mean-field equations for an optical system with saturable gain are given by [3]
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= F (A, 2A) + DA , |
(12.8a) |
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∂D |
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(12.8b) |
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∂t |
where A(r, t) is the optical field (order parameter) and D(r, t) is the gain field (e.g. the population inversion). The operator F (A, 2A) is a given nonlinear and nonlocal function of the order parameter A(r, t), d is the di usion coe cient for the saturable gain, and γ is its relaxation rate. The complex conjugate equation of (12.8a) must also be taken into account when the optical field is complex (if di raction or focusing/defocusing nonlinearities are present in the function F (A, 2A)).
For simplicity, it is assumed below that the gain relaxation is fast (γ = O(1/ε), with ε 1), and the gain variable D can be adiabatically eliminated from (12.8b) by requiring that ∂D/∂t = 0. However, as numerical calculations show, the main conclusions are valid even for moderate gain relaxation, i.e. γ = O(1). The adiabatic elimination from (12.8b), neglecting gain di usion
(d = 0), is straightforward, and gives |
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D = |
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1 + |A| |
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In general (for d = 0), the adiabatic elimination requires the inversion of
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N = 1 + |A|2 − d 2 , |
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since (12.8b) can be written, in the stationary case, as ND = D0. The inversion can be performed for small di usion, assuming that d 2 = O(ε) and all the other variables are of O(1), yielding
N−1D = |
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1 + |A| |
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1 + |A|2 |
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(12.11)
12.2 Laser with Di using Gain |
173 |
where the Laplace operator acts on the variables to the right of it. It is easy to verify that N−1ND0 = D0 1 + O(ε2) , which confirms the validity of the inverse operator (12.11) at O(ε).
For a spatially homogeneous pump parameter D0, the last term on the right-hand side of (12.11) vanishes, and the population inversion becomes
D = |
D0 |
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1 + |A| |
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Inserting (12.12) into (12.8a), we finally obtain the order parameter equa-
tion |
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∂A |
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D A |
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where F (A, |
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2A) = F (A, |
2A) + D |
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2). The last term on the right- |
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hand side of (12.13) is due to the di usion of the saturable gain.
Equation (12.13) will be used as a basis to investigate how the gain diffusion a ects the stability of the homogeneous solutions of that equation.
Linearization of (12.13) around the homogeneous stationary solution (which now depends on the explicit form of F , and is assumed to be real-
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generality), with perturbations of the form A = |
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A¯ + a |
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ikr), leads to |
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exp(λt+ikr) + a exp(λt |
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λa = La + Da, |
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(12.14) |
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where a = (a1, a2)T is the column vector of the perturbation amplitudes. L is the linear evolution matrix generated by the nondi usive part of (12.13),
L = |
δF /δa1 |
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and D is the perturbation matrix due to gain di usion, |
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D = |
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A |
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A useful representation can be found by rewriting (12.14)–(12.16) in terms of the new basis a± = a1 ± a2 (corresponding to perturbations of the amplitude and the phase, respectively), in which one obtains, instead of (12.16),
D = |
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(12.17) |
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From this general analysis, we can draw some conclusions:
174 12 Turing Patterns in Nonlinear Optics
1.The sum of the Lyapunov exponents is always equal to the trace of the
linear evolution matrix. The perturbation (12.17) increases the sum of the Lyapunov exponents by an amount 2dk2D0A2/ 1 + A2 3. This indicates that, overall, the gain di usion works as “antidi usion” of the order parameter, and spatial components with nonzero transverse wavenumbers (o -axis modes) may be amplified because of gain di usion.
2.If the amplitude and the phase of the order parameter A are decoupled from one another, then the gain di usion a ects only the amplitude perturbations. Therefore the gain di usion always increases the amplitude modulations, and as a consequence may stabilize spatial solitons. On the contrary, it does not a ect purely phase perturbations at all.
3.If the amplitude and phase perturbations of the order parameter A are coupled, then the eigenvalues are complex and form a conjugate pair, i.e. λ1,2 = λRe±iλIm. The sum of the eigenvalues is proportional to the real part, i.e. λ1 + λ2 = 2λRe. Therefore (12.17) indicates also the destabilization of coupled amplitude and phase perturbations. The gain di u- sion thus increases (or initiates) a modulational instability of oscillatory (Hopf) type.
4.In the case of bistability, the gain di usion a ects predominantly the upper bistability branch: the coe cient of the e ective “antidi usion” of the order parameter 2dk2D0A2/(1 + A2)3 depends on the intensity of the optical field, and is evidently larger for the upper branch.
The above conclusions are now set out in detail for the case of a bistable laser (a laser with an intracavity saturable absorber).
12.2.2 Laser with Saturable Absorber
We consider first the simplest case of a monostable laser (with linear losses), represented by
F (A, 2A) = −A + ia 2A + g 4A . |
(12.18) |
A linear stability analysis of the full system ((12.8) and (12.18)) shows that, although the λ-branch related to the amplitude perturbation is shifted upward, its maximum value can never become positive. As a consequence, amplitude modulations (due to lateral boundaries or other reasons) can be enhanced, but never cause absolute instabilities. Some bistability mechanism is required to reach an instability. As an example, we consider the case of a laser with a saturable absorber, discussed in Chap. 9. The functional F (A, 2A) is now given by
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A) = −A − |
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A − g A , |
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1 + |A|2 /Is |
where α0 is the coe cient of the unsaturated losses and Is is the saturation intensity. Again, zero detuning is assumed in (12.19).
12.2 Laser with Di using Gain |
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For g = 0, one has a purely di ractive case, as studied by Rosanov [4]. For a = 0, the purely di usive case is obtained instead. In optics, the purely di usive case can be realized using a self-imaging resonator, as described in Chap. 6.
First we investigate the purely di usive case, where the amplitude and phase perturbations are decoupled. In this case we have
∂t |
= 1 + |A|2 |
− A − 1 + |A|2 |
/Is |
− g 4A + 1 + |A|2 d 2 |
1 + |A|2 . |
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A linear stability analysis of the homogeneous upper-branch solution of (12.20) gives
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2α0(A /Is)
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1 + A2/Is
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for amplitude perturbations. The phase perturbations are not a ected by the gain di usion in this purely di usive case.
A family of plots of (12.21) is given in Fig. 12.1a, showing the modulational instability, which appears and grows with increasing gain di usion d. For su ciently large gain di usion, the upper branch can be modulationally unstable.
In order to test whether the above procedure of operator inversion reveals the correct results, a linear stability analysis of the full problem ((12.8) and (12.19)) was also performed. Figure 12.1b shows the results of the stability analysis of the full system. Evidently, the instability spectra for small gain di usion and small transverse wavenumbers coincide well in the two cases (the smallness parameter in the adiabatic elimination (12.11)–(12.13) is indeed d 2 = −dk2 = O(ε)). Discrepancies appear for relatively large values of the gain di usion, leading to di erent quantitative (but not di erent qualitative) results.
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Fig. 12.1. Growth rate of a perturbation as a function of the transverse wavenumber for di erent values of gain di usion d, obtained from a linear stability analysis of (a) the simplified model (12.20) and (b) of the full system (12.8). The parameters are D0 = 2.8, γ = 5, Is = 0.1 and α0 = 5.0