Файл: Staliunas K., Sanchez-Morcillo V.J. (eds.) Transverse Patterns in Nonlinear Optical Resonators(ST.pdf
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52 3 Order Parameter Equations for Other Nonlinear Resonators
degenerate model. However, this is not true in general, owing to the complex character of the fields.
We start the analysis with the simpler, degenerate case.
3.2 The Real Swift–Hohenberg Equation for DOPOs
In order to simplify the analysis of this section, we make the following changes of variables in the model (3.2):
A0 = E − (1 + iω0)X , |
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(3.3a) |
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A1 |
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1 + ω2Y , |
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(3.3b) |
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E = (1 + iω0)E . |
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(3.3c) |
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The DOPO model now reads |
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∂X |
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γ (1+iω0) |
X + Y 2 |
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+ i |
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2X , |
(3.4a) |
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∂Y |
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∂t |
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= − (1+iω1) Y + [E + (1 − iω0)X] Y + i 2Y , |
(3.4b) |
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∂t |
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where the time is normalized to γ1, the space variables are to a1, γ = γ0/γ1 and we have used a0 = a1/2.
In this new representation, the simplest stationary solution takes the form
X = Y = 0 , |
(3.5) |
which is actually the trivial (nonlasing) solution.
3.2.1 Linear Stability Analysis
Next a stability analysis of the trivial solution (3.5) is performed against space-dependent perturbations, with arbitrary wavenumber k. The linearization of the system leads to the following eigenvalues:
λ0 |
= −γ 1 ± i ω0 + |
k2 |
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(3.6) |
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λ1 |
= −1 ± |
E2 − (ω1 + k2)2 |
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(3.7) |
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Clearly, λ0 has a negative real part for any value of the perturbation wavenumber k. In contrast, one root of λ1 becomes positive for a given pump
value, indicating |
the presence of a bifurcation at |
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EB(k) = |
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+ (ω1 + k2)2 . |
(3.8) |
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The emission threshold corresponds to the minimum value of EB(k), which occurs at a critical wavenumber k = kc. Identically to the case of the laser, the bifurcation depends on the sign of the detuning:
3.2 The Real Swift–Hohenberg Equation for DOPOs |
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1.For ω1 > 0, the instability occurs at kc = 0, corresponding to homogeneous emission. Thus, a positive detuning implies that no patterns (no
modes of the resonator) are excited. |
√−ω1, corresponding to a |
2. For ω1 < 0, the instability occurs at kc = |
pattern-forming instability (conical emission).
The pump threshold is di erent for di erent signs of the detuning. From
(3.8) it follows that E0 = EB(kc) = 1 + ω21 for a positive detuning, while E0 = 1 for a negative detuning. The situation in this respect is identical to that in lasers.
3.2.2 Scales
We use the multiscale expansion technique described in Sect.2.3.2 to derive an order parameter equation for a DOPO. Obviously, adiabatic elimination is also possible and leads to the same result [2]. The first step consists in the determination of the proper scalings.
We make again the near-to-threshold assumption,
E = E0 + ε2E2 . |
(3.9) |
The near-to-resonance assumption, |
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a1 2 − ω1 = εΘ , |
(3.10) |
unlike the case for lasers, is not always valid. It is valid for self-imaging resonators (see Chaps. 5 and 11), which allows one to obtain independent values of the di raction coe cients for both waves. However, for plane mirror resonators, strictly one should assume that both the detuning and the di raction are small:
a0 2 O(ε) , ω1 O(ε) . |
(3.11) |
To find the characteristic scale of the temporal evolution, we investigate how the eigenvalue behaves under the above assumptions. Substitution of (3.9)–(3.11) in (3.7) and expanding into Taylor series leads to
λ1 |
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(E − E0) − 2 |
ω1 + k2 |
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+ O(ε4) , |
(3.12) |
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which is valid for both signs of the detuning. In particular, the largest eigenvalue is always λ1 (kc) = E − E0 O(ε2), which suggests the introduction of a slow timescale T , given by
T = ε2t . |
(3.13) |
The linear stability analysis does not predict any particular order of magnitude for the pump detuning. Therefore, the pump detuning can in principle
54 3 Order Parameter Equations for Other Nonlinear Resonators
be chosen freely. However, we can use a property of the homogeneous solution of the DOPO model to obtain some useful information. As shown in [3], the homogeneous, nontrivial solution of the DOPO model shows bistable behavior for ω0ω1 > 1. As the order of magnitude of ω1 is required in our analysis to be O(ε), we can consider two main cases:
1.ω0 O(1), covering only a monostable situation, and
2.ω0 O(ε−1), covering also bistable situations.
In the following, we treat these two cases separately.
3.2.3 Derivation of the OPE
Consider an expansion of the fields in the form
∞∞
X = εnxn , Y = |
εn yn , |
(3.14) |
n=1 |
n=1 |
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together with the scalings (3.9), (3.10), (3.12) and (3.13), and either (a) moderate or (b) large pump detuning. Substitution in (3.4) leads to a system of equations to be solved at each order.
(a) Moderate Pump Detuning. At O(ε), we find the solution |
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x1 = 0 , |
(3.15) |
together with the relation |
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y1 = y1 , |
(3.16) |
i.e. the signal field is, in the lowest order, real-valued. At O(ε2), the equations read
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x2 = −y12 , |
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(3.17a) |
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y2 − y2 = −iΘy1 . |
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(3.17b) |
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At O(ε3), only the equation for the signal field is relevant, which reads |
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∂y1 |
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y |
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+ y |
+ E y |
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+ (1 |
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iω |
) x y + i Θy |
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(3.18) |
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∂T |
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The solvability of (3.18) can be checked by adding it to its complex conjugate, in order to eliminate the explicit third-order contributions. Then, taking into account (3.16) and (3.17), after some algebra, we find
∂y1 |
= E2y1 − y13 − |
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Θ2y1 |
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(3.19) |
∂T |
2 |
3.3 The Complex Swift–Hohenberg Equation for OPOs |
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which can be written in terms of the initial parameters. If we define the order parameter as A = εy1, (3.19) leads to
∂A |
= (E − 1)A − A3 − |
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ω1 − 2 |
2 |
A , |
(3.20) |
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∂t |
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which is the real Swift–Hohenberg (RSH) equation for the order parameter A. The RSH equation was first derived in a hydrodynamic context [4], and was later used to describe several nonlinear optical systems, such as optical bistability and four-wave mixing [5].
(b) Large Pump Detuning. We repeat the derivation now, but using the scalings (3.9)–(3.11) together with ω0 = Ω0/ε. From the first and second order, we obtain (3.15) and (3.17a) again, and also
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x3 = −2y1y2 , |
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(3.21a) |
y2 − y2 = −i Θ + Ω0y12 |
y1 . |
(3.21b) |
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results in |
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The third order |
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∂y1 |
= −y3 + y3 + iΩ0 |
(2y2 + y2) + E2y1 − y13 + iΘy2 . |
(3.22) |
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∂T |
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By adding (3.22) to its complex conjugate, and using (3.21) we finally obtain
∂y1 |
= E2y1 − y13 − |
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Θ + Ω0y12 |
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y1 |
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(3.23) |
∂T |
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which, expressed in terms of the original parameters, results in the following order parameter equation:
∂A |
= (E − 1)A − A3 − |
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ω1 − 2 − ω0A2 |
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A , |
(3.24) |
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∂t |
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where A is the signal amplitude to leading order.
Note that (3.24) reduces to (3.20) when ω0 is small. The term appearing at large ω0 is responsible for the intensity-dependent wavenumber selection, corresponding to a spatial nonlinear resonance. Many important features of pattern formation are related to this e ect, which will be discussed in Chap. 10.
3.3 The Complex Swift–Hohenberg Equation for OPOs
Again, it is convenient to make some changes in the model before starting the multiscale expansion procedure. The derivation is simplified if we apply to (3.1) the changes (3.3a) and (3.3c), together with