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60 3 Order Parameter Equations for Other Nonlinear Resonators

where Ω = ωp − ωs and q = kp − ks.

In the mean-field limit, the equations describing the evolution of the signal wave and the refractive index are (the details of the derivation can be found

where A = As/Ap is the normalized signal field, ns is the saturation value of the index grating, β is the detuning of the resonator frequency from the center of the gain line, and κ and γ are the decay rates of the photon and index gratings, respectively. Usually, the condition γ κ holds, which allows the adiabatic-elimination of the optical field. In the next section, the adiabatic elimination technique is used to derive an order parameter equation for PROs.

3.4.2 Adiabatic Elimination and Operator Inversion

The envelope of the refractive-index grating can be expressed in terms of the optical field by assuming that the field is a fast-relaxing variable, i.e. ∂A/∂t = 0. In this case

Substituting (3.52) in (3.51), and letting the di erential operator act on

where τ = γt is a normalized time.

The stationary solution of (3.53) is A (r, t) = A0 exp (ikr), where A0 = ns/2 − 1 and ak2 = 1 − β. This means that certain spatial modes with wavenumbers k proportional to the resonator detuning are favored. Note the presence of a constant frequency shift β = 1, which is di erent from the case of a laser.

The di erential operator in (3.53) can be expanded in a Taylor series,

This is valid when the condition β − 1 − a 2 1 holds. As can be shown by an analysis of (3.53), this is the case for generation near the threshold, when |A| 1, and thus the expansion is justified.

The expansion (3.54), truncated at the second order, and inserted into

keep the analogy with lasers, we have defined ω = β − 1.

Retaining in (3.55) only the terms at the leading order and applying the cubic approximation X ≈ (2/ns) A |A|2, we finally obtain

which is again the CSH equation. However, there is an important di erence between (3.56) and the CSH equations derived for lasers and OPOs: the coe cient of the nonlinear term is complex, which causes self-defocusing in this system.

3.5 Phenomenological Derivation

of Order Parameter Equations

The CSH equation (3.47) was first derived for lasers [8]. Afterwards the same equation was derived for photorefractive oscillators [7] and for nondegenerate optical parametric oscillators [6]. The fact that the same order parameter equation has been derived for di erent types of nonlinear optical system hints at its universality. Its universality becomes evident when we derive the CSH equation phenomenologically from general symmetry assumptions, not restricting ourselves to a specific physical system. Let us assume for this purpose that an isotropic physical system characterized by a complex order parameter loses its stability at a nonzero wavenumber k0. This means that the maximum of the real part of the dominating (largest and positive) instability branch crosses the λ = 0 axis at the wavenumber k0, as illustrated in Fig. 3.1. The dominating instability branch can be approximated by a parabola at its maximum (near to the points where it crosses the λ = 0 axis),

62 3 Order Parameter Equations for Other Nonlinear Resonators

(a)

(b)

Fig. 3.1. (a) Real part of the dominating Lyapunov exponents as a function of the transverse wavenumber (dashed curve), and the parabolic approximation (solid curve). (b) Imaginary part of the dominating Lyapunov exponent (dashed curve) and its linear approximation (solid line). The dashed curves are generic (they are free-hand curves). It is significant that the real part of the dominating Lyapunov exponent just crosses the λ = 0 axis from below

where p is a control parameter. The spatial isotropy implies that λ depends on k2 = |k|2 and not on the first power of k. The imaginary part of λ can be approximated by the line

central frequency of the gain line of the nonlinear optical system.

If we now substitute λ by the temporal derivative ∂/∂t (thus returning from the frequency spectrum to the evolution in time), and the wavenumber k by the di erential operator −i (thus returning from the spatial spectrum to the distributions in space), then the linear part of the order parameter

where A(r, t) is an eigenvector associated with the relevant (most unstable) λ-branch and plays the role of an order parameter.

It remains to close the linear equation (3.59) using some nonlinearity, in order to prevent the infinite growth of unstable modes. If we assume the invariance of the field phase, we obtain a cubic, saturating, nonlinear term A |A|2 in the lowest order. In fact, the terms pA − A |A|2 represent a normal

References 63

form of the Hopf bifurcation, the bifurcation preserving the invariance of the phase of the order parameter. This closure of the linear equation (3.59) leads to

which is exactly equivalent to (3.47).

In such a way the CSH equation can be derived phenomenologically as a universal model that includes, in the lowest-order approximation, all the significant ingredients of a nonlinear optical system: (a) the finite width of the gain line, (b) spatial isotropy and (c) phase invariance. It is a normal form (a minimum equation) for a tunable system with a finite gain line width characterized by a Hopf bifurcation.

The above phenomenological derivation of (3.47) allows one to consider it as not only a simplified model equation for a specific physical system, such as a class A or class C laser, which is valid in a particular parameter range, but also as a universal model equation that describes a class of patterns and a class of phenomena in nonlinear optics in the lowest order of approximation.

In the same spirit, one can derive phenomenologically the real Swift–Ho- henberg (RSH) equation, which was derived above for a DOPO. DOPOs are characterized not by a Hopf but by a pitchfork bifurcation (a DOPO favors two values of the phase, di ering by π). The normal form of a pitchfork bifurcation is pA − A3, where A is the real-valued order parameter. Adding the spatially dependent term, one obtains

which is the RSH equation. The RSH equation is actually valid not only for DOPOs, but also for degenerate four-wave mixers, for systems showing optical bistability [5], and for other systems showing phase squeezing.

References

1.G.L. Oppo, M. Brambilla, D. Camesasca, A. Gatti and L.A. Lugiato, Spatio temporal dynamics of optical parametric oscillators, J. Mod. Opt. 41, 1151 (1994). 51

2.K. Staliunas, Transverse pattern formation in optical parametric oscillators, J. Mod. Opt. 42, 1261 (1995). 53

3.L.A. Lugiato, C. Oldano, C. Fabre, E. Giacobino and R. Horowicz, Bistability, self-pulsing and chaos in optical parametric oscillators, Nuovo Cimento 10D, 959 (1988). 54

4.J.B. Swift and P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15, 319 (1977). 55