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

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30 1 Introduction

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34 2 Order Parameter Equations for Lasers

Chap. 7, since this class of laser leads to more complicated OPEs, and to richer dynamics than those of class A and class C lasers.

2.1 Model of a Laser

Our starting point is the semiclassical Maxwell–Bloch (MB) equation system, which describes many types of lasers with transverse degrees of freedom:

The complex fields E(r, t) and P (r, t) are the envelopes of the electromagnetic (optical) field and of the polarization, and D(r, t) is the real valued field of the population inversion, which in the absence of stimulated radiation is equal to its unsaturated value D0(r). κ is the relaxation rate of the optical field in the resonator due to the (small) transmittivity of the mirrors and to linear losses in the resonator; γ and γ are the decay rates of the polarization and population inversion, respectively. Finally, ω is the resonator detuning (the detuning of the resonance frequency of the corresponding longitudinal mode with respect to the center of the gain line).

It is assumed that the optical field E(r, t) is linearly polarized, and the gain line is homogeneously broadened. It is also assumed that only one longitudinal mode family is excited. Otherwise, there would be a dependence on the longitudinal coordinate z, and not only a dependence on the time t and on the transverse coordinates r = (x, y) as in (2.1).

The system (2.1) describes a laser with multiple transverse modes but a single longitudinal mode. The evolution in time is assumed to occur on a timescale much slower than the round-trip time of the light in the cavity; otherwise, the assumption of a single longitudinal mode would be invalid.

The di raction term is related to the spatial degrees of freedom. This term, being nonlocal, couples the field throughout the cross section of the laser, and is responsible for the collective behavior of the laser radiation.

The simplest limit of the Maxwell–Bloch system (2.1) is the class A laser, in which the polarization and population inversion are fast compared with the optical field in the resonator. In this limit, sometimes called the “good cavity limit”, the fast material variables can be adiabatically eliminated, and a relatively simple order parameter equation can be obtained. A straightforward adiabatic elimination has been performed in this case in [2]. Although many pattern-forming properties of lasers are lost in this adiabatic elimination, let us start from this procedure.

It is assumed that the material variables decay fast, i.e. κ/γ = O(ε) and κ/γ = O(ε), where ε is a smallness parameter, and that the temporal derivatives of all variables have finite values, i.e. ∂E/∂t ∂P/∂t ∂D/∂t = O(1). Multiplying both sides of (2.1b) and (2.1c) by κ/γ and κ/γ , respectively, we obtain the result that the left-hand sides of both equations are of order O(ε). Keeping only the terms of zero order O(1), we can eliminate the material variables from (2.1b) and (2.1c), and obtain

The expressions (2.2) imply that the material variables P and D follow instantaneously, or adiabatically, the changes of the field variable E. Inserting (2.2) into (2.1a) we obtain a single equation for the field,

where τ = κt is a slow time. Close to the emission threshold p = (D0 − 1) 1, the emitted fields are relatively weak, |E|2 1, which allows a cubic approximation for the nonlinear term in (2.3),

this is the complex Ginzburg–Landau equation. In (2.4), p is the balance between the gain and loss of the laser, and is a criticality parameter of the CGL.

The CGL equation is a crude approximation for a laser. For instance, the selection of transverse wavenumber (transverse mode) is not accounted for by (2.4). A linear stability analysis of the zero (nonlasing) solution of (2.4) leads to equal growth exponents of all the components of the spatial spectrum, which can be easily checked by inserting a test solution in the form of a tilted wave, E(r, t) = e exp(ik · r + λt), with a small amplitude e, linearizing it with respect to e, and calculating the exponent λ. Here k = k is the transverse wavenumber, or, in other words, the transverse component of the wavevector tilted with respect to the optical axis of the resonator. The value of λ is independent of the transverse wavenumber k = |k| and is equal to λ = p = D0 − 1.

It is well known, however, that lasers emit particular transverse modes (transverse wavenumbers) that depend on the length of the resonator. When the maximum of the gain line coincides with a particular transverse mode family (Fig. 2.1), the corresponding transverse mode is favored, and grows the fastest. This tunability property of the laser is lost in the derivation of (2.4). It is not di cult to understand why spatial-frequency selection is absent in (2.4): the derivation assumes, among other things, the condition γ /κ → ∞,