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

In (2.16), the normalizing coe cient has been chosen in such a way, that the expressions given later for the nonlinear terms simplify maximally.

The full nonlinear equations (2.1) in the matrix representation are

where E = (E, P )T, and (2.17b) has been written in a scalar form. The nonlinear evolution matrix is

00

Now we multiply (2.17a) by the transformation matrix (as in (2.14)) in order to change to the new variables A and B; also, we express the old field and polarization variables E and P in terms of the new ones in (2.17b). The nonlinear evolution matrix in the new basis is

Adiabatic elimination can be performed if one λ-branch (namely the λ1 branch) dominates, and the other two branches lie deep below the zero axis.

44 2 Order Parameter Equations for Lasers

In this case, from (2.20b), the equation for the (enslaved) variable of the population inversion D is

Assuming a near-threshold condition, together with the “close-to-resonance” condition ∆ω = O(ε), we obtain λ2 = −2κ in the lowest order. The enslaved variable B is negligibly small compared with its master variable A, i.e. |B| |A|, and can be eliminated adiabatically from (2.22) to obtain B = (1/4) |A|2 A, which justifies the assumption about the smallness of B close to the threshold.

There now remains the equation for the order parameter A, which is associated with the unstable λ1 branch. From (2.20), we obtain

which, taking into account the smallness of B and using the expression (2.10) for λ1, simplifies to

Expanding the square root in a Taylor series (assuming the “near-threshold” and “close-to-resonance” conditions discussed above) we obtain

This is the final result, the order parameter equation, which captures the essential features of the nonlinear dynamics of the laser under the assumptions made here. For arbitrary values of κ and γ (but assuming that both parameters are of O(1)) we obtain the OPE for the general case of a class C laser,

where τ = tκγ /(κ + γ ) is a normalized time.

Let us now repeat all assumptions used to derive (2.26):

•“near-threshold” condition: p = D0 − 1 = O(ε2). It follows from this condition that the field amplitude A is small: A = O(ε).

•“close-to-resonance” condition: ∆ω = a 2 − ω = O(ε).

•Class C laser assumption: κ, γ and γ = O(1). If γ is small (the case of a class B laser), then the adiabatic elimination of the population inversion is impossible, and one ends up with a system of two equations and two order parameters. This case will be investigated in Chap. 7.

All the simplifications used here follow from the above assumptions. For instance, the enslaved parameter B was neglected because of its smallness. Indeed, from the expression B = (1/4) |A|2 A, it follows that B = O(ε3), and thus this neglect is justified.

The terms on the right-hand side of (2.26) are of the third order of smallness, except for the di raction term, which is of the second order of smallness. Consequently, the evolution occurs on two timescales: the evolution due to di raction (e.g. the beating of the transverse modes of the laser) occurs on a slow timescale T1 = O(1/ε), and the dynamics related to the linear growth of the fields and nonlinear saturation (e.g. the buildup of the radiation in the laser) occur on an even slower temporal scale T2 = O(1/ε2).

The detuning term was considered as a scalar during the derivation of the OPE (2.26), although it is an operator. This detuning operator does not commute with the order parameter A(r, t), nor with the nonlinearities. If we take into account this noncommutativity, the nonlinear term in the OPE (2.25) becomes

where the operators act on the variables to the right of them. Now, calculating the nonlinear term (2.27) and taking into account of the above smallness conditions (retaining the terms of O(ε3)), we obtain N = −(κ/2) |A|2 A, which leads again to (2.25) and (2.26).

The OPE (2.25), and also (2.26), is called the complex Swift–Hohenberg equation, owing to its similarity to the usual real Swift–Hohenberg equation [5].

Equation (2.26) retains all the ingredients of spatial pattern formation in lasers. One important property of the radiation in lasers is its di raction. The second term on the right-hand side accounts for that. The third term on the right-hand side describes the spatial-frequency selection, a phenomenon essential for the correct description of narrow-gain-line lasers. In many such lasers, selection of the spatial frequency (transverse mode) is possible by tuning the length of the resonator: particular transverse modes fall under the gain line and thus can be excited. Owing to the spatial-frequency selection term, the maximum amplification occurs at a nonzero transverse wavenumber k2 = −ω/a, which depends on the detuning ω. This means that a laser with

46 2 Order Parameter Equations for Lasers

negative detuning emits waves at an angle to the optical axis of the resonator (conical emission). Such a detuning-caused pattern-forming instability of lasers was first predicted in [6].

The first and last terms on the right-hand side of (2.26) give the normal form of a supercritical Hopf bifurcation. When the control parameter p = D0 − 1 goes through zero a bifurcation occurs, bringing the system from a stable point corresponding to the nonlasing solution A = 0 to a ring corresponding to the lasing solution |A|2 = p, characterized by a fixed amplitude but arbitrary phase.

2.3.2 Multiple-Scale Expansion

Another method that allows one to derive the OPE is the multiple-scale expansion technique, widely used in nonlinear analysis. The starting point is again a linear stability analysis, but the evolution equation of the order parameter is found as a solvability condition.

This technique consists of the following steps:

1.The relevant variables and parameters of the system are expressed in terms of a smallness parameter ε. This allows one to write the fields as an asymptotic expansion,

n=1

2.The original equations are expanded, and the coe cients of powers of ε are gathered. At order n, the equation has the form Lvn = gn, which is linear in vn, where gn contains the nonlinear interactions and variations of the fields at lower orders, and L is a linear operator.

3.A solvability condition is applied at some order n, to require the exis-

tence of solutions. This is done by requiring that gn be orthogonal to the solutions of the adjoint homogeneous problem, L vn = 0. This process is also known as the Fredholm alternative theorem.

4.Finally, at a given order, a closed equation is obtained for the evolution of one single variable, namely the order parameter.

In the following, we apply this method to the Maxwell–Bloch equations (2.1) [4].

First, we assume that the near-to-resonance condition holds, requiring that ∆ω = a 2 − ω be a small quantity:

In the original paper [4], ω and 2 were both required to be small. This restricts seriously the validity of the order parameter equation. Here we note that requiring only ∆ω to be small, leads to the same result, and this allows