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

36 2 Order Parameter Equations for Lasers

Fig. 2.1. Transverse modes of the resonator and the gain line of the amplifying medium of a laser. In the resonator length is varied, the transverse modes are shifted with respect to the gain line, which allows the tuning of the modes. By increasing the resonator length, one tunes to higher-order transverse mode families (a). This corresponds to a negative detuning parameter ω in (2.1). By decreasing the resonator length, one excites the lower-order transverse mode families (b). This corresponds to a positive detuning

which means that the gain line is infinitely broad. An infinitely broad gain line obviously cannot cause a transverse frequency selection. In order to account for the spatial-frequency selection a more sophisticated derivation of the laser OPE is required, which is the subject of the following sections.

To continue with the derivation of a more precise OPE a linear stability analysis of the laser equations is useful.

2.2 Linear Stability Analysis

A standard technique is applied here to investigate the stability of the nonlasing solution of (2.1), given by E(r, t) = 0, P (r, t) = 0, D(r, t) = D0. By perturbing this trivial zero solution by E(r, t) = e exp(ik · r + λt), P (r, t) = p exp(ik · r + λt) and D(r, t) = D0 + d exp(ik · r + λt), inserting it into (2.1) and gathering the linear terms with respect to e, p and d, we obtain

The last equation in (2.5) is not coupled to the rest, and therefore the calculation of one λ-branch is trivial: λ3 = −γ . The solution of the secular equation (the solvability condition of (2.5) with respect to e and p) gives two other branches of the growth exponent:

λ1,2 (∆ω) = −κ + γ + iκ ∆ω ± 1 (κ − γ + iκ ∆ω)2 + 4γ κD0 . (2.6) 2 2

Fig. 2.2. The real parts of the three Lyapunov growth exponents depending on the transverse wavenumber for the MB equations (2.1) with κ = 1, γ = 1.2 and γ = 0.4. The λ3 branch is associated with the decay of the population inversion, and is the horizontal straight line. b) The real and imaginary part of the most relevant (upper) branch of the Lyapunov growth exponents (2.6) - dashed curves, and their Taylor expansions (2.8) - solid curves.

Here ∆ω = ω + ak2 is proportional to the deviation of the mode with transverse wavenumber k from its resonant value, ωres = −ak2. Figure 2.2 illustrates the dependence given by (2.6).

From (2.6), the threshold for the laser emission (which occurs when λ = 0)

is

A simplification of (2.1) is possible when only one λ-branch is relevant to the dynamics of the system. This occurs, in particular, for class A and class C lasers close to the emission threshold. In this case the other two branches (the eigenvalue given by (2.6) with the negative sign, and also the straight line λ = −γ associated with the relaxation of population inversion) lie deep below the zero axis, and the dynamics related to these branches are enslaved by the dynamics related to the upper branch. We can expand the upper λ-branch in a Taylor series around its maximum, which gives

This is plotted in Fig. 2.2b as the solid curves.

The growth rate (2.8) is obtained by assuming that ∆ω is of O(ε) and p = (D0 − 1) is of O(ε2). In this case the truncation of the Taylor series at O(ε3) leads to (2.8). It is also possible to perform the Taylor expansion under di erent smallness assumptions, which leads to slightly di erent expressions.

The linear stability analysis predicts the initial stage of evolution of the radiation from the initial (thermal or quantum) noise. To illustrate the dynamics predicted by (2.8), the Maxwell–Bloch system (2.1) was integrated

38 2 Order Parameter Equations for Lasers

numerically, starting from a random field distribution in space. A series of plots illustrating the evolution of the field in the linear stage is given in Figs. 2.3 and 2.4. A discrimination against nonresonant components in the spatial spectrum is clearly seen: either a central spot (Fig. 2.3) or a resonant ring (Fig. 2.4) emerges from the initial broadband spatial spectrum, depending on the detuning of the resonator. The radius of the resonant ring is given by k2 = −ω/a, in accordance with the results of the linear stability analysis (2.6). In the spatial domain, structures with a particular spatial scale develop, which is related to the radius of the resonant ring; this scale is

that describes the linear stage of the field evolution. Recall that, in the linear stability analysis, one substitutes the time evolution operator ∂/∂t by λ, and the Laplace operator 2 by −k2. The opposite substitution (i.e. the substitution in (2.8) of the algebraic variables by their corresponding operators) leads to

where A(r, τ) is the order parameter related to the optical field in the laser resonator (the relation between A(r, τ) and E(r, t) is obtained in the next section), and τ is a normalized time, τ = tκγ /(κ + γ ).

The last plots in Figs. 2.3 and 2.4 correspond to the nonlinear stage of the evolution. Here the wavevectors from the resonant spot and from the resonant ring, respectively, start to compete. Also, a nonlinear broadening of the resonant ring (Fig. 2.4) or of the central spot (Fig. 2.3) occurs. The ring in the spatial spectral domain (Fig. 2.4) can split into a few spots: one spot corresponds to a single tilted wave; two symmetrically placed spots correspond to two counterpropagating tilted waves; several spots placed irregularly on the ring correspond to domains of di erently directed tilted waves; and four spots correspond to a more fundamental pattern, the cross-roll, or square vortex lattice pattern. The nonlinear patterns will be discussed in detail in the following chapters. The linear theory can say nothing about the symmetry of the nonlinear pattern: the last plots in Figs. 2.3 and 2.4 are beyond the predictions of the linear order parameter equation (2.8). From the linear theory (linear stability analysis), one can learn only that the most favored modes (or wavenumbers) have a particular value that depends on the detuning. The spatial spectral components grow to infinity, since there is no mechanism to prevent their exponential growth in this linear theory. There is no competition between wavevectors in the framework of the linear theory. To retrieve the correct nonlinear picture of the evolution, one needs to close the linear evolution equation (2.8) with some saturating nonlinear terms.

This closure is performed in the following sections in two di erent ways, both using results from the linear stability analysis. One possibility is the

adiabatic elimination technique [2, 3]. The other is the multiscale expansion technique, as developed in [4] for the laser.

2.3 Derivation of the Laser Order Parameter Equation

The order parameter equation for a laser is derived in this section. For completeness, two techniques of derivation are used, adiabatic elimination and the multiscale expansion, both leading to the same result. Physically, both derivations have the same purpose: to get rid of unnecessary degrees of freedom. This is illustrated in Fig. 2.5, where it is symbolically shown how the dynamics in the three-dimensional phase space can be reduced to dynamics in a two-dimensional space by a suitable transformation of the coordinate system. In general, the derivation of an order parameter equation is always related to a reduction of the dimension of the original problem: Rn → Rm, with m < n.

Fig. 2.5. A dimension reduction is achieved by a suitable transformation of the coordinate system

2.3.1 Adiabatic Elimination

The adiabatic-elimination (AE) technique consists of the following steps:

1.Linear stability analysis. In this stage, the eigenvalues are calculated and the corresponding eigenvectors are determined. The eigenvectors are mutually orthogonal with respect to the linear part of the equation system.

2.The initial nonlinear equation system is rewritten in the basis of the eigenvectors of the corresponding linearized problem. Owing to the nonlinear terms, the eigenvectors are mutually coupled.

3.The eigenvectors with negative eigenvalues are adiabatically eliminated from the corresponding equations. It is assumed that the eigenvectors with negative eigenvalues are dominated by the eigenvectors with positive eigenvalues.

42 2 Order Parameter Equations for Lasers

This procedure leads to an order parameter equation. We illustrate how the adiabatic-elimination technique described above works for the special case of a class C laser, when the relaxation rates for the optical field and the polarization are equal, i.e. κ = γ . This simplifies the problem from an algebraic point of view, without losing generality.

The eigenvalues of the linearized problem (2.6), under the assumption κ = γ , simplify to

where ∆ω = ω − a 2 is an operator, unlike the case for (2.6), where the Laplace operator 2 was substituted by −k2.

The linear stability analysis can be rewritten in a more convenient matrix form. In the matrix representation, the linearized equations (2.5a) and (2.5b) are expressed in the following way:

is the linear evolution matrix. The linear stability analysis is nothing but a procedure of diagonalization of the linear evolution matrix L. Multiplying (2.11) from the left by the transformation matrix S (to be determined below), and inserting formally the unit matrix I = S−1S between L and e in the right-hand side of (2.11), we obtain

consisting of the eigenvalues λ1,2, if the transformation matrix is adjoint to

the eigenvector matrix V , i.e. S = V −1 T. The eigenvector matrix V transeigenvectors

forms the coordinate system to one with axes directed along the

of the system; the coordinates in this new coordinate system A = (A, B)T

are related to the old coordinates e = (e, p)T by A = Se, and, vice versa, e = S−1A.

Specifically, the matrix of eigenvectors is