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

102 6 Resonators with Curved Mirrors

Fig. 6.9. Left : a transverse pattern (near field at the left, far field at the right ) recorded experimentally in a PRO with a nearly confocal resonator. The intracavity aperture was completely open. Right: schematic interpretation of the Fresnel rings in the PRO resonator. The PRO emits in a narrow frequency band, which is related to the wavenumber of the emitted light k = ω/c. The light is radiated at angles such that the longitudinal component of the wavevector matches the longitudinal-mode condition of the resonator

To realize the case of a single-longitudinal mode (but with multiple transverse modes) describable by the CSH equation, the emission was restricted to a single ring by an aperture in the far-field plane. Under these conditions the PRO displayed the typical patterns predicted by CSH equation, which depended on the resonator tuning.

References

1.G.J. de Valc´arcel, Order-parameter equations for transverse pattern formation in nonlinear optical systems with nonplanar resonators, Phys. Rev. A 56, 1542 (1997). 92

2.M. Brambilla, F. Battipede, L.A. Lugiato, V. Penna, F. Prati, C. Tamm and C.O.Weiss, Transverse laser patterns. I. Phase singularity crystals, Phys. Rev. A 43, 5090 (1991). 95

3.K. Staliunas, G. Slekys and C.O. Weiss, Nonlinear pattern formation in active optical systems: shocks, domains of tilted waves, and cross-roll patterns, Phys. Rev. Lett. 79, 2658 (1997). 96

104 7 The Restless Vortex

separately from the equation of the population inversion (2.1c), and diagonalize the system formed by (2.1a) and (2.1b), considering D(r, t) as a parameter. This leads to the following growth rate of perturbations:

Here, as in Chap. 2, the particular case κ = γ is considered for simplicity. Generalization to arbitrary values of κ and γ (but keeping O(κ) = O(γ )) is straightforward. The eigenvector system associated with (7.2) is, in matrix form,

The expressions for the eigenvalues (7.2), the eigenvectors (7.3) and the transformation matrix (7.4) are analogous to the expressions (2.10), (2.15) and (2.16) for class A and class C lasers in Chap. 2. The di erence is that here

atimeand space-dependent population inversion D(r, t) appears instead of

ahomogeneous, stationary population inversion D0.

Now, we define a new set variables A = (A, B)T, related to the old ones by the transformation A = SE and viceversa, E = S−1A.

The equations for the field and polarization can now be rewritten, in terms

As follows from (7.5b), the stable variable B decays rapidly to zero and can be neglected. Now, expanding the term with the square root in (7.5a) and (7.6) as a Taylor series, assuming the near-threshold condition p = D − 1 = O(ε2) and the near-to-resonance condition ∆ω = O(ε), we obtain

If we remove the assumption of κ = γ (but assume that these two coefficients are of the same order, O(1)) we obtain the order parameter equation for the general case of a class B laser:

where we recall that the variables appearing in (7.8) are related to the original ones through the relations τ = tκγ /(κ + γ ), γ = (κ + γ )γ /κγ , and a

normalized field intensity κ |A|2 → |A|2 has been used.

The two coupled equations (7.8) have also been derived in [1] using the multiscale expansion technique. They are the basic equations used for the study of the spatio-temporal dynamics in class B lasers throughout this chapter.

During the diagonalization procedure, the time-dependent transformation matrix S was used, which obviously does not commute with the operator of the time derivative ∂/∂t. In fact, the commutation of these operators was assumed in obtaining (7.8). Assuming the above near-threshold condition D − 1 = O(ε2), however, we obtain the result that the commutator is of higher order of smallness than the rest of the terms in (7.8), and thus can be neglected. Therefore, in the framework of the near-threshold assumption used here, the system (7.8) is valid for a class B laser.

7.2 Single Vortex

A numerical study of a single vortex in a class B laser (in the framework of (7.8), and also in the framework of the complete Maxwell–Bloch equations (2.1)) reveals a surprising result: a single, isolated vortex, placed in a homogeneous background, is not stationary as in a class A laser, but shows a self-induced, permanent motion. For some values of the parameters, the motion resembles a “stochastic meandering” of the vortex core. Sometimes the motion is circular or flower-like. In order to understand the origin of this self-induced motion we inspect the distributions of the optical field A and of the population inversion D at the vortex core.

Figure 7.1a shows the fields along a line crossing the center of a stationary vortex. At the vortex center the order parameter A, which is proportional to the optical field, is zero. As the population inversion does not decay in the

106 7 The Restless Vortex

Fig. 7.1. Electric-field modulus (solid line) and population inversion (dashed line), corresponding to (a) a stationary optical vortex, and (b) a vortex moving to the right, on the axis along which the vortex moves in (b). The changes of the field are shown by arrows

absence of an optical field, an inversion maximum builds at the location of the optical vortex.

If we perturb the envelope of the field by shifting the “optical part” of the vortex, as shown in Fig. 7.1b, restoring forces appear, represented by vertical arrows in the figure. The inversion builds up at the new location of the zero of the optical field. In this way, the inversion profile tends to follow the motion of the “optical part” of the vortex. The restoring forces on the optical fields are such that the shifted zero of the field is pushed further in the same direction: the trailing slope is amplified more since the inversion is larger there, and the leading edge is attenuated more, since the inversion is smaller there.

As a consequence, the zero of the optical field moves away from the maximum of the population inversion. Its escape velocity is proportional to κ, the buildup rate of the optical field. The population inversion follows the escaping zero of the field. Its maximum velocity is proportional to the buildup rate of the population inversion, γ . If the population inversion is faster than the optical field, i.e. γ ≥ κ, its maximum cannot escape the zero of the optical field, and thus the location of the optical vortex is stabilized. In the case of a class B laser, where γ ≤ κ, the population inversion relaxes more slowly than the field, and the race between the zero of the optical field and the maximum of the population inversion continues forever.

The velocity of the vortex motion can be estimated by assuming that the laser field at a particular location goes through half a period of a relaxation oscillation during the passage of a restless vortex through it. Since the ra-

dius of the vortex is proportional to r0 ≈ a/ (D0 − 1), and the relaxation

oscillations have a frequency ωrel = 2γ(D0 − 1) − γ2/4, the velocity of the self-induced motion estimated in this way is