Ray-wave correspondence in the nonlinear description of stadium-cavity lasers

Abstract

We show that the solution of fully nonlinear lasing equations for stadium cavities exhibits a highly directional emission pattern. This directionality can be well explained by a ray-dynamical model, where the dominant ray-escape dynamics is governed by the unstable manifolds of the unstable short periodic orbits for the stadium cavity. Investigating the cold-cavity modes relevant for the lasing, we found that all of the high-Q modes have the emission directionality corresponding to that of the ray-dynamical model.Establishing a correspondence between the ray ͑or classi-cal͒ picture and the wave ͑or quantum͒ picture has been a fundamental problem in the field of wave ͑or quantum͒ chaos ͓1͔. One encounters this problem when trying to understand the emission properties from two-dimensional ͑2D͒ microcavity lasers. In such lasers, as a way to extract highly directional emission, it has been proposed to deform the cavity shape smoothly from perfect circularity ͓2-6͔. The result is that rays start to exhibit a variety of dynamics from integrable to strongly chaotic, which is tunable by the deformation.The ray picture has been providing a simple and intuitive method to explain experimental observations of emission directionality. For instance, emission directionality has been associated with the existence of a periodic orbit with a particular geometry ͓6,7͔, drastic shape dependence of emission directionality has been successfully explained by the difference of phase-space structure ͓8͔, and the far-field intensity patterns have been closely reproduced by ray-tracing simulations ͓8-10͔.Among various cavity shapes, the stadium is a simple geometry for which ray dynamics has been rigorously proven to become strongly chaotic ͓11͔. For almost all initial conditions, a ray trajectory explores the entire phase space uniformly. Even for such a strongly chaotic cavity, if one considers refractive emission of light due to the dielectric nature of the cavity, the emission pattern can become highly directional. Namely, strongly chaotic dynamics and highly directional emission are compatible, as was demonstrated by Schwefel et al. ͓8͔, who associated this property with escape dynamics dominated by flow in phase space along the unstable manifolds of the unstable short periodic orbits of a chaotic system.In this paper, we report further evidence for the ability of a ray-dynamical model to describe the lasing states of twodimensional microcavities. Earlier work has focused on establishing a relationship between the ray model and a few quasibound-state solutions of the linear wave equation, without pumping or gain. Which modes to choose for comparison in this case has an intrinsic arbitrariness, although plausibility arguments can be made based on their Q values. Here we show that the solution of the full nonlinear lasing equations for a stadium cavity, uniquely determined by the pumping conditions, has highly directional emission in good agreement with the ray model. This is one of the first pieces of evidence that the multimode solut...

Two‐dimensional microcavity lasers

“…In this case, the ray model predicts a highly directional emission pattern. Moreover, wave analysis reveals that every low-loss resonance has a far-field pattern closely corresponding to that of the ray model [48,49,119]. A typical far-field pattern of a low-loss resonance is plotted by a thin curve in Fig.…”

Inferring periodic orbits from spectra of simply shaped microlasers

“…[1]) it is widely noticed and accepted at least for modes with small losses (cf. [14,15,16]). For each individual lasing mode, the C m coefficients could be determined only after the solution of the full Maxwell-Bloch equations.…”

“…The derivation of this trace formula assumes only the semi-classical approximation (|k|R ≫ 1) and can be done in a way similar to that of the billiard case (see e.g. [27]), leading to an expression closed to (16) …”

Fractal Weyl law for chaotic microcavities: Fresnel’s laws imply multifractal scattering

“…We choose the case L = R which is the most chaotic one [44]. The spatial structure of optical modes in microstadiums has been extensively studied in the context of ray-wave correspondence in open systems [45,46,47,48,49]. Our aim is to study the spectral properties of modes in such a kind of cavity.…”