Mesoscopic Physics and Quantum Chaos

Mesoscopic Optics

Jens U.Nöckel

A worldwide rush is on to develop integrated optical components, aiming to increase data transmission rates and volumes in fiber optic networks in response to exponentially increasing demand. Of central importance to this 1.3 trillion US dollar market sector are dielectric microcavity resonators for use in photonic applications such as filters, multiplexers and lasers. Using semiclassical methods and tools from nonlinear dynamics, we analyze the spectral and emission characteristics of such open resonators.

Waveguiding and long-lived resonator modes

Fig.1 Extremely small, high-quality resonators can be fabricated by exploiting total internal reflection at the interface between a circularly symmetric dielectric and the surrounding air; see Fig.1 which shows the cross-sectional intensity distribution of a long-lived photonic mode in a dielectric disk. Classically forbidden tunnel leakage across the rim (black line) leads only to minute outcoupling losses which moreover are isotropic. Coupling to such a resonator is very inefficient unless another optical component is brought extremely close to the rim so as to overlap with the evanescent near field. This is not amenable to highly reproducible device fabrication.

An alternative and potentially much more efficient coupling mechanism is introduced when the rotational invariance of the resonator is broken, thus creating preferred emission directions, cf. Fig.2. Shown in false-color is the intensity of a long-lived mode as obtained from exact solution of Maxwell's equations. The green lobes extending into the far-field indicate the preferred directions along which coupling into or out of the resonator is now possible with great efficiency.

Fig. 1

Fig.2 Shown at the bottom of Fig.2 is the classical ray picture corresponding to the wave solution depicted above. As indicated by the arrow, rays escape according to Snell's law after a number of total internal reflections because the angle of incidence is not conserved in the oval cavity. The wave solution mimicks this classical behavior and hence the emission directions as well as the lifetime of the mode can be predicted with the help of the ray picture. To establish the connection between rays and waves, semiclassical methods are required.

Fig. 2

Phase space structure in the emission profile

The classical ray dynamics in general deformed resonators exhibits a coexistence of periodic, quasiperiodic and chaotic trajectories which is difficult to disentangle by ray-tracing. A more structured representation of the multitude of different trajectories can be obtained by making phase-space portraits. In this Poincaré section, black dots represent individual reflections of rays at the boundary, giving the position (polar angle $\phi$ ) and angle of incidence $\chi$. This representation (Fig. 3) of an orbit as a sequence of points ($\phi$, sin$\chi$) contains more information than is actually needed to reconstruct a given trajectory, but it reveals the hidden structure of the phase space. Chaotic orbits generate the grainy region filling most of the area in this plot. Island structure ("solid" lines) belongs to non-chaotic, regular motion. Emission properties of the resonator can be related to transport in this anisotropic phase space.
Superimposed onto the Poincaré section is a false-color density representation of a particular resonator mode.

Fig. 3

The high intensity around a chain of four islands of stability corresponds to a resonator mode associated with the corresponding stable motion. Its real-space intensity is shown in Fig. 4, where a bowtie-shaped pattern is apparent. The focussing inside and outside of the resonator is a desirable property that is achieved here by intentionally inducing the transition to partially chaotic ray dynamics. Oval lasers like the one shown here have been fabricated at Bell Labs and at less than 100 µm diameter (5.2 µm wavelength) delivered roughly 3 orders of magnitude more power than a comparable circular (non-chaotic) cavity.

Fig. 4

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