Modes of wave-chaotic dielectric resonators

A promising approach to making compact and high-Q optical resonators is to base them on "totally internally reflected" (TIR) modes of dielectric microstructures. Such devices have received considerable attention as versatile components for integrated optics and for low-threshold micron-scale semiconductor lasers (Chang and Campillo [1996], Yamamoto and Slusher [1993]). The interest in such resonators for applications and for fundamental optical physics has motivated the extension of optical resonator theory to describe such systems. All optical resonators are open systems described by modes characterized by both a central frequency and a width (their ratio giving the Q-factor of the mode). In a mirror-based resonator the set of resonant frequencies is determined by the optical path length for one round trip along a path determined by the mirrors within the resonator; the width is determined by the reflectivity of the mirrors, diffraction at the mirror edges and absorption loss within the resonator. Accurate analytic formulas can be found for the resonator frequencies and for the electric field distribution of each mode using the methods of Gaussian optics (Siegman [1986]). The modes are characterized by one longitudinal and two transverse mode indices (in three dimensions). These mode indices play the same role mathematically for the electromagnetic wave equation as do good quantum numbers in characterizing solutions of the wave equation of quantum mechanics. For constructing optical resonators on the micron scale, using total internal reflection from a dielectric interface for optical confinement is convenient as it simplifies the fabrication process. However such dielectric resonators define no specific optical path length; many different and potentially nonclosed ray trajectories can be confined within the resonator. An important point, emphasized in the current work, is that in such resonators there typically exist many narrow resonances characterized by their frequency and width, but such resonances often cannot be characterized by any further modes indices. This is the analog of a quantum system in which there are no good quantum numbers except for the energy. We shall see that the way to determine whether a given mode has additional mode indices (other than the frequency), is to determine whether it corresponds to regular or chaotic ray motion. We will present below an efficient numerical method for calculating all of the resonances of a large class of dielectric resonators; we will also describe the surface of section and Husimi-Poincaré projection method to determine the ray dynamics corresponding to such a mode. Although both DBR-based and edge-emitting optical resonators rely on reflectivity from a dielectric interface (at normal incidence), we will use the term dielectric resonator (DR) to refer to resonators that rely on the high reflectivity of dielectric bodies to radiation incident from within the dielectric near the critical angle for total internal reflection. This is the only class of resonators we will treat below. We immediately point out that TIR solutions of the wave equation exist only for infinite flat dielectric interfaces; any curvature or finite extent of the dielectric will allow evanescent leakage of propagating radiation from the optically more dense to the less dense medium. As a dielectric resonator is a finite dielectric body embedded in air (or in a lower-index medium) it will of necessity allow some evanescent leakage of all modes, even those which from ray analysis appear to be totally internally reflected. A very large range of shapes for DRs have been studied during the recent years. By far the most widely studied are rotationally symmetric structures such as spheres, cylinders and disks. In this case the wave equation is separable and the solutions can be written in terms of special functions carrying two or three mode indices (neglecting finite size in the axial direction for the cylinders and disks). The narrow (long-lived) resonances correspond to ray trajectories circling around the symmetry axis near the boundary with angle of incidence above total internal reflection; these solutions are often referred to as "whispering-gallery" (WG) modes or morphology-dependent resonances. In this case, owing to the separability of the problem, it is straightforward to evaluate the violation of total internal reflection, which may be interpreted as the tunneling of waves through the angular momentum barrier (Johnson [1993], Nöckel [1997]). Micron-scale, high-Q microlasers were fabricated in the mid-1980s and early 1990s based on such cylindrical or disk-shaped ( Q ∼ 10⁴ - 10⁵) (McCall, Levi, Slusher, Pearton and Logan [1992], Slusher, Levi, Mohideen, McCall, Pearton and Logan [1993], Levi, Slusher, McCall, Tanbunek, Coblentz and Pearton [1992]), and spherical ( Q ∼ 10⁸ - 10¹²) (Collot, Lefevreseguin, Brune, Raimond and Haroche [1993]) dielectric resonators. However, the very high Q-value makes these resonators unsuitable for microlaser applications, because such lasers invariably provide low-output power and furthermore, unless additional guiding elements are used, the lasing output is emitted isotropically. As early as 1994, Nöckel, Stone and Chang proposed to study dielectric resonators based on smooth deformations of cylinders or spheres which were referred to as "asymmetric resonant cavities" (ARCs). The idea was to attempt to combine the high Q provided by near total internal reflection with a breaking of rotational symmetry, leading to directional emission and improved output coupling. General principles of nonlinear dynamics applied to the ray motion (to be reviewed below) suggested that there would be only a gradual degradation of the high-Q modes, and one might be able to obtain directional emission from deformed whispering-gallery modes. Experimental (Nöckel, Stone, Chen, Grossman and Chang [1996], Mekis, Nöckel, Chen, Stone and Chang [1995], Chang, Chang, Stone and Nöckel [2000]) and theoretical (Nöckel and Stone [1997]) work following that initial suggestion has confirmed this idea, although the important modes are not always of the whispering-gallery type (Gmachl, Capasso, Narimanov, Nöckel, Stone, Faist, Sivco and Cho [1998], Gianordoli, Hvozdara, Strasser, Schrenk, Faist and Gornik [2000], Gmachl, Narimanov, Capasso, Baillargeon and Cho [2002], Rex, Türeci, Schwefel, Chang and Stone [2002], Lee, Lee, Chang, Moon, Kim and An [2002]). The calculation of the modal properties of deformed cylindrical and spherical resonators presents a much more challenging theoretical problem. Unless the boundary of the resonator corresponds to a constant coordinate surface of some orthogonal coordinate system, the resulting partial differential equation will not be solvable by separation of variables. The only relevant separable case is an exactly elliptical deformation of the boundary, which turns out to be unrepresentative of generic smooth deformations. Using perturbation theory to evaluate the new modes based on those of the cylindrical or spherical case is also impractical, as for interesting deformations and typical resonator dimensions (tens of microns or larger) the effect of the deformation is too large for the modes of interest to be treated by perturbation theory. The small parameter in the problem for attempting approximate solutions is the ratio of the wavelength to the perimeter, λ / 2 π R = ( k R )^-1. Eikonal methods (Kravtsov and Orlov [1999]) and Gaussian optical methods (Siegman [1986]) both rely on the short-wavelength limit to find approximate solutions. The Gaussian optical method can be used to find a subset of the solutions for generic ARCs, those associated with stable periodic ray orbits, as explained in detail by Türeci, Schwefel, Stone and Narimanov [2002]. The eikonal method can also be used to find a subset of ARC modes, if one has a good approximate expression for a local constant of motion; an example is the adiabatic approximation used by Nöckel and Stone [1997]. However a large fraction of ARC modes cannot be described by either of these methods. The breakdown of the Gaussian optical method is easily understood, as a fully chaotic system will have only unstable periodic orbits, and no consistent solution can be obtained by the Gaussian method based on unstable periodic ray orbits (Türeci, Schwefel, Stone and Narimanov [2002]). The failure of eikonal methods for generic shapes is more subtle, and arises from the possibility of chaotic ray motion in a finite fraction of phase space. Optics textbooks, and even standard research references, often treat the eikonal method as being of completely general applicability; we therefore will devote the next section of this chapter to an explanation of the failure of eikonal methods for resonators with arbitrary smooth boundaries. In Section 3 we describe the phase-space methods which indicate that this failure is generic. In Section 4 we present the formulation of the resonance problem, and Section 5 covers the reduction of the Maxwell equations to the Helmholtz equation for the resonators we study. The failure of all standard short-wavelength approximation methods to describe the solution of wave equations in finite domains with arbitrary smooth boundaries has led to the problem of "quantizing chaos" in the context of the Schrödinger equation and the Helmholtz equation (although our problem is somewhat different due to the dielectric boundary conditions on this equation). Although substantial progress has been made using periodic-orbit methods in obtaining approximations for the density of states of fully chaotic systems, these methods do not yield individual solutions of the wave equation.