Hexagonal-shaped microcrystal lasers:

effects of corners and coupling

JAN WIERSIG

A novel class of microlasers based on nanoporous molecular sieve host-guest systems has been fabricated recently [1,2]. Organic dye guest molecules were inserted into the channel pores of a zeolitic microcrystal host. The aluminophosphate-crystals grow with natural hexagonal-shaped boundaries; see Fig. 1a. In terms of pump energy needed to reach lasing threshold these microlasers are comparable to semiconductor based vertical-cavity surface-emitting lasers (VCSELs). This makes them a promising candidate for future applications as e.g. optical communication devices. In collaboration with experimental physicists of the Technical University of Darmstadt and chemists of the University of Bremen (Volkswagen Foundation Project ``Molekularsieblaser-Konglomerate im Infraroten'') we study these microlasers in more detail. In particular, we try to improve their properties by coupling them to each other and/or to passive high-quality microresonators. Our theoretical approach is illustrated in the sequence of Figs. 1a-c. The real microlasers (Fig. 1a) are modelled by two-dimensional passive dielectric resonators using both numerical simulations of the full wave equations (Fig. 1b) and a semiclassical ray model (Fig. 1c).

a) b) c)

Figure 1: (a) Hexagonal microcrystal laser, diameter is 4..50 µm and wavelength is 600..800 nm. (b) False-colour representation of the near-field intensity pattern of a long-lived resonant mode calculated numerically from the full wave equations. Intensity is higher for redder colours, and vanishes in the dark regions. (c) Ray model. Red polygon marks a member of the family of whispering-gallery rays, other members are obtained by shifting the ray along the boundary. Arrows indicate emission due to boundary waves (black) and pseudointegrable dynamics (blue).

The resonant modes of dielectric cavities can be calculated analytically by means of separation of variables only for special geometries, like the circular cavity. In general, numerical methods are needed. Frequently used is the wave-matching method [3,4] which is suitable for sufficiently slight deformations of an isolated circular cavity. To treat coupled cavities with sharp corners, we have invented a variant of the boundary element method (BEM) [5].

Figure 1b shows a TM polarized resonant mode of an isolated hexagonal cavity with low refractive index calculated with the BEM. The light is concentrated along the boundary (whispering-gallery mode) and escapes predominantly at the corners. The emission directionality is highly anisotropic, a desired property in optical applications. These results are in agreement with the experiments and with earlier numerical results on rounded hexagons obtained from the wave-matching method [2]. However, we have observed that the modes in the hexagon with sharp corners and those in rounded hexagons differ substantially with respect to their lifetime and the exact emission direction. Apart from this unexpected sensitivity to rounding we found another interesting result: if the wavelength is fixed then the Q-factor (a measure of the quality of the mode) is approximately proportional to the diameter squared. Based on this finding we predict that the lasing threshold can be reduced considerably by increasing the size of the resonator.

The semiclassical (short-wavelength) approximation is applied in the field of quantum chaos to relate quantum (wave) dynamics to their underlying classical (ray) dynamics. Most research efforts have been focused on closed resonators, so-called billiards. Rational polygonal billiards (all angles between sides are rationally related to ) like the hexagon have, in general, peculiar properties due to the presence of corners, e.g. the classical dynamics is neither chaotic nor integrable but instead pseudointegrable [6,7], classical Fourier spectra have multifractal properties [8], the quantum spectrum obeys critical statistics [9], and the quantum-classical correspondence is exotic [10]. Many concepts of quantum chaos have been carried over to open resonators, e.g. scars [11,12]. However, semiclassical approximations have been introduced only in the case of smooth boundaries [3,4].

In collaboration with J. U. Nöckel from the University of Oregon we develop a semiclassical ray model for hexagonal-shaped cavities which later might be extended to any rational polygon. According to geometric optics, a family of periodic rays (whispering-gallery rays) with identical lengths is confined within the hexagonal resonator by total internal reflection at the facets; see Fig. 1c. Responsible for emission of light are wave effects. We have identified three different wave effects, all of which are related to the corners. The most obvious one is diffraction at corners. The next one is propagation of boundary waves and their separation from the boundary at the corners as illustrated in Fig. 1c. The third effect is that waves smear out classical phase-space structures. To take this into account one has to choose a small distribution of rays localized around the whispering-gallery rays as initial conditions rather than taking the whispering-gallery rays itself. As a consequence of the pseudointegrable dynamics the initial rays diverge slowly from the periodic rays until they fully separate from them at a corner leading directly to refractive escape as depicted in Fig. 1c. Our preliminary semiclassical model includes the boundary-wave and the pseudointegrable mechanism, whereas corner diffraction is ignored. The model allows for analytic evaluation of many quantities of interest, including the frequency spacing, the lifetimes, and the Q-factors. Moreover, it explains the sensitivity to rounding of the corners. In the future, we will include corner diffraction which seems to be important for the far-field pattern.

In Fig. 2a we see an example of two coupled hexagonal-shaped resonators. Careful comparison with the isolated hexagon demonstrates that the emission directionality has improved, whereas the Q-factor has remained unchanged. Figure 2b shows a hexagonal-shaped resonator coupled to a circular high-quality resonator. It turns out that the Q-factor of the mode in the composite system has been enhanced by a factor of if compared to the corresponding mode in the isolated hexagon. In the experiments, silica spheres will be used rather than circular cylinders. Nevertheless, we believe that the 2D calculations will give deep insights, not only relevant for the experiments but also for the theory of dielectric resonators in general.

a) b)

Figure 2: Near-field intensity pattern of coupled microresonators calculated numerically from the full wave equations.

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