Shedding light on periodic orbits in triangular organic micro-billiard lasers

Abstract

The resonances of optical micro-resonators can be related to the periodic orbits of the corresponding classical billiard via a semiclassical trace formula [1]. This formula connects the resonance frequencies with the periodic orbits existing in the corresponding billiard and their properties like length, refractive losses and stability. The trace formula thus allows to compute the resonance frequencies approximately using only quantities from classical mechanics which can often be obtained with relative ease.For the case of general triangle billiards, however, the computation of periodic orbits remains an open mathematical problem [2], and the calculation of the modes of triangular resonators is also possible only for special cases [3]. We therefore adopt the inverse approach, i.e. we start from the resonant modes that we obtain experimentally from organic microlasers, and deduce from these the periodic orbits existing in the corresponding billiards. Indeed, the connection between periodic orbits and the lasing modes of flat organic microlasers has been demonstrated experimentally for cavities of various shapes [1,4]. We fabricate microlasers made of a polymer (PMMA) doped with an organic laser dye (e.g. DCM) using electron-beam lithography. The dye is homogeneously spread in the resonator volume [see the yellow colour of the optical photograph in Fig. 1(b)] and it is pumped optically by a pulsed 532nm laser. The thickness of the cavities is about one wavelength [see Fig. 1(a)] so that they can be approximated as two-dimensional systems. On the other hand, the transverse extension is big enough (~300 μm) so that they can be treated with semiclassical methods like trace formulas. Different triangular cavities are investigated corresponding to different mathematical classes of triangles. The lasing spectrum and far-field patterns of the microlasers are measured. The lasing spectra consist of sharp resonances that are equally spaced [see Fig. 1(d)]. Their free spectral range is directly connected to the length of the underlying periodic orbit [4]. The angular distribution of the lasing emission is not isotropic but concentrated in a few specific directions [see Fig. 1(c)]. These can also help to identify the corresponding periodic orbit. For example, the major emission directions in Fig. 1(c) are the directions into which the orbit indicated by the solid line in Fig. 1(b) is refracted (indicated by the dashed lines). Thus the experimental observations allow to identify the periodic orbit on which the lasing modes are based.In conclusion, we investigated the periodic orbits in different triangle billiards using organic microlasers. The periodic orbits sustaining the lasing modes are determined from the experiments and compared to the predictions by the trace formula. Different kinds of periodic orbits (in-family/isolated, diffractive, ...) were observed for various shapes of triangles. Interesting cases are reported where respectively none or several competing orbits were identified.