Playing billiard in laser micro-cavities: a test-bed for semi-classical physics

The generic “billiard problem” is a well-known paradigm of nonlinear mathematical physics, which connects to deep issues in quantum and wave physics all the way to quantum chaos. It can be implemented in mechanics, optics or electromagnetism, either in classical or quantum mechanics, depending on experimental configurations and on the billiard length-scale. The elusive borders between wave and geometric optics on the one hand, and between quantum and classical mechanics on the other, exhibit deep analogies, which can be both addressed in actual billiard-like physical systems. We will show the relevance in this context of micro-billiard shaped lasers (1-4), thanks to new experimental and technological advances in the realm of polymer based photonics at micron and submicron scales. In such configurations, confinement of light in resonators can be considered by analogy with that of a quantum particle in a well (the 2-D quantum or wave billiard). Spatially distributed modes can be connected to classical orbits within the frame of semi-classical physics approximations, by use of the celebrated “trace theorem”, herein generalized to open and chaotic cavities. A beneficial feature of dielectric cavities, in contrast with their more investigated metallised contour counterparts, is the ability for the electromagnetic wave to spread-out of the cavity by refraction, diffraction, evanescent tunnelling or a combination of these, allowing to simplify the otherwise hidden physics and eventually lending itself to sensor applications. However, such “open cavities” are more challenging from a theoretical and modelling point of view, giving raise to non-Hermitian operators and imaginary eigenvalues accounting for a finite modal excitation life-time in lossy cavities. Analytical solutions such as available for the metallized 2-D rectangle are not valid for the equivalent open structures which demand to resort either to full-fledged solutions of the Maxwell-Helmholtz equations with continuity conditions on the contour, or to application of semi-classical orbit methods. We will show consistence between those two avenues and experimental findings. Particular attention will be paid to recent advances: systematic investigations of triangular cavities (5) and extension to 3-D micro-billiards (6). The role of contour singularities will be evidenced and the related diffractive orbits pin-pointed. The technological precision required for such studies is reached by advanced e-beam patterning methods applied to dye-doped polymer structures, down to the required nano-scale level of precision.
Our investigations illustrate an approach whereby, contrary to the more academic pathway from upstream mathematical predictions down to experimental applications, experimental findings may help provide guidelines towards otherwise elusive mathematical problems, such as the diffraction of light by a dielectric prism that the lasing property allows to illuminate from its inside. 
This is being performed at LPQM/ENS Cachan, together with Clément Lafargue (Ph.D. student), Stefan Bittner (postdoctoral) and Mélanie Lebental (assistant professor) within our “microlaser and nonlinear dynamics” research group.

 (1) Directional emission of stadium shaped micro-lasers, M.Lebental, J.S.Lauret, J.Zyss, C.Schmidt, E.Bogomolny, Phys. Rev. A 75, 033806 (2007)
 (2) Inferring periodic orbits from spectra of shaped micro-lasers, M.Lebental, N.Djellali, C.Arnaud, 
J.-S.Lauret, J.Zyss, R.Dubertrand, C.Schmit, E.Bougomolny, , Phys Rev. A, 76 023830 (2007).
 (3) Organic Micro-Lasers: A New Avenue onto Wave Chaos Physics, M.Lebental, E.Bogomolny and J.Zyss, Chapter 6, pp. 317-353 in “Practical Applications of Microresonnators in Optics and Photonics”, Andrey Matsko editor,(CRC Press, Boca Raton, 2009)
(4) Trace formula for dielectric cavities. II. Regular, pseudointegrable, and chaotic examples E.Bogomolny, N.Djellali, R.Dubertrand, I.Gozhyk, M.Lebental, C.Schmit, C.Ulysse and J.Zyss, Phys.Rev. E 83, 036208 (2011)
(5) Localized lasing modes of triangular organic microlasers, C. Lafargue, M. Lebental, A. Grigis, C. Ulysse, I. Gozhyk, N. Djellali, 
J. Zyss and S. Bittner, Phys.Rev.E 90, 052922 (2014)
(6) Three-dimensional organic microlasers with low lasing thresholds fabricated by multiphoton and UV lithography
Vincent W. Chen, Nina Sobeshchuk, Clément Lafargue, Eric S. Mansfield, Jeannie Yom, Lucas R. Johnstone, Joel M. Hales, Stefan Bittner, Séverin Charpignon, David Ulbricht, Joseph Lautru, Igor Denisyuk, Joseph Zyss, Joseph W.Perry and Melanie Lebental
Optics Express 22 (10), 12316-12326 (2014)