When do spheroids become spheres?

When do spheroids become spheres? In optics, it depends on the wavelength!
You think you have a spherical silica resonator? Watch out for non-perturbative optical effects even at seemingly minute deformations where ray chaos plays no significant role!

For an overview of our work on deformed spheroidal resonators, go to the droplets page.

Whispering-gallery modes can support lasing in extremely small dielectric resonators. However, even glass spheres of roughly 1mm diameter - a macroscopically large size compared to the wavelength, - have long ago been observed to lase on such modes [C.G.B. Garrett et al., Phys. Rev. 124, p. 1807 (1961)].

In this large-size limit one can explore the interesting double limit ε→0 and λ/R→0. Here, ε is the deformation (in percent) away from the circular cross section, and λ is the wavelength, which should be compared to the mean cavity radius R.

That is the point of view we pursue with a series of non-spherical resonators of mean radius R≈100μm. The index of refraction of such fused-silica spheroids is n=1.45, and in contrast to earlier experiments on lasing droplets they contain no amplifying dopants.

Above:

A pin-wheel-like emission pattern from the near-circular cross section of a microsphere. The radiation in the far-field looks almost symmetric under rotation by 90°.

Parameters: Deformation ε = 3.4% Size parameter kR = 112.452 Decay width Im(kR)=-0.0005

Below:

Husimi projection for the mode above. High intensity is restricted to a region filled with invariant curves belonging to regular whispering-gallery orbits. Since this keeps the rays confined above the classical escape condition (sinχ<1/n, shown as the purple horizontal line), the escape must occur by tunneling: this happens most strongly at the 4 values of φ where the invariant curves dip toward the escape window.

Above:

The same mode as on the left, but followed to a slightly more deformed cross section. The radiation pattern is less symmetric, with much stronger emission in the directions corresponding to dynamic eclipsing.

Parameters: Deformation ε = 6.5% Size parameter kR = 112.063 Decay width Im(kR)=-0.024

Below:

The Husimi projection for the mode above illustrates that classically allowed ray escape is taking place. Although the mode is partially supported by the non-chaotic, unbroken whispering gallery curves above the 4-bounce island chain, the escape line is reached primarily by overlap with classically escaping regions of the phase space.

How can emission patterns have fewer symmetries than the emitting cavity?

The "pinwheel" shown above doesn't have reflection symmetry. This has arisen in another paper published shortly after the work described here, by Harayama et al. ["Asymmetric Stationary Lasing Patterns in 2D Symmetric Microcavities ", PRL 91, 073903 (2003)]. In my own work, this has previously happened in several different contexts:

The basic idea is that all the symmetries of the cavity are indeed reflected in the quasibound states, but when some of these are nearly degenerate (to within their intrinsic linewidth) one can observe arbitrary superpositions that break some of the spatial symmetries. In our case, standing-waves of different parity under reflection combine into travelling waves.

The travelling-wave superpositions whose wave patterns are plotted above are adapted to the experimental excitation conditions. I originally came up with this type of travelling-wave mode to explain the observed emission directionality in experiments by Claire Gmachl et al., in an earlier collaboration (unpublished). In lasers, such travelling waves can occur due to spontaneous symmetry breaking, and that caused me to use the same type of superpositions in modelling hexagonal zeolite-dye microlasers.

The relationship between asymmetric emission, time-reversal and reflection symmetry was a main topic of my invited talk at the Symposium on semiclassical methods in atomic and molecular physics in Bonn, April 3, 2000.

In the purely "elastic" experiments described here, the travelling waves are excited because the coupling prism injects waves with only one sense of circulation into the whispering-gallery region of the spheroid.

There is more discussion of the difference between standing and traveling waves in open resonators on this page dealing with rotationally symmetric geometries.