Frequency splitting of polarization eigenmodes in microscopic Fabry–Perot cavities

Abstract

We study the frequency splitting of the polarization eigenmodes of the fundamental transverse mode in CO 2 laser-machined, high-finesse optical Fabry-Perot cavities and investigate the influence of the geometry of the cavity mirrors. Their highly reflective surfaces are typically not rotationally symmetric but have slightly different radii of curvature along two principal axes. We observe that the eccentricity of such elliptical mirrors lifts the degeneracy of the polarization eigenmodes. The impact of the eccentricity increases for smaller radii of curvature. A model derived from corrections to the paraxial resonator theory is in excellent agreement with the measurements, showing that geometric effects are the main source of the frequency splitting of polarization modes for the type of microscopic cavity studied here. By rotating one of the mirrors around the cavity axis, the splitting can be tuned. In the case of an identical differential phase shift per mirror, it can even be eliminated, despite a nonvanishing eccentricity of each mirror. We expect our results to have important implications for many experiments in cavity quantum electrodynamics, where Fabry-Perot cavities with small mode volumes are required. [33,34] to applications in quantum information processing, such as the efficient and coherent coupling of atomic states to the polarization of single photons [35]. The latter requires degeneracy of the polarization eigenmodes, which has been achieved for Fabry-Perot cavities built from superpolished mirror substrates [36]. Microfabricated cavities, however, can have increased frequency splittings between polarization eigenmodes, as was first observed in CO 2 laser-machined resonators [10,20]. If the splitting is on the order of the linewidth of the cavity, there can be detrimental effects on all kinds of experiments [37][38][39]. There are two strategies for dealing with the splitting in cavities: to minimize it until it becomes negligible, or to increase it such that the two polarization modes are well separated [38]. In either case, it is necessary to understand and control this splitting.Two potential sources of the splitting of polarization eigenmodes in a Fabry-Perot cavity can be distinguished. The first one is birefringence of the mirror materials, usually attributed to mechanical stress [39,40]. Combined with a finite penetration depth, this leads to a polarization-dependent phase shift upon reflection. The second source is directly related to the cavity geometry. Its existence is not evident from the usual paraxial resonator theory, in which the cavity field and its resonances are described by a scalar mode function that is independent of the polarization. The paraxial theory does describe the polarization-independent splitting of higher-order transverse modes of equal order in a cavity with elliptical mirrors, but it cannot account for an additional splitting of each of these modes into a doublet via the polarization degree of freedom. Any splitting of the polarization modes thus has to originat...

Fabrication of mirror templates in silica with micron-sized radii of curvature

“…In Figure 4(c), two adjacent longitudinal modes (q, 0) are recorded, each split into a linearpolarized fine structure likely originating from the slightly elliptical mirror surface. 24 For the TEM 400 mode, we demonstrate a large quality factor of Q ¼ 10 5 and high finesse F ¼ 2:5 Â 10 4 , very well suitable for the target applications in cavity quantum electrodynamics. The measured finesse is close to the one expected from the bare mirror reflectivity (99.9925%, measured on a fused silica witness sample by the manufacturer).…”

Optomechanics with a polarization nondegenerate cavity

“…A clear twofold symmetry is present, with a minimum ROC of about 1 mm and a maximum ROC of about 4 mm. Using these numbers together with the recently published work by Uphoff et al [26], a polarization splitting of about 60 kHz for the fundamental mode is expected based on the parameters of the setup.…”

“…With some small modifications to the trampoline resonator design, the mode splitting can be pushed from 83 to about 100 kHz. Since the mode splitting scales inversely with cavity length [26], reducing the cavity length by a factor of five results in the desired mode splitting. This will also increase the optomechanical coupling strength g 0 to about 2π × 8 rad/s.…”

High mechanical bandwidth fiber-coupled Fabry-Perot cavity

“…The modulated beam passes through a circulator (C) and couples to the cavity through the fiber mirror. The degeneracy of the two linear polarization states of the fundamental cavity mode is lifted by ∼100 MHz, likely due to a combination of fiber mirror ellipticity and birefringence [29]. The polarization is adjusted to select only one of these modes using a fiber polarizer (P), which we correct on an hourly basis as the strain in the fibers changes with lab temperature (typical temperature excursions < 1 • C).…”