Degenerate perturbation theory describing the mixing of orbital angular momentum modes in Fabry-Perot cavity resonators

We present an analytic perturbation theory which extends the paraxial approximation for a common cylindrically symmetric stable optical resonator and incorporates the differential, polarization-dependent reflectivity of a Bragg mirror. The degeneracy of Laguerre-Gauss modes with distinct orbital angular momentum (OAM) and polarization, but identical transverse order $N$, will become observably lifted at sufficiently small size and high finesse. The resulting paraxial eigenmodes possess two distinct OAM components, the fractional composition subtly depending on mirror structure.

Figure 1

(Color online) (a) Planoconcave model geometry. The top mirror radius of curvature $R$ is $100\mu \mathrm{m}$ in the examples. (b) Wave number splitting $\Delta k\equiv k_2-k_1=\delta k_2-\delta k_1$ from numerical data at $L=25\mu \mathrm{m}$, $\lambda \approx 400\mathrm{nm}$, indicating the avoided crossing between a mixable pair of modes. (c) Numerical results (symbols) for the mode mixing angle $\alpha$ closely fit the (solid) curve $\alpha =\arctan [({ϵ}_p-{ϵ}_s)∕\Delta ϵ-b]∕2$. Different symbols (colors) represent data series at $L=2.5–25\mu \mathrm{m}$ (in steps of $2.5\mu \mathrm{m}$) with $kR\approx 780$ to 1631. $\Delta ϵ$ and $b$ are fit here for each series.

Figure 2

(Color online) Mode-coupling width $\Delta ϵ$ from fit (symbols) compared to perturbative prediction (solid line), plotted as a function of the cavity length $L$ at fixed $R=100\mu \mathrm{m}$. Error bars were obtained from the fits of Fig. 1c. Because the analytic equation (10) is wavelength independent, all data for $\lambda \approx 400\mathrm{nm}$ and $\lambda \approx 800\mathrm{nm}$ fall onto the same curve.