Advanced mirror shapes for mode enhancement in plano-concave cavities

TL;DR

The paper addresses the challenge of achieving strong emitter–cavity coupling in plano-concave Fabry-Pérot cavities by introducing shaped non-planar mirrors. It defines an emitter-independent metric, $C_{ ext{int}}^0 = \frac{6 \lambda^2}{\pi^2 w_e^2 \mathcal{L}_{\text{int}}}$, to quantify focusing efficiency and compares limits for concave–concae and plano–concave geometries, highlighting clipping losses and misalignment sensitivity. Through mode-mixing simulations, the authors show that simple, manufacturable mirror shapes (Gaussian, dual-curvature, spline) can realize substantial, often order-of-magnitude, improvements in $C_{ ext{int}}^0$ for plano-concave cavities, approaching CC performance without sacrificing alignment tolerance. These results yield a practical framework for designing shaped plano-concave cavities, enabling more stable, scalable emitter–cavity systems for quantum technologies such as single-photon sources and qubit readout.

Abstract

Optical cavities are frequently used in quantum technologies to enhance light matter interactions, with applications including single photon generation and entanglement of distant emitters. The Fabry-Pérot resonator is a popular choice for its high optical access and large emitter-mirror separation. A typical configuration, particularly for emitters that should not be placed close to the mirror surface like trapped ions and Rydberg atoms, features two spherical mirrors placed around a central emitter, but this arrangement can put demanding requirements on the mirror alignment. In contrast, plano-concave cavities are tolerant to mirror misalignment and only require the manufacture of one curved mirror, but have limited ability to focus light in the centre of the cavity. Here we show how mirror shaping can overcome this limitation of plano-concave cavities while preserving the key advantages. We demonstrate through numerical simulations that simple mirror shaping can increase coupling between a plano-concave cavity and a central emitter by an order of magnitude, even rivalling misalignment-sensitive concave-concave counterparts for achievable interaction strength. We use these observations to establish the conditions under which plano-concave cavities with shaped mirrors could improve the performance and practicality of emitter-cavity systems.

Figures (5)

• Figure 1: Results of retroreflective optimisation for a cavity with the example geometry ($L=$ 1mm, $\lambda=$ 1033nm, $D=$ 300µm, $\mathcal{L}_{\mathrm{abs}}=$ 20ppm). a) Surface profile of the (black) best spherical mirror and (blue) retroreflective optimised mirror with (red) residuals overlaid. b) Performance of a plano-concave cavity with the optimised non-planar mirror as a function of length change from the nominal value. The black line marks $C_{\mathrm{int}}^{0,\mathrm{PC}}$ over the length scan. c) The running-wave intensity (linear scale on top, log scale bottom) of the (blue) cavity eigenmode and (black) best spherical mode $\epsilon^{\mathrm{PC}}_{\mathrm{opt. sph.}}$ in the i) emitter and ii) mirror planes respectively. The vertical lines in ii) represent the edge of the mirror. d) Mode intensity in the $xz$-plane ($y=0$) for i) $\epsilon^{\mathrm{PC}}_{\mathrm{opt. sph.}}$ and ii) the optimised mode. The planar mirror is at $z=0$ (left) and the non-planar mirror at $z=L$ (right). The cyan cross marks the emitter position.
• Figure 2: Example mirror profiles and resulting high-performing cavities for the example geometry ($L=$ 1mm, $\lambda=$ 1033nm, $D=$ 300µm, $\mathcal{L}_{\mathrm{abs}}=$ 20ppm). a) Example mirror shapes with i) Gaussian, ii) dual-curvature, and iii) 3-point spline designs (blue) compared to the best spherical mirror (black). In ai) annotations mark the central radius $R$ and waist $w_G$ of the Gaussian profile (labelled as $\frac{1}{2}w_G$ because the Gaussian waist exceeds half the mirror diameter). In aii) annotations mark the four parameters of the dual curvature mirror (central $R_c$ and outer $R_o$ radii of curvature, and transition radius $\delta_\sigma$ and width $w_\sigma$). In aiii) the crosses mark the coordinates of the chosen spline points. b) Mode intensities in the $xz$-plane ($y=0$) for cavities with the Gaussian, dual curvature, and 3-point spline mirrors shown in a), where cyan crosses mark the emitter positions. c) Mode intensity in the emitter plane and d) $C_{\mathrm{int}}^0$ as a function of length change from the nominal value for the surface profiles in a), and for the optimised spherical mirror and the retroreflective optimised mirror shown in Fig. \ref{['fig: retroreflective_optimisation_example']}.
• Figure 3: Achievable $C_{\mathrm{int}}^0$ as a function of mirror diameter for different surface profile designs. The cavities have $\lambda=$ 1033nm, $L=$ 1mm, and $\mathcal{L}_{\mathrm{abs}}=$ 20ppm. a) Comparison of mirror shaping in plano-concave cavities. The different lines represent shaping strategies labelled in the legend, where 'PC' is shorthand for plano-concave. The black, dashed, perfectly horizontal line marks $C_{\mathrm{int}}^{0,\mathrm{PC}}$ (Eq. (\ref{['eq: cint pc limit']})). b) Comparison of plano-concave designs to concave-concave designs, where 'CC' in the legend denotes 'concave-concave'. For the concave-concave cavities with transverse misalignment, the misalignment is labelled on the corresponding line. The blue dashed line graphs $C_{\mathrm{int}}^{0,\mathrm{CC}}$ (Eq. (\ref{['eq: cint concave concave large diameter limit']})).
• Figure 4: Sensitivity of few-parameter mirrors to errors in their shaping parameters for the example cavity geometry ($L=$ 1mm, $\lambda=$ 1033nm, $D=$ 300µm, $\mathcal{L}_{\mathrm{abs}}=$ 20ppm). a) Sensitivity of $C_{\mathrm{int}}^0$ to the parameters of the Gaussian mirror, where the black cross marks the example Gaussian mirror in Fig. \ref{['fig: designed surfaces']}. b-c) Sensitivity of $C_{\mathrm{int}}^0$ to the b) radial and c) sigmoid parameters of the dual curvature mirror. The black crosses collectively mark the example dual curvature mirror in Fig. \ref{['fig: designed surfaces']}, with the black cross in b) marking the static radial parameters for panel c), and vice versa. For all figures, the dotted black line marks $R^{\mathrm{PC}}_{\mathrm{opt. sph.}}$, and the cyan contour $C_{\mathrm{int}}^{0,\mathrm{PC}}$ (Eq. (\ref{['eq: cint pc limit']})).
• Figure 5: Diagram of the clipping loss calculation for a misaligned concave-concave cavity with length $L$ and mirror misalignment $M$ used for results in Fig. \ref{['fig: master scan combined']}b of the main text. The mode (thick red line, with thin red lines indicating the $1/e$ power waist) predicted by classical theory propagates along the line between the centres (black dots) of the circles that define the two mirrors. The point on the edge of the mirror where the mode has highest intensity is $P_{\mathrm{clip}}$. The clipping plane $\Pi_{\mathrm{clip}}$ is defined as the plane perpendicular to the mode running through $P_\mathrm{clip}$. The power retained in the mode upon reflection is calculated by integrating the intensity profile of the mode in $\Pi_{\mathrm{clip}}$ (yellow Gaussian) over the mirror aperture rotated into this plane (cyan oval).