Retroreflective surface optimisation for optical cavities with custom mirror profiles

TL;DR

The paper tackles the challenge of optimizing Fabry-Pérot cavity mirrors to maximize emitter–photon coupling by adopting a two-stage approach: first optimise the intracavity mode within a Laguerre-Gauss/Hermite-Gauss basis to boost a chosen performance metric, then design a mirror topography that retroreflects this target mode. This method yields substantial improvements over conventional spherical mirrors, with a typical gain of about $3$ in internal cooperativity for a central emitter and potential improvements exceeding an order of magnitude for configurations with multiple emitters, as the optimized mode concentrates intensity more effectively at emitter positions. The framework relies on the paraxial approximation and accurate mode-mixing calculations to construct mirror surfaces, while also addressing practical considerations such as clipping losses, phase vortices, and manufacturability. The results point to strong potential for shaped mirrors to enhance quantum networking, scalable quantum computation, and sensing, and they provide a systematic way to assess where mirror shaping offers the most benefit and how to translate it into realistic designs.

Abstract

Coupling an emitter to a Fabry-Pérot optical cavity can provide a coherent and strong light-matter interface whose performance in a variety of applications depends critically on the emitter-photon coupling strength. Altering the typically spherical profiles of the cavity mirrors can improve this coupling strength, but directly optimising the mirror shape is numerically challenging as the multidimensional parameter space features many local optima. Here, we develop a two-step method to optimise mirror surface profiles while avoiding these issues. First, we optimise the target cavity eigenmode for the chosen application directly, and second, we construct the mirror surfaces to retroreflect this optimised target mode at both ends of the cavity. We apply our procedure to different emitter-cavity coupling scenarios. We show that mirror shaping can increase the cooperativity of coupling to a central emitter by a factor of approximately 3 across a wide range of geometries, and that, for coupling two or more emitters to a single cavity mode, the improvement factors can far exceed an order of magnitude.

Figures (10)

• Figure 1: Example and summary of improvements in $C_{\mathrm{int}}$ achievable with retroreflective optimisation for a single emitter in the centre of an optical cavity. The example cavity has a length of 500µm, a mirror diameter of 70µm, resonant wavelength of 854nm, and non-clipping internal loss of 20ppm. a) The $C_{\mathrm{int}}$ as a function of nominal mirror radius of curvature $R$ for (black) a spherical mirror and (dashed purple) the mirrors from the ansatz method, where the number of superposed states is labeled around the peak of the corresponding line. Horizontal lines depict the maximum $C_{\mathrm{int}}$ achievable with a (dashed black) spherical mirror, and (blue) retroreflective optimised mirror. b) The surface profile of the best spherical mirror (black) and the optimised mirror (blue), with the residuals overlaid (red). c) Comparison of the mode intensities (linear scale in top row, log-10 scale in bottom row with gridlines indicating decades) for the best spherical mirror cavity (black) and the optimised cavity (blue) in the i) central transverse plane containing the emitter and ii) mirror transverse plane, where the vertical lines mark the mirror edges. d) Mode intensities of the i) best spherical mirror cavity, ii) retroreflective-optimised cavity in an $xz$ cross-section of the mode ($y=0$). The cyan cross indicates the emitter position. e) The maximum $C_{\mathrm{int}}$ achievable with spherical mirrors and optimised mirrors. The white dashed line depicts $D_{\mathrm{crit}}$. f) The improvement in $C_{\mathrm{int}}$ from choosing the optimised surface over the optimised spherical surface. No data shown where $D<D_{\mathrm{crit}}$, as these cavities do not have high performance. The green crosses in ei), eii), and f) show the configuration exemplified in panels a)-d).
• Figure 2: Example of improvements in $C_{\mathrm{int}}$ achievable with retroreflective optimisation for a single emitter displaced from the centre of an asymmetric cavity geometry along the cavity axis. The example cavity has a length of 500µm, left and right mirror diameters of 75µm and 125µm respectively, resonant wavelength of 854nm, non-clipping internal loss of 20ppm, and an emitter placed 125µm to the right of the cavity centre. a) The $C_{\mathrm{int}}$ for a cavity with spherical mirrors, where the curvatures of the two mirrors are set implicitly by the central waist size and position of the corresponding Gaussian beam family. The green vertical line shows the position of the emitter, and the orange cross the spherical mirror configuration with the highest $C_{\mathrm{int}}$. b) The surface profile of the best spherical mirror (black) and the optimised mirror (blue), with the residuals overlaid (red) for the i) left and ii) right mirror. c) Mode intensities of the i) best spherical mirror cavity, ii) retroreflective-optimised cavity in an $xz$ cross-section of the mode ($y=0$). The cyan cross indicates the emitter position. d) Comparison of the mode intensities (linear scale in top row, log-10 scale in bottom row with gridlines indicating decades) for the best spherical mirror cavity (black) and the optimised cavity (blue) in the i) central transverse plane containing the emitter ii) transverse plane of the left mirror iii) transverse plane of the right mirror, where the vertical lines in ii) and iii) mark the mirror edges.
• Figure 3: Example of improvements in $C_{\mathrm{int}} /L^{-1}_{\mathrm{y, eff}}$ achievable with retroreflective optimisation for a single emitter displaced from the centre of a cavity geometry in a transverse direction, where the other direction is assumed large. The example cavity has a length of 1mm, mirror diameter (full mirror length) of 140µm, resonant wavelength of 854nm, non-clipping internal loss of 20ppm, and where the emitter is displaced from the centre of the cavity in the transverse direction by row I) 10µm row II) 50µm. a) The $C_{\mathrm{int}}/ L^{-1}_{\mathrm{y, eff}}$ for a cavity with circular mirrors, where the curvature is set implicitly by the central waist size and position of the corresponding Gaussian beam family. The green horizontal line shows the position of the emitter, with the orange cross indicating the best mirror curvature parameters, and orange dotted line the $x$-coordinate of the centre of the best circular mirror. In IIa), the white dotted lines indicate values of $w_{\mathrm{res}}^{q,p}$, where the corresponding fraction labels $q/p$, the transverse mode splitting to free spectral range ratio of the degeneracy. b) Mode intensities (per unit length) of the (top) best circular mirror cavity, (bottom) retroreflective-optimised cavity in the $xz$ cross-section of the mode. The cyan cross indicates the emitter position. c) The surface profile of the best spherical mirror (black) and the optimised mirror (blue), with the residuals overlaid (red). The green solid vertical line indicates the transverse position of the emitter, and the orange dotted line the centre of the best circular mirror.
• Figure 4: Schematic diagrams of the cavity geometries discussed in a) Sec. \ref{['subsec: two axial emitters']} and b) Sec. \ref{['subsec: emitters on line']}. a) Two emitters separated by $s$ are placed symmetrically on the axis of a cavity of length $L$ with mirror diameter $D$. b) An array of $N_e=5$ emitters of length $L_e$, with the separation between emitters again $s$.
• Figure 5: Example and summary of improvements in $C_{\mathrm{int}}^{\mathrm{min}}$ achievable with retroreflective optimisation for two emitters placed on the axis of the cavity, symmetrically about the central point (see Fig. \ref{['fig: multi emitter cavity geometries']}a). The example cavity (panels a-d) has length 2.8mm, mirror diameter 0.56mm, resonant wavelength 854nm, emitter separation 1.12mm, and non-clipping internal loss 20ppm. a) $C_{\mathrm{int}}^{\mathrm{min}}$ across the two emitters for (black line) two identical spherical mirrors as a function of mirror curvature $1/R$ with (dashed black) expected optimum $C_{\mathrm{int}}^{\mathrm{min}}$ and $1/R$ from Eq. (\ref{['eq: best spherical parameters two emitters']}), and (horizontal blue line) for retroreflective optimised mirrors. b) Surface profile of the (black) best spherical mirror and (blue) optimised mirror, with (red) residuals overlaid. The optimised surface is fit to a profile with an abrupt transition (dashed green vertical line) between an inner (dotted green line) and outer (dashed green line) spherical profile. c) Mode intensities (linear scale in top row, log-10 scale in bottom row with gridlines indicating decades) for the best spherical mirror cavity (black) and the optimised cavity (blue) in i) a transverse plane containing one of the emitters and ii) the mirror transverse plane, with vertical lines marking the edges of the mirror. d) Mode intensities of the i) best spherical mirror cavity, ii) retroreflective-optimised cavity in an $xz$ cross-section of the mode ($y=0$) with (cyan crosses) the two emitter positions. In ii) The surface fit from b) is indicated through green arrows to show the scale of the central (outer) curvature $R_c$ ($R_o$) and green dashed lines to show the focal points of the central curvature from the transition radius, and the outer curvature from the mirror radius. e) $C_{\mathrm{int}}^{\mathrm{min}}$ across the two emitters for spherical mirrors and optimised mirrors. The white dashed diagonal line depicts $s_{\mathrm{crit}}$, and the vertical line delineates $D<D_{\mathrm{crit}}$ (left) from $D>D_{\mathrm{crit}}$ (right). f) Improvement in the $C_{\mathrm{int}}^{\mathrm{min}}$ value from retroreflective optimisation. No data shown where $D<D_{\mathrm{crit}}$. The green crosses in ei), eii), and f) show the configuration used for panels a)-d).