High-Precision Lunar Corner-Cube Retroreflectors: A Wave-Optics Perspective

TL;DR

The paper develops a comprehensive wave-optics model for lunar CCRs in the 80–110 mm aperture range, incorporating realistic WFEs, diffraction, and velocity aberration. It demonstrates a strong diameter–offset coupling: larger apertures yield higher on-axis flux only under ideal conditions, but suffer flux loss under modest velocity offsets, whereas mid-sized apertures are more robust. Compared with solid fused-silica CCRs, hollow SiC designs offer similar reflectivity, substantially reduced mass, and improved thermal stability, especially at 1064 nm where relative phase errors are smaller. A dual hollow SiC CCR configuration at 100 mm, deployed on a lander platform with a 0.5 m baseline, provides sub-mm LLR performance with improved misalignment tolerance and differential ranging capability, making hollow CCRs a compelling choice for next-generation lunar missions and fundamental physics tests.

Abstract

High-precision corner-cube retroreflectors (CCRs) are critical for advanced lunar laser ranging (LLR) because they enable sub-millimeter-scale measurements of the Earth-Moon distance -- a level of precision essential for rigorous tests of relativistic gravitation and for advancing our understanding of lunar geophysics. In this work, we develop a comprehensive two-dimensional Fourier-optics model for single CCRs with apertures ranging from 80-110 mm. Our model incorporates realistic thermal-mechanical wavefront errors, detailed diffraction effects, and velocity aberration offsets. Our analysis reveals a strong coupling between aperture size and aberration angular offset: while larger CCRs deliver high on-axis flux under near-ideal conditions, their narrow diffraction lobes suffer significant flux loss at moderate aberration offsets, thereby favoring smaller apertures with broader main lobes. Furthermore, comparisons between solid fused-silica and hollow silicon-carbide (SiC) CCRs show that hollow designs not only achieve competitive or superior photon return -- particularly at 1064 nm, where phase errors are relatively reduced -- but also offer nearly an order-of-magnitude mass reduction for the same aperture sizes. These results establish a robust quantitative framework for optimizing CCR designs to perform at the sub-millimeter level under realistic lunar conditions and underscore the advantages of precision hollow SiC CCRs for next-generation LLR operations.

Figures (5)

• Figure 1: The far‐field diffraction pattern of a circular CCR with a 100 mm aperture is plotted on a linear scale (left) and on a logarithmic scale (right). The smooth line corresponds to 532 nm illumination, while the dashed line represents 1064 nm illumination. For 532 nm, the half-power point occurs at $\theta = 2.737\,\mu$rad and the first null is observed at $\theta = 6.485\,\mu$rad. For 1064 nm, the half-power point is at $\theta = 5.474\,\mu$rad and the first null is at $\theta = 12.969\,\mu$rad.
• Figure 2: The far‐field diffraction pattern of a circular CCR with various apertures as a function of the velocity aberration offset, $\alpha$, as given by (\ref{['eq:abber_Airy']}). Solid lines correspond to $\lambda = 532$ nm, while dashed lines are for $\lambda = 1064$ nm. From top to bottom, the curves represent CCRs with diameters ranging from 80 mm to 110 mm. Left is linear scale; right is log scale for the angular offset range of $\sim$ 3.8--7.3 $\mu$rad (i.e., $\sim$ 0.8--1.5$"$) -- region where existing LLR stations are affected by velocity aberration Williams-etal:2023.
• Figure 3: Normalized flux for an ideal CCR (WFE = 0 nm, $\rho = 1$). Solid lines correspond to $\lambda = 532$ nm, while dashed lines correspond to $\lambda = 1064$ nm. Left is linear scale; right is log scale, focusing on the region with $\alpha\in[3.8,7.5]\,\mu$rad. From top to bottom, the curves represent CCRs with diameters ranging from 110 mm to 80 mm. Compared to Fig. \ref{['fig:Airy-zeros-D']}, this figure also shows that the flux scales with aperture size as $\propto D^4$, see (\ref{['eq:photon-return-alpha2']}). Within the typical velocity-aberration range $\alpha \in [3.8,7.5]\,\mu\mathrm{rad}$, smaller apertures outperform larger ones at $532\,\mathrm{nm}$, whereas at $1064\,\mathrm{nm}$, larger apertures exhibit higher on-axis flux.
• Figure 4: Left: Normalized flux anticipated from CCRs at $\lambda=532\,{\rm nm}$ with WFEs from Table \ref{['tab:wfe-strehl-realistic']} and reflectivities, $\rho = 0.92$ for solid and $\rho = 0.95$ for hollow designs. Solid lines correspond to solid CCRs, while dashed lines are for hollow ones. Right: the same plot, but in logarithmic scale, while zooming in the range with typical values for LLR velocity aberrations at 532 nm.
• Figure 6: The normalized intensity of the returned signal as a function of beam incidence angle differs notably between solid fused-silica ($n=1.455$, solid line) and hollow ($n=1$, dotted line) CCRs. Solid fused-silica CCRs retain 50% efficiency up to $\sim13^\circ$ incidence, declining to zero near $45^\circ$. In contrast, hollow CCRs exhibit a narrower acceptance, maintaining 50% efficiency only up to about $10^\circ$ and reaching zero around $31^\circ$. Consequently, hollow CCRs require more stringent alignment or co-boresighted reflector configurations to sustain robust performance under lunar libration and typical operational misalignments.