High-Precision Amplitude-Modulated Continuous-Wave Lunar Laser Ranging

TL;DR

This work develops a complete AM-CW metrology framework for high-power Lunar Laser Ranging (LLR) that replaces time-of-flight with RF-phase metrology on a bright CW optical carrier. By modeling the total one-way path $R_{\rm tot}=R_{\rm geom}+R_{\rm atm}+R_{\rm inst}$ and defining observables $R_{2\mathrm{w}}$ and $v_r$ (plus differential counterparts), the authors derive a unified covariance ${\bf C}(T)$ decomposed into photon, atmospheric, instrumental, oscillator, and nonlinearity contributions, enabling direct translation into subsystem requirements. Using a 1 kW, 1064 nm CW transmitter on a 1 m telescope and 10 cm CCRs, the photon-shot floor on 2-way range is a few tens of micrometers over ~100 s, with atmospheric and instrumental terms pushing total absolute range precision to about $\sim 0.08$ mm and range-rate sensitivities of $\sim 0.1$–$1$ $\mu$m s$^{-1}$; differential AM-CW LLR between CCRs separated by ~1000 km yields $\Delta R$ and $\Delta v_r$ at the tens of micrometers and sub-$\mu$m/s levels. The framework defines three operating cases and a concrete hardware/CONOPS plan to push absolute and differential LLR into the sub-0.1 mm and tens-of-μm differential regimes, with potential scientific impact including ultra-slow tidal dissipation studies, core-mantle coupling, and detection of very-low-frequency gravitational waves.

Abstract

Lunar laser ranging (LLR) currently delivers mm-class tests of relativistic gravity and the lunar interior, but further gains are limited by photon-starved pulsed systems, array-induced pulse broadening, and atmospheric variability. Pushing into the sub-0.1mm regime is required to probe ultra-slow tidal dissipation, long-period librations, uHz-band gravitational waves. We develop a complete amplitude-modulated continuous-wave (AM-CW) framework for high-power LLR, including measurement models, estimators, a unified covariance description, and an observatory CONOPS. CW LLR is treated as RF phase metrology on a bright CW optical carrier: 2-way range and 1-way line-of-sight range-rate obtained from the phase and phase slope of a GHz-class modulation tone. The joint estimators are characterized by a covariance matrix decomposed into photon, atmospheric, instrumental, oscillator, nonlinearity contributions, mapped into subsystem requirements, including a closed-form range-rate bias from modulation-frequency slew across the ~2.56s round trip and synthetic-wavelength constraints on multi-tone nonlinearity. Using realistic photon budgets for a 1 kW, 1064 nm CW transmitter on a 1-m telescope ranging to 10cm corner-cube retroreflectors, we show that the photon-statistical floor for 2-way range at a GHz-class modulation frequency is at the level of a few-10 um in ~100s. With allocations to residual atmosphere and instrument, the resulting single-station 2-way precision is ~80 um, with range-rate sensitivities of ~0.1-1 um/s. Differential LLR between reflectors separated on the Moon by ~1000km can deliver 20-50 um and few (0.1-1) um/s differential range and range-rate. These precisions are realized across three representative regimes that span detected photon rates of (5-7) x 10^3-10^5 s^{-1} and are translated directly into hardware and CONOPS requirements for next-generation high-power AM-CW LLR.

Figures (4)

• Figure 1: Photon-limited two-way range uncertainty $\sigma_{R,\mathrm{shot}}(T)$, from (\ref{['eq:shot']}) and (\ref{['eq:lockin_snr']}) with the values from Table \ref{['tab:link-budget']}, as a function of integration time $T$ for the three representative photon-flux regimes (Cases A--C) used in the link and covariance analysis. The curves assume a single GHz-class modulation tone on a bright CW carrier with negligible background, so that all three cases follow the expected $\sigma_{R,\mathrm{shot}}\propto T^{-1/2}$ scaling; the vertical separation reflects the different detected photon rates in each regime.
• Figure 2: Expected one-way line-of-sight range-rate uncertainties as a function of integration time $T$ for the absolute observable $\sigma_{v_r}(T)$ and the differential observable $\sigma_{\Delta v_r}(T)$ (from (\ref{['eq:diffbudget-rr']})) between nearby reflectors. Both curves correspond to the Case B range allocations and use the slope-estimator relation $\sigma_{v_r}(T)\simeq\sigma_R(T)/T$ (using (\ref{['eq:budget']})) applied to the underlying range budgets. Differential range-rate benefits from common-mode cancellation of atmospheric and instrumental terms between lines of sight, leading to systematically lower $\sigma_{\Delta v_r}(T)$ at fixed $T$ in the $10^2$--$10^3\,\mathrm{s}$ regime.
• Figure 3: Simplified schematic of the RF-coherent high-power AM--CW LLR station. A hydrogen maser and RF synthesis stage generate the modulation tones $\{f_{m,i}\}$, which drive a LiNbO$_3$ MZM and MOPA chain to impose amplitude modulation with index $a_m$ on a 1064 nm CW beam at power $P_0\sim 1$ kW. The transmit/receive telescope launches the beam to the lunar 10 cm CCRs and collects the weak, delayed return, which is filtered, detected, and digitized by the receive chain. Digital lock--in processing and estimators on the precision tone, aided by a short internal reference path that supplies $\phi_{\rm inst}(t)$, produce the observables $R_{2\mathrm{w}}(T)$, $v_r(T)$, $\Delta R_{2\mathrm{w}}(T)$, and $\Delta v_r(T)$.
• Figure 4: Representative absolute two-way range error budget $\sigma_R(T)$ for the dedicated AM--CW facility (Case B), from (\ref{['eq:budget']}) and the values from Table \ref{['tab:link-budget']}. The photon, atmosphere, and instrument contributions are shown separately together with their root-sum-square total. Photon and atmospheric terms follow the $T^{-1/2}$ scaling implied by the shot-noise and Kolmogorov models, while the instrumental term is approximately constant over the $30$--$100~\mathrm{s}$ band. The total curve illustrates the transition from a photon/atmosphere-dominated regime at short $T$ to an instrumental floor at longer integrations, consistent with the scalar allocations used in the range covariance.