Lunar Laser Ranging with High-Power CW Lasers

Slava G. Turyshev

REVIEW 4 major objections 6 minor 111 references

Lunar Laser Ranging with High-Power CW Lasers

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A kilowatt continuous-wave laser at 1064 nm can collect about 5,884 photons/s from a 10 cm lunar corner-cube reflector and reach ~44 micrometer single-range precision in 100 s.

desk verdict A useful and transparent link budget for CW LLR, but the precision claims — especially the differential tens-of-micrometer numbers — are not supported by the paper's own error equations. read the letter →

arxiv 2502.02796 v1 pith:K5LXDSXT submitted 2025-02-05 astro-ph.IM

Slava G. Turyshev This is my paper · ORCID

classification astro-ph.IM

keywords continuous-wavelaserranginglunarcorner-cuberetroreflectorshot-noiselimitdifferentialatmosphericturbulencelinkbudgetphasemetrology

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that replacing pulsed lasers with a kilowatt-class continuous-wave (CW) beam removes the main obstacle to using small, next-generation corner-cube reflectors on the Moon. With a 1 kW laser at 1064 nm and a 1 m telescope, the link budget gives about $5.9 \times 10^3$ detected photons per second from a 10 cm reflector, roughly three orders of magnitude more than a pulsed station would detect. The resulting shot-noise-limited range precision is about $44\,\mu$m in 100 s at a 1 GHz modulation frequency, and alternating between two reflectors ~1000 km apart yields $\sim$32–43 $\mu$m RMS in differential mode. If correct, this would take lunar laser ranging from millimeter-level to tens-of-micrometer-level accuracy, sharpening tests of general relativity, lunar interior models, and reference-frame ties for lunar exploration.

What carries the argument

The carrying objects are the two-way link-budget equation and the shot-noise phase-error relation, plus a Kolmogorov turbulence scaling for the differential mode. The link budget, Eq. (24), combines the laser's photon emission rate, two-way atmospheric and telescope efficiencies, and the area ratios of the reflector to the lunar beam footprint and of the telescope to the Earth return footprint, yielding detected photons per second. The phase measurement at modulation frequency $f_m$ then converts signal-to-noise ratio into range uncertainty via $\delta R = (c/4\pi f_m)/\mathrm{SNR}$. For differential ranging, Eq. (37) estimates uncorrelated path errors between two lines of sight separated by about $0.15^\circ$ from the isoplanatic angle (the angular separation over which wavefront errors stay correlated) using the $(5/6)$-power scaling, giving 5–20 $\mu$m instantaneously and 10–30 $\mu$m after about 100 s of averaging.

What would settle it

A field experiment that alternately ranges to two lunar reflectors separated by ~1000 km with a 1 kW CW beam, integrating about 100 s per reflector, and measures the differential range RMS would settle the claim. If the observed RMS stays above about $100\,\mu$m, or if the turbulence contribution fails to follow the $(5/6)$-power scaling and $\sqrt{T/t_0}$ averaging, the differential result is not achievable. A simpler check would be a direct measurement of uncorrelated turbulence on two $0.15^\circ$-separated lines of sight at $r_0=20$ cm.

Watch

Extended reading notes

Core claim

The central claim is that a photon-rich CW ranging system can keep lunar laser ranging precise even when the retroreflector is shrunk to ~10 cm to reduce intrinsic range errors. Using a 1 kW beam at 1064 nm, a 1 m telescope, a 10 cm corner cube, and seeing with Fried parameter $r_0=20$ cm, the paper computes a return of about 5,884 photons/s. At a 1 GHz modulation frequency and 100 s integration, that flux puts the shot-noise floor at about $44\,\mu$m for a single range, and about $0.4\,\mu$m/s for range rate. By toggling between two reflectors separated by ~1000 km, common-mode station errors cancel and the paper budgets the differential RMS at 32–43 $\mu$m, with the residual set mostly by uncorrelated atmospheric turbulence. The paper presents this as a scalable path from the current few-millimeter regime toward sub-0.1 mm and tens-of-micrometer lunar ranging.

Load-bearing premise

The differential tens-of-micrometer claim rests on the assumption that uncorrelated atmospheric turbulence between two lines of sight separated by about $0.15^\circ$ can be averaged down to 10–30 $\mu$m over about 100 s of rapid toggling; if turbulence decorrelation behaves differently, the differential precision is not supported.

Editorial extensions

If this is right

A 1 kW CW station can range to a single 10 cm lunar reflector at about 5,884 photons/s, making next-generation compact reflectors viable despite their small area.
Single-range shot-noise precision reaches about 44 $\mu$m in 100 s at 1 GHz modulation, with corresponding range-rate precision about 0.4 $\mu$m/s.
Differential ranging between reflectors about 1000 km apart reaches 32–43 $\mu$m RMS, enabling tests of the equivalence principle, lunar tidal deformation, and reference-frame ties at tens-of-micrometer level.
Because the photon flux is orders of magnitude higher than pulsed systems, narrowband filtering and turbulence averaging can keep SNR high even under bright-sky conditions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The turbulence-averaging term in the differential budget is the most uncertain piece; a dedicated two-reflector toggle test would show whether real atmospheric decorrelation reaches the modeled 10–30 $\mu$m floor or stays higher.
If tens-of-micrometer differential ranging is achieved, a single station could track lunar libration and tidal deformation at scales that may constrain the lunar core and mantle properties beyond current models.
The same photon-rich link suggests that high-cadence daytime or full-Moon ranging may become practical, increasing LLR data volume rather than remaining limited to dark-sky windows.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The useful new content here is the worked link budget and the consolidated error tables for a 1 kW CW LLR system ranging to small 10 cm CCRs. The underlying CW/differential idea goes back to the author's own workshop reports [15,16], so the novelty is in the quantitative engineering, not the concept. The link-budget core is transparent and standard; I checked the photon return (~5,884/s) and the shot-noise formula, and they are fine. The comparison with APOLLO/OCA fluxes is a useful way to frame the problem, and the citation pattern is mostly mainstream LLR literature. The soft spots are in the precision claims, and they are not minor. On the same page the paper computes a 43.6 micron shot-noise floor from SNR 547 and then says sub-30 micron is achievable; the 43.6 micron number is the one that follows from the stated background. In the differential section, Eq. (37) with the paper's own values gives ~1-3 micron after 100 s of averaging, not the 10-30 micron entered in Table IX. The stress-test note is right about a larger problem: the differential error budget omits the quasi-static tropospheric delay difference between two lines of sight separated by ~0.15 degrees. The two paths have different mean airmass, so even after modeling there is a near-constant millimeter-scale differential delay that will not average down with sqrt(T/t0). That omission is load-bearing for the 32-43 micron differential claim. The absolute RSS budget of 320-550 micron also sits awkwardly with the abstract's 'sub-0.1 mm' language; the paper should say what is actually being claimed. Who gets value: someone scoping high-power CW LLR and wanting a first-cut link budget. The differential tens-of-micrometer claims should be treated as an aspiration, not a demonstrated result. Recommendation: send to peer review because the engineering concept and link budget deserve referee time, but require major revision of the error budget and precision claims. This is a conditional accept / major revision, not a desk reject.

Referee Report

4 major / 6 minor

Summary. The paper proposes a high-power continuous-wave (CW) lunar laser ranging system: a 1 kW, 1064 nm laser transmitted through a 1 m telescope to a next-generation 10 cm corner-cube reflector, with the return detected by a single-photon detector and the round-trip range encoded in the phase of an RF amplitude modulation. It derives a link budget (Sec. IV) yielding about 5.88e3 detected photons/s, a shot-noise-limited single-range precision of about 43.6 um in 100 s at a 1 GHz modulation frequency (Sec. V.B.4), and a differential mode in which alternating between two reflectors ~1000 km apart gives a combined RMS error of 31.8-42.9 um (Table IX). The paper argues that this photon-rich approach overcomes the low return rates of small CCRs and enables tens-of-micrometer differential measurements for gravitational and geodetic applications.

Significance. If correct, the proposal is scientifically significant: it would advance LLR from millimeter normal-point precision to tens-of-micrometer differential precision, with applications to tests of general relativity, lunar interior studies, and selenodesy. The link-budget calculation is transparent and reproducible; Eq. (24) with the component efficiencies in Eq. (19) yields the quoted photon rate, and the shot-noise expression in Eq. (11) is standard. The paper also provides a detailed accounting of thermal, mechanical, geophysical, and calibration errors, and it does not rely on fitted constants to manufacture the central photon-flux result. The main weaknesses are internal inconsistencies in the claimed precision numbers and a missing quasi-static differential tropospheric term; these undermine the headline numbers as currently stated but are addressable within the manuscript's scope.

major comments (4)

[Sec. V.B.4, Eq. (30)] The text states that sub-30 um range precision is achievable under shot-noise-limited conditions, but the calculation immediately above gives delta_R = 2.387 cm / 547 ~ 43.6 um for T = 100 s and f_m = 1 GHz. To reach 30 um with the stated SNR_1s ~ 54.7 would require T ~ 212 s, not 100 s. The precision claims in this section, and the downstream tens-of-micrometer statements in Sec. VI, should be made consistent with the computed value.
[Sec. VI.C.2, Eq. (37), Table IX] Equation (37) gives instantaneous differential turbulence errors of about 5-20 um for delta_theta = 400-1800 arcsec, starting from ~106 nm at theta_iso = 4 arcsec. With the stated averaging law sigma(T) = sigma(t0)/sqrt(T/t0) and t0 = 1-5 s, averaging over T = 100 s gives a reduction by a factor of 4.5-10, i.e., roughly 0.5-4.4 um, not 10-30 um. Table IX's 'Uncorrelated Turbulence (delta_turb) 10-30' is therefore not supported by the paper's own formula. In addition, Eq. (37) is an extrapolation of isoplanatic scaling far beyond its usual small-angle validity (delta_theta/theta_iso ~ 100-400); the authors should justify this scaling or present an alternative model. As written, the 31.8-42.9 um differential result in Table IX rests on an internally inconsistent turbulence treatment.
[Sec. VI.C and Table IX] The differential error budget omits the quasi-static difference in mean tropospheric delay between the two lines of sight. For reflectors separated by 1000 km, delta_theta ~ 0.15 deg, and at 45 deg elevation the airmass difference is about 0.26%, corresponding to a two-way delay difference of several millimeters. This term is not zero-mean and is not reduced by the sqrt(T/t0) averaging in Eq. (37); it must be modeled or calibrated. Section V.A.2 itself reports residual dispersive path delays of 500-800 um after dual-wavelength correction, which is far above the 10-30 um turbulence entry in Table IX. Unless this quasi-static differential delay is included in Table IX and shown to be calibrated to below roughly 10 um, the claimed 32-43 um differential precision is not supported.
[Sec. V.A.2 and Table VII] Section V.A.2 states that dual-wavelength ranging reduces dispersive atmospheric errors to residual delays of 500-800 um. However, Table VII's root-sum-square budget for the same system lists no dispersive term and reports a total RMS of only 320-550 um, with atmospheric turbulence at 300-500 um. If the 500-800 um residual is a real two-way error, it dominates the absolute error budget and contradicts the table's total. The authors should include this term in the RSS budget or explain why it is not applicable to the CW configuration.

minor comments (6)

[Table IV and Eq. (19)] Table IV's caption reports eta_eff ~ 0.27, but Eq. (19) evaluates to eta_eff ~ 0.194 with the listed component values, and Table V uses 0.194; these values should be aligned.
[Sec. IV.C and Table VI] The text states that replacing APOLLO's pulsed source with a 1 kW CW laser could reach about 3.1e5 photons/s from the Apollo 15 array, but Table VI lists about 3.68e4 for that entry; one of these numbers is incorrect.
[Sec. IV.D and Table IV] Section IV.D refers to eta_CCR = 0.6 for hollow CCRs, while Table IV assumes eta_CCR = 0.7; the baseline value should be stated consistently.
[Eq. (17)] Equation (17) defines the FM unambiguous range as d_max ~ c/(2 beta f_b), but f_b itself depends on range, making the expression circular; the FM ambiguity should be stated in terms of chirp bandwidth and chirp duration.
[Sec. V.B.4 and Sec. VI] The paper uses 'sub-30 um' and 'tens-of-micrometer' in ways that are not tied to specific integration times and modulation frequencies; precision claims should be stated with explicit assumptions throughout.
[References [15,16]] References [15] and [16] are the only support cited for the feasibility of the proposed CW approach and are self-citations to workshop proceedings; independent or peer-reviewed support would strengthen the proposal.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the photon-return and precision claims are propagated from standard link equations and noise statistics; self-citations [15,16] are minor and not load-bearing.

full rationale

The central quantitative chain is self-contained. The photon-return rate in Table V (approximately 5.88e3 photons/s) follows from the standard CW link equation, Eq. (24), using explicitly listed assumptions: 1 kW at 1064 nm, a 1 m telescope, a 10 cm CCR, beam divergences from Eqs. (22)-(23), and component efficiencies from Table IV. The single-range precision of about 43.6 um for 100 s is then obtained by propagating that flux through the SNR definition, Eq. (29), and the AM phase-error relation, Eqs. (8)-(11). No fitted parameter is renamed as a prediction: the efficiency values are assumed inputs, and the precision is a derived output. The differential error budget (Table IX) combines station-level terms (Table VIII), two-channel shot noise from Eqs. (35)-(36), and the turbulent differential path error from Eq. (37). The Kolmogorov scaling in Eq. (37) is attributed to the external turbulence literature [56-58], not to the present authors, and even if the averaging assumption is physically optimistic, that is a correctness or modeling risk rather than circularity. Self-citations [15,16] appear as pointers to earlier proposals and pilot-program ideas, but the quantitative derivation in this paper does not rely on them for its central conclusions. Table IX itself flags the omission of lunar-dynamics, libration, and tidal-dissipation modeling errors, which is an acknowledged limitation rather than a circular step. No instance was found in which an output quantity is defined in terms of the quantity it is claimed to predict.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central estimates rest on standard laser-ranging formulas plus a small set of hand-picked system parameters (efficiency, seeing, modulation frequency, background, turbulence averaging). No independent data are used to validate the assumed return flux or the turbulence reduction.

free parameters (5)

Total round-trip efficiency η_eff = 0.194 (up to 0.35-0.40 in upgraded designs)
Assembled in Table IV from assumed telescope, atmosphere, CCR, aberration, and detector efficiencies; directly multiplies the photon-return rate in Eq. (24), so the central precision claims scale linearly with it.
Fried parameter r0 = 20 cm at 1064 nm
Chosen as 'good seeing'; enters the outgoing beam divergence (Eq. 22), spot size on the Moon, and hence photon flux. Different sites or weather would change the predicted return rate.
Modulation frequency fm = 1 GHz
Design choice that sets the cm-to-μm conversion factor c/(4π f_m) in Eq. (8); the shot-noise-limited range precision is inversely proportional to f_m.
Background photon flux N_noise = 5.7e3 s^-1 (full Moon worst case)
Estimated from lunar phase and filter bandwidth (Sec. V.B.2); used in SNR (Eq. 29) and in the 43.6 μm shot-noise estimate, so it directly controls whether 'sub-30 μm' is reached.
Turbulence averaging time t0 and averaging law = t0 = 1-5 s; σ ∝ (T/t0)^(-1/2)
Assumed to reduce uncorrelated turbulence over T = 100 s in Sec. VI.C; the paper's own numbers give a smaller residual than the 10-30 μm entered in Table IX.

assumptions (4)

domain assumption Kolmogorov turbulence statistics describe atmospheric path-delay fluctuations, including the (Δθ/θ_iso)^(5/6) scaling and T^-1/2 averaging.
Invoked in Sec. VI.C (Eq. 37) to estimate the uncorrelated differential turbulence term, the key limit on the differential tens-of-micrometer claim.
domain assumption A single 10 cm CCR behaves as a diffraction-limited retroreflector with far-field divergence θ_return ≈ 1.22λ/d_CCR and efficiency η_CCR = 0.7.
Used in Eqs. (23)-(25) to compute the Earth-side spot and photon return; real CCRs have velocity aberration, thermal, and coating effects not fully captured.
standard math Phase estimation with coherent detection is limited by photon shot noise as δφ ≈ 1/SNR and beat-frequency error δf_b ≈ 1/(T_chirp SNR).
Standard results from estimation theory [42-44], used in Secs. III.B.2 and III.C.2; assumed valid for the lunar return signal despite unknown optical coherence.
domain assumption A 1 kW CW beam can be amplitude- or frequency-modulated to 10 GHz with sub-ps phase stability and delivered through a 1 m telescope.
Explicitly listed as a system requirement in Appendix B, not demonstrated; the entire ranging concept depends on this engineering capability.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Lunar Laser Ranging with High-Power CW Lasers." pith.science (2026). https://pith.science/paper/K5LXDSXT

@misc{pith2026250202796,
  author       = {Pith},
  title        = {Pith review of: Lunar Laser Ranging with High-Power CW Lasers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5LXDSXT}},
  note         = {Machine review of arXiv:2502.02796}
}

read the original abstract

We present a high-power continuous-wave (CW) lunar laser ranging (LLR) technique that has the potential to significantly improve Earth--Moon distance measurements. Using a 1 kW CW laser at 1064 nm and a 1 m-aperture telescope as an example, we develop a detailed link budget and analyze the prevailing noise sources to assess system performance when ranging to next-generation ~10 cm corner-cube retroreflectors (CCRs). Unlike legacy arrays, these smaller CCRs are designed to yield lower intrinsic range errors, yet their reduced reflective area results in lower photon return rates, posing challenges for pulsed LLR systems. The photon-rich CW approach, by providing continuous high-power illumination, overcomes this limitation, reducing shot noise and enabling sustained millimeter-level ranging with a pathway to sub-0.1 mm precision. Furthermore, by alternating measurements between widely separated lunar reflectors, differential LLR mitigates common-mode station errors to achieve tens-of-micrometer precision, limited primarily by uncorrelated atmospheric turbulence. This scalable approach -- integrating high-power CW lasers, narrowband filtering, and rapid atmospheric turbulence averaging -- enables next-generation gravitational tests, precision lunar geodesy, and improved lunar reference frames in support of planetary exploration.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

111 extracted references · 71 canonical work pages

[1]

organized as follows:

Coarse-to-Fine Frequency (AM) or Slope (FM) Approach A typical multi-frequency AM protocol may use three or more discr ete RF tones f1, f2, f3, . . . organized as follows:
[2]

Measuring the round-trip phase at f1 localizes the range rEM to a bin of roughly a few meters (modulo the coarse-wavelength spa cing)

Coarse Tone (f1 ∼ 10–50 MHz): Provides a synthetic wavelength of c/(2f1) ≈ 3–15 m. Measuring the round-trip phase at f1 localizes the range rEM to a bin of roughly a few meters (modulo the coarse-wavelength spa cing). 12
[3]

Phase unwrapping between f1 and f2 yields a unique solution over the full Earth–Moon baseline

Intermediate Tone (f2 ∼ 100–500 MHz): Reﬁnes the distance to the centimeter or millimeter le vel since each full 2 π cycle now corresponds to a shorter distance of c/(2f2). Phase unwrapping between f1 and f2 yields a unique solution over the full Earth–Moon baseline
[4]

Equation ( 5) (for AM) or Equation ( 15) (for FM) then translates the measured phase or beat frequenc y into a ﬁne distance rEM, fine

Fine Tone (f3 ∼ 1–10 GHz): Finally, a high-frequency tone locks in sub-mm or even te ns-of-µm precision once the coarse and intermediate phases have identiﬁed the corre ct ambiguity bin. Equation ( 5) (for AM) or Equation ( 15) (for FM) then translates the measured phase or beat frequenc y into a ﬁne distance rEM, fine. Mathematically, if φi denotes the m...
[5]

Multi-Chirp or Multi-Slope FM Variant An FM LLR station may replace discrete RF tones with several chirp s weeps characterized by slopes β1, β2, . . .. Each chirp sweep produces a beat frequency that encodes the ro und-trip delay to the Moon. From ( 14), for a chirp with slope βi, the beat frequency is given by fb,i = βi τ = βi 2rEM c . A typical multi-ch...
[6]

Gentle Chirp (β1): Provides a large unambiguous range of c/(2β1), spanning thousands of kilometers, and localizes rEM to a bin of roughly a few meters
[7]

Phase unwrapping between β1 and β2 yields a unique solution

Intermediate Slope (β2): Reﬁnes the range to the centimeter or millimeter level since each f ull 2 π cycle corre- sponds to a shorter distance c/(2β2). Phase unwrapping between β1 and β2 yields a unique solution
[8]

Here, the chirp rate is deﬁned as βi = B/Tchirp, where B is the chirp bandwidth and Tchirp is the duration of the chirp

Steep Slope (β3): Achieves sub-mm precision once the correct ambiguity bin is identiﬁ ed; the measured beat frequency is then converted into a ﬁne distance rEM, fine via Equation ( 5) (for AM) or Equation ( 15) (for FM). Here, the chirp rate is deﬁned as βi = B/Tchirp, where B is the chirp bandwidth and Tchirp is the duration of the chirp. High linearity ...

Show all 111 references

[9]

With proper phase unwrapping over time, any constant 2 π oﬀset is eliminated upon diﬀerentiation, resulting in a continuous and unambiguous phase derivative

Range-Rate as a New LLR Observable Although absolute phase measurements are inherently ambiguous m odulo 2 π — leading to an integer ambigu- ity in the absolute range rEM — the range-rate is determined from the time derivative of the phas e. With proper phase unwrapping over t...
[10]

This requires precise cont rol over phase measurements and beat-frequency calculations to avoid integer- cycle errors in the ﬁnal range

Distance Unambiguous Resolution: Multi-frequency schemes (in AM) or multi-chirp schemes (in FM) must be designed to span the ∼ 4 × 108 m Earth–Moon baseline without ambiguity. This requires precise cont rol over phase measurements and beat-frequency calculations to avoid integ...
[11]

Reference Stability: Achieving sub-ps timing stability over the 2.56 s round-trip delay requ ires oscillator fractional drifts on the order of ∼ 10−13. Hydrogen masers or optical frequency combs can meet this requ irement, but calibration oﬀsets or frequency drifts must be con...
[12]

Real-Time Doppler Monitoring: The Moon’s ±1 km/s radial velocity induces Doppler shifts that vary over the 2.56 s round-trip, potentially shifting the measured phase or beat f requency by tens of kHz. Both hardware-level corrections and post-processing adjustments—based on acc...
[13]

The use of low-expansion materials and rigorous environmen tal control is essential to ensure consistent performance

Mechanical and Thermal Stability: Even small thermal expansions or mechanical drifts can introduce range errors at the tens-of- µm scale. The use of low-expansion materials and rigorous environmen tal control is essential to ensure consistent performance. Under optimal conditi...
[14]

Consequently, a 1 kW beam emits Plaser Ephoton = 1000 J/s 1.87 × 10−19 J ≈ 5.35 × 1021 photons s−1

Laser Emission Rate and Single-Photon Energy A single photon at λ = 1064 nm carries Ephoton = hc λ ≈ 1.87 × 10−19 J, (20) where h is Planck’s constant and c is the speed of light. Consequently, a 1 kW beam emits Plaser Ephoton = 1000 J/s 1.87 × 10−19 J ≈ 5.35 × 1021 photons s−...
[15]

(22) For an Earth–Moon center-to-center distance rEM ≈ 3.84×108 m, the beam spot on the Moon attains kilometer-scale dimensions

Outgoing and Return Footprints The total beam divergence accounts for both diﬀraction and wave front distortions from atmospheric turbulence: θdiv = √ ( 1.22λ dtel ) 2 + ( λ r0 ) 2 ≈ 5.48 × 10−6 rad. (22) For an Earth–Moon center-to-center distance rEM ≈ 3.84×108 m, the beam s...
[16]

(25) Table V summarizes the estimates of each term in the link equation ( 24)

Link-Budget Formula Combining the beam geometry, two-way transmission, and collection aperture yields an approximate ﬂux at the detector [ 18, 20]: ˙Nphotons = Plaser Ephoton × ηeﬀ × ACCR Aspot, Moon × Atel Aspot, Earth , (24) where ACCR = π ( 1 2 dCCR ) 2 , A tel = π ( 1 2 dt...
[17]

Multi-MHz repetition rates to accumulate 10 7–108 photons during each measurement
[18]

Ultra-stable amplitude and minimal pulse jitter to reduce range wa lk errors
[19]

These requirements are technically feasible but impose signiﬁcant en gineering challenges

Advanced short-pulse electronics capable of isolating single-pho ton events with sub-ps resolution. These requirements are technically feasible but impose signiﬁcant en gineering challenges. By contrast, CW systems inherently provide continuous illumination, leveraging multi-f...
[20]

Instabilities in fm over the integration time T introduce systematic range errors, scaling as: ∆ Rdrift = c ∆ fm 4π f 2m

Oscillator Frequency Drift The range observable in CW LLR systems depends critically on the rou nd-trip phase shift of an RF modulation at frequency fm. Instabilities in fm over the integration time T introduce systematic range errors, scaling as: ∆ Rdrift = c ∆ fm 4π f 2m . (...
[21]

Diﬀer- ential refraction between these wavelengths provides estimates of column water vapor and refractive index gradients [18, 27, 48]

Dual-Wavelength Ranging for Atmospheric Dispersion Dispersive atmospheric eﬀects are mitigated by operating at two wa velengths (e.g., 1064 nm and 532 nm). Diﬀer- ential refraction between these wavelengths provides estimates of column water vapor and refractive index gradient...
[22]

In the signal dominated regime, the range precisio n depends on the detected photon ﬂux ˙Nphotons and the integration time T : ∆ Rshot = c 4πfm 1√ ˙Nphotons T

Shot Noise and Detector Timing Shot noise imposes a fundamental limit on range precision in LLR syste ms, arising from the Poisson statistics of photon arrivals. In the signal dominated regime, the range precisio n depends on the detected photon ﬂux ˙Nphotons and the integrati...
[23]

Thermal expansion in low-expansion materials like Zerodur an d Invar is ∼ 1 µm/m/◦C [ 5]

Mechanical and Thermal Drifts, Geophysical Motions Achieving sub-millimeter LLR precision requires strict control over t hermal, mechanical, and environmental factors, especially under favorable conditions ( r0 ∼ 20 cm). Thermal expansion in low-expansion materials like Zerodu...
[24]

IV B, the detected signal ﬂux ( ˙Nsig) returning from the Moon depends on system parameters such as telescope aperture, beam divergence, CCR size, and atmo spheric conditions

Signal Photon Flux from 10 cm CCR As described in Sec. IV B, the detected signal ﬂux ( ˙Nsig) returning from the Moon depends on system parameters such as telescope aperture, beam divergence, CCR size, and atmo spheric conditions. Using the values from Table V we see that the ...
[25]

These terms are summarized below:

Noise Terms in the Telescope and Detector The overall SNR is aﬀected by several noise sources, including back ground photons, detector dark counts, and electronic noise. These terms are summarized below:
[26]

For nighttime conditions with narrowband ﬁltering (e.g., 1– 30 GHz passband) and a few-arcsecond ﬁeld of view (FOV), the background photon rate is typically: ˙Nbkg ∼ 102–103 s−1

Sky/Background Photons ( ˙Nbkg): Background ﬂux arises from scattered moonlight, natural airg low, and terrestrial radiance. For nighttime conditions with narrowband ﬁltering (e.g., 1– 30 GHz passband) and a few-arcsecond ﬁeld of view (FOV), the background photon rate is typic...
[27]

For su perconducting nanowire single- photon detectors (SNSPDs), dark counts can be as low as ∼ 10 s−1 under cryogenic cooling [ 33]

Dark Counts ( ˙Ndark): Dark counts result from internal detector processes. For su perconducting nanowire single- photon detectors (SNSPDs), dark counts can be as low as ∼ 10 s−1 under cryogenic cooling [ 33]. Avalanche photodiodes (APDs) at room temperature exhibit higher dar...
[28]

The total noise ﬂux is given by: ˙Nnoise = ˙Nbkg + ˙Ndark + ˙Nread

Readout Noise ( ˙Nread): Electronic noise from readout systems, including ampliﬁers and dig itizers, adds uncertainty but remains minimal in well-designed systems. The total noise ﬂux is given by: ˙Nnoise = ˙Nbkg + ˙Ndark + ˙Nread. (28) Under optimized conditions, this totals ...
[29]

For ˙Nsig ∼ 5.88 × 103 s−1 and ˙Nnoise ∼ 5.71 × 103 s−1, the per-second SNR is calculated as: SNR1 s ≈ 5.88 × 103 √ 5.88 × 103 + 5.71 × 103 ≈ 54.7

Combining Fluxes into a Shot-Noise SNR The total SNR for a CW LLR system, derived from ( 10), is given by: SNRtotal = ˙Nsig √ T√ ˙Nsig + ˙Nnoise ≡ SNR1 s √ T 1 s, (29) where ˙Nsig is the signal photon ﬂux, ˙Nnoise is the total noise photon ﬂux, and T is the integration time. F...
[30]

Substituting: ∆ R = 2.387 cm 547 ≈ 43.6 µm

Range and Range-Rate Precision in the Presence of Backgro und Noise Using the total SNR, the shot-noise-limited range uncertainty in th e presence of background is given by ( 11) as: ∆ R = (c/4πfm)/SNRtotal, where fm = 1 GHz, c = 3 × 108 m/s, and SNR total ∼ 547. Substituting:...
[31]

These largely cancel in the diﬀerential range observable ( ρ ≈ 1), though residuals persist due to imperfect correlation, localized eﬀects, and unmodeled disturbances

Station-Based (Common-Mode) Terms Station-level errors—oscillator drifts ( δosc), thermal expansions ( δtherm), mechanical vibrations ( δmech), geophysical motions ( δgeo), seismic activity ( δseismic), wind shear ( δwind), and calibration oﬀsets ( δstat)—are highly correlated...
[32]

diﬀerential

Shot-Noise Limits for Two CW Channels Shot noise contributes independently to each path in diﬀerential LL R. From (34), the diﬀerential shot noise is given: σshot, diﬀ = √ σ2(ε1) + σ2(ε2), (35) where ε1 and ε2 represent the shot-noise contributions from the two paths. Ass umin...
[33]

Mitigation Strategies: High-Power Lasers and Narrowban d Filters A kW-class CW laser system operating at dual wavelengths (e.g., 106 4 nm and 532 nm) provides continuous high photon ﬂux, overcoming the low duty cycles of pulsed systems. The primary beneﬁts include [ 15, 16]: •...
[34]

At near-IR wavelengths ( λ ∼ 1 µm), the isoplanatic angle θiso is typically 4–5 ′′ at high-altitude observatories

Kolmogorov Scaling and Turbulence Averaging Atmospheric turbulence introduces refractive index ﬂuctuations that follow predictable statistical properties de- scribed by Kolmogorov turbulence theory [ 56–58]. At near-IR wavelengths ( λ ∼ 1 µm), the isoplanatic angle θiso is typ...
[35]

photon-rich

Rapid Toggling and Statistical Consistency Frequent toggling ( ∼ 1–2 s) ensures atmosphere-induced errors between reﬂectors remain statistically independent. Combined with turbulence averaging, this enables diﬀerential error s to approach ∼ 30 µm under optimal conditions. Obse...
[36]

J. E. Faller and E. J. Wampler, Scientiﬁc American 223, 38 (1970)

1970
[37]

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. R icklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, et al., Science 265, 482 (1994)

1994
[38]

J. G. Williams, S. G. Turyshev, and D. H. Boggs, Phys. Rev. Lett. 93, 261101 (2004)

2004
[39]

J. G. Williams, S. G. Turyshev, and D. H. Boggs, IJMPD 18, 1129 (2009)

2009
[40]

J. G. Williams, S. G. Turyshev, and D. H. Boggs, CQG 29, 184004 (2012)

2012
[41]

J. G. Williams, in 16th International Workshop on Laser Ranging (2008), p. 17

2008
[42]

J. G. Williams, A. S. Konopliv, D. H. Boggs, R. S. Park, D.- N. Yuan, F. G. Lemoine, S. Goossens, E. Mazarico, F. Nimmo, R. C. Weber, et al., JGR (Planets) 119, 1546 (2014)

2014
[43]

Briaud, C

A. Briaud, C. Ganino, A. Fienga, A. M´ emin, and N. Rambaux , Nature 617, 743 (2023)

2023
[44]

S. G. Turyshev, Ann. Rev. Nucl. Part. Sci. 58, 207 (2008)

2008
[45]

Zhang, J

M. Zhang, J. M¨ uller, L. Biskupek, and V. V. Singh, Astro n. Asrophys. 659, A148 (2022)

2022
[46]

A. S. Konopliv, R. S. Park, D.-N. Yuan, S. W. Asmar, M. M. W atkins, J. G. Williams, E. Fahnestock, G. Kruizinga, M. Paik, D. Strekalov, et al., J. Geophys. Res. (Planets) 118, 1415 (2013)

2013
[47]

M. T. Zuber, D. E. Smith, M. M. Watkins, S. W. Asmar, A. S. K onopliv, F. G. Lemoine, H. J. Melosh, G. A. Neumann, R. J. Phillips, S. C. Solomon, et al., Science 339, 668 (2013)

2013
[48]

Blas and A

D. Blas and A. C. Jenkins, Phys. Rev. Lett. 128, 101103 (2022)

2022
[49]

S. G. Turyshev, M. Shao, I. Hahn, D. Blas, and A. C. Jenkin s, in LPI Contributions (2024), vol. 3065, p. 5001

2024
[50]

M. Shao, S. G. Turyshev, I. Hanh, and R. Trahan, in Proceedings of the 21st International Workshop on Laser Ran ging, Canberra, Australia, 5-9 November 2018 (IILRS, 2018)

2018
[51]

S. G. Turyshev, M. Shao, I. Hanh, J. G. Williams, and R. Tr ahan, in Proceedings of the 21st International Workshop on Laser Ranging, Canberra, Australia, 5-9 November 2018 (IILRS, 2018)

2018
[52]

T. W. Murphy, Jr., E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt , A. E. Orin, J. D. Strasburg, et al., PASP 120, 20 (2008)

2008
[53]

Samain, P

E. Samain, P. Veillet, C. Fridelance, and et al., Astron . Astrophys. 336, L17 (1998)

1998
[54]

T. W. Murphy, Jr., Rep. Progr. Phys. 76, 076901 (2013)

2013
[55]

J. J. Degnan, Millimeter Accuracy Satellite Laser Ranging: a Review (AGU Geodynamics Series, 1993), vol. 25, pp. 133–162

1993
[56]

T. W. Murphy, Jr., E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, Icarus 208, 31 (2010)

2010
[57]

J. B. R. Battat, T. W. Murphy, E. G. Adelberger, B. Gilles pie, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, K. Nordtved t, A. E. Orin, C. W. Stubbs, et al., PASP 121, 29 (2009)

2009
[58]

Lisdat, G

C. Lisdat, G. Grosche, N. Quintin, C. Shi, B. Lipphardt, C. Tamm, A. Al-Masoudi, F. Fitzek, T. B¨ uckle, J. Jeon, et al. , Nature Communications 7, 12443 (2016)

2016
[59]

R. S. Park, W. M. Folkner, J. G. Williams, and D. H. Boggs, Astron. J. 161, 105 (2021)

2021
[60]

J. G. Williams, D. H. Boggs, and J. T. Ratcliﬀ, JGR (Plane ts) 119, 1546 (2014)

2014
[61]

T. W. Murphy, Jr., E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, N. H. Johnson, R. J. McMillan, C. W. Stubbs, and H. E. Swanson, CQG 29, 184005 (2012)

2012
[62]

Mendes and E

V. Mendes and E. Pavlis, Geophysical Research Letters 31, L14602 (2004)

2004
[63]

S. G. Turyshev, J. G. Williams, W. M. Folkner, G. M. Gutt, R. T. Baran, R. C. Hein, R. P. Somawardhana, J. A. Lipa, and S. Wang, Experimental Astronomy 36, 105 (2013)

2013
[64]

F. H. Lyard, D. J. Allain, M. Cancet, L. Carr` ere, and N. P icot, Ocean Science 17, 615 (2021)

2021
[65]

Vokrouhlick´ y, J

D. Vokrouhlick´ y, J. M¨ uller, D. Dirkx, and et al., Space Science Reviews 215, 16 (2019)

2019
[66]

Altamimi, P

Z. Altamimi, P. Rebischung, L. M´ etivier, and X. Collil ieux, JGR: Solid Earth 121, 6109 (2016). 28

2016
[67]

Sakamoto, Y

M. Sakamoto, Y. Sato, and Y. Imamura, Optics Letters 39, 1581 (2014)

2014
[68]

Marsili, V

F. Marsili, V. B. Verma, J. A. Stern, and et al., Nature Ph otonics 7, 210 (2013)

2013
[69]

Fienga, P

A. Fienga, P. Deram, A. Di Ruscio, V. Viswanathan, J. I. B . Camargo, L. Bernus, M. Gastineau, and J. Laskar, Notes Scientiﬁques et Techniques de l’Institut de Mecanique Cele ste 110 (2021)

2021
[70]

S. G. Turyshev and J. G. Williams, IJMPD 16, 2165 (2007)

2007
[71]

Currie, S

D. Currie, S. Dell’Agnello, and G. Delle Monache, Acta A stronautica 68, 667 (2011)

2011
[72]

Samain, P

E. Samain, P. Guillemot, C. Courde, and L. Torre, in 16th International Workshop on Laser Ranging (2009), URL https://ilrs.gsfc.nasa.gov/lw16/docs/papers/new_4_Samain_p.pdf

2009
[73]

Courde, J

C. Courde, J. M. Torre, E. Samain, G. Martinot-Lagarde, M. Aimar, D. Albanese, P. Exertier, A. Fienga, H. Mariey, G. Metris, et al., Astron. Astrophys. 602, A90 (2017)

2017
[74]

S. L. Campbell, R. B. Hutson, G. E. Marti, A. Goban, N. Dar kwah Oppong, R. L. McNally, L. Sonderhouse, J. M. Robinson, W. Zhang, B. J. Bloom, et al., Science 358, 90 (2017)

2017
[75]

S. G. Turyshev, W. Farr, W. M. Folkner, A. R. Girerd, H. He mmati, T. W. Murphy, J. G. Williams, and J. J. Degnan, Experimental Astronomy 28, 209 (2010)

2010
[76]

J. G. Williams, Cel. Mech. Dyn. Astron. 130, 63 (2018)

2018
[77]

D. C. Rife and R. R. Boorstyn, IEEE Transactions on Infor mation Theory 20, 591 (1974)

1974
[78]

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimatio n Theory (Prentice Hall, 1993)

1993
[79]

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley-Interscience, 2001)

2001
[80]

T. W. Murphy, Jr., E. G. Adelberger, J. B. Battat, C. D. Ho yle, E. L. Michelsen, C. W. Stubbs, and H. E. Swanson, in Proceedings of the 15st International Workshop on Laser Ran ging, Canberra, Australia, 15-20 October 2006 (IILRS, 2006)

2006
[81]

T. W. Murphy Jr., E. G. Adelberger, J. B. R. Battat, C. D. H oyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, Icarus 208, 31 (2010)

2010
[82]

H. R. Telle, B. Lipphardt, and J. Stenger, Applied Physi cs B 69, 327 (1999)

1999
[83]

R. R. Beland, in The Infrared & Electro-Optical Systems Handbook , edited by F. G. Smith (SPIE Press, 1993), vol. 2, pp. 157–232

1993
[84]

Dro˙ zd˙ zewski and K

M. Dro˙ zd˙ zewski and K. So´ snica, J. Geodesy95, 100 (2021)

2021
[85]

Arnold, A

D. Arnold, A. Couhert, O. Montenbruck, C. Kobel, E. Saqu et, H. Peter, F. Mercier, and A. J¨ aggi, in 44th COSPAR Scientiﬁc Assembly. Held 16-24 July (2022), vol. 44, p. 3396

2022
[86]

Petit and B

G. Petit and B. Luzum, IERS Conventions (2010) , no. 36 in IERS Technical Note (International Earth Rotatio n and Reference Systems Service (IERS), Frankfurt am Main, Germa ny, 2010)

2010
[87]

E. C. Pavlis, V. Luceri, M. Kuzmicz-Cieslak, M. Pirri, K . Evans, and G. Bianco, in EGU General Assembly Conference Abstracts (2018), EGU General Assembly Conference Abstracts, p. 9216

2018
[88]

H. H. Kieﬀer and T. C. Stone, Astron. J. 129, 2887 (2005)

2005
[89]

R. T. Walker, M. K. Barker, E. Mazarico, X. Sun, G. A. Neum ann, D. E. Smith, J. W. Head, and M. T. Zuber, The Planetary Science Journal 5 (2024)

2024
[90]

Zhang, J

M. Zhang, J. M¨ uller, and L. Biskupek, Astron. Asrophys . 681, A5 (2024)

2024
[91]

D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966)

1966
[92]

Roddier, Progress in Optics 19, 281 (1981)

F. Roddier, Progress in Optics 19, 281 (1981)

1981
[93]

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, New York, 1998)

1998
[94]

S. G. Turyshev, M. Shao, and I. Hahn, Fundamental physics and lunar science investigations with advanced lunar laser ranging (2021), NAS Decadal Survey on Biological and Physical Scien ces (BPS) Research in Space 2023-2032

2021
[95]

J. G. Williams, Geophys. Res. Let. 34, L03202 (2007)

2007
[96]

S. G. Turyshev, J. G. Williams, D. H. Boggs, and R. S. Park (2024), arXiv:2406.16147 [astro-ph.EP]

2024 arXiv
[97]

S. G. Turyshev, M. Shao, and I. Hahn, Night-time power delivery to lunar outposts with a high-pow er ground-based laser array (2021), NAS Decadal Survey on Biological and Physical Scien ces (BPS) Research in Space 2023-2032. Appendix A: Calibration and Maintenance High-precisi...

2021
[98]

Key methods include:

Daily and Continuous Calibration LLR facilities typically implement daily calibration routines, supplemented by continuous checkouts at hourly or sub-hourly intervals. Key methods include:
[99]

Short-Distance Reference Target: A stable local retroreﬂector (e.g., a CCR or precision cavity) positio ned at a known baseline (10–100 m) from the main telescope serves as a zero -point reference. Regular ranging to this 29 target—performed every 1–2 hours under variable cond...
[100]

Since drifts in fm can lead to cm- to mm-scale range biases (see Sec

GPS or Frequency-Comb Tie: The station’s local oscillator, which determines the CW modulation fre quency fm or chirp rate β, is phase-locked to an external reference (such as a GPS-disciplin ed clock or optical frequency comb). Since drifts in fm can lead to cm- to mm-scale ra...
[101]

Calibration re-baselines the system, keeping range oﬀsets below 1 m m—and even to tens of µm with precise surveying

Telescope Alignment Tracking: Internal metrology beams or star-based auto-collimators period ically verify align- ment and focus, preventing sub-arcsec misalignments from degra ding photon return or introducing timing errors. Calibration re-baselines the system, keeping range ...
[102]

Maintaining Sub- µ m/s Stability Daily calibration may not suﬃce when environmental variations are ra pid. To achieve sub- µm/s stability during live observations, it is necessary to: • Thermal Control: Maintain the optical assembly (optical bench, relay optics, detect or hous...
[103]

Annual tie surveys detect tectonic shifts or pier settling at the su b-millimeter level

External Ties to Metrological Networks Anchoring LLR data to global geodetic frames requires station pos itions accurate to within ≲ 1 mm in every axis: • Local Surveys: Instruments such as GNSS antennas, satellite laser ranging (SLR) systems, or VLBI radio telescopes provide ...
[104]

• Stable Conditions: During consistently calm nighttime periods with minimal dome seeing, on ce-daily calibration and pre/post-session checks are typically suﬃcient

Calibration Frequency and Adaptation Environmental conditions dictate the optimal calibration frequenc y: • Rapid Weather Changes: Under unstable conditions (e.g., large temperature swings or high win ds), calibrating every 1–2 hours is recommended. • Stable Conditions: During...
[105]

High-Power Laser System The laser system is the cornerstone of CW LLR, providing the high ph oton ﬂux required to suppress shot noise and achieve sub-mm precision: • Wavelength (1064 nm): Widely used in Nd:YAG and ﬁber lasers, this wavelength oﬀers high atmo spheric trans- mis...
[106]

Larger apertures boost photon ﬂux but require tighter therma l control

Telescope System The telescope must both transmit a high-power beam and eﬃciently c ollect the faint returns: • Aperture (1–2 m): A 1 m Ritchey–Chr´ etien design strikes a balance between photon co llection eﬃciency, me- chanical complexity, and cost [ 18]. Larger apertures bo...
[107]

• High-Damage-Threshold Coatings: Coatings must withstand /greaterorsimilar1 kW power, and active cooling prevents wave- front distortions or lensing [ 18]

Beam Expander Beam expanders are used to reduce near-ﬁeld intensity and improv e the far-ﬁeld spot size: • Expansion Ratio (10–20 ×): Adjustable beam diameters (10–20 cm) help mitigate optic damage at high power levels and counteract atmospheric seeing eﬀects. • High-Damage-Th...
[108]

31 • Avalanche Photodiodes (APDs): APDs are easier to operate at moderate temperatures ( −30◦C) but typically have higher dark counts and lower quantum eﬃciency compared to S NSPDs

Detector System The detector system determines the ultimate timing precision and sh ot-noise limit: • Superconducting Nanowire SPDs (SNSPDs): These detectors achieve > 80% quantum eﬃciency at 1064 nm, with dark counts below 10 s −1 and timing jitter of 20–50 ps [ 33]. 31 • Ava...
[109]

Multi-second averagin g enhances precision while suppressing random noise [ 22]

Timing and Control Electronics Ultra-stable timing electronics are essential for resolving small pha se shifts over multi-second integrations: • Time-to-Digital Converters (TDCs): Sub-10 ps bin resolution minimizes timing error. Multi-second averagin g enhances precision while...
[110]

• Vibration Isolation: High-stiﬀness piers and passive or active damping systems minimize me chanical noise [ 17]

Environmental Control and Safety Systems A high-power CW LLR system requires precise environmental contr ols and robust safety mechanisms: • Thermal Regulation: Dome interiors and optical components must be maintained within ±0.01◦C to limit thermal expansion to sub- µm levels...
[111]

Adaptive modes or pause s protect the system under poor conditions

Control Software Advanced control software is essential for real-time operation a nd diagnostics: • Dynamic Scheduling: Automated scripts integrate real-time ephemerides, atmospheric data, and calibration results to optimize observing sequences. Adaptive modes or pause s prote...

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

organized as follows:

Coarse-to-Fine Frequency (AM) or Slope (FM) Approach A typical multi-frequency AM protocol may use three or more discr ete RF tones f1, f2, f3, . . . organized as follows:

[2] [2]

Measuring the round-trip phase at f1 localizes the range rEM to a bin of roughly a few meters (modulo the coarse-wavelength spa cing)

Coarse Tone (f1 ∼ 10–50 MHz): Provides a synthetic wavelength of c/(2f1) ≈ 3–15 m. Measuring the round-trip phase at f1 localizes the range rEM to a bin of roughly a few meters (modulo the coarse-wavelength spa cing). 12

[3] [3]

Phase unwrapping between f1 and f2 yields a unique solution over the full Earth–Moon baseline

Intermediate Tone (f2 ∼ 100–500 MHz): Reﬁnes the distance to the centimeter or millimeter le vel since each full 2 π cycle now corresponds to a shorter distance of c/(2f2). Phase unwrapping between f1 and f2 yields a unique solution over the full Earth–Moon baseline

[4] [4]

Equation ( 5) (for AM) or Equation ( 15) (for FM) then translates the measured phase or beat frequenc y into a ﬁne distance rEM, fine

Fine Tone (f3 ∼ 1–10 GHz): Finally, a high-frequency tone locks in sub-mm or even te ns-of-µm precision once the coarse and intermediate phases have identiﬁed the corre ct ambiguity bin. Equation ( 5) (for AM) or Equation ( 15) (for FM) then translates the measured phase or beat frequenc y into a ﬁne distance rEM, fine. Mathematically, if φi denotes the m...

[5] [5]

Multi-Chirp or Multi-Slope FM Variant An FM LLR station may replace discrete RF tones with several chirp s weeps characterized by slopes β1, β2, . . .. Each chirp sweep produces a beat frequency that encodes the ro und-trip delay to the Moon. From ( 14), for a chirp with slope βi, the beat frequency is given by fb,i = βi τ = βi 2rEM c . A typical multi-ch...

[6] [6]

Gentle Chirp (β1): Provides a large unambiguous range of c/(2β1), spanning thousands of kilometers, and localizes rEM to a bin of roughly a few meters

[7] [7]

Phase unwrapping between β1 and β2 yields a unique solution

Intermediate Slope (β2): Reﬁnes the range to the centimeter or millimeter level since each f ull 2 π cycle corre- sponds to a shorter distance c/(2β2). Phase unwrapping between β1 and β2 yields a unique solution

[8] [8]

Here, the chirp rate is deﬁned as βi = B/Tchirp, where B is the chirp bandwidth and Tchirp is the duration of the chirp

Steep Slope (β3): Achieves sub-mm precision once the correct ambiguity bin is identiﬁ ed; the measured beat frequency is then converted into a ﬁne distance rEM, fine via Equation ( 5) (for AM) or Equation ( 15) (for FM). Here, the chirp rate is deﬁned as βi = B/Tchirp, where B is the chirp bandwidth and Tchirp is the duration of the chirp. High linearity ...