Abstract

The Laser Retroreflector Array for Lunar Landers (LRALL) is a small optical instrument designed to provide a target for precision laser ranging from a spacecraft in lunar orbit, enabling geolocation of the lander and its instrument suite and establishing a fiducial maker on the lunar surface. Here we describe the optical performance of LRALL at visible and near-infrared wavelengths. Individual corner cube reflectors (CCRs) within LRALL were tested for surface flatness and dihedral angle values. We also imaged the far-field diffraction patterns of individual CCRs as well as the entire retroreflector array over the range of possible incident angles to extract the optical cross section as a function of viewing angle. We also measured the optical properties of one of the CCRs over the lunar temperature range (100–380 K) and found no significant temperature-dependent variance. The test results show LRALL meets the design criteria and can be ranged to elevation angles above 30° with respect to the instrument base from an orbital laser altimeter such as the Lunar Orbiter Laser Altimeter on the Lunar Reconnaissance Orbiter. This work summarizes the test data and serves as a guide for future laser ranging to these retroreflector arrays.

1. INTRODUCTION

Laser retroreflector arrays (LRAs) on the Moon and other planetary bodies can act as fiducial markers for decades of laser ranging. The Apollo and Lunokhod LRAs on the lunar surface [1,2] have significantly contributed to our knowledge of the Moon [25]. The recent development of lunar missions from government space agencies and commercial and private partners [6] offers a unique opportunity to support lunar science and exploration through the deployment of small LRAs on lunar landers. Placement of a small LRA on the deck of a lander or rover enables tracking with an orbital laser altimeter with a precision on the order of centimeters. An optical marker observable from orbit mounted alongside a suite of scientific instruments on the surface enables geolocation of those instruments in the lunar geodetic frame. In addition, LRAs can in turn be used to validate geolocation and ranging performance of orbital laser altimeters, as was done in support of the Advanced Topographic Laser Altimeter System (ATLAS) on the Ice, Cloud and land Elevation Satellite-2 (ICESat-2) mission [7]. Knowledge of the precise location of the array and the lander on the lunar surface is beneficial for orbit reconstruction, navigation, and broader exploration goals. Finally, optical markers such as LRAs can support precision autonomous navigation and landing without special demand for the lighting conditions, as was accomplished by the Hayabusa mission to asteroid 25143 Itokawa [8] and the Hayabusa2 missions to asteroid 162173 Ryugu [9].

The Laser Retroreflector Array for Lunar Landers (LRALL) is an instrument designed to provide a high-gain optical target that can be ranged to with a lunar-orbiting laser altimeter from any angle above 30° from the mounting plane. These low-mass, small instruments (Table 1) were designed and tested for decades of lifetime on the lunar surface. They can operate over the entire lunar day and night and are completely passive; they require no power, communication, or thermal control. LRALL was integrated with the SpaceIL Beresheet spacecraft prior to its 21 February 2019, launch and became the first NASA instrument to fly on a non-governmental mission to the Moon. Similar small LRAs were placed on the ExoMars and InSight landers [10]. For further discussion of the design, environmental testing, and future uses of LRALL please see our companion publication [11].

 figure: Fig. 1.

Fig. 1. LRALL. (a) Array with eight metal-coated CCRs. (b) Array with eight TIR CCRs. (c) Illustration showing LRALL geometry and optical reference point. CCR illustration is not to scale. CG, center of gravity.

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Tables Icon

Table 1. LRALL Instrument Parameters

Here we describe the optical performance of LRALL. Our optical test program included measurement tests from the qualification of retroreflector arrays for satellite laser ranging (SLR) [12,13]. However, our test program extended beyond previous characterization to measure returns at multiple wavelengths, track performance over the lunar temperature range, and perform a detailed study of the interference effects of multiple-cube returns. We tested two versions of LRALL: one with metal-coated corner cube reflectors (CCRs), and one with CCRs that rely on total internal reflection (TIR). First, we will present the mechanical and optical design of LRALL and discuss its similarities and differences to other lunar retroreflector arrays. Following this, the optical testing is presented in four sections. In each section, the test setup will be described first, followed by the test results and analysis. The four test sections are as follows: interferometric testing of individual CCRs for surface flatness and dihedral angle measurements, tests of retroreflector optical performance in vacuum over the expected lunar surface temperature range (100–380 K), far-field diffraction experiments at visible (532 nm) and near-infrared (1064 nm) wavelengths of both individual CCRs and the entire LRA to measure the optical cross section (OCS) and investigate interference effects, and return pulse waveform tests to determine the pulse-spreading characteristics of the LRA.

2. INSTRUMENT DESIGN

The LRALL instrument comprises eight unspoiled fused silica CCRs (circularly cut trihedral prisms) bonded to a chromate conversion-coated aluminum shell, as shown in Fig. 1. The eight CCRs are set into the shell in a pattern of two concentric rings of four cubes each: one ring with a CCR surface normal at 20° from the base normal and one ring at 40° from the base normal [Fig. 1(c)]. The orientation of each CCR with respect to the overall array was randomized. Two versions of the instrument are described here: one in which the CCRs have aluminum-coated (i.e., aluminum-plated or metal-coated) back faces, and one in which the CCRs are uncoated and rely on TIR. The instrument parameters are given in Table 1.

The position of the optical reference point of the array with respect to the center of gravity and the base of the array is shown in Fig. 1(c). The optical reference point is defined as the intersection of the optical axes of the eight individual CCRs. The center of gravity of the array is co-located with the axis of rotational symmetry of the array but offset by 6.73 mm above the basal plane of the array. As the center of gravity and optical reference point are not co-located, an elevation-dependent correction is needed to translate the range to a position with respect to the center of gravity or the base of the array.

LRALL is designed to be mounted on landed spacecraft and ranged to from orbit (up to 300 km away) and not from Earth; these arrays are too small to be ranged to from Earth with current technology. LRALL has significant design differences from previous lunar retroreflector arrays to accommodate ranging from an orbiting spacecraft. The most prominent changes are the size of the CCRs and the non-planar design of the array.

The CCRs set into the LRA have a front-face clear aperture diameter of 1.27 cm (0.5 in.), which results in a significantly smaller cross-sectional area than the 3.8 cm CCRs in the Apollo arrays, though the smaller CCRs offer several advantages. First, the smaller CCRs have a lower mass and can be packed tightly to form a small array suited for smaller, lighter lunar landers. In addition, smaller CCRs are less susceptible to solar-induced thermal gradients that may affect the performance of larger CCRs [14,15]. For the coated CCRs, absorption of incident light by the metal coatings can create thermal gradients within the CCR and lead to phase distortions and a subsequent temporal instability and a reduction in OCS. The CCRs in the TIR version of LRALL are uncoated and consequently are less susceptible to degradation over time [11,14]. Uncoated glass surfaces provide improved longevity on the lunar surface, crucial for the goal of creating and sustaining lunar geodetic reference points. Spacecraft integration involves securing one or more arrays to a spacecraft deck with the appropriate fasteners in a location that gives the largest unimpeded field of view of the sky for the array(s).

3. DIHEDRAL ANGLE ERROR AND SURFACE FLATNESS OF CCRs

Two primary optical figures of merit for a CCR are the surface flatness and the dihedral angle error (DAE) or deviation in angle from 90° between the three facets of the trihedral prism. A rough surface or large DAE will spread the retroreflected beam and reduce the overall optical gain of the instrument for this application. The design specification for LRALL was for each CCR to have a surface flatness of ${1/10}\;\lambda$ at 532 nm and exhibit an average DAE within $\pm {0.5}\;{\rm arc}\;{\rm sec}$. This is in contrast to small LRAs for SLR in low-Earth orbit where one of the dihedral angles is often spoiled to account for the velocity aberration. The performance of each CCR was measured using a laser interferometer (Zygo Verifire) operating at 633 nm. The LRA was mounted in front of the interferometer such that a single CCR in the array was near normal to the incident laser, and all other CCRs were blocked. Each CCR was measured four times. All CCRs on all LRAs were tested and showed similar results, meeting the specifications for DAE and surface flatness. Figure 2 shows representative results for the DAEs from one of the LRAs.

 figure: Fig. 2.

Fig. 2. DAE measurements of CCRs for a representative LRALL instrument. Angles 1, 2, and 3, corresponding to the intersections of the three prism facets, were designated prior to test and held consistent throughout testing.

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4. DAE VARIANCE OVER THE LUNAR TEMPERATURE RANGE

A. Temperature Test Motivation

The thermal extremes experienced on the lunar surface over the course of the diurnal cycle—from 95 to 385 K at the equator [16]—can induce thermal gradients in both coated and uncoated CCRs [14,15,17], though metal-coated CCRs are more susceptible to this effect. This effect can lead to reduced OCS for gradients as small as 4 K from the front surface to the rear vertex [18]. To test whether the 1.27 cm CCRs in LRALL would suffer degraded optical performance over the expected lunar temperature range, we mounted the array inside a cryogenic vacuum Dewar behind an optically transparent, wedged window. We then varied the temperature of the baseplate of the Dewar as we measured the CCR DAEs with a Zygo Verifire laser interferometer. Changes in the interference pattern from the CCR (due to thermal gradients in the CCR or other effects) would thus manifest as higher DAE variance.

B. Temperature Test Setup

We attached the LRA to the Dewar mounting plate with four M2.5 socket cap fasteners using 2.6–3.0 ft lb of torque to simulate integration conditions. The Dewar was evacuated to a vacuum level of 0.03 Torr prior to cooling to prevent frost formation on the array. The array temperature was monitored with a silicon diode thermocouple fixed within 2 mm of the LRA on the aluminum mounting plate. Heating from cryogenic temperatures was performed via a 100 W cartridge heater fixed to the rear of the mounting plate. Measurements were obtained from 100 to 380 K in 20 K steps. The heating rate was held at 3 K per minute between temperature steps.

 figure: Fig. 3.

Fig. 3. Measured retroreflecting properties of a TIR CCR over the expected lunar surface temperature range. Angles 1, 2, and 3, corresponding to the intersections of the three facets, were designated prior to test and held consistent throughout testing.

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The LRA heating in the test configuration was conductive via the array shell to the outer front surface of the CCR and radiative from the baseplate to the back of the CCRs, which differs from the front-surface radiative heating that the CCRs will experience on the lunar surface. In our case, the inability to directly measure the 1.27 cm CCR temperature in real time, as well as space limitations of the cryogenic Dewar needed to reach the lower end of the lunar temperature range while measuring the DAE, precluded the use of a solar simulator. We believe the thermal conditions in our test setup represent a more extreme test environment than the lunar surface, as the heating was more localized and rapid than under surface conditions. Temperature measurements were obtained during heating of the system, which began after cooling of the array with liquid nitrogen to 100 K. The measurement temperatures were held with a proportional-integral-derivative (PID) heating loop for roughly one minute at each measurement temperature during which the interferometric measurements were obtained.

C. Temperature Test Results

We observed small variations in the DAEs over the lunar surface temperature range, and the three angles in the CCR exhibited different trends from one another (Fig. 3). Measurements of the DAEs taken at room temperature prior to cooling were ${-}{0.19}\;{\rm arc}\;{\rm sec}$, 0.61 arc sec, and 0.14 arc sec for angles 1, 2, and 3, respectively. Thus, for the entire observed temperature range, all three DAEs varied over a range near the tolerance of the CCRs ($\pm {0.5}\;{\rm arc}\;{\rm sec}$). This suggests that no strong axial or radial thermal gradients were present in the 1.27 cm CCRs that would impact optical performance under test conditions. On the lunar surface, solar illumination can induce thermal gradients in the CCRs due to thermal breakthrough (loss of TIR) of the TIR CCRs at angles above ${\sim}{17}^\circ$ and conduction through the aluminum shell [19]. However, the small size of the CCRs in LRALL (1/3 the diameter of the Apollo CCRs) should reduce the magnitude of thermal gradients while on the lunar lander deck. Some margin also exists in the link for the potential of thermally induced reduction in OCS based on our calculation of the maximum ranging distance of 170 km for the Lunar Orbiter Laser Altimeter (LOLA) [11]. The non-planar design of the array should also enable ranging to multiple CCRs in the array during a single pass should thermal breakthrough at certain times of day reduce the OCS of a portion of the array.

 figure: Fig. 4.

Fig. 4. FFDP optical test setup.

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5. FAR-FIELD DIFFRACTION IMAGING AND OPTICAL CROSS-SECTION MEASUREMENTS OF INDIVIDUAL CCRs AND OF THE ARRAY

A. Test Setup and Discussion of OCS

The far-field diffraction pattern (FFDP) is the spatial and intensity distribution of light returned to an interrogating laser source from an illuminated CCR under Fraunhofer conditions (i.e., the distance from the source to the CCR is much greater than the aperture diameter). A theoretical, perfect reflecting circular surface illuminated at normal incidence has a FFDP that is the well-known two-dimensional Airy function. Far-field patterns of individual CCRs and the LRA were obtained at wavelengths of 532 and 1064 nm using a 2.5 m focal length parabolic reflective collimator. The collimator test setup is shown in Fig. 4(a). The array was mounted to a dual-axis rotation stage to allow for imaging over a range of viewing angles [Fig. 4(b)]. We placed a circular aperture between the beam splitter and first mirror that spatially filtered the beam to ½ the initial beam diameter. The beam diameter at the LRA was measured to be 75 mm with a wavefront quality of $\lambda /{8}$. A linear polarizer was placed in front of the laser source to control the laser polarization with respect to the array. Unless otherwise stated (as in Section 5.C), the polarization state for all tests was horizontal with respect to the LRA oriented at azimuth = 0, which is the array orientation shown in the inset of Fig. 2. Individual CCR FFDPs were obtained by blocking surrounding cubes in the array, ensuring returns from only the CCR under test.

The quantitative optical return from a CCR can be described by its OCS (denoted as $\sigma$). The OCS is intrinsic to a CCR and is given here in units of million square meters. It can be intuitively understood to be the equivalent area of a Lambertian surface that returns the same optical signal as the retroreflector(s). The peak OCS of a CCR under normal incidence is given by [12]

$$\sigma = \frac{{\rho \cdot{A^2}\cdot4\pi}}{{{\lambda ^2}}},$$
where $\rho$ is the reflectivity of the CCR (including surface and internal reflections), $A$ is the cross-sectional area of the CCR front face, and $\lambda$ is the wavelength of light illuminating the CCR. When illuminated at off-normal incidence, the peak OCS is reduced by a factor $\eta$ based on the angle of incidence ${\theta _i}$ and index of refraction $n$ of the glass given by [20]
$$\begin{split}\eta &= {\left[{\frac{{2\cos ({{\theta _i}} )}}{\pi}} \right]^2}\cdot\left\{{\sin^{- 1}}\left[{\sqrt {1 - 2\tan^{2}\left({{\sin^{- 1}}\left({\frac{{\sin ({{\theta _i}} )}}{n}} \right)} \right)}}\! \right]\right.\\&\quad - \left.\sqrt 2 \tan\left[{{\sin^{- 1}}\left({\frac{{\sin ({{\theta _i}} )}}{n}} \right)} \right] \right\}^2.\end{split}$$

For TIR CCRs, polarization effects further reduce the peak OCS. Using this framework, the quantitative OCS values presented here were calibrated using the OCS from a 1.27 cm diameter protected-gold mirror with a reflectance calibrated at each measurement wavelength.

As a multi-reflector array, the optical return from LRALL is a function of interfering returns from one to four individual retroreflectors. In the interest of completeness and to aid analysis and modeling of future laser ranging efforts, we will first discuss the results for an individual CCR, which provides simpler results from which the more complicated LRALL results can be contextualized and interpreted.

B. Single-CCR Optical Performance

FFDPs of individual CCRs in the array were obtained with an opaque plastic mask placed around the CCR under test to block light from any adjacent reflector. Diffraction patterns were captured with a ${2048} \times {2048}$-pixel complementary metal-oxide-semiconductor (CMOS) camera (Basler acA2500-14 gm) at 12 bit depth with a pixel size corresponding to ${2.15}\;\unicode{x00B5}{\rm rad} \times {2.15}\;\unicode{x00B5}{\rm rad}$ in reciprocal space. Each FFDP was compared to a calibrated, protected-gold mirror for which the peak OCS was calculated using Eq. (1). The laser and camera settings were held constant between measurements of the CCRs and this reference mirror. This process enabled a pixel-by-pixel conversion from pixel intensity to OCS for each FFDP. After this normalization, the highest-intensity pixel in each FFDP was set to zero beam deviation for single-CCR FFDPs. Figure 5 shows representative results from the metal-coated [Figs. 5(a) and 5(b)] and TIR [Figs. 5(c) and (5d)] CCRs at normal incidence to the CCR.

 figure: Fig. 5.

Fig. 5. FFDPs of individual CCRs. The patterns from (a) and (b) the metal-coated and (c) and (d) TIR CCRs were obtained under uniform illumination with collimated (a) and (c) 532 nm and (b) and (d) 1064 nm laser light. The axes correspond to beam deviation from the source as measured on the calibrated detector. The color scales correspond to OCS and are given by the scales to the right of each frame.

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The metal-coated CCRs exhibited a circularly symmetric optical return similar to a classical Airy pattern, as predicted from diffraction theory. In addition, the size of the Airy disk scales linearly in reciprocal space with the wavelength. Measurements of the TIR retroreflectors using linearly polarized light exhibited well-known polarization effects; the sixfold symmetry due to the real and reflected back faces of the CCR can be seen in the FFDPs as each reflection within the cube introduces an element of elliptical polarization [21,22]. Thus, the contributions from all six unique paths through the CCR, each introducing a different phase shift, interfere to generate the six sidelobes seen in the bottom of Fig. 5. This effect reduces the central peak of the FFDP to 26.4% of what it would be for a perfect reflector of the same area and shape.

The numerical results for the coated and TIR CCRs at both measurement wavelengths are given in Table 2. The stated variances correspond to 1σ from five independent measurements at normal incidence. The coated CCR peak OCS values compared to the perfect reflector case (based on Eq. 1) are 72% and 74% for 532 and 1064 nm, respectively. This difference can be explained by the aluminum coating on the cubes, which is 91% reflective and thus induces a 25% reflectance loss through the CCR for three reflections.

Tables Icon

Table 2. Optical Test Results from FFDPs of Individual CCRsa

For the TIR CCRs, the measured peak OCS was smaller than that expected from theory. The central irradiance of a TIR retroreflector with no dihedral angle offset is expected to be 26% that of a perfect reflector; however, we measured values of 14% and 15% at 532 and 1064 nm, respectively. Surface reflection losses for the TIR CCRs were measured as ${\sim}{2}\%$ per pass, which brings the expected peak OCS to 22% that of a perfect reflector. This unexpected lower peak OCS for the TIR CCRs is most likely due to the laboratory measurement setup (e.g.,  laser intensity fluctuations between calibration and measurement, air currents or turbulence changing the spatial laser profile, slight misalignment of the CCR mask) as the interferometer results show a low DAE and surface flatness that would predict near-theoretical CCR performance.

 figure: Fig. 6.

Fig. 6. Normalized peak OCS of metal-coated and TIR CCRs as a function of incident angle at 1064 nm. Normalized peak OCS is defined as the OCS returned to the light source normalized to the OCS at normal incidence for each type of CCR.

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 figure: Fig. 7.

Fig. 7. Polarization dependence of far-field patterns from TIR CCRs. Far-field patterns were obtained from illumination at (a)–(c) 532 nm (outlined in green) and (d)–(f) 1064 nm (outlined in red). The frame axes correspond to beam deviation in µrad and are the same for all images. (insets) The polarization axes for the three polarizations tested (black arrows) with respect to the CCR. Edges of the facets of the CCR are shown in red. The color scale corresponds to OCS with warmer colors denoting higher OCS, and cooler colors denoting a lower OCS. The OCS scale for (a)–(c) is given by the colorbar next to (c) and the scale for (d)–(f) is given by the colorbar next to (f).

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In addition to testing the performance at normal incidence, the peak OCS of individual CCRs was tracked as a function of incidence angle to determine the range of angles from which successful ranging can occur. Figure 6 shows the normalized peak OCS as a function of incident angle for coated and TIR CCRs. The coated and TIR retroreflectors exhibited similar retroreflective behavior; both retroreflectors showed a reduction of peak OCS due to the angle-dependent reduction in visible cross section of the CCR. The drop to 50% OCS occurred at ${\sim}{12}^\circ$ from normal incidence. The TIR CCRs showed a reduced relative return at angles above ${\sim}{15}^\circ$ corresponding to the onset of the TIR breakthrough effect, or the loss of TIR and subsequent increase in CCR optical transmission. The experimental results are compared to the predicted falloff due to the reduction in the effective area [Eq. (2)], which matches well for the metal-coated CCRs.

C. Polarization Sensitivity of TIR CCRs

Understanding the polarization sensitivity of the TIR version of LRALL may be useful to interpreting future orbital ranging measurements as solid-state laser transmitters comprise the bulk of planetary laser altimeters and emit a linearly polarized beam. We captured FFDPs from single-TIR CCRs at 532 and 1064 nm for three incident polarization states. The results are shown in Fig. 7, with the polarization states indicated with respect to the CCR symmetry in the insets. Rotation of the input polarization led to changes in the pattern structure and amplitude of the optical return for both wavelengths. The six outer spots rotated counter to corresponding changes in the input polarization, and the outer lobes were preferentially excited when the polarization axis closely aligned with the lobe locations [21,23]. One notable case was the horizontal polarization case at 532 nm [Fig. 7(c)], which exhibited a reduced optical return across the entire pattern. As this result was not replicable at 1064 nm, we conclude that it is an artifact of the visible light measurement setup and not intrinsic to the CCR performance. The region of the FFDP relevant for ranging from lunar orbit is within 10 µrad of the center (due to velocity aberration), thus the sidelobes should not affect ranging efforts under single-CCR conditions.

D. Array Reference Frame

To clarify discussion of the optical test results from the full array, we will briefly describe our orientation terminology. As a hemispherical object, vectors incident to the LRA may be described by two angles (as in a spherical coordinate system), which we have termed the azimuth and elevation (Fig. 8) that are defined with respect to an LRA-centric reference frame.

 figure: Fig. 8.

Fig. 8. LRA schematic describing angle terminology overlaid on a top-down view of the LRA.

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The angle termed azimuth refers to rotation about an axis running through the center of the array from the top to the base (Fig. 8, left), with zero azimuth defined by the center of one of the four outermost CCRs. The angle termed elevation refers to rotation about an axis perpendicular to the azimuth axis and running from 0° (corresponding to viewing the array edge on) to 90° (corresponding to viewing the array from directly above, as is shown in Fig. 8). Thus, any incident vector can be uniquely defined by its azimuth (0° to 360°) and elevation (0° to 90°).

E. LRALL FFDPs and OCS

We performed a series of optical tests of both the coated and TIR LRALL instruments to measure the optical return over a broad range of incident angles. With these results, we hope to aid and give context to future laser ranging measurements to these instruments on the lunar surface. Using the measurement setup described in Section 5.A, FFDPs were obtained at one-degree angular steps from 0 to 90° in azimuth and 10° to 90° in elevation under constant illumination conditions, for a total of 7200 distinct FFDPs per array tested, per wavelength. As the array contains fourfold rotational symmetry, measuring one quarter of the hemisphere will yield information on all observable angles, assuming the individual CCRs within an array perform similarly.

A subset of FFDPs for the TIR version of LRALL is shown in Fig. 9 in equirectangular projection with the color scale corresponding to OCS. The highest optical return occurred at 60° to 80° in elevation as is apparent from the bright central lobes at those angles. FFDPs in that elevation range also exhibited interference fringes as returns from multiple CCRs contributed to the diffraction patterns. This multiple-CCR interference causes the intensity in those patterns to change as a function of time due to wavelength-scale variations in array position and incident angle. The pattern obtained at normal incidence to the array was notably dim, and that pattern (as well as the other patterns in this high-elevation interference regime) did not qualitatively resemble the single-cube FFDPs shown in Fig. 5. At elevations below 50°, the patterns resembled those of single CCRs at various incident angles. The orientation-dependent polarization effects of the CCRs described in Section 5.C, when combined with the interreference effects shown here, complicate precise prediction of the instantaneous cross section.

 figure: Fig. 9.

Fig. 9. Montage of FFDPs from LRALL (TIR CCRs) in equirectangular projection. The color scale is shown in the upper left and corresponds to relative OCS, with warmer colors corresponding to higher optical return. FFDPs were all obtained under identical imaging and illumination conditions.

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Due to the size of the LOLA receiver and the ranging distance from orbit (nominally 50 km), the LOLA receiver (or a similar future altimeter in lunar orbit) will only intersect and measure a small portion of the retroreflected FFDP (i.e., a 2.8 µrad cone for LOLA). Thus, the return from a single pixel (size: ${2.2}\;\unicode{x00B5}{\rm rad} \times {2.2}\;\unicode{x00B5}{\rm rad}$) of our measured FFDP will determine the received optical signal from orbit, which we label the lunar orbit optical cross section (LOOCS).

To extract this pertinent parameter from our laboratory measurements, we averaged the OCS at the pixels nearest $\pm {11}\;\unicode{x00B5}{\rm rad}$ from the zero-deviation pixel. This is the nearest approximation to the optical return that LOLA would observe due to velocity aberration in a 50 km circular orbit at each flyover azimuth (i.e., the values in these maps are not azimuthally averaged). For the FFDPs involving returns from more than one CCR, the pixel location defined as zero deviation was determined from measurements of the masked, individual CCRs ensuring proper extraction of the LOOCS (which is not always the pixel with the highest optical return due to interference effects). This zero-velocity-aberration pixel determination was made at each wavelength to account for variations in the measurement system at the different wavelengths. First, we will present the results for the aluminum-coated LRA followed by the TIR version of LRALL. The LOOCS maps constructed from the experiments performed at 532 and 1064 nm are shown in Fig. 10 for the coated version of LRALL and Fig. 11 for the TIR version.

 figure: Fig. 10.

Fig. 10. Map of optical return from the retroreflector array as a function of incident angle for the LRA with metal-coated CCRs. (a) Map created from FFDPs using 532 nm illumination. (b) Map created from FFDPs using 1064 nm illumination. The color scales correspond to the LOOCS with warmer, lighter colors denoting higher optical return. Elevation and azimuth angles refer to terminology defined in Fig. 8. FFDPs were obtained in one-degree intervals in both elevation and azimuth for the ranges shown; each pixel of the map represents a distinct FFDP from which the LOOCS was extracted. The pixel-to-pixel variation is real and is a result of sampling a complex interference pattern.

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 figure: Fig. 11.

Fig. 11. Map of optical return from retroreflector array as a function of incident angle for the TIR version of LRALL. Following the conventions of Fig. 10, (a) and (b) correspond to 532 and 1064 nm, respectively. The pixel-to-pixel variation is real and is a result of sampling a complex interference pattern.

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The first result of note is the similarity of the performance between wavelengths. As demonstrated in Fig. 5, illuminating a retroreflector with differing wavelengths will shrink or expand the FFDP in angular space based on the wavelength per diffraction theory. Similarly, the reduced OCS at 1064 nm compared to 532 nm is expected from Eq. (1). Less predictable are the subtle variations in the structure of the maps. Importantly, the structure is defined by significant returns above 30° elevation at all azimuths, which shows LRALL meets the measurement objectives for ranging from the full sky above 30°. These results suggest that future laser altimeters operating at longer wavelengths in the near or mid-infrared would be suitable for ranging to the coated version of LRALL from an orbital platform, with corresponding OCS values that roughly scale as ${\lambda ^{- 1/2}}$ and proportionally with the reflectivity of the metal coating.

Fine structure can be seen in the form of diagonal dark features running from high elevations to low and from 0° and 90° to 45° in azimuth. These bands are the high-elevation angles furthest from any CCRs, as the outer ring of cubes is located at azimuth values of 0°, 90°, 180°, and 270°. The LOOCS variation between adjacent pixels is particularly striking and is due to time-dependent interference of retroreflected light from multiple CCRs. The magnitude of the highest LOOCS is about 1.5 to 2 times higher than the peak OCS of a single CCR (Fig. 5), which confirms that constructive interference is a factor in determining the LOOCS.

The LOOCS map for the TIR version of LRALL is distinct in several ways from that of the coated array. First, the LOOCS declines sharply at elevations above 80° and peaks strongly near 70° elevation across all azimuths. This optical “dead spot” at high elevations near normal incidence to the array is due to the slightly narrower retroreflection angular range of TIR CCRs (Fig. 6) as well as interference of the sidelobe returns from the four inner-ring CCRs. The diagonal dark features observed in the coated array are more distinct here, stemming from the layout of the cubes within the array. The pixel-to-pixel variations are strongest at incident angles near 70° elevation and azimuth of 0° and 90° where two of the four inner-ring cubes are contributing equally to the optical return, and small shifts in the laser or array tilt of fractions of a wavelength can modify the amount of constructive or destructive interference. Again, the highest LOOCS values are 1.5 to 2 times the peak OCS from a single CCR, which indicates constructive interference from multiple reflectors affecting the optical return.

 figure: Fig. 12.

Fig. 12. Azimuthally averaged optical performance of LRALL as a function of elevation angle. The dark blue and dark red lines denote the azimuthal average value for each elevation, whereas the light blue and light red shaded regions denote the $1 \sigma$ variance at each elevation from the population of all azimuth angles.

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The maps presented in Figs. 10 and 11 can be used to calculate the expected return signal from an orbital laser altimeter system. If the orientation of the lander (and thus the array) is known with respect to the spacecraft orientation, the azimuth and elevation of the array during ranging can be calculated a priori to within the uncertainty of the spacecraft position. If the orientation of the lander or the array is not known precisely, the azimuth-averaged return can be used to calculate the expected return from a laser altimeter. We note that the FFDP measurements presented here were performed with a continuous-wave laser system. The time-varying interference effects will make precise shot-to-shot calculation of the OCS for an orbiting laser altimeter difficult. As is done for SLR measurements, multiple returns from the array can be used to build up a statistical description of the return rate and precise range.

F. Azimuthal-Averaged OCS

The azimuth-averaged LOOCS and variance as a function of elevation, generated from the maps above, is shown in Fig. 12 for the metal-coated and TIR version of LRALL at the two measurement wavelengths. Peak OCSs are consistent with the presented LOOCS results (to within 5%).

The layout of the CCRs within the array, with an inner ring of cubes at 20° off nadir (elevation: 70°) and an outer ring 40° off nadir, is apparent from the azimuthal averages. Similar features from the LOOCS maps are also resolvable, including the high-elevation “dead zone” in the TIR version, as well as multi-cube interference leading to higher variance but also higher observed optical return.

Due to the use of only the TIR version of LRALL on the surface, further discussion will be focused on only those instruments. The empirical azimuth-averaged OCS for the TIR version of LRALL can be described at elevation angles ($el.$) from 10° to 90° with a two-Gaussian model:

$$\!{\rm OCS}({\lambda ,el.} ) = {B_{i,\lambda}}\cdot{e^{- \frac{{{{({el. - {C_{i,\lambda}}} )}^2}}}{{2\sigma _{i,\lambda}^2}}}} + {B_{o,\lambda}}\cdot {e^{- \frac{{{{({el. - {C_{o,\lambda}}} )}^2}}}{{2\sigma _{o,\lambda}^2}}}},\!$$
where ${B_{i,\lambda}}$ is the wavelength-dependent maximum OCS at the inner-ring elevation, ${C_{i,\lambda}}$ is the elevation of the inner-ring feature, and ${\sigma _{i,\lambda}}$ is the width of the inner-ring feature. Similarly, ${B_{o,\lambda}}$ is the maximum OCS at the outer-ring elevation, ${C_{o,\lambda}}$ is the elevation of the outer-ring feature, and ${\sigma _{o,\lambda}}$ is the width of the outer-ring feature. Table 3 lists the best-fit parameters for the two-Gaussian model given in the equation above computed by running a Nelder–Mead algorithm [24], with initial values extracted manually from the data. Figure 13 shows the measurement results and two-Gaussian model fit as well as the residuals for both wavelengths.
Tables Icon

Table 3. Fitting Parameters for Two-Gaussian Model from Azimuthal-Averaged Data from TIR Retroreflector Array

 figure: Fig. 13.

Fig. 13. Best-fit functions of the two-Gaussian model. (a) Measurements and model fit to the azimuthal-averaged LOOCS as a function of elevation angle for the TIR version of LRALL. (b) Residuals from the fitting process.

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The OCSs at a ranging elevation of 70° are ${0.108}\;{\rm million}\;{{\rm m}^2}$ and ${0.0382}\;{\rm million}\;{{\rm m}^2}$ at 532 and 1064 nm, respectively. This high optical gain means LRALL will return ${\sim}{1000}$ times more photons per shot than measurements taken from the lunar surface by LOLA from a 50 km altitude. This allows LRALL to be ranged to from ranges of hundreds of kilometers by LOLA or future laser altimeters.

G. Suitability of LRALL for Short-Wave Infrared and Mid-Infrared Laser Altimeters

Looking toward the future, LRALL can be ranged to by a wide array of laser systems in visible and infrared wavelengths. We measured the total hemispherical reflectance spectrum of one of the CCRs via spectrometer, the results of which are shown in Fig. 14. The reflectance measurements were obtained at ${2}\;{{\rm cm}^{- 1}}$ resolution. Hydroxyl impurities in the fused silica glass cause absorption features near 1.4, 2.2, and 2.7 µm, but optical losses notwithstanding, the CCR can be used with interrogating lasers from the visible through 3.5 µm.

 figure: Fig. 14.

Fig. 14. Reflectance spectrum of LRALL CCR. Reflectance is shown normalized to the measured value at 1064 nm.

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6. LRALL RETURN PULSE WAVEFORMS AND TIME DELAY

Precision ranging to LRALL via orbital laser altimeter requires knowledge of the effects of the array on the laser pulse shape [13]. We have designed and performed tests to measure the pulsed laser return from the LRA to check for pulse spreading or distortion by the array as well as to test for range changes as a function of incident angle that could limit the range measurement resolution. A collimation setup similar to the one in Fig. 4(a) was constructed, although the target return collimation system used a refractive rather than reflective collimation scheme. We used a 1064 nm fiber-coupled, pulsed laser source (PicoQuant GmbH). The measured pulse width was 48 ps (Gaussian rms width) from the reference mirror, including the effects of the photodiode (Thorlabs, 8 GHz bandwidth) and oscilloscope (13 GHz electrical bandwidth).

We captured the return waveforms at three laser incidence angles: first the array was tilted such that the beam was at normal incidence to Cube 2 (see Figs. 2 and 15 insets). Second, the array was tilted such that the beam was at normal incidence to the adjacent cube (Cube 3). Finally, the array was illuminated at an angle between Cubes 2 and 3 (see Fig. 15 inset). The results of the experiments are shown in Fig. 15.

 figure: Fig. 15.

Fig. 15. Pulsed laser waveforms retroreflected from LRALL. The colored circles correspond to the waveforms measured at the three retroreflector array angles as well as a reference mirror waveform. The black lines correspond to Gaussian fits to the measured waveforms, with the Gaussian rms width listed above each waveform. The given time is relative to the start of the Cube 2 normal pulse, and the waveforms are offset vertically for clarity. The gray lines denote the centroid positions of the Cube 2 and Cube 3 Gaussian fits. (inset) A schematic of the array showing the array region that was closest to normal incidence for the three measurements.

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The pulse shape was unchanged by the array as measured on the system described here, with a Gaussian rms width of 46–49 ps compared to 48 ps when using a reference mirror. The delay time between Cubes 2 and 3 as measured from the Gaussian centroid positions was 39 ps, which corresponds to 6 mm in range offset. This range offset between Cubes 2 and 3 is larger than expected based on the LRA design (${\sim}{1}\;{\rm mm}$). We note, however, that the measured value is not strictly the variance in the optical reference point between cubes but is also affected by the offset of the rotation axis with respect to the array (i.e., the overall path length change in the laboratory frame). This result does set an upper bound on the range difference between these cubes and indicates that ranging to a single-centimeter level should be possible with LRALL.

7. SUMMARY

Here we have reported the optical performance of LRALL as determined prior to spacecraft integration. The CCRs each have a surface flatness of less than ${1/10}\;\lambda$ at 532 nm and have DAEs of less than 0.5 arc sec. The DAEs did not vary considerably over the tested temperature range of 100–380 K for a TIR CCR, indicating that significant thermal gradients that would impact performance are not expected under lunar conditions. The individual CCRs also demonstrated the expected far-field behavior, although with a reduced peak OCS for the TIR CCRs. Tests of the full array showed the highest OCS at 70° elevation as well as a local maximum at 43° elevation corresponding to the two rings of CCRs. Interference effects between the cubes played a significant role in affecting the optical return, increasing the variance in the return. The results show that LRALL can be ranged to from angles above 30° elevation from an orbiting laser altimeter operating at wavelengths from the visible to ${\sim}{3.5}\;{\unicode{x00B5}{\rm m}}$. Finally, the arrays did not measurably affect the return pulse waveform, and inter-cube range offsets showed an upper bound of 6 mm.

The metal-coated version of LRALL underwent space qualification and optical testing at Goddard Space Flight Center in 2018 and was integrated with the SpaceIL Beresheet lander in November 2018. Uncoated, TIR versions of LRALL were also space qualified at Goddard Space Flight Center, one of which was integrated in April 2019 with the Indian Space Research Organization (ISRO) Vikram lander, a part of the Chandrayaan-2 mission.

Additional LRALL units are expected to be integrated with lunar lander missions with the purpose of establishing a network of permanent fiducial markers on the lunar surface as well as enabling precision geolocation of the lander (and its instrument suite) from orbit. Though designed for use on the lunar surface, the LRALL concept can be readily applied to other planetary bodies (e.g., icy moons, asteroids, cometary cores, rocky planets, etc.) and other types of spacecraft (e.g., rovers, rotorcraft, hoppers, boats, etc.) and can be used in support of human exploration on the Moon and Mars.

Funding

National Aeronautics and Space Administration [Commercial Lunar Payload Services (CLPS), NASA Postdoctoral Program].

Acknowledgment

The authors would like to thank Pete Dogoda for his assistance in the setup of the FFDP measurement system and Scott Wetzel and Paul Christopoulos of KBRwyle Technology Solutions, LLC, for the fabrication and test of the LRAs.

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994). [CrossRef]  

2. J. G. Williams, S. G. Turyshev, D. H. Boggs, and J. T. Ratcliff, “Lunar laser ranging science: Gravitational physics and lunar interior and geodesy,” Adv. Space Res. 37, 67–71 (2006). [CrossRef]  

3. A. Khan, K. Mosegaard, J. G. Williams, and P. Lognonné, “Does the Moon possess a molten core? Probing the deep lunar interior using results from LLR and Lunar Prospector,” J. Geophys. Res. Planets 109, E09007 (2004). [CrossRef]  

4. I. Matsuyama, F. Nimmo, J. T. Keane, N. H. Chan, G. J. Taylor, M. A. Wieczorek, W. S. Kiefer, and J. G. Williams, “GRAIL, LLR, and LOLA constraints on the interior structure of the Moon,” Geophys. Res. Lett. 43, 8365–8375 (2016). [CrossRef]  

5. J. G. Williams and D. H. Boggs, “Tides on the Moon: Theory and determination of dissipation,” J. Geophys. Res. Planets 120, 689–724 (2015). [CrossRef]  

6. E. R. Jawin, S. N. Valencia, R. N. Watkins, J. M. Crowell, C. R. Neal, and G. Schmidt, “Lunar science for Landed Missions Workshop findings report,” Earth Space Sci. 6, 2–40 (2018). [CrossRef]  

7. L. A. Magruder and K. M. Brunt, “Performance analysis of airborne photon-counting lidar data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 56, 2911–2918 (2018). [CrossRef]  

8. T. Yoshimitsu, J. Kawaguchi, T. Hashimoto, T. Kubota, M. Uo, H. Morita, and K. Shirakawa, “Hayabusa—Final autonomous descent and landing based on target marker tracking,” Acta Astronaut. 65, 657–665 (2009). [CrossRef]  

9. T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017). [CrossRef]  

10. S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017). [CrossRef]  

11. X. Sun, D. E. Smith, E. D. Hoffman, S. W. Wake, D. R. Cremons, E. Mazarico, J.-M. Lauenstein, M. T. Zuber, and E. C. Aaron, “Small and lightweight laser retro-reflector arrays for lunar landers,” Appl. Opt. 58, 9259–9266 (2019). [CrossRef]  

12. J. J. Degnan, “Millimeter accuracy satellite laser ranging: A review,” in Contributions of Space Geodesy to Geodynamics: Technology (American Geophysical Union, 1993), pp. 133–162.

13. P. O. Minott, T. W. Zagwodzki, T. Varghese, and M. Seldon, Prelaunch Optical Characterization of the Laser Geodynamic Satellite (LAGEOS 2) (NASA, 1993).

14. T. W. Murphy, 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, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010). [CrossRef]  

15. T. W. Murphy, “Lunar laser ranging: The millimeter challenge,” Rep. Prog. Phys. 76, 076901 (2013). [CrossRef]  

16. A. R. Vasavada, J. L. Bandfield, B. T. Greenhagen, P. O. Hayne, M. A. Siegler, J.-P. Williams, and D. A. Paige, “Lunar equatorial surface temperatures and regolith properties from the Diviner Lunar Radiometer Experiment,” J. Geophys. Res. Planets 117, E00H18 (2012). [CrossRef]  

17. 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, “Corner-cube retro-reflector instrument for advanced lunar laser ranging,” Exp. Astron. 36, 105–135 (2013). [CrossRef]  

18. S. D. Goodrow and T. W. Murphy, “Effects of thermal gradients on total internal reflection corner cubes,” Appl. Opt. 51, 8793–8799 (2012). [CrossRef]  

19. S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011). [CrossRef]  

20. P. O. Minott, Design of Retrodirector Arrays for Laser Ranging of Satellites (NASA, 1974).

21. T. W. Murphy and S. D. Goodrow, “Polarization and far-field diffraction patterns of total internal reflection corner cubes,” Appl. Opt. 52, 117–126 (2013). [CrossRef]  

22. J. Liu and R. M. A. Azzam, “Polarization properties of corner-cube retroreflectors: Theory and experiment,” Appl. Opt. 36, 1553–1559 (1997). [CrossRef]  

23. M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetallized faces,” Opt. Spectrosc. 107, 201–206 (2009). [CrossRef]  

24. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998). [CrossRef]  

References

  • View by:

  1. J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
    [Crossref]
  2. J. G. Williams, S. G. Turyshev, D. H. Boggs, and J. T. Ratcliff, “Lunar laser ranging science: Gravitational physics and lunar interior and geodesy,” Adv. Space Res. 37, 67–71 (2006).
    [Crossref]
  3. A. Khan, K. Mosegaard, J. G. Williams, and P. Lognonné, “Does the Moon possess a molten core? Probing the deep lunar interior using results from LLR and Lunar Prospector,” J. Geophys. Res. Planets 109, E09007 (2004).
    [Crossref]
  4. I. Matsuyama, F. Nimmo, J. T. Keane, N. H. Chan, G. J. Taylor, M. A. Wieczorek, W. S. Kiefer, and J. G. Williams, “GRAIL, LLR, and LOLA constraints on the interior structure of the Moon,” Geophys. Res. Lett. 43, 8365–8375 (2016).
    [Crossref]
  5. J. G. Williams and D. H. Boggs, “Tides on the Moon: Theory and determination of dissipation,” J. Geophys. Res. Planets 120, 689–724 (2015).
    [Crossref]
  6. E. R. Jawin, S. N. Valencia, R. N. Watkins, J. M. Crowell, C. R. Neal, and G. Schmidt, “Lunar science for Landed Missions Workshop findings report,” Earth Space Sci. 6, 2–40 (2018).
    [Crossref]
  7. L. A. Magruder and K. M. Brunt, “Performance analysis of airborne photon-counting lidar data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 56, 2911–2918 (2018).
    [Crossref]
  8. T. Yoshimitsu, J. Kawaguchi, T. Hashimoto, T. Kubota, M. Uo, H. Morita, and K. Shirakawa, “Hayabusa—Final autonomous descent and landing based on target marker tracking,” Acta Astronaut. 65, 657–665 (2009).
    [Crossref]
  9. T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
    [Crossref]
  10. S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
    [Crossref]
  11. X. Sun, D. E. Smith, E. D. Hoffman, S. W. Wake, D. R. Cremons, E. Mazarico, J.-M. Lauenstein, M. T. Zuber, and E. C. Aaron, “Small and lightweight laser retro-reflector arrays for lunar landers,” Appl. Opt. 58, 9259–9266 (2019).
    [Crossref]
  12. J. J. Degnan, “Millimeter accuracy satellite laser ranging: A review,” in Contributions of Space Geodesy to Geodynamics: Technology (American Geophysical Union, 1993), pp. 133–162.
  13. P. O. Minott, T. W. Zagwodzki, T. Varghese, and M. Seldon, Prelaunch Optical Characterization of the Laser Geodynamic Satellite (LAGEOS 2) (NASA, 1993).
  14. T. W. Murphy, 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, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
    [Crossref]
  15. T. W. Murphy, “Lunar laser ranging: The millimeter challenge,” Rep. Prog. Phys. 76, 076901 (2013).
    [Crossref]
  16. A. R. Vasavada, J. L. Bandfield, B. T. Greenhagen, P. O. Hayne, M. A. Siegler, J.-P. Williams, and D. A. Paige, “Lunar equatorial surface temperatures and regolith properties from the Diviner Lunar Radiometer Experiment,” J. Geophys. Res. Planets 117, E00H18 (2012).
    [Crossref]
  17. 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, “Corner-cube retro-reflector instrument for advanced lunar laser ranging,” Exp. Astron. 36, 105–135 (2013).
    [Crossref]
  18. S. D. Goodrow and T. W. Murphy, “Effects of thermal gradients on total internal reflection corner cubes,” Appl. Opt. 51, 8793–8799 (2012).
    [Crossref]
  19. S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
    [Crossref]
  20. P. O. Minott, Design of Retrodirector Arrays for Laser Ranging of Satellites (NASA, 1974).
  21. T. W. Murphy and S. D. Goodrow, “Polarization and far-field diffraction patterns of total internal reflection corner cubes,” Appl. Opt. 52, 117–126 (2013).
    [Crossref]
  22. J. Liu and R. M. A. Azzam, “Polarization properties of corner-cube retroreflectors: Theory and experiment,” Appl. Opt. 36, 1553–1559 (1997).
    [Crossref]
  23. M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetallized faces,” Opt. Spectrosc. 107, 201–206 (2009).
    [Crossref]
  24. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
    [Crossref]

2019 (1)

2018 (2)

E. R. Jawin, S. N. Valencia, R. N. Watkins, J. M. Crowell, C. R. Neal, and G. Schmidt, “Lunar science for Landed Missions Workshop findings report,” Earth Space Sci. 6, 2–40 (2018).
[Crossref]

L. A. Magruder and K. M. Brunt, “Performance analysis of airborne photon-counting lidar data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 56, 2911–2918 (2018).
[Crossref]

2017 (2)

T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
[Crossref]

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
[Crossref]

2016 (1)

I. Matsuyama, F. Nimmo, J. T. Keane, N. H. Chan, G. J. Taylor, M. A. Wieczorek, W. S. Kiefer, and J. G. Williams, “GRAIL, LLR, and LOLA constraints on the interior structure of the Moon,” Geophys. Res. Lett. 43, 8365–8375 (2016).
[Crossref]

2015 (1)

J. G. Williams and D. H. Boggs, “Tides on the Moon: Theory and determination of dissipation,” J. Geophys. Res. Planets 120, 689–724 (2015).
[Crossref]

2013 (3)

T. W. Murphy, “Lunar laser ranging: The millimeter challenge,” Rep. Prog. Phys. 76, 076901 (2013).
[Crossref]

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, “Corner-cube retro-reflector instrument for advanced lunar laser ranging,” Exp. Astron. 36, 105–135 (2013).
[Crossref]

T. W. Murphy and S. D. Goodrow, “Polarization and far-field diffraction patterns of total internal reflection corner cubes,” Appl. Opt. 52, 117–126 (2013).
[Crossref]

2012 (2)

S. D. Goodrow and T. W. Murphy, “Effects of thermal gradients on total internal reflection corner cubes,” Appl. Opt. 51, 8793–8799 (2012).
[Crossref]

A. R. Vasavada, J. L. Bandfield, B. T. Greenhagen, P. O. Hayne, M. A. Siegler, J.-P. Williams, and D. A. Paige, “Lunar equatorial surface temperatures and regolith properties from the Diviner Lunar Radiometer Experiment,” J. Geophys. Res. Planets 117, E00H18 (2012).
[Crossref]

2011 (1)

S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
[Crossref]

2010 (1)

T. W. Murphy, 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, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[Crossref]

2009 (2)

T. Yoshimitsu, J. Kawaguchi, T. Hashimoto, T. Kubota, M. Uo, H. Morita, and K. Shirakawa, “Hayabusa—Final autonomous descent and landing based on target marker tracking,” Acta Astronaut. 65, 657–665 (2009).
[Crossref]

M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetallized faces,” Opt. Spectrosc. 107, 201–206 (2009).
[Crossref]

2006 (1)

J. G. Williams, S. G. Turyshev, D. H. Boggs, and J. T. Ratcliff, “Lunar laser ranging science: Gravitational physics and lunar interior and geodesy,” Adv. Space Res. 37, 67–71 (2006).
[Crossref]

2004 (1)

A. Khan, K. Mosegaard, J. G. Williams, and P. Lognonné, “Does the Moon possess a molten core? Probing the deep lunar interior using results from LLR and Lunar Prospector,” J. Geophys. Res. Planets 109, E09007 (2004).
[Crossref]

1998 (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

1997 (1)

1994 (1)

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[Crossref]

Aaron, E. C.

Adelberger, E. G.

T. W. Murphy, 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, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[Crossref]

Alley, C. O.

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S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
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S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
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Cantone, C.

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
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S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
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S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
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S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
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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, “Corner-cube retro-reflector instrument for advanced lunar laser ranging,” Exp. Astron. 36, 105–135 (2013).
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A. R. Vasavada, J. L. Bandfield, B. T. Greenhagen, P. O. Hayne, M. A. Siegler, J.-P. Williams, and D. A. Paige, “Lunar equatorial surface temperatures and regolith properties from the Diviner Lunar Radiometer Experiment,” J. Geophys. Res. Planets 117, E00H18 (2012).
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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, “Corner-cube retro-reflector instrument for advanced lunar laser ranging,” Exp. Astron. 36, 105–135 (2013).
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S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
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I. Matsuyama, F. Nimmo, J. T. Keane, N. H. Chan, G. J. Taylor, M. A. Wieczorek, W. S. Kiefer, and J. G. Williams, “GRAIL, LLR, and LOLA constraints on the interior structure of the Moon,” Geophys. Res. Lett. 43, 8365–8375 (2016).
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T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
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Lognonné, P.

A. Khan, K. Mosegaard, J. G. Williams, and P. Lognonné, “Does the Moon possess a molten core? Probing the deep lunar interior using results from LLR and Lunar Prospector,” J. Geophys. Res. Planets 109, E09007 (2004).
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S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
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S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
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S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
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L. A. Magruder and K. M. Brunt, “Performance analysis of airborne photon-counting lidar data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 56, 2911–2918 (2018).
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Maiello, M.

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
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S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
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Marchetti, E.

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
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S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
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S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
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I. Matsuyama, F. Nimmo, J. T. Keane, N. H. Chan, G. J. Taylor, M. A. Wieczorek, W. S. Kiefer, and J. G. Williams, “GRAIL, LLR, and LOLA constraints on the interior structure of the Moon,” Geophys. Res. Lett. 43, 8365–8375 (2016).
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McGarry, J. F.

S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
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T. W. Murphy, 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, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
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T. W. Murphy, 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, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
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T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
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P. O. Minott, T. W. Zagwodzki, T. Varghese, and M. Seldon, Prelaunch Optical Characterization of the Laser Geodynamic Satellite (LAGEOS 2) (NASA, 1993).

P. O. Minott, Design of Retrodirector Arrays for Laser Ranging of Satellites (NASA, 1974).

Mita, M.

T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
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T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
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Mondaini, C.

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
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Morita, H.

T. Yoshimitsu, J. Kawaguchi, T. Hashimoto, T. Kubota, M. Uo, H. Morita, and K. Shirakawa, “Hayabusa—Final autonomous descent and landing based on target marker tracking,” Acta Astronaut. 65, 657–665 (2009).
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Mosegaard, K.

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Mugnuolo, R.

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
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T. W. Murphy and S. D. Goodrow, “Polarization and far-field diffraction patterns of total internal reflection corner cubes,” Appl. Opt. 52, 117–126 (2013).
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T. W. Murphy, “Lunar laser ranging: The millimeter challenge,” Rep. Prog. Phys. 76, 076901 (2013).
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S. D. Goodrow and T. W. Murphy, “Effects of thermal gradients on total internal reflection corner cubes,” Appl. Opt. 51, 8793–8799 (2012).
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T. W. Murphy, 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, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
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Namiki, N.

T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
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Neal, C. R.

E. R. Jawin, S. N. Valencia, R. N. Watkins, J. M. Crowell, C. R. Neal, and G. Schmidt, “Lunar science for Landed Missions Workshop findings report,” Earth Space Sci. 6, 2–40 (2018).
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Newhall, X. X.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
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Nimmo, F.

I. Matsuyama, F. Nimmo, J. T. Keane, N. H. Chan, G. J. Taylor, M. A. Wieczorek, W. S. Kiefer, and J. G. Williams, “GRAIL, LLR, and LOLA constraints on the interior structure of the Moon,” Geophys. Res. Lett. 43, 8365–8375 (2016).
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Noda, H.

T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
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Paige, D. A.

A. R. Vasavada, J. L. Bandfield, B. T. Greenhagen, P. O. Hayne, M. A. Siegler, J.-P. Williams, and D. A. Paige, “Lunar equatorial surface temperatures and regolith properties from the Diviner Lunar Radiometer Experiment,” J. Geophys. Res. Planets 117, E00H18 (2012).
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Patrizi, G.

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
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Pearlman, M. R.

S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
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Porcelli, L.

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
[Crossref]

S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
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Ratcliff, J. T.

J. G. Williams, S. G. Turyshev, D. H. Boggs, and J. T. Ratcliff, “Lunar laser ranging science: Gravitational physics and lunar interior and geodesy,” Adv. Space Res. 37, 67–71 (2006).
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J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
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J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
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Ries, J. G.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
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M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetallized faces,” Opt. Spectrosc. 107, 201–206 (2009).
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Salvatori, L.

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
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Samad, R. L.

T. W. Murphy, 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, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
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Schmidt, G.

E. R. Jawin, S. N. Valencia, R. N. Watkins, J. M. Crowell, C. R. Neal, and G. Schmidt, “Lunar science for Landed Missions Workshop findings report,” Earth Space Sci. 6, 2–40 (2018).
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Sciarretta, C.

S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
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Seldon, M.

P. O. Minott, T. W. Zagwodzki, T. Varghese, and M. Seldon, Prelaunch Optical Characterization of the Laser Geodynamic Satellite (LAGEOS 2) (NASA, 1993).

Senshu, H.

T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
[Crossref]

Shelus, P. J.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
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Shiina, T.

T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
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Shirakawa, K.

T. Yoshimitsu, J. Kawaguchi, T. Hashimoto, T. Kubota, M. Uo, H. Morita, and K. Shirakawa, “Hayabusa—Final autonomous descent and landing based on target marker tracking,” Acta Astronaut. 65, 657–665 (2009).
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Siegler, M. A.

A. R. Vasavada, J. L. Bandfield, B. T. Greenhagen, P. O. Hayne, M. A. Siegler, J.-P. Williams, and D. A. Paige, “Lunar equatorial surface temperatures and regolith properties from the Diviner Lunar Radiometer Experiment,” J. Geophys. Res. Planets 117, E00H18 (2012).
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Smith, D. E.

Sokolov, A. L.

M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetallized faces,” Opt. Spectrosc. 107, 201–206 (2009).
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Somawardhana, R. P.

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, “Corner-cube retro-reflector instrument for advanced lunar laser ranging,” Exp. Astron. 36, 105–135 (2013).
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Stubbs, C. W.

T. W. Murphy, 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, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[Crossref]

Sun, X.

Swanson, H. E.

T. W. Murphy, 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, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[Crossref]

Tauraso, R.

S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
[Crossref]

Taylor, G. J.

I. Matsuyama, F. Nimmo, J. T. Keane, N. H. Chan, G. J. Taylor, M. A. Wieczorek, W. S. Kiefer, and J. G. Williams, “GRAIL, LLR, and LOLA constraints on the interior structure of the Moon,” Geophys. Res. Lett. 43, 8365–8375 (2016).
[Crossref]

Terui, F.

T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
[Crossref]

Tibuzzi, M.

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
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Turyshev, S. G.

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, “Corner-cube retro-reflector instrument for advanced lunar laser ranging,” Exp. Astron. 36, 105–135 (2013).
[Crossref]

J. G. Williams, S. G. Turyshev, D. H. Boggs, and J. T. Ratcliff, “Lunar laser ranging science: Gravitational physics and lunar interior and geodesy,” Adv. Space Res. 37, 67–71 (2006).
[Crossref]

Tuscano, P.

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
[Crossref]

Uo, M.

T. Yoshimitsu, J. Kawaguchi, T. Hashimoto, T. Kubota, M. Uo, H. Morita, and K. Shirakawa, “Hayabusa—Final autonomous descent and landing based on target marker tracking,” Acta Astronaut. 65, 657–665 (2009).
[Crossref]

Valencia, S. N.

E. R. Jawin, S. N. Valencia, R. N. Watkins, J. M. Crowell, C. R. Neal, and G. Schmidt, “Lunar science for Landed Missions Workshop findings report,” Earth Space Sci. 6, 2–40 (2018).
[Crossref]

Varghese, T.

P. O. Minott, T. W. Zagwodzki, T. Varghese, and M. Seldon, Prelaunch Optical Characterization of the Laser Geodynamic Satellite (LAGEOS 2) (NASA, 1993).

Vasavada, A. R.

A. R. Vasavada, J. L. Bandfield, B. T. Greenhagen, P. O. Hayne, M. A. Siegler, J.-P. Williams, and D. A. Paige, “Lunar equatorial surface temperatures and regolith properties from the Diviner Lunar Radiometer Experiment,” J. Geophys. Res. Planets 117, E00H18 (2012).
[Crossref]

Veillet, C.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[Crossref]

Vittori, R.

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
[Crossref]

S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
[Crossref]

Wake, S. W.

Wang, S.

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, “Corner-cube retro-reflector instrument for advanced lunar laser ranging,” Exp. Astron. 36, 105–135 (2013).
[Crossref]

Watkins, R. N.

E. R. Jawin, S. N. Valencia, R. N. Watkins, J. M. Crowell, C. R. Neal, and G. Schmidt, “Lunar science for Landed Missions Workshop findings report,” Earth Space Sci. 6, 2–40 (2018).
[Crossref]

Whipple, A. L.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[Crossref]

Wiant, J. R.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[Crossref]

Wieczorek, M. A.

I. Matsuyama, F. Nimmo, J. T. Keane, N. H. Chan, G. J. Taylor, M. A. Wieczorek, W. S. Kiefer, and J. G. Williams, “GRAIL, LLR, and LOLA constraints on the interior structure of the Moon,” Geophys. Res. Lett. 43, 8365–8375 (2016).
[Crossref]

Williams, J. G.

I. Matsuyama, F. Nimmo, J. T. Keane, N. H. Chan, G. J. Taylor, M. A. Wieczorek, W. S. Kiefer, and J. G. Williams, “GRAIL, LLR, and LOLA constraints on the interior structure of the Moon,” Geophys. Res. Lett. 43, 8365–8375 (2016).
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J. G. Williams and D. H. Boggs, “Tides on the Moon: Theory and determination of dissipation,” J. Geophys. Res. Planets 120, 689–724 (2015).
[Crossref]

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, “Corner-cube retro-reflector instrument for advanced lunar laser ranging,” Exp. Astron. 36, 105–135 (2013).
[Crossref]

J. G. Williams, S. G. Turyshev, D. H. Boggs, and J. T. Ratcliff, “Lunar laser ranging science: Gravitational physics and lunar interior and geodesy,” Adv. Space Res. 37, 67–71 (2006).
[Crossref]

A. Khan, K. Mosegaard, J. G. Williams, and P. Lognonné, “Does the Moon possess a molten core? Probing the deep lunar interior using results from LLR and Lunar Prospector,” J. Geophys. Res. Planets 109, E09007 (2004).
[Crossref]

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[Crossref]

Williams, J.-P.

A. R. Vasavada, J. L. Bandfield, B. T. Greenhagen, P. O. Hayne, M. A. Siegler, J.-P. Williams, and D. A. Paige, “Lunar equatorial surface temperatures and regolith properties from the Diviner Lunar Radiometer Experiment,” J. Geophys. Res. Planets 117, E00H18 (2012).
[Crossref]

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

Yamada, R.

T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
[Crossref]

Yoder, C. F.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[Crossref]

Yoshimitsu, T.

T. Yoshimitsu, J. Kawaguchi, T. Hashimoto, T. Kubota, M. Uo, H. Morita, and K. Shirakawa, “Hayabusa—Final autonomous descent and landing based on target marker tracking,” Acta Astronaut. 65, 657–665 (2009).
[Crossref]

Zagwodzki, T. W.

S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
[Crossref]

P. O. Minott, T. W. Zagwodzki, T. Varghese, and M. Seldon, Prelaunch Optical Characterization of the Laser Geodynamic Satellite (LAGEOS 2) (NASA, 1993).

Zerbini, S.

S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
[Crossref]

Zuber, M. T.

Acta Astronaut. (1)

T. Yoshimitsu, J. Kawaguchi, T. Hashimoto, T. Kubota, M. Uo, H. Morita, and K. Shirakawa, “Hayabusa—Final autonomous descent and landing based on target marker tracking,” Acta Astronaut. 65, 657–665 (2009).
[Crossref]

Adv. Space Res. (3)

J. G. Williams, S. G. Turyshev, D. H. Boggs, and J. T. Ratcliff, “Lunar laser ranging science: Gravitational physics and lunar interior and geodesy,” Adv. Space Res. 37, 67–71 (2006).
[Crossref]

S. Dell’Agnello, G. Delle Monache, L. Porcelli, A. Boni, S. Contessa, E. Ciocci, M. Martini, M. Tibuzzi, N. Intaglietta, L. Salvatori, P. Tuscano, G. Patrizi, C. Mondaini, C. Lops, R. Vittori, M. Maiello, E. Flamini, E. Marchetti, G. Bianco, R. Mugnuolo, and C. Cantone, “INRRI-EDM/2016: The first laser retroreflector on the surface of Mars,” Adv. Space Res. 59, 645–655 (2017).
[Crossref]

S. Dell’Agnello, G. O. Delle Monache, D. G. Currie, R. Vittori, C. Cantone, M. Garattini, A. Boni, M. Martini, C. Lops, N. Intaglietta, R. Tauraso, D. A. Arnold, M. R. Pearlman, G. Bianco, S. Zerbini, M. Maiello, S. Berardi, L. Porcelli, C. O. Alley, J. F. McGarry, C. Sciarretta, V. Luceri, and T. W. Zagwodzki, “Creation of the new industry-standard space test of laser retroreflectors for the GNSS and LAGEOS,” Adv. Space Res. 47, 822–842 (2011).
[Crossref]

Appl. Opt. (4)

Earth Space Sci. (1)

E. R. Jawin, S. N. Valencia, R. N. Watkins, J. M. Crowell, C. R. Neal, and G. Schmidt, “Lunar science for Landed Missions Workshop findings report,” Earth Space Sci. 6, 2–40 (2018).
[Crossref]

Exp. Astron. (1)

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, “Corner-cube retro-reflector instrument for advanced lunar laser ranging,” Exp. Astron. 36, 105–135 (2013).
[Crossref]

Geophys. Res. Lett. (1)

I. Matsuyama, F. Nimmo, J. T. Keane, N. H. Chan, G. J. Taylor, M. A. Wieczorek, W. S. Kiefer, and J. G. Williams, “GRAIL, LLR, and LOLA constraints on the interior structure of the Moon,” Geophys. Res. Lett. 43, 8365–8375 (2016).
[Crossref]

Icarus (1)

T. W. Murphy, 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, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (1)

L. A. Magruder and K. M. Brunt, “Performance analysis of airborne photon-counting lidar data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 56, 2911–2918 (2018).
[Crossref]

J. Geophys. Res. Planets (3)

J. G. Williams and D. H. Boggs, “Tides on the Moon: Theory and determination of dissipation,” J. Geophys. Res. Planets 120, 689–724 (2015).
[Crossref]

A. Khan, K. Mosegaard, J. G. Williams, and P. Lognonné, “Does the Moon possess a molten core? Probing the deep lunar interior using results from LLR and Lunar Prospector,” J. Geophys. Res. Planets 109, E09007 (2004).
[Crossref]

A. R. Vasavada, J. L. Bandfield, B. T. Greenhagen, P. O. Hayne, M. A. Siegler, J.-P. Williams, and D. A. Paige, “Lunar equatorial surface temperatures and regolith properties from the Diviner Lunar Radiometer Experiment,” J. Geophys. Res. Planets 117, E00H18 (2012).
[Crossref]

Opt. Spectrosc. (1)

M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetallized faces,” Opt. Spectrosc. 107, 201–206 (2009).
[Crossref]

Rep. Prog. Phys. (1)

T. W. Murphy, “Lunar laser ranging: The millimeter challenge,” Rep. Prog. Phys. 76, 076901 (2013).
[Crossref]

Science (1)

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, and C. F. Yoder, “Lunar laser ranging: A continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[Crossref]

SIAM J. Optim. (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

Space Sci. Rev. (1)

T. Mizuno, T. Kase, T. Shiina, M. Mita, N. Namiki, H. Senshu, R. Yamada, H. Noda, H. Kunimori, N. Hirata, F. Terui, and Y. Mimasu, “Development of the laser altimeter (LIDAR) for Hayabusa2,” Space Sci. Rev. 208, 33–47 (2017).
[Crossref]

Other (3)

J. J. Degnan, “Millimeter accuracy satellite laser ranging: A review,” in Contributions of Space Geodesy to Geodynamics: Technology (American Geophysical Union, 1993), pp. 133–162.

P. O. Minott, T. W. Zagwodzki, T. Varghese, and M. Seldon, Prelaunch Optical Characterization of the Laser Geodynamic Satellite (LAGEOS 2) (NASA, 1993).

P. O. Minott, Design of Retrodirector Arrays for Laser Ranging of Satellites (NASA, 1974).

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Figures (15)

Fig. 1.
Fig. 1. LRALL. (a) Array with eight metal-coated CCRs. (b) Array with eight TIR CCRs. (c) Illustration showing LRALL geometry and optical reference point. CCR illustration is not to scale. CG, center of gravity.
Fig. 2.
Fig. 2. DAE measurements of CCRs for a representative LRALL instrument. Angles 1, 2, and 3, corresponding to the intersections of the three prism facets, were designated prior to test and held consistent throughout testing.
Fig. 3.
Fig. 3. Measured retroreflecting properties of a TIR CCR over the expected lunar surface temperature range. Angles 1, 2, and 3, corresponding to the intersections of the three facets, were designated prior to test and held consistent throughout testing.
Fig. 4.
Fig. 4. FFDP optical test setup.
Fig. 5.
Fig. 5. FFDPs of individual CCRs. The patterns from (a) and (b) the metal-coated and (c) and (d) TIR CCRs were obtained under uniform illumination with collimated (a) and (c) 532 nm and (b) and (d) 1064 nm laser light. The axes correspond to beam deviation from the source as measured on the calibrated detector. The color scales correspond to OCS and are given by the scales to the right of each frame.
Fig. 6.
Fig. 6. Normalized peak OCS of metal-coated and TIR CCRs as a function of incident angle at 1064 nm. Normalized peak OCS is defined as the OCS returned to the light source normalized to the OCS at normal incidence for each type of CCR.
Fig. 7.
Fig. 7. Polarization dependence of far-field patterns from TIR CCRs. Far-field patterns were obtained from illumination at (a)–(c) 532 nm (outlined in green) and (d)–(f) 1064 nm (outlined in red). The frame axes correspond to beam deviation in µrad and are the same for all images. (insets) The polarization axes for the three polarizations tested (black arrows) with respect to the CCR. Edges of the facets of the CCR are shown in red. The color scale corresponds to OCS with warmer colors denoting higher OCS, and cooler colors denoting a lower OCS. The OCS scale for (a)–(c) is given by the colorbar next to (c) and the scale for (d)–(f) is given by the colorbar next to (f).
Fig. 8.
Fig. 8. LRA schematic describing angle terminology overlaid on a top-down view of the LRA.
Fig. 9.
Fig. 9. Montage of FFDPs from LRALL (TIR CCRs) in equirectangular projection. The color scale is shown in the upper left and corresponds to relative OCS, with warmer colors corresponding to higher optical return. FFDPs were all obtained under identical imaging and illumination conditions.
Fig. 10.
Fig. 10. Map of optical return from the retroreflector array as a function of incident angle for the LRA with metal-coated CCRs. (a) Map created from FFDPs using 532 nm illumination. (b) Map created from FFDPs using 1064 nm illumination. The color scales correspond to the LOOCS with warmer, lighter colors denoting higher optical return. Elevation and azimuth angles refer to terminology defined in Fig. 8. FFDPs were obtained in one-degree intervals in both elevation and azimuth for the ranges shown; each pixel of the map represents a distinct FFDP from which the LOOCS was extracted. The pixel-to-pixel variation is real and is a result of sampling a complex interference pattern.
Fig. 11.
Fig. 11. Map of optical return from retroreflector array as a function of incident angle for the TIR version of LRALL. Following the conventions of Fig. 10, (a) and (b) correspond to 532 and 1064 nm, respectively. The pixel-to-pixel variation is real and is a result of sampling a complex interference pattern.
Fig. 12.
Fig. 12. Azimuthally averaged optical performance of LRALL as a function of elevation angle. The dark blue and dark red lines denote the azimuthal average value for each elevation, whereas the light blue and light red shaded regions denote the $1 \sigma$ variance at each elevation from the population of all azimuth angles.
Fig. 13.
Fig. 13. Best-fit functions of the two-Gaussian model. (a) Measurements and model fit to the azimuthal-averaged LOOCS as a function of elevation angle for the TIR version of LRALL. (b) Residuals from the fitting process.
Fig. 14.
Fig. 14. Reflectance spectrum of LRALL CCR. Reflectance is shown normalized to the measured value at 1064 nm.
Fig. 15.
Fig. 15. Pulsed laser waveforms retroreflected from LRALL. The colored circles correspond to the waveforms measured at the three retroreflector array angles as well as a reference mirror waveform. The black lines correspond to Gaussian fits to the measured waveforms, with the Gaussian rms width listed above each waveform. The given time is relative to the start of the Cube 2 normal pulse, and the waveforms are offset vertically for clarity. The gray lines denote the centroid positions of the Cube 2 and Cube 3 Gaussian fits. (inset) A schematic of the array showing the array region that was closest to normal incidence for the three measurements.

Tables (3)

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Table 1. LRALL Instrument Parameters

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Table 2. Optical Test Results from FFDPs of Individual CCRsa

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Table 3. Fitting Parameters for Two-Gaussian Model from Azimuthal-Averaged Data from TIR Retroreflector Array

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

σ = ρ A 2 4 π λ 2 ,
η = [ 2 cos ( θ i ) π ] 2 { sin 1 [ 1 2 tan 2 ( sin 1 ( sin ( θ i ) n ) ) ] 2 tan [ sin 1 ( sin ( θ i ) n ) ] } 2 .
O C S ( λ , e l . ) = B i , λ e ( e l . C i , λ ) 2 2 σ i , λ 2 + B o , λ e ( e l . C o , λ ) 2 2 σ o , λ 2 ,

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