## 1.

## Introduction

Satellite laser retroreflector (LRR) is a solid tetrahedron composed of three mutually perpendicular reflecting surfaces and one front face, which is the indispensable component in satellite laser ranging to reflect the laser beams directly back to the ground station.^{1} For the purpose of preventing it from being damaged during handling, LRR is usually manufactured with mounting tabs to hold it conveniently into the protective housing.^{2}3.^{–}^{4} However, due to the protection manners, practical solid retroreflector designs require mounting structures that result in their reflecting surfaces being recessed with respect to their front window faces. With the structural recession effect, the active retroreflecting region (ARR) of LRR is diminished for the oblique incidence of laser beams. Considering the performance of LRR, including the received optical signal intensity (ROSI), is subject to the ARR, the analysis of structural recession effect on ROSI becomes more important.

In the past, the effect of structural recession on ARR has been expounded qualitatively.^{5} Evidently, investigating the relationship between structural recession size and the LRR performance quantitatively is not enough. In this paper, we apply a ray tracing method to calculate the ARR. The influence of structural recession on the mathematical model of ROSI is then described. Combined with the parameters of an HY-2A satellite LRR, the direct and indirect effects of structural recession on ROSI and range correction of LRR array are studied by computer simulation.

## 2.

## Active Retroreflecting Region Model

The ARR is an aperture on the front face of LRR, which is established by restricting the positions of a ray on the LRR surface.

The schematic of the LRR mounting assembly is illustrated in Fig. 1. The LRR is held by a ring assembly within a cylindrical shield. The upper and lower mounting rings fit over tabs to encircle the LRR. A retainer ring or encloser (RoE) on the top of upper mounting rings presses the LRR assembly against a structural frame. Taking into consideration that the reflecting faces of LRR lie on the bottom of the upper mounting rings, the LRR can be treated as structural recession. Figure 1 shows that the structural recession parameters include the distance ${d}_{2}$ from the top of the tab to the front face of LRR, and the distance ${d}_{1}$ from the front face of the LRR to the RoE.

The reflecting face coordinate system (RCS) ($x,y,z$) used to describe the geometry of the LRR is shown in Fig. 2. For a better indication of each direction of incident ray, we define the front face coordinate system (FCS) (${x}^{\prime},{y}^{\prime},{z}^{\prime}$), where the ${z}^{\prime}$ axis is the symmetry of the LRR, and ${y}^{\prime}$ axis is the projection of the $y$-axis upon the front face.

The coordinate system transformation relationship between RCS and FCS can be expressed by Ref. 6

## (1)

$$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=M\left[\begin{array}{c}{x}^{\prime}\\ {y}^{\prime}\\ {z}^{\prime}\end{array}\right]+N=\left(\begin{array}{ccc}\frac{\sqrt{2}}{2}& -\frac{\sqrt{6}}{6}& \frac{\sqrt{3}}{3}\\ 0& \frac{\sqrt{6}}{3}& \frac{\sqrt{3}}{3}\\ -\frac{\sqrt{2}}{2}& -\frac{\sqrt{6}}{6}& \frac{\sqrt{3}}{3}\end{array}\right)\left[\begin{array}{c}{x}^{\prime}\\ {y}^{\prime}\\ {z}^{\prime}\end{array}\right]+\left[\begin{array}{c}\frac{L+{d}_{1}+{d}_{2}}{\sqrt{3}}\\ \frac{L+{d}_{1}+{d}_{2}}{\sqrt{3}}\\ \frac{L+{d}_{1}+{d}_{2}}{\sqrt{3}}\end{array}\right],$$If the incident ray with the incident angle $\phi $ and the azimuth angle $\theta $ is incident on the LRR, the direction cosine ${R}_{0}$ of the incident ray in the FCS is written by

## (2)

$${R}_{0}=({R}_{0x},{R}_{0y},{R}_{0z})=(\mathrm{sin}\text{\hspace{0.17em}}\phi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta ,\mathrm{sin}\text{\hspace{0.17em}}\phi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta ,\mathrm{cos}\text{\hspace{0.17em}}\phi ).$$Then, the incident ray vector can be rewritten as $R=\phantom{\rule{0ex}{0ex}}M{R}_{0}$ in the RCS. The incident ray undergoes two refractions at the front face and three reflections at the reflecting face. The position and direction at each face can be traced by a ray tracing approach. There are six different orders of reflections determined by where the ray strikes the LRR.

We consider the case in which the incident ray is refracted by the front face and reflected successively by the $y-z$, $x-z$, and $x-y$ planes, then refracted again by the front face. According to the vector form of the refraction, if we ignore the effect of the LRR manufacture error on the direction of ray, we can obtain the direction cosine of the refracted and reflected ray in the RCS as

## (3)

$$Rr=(\alpha ,\beta ,\gamma )=nR+\sqrt{1-{n}^{2}{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\phi}-n\text{\hspace{0.17em}}\mathrm{cos}\phi ,\phantom{\rule{0ex}{0ex}}Rx=(-\alpha ,\beta ,\gamma ),Ry=(-\alpha ,-\beta ,\gamma ),Rz=(-\alpha ,-\beta ,-\gamma ),$$Because the active reflecting region is represented in the FCS, we assume that the coordinate of the incidence point is (${x}_{1}^{\prime},{y}_{1}^{\prime},0$). For ease of calculation, we get another coordinate from using Eq. (1) with (${x}_{1},{y}_{1},{z}_{1}$). Therefore, we can calculate the positions ($0,{y}_{2},{z}_{2}$), (${x}_{3},0,{z}_{3}$), (${x}_{4},{y}_{4},0$, and (${x}_{5},{y}_{5},{z}_{5}$) on the LRR surface by using the direction cosines, incident point coordinates of the ray, and equations for the LRR surface. To give a better restrictive description of the leaving point (${x}_{5},{y}_{5},{z}_{5}$), we convert its coordinate from the RCS to FCS, and it can be expressed as (${x}_{5}^{\prime},{y}_{5}^{\prime}0$).

The positions on the front face should be distributed within the circular front face, except for the incident and outgoing rays that passed through them cannot intersect with the RoE. The positions on the reflecting face must be located within the active reflection region, which is the same distribution form for the three reflecting faces. In terms of a schematic of the LRR, Fig. 3 illustrates the region boundary condition (RBC) for the positions on the front face and $x\u2013y$ reflecting face.

Based on the RBC for the positions with the above-mentioned reflections order, we can express the first ARR as

## (4)

$${Q}_{1}=\{({x}_{1}^{\prime},{y}_{1}^{\prime})\in {B}_{f}\phantom{\rule[-0.0ex]{1em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1em}{0.0ex}}({x}_{5}^{\prime},{y}_{5}^{\prime})\in {B}_{f}\phantom{\rule[-0.0ex]{1em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1em}{0.0ex}}\phantom{\rule{0ex}{0ex}}({x}_{4},{y}_{4})\in {B}_{xy}\phantom{\rule[-0.0ex]{1em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1em}{0.0ex}}({y}_{2},{z}_{2})\in {B}_{yz}\phantom{\rule[-0.0ex]{1em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1em}{0.0ex}}\phantom{\rule{0ex}{0ex}}({x}_{3},{z}_{3})\in {B}_{xz}\},$$We can calculate the other five ARRs in the same manner for the remaining orders of reflections, which can be denoted as $Qj$ ($j=2,3,\dots ,6$).

## 3.

## Effect of Structural Recession on the Performance of a Satellite Laser Retroreflector

In a satellite laser ranging system, LRR is used to retroreflect a laser beam transmitted by a ground station, and support the ground station to implement the satellite laser ranging.^{7} The performance of satellite LRR can be regarded as the ROSI zby the ground station. A schematic of the system can be described in Fig. 4.

Due to the relative velocity between the satellite and ground station, the reflected laser beam shifts with the slightest angle aberration, which can be given approximately as^{8}

## (5)

$$\mathrm{\Delta}\phi =\frac{2}{c}\sqrt{\frac{{R}_{e}^{2}g}{{R}_{e}+h}}\mathrm{cos}\text{\hspace{0.17em}}\phi .$$Here, ${R}_{e}$ is the Earth radius, $g=9.8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{m}/{\mathrm{s}}^{2}$ is the gravitational acceleration at the surface, $h$ is the satellite height above sea level, and $c$ is the velocity of light. The plane coordinate of the velocity aberration position is written as $(h\mathrm{\Delta}\phi ,0)$.

Given that the distance between the LRR and ground station is far greater than the ARR, the ROSI is treated as far-field diffraction intensity (FFDI) at the velocity aberration position. According to Kirchhoff’s diffraction equation, we can solve the ROSI approximately by the following expression:

## (6)

$$I(\phi ,\theta ,{d}_{1},{d}_{2})={|\frac{1}{\lambda i}\sum _{j=1}^{6}\underset{Qj}{\iint}\frac{{U}_{t}}{\rho}\mathrm{exp}\{i[(2\pi /\lambda )\rho +W]\}\mathrm{d}{x}^{\prime}\mathrm{d}{y}^{\prime}|}^{2},$$^{9}

The ROSI is a function of the optical incidence condition and LRR structural recession size in the case that other parameters of LRR are known. To expatiate the effect of structural recession, we introduce a new factor $\eta $, which represents the decrease in the ROSI due to ${d}_{1}$ and ${d}_{2}$. On the basis of Eq. (6), $\eta $ can be described as

## (7)

$$\eta (\phi ,\theta ,{d}_{1},{d}_{2})=\frac{I(\phi ,\theta ,{d}_{1},{d}_{2})}{I(\phi ,\theta ,0,0)}.$$The attenuation coefficient $\eta $ is determined by the optical incidence condition ($\phi ,\theta $) and structural recession sizes, which falls in the range of 0–1.

## 4.

## Simulation for the Effect of Structural Recession on the ROSI

The HY-2A satellite is a marine remote sensing satellite developed by China, which is equipped with the LRR array as a cooperative target for satellite laser ranging.^{10} The HY-2A satellite LRR array consists of nine LRRs, which are equally mounted on a regular 48-deg pyramid frame with one nadir-looking LRR in the center. The structure of the HY-2A satellite LRR array is shown in Fig. 5.

Each LRR is made from fused quartz glass, whose reflecting faces are silver-coated. Except for structural recession sizes, other parameters of the LRR are indicated in Table 1, which are the initial conditions for calculating the ROSI.

## Table 1

Parameters of the HY-2A satellite laser retroreflector.

Radius of the front face (mm) | 16.5 |

Vertex length (mm) | 23.3 |

Tabs height (mm) | 5 |

Tabs width (mm) | 2 |

Dihedral angle offset (arcsec) | 1.6” |

Surface flatness | λ/10 (λ=632.8 nm) |

Index of refraction | 1.461 |

According to the parameters of the LRR and ARR model, the distributions of the ARR with different LRR structural recession sizes and an optical incidence condition (30 deg, 0 deg) are illustrated in Fig. 6. The $d/L$ ratio is applied to generalize the calculation results for all values of $L$.

Figure 6 shows that ARRs are composed of six sections that correspond with six orders of reflections, which are indicated as six colors. The shape of all ARRs approaches an ellipse. With the increment of structural recession sizes, the dimension of ARRs diminishes gradually. The reduction effect of the structural recession parameter ${d}_{1}$ on the ARRs is evidently much stronger than structural recession parameter ${d}_{2}$.

Considering that the ARRs vary with the appearance of structural recession, the attenuation coefficient $\eta $ associated with the ROSI will present a corresponding change. The simulation in Fig. 7 shows the distribution regularities of $\eta $ with different structural recession sizes for three optical incidence conditions (10 deg, 0 deg), (20 deg, 0 deg), and (30 deg, 0 deg).

We observe that $\eta $ falls off rapidly due to the augment of structural recession size for the above three incidence conditions. For any given structural recession size, the value of $\eta $ falls more quickly with an increasing incident angle. By comparing two groups of $\eta $ distribution curves in Fig. 7(a) and 7(b), we have obtained that structural recession size ${d}_{1}$ has a stronger effect on the fall-off extent of $\eta $ when compared to structural recession size ${d}_{2}$ at the same incident angle.

The comprehensive effect of structural recession parameters ${d}_{1}$ and ${d}_{2}$ on the attenuation coefficient $\eta $ with an incidence condition (20 deg, 0 deg) is simulated in Fig. 8. The distribution of $\eta $ can be described approximately as slanted plane. The slope in the ${d}_{1}$ direction is obviously bigger than the ${d}_{2}$ direction, which validates again, that the influence degree of ${d}_{1}$ on $\eta $ is greater than ${d}_{2}$. For the purpose of better assembling the HY-2A LRR, the structural recession sizes of ${d}_{1}$ and ${d}_{2}$ are chosen as 2.5 and 2.8 mm, respectively. We get the relationship curve between incident angle and attenuation coefficient $\eta $ in Fig. 9. The attenuation coefficient $\eta $ decreases rapidly with an increasing incident angle. The value of $\eta $ is almost equal to zero as a result of the restriction of structural recession effect when incident angle is 40 deg. The range correction is the distance of the centroid of ROSI from the center of the satellite LRR array, which can be given by Ref. 11

## (8)

$$S(\varphi ,\mathrm{\Theta},{d}_{1},{d}_{2})=\frac{\sum _{m=1}^{\mathrm{num}}{S}_{m}I({\phi}_{m},{\theta}_{m},{d}_{1},{d}_{2})}{\sum _{m=1}^{\mathrm{num}}I({\phi}_{m},{\theta}_{m},{d}_{1},{d}_{2})},$$The symbol $\mathrm{\Delta}{S}_{m}$ represents the distance between the incidence point and reflection point for $m$’th LRR.^{12} According to the definition of the distance ${S}_{m}$, it can be calculated by using a general model:^{13}

## (9)

$${S}_{m}={b}_{m}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\phi}_{m}-L\sqrt{{n}^{2}-{\mathrm{sin}}^{2}\text{\hspace{0.17em}}{\phi}_{m}},$$Considering the variety of the ROSI by the structural recession effect, the range correction of the LRR array will present a few undulations. In the general case, if we ignore the structural recession effect on ${S}_{m}$, we may use the following expression to indicate the differences of range correction as

## (10)

$$\mathrm{\Delta}e(\varphi ,\mathrm{\Theta},{d}_{1},{d}_{2})=S(\varphi ,\mathrm{\Theta},0,0)-S(\varphi ,\mathrm{\Theta},{d}_{1},{d}_{2}).$$In terms of Eq. (10) and the HY-2A LRR array parameters, Fig. 11 shows the contour distribution of the difference of range correction. In Fig. 11, the numbers around the circumference and radial direction represent the values of $\mathrm{\Theta}$ and $\mathrm{\Phi}$.

The contour distribution of the differences of range correction is irregular with the change of $\mathrm{\Theta}$ and $\mathrm{\Phi}$. The differences of range correction are confined between -1.81 and 9.36 mm. As for satellite laser ranging with millimeter accuracy, the differences cannot be neglected. It demonstrates the importance of analyzing the structural recession effect on the ROSI.

## 5.

## Conclusion

We use the ray tracing method and FFDI theory of an LRR to investigate the structural recession effect on the performance of an LRR array. The simulations on the ARR and ROSI demonstrate that structural recession parameters have strong effects on them, which result in reducing the ARR and ROSI with the increment of structural recession size. The extent of reduction depends on the optical incidence condition. A greater incident angle should bring about further reduction of the ROSI. On the basis of the analysis results, we obtain the difference of range correction when the structural recession effect on the ROSI is considered for the HY-2A satellite LRR array. The results show that the structural recession effect cannot be neglected when calculating the range correction for an LRR array.

## References

## Biography

**Hui Zhou** received his BE in mechanical engineering in 2001, and his PhD in laser remote sensing in 2007 from Wuhan University, China. His current research focuses on the signal process for a satellite laser altimeter and design for a satellite LRR array.

**Song Li** received her BE in 1986 and ME in 1989 in optical engineering from Wuhan Technical University of Surveying and Mapping, China, and her PhD in surveying engineering in 2002 from Wuhan University, China. Her current research focuses on photoelectric detection, laser interferometry, and laser atmospheric detection.