The radar equation for laser ranging of satellites is described and the effect of the velocity aberration explained. Equations for the cross sections of cube corners and arrays of cube corners are derived. Interference effects on the distribution of the array cross section and upon range error are described. Tolerance requirements for cube corners are briefly outlined

Minott, P. O.

English

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' DESIGN OF RETRODIRECTOR
" ARRAYS FOR LASER
RANGING OF SATELLITES -..
., J / .
. (NASA-TS-X-?0657) DESIGN OF EETPODIEECTO_ N7_-26306
ARRAYS FOR LASER RANGING OF SATELLITES
(NASA) 20 p HC $4.00 CSCL 22C Unclas
G3/30 _0973
_' _ PETER O. MINOTT _ '
/
.. .,.
-, MARCH1974 "
..,
i
• ..... GODDARD_PACEFLIGHTCENTER ,,,,,, ,
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, GREENBELT,MARYLAND
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https://ntrs.nasa.gov/search.jsp?R=19740018193 2020-03-23T09:08:56+00:00Z
s/
DESIGN OF RETRODIRECTOR ARRAYS
FOR LASER RANGING OF SATELLITES
Peter O. Minott
March 1974
Goddard Space Flight Center
Greenbelt, Maryland 20771 o_
_{.
!
/SUMMARY
• The radar equation for laser ranging of satellites is described and the effect
of the velocity aberration explained. Equations for the cross sections of cube
corners and arrays of cube corners are derived. Interference effects on the
distribution of the array cross section and upon range error are described.
Tolerance requirements for cube corners are briefly outlined.
iii
f! i
CONTENTS
Page
SUMMARY ........................................ iii
INTRODUCTION .................................... 1
, THEORY ......................................... 1
MANUFACTURING TOLERANCES ........................ 7
RANDOM ERRORS ................................... 7
i
" CONC LUSION 10• * t q* • • • • • • • • • • • • • • • • • • , • • D • • • • • • • • • • • •
INTRODUCTION
Optical retroreflectors of the type ltnown colloquially as cube corners have _
been used to increase the reflectivity of passive targets for at least one hundred
years. Probably the most familiar application is the molded plastic retrodirector
array in the tail-light of nearly every car produced in the last 40 years. In 1962
the first satellite (Beacon Explorer A) equipped with a retroreflector array for
laser ranging was launched by NASA. Since then, laser ranging has been de-
veloped to the point where it is the most precise method of orbit determination
in existence. Because of this precision, it has become one of the primary tools
of the geodocist in the study of the earth's gravity field.
Due to the high cost of the precision cube corner prisms required for the
arrays and the desire to obtain maximum laser echoes with the available tracking
equipment, techniques have been developed to maximize the effective cross
section of the array. The purpose of this paper is to introduce some of these
procedures to persons not familiar with laser tracking.
THEORY
Consider the basic radar situation shown in Figure 1. The transmitter
radiates a power PT with an antenna of gain GT to a target at range R. The
target with a cross section _* reflects radiation back to a receiver antenna with
gain G_. Neglecting losses the echo strength may be determined by the classical
radar equation.
d
s- (D
(4,)a 1_
t
1
If the target is a flat reflective plate oriented normal to the incident wave, the
radar cross section is
J
4vA 2
o =-- (2)
_2
where A is the area of the plate and _ is the wavelength. A cube corner reflector
made of three mutually perpendicular conductive flats will produce the same
cross section but will eliminate the need for precise orientation of the target.
For reasons which will be explained later, a single large cube corner cannot be
used as a satellite retroreflector and, therefore, arrays of many small cube
corner prisms are used to increase the target cross section. The reason for the
use of prisms rather than the triple mirror type of cube corner lies in their
greater simplicity of fabrication, great strength and improved performance at
oblique angles of incidence.
When a cube corner prism is tilted so that the angle between the normal to
the entrance pupil and the incident wavefront normal is i, the effective area
becomes
A(i)= 2a2 [sin'l/_-_'/_tani'] cosi (3)
where
t_ = 1/1 - 2tan2i '
i -- angle of incidence
t' = refractedangleofinctdence=sin-1 I_ 1
n = index of refraction of prism
i
1
2
The geometry of the cube corner is shown in Figure 2. Using this expression,
we may plot the radar cross section as a tunction of incidence angle utilizing
Equation 2. In Figure 3, this has been done for a fused silica cube corner prism
with an index of refraction of 1.46 and for a prism with a refractive index of
unity which corresponds to a triple mirror type cube corner. The superiority
of the prism over the triple mirror type is clearly shown for oblique incidence
angles. Greater improvements in acceptance angle can be obtained with higher
refractive indexes. However, high purity fused silica has proven to be the only
satisfactory prism material due to its high radiation browning resistance and its
low rate of change of refractive index with temperature.
An array of cube corners has an average cross section of
,!
N
=4.____ [A(i)]_ (4)
m--1
Because the prisms cannot be placed with respect to each other to optical
tolera_.ces, they act as a randomly phased array with an exponentially distributed
cross section. The probability densit_ becomes
p(_) -_1_ e._/_ (5)
:::r
as shown in Figure 4. The same random phasing causes a random ranging
error with a Rician distribution.
Most of the satellites used in laser ranging experiments are gravity gradient
stabilized and in near earth orbits. The retroreflector arrays are normally
located on the earth facing side; and in order to obtain a good cross section at _ .. .
low elevation angles, the arrays are configured on spherical, conical or pyra-
midal surfaces. The cross section may be computed by establishing a spherical
coordinate system around the satellite gravity gradient axis. The orientation
of th_ cube corner (i.e., the outward vector normal to its entrance pupil) may
then be specified by its polar coordinate angles 0N and _b_ as shown in Figure 5.
The line from the laser transmitter to the satellite is used as an azimuth
reference so that the satellite/transmitter vector may be specified by the polar
angle eL alone. Once defined in this manner, the incidence angle i for each cube
corner is established by the following relation
cosi - sineNsineLcos¢_+cose__oseL _6)
Equations 3 and 4 may then be used to compute the array cross section.
To this point, we have considered the transmitter/target relationship to be
stationary with no relative motion. We will now consider the effects of the
satellite motion on the radar cross section. It is this effect which places a
limit on the maximum size cube corner which may be used.
In the case of the stationary target and a cube corner with circular entrance
pupil oriented for an incidence angle of zero, the reflected beam intensity may
be described by the familiar Airy equation
kaa .J _.2
where
k = wave number - 2-/A
a = radius of entrance pupil _'_'£ ..
J
4
W,'_ = angle off transmitter/target axis
f
ER = total reflected energy I
Using the basic definition of cross section
477F Powe.r/Unit Solid&_2glc ,].2.
Power Rece ived/IJn i t Areal (8)k
Equation 7 becomes
or(a) = L kaa
_ (9)
L j
If the cube corner is moving with a velocity V normal to _he transmitter/target
line of sight, a Bradley or velocity aberration effect causes the reflected beam
pattern to be angularly displaced by an amount
a = 2V/C (10)
where c is the speed of light. This places an upper limit on the practical size
of the cube corner, since as the entrance pupil radius is increased, the central
lobe of the reflected intensity will decrease in angular extent until because of
the velocity aberration affect, it no longer illuminates the receiver. On the d
other hand, small values for the entrance pupil radius spread the central lobe
over too large an area so that little energy can be collected by the receiver.
5
$
Clearly, ther_ is an optimum s_ze which may be computed by differentiating
Equation 9 with respect to karl, setting the result equal to zero and substituting
2__Yfor _.. Doing this, we obtain for the optimum entrance pupil radius (Figure 6)C
a : 1.85 c E (11)
4vv
Under this condition, the receiver will be located on the side of the central lobe
of the reflected cube corner intensity pattern at a point where the intensity is
40% of the peak intensity.
For a circular orbit of altitude h, the maximum velocity aberration is
a =21_ (12)
where R_ = radius of the earth
g - acceleration of gravity
Therefore, the optimum cube corner size becomes
1.85C_ R]_+ 17 (13)
a - 4_r VR_K
as shown in Figure 7. If the cube corner size is selected in this manner, the
cross section of each cube corner in the array can never decrease below 40%
of the value computed through Equation 3 and can, of course, never exceed
Equation 3.
The actual value of the array cross section will be somewhere between these
limits, but its value will depend upon the orientation of the spacecraft with respect
to the transmitter, the orientation of each of the cube corners on the spacecraft,
and the wavolength. Since calculation of these values for all possible conditions
would be impractical and because there are large variations from the average
cross section due to its exponential distribution, the average cross section is
normally set 2 dB below its value under stationary conditions. When this is done,
the maximum error is only _- 2 dB.
MANUFACTURING TOLERANCES
In order to obtain cube corners which perform as predicted by the above
equations, the prisms must be essentially perfect, i.e., diffraction limited.
Using the Airy criterion, this requires that the angles between the reflective
faces be perpendicular to a tolerance of less than
< 0.374 _ (14)
2 "qa
Surfaces must be flat to better than k,/10, and material of the highest homogeneity
must be employed. Material homogeneity is the greatest problem in manufacture
of the prisms and usually limits the maximum size of a cube corner to about
5 cm diameter due to the difficulty of obtaining large homogeneous blocks of
material. The material is normally tested by polishing it into a cube before
t
manufacture of the cube corner and tested in three mutually perpendicular
directions using an interferometer.
RANDOM ERRORS
The intensity at the center of the reflected beam from an actual optical d
system compared to an ideal system is defined as the Strehl ratio S R. If the
errors in the reflected wavefront are assumed small relative to a wavelength,
and a normal-random variable _ "'
7
(27A_ 21- "E'} (15)
j,.
where _ is the standard deviation. Then by inversion
Cr2W: (_-_) 2 (1 -Sa) (16)
If, in turn, the errors in each of the five optical surfaces of the cube are assumed
:
normal random variables, and if the material is assumed to add a normal wave-
front error distribution, then if all of these error sources are independent
c_2 _ +o.2. __2 2 2= n *' ml+crr_ +_r_ +cr_ (17)
, Where the subscripts refer to total, base, homogeneity, mirror #1, mirror #2,
etc. If all surfaces are polished with equal care
and
o.2 >>o-_ (19)I11
So that "
i
To provide satisfactory performance therefore
3,-r2 h2 < -,m+_ : (1-s.) 121)
I
8
wObviously :increased polishing precision can be traded off to compensate for
poor material and vice versa, but also in any case
o2 < (22)
m "-
and
(1-s R) (2a)
For a Strehl ratio of 0.50 (3 dB loss in signal)
_r -< _ (24)
m- 15.3
_h< _ (25)
= 8.---9
If we assume that material inhomogeneity and polishing errors each contribute
an equal amount to the total error then
crm <___)_ (26)30.8
and
}_ (27)
,, Crh < 1-_, 8
It is thus seen that the fabrication tolerances are quite severe.
9
° C ONC LUSION
The previous discussion has outlined several of the basic problems Involved
• o in design of laser retroreflector arrays. It is not intended to be a complete
design procedure but rather an introductory text for the reader unfamiliar with
_ the design procedure. The paper has outlined the importance of using the proper
_:. terminology for the range equation and the distinction between "cross section"
and area.
: The most common misconception in the design of retrorefleetor arrays in
_., that area is the only important factor in obtaining strong echoes. In actual
.... practice, this is not true. Beam divergence, rather plays the key role in obtain-
_ Lug strong echoes. It is for this reason that for equal area arrays, the array
i_: with the larger individual cube corner provides a much larger echo. Future
improvements in cube corner array design lie in development of larger cube
• _ corners and improved methods for compensation of the velocity aberration.
o ........ ,
d_
10
TARGET
I
pRANSMITTER/RECEIVER /
T, GTGRX2 _ +
Figure 1. Basic Radar System
11
12
i00
80
6O
6-
4O
N':1.46
20 N':1.00
0 I
0 10 20 30 40 50
ANGLE OF INCIDENCE - DEGREES
Figure 3. Normali_.ed Cube Corner Cross Section.
13
100 10
" _ P (<o-) 88o 5(f)
-r-t_ O
t-- ,=:: 60 dP (o') r_
>'Z _---- _o3
! _o - o.F
4 W_V-- UJ
m :D 40 -<
< _j
t_ < CD
O> m
tw Z
_ 2 u',
o 20 __
"_-J .<b,,,
v rn
-- 0
0
-30 -20 -i0 o'500 i0 20
RELATIVECROSSSECTIONIN dB
(referred to averagecross section 5")
Figure 4. Exponential Distribution of Cross Section.
t
14
!S/C POLAR
AXIS
{_NCIDENT
.___Or / LIGHT
O_- O_
_ I
/ "._7_L,g/Y-*._°o.
A
I
Figure 5. Coordinate System for Cube Corner Orientation.
P
r
?
'1
15
1.0 i j
0.9
o 0.8-
>- 0.7-
F-
oO
z 0.6-
"' IMUM VALUE OF k a eI-.
Z 0.5 - IN PRESENCE OF
w VELOCITY ABBERATION OF a
> 0.4
I-"-
<: 0.3-
...I
I..U
0.2-
0.1 --
0 I I
0 1 2 3 4 5 6
kaa
Figure 6. Fraunhofer Diffraction from a Circular Cube
Corner of Radius a.
t
16 ,
3117731VS O-S030
0 oO _0 _ o4 ,,-4
r-,l
lala-Snlava _I3NNO03ano _ ,
I
17
.......... M ..............
t.j 7 -. + ;,,


Design of retrodirector arrays for laser ranging of satellites

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