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Commodore R. H. Schumaker David A. SchradySuperintendent Provost
This thesis prepared in conjunction with researchsupported in part by Defense Advanced Research Projects
Agency under DARPA Order 4035.
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Released as aTechnical Report by:
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UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (IWhen Data Entered)
20. Abstract continued
orientations are resolved and the lens studies for a rangeof orientations and strengths of gradient. The best lens isthen used in conjunction with the mirror and detector in aray-tracing scheme. Such a system was found to be feasibleand worthy of further study.
N 0102- LF-
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(IN(, A---, T EDSIECURITY CLASSIFI#CATION OF THIS PmAGEr(Wheff Data EnIS-e,'
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Approved for public release; distribution unlimited.
SUPERSONIC MISSILE SEEKER LENS SYSTEMUSING GRADIENT INDEX MATERIAL
FOR FIRST ELEMENT LENS
by
David S. DavidsonCaptain, Canadian Forces
B. Eng (Engineering Physics), Royal Military College, 1976
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN PHYSICS
and
MASTER OF SCIENCE IN ENGINEERING SCIENCE
from the
NAVAL POSTGRADUATE SCHOOL
September 1984
Author: *David S. Davidson
Approved by: __ __A. E. Fuhs, Thesis Advisor
E. A. Milne, Second Reader
G. E. Schacher, ChairmanDepartment of Physics
M. F. Platzer, ChairmanDepartment of Aeronautics
/ J..,1. DyerDean of Science and Engineering
3
" ... •.
ABSTRACT
The design of a supersonic missile seeker lens system
using an aerodynamically efficient lens, a scanning mirror
and detector was accomplished through the use of ray-
tracing routines. The design of the first element lens
uses a gradient in refractive index to overcome the severe
handicap that the pointed shape imposes. This problem has
been previously solved for positive orientations of the
center of symmetry of the Gradient Refractive Index (GRIN)
material with respect to the lens. In this study- the
negative orientations are resolved and the lens studies for
a range of orientations and strengths of gradient. The
best lens is then used in conjunction with the mirror and
detector in a ray-tracing scheme. Such a system was found
to be feasible and worthy of further study. / -
4
TABLE OF CONTENTS
I. INTRODUCTION................ .. .. .. 15
A. OUTLINE OF LENS DESIGN TO DATF.........15
3. GRADIENT REFRACTIVE INDEX (GRIN)........15
C. AIM AND INTENT OF THIS STUDY ............ 16
II. GRIN LENS DESIGN..................18
A. GRIN LENS DESIGN METHODOLOGY............13
B. GRIN RAY PATH DESCRIPTION...........19
C. GRIN LENS GEOMETRY..............22
D. RAY MATCHING .......... .. ... ... 24
III. NEAR-RADIAL LINES . . .. .. .. .. .. ... .. 30
A. DESCRIPTION OF DIFFICULTY . .. .. .. .. .. 30
B. PRELIMINARY WORK - IMPROVING THE ITERATION
SCHEME......... . .. .. .. .. .. .31
C. PRELIMINARY WORK - DOUBLE PRECISION......36
D. PRELIMINARY WORK - CHECKING ACCURACY ..... 36
E. FIRST PROPOSED SOLUTION - STRAIGHT LINES . .38
F. SECOND PROPOSED SOLUTION - REDUCED EQUATION .38
G. THIRD PROPOSED SOLUTION - MARCHAND METHOD .45
H. FOURTH PROPOSED SOLUTION - ANGLE IN TERMS
OF RADIUS....................46
IV. GRADIENT INDEX SEEKER LENS RAY TRACE RESULTS . .53
A. FRONT SURFACE ACCURACY..............53
B. SPOT SIZE RESULTS................53
5
C. SELECTION OF A BEST LENS............53
V. MIRROR AND DETECTOR SYSTEM............57
A. DESCRIPTION OF SYSTEM..............57
B. LENS TO DETECTOR RAY TRACE..........58
1. Ray Direction Exiting Lens........58
2. Point of Intersection on mirror......53
3. Ray Direction Exiting Mirror . .. .. . .61
4. Ray Count on Detector..........................6
C. MIRROR AND DETECTOR SYSTEM RESULTS ...... 65
VI. CONCLUSIONS AND RECOMMENDATIONS...........68
A. CONCLUSIONS....................68
B. RECOMMENDATIONS.................69
APPENDIX A: PROGRAM GISL LISTING.............70
APPENDIX B: PROGRAM CHECK LISTING............93
APPENDIX C: PROGRAM DETPLOT LISTING . .. .. .. .. .. 103
APPENDIX D: DETECTOR SIGNAL..............107
LIST OF REFE~RENCES....................117
INITrIAL DISTRIBUTION LIST..................118
6
LIST OF TABLES
I.Front Surface Accuracy ................. 55
7
LIST OF FIGURES
2.1 GRIN Lens Coordinate System and Parameters
(fromnCarr)....................20
2.2 GRIN Lens and Ray Geometry............23
2.3 Ray Path Showing Improper Matching and Minimum
Radius.....................26
2.4 Ray Path Radius and Angle.............27
2 .5 Ray Matching Regions..............29
3.1 Ray Trace Showing Region of Radial Lines .... 3
3.2 Newton-Raphson Iteration Geometry ......... 34
3 .3 .Newton-Raphson Logic Flowchart.........35
3.4 Geometry of Accuracy Check - Forward Ray Trace .39
3 .5 Expanded View of Front Surface - Straiqnt Line
Method.....................41
3 .6 Reduced Equation Implementation ......... 44
3 .7 Radius and Angle Relation.............48
3.3 Angle in Terms of Radius Iteration Logic
Flowchart.....................50
3.9 Angle in Terms of Radius Iteration Geometry . . .51
3.10 Diverging Iteration - Forward Ray Trace. .... 52
4.1 Spot Size Results - ALFAP = 0 .3 Radians. .... 56
5 .1 Lens, Mirror and Detector System........59
5.2 Detector Signal.................60
5 .3 Reflertion of Ray on Mirror ........... 63
5 .4 Mirror Tilt for On-Axis Spot and Angle of
Incidence.....................67
LIST OF SYMBOLS
SYMBOL FORTRAN DEFINITION UNITSEQUIVALENT
a A index function parameter nondimensional
arg ARG argument of arcsine nondimensional
function in GRIN rayequation
arg argument of initial nondimensional.9 arcsine function in GRIN
ray equation
b B index function parameter nondimensional
c C ray matching constant radians
Dim Dl ray geometrical distance nondimensional
from lens to mirror
Dnd D2 ray geometrical distance nondimensionalfrom mirror to detector
DT DETP position of detector nondimensionalalong x-axis
e E GRIN ray trace constant nondimensional
K CK x-direction cosine of nondimensional
ray from lens to mirror
K' CKK x-direction cosine of nondimensionalray from mirror todetector
- z-direction cosine of nondimensionalray at initial point
L CL y-direction cosine ofray from lens to mirror nondimensional
L' CLL y-direction cosine of nond imensionalray from mirror to
detector
10
M CM z-direction cosine of nondimensional
ray from lens to mirror
M' CLVIM z-direction cosine of nondimensional
ray from mirror todetector
M - Marchand reducedO integral expression
MP MIRP position of mirror pivot nondigensionalpoint on x-axis
n - index of refraction nondimensional
n - index of refraction at nondimensional0 initial point of GRIN ray
n vector normal to mirrorplane
* n. - x-component of vector nondilnensional
normal to mirror plane
ny - y-component of vector nondimensional
normal to mirror plane
p x-direction cosine of GRIN nondimensional0 ray at initial point
q 0 y-direction cosine of GRIN nondimensionalray at initial point
r RAD radius to ray from center nond inensionalof symmetry
r RO radius to ray at initial nondimensional0 point
r' RP first guess radius in nondimensional
iteration scheme
r- second guess radius in nondimensionaliteration scheme
* r"' - third guess radius in nondimensionaliteration scheme
rfs radius to front surface nondinensional
• ii
rfs derivative with respect to nondimensionalangle of radius to frontsurface
rgr radius to GRIN ray nondimensional
rgr derivative with respect nondimensionalto angle of radius toGRIN ray
r i ray vector incident to nondimensional-nirror
rxi, components of ray vector nondimensionalryi , rzi incident to mirror
4
rr ray vector reflected offmirror
rxr, components of ray nondimensionalryr , rzr reflected off mirror
R R radius from x-axis to nondimensionaloutermost point of innersurface
Rz Rzero radius to edge of GRIN nondimensionalmaterial
U U angle of incident radiansradiation for lensconstruction
xd XD x-component of ray at nondimensionaldetector
xm XM x-component of ray at nondimensionalmirror
x XO x-component of ray at nondimensional0 lens inner surface
Yd YD y-component of ray at nondimensionaldetector
Ym YM y-component of ray at nondimensionalmirror
12
--- --.-.,,-*-*.. . . ....
Y YO y-component of ray at nondimensional0 lens inner surface
zd ZD z-component of ray at nondimensionaldetector
Zi ZM z-component of ray at nondimensionalmirror
z ZO z-component of ray nondimensional0 lens inner surface
: EPSLON sign function of GRIN rayequa tion
PSI angle between GRIN ray and radiansradial arm from center ofsymme try
10 PSIO angle between GRIN ray and radians0 radial arm at initial
point
THETA angle from x-axis to radiansradius
THETO angle from x-axis to radians0 initial point
OL angle for GRIN ray part radiansto left of part of minimumrad ius
OR angle for GRIN ray part to radiansright of point of minimumrad ius
o' THETAP angle of first guess in radiansiteration scheme
-"angle to sedonc guess in radiansiteration scheme
"'angle of third guess in radiansiteration scheme
9m THETM angle from x-axis to plane radiansof mirror
13
.
ACKNOWLEDGEMENTS
The author would like to thank Distinguished Professor
Allen E. Fuhs for his guidance and perserverance during the
writing of this thesis.
The author would also like to express his gratitude to
his wife Johanne for her support during this period.
14
I. INTRODUCTION
iA. OUTLINE OF LENS DESIGN TO DATE
The design of a 'pointed' aerodynamically efficient
missile seeker lens has been the topic of a number of
studies. Frazier [Ref. 11, and Terrell [Ref. 2], first
studied the idea of using a pointed lens as a means of
obtaining some degree of aerodynamnic efficiency. They also
introduced the idea of using Gradient Refractive Index
(GRIN) material in order to overcome the difficulties in
imaging imposed by the odd shape. Amichai [Ref. 3], and
Carr [Ref. 4], studied the GRIN concept in more detail,
Amichai by defining the outer surface as a cone and solving~
for the inner surface and Carr by defining th~e inner
surface as a cone and solving for the outer. The main
reference for this author's study was drawn from Carr's
work and a review of that thesis would be helpful if the
reader seeks a more detailed background from that given in
the next few pages.
B. GRADIENT REFRACTIVE INDEX (GRIN)
Although the theory for GRIN has been developed and
known for some time it was not until fairly recently that
materials have become available that allow the theory to be
put into practice. A gradient in index will cause light
rays to bend in an otherwise homogeneous material. Thus,
15
the lens designer now has a new dimension of flexibility
which will allow him to solve problems that could not be
ri formerly solved. A proper selection of the manner in which
the gradient varies is also important. The designer would
normally select certain degrees of symmetry. Three are
most common; axial, radial or spherical gradients.
GRIN material in the shape of rods now commonly use
axial and radial gradients in such devices as fiber-optic
cables and photocopying machines. Spherical gradients are
not available because spherical gradient material has not
yet been produced. Spherical gradients do offer an
advantage in that they lend themselves more easily to
mathematical description since closed form solutions for
the ray paths can be obtained.
A good introduction to gradients and the materials
currently available for production of large scale GRIN lens
systems is provided by Moore [Ref. 61, and Light [Ref. 7].
These also give the reader an idea of the advantages that
may be realized along with the problems associated witi~ the
design and production of such items.
C. AIM AND INTENT OF THIS STUDY
The aim of this study is to complete the design of the
* Gradient Index Seeker Lens (GISL) initiated by Carr. This
design uses a spherical gradient and 'pointed' lens shape.
This study does not determine the aerodynamic efficiency of
16
the design but rather ensures that the lens has a
generally pointed character that will also allow some
degree of imaging for off-axis objects. The spherical
gradient is chosen since it seems to fit the geometrical
shape of the lens and also since the three-dimensional
analysis of rays is much simpler.
This design is accomplished through the use of ray
tracing trchniques. Thus, no actual study of the lens
material nor the manner in which such a lens could be
manufactured is intended. It is the hope of the author
that studies of this nature will provide the impetus for
the development of suitable GRIN material with spherical
gradients.
Once the lens design was completed a 'best' lens was
chosen and a simple detector system applied to it to see if
precise definition of off-axis targets is possible.
17
II. GRIN LENS DESIGN
A. GRIN LENS DESIGN M~ETdODOLOGY
The method by which the GRIN Lens was designed in
Carr's thesis [Ref. 41, is the subject of this section. A
cross-section view of the top half of the lens is shown in
Figure 2.1.
As can be seen, the lens is designed by defining the
inner surface as a cone and then constructing the outer
surface so that a backwards ray trace from a point source
will result in parallel lines emerging. This is done by
tracing rays from the back edge of the lens towards the
front surface and finding the point of intersection of the
ray and front surface line. The front surface line is
found by using the previous ray solution as described
below. The intersection of the ray and the line is
determined using an iteration procedure. Once a solution
is found the front surface line that will cause the ray to
emerge parallel with all other rays is deter-nined. This
front surface line is then used to find a point solution
for the next ray. twout 1000 rays are traced starting at
the outside edge of the inner cone and working inwards
towards the center. Details of the iteration process is
included in later sections.
18
Once the lens outside surface has been constructed the
lens is then studied to see how well it will image three
dimensional skew rays that enter the lens at varying
degrees of incidence. To accomplish this, a grid is placed
in front ot the lens and then rays are traced from the
intersection points of the grid through the lens and onto
the image plane. The grid is then tilted at successive
angles and the resulting 'spots' on the image plane
recorded for each case. The manner in which these spots
move on the image plane and their size are the major
performance characteristics of the lens.
B. GRIN RAY PATH DESCRIPTION
Following Marchand [Ref. 5], GRIN ray paths in
spherical gradients are found through the solution of the
appropriate line integral:
+ e r dr (2.1)0 r e 0 r(n2 r 2- e2)1/2
where
e = nrsin = n0 r 0 sin 0 (2.2)
and
It should be noted that the value of is -1.0 if tile
radius is decreasing with angle and +1.0 if the radius is
increasing with angle.
19
U-
C)
C)
W L)
L) C)
(nC
0
D0
200
The choice of index gradient is instrumental in
allowi.,g the integral to be solved in closed form. In this
study the index is defined as:
n 2 = (a + br 2/Rz 2 )
The constant a defines the strength of the index at the
origin and the constant b defines the strength of the
gradient and whether it is negative or positive.
As can be seen in Figure 2.2, the ray path is described
by the constant e in conjunction with the parameter
the angle between the radial arm of the coordinate system
and the direction of the ray at any point.
The solution of the integral in equation 2.1 results in
the following expression:j- sinIar 0 - -
) 0 2- (argo - sin- (arg)' (2.3)
where
a- 2e/r 02
arg oa0 (a 2 + 4be 2 /R 2)1/2
and
a - 2e/r 22r 2 2 2 7-
(a + 4be /Rz2)1/2
or, expressed in terms of the radius,
21
r /2 el
-I'a+(a 2+4be 2 /R 2) sin [-2c(- 0 ) - sin 1 (arg0 ) ] 1/2
z '
(2.4)
C. GRIN LENS GEOMETRY
A factor which must be considered is the manner in
which the lens is cut out of the GRIN material. Figure 2.2
shows an orientation of the material with respect to the
lens such that the center of symmetry of the material is
inside the lens. Carr has defined this as a negative value
of OB, the distance from the point 0 to the point B.
Carr was able to work out the lens problem for positive
values of this parameter only, thus the major portion of
this study is to develop methods that enable solutions for
negative values of OB. In so doing, two factors have to
be overcome:
a) ray matching - When the ray path radius initially
decreases and then increases with change in angle the ray
path equation !nust be matched at the point of ninimum
rad ius.
b) near-radial lines - Rays that show very little
curvature are difficult to describe in terms of radius anJ
angle and these cause problems in the iterative solution
scheme.
22
I
LL
CfC)
ww
L))
z !w 0 )
00OLL Z
23
.-.. ~~~ ~~ .~-~-~-- .-.- - . . . . . .
The solution to the ray matching problem is relatively
simple and is discussed in the next section. The
resolution of near-radial lines is a more complex task and
is covered in the following chapter.
D. RAY MIATCHING
As stated previously the ray path equation changes
signs as the radius of the path decreases anJ then
increases. Such a path is shown in Figure 2.3. Simply
changing sign in the equation as the point of minimum
radius is passed does not produce the desired result, but
rather has the effect shown in Figures 2.3 and 2.4.
The resolution of this problem is fairly simple. At
the point of minimum radius the arcsine function in
equation 2.3 containing this radius (arg) reduces to - /2.
This is due to the fact that at this point the value of
is -/2. Thus:
sin 1 = 1
and
e2 =n2r 2
Thus:
2 2a_2n2 a-2(a+br /Rz
2 2 2 2 172 2 2 2 2 2 1(a2+4bn r /R / (a +4b(a+br 2 /Rz r2/Rz
24
. . . .. . . . . . . . . . . . . .. . ''
and
sin - l (arg) =-/2
If the radius is initially decreasing as shown in
Figure 2.3, we can find the angle of the minimum radius.
We call this the critical angle or L:
0L = 9 -0 si& 1 (arg0 ) + 7/2
The positive and the neqative forms of the equation
should both contain the point of minimum radius, in other
words, by approaching the point from the left or the right
the same result should be obtained. The fact that this
does not happen is shown in Figure 2.3 and reveals the need
to match the rays at the point of minimum radius.
This matching is achieved by simply adding a constant
to either the positive or the negative form of equation 2.3
and then matching at the point of minimum radius in the
following manner:
OR = OL + c
thus
L i Tr/2+sin- (arg )l = + -1(arg + c
The constant c can then be found:
c = - I/2+sin (arg0 )
*• This constant must now be applied for all points past
the point of minimum radius; in our example all points in
25
.. .. ..
jI 2.0
X 1.0
CC)
0.5
0.0 1
00.
Fig~ure 2.3 Ray Path Showing Improper Matching an'f
r*inirnum Radius.
26
4.
3.
1.~ra witwhtummatching
0.0 0.5 1.0 1.5 2.0
RO I US
Figure 2.4 Raj, Path Radius and Angle.
27
the increasing radius portion of the ray must be found
using the form:
0 + i- Isin 1 (argo) = sin l(ary)l + c
In the iteration scheme the ray path equation must be
more carefully handled. First, the critical angle and the
constant c must be found. Second, the angle of the
ray must be checked to see if it has passed the critical
angle. If it has then the constant c is added to#A
equation 2.3 and the point recalculated. The details of
the iteration methods will be described in the following
chapter.
For further clarification Figure 2.5 shows the regions
in which ray matching is required for a certain position of
the center of symmetry. Note also that were the value of
OB positive then all rays in the lens would never reach aA
*minimum radius point. Thus ray matching is a problem
unique to the negative OB case.
28
~28
*
2.0
oSrMB - -0.1000
A 2.2500000
8 -0.9843750
1.5
0c-n0.5
\ 2o~s /, 3 locus of:.
minimum radius
/" I
-4
0.0 I I , ;-1.0 -0.5 -0.0 0.5
x RXIS
Figure 2.5 Ray MNatching Regions.
29
• 1 ,- . .
III. NEAR-RADIAL LINES
A. DESCRIPTION OF DIFFICULTY
As mentioned previously, the ray paths that are nearly
radial are difficult to describe mathematically in terins of
radius and angle. This is due to the simple reason that a
straight line originating from the center of the coordinate
system has no change in angle and an undefinable chanye in
rad ius.
The region in which this problem occurs in the lens
design is shown in Figure 3.1. As can be seen, near-radial
lines mean that the angle and thus the constant e
will be. very small, so small as to cause equation 2.4 to
approach a singularity and thus break down. In the
iterative scheme, as the front surface solutions are found
froma the outside edge of the lens inwards, the values of
j and e become increasingly smaller if the center of
symmetry is inside the lens (OB negative). When this
region is reached, equation 2.4 reduces the r,3 for all
p angles and futher construction of the front surface is
rendered impossible. This effect is shown in Figure 3.1.
Of note is the fact that if the radial line region is
passed successfully, the rays begin to curve in the
opposite sense, that is they are always convex to the
30
radial lines in the case of a negative gradient and concave
to the radial lines if a positive gradient is used.
The resolution of the near-radial lines proved to be a
difficult task and became a major portion of this study.
The manner in which it was overcome is the subject of the
following sections.
B. PRELIMINARY WORK - IMPROVING THE ITERATION SCHEME
The first step in the resolution of the problem was to
simplify the iteration scheme. In his work Carr developed
a scheme based on the geometry of the rays and coordinate
system [Ref. 4], which worked well for all positive values
of OB (that is, for all orientations of the center of
symmetry to the left of the inside surface cone. See
Figure 2.2). It was decided instead to use the Newton-
Raphson approach, which is simpler in concept and not
dependent on geometry.
Carr used the Newton-Raphson method in the second
portion of the design involving the skew ray trace, but he
did not use it for the lens construction.
The manner in which this scheme works is now described
so that the reader will be able to better understand trie
concept of how all iterative methods work for this design,
as well as how the ray matching previously described fits
in.
31
2.0OSYMB - -0.5000
A - 2.2500000
B - -0.9843750
1.5
, 1.0
front surface showing ,
affect of singularity
0.5
region of
near-radial lines
0.0-1.0 -0.5 -0.0 0.5 . "
X RXIS
Figure 3.1 Ray Trace Showing Region of Radial Lines and
Effect of Singularity,.
32
The geometry associated with this plan is shown in
Figure 3.2 and a flowichart outlining the iteration logic is
shown as Figure 3.3. As can be seen with reference to the
two figures, an initial point is first chosen by drawing a
straight line from the inside surface intercept to the
slope of the front surface obtained from a previous ray
path solution. Newton-Raphson seeks to find the intercept
of this front surface line with the curved ray path. This
is done by using the initial angle to find the radius anJ
derivative with respect to angle for each line and then
calculating a new angle according to the relation:
r -r
fS rgr
This new angle is then used to recalculate the radii
and derivatives and thus an iterative loop if formed. Once
the difference in angle is small enough the iteration is
stopped and the solution considered to be found.
Ray matching is also employed in this scheme. As each
new angle is calculated, it is checked to see if the
critical angle has been passed. If it has, the constant c
of equation 2.5 is added to the angle and thus the proper
radius and derivative will be then found.
Once a solution has been found it is used to find a new
front surface slope by determniningj that slope which will
33
B 0
Figure 3.2 Newton-Raphson Iteration Geometry.
34
start
calculate lineccnstants
find Itialpotit then find
theta'
add c to theta'and change sign
for epsilon
find r and rdotfrom' grinequati on
find r and rdot
from fronsurface equation
fInd new anglefromt
Newt on-Ra Dhsonequation
Figure 3 .3 Newton-Raphson Logic eiowchart.
35
cause the refracted ray to emerge parallel to the axis. it
is this slope that is used to find the solution for the
next ray in the scheme.
When this new iterative procedure was implemented it
was found to work efficiently for both the positive and
negative gradients, but it did not resolve the problem of
near-radial lines.
The reader should refer to Appendix A for a listing of
program GISL, which is the ray tracing program originally
written by Carr and modified by the author.
C. PRELIMINARY WORK - DOUBLE PRECISION
The next avenue that was explored was to place the ray
tracing scheime in double precision. This was considered
necessary since very small differences in angle were
required in the region of the radial lines.
Once this was installed in the routine it was found to
improve significantly the accuracy of the calculations and
succeeded in delaying the point at which the iterations
broke down. It did not solve the crux of the difficulty
however.
D. PRELIMINARY WORK -CHECKING ACCURACY
At this point it was decided to develop a scheme to
check the accuracy of the front surface solution. The
manner in which this was done is shown in Figure 3.4
36
The ray trace was coinpleted in a forwar,] direction from
the front surface to the focal point in a two-dimensional
fashion. These rays were caused to originate from the mid
point of each set of points that define the front surface.
The slope of the front surface for each of these rays was
taken to be the line drawn between the two ad3acent points
on the front surface. In this fashion the worst case was
chosen since the skew ray trace used to determine the
performance of the lens uses incident rays that can hit the
front surface at any point.
These rays were then traced through the lens and back
towards the focal point. The x-coordinate of the focal
point was defined and the y-coordinate calculated fromn the
ray trace to see how close it would come to the
y-coordinate of the focal point originally used to start
construction of the front surface. A listing of this
routine is shown as Appendix B.
Another benefit of this two-dimensional ray trace was
that it allowed the author to confirm the methods by which
a forwards ray trace is done. In this manner ray matching
and sign convention in angles was checked. Also this ray
trace is different in that a solution between the ray path
in the lens and the back surface, rather than the front
surface, was now required in the iterative solution scheme
and this could now be verified to ensure that it was
properly done.
37
7.- -
Another method of checking the accuracy of the sol,,tio:n
would be to study the front surface smoothness in the
region of near-radial lines. Although not as precise as
the above method this idea allowed the author to confirm
that the front surface was sufficiently smooth. By this
means it was also possible to see a physical representation
of the lens in the region of the singularity.
E. FIRST PROPOSED SOLUTION - STRAIGHT LINES
The first proposed method of solving the mathematical
difficulty was to simply replace the near-radial lines with
completely straight lines just before the singularity was
reached. This could be done by following the decrease of
the parameter ~.until it became small enough to ensure
that the lines were nearly straight.
This idea was implemented but it was found not to be
sufficiently accurate. It must be emphasized that an
inaccuracy in even a single point could not be tolerated
since this would mean that all succeeding points in the
iterative solution scheme would carry that error as well.
For these reasons the straight line solution was
rejected and other means of solvinq the difficulty were
sought.
F. SECOND PROPOSED SOLUTION - REDUCED EQUATIONS
The second method that was use3d to resolve the problem
of near-radial lines was to reduce equation 2.4 by
33
start of trace
end of trace
Figure 3.4 Geometry of Accuracy Check -Forwards Ray Trace.
39
employing small angle aproximations, binomial expansions
and curve fitting. The details of how this was ]one
follows.
Recalling equation 2.3:
1 -= 0- sin (arg 0 ) - sin (arg)02
The arguments of these arcsine functions were reworked
as follows:
a-2(a+b') 0 2
argo = 2+4b , (a+b, 2)1/2
andia-2 (a+b ),, 0 2 r0 2/r 2
arg/(a2 +4b'(a+b'),!'0
2)1 L
where
0 = sin.,,,0
b= br 02/Rz 2
Note the angle 0 is very small.
These arguments were then reduced using a binomial
expansion:
2(a+b')2arg =i- 2
a
40
-
. .. . . . . . . . . . . ." " . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
FRONT SURFACE EXPANOED VIEW7
0SYMB -0.50
A = 2.2500000
8 = -0.984375
0.23
U-)
X
CI
0.22
0 . 2 1 I i I J I I i I I I I I I I I
--0.210 -0.235 -0.230 -0. 225 0.220 I.1 ..
X AX[1
Figure 3.5 Expanded View of Front Surfac,.-- Straight Line
Method.
41
.& . . : '.. -." . . . ".. -' .. .. . . . ." . , . - . . -• . . . -. . ''"" L ' - ' - .. -
and
22r02(a+b') !,O(-T- a+b')r
arg=l- 2
2Again the 0 2 term ensures that the expansion is valid.
A function of tie type arcsin(x) where x is known to
be close to the value I can be curve fitted into the form:
sin-l(x) = /2 - [2(1- x)11/2
This equation applies to both arcsine functions as the 02
term ensures that the argument is close to 1.
When placed in the equation the arguments result in the
following final expression:
2b , , t)(O2 1/2,
S= e0 +-- f(a+b') - [(a+b')(---. a+b)]S0 2 2r
In terms of radius:
r = r (a(a+b') )l/2
- - (a+b)] 2 - b,(a+b)
The reduced equation was then implemented in the
iterative front surface solution scheme by constantly
checking it against the complete equation. When the
42
difference between the two became small, the reduced
equation was subsequently used in solving for the front,
dsurface. The reduced equation was then turned off when the
rays had progressed an equal distance on the other side of
the singularity. See Figure 3.2 for clarification.
The equation worked well in resolving the front surface
for the lens construction and did produce a smooth Surface
similar to that shown in Figure 3.5.
b It did not do well when the accuracy program check was
applied. One reason is that although it was easy to
implement the equation for the lens construction it was not
so easy to complete the forwards trace since these rays hit
the front surface randomly making it difficult to know when
to use the reduced equation. In practice the reduced
equation could be implemented when the absolute value of
the parameter 0 became smaller than that used in the
lens construction phase during the time that the reduce(]
equation was used. There was then no way of telling how
accurate the front surface was, in other words, there was
no inethod to show how the rays would physically react in
the region of the singularity. Thus the reduced equation
method would work but it is not an entirely satisfactory
solution.
43
region where reducedequation used
0 ,-inner surface
B 0ot F
Figure 3.6 Reduced Equation Implementation.
44
Program CHECK was tried without using the reduced
equation but disastrous results were obtained for those
points in the region of the singularity.
For these reasons it was decided that another means of
resolving the problem was required.
G. THIRD PROPOSED SOLUTION - MARCHAND METHOD
The third proposed solution is drawn from Marchand
[Ref. 5] p. 27, in which he proposes a means of improving
the ray trace formula so that nearly radial lines will be
more easily described.
The method is based on redefining variables as
follows:
e = eM0 (3.1)
where
M = jr dr0 r0 r(n2 r2 e2 1/2
Marchand describes the ray path using the parameters
and , such that:
= sin e/sin,,( (3.2)
a = COS9 - ?COS;,O
This enables the coordinates of the ray to be described
as:
x r~ax0 0x = r{cr- + 2-)..
45
. .~ .• .. .
y = r( + L)
z0 0
z = r ( -0 + no
Upon substitution of equations 2.2 and 3.2, equation
3.1 becomes:
= n0 r 0 M0 sinc
The parameter can now be more accurately solved for
small angles of using a Maclaurin series for the sinc
func-ion.
This method was then implemented in program GISL using
an iteration scheme based on Cartesian rather than polar
coordinates.
This method was not found to improve the accuracy of the
ray trace in the region of near radial lines and was not
adopted.
One reason for this may lie in the fact that in our
solution to the integral in equation 2.1 the value e no
longer appears at the front of the expression. Thus there
is no advantage to redefining the variable as shown in
equation 3.1.
H. FOURrH PROPOSED SOLUTION - ANGLE IN TERMS OF RADIUS
The fourth method and the one that proved successful
involved the solution of the front surface slope and the
46
. . . . .
GRIN ray using the form of the ray trace equation shown in
equation 2.3 rather than that of equation 2.4.
The reason for this choice is shown in Figure 3.7.
This figure shows that when the rays are nearly radial a
small change in angle will result in a huge change in
radius. Thus the form of the ray tace equation using r(
is inherently unstable. Previously we have used this forff
in order to .iake use of the Newton-Raphson technique. It
was thought that if the iteration scheme could be reworked
so that the form ,(r) of the equation could be used then
better results might be obtained.
A suitable iteration method was found and is
illustrated in Figures 3 .8 and 3 .9.
The figures show that an initial point was chosen in
the same manner as before, that is by drawing a straight
line from the ray refracted at the inner surface out to the
front surface slope. This point provides the radius for
equation 2.4 so that a new angle can be calculated. Tis
angle is then used to find the corresponding radius out to
the front surface slope. This new radius is then used in
equation 2.4 to find the next angle and thus an iterative
loop is formed.
Thus a simple scheme was found. It should be noted
however that this method will only converge when the
initial angle is greater than 90 degrees. The reason for
this lies in the geometry and is shown in Figure 3.10.
47
-
w ~
z
C1aCD)
48
Thus the iteration will only converge when the rays are in
the region of the nearly radial lines. This does not cause
a difficulty since the Newton-Raphson method, modified to
include ray matching, can find solutions efficiently in the
region where the initial angle is less than 90 degrees.
Figure 3.10 is also drawn to show the reader an example
of a forward ray trace.
This simple scheme was implemented and found to work
successfully for both the lens construction and the skew
ray trace; see [Ref. 8].
49
.J
start
calculate lline constsrt
Figure~~~~~~~ 3 8 ng e i Te m f In ta dl s I e at o o i
Flowchatet.
50cuat
locus ofequal radius
first guess
seconduguess
thirduess g i ray
r'"
B 0
Figure 3.9 Angle in Terms of Radius Iteration Geometry.
51
..:L. . . . . . . . . . . . . ... : , t L. . A S' . ..... .. . . . . : .
thirdguess
grin ray path
f irstsolution .
'11 1 of
0
Figure 3.10 Diverging IteraLion - Forward Ray Trace.
52
IV. GRADIENT INDEX SEEKER LENS RAY TRACE2 RESULTS
A. FRONT SURFACE ACCURACY
As mentioned previously, the front surface of the
Gradient Index Seeker Lens was checKed for accuracy by
completing a ray trace from the midpoint of each set of
solutions. A sample of the output froma program Check is
shown in Table I so that the reader will have an indication
of the accuracy of the ray trace in the region of near-
radial lines. Note that the most radial lines occur when
the absolute value of the angle *; is smallest.
B. SPOT SIZE RESULTS
The results of program GISL showing the variation of
spot size are shown in Figure 4.1. This figure shows how
the spot size varies as gradient strengths and orientations
of the center of symmetry are changed for a median angle of
incidence of 0.3 radians.
Note that all orientations of the center of symmetry
are now included. The studious reader should refer to
Carr's work [Ref. 4], to compare results.
C. SELECTION OF A BEST LENS
The best lens was selected by determining those
parameters that would result in a small spot size and also
-ause the least number of rays to be reflected or refracted
53
outside the missile diameter. Figuce 4.1 shows that these
criterion were not easily discernable and a judgemental
decision was required.
The reader should note that a value of 2.25 was chosen
- -for the parameter a. This was done since the resulting
value of 1.5 for the index of refraction at the center of
symmetry is common to a large number of possible lens
materials. Also of note is the fact that the focal point
tM was chosen to be 2.0 units along the x-axis from the inner
surface cone. This was done since this value resulted in
smaller spot sizes generally.
0 With these criterion in mind a value of -0.2 for 0i6
and a 5% negative gradient were chosen as parameters for
the best lens design.
A more detailed study of the lens parameters could be
accomplished with use of an optimization program in which a
large number of parameters could be varied in order to find
the best lens. This would achieve a more satisfactory
result than the one shown.
54
.~~~~ ~ ~ ~ ...
TABLE I
FRONT SURFACE ACCURACY
M~ine Number I Angle .0 Error in y- I number of
I j (Radians) Icoordinate I iterationsI
I 895 I 0.0411 0 0.0008 I4I 896 0 0.0390 0 0.0007 I4
897 0 0.0369 0 0.0003 I483 0 0.0349 0 0.0009 I 4
I 899 0 0.0328 0 0.0007 I4I 900 I 0.0308 I 0.0009 I4I 901 I 0.0288 0 0.0008 I4I 902 0 0.0268 0 0.0008 j4
903 I 0.0247 I 0.0009 4I 904 I 0.0227 I 0.0009 I3I 905 I 0.0207 I 0.0008 I3II 906 I 0.0137 I 0.0009 I3I 907 0 0.0167 0 0.0008 3
*I 908 0 0.0147 0 0.0003 3I 909 0 0.0127 0 0.0010 I3II 910 0 0.0107 0 0.0009 I3II 911 0 0.0087 0 0.0008 I3II 912 0 0.0067 0 0.0010 I3II 913 0 0.0048 0 0.0009 I3II 914 0 0.0028 0 0.0010 I2II 915 0 0.0008 0 0.0010 I2I 916 I -0 .0010 0 0.0010 I2II 917 I -0 .0030 0 0010o 3I 918 I -0 .0050 0 0.0010 I3II 919 I -0 .0069 0 0.0309 I3II 920 I -0 .0089 0 0.0011 I3II 921 I -0 .0108 I 0.0010 I3II 922 I -0 .0127 0 .0010 I3II 923 I -0 .0147 0 .0009 I3II 924 I -0 .0166 0 0.0011 I3II 925 I -0 .0185 0 0.0010 I3I 926 I -0 .0204 I 0.0011 4II 927 I -0 .0223 0 0.0011 I4II 928 -0 -. 0243 0 0.0010 I 4I
929 I -0.0261 I 0.0012 I4II 930 I -0 .0281 I 0.0911 I4I
*I 931 I -0 .0300 0 0.0012 I4II 932 I -0 .0313 0 0.0012 I4I 933 I -0 .0337 0 0.0012 I4II 934 I -0 .0356 0 0.0011 14I 935 I -0 .0375 0 0.0013 I4I
55
o ~ ~ r o r: a, o,* S % 0 * 0 * * s ~
, t N 't , - N, -o * ,* 6% N 6 0 *
ao, a4 4 4 4*, *4 v In 4
-,~ ~~~~ -, -, -% - -* - -, - -* 4 0
co" .4 4 * oo v N Ua Do o * S* It 4 N
NW NW 4N 4 N 4 r:6 . 6 4, 'I4, 4 W -!"l
A7 -0 -W -- - 4 - V --? -%~ -~? - ~ -i -'U - j -r -4 - ---
'4c, N% 6 C4 -4 -*40 N % 4 O w m? I? mD? a!s5~
NN~ ~~~~~~ NN N-! O 4 V 40 4N N% N * 4 - . -, I'
Na -, -- - - - - -% - - - -N6% 065 % 0co c,0 656 a,0 0 0 OW N Nr. 0 ,4UUN0 . N -4 4 oz S , V? 4 N ,"
- -, -N - - --4 - -- - -' - - -
-4 m m a% N, mm mm4'- I 0r.4 C, r4- 4 N m, c, N, 4z mm A co% a, ,0
m 0o
N -4 4 4C-%46 c 46 U,6% 4,6 cc % 0 0 06 % 6 5 0 lo 4 5 0 6 S NN N N ...4 N .- 4 -! 4, O%-4 6 ,6 6
-o xc r- '
4 4 -4 N - -4 -- N 4, n In o,. 4 -4 0,S ?-4 --
40 *e oo '.4 %c *4 '65 6% o '5o 'co co o c
(4 m1
az - -! o t ! ,
-4 0 4 t*. 4
40~~~~c N0 a% km0 4'- 44 , 4% 4 N 4 * S O
-, 0, w4 U, Un, U , 4U
O 0' 0 0 0 o 4 '-4 -4 v~ 0n :
m, In In 4, W * , 4 4 W l 6%
*-4o o- o- o- -- - - * . 4 *4 04 G C
0 N N, N, U, U, U, U, U, w N . z
4, 4 -I4 -M U, 4- o, c, 4, U 0 *(-
4o -4 -
1- 4, 4, U, U, U, -' -4 -4 6% 4 Cl)
0n 0 o 0 o 0 4-1 Ii 0
lo
O 0 4, 4, U V, U, m o
O 2 0 o 0 4 o% 0o 0 0o 0o 0 0 -o 0 -4 -. 0n~4.4 .. 4 N6 N6% U,- U,4 0. .44 ?I SDI 04 S-I 0-4 -4
-~ ~ -: 0 0 o 4, 4n, U , 4
64 1%4 N4- 06 4- U,- -I 44 6- D 64 -I % 4.
N 0 0 0 0 0-4 -1 -4
.4 .4 . .. . N . . . .D . . , 0 6 )
V. MIRROR AND DETECTOR SYSTEM
A. DESCRIPTION OF SYSTEM
A scanning mirror and a detector were added to *the GRIN
best lens in order to determine if precise definition of
off-axis targets would be possible. The design of these
two new elements is shown in Figure 5.1.
The mirror is made to 'nod' at a high rate and thus
scans the spot across the detector. The detector consists
of two sections, as shown in Figure 5.1. Thus, when the
spot is scanned across the two plates a signal from each
detector will be generated and will result in the form
shown in Figure 5.2. This arrangement will allow an
accurate measurement of the mirror an~gle for which the spot
crosses the axis. The electronics of the detector signal
processor could then be arranged to relate the mirror angle
to the angle of incident radiation so that the precise
line-of-sight angle of an off-axis target will be known.
Note that this is a simple detector systemn that will
only resolve angles in one plane. A more sophisticated
system along the same lines could be designed with a four-
element detector and a more intricate method of scanningj
the spot across the detector. It was decided to keep the
mirror and detector system as simple as pos3sible since the
57
. 0-2-.t. A .. . . . . . . . . . -. . . ..
aim of this work is only to show that a precise angle
measuring device is feasible.
B. LENS TO DETECTOR RAY TRACE
This section describes the mathematics of the ray trace
from the lens to the mirror and then to the detector. A
listing of program DETPLOT which accomnplishes this task is
included as Appendix C.
The ray trace consists of the following parts:
a. ray direction exiting lens
b. point of intersection on mirror
c. ray direction exiting mirror
d. detector ray count
1. Ray Direction Exiting Lens
The ray direction exiting the lens is derived from
the direction cosines of the ray as calculated by program
GISL. This data is used as an input to program DETPLOT
along with the coordinates of the point at which the ray
left the lens.
2. Point of Intersection on Mirror
The point of intersection of the ray and mirror may
be described by the following expressions:
xm = DmK + x0(5.1)
Ym = D mL + y0 (5.2)
Zm DImM + z0 (5.3)
58
41
44
00
0 >0
59U
0
-~~.1
0
-o
600
Ca
O~. 9 OP OC .01
ILNflOJ AVU
Figure 5.2 Detector Signal. .
60
. ..
Since the surface of the mirror is tilted along the
z-axis it can be described by the equations:
Ym = tan9M(xm - MP) (5.4)
Ym 2 cos 2 3m + Zm2 < R (5 .5
Where equation 5.4 describes a tilted plane and
equation 5.5 limits the plane to the mirror diameter.
Neglecting equation 5.5 for a moment, equations
5.1, 5.2 and 5.4 may be combined to solve for DoM . The
result is:
tan m (x 0- MP)- Y0D = m P)-y
Pm L- K tan6 m
Now that DPm is known, equations 5.1, 5.2 and 5.3
may be used to find the coordinates of the mirror
intersection point. These are then checked with equation
5.5 to make sure they are not outside the mirror diameter.
3. Ray Direction Exiting Mirror
The solution of the ray direction leaving the
mirror is best understood vectorially. See Figure 5.3.
Reflection off a mirror is governed by two factors.
Firstly, the angles of incidence and reflection must equal.
Using vectors this is expressed as:
Sr. =- n. r1r
In component form:
n r . + nr nr r n nr(56x xi y yi xr y yr (5.6)
61
Secondly, the ray is reflected in the same plane as
the incident ray. This may be expressed vectorially by !using the cross product to generate the vector np as
shown in Figure 5.3. Thus:
np = rixn = rrxn
In component form:
ex(-nyrzi) + ey(-nxrzi) + ez(nyrxi-nxry i ) =
ex(-nyrzr) + ey(-nxrzr) + ez(nyrxr - nxryr)
Three equations result:
-nyrzi = -ny rzr
-nxrzi = -nxrzr
nyrxi - nxryi = nyrxr - nxryr (5.7)
The first two results are not unexpected as the
mirror should not effect the z-component of the ray.
Equations 5.6 and 5.7 may now be combined to find
the x- and y-components of the ray. When this is done the
two following expressions result:
(n 2 2 2nnr y -n ) xy
xr (n 2 + n2 xi - 2 + n 2 yix y x y
(n 2 n 2 2n nx y x y
yr (n2 + n2 )yi ( + 2 xix y x y
62
i . . . _ _. .. . / : : i .i . -::. _ : . . . -.- :." . . /., •: ;:-- : :: :- :• :: :-.;
mirror plane
I/
Figure 5.3 Peflection of Ray on Mirror.
63 1
7.c1
Since the vector n is entirely in the x-y plane,
its components nay be related to mirror angle as follows:
n X = -sin m
ny = COS m
When the above equations, along with well known
trigonometric identities are used the following expressions
will be found.
rxr = cos(2 m)rxi + sin(2 m)ryi
ryr = -cos(2 m)ryi + sin(2 m)rxi
Thus these equations represent the change in direction for
the exiting ray and form two of its direction cosines, the
third being rzr.
4. Ray Count on Detector
The coordinates of the ray at the detector are
described by the expressions:
xd = Dnd K' + x.
Yd = Dnd L' + Ym
zd = Dnd 1' + zm
Since xd is used to define the position of the
detector, it is known. Thus the value Dnd may be solved
and the corresponding y and z-coiponents may be found.
The ray is then checked to make sure it is within
the detector diameter. If it is the y-coordinate is
checked to find out if it is negative or positive, and thus .-
64
whether the ray has struck the upper or lower half of the
detector. The ray count is then incremented accordingly.
Program DETPLOT traces all rays for each increment
of mirror angle and thus generates plots such as that shown
in Figure 5.2.
C. MIRROR AND DETECTOR SYSTEM RESULTS
Appendix D shows the results of the detector signals as
a function of mirror angle at angles of incidence froin 0 to
40 degrees. For these the mirror pivot point was placed at
1.4 units along the x-axis from the inner surface cone
since this was the smallest distance that would still allow
full movement of the scanning mirror. The focal length was
adjusted to 2.8 units for the lens design calculations so
that the spot would be imaged as small as possible on the
detector. Figure 5.4 summarizes the results shown in
Appendix D. As can be seen, the smaller angles of
incidence (those in the range 0 to 16 degrees) result in a
linear relation between angle of incidence and mirror tilt
angle, corrc3ponding to an on-axis spot. The larger
values of the angle of incidence resulted in a more random
relation between these two quantities.
Appendix D also shows that the error in determining
mirror angle increases as the angle of incidence increasas.
This is due to the fact that each successive plot is less
distinct and thus a precise measure of the angle becomes
65
..
more difficult. Although this effect shows a deterioration
in the lens system performance, it is not harmful since, as
the missile corrects itself, the measurement of angle will
become more accurate.
For these reasons this simple system is considered to
be a viable means of target detection and shows that the
GISL can be a successful missile seeker lens.
66
.0
.. . . . . . .. ... . ./
C
/ cc
.. ..*. . . . . . . . . . . .. ..
... .. .. ... .. .. .. .. .. . . .0'
.. . .. . .. . . . .. .
. ... .. .. . .. ... . .. .. .. .. ... .. .. .. .. ..
. . . . . . . . . . . . .. .. ...
ot 9 ac z 0z of Z 9 t 0
ajN~aJN1 i sm~0
Figue 5.4 Mrro Til fo OnAxisSpo andAnge o
Incidence
670
VI. CONCLUSIONS AND RECOMIMENDATIONS
aA. CONCLUSIONS
Two objectives were accomplished in the course of this
study. Firstly, the Gradient Index Seeker Lens design, as
initiated by Carr [Ref. 41, was completed in that all
orientations of the center of symmetry with respect to the
lens were made possible to describe mathematically. The
results obtained for the spot size on the image plane
showed trends that indicated the parameters that would
produce a 'best' lens, but precise definition of these
values was not possible.
Secondly, it was shown that a scanning mirror and
detector system applied to the lens could be described
using ray-trace techniques and also that such a system
could be used to provide precise angle measurements of
off-axis targets at angles less than 16 degrees. For
larger values of angle, error in definition became
increasingly greater.
Specifically, then, the following conclusions are nade:
a) The Gradient Index Seeker Lens can be described
using ray-trace techniques and can be shown to provide
adequate imaging of off-axis objects.
b) A scanning mirror and detector system can be used
to provide precise angle measurements.
68
0
B. RECOMMENDATIONS
The following recommendations for further study are
made:
a) An optimization program should be applied to
program GISL to study the variations of all important
parameters.
b) A detector system capable of detection in all
directions should be designed.
c) The manner in which signal processing of the
detector signals can be carried out should be studied.
d) A study of Gradient Index material and the methods
by which a lens could be manufactured should be undertaken.
69
• I
APPENDIX A
CA.
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LIST OF REFERENCES
1. Frazier, Robert L., Exterior Ballistics, Guidance Laws
and Optics for a Gun-Launched Missile, M. S. Thesis,Naval Postgraduate School, Monterey, California,Decenber 1980.
2. Terrell, James M., Conical Lens for a 5'"/54 GunLaunched Missile, M. S. Thesis, Naval Postgraduate
School, Monterey, California, June 1981.
3. Amichai, Oded, Sharp Nose Lens Design Using RefractiveIndex Gradient, Naval Postgraduate School ContractorReport NPS67-82-003CR, Monterey, California,June 1982.
4. Carr, Herbert M., Aerodynamically Efficient GradientRefractive Index Missile Seeker Lens, M. S. Thesis,Naval Postgraduate School, Monterey, California,
October 1982.
5. Marchand, Erich W., Gradient Index Optics, AcademicPress, 1978.
6. U. S. Army Missile Command Technical Report RG-CR-32-6,Gradient Index Lens Research-Final Report, by D. T.Moore, p. 36, 19 October 1981.
7. U. S. Army Missile Command Technical Report H8 3-3-B020-1, Gradient Index Optics Application by R. L.
Light, 30 September 1983.
8. D. S. Davidson and A. E. Fuhs, Tracing Nearly RadialLines in a Spherically Symmetric Gradient Index Lens,Applied Optics, Letter to Editor (in press).
117
"-'--""---'" " -''-- " -- -- ' '. i..a..,,m i v- ' t
. . ...- ,, .... i....." .............................
INITIAL DISTRIBUTION LIST
No. Copies
1. Defense Technical Information Center 2Cameron StationAlexandria, Virginia 22314
2. Library, Code 0142 2Naval Postgraduate SchoolMonterey, California 93943
3. DiLectorate of Land Armament and Electronics 2Engineering and MaintenanceAttn: Colonel B. L. CodeNational Defense HeadquartersOttawa, Ontario, Canada KlA 0K2
4. Distinguished Professor A. E. Fuhs 4Code 67FuDepartment of AeronauticsNaval Postgraduate SchoolMonterey, California 93943
5. Col. R. Larriva, USMC 2Defense Advanced Research Projects Agency1400 Wilson Boulevard
Arlington, Virginia 22209
6. Charles R. ChristensenResearch Directorate U. S. Army Missile Command(DRSMI- RRO)Redstone Arsenal, Alabama 35898
7. Commander, U. S. Army Missile Command(DRSMI- ROA)Redstone Arsenal, Alabama 35898Attn: CAPT H. Carr
8. Professor A. E. MilneCode 61MnNaval Postgraduate SchoolMonterey California 93943
9. Professor D. T. MooreUniversity of RochesterRochester, New York 14627
118
. ..i
10. Dr. Lloyd Smith 1Code 3205Naval Weapons CenterChina LaKe, California 93535
Ii. Directorate of Land Armament and Electronics 2Engineering and MaintenanceAttn: Capt. D. DavidsonNational Defense HeadquartersOttawa, Ontario, Canada KlA OK2
12. Directorate of Personnel Education 1and Development (DPED)National Defense HeadquartersOttawa, Ontario, Canada KlA OK2
119
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