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273 611
AD
m/e

ARMED SERVICES TECHNICAL INFORMATION AGENCY

ARLINGTON HALL STATION
ARLINGTON 12, VIRGINIA

UNCLASSIFIED
NOTICE: When government or other drawings, speci-
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 AFCRL 62-18

273 611
7 Wide -angie Microwave Lens for
Line Source Applications

W. ROTMAN
R.F. TURNER

JANUARY 1962

ELECTRONICS RESEARCH DIRECTORATE

AIR FORCE CAMBRIDGE RESEARCH LABORATORIES
OFFICE OF AEROSPACE RESEARCH
UNITED STATES AIR FORCE
BEDFORD MASSACHUSETTS
AFCRL 62-I1

Wide -angle Microwave Lens for

Line Source Applications
W. ROTMAN
R.F. TURNER

PROJECT 4600
TASK 46008

JANUARY 1962

ELECTROMAGNETIC RADIATION LABORATORY

ELECTRONICS RESEARCH DIRECTORATE
AIR FORCE CAMBRIDGE RESEARCH LABORATORIES
OFFICE OF AEROSPACE RESEARCH
UNITED STATES AIR FORCE
BEDFORD MASSACHUSETTS
 ABSTRACT
The design equations for a constrained wide-angle, two-
dimensional microwave lens have been derived for the special
case in which the front lens face is straight and in which lens
elements can connect arbitrary points on the two lens contours.
A phase analysis indicates that this lens design has very small
coma aberrations and is capable of generating beams a fraction
of a degree in width.
Criteria are developed for selection of optimum lens
parameters for specific applications.
An experimental model in which the lens elements consist
of coaxial cables was constructed to demonstrate techniques of
fabrication. Radiation patterns indicated the expected character-
istics. The application of these principles to the design of sym-
metrical three-dimensional lenses is briefly indicated.
Tables of lens contour parameters and path length aberrations
are presented for the specific case of a scan angle a of 30 0.

ii
 CONTENTS
1. Introduction 1
2. Theory 2
2. 1 Derivation of Design Equations 2
2. 2 Selection of Optimum Parameters 8
2. 3 Lens Contour and Phase Error Calculations 10
3. Experimental Studies 16
3. 1 Wide-Angle Lens Design 16
3. 2 Radiation Characteristics of Lens Model 23
4. Conclusions 27
Appendix A: Lens Contour Calculations 29

Appendix B: Phase Error Calculations 40

TABLES
Table 1. Lens Contour
a= 300, and rParameters
= 0.597 for g = 1.137, 29
Table 2. Lens Contour Parameters for g = 0.90
<0.05>1.200 and a= 300 33
A)g= 0.900(r= 3.70); (b)g= 0.950
r= 1.53); () g= 1.000 (r= 1.00);
j) g= 1.050(r= 0.771); (e) g= 1.100
(r = 0.651); (f) g = 1.150 (r = 0.582);
(g) g= 1.200 (r= 0.541)
Table 3. Normalized Path Length Errors, A I, for
Microwave Lens a= 300 41
(a) g= 0.900; (b) g= 0.950; (c) g= 1.000;
Sg= 1.050; () g = 1.100; () g = 1.150;
(g) g= 1.200
Table 4. Normalized Path Length Errors A for
Microwave Lens a = 300 and g = 1. 137 55
References 56

V
 ILLUSTRATIONS
Figure Page
1. Parallel-Plate Microwave Lens 3
2. Ray Trace Diagram for Two-Dimensional Electromag-
netic Lens 4
3. Microwave Lens Parameters 9
cc)G = Lens
4, Microwave 1. 137F; (d) G = (a)
Contours, G = 1. 10F# (.? G = F;
1. 20F 11
5. Path Length Errors in Microwave Lens, (g) G = F;
(.b) G = 1. 10F; (c) = 1. 13T 13
6. Parallel-Plate Lens as Line Source Feed for Reflector 18
7. Experimental Microwave Lens Antenna (Front View) 21
8. Experimental Microwave Lens Antenna (Rear View) 22
9. H-Plane Radiation Patterns of Microwave Lens Antenna,
(a) 0 = 00 (On-Axis); (b_)0 = 150; (g) 0 = 300 24
10. Microwave Lens Antenna with Phase Shifters in Lens
Elements 25
11. Radiation Patterns of Microwave Lens Antenna with
Phase Shifters in Lens Elements 26

vii
 WIDE-ANGLE MICROWAVE LENS FOR
LINE SOURCE APPLICATIONS

1. INTRODUCTION
The wide-angle scanning characteristics of two-dimensional micro-
wave lenses have been extensively investigated 1 and applied to the design
of radar antennas. Ruze 2 has shown, for example, that constrained
lenses of the no-second-order type are capable of generating 10 beam-
widths and of scanning these beams over angles as great as one hundred
beamwldths. The no- second- order designation refers to a lens with two
perfect off-axis symmetrical focal points and an on-axis focal point for
which the second- order phase deviation is zero. For no-second-order
lenses both front and back lens faces are curved.
Ruze also discusses the design of a straight-front-face lens re-
quired for line source applications as the primary illuminator for a para-
bolic cylindrical reflector or as the feed for a rectangular planar array.
This straight-face lens has excellent scanning characteristics since both
second and third order coma may be almost eliminated by proper de-
focussing. It has two perfect, symmetrical, off-axis focal points and a
highly corrected on-axis focal point. For very narrow beam antennas,
however, its higher order coma aberrations may still be objectiunable.
A further improvement in coma aberrations may be achieved by
applying general lens design principles, developed by Gent et al. 3, 4
to the special case of the straight-front-face lens. This results in a
lens design with three perfect focal points, two symmetrically located
off-axis and one on-axis. It is the purpose of this.paper to obtain the
design equations for the improved straight-front-face lens from Gent's
generalized equations, to evaluate its phase aberrations and scanning
capabilities, and to demonstrate fabrication techniques applicable to this
type of design.

Received for publication September 1, 1961

2.

2. THEORY
2. 1 Derivation of Design Equations
A two-dimensional schematic representation of the straight-front-
face lens is shown in Fig. 1. The basic external difference
between this lens and Ruze's design is that the lens elements are
lengths of coaxial transmission line, rather than waveguide. This
permits the connection of arbitrary points on the front and rear surface
of the lens so that corresponding front and rear surface distances,
N and Y, for a single lens element, are not necessarily equal (as they
are in Ruze's design). This additional degree of freedom permits
specifying four independent conditions to determine the lens uniquely,
rather than the three conditions that were available to Ruze. In the
present lens design, these conditions were selected as the straight-
front face, the two symmetrical off-axis focal points, and an on-axis
focus. The ends of the coaxial cables that form the transmission line
elements of the microwave lens are connected directly to a straight
line of radiators to form a line source.
The formulation of the lens design equations and notation follows
3
that of Gent. In Fig. 2, the lens surfaces are shown two-dimensionally
by the cross sections Z 1 and Z 2 . The first contour, Z I, determines
the position of the probe transitions between the parallel plates and the
coaxial cables. The second contour, 2 2' is straight and defined by the
location of the radiating elements that comprise the line source. Corres-
ponding elements on contours Z, and Z2 are connected by a transmisson
line TL.
The contour Z1 is defined by the two coordinates (X, Y) that are
measured relative to a point 01 on the central axis of the lens. Points
on the straight contour Z2 are similarly determined by the single co-
ordinate N, measured relative to the point 02 on the axis. The points
01 and 02 lie on contours Z and Z2 respectively and are connected by
a transmission line TL0 of electrical length Wo . The point P, defined
by the coordinates X and Y, is a typical probe element in Z2i and is con-
nected to point Q, which lies on Z 2 and is defined by the coordinate N,
by the transmission line TL of electrical length W. The three quantities
X, Y, and W can be chosen at will; thus this straight-front-face lens has
 3

INPUT PARALLEL STRAIGHT FRONT LENS

HORN PLATE FACE: LINEAR ARRAY
+ N OF RADIATORS

FCACLE

TOP VIEW FC

INPUT PARALLEL
HORN PLATES

SECTION A -A
FIG. 1. Parallel-plate microwave lens.
4

P(XY)~~ I 21C

c~(-Fcosa,Fsina) electrical K

/Section of
wovef rant

FIGtrce
2.Ra iaramfortw-diensoLelcrmgtilns
 5
three degrees of freedom. Other types of lenses, (including Ruze's
design in which Y= N) have at least one less degree of freedom.
Actual values for the three degrees of freedom will now be
selected to obtain wide-angle scanning characteristics. These design
parameters include (Fig. 2) two symmetrical off-axis focal points,
F 1 and F2, and one on-axis focal point, G, having coordinates (-F cos a.,
F sin a), (-F cos a, -F sin a), and (G,O) respectively relative to the point 0.
A ray through the lens at the origin is represented by F 1 0 1 0 2 M and
F 1 PQK represents any other typical ray.
The lens is now designed so that the three focal points F 1 , F 2 ,
and G give perfectly collimated beams of radiation at angles to the axis
of - a, + a, and 00 respectively.
In our special case Gent's 3 equations for the optical path-length
conditions for path-length equality between a general ray and the ray
through the origin are:

(F 1 P)+W+N sin a=F+W O , (1)

(F 2 P)+W-Nsinc=F+W o , (2)
and
(G 1 P)+W=G+ W o , (3)
where
(F 1 P) 2 =F 2 +X 2 +y 2 +2FXcos a-2FYsina 3 (1)

(F P) 2 F 2 +X 2 +y 2 + 2FX cos a+ 2 FY sin a, (2a

(G 1 P) 2 (G+ X) 2 +y
2 (3a)
and

F 1 P j F 2 P, and G 1 P represent path lengths from focal points

Fly F 2 , and G respectively to the rear surface of the lens.
We now normalize relative to the focal length F by defining a
new set of parameters
11=N/F) x=X/F, y=Y/F,

W-W
W= F g=G/F.
6

Also
ao =cos a, b,= sin a.
Equations (1a) to (3a) may then be written
(F1 X) 2 2
F 2
1+x 2+y 2+ 2ax - 2 boy,
F2 - (M)

(F 22P) 2 2
F +x +y2 + 2aox + 2boy,
(2D)

and
FGP2 =(g +x) 2 +yy2 (3Db)
F2 2

Combining the normalized Eqs. (1) and (lb)

(F 1 P) 22 2
F2 F =(1 -w-bo1 ) 2
- &-v-.)

= 1+w 2+ bo 2 -2bio + 2bow j (1-0

= 1+x 2 +y 2 + 2aox- 2boy.

Since the off-axis focal points are symmetrically located, the lens
surfaces must also be symmetrical about the center axis. This means
that, if q is replaced by - 11 and y by -y3 Eq. (lc) remains unchanged.
Equation (1c) can therefore be separated into two independent equations;
one contains only odd powers of y and il while the other contains the
remaining terms. Thus,

- 2b o 11 + 2bow= - 2boy
or (4)
y = (1-w).

-1
 7

Also
x 2 + 2 + 2aox = w 2 + bo2 I22w. (5)

Equations (3) and (3b), relating to the on-axis focus, may likewise
be written

(GP) (g _ w) 2 = (g +x) 2 + y2 (3c)

2
F
or
x 2 + y 2 + 2gx = w 2 - 2gw. (6)

Equations (5) and (6) can be combined to give the following re-
lation between w and 11
aw2+ bw + c 00 (7)

where
a= - _ io

c= 22
1bo -b1 2 2]

ao -4(g - ad T)

Equation (7) is a quadratic equation in w whose solution is

w- -b+b 2 _ 4ac

This completes the solution for the lens design. For fixed values

of a and g, w can be computed as a function of 11 from Eq. (7). These

values of w and ' may then be substituted into Eqs. (4) and (6) to deter-
mine x and y and complete the specification of the lens dimensions.
8
2.2 Selection of Optimum Parameters
The design procedure, as outlined, gives a lens which has three
perfect focal points, corresponding to the angles + a and 00. For wide
angle scanning, the lens must focus well, not only at these three points,
but also at intermediate angles along some focal arc. The basic lens
equations do not indicate how to select the remaining variables, such as
the factor g (ratio of on-axis focal length G: to off-axis focal length F)
to minimize the overall phase aberrations.
A clue to the optimum value of the parameter g may be obtained
from Ruze's phase error analysis of an electromagnetic wave lens under
the restrictions that y = r. His basic design assumes a lens in which
the focal arc is a portion of a circle of radius F, centered at the origin
of the surface Z1. For the special case of a straight-front-face lens,
he shows that minimum coma and overall phase error is obtained by de-
focussing the feed from the assumed focal arc by an amount equal to
1/2 (a 2 _ 2) F where 6 is the intermediate angle at which correction Is
desired. With this defocussing the residual aberrations are quite small
and the lens can scan a narrow beam over wide angles.
Under the above conditions, Gent's (y ) and Ruze's (y= r)designs
for a straight-front-face lens should have closely related parameters
since in the former case the on-axis aberrations are at zero while in the
latter case these aberrations are at a minimum. It would therefore seem
reasonable to select a value of gin our derivation that corresponds to
this same amount of defocussing (0 = 00). This establishes the optimum
value of g as:

_G 1+c2
1+ 2" (8)

The focal arc is now selected (Fig. 3) as a segment of a circle of

radius R, whose center lies on the axis of symmetry of the lens and which
passes through the two symmetrical off-axis and the one on-axis focal
points. The phase error from any point on this focal arc (expressed as

-6
 .£

1
ic a
L-
/ 0)-

cr 04

Li /
4
/0,j~ 04

xL T
10

the difference in path length between the central ray and any other ray)
may be shown to be:

Al = L= (h2 +x2+y2+ 2hx cosO - 2hy sin)2 - h+ w+ sinO (9)

where
AL =path length error,
h H/ F = normalized distance from point on focal arc to origin
01 of surface Z1 . H is determined from the triangle with
sides R, H, and G-R and with included angle (Fig. 3).
0 = angle between central axis and point on focal arc.
R = radius of focal arc (determined by the three points G, F, and
F 2 on the arc).
2.3 Lens Contour and Phase Error Calculations
The preceding section indicated that an optimum set of lens para-
meters exists, in the sense that they provide minimum phase aberrations
over a prescribed range of angles. This selection of parameters assumes
that the correction for second and third order coma terms also results
in the minimization of the higher order aberrations. Since this is not
obvious, an investigation was conducted to determine the phase errors
in lenses of this type for parameters that may differ from the optimum
value.
The total scan angle, 2a, for the lens was fixed at 600 (a= 300),
as representative of a wide-angle lens system. From Eq. (8), the
optimum on-axis focal point is given by g= 1. 137. Accordingly, calcu-
lations were made on a digital computer of the normalized lens contour
parameters, y, x, and w and also of the normalized phase error, A4
from Eqs. (4), (6), (7), and (9) for the following range of values:
g= 0.90 <0. 05 >1. 20, and 1.137.
0= ± 50 ±150 ±25o ±35%
r = 0 <0.25 > 0. 80.
For = 00 and±300, At, is zero.
Selected lens contour curves are shown in Figs. 4a to d for
g = 1.00, 1. 10, 1. 137, and 1. 20. Their tabular values are given in
 170Y 17,y

Dimensions normalized with distance 1.

to off -atils perfect focal point
.6 .6

.5 .5

4*.4 .4
.3 -3

FF
On-ais pirft . 0.
.,ffa point0.4
x G~ 0 -.--
-12 -1.1 -1.0 -.9 -8 -. 6 n5 -4 -3I 2n -. -1-2 -11 -1.0 -.8 -.8 -.7 6- -4 -'

k , " uf. -.2 -2

9-aOuter lens surface 21--

d'-.4
-.4

surface .6r .

Contur for .
7 y lens
Ft. .6
F,(Ruzis design)
5

-4 .

F .3 F

0.l OF0.
x - -tl -1.0 -.8 .8 -.7 -. 1I n -3
-62 5i - ~ 0O -.
1.65.0 -. 4F
It 1.0 -5~ e 7 .6 -.5 -.4 .3 -2 -.1
0
-1

-.2

F -3F

-4.

Fe Ire5

(c)

FIG. 4. Microwave lens contours, (_4) G = 1. 10F; jP G =F;

12

Appendix A for g = 0.90 <0.05> 1.20 and 1.137. The light lines
between the rear and front lens contours indicate corresponding
values of y and i which are the junction points for the coaxial lens
elements.
A comparison is made in Fig. 4c between the rear lens con-
tour for the Gent (y ? i) and the Ruze (y = 11) designs for the opti-
mized value of g = 1. 137. In the latter case the straight-front-face
lens equations are
222
x 2 +ao y 2 + 2aox =0, (10)

w=0, and 9=y.

The rear lens contour is therefore elliptical and does not depend on
the value of g. It should be noted that the lens contours for both
y / i and y = 1 conditions practically coincide for values of T1 less
than 0.65. This agrees with our original assumption.
The lens contour for which g = 1.00 (Fig. 4b) is also of interest
in that it is the onJly case for which the focal arc is centered at the
vertex of the rear lens face and for which the central ray paths from
all points on the focal arc are equal in length. Such considerations
may be of importance in monopulse applications. This lens has been
extensively investigated by Hatcher 5 who showed that its inner con-
tour could be approximated by a segment of a circle. Its phase aber-
rations, however, are considerably poorer than those for the
optimum design.
Phase aberrations, expressed in terms of the normalized dif-
ference in optical path length relative to the central ray A l, are
tabulated in Appendix B for g = 0.90<0.05>1.20 and 1.137, and shown
graphically in Fig. 5a to c for g = 1. 00, 1. 10, and 1. 137. It can be
seen that, for values of il less than t0. 53, g = 1. 137 is indeed optimum
in the sense that the path length error, A 1, remains below +. 0.0001
for all angles of scan up to +350. If the permissible path length error,
A, can be as great as -0.0005, however, a lens with g = 1. 10 may be
more suitable since it would permit apertures up to T1 = 0.8. Note that
the path length errors for the g = 1.00 lens have essentially even sym-
 13

'
j . __ , __ ,0

I'

.... . : ... . " - -1

"i - ______ - II

j ,. Q....

/ . ,4I
__ __ / __

! I I " , I +"

_____ ____ -; A
14
metry with respect to y] while those for g = 1. 10 and g = 1. 137 have odd
symmetry. (The path length error curves show only positive values
for e while I takes on positive and negative values. Alternatively,
both positive and negative values of e could be used with only positive
values of Tb)
The minimum beamwidth obtainable from a microwave lens is
determined by the size of the aperture, the operating wavelength, and
the maximum permissible phase or path-length aberrations. The
beamwidth for a rectangular aperture antenna with a cosine illumina-
tion taper and 23-db sidelobes (typical of the usual antenna practice) is
given by
HPBW=690x (11)
D
where
H P B W is the half-power beamwidth,

D (= 2 Nmaxcos 0) is the projected aperture,

X is the wavelength, and
Nmax is one-half of the physical aperture.
For minimum beamwidth, the maximum path-length error
deviation (- 2 A Lmax) cannot exceed about 1/4 % without adversely affect-
ing the sidelobe level.
Thus
(AL)max- (12)
8
Also
D= 2 NmaxCos e (13)
Combining Eqs. (11), (12), and (13)
m (Cos
14)
4 (AL)max

and the minimum possible beamwidth for a given lens is

(HPBW)m (A)max x 2760. (15)

flmaxCOs e
 * 15
As a typical example, the following parameters are selected to give
the minimum beamwidth:
g =1.137,

Ji 'max 0. 55,

G/ -
G/2max = . 035,

-30°<0 <+ 30 ° ,
and
(A)max = 0.00013 (from Fig. 5c and Appendix B.
Then
(HP BW)min = 0.0650 (For 0 =00)
=0.0750 (For 0 =300).
Thus, we can scan a beam of less than 0. 10 over 600 for a total scan
angle of greater than 600 beamwidths.
As a second example, we will try to minimize the ratio G/D
(focal length to aperture ratio) at the expense of a somewhat greater
minimum beamwidth.
The following parameters are therefore chosen:
g=1. 100,
()max = 0.0005,
-300 <0<+300,
max w=0.80
G/D =0.687(For 0=00),
and
<HPBW)mtn=0.l 8 0(0 =00)
=0.210 (0 = 300).
For both these examples, the scan angle 0 can be extended to
+35 0 with very little deterioration of radiation pattern. Note, also,
that defocussing from the assumed circular focal arc to some other
noncircular contour that also passes through F 1 , F 2 , and G does not
accomplish much in reducing coma aberrations (except for the case
16
of g = 1.00) since defocussing primarily affects the even- order aber-
rations whereas the residual phase aberrations are primarily of odd
order.
The g = 1. 137 lens contour is thus optimum in the sense that it
minimizes the obtainable beamwidth for a reasonable F/D ratio. The
value of g = 1. 10 seems more suitable, however, if the F/D ratio must
be minimized and if the permissible beamwidth is greater than 1/40.
Either of these two designs should be equally applicable, however, to
the majority of lens design problems.
3. EXPERIMENTAL STUDIES
3. 1 Wide-Angle Lens Design
The analysis of the previous sections has shown that the design of
wide-angle microwave lenses, with beamwidths on the order of fractions
of a degree, is theoretically feasible. A two-dimensional model of such
a microwave lens was constructed to determine the problems inherent
in this design. Design specifications include the following parameters:
fo = 3.0 Gc (design frequency),
g = 1.137,
lmax = 0. 60,

D/% =18,
emax = 300,
HPBW = 3 0 (Nominal Half-Power Beamwidth),
Primary Illuminator- Open- Ended Waveguide Horn,
Lens Elements-RG-9/U Coaxial Cables.
Refer to Fig. 4c for the lens contour and to Fig. 5c for the normalized
path length error At.
The beamwidth for this lens was chosen to keep its physical dimen-
sions within reasonable limits. On the other hand, the theoretical phase
errors inherent in this model are so small that they cannot be detected
by their effect on the radiation pattern or other electrical characteris-
tics of the lens. For example, either the Ruze (y = 11) or Gent design,
for g = 1.00, 1.10 or 1. 137 (Figs. 4 and 5) results in lenses with equiva-
lent electrical performance when the beamwidth is approximately 30 .
 17
Our objective is not to obtain an experimental comparison between the
competitive Ruze and Gent lens designs,. but rather to demonstrate
techniques unique to the construction of microwave lenses with variably
spaced coaxial lens elements.
The microwave lens model (Fig. 6) uses the parallel plate and
coaxial TEM modes to obtain maximum bandwidth. Microwave radia-
tion from the primary horn illuminators, located along the focal arc,
propagates between the parallel plates to the reflector-backed probes
that form the rear lens contour. These probes extract the energy from
the parallel-plate region and feed it into the coaxial cable lens elements
which, in turn, excite the probes on the straight-front-lens contour.
These front probes form a linear array between a second set of parallel
plates that radiate into space through a short TEM horn transition. The
lens in Fig. 6 is shown feeding a cylindrical parabolic reflector that
collimates the beam in the elevation plane.
The primary horn illuminator is designed for a prescribed ampli-
tude distribution along the front lens face by selection of its aperture
dimensions and orientation. Its required radiation pattern between the
parallel plates is first determined graphically by ray-tracing, equating
the power radiated per unit length along the front lens face to the angu-
lar sector subtended by this power flow at the horn position on the focal
arc. The aperture and orientation of the horn illuminator are then se-
lected7 to give the closest approximation to the desired primary pattern.
The realizable lens illumination differs somewhat from the theoretical
value because the required primary pattern is not symmetrical (except
f or the on-axis position e = 00) and does not conform exactly to the
physically realizable patterns obtained from a uniformly-phased horn
illuminator. The farfield radiation pattern is, however, not very sensi-
tive to small changes in illumination taper. Since the input horn's
directivity is a function of its position along the focal arc, its dimensions
and orientation depend upon its focal position. Also, the peak of radia-
tion is not, in general, in the direction of the vertex of the lens. For
example, the required horn parameters in the present model vary from
an aperture of 1. 2% (primary HPBW = 41.0) for the on-axis focal point
18

(I))
ww z
WZZ
I-8
-S
o<

z 2
zoLaa. j
J0~~ ir CA:'.

wo
2)
U)

0 4
W

W)
0
m-

z w
 19

to 1.45% (primary HPBW = 340) for the 0 = 300 focal point. Since
several stationary horn illuminators are used in the experimental
model, this change in horn dimensions causes no difficulty. If the
beam scanning were accomplished by moving a single input horn along
the focal arc, a compromise aperture dimension (for example, 1. 3%)
would have to be selected.
The dimensions and spacing of the probes must be chosen to
ensure adequate coupling of the lens elements to the parallel plate
structures over a wide range of incident angles. The performance of
each probe is affected by mutual coupling to its neighbors. The prob-
lem of analyzing the behavior of a set of probes located along a curved
contour and excited by a curved wavefront is quite difficult. An approxi-
mate evaluation can be obtained, however, by considering the perform-
ance of an infinitely long linear array of uniformly spaced probes
phased to radiate a plane wave front at an angle, y, normal to the array.
The relation between this angle of radiation and the electrical phasing of
the probes, Yp, is given by

-)- siny ± 27cn n = 0, 1, 2, 3, and so forth, (16)

0

where d is the spacing between probes.

Equation (16) has multivalued solutions for y when d/ X0 is large,
corresponding to the generation of secondary maxima in the diffraction
pattern. This condition must be avoided by restricting the probe spac-
ing to values of d/X 0 given by
d/ X 1
'0< Isin yo J+ 1, (17)
where y is the solution of Eq. (16) with n= 0. This relation assures
that Eq. (16) is single-valued in y and has a real solution for n = 0 only.
The probe spacing is therefore restricted to spacings of less than X for
broadside radiation (y 0 = 0* ) and to 1/2 X for endfire radiation
(YO = 900).
20
The restrictions of Eq. (17) are somewhat more stringent than
required for the experimental model since this equation applies to a
set of omnidirectional probes. The limited range of the angle of
radiation, y, restricted by the dipole-like element pattern of the
reflector-backed probes to within about +_600, permits a probe spac-
ing of about 0. 65 X0 rather than the value of 0. 535%o indicated for a
maximum value for y of 600. The probe spacing along the straight-
front-lens face, held at a constant value of 0. 50%, places the coaxial
elements at equal increments of the parameter r. This results in
unequal spacing of the coaxial elements and probes along the rear lens
face (since y = r). This spacing is always less than the permissible
probe spacing of 0.65?.
The probe dimension is determined. by constructing a section of
an array of probes, backed by a straight reflecting surface, between
parallel plates. Tests wereinade of the input impedance of a single
probe when all the other probes were terminated in matched loads and
also of the power reflected from this array for angles of incidence of
a plane wave of up to 600. It was found that the best probe impedance
match and the minimum reflected power occur at the same values of
probe parameters. The probes penetrate 0. 15% and are located about
0. 21 X from the reflector; the probe spacing and the parallel plate spac-
ing are 0. 5. and 0. 375% respectively. It should be noted that the spac-
ing between probes determines the probe penetration and distance from
the reflecting surface.
The theoretical lens contours coincide with the reflecting surface
behind the probes, rather than with the position of the probes them-
selves, since the phase center of radiation for a probe imaged in a
ground plane is located at the surface of the reflector. The reflecting
surface for the rear lens contour was therefore determined from the
theoretical values of x and y. The probe position is located 0.21?.
from this contour (measured along its normal). Front and rear views
of the completed microwave lens are shown in Figs. 7 and 8. Several
horns are used simultaneously to give a number of independent beams.
The flexibility of the coaxial cable lens elements permits displacement
 21

M
0

C)l
z 0
0 w
-J -

IL

00

ClD

I-
col

wr

01 Cou
o
CL

a I1

c-a
w
z 0
Cl)
22 c
0
0 z 0
0 (D
z Q w
o 0)
z w
U))
zw
CL
LLJ -i
a)
I>
LL C'
I-r
ma

ca)

U))

Li-
 23
of the line source, which forms the straight front face of the lens,
from the rest of the lens structure.
3.2 Radiation Characteristics of Lens Model
Many radiation patterns of this microwave lens were measured
for different angular positions and sizes of the horn illuminators and
for focussing adjustments over a range of frequencies. Some repre-
sentative patterns are shown in Figs. 9 a to c for a frequency of
3 Gc and for e = 0*, 150, and 300. The horn was located on the focal
arc and had an aperture of 1. 45%. In all cases the position of best
focus was found to be within one inch of the focal arc and was not par-
ticularly critical. The radiation patterns were also measured over
the frequency range of 2. 8 to 3. 2 Gc with no significant deterioration
of performance. The measured sidelobe levels, as high as -18 db,
are somewhat inferior to the -22 db design value. This deviation is
probably caused by a nonoptimum selection of horn aperture size
(which would also account for the measured beamwidth being some-
what smaller than the theoretical value), as well as by constructional
tolerances.
An alternate technique of scanning the antenna beam of a wide-
angle microwave lens is the introduction of a linearly progressive,
time-variable phase delay in the coaxial lens elements. This in turn
causes a progressive phase shift between radiating probes of the
straight front face of the lens and a corresponding shift in the position
of the radiation pattern's peak. Thus, by putting phase shifters in the
coaxial cables of the lens, a beam from a stationary input horn illumi-
nator may be scanned over a limited region of space.
The original concept for the antenna studies described in this
paper was that of a microwave lens that would combine a series of
angularly spaced, stationary input horn illuminators with coaxial phase
shifters in the lens elements. Although the beam generated through the
lens by each horn points in a different direction, all the beams can be
moved simultaneously by varying the phase shifters with progressively
increasing phase. Thus, by spacing the horns at the proper angular
24

'!it I'i !p I I, 1- Cd)'"J

Hill i . ,1 1,1

l';.. -- T

-5 -2N
4 !il 1.24 3

Ti i JA
~~IT~U 1.'L V Ij
L.'
4-2 b

-3 -4
. - ' 1- . 6 12-' 1 824 0

j+1~~ 4.FT1

1-l;:, r -H"

T-.. .. ... . T e A lrhfl 4-1

1:1 17' r I-
441 7'T 11' III
it It1111 i-

.... . . ~ - X: - ;P 7

. ........ ... ?.
.. -

I:4 Is:10 25
U14-~
FIG. ralatio
9 H plne pattrns o len ane,
Nicrwav
(.a~~~~;
b .10 ~0=3g 0(nAi) 7
 25

w
cr.,
w LL
(.

z c,,

IL _ _

-Jn

CO)

L
26

r= (
-F-4 Md
-A*-3["
31jj*

LN

*Q- ~O
N N Y IJO N

i~a)

pip,

r--

OPd

j - -: I,
 27
intervals, the entire field of view of the lens may be covered by
several independent beams, each of which scans only a small sector.
This antenna concept was designated as the Multiple-Beam Interval
Scanner (MUBIS) system. Although its development has been delayed
(replaced by a competitive antenna system 8 with a wide-angle micro-
wave lens and coaxial organ-pipe scanners), a study was made of the
action of the coaxial phase shifters in beam scanning. Figure 10
shows the wide-angle microwave lens model, modified by adding
trombone-type coaxial phase shifters (line stretchers) to each of the
coaxial cable lens elements. The phase shifters are adjusted to in-
crease the cable lengths linearly along the face of the lens. A set cf
radiation patterns is shown in Fig. 11 for the case in which the input
horn is located at 300. The line stretchers are adjusted to scan the
beam ± 120 about this central value in 30 steps. Only the region in
the vicinity of the peak of each radiation pattern is shown. It can be
seen that the gain of the antenna remains fairly constant and that the
pattern shape does not deteriorate severely even for these large angles
(420 maximum) from the normal. Similar patterns have been obtained
with the input horn set at 00 and 150.
4. CONCLUSIONS
The design principles presented in this report for two-dimensional
lens structures may also be applied to three-dimensional lenses. The
additional degree of freedom obtained by allowing nonuniform lens-
element spacing (y 9 1), instead of the more conventional uniform
spacing (y = ij), permits the specification of two symmetrical off-axis
and one on-axis focal points for a two-dimensional straight-front-face
lens. In the Ruze (y = 11) design the inner lens contour is uniquely de-
termined by specifying the scan angle a. It is also known 1 that no so-
lution is possible for the design of a three-dimensional, axially sym-
metrical lens if the Ruze constraints are applied. The inner contour
of the Gent (y 3 q) two-dimensional lens design is however determined,
not only by the required scan angle a, but also by the normalized on-
axis focal length, g. Figure 4 shows that the shape of the inner lens
contour depends strongly upon the parameter g.
28
A three-dimensional straight-front-face lens design may be derived
as a figure of revolution of an appropriate two-dimensional lens contour.
The parameter g that controls the inner lens contour should be selected
to minimize the lens' astigmatism, as measured in the principal plane
orthogonal to that of the selected focal points. A three-dimensional
phase-error analysis would be required to assure that aberrations
are within permissible tolerances over the entire lens surface.
It is also possible that the basic Gent lens equations may contain
a solution for the design of a three-dimensional lens with two perfect
and symmetrical off-axis focal points for which both lens surfaces can
be nonplanar. Since these three-dimensional lens designs have not
been extensively investigated under the present study, no numerical
solutions are available.
The analysis presented in this paper has shown that microwave
lenses that are capable of scanning wide-angle sectors with beamwidths
on the order of a fraction of a degree are theoretically feasible and
also practical to construct. These line sources may be used as primary
feeds for parabolic cylindrical reflectors or planar arrays in large
radar antenna systems.
 29

Appendix A

LENS CONTOUR. CALCULATIONS

The variables x, y, and w that specify the lens contour have

been computed as a function of ij for the straight-front-face lens
design from Eqs. (4), (6), and (7) of the text and are presented
below for the following range of parameters
a= 300,
g =-1.137; 1 =0<0.01>0.75 (Table 1)
and
g =0.90 0 05> 1.20; 0 <0.05> 0.80 (Table 2).
The normalized radius of curvature, r = R/ F, of the focal arc is
also given for each value of g.
The lens contour parameters for g = 1.00, 1.10, 1. 137, and
1. 200 are also shown graphically in Fig. 4.

Table 1.
Lens Contour Parameters for g = 1.137, .a= 300, and r = 0.597

1 w -x y
0.00 0.00000 0.00000 0.00000
0.01 0.00000 0.00005 0.01000
0.02 0.00002 0.00019 0.02000
30

Table 1. (Contd
w -x y
0.03 0.00004 0.00043 0.03000
0.04 0.00007 0.00077 0.04000
0.05 0.00011 0.00121 0.04999
0.06 0.00016 0.00174 0.05999
0.07 0.00021 0.00237 0.06999
0.08 0.00027 0.00309 0.07998
0.09 0.00034 0.00391 0.08997
0.10 0.00042 0.00483 0.09996
0.11 0.00051 0.00584 0.10994
0.12 0.00060 0.00695 0.11993
0.13 0.00070 0.00815 0.12991
0.14 0.00080 0.00945 0.13989
0.15 0.00091 0.01084 0.14986
0.16 0.00103 0.01233 0.15984
0.17 0.00115 0.01391 0.16980
0.18 0.00127 0.01559 0.17977
0.19 0.00140 0.01736 0.18973
0.20 0.00153 0.01922 0.19969
0.21 0.00166 0.02118 0.20965
0.22 0.00179 0.02323 0.21961
0.23 0.00192 0.02537 0.22956
0.24 0.00205 0.02761 0.23951
0.25 0.00217 0.02993 0.24946
 31
Table 1. (ontd)
w -x y
0.26 0.00230 0.03234 0.25940
0.27 0.00241 0.03485 0.26935
0.28 0.00253 0.03744 0.27929
0.29 0.00263 0.04013 0.28924
0.30 0.00273 0.04290 0.29918
0.31 0.00281 0.04575 0.30913
0.32 0.00289 0.04870 0.3190.8
0.33 0.00294 0.05172 0.32903
0. 34 0.00298 0.05484 0. 33899
0.35 0.00301 0.05803 0.34895
0.36 0.00301 0.06130 0.35892
0.37 0.00298 0.06466 0. 36890
0.38 0.00293 0.06809 0.37889
0.39 0.00284 0.07160 0.38890
0.40 0.00273 0.07519 0.39891
0.41 0.00257 0.07884 0.40895
0.42 0.00237 0.08257 0.41900

0.43 0.00213 0.08637 0.42909

0.44 0.00183 0.09023 0.43920
0.45 0.00147 0.09416 0.44934
0.46 0.00105 0.09814 0.45952
0.47 0.00056 0.10218 0.46974
0.48 -0.00001 0.10628 0. 48000
0.49 -0.00066 0. 11042 0.49032
32

Table 1. (Contd)
w -x y
0.50 -0.00142 0.11461 0.50071
0.51 -0.00227 0.11883 0.51116
0,52 -0.00325 0.12309 0,52169
0.53 -0.00435 0.12738 0.53231
0,54 -0.00560 0.13168 0.54302
0.55 -0.00701 0.13600 0.55385
0.56 -0.00859 0. 14032 0.56481
0.57 -0.01038 0.14463 0.57592
0.58 -0. 01238 0. 14892 0.58718
0.59 -0.01464 0.15318 0.59864
0.60 -0.01717 0.15739 0.61030
0.61 -0.02001 0.16153 0,62221
0.62 -0.02321 0.16559 0.63439
0.63 -0.02681 0. 16954 0.64689
0.64 -0.03086 0. 17335 0.65975
0.65 -0.03543 0. 17699 0067303
0.66 -0. 04060 0.18042 0.68679
0.67 -0.04645 0. 18359 0.70112
* 0.68 -0.05910 0.18646 0.71611
0.69 -0.06068 0.18895 0. 73187
0.70. -0.06935 0,19097 0.74855
0.71 -0.07932 0.19243 0.76632
0.72 -0.09085. 0,19320 0.78541
0,73 -0. 10427 0.19311 0.80611
 33
Table 1. (Contd)

I w -x y
0.74 -0.12000 0.19194 0.82880
0.75 -0.13861 0.18940 0.85395
0.80 -0.3172 0.1349 1.054

Table 2.
Lens Contour Parameters for g = 0.90<0.05>1. 200 and a= 300

Table 2a. g= 0.900 (r- 3.70)

w -x y
0.00 -0.0004 0.0000 0.0000
0.05 0.0020 0.0034 0.0499.
0.10 0.0079 0.0135 0.0992
0.15 0.0179 0.0302 0. 1473
0.20 0.0318 0.0536 0.1936
0.25 0.0497 0.0836 0. 2376
0.30 0.0717 0.1200 0.2785
0.35 0.0979 0.1626 0.3157
0.40 0.1282 0.2114 0.3487
0.45 0.1627 0.2662 0.3768
0. 50 0. 2015 0. 3268 0. 3993
0.55 0.2446 0.3931 0.4155
0.60 0. 2920 0.4650 0.4248
0.65 0.3437 0-5429 0.4266
0.70 0.3991 0.6280 0.4206
0.75 0.4567 0.7254 0.4075
0.80 0.4988 0.8864 0.4009
34

Table 2b. g= 0.950 (r = 1.53)

w -x y
0.00 0.0000 0.0000 0.0000
0.05 0.0015 0.0028 0.0499
0.10 0.0060 0.0113 0.0994
0.15 0.0136 0.0254 0.1480
0.20 0.0243 0.0451 0.1951
0.25 0.0381 0.0704 0.2405
0.30 0.0551 0.1012 0.2835
0.35 0.0754 0.1375 0.3236
0.40 0.0991 0.1792 0.3604
0.45 0.1263 0.2262 0.3932
0.50 0.1572 0.2785 0.4214
0.55 0.1920 0.3360 0.4444
0.60 0.2308 0.3984 0.4615
0.65 0.2739 0.4659 0.4720
0.70 0.2312 0.5381 0.4752
0.75 0.3727 0.6154 0.4705
0.80 0.4272 0.6983 0.4583
 35

Table 2c. g = 1.000 (r = 1. 00)

TI w -x y

0.00 0.0000 0.0000 0.0000

0.05 0.0011 0.0023 0.0500

0.10 0.0043 0.0093 0.099.6
0.15 0.0098 0.0210 0.1485
0.20 0.0175 0.0373 0.1965
0.25 0.0274 0.0583 0.2431
0.30 0.0397 0.0840 0.2881
0.35 0.0545 0.1143 0.3309
0.40 0.0718 0.1493 0.3713
0.45 0.0918 0.1889 0.4087
0.50 0.1146 0.2333 0.4427
0.55 0.1406 0.2822 0.4727
0.60 0.1699 0.3359 0,4981
0.65 0.2028 0.3942 0.5182
0.70 0.2399 0.4572 0.5321
0.75 0.2816 0.5248 0.5388
0.80 0.3285 0.5971 0.5372
36

Table 2d. g - 1.050 (r = 0.771)

w -x y
0.00 0-.0008 0.0002 .0.0000
0.05 0.0007 0.0019 0.0500
0.10 0.0028 0.0076 0,0997
0.15 0.0063 0.0170 0.1491
0.20 0.0112 0.0302 0.1978
0.25 0.0176 0.0473 0.2456
0.30 0.0255 0.0681 0.2924
0.35 0.0349 0.0927 0.3378
0.40 0.0458 0.1212 0.3817
0645 0.0584 0.1534 0.4237
0.50 0.0726 0.1896 0.4637
0.55 0.0886 0.2296 0.5013
0.60 0.1064 0. 2735 0.5362
0.65 0.1261 0.3213 0.5681
0.70 0.1476 0.3731 0.5967
0.75 0.1710 0.4287 0.6217
0.80 0.1957 0.4880 0.6434
 37

Table 2e. g= . 100(r= 0.651)

T w -x y

0.00 0.0009 0.0000 0.0000

0.05 0.0003 0.0015 0.0500
0.10 0.0014 0.0059 0.0999
0.15 0.0031 0.0134 0.1495
0.20 0.0055 0.0237 0.1989
0.25 0.0085 0.0370 0.2479
0.30 0.0121 0.0533 0.2964
0.35 0.0162 0.0724 0.3443
0.40 0.0208 0.0944 0.3917
0.45 0.0256 0.1191 0.4385
0.50 0.0304 0.1465 0.4848
0.55 0.0347 0.1765 0.5309
0.60 0.0381 0.2086 0.5772
G.65 0.0391 0.2424 0.6246
0.70 0.0354 0.2769 0.6753
0.75 0.0216 0.3097 0.7338
0.80 -0.0175 0.3346 0.8136
38

Table 2f . g= 1. 150 (r = 0. 582)

w -x y
0.00 0.0000 0.0000 0.0000
0.05 0.0000 0.0011 0.0500
0.10 0.0001 0.0045 0.1000
0.15 0.0002 0.0100 0.1500
0.20 0.0002 0.0177 0.2000
0. 25 0.0000 0.0275 0. 2500
0.30 -0.0005 0.0394 0.3001
0.35 -0.0015 0.0531 0.3505
0.40 -0.0035 0.0686 0.4014
0.45 -0. 0070 0.0855 0. 4531
0.50 -0.0127 0.1033 0.5064
0.55 -0.0222 0.1214 0.5622
0.60 -0.0378 0.1385 0.6227
0.65 -0.0644 0.1520 0.6918
0.70 -0.1126 0.1562 0.7789
0.75 -0.2127 0.1352 0.9095
0.80 -0.4990 0.0181 1.199
 39

Table 2g. g 1. 200 (r = 0.541)

w -x y
.00 +0.0006 0.0003 0.0000
.05 -0.0003 0.0008 0.0500
.10 -0.0011 0.0031 0.1001
.15 -0.0025 0.0069 0.1504
.20 -0.0048 0.0121 0.2010
.25 -0.0079 0.0186 0.2520
.30 -0.0124 0.0263 0.3037
.35 -0.0186 0.0347 0.3565
.40 -0.0273 0.0435 0.4109
.45 -0.0396 0.0521 0.4678
.50 -0.0572 0.0593 0.5286
.55 -0.0836 0.0632 0.5960
.60 -0.1247 0.0601 0.6748
.65 -0. 1944 0.0417 0.7764
.70 -0,3307 -0.0146 0.9315
075 -0.6998 -0.2085 1.275

.80 -25.45 -15.00 21.16

40

Appendix B

PHASE ERROR CALCULATIONS

The normalized path length errors, At, that specify the

aberrations of the lens, have been computed from Eq. (9) and
the lens contour parameters and are given below for the following
ranges of variables
c= 300
] = 0.00<0.05>0.80,
g = 0.90<0.05>1.20 (Table 3.),
and
g = 1. 137 (Table 4.),
e = ±.501 +.1001 +.150,£200, £250, £350, ±400.
The values of L for e = +-a (+_301) and e = 00 are zero.
Curves of Al versus q are shown in Fig. 5 for g = 1.00,
1.10, and 1.137.
Note that only positive values of 0 are plotted in Fig. 5 with
both positive and negative values of . In Tables 3 and 4, the
equivalent representation of only positive values of q are shown
with both positive and negative values of 0.
 41

Table 3.
Normalized Path Length Errors, Al, for Microwave Lens a= 300
Table 3a g= 0.900
0 = 50 0 = 100 0 = 150 0= 200 0 = 25' 0 = 350 0 = 400
0.00 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000
0.05 -0.000003 -0.000008 -0.000014 -0.000017 -0.000014 -0.000030 0.000081
0.10 -0.000015 -0.000039 -0.000065 -0.000080 -0.000065 -0.000135 0.000360
0.15 -0.000042 -0.000106 -0.000172 -0.000204 -0.000164 -0.000337 0.000896
0.20 -0.000092 -0.000223 -0.000352 -0.000415 -0.000329 -0.000668 0.001767
0.25 -0.000173 -0.000408 -0.000632 -0.000737 -0.000580 -0.001168 0.003076
0.30 -0.000296 -0.000682 -0.001044 -0.001207 -0.000945 -0.001892 0.004962
0.35 -0.000473 -0.001074 -0.001630 -0.001876 -0.001464 -0.002914 0.007615
0.40 -0.000721 -0.001622 -0.002450 -0.002811 -0.002191 -0.004345 0.011310
0.45 -0.001064 -0.002380 -0.003587 -0.004116 -0.003210 -0.006351 0.016460
0.50 -0.001534 -0.003425 -0.005167 -0.005946 -0.004650 -0.009192 0.023700
0.55 -0.002173 -0.004872 -0.007389 -0.008557 -0.006728 -0.013300 0.034040
0.60 -0.003066 -0.006907 -0.010580 -0.012390 -0.009839 -0.019450 0.049160
0.65 -0.004307 -0.009836 -0.015340 -0.018330 -0.014790 -0.029120 0.071970
0.70 -0.006077 -0.014210 -0.022870 -0.028300 -0.023510 -0.045350 0.107300
0.75 -0.008639 -0.021020 -0.035870 -0.047920 -0.042440 -0.074340 0.162000
0.80 -0.011520 -0.029640 -0.055880 -0.091820 -0.118800 -0.116600 0.228300
 42

Table 3a. (Contd)

1 0 = -50 -100 0 =-150 -200 e =-,5 0 0 = -350O =-400
0.00 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000
0.05 -0.000001 -0.000004 -0.000009 -0.000013 -0.000011 0.000024 0.000067
0.10 0.000001 -0.000011 -0.000029 -0.000042 -0.000038 0.000087 0.000241
0.15 0.000010 -0.000010 -0.000046 -0.000076 -0.000072 0.000175 0.000489
0.20 0.000033 0.000010 -0.000048 -0.000103 -0.000106 0.000275 0.000780
0.25 0.000077 0.000059 -0.000022 -0.000110 -0.000131 0.000375 0.001085
0.30 0.000149 0.000150 0.000045 -0.000086 -0.000141 0.000464 0.001379
0.35 0.000259 0.000296 0.000169 -0.000018 -0.000127 0.000532 0.001636
0.40 0.000417 0.000516 0.000369 0.000111 -0.000080 0.000565 0.001829
0.45 0.000639 0.000830 0.000666 0.000318 0.000012 0.000551 0.001930
0.50 0.000943 0.001265 0.001087 0.060625 0.000159 0.000475 0.001908
0.55 0.001355 0.001858 0.001667 0.001058 0.000376 0.000322 Q.001730
0.60 0.001910 0.002654 0.002450 0.001651 0.000681 0.0000113 0.001357
0.65 0.002657 0.003714 0.003490 0.002442 0.001095 -0.000293 0.000751
0.70 0.003661 0.005118 0.004854 0.003476 0.001638 -0.000793 -0.000122
0.75 0.004992 0.006930 0.006584 0.004773 0.002315 -0.001420 -0.001236
0.80 0.006284 0.008593 0.008086 0.005834 0.002830 -0.001791 -0.001736
 43

Table 3b. g = 0.950

0=5 ° 0= 100 0 = 15 0 =200 = 250 0 = 350 0 = 400
0.00 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000
0.05 -0.000001 -0.000005 -0.000009 -0.000011 -0.000009 0.000020 0.000054.
0.10 -0.000007 -0.000022 -0.000038 -0.000048 -0.000040 0.000086 0.000232
0.15 -0.000020 -0.000056 -0.000095 -0.000118 -0.000097 0.000208 0.000563
0.20 -0.000042 -0.000112 -0.000188 -0.000230 -0.000188 0.000401 0.001083
0.25 -0.000075 -0.000197 -0.000324 -0.000394 -0.000321 0.000682 0.001843
0.30 -0.000125 -0.000318 -0.000517 -0.000624 -0.000507 0.001076 0.002906
0.35 -0.000195 -0.000484 -0.000780 -0.000939 -0.000760 0.001616 0.004364
0.40 -0.000290 -0.000709 -0.001135 -0.001361 -0.001102 0.002347 0.006345
0.45 -0.000418 -0.001009 -0.001607 -0.001926 -0.001561 0.003339 0.009040
0.50 -0.000588 -0.001408 -0.002235 -0.002682 -0.002180 0.004696 0.012740
0.55 -0.000813 -0.001936 -0.003075 -0.003702 -0.003025 0.006585 0.017910
0.60 -0.001111 -0.002641 -0.004209 -0.005101 -0.004204 0.009291 0.025340
0.65 -0.001510 -0.003595 -0.005771 -0.007071 -0.005905 0.013340 0.036460
0.70 -0.002043 -0.004910 -0.007984 -0.009959 -0.008490 0.019780 0.053960
0.75 -0.002786 -0.006771 -0.011250 -0.014450 -0.012750 0.031000 0.083200
0.80 -0.003847 -0.009471 -0.016320 -0.022070 -0.020770 0.052880 0.133600
 44

Table 3b. (Contd)

0 =-50 0 = -100 0 =-150 6 =-200 =-250 0 =-350 9 =-400

0.00 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

0.05 -0.000001 -0.000003 -0.000007 -0.000009 -0.000008 0.000017 0.000047
0.10 -0.000002 -0.000011 -0.000024 -0.000033 -0.000029 0.000065 0.000177
0.15 -0.000001 -0.000020 -0.000048 -0.000067 -0.000059 0.000136 0.000373
0.20 0.000004 -0.000026 -0.000073 -0.000108 -0.000097 0.000226 0.000624
0.25 0.000014 -0.000026 -0.000094 -0.000150 -0.000138 0.000330 0.000917
0.30 0.000034 -0.000015 -0.000108 -0.000189 -0.000181 0.000444 0.001242
0.35 0.000065 0.000012 -0.000110 -0.000221 -0.000222 0.000564 0.001589
0.40 0.000112 0.0000.9 -0.000093 -0.000241 -0.000259 0.000686 0.001950
0.45 0.000178 0.000133 -0.000051 -0.000245 -0.000288 0.000806 0.002315
0.50 0.000269 0.000243 0.000024 -0.000225 -0.000306 0.000919 0.002674
0,55 0.000393 0.000397 0.000142 -0.000173 -0.000308 0.001020 0.003017
0.60 0.000559 0.000609 0.000316 -0.000080 -0.000289 0.001105 0.003334
0.65 0.000780 0.000897 0.000560 0.000065 -0.000244 0.001166 0.003611
0.70 0.001073 0.001281 0.000895 0.000276 -0.000165 0.001197 0.003837
0.75 0.001462 0.001791 0.001344 0.000570 -0.000044 0.001192 0.003998
0.80 0.001973 0.002451 0.001926 0.000956 0.000121 0.001150 0.004098
 45

Table 3c. g = 1.000

0 = 50 10* = 150 0 = 200 0 = 250 = 350 =400
0.00 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.05 -0.000001 -0.000002 -0.000004 -0.000006 -0.000005 0.000011 0.000030
0.10 -0.000003 -0.000009 -0.000018 -0.000023 -0.000020 0.000045 0.000126
0.15 -0.000006 -0.000021 -0.000040 -0.000053 -0.000046 0.000106 0.000294
0.20 -0.000010 -0.000038 -0.000073 -0.000097 -0.000084 0.000195 0.000548
0.25 -0.000016 -0.000061 -0.000117 -0.000157 -0.000136 0.000319 0.000900
0.30 -0.000024 -0.000089 -0.000174 -0.000233 -0.000204 0.000484 0.001370
0.35 -0.000033 -0.000125 -0.000244 -0.000330 -0.000290 0.000697 0.001986
0.40 -0.000044 -0.000168 -0.000330 -0.000449 -0.000398 0.000970 0.002783
0.45 -0.000058 -0.000220 -0.000435 -0.000597 -0.000532 0.001319 0.003815

0.50 -0.000074 -0.000283 -0.000563 -0.000778 -0.000700 0.001768 0.005161

0.55 -0.000092 -0.000357 -0.000718 -0.001002 -0.000911 0.002352 0.006946
0.60 -0.000115 -0.000448 -0.000909 -0.001281 -0.001179 0.003130 0.009374
0.65 -0.000141 -0.000557 -0.001144 -0.001635 -0.001528 0.004199 0.012810
0.70 -0.000174 -0.000693 -0.001442 -0.002095 -0.001996 0.005746 0.017960
0.75 -0.000213 -0.000864 -0.001829 -0.002713 -0.002653 0.008156 0.026370
0.80 -0.000264 -0.001085 -0.002352 -0.003590 -0.003643 0.012400 0.042000
 46

Table 3c (Contd)
0 = -35 ° 0 =-400
6 = -50 0 =-100 8 =-150 0 = -200 e =-250

0.00 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

0.05 -0.000001 -0.000002 -0.000004 -0.000005 -0.000005 0.000010 0.000028
0.10 -0.000002 -0.000009 -0.000017 -0.000022 -0.000018 0.000040 0.000110
0.15 -0.000006 -0.000020 -0.000037 -0.000048 -0.000040 0.000089 0.000242
0.20 -0.000010 -0.000035 -0.000066 -0.000085 -0.000071 0.000155 0.000421
0.25 -0.000016 -0.000056 -0.000103 -0.000131 -0.000109 0.000237 0.000645
0.30 -0.000023 -0.000080 -0.000148 -0.000188 -0.000156 0.000337 0.000913
0.35 -0.000031 -0.000110 -0.000201 -0.000256 -0.000211 0.000453 0.001223
0.40 -0.000041 -0.000145 -0.000264 -0.000334 -0.000275 0.000586 .0.001578
0.45 -0.000053 -0.000185 -0.000336 -0.000424 -0.000348 0.000736 0.001976
0.50 -0.000067 -0.000232 -0.000419 -0.000526 -0.000430 0.000904 0.002420
0.55 -0.000083 -0.000286 -0.000514 -0.000642 -0.000523 0.001091 0.002913
0.60 -0.000101 -0.000348 -0.000622 -0.000773 -0.000327 0.001299 0.003458

0.65 -0.000123 -0.000419 -0.000745 -0.000921 -0.000744 0.001530 0.004060

0.70 -0.000148 -0.000502 -0.000886 -0.001090 -0.000876 0.001787 0.004727
0.75 -0.000178 -0.000598 -0.001050 -0.001283 -0.001026 0.002074 0.005438
0.80 -0.000214 -0.000713 -0.001241 -0.001506 -0.001197 0.002397 0.006296

&
 47

Table 3d. g= 1.050

9= 50 9=oo = 150 0 = 20'0 = 250 03o = 400
0.00 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000
0.05 0.000000 -0.000001 -0.000001 -0.000002 -0.000002 0.000004 0.000012
.10 0.000000 -0.000001 -0.000004 -0.000006 -0.000006 0.000016 0.000046
0.15 0.000002 -0.000001 -0.000006 -0.000011 -0.000012 0.000033 0.000099
0.20 0.000005 0.000003 -0.000006 -0.000015 -0.000017 0.000054 0.000168
0.25 0.000012 0.000012 -0.000000 -0.000015 -0.000022 0.000078 0.000250
0.30 0.000024 0.000028 0.000014 -0.000009 -0.000023 0.000101 0.000338
0.35 0.000040 0.000053 0.000038 0.000007 -0.000019 0.000121 0.000427
0.40 0.000064 0.000090 0.000077 0.000036 -0.000007 0.000134 0.000509
0.45 0.000096 0.000143 0.000135 0.000082 0.000017 0.000134 0.000574
0.50 0.000139 0.000215 0.000217 0.000152 0.000055 0.000115 0.000606
0.55 0.000194 0.000310 0.000329 0.000252 0.000114 0.000068 0.000585
0.60 0.000264 0.000435 0.000480 0.000391 0.000200 -0.000021 0.000477
0.65 0.000353 0.000596 0.000679 0.000580 0.000323 -0.000171 0.000235
0.70 0.000464 0.000801 0.000939 0.000836 0.000494 -0.000411 -0.000222
0.75 0.000601 0.001058 0.001274 0.001174 0.000730 -0.000787 -0.001031
0.80 0.000767 0.001376 0.001696 0.001613 0.011049 -0.001362 -0.002428
 48

Table 3d. (Contd)

l O=-50 6=-100 -=_15o -=-20o0=-250 =_350 0=_400

0.00 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000
0.05 -0.000000 -0.00000i -0.000002 -0.000002 -0.000002 0.000005 0.000013
0.10 -0.000002 -0.000005 -0.000009 -0.000011 -0.000009 0.000019 0.000053
0.15 -0.000006 -0.000014 -0.000023 -0.000027 -0.000021 0.000045 0.000123
0.20 -0.000013 -0.000030 -0.000045 -0.000052 -0.000041 0.000083 0.000225
0.25 -0.000024 -0.000052 -0.000078 -0.000088 -0.000068 0.000136 0.000362
0.30 -0.000039 -0.000085 -0.000123 -0.000137 -0.000104 0.000203 0.000537
0.35 -0.000061 -0.000128 -0.000183 -0.000200 -0.000150 0.000287 0.000753
0.40 -0.000090 -0.000185 -0.000259 -0.000279 -0.000207 0.000389 0.001012
0.45 -0.000127 -0.000257 -0.000355 -0.000377 -0.000277 0.000511 0.001319
0.50 -0.000175 -0.000347 -0.000472 -0.000497 -0.000361 0.000655 0.001678
0.55 -0.000234 -0.000458 -0.000615 -0.000640 -0.000460 0.000823 0.002094
0.60 -0.000307 -0.000593 -0.000787 -0.000810 -0.000578 0.001017 0.002571
0.65 -0.000396 -0.000756 -0.000992 -0.001011 -0.000715 0.001240 0.003116
0.70 -0.000504 -0.000951 -0.001234 -0.001247 -0.000874 0.001495 0.003734
0.75 -0.000634 -0.001181 -0.001518 -0.001520 -0.001057 0.001784 0.004429
0.80 -0.000786 -0.001448 -0.001844 -0.001831 -0.001264 0.002106 0.006200
 49

Table 3e. g = 1.100

q 0 = 50 0 = 100 0 = 150 0 = 200 0 = 250 0 = 350 0 = 400

0.00 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

0.05 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000001
0.10 0.000001 0.000002 0.000002 0.000002 0.000001 -0.0000011-0.000000

0.15 0.000003 0.000006 0.000007 0.000007 0.000004 -0.000005 -0.000009

0.20 0.000008 0.000015 0.000018 0.000017 0.000011 -0.000016 -0.000033
0.25 0.000016 0.000029 0.000037 0.000035 0.000024 -0.000035 -0.000078
0.30 0.000028 0.000051 0.000065 0.000063 0.000043 -0.000067 -0.000154
0.35 0.000045 0.000082 0.000104 0.000103 0.000070 -0.000113 -0.000268
0.40 0.000066 0.000122 0.000156 0.000155 0.000107 -0.000177 -0.000430
0.45 0.000092 0.000172 0.000221 0.000222 0.000154 -0.000262 -0.000645
0.50 0.000123 0.000230 0.000297 0.000300 0.000211 -0.000365 -0.000913
0.55 0.000156 0.000292 0.000380 0.000387 0.000274 -0.000482 -0.001223
0.60 0.000187 0.000352 0.000460 0.000469 0.000334 -0.000596 -0.001529
0.65 0.000208 0.000392 0.000511 0.000522 0.000372 -0.000666 -0.001711
0.70 0.000201 0.000374 0.000483 0.000488 0.000342 -0.000593 -0.001482
0.75 0.000123 0.000216 0.000260 0.000240 0.000148 -0.000152 -0.000163
0.80 -0.000129 -0.000292 -0.000452 -0.000546 -0.000462 0.001201 0.003835
 50

Table 3e. (Contd)

q e =-55= -100 e=-150 0 =-200 e=-250 0 =-350 0 =-400
0.00 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.05 0.000000 0.000000 0.000000 -0.000001 0.000000 0.000001 0.000002
0.10 -0.000001 -0.000002 -0.000003 -0.000003 -0.000002 0.000005 0.000013
0.15 -0.000004 -0.000007 -0.000009 -0.000010 -0.000007 0.000013 0.000035
0.20 -0.000009 -0.000016 -0.000021 -0.000022 -0.000015 0.000028 0.000072
0.25 -0.000017 -0.000031 -0.000040 -0.000040 -0.000028 0.000050 0.000128
0.30 -0.000028 -0.000052 -0.000067 -0.000067 -0.000046 0.000080 0.000205
0.35 -0.000044 -0.000081 -0.000103 -0.000102 -0.000070 -0.000120 0.000303
0.40 -0.000065 -0.000118 -0.000148 -0.000146 -0.000101 0.000169 0.000424
0.45 -0.000090 -0.000162 -0.000203 -0.000199 -0.000136 0.000227 0.000566
0.50 -0.000118 -0.000213 -0.000266 -0.000280 -0.000177 0.000293 0.000727
0.55 -0.000149 -0.000267 -0.000333 -0.000324 -0.000221 0.000363 0.000897
0.60 -0.000178 -0.000319 -0.000397 -0.000386 -0.000263 0.000430 0.001062
0.65 -0.000199 -0.000356 -0.000443 -0.000432 -0.000294 0.000483 0.001192
0.70 -0.000194 -0.000351 -0.000440 -0.000432 -0.000296 0.000493 0.001227
0.75 -0.000131 -0.000246 -0.000319 -0.000324 -0.000229 0.000404 0.001033
0.80 0.000081 0.000110 0.000096 0.000055 0.000011 0.000066 0.000267
 51

Table 3f. g = 1.150

=50 =100 0 = 150 0 = 20'0 = 250 0 = 350 0= 400
0.00 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.05 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000001
0.10 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000001 -0.000002
0.15 0.000000 0.000001 0.000001 0.000001 0.000001 -0.000002 -0.000004
0.20 0.000000 0.000001 0.000001 0.000001 0.000001 -0.000002 -0.000006
0.25 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.30 -0.000002 -0.000004 -0.000005 -0.000006 -0.000004 0.000009 0.000026
0.35 -0.000007 -0.000013 -0.000018 -0.000020 -0.000015 0.000032 0.000092

0.40 -0.000017 -0.000034 -0.000048 -0.000052 -0.000040 0.000084 0.000238

0.45 -0.000038 -0.000075 -0.000105 -0.000114 -0.000088 0.000181 0.000531
0.50 -0.000076 -0.000152 -0.000211 -0.000230 -0.000177 0.000378 0.001088
0.55 -0.000144 -0.000288 -0.000401 -0.000439 -0.000337 0.000728 0.002111
0.60 -0.000264 -0.000529 -0.000736 -0.000807 -0.000621 0.001352 0.003946
0.65 -0.000477 -0.000954 -0.001330 -0.001458 -0.001125 0.002458 0.007195
0.70 -0.000864 -0.001727 -0.002405 -0.002636 -0.002032 0.004422 0.012890
0.75 -0.001616 -0.003219 -0.004467 -0.004872 -0.003733 0.007951 0.022710
0.80 -0.003314 -0.006540 -0.008964 -0.009628 -0.007233 0.014510 0.039610
 52

Table 3f. (Contd)

0 =-50 e =-100 =-150 0 =-20 e= -250 0 =-350 0 =-400
0.00 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000
0.05 0.000000 -0.000000 -0.000600 -0.000000 0.000006 0.000000 0.000000
0.10 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000001
0.15 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000
0.20 0.000000 -0.000001 -0.000001 -0.000001 0.000000 0.000000 0.000001
0.25 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000001
0M30 0.000001 0.000002 0.000003 0.000002 0.000001 -0.000001 -0.000003
0.35 0.000005 0.000009 0.000010 0.000009 0.000006 -0.000007 -0.000015
0.40 0.000014 0.000024 0.000028 0.000025 0.000016 -0.000021 -0.000045
0.45 0.000032 0.000053 0.000062 0.000056 0.000036 -0.000049 -0.000108
0.50 0.000064 0.000108 0.000126 0.000115 0.000073 -0.000102 -0.000230
0.55 0.000122 0.000206 0.000242 0.000221 0.000141 -0.000199 -0.000451
0.60 0.000224 0.000379 0.000445 0.000408 0.000260 -0.000371 -0,000845
0.65 0.000405 0.000685 0.000805 0.000739 0.000473 -0.000677 -0.001548
0.70 0.000734 0.001244 0.001463 0.001345 0.000861 -0.001239 -0.002840
0.75 0.001379 0.002342 0.002758 0.002540 0.001630 -0.002352 -0.005404
0.80 0.002871 0.004901 0.005799 0.005360 0.003450 -0.005003 -0.011510
 53

Table 3g. g = 1. 200

0 = 50 8 = 100 0 = 15' 8 = 20' 0 = 250 0 = 350 0 = 400
0.00 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.05 0.000000 -0.000001 -0.000002 -0.000002 -0.000002 0.000004 0.000014
0.10 -0.000002 -0.000005 -0.000008 -0.000010 -0.000008 0.000020 0.000066
0.15 -0.000007 -0.000015 -0.000023 -0.000028 -0.000023 0.000056 0.000175
0.20 -0.000016 -0.000035 -0.000052 -0.000061 -0.000049 0.000120 0.000372
0.25 -0.000033 -0.000069 -0.000102 -0.000117 -0.000094 0.000225 0.000698
0.30 -0.000060 -0.000125 -0.000182 -0.000207 -0.000166 0.000392 0.0012.17
0.35 -0.000103 -0.000213 -0.000307 -0.000347 -0.000276 0.000653 0.002025
0.40 -0.000169 -0.000348 -0.000499 -0.000562 -0.000446 0.001051 0.003268
0.45 -0.000272 -0.000555 -0.000792 -0.000889 -0.000705 0.001659 0.005165
0.50 -0.000428 -0.000872 -0.001239 -0.001388 -0.001099 0.002585 0.008053
0.55 -0.000671 -0.001362 -0.001929 -0.002157 -0.001704 0.003999 0.012440
0.60 -0.001054 -0.002134 -0.003016 -0.003363 -0.002650. 0.006172 0.019060
0.035 -0.001685 -0.003399 -0.004785 -0.005314 -0.004166 0.009546 0.029020
0.70 -0.002798 -0.005612 -0.007849 -0.008647 -0.006710 0.014860 0.043850
0.75 -0.005063 -0.010040 -0.013840 -0.014990 -0.011370 0.023540 0.065970
0.80 -0.013460 -0.025470 -0.033260 -0.033820 -0.023890 0.041660 0.105600
 54

Table 3g. (Contd)

0 =-50 =-100 0 =-150 0 =-200 =250 0 =-35 ° 0 -40
0.00 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.05 0.000000 0.000000 -0.000001 -0.000001 -0.000001 0.000003 0.000010
0.10 0.000001 0.000001 -0.000001 -0.000002 -0.000003 0.000009 0.000031
0.15 0.000004 0.000005 0.000003 0.000000 -0.000003 0.000015 0.000057
0.20 0.000011 0.000015 0.000013 0.000007 0.000001 0.000017 0.000075
0.25 0.000023 0.000035 0.000035 0.000025 0.000010 0.000011 0.000077
0.30 0.000045 0.000070 0.000074 0.000059 0.000030 -0.000009 0.000047
0.35 0.000080 0.000128 0.000139 0.000116 0.000065 -0.000051 -0.000034
0.40 0.000135 0.000219 0.000244 0.000210 0.000123 -0.000126 -0.000192
mp

0.45 0.000219 0.000360 0.000408 0.000359 0.000216 -0.000253 -0.000469

0.50 0.000350 0.000579 0.000664 0.000591 0.000364 -0.000460 -0.000932
0.55 0.000553 0.000922 0.001064 0.000958 0.000598 -0.000793 -0.001690
0.60 0.000876 0.001467 0.001704 0.001546 0.000975 -0.001336 -0.002937
0.65 0.001410 0.002373 0.002771 0.002528 0.001606 -0.002252 -0.005055
0.70 0.002364 0.003996 0.004689 0.004300 0.002748 -0.003920 -0.008922
0.75 0.004354 0.007413 0.008752 0.008073 0.005188 -0.007495 -0.017210
0.80 0.012440 0.021760 0.026230 0.024570 0.015940 -0.023130 -0.052810
 55

Table 4.
Normalized Path Length Errors Al, for Microwave Lens - a= 300 and g = 1.137.
0 = -50 6 = -i0o = -15' 0 = -200 = -250 9 -35' 0 = -400
0.00 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.05 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000001
0.10 0.000000 -0.000001 -0.000001 0.000000 0.000000 0.000000 0.000000
0.15 -0.000001 -0.000002 -0.000002 -0.000002 -0.000001 0.000001 0.000002
0.20 -0.000003 -0.000005 -0.000005 -0.000005 -0.000003 0.000004 0.000009
0.25 -0.000005 -0.000009 -0.000010 -0.000009 -0.000006 0.000008 0.000020
0.30 -0.000008 -0.000013 -0.000016 -0.000015 -0.000010 0.000014 0.000034
0.35 -0.000010 -0.000018 -0.000021 -0.000020 -0.000013 0.000020 0.000048
0.40 -0.000011 -0.000019 -0.000023 -0.000022 -0.000014 0.000023 0.000057
0.45 -0.000007 -0.000012 -0.000016 -0.000015 -0.000010 0.000019 0.000049
0.50 0.000006 0.000009 0.000009 0.000007 0.000003 0.000000 0.000008
0.55 0.000036 0.000059 0.000067 0.000060 0.000037 -0.000047 -0.000098
0.60 0.000095 0.000160 0.000186 0.00018. 0.000106 -0.000145 -0.000322
0.65 0.000210 0.000354 0.000414 0.000378 0.000240 -0.000337 -0.000761
0.70 0.000431 0.000729 0.000855 0.000783 0.000500 -0.000710 -0.001616
0.75 0.000875 0.001483 0.001743 0.001601 0.001024 -0.001437 -0.003355
0.80 0.001883 0.003205 0.003781 0.003485 0.002237 -0.003226 -0.007403

= 5' = 0° 0 = 150 0 = 200 0 = 250 8 = 350 0 = 40

0.00 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.05 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000001 -0.000001
0.10 0.000000 0.000001 0.000002 0.000002 0.000001 -0.000003 -0.000007
0.15 0.000002 0.000003 0.000004 0.000005 0.000004 -0.000007 -0.000019
0.20 0.000003 0.000007 0.000009 0.000010 0.000008 -0.000015 -0.000040
0.25 0.000006 0.000012 0.000016 0.000017 0.000013 -0.000025 -0.000068
0.30 0.000009 0.000018 0.000024 0.000025 0.000019 -0.000037 -0.000099
0.35 0.000011 0.000022 0.000030 0.000032 0.000024 -0.000046 -0.000123
0.40 0.000012 0.000023 0.000030 0.000032 0.000023 -0.000045 -0,000119
0.45 0.000007 0.000013 0.000016 0.000016 0.000011 -0.000019 -0.000045
0.50 -0.000008 -0.000018 -0.000027 -0.000031 -0.000025 0.000058 0.000175
0.55 -0.000043 -0.000088 -0.000123 -0.000136 -0.000106 0.000233 0.000680
0.60 -0.000114 -0.000228 -0.000319 -0.000351 -0.000271 0.000591 0.001725
0.65 -0.000249 -0.000499 -0.000697 -0.000765 -0.000591 0.001290 0.003772
0.70 -0.000510 -0.001020 -0.001422 -0.001560 -0.001204 0.002624 0.007659
0.75 -0.001029 -0.002055 -0.002858 -0.003125 -0.002402 0.005164 0.014880
0.80 -0.002191 -0.004344 -0.005988 -0.006476 -0.004907 0.010100, 0.028070
56

RE FERENCES
1. Final Engineering Report on Investigation of Variable Index of
Refraction Lenses, Sperry Engineering Report No. 5224-1233,
Signal Corps Contract No. DA 33-039-sc-15323, Sperry
Gyroscope Co. Great Neck, New York, September 1952.
2. J. RUZE, Wide-Angle Metal-Plate Optics, Proc. IRE 38:53-58,
January 1950; Also CFS Report No. E5043, Cambridge Field
Station, Cambridge, Mass., March 1949.
3. S.S.D. JONES, H. GENT, and A. A.L. BROWNE, Improvements
In or Relating to Electromagnetic-Wave Lens and Mirror Systems,
British Provisional Patent Specification No. 25923/53, August 1953.

4. H. GENT, The Bootlace Aerial Royal Radar Establishment Journal,

Malvern, Worcester, October 1957.

5. B. R. HATCHER, Multiple- Beam interval Scanning Antenna Feasibility

and Development Study - Final Engineering Report, Rpt No. AFCRC-
TR-60-146, Contract AF19(604)-5202, Chu Associates, Littleton,
Mass., May 1960.

6. W. ROTMAN, A Study of Microwave Double-Layer Pillboxes:

Part I, Line Source Radiators, Tech Rpt AFCRC-TR- 102,
Appendix A, Sec. 3, pp 41-42, Air Force Cambridge Research
Center, Cambridge, Mass., July 1954.
7. G. C. SOUTHWORTH, Principles and Applications of Waveguide
Transmission, Sec. 10.1, pp. 402-411, D.Van Nostrand Co.,Inc,
New York, N. Y. (1950).

8. L. BLAISDELL, Multiple Beam Interval Scanner Antenna System

Interim Engineering Report, Contract AF19(604)-7385, Sylvania
Electronics System Lab., Waltham, Mass., November 1930.
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