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Performance of coherently illuminated multiaperture optical and microwave systems

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Order Num ber 8917311
Performance o f coherently illuminated m ulti-aperture optical
and microwave system s
W atson, Steven M anley, P h .D .
The Union for Experimenting Colleges and Universities, 1989
UMI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
PERFORMANCE OF COHERENTLY ILLUMINATED MULTI-APERTURE
OPTICAL AND MICROWAVE SYSTEMS
DISSERTATION
Presented to the Faculty of the Union Graduate School
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
Steven M. Watson, B.S., M.S.
Major, USAF
February 1989
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Table of Contents
Page
List of Figures..................................................
iii
Abstract.........................................................
vil
I. Introduction.................................................
II. Theory......................................................
1
7
A.
B.
Multi-Aperture Impulse Response
:.....................
Impulse Response for Concentric Rings of Apertures.......
7
12
III. Results.....................................................
21
A.
B.
Computer Program.........................................
Optical Analysis.........................................
21
21
1. Transmittance Area Comparisons.......................
2. Far-Field Diffraction Patterns.......................
3. Central Lobe Irradiance Comparisons..................
4. Central Lobe Dimensions..............................
5. Side Lobe Maxima Comparisons........................
6. Radial Energy Distributions.........................
22
25
32
35
36
39
Microwave Analysis......................................
43
1.
2.
Far-Field Diffraction Patterns.......................
Side Lobe Comparisons Vith and Vithout Central
Obscurations........................................
Central Lobe Irradiance Ratios.......................
Jitter Analysis.....................................
Steerability........................................
Non transmit ting Elements............................
43
Conclusions.................................................
64
Bibliography.....................................................
67
Appendix A - Computer Program....................................
71
Appendix B - Experimental Design
85
C.
3.
4.
5.
6.
IV.
..........................
ii
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47
51
53
57
61
List of Figures
Figure
Page
la.
Single large optic...................................
lb.
Six aperture system withequivalentdiameter
ofsingle large
optic................................................
2
2a.
Impulse response of a single large optic...................
3
2b.
Impulse response of a six aperture system of equivalent
diameter..................................................
3
Field amplitude generated by outer thin ring of thedepicted
single large aperture system..............................
5
Field samplitude generated by the summation of the field
amplitudes from the outer and middle rings.................
5
3c.
Resultant field amplitude from summation of all three rings.
5
4.
Configuration for observing the impulse response for a
multi-aperture system.....................................
8
Example of multi-aperture system with an aperture-origin
separation of pR and subaperture radius of a ...............
9
3a.
3b.
5.
2
6a.
Irradiance pattern for a single aperture of a multi-aperture
system..............................................
13
6b.
Six aperture system composed of identical single apertures..
13
7a.
Far-field irradiance pattern for a six aperture system with
aperture-origin separation of 2.00a........................
14
Far-field irradiance pattern for a six aperture system with
aperture-origin separation of 3.00a........................
14
Far-field irradiance pattern for a six aperture system with
aperture-origin separation of 4.00a........................
14
A 19 aperture system illustrating the distance (R) from the
origin to the circle on which the centers of each aperture
of the outside ring were placed............................
16
7b.
7c.
8.
9a.
9b.
10a.
Seven aperture system comparing the subaperture radii with
aref......................................................
17
19 aperture system comparing the subaperture radii with
a *......................................................
ref
17
Single large aperture......................................
23
iii
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10b.
10c.
lOd.
lOe.
lOf.
11.
12.
19 aperture system vith an equivalent diameter equal to
that of the single large aperture..........................
23
19 aperture system vith outside ring rotated 6.75° vith
respect to inner ring vith an equivalent diameter equal to
that of the single large aperture..........................
23
37 aperture system vith an equivalent diameter equal to
that of the single large aperture..........................
23
61 aperture system vith an equivalent diameter equal to
that of the single large aperture..........................
23
91 aperture system vith an equivalent diameter equal to
that of the single large aperture..........................
23
Transmittance of each multi-aperture system referenced to
the single large aperture of equivalent diameter...........
24
Theoretical and experimental far-field diffraction patterns
for the single large aperture of normalized radius *
aref......................................................
13.
14.
15.
16.
Theoretical and experimental far-field diffraction patterns
for the 19 aperture system of total normalized radius aref....................................................
27
Theoretical and experimental far-field diffraction
patterns for the 19 aperture system, vith outside ring of
apertures rotated 6.75 vith respect to inner ring, of
...........................
total normalized radius -
28
Theoretical and experimental far-field diffraction
patterns for the 37 aperture system of total normalized
radius « a ,............................................
ref
18.
19a.
19b.
20a.
29
Theoretical and experimental far-field diffraction patterns
for the 61 aperture system of total normalized radius aref....................................................
17.
26
Theoretical and experimental far-field diffraction patterns
for the 91 aperture system of total normalized radius *
a
....................................................
ref
30
31
Comparison of secondary lobe maxima (0 * theoretical values;
X * experimental values vith associated uncertainties)
37
Comparison of maximum side lobe values for the 19 rotated
aperture system...........................................
38
Comparison of maximum side lobe values for the 19 closepacked aperture system.....................................
38
Field amplitude of outer ring of 19 aperture system........
40
iv
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20b.
Summation of field amplitudes from inner and outer rings of
19 aperture system
AO
Resultant field amplitude from summation of all 19
apertures.................................................
AO
Radial Energy Distribution plot for single large aperture,
19, 37, 61, and 91 multi-aperture systems.................
A2
Comparison of percentage of irradiance contained in central
lobes of the single large aperture, 19, 19R (19 aperture
system vith outside ring rotated 6.75 vith respect to
inner ring), 37, 61, and 91 aperture systems (0 theoretical values; X « experimental values vith
associated uncertainties)................................
A2
22.
Arrays for microvave analysis..,........ -
A5
23.
Single large aperture far-field diffraction pattern in
units of dB..............................................
A6
Far-field diffraction patterns for the A(square) element
array....................................................
A8
24b.
Far-field diffraction
48
24c.
Far-field diffraction patterns for the 19 element
(rotated) array...................................
20c.
21a.
21b.
24a.
25a.
25b.
26.
27a.
27b.
27c.
28.
29a.
patterns for
the 7 element array....
y
48
Far-field diffraction patterns generated by the 19 element
(rotated) array vithout central obscurations in each
subelement..............................................
49
Far-field diffraction patterns generated by the 19 element
(rotated) array vith central obscurations in each
subelement..............................................
49
Comparison of maximum side lobe values (0 - vithout
obscuration, X = vith 20X obscuration) for the analyzed
arrays..................................................
50
Far-field diffraction pattern of a 7 element system
vithout dephasing.......................................
55
Far-field diffraction pattern of a 7 element system vith
the outer ring of elements shifted in phase by X/10 vith
respect to thecentral element...........................
55
Far-field diffraction pattern of 7 element system vith
tvo elements of the outer ring (separated by one element)
dephased by X/10 vith respect to the remaining elements—
55
Far-field diffraction pattern for the 19 element (rotated)
array experiencing a 3 a jitter value of .IX.............
56
Side lobe irradiances generated by the multi-element
v
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29b.
30a.
30b.
31a.
31b.
31c.
31d.
32a.
32b.
B-l.
systems experiencing jitter.............................
58
Maximum allowable jitter values for each multi-element
system which will maintain side lobes equal to or less
than that of a perfectly transmitting single antenna
58
Far-field diffraction patterns of the 19 element (rotated)
system with the following phase steering magnitudes:
O.OX, 0.0X — X, and 0.0X —
2 X........................
60
Far-field diffraction patterns of the 91 element systems
with the following phase steering magnitudes: O.OX,
0.0X — X, and O.OX — 2 X ..............................
60
The central lobe (X) and maximum side lobe (0) gains for
various values of phase steering (steering magnitudes)
for the 19 element (rotated) system......................
62
The central lobe (X) and maximum side lobe (0) gains for
various values of phase steering (steering magnitudes)
for the 37 element system...............................
62
The central lobe (X) and maximum side lobe (0) gains for
various values of phase steering (steering magnitudes)
for the 61 element system...............................
62
The central lobe (X) and maximum side lobe (0) gains for
various values of phase steering (steering magnitudes)
for the 91 element system...............................
62
Far-field antenna patterns for the 19 element (rotated)
system with a single nonfunctioning element located in
the inner ring..........................................
63
Far-field antenna patterns for the 19 element (rotated)
system with a single nonfunctioning element located in
the outer ring..........................................
63
Experimental configuration..............................
88
vi
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Abstract
Many o£ the current designs of multi-aperture optical and microwave
systems generate impulse responses which exhibit large side lobe
irradiances.
Those systems which are designed to minimize side lobe
irradiances suffer large increases in the radial dimensions of the central
lobes.
The purpose of this research was to design and analyze
multi-aperture systems which had impulse responses which exhibited side
lobe irradiances less than that of the Airy pattern and central lobe
widths less than or equal to that of a single large aperture of an
equivalent diameter. Multi-aperture systems composed of 19, 37, 61, and 91
apertures satisfied these performance criteria.
However, the amount of
energy in the central lobes of these multi-aperture systems was less than
that of an equivalent single large aperture.
Further analysis indicated
that central obscurations in each subaperture lowered the maximum side
lobe irradiance in the 19 aperture arrays.
Random dephasing of the
antennas caused degraded performance of the arrays.
The ability to steer
the arrays using phase variations was investigated.
The effects of a
nontransmitting antenna in the 19 aperture array vas studied.
v ii
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I.
Introduction
There are many astrophysical observations that require large diameter
optics.
The weight and size of these optical systems pose restrictions
for transportation into the space environment.
this dilemma are multi-aperture optical systems.
A possible solution to
A six aperture system
vith an equivalent diameter of a single large optic is illustrated in
Figure 1.
Transportation of these systems into orbit would be less
restrictive.
However, the imaging properties of these systems need to be
investigated.
Figure 2 illustrates the far-field diffraction patterns of the
impulse responses of a single large optic and a six aperture system vith
an equivalent diameter^.
The multi-aperture system impulse response
exhibits relatively large side lobes as compared to the standard Airy
pattern.
Depending on the configuration of subapertures of the
multi-aperture system, the side lobes can become extremely large.
The purpose of this research was to design and analyze a
multi-aperture system which "performed" as well or better than a single
large aperture system of equivalent diameter and allowed a maximum
transmittance of irradiance.
The performance of an optical system in this
research was determined by evaluating the far-field impulse response of
each multi-aperture system.
was based on two criteria.
The performance of the multi-aperture systems
The performance was considered good if the
central lobe width of the imaged point source was equal to or less than
that of the single large aperture of equivalent diameter.
Also, good
performance was characterized by secondary irradiance maxima which were
1
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Fig. 1.
a) Single large optic and b) six aperture system.
2
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'1
Fig. 2. Impulse response of a) a single large optic and b) a six aperture
system of equivalent diameter. The aperture configuration of each system
is depicted in the upper right-hand corner of each plot.
less than 0.0175 (Airy pattern secondary maximum)*.
Another factor that
was considered, in addition to reducing side lobe maxima, was the amount
of transmittance that was permitted through the multi-aperture system as
compared to a single large aperture of equivalent diameter.
Toraldo Di Francia^ examined the reduction of side lobes in proximity
to the central lobe while narrowing the central lobe of the far-field
diffraction patterns of a single aperture.
Be accomplished this by
stopping all the light impinging on the aperture except for a thin ring
located at the extreme outer edge of the aperture.
Adding thin rings at
the appropriate locations and with the correct transmittance coefficients
on this aperture caused the side lobes to be reduced for an increasingly
larger field of view.
However, as more thin rings were added, a smaller
percentage of irradiance was contained within the central lobe.
The
remaining irradiance was forced into extremely large side lobes (many
times larger than the central lobe) at the edge of the field of view.
Figure 3 illustrates the idea of the addition of complex fields
generated by concentric rings to reduce side lobes.
The single aperture,
as depicted in the upper right corner of the figure, has zero
transmittance except for three concentric thin rings of equal width and
transmittance of 1.00.
Figure 3a is the complex field amplitude generated
by the outer thin ring.
The side lobes exhibit large field amplitudes as
compared to the central lobe.
Adding the field amplitude from the middle
ring to that of the outer ring (Figure 3b) causes the resultant side lobes
to decrease in magnitude.
The addition of the fields from all the rings
(Figure 3c) causes a greater decrease in the side lobe field amplitudes.
However, the width of the central lobe produced by the outer ring is less
than that of the summation of all of the rings.
The research reported here extended the idea of the summation of
4
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OUTER RING
05-
<
-0.5
-10
-5
0
s
10
u
a
OUTER+MIDDLE+INNER RMGS
FIELD AMPUTUDE
OUTER + MIDDLE RINGS
0.5-
-0.5
-10
-0.5
-5
0
5
10
-1 0
-5
0
5
10
u
U
Fig. 3. a) Field amplitude generated by outer thin ring of the depicted
single large aperture system, b) summation of the field amplitudes from
the outer and middle rings, and c) resultant field amplitude from
summation of all three rings.
5
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complex field amplitudes generated by concentric rings to multi-aperture
systems.
The approach consisted of designing multi-aperture systems
composed of concentric rings of identical circular subapertures which was
analogous to the concentric thin rings on a single large aperture.
The
results of this research indicated that particular designs of
multi-aperture systems met and, in some cases, exceeded the performance of
a single large aperture system vith an equivalent diameter for the above
criteria.
These systems were comprised of 19, 37, 61, and 91 circular
subapertures.
These subapertures were arranged in expanding concentric
rings about a common origin.
In all cases, the central lobe widths were
equal to that of a single large aperture with an equivalent diameter.
This was an expected result since the central lobe width of a
multi-aperture array is characterized by its largest dimension.^ In
addition, the secondary maxima of the impulse responses for each
multi-aperture systems was less than that of the Airy pattern.
in all cases
However,
the percentage of irradiance in the central lobes of these
systems was less than that of the Airy pattern.
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II.
Theory
The measures of performance of a multi-aperture system, fur the
purposes of this research, were the width of the central lobes and the
maximum irradiances of the side lobes of the far-field diffraction
patterns.
In order to determine the performance of these systems, it was
necessary to model and analyze the impulse responses of each
multi-aperture system.
The theoretical analysis consisted of a derivation
of the generalized form of the impulse response for any multi-aperture
system.
This analysis was taken a step further to derive the complex
field amplitude for multiple rings of identical circular apertures.
The
primary assumption in the analysis was that the aperture systems were
illuminated by monochromatic light.
Multi-Aperture Impulse Response
Figure 4 depicts the configuration used to model a multi-aperture
1 3
system ’ .
In this analysis, the field produced by a single point source
is propagated from the object plane, through the aperture plane, to the
image plane.
The subapertures which comprise the multi-aperture systems are
identical circular apertures.
multi-aperture system.
Figure 5 illustrates one form of a
This particular arrangement of six apertures
illustrates the variables used in the calculation of the impulse response
for all of the multi-aperture systems.
In this analysis "a" was the
radius of each of the subapertures; N was equal to the number of apertures
in the system; x
n
and y
n
described the location of the centers of the n 1*1
7
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/
object
plane
collimating
lens
aperture
lens
plane
observation
(im age)
plane
Pig. 4. Configuration for observing the impulse response for a
multi-aperture system.
8
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►
2a
Fig. 5. Example of multi-aperture system with an aperture-origin
separation of pfl and subaperture radius of a.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
x
subaperture; 0n vas the angle in degrees from the x axis; and pR was the
distance o£ the n**1 subaperture from the origin of the system and vas
expressed in terms of multiples of the subaperture radius.
The pupil function of a multi-aperture function can be expressed as
the convolution of one of the apertures vith the delta functions vhich
describe the location of the centers of each aperture.
As a result, the
generalized pupil function can be vritten as
N
P(x,y) - circ[r/a] * } 6(x - Pncos9
n=l
,y - p sin9 )
(1)
where circ[r/a] « 1 if r/a is less than or equal to 1, otherwise, it
equals 0.
The circ function describes a single aperture of the
multi-aperture system where a is the radius of the circular aperture and
,
.
r-(x 2+y2 )1/2
The impulse response, h(x^,y^), is defined to be the Fourier
8
1 3 11 12
transform of the exit pupil . Therefore, the impulse response is ' ’
’
h(x.,y.) = F{P(x,y)}
1 1
N
= F{circ[r/al) x F{ ]>S(x _ pncosGn ,y - pnsin0n )}
n=l
=
aJ1 {2na[(xi/Af)2 + (y./Af)2 J1/2}
l(x./X£)
,\£.2 ,1/2
+ (y,/Af) J
N
x5expl-i2x (x.p cos©
1
11 + y.p
X 11sin© II)/Xf)
1 11
where F{ } denotes the Fourier transform.
factors have been discarded.
(2)
Nonessential proportionality
Note that for each delta function, which
describes the location of the center of a particular subaperture in the
aperture plane, there corresponds a plane wave traveling from the
subaperture to the focal point.
10
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Letting
u - 2anXj
(3)
Xf
and
v - 2anyi
(4)
Xf
the impulse response can be expressed as
h(u,v) = 2na2J1t(u2 + v2)1/2]
7~2
2717?
(u + v )
N
x 5exPl-i(upncos0_ + vp sin6n)/a]
__ 1
n
n
n
n
n«l
(5>
This impulse response can be further simplified by expressing the
image coordinates as
u = qcost.
v »
qsin+
(6)
where q = the radial coordinate in the image plane and ^ = the angular
position of the image plane coordinate.
Hence, the impulse response can
be expressed as:
h(qf$) =
2
—
=
N
2na J«[q]
^exp[-ip
a
«
nq(cos4cos9n + sin+sinOn )/a]
—
2
N
2Jia J1 Iql ^expl-ip qcos(*-0 )/a]
— q
n=l
n
n
(7)
The envelope function of this impulse response is
11
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2rea2J1 Iq]
q
(8 )
which is a scaled version of the impulse response for a single subaperture
of the multi-aperture system.
This envelope function is modulated by
cosine fringes which arise from the addition of the complex exponentials
which describe the plane waves.
The frequency and direction of the cosine
fringes is a function of the aperture-origin spacing, pn , and the angular
spacing of subapertures, ©n , respectively.
Figure 6 illustrates the modulation effect.
Figure 6a is the impulse
response of any one of the subapertures which comprise the six aperture
system depicted in the top right hand corner of Figure 6b.
Figure 6b
illustrates the modulation of the Airy pattern caused by the addition of
subapertures to the single subaperture.
It is this modulation which
causes the side lobes to increase in irradiance and cause many
multi-aperture systems to perform poorly when compared to a single large
aperture of an equivalent diameter.
Vhen the apertures were moved further
apart, the side lobes increased in amplitude.
Figures 7a through c
illustrate this phenomenon for a six aperture system vith aperture-origin
separations of 2.00a, 3.00a, and 4.00a, respectively.
Impulse Response for Concentric Rings of Apertures
Equation (7) is the generalized expression for the impulse response
of any multi-aperture system which contains identical circular apertures.
This equation is used to form a subset of equations which describe the
impulse responses of multiple rings of identical circular apertures.
The multi-aperture systems examined were comprised of concentric
rings of identical circular subapertures vith a single subaperture at the
12
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 6. Irradiance pattern for a) single aperture of a eulti-aperture
system and b) six aperture system composed of these identical single
apertures.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
oo
oo
■n
X
U
oo
vT
L
oo
7
Pig. 7. Far-field irradiance patterns for a six aperture systea with
aperture-origin separations of a) 2.00a, b) 3.00a, and c) 4.00a.
I
origin.
The single aperture was surrounded by six apertures which just
contacted each other and the center aperture (Figure 8).
The radial
distance of the center of each of the subapertures comprising the next
ring of subapertures was determined by placing an aperture along an axis
which passed through the center of the center aperture and an aperture in
the six aperture ring.
The first aperture, along the horizontal axis, of
the next ring just contacted the aperture of the six aperture ring.
The
distance from the origin to the center of the new aperture (depicted as R
on the figure) defined the radial distance of the circle for the placement
of the centers of the remaining apertures of the new ring.
apertures were symmetrically located along the circle.
These
All of the
multi-aperture systems were constrained to fit inside of a single large
aperture of constant radius aref
As a result, the subaperture radii of
the multi-aperture systems had to be adjusted appropriately.
The following is a derivation of a generalized expression for systems
of multi-apertures arranged in concentric rings which just fit inside a
single large aperture of radius aref
To find the expression for the generalized subaperture radii for any
concentric ring system, let a ■ the subaperture radius of the apertures
which comprised the multi-aperture system.
Using geometric arguments and
referring to Figures 9a and b, the radius of each subaperture comprising
the concentric ring multi-aperture system may be described as:
a ■
— ^
---(2)(rings)+l
(9)
where rings « number of rings which comprise a multi-aperture system.
The radial distance from the origin to the circle on which the
centers of the subapertures of a particular ring are placed, pm can be
15
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Fig. 8. A 19 aperture system illustrating the distance (R) from the
origin to the circle on vhich the centers of each aperture of the outside
ring vere placed.
16
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'ref
A
Fig. 9. a) Seven and b) 19 aperture systems comparing the subaperture
radii vith a c.
ret
I
I
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generalized.
"m" represents a particular ring beginning with m«l as the
ring closest to the origin at a subaperture radius of 2a; m-2 as the next
ring with a radial distance of 4a, etc.
For any ring, pn may be expressed
as:
Pi * 2a
p2 - 4a
« 6a
pn = 2raa
(10)
Substituting equation (9) into (10) yields:
Pm "
2maref
(2)(rings)+l
(11)
For each ring, there is an associated number of apertures that can
fit without overlap.
The number of apertures for each ring can be
described by:
#ring. * 6 - (1)(6)
#ringi - 12 - (2)(6)
Kringj - 18 - (3)(6)
#ringm
*
m6
(12)
There is also a definite relationship between each ring and the
apertures that comprise that ring.
Since each larger ring has six more
apertures than the preceeding ring, the angular spacing between each
aperture in each ring can be expressed as:
Bring. = 360°/6 = 60°
Bring- = 360°/12 = 30°
Bring^ = 360°/18 = 20°
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
©ring
m
- 360°/m6 ■ 60°/m
(13)
Therefore, the location of each aperture in a particular ring may be
generalized as:
©1 « 0,
0- * 9, + 60/m
©^ « ©J + 29-
©J - ©J + 3©j
0m6 * 9 i + [n>6-i]©2
(14)
vhere ©i is the angular location of any single aperture in a particular
ring.
Substituting equations (9) and (10) into equation (7), the impulse
response for each ring is:
0
6
h,(q,+) = 2na J,[q] 2®xPl-i2aqcos(4>-© )/a]
--------- n*l
q
2
= 2na
6
f J,[ql
2exp[-i2qcos(*-© ))
—
—
y n*l
n
[(2)(rings)+l] q
2
(15)
12
h2(q»+) * 2na J1[qJ ^exp[-i4aqcos($-©n)/a)
-------- n=l
q
* 2ita
2
12
f J.[q]
^exp[-i4qcos( *-© )]
--- * n=l
n
t(2)(rings)+l] q
(16)
2
2m
h (q, ♦) = 2 na J.fq] 2«XP [~12mclcos C♦-© )/a]
-------- n=l
19
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2naref Jllq l
2exp[-i2mqcos(*-0n )]
(17)
I(2)(rings)+1] q
which describes the impulse response generated by each ring composed of
identical circular subapertures.
The frequency of the complex field
amplitudes generated by each ring is determined by the argument of each
exponential.
For the rings that are further from the origin, i.e. as 2m
increases, the impulse responses exhibit higher frequencies.
However, the
rings located in proximity to the origin produce lower frequency field
amplitudes.
The total impulse response of the multi-aperture system is a
summation of the impulse responses (complex field amplitudes) of each of
the rings.
This coherent summation of impulse responses yields the
following complex field amplitude at the image plane:
U(q> ♦) = hQ(q,4>) + h^q,*) + h2(q,$) +
+ ^(q,*)
(18)
where hg(q,$) is the impulse response for the single aperture located at
the origin.
Q
The irradiance for each diffraction pattern is expressed as:
I(q»♦) - U(q,♦)U*(q,♦)
(19)
Equations (17), (18), and (19) formed the basis for the calculations
of the far-field diffraction patterns which, in turn, were used to
determine the performance of the concentric ring multi-aperture systems.
20
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III.
Results
The following is a discussion and analysis of the results of the
calculated and experimentally obtained diffraction patterns of the
concentric ring multi-aperture systems.
Using these results, the
performance of each system was evaluated and compared to that of a single
large aperture of equivalent diameter.
COMPUTER PROGRAM
The Fortran computer program that was used to compute the diffraction
patterns is listed in Appendix A.
The first portion of the program
computes the diffraction patterns of the envelope function from a point
source using the IMSL^ subroutine MMBSJ1 to calculate the required Bessel
functions.
This was followed by the calculation of the complex
exponential functions generated by the various apertures in each system.
The final portion of the program finds and stores the maximum side lobe
values and computes the radial energy distribution values of each
diffraction pattern using IMSL^ subroutine DBCQOU.
OPTICAL ANALYSIS
The purpose of this research was to design and analyze multi-aperture
systems which had impulse responses characterized by maximum side lobe
irradiances which were less than that of a single large aperture; i.e.
0.0175.
In addition, the resulting diffraction patterns would have
central lobe widths which were equal to or less than that of a single
large aperture of equivalent diameter.
Although not considered one of the
21
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performance criteria, the percentage of irradiance in the central lobe of
these patterns vas evaluated.
Figures 10b through f depict the multi-aperture systems which were
designed and analyzed.
Each multi-aperture system vas comprised of
concentric rings of identical subapertures with a single aperture at the
origin of the system.
The multi-aperture systems vere comprised of 19,
37, 61, and 91 subapertures, respectively.
system.
Figure 10b vas the 19 aperture
Figure 10c vas the 19 aperture system with the outer ring of 12
apertures rotated 6.75° vith respect to the inner ring (Tschunko and
Sheehan^ examined the MTF and radial energy distribution of a similar 19
aperture system).
Geometrical constraints dictated the number and
location of subapertures which comprised each ring of the various systems.
Figures lOd, e, and f vere the 37, 61, and 91 aperture systems,
respectively.
In each case the radius of each multi-aperture system was
equal to that of the single large aperture of Figure 10a.
As a result,
each of these systems could just fit inside of the single large aperture.
Transmittance Area Comparisons
Since the amount of energy which passed through each aperture system
vas of importance, the transmittance area of each aperture system vas
calculated and is depicted in Figure 11.
All values vere normalized such
that the single large aperture vith an equivalent diameter had 100Z
transmittance.
Using equation (9) and geometric analysis, the total area
of transmittance, T, for each aperture system was:
T19 = n(aref)
19
= Tref (-760)
(20)
25
22
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•••••
• • • • • • •
• • M M M
• • • • • • • • •
V&V.V
• • • • • • • • •
••••••••••
•••••••••••
• • • • • • • • • •
• • • • • • • • •
Fig. 10. a) Single large aperture, b) 19 aperture system, c) 19 aperture
system vith outside ring rotated 6.75 vith respect to inner ring, d) 37
aperture system, e) 61 aperture system, and f) 91 aperture system. All
systems have an equivalent diameter equal to that of the single large
aperture.
23
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TRANSMITTANCE AREA COMPARISON
100 H
80
a
-
£
<
a
a
<
60
a
a
-
o
z
g
40
-
CO
z
<
q
:
20
-
\
APERTURE SYSTEM
Pig. 11. Transmittance of each multi-aperture system referenced to the
single large aperture of equivalent diameter.
24
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
T37 *
n(are£>2 37
- T ref ('755)
(21)
* T ref ('753)
(22)
- T ref (.752)
(23)
49
T6i -
"<aref>2 ^
81
T91 -
K(aref)2
121
where
= *(aref)
2
which is the transmittance area of the single large
aperture and T^g, T yj, Tg^, and Tgj are the transmittance areas of the 19,
37, 61, and 91 aperture systems, respectively.
The 19 aperture system had
the greatest amount of transmittance, 76.OX, vith respect to the single
large aperture.
Far-Field Diffraction Patterns
Figures 12 through 17 are the calculated and experimental diffraction
patterns for each of the depicted aperture systems. For each aperture
system, the calculated modulus (the square root of the irradiance) of the
far-field diffraction pattern (impulse response) is displayed.
The
modulus is displayed for each aperture system in order to enhance the
structure of the side lobes.
The upper left-hand corner of these figures
represents the aperture system which generated the associated impulse
response.
The photograph in the upper right-hand corner is the
experimental diffraction pattern for the same aperture system.
Refer to
Appendix B for a description of the experimental configuration used to
generate and analyze the diffraction patterns.
The maximum central lobe value for each diffraction pattern was
normalized to value of 1.00.
Since a certain percentage of energy vas
25
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}2 ‘i T^eoretical and experimental far-field diffraction patterns for
the single large aperture of normalized radius - a
.
Patterns for
ref
26
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 13. Theoretical and experimental far-field diffraction patterns
the 19 aperture system of total normalized radius * aref*
27
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 14. Theoretical and experimental far-field diffraction patterns for
the 19 aperture system, vith outside ring of apertures rotated 6.75 vith
respect to inner ring, of total normalized radius » a ref
28
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
‘ref29
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
Fig. 16. Theoretical and experimental far-field diffraction patterns for
the 61 aperture system of total normalized radius - aref*
30
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Fig. 17. Theoretical and experimental far-field diffraction patterns for
the 91 aperture system of total normalized radius ■ aref
31
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
stopped at the multi-aperture systems, a theoretical examination vas
conducted to determine the maximum central lobe irradiance values o£ each
multi-aperture system.
A generalized expression is derived for any system
containing concentric rings of identical circular subapertures.
Central Lobe Irradiance Comparisons
For the single large aperture of radius - aref> the impulse response,
assuming unit amplitude plane vave illumination, (referring to equation
(17)), is:
Urer
,(q)
* 2n(aref)2J1(q)
rei
1 ---
(24)
2
Substituting T re^ for X(arg^) , the value of the impulse response at q
0.0, the maximum central lobe value, is:
Uref<°’°> - ^ r e f
" Tref
<25>
2
This central lobe value of the complex field amplitude is equal to the
transmittance area of the single large aperture of radius
The irradiance of the central lobe is:
Iref(0,0) - Uref(0,0)Uref*(0,0) = (Tref)2
(26)
Examining the 19 aperture system comprised of aperturesvith radii
equal to that described in equation (9), the complex field amplitude
32
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(equation (15)) yields:
U19(q,*) -
9
2naref
^
19
Jil<ll
I
1 + exp[-i2qcos(*-0n)]
q
+ exp(-i4qcos(^-0n )l
(27)
Substituting q * 0.0 yields the folloving impulse response:
U19(0,0) - it(are£)2(19)
25
-
T19
(28)
which is the transmittance area for the 19 aperture system.
The maximum central lobe irradiance for the 19 aperture system is:
I19(0,0) = U19(0,0)U19*(0,0)
(29)
Substituting equation (30) into (31) yields:
I19(0,0) = (T19)2
(30)
which demonstrates that the maximum central lobe value for the 19 aperture
system is equal to the square of the transmittance area (assuming unit
amplitude plane wave illumination).
Substituting equation (20) into equation (30) results in:
Iig(0,0) =
-
(Tref)2(.760)2
(Tref)2(.58)
(31)
The ratio of the central lobe values of the single large aperture and
33
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19 aperture system yields:
(32)
which is equal to the square of the transmittance value indicated in
Figure 11.
This analysis can be generalized for any system of multi-aperture
systems composed of concentric rings of identical circular apertures.
Unit amplitude plane wave illumination is assumed.
Combining equations
(17) and (18) and expanding, as in equation (27), results in:
#sub
1 + exp[-i2qcos($-0n)J
+ exp[-i4qcos(9-0n)J
„
(33)
where #sub is the number of subapertures in the system and Tsujj is equal
to the transmittance area of any single subaperture.
Setting q = ♦ = 0.0,
the field amplitude becomes:
u#Sub<°-°> • W
tsub>
(34)
^#sub
where T
transmittance area of a single subaperture and T
transmittance area for the generalized multi-aperture system.
The
irradiance becomes:
2
(35)
I
34
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
vhich indicates that the maximum central lobe irradiance value of any
multi-aperture system composed of concentric rings of identical circular
subapertures is equal to the square of the transmittance area of that
particular system.
Using these same arguments, the maximum central lobe irradiances of
the other multi-aperture systems examined in this research, as compared to
that of a single large aperture of equivalent diameter, equaled the
percentage of transmittance area of the respective systems.
Central Lobe Dimensions
Each of the diffraction patterns vas analyzed to determine the
central lobe width and compared to the central lobe width of the single
large aperture impulse response.
The theoretical and experimental data
indicated that the central lobe widths vere equal.
Harvey** et. al.
arrived at the same conclusions using close-packed synthetic-aperture
systems composed of 3, 7, and 19 subapertures vhich had the same
equivalent diameter as a single large aperture.
The central lobe
diameters generated by each aperture system were:
D1
°19
s
6.2- .3mm
a
6.3- .3mm
6.2- .3mm
°19R D37
°61
°91
where
» 6.1- .3mm
s
6.2- .3mm
6.2- .3mm
= central lobe width generated by the single large aperture and
35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
°19’ °19R' °37’ °61’ °91 are the central lobe widths generated by the 19,
19 rotated, 37, 61, and 91 aperture systems, respectively.
Side Lobe Maxima Comparisons
The side lobe patterns and maxima vere also analyzed.
The details of
the side lobe structure are depicted in the calculated modulus diffraction
patterns.
As the number of subapertures increased from 19, the structured
side lobes, evidenced by the pronounced peaks, moved further from the
central lobe.
The irradiance of these structured lobes decreased as the
number of apertures increased.
As the number of subapertures approached
91, the diffraction patterns more closely resembled that of the Airy
pattern depicted in Figure 12.
The irradiance of the side lobe maxima vere measured for each
diffraction pattern (refer to Appendix B for the experimental design).
Figure 18 depicts the results of this analysis.
In all cases, the side
lobe maxima of the multi-aperture systems vere less than the Airy pattern
value of 0.0175.
For the 19 aperture systems, the maximum side lobe
values vere located at the prominent peaks of the structured side lobes.
The 19 aperture system vith the outer ring of apertures rotated 6.75° vith
respect to the inner ring exhibited the lovest side lobe maxima (the value
of 6.75° vas determined empirically).
The addition of the fields from the
tvo rings of apertures, vith a relative rotation betveen the rings, and
the center aperture vas such that the irradiance of the structured side
lobes vere reduced. (It should be noted that Harvey^ et. al. examined the
far-field diffraction pattern of a 19 subapertrue close-packed
synthetic-aperture system as depicted in Figure 19b.
Even though this
system is similar to the 19 subaperture system examined in this research,
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
SECONDARY LOBE COMPARISON
0.020
LU
o
z
5
0.015
-
<
Qd
C£
Ld
CD
0.010
-
3
oE
2
z
o
0.005
-
3
CO
0.000
APERTURE SYSTEM
Pig. 18. Comparison of secondary lobe maxima (0 - theoretical values
experimental values vith associated uncertainties).
37
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
00
SIDE LOBE M AX = 0.0136
a
SIDE LOBE M AX = 0.0219
b
Pig. 19. Comparison of max side lobe values for the a) 19 rotated and b)
19 close-packed aperture systems.
the close-packed system yields a secondary maxima of 0.0219.
This maxima
is 61% greater than that of the 19 aperture system vith the outer ring
rotated 6.75°).
As the number of apertures increased, the irradiance of the side lobe
disk nearest the central lobe increased.
For the 37, 61, and 91 aperture
systems, the maximum side lobe values vere located in these disks.
As is
illustrated in Figure 18, as the number of apertures increased, the value
of the first side lobe disk approached the value of the first side lobe
disk of the Airy pattern.
Figures 20a through c illustrate the reason for the reduced maximum
side lobe irradiance values for these particular configurations of the
subapertures.
The 19 aperture system, depicted in the upper right-hand
corner, vas used to illustrate this effect.
Figure 20a is the calculated
field amplitude in the far-field due to the outer ring composed of 12
apertures.
Note the large amplitude of the side lobes.
Figure 20b
represents the addition of the field amplitudes from the outer (12
apertures) and inner (6 apertures) rings.
The amplitudes of the side
lobes vere substantially reduced compared to those generated by the outer
ring alone.
apertures.
Figure 20c is the field amplitude generated by all 19
The side lobe heights are considerably less than that
generated solely by the outer ring.
Radial Energy Distributions
The next portion of the research consisted of an examination of the
percentage of energy contained vithin the central lobe of each of the
impulse responses generated by the multi-aperture systems.
Figure 21
represents the theoretical results of the calculated radial energy
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
OUTER RING
-05
5
0
U
s
O
OUTER+ NCR RNGS
05-
-05-1
-0
r
-5
0
u
b
5
o
19 APERTURES
05-
0
U
5
c
Pig. 20. a) Field amplitude of outer ring of 19 aperture system, b)
summation of field amplitudes from inner and outer rings of 19 aperture
system, and c) resultant field amplitude from summation of all 19
apertures.
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0
distributions for the single large aperture and the multi-aperture
systems.
In each case, the radial energy distributions were computed and
normalized for each of the theoretical diffraction patterns in Figures 12
through 17.
The results depicted in Figure 21a indicated that as the number of
apertures increased, the percentage of irradiance contained in the
structured side lobes decreased.
The rapid increase of the slope of the
radial energy curves, past the location of the first zero, indicated the
radial distance where the structured side lobes were located.
As depicted
in Figure 21a, the structured side lobes for the 19 aperture system, as
compared with the other multi-aperture systems, vere closer to the central
lobe.
This observation can be confirmed by noting structured side lobe
locations in Figures 13 through 17.
As the number of apertures increased, the percentage of irradiance in
the side lobes decreased.
As a result, the percentage of irradiance that
was contained within the central lobes necessarily increased.
This is
illustrated in Figure 21a as the percentage of irradiance located from 0.0
radial distance to the location of the first zero.
For instance, the
radial energy distribution curve for the 91 aperture system closely
approximated that of the single large aperture system.
However, the
curves for the 19, 37, and 61 aperture systems exhibited a larger
percentage of irradiance in the side lobes.
Figure 21b depicts the theoretical and experimental results of
measuring the percentage of irradiance contained within the central lobes
of each multi-aperture system.
The results indicated that the percentage
of irradiance of energy within the central lobe of each of the
multi-aperture diffraction patterns was less than that of a single large
aperture.
Generally, the percentage of irradiance in the central lobes
41
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
% OF IRRADIANCE IN CENTRAL LOBES
RADIAL ENERGY DISTRIBUTION
100-1
lOO-i
X
80-
37
80-
60O
O
z
<
Q
<
§
40-
40-
C£
cm
fs>
20
LOCATION OF FIRST ZERO
-
0
8
4
6
2
RADIAL DISTANCE
20
10
-T "
\
— i--- r
<o~
aperture system
Fig* 21. a) Radial energy distribution plot for single large aperture,
19, 37, 61, and 91 aulti-aperture systems, b) Comparison of percentage of
irradiance contained in central lobes of the single large aperture, 19,
19R <19 aperture system with outside ring rotated 6.75 with respect to
inner ring), 37, 61, and 91 aperture systems (0 - theoretical values; X experimental values vith associated uncertainties).
increased as the number of apertures in the multi-aperture systems
increased.
The percentage of irradiance within the central lobes of the
19 and 37 aperture systems was 73.1% and 73.2X, respectively.
percentage for the 61 aperture system increased to 77.5X.
The
The central
lobe of the 91 aperture system contained 82.8Z of the irradiance for the
field of view examined.
This closely approximated the irradiance
contained within the central lobe of the single large aperture (83.8X)
with an equivalent diameter.
MICROVAVE ANALYSIS
The assumption that prevailed in this analysis vas that the
multi-aperture systems were illuminated with monochromatic light.
This
can be seen in the generalization of the image plane coordinates for the
impulse responses.
The generalizations were:
u = 2naxj
and
v * 2nay^
Af
Af
(36)
where (Xj,y^) were the image plane coordinates, A was the wavelength, and
f the distance to the Fourier transform plane (the image plane in this
analysis).
Using this generalization allows one to utilize this analysis
for any wavelength.
Far-Field Diffraction Patterns
The next portion of this research examined the theoretical
diffraction patterns of arrays composed of three to 91 elements
(antennas).
Far-field calculations and side lobe comparisons were
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
performed for these arrays with and without obscurations. Dephasing of
individual elements (jitter) and steerability vere examined for the 19
(rotated), 37, 61, and 91 element systems.
Finally, the 19 rotated system
vas examined for its far-field performance vith
nonfunctioning elements.
The arrays that vere examined are depicted in Figure 22.
The 19,
19R, 37, 61, and 91 element systems are the same as those examined earlier
in this research.
It has been assumed that there vas no coupling between
the individual elements and that each element vas at least 10 wavelengths
in diameter.
This last assumption ensures that there are minimal effects
on the diffraction patterns due to edge effects^.
The far-field
diffraction patterns vere calculated using similar algorithms (equations
17, 18, and 19) as those for the optical diffraction patterns.
However,
since the results of microwave measurements are normally presented in
terms of decibels (dB), the normalized irradiance values (I) vere
converted using the following equation:
IdB = 10 Log10(I)
(37)
These relative irradiance values were analyzed to determine the
performance of the multi-element microwave arrays.
Figure 23 is the diffraction pattern for a single large element
plotted in units of dB.
The first sidelobe is the largest at 17.6 dB
below that of the mainlobe (because of the low sampling rate along the U
axis, the lobe structure will appear to be jagged.
would smooth the lobes).
symmetric.
A higher sampling rate
The single aperture pattern is circularly
However, as has been indicated before, the multi-element
patterns are not.
The succeeding multi-aperture gain patterns are
displayed in this format.
Representative patterns are shown in Figures
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1 element
3 elements
4 elements
4 elements
(square)
MM
V.V
7 elements
19 elements
19 elements
(rotated)
••••••••••
•••••••••••
••••••••••
37 elements
Fig. 22.
61 elements
91 elements
Arrays for nictovave analysis.
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
SINGLE LARGE APERTURE
CD
-17.6 dB
I—
(/I
-20
LU
-30
>
Ld
-40 H
QC
■5
0
5
U (X /D )
Fig. 23.
dB.
Single large aperture far-field diffraction pattern in
46
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
24a, b, and c.
These are the Car-field patterns for the 4(square), 7, and
19 element (rotated) arrays, respectively.
Each figure represents a
planar cut of the pattern which includes the largest sidelobe.
Side Lobe Comparisons With and Without Central Obscurations
This portion of the analysis consisted of examining the relative
intensities of the side lobes
without
aperture blocking.
of the multi-element arrays with and
Blocking simulates the addition of feed horns
or Cassegrain secondary reflectors to the central portion of each of the
elements that comprised an array (the obscured portion of each aperture
has a transmittance = 0.0).
As an upper limit, a 20% central obstruction
(20% radius of unobstructed element) was chosen as a reasonable upper
limit for obscuration due to the feedhorn or secondary reflector.
Figures
25a and b depict the far-field diffraction patterns generated by the 19
element (rotated) array with and without the element blockage.
It is
interesting to note that the central obstructions caused the maximum
sidelobe irradiances to decrease to -18.9 dB from the unobstructed value
of -18.7 dB (in each case, the central lobe gain was normalized to a value
of 0.0 dB.
As a result, the side lobe values vere measured relative to
the central lobe value generated by each particular array.).
The central
obstruction modulated the envelope function, generated by a single element
of the array, such that the peripheral sidelobes contained within the
envelope function vere reduced.
However, the sidelobes nearest the
mainlobe vere unaffected.
Figure 26 is a summary of the results of this analysis for the arrays
with and without obscurations.
Generally, the central obstructions
lowered the maximum sidelobes.
However, the 3, 4, 4 square, and 7
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
O
-o
§
<0
4»
u
CO
9
O'
W
(8P) AJJSN31M 3Ali\rOM
4>
JS
u
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c
u
4P
(0
a
c •
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CO
«-» CO
U t4
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48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19 APS ROWED - 2055 08S0URED
19 APS ROTATED
<8
-*>-
16.7 dB
-
20-
vO
-30
-SO
-10
u (A/D)
-5
u(X/D)
a
Fiff 25. Far-field diffraction patterns generated by the 19 element
(rotated) array a) without and b) with central obscurations in each
subeleaent.
MAX SIDE LOBE COMPARISON
O = WITHOUT OBSCURATIONS
X = WITH OBSCURATIONS
CD
tz
(/)
LxJ
DESIRED MAX SIDE LOBE VALUE
(UPPER LIMIT)
CtZ
Ld
CD
-15-
Q
Q
i>—
O —
CO
-20 4
NUMBER OF APERTURES
X^'vith ? o n K ^ SOn,?f “a*inu“ side lobe values (0 - without obscuration,
with 20* obscuration) for the analyzed arrays.
50
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aperture systems continued to exhibit side lobe irradiances greater than
-17.6 dB.
On the other hand, the 37, 61, and 91 aperture systems did not
have reduced side lobes but still exhibited side lobes less than -17.6 dB.
In these cases, the maximum sidelobe values vere located immediately next
to the mainlobes (refer to the diffraction patterns in Figures 15, 16, and
17).
As explained above, the central obstructions had no effect on the
proximity sidelobes (sidelobes nearest the mainlobe).
As a result, the
maximum sidelobe values did not change.
Central Lobe Irradiance Ratios
The next portion of the microwave analysis was an examination of the
central peak irradiance ratios of the examined arrays.
The central peak
ratios vere computed using equation (35) and:
w°>°>
Iref(0,0)
(38)
vhere Iref(O,0) is the peak irradiance of the corresponding single large
aperture system.
Table 1 displays the irradiance ratios for the arrays
considered.
The 7 element system exhibited the largest central lobe peak
irradiance ratio whereas the 4 element array displayed the lovest ratio.
5i
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Table 1
Peak Irradiance Ratios for Multi-Element Arrays
AFray.
Ilsub(0,0)/Iref(0,0)
Single
3
4
4(square)
7
19
37
61
91
1.000
0.417
0.197
0.471
0.605
0.578
0.570
0.567
0.566
The following is a mathematical analysis of the effect on the peak
irradiance values for multi-aperture systems with central obscurations.
For this analysis, let a - the radius of an element of an array and b =
the radius, expressed as a fraction of the element radius, of the central
obstruction.
The central obstruction has a transmittance of 0.0.
The function which describes an array of apertures with central
obscurations of radius b is:
N
F{Circ(r/a) - Circ(r/b)}F{ I S(x-x , y-y )}
n«l
(39)
The impulse response (complex field amplitude) of this system can be
expressed as (using equation (7) as a reference):
U(q,40 =
2rta2J 1[ql - 2na2b2J ]l[bq]
q
q
N
x J exp[-ipnqcos(*-0n)/a]
(40)
where q = the radial coordinate in the image plane, $ = the angular
position of the image plane coordinate, a = radius of the unobscured
52
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aperture, and b » radius o£ the central obscuration o£ each aperture.
At the location o£ the central lobe maximum, q » ♦ ■ 0.0 vhich yields
the field:
U(0,0) » [na
2
2
2
- b na ] x [tsubapertures in the system]
(41)
where the ftsubapertures in the array is equal to the exponential term in
equation (40) when the argument is equal to 0.0.
2
If Tgub is substituted for na
(the transmittance area of a single
aperture of the array), equation (41) becomes:
2
U(0,0) =
lTsujj - b Tsujj] x [#subapertures in the array]
U(0,0) »
^sub^ “ ^ ) x l#subapertures in the array]
2
(42)
Therefore, the irradiance of the central lobe maximum is:
1(0,0) = U(0,0)U*(0,0)
,2...
,
.2,, .2.2
= (Tsub) (tsubapertures in the array) (1-b )
>>
- (Ttsub>2(1
- b2>2
2,,
-.2
.4,
* <T#sub> (1 - 2b + b >
<*3)
where T#suj, = (Tsuj))(#subapertures in the array) which is the
transmittance area of the unobstructed array.
Jitter Analysis
The far-field patterns generated by the multi-element arrays are
sensitive to the relative dephasing of the array elements
17-22
.
To
53
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
illustrate this dephasing, the outer ring of apertures of the 7 aperture
system vas shifted in phase by X/10 with respect to the central element.
A close examination of Figures 27a and b indicate that the relative
dephasing caused the sidelobe patterns to symmetrically increase in
amplitude and shift in position.
Dephasing two of the outer ring elements
(separated by 1 element) by X/10 with respect to the rest of the array had
an adverse affect on the far-field diffraction pattern (Figure 27c).
The
sidelobe patterns exhibited a marked asymmetry with an increase in
amplitude.
Calculations vere then performed to analyze the random
element-to-element jitter for the 19 (rotated), 37, 61, and 91 element
arrays.
Jitter, in this context, is defined as the random shifting
(phase) of the elements vith respect to each other into and out of the
aperture plane.
The calculations vere performed using:
N
UN (u»v) = U(u,v)x Jexp[-ip (ucos0 +vsin0 )/a)exp(i* )
w
_ i
n
n
n
n
n=l
(44)
where U(u,v) is the impulse response generated by any single subaperture
of the array and $n is the phase associated vith each element.
The phase
values for each element, in terms of X, vere randomly generated using a
Gaussian distribution and ranged betveen the folloving values:
-0.0125X
-0.0250X
-0.0375X
-0.0500X
—
—
—
—
+0.0125X
+0.0250X
+0.0375X
+0.0500X
vhere the extreme ranges are the 3 a values.
Figure 28 illustrates the effect of the jitter on the far-field
diffraction pattern on the 19 element (rotated) array.
The effect is
54
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-lA
‘AJJSN31N 3AU.VT3S
rfi
8P ‘AUSNliNI BAlJLVBa
Pig. 27.
Far-field diffraction pattern of a 7 element system a) without
dephasing, b) vith the outer ring of elements shifted in phase by X/10
vith respect to the central element, and c) vith two elements of the outer
ring (separated by one element) dephased by X/10 vith respect to the
remaining elements.
o
I
aP'AUSNUM 3AliVT3a
55
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
RELATIVE INTENSITY (dB)
19 APS ROTATED: 3 a JITTER = .1A
-10-
U (V D )
Fig. 28. Far-field diffraction pattern for the 19 element (rotated) array
experiencing a 3 a jitter value of .IX.
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
manifested by asymmetry and increased amplitude of the sidelobe patterns.
Figure 29a is a compilation of maximum side lobe values generated by
the 19 (rotated), 37, 61, and 91 element systems experiencing varying
degrees of jitter.
For each range of jitter values, the maximum side lobe
value vas determined 10 times.
lobe irradiances.
The RMS value vas computed from these side
The x axis reflects the total range of the 3 o values
that vere mentioned previously.
As seen in Figure 29a, in all cases, the
magnitude of the maximum sidelobe increased as the jitter value increased.
The 19 element (rotated) system exhibited the lovest side lobes for each
jitter value up to a jitter value of approximately .075X.
However, it is
evident that the rate at vhich the side lobes grew vas inversely
proportional to the number of elements in the system.
This is apparent in
the slopes of the curves, particularly betveen a 3 a jitter values of
.050A and .100X.
Figure 29b demonstrates the amount of jitter that could be
experienced by each system and generate side lobe irradiances vhich vere
equal to or less than that of a single perfectly transmitting antenna
(-17.6 dB).
The 19 element (rotated) system vas able to vithstand .073A
maximum jitter and maintain side lobe values equal to -17.6 dB.
However,
the 91 element system vas able to experience only .048X maximum jitter
before it produced side lobes greater than -17.6 dB.
Steerability
The next portion of this research vas an examination of the ability
to steer the central lobes using phase variations of the individual
elements of each of the multi-element systems.
Crockett and Strange
23
examined the ability to steer the central lobe of a seven element
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CD
■-o5
O
Q.
C
o
CD
Q.
with permission of the copyright owner. Further reproduction prohibited without permission.
Jll IL K
MAX ALLOWABLE JITTER
-I6-1
0.08
,<
0.07-
$
0.06-
QC
£
L/l
00
19-
19R
£
0.05-
0.04 1 1
0.0
.025
.050
.075
3a MAXIMUM JITTER (X )
a
.100
i
19R
1---- 1
37
61
r
91
APERTURE SYSTEM
b
Fig. 29. a) Side lobe irradiances generated by the multi-element systems
experiencing jitter, b) Maximum allowable jitter values for each
multi-element system which will maintain side lobes equal to or less than
that of a perfectly transmitting single antenna.
close-packed system by smoothly and linearily varying the phase across its
face.
This researach utilizes the same steering technique where each
element vas assigned a discrete phase value.
The magnitudes of the phase
difference, i.e. the steering magnitudes, across the face of these systems
vere:
0.0X
0.0X
0.0X
0.0X
—
—
—
—
X/2
X
3 X/2
2X
where the 0.0X value vas at one edge of the element system and the maximum
phase magnitude value is located at the opposite edge.
Equation (44) vas
used to calculate the far-field antenna patterns for this steering
technique.
Figures 30a and b illustrate the effects of steering the central lobe
of the 19 (rotated) and 91 element systems using the technique described
above.
The first plot of each series is the ideal far-field diffraction
pattern for each system without attempting to phase steer the central
lobe.
The second plot employs the 0.0X to X phase difference across the
face of the multi-element system.
Finally, the third plot of each series
illustrates the 0.0X to 2X steering magnitude.
The dashed curves on the
second and third plots are the envelope functions vhich vere generated by
a single element of each respective system.
These plots indicate that the
movement and relative gain of the central lobes vere constrained by the
envelopes of each system.
In addition, the further the central lobe vas
moved from the optical axis, the greater the gain of the side lobes.
Since the central lobe dimension of envelope function of the 19 element
(rotated) system vas considerably narrower than that of the 91 element
system, the central lobe experienced an accelerated gain loss as compared
to the 91 system.
59
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CD
■-o5
O
Q.
C
o
CD
Q.
with permission of the copyright owner. Further reproduction prohibited without permission.
19 APS ROTATED
0 - 2A
0 - A
-
a
20-
I
-3 0 -4 0 -50
-O
-50
-5
0
5
u(vt>)
U(VD)
O'
91 APERTURES
o
0-2A
0 - A
«s>
-
-3 0 -
-O
-
-5
0
s
U(X/D)
10-
20-
-3 0 -
-3 0 -
8
-so
20 -
-
_40'
-50
-O
-5
0
5
n
-5 0
-10
-5
U(X/D)
Fig. 30. Far-field diffraction patterns of the a) 19 (rotated) and b) 91
element sytsems vith the following phase steering magnitudes: 0.0X, 0.0X
— X, and 0.0X — 2X (dashed curve «. envelope function generated by a
single antenna of each respective array).
In order to compare the performance of the different multi-element
systems, the central lobe and maximum side lobe gains vere recorded for
each value of steering magnitude.
The results of this analysis are
contained in Figures 31a, b, c, and d for the 19 (rotated), 37, 61, and 91
multi-element systems, respectively.
These plots illustrate that as the
number of elements increased, vith a corresponding decrease in the size of
the individual elements, the central lobe gain decreased more slovly as
the steering magnitude increased.
lobe values increased vas less.
As a result, the rate at vhich the side
This is a result of the increased central
lobe vidths of the envelope function generated by the increasingly smaller
element dimensions.
As explained earlier, these envelope functions
constrained the movement of the central lobes.
Nontransmitting Elements
The final portion of this research examined the
performance of the
19 element (rotated) array vith one of the 19 elements nonfunctioning.
The far-field antenna patterns vere examined vhen each of the 19 elements
had, in turn, zero transmittance.
The performance of the system vas
determined using the maximum side lobe values.
The maximum sidelobe
values ranged from -15.85 dB to -15.43 dB vith an RNS value of -15.68 dB.
Figures 32a and b illustrate the modulus of the far-field diffraction
patterns for the cases of a nonfunctioning element in the inner and outer
rings, respectively.
the side lobe values.
In both cases, the patterns exhibited an increase in
The most noticeable effects occurred in the side
lobes vhich vere in proximity to the central lobe.
61
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19R APS - STEERABLfTY
37 APS - STEERABUTY
x
x
-5-
-O
-
8-
-
-20
8-
I
-20
I
!-- 1-
0.0 0.5 tO 15 2.0
S1EERNG MAGWTUX (X)
STEERNG MACMTUOC (X )
91 APS - STEERABILITY
61 APS - SnERABUJTY
“X
-* ---v—
ST
sr
-5
-5 -
-20
I
0.0 o!s to 15 10
I1 T
0.0 0 8
f ""1
LO
18
-20
I
0.0 0 8
2.0
tO
18
2.0
STEERNG MAGMDJOC (X )
STtERNG MXGMTUX (X)
c
Fig. 31. The central lobe (X) and maximum side lobe (0) gains for various
values of phase steering (steering magnitudes) for the a) 19 (rotated), b)
37, c) 61, and d) 91 element systems.
62
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
1
O'
u
Fig. 32. Far-field antenna patterns for the 19 element (rotated) system
with a single nonfunctioning element located in the a) inner ring and b)
outer ring.
IV. Conclusions
This research examined multi-aperture systems vhich, according to tvo
reasonable engineering criteria, could perform as veil as a single large
aperture vith an equivalent diameter.
The multi-aperture systems examined
vere composed of 19, 37, 61, and 91 circular apertures arranged in
concentric circles.
Due to geometrical considerations, these systems alloved
approximately 76% irradiance transmittance as compared to the 100X
transmittance of a single large aperture of equivalent diameter.
The
secondary maxima for each of these systems vas less than that of the
single large aperture.
The 19 aperture system vith the outside ring
rotated 6.75° exhibited the lovest maximum side lobes vith a value of
0.0136.
In all cases, the impulse response of the multi-aperture systems
had central lobe vidths equal to that of the single large aperture system.
Hovever, The percentage of irradiance contained by the central lobes
generated by the multi-aperture systems vas less than that of the single
large aperture.
The 19 and 37 aperture systems exhibited the lovest
percentage of central lobe irradiance vhile the 61 and 91 aperture systems
exhibited increased central lobe irradiance values.
The percentage of
irradiance within the central lobe of the 91 aperture system closely
approximated that of the single large aperture system.
This research also analyzed these multi-element systems in terms of
microwave transmitting devices.
The far-field patterns vere examined for
the effects of central obscurations
in each of the subelements, random
phase fluctuations (jitter), steerability using a smoothly and linearly
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
phase difference across the face of each system, and nonfunctioning
elements.
The results indicated that for central obscurations, side lobe
values actually were reduced for the 19 and 19 (rotated) element systems
when compared to the perfectly transmitting system.
Vhen jitter was
introduced into each of the systems, the side lobe values increased and
the far-field antenna patterns exhibited asymmetry.
The 19 element
(rotated) system vas able to sustain the greatest amount of jitter and
continue to exhibit side lobe values equal to or less than -17.6 dB.
The
multi-element systems are capable of being phase steered, hovever, the
side lobe values began to increase rapidly with a corresponding decrease
in central lobe gain.
The 91 aperture system exhibited a superior
steering ablility when compared to the other multi-element systems.
Finally, single nontransmitting elements in the 19 element (rotated)
system caused asymmetry in the far-field antenna patterns with increased
side lobe maxima.
It has been demonstrated that multi-aperture (antenna) systems can
exhibit large side lobes. This analysis has demonstrated that there are
multi-telescope/antenna
designs which actually reduce side lobes below
that of a single large telescope (antenna). When used for imaging
applications, such as Ladar, Radar, and astrophysical observations,
reduction of the side lobes could improve the two point resolution of
these systems.
The antenna arrays discussed in this reseach could also be
utilized for power transmission.
In addition to the multi-aperture approach to this research, one
could interpret these systems as single large apertures that have been
apodised.
Reducing the transmittance of certain small portions of the
aperture to 0.0 will reduce the side lobe values as compared to that of a
single large aperture with no apodization and improve the imaging
65
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properties.
66
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Bibliography
1.
Watson, S.M., Mills, J.P., and Rogers, S.K. "Two-Point Resolution
Criterion for Multi-Aperture Optical Telescopes," Journal of the Optical
Society of America A , Vol. 5, No. 6, (Jun 1988).
2.
Shack, R.V., Ramcourt, J.D., and Morrow, H.
Six-Element Synthetic Aperture," Applied Optics.
3.
Fender, J.S.
"Synthetic Apertures:
"Effects of Dilution on a
10, 257-259 (1971).
An Overview," Synthetic
ApertureSystems, Proc. SPIE, 4AO, 2-7 (1983).
4.
O'Neill, G.K.
"A High Resolution Orbiting Telescope," National
Academy of Sciences, Synthetic Aperture Optics, Vol. 2, Woods Hole Summary
Study (National Academy of Sciences - National Research Council Advisory
Committee to the AFSC), Washington, D.C. (1967).
5.
Sintsov V.N. and Zapryagaev, A.F.
"Aperture Synthesis in Optics,"
Usp. Fiz. Nauk 114, 655-676 (December 1974).
6.
Kwong, R.
"Analytic Expression for the MTF of an Array of Circular
Unaberrated Phased Apertures," Applied Optics, Vol. 27, No. 10 (15 May
1988).
7.
Born, M. and Wolf, E.
Principles of Optics. New York:
Pergamon
Press, 1980.
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8.
Toraldo Di Francia, G.
"Super-Gain Antennas and Optical Resolving
Power," Suppltemento A1 Vol.IX, Series IX Del Nuovo Cimento, 426, 1952.
9.
Goodman, Joseph V.
Introduction to Fourier Optics.
San Francisco:
McGraw-Hill, 1968.
10. IMSL Library.
Library.
Computer Mathematics and Statistics Fortran Subroutine
Houston:
IMSL, Inc., 1984.
11. Harvey, J.E., MacFarlane, M.J., and Forgham, J.L.
Performance of Ranging Telescopes:
"Design and
Monolithic Versus Synthetic Aperture,"
Optical Engineering, Vol. 24, No. 1, 183-188 (Jan/Feb 1985).
12. Tschunko, J.F.A. and Sheehan, P.J.
"Aperture Configuration and
Imaging Performance," Applied Optics.
10, No. 6. 1432 - 1438, June
(1971).
13. Harvey, J.E., MacFarlane, M.J., and Forgham, J.L.
Performance of Ranging Telescopes:
"Design and
Monolithic Versus Synthetic Aperture,"
Optical Engineering, Vol. 24, No. 1, 183-188 (Jan/Feb 1985).
14. Chow, W.V.
"Far-Field Intensity of a Partially Locked Optical Phased
Array," Applied Optics.
23, 4332-4338 (1984).
15. Mills, James P., The Effects of Aberrations and Apodisation on the
Performance of Coherent Imaging Systems, PhD Dissertation,
University of
Rochester, New York, 1984.
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
16. Taylor, C.A. and Thompson, B.J.
"Some Improvements in the Operation
of the Optical Diffractometer," J. of Scientific Instruments, 43, 439,
1957.
17. Palma, G.E. and Townsend, S.S.
"Performance and Phasing of Multiline
Synthetic Apertures," Synthetic Aperture Systems, Proc. SPIE, 440, 68-76
(1983).
18. Vaite, T. and Sun, K.
"Physical Understanding of Synthetic Aperture
Arrays Via Simple Models," Synthetic Aperture Systems, Proc. SPIE, 440,
52-55 (1983).
19. Crockett, G.A. and Strange, D.A.
"Computer Model for Evaluating
Synthetic Aperture Propagation," Synthetic Aperture Systems, Proc. SPIE,
440, 77-84 (1983).
20. Harvey, J.E., Silverglate, P.R., and Vissinger, A.B.
"Optical
Performance of Synthetic Aperture Telescope Configurations," Southwest
Conference on Optics, Proc. SPIE, 540, 110-118 (1985).
21. Butts, R.R.
"Effects of Piston and Tilt Errors on the Performance of
Multiple Mirror Telescopes," Wavefront Distortions in Power Optics, 293,
85-89 (1981).
22. Shellan, J.B.
"Phased-Array Performance Degradation Due to Mirror
Misfigures, Piston Errors, Jitter, and Polarization Errors," J. Opt. Soc.
Am. A, Vol. 2, No. 4, 555-567 (April 1985).
69
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23. Crockett, G.A. and Strange, D.A.
"Computer Model for Evaluating
Synthetic Aperture Propagation," Synthetic Aperture Systems, Proc. SPIE,
440, 77-84 (1983).
I
70
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APPENDIX A
COMPUTER PROGRAM
71
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C
PARAMETER ANNRAD r RADIUS OF CENTRAL OBSTRUCTION.
C
WHEN ANNRAD=SUBRAD, THE ENTIREAPERTURE ISOBSTRUCTED.
C
C PARAMETER PHASE = THE PHASE, IN RADIANS, OF THE OBSTRUCTING APERTURE.
C
THETA IS USED IN THE EXPRESSION, EXP(i*th«t«), WHICH
C
DESCRIBES THE PHASE OF THE OBSTRUCTING APERTURE.
C
WHEN PHASE =0.0, EXP(i.phase) IS SET
C
=0.0.
THIS ESSENTIALLY REMOVES THE PHASE ANNULUS
C
FROM THE APERTURE.
C
C PARAMETER NUM = THE NUMBER OF APERTURES COMPRISING THE SYSTEM.
C
C
PARAMETER TOTRING = TOTAL # RINGS OF APERTURES IN THESYSTEM.
THE CENTER
C
APERTURE IS COUNTED AS A RING.
C
C PARAMETER NUMAPS = # APERTURES IN THE PARTICULAR RING
C
CPARAMETER
DECREE = THE ANGLE IN DECREES, TO DESCRIBE THE LOCATION OF THE
C
THE FIRST APERTURE OF THE PARTICULAR RING. THIS CAN
C
BE THE LOCATION OF ANY SINGLE APERTURE IN THE RING
C
C PARAMETER DELTADEG = THE ANGULAR SEPARATION BETWEEN EACH APERTURE IN A RING
C
C PARAMETER DIS = DISTANCE OF THE APERTURES FROM THE ORIGIN IN THE
C
PARTICULAR RING. ALL APERTURES IN THE RING MUST BE THE
C
DISTANCE •DIS* FROM THE ORIGIN
C
C PARAMETER DNORM = NORMALIZATION FACTOR FOR THE FINAL
C
OUTPUT OF THE INTENSITY
C
37 FORMAT(I3,3X,F10.6)
38 FORMAT(F10.6.3X,13)
39 FORMAT(13)
40 FORMAT (F10.6)
41FORMAT(I3,3X,I3,3X,F10.4,3X,F10.4)
42 FORMAT(13,3X,I3,3X,F10.4)
44 FORMAT(I3,3X,F10.4,3X,F10.4)
60 FORMAT(' PERFORMING COHERENT IMAGING')
61 FORMAT(■ PERFORMING INCOHERENT IMAGING’)
66 FORMAT('0VALUE OF SECONDARY SIDE LOBE IN R.H.S. PLANE IS')
60 FORMAT('0VALUE OF SECONOARY SIDE LOBE IN L.H.S. PLANE IS')
61 FORMAT(’ MAX SECONDARY PEAK IS CREATER THAN PRIMARY
A PEAK ON R.H.S. OF PLANE’)
82 FORMAT(’ MAX SECONDARY PEAK IS GREATER THAN PRIMARY
A PEAK ON L.H.S. OF PLANE’)
63 FORMAT(’ MAX SECONDARY VALUE IS AT (24,24)’)
64 FORMAT(’0POINT SEPARATION = \F10.6)
66
FORMAT(’0
INTENSITY AT (24,24) = ’.F10.6)
66
FORMAT(’0R.H.S. PRIMARY PEAK VALUE = ’,F10.6)
87
FORMAT('0 L .H .S . PRIMARY PEAK VALUE = \F10.6)
70
FORMAT(’0THE IRRADIANCE VALUE USED TO NORMALIZE THE
A MAX IRRADIANCE VALUE TO 1.00 IS’)
71 FORMAT('0COMPUTING THE ENERGY UNDER THE DIFFRACTION PATTERN’)
72 FORMAT(’0NO ENERGY CALCULATIONS BEING ACCOMPLISHED’)
73 FORMAT(’0THE MULTIPLICATIVE FACTOR FOR THE FOV = \ F 10.8)
74 FORMAT(’0COMPUTING THE MODULUS’)
C
PI=3.1416927
IF (COHER.E Q .60)WRITE(•,60)
IF (COHER.ER.61)WRITE(•,51)
C
C COMPUTE THE PHASE TERM FOR THE PHASE ANNULUS
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
TA = CEXP(CMPLX(0.0.PHASE))
IF (PHASE.E Q .0.00)TA s 0.00
C
C READ IN VALUES OF THE LOCATION OF THE APERTURES
C
READ(S,39)TOTRINC
NUMI = 1
DO 6000 1=1,TOTRINC
READ(6,39)NUMAPS
READ(6,40)DIS
READ(5,40)DEGREE
READ(6,40)DELTADEG
IF(NUMI.EQ.1)G0 TO 5003
NUMAPS=NUMI ♦ NUMAPS - 1
5003
CONTINUE
DO 6002 I1=NUMI,NUMAPS
DEGREE = DEGREE * DELTADEG
NX(I1)=C0S(DEGREE*PI/180.0)
NY(I1)=SIN(DEGREE*PI/180.0)
DIST(II) = DIS
PRINT*,I1,DIST(I1).DEGREE
IF(II.EQ.NUMAPS)GO TO 5004
6002
CONTINUE
6004
NUMI = II ♦ 1
6000
CONTINUE
C
C INPUT VALUES OF POINT SEPARATION AND INITIALIZE CENTMIN
C
1100 = 0
CENTMIN=40
FMULT = 0 . 4
WRITE(*, 73) FMULT
C
Ci i i ) i i i ! i i j i i i i i i i ; ; ! ; i |
; i i i i ; | ) { I |
; i i I i i ; | i i I | I i i i i i i i
C 'COMMENT'OUT'NEXT LINE IF'NOT ’F I W I N G 2 Di M 'INTENSITY V S !
C
POINT SEPARATION PLOTS
C
C
DO 1000 PTSEP=0.0,.6,.05
1100 = 1100 * 1
d I l l l l I l l I I l l I l M l l I l l l l l I I | I l 1i ! ! ! !
I I i i
l l !{ I i I I | I I | | | !|
|
C COMMENT'OUT'NEXT LINE’IF‘FINDING'2 DIM INTENSITY V S .
C POINT SEPARATION PLOTS
C
C
PTSEP=0.1936
WRITE(•,64)PTSEP
DO 276 1=1,48
DO 280 J=1,48
Y=(J - 24.0)*FMULT
X=(I ♦ (24.0 - 2.0 • I))*FMULT
IF (X.EQ.0.00000000)X=0.000001
I COMPUTE ARGUMENT FOR FIRST BESSEL FUNCTION (ENVELOPE FUNCTION)
Z= SUBRAD • 5QRT((X-PI*PTSEP)**2.0 ♦ Y**2.0)
CALL UP MMBSJ1 (IMSL) TO COMPUTE THE BESSEL FUNCTION
DUE TO ORIGINAL UNOBSTRUCTED APERTURE FOR POINT SOURCE AT -PTSEP.
CALL MMBSJ1 (Z ,IER)
E = MMBSJ1(Z,IER)
E = (E/Z) • (SUBRAD**2.0)
IF (ANNRAD.E Q .0.00)GO TO 200
C
C
CALL UP MMBSJ1 TO COMPUTE BESSEL FUNCTION FOR THE
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
C
C
OBSTRUCTING APERTURE FOR POINT SOURCE AT -PTSEP.
Z1 =
CALL
El
El
(ANNRAD*Z)/SUBRAD
MMBSJ1 (Zl.IER)
= MMBSJ1 (Zl,IER)
= (El/Zl) • (ANNRAD**2.0)
C
C COMPUTE FUNCTIONS DUE TO LOCATION AND SPACING OF
C
THE fl APERTURES FOR POINT SOURCE AT -PTSEP.
C
200
S=CMPLX(0.0,0.0)
DO 100 11=1,NUM
XI = -((X»NX(I1)*DIST(I1)) - (NX(II)*PI»DIST(II)*PTSEP))
Y1 = -Y*NY(I1)*DIST(II)
Ci i i i i i i
vl
l-n
i i
i i i | I
i i
i i i I
i i
I i i i
i i
I I i i i
i i
i i i
i i
i i i i i i i i i i i
C COMMENT OUT NEXT LINE IF LOOKING AT FIELDS ONLY
C
C(I1)=CEXP(CMPLX(0.0,XI ♦ Yl))
C
CM!!!!!!!!!!! USE THIS NEXT LINE FOR COMPUTING FIELDS ONLY!!!!!!!!!!!!
C
C
G (I1) = COS(Xl +Y1)
C
100
S = S
G(I1)
C
C!!!(COMMENT OUT NEXT LINE IF LOOKING AT FIELDS ONLY!!!!!!!!
C
S2=CABS(S)
C
C !!!!!!!!!!!USE NEXT LINE ONLY IF LOOKING AT FIELDS!!!!!!!!!!!!!!!!!!
Q
C
S2 = S
C
C THE FIELD AMPLITUDE DUE TO THE POINT SOURCE LOCATED AT -PTSEP IS:
C
CONTINUE
IF (ANNRAD.EQ.0.00)El=0.00
A(I,J) = (E-(El*(1.00-TA))) • S2
IF (COHER.EQ.S0)GO TO 700
C
C THE INTENSITY DUE TO THE POINT SOURCE LOCATED AT -PTSEP IS:
C
(THIS NEXT LINE IS USED FOR INCOHERENT IMAGING ONLY)
C n !!!!!!!!!!!!!<!!!!!!!M !!!!!!!!!!!!!! M !!!!!!!!!!!
C
C
IF WANT TO LOOK AT FIELD ONLY COMMENT OUT NEXT LINE
A(I,J) = A (I,J) **2
C
C i I I
i
i
i I I I I I i ! i 1 i i i i i I ! i I i ■! i i ! ! I I i i i j ! i i i i i ! M
i ! ! i ! ! I i j ! ! j ! ! !
C IF NEED TO OBSERVE THE MODULUSi TAKE COMMENT OFF NEXT 2 LINES
C IF NEED TO LOOK AT FIELD ONLY COMMENT OUT NEXT LINE
A (I,J) = SQRT(A(I, J))
WRITE(•,74)
C
700
CONTINUE
C
C IF NEED TO OBSERVE MODULUS FOR COHERENT IMAGING,TAKE COMMENT
C OFF OF THE NEXT LINE
C
A (I,J) = ABS (A (I,J) )
IF (PTSEP.EQ.0.00) GO TO 28
C
C COMPUTE THE ARGUMENT FOR THE SECOND BESSEL FUNCTION (ENVELOPE FUNCTION)
C
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
O'
26
P = SUBRAD • SQRT ((X-»PI*PTSEP) **2 .0 ♦ Y**2.0)
C
C CALL UP MMBSJ1 (IMSL) TO COMPUTE BESSEL FUNCTION DUE TO
C
ORIGINAL UNOBSTRUCTED APERTURE FOR POINT SOURCE AT *PTSEP.
C
CALL MMBSJ1 (P,IER)
Fll = MMBSJ1(P,IER)
Fll = (FI1/P) • (SUBRAD**2.0)
IF (ANNRAD.E Q .0.00)GO TO 3
C
C CALL UP MMBSJ1 TO COMPUTE BESSEL FUNCTION FOR THE
C
OBSTRUCTING APERTURE FOR POINT SOURCE AT +PTSEP.
C
PI = (ANNRAD • P)/SUBRAD
CALL MMBSJ1 (Pi,IER)
FI = MMBSJ1 (Pi,IER)
FI = (Fl/Pl) • (ANNRA0*«2.0)
C
C COMPUTE FUNCTION DUE TO THE LOCATION AND SPACING OF THE
C
SIX APERTURES FOR POINT SOURCE AT »PTSEP.
C
3
V = CMPLX(0.0,0.0)
DO 125 11=1,NUM
X1=-((X.NX(I1).DIST(I1)) ♦ (NX(II)*PI*DIST(11)*PTSEP))
Y1 = -Y*NY(II)*01ST(II)
C(I1) = CEXP(CMPLX(0.0,XI + Yl))
125
V = V ♦ 0(11)
V1=CABS(V)
C
C THE FIELD AMPLITUDE DUE TO THE POINT SOURCE LOCATED AT -»PTSEP IS:
C
CONTINUE
IF (ANNRAD.EQ.0.00)F1=0.00
8(1, J) = (Fll - (FI* (1 .00-TA) )) * VI
IF (COHER.EQ.60)CO TO 701
C
C THE INTENSITY DUE TO THE POINT SOURcE LOCATED AT +PTSEP IS:
C
(USE THE NEXT LINE FOR INCOHERENT IMAGING ONLY)
C
B(I, J) = 8(1, J) **2
701
CONTINUE
C
C
THE TOTAL INTENSITY DUETOBOTHPOINT
SOURCES IS(FOR INCOHERENT):
C
THE TOTAL FIELO AMPLITUDEDUE TO BOTH POINTSOURCESIS (FOR COHERENT):
C
28
IF (PTSEP.E Q .0.00)B(I,J)=0.00
C
C IF NEED TO LOOK AT FIELD FROM A(I,J) ONLY, TAKE COMMENT OFF OF NEXT
C
LINE AND PLACE COMMENT ON A(I,J)=0.0
C
C
B(I,J)=0.0
C
C IF NEED TO LOOK AT FIELD FROM B(I,J) ONLY, TAKE COMMENT OFF OF NEXT
C
LINE AND PLACE COMMENT ON B(I,J)=0.0
C
C
A(I,J)=0.0
C(I,J) = A (I,J) ♦ B(I,J)
IF (COHER.E Q .50)GO TO 702
C
C THE INTENSITY IN THE IMAGE PLANE IS (USE FOR INCOHERENT):
C
INT(I,J)=C(I,J)/DNORM
IF (COHER.EQ.61)GO TO 280
C
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
C USE THE NEXT LINE FOR COHERENT IMAGING ONLY. THIS LINE WILL PROVIDE
C
INTENSITY INFORMATION
C
C702
INT(I,J) =C (I,J) •*2/DN0RM
C! I!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!'!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !
C IF NEED TO LOOK AT FIELD AMPLITUDE, PLACE COMMENT ON LINE ABOVE AND TAKE
C COMMENT OFF OF THE NEXT LINE. USE FOR MODULUS CALCULATIONS!!!!!!!
C
702
INT(I ,J) =C (I,J) /DNORM
C
280
CONTINUE
276
CONTINUE
C
C FIND THE MAXIMUM VALUE OF INTENSITY
C
H=INT(1,1)
DO 10 1=1,48
DO 20 J=1,48
IF (INT(I,J).G T .H)H=INT(I, J)
20
CONTINUE
10
CONTINUE
WRITE (*,70)
PRINT*,H
C
C NORMALIZE INTENTSITY TO A MAXIMUM VALUE OF 1.00
C
DO 16 1=1,48
DO 17 J=1,48
C
C ii i i ; ; I j m | I I ; j i ; | ) I I I I j i I | I i i i | i j ; i j I i ; I ; j j ! I IjI i
'J
c COMMENT’OUT NEXT LINE IF'NORMAL12INC VALUES TO A REFERENCE
C
FROM ANOTHER APERTURE SYSTEM
C
INT(I,J)=INT(I,J)/H
C
C iiiiiiiiiiii iii iiiiiiiiiiiiiiii!iiiiiii
i
iiijilliii
C ‘USE ’NEXT'LINE'NORMALIZING'INTENSITY'VALUES T O ’A REFERENCE VALUE
C
C
INT(I,J) = INT(I,J)/8.999998
CNEXT LINE CALCULATES THE DB VALUES.
C
INT(I,J) = 10.0 • ALOG10(INT(I,J))
C
C I I I I 11 I I l l l l I I l I I I 11 I I I ! i I ! I I j I I I i i I M M I i ! i | j I f I i i
C NEXT LINE SETS VALUES FOR THRESHOLD VALUES OF INTENSITY
C WHEN PLOTTING EXAMPLES,ONLY. COMMENT OUT THIS NEXT LINE
C IF PERFORMING ACTUAL CALCULATIONS
C
C
IF(INT(I,J).LT..6)INT(I,J)=.5
C
CI !IIIIII|III•IM Ii;IIiiiii!II!!!!!!<!!!!!!!!!!!!!!!1!
C j!!!!!!!■!!!!!!!!!!!!!!!!!!!!H !!!!!!!!!!!!!!!!!!!!!!
C COMMENT OUT NEXT LINE IF FINDING 2 DIM INTENSITY VS.
C
POINT SEPARATION PLOTS
C
WRITE(15,42)I ,J,INT(I,J)
17
CONTINUE
15
CONTINUE
170 = 0
C
C i i i i i i i i i I I i ! ! ! ! ! ! ( I ( ( i !| I i | |I i i j i i (i ( i (! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !
!
C THE NEXT’SECTION'COMPUTES'THEENERGY UNDER THE DIFFRACTIONPATTERN
C
C IF DO NOT WANT TO COMPUTE THE ENERGY, TAKE COMMENT OFF THE NEXT LINE
C
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
170 = 70
IF(170.EQ.70)CO TO 8X00
WRITE(*,71)
110
=
0
C INPUT
X AND Y DIMENSIONS
NNX = 48
NNY = 48
C COMPUTE X AND Y VALUES AND READ THE VALUES OF INT(X,Y) FOR EACH X,Y.
DO 6000 II = 1,48
110 = 110 * 1
^
cs
DO 6001 JJ = 1,48
X X (110) = -(110 ♦ (24.0 - 2.0 • 110))*FMULT
Y Y (JJ) = (JJ - 24.0)*FMULT
F(I10,JJ) = INT(110,JJ)
C
PRINT*,110,JJ
C
PRINT.,110,JJ,F(I10,JJ)
6001
CONTINUE
6000
CONTINUE
C INPUT THE EXACT DECLARED ROW (X) DIMENSION OF INT(X,Y)
IFD=200
C BEGIN THE RADIAL ENERGY CALCULATIONS
C
FM = FMULT/2.0
C COMPUTE THE VALUE FOR DETERMINING WHETHER TO USE THE DATA
C FIELD SURROUNOING THE DATA POINT
RADI = SQRT(2.0*(FM..2.0))
•RAD = 0.0
C DETERMINE THE NUMBER OF ITERATIONS(NUMBER OF POINTS TO BE PLOTTED
C ON THE CHART.
NREF = (YY (48)/FMULT) *2 - 2
C BEGIN THE CALCULATIONS
DO 8020 I = 1, NREF
150 = 1
IF(I.E Q .1)GO TO 8200
C COMPUTE THE LENGTH OF THE RADIUS FOR SAMPLING THE DATA FIELD
RAD = RAD * (FMULT)/2.0
IF(RAD.G T .Y Y (48))GO TO 8050
DO 8030 110 * 1,48
DO 8040 J10 = 1,48
C COMPUTE THE LOWER LIMIT FOR THE XX COORDINATE
AA = XX(I10) - FM
C COMPUTE THE UPPER LIMIT FOR THE XX COORDINATE
BB = XX(I10) * FM
C COMPUTE THE LOWER LIMIT FOR THE YY COORDINATE
CC = YY(J10) - FM
C COMPUTE THE UPPER LIMIT FOR THE YY COORDINATE
DD = Y Y (J10) ♦ FM
C COMPUTE THE PRESENT DATA POINT DISTANCE FROM THE ORIGIN (24,24)
RADILOC = SQRT(XX(I10)*«2 ♦ YY(J10).*2)
C DETERMINE IF ONE IS TO USE THIS DATA POINT LOCATION FOR ENERGY CALC.
IF((RADILOC.GT.RADfRADl).OR.(RADILOC.LT.RAD-RADl))GO TO 8040
C COMPUTE THE ENERGY(VOLUME) UNDER THE DIFFRACTION PATTERN AT THIS
C PARTICULAR POINT LIMITED BY THE ABOVE LIMITS
CALL DBCQDU(F,IFD,X X ,NN X ,Y Y ,NNY,A A ,B B ,C C ,D D ,Q ,W K ,IER)
150 = 150 ♦ 1
C ADD VOLUME ELEMENTS APPLICABLE TO THE PARTICULAR RADIUS OF INTEREST
QQA(I50) = QQA(I60 - 1) ♦ Q
QQ(I) = QQA(I60)
GO TO 8040
8200
QQA(I) =0.0
QQ(I) = 0.0
PRINT*,I ,RAD,QQ(I)
GO TO 8020
8040
CONTINUE
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
]
8030
CONTINUE
C SUM THE ENERGY VALUES FROM EACH RADIUS TO THE PREVIOUS RADIUS
QQ(I) * QQ(I-l) ♦ QQ(I)
PRINT*,I ,RAD,QQ(I)
8020
CONTINUE
C NORMALIZE THE MAXIMUM VALUE OF ENERGY TO 1.00
QQMAX = QQ(I-l)
PRINT*,1-1,QQMAX
DO 80S0 I = 1,NREF
QQ(I) = QQ(I) /QQMAX
C
PRINT»,I,QQ(I)
WRITE(31,44)I,qq(I)
8060
6100
CONTINUE
CONTINUE
PRINT* .qQMAX
C iIIIIiI IIIiiI I11MiI IiI>iiiI I|I|!If!I||{ |!!! iIiI!| iii!
-j
o
C F I W THE MAX INTENSITY ON r !h !s ! OF PLANE
C
H1=INT(1,1)
DO 11 1=1,24
00 21 J=l,48
IF (1NT(I,J).GT.H1)H1=INT(I,J)
IF (HI.Eq.INT(I,J))K1=I
IF (HI.Eq.INT(I,J))L1=J
21
CONTINUE
11
CONTINUE
C
C FIND THE POINT SEPARATION WHEN THE SUMMATION OF THE TWO
C PRIMARY PEAKS REACHES A MINIMUM. THIS NEXT PROCEDURE PERFORMS
C THIS FUNCTION. CENTMIN IS THE VALUE OF 1100 (POINT SEPARATION)
C WHERE THE SUMMATION OF THE TWO PRIMARY PEAKS IS A MIN.
c
CENT(1100)=INT(24,24)
IF((I100.Eq.1) .OR.(I100.OT.CENTMIN))GO TO 1281
DO 1270 160=2,1100
IF(CENT(160).GT .CENT(160-1))CENTMIN=160-1
CONTINUE
CONTINUE
1270
1281
C
C CHECK IF HAVE A MAX INTENSITY (MAX SECONDARY INTENSITY)
C
AT INT(24,24) WHEN THE PRIMARY PEAKS HAVE CEASED TO
C
ADD TO ONE ANOTHER.
C
IF((I100.GT.CENTMIN).AND.(INT(24,24).CE..999))G0 TO 1806
C
C NEXT LINE WILL EXECUTE IF SECONDARY MAX PEAK(NOT
C
INCLUDING THE POSSIBLE MAX SECONDARY PEAK AT INT(24,24))
C
> PRIMARY PEAK
C
IF (L1.NE.24)C0 TO 1050
PRIM(1100)=H1
GO TO 1100
C
C FIND MAX OF PRIMARY PEAK ON R.H.S. OF PLANE WHEN SECONOARY
C
PEAK (NOT INCLUDING A MAX SECONOARY PEAK LOCATED AT INT(24,24))
C
IS GREATER THAN PRIMARY PEAK
C
1060 H60=INT(1,24)
DO 1061 1=1,24
IF(INT(I,24).GT.H60)H60=INT(1,24)
1061 CONTINUE
PRIM(I100)=H60
C
WRITE(*,61)
C
WRITE(*,86)H60
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
c
»
°
C1100 IF ((K1.EQ.24).AND.(L1.EQ.24).AND.(PTSEP.GT.0.0))WRITE(*,61)
1100
CONTINUE
C
C FIND THE MAX INTENSITY ON L.H.S. OF PLANE
C
H2=INT(1,1)
C
00 12 1=24,46
C
DO 22 J=l,48
C
IF (INT(I,J).GT.H2)H2=INT(I,J)
C
IF (H2.EQ.INT(I,J))K2=I
C
IF (H2 .EQ.INT (I,J) )L2=J
C22
CONTINUE
C12
CONTINUE
C
IF (L2.NE.24)GO TO 1150
C
GO TO 1200
C
C FINO MAX OF PRIMARY PEAK ON L.H.S. OF PLANE WHEN THE
C SECONOARY PEAK IS GREATER THAN THE PRIMARY PEAK
C
C1160 H51=INT(48,24)
C
DO 1151 1=48,24,-1
C
IF(INT(I,24).G T .H61)H51=INT(I,24)
C1161 CONTINUE
C
WRITE(* ,62)
C
WRITE (•, 87) H61
C
C1200
CONTINUE
C
IF((K2.EQ.24).AND.(L2.EQ.24).AND.(PTSEP.GT.0.0))WRITE(.,62)
C
180S CONTINUE
C
C FIND THE LOCATION OF THE FIRST MIN TO THE RIGHT OF THE
C
PRIMARY PEAK IN THE RIGHT PLANE
C
DO 401 I3=K1,1,-1
IF (INT(13-1,LI).G T .INT(13,LI))GO TO 402
401
CONTINUE
402
IMIN1R=I3
C
C IIMIN1R IS THE VALUE OF THE MIN
C
IIMIN1R=INT(13,L1)
C
C FINO THE LOCATION OF THE FIRST MIN ABOVE THE PRIMARY PEAK
C
IN THE RIGHT HAND PLANE
C
DO 403 J3=L1,48
IF (INT(K1,J3*l).GT.INT(K1,J3))GO TO 404
403
CONTINUE
404
JMIN2R=J3
C
C
FINO THE LOCATION OF THE FIRST MIN TO THE LEFT OF THE PRIMARY
C
MAX IN THE RIGHT HANO PLANE
C
00 405 I3=K1,24
IF (INT(13*1,LI).GT.INT(13,LI))CO TO 406
405
CONTINUE
406
IMIN3R=I3
C
C
FIND THE LOCATION OF THE FIRST MIN BELOWTHEPRIMARY
C
MAX IN THE RIGHT HANO PLANE
C
DO 407 J3=L1,1,-1
IF (INT(K1,J3-1).G T .INT(K1,J3))CO TO 408
407
CONTINUE
408
JMIN4R=J3
C
C FIND THE SECONOARY MAX INTENSITY IN THE R.H.S. OF PLANE
C
H3=INT(1,1)
DO 13 1=1,24
DO 23 J=l,48
13=1
J3=J
IF (CCI3.LT.IMIN3R).AND.CI3.GT.IMIN1R)) .AND.
k
((J3.L T .JMIN2R).AND.(J3.G T .JMIN4R)))GO TO 23
IF (INT(13,J3).C T .H3)GO TO 1800
GO TO 23
1800 H3=INT(13,J3)
IRSECMAX=I3
23
CONTINUE
13
CONTINUE
IF(PTSEP.E q .0.0)H69=H3
C
C IF HAVE A MAX SECONDARY INTENSITY AT INT(24,24) WHEN POINT
C SEPARATION IS SUCH THAT PRIMARY PEAKS ARE NOT SUMMING AT INT(24,24)
C EXECUTE NEXT LINE
C
IF((1100.GT.CENTMIN).AND.(INT(24,24).GE..999))GO TO 1810
C
C IF THE POINT SEPARATION OF THE TWO POINTS IS SUCH THAT THE PRIMARY
C
PEAKS ARE SUMMING AT INT(24,24), EXECUTE THE NEXT LINE
C
IF(1100.LE.CENTMIN)GO TO 1695
GO TO 1598
1810 PRIM(I100)=H3
C
WRITE(*,66)
C
PRINT*,INT(24,24)
SEC(1100)=1NT(24,24)
GO TO 1S99
C
C FIND SECONDARY MAX INTENSITY IN RHS PLANE IF 2 PRIMARY PEAKS
C
ARE SUMMING AT INT(24,24);I .E .THERE IS A CONTRIBUTION FROM
C THE SUMMATION OF THE TWO PRIMARY PEAKS.
C
C FINO FIRST MIN TO RIGHT OF THE PRIMARY PEAK IN RHS
C
PLANE IF THE TWO PRIMARY PEAKS ARE SUMMING
C
1595 CONTINUE
IF((I100.LE.CENTMIN).AND.(INT(24,24).LT..999))IRSECMAX=Kl
IF((I100.LE.CENTMIN).AND.(INT(24,24).LT..999))PRIM(I100)=H1
DO 1475 I3=IRSECMAX,1,-1
IF (INT(13-1,LI).GT.INT(13,L1))GO TO 1478
1475 CONTINUE
1478 I2MIN1R=I3
C
PRINT*,PRIM(I100)
C
C FIND THE SECONDARY MAX INTENSITY IN RHS PLANE IF THERE IS A
C CONTRIBUTION DUE TO THE SUMMING OF THE PRIMARY PEAKS.
C
H3=INT(1,1)
DO 1700 1=1,24
DO 1705 J=1,48
13=1
J3=J
IF(((13.C T .I2MIN1R).AND.(13.L E .24)).AND.
k
((J3.LT.JMIN2R).ANO.(J3.GT.JMIN4R)))G0 TO 1705
IF(INT(13,J3).G T .H3)H3=INT(13,J3)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
1705 CONTINUE
1700
C
1598
C
C
CONTINUE
CONTINUE
WRITE(*,65)
PRINT*,H3
SEC(1100)=H3
CONTINUE
1699
C
C FIND THE LOCATION OF THE FIRST MIN TO THE RIGHT OF THE PRIMARY
C
PEAK IN THE LEFT HANO PLANE
C
C
DO 410 I4=K2,24,-1
C
IF (INT(I4-1,L2).GT.INT(I4,L2))G0 TO 411
C410
CONTINUE
C411
IMIN1L=I4
C
C
FIND THE LOCATION OF THE FIRSTMIN ABOVE THE
C
MAX IN THE LEFT HAND PLANE
PRIMARY
C
Co
M
C
DO 412 J4=L2,48
C
IF (INT(K2, J4 +1).GT.INT(K2,J4))G0 TO 413
C412
CONTINUE
C413
JMIN2L=J4
C
C
FIND THE LOCATION OF THE FIRSTMIN TO THE LEFT OF THE PRIMARY
C
MAX IN THE LEFT HANO PLANE
C
C
DO 414 I4=K2,48
C
IF (INT(I4+1,L2).GT.INT(I4,L2))GO TO 415
C414 CONTINUE
C416 IMIN3L=I4
C
CFIND THE LOCATION OF THE FIRST
MIN BELOW THE
PRIMARY
C
MAX IN THE LEFT HANO PLANE
C
C
00 416 J4=L2,1,-1
C
IF (INT(K2,J4-1).CT.INT(K2,J4))GO TO 417
C416
CONTINUE
C417
JMIN4L=J4
C
C FIND THE SECONOARY MAX INTENSITY IN L.H. PLANE
C
C
H4=INT(1,1)
C
DO 14 1=24,48
C
DO 24 J=l,48
C
14=1
C
J4=J
C
IF (((I4.LT.IMIN3L).AND.(14.GT.IMIN1L)) .AND.
C
k
((J4 .LT.JMIN2L).AND.(J4.GT.JMIN4L)))GO TO 24
C
IF (INT(14,J4).GT.H4)H4=INT(14,J4)
C24
CONTINUE
C14
CONTINUE
C
WRITE(*,«0)
C
PRINT*,H4
C
IF ( (H3 .T.Q. INT(24 ,24) ) .AND .(H4 .EQ.INT(24 ,24) ) )WRITE(* ,63)
C
WRITE(*,»S)INT(24,24)
C
CiI iiiiI iiM iii'i1iI■iiIM I!;!!IM •I{!!!<!!■! !!!!!!!! !!!!!!!! !
C TAKE OFF‘COMMENT’O N 'NEXT LINE IF WANT APERTURE SEPARATION VS
C POINT SEPARATION 2-DIM PLOTS
C
C1000 CONTINUE
C
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
C FIND THE LOCATION (POINT SEPARATION) WHERE THE SUMMATION OF THE
C TWO PRIMARY PEAKS DECREASES FROM 1.00
C
00 1300 I10=1,CENTMIN
IF(CENT(110).LT.1.0)GO TO 1301
CONTINUE
INIT=I10
1300
1301
C
C WRITE INT(24,24) VALUES FROM EACH POINT SEPARATION WHERE
C
THE TWO PRIMARY PEAKS SUMMED TOGETHER
C
DO 1325 110=1,CENTMIN
WRITE(18,44)I10,CENT(110)
1325
CONTINUE
C
C WRITE PRIMARY PEAK VALUES FROM EACH POINT SEPARATION TO A FILE
C
STARTING FROM X LOCATION INIT
C
DO 1400 I20=INIT,1100
WRITE(19,44)120,PRIM(120)
1400
CONTINUE
C
C WRITE SECONDARY MAX VALUE FROM EACH POINT SEPARATION TO A FILE
C
DO 1450 130=1,1100
IF((130.EQ.1).AND.(PTSEP.EQ.0.0))SEC(130)=H89
WRITE(20,44)130,SEC(130)
1450
CONTINUE
C
C FIND POINT SEPARATION WHERE THRESHOLD VALUE IS MET OR EXCEEDED
CD
c
w
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
2030
2035
DO 1960 14=1100,1,-1
IF((SEC(14).GE..1).OR.(CENT(14).GE..1))G0
CONTINUE
WRITE(21,38)DIS,14
DO 1970 14=1100,1,-1
IF((SEC(I4).GE..2).OR.(CENT(I4).GE.,2))G0
CONTINUE
WRITE(22,36)DIS,14
DO 1980 14=1100,1,-1
IF((SEC(I4).GE..3).OR.(CENT(I4).GE.,3))G0
CONTINUE
WRITE(23,38)DIS,14
DO 1990 14=1100,1,-1
IF((SEC(I4).GE..4).OR.(CENT(I4).GE..4))G0
CONTINUE
WRITE(24,38)DIS,14
DO 2000 14=1100,1,-1
IF((SEC(14).GE..5).OR.(CENT(14),GE..6))GO
CONTINUE
WRITE(25,38)DIS,14
DO 2010 14=1100,1,-1
IF((SEC(I4).GE..6) .OR.(CENT(I4).GE.,6))G0
CONTINUE
WRITE(26,38)DIS,14
DO 2020 14=1100,1,-1
IF((SEC(14).GE ..7)-OR.(CENT(14).GE..7))GO
CONTINUE
WRITE(27,38)D1S,14
DO 2030 14=1100,1,-1
IF((SEC(14).GE..8).OR.(CENT(14).CE..8))CO
CONTINUE
WRITE(28,38)DIS,14
DO 2040 14=1100,1,-1
TO 1965
TO 1975
TO 1985
TO 1995
TO 2005
TO 2015
TO 2025
TO 2035
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
00
F~
IF((SEC(I4).GE..9).OR.(CENT(I4).GE.,9))G0 TO 2046
2040
CONTINUE
2046
WRITE(29,38)DIS,14
C WRITE CENTRAL SLICE OF INTENSITY; I.E. A ONE DIMENSIONAL VIEW.
C THE FIRST L00P(J=1,48) VIEWS THE CENTRAL SLICE ALONG THE Y AXIS(X=0.0)
C
DO 426 J=l,48
WRITE(17,44)J,INT(24, J)
426
CONTINUE
C
C THIS SLICE VIEWS ALONG THE X AXIS (Y=0.0)
C
DO 426 1=1,48
WRITE(16,44)I,INT(I,24)
425
CONTINUE
CLOSE(UNIT=16)
CLOSE(UNIT=18)
CLOSE(UNIT=17)
CLOSE(UNIT=18)
CLOSE(UNIT=19)
CLOSE(UNIT=20)
CLOSE(UNIT=21)
CLOSE(UNIT=22)
CLOSE(UNIT=23)
CLOSE(UNIT=24)
CLOSE(UNIT=25)
CLOSE(UNIT=26)
CLOSE(UNIT=27)
CLOSE(UNIT=28)
CLOSE(UNIT=29)
CLOSE (UNIT=30)
CLOSE (UNIT=31)
STOP
END
APPENDIX B
EXPERIMENTAL DESIGN
85
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
Experimental Design
Figure B-l is a representation of the experimental configuration used
to photograph the diffraction patterns, measure the secondary lobe
irradiances, and measure the percentage of irradiance in the central lobes
generated by the multi-aperture systems.
This configuration was aligned
using the methods and techniques described in Mills
8
9
and Taylor .
The light source for the experiments was a 50 milliwatt Helium-Neon
laser.
The laser beam passed through a 10X microscope objective followed
by a spatial filter (10 micron pinhole) which functioned as a single point
source.
The spherical wavefront that was generated by the pinhole passed
through absorbing filters.
These neutral density filters were used to
reduce the irradiance at the detector in order to prevent saturation.
filtered spherical wave continued through a doublet.
The
This doublet
functioned as a collimating lens to produce a plane wave.
The plane wave passed through the multi-aperture mask (photographic
plates).
The masks were made in several steps.
inscribed on sheets of rubylith.
outlines was removed.
The aperture designs were
The rubylith surrounding the aperture
The sheets were placed on a light board and
photographed on 2X2 inch glass plates.
clear transmitting apertures.
The developed plates featured
In all cases, the equivalent diameter of
all aperture systems vas 0.5 inch.
Each photographic mask was placed in a
liquid gate; i.e., an optical flat with index of refraction fluid vas
placed on each side of the mask, so that aberrations due to the
photographic glass could be minimized.
After the plane wave passed through the mask, the resulting field vas
focused by the next doublet.
The resulting irradiance pattern vas
86
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
magnified by a 20X microscope objective.
This image vas then photographed
or analyzed using a linear detector array (I-Scan 256 element line scan
sensor) and a FND-100 detector.
An adjustable iris vas interposed betveen
the image plane and the FND-100 in order to stop incremental portions of
the diffraction patterns in order to determine the percentage of
irradiance in the central lobes.
87
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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