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Geometrical theory of aberrations for classical offset reflector antennas and telescopes

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Geometrical theory of aberrations for classical offset reflector antennas and telescopes

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Content GEOMETRICAL THEORY OF ABERRATIONS FOR CLASSICAL OFFSET
REFLECTOR ANTENNAS AND TELESCOPES
by
Seunghyuk Chang
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
December 2003
Copyright 2003 Seunghyuk Chang
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UMI Number: 3133247
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UNIVERSITY OF SOUTHERN CALIFORNIA
THE GRADUATE SCHOOL
UNIVERSITY PARK
LOS ANGELES, CALIFORNIA 90089-1695
This dissertation, written by
________ Seunghyuk Cliang
under the direction o f h i s dissertation committee, and
approved by all its members, has been presented to and
accepted by the Director o f Graduate and Professional
Programs, in partial fulfillment of the requirements for the
degree o f
DOCTOR OF PHILOSOPHY
Date Decem ber 1 7 , 2003
Dissertation Committe<
Chair
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ACKNOWLEDGMENTS
I would like to thank the following people for their help making this work possible.
. My academic advisor, Professor Aluizio Prata, Jr., for not only introducing
me the aberration theory but also providing me his earlier effort. His help
is not limited to academic. I will never forget the time we spent together
watching stars at Mt. Pinos.
. Professors Hans H. Kuehl and John S. Nodvik, members of the dissertation
committee.
. Somsak Datthanasombat, my friend, for his true friendship.
. Cheolwooo Kim, for his advice.
. Michael Barclay for assembling the foundation of the thesis-style option
used in typesetting this dissertation.
. My children, Eunji, and Jeekyu for loving me. I am sorry for not being a
good father.
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My wife, Hyun-Jung Kwak for sacrificing her life supporting me. I am really
sorry for not being able to keep my promise to finish my study early.
Finally, and most importantly, my parents, Sang-Hyon Chang and Jeong-
Hye Yoon for their support during my study. They patiently waited until I
finish my Ph.D.
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iv
TABLE OF CONTENTS
A c k n o w le d g m e n ts ................... ...................................
List o f T a b le s ....................................................................
List o f F i g u r e s ...................... ..........................................................................
A b s t r a c t ..............................................................................................................
C h a p te r 1 IN T R O D U C T IO N
C h a p te r 2 C O N IC S E C T IO N S O F R E V O L U T IO N
2.1 Generalized Vertex Equation ............................................. ................
2.2 Series Expansion of Conic Section S urfaces.......................................
C h a p te r 3 O P T IC A L PA T H L E N G T H
3.1 Concave Ellipsoidal Mirror with a Real Image .............
3.2 Concave Paraboloidal Mirror with an Object at Infinity .............
3.3 Convex Paraboloidal Mirror with an Object at I n f in ity ................
3.4 Convex or Concave Hyperboloidal Mirror with a Virtual Object .
3.5 General Path Length Expression..........................................................
C h a p te r 4 A B E R R A T IO N S O F S IN G L E M IR R O R SY STE M S
4.1 A stigm atism .......................................................................................
4.2 Coma ....................................................... ...................
C h a p te r 5 A P E R T U R E S T O P AND ITS E F F E C T O N A B E R R A
T IO N S
5.1 Scan and Incidence Angle Changes by Aperture S t o p ...................
5.2 Astigmatism and Coma with an Displaced Aperture S to p .............
C h a p te r 6 A B E R R A T IO N S O F C L A SSIC A L O F F -A X IS T W O -
M IR R O R SY STE M S
6.1 The Geometry of Two-Mirror Systems .............................
6.2 Elimination of Linear Astigmatism and Rusch’s Condition ....
6.3 Coma ....................................... ......................
6.4 Third Order Astigmatism .................................................
6.5 Case Study ....................................................................... ......................
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vi
v ii
ix
1
8
8
12
15
18
22
23
24
26
28
28
34
37
37
45
48
48
52
55
60
61
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V
C h a p te r 7 C O N C L U SIO N 66
B ib lio g rap h y 69
A p p e n d ix A D E R IV A TIO N O F U SE FU L EQ U A T IO N S 72
A .l Equivalent Forms of Equations for Elimination of Linear Astigma
tism ........................................................................................................... 72
A.2 Equivalent Paraboloid and Elimination of Linear Astigmatism . . 74
A.3 Focal Length of Off-Axis Two-Mirror Systems . . . . . . . . . . . 75
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vi
LIST OF TABLES
2.1 Types of conic sections.................. 10
6.1 Data of the case study. See Sec. 7 for the description of i?1 ; itfo h,
i2, ii, and i2................................................................................................ 63
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vii
LIST OF FIGURES
1.1 Representative classical offset dual-reflector s y s te m ............... 3
2.1 Conic section coordinate systems........................................................ . . 9
3.1 Five relevant combinations of conic mirrors, objects, and images. . 16
3.2 Concave ellipsoidal mirror with a real image. . . . . . . . . . . . . 18
3.3 Concave paraboloidal mirror with an object at in fin ity .......... 22
3.4 Convex paraboloidal mirror with an object at infinity.................... . . 23
3.5 Convex hyperboloidal mirror with a virtual object.................... 25
3.6 Concave hyperboloidal mirror with a virtual object................... 25
4.1 Tilted image planes of off-axis paraboloidal mirror.................... 32
4.2 Ray tracing result of an off-axis paraboloidal mirror. Generated
using OSLO LT from Sinclair Optics, Inc........................ 33
5.1 Aperture stop and relevant rays..................................................... 38
6.1 Classical off-axis two-mirror telescopes, (a) Cassegrain with a con
vex secondary mirror, (b) Gregorian, (c) Cassegrain with a concave
secondary mirror, (d) Inverse Cassegrain..................................... 49
6.2 The geometry of a classical off-axis two-mirror system. . . . . . . 50
6.3 The equivalent paraboloid of the offset Cassegrain system ..... 54
6.4 Relations between infinitesimal lengths of the primary and secondary
mirrors. . ....................................... 56
6.5 Side view of the case study off-axis Cassegrain system............. 62
6.6 Astigmatic image surfaces of example telescope, (a) Vertical field
(tja = 0). (b) Horizontal field (xa = 0). Generated using OSLO LT
[17]............................................................. 64
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viii
6.7 Spot diagrams, (a) 1 meter diameter //2 case study telescope, (b)
Single 1 meter diameter f /2 on-axis paraboloidal mirror. Generated
using OSLO LT [17] . * ............................................................... 64
A .l Parameters of off-axis two-mirror systems.................................. 74
A.2 Small length on projected aperture corresponding to a small angle of
the system focus, (a) Compensated two-mirror system, (b) On-axis
paraboloid. ............................................. 77
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ABSTRACT
Classical offset reflector antennas and telescopes are single- or multi-mirror imaging
system of which each two adjacent mirrors share a focus. This optical imaging
system generate a geometrically perfect point focus of incoming rays parallel to
the primary mirror axis and vice versa. However, unlike axially-symmetric system,
axes of mirrors of an offset system do not coincide with each other. Hence, a
conventional aberration theory can not be applied to study offset systems and the
geometrical theory of aberrations of classical offset reflector antennas and telescopes
is derived in this dissertation. A convenient equation suitable to describe off-axis
conic mirrors are derived first. Then, the optical path length concept is applied to
obtain aberrations of single off-axis conic mirrors. It is shown that the dominant
aberration of a single off-axis conic mirror is linear astigmatism, which is second
order and hence larger than conventional third order aberrations. A method to
deal with the effects of a displaced aperture stop in offset systems is developed
next. It is then applied to the derivation of aberrations of classical offset two-
mirror systems. It is shown that linear astigmatism of a classical offset two-mirror
system can be eliminated to yield a compensated system by properly selecting the
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angle between primary and secondary mirrors. It is also shown that third order
coma of a compensated classical offset two-mirror system is identical to the coma
of a single axially-symmetric paraboloidal mirror having the same focal length as
the system. Hence, the practical geoinetric-optics performance of a compensated
classical offset two-mirror system is equivalent to an axially-symmetric system. The
developed theory is verified by designing a fast (i.e., //2 ) compact unobstructed
classical offset Cassegrain system.
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C hapter 1
INTRODUCTION
An increasing number of wide field-of-view antennas and telescopes (e.g., imaging,
scanning, multibeam, etc.) are being made based on classical offset dual-reflector
geometries. The field-of-view performance of these antennas are determined mostly
by the geometrical aberrations of the system. In spite of this, the aberration theory
essential to properly design these antennas remains relatively undeveloped. In the
optical telescope field, most reflecting systems have been made in axially-symmetric
forms although more and more interests are being given to off-axis telescopes. How
ever, axially-symmetric reflecting telescopes have unavoidable obstructions that re
sult in degraded images due to scattering. In fact, it is a large central obstruction
that makes the Schwarzschild aplanatic and Couder anastigmatic telescopes im
practical despite their excellent performance [26]. This problem can be eliminated
in unobstructed reflective optical systems without rotational symmetry. Although
significant works have been conducted on aberrations of these systems [5, 19, 23],
designing a general off-axis system is not a simple task due to the complexity related
to asymmetry. Of the non-axisymmetric system family, plane-symmetric system has
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2
been known to be most practical since, when compared to axially-symmetric sys
tems, they offer manageable mechanical and analytical complexity increases [23, 22].
However, it is still not easy to represent the aberration coefficients of the plane-
symmetric systems in terms of system parameters—only approximate formulas for
aberration coefficients of spherical and slightly aspheric surfaces were previously
obtained in closed forms [23].
A confocal or classical off-axis system is a special case of a plane-symmetric
system. When a point object is located at the focus of the first mirror (or at
infinity on the first mirror axis, if the first mirror is a paraboloid), a classical
system produces a perfect image at the focus of the last mirror since every two
adjacent conic mirrors share a focus. The number of aberration terms of classical
off-axis system is dramatically reduced from general plane-symmetric systems [23].
Furthermore, as it will be shown in this work, some aberration coefficients of a
classical off-axis system can be represented exactly in terms of system parameters
such as vertex curvature, conic constant, etc. Figure 1.1 shows a two-mirror classical
off-axis system. Note that the mirrors of a off-axis system do not need to have
a common axis for a perfect image at the system focus. Although in general this
system suffers from large field aberrations, it will be shown that tilting the secondary
mirror axis with respect to the primary mirror axis can reduce (or even eliminate)
some of the aberrations, producing systems with outstanding performance.
In this dissertation the aberrations of reflecting systems (either antennas or
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3
v
IMAGE ✓ ^ \
■7 FOCUS i
FOCUS \
FOCUS PRIMARY
OBJECT AX1S
Figure 1.1: Representative classical offset dual-reflector system
telescopes) are studied. The main trust of this work is not on predicting the perfor
mance of a given system—many numerical tools are available for this task. Instead,
the emphasis is in analytical expressions that yield design insight for producing sys
tems with improved field of view.
The study of the aberrations of axially symmetric imaging systems started with
Seidel, who developed the first systematic third order aberration theory. He derived
the optical path length between an object and image and expanded it in powers
of the field angle of the object and the coordinates of the ray’s passing point on
the lens. He considered up to fourth order of the field angle and the coordinates
combined. Since the transverse aberration is the derivative of the optical path
length, it has one less degree than the optical path length itself. That is the reason
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4
that Seidel’s aberration theory is called the third order aberration theory. Third
order theory is well explained in Born and Wolf [2]. It was Schwarzschild who
applied Seidel’s theory to the two-rnirror telescopes [28].
In this dissertation the aberration theory of classical offset reflective imaging
system is derived based on the optical path length concept. Since geometrical op
tics is a high-frequency approximation, this aberration theory can be applied to
reflector systems operating at any frequency. An excellent work on the aberra
tions of reflecting optics is [24]. This book explains in detail the derivation of the
aberration coefficients of axially symmetric reflecting systems, as well as analyzes
the characteristics of many widely used systems. The basic method used to derive
the aberration theory presented in this dissertation, as well as most of the nota
tion used, follows [24], However, some serious deviations and generalization were
necessary and unavoidable when dealing with off-axis systems.
Seidel’s third order aberration theory considers five aberration types: spherical,
coma, astigmatism, field curvature, and distortion [2]. However, the aberration
theory presented in this dissertation is limited to coma and astigmatism, since
these are the aberrations that cause image defect in classical systems. Spherical
aberration does not exist in the classical systems considered In this work. Hence, it
is not considered here although it Is present and causes image defects in non-classical
optical systems. Also, due to improved capability of modern computers, distortion
can easily be corrected through image processing. Although current charge coupled
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5
device (CCD), used as detector at focal plane, is flat and hence records blurred
images when field curvature is present, flexible CCD might be developed in the
future to compensate field curvature. Higher-order aberrations are not considered
here since they are not useful for design due to their complexity. Exact ray tracing
is easily possible with modern computers these days. Third order theory is sufficient
to understand the aberration characteristics of a system.
Dragone studied the performance loss of reflector antennas due to aberrations
caused by displaced feeds [11, 12}. However, his study is limited to the first- order
of field angle since he used optical path length of objects located at system focus to
approximate the optical path length variance of object displaced from focus. The
aberration theory presented here is derived by expanding the actual path length
associated with a ray reflected by a mirror in powers of its field angle and the
coordinates of the reflection point. Therefore, aberrations having higher order of
filed angle such as third order astigmatism can also be obtained.
We start from the general equation of an arbitrary conic mirror and from it
derive the aberrations of a single mirror. Aberrations of two-mirror systems, which
are the most commonly used systems, are considered next. It is possible to extend
the presented theory to any number of mirrors even though this was not done
here. Whenever appropriate, the presented theory was verified using a ray-tracing
program. Although several suitable ray-tracing programs are currently available,
in this work OSLO LT was used for all ray-tracing tasks [17].
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In Chap. 2, the vertex equation for an off-axis portion of a conic section of rev
olution is derived from the conventional vertex equation using a coordinate trans
formation [24]. The equation is then expanded to the third order in powers of the
transformed coordinates to cast it in a convenient form for computing of the optical
path length.
In Chap. 3, the optical path length between the object and image of an arbitrary
ray reflected by a conic mirror is calculated and expanded in powers of the field
angle of the object and the coordinates of the reflecting point on the mirror. The
calculation is performed for a concave ellipsoidal mirror with a real image first and
then extended to an arbitrary conic mirror by considering various types of mirror
and object combinations.
In Chap. 4, the astigmatism and coma aberrations of a single conic mirror are
discussed. The second order astigmatism is shown to be the dominant aberration
of offset systems. The study of third order coma is done next.
In Chap. 5, the effect of the unavoidable aperture stop is studied. Aperture
stops occasionally do not coincide with the rim of a particular reflecting mirror
(for example, sub reflector of most dual reflector antennas has its aperture', stop
located at the main reflector). Hence, the effect of displaced aperture stop must be
considered when determining the aberrations of two-mirror systems.
In Chap. 6, the aberrations of classical offset two mirror systems are derived.
It is established that the second order astigmatism can be eliminated by tilting
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7
the secondary mirror axis with respect to the primary mirror axis. The equation
that cancels second order astigmatism is derived and shown to be equivalent to the
Rusch’s condition [20]. Coma and third order astigmatism are derived next. It is
shown that the coma aberration of a classical two-mirror system with zero second
order astigmatism is identical to that of the equivalent paraboloid. The result
obtained in this chapter is proved by a fast (i.e., /.2) compact obstruction-free
classical off-axis Cassegrain case study.
In Chap. 7, summary of the presented work and planned future study will be
given with some remarks.
Throughout this dissertation the notation 0(u) will be used to represent powers
not shown explicitly on a given Taylor series expansion. In this expression u rep
resents any variable or combination of variables. For example, in the hypothetical
expression
f(x , y) = C0 + Cxx + C2y + C3x2 + C4xy + C5y2 + 0(xn*yny) , (1.1)
with nx - 1 - ny = 3, the 0(xnxyny) represents ail terms with powers nx + ny > 3.
Also, the sign convention for object and image distance is determined such that
the distance is positive (negative) when it is real (imaginary).
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C hapter 2
CONIC SECTIONS OF REVOLUTION
In order to study the imaging characteristics of conic sections their surfaces must be
properly represented mathematically. In this chapter an appropriate equation for
generally describing off-axis conic sections in presented. In addition to this, a series
expansion of this equation, in a form convenient to study third order aberrations,
is developed.
2.1 Generalized Vertex Equation
All the conic sections of relevance to this work can be conveniently represented by
a single equation, which is known as the vertex equation of a conic section [24],
namely
p'2 - 2Hz' + (1 + K)z'2 = 0. (2.1)
In this expression R is the vertex radius of curvature and K is the conic constant
(also called the Schwarzschild constant). The p\ z' cylindrical coordinate system
used in this equation is related to the x', y\ z' Cartesian coordinate system of
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9
X'
V E R T E X
CONIC SECTION
Figure 2.1: Conic section coordinate systems.
Fig. 2.1 through
p’2 = x '2 + i/2 . (2.2)
R is assumed positive (negative) when the center of curvature is on the positive
(negative) T-axis, and K is given by
K = — e2 , (2.3)
where e is the eccentricity. As indicated in Tab. 2.1, depending on the particular
choice of K, all types of conic sections can be represented. Substituting Eq. (2.2)
in Eq. (2.1) one obtains
x'2 + y'2 - 2Rz' + (1 + K)z'2 = 0. (2.4)
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10
conic section conic constant eccentricity
hyperboloid K < - 1 e > 1
paraboloid K = - 1 e = 1
prolate ellipsoid - 1 < K < 0 0 < e < 1
sphere K = 0 e = 0
oblate ellipsoid K > 0 e < 0
Table 2.1: Types of conic sections.
Although Eq. (2.4) is adequate for an on-axis mirror, it is more convenient to use a
localized coordinate system when dealing with an off-axis section. To this effect a
local x-y-z Cartesian coordinate system is here introduced to describe the off-axis
section near the reference point with x' — Xq, y' = 0, and z' = z '0. The z-axis of this
system is normal to the surface at the reference point, as shown in Fig. 2.1. The
angle B q between the z and z'-axes is assumed positive (negative) if the smallest
rotation that takes the z-axis to the z'-axis is counterclockwise (clockwise). With
this convention the particular 0O shown in Fig. 2.1 is positive. The relation between
these primed and unprirned coordinate systems [1 ] is
x' = x cos d0 — z sin 0O + x 'Q
y’ — y (2-5)
z ‘ = x sin 0q T z cos 60 + z 'Q .
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Substituting Eq. (2.5) in Eq. (2.4) yields
(1-1- K cos2 8q)z2 + [xK sin 20o — 2 sin60x'0 + 2(1 4- K)z'0 cos0o — 2Rcos0o}z
+ (1 + K sin2 6q)x2 + 2[x’ q cos 80 — R sin 80 + (1 + K)z'0 sin 60}x + y2
+ [x'q — 2Rz'q + (1 + K)z'q ] = 0. (2.6)
Now observe that
x 'o ~ 2R z'q + (1 + K ) zq2 = 0, (2.7)
since the point (x'0, 0, z'0) satisfy Eq. (2.4). Hence, the last term of Eq. (2.6) is equal
to zero. Furthermore, implicit differentiation of Eq. (2.4) with respect to x' yields
X' - R % + {1 + K ) i ' % = 0 ' ' 28)
where (from Fig. 2.1)
Qz!
— = tan 0O • (2-9)
C,X x '= x ’ 0,y'= 0
Substituting x’ — x'0, yl = 0, z' = z'Q , and the above equation, in Eq. (2.8) produces
x’ Q cos0O — Rsin 0o + (1 + K)z'0 sin0o = 0. (2.10)
The fourth term of Eq. (2.6) is then also equal to zero. Finally, solving Eqs. (2.7)
and (2.10) for x'0 and z'0 to obtain
X j= / 5ingb . (2.11)
\J 1 T K sin 0O
R (l cos 0 0 \ . .
Z° 1 + K I y/i + ibsin20o j
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12
and substituting these results in Eq. (2.6), the vertex equation of an off-axis section
of conic can be conveniently cast as
(1 + K cos2 6o)z2 + I x K sin 290
2 R
z + (1 + K sin2 9q)x2 + y2 = 0 .
\ / l + K sin2 90
(2.13)
The above equation conveniently describes the conic section of revolution near the
reference point (x'0, 0, z'Q ), in terms of the local x-y-z Cartesian coordinate system.
Since the reference point has y'0 = 0, as expected this equation is symmetric in y
and asymmetric in x.
2.2 Series Expansion of Conic Section Surfaces
To obtain a form suitable for deriving the Seidel aberrations [2], which requires
powers of x and y, Eq. (2.13) is expanded in a power series of x and y using the
Taylor expansion [1 ] as
where nx + ny = 4. As discussed in the introduction, the symbol 0(xnxyUy) repre
sents all powers of x and y with a combined value equal to or greater than 4. Since
+ 0(xnxyUy), (2.14)
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13
in this work we are only concerned with the third order aberrations, these terms
do not need to be expressed explicitly. Differentiating Eq. (2.13) readily yields the
derivatives needed in the above equation, namely
<9z(Q,0)
dx
dz{ 0, 0)
= 0, (2.15)
= 0, (2.16)
dy
d2z( 0,0) (1 + A'sin2
dx2 R
d2z(0,0)
dxdy
< 9 2z(0, 0) (1 + A'sin2
dy2 R
d3z(0,0) 3K sin 20o(l + K sin2 ( 9 0)2
_____ = —
a3 z(o , o)
d2xdy
d3z( 0,0) K sin 2 6 * o (1 + K sin2 9{
(2.17)
= 0, (2.18)
1
(2.19)
(2.20)
= 0 , (2.21)
(2 .22)
dxdy2 2 R?
= 0. (2.23)
oyA
Substituting Eqs. (2.15)-(2.23) in Eq. (2.14) yields the generalized conic section
vertex equation, expanded near the point x — y = z = 0, as
z = Q j \x2 + a2y2 + dsx3 + C L dX y“ + 0 (xnxynv j , (2.24)
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14
where
(1 + K sin2 9q Y ‘
ai = n---
f 2.254
tt2 = (i + (226)
K sin 2^o (1 + K sin2 90)2
“ 3 = ---------------------- i B i ------------------------ ■ <2 -2 7 )
K sin 20o(1 + K sin2 90) ,n
“4 = w ---------------! (2’28)
and n = nx ~ h ny = 4. Note that fourth order terms (i.e., terms with nx + ny = 4)
are not explicitly shown in the expansion given by Eq. (2.24) because they are
associated with spherical aberration [24], which is absent in classical systems.
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15
C h a p te r 3
OPTICAL PATH LENGTH
Wavefront and ray aberrations can be derived using the optical path length (OPL)
between the object and the image [24]. When all the rays travelling from the
object to the image have the same OPL, the optical system will produce; a perfect
image. This is the case of conic mirrors with objects located at their foci. Since
the distance between the two foci (via an arbitrary point on the conic surface)
is constant, a conic mirror will generate a perfect image at a focus when a point
object is located at the other focus [4]. However, when an object is displaced
from the focus, the OPL between the object and image becomes dependent on the
reflection point location, and the conic mirror will then produce a blurred image.
This phenomenon is called aberration, and its magnitude can be determined from
the difference in OPL between the rays * 2 ], The purpose of this chapter is to obtain
an analytical expression for the OPL when objects are located in the vicinity of
the foci of conic mirrors. Depending on the particular type of conic section used
(i.e.,paraboloid, hyperboloid, or ellipsoid) and type of object (i.e., real or virtual),
the five different situations depicted in Fig. 3.1 are of interest in classical two-mirror
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O B JE C T
F O C U S
- ® F O C U S
IM A G E
(a) Concave ellipsoidal mirror (b) Concave paraboloidal mirror
✓ /
\
F O C U S
F O C U S
F O C U S AXIS AXIS O B JE C T
(VIRTUAL)
IM A G E
(VIRTUAL)
IM A G E
(c) Convex paraboloidal mirror (d) Convex hyperboloidal mirror
F O C U S A XIS
F O C U S
O B JE C T
IM A G E
(VIRTUAL)
(e) C oncave hyperboloidal mirror
Figure 3.1: Five relevant combinations of conic mirrors, objects, and images.
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17
systems:
1. Concave ellipsoidal mirror with a real object. The secondary mirror of the
Gregorian and inverse Cassegrain systems are examples of this situation.
2. Concave paraboloidal mirror with an object at infinity. The primary mirror of
all classical two-mirror systems that operate with collimated radiation, except
for the inverse Cassegrain, fall in this category.
3. Convex paraboloidal mirror with an object at infinity. The primary mirror of
an inverse Cassegrain system is an example of this case.
4. Convex hyperboloidal mirror with a virtual object. The secondary mirror of
a Cassegrain system fall in this category.
5. Concave hyperboloidal mirror with a virtual object. The secondary mirror of
a Dragonian system is an example of this situation.
Note that other combinations exist. However, they have not been listed above
because they are not used in the classical two-mirror systems considered in this
work. The above five situation are considered individually in the following sections.
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18
APERTURE STO P
CENTER POINT
FOCUS
FOCUS
Figure 3.2: Concave ellipsoidal mirror with a real image.
3.1 Concave Ellipsoidal Mirror with a Real Image
Assuming an object slightly displaced from the concave ellipsoidal mirror focus, as
shown in Fig. 3.2, the corresponding OPL is given by
OPL = d + d!, (3.1)
where d and d' are the distances from an arbitrary point (x, y, z) on the conic surface
to the object (x0, y0, z0) and image (xi,yi, z% ) points, respectively, given by
d = + y j(x - x0) 2 + ( y - y0)‘ 2 + {z - z0f , (3.2)
d' = + \ / { x - Xi)2 + ( y - yi)2 + (z~ Zip-. (3.3)
For simplicity, from now on only objects lying on the x-z plane (i.e., y0 = 0) are
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19
considered. Although this causes loss of generality, the subsequent material of this
dissertation will demonstrate that sufficient detail is still left in the formulation to
yield very significant results. The reference ray (shown in Fig. 3.2) is defined as
the ray that emanates from the focus near the object and reflects at the chief ray’s
reflection point. The chief ray (also shown in Fig. 3.2) goes from the object to its
image, passing through the aperture stop center point. Note that the path lengths
s and s' are defined by the chief ray. Also note that the numerical value of s and
s' are positive, since they are both real as mentioned in the introduction. Since
yo = 0 , symmetry indicates that ?/»=0 and the object and image coordinates (i.e.,
x'o, yo and ap ? /* , respectively) are given by
xa = s sin (A ,. + 0)
(3.4)
£ 0 = s cos (0S + 9)
and
Xi = —s' sin(0 s + 9)
(3.5)
z; = s' cos(6S + 0),
where 9S and 9 are the incidence angle of the reference ray and the field angle of the
object, respectively. The angle 6S is assumed positive (negative) if ko ■ x is negative
(positive), where ko is the vector wavenumber of the reference ray. We now expand
the OPL in terms of the coordinates of the reflection point to obtain a suitable
form for deriving the Seidel aberrations. Recalling that
(s2)5 = s , (3.6)
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2 0
the substitution of Eqs. (2.24) and (3.4) into Eq. (3.2) yields
d = s( 1 - , (3.7)
where
2 2
fi = —2 sin^s + 0) — — [1 ~ 2 eq cos(0s + 0 )] — — [1 — 2 a2s cos(0 A - + 0)]
s s s
3 2
+ — 2 a 3 c o s (0s + 0) + — — — 2 ^ 4 cos(0s + 0) + 0(xnxyn ,J) (3.8)
s s
and nx + n s = 4. Equation (3.7) can be expanded using the binomial expansion [1 ]
,, .,2 ,,3
(i-,0* = i - S - £ - £ + < V )
(3.9)
to yield
x
d = s — x sin(0s + 0) + — cos(0 s + 0)
CO s(0A + 0)
— 2 ai
+
1/
2 a2 cos(0 s + 0)
cos(0 A 0 ) \ 2 a3
xy
2 « 4 cos(0s + 0 )
sin(0 s + 0)
sin (0* + 0 )
cos(0 s + 9)
2 a i
2 a2 cos(0s + 0 )
(3.10)
Similarly, recalling that
(s'2) 5 = s' (3.11)
the substitution of Eqs. (2.24) and (3.5) into Eq. (3.3) yields
d! = s'(l — y!)* (3.12)
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2 1
where
/i
- —2sm{9s + 9) - ^-[1 - 2ax cos(0s + 9)] - ~ r[l — 2a2s' cos(6s + 0)]
s s s "
(3.13)
x
sr
r
sn
Q 9
T Til
— 2a3 cos(0s + 9) 4---- — 2 « 4 cos(0s + (9 ) 4- 0{xnxyny) .
s' " s'
The binomial expansion can then be used again to yield the expansion of d! as
x
d' = s' + x sin(9s + 9) + — cos (9S + 6)
cos(0s + 9)
2 ai
y2
+ —
2
X'
- - 2a2 cos(0s + 9)
sin(0s + 9)
— cos[9S + 9) |2 a 3 +
xy2 f sin(9s + 9)
2 ~ | cos(^s + 9) H p
+ 0 (xn*yny) .
cos (9S + 9)
1
(3.14)
— — 2 a2 cos(0 s + 6)
Substituting Eqs. (3.10) and (3.14) in Eq. (3.1) finally yields the OPL as
OPL = s + s' + Axx 2 + A[y2 + A 2xa + A'2xy2 + 0{xrixyny) (3.15)
where
Ax
A'
A‘ 2
- cos(6s + 9)
1 1
_
s s'
cos(9s + 9) ^ — I — -
- 4 a 2 cos(£fi. + 9)
p cos(0S + 9) | sin(0s + 0) Q - p
A' =
- 4 a 3} ,
■ sin(0s + 0)
2 ' y s s'
— 2 04 cos(0, + 9)
1 1
- + —
s s'
4ai
cos(6fi + 9) ( p + p
— 2 a2 cos ($0 . + 6 )
(3.16)
(3.17)
— 2 ax
(3.18)
(3.19)
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2 2
REFERNECE RAY
FOCUS
AXiS
IMAGE
Figure 3.3: Concave paraboloidal mirror with an object at infinity,
and nx + ny — 4. In the above expressions a^s are given by Eqs. (2.25)-(2.28).
3.2 Concave Paraboloidal Mirror with an Object at Infinity
This situation can be regarded as a limiting case of the concave ellipsoidal mirror
with a real image, treated In Sec. 3.1, since a paraboloid can be considered to be
an ellipsoid with a focus at infinity (Fig. 3.3). Since this basic idea can then be
used to transform an ellipsoid into a paraboloid, all the equations of Sec. 3.1 can
be applied for the case of a concave paraboloidal mirror with an object at infinity,
provided that the substitutions s = oo and K = — 1 are made.
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23
REFERENCE RAY
FOCUS
Jj
AXIS
IM A G E
(VIRTUAL)
Figure 3.4: Convex paraboloidal m irror with an object at infinity.
3.3 Convex Paraboloidal Mirror with an Object at Infinity
In this case the image of the mirror is located behind its surface, and hence becomes
the virtual image shown in Fig. 3.4. Since the image is now virtual, the OPL must
be given by (apart from a non-important constant)
instead of by Eq. (3.1). Note that Eq. (3.11) was used in the derivation of d!
in Sec. 3.1 [i.e., Eq. (3.12)] because s' was positive. However, since in the present
section the image is located behind the mirror, and hence s' is negative, the equality
OPL = d - d ' , (3.20)
(s,2)t = - s ',
(3.21)
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24
must be used instead of Eq. (3.11). W ith this in mind the expression for d' is now
given by
d '= s ’(l-fj!)z , (3.22)
instead of Eq. (3.12). Note that, since s is still positive, as it was in Sec. 3.1, the
expression for d is still given by Eq. (3.7). Observe that there is now a negative
sign in front of s' in Eq. (3.22). However, since this negative sign is cancelled out
by the negative sign in front of d' in Eq. (3.20), for the current geometry the OPL
is still given by Eqs. (3.15)-(3.19).
3.4 Convex or Concave Hyperboloidal Mirror with a Vir
tual Object
These two cases are shown in Figs. 3.5 and 3.6. Since the object is now located
behind the mirror surface, the OPL must be computed using (apart from a non-
import ant constant)
OPL = - d + d', (3.23)
instead of Eq. (3.1). Similarly as in section 3.3, since s is negative, the equality
(s2)5 = - s (3.24)
must be used in the derivation of d instead of Eq. (3.6). The expression for d is
then given by
d = — s(l - fip . (3.25)
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25
C o
AXIS
FOCUS FOCUS
OBJECT
(VIRTUAL)
IMAGE
Figure 3.5: Convex hyperboloidal m irror w ith a virtual object.
to
rn
AXIS
FOCUS FOCUS
OBJECT
(VIRTUAL)
IMAGE
Figure 3.6: Concave hyperboloidal m irror w ith a virtual object.
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26
Note th at Eq. (3.12) is still valid since s' is positive. As in the previous section, the
negative sign in front of s [in Eq. (3.25)] is cancelled out hv the negative sign in front
of d [in Eq. (3.23)]. As a consequence, the OPL is still given by Eqs. (3.15)-(3.19).
3.5 General Path Length Expression
It was shown from Sec. 3.1 through Sec. 3.4 that, regardless of the situations con
sidered, the OPL expression is always given by Eqs. (3.15)-(3.19), namely
OPL = s + A + Ajx2 + A\y 2 + A 2x 3 + A'2xy2 + 0{xn*yn») (3.26)
where
Ai = - cos(#s + 6)
A[
A2
2 ^ s ' f o S ( 8 s + 0 ) Q + i
\ 0 + ? ) _4fl2c o s ^ s + ^
1 f ................(1 1
— 4 c< q
A' =
- cos(0s + 0) |s in (6* s + 0) - —
— 4 0.3} ,
1 S in(0, + 0) ( 1 - 4 ) [ ( ) + t
— 2 « 4 cos(0s + 6),
cos(0 s + 0) f — i — -
\ s s'
2 n2 cos(0 s . -f- 0 )
(3.27)
(3/28)
— 2 a,\
(3.29)
(3.30)
and nx + ny — 4. Hence, the above OPL expression can be used for any cases
of classical two-mirror systems, provided that the numerical value of s and s' are
positive (negative) if they are real (virtual), and 0S Is positive (negative) If • x is
negative (positive) (see Pig. 3.2).
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27
Before proceeding into the next chapter, at this point it is convenient to derive
an equation relating < ? 0 and 0S. Considering an object at the focus (i.e., 9 = 0), the
OPL = s + s' (all higher order terms must be zero). Hence, forcing A\ = 0 and
A\ = 0 in Eqs. (3.16) and (3.17) produces
. (1 1 \ 2(1 + K sin2 ( 9 0)i
“ ■«* U + ? - J R ° ’ (3'3 1 )
1
1 1 \ 2(1 + K sin 0O) 2
cos 9S - + — 1 ------------- cos 6S = 0. (3.32)
\ s s' / R
Subtracting these two equations yields the desired useful relation between 90 and
9S 1 namely
K sin2 90 = sin2 6S . (3.33)
Note that, although the above equation was derived considering the case with 0 = 0,
it is also valid for 9 ^ 0 since 0O and 0S are independent of 9.
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28
C hapter 4
ABERRATIONS OF SINGLE MIRROR SYSTEMS
In this chapter each term of Eq. (3.15) is considered individually to determine the
aberrations of a single mirror systems. The second order and third order terms of
Eq. (3.15) give astigmatism, field curvature, and coma [24]:
OPL — s + s' + A 1x 2 + A'ly2 + A 2x3 + A'xy 2 + 0(xn*yn*). (4.1)
^ v v ....................................... ............ ......s
^ ^r u V ^ '--------------- : -
Astigmatism and Coma
Field Curvature
Closed form practical expressions for the above four terms, as they appear in single
mirror systems, are derived. From these expressions it is shown that the dominant
aberration of a single off-axis mirror is linear astigmatism [23]. Expressions for
third order coma and astigmatism are also derived and discussed.
4.1 Astigmatism
Let’s consider Eq. (3.15) once more, namely
OPL = s + s’ + 0{xn*ynv ) , (4.2)
=£0
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29
where now nx + ny = 2. If the A\ and A'x coefficients could somehow be made equal
to zero, one would have
OPL = s + s' + 0{xn*ynv), (4.3)
where nx + ny is now equal to 3. In this situation the image quality would improve
since the OPL will significantly deviates from s + s' for larger values of x and y.
However, as will be seen below, the A\ and A\ can not be unconditionally made
equal to zero. Nevertheless a good tradeoff is still possible. To see this consider
only tangential and sagittal rays (i.e., rays lying on the y = 0 and x = 0 planes,
respectively.) In these cases, Eq. (3.17) reduces to
OPL\t = s + s' + Aix 2 + 0(x 3
OPL\s = s + s' + A[y2 + 0(y 3) ,
(4.4)
(4.5)
for the tangential and sagittal rays, respectively.
Convenient expressions for A\ and A\ can be obtained substituting Eqs. (2.25),
(2.26) and (3.33) in Eqs. (3.16) and (3.17) to yield
A 1 = - cos {9 a + 0) — ! — - j cos(0s + 6) —
O O ' I
s s’
2 cos3 1
R
A ' — -
- o
s s'
2 cos3 9S cos(6b + 0)
R
(4.6)
(4.7)
These expressions indicate that Ai and A[ can only be made zero by properly
selecting the image distance s'. It is then impossible to simultaneously zero Ai and
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30
A{. Hence, making A\ — 0 and A\ = 0 yields the so-called tangential and sagittal
astigmatic image locations s't and s' 8 as
1 2 cos3 0.
' S
s't Rcos(9s + 9) s ’
1 2 cos cos(#s + 9) 1
(4.8)
(4.9)
s( R s ’
respectively. In other words, if the image distances s' = s't (s' ~ s's) the tangential
(sagittal) rays will come to a perfect focus [within the third order approximation
indicated in Eqs. (4.4) and (4.5)]. The presence of A 1 and A[ then produces an
astigmatic image [2]. To understand the behavior of the tangential and sagittal
image locations with field angel 9, it is convenient to expand the above equations
in power series of 9. Using the appropriate trigonometric series expansions [13] we
obtain
1* Z )
sin(9s + 9) = sin 9S + 9 cos 9S — ( 9 2^ — F 0(83) , (4.10)
PQ C
cos(0, + 9) = cos 6S — 9 sin 9S — 92—— — + 0(93) . (4.11)
Substituting Eq. (4.11) in Eqs. (4.8) and (4.9) yields
1 = l a k h _ 1 + „ h h h + L t h f h + , ( 4 . 1 2 )
st R s R R
i = 5in 2 ? i _ < j2 ' ^ j L + o m (413)
s's R s R R y ’ v '
Observe that, since 0S = 0 for on-axis systems, the first order terms (i.e., 9 to the
first power) of Eqs. (4.12) and (4.13) are unique to off-axis mirrors. These terms
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31
represent the obliqueness of the two astigmatic image surfaces causing linear astig
matism [23]. Note that linear astigmatism is the lowest order aberration present in
off-axis mirrors and hence gives the dominant image degradation.
Subtracting Eq. (4.13) from Eq. (4.12) yields the astigmatic images’ separation
A s' as
A s' s' - s', 2 sin 20, 2 2 „
T 7 = - V r 1 = A - °+nd + °(°) ■ (4-14)
stss SWS R R
The 92 term of Eq. (4.14) causes the two astigmatic image surfaces to have differ
ent curvatures, and hence gives third order astigmatism [24]. Although in a single
off-axis mirror the impact of third order astigmatism is small compared to lin
ear astigmatism , it becomes more significant in compensated two-mirror systems,
where the linear astigmatism is eliminated. Compensated two-mirror system will
be studied later in this dissertation.
As a specific example, consider an off-axis section of a paraboloidal mirror shown
in Fig. 4.1. Since its object point is located at infinity, substitution of s = oo in
Eqs. (4.12) and (4.13) yields
- 7 = - 7(1 + 0ta,n9a) + 0(93) , (4.15)
■ st so
1 = i ( l _ 0 t a n 0 s) + 0 ($3), (4.16)
Ss S0
where s(} is the distance from the off-axis paraboloidal mirror to its focus (measured
along the reference ray) and given by
± = (4 ,7 )
Sn I t
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32
CHIEF RAV
R EF ERE NC E RAY
SAGITTAL
IM AGE PLANE
AXIS
TA N G EN TIA L
IM AGE PLANE
Figure 4.1: T ilted image planes of off-axis paraboloidal m irror.
Equations (4.15) and (4.16) show that the tangential and sagittal image surfaces do
not coincide and that they are tilted from the plane normal to the reference ray by
an angle 0S in opposite directions (see Fig. 4.1). As a specific example, ray tracing
results for an off-axis paraboloidal mirror of D=100 mm and F=300 mm with 1°
of field angle is shown in Fig. 4.2. The tilted image planes and the dominant
second order astigmatism can be clearly seen: note the line images produced at
the tangential and sagittal image locations at focus shift — 4 mm and +4 mm,
respectively.
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33
PARABOLOIDAL MIRROR
REFERENCE
100mm
RAY
FOCAL
RAY 10° mm
TANGENTIAL SAGITTAL
IMAGE LOCATION
5mm
- 1.0
(a)
TOWARDS MIRROR
(b)
0 = 1.0°
0 =0.7°
6 = 0°
I
—i* | 2mm [*—
0=— 0.7
° £
I s-
0=-l.O j
-4mm -2mm 0mm +2m m +4mm
FOCAL PLANE SHIFT ALONG THE REFERENCE RAY
(C )
Figure 4.2: Ray tracing result of an off-axis paraboloidal mirror. G enerated using OSLO
LT from Sinclair Optics, Inc.
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34
4.2 Coma
The third order terms of Eq. (3.15) can be shown to be associated with the coina
aberration, and the first order terms of the expansion of A 2 and A'2 in powers of
6 give the third order coma coefficients [24], The rays on the tangential plane
(i.e., y = 0) give the tangential coma. Since the tangential plane is the plane of
mechanical symmetry, these tangential rays always remain on the tangential plane
after reflection, and hence the study of the tangential coma is relatively simple.
Note that general sets of rays on a plane other than the tangential plane do not
remain on a plane after the reflection. This makes the decision of which image
location to use in the study of the coma aberration unclear, and hence a general
analysis of the coma aberration of these rays is highly complicated. Therefore, only
the tangential coma will be considered in this work. Although this obviously causes
a loss of generality, sufficient significance is still left in the presented formulation.
Observe that, for the rays on the tangential plane, the contribution of A' 2 is
not present (since y = 0 in the OPL equation). Hence, to obtain the third order
tangential coma coefficient, only A 2 needs to be expanded in powers of 0. To this
effect Eqs. (2.25) and (2.27) are substituted in Eq. (3.18) to yield
A '2 — — — cos(6tj + 0) / sin(0 s + 0) ( — — — ^ cos(0 s + 0) ^
Z „ ^ (4.18)
K sin 2d0(1 + K sin 0O )
4-----------
(1 + K sin 0O) 2
R R 2
Since 0S and 9q are related by Eq. (3.33), it is convenient to have only one of them
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35
in Eq. (4.18). Using then Eq. (3.33) we obtain
cos2 0q = 1 — sin2 0o
K + sin'
(4.19)
which yields, after it is multiplied by Eq. (3.33),
2 sin 0S \J— K — sin2 0S
sin 2 0 q =
K
(4.20)
Now, substituting Eqs. (3.33) and (4.20) in Eq. (4.18) we obtain the desired A2 in
terms of 0S only as
1 - u n 1
A 2 = - - cos(0s + 0) < sin(0 s + 9) I —
J L I \ S S
cos(0 s + 0) ( — + -
cos3 6S
R
+
2 sin 0S cos4 9S\ ] —K — sin2 1
W 2
(4.21)
As discussed in the previous chapter, the geometrical aberrations are obtained from
the OPL between the object and image. Since the image location for the tangential
rays is tangential astigmatic image location, s' = s't is substituted in Eq. (4.21) to
study coma aberration. Although this choice of image location might not minimize
coma aberrations, we can isolate coma from linear astigmatism since A\ = 0 at the
tangential image location. Hence, Eq. (4.8) is substituted in (4.21) to yield
A2 = cos(0s + 0) |sin (0 s + 0)
1 cos3 0 ,
s R , cos(0s + 9)
sin 9S cos4 0S \J—K — sin2
+ _
Substituting Eqs. (4.10) and (4.11) in (4.22), we finally obtain
cos3 9S
R
(4.22)
cos3 0S ( . ( 1 1
A2 = — - — < sm 0S cos 9S I -
R 1 Vs h)
e cos2 6S I - - 1 - sin2 9 ,
So
+ 0 ( f ) , (4.23)
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36
where sq is the distance from the reflection point of the chief ray to the focus near
the object and given by [4 ]
As a specific example, an off-axis section of a paraboloidal mirror is considered
again. Substituting s = oo and K = — 1 in the above equations yields
When 6S = 0, the above coma coefficient can be shown to be consistent with the
corresponding coefficient of the on-axis paraboloidal mirror [24], but deviates from
it as 6S increases. However, as mentioned in the previous section, the dominant
aberration of an off-axis mirror is the linear astigmatism. Therefore, the coma
aberration is less important in a single off-axis mirror and will be discussed later in
association with two-mirror telescopes where the linear astigmatism is eliminated.
Third order coma is the dominant aberration for those systems.
1
R
(4.25)
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37
C h a p te r 5
APERTURE STOP AND ITS EFFECT ON
ABERRATIONS
In the previous sections it was implicitly assumed that the aperture stop center
point coincides with the x-y-z reference coordinate system origin. However, in many
situations the aperture stop is displaced from the reflecting surface. In dual reflector
antennas or two-mirror telescopes the primary or secondary mirror becomes the
aperture stop of the system unless a physical aperture stop is present in the system.
When the aperture stop is displaced from the reflecting surface, the aberration
coefficient of the mirror also changes [24], In this chapter, the aberration coefficients
altered by the displaced aperture stop are explicitly calculated.
5.1 Scan and Incidence Angle Changes by A perture Stop
Figure 5.1 shows a mirror system where the aperture stop center point is displaced
from the mirror surface and located at the point O. In this case, the reflection point
D of the chief ray and the incidence angle 6S of the reference ray depend on the
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38
o
APERTURE STOP
CENTER POINT
OBJECT
IMAGE 0 A
FOCUS
B®
FOCUS
(a) A perture stop an d relevant rays.
V
A
|b} Part of (a) with a d d itio n a l lines
Figure 5.1: A perture stop and relevant rays.
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39
location of the object. However, an off-axis mirror system can be more conveniently
described using a ray that is independent of the object location and plays a similar
role as the optical axis (i.e., axis of symmetry) of axially-symmetric systems. Hence,
the optical axis ray (OAR) shown in Fig. 5.1 is introduced [23]. The OAR is defined
as the ray going from one focus to another, while passing through the aperture stop
center point. In the previous section the aberration coefficients of a single mirror
system was expressed in terms of the parameters associated with the chief ray’s
reflection point (i.e., 9S , 0, and s0 in Fig. 5.1). However, multi-mirror systems
are best described by the parameters, dg, V b and po, which are associated with the
OAR and the aperture stop. Hence, expressions for 0S, 9, and s0 in terms of ijjs, 0,
and po are desired. The purpose of this section is deriving equations relating these
parameters, which are necessary to obtain the aberration coefficients in existence
of a displaced aperture stop.
The angles 8S and 9 can be expressed in terms of and A by considering various
triangles (A) shown in Fig. 5.1. The angle 9 is computed first. Observing from
indicates that 9 can be obtained by computing $ { , . Since A BDE and A ODE share
a common side DE one has
A OBD that
6 = A - 9b ■
(5.1)
OE tan 0 = BE taxi 9b . (5.2)
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40
Substituting OE = W + h and BE = p0 + h in the above equation yields
(W + h) tan ip — (p0 + h) tan 61,. (5.3)
The distance a in the above equation can be expressed in other parameters. Con
sidering A CDE yields
h = DE tan 7 , (5.4)
where 7 = ZCDE. Since DE is also a side of A ODE one has
D E = ( W + h)tmit/j. (5.5)
Substituting Eq. (5.5) in (5.4) yields
h = (W + h) tan ip tan 7 . (5.6)
Solving the above equation for h we obtain
h = W tail tail 7 ^
1 — ta n -0 tan 7
Finally, substituting Eq. (5.7) in (5.3) yields 0b as
O h = arctan
W \ tan il)
(5.8)
This 0b expression still contains 7 , which can be expressed in other angles as shown
below. A closer look of the A ODE region is necessary to compute 7 . Figure 5.1 (b)
shows this region with some additional lines and points. The line CQ is tangent
to the inirror surface at the point C. Also, the line DQ is perpendicular to CQ.
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41
The lengths of CQ and DQ are given by u and v, respectively. The point P is the
intersection of CQ and DE. Since OE makes an angle (tt/2 — by) with CQ, /.CPE
is obtained by considering A C P E as
EC PE = ^ - ZPCE = Qs . (5.9)
The angle between CD and CQ is denoted by ( f > (i.e., < j > = EDCQ). Considering
A C D P we can write
7 = Qs -(j). (5.10)
Now, it will be shown that < f > = 0(C) by the argument explained below. Since CQ
is tangent to the mirror surface at the point C , the length v is given by
v = 0(u2) . (5.11)
Also, considering ACDQ we obtain
< j> = arctan ^ . (5.12)
Substituting Eq. (5.11) in (5.12) and expanding the result in powers of vfu yield
C = C(u). (5.13)
Note from AC D Q that u is smaller than CD, which is 0(C)- Hence, it can be
written that u = 0(C)- Substituting this in Eq. (5.13) yields
C = 0(C) ■ (5.14)
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42
Observing Eqs. (5.10) and (5.14) we obtain
7 = % b a + 0 ( 0 ). (5.15)
Now, substituting the above equation in Eq. (5.8) and expanding the result in
powers of 0 produce
W W ( W \
0b = ^ \- % p 2— | 1 j ta n ips + 0('tp3) . (5.16)
Po Po \ Po J
Substituting Eq. (5.16) in (5.1) we finally arrive at
9 = ip ( 1 ------- | — ijA— | 1 j tan ips + 0 ( 0 3) . (5.17)
V Po J Po \ Po J
9S is considered next. Observe from A A B D that
29s + {ZABC - 9b) + {ABAC + 9a) = tt . (5.18)
The above equation indicates that 9S can be obtained by computing 9a since 9b is
already calculated in Eq. (5.16). Note that A ACD and A B C D share a common
side CD. This implies that if 6b is expressed in parameters of A B C D , 9a can be
obtained by substituting corresponding parameters of A A C D in the 6b expression.
Hence, 9b is considered again first. Since A B D E and A C D E share a common side
DE, one has
(po + h) tan 6b = CD cos 7 . (5.19)
Substituting Eq. (5.15) in the above equation and noting that
cos [A ? + = cos ips + 0('tp), (5.20)
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43
we obtain
9b = arctan { — — [cos 'ip s + 0(ip)]1 . (5.21)
IPo + h J j
Also note from Eq. (5.7) that a = Opip). Substituting this in Eq. (5.21) and
expanding the result in powers of pj finally yield
C D
9b — cos'ips + O(ip). (5.22)
Po
Now, the expression for 9a can be obtained by substituting 9a — > O b and p'0 — > po in
the above equation as
CD
9a = -~r cos + 0(ip). (5.23)
Po
Dividing Eq. (5.23) by Eq. (5.22) and observing from Eq. (5.16) that O b = 0(ip)
yield
9a = O b^ + 0 (P j2) . (5.24)
Po
Substituting Eq. (5.16) in Eq. (5.24) then produces
W
9a = tp— + 0('lp2). (5.25)
Po
Finally, 9S is obtained by substituting Eqs. (5.16) and (5.25) in Eq. (5.18) and
noting from A A B C that A A B C + A B A C — t v — 2 tps as
W ( 1 1
9s = A + v i - - - ^ ) ^ + 0(tp2). (5.26)
2 VPo Po/
The distance between aperture stop and the mirror surface along the chief ray,
which is denoted by Wc, can also be expressed by other parameters. Considering
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44
A ODE one has
Wcmsip — W + h. (5.27)
Substituting Eq. (5.7) in the above equation we obtain
W
Wc = ----- . (5.28)
cosb’(l — tan ip tan 7 )
Equation (5.15) is substituted in the above equation and the result is expanded in
powers of ip to produce
Wc = W (1 + E tan i/d) + 0(V;2) • (5.29)
This Wc is needed later in the calculations leading to the aberrations of two-mirror
systems. The distance between the chief ray’s reflection point and the focus near
the object, po can be obtained similarly. Considering A BDE yields
s0 cos 9b — po + h . (5.30)
Substituting Eq. (5.7) in the above equation and rearranging its terms, we obtain
1 = cos^ f5 3n
sq W tan ip tan 7
Po
1 — tan 'ip tan 7
Finally, substituting Eqs. (5.15) and (5.16) in the above equation and expanding
the result in powers of % /j, we arrive at
— = — — pi— rr tan p:s + G(pJ2) . (5.32)
S o Po Po
Alternatively, the above equation can also be obtained observing the similarity
between A ODE and A BDE and substituting so — > Wc, po W, and P & -a tp in
Eq. (5.29).
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45
5.2 Astigm atism and Coma with an Displaced Aperture
The relations obtained in the previous section can be applied to describe the aber
ration coefficients in terms of the parameters associated with the aperture stop
and OAR. Before proceeding, we first compute several elementary terms appearing
repeatedly in the equations. Using appropriate trigonometric series expansions [13]
we can show from Eq. (5.26) that
Now, the coma coefficient A2, represented in terms of ips, td and pQ , is obtained by
susbtituting Eqs. (5.17), (5.32), and (5.33)-(5.36) in Eq. (4.23) as
Stop
sin 20 s — sin 2ips + ipW — - - t ! c o s 2tps + 0(ip2)
Po PoJ
(5.33)
1 JL
Po Po
(5.34)
(5.35)
Equation (5.35) can be used to yield
y/cos2 0S — K = \Jcos2 ips — K — ip
+ 0 ( u 2) •
(5.36)
(5.37)
+
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46
Also, substituting Eqs. (5.17), (5.32), and (5.33)-(5.36) in Eqs. (4.12) and (4.13)
yields the tangential and sagittal astigmatic image locations represented in terms
of tps, ip, and po as
1 2 cos2 ips 1 , ^ sin ips
7 =
I
+ip2 — [b2{ 1 + sin2 ’ ips) + 2ab cos 2ips — 2 a2 cos 2'tp s + d sin 2ips]
R
+0{ip3) , (5.38)
1 2 cos2 ips 1 sin
ip (2a + & )-
s’ s R st R
—'ip 2~ = t [b2 cos2 ips + 2 a6 cos 2ijjs + 2 a 2 cos 2 ^ s + d sin 2ips\
R
+0(ip3) , (5.39)
where
a = (5-40)
2 \do do/
W 7
6 = 1 ------- , (5.41)
do
W 7 / W \
d — ( 1 j tan 6S . (5.42)
d o V d o /
In the expressions of Eqs. (5.38) and (5.39), st and s3 are the object locations cor
responding the tangential and sagittal astigmatic images. Observe that, although
in a single-mirror system st = ss, in the secondary mirror of a two-mirror system
st and ss are in general different since the tangential (sagittal) astigmatic image
of the primary mirror is the object producing the tangential (sagittal) astigmatic
image of the secondary mirror. Subtracting Eq. (5.39) from Eq. (5.38), we obtain
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47
the astigmatic image separation expanded to the third order of 0 as
s'ts's \$t ssJ \ Po J R
R \ po J
W \ u/ f 1 1 A o / 2 M / • 2 ,
— + W [ --------- - j cos 2 % b s— sm 'if) s
P o J \ P o P o ) Po
+0{iP3) . (5.43)
Using the expressions for coma coefficient [Eq. (5.37)] and the astigmatic image sep
aration [Eq. (5.43)] we can derive the aberrations of a two-mirror system whenever
a displaced aperture stop must be taken into account. The aberrations of classical
off-axis two-mirror telescopes are considered in the following chapter.
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48
C hapter 6
ABERRATIONS OF CLASSICAL OFF-AXIS
TWO-MIRROR SYSTEMS
The aberrations of multi-mirror systems can be obtained applying the aberrations
of single mirror systems to each mirror consecutively. Aberrations of two-mirror sys
tems with the aperture stop located at the primary mirror are considered here since
they are the most frequently used multi-mirror systems. However, the methodology
developed in this chapter can be applied to the study of classical systems with any
number of conic mirrors. We start from the discussion of two-mirror system ge
ometry and continue to obtain aberration coefficient as well as their minimization
condition. To verify the results obtained, a case study is also presented.
6.1 The Geometry of Two-Mirror Systems
Figure 6 .1 shows four different types of classical off-axis two-mirror telescopes. The
first two systems are the off-axis variations of the well-known classical Cassegrain
and Gregorian telescopes. The other two geometries are less common. They utilize
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49
P R IM A R Y
P A R E N T M IR R O R ,
(P A R A B O L O ID ) /
P R IM A R Y P A R E N T
M IR R O R A X IS /)
S E C O N D A R Y
P A R E N T M IR R O R
(H Y P E R B O L O ID )
P R IM A R Y
P A R E N T M IR R O R
(P A R A B O L O ID )
v _
P R IM A R Y P A R E N T M IR R O R A X IS
S E C O N D A R Y
P A R E N T M IR R O R
(E L L IP S O ID )
S E C O N D A R Y P A R E N T M I R R O R A X IS
(a) (b)
----------------- S E C O N D A R Y
\ P A R E N T M IR R O R
(H Y P E R B O L O ID )
'X
P R IM A R Y P A R E N T M IR R O R
(P A R A B O L O ID )
P R IM A R Y P A R E N T M IR R O R A X IS
( 0 )
S E C O N D A R Y P A R E N T M IR R O R (E L L IP S O ID )
P R IM A R Y
v P A R E N T M IR R O R
\ (P A R A B O L O ID )
P R IM A R Y P A R E N T M IR R O R A X IS
(d)
Figure 6.1: Classical off-axis two-mirror telescopes, (a) Cassegrain with a convex sec
ondary mirror, (b) Gregorian, (c) Cassegrain with a concave secondary mirror, (d) Inverse
Cassegrain.
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50
PARABOLOID
OPTICAL AXIS RAY
HYPERBOLOID
FOCUS
FOCUS • IMAGE
Figure 8.2: The geometry of a classical off-axis two-mirror system.
concave secondary mirrors to produce short system focal ratios and a compact
geometry. To obtain equations capable of handling any of these four types of
classical two-mirror systems a parameter 6, corresponding to the number of foci
between the primary and secondary mirrors, is introduced. One can see that 5 = 1
for Gregorian and 5 = 0 for the other three types of two-mirror telescopes.
Figure 6.2 shows the basic relevant parameters of classical off-axis two-mirror
telescopes. £ is the distance between the primary and secondary mirrors. £2 is the
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51
distance between the secondary mirror and the focus of the primary mirror. l\
is the distance between the focus of the primary mirror and the primary mirror
itself, given by l\ = £ — i2. These £\ and l2 become negative if the focus of the
primary mirror is located behind the primary and secondary mirrors, respectively.
£q is the distance between the secondary mirror and the system focus. All these
distances are measured along the OAR. The incidence angles of the OAR at the
primary and secondary mirrors are given by A and i2, respectively. Also, the vertex
radius of curvature of the primary and secondary mirrors are denoted by R\ and i?,2,
respectively. The primary and secondary mirrors have their own local xp-yp-zp and
xs-ys-zs coordinate systems, respectively. The projected aperture (i.e., aperture
seen looking from infinity along the primary parent mirror axis) also has its own
local xa-ya-za coordinate system with za-axis aligned along the optical axis ray
toward infinity. To describe the aberrations near the system focus another local
Xf-yf-Zf coordinate system is used. The orientation of the xp-yp-zp and xs-ys-zs
coordinate systems is determined in the following way. Consider a ray parallel to
the OAR incoming from infinity at hight xa from the OAR, and let xp and xs be
the x-coordinates of the reflection points of this ray on the primary and secondary
mirrors, respectively. Then, the orientation of xp and xs is chosen to make xaxp > 0
and xaxs > 0. The Xf-yf-Zf coordinate system has its origin at the system focus and
the z is aligned with the OAR going towards the secondary mirror. The orientation
of the Xf-yf-Zf coordinate system is selected such that the chief ray with positive
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52
o j intersects the positive ay-axis.
6.2 Elimination of Linear Astigmatism and Rusch’s Condi
tion
This section deals with linear astigmatism, which is the dominant aberration of
classical off-axis two-mirror systems. The astigmatic images’ separation As2 of the
secondary mirror, expanded in powers of the field angle of the object for the system
can be used to quantify the linear astigmatism of a two-mirror system [24], Let u
be the field angle of the system and assume that the aperture stop is located at the
primary. Then, As2 can be derived substituting ijj — cu , tps = i2, po = £2, R = R2
and W = t in Eq. (5.5) as
A 4 0 h sin 2*2 ( 1 1 \ n ( 2, ,R u
= ~ 2ujT —5----------— “ 7 “ + ° ( u ) > I6-1)
s's2S t2 ^2 R
where s't2 and s's2 are the tangential and sagittal astigmatic image locations of
the secondary mirror with corresponding object locations s t2 and ss2, respectively.
Since the aperture stop is located at the primary mirror, the object locations st2
and ss2 are given by
si2 = WC- s'n , (6 .2 )
S s2 = WC- s' s l, (6.3)
where s'n and s(vl are the tangential and sagittal image locations of the primary
mirror, respectively. Referring to Fig. 6.2 and substituting Eqs. (4.15), (4.16), and
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53
(5.29) in Eqs. (6.2) and (6.3), we obtain
JL - 1
Sf2 ^2
1 1
— — 7" + w
Ss2. 4
4 , . X 1
— ^(tanij + ta m 2) + y tan t2
L.4 4
4 , . N 1
—2 (tan?i — tan * 2) — — tan z 2
+ 0 (cu2) ,
+ 0(w 2) .
(6.4)
(6.5)
Finally, substituting Eqs. (6.4) and (6.5) in Eq. (6.1), a convenient form for the
astigmatic images’ separation As'(= As'2) of the classical off-axis two-mirror system
is obtained as
A s' o 4 ( 4 • o-
= 2 u i — - 1 — sm 2*i
sn s's 2 4 2 \ R i
2 sin 2*2 ) + 0(u>2) .
R
(6.6)
To arrive at the above result the equation
Ri
2 cos2 i 1
(6.7)
was used, which can be derived from the equation of a paraboloid. Equation (6 .6 )
indicates that the linear astigmatism of classical off-axis two-mirror systems can be
eliminated by imposing the condition
4 4
— sin 2R — — sm 2 i2 •
Ri
(6,8)
This is the most important condition for classical off-axis two-mirror systems since
it makes the performance of them comparable to on-axis systems.
In fact, the above equation has previously been obtained, in a somewhat different
form and for a different application, by Rusch et al, [20]. They derived the aperture
field distribution of classical off-axis Cassegrain with a convex secondary mirror
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54
EQUIVALENT
PARABOLOID
PARABOLOID
HYPERBOLOID
PARABOLOID AXIS
FOCUS
FOCUS
Figure 6.3: The equivalent paraboloid of the offset Cassegrain system
and Gregorian systems and obtained a paraboloidal mirror that produces identical
aperture field distribution (the equivalent paraboloid, see Fig. 6.3). It was then
shown that it is possible to obtain an on-axis equivalent paraboloid by satisfying
P ( e - 1 '
tan — = I ------
2 le + 1
tan
P — 0 Q
(6.9)
where P is the angle between the primary and secondary mirror axes, e is the eccen
tricity of the secondary mirror, and 8q is the off-axis angle of the primary mirror.
In the cases of a classical off-axis Cassegrain with a concave secondary mirror and
inverse Cassegrain systems, the condition for an on-axis equivalent paraboloid is
t P f e + 1 Y P - 6 o
tan — = ------ tan —— —
2 V e - 1 J 2
(6.10)
A similar equation was also derived for confocal two ellipsoidal mirror systems
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55
by Stavroudis [25]. He obtained a pseudoaxis about which all other rays passing
through the systems focus are symmetric. Since Eq. (6 .8 ) can be shown (see Ap
pendix A) to be algebraically equivalent to Eq. (6.9) or (6.10), a classical off-axis
two-mirror telescope with an on-axis equivalent paraboloid will have zero linear
astigmatism. It is possible to satisfy Eq. (6 .8 ) in any classical off-axis two mirror
system [3, 6 , 8 ].
6.3 Coma
In the previous section the condition for zero linear astigmatism was derived. Since
the linear astigmatism must be eliminated for adequate field-of-view performance,
we assume that Eq. (6 .8 ) is then satisfied from this point onwards. Two conve
nient parameters to characterize the magnitude of aberrations are the transverse
and angular aberrations along the x j axis, which are denoted by TAX f and AAX },
respectively. These parameters are given by [24]
In these equations / is the system focal length (i.e., / is the effective focal length
of the telescope) and A is the optical path difference (OPD) between a ray and the
chief ray of the system. Computing the OPD of an off-axis two-mirror telescope,
which corresponds to coina term, the angular tangential coma (ATC) of the system
(6 .12)
(6,11)
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56
dx-
25
PRIMARY MIRROR FOCUS
O
Figure 6.4: Relations between infinitesimal lengths of the primary and secondary mirrors,
can be obtained as [24]
£0 d
ATC = (A2pXPs + A2gxs3) (6.13)
/ dxf
where the subscripts p and s represent the primary and secondary mirrors, re
spectively. To perform the differentiations indicated in the above equation, rela
tions between Xf-ijf-Zf, xp-yp-zp, and xs-ys-zs coordinate systems (see Fig. 6.2) are
needed. These relations can be approximately derived considering the behavior of
rays which are converging to the primary focus and located near the OAR as shown
in Fig. 6.4. In this figure the OAR and a ray near the OAR (i.e., OF) with the
small portions of the primary and secondary mirrors between these two rays are
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57
shown. Note that, although the lengths dxx, dx2, dxp, and dxs are exaggerated in
the figure for clarity, they are infinitesimal. Then, A A B C ce 90°, A D E F ~ 90°.
W ith this in mind, considering A D E F and AABC yields
dxp 1 ,
7T (6 . i4)
dxi costal
and
~ t~ ~ ~ ~ cos 9s2 , (6.15)
axs
respectively. Also, noting that A O AB and A O D E are similar having a common
vertex O, the relations between dx% and dx2 can be obtained as
dxi ~ £l (p~\a\
- ~ T ’ (6-16)
dx 2 £ %
Now, the relation between dxp and dxs is obtained using the chain rule [1] as
dxp dxp dxi dx2
dxs dx 1 dx2 dxs
Substituting Eqs. (6.14)-(6.16) in Eq. (6.17) finally yields
dxp 11 COS 2-2
(6.17)
dxs £2 cos ix
Also, observing Fig. 6.2 we obtain
(6.18)
P - = H ) m - W (6-19)
dXf COS *2
dxp 1
(6 .20)
dxa cos
Now, all these relations between coordinate systems are applied to manipulate
Eq. (6.13). First, using the chain rule Eq. (6.13) is converted to a derivative with
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58
respect to xs as
* r -r -, A ) dX q d / q C f\
ATC = — — ---- ; (A2 pXp + A2sXs ) (6.21)
/ d L x / day
Substituting Eq. (6.19) in the above equation and performing the differentiation
yields
- . . ‘ Afn ( _
(6 .22)
2p^p ^ ATC = ( - l f + V 6
/ C O S « 2 \
To substitute xa = D / 2 in the above equation, xp and xs are needed to be converted
to xa. This conversion can be obtained approximately as
dxs
rp rp
Jr$ ----- 7 • * ' ! ,
axp
rp C N J
'A 'P --
dxv
dx a
(6.23)
(6.24)
Substituting Eqs. (6.18), (6.20), (6.23) and (6.24) in Eq. (6.22) yields
ATC = (~l)m \ d 2j 2 p
1 h
COS3 i< 2
A2s
COS3 *2 U l
(6.25)
A-ip is obtained by substituting 0S = ix, 0 — uj, and R = R i in Eq. (4.25) as
2 + 0 (cu2) . (6.26)
cos5b
A 2 p = -w - 1 ^ M
Ri
The tangential astigmatic object location is used to compute A2s. Substituting
ip = u > , ips — * 2, Po = R = R2, W = £, and Eq. (6.4) in Eq. (5.37) yields
£ ■ 2 x
Aog — — U J
cos5 i2
Ri
Q 2 ) (
ta n ij ta n i 2 — ~ + 1 ! + 0(cv ). (6.27)
Since the system is assumed to have zero second order astigmatism, Eq. (6 .8 ) can
be used to manipulate A2s. Several equivalent forms of Eq. (6 .8 ) can be derived
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59
(see Appendix B) as
£2 tan A
i ?2 sin 2*2 ’
(6.28)
tan A = — - tan A , (6.29)
£q
cos2 % 2 cos2ii {£1 l i \ . .
“ R r = _ R r U v ■ (0-30)
± = 1( 1+ 1 ^ . J _ £ i ( _ (e 31)
R 2 2 \ £ 0 £ 0 + £ 2 ) R i £2 \ £ 0 + £ 2 ) ' { ' j
Substituting Eqs. (6 .8 ) and (A.27)-(6.31) in Eq. (A.26) yields
1 3 • 4 A 2 * 6 2
- U ! COS %2— o— 1 —
4 2i 0% V ^2
A2 , = - u cos i2- j - 1 - — -4-f- + O (o /). (6.32)
Also, it can be shown that (see Appendix C)
/ = ( _ i )i+1« i {6 33)
Finally, substituting Eqs. (A.25), (6.32) and (6.33) in Eq. (6.25), ATC of the system
is obtained to be
a t c = m J 7 W + 0 ^ - (6-34>
Note that the derived coma aberration is identical to that of an on-axis paraboloidal
mirror with the same focal ratio as the telescope [24], Therefore, the practical geo
metric optical performance of an off-axis telescope that satisfies Eq. (6 .8 ) is equiv
alent to an on-axis system since the third order coma is the dominant aberration
for these systems.
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60
6 . 4 T h i r d Order Astigm atism
The third order astigmatism can be obtained by expanding the astigmatic image
separation to the second order of 0. Substituting ijj = uj, A = H, Po = £2, R = R2,
and W — £ and
s0 =
(6.35)
in Eq. (5.43) yields
3t2 S's2
~ - 2 6
£\ sin 2 i 2
h R2
2
R2 w i } 2 - ^ ) e i cos^
£1
H -~W tan * o sin 2*2
£2
_ (2 — L
\ St2 Ss2
where st2 and s s2 are given by Eqs. (6.2) and (6.3), respectively. Hence,
(6.36)
1 1
4 i -
s a sa (W' - s'a ) (W - < ,)
(6.37)
Substituting s = 0 0 in Eqs. (4.8) and (4.9) and expanding to the second order of
0, we obtain
% ~
ysl
r ( 1 - w tanii - + 0(u3) ,
2 cos2*i \ 2 1
Ri
2 cos2 i\
1 + uj tan ii + uj2 ( ^ + tan 2 q
(6.38)
+ 0 {u j3) . (6.39)
We now substitute the above equations and Eq. (5.29) in Eq. (6.37) to obtain
1 1 _ £1
St2 SS 2 £22
2cutan i\ + a ; 2 ( 1 — 4— ta n q tan *2 + tan 2 ix
£2
+ 0 (w 3) . (6.40)
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61
Finally, substituting the above equation and Eq. (6 .8 ) in Eq. (6.36), we obtain the
astigmatic image separation As' of the system as
f
As' ~ 02 — \f(l — tan A ta n i2) — W] . (6.41)
to L
This equation is consistent with an on-axis system when A = A = 0 [24].
6.5 Case Study
It was shown in the previous sections that the linear astigmatism can be elimi
nated in classical off-axis two mirror systems. Furthermore, the third order coma
of systems without linear astigmatism was shown to be identical to that of on-axis
systems. To verify these results, a classical off-axis Cassegrain telescope with a
concave secondary mirror was designed and analyzed. In order to have no linear
astigmatism and obstruction, this type of telescope requires a large tilt angle be
tween the primary and secondary parent mirror axes. However, it is capable of
producing very fast focal ratios [6]. Although an obstruction-free zero linear astig
matism off-axis Cassegrain with a convex secondary mirror or Gregorian system can
be designed with a small tilt angle [3], the example here presented is more appropri
ate to verify our aberration theory due to its extreme off-axis property. Figure 6.5
shows the side view and parameters of the Cassegrain case study having a final focal
ratio f/2. The primary mirror is a 1 m diameter f /4.56 off-axis section of a 7.99
m diameter //0 .4 7 paraboloid parent mirror. This system is designed to have no
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62
lm diam eter f/4 ,56
Off-Axis Mirror <
O
o
o
y 1
M2 S eco n d ary \
Parent Mirror
{Hyperboloid} \ ° q
to
G i
■ * $
c o
7.99 m d iam eter i/0.47
■ Primary Parent Mirror
(Paraboloid)
Primary Parent Mirror Axis
Figure 6.5: Side view of the case study off-axis Cassegrain system.
linear astigmatism [Eq. (6 .8 ) is satisfied]. The detailed data of the telescope is given
in Table 6.5. The astigmatic image surfaces for the objects on the vertical (ya = 0)
and horizontal (xa = 0) planes are shown in Fig. 6.5 (a) and (b), respectively. The
slopes of the tangential and sagittal image surfaces coincide with each other at the
origin showing that linear astigmatism has been eliminated. Figures 6.5 (a) shows
the representative spot diagrams of objects with 0.1 degree field angle. The spot
diagram of a 1 m f/2 on-axis paraboloid mirror is shown in Fig. 6.5 (b) for com
parison. It is clear from Fig. 6.5 that, as the previously derived theory predicted,
the coma aberration of the off-axis Cassegrain example is identical to that of the
paraboloid mirror with focal length identical to the final focal length of the off-axis
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
D = 1 0 0 0 mill
f (final telescope focal length) = 2 0 0 0 mm
fi (primary parent mirror focal length) = 3747.4774 mm
i?.i = 7494.9548 mm
Ki (conic constant of primary parent mirror) = - 1
h
= 25°
ii
= 4562.3395 mm
i?2
= 4910.5514 mm
K ‘ 2 (conic constant of secondary parent mirror) = -5.9088060
2 c (interfocal distance of secondary parent mirror) = 4863.3357 mm
h
= - 2 0 °
4
= -3562.3395 mm
Table 6.1: Data of the case study. See Sec. 7 for the description of Ri, R2, * 1, ^1, and
4 -
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64
O .ld eg
TANGENTIAL ’
IMAGE LOCATION
TANGENTIAL
IMAGE LOCATION
SAGITTAL
IMAGE LOCATION
- 5 |i m +5gm -5um
+ 5pim
( a )
(b)
Figure 6.6: Astigm atic image surfaces of example telescope, (a) Vertical field (ya = 0).
(b) Horizontal field (xa = 0). G enerated using OSLO LT [17]
20 ARC SEC
0.1 X0.1
< f 'V— -X
^ ( J 20 ARC SEC
(a) (b)
Figure 6.7: Spot diagrams, (a) 1 m eter diam eter f / 2 case study telescope, (b) Single 1
m eter diam eter f / 2 on-axis paraboloidal mirror. G enerated using OSLO LT [17]
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65
Cassegrain. Also note that, although only objects on the vertical plane were consid
ered in th e presented aberration theory, the spot diagrams of the example indicate
that classical off-axis systems with zero linear astigmatism have axisymmetric coma
aberrations about the center of field. Elimination of linear astigmatism can then
be said to yield a compensated system—a system that behaves practically like an
axially-symmetric system from an aberration stand point.
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66
C h a p te r 7
CONCLUSION
Since their invention, most geometric optical elements (i.e.. lenses and mirrors)
were made in axially symmetric forms. This is a natural choice since, intuitively
and in reality, axially symmetric optical elements suffer from less aberrations com
pared to asymmetric elements. For lenses this is a best choice unless it is required
to compensate aberrations caused by preceding asymmetric elements. However,
utilizing axially symmetric mirrors causes a serious unavoidable problem: subse
quent optical elements inevitably block the rays incoming to the mirror since an
axially symmetric mirror reflects back the rays into the their incoming direction.
This problem can be avoided using asymmetric mirrors. As a class of asymmetric
mirrors, an off-axis section of axially symmetric conic mirrors (offset conic mirror)
have been proved most practical since they still possess capability of their parent
mirrors: they produce a perfect focus at their field center. However, a single off
set conic mirror has been known to suffer from large field aberrations and it was
indeed proved in this dissertation that the dominant aberration of a single offset
conic mirror is linear astigmatism, which is larger than any third order aberrations.
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Offset conic mirrors have been widely used in offset reflector antennas because
most reflector antennas have feed(s) at their focus and hence concerns only the
field center. However, more and more interests are being applied towards imag
ing and scanning antennas, where field aberrations are also important. In optical
frequency, almost all reflecting telescopes have been built using axially-symmetric
(on-axis) mirrors. Performance of a finite field of view has been always an important
factor for reflecting optical telescopes and it has been believed that utilizing asym
metric mirrors causes large field aberrations. Furthermore, manufacturing precise
off-axis mirrors suitable for optical frequency was much more difficult than reflector
antennas unless impossible. However, improved technology of these days makes it
feasible to produce off-axis mirrors that is precise enough to yield a diffraction lim
ited image in optical frequency. Off-axis mirrors are now widely used for telescopes
that employ segmented mirrors [26].
In spite of all these technological evolutions, off-axis reflecting telescopes are ex
tremely rare and only used for applications where suppression of scattering is more
important than field-of-view performance such as planets and solar observations.
Now, since it was shown in this dissertation that the off-axis specific aberration (lin
ear astigmatism) can be eliminated and performance of the compensated off-axis
system is equivalent to an on-axis system, we don’ t need to stick to conventional
on-axis system any more. Utilizing off-axis mirrors will enable new possibilities in
optical design which have never been explored before.
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68
However, real revolution in reflective optical system will be achieved by one
further progress beyond this dissertation: off-axis aplanatic system. Although a
compensated classical off-axis system is equivalent to an on-axis paraboloidal mir
ror having the same focal length, it is desirable to suppress third order coma for
wider field-of-view performance (i.e., aplanatic system). Most modern optical tele
scopes including Hubble Space Telescope have been built as an aplanatic system
[24, 26]. On-axis aplanatic systems is achieved by deviating from classical systems.
Similar approach might be possible to obtain off-axis aplanatic systems. In fact,
in the beginning of this work, the author intended to develop an aberration theory
for general off-axis systems. However, the complexity related to off-axis systems
constrained him to focus on classical systems. Nevertheless, this author will con
tinue to work on off-axis aplanatic systems to achieve further progress in reflective
optics.
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69
Bibliography
[1] G. B. Arfken and H-J Weber, Mathematical Methods for Physicists, Academic,
New York, 2000.
[2] M. Born and E. Wolf, Principles of Optics, Cambridge, Oxford, 1980.
[3] K. W. Brown and A. Prata, Jr., “A design procedure for classical offset dual
reflector antennas with circular apertures,” IEEE Trans. Antennas Propagat.,
vol. 42, no. 8 , Aug. 1994.
[4] H. P. Brueggemann, Conic Mirrors, The Focal Press, London, 1968.
[5] R. A. Buchroeder, “Tilted component optical systems,” Ph.D. thesis, Optical
Science Center, University of Arizoan, Tucson, Arizona, 1976.
[6] S. Chang and A. Prata, Jr., “A design procedure for classical offset Dragonian
antennas with circular apertures,” in IEEE Antennas Propagat. Soc. Symp.
Dig., Orlando, FL, vol. 2, pp. 1140-1143, Aug. 1999.
[7] S. Chang and A. Prata, Jr., “The geometrical theory of aberrations of classical
offset dual-reflector antennas” in IEEE Antennas Propagat. Soc. Symp. Dig.,
Boston, MA, vol. 1, pp. 530-533, Jul. 2001.
[8] S. Chang and A. Prata, Jr., “A design procedure for ealssical offset inverse
Cassegrain antennas with circular apertures” in IEEE Antennas Propagat. Soc.
Symp. Dig., Boston, MA, vol. 1, pp.534-537, Jul. 2001.
[9 ] T-S. Chu and R. H . Turrin, “Depolarization properties of offset reflector anten
nas,” IEEE Trans. Antennas Propagat., vol. AP-21, no. 3, pp. 339-345, May
1973.
[10] C. Dragone, “Offset multibeamreflector antennas with perfect pattern symme
try and polarization discrimination,” Bell Syst. Tech. J., vol. 57, pp. 2663-2684,
1978.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
[11] C. Dragone, “First-order treatment of aberrations in Cassegrain and Gregorian
antennas,” IEEE Trans. Antennas Propagat., vol. A P - 3 0 , no. 3, pp. 331-339,
May 1982.
[1 2 ] C. Dragone, “First-order correction of aberrations in Cassegrain and Gregorian
antennas,” IEEE Trans. Antennas Propagat, vol. AP-31, no. 5, pp. 764-775,
Sep. 1983.
[13] H. B. Dwight, Tables of Integrals and Other Mathematical Data, New York,
The Macmillan Company, 1964.
[14] R. Jprgensen, P. Balling, and W. J. English, “Dual offset reflector rnultibeam
antenna for international communications satellite applications,” IEEE Trans.
Antennas Propagat., vol. AP-33, no. 12, pp. 1304-1312, Dec. 1985.
[15] S. Makino, Y. Kobayashi, and T. Kataki, “Front fed Cassegrain type multi-
beam antenna,” in IEEE Antennas Propagat Soc. Symp. Dig., Vancouver,
Canada, pp. 341-344, Jun. 1985.
[16] G. Moretto and J. R. Kuhn, “Off-aixs systems for 4-m class telescopes,” Appl.
Opt. vol. 37, p.p. 3539-3546, 1998.
[17] OSLO, Lambda Research Corporation, 80 Taylor Street, Littleton, MA 01460-
4400, http://www.lambdares.coin, 2002.
[18] A. Prata, Jr., M. D . Thompson, and H. G. Pascalar, “A compact high-
performance dual-reflector millimeter-wave imaging antenna with a 20 x 20
degrees square field of view,” in IEEE Antennas Propagat. Soc. Symp. Dig.,
Seattle, WA, pp. 2050-2-53, Jun. 1994.
[19] J. R. Rojers, “Vector aberration theory and the design of off-axis systems,”
in International Lens Design Conference, W. H. Taylor and D. T. Moore, eds.
Proc. SPIE 554, pp. 76-81, 1985.
[20] W. V. T. Rusch, A. Prata, Jr., Y. Rahmat-Samii, and R. Shore, “Derivation
and application of the equivalent paraboloid for classical offset Cassegrain and
Gregorian antennas,” IEEE Trans. Antennas Propagat., vol. AP-38, no. 8 , pp.
1141-1149, Aug. 1990.
[21] H. G. J. Rutten and M. A. M. van Venrooij, Telescope Optics, Wilhnan-Bell,
Richmond, VA, 1999.
[22] P. J . Sands, “Aberration coefficients of plane symmetric systems,” J . Opt. Soc.
Am. vol. 62, pp. 1211-1220, 1972.
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71
[23] 3. M. Sasian, “How to approach the design of a bilateral symmetric optical
system ,” Optical Engineering vol. 33, pp. 2045-2061, 1994.
[24] D. 3. Schroeder, Astronomical Optics, Academic, San Diego, 2000.
[25] O. N. Stavroudis, “Confocal prolate spheroids in an off-axis system,” J. Opt.
Soc. Am. A, vol. 9, pp. 2083-2088, 1992.
[26] R. N. Wilson, Reflecting Telescope Optics, Springer, Berlin, 1996.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
72
A p p en d ix A
DERIVATION OF USEFUL EQUATIONS
A .l Equivalent Forms of Equations for Elimination of Lin
ear Astigmatism
The condition to eliminate linear astigmatism in classical off-axis two-mirror sys
tems can be manipulated to yield several equivalent forms. Substituting the equa
tions for the primary mirror,
Ry = 2 /j (A.l)
and
h = - T (A.2)
C Q S ^ l\
in Eq. (6 .8 ), we obtain
12 tan i\
J?2 s in 2 i2
Also we know from the equation of hyperboloid [4] that
(A.3)
A ) + L — 2a , (A.4)
J? 2 = a (l — e2) , (A.5)
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73
IqI% cos2 % 2 = a2(l — e2) , (A.6 )
where e is the eccentricity of the secondary mirror and a is given by
a = - , (A.7)
e
with 2c being the interfocal distance of the secondary mirror. Substituting Eqs.(A.4)
and (A.5) in Eq. (A.6 ), we obtain
2 £ ( / 2 2 • /a C l
= 4 + < 2C0S *2 ' ( 8)
We now substitute Eq. (A.8) in Eq. (6.8) to obtain
. £ 0 + f-2 r \ n\
tan ii = — —— tan ?2 , (A.9)
Multiplying Eq. (6 .8 ) by Eq. (A.9), we obtain
sin2 ii f £ i \ ( £0 \ sin2 i2 ^ ^
Ri \£%/ \£q + £2/ R -2
Also, dividing Eq. (6 .8 ) by Eq. (A.9) yields
R i V o h ) R
We now add Eqs. (A.10) and (A.11) together to obtain
A " K ! + « A c ) + ( ! ) ( 4 T 4 ) ' (A,12)
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74
PARABOLOID
OPTICAL AXIS RAY
HYPERBOLOID
y \ " ' FOCUS
FOCUS
F ig u r e A . l : P a r a m e t e r s o f o ff -a x is tw o - m ir r o r s y s te m s .
A.2 Equivalent Paraboloid and Elimination of Linear Astig
matism
In this section equivalence between Eqs. (6 .8 ) and (6.9) is derived. Referring to
Fig. A. 1 and equations of conic sections one has [4 ]
2R = - 0 O , (A.13)
g (1 — e2)
—
e cos 7 + 1 ’
(A.14)
e sin 7 , . .
sm «2 = , - , (A.15)
\J\ + e2 + 2 e cos 7
1 + ecos 7 ^
COS — ■ ■ ■ y v . (A. 16)
y'l + e2 + 2 e cos 7
Substituting Eqs. (A.l), (A.2), (A.5), (A.7), (A.13)-(A.16) in Eq. (6 .8 ) yields
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Referring to Fig. A.l again one sees
0O = /? - 7 . (A.18)
Also, using the half angle formula [13], one obtains
sin 7 = 2 cos2 — tan ^ . (A.19)
2 2
Now, substituting the above two equations in Eq. (A.17) yields
f 1 + e2 + 2e cos 7 — 4e2 cos2 ^ j tan ^
\ Z ' Z
= f l + e2 + 2 e cos 7 + 4e sin2 ^ tan ^ . (A.20)
\ Z t / Z *
The above equation can be manipulated using Eq. (A. 18) and
cos 7 = 2 cos2 ^ — 1 (A.21)
Z
to yield
tan f ^ tan - , (A.2 2 )
Hence, the equivalence between Eq. (6 .8 ) and (6.9) has been proved.
A.3 Focal Length of Off-Axis Two-Mirror Systems
This section deals the focal length of compensated off-axis two-inirror systems [i.e.,
systems that satisfy Eq. (6 .8 )]. A small length dxa on the projected aperture of a
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76
compensated two-mirror system, which corresponds to a small angle dO seen from
the system focus, is computed and compared with an on-axis paraboloid to obtain
the focal length of the two-mirror system. Referring to Fig. A.2 (a) yields a small
distance dxs on the secondary mirror corresponding to d6 as
dXe =
i,o
dO.
COS
(A.23)
Using the chain rule we also obtain
, dxa dxp ,
dxa = ---- — 1 - dxs.
dxp dxs
(A. 24)
Substituting Eqs. (6.18) and (6.20) in the above equation yields
dxa
i§l\
dB. (A.25)
In the case of an on-axis paraboloid [Fig. A.2 (b)], dxa is simply given by
dxa = fdff. (A.26)
Comparing Eqs. (A.25) and (A.26) we obtain the focal length / of a compensated
two-inirror system as
/ =
M i
(A.27)
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77
dx.
dx,
PA RA BO LO ID
OPTICAL AXIS R A Y
HYPERBOLOID
de
FOCUS
FOCUS
(a)
PARABOLOID
dO
(b)
Figure A,2: Small length on projected aperture corresponding to a small angle of the
system focus, (a) Compensated two-mirror system, (b) On-axis paraboloid.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Asset Metadata

Creator Chang, Seunghyuk (author)

Core Title Geometrical theory of aberrations for classical offset reflector antennas and telescopes

School Graduate School

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, electronics and electrical,OAI-PMH Harvest,physics, optics

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Prata, Aluizio (committee chair), Kuehl, Hans H. (committee member), Nodvik, John S. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-476865

Unique identifier UC11335116

Identifier 3133247.pdf (filename),usctheses-c16-476865 (legacy record id)

Legacy Identifier 3133247.pdf

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Document Type Dissertation

Rights Chang, Seunghyuk

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

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