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Design of a zoom condenser system
Item Type text; Thesis-Reproduction (electronic)
Authors Chen, Muh-fa, 1942-
Publisher The University of Arizona.
Rights Copyright © is held by the author. Digital access to this materialis made possible by the University Libraries, University of Arizona.Further transmission, reproduction or presentation (such aspublic display or performance) of protected items is prohibitedexcept with permission of the author.
Download date 18/03/2021 23:37:13
Link to Item http://hdl.handle.net/10150/348101
DESIGN OF A ZOOM CONDENSER SYSTEM
By
Muh-fa Chen
A Thesis Submitted to the Faculty of the
COMMITTEE ON OPTICAL SCIENCES (GRADUATE)
In Partial Fulfillment of the Requirements For the Degree of
MASTER OF SCIENCE
In The Graduate College
THE UNIVERSITY OF ARIZONA
1977
STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.
SIGNED
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
- 7 7R. R. SHANNON
Professor of Optical SciencesDate
ACKNOWLEDGMENTS
The author would like to express his sincere appreciation to his
thesis director. Professor R. R. Shannon, for his guidance and encour
agement. The comments and corrections received from Dr. P. N. Slater
and Dr. J. C. Wyant, the other members of his committee, are also
appreciated. Thanks also go to M. Ruda, G. Lawrence, Dr. G. Hopkins, II,
and Dr. R. Buchroeder for their help in the.use of ACCOS programs and
some valuable discussions.
The ACCOS programs used in this study were run on a Control Data
Corporation 6400- computer of The University of Arizona. The computing
fund was provided by the,State of Arizona.
TABLE OF CONTENTS
. . Page
LIST OF ILLUSTRATIONS. . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF TABLES ....................................... . vii
ABSTRACT ........... .-. . viii
1. INTRODUCTION . . .................... 1
2. . THEORY OF ILLUMINATION 3
The Inverse Square Law . ., ......... 3The Cosine Law of Illumination 4Illumination Produced by a Lambertian Source . . . ........ 6Luminance of Optical Images ......... 9Illumination by Projection . . 11
3. REQUIREMENT FOR CONSTANT ILLUMINATION....................... . . 15
General Consideration. ....................... 15Description of the Problem ............................. 16
4. 'FIRST-ORDER ZOOM LENS DESIGN . ........... . . . 19
Thin Lens Solution ..................... 19Thick Lens Solution. 27
5. ABERRATION CONSIDERATIONS. . .......... 33
Spherical Aberration (SA)........... 33Coma (CMA) ........... . . 34Astigmatism (AST). ............... 34Field Curvature (FCj . . . . . . . . . . . . . . . . . . . . . . 34Distortion (DIST). . . . . . . . . . . . . . . 35Chromatic Aberration . . . . . . . i............................35
6. ABERRATION DESIGN. ............ . . 38
7. RESULTS AND DISCUSSION ................................ 54
iv
V
TABLE OF CONTENTS--Continued
Page
APPENDIX A: FORMULAE OF FIRST ORDER SYSTEM PARAMETERS. . . . . . 62
APPENDIX B: PROPERTIES OF BK7 AND F2 GLASSES........... 65
APPENDIX C: FORD TABLES. ...................... 68
APPENDIX D: STOP SHIFT EQUATIONS.......... ........... . 70
REFERENCES......... 72
LIST.OF ILLUSTRATIONS
Figure Page
2.1. The Inverse Square Law............ 3
2.2. Cos3 0 Law. . ............. 4
2.3. Illumination from a Finite Source . . . . . ...... . . . . . 5
2.4. Illumination from a Circular Disc Source. . ............. 7
2.5. Aplanatic Optical Imaging System. . . . . . . . . . . . . . . 9
2.6. An Optical Projection System. . . . . . . . . . . . . . . . . 11
3.1. A Zoom Condenser System ....................... . . . . . . 16
4.1. A Zoom Lens Configuration............... . 20
4.2. First-Order Thin Lens Solution...............................22
4.3. A Thick Lens. ......... 27
4.4. Lens Parameters' Relation 29
4.5. Schematic of First-Order Thick Lens Solution(1) . . . . . . . 30
4.6. Schematic of System Configuration ............... . . . . . 30 .
6.1. Ray Aberrations (Starting Value).............................39
6.2. Ray Aberrations (SA Removed). . . . . . . ................. 43
6.3. Ray Aberrations (Distortion Corrected) at -5x . . . . . . . . 45
6.4. Ray Aberrations (PSA3 Reduced) at - 5 x . ............. 46
6.5. Ray Aberrations (Final Version) . . . . . . . . . 47
7.1. Elements of a Zoom Condenser......... 55
7.2. Transverse Ray Aberrations at Reference Wavelength 587.6 nm.. 57
7.3. Field Size versus Magnification . . . ......... . . . . . .60vi
LIST OF TABLES
Table Page
3.1. First-Order Parameters of a Zoom Condenser System. . . . . . . 18
4.1. Summary of First-Order. Thin Lens Solution.................... 21
4.2. Lens Data of First-Order Thick Lens Solution (2) . . . . . . . 31
4.3. System Parameters of First-Order Thick Lens Solution (2) . . . 32
5.1. Summary of Third-Order Aberrations . .................. 37
7.1. Lens Data and Aspheric Surface Data of Final Version......... 56
vii
ABSTRACT
Design of a five-element (P-PP-N-P) zoom condenser system with
a zoom ratio of 10 to 1 is presented. The system stop is located at the
last surface of the last element, its effective size is varied from 2.5
mm to 2.5 mm in radius by an optical method. The image field (entrance
pupil of the projection system) varies from 25 mm to 2.5 mm in radius.
Two aspheric surfaces are employed to correct distortion and spherical
aberration.
The goal was to design a zoom condenser to match a hypothetical
zoom projection system.
CHAPTER 1
INTRODUCTION
The function of a condenser is to collect as much light as
possible from a source and direct it through the entrance pupil of a
projection optical system. The illumination produced by an optical
system is then determined by its angle of illumination. Therefore,
if such an optical system is working in a certain zoom range, its
illumination will vary from magnification to magnification, unless
some method is employed to compensate this variation.
Conventionally, an adjustable (in size) aperture stop is used
to control the total amount of light passing through the optical system
and this can be referred to as a "mechanical adjustment."
We will call the other method to be discussed here an "optical
adjustment." In this method we can either (a) have the exit pupil
stationary both in size and location, or (b) keep the entrance pupil
stationary and let the exit pupil vary in size and position. In the
former case, we have the optical system designed so that its aperture
stop is at the last surface next to the image plane. The only require
ment then is to zoom the illumination optics such that the entrance
pupil of the optical system is always properly filled. The entrance
pupil of the illumination pptics will then vary in size and shift in
position as the system is zoomed. In the second case, the entrance
2
pupil is fixed in place and the condenser is zoomed to the proper
magnification according to the operation of the optical system. This
scheme is investigated in this study. This approach presumes, of course,
that a constant Lagrange invariant, nhu, is maintained by the projection
optics. This is, in fact, a very reasonable assumption to make, when
feasibility of the illumination optics is to be discussed as a separate
topic.
Chapter II reviews the theory of illumination. The problem is
then defined in Chapter III. Chapter IV describes the approach of first
order design. Aberrations are briefly discussed in Chapter V to suit
our interest. Corrections of aberrations by using ACCOS programs are
given in Chapter VI. To conclude this study, the system is then
evaluated in Chapter VII.
CHAPTER 2
THEORY OF ILLUMINATION
"The illumination at a point of a surface is the quotient of the
luminous flux incident on an infinitesimal element of surface containing
the point under consideration by the area of this element (symbol E)."
This definition is given by Walsh (1965). Hence we may define illumina
tion as the amount of light falling upon unit area of surface provided
that light is uniformly distributed over the surface considered. It is
' expressed, for example, as lumens per square foot (i.e., footcandle).
The following concepts are essential to the following discussions:
The Inverse. Square- Law -
Consider a point source of constant intensity I (Fig. 2.1), the
illumination on a surface of circular cross section area A at a distance
S, by definition, can be written as
ea = ^ C2-11
Fig. 2.1. The Inverse Square Law.
3
where fi is the solid angle subtended by the illuminated area at the
point source and for small cone angles,
$ =
thus (2.2)
ea = y
this is the Inverse Square Law.
The.Cosine Law of Illumination
When radiation is incident at an angle 0 (Fig. 2.2), the pro
jected receiving area normal to the incident direction is
pointsource
Fig. 2.2; Cos3 9 Law .
reduced by the factor cos 8, the distance from the source to the
illuminated surface is increased by the same factor cos 9. Therefore,
the solid angle subtended by the receiving area is reduced by a factor
cos3 0 and the corresponding illumination at an angle 0., from Eq. (2.1)
is related to the on-axis illumination E by' . o
Eg = Eo cos3 0 (2.3)
However, we have to consider the illumination produced by a
finite source. Let its elementary area be dA% and the elementary
illuminated area be dA^, 8% and 02 are angles between the normals to
these surfaces and the line joining the surface elements (Fig. 2.3).
Then the amount of light, dF, emitted by the source element in the
direction 0% is
dF =:• B(dA% cos 0%) dwg
dA.dA
Fig. 2.3, Illumination from a Finite Source .
where B is the source luminance (or brightness in terms of visual
effect) and dti)2 is the solid angle subtended:by the receiver (dA2) at
the source, that is
6
dAgcos 02dtoo = - --
S2
therefore the illumination at dAg becomes
an B dAj cos 0! dm2 'dE = dA^ = dAf --- = B dt°l cos 02 (2.4)
dA% cos 8%where dw% = r-— ^ is the solid angle subtended by the source
' s 2 .element at the receiver. Hence the illumination from a small source is
proportional to the cosine of the angle of incidence.
Illumination Produced by a Lambertian Source
Experiments have shown that most extended sources, for moderate
values of 8%, follow Lambert's Cosine Law. That is the luminous
intensity, I, falls off as the cosine of the angle of emission, or in
mathematical equivalent form
% = I0 cos 01 - (2.5)
this relationship is known as the Cosine Law of Emission. Hence the
luminance in the direction 0% is
102 : I0 cos 0i I0B01 AQl Aq COS 01 ■ ; A0 B0
where A ^ is the projected area normal to the direction of emission.
Therefore its luminance is independent of the viewing direction.
7
Now for an extended source which follows Lambert's Cosine Law
(i.e., a Lambertian Source), we will calculate the illumination
produced at a point X (Fig. 2.4). For simplicity, we consider the
case of a circularly symmetric source.
dA
Fig. 2.4. Illumination from a Circular Disc Source.
Point X is in a plane parallel to the source disc, then
0! = 02 = 0
4) becomes, '
dEe = BdA cos 0 COS 0 = B ^
s2(S/cos 8)2
Thus at off-axis points, the illumination falls off as the fourth
power of the cosine of the angle measured from axis.
From the geometry, the same illumination is produced by each
incremental area making a ring of radius r and width dr, then the area
of this ring is
dA = 2 it r dr
and the total illumination from this ring can be written as
dE = B 2 n r dr cos i
but r = S tan0i, dr = S sec20 d0
then dE = L j jrCS tan.8) (S sec^ de) cos^e = 2lrB sin6 cos6 de
S2
integrating over the entire surface, we obtain the illumination at
point X produced by the Lambertian disc source
9E = f dE = 2ttB sin0 cos0 d0 = ttB sin20 (2.6)C JA Jo
where subscript c denotes circular source.
In the case of non-circular source, an approximation can be
made by noting that the solid angle subtended by the source from X is
0 = 2tt (1 - cos0) = 2tt — -5-in 91 + COS0
for small angle 0,
ft - a) = irsin20
Therefore the illumination produced by a diffuse source of luminance
B at a point from which the source area subtends a solid angle oj is
E = Boo ( 2 .
Note that Eq. (2.4) would have the same form if a Lambertian source
(B = constant) and small angle of 0 were assumed.
Luminance of Optical Images
Consider an aplanatic optical system imaging an elementary
area dA of a Lambertian source at dA' as shown in Fig. 2.5.
dA dA'-h
principal surfaces
Fig. 2.5. Aplanatic Optical Imaging System.
10
Then the flux intercepted by area P on the first principal surface can
be written as
dF = B — dA cos 6S2
while the luminance at A' can be expressed as
B' = T (B — dA cosQ) (2.8)S2 (P'/S'2) dA' cos6'
where T = transmission factor of optical system
The sine condition gives:
nhsin6 = n'h'sine’
where h and h' are object and image heights, 8 and 6' are corresponding
marginal ray angles. Squaring both sides:
n2 h2 sin29 = n'2 h '2 sin26'
Remembering that
Aj_ _ hj2 _ (S '/n*)2 = n2 S'2A h2 (S/n)2 n'2 S2
Also note that principal surfaces are images of unit magnification, or
P cos8 = P' cos9'
Eq. (2.8) can then be simplified as follows:
B' = T B (n'/n)2 (2.9)
At image plane A', the corresponding illumination is
E' = T B (n'/n)2 tt sin2e' (2.10)
11
where 9' is the half angle subtended by the exit pupil of the system at
the axial image point.
When both object and image lie in media of the same refractive
index, Eq. 2.10 becomes
E' = T B tt sin2 0 ’ (2.11)
for a circular exit pupil, and
E' = T B go (2.12)
for non-circular exit pupils and small values of 9’.
Since the flux is confined to a cone of solid angle tt sin2 9',
outside this cone there is no luminance at all.
Illumination by Projection
Fig. 2.6 shows the conventional projection system in air. The
condenser images the source into the entrance pupil of the projection
lens so that the lens aperture has the same luminance as the source
excluding the transmission factor involved.
projectionlenscondenser
Fig. 2.6. An Optical Projection System.
12
Consider a small area (not shown in the figure) in the plane
of the transparency which has a luminance Then the amount of light
arriving at the entrance pupil of the projection system is
'1 = £2
where # 2 = area of source image
£ = distance from transparency to entrance pupil of projection
system.
Assuming a\ is a small circular area of radius r, then
a 1 = ir r|
and
tt r^ a.2 r^'l =' — . =ir B1&2 — = tt B1a2 sin2 m 1 (2.13)
where u' is the marginal ray angle. .
From previous analysis, we know that the luminance of the
source image is equal to T B, where T is the transmission factor of
the condenser and B is luminance of the source. Hence the amount of
light passing through the condenser and reaching the entrance pupil
of the projection system, following Eq. (2.11), is
F = 7T T B a2 sin2 u ' o c z (2.14)
Equating Fi = F we have
Bi = T B 1 c
Therefore the effective luminance of the transparency is equal to that of
the source multiplied by the proper transmission factor of the optics
considered.
Thus the illumination on the screen can be written as
E = T ir B sin2 8' = T B co (2.15)
where B = luminance of the source
T = transmission factor of the system
6 * = half angle subtended by the exit pupil from the screen
Let D = diameter of projection lens
£' = distance from projection lens to screen
= focal length of projection lens
£c = separation between source and condenser
5 = lamp filament diameter (circular shape assumed)
the magnification of the condenser can then be written as
Mc = £ji/Zc = D/6
Similarly, for a long projection distance, the magnification of
projection lens is
■ 14In the above formulation, we have assumed that plane of the transparency
is close to the condenser (i.e., f^ = t ) . The solid angle subtended
by the projection lens is
irD2/4 it ^ c 62 tt .62to = — --- — - — ---- — - — ----V 2 m / f£2 4 M£2 £c2
For a system free of vignetting, the illumination on the screen can be
written as
Therefore the image illumination varies when the magnification of the
projection system changes.
CHAPTER 3
REQUIREMENT FOR CONSTANT ILLUMINATION
General Consideration
In Chapter 2, Illumination by Projection, we have shown that
the illumination on an image plane is
E. = T ir B sin2 6' (3.1)
where 8' is a measure of the system magnification. Thus the illumination
changes as the magnification varies. To keep the illumination constant,
we can either (1) vary the aperture size accordingly through a mechanical
device, or (2) use a zoom condenser system. The first method has been
widely used and will not be discussed here. We will therefore concen
trate on the second method.
In the paraxial region, Eq. (3.1) becomes
E = TT T B 8'2
then the total flux passing through the system is
F = E-A = T tt B G'2 ir h'2 = ir2 T B (6'h')2
where A = irh12 - circular image area.
Here we recognize that 8'h' is equal to the optical invariant, H,
hence the total flux transmitted by the system is proportional to the
15
16optical invariant squared. Therefore, if the flux transmitted is kept
constant, the illumination will be constant so long as the image stays
the same size. This implies that we should have a single value of the
optical invariant, H, throughout the system at all magnifications. Our
task in then to match the size of the entrance pupil (of the projection
system), at every magnification, to that of the source image. In other
words, the effective size of the entrance pupil is equal to the image
size of the source. This will ensure that the total amount of flux
which gets through the system stays constant.
Description of the Problem
With a given optical invariant H and image height h’, 9' is
fixed. Either the pupil diameter or its location will specify the
system completely. Fig. 3.1 gives one configuration of the system: 0
is the source element, 1 through 5 are condenser elements (among them
2, 3, and 4 are moving parts), surface 6 is the image plane or entrance
stop entrance(condenser) pupil
Fig. 3.1. A Zoom Condenser System-
17
pupil of the projection system, the transparency lies between 5 and 6
and in general, close to element 5. Hence the image of the source
formed on surface 6 varies in size at different magnifications. We will
attempt to design a set of zoom lenses with the following parameters:
zoom ratio: 10 to 1
tc: 200 mmD
total field size (diameter of image): 5 mm to 50 mm
corresponding F/no: F/40 to F/4
diameter of lamp filament: 10 mm
Hence'we calculate the optical invariant as
H = n hp Up = (1) (50/2) (-1/80) = -5/16
and axial incident ray angle
uo = H/ho = (-5/16)/(-5) = 1/16.
Table 3.1 gives some of the parameters as calculated at different
magnifications.
18Table 3.1. First-Order Parameters of a Zoom Condenser System*
mc ; V us ►dcII of'
' v 2 v 2 yP ^-5 -1/80 1/8 50 mm 2.5 mm
-4 -1/64 1/10 40 3.125-3 -1/48 . 3/40 ' 30 4.167-2 -1/32. 1/20 20. 6.25
-1.6 -5/128 1/25 16 7.813-1 -1/16 1/40 10 . 12.5
-1/2 -1/8 1/80 5 25
* ■ . :mc - magnification of condenser system. Subscripts p indicate pupil
of projection system, others have their usual meanings.
CHAPTER 4
FIRST-ORDER ZOOM LENS DESIGN
Thin Lens Solution
A. D. Clark (1973) gave a good review of zoom lenses. In his
monograph, he described different methods to approach a first-order
solution. As usual, we break the system into three groups: (a) fixed
front group, (b) zooming group, and (c) fixed rear group. After
trying several configurations, including a telescopic zooming group,
we found that the following approach is feasible.
First, we start with the fixed front and rear groups for one
extreme magnification, adding a zooming group in between them to obtain
other magnifications. We also deliberately extend the magnification
range so that a certain extra range.allowance is provided. This will
relieve a possible difficulty when we get into the thick lens solution.
As shown in Fig. 4.1, each group may have one or more lens
elements as required. For simplicity, we let the axial ray travelling
within the fixed rear group be collimated. That is y^ = y^, and t^
is rather arbitrary except for chief ray considerations. Thus in the
absence of lenses 2 and 3 (their powers are equal in magnitude and
opposite in sign), we can determine the powers of lenses 1 and 4. The
power of lens 5 is then fixed by the back focal, distance of the total
19
20
system. The powers of lenses 2 and 3 are calculated by applying
paraxial refraction and transfer equations to the other extreme
magnification. In principle, we can start with either end magnifica
tion. In our formulation, however, physical solutions can be obtained
only if we start with the lower end magnification (-0.5x).
fixed front group
source
zooming group fixed rear group
5image of source
Fig. 4.1. A Zoom Lens Configuration.
To expedite such lengthy calculations, we worked out general
equations which fulfilled the prescribed restrictions (Appendix A)
We then determined if there was a possible first order solution with
certain input parameters. A Hewlett-Packard 9100B calculator was used
to handle these calculations.
After we had reasonable lens powers, we determined the spacings
between the lenses at any magnification within the magnification range
21
being considered. Paraxial marginal and chief rays were then traced and
lens diameters defined. Table 4.1 and Fig. 4.2a summarize and tabulate
our thin lens solution; Figs. 4.2b through 4.2e continue the summary of
the solution.
Table 4.1. Summary of First-Order Thin Lens Solution
Powers Lens Dia- . meters
Spacings
*i- * 0.004 333 333 d1 = 146.5 t0 = 1000
0.005 631 821 dg = 166 t4 = 1
*3 - 0.005 631 821 d3 = 33.5 tg = 200
*4 * *5 *
0.001 287
0.005
879 d4 = d5 = 25-
”c ' tl V . t3 . ^5*-5 54.62899 535.37101 10 2.5
-4 60.94819 466.87913 ' 72.17268 3.125
-3 . 70.20799 387.05016 142.74185 4.167
-2 85.74148 287.80860 226.44992 6.25
-1.6 95.7330 238.49562 265.77138 7.8125
-1 120.80448 143.65512- 335.54040 12.5
-1/2 170.39106 16.92992 412.67902 25-
* y5 = radius of exit pupil; all linear dimensions in mm.
22
Fig. 4.2. First-Order Thin Lens Solution
a. Spacings
magn
ification
23
Zoom Element Movements5
-4
1 0 0 mm3
-2
- 1
Fig. 4.2. Continued. First-Order Thin Lens Solution,
b. Zoom Element Movements.
24
paraxial ray trace
marginal ray
chief ray
marginal rayc
'chief ray
Fig. 4.2. Continued. First-Order Thin Lens Solution.
c. Paraxial Ray Trace, = -5, and = -4.
25
paraxial ray trace
marginal ray
chief ray
marginal ray
chief ray
Fig. 4.2. Continued. First-Order Thin Lens Solution.
d. Paraxial Ray Trace, m^ = -3, and = -2.
26
paraxial ray trace
m — -1,6
marginal ray
chief ray
marginal ray
chief ray
m c — - 1 / 2
marginal ray
chief ray
Fig. 4.2. Continued. First-Order Thin Lens Solution.
e. Paraxial Ray Trace, = -1.6, = -1, and m^ - 1/2.
27
Thick Lens Solution
To be a real system, thickness must be inserted into the lenses
we designed. The power of a thick lens can be calculated as:
(j) = ( n - l ) ( c 1- c 2 + ^ p t c 1c 2 ) (4.1)
As shown in Fig. 4.3, c% and c2 are curvatures (with proper signs), t is
thickness and n is the refractive index of the lens material; s and s'
locate the two principal points (P and Pf) which are essential in
relating thin lens to thick lens solutions. They can be obtained as:
t 4*2 s ’ = " n F
where (j) = (n-1) Cj (j)2 — -(n-1) c2.c
With an arbitrary curvature ratio K = Eq. (4.1) becomes,
c 2 =
-n(K-1)2(n-1)tK 1 - / 1 40tK
n(K-l) 2
-c
- s ’y
Fig. 4.3. A Thick Lens.
Fig. 4.4 gives us some idea of how these parameters change. Alterna
tively, s and s' can be determined by tracing a parallel ray from either
side. of a lens successively.
In picking glasses for our application, we need to consider
their thermal properties in addition to their optical properties.
Essentially, we want low-expansion, high thermal conductivity and high
heat capacity. From the Schott Glass Table we find that BK7 and
F2 are acceptable and popular ones. Their important properties are
listed in Appendix B . Schott Optical Glass (1966).
Bending of lens V is set to yield minimum primary spherical
aberration and coma contributions. For others, their shapes are
arbitrarily defined. Fig. 4.5 gives a system schematic picture. Thick
ness of lenses are checked against their required clear apertures defined
by paraxial ray traces.
Since lens II would need about 40 mm thickness to maintain its
necessary clear aperture as a singlet, we break it into two in contact.
Lenses IV and V have fixed spacing between them and they therefore can
be combined as one. Finally, we end up with a slightly different five-
element design as shown in Fig. 4.6.
Table 4.2 gives lens data of first-order thick lens solution (2)
and Table 4.3 gives system parameters of first-order thick lens
solution (2).
X CO
29
C2/<f
p =n =1.51680
negative power positivepower
—2
positive power
negativepower
--2
Fig. 4.4. Lens Parameters' Relation
30
lamp
lens Iglass BK7 (Schott)thickness 21 (mm)
IIBK7
40
n
i nF2
IVBK7
nV VJ
stop
V3K7
5.5
lampimage
Fig. 4.5. Schematic of First-Order Thick Lens Solution (1).
V7A
rV
3 V V 8 4 5 6 7
10 11
Fig. 4.6. Schematic of System Configuration
31
Table 4.2. Lens Data of First-Order Thick Lens Solution (2)
Surface Curvature Thickness Glass Diameter(mm""1) (mm) (Schott) (mm)
23 .
0-8.384932E-03 21 BK7 132
45
2.600000E-03-2.600000E-03 20 BK7 . 164
67
2.600000E-03 -3.430630E-03 20 BK7 164
8 -4.521963E-03 g F2 369 4.521963E-03
1011
1.216700E-02 0 6 BK7 SO
32
Table 4.3. System Parameters of First-Order Thick Lens Solution (2)
mc TH,3 TH,7 TH, 9
-5 40.18232 521.36229 8.46347
-4 46.50152 452.87041 70.63615
-3 55.76132 373.04144 141.20532
-2 71.29481 273.79988 224.91339
-1.6 81.28633 224.48690 264.23485
-1 106.35782 129.64640 334.00387
-1/2 155.94440 2.92119 411.14249
TH,0 = 86.155063
TH,5 - 0
TH,11 = 196.0443
CHAPTER 5
ABERRATION CONSIDERATIONS
The success of our first-order solution depends solely upon
the accuracy of our calculations. However, its results are applicable
only to a small region close to the optical axis (paraxial region) and
at the-single wavelength used in our computation. Departure of rays
(by exact trigonometric raytrace) from paraxial image point yields
aberrations. There are five primary monochromatic aberrations and two
chromatic errors. For example, W. T. Weiford (1974) offered an
excellent treatment of these aberrations in his book. In this section,
we will briefly describe these aberrations in a manner to suit our
interest.
Spherical Aberration (SA)
This is the variation of focus with aperture. It depends on
ray height, object position, and bending of lenses. A point source
image suffering from SA has a bright dot surrounded by a halo of light.
The contrast of an extended image will be softened and image details
blurred. It is less important for non-imaging systems. However,
excessive SA may result in dark areas or rings because the effect of
vignetting may appear on the screen, especially when the source image
completely fills the projection lens aperture'.
33
Coma (CMA)
This is one of the off-axis imaging.defects which causes
variation of magnification with aperture. It is a rather undesirable
defect because of its non-symmetrical feature, the results being non-
uniform in energy distribution. The shape of the lens and its stop
location are two factors which govern this sort of aberration.
Astigmatism (AST)
Astigmatism occurs when tangential and sagittal foci do not
coincide. The image of a point source becomes two separate lines
perpendicular to each other. An extended image would compose ellip
tical patches of light. A circular patch can be obtained by defocusing
the image plane so it is located midway between the tangential and
sagittal foci. Bending and stop shifts are employed in aberration
correction.
Field Curv ature (FC)
An aberration which gives curved image field even in the absence
of astigmatism. Its existence has an additive effect to astigmatism
if the latter does not vanish. Therefore the two are normally treated
together. In some instances, FC can be used to balance against higher
order aberrations. For example, in a triplet design, residual FC is
left in to balance backward curving higher order sagittal field curva
ture and oblique spherical aberration (OBSA). The power of the elements
and refractive indices are determining factors of this defect. This
aberration is of little importance in condenser design.
Distortion (DIST)
A defect which results in variation of magnification with field
angle and shift of image proportional to the cube of the Gaussian image
height. The image of a line object which does not go through axis is
curved. This defect is usually corrected by shifting the stop,
although it can also be done at the expense of other aberrations. In
this condenser design, distortion must be controlled to obtain full
illumination.
Chromatic Aberration
This is the result of inherent properties of the glasses used.
Because refractive index varies with wavelength, unless the system is
operated at the single designed wavelength, both focus and magnifica
tion change with wavelength. These are referred to as longitudinal (or
axial) and lateral chromatic aberration, respectively. A point image
will have different color compositions,at different focal positions.
On the image plane, color rings will show up because each color has its
own chief ray. The image of an extended object near the edge of the
field and illuminated with white light will be surrounded by colored
fringes. In practice, corrections are achieved at two or three wave-
lengths by achromatizing or apochromatizing each element of the system.
Lateral color can also be reduced or eliminated by a stop shift. For
a thin lens, when the stop is at the lens, there is no lateral color
or distortion.
36
Of the above five monochromatic aberrationsspherical aberra
tion, coma, and astigmatism affect image sharpness, while field curvature
and distortion change the geometry of the image. Strictly speaking,
any of these aberrations will result in a loss of light. Yet for a
condenser system, larger amounts of aberration can be tolerated without
losing its function of collecting light from a lamp and directing it
toward the aperture of the projection lens.
From our discussion above, we may conclude that spherical
aberration and chromatic error are more important than others in a
condenser system.
It is for simplicity of discussion that we broke the imaging
defects into five monochromatic aberrations and two chromatic errors.
In a real system, however, more than one aberration exists at the same
time. Also, higher order aberrations may dominate in certain situations.
The primary monochromatic aberrations are also wavelength dependent.
Therefore, it is too complicated to express aberrations analytically.
In Table 5.1 we summarize' the above discussed aberrations with different
notations cited.
37
Table 5.1. Summary of Third-Order Aberrations
Aberrations Seidel Sums ACCOS IV (transverse
ray aberration)
WavefrontAberrationCoefficient
aberration SI = ' ? S A 3 = 2 ^ SI Wo40 =
coma Sn = - IAB yA$ CMA3 = STT W, „ = a(CMA3)2n'u' II 131
astigmatism (n) AST3 2n'u' ^III ^222 a(A.ST3)
fieldcurvature
S;v=- I PTZ3 = 1 STv = - (PTZ3)-2n'u' IV 220 2(AST3)
distortion S^ =- I (j)
PH2-B2yA(^)
MSS = 2 ^ Sv *311 = *(0133)
axialcolor
Cj = yniA(^) PAG =n f u ’2
lateralcolor cn = PLC = 'II
r fn 'u
where: A = ni = n'i B = ni = n ’i 1,11
. rdn. dn’ dnAC^ = — - 5T
p = CA(I) = c(i - 1) H = Lagrange invariant = nhu
CHAPTER 6
ABERRATION DESIGN
Our first order design gives powers, spacings, clear apertures
and angles of incidence and emergence. Therefore the variables left
over are refractive indices and lens shape factors. Once the glasses
are chosen, we can only change the shapes of lenses (bending) and/or
move the stop around to correct aberrations.
First we put our thick lens data (Table 3) into ACCOS IV
(ACCOS-GOALS, 1967) program for analysis. From the output FORD tables
(Appendix C) we notice that
(a) distortion and third order pupil spherical aberration are
huge at higher magnifications,
(b) chromatic errors are tremendous at higher magnifications,
(c) third order spherical aberration is relatively large at lower
magnifications.
Ray fans of both extreme and mid-magnifications are shown in Fig- 6.1.
Since our stop is fixed at surface 11, stop shift is not available for
aberration correction. Our attempt to reduce distortion was unsuccess
ful when, restricted to spherical surfaces. Because distortion is
related to pupil spherical aberration, elimination of the latter will
also reduce the former. From our inspection, we find that surface
7 (or 4, 5, 6 ) is a proper one to be aspherized. Since in that lens
38
39
tangential sagittal
full field
-5 mm _ -5 mm
aberrations relative to y plus
25.060— — — -- 24.829------- 24.719y 25.000
(5 8 7 . 6 nm) (486.1 nm) (656.3 nm) (587.6 nm)
r 5 mmon axis
_ -5 mm
Fig. 6.1. Ray Aberrations (Starting Value),
a. -5x.
40
\sagittal
“ 0 . 5 rrun
full field
L-0.5mm
— 0 . 5 mm
\\
aberrations relative to y plus
-------- 0.936 (587.6 nm) 1.375 (486.1 nm)-------- 0.718 (656.3 nm)y 8 . 0 0 0 (5 8 7 . 6 nm)
0 . 5 mm
on axis
0 . 5 mm
Fig. 6.1. Continued. Ray Aberrations (Starting Value),
b. -1.6x.
41
tangential
r 50 mm
full field
L - 5 0 mra
aberrations relative to \
sagittal
y plus0 . 0 0 0 (5 8 7 . 6 nm)
-0.004 (486.1 nm)-0 . 0 0 0 (6 5 6 . 3 nm)2.500 (587.6 nm)
50 mm
on axis
\
Fig. 6.1. Continued. Ray Aberrations (Starting Value),
c. -l/2x.
42
group chief ray heights are huge and incident angles are unusually large
compared to those at other Surfaces. This can also be seen from the
stop shift equations: (Appendix D). It is advisable to add aspherics
where the chief ray height is greatest to correct distortion with least
effect on spherical aberration. On the contrary, to correct spherical
aberration, the aspheric should be added at the surface where the chief
ray goes through the axis (stop location).as is usually done for the
Schmidt system. Figs. 6.2 a and b show the ray aberrations at extreme
magnifications when spherical aberration is removed. Fig. 6.3 shows ray
fans at -5x with distortion corrected. Shown in Fig. 6.4 are those with
PSAS reducted. Up to this step, we see that astigmatism is relatively
large, so we try to reduce it. This step of the correction leads us to
the results shown in Figs. 6.5 a through g. Our first response to these
curves may suggest defocus for the next step. When we defocus, however,
we gain the image quality at the expense of system efficiency. This
further step is unlikely to be justified in a non-imaging condenser
system.
The next point to be noted is chromatic aberration. Since it is
controlled by glass dispersion, color control is not feasible on a
balancing basis in a zoom system. If we really want to correct it, we
are forced to achromatize or apochromatize each of the lens groups.
Again this may not be justified by its increased cost. We would rather
leave our aberration design at this stage and analyze our system in
Chapter 7.
43
tangential sagittal
r 0.5 mmfull field
- -0 . 5 ram 0.5 mm
r 0 . 5 mm
0 . 7 field
_-0 .5 mm
r 0 . 5 mm
--0.5 mm
y
field
aberrations relative to y plus
0 . 0 0 0 0 . 0 0 0 (587.6 nm)-0.004 -0.003 (436.1 nm)-0 . 0 0 0 -0 . 0 0 0 (6 5 6 , 3 nm)2.500 1.750 (587.6 nm)full 0 . 7
on axis
--0 .5 mm
Fig. 6.2. Ray Aberrations (SA Removed).
a. - l/2x
44
tangential sagittal
5 mm
full field
— 5 mm
aberrations relative to y plus
28.567 (587.6 nra) 30.506 (486.1 nm) 27.540 (656.3 nm)
2 5 . 0 0 0 (5 8 7 . 6 nra)
5 mm
-5 mra
5 mm
on axis
X \ XX
-5 mm
Fig. 6.2. Continued. Ray Aberrations (SA Removed),
b. -5x.
45
tangential
r 0 .5 mm
full field
mm
sagittal
\ \ \\
0 . 7 field
u -0 , 5 mm
aberrations relative to y plus
- -0 . 2 4 6 -0.018 (5 8 7 . 6 nm)2 . 7 8 8 (486.1 nm)
-1 . 2 1 1 (65.6.3 nm)y 25.000 17.500 (587.6 nm)
field full 0.7
— — 4.094— - — — —1.471
r-0 . 5 mm
..-0 . 5 mmX
-0 , 5 mm
on axis
0 <=-
Fig. 6.3. Ray Aberrations [Distortion Corrected) at -5x.
46
tangential
full field
mm
sagittal
0 . 7 field
- -0 . 5 mm
r-0.5 mm
x
P- 0.5 mm
on axis
aberrations relative to y plus
0.157 -0.114 (5 8 7 . 6 nra)------- 4.001 2.657 (436.1 nra) I .559 -1.342 (656.3 nm) 0y 2 5 . 0 0 0 1 7 . 5 0 0 (5 8 7 . 6 nm)
field full 0.7
_-0 .5 mm
Fig. 6.4. Ray Aberrations (PSAS Reduced) at - 5x.
47\\ tangential sagittal\
full field
_ -0 . 5 mm
r- 0.5 mm
--0 . 5 mm
X X X XX X
— v . mill
0.7 field
1_________ __ -
X X
_ -0.5 mm
aberrations relative to y plus
-------- 1.383 -0.672 (587,,6 nm)------ 2,477 2.099 (486,,1 nm)— * — * — —3.104 -1 . 9 0 1 (6 5 6 ,.5 nm)
y 2 5 . 0 0 0 17.500 (587,,6 nm)field full 0.7
_ 0 . 5 mm
_ -0 . 5 mm
5on axis
_ -0.5 mm
Fig. 6.5. Ray Aberrations (Final Version)
a. -5x.
48
tangential
r- 0 . 5 m m
sagittal
full field
_ -0 . 5 mm
0 . 5 mm
0 . 5 mm
r 0 . 5 mm
0 . 7 field
_-0 . 5 mm
r- 0.5 mm
y
field
aberrations relative to y plus
-2.536 -1 . 1 1 0 (5 8 7 . 6 nm)0.451 1.048 (486.1 nm)
-3.954 -2.083 (656.3 nm)2 0 . 0 0 0 14.000 (537.6 nm)full 0 . 7
on axis
0
Fig. 6.5. Continued. Ray Aberrations (Final Version)
b. -4x.
full field
— 0 . 5 nun
/zr 0 . 5 mm
0.7 field
- -0 . 5 mm
aberrations relative to y plus
-2.654 -1.015 (587.6 nm)— — — — —0 .6 9B -0.552 (4 8 6 . 1 nm)— ---— —5.556 -1.645 (656.5 nm)
y 1 5 . 0 0 0 1 0 . 0 0 0 (587.6 nm)field full 0.7
sagittal
r 0 . 5 mm
L -0 . 5 mm
0 . 5 mm
- -0 . 5 mm
- 0 . 5 mm
on axis
0 . 5 mm
Fig. 6.5. Continued. Ray Aberrations (Final Version).
50
tangential sagittal
0.5 mm
full field
L -0.5 mm
r 0.5 mm
0 . 7 field
- -0 . 5 mm
L -0.5 mm
r 0 . 5 mm
-0 . 5 mm
aberrations relative to
y
field
y plus-1.157 . -0.410 (537.6 nm)-0 . 2 7 6 -0.194 (4 8 6 . 1 nm)-1.572 -0.695 (656.3 nm)1 0 . 0 0 0 7 . 0 0 0 (587.6 nm)full 0.7
0 . 5 mmon axis
--0.5 mm
Fig. 6.5. Continued. Ray Aberrations (Final Version)
d. -2x.
tangential sagittal
full field
-0 . 5 mm
0 . 5 mm0 . 7 field
-0 . 5 mm
aberrations relative to y plus
-0 . 5 6 2 -0 . 1 9 6 (587.6 nm)— — — — —0.04-0 0.162 (486.1 nm)— -— * —— —0.313 -0 . 3 8 6 (656.5 nm)y 8 . 0 0 0 5.600 (587.6 nm)field full 0.7
r* 0.5 mm
L- -0 . 5 mm
0 . 5 mm
_ -0 . 5 mm
r 0 . 5 mm
on axis
- -0 . 5 mm
Fig. 6.5. Continued. Ray Aberrations (Final Version).
52
tangential sagittal
0.5 mm
0 . 5 mm
0 . 5 mmfull field
0 . 5 mm
-.0 . 5 mm0.7 field
- -0 . 5 mm
aberrations relative to y plus
0.082 -0.028 (537.6 nm)— — — — 0,053— ----—0.154y 5 . 0 0 0
field full
0.069 (486.1 nm)-0.078 (656.3 nm)3.500 (587.6 nm)0.7
0 . 5 mm
_ -0.5 mm
0 . 5 mm
on axis
■0 . 5 mm
Fig. 6.5. Continued. Ray Aberrations (Final Version).
f. -Ix.
53
tangential sagittal
r* 0 . 5 mm
full field
0 . 5 ram U -0.5 ram
p 0 . 5 ram
0.7 field
-0.5 ram
0 . 5 ram
_ -0 . 5 ram
aberrations relative to y plus
- -0 . 0 0 2 -0 . 0 0 0 (5 8 7 . 6 nm)-0.003 (486.1 nm)-0 . 0 0 1 (6 5 6 . 3 nm)1.750 (537.6 nm)0.7
■ — — — — 0.006 —- — - — —0.003 y 2.500field full
on axis
-0.5 mm
Fig. 6.5. Continued. Ray Aberrations (Final Version)
g. -l/2x.
CHAPTER 7
RESULTS AND DISCUSSION
In Fig. 7.1 we show the scaled drawing of the condenser elements
and, in Table 7.1, the lens data of the final version. The primary ray
aberrations as a function of magnification are given in Fig. 7.2. Spac-
ings between elements are the same as those given in Table 4.3. Real
ray aberrations have been depicted in Fig. 6.5, a through g. For a
condenser system. We are more concerned about its function of collecting
light from the lamp directing it toward the aperture of the projection
lens. In this design we are aiming at constant illumination on the screen.
We are therefore interested in the amount of light transmitted as well
as its color structure. For analysis purposes, we trace rays backwards
from the edge of the field (image plane of condenser or entrance pupil
of projection system) through the edge of the exit pupil, one through
the top and one through the bottom, with another ray traced through one
edge (equivalent to two) in the sagittal plane. Due to aberrations,
these rays will not meet at the same point in the object plane (of source
filament). They will then define the limiting field boundary from
which light can get through different parts of the exit pupil (of
condenser) and reach the entrance pupil of the projection system.
Let the contribution through different parts of the pupil be equal, then
54
55
I I II I
I V
Full Scale
Fig. 7.1. Elements of a Zoom Condenser
Table 7.1. Lens Data and Aspheric Surface Data of Final Version
Lens
I
II
III
IV
V
)
SurfaceNumber
7
SurfaceNumber
Curvature(1/mm)
Conic ThicknessConstant (K) (mm)
23
45
-8.384932E-03
2.600000E-03.-2.600000E-03
00
00
21
20
6 2.600000E-037 -3.430630E-03
0■3.29515E-00 20
8 -4.521963E-039 4.521963E-03
00
10 1.216700E-0211 0
•7.43208E-010
Aspheric Surface Data
AD AE: AF . AG
6.67680E-08 -1.15144E-12 1.65433E-22 2.23180E-26
Aspheric surface profile in the y-Z plane:
cy5
1 + /+ (AD)^y + (AE)6y + (AF)sy + (AG)y10
1-(K+l)c2y2
57
04 0 CMA3.02 0 .02 SA3
AST3 + AST5
AST31 2 30
Fig. 7.2. Transverse Ray Aberrations at ReferenceWavelength 587.6 nm.
58
DIS3 + DIS5, PTZ3 + PTZ5
- 2
-1
5 0 DIS3
—4
- 2
-1
-.05 0 PTZ3
-4
-2
- 1
0 5 PLC
-4
- 2
- 1
-.5 0 PAG
Fig. 7.2. Continued. Transverse Ray Aberrations.
59
we average out these four values and plot them in Fig. 7.3. Since our
half field size is 5 nm (the aperture of the projection optics will be.
underfilled for those greater than 5 mm). On the other hand, on those
smaller than 5 mm, image of the source will overfill the aperture and
light is not fully utilized. Referring to Fig. 7.3, we see that at one
end of magnification (-l/2x), the aperture is underfilled and almost no
color errors show up. As we go on to higher magnifications, we have
different degrees of underfill and/or overfill at different wavelengths.
For example, at -3x, there are about 60% and 87% of filled aperture for
red, green and blue light respectively. That means different colors will
go through different parts of the entrance pupil of the projection
system. All aberrations will be complicated by such additional "wave
length" dependent feature. This is a father undesirable result, .unless
the projection system is designed to compensate for it.
Aside from the monochromatic aberrations, for an overfilled
aperture (all colors) we will not have any color ring structures. It
seems that we should have larger lamp filament (radius 6.5 mm or larger)
so that we can overfill the aperture at all magnifications. However,
this is contrary to our desire to have equal amounts of energy reaching
the aperture Of the proj ection system. Therefore we have to make a
compromise and let the filament stay the same size, so that differences
in energy on the screen are small, and the color ring structures are small
and close to the outer part of the aperture. For a filament of 5.25 mm
in radius, the corresponding percentages of transmitted energy are:
field
size at
lamp
plane
(mm)
exit pupil of condenser
image plane of source
Rays traced backwards from edge of fieldone thru top of pupil in the tangential plane one thru bottom of pupil in the tangential plane one thru edge of pupil in the sagittal plane
'aperture underfilled
6
587.6 nmX5
aperture overfilledN 486.1 nm
4
- 1 -2 3 -4magnification of condenser mc
Fig. 7.3. Field Size versus Magnification.
587.6 ran under under under under under 97% 97%
486.1 ran 64% 83% under 100% 94% 91% 97%
656.3 ran under under under under under 100% 97%
For those designated as "under", we mean that the aperture
is underfilled. In this case, color fringes will show up towards the
edge of the aperture. We may limit the image field of the condenser
system (stop down the aperture of the projection system) to chop off
these fringe structures. Also we should note that the filament is
generally rectangular or square, and the projection lens pupil is
circular. Thus, only a nominal fit is really possible.
From our analysis above, we perhaps obtain an appreciation for
the■difficulty in designing a high-ratio zoom condenser. We conclude
that even for not a "perfect" design, we can be much better off than
with a non-zoom condenser. Without a zoom condenser, for a magnifica
tion ratio of 10 to 1 , the illumination-will have a ratio of 1 to 100 at
the two extreme magnifications. We are sure that further improvement
can be achieved if more effort is devoted to the design. In any event,
this is a good starting point for a detailed design.
APPENDIX A
FORMULAE OF FIRST-ORDER SYSTEM PARAMETERS
The following graphs and calculations present a derivation
of the important first order formulae. The format used is consistent
with the approach and sign convention used in courses at the Optical
Sciences Center.
(a) At one extreme magnification: ^ and <f>4 determined
<K
2 _ + I
*0 fH-^o^oK T ~
H1 uo to
62
63
(b) At another extreme magnification: $ 2 = -<t>3 determined
u
1 2
- < P l - < p 2
f.-1 2~ 1 3
UoU-Vh)t u o o Y2
t 2
U2 *
2
-@3
t 3
h<j)4
4-<J>4
V2 = V o + t2 " t3 u0 (1 - 1u2» = UQ (1 - tQ(j)i) = Y2<j)2
Y3 = Y2 + t2U2 ’
u3f = u2’ - Y3(f>3
y4 = h = Y3 + t3 h <i>4
let <t>3 = -<j)2
<t>2 = --------------------------------------h(l-t3<})Lf) - touo -
64
(c) at any arbitrary magnification: t2 and t3 determined
21
E - B t 3
- 6 ± /g2 - 4ay 2a
E = (t>2 (d + H) -a
m = uq/u5 ’
a = (tQ(#)1-l) - H ((()4 + (j)3)
b = BC = H ^ 4 ^ 3
d = Eti. [ -H
B = + H^4 ^ 3 = A + C
where: a = CB
B = aB - bA - CE
y = Ad - aE
APPENDIX B
PROPERTIES OF BK7 AND F2 GLASSES
The following properties of BK7 and F2 glasses are from the
Schott Glass Catalog (1966).
65
BK7 - 517642Refractive _ , Index "d 1.51680
Refractive Indices for A lnm l
1014,0 852.1 7C6.5 656.3 643.8 539.3 587.6 546.1 486.1 480.0 435,6 404.7 365.0
Vn p -n c d 64.17 "t ns nr nC n c nD nd ne nF np- n9 nh nl
1.50731 1.50981 1 51289 1.51432 1.51472 1.51673 1.51680 1.51872 1.52238 1.52233 1.52669 1.53024 1.53626
Dispersion n p -n g 0.005054 Relative Partial Disp irslons
Refractive _ Index °e
n -1--------------— v en p .-n c *
1.51872
63.98
ns -n ,
lip - OQ
0.3097
nr -n 9
" p -n C03833
nC ' nrn p -n c0.1774
nd - r,C n p -n c
0.3075
ne ' nd n p -n c
02386
nF -n e
n p -n c0.4539
ng *np
n p -n c05350
nh ' ngn p -n c0.4413
n, -n h
n p -n c
0.7478
ns * nt n p - nc -
03075
nr - n s
np-nc0.3806
r>C - nr np-nc0.2252
nd-ncn p - n c02565
ne -n d
n p - n c0 2370
np --ne
np. - n c0 5C65
ng-nF;np. . n c04755
nh -n g
n p - n c0.4232
n, -n h
np. - nc* 0.7427Dispersion nF' -•'C ' 0.003110
Constants of D ispersion Formula Temperature Coefficients of Refractive Index
A . 1 A , A , A , A , A , Range of temperature
relative x 10 8 / °C ~ absolute x 106 / CC
2 .2 7 1 2 9 2 9 1 -1 0 1 0 8 0 7 7 1 0 ' 1 .0 5 9 2 5 0 9 -1 0 ' 2 C81C955 -10 * -7 .6 4 7 2 5 3 8 -1 0 * j 4 .9240091 1 0 ' r c i C d • F' 9 c- d e F" 9-'.Ci to - 2 0 1 0
2 3
2 1
~ ~ n ~
2.2 2 5 2.7 C 2 0 3 0 4 0 6 0 3
- 2 0 to C 2 6 2 9 3 2 OC 0 7 0 9 1 2 1 4
A P c , - A n? - ‘ -n p - n c
A P c . _ A r c : " =n p - n c
APF.0 - A - n F -ne- np n c
0 to 4 20 2 3 2 5 2 5 2 3 3 3 1 1 1 2 1.3 1 6 1 9
4 20 to + 40 2 6 2 0 3 0 3 3 3 C
T e T1 5 1 6 1.7 1 3 2 2
+ 40 to 4 GO 2 3 2 0 3 0 3 3 1 7 1 8 1 9 2 2 2 5
0 .0 2 1 0 0.0 0 6 3 — 0 0009 — 0.00C8 0 0029 + CO to 4 CO 2 0 2.9 3 0 3 3 3 5 1 7 1 9 2 0 2 3 2 5
Density Bubblequality
Resistance to climatic vaiiations
Resistivity to staining
Cocllicienl of linear thermal expansion
(Tao'to I r+ a r to + ;o"C ) 1 + 3 0 0 - 0
ex 10'/°C |~ e x 1 0 '. C
Transformation
lempe-rnlure
Meanspecific
heal
Thermalconduc
tivity
Young'sModulus
Modulus of rigidity
PoissonRatio
Specialcharacter
istics
[ c m > ]Group group group Tg ( ‘ C l col n
L 9 “C Jl~ kcal "L"mh ‘ C j
r>-il mrn'Jr kpLmin'J - -
2.51 0 2 0..... .71 83 559 0 .2 0 5
(bol 20 C)0.953
(bol20°C)8310 3440 0 .206 —
Internal Transmittance T; of an Average M elt
A Inml 280 200 SCO 310 320 330 340 350 360 370 330 390 400 420 440 460 480 500 540 580 620 660 700
rj at 5 mm thlcknosa 0 0 6 0.30 0 6 3 U.C05 ooo; C.S55 0 972 0 934 0.992 0.905 0 9 % 0.097 0937 0.998 0.993 0.998 0993 0.998 0 993 0 993 0930 0993
T< at 25 mm thickness 0.10 0 34 061 0 7 0 0.87 0.92 0 0 6 0.975 0960 0.984 0937 0933 0.990 0.991 0.991 0.933 0994 0 935 0 995 0 995
6 (BK) SCHOTT No. 3050/63
F 2 - 620364FW .-actlreIndex nd
nd - t
Dlrpsralon n p -n g
RefractiveIndex
np> - nQ'
Dieperslon nF’ - n C'
1.620C4
0.017050
1.624C3
36 11
Refractive Indices for X (nml
1014,0 852,1 706.5 656,3 643.8 589.3 587,6 546.1 486.1 480,0 435,8 404,7 365.0
nt ns nr nC nc - nD nd ne nF np ng nh nl
1 60280 1 60672 1 61227 1.61503 1.61582 1.61989 1.62004 1.62408 1.63208 1.63310 1.64202 1.65063 1.66621
Relative Partial Dispersions
n3 -r.,
nF * nC02200
nr ' ns
n F " nC0.3255
nC ‘ nr
° F " nC0.1622
nd ‘ nC
nF ‘ nc0.2637
ne ‘ nd
nF ' nC0.2370
nF - ne
nF " nC 0 4693
ng ‘ nF
nF ‘ nC0.5826
nh;_ngnp-nQ
0.5054
n, -n h
nF -n c
0.9133
ns - ntn p -n c -0.2268
nr * ns n p .-n c 0.3210
nC * nr
nr ' * nC02054
nd - n c 1np. - o q .
0.2443
nc - nd np.-nQ . 0.2333
nF, ' ne np.-nQ. 0 5219
ng ; nF; np. - o q . 05157
np.-nQ.
0.4956
n, -n h
Op. • Oq .
0.9015
Constants of Dispersion Formula Temperature Coefficients of Refractive Index
A , A , A , A , A . A , Range of temperature
r e la t iv e * '0« /"C j j absolute * 106/ CC
2.555-1063 -6 8746150-10> 2.2494:67 10 ' 8.6924972 10« r2 4011704 -10 ‘ 4 6305169 -104 r c i C d o F- 9 C d e F‘ 01 in n - - 4 0 to - 2 0
- 2 0 to 0
- p Q.t - i ! APC,s - A - — — np-nQ np - nQ
3P Fie - A - F -'n-enp-nQ *P|-»-A5r4
0 to + 2 0 3 6 3 9 4 3 5.1 5 9 1 6 1.9 2 4 3 5 4 7
+ 20 to + 4 0 3 6 3 9 4 3 5.1 5.9 2.5 26 3 1 4 0 4 9
+ 40 to + 6 0 31. 3 9 4 3 51 5.9
5 9
2 7 3 0 3.4 4 2 5 20 | 0 0 0 0 + 60 to + 60 3 C 3 9 4 3 5.1 2 6 31 3 5 4 4 5 3
Density Bubblequnlty
Resistance to climatic variations
Resistivity to staining
Coefficiethcimol
unb’1 to + 70=0
it of linear expansion” ( • - jo 0 to
+ 300fC)
T rans- formation
temperature
Meanspecific
heat
Thermalconduc
tivity
Young'sModulus
Modulus of rigidity
PoissonRatio
Specialcharacter
istics
L-il flroup group gioup ux lO '/'C ex 1 0 l / eC I g f C l [,“=] p e e . I !_ mh G J [rnm ' ] - -
3 Cl 0 1 0 82 93 432 0 123(bei20'C| lo d
** 5910 2410 0.225 -
Internal Transmittance of an Average M elt
* tnm! 260 290 300 310 320 330 340 350 360 370 360 390 400 420 440 460 480 500 £40 550 620 660 700
q el 5 mm thickness 0.01 0.10 0 49 0 80 0.30 0 9 6 0962 OSES 0.99 0.993 0.595 0.956 0 957 O.S97 0 997 0.995 0.9S8 0.559 0929 0 996
r, at 25 mm thickness 031 0 5 0 0.79 0.91 0 9 4 0 96 0 97 0.98 0.983 0 985 0.987 0988 0 990 0 391 0 993 0 993 0 992
2 IF) S C H O H No. 3050/66
APPENDIX C
FORD TABLES
Reproduction of aberration tables from the AGCOS program.
■ Final
Transverse aberrations at wavelength 1
.027470 -.040702 .005454 -.002360 -.002878 -l/2x
.001090 .001216 -.000033 .000014 -.000024
.000062 -.438130 -.001697 -.301139 -.002219>000243 .000217 -.000193 .000171 .000239.000117 -.000126 .016167 .018476
.010220 -.048138 .151320 -.580597 -.009209 -1.6x
.000060 .000680 -.003098 .018549 .003173
.000000 -.309144 .760802 -.212695 .523976
.004862 -.002565 -.000878 -.000865 -.000822 "
.003593 .001269 -1.464297 .400206
„004494 .012469 .080100 -2.412003 -.028779 -5x.000003 .000146 -.081192 .862273 .021022.000000 -.614206 5.805360 -.424418 4.020659.068193 -.005594 -.000895 -.002722 -.001978.041571 .026622 -.685635 -.091481
SA3 CMA3 AST3 DIS3 PTZ3SA5 CMA5 AST5 DIS5 PTZ5SA7 PAG PLC SAC SLGECOMA TOBSA SOBSA Ml M3N1 N2 PSA3 PCMA3
69
Starting Value
.000171 -.000004
-1.080804 .079135
-.006951 -.003568 -.002391
-.057302 .001056
-.000005 -.038407 -.027739
-.063539 .000228
-.000003 -.803984 -.492042
-.026160-.008113-.438130.001915
-.001177
.075172-.012825-.309144.039736
-.010668.530424
-.007733 -.614206 .107698
-.311943
-.000143 -.000027 -.001697 .000583 .027389
-.284155-.000185.760802.014348
3.792201
-4.544378 .678697 .
5.805363 .024870
35.479253
-.301139.000238
-.006328
.965754 -.017042, -.212695 .007963
-1.080055
38.893542 -6.136281 -.424418 .040060
-4.140385
-.002878-.000030-.002219.001095
-.009209-.005961.523976.017425
-.028779 -.075807 4.020661 .042768 .042768
-l/2.x
-1 . 6x
-5x
APPENDIX D
STOP SHIFT EQUATIONS
The following stop shift equations are as presented in courses
at the Optical Sciences Center.
°pD = | Sl(Px2+Py2) + 7 SH npy(px2 + py2)
4 n2^ 3SIII + SIV')py2 + ^III + SIV^px2-*
shiftedpupil
originalpupil
where: p x, .are normalized pupil coordinates
n is relative field height
New OPD* = OPD (p^,p + p *n)
, = I SI* py4 + bSIl\ 3 + + SIV*^ 2v
+ 2 n3SV*py70
Then
REFERENCES
ACCOS/GOALS, Version II, Scientific Calculations, Inc., Rochester,New York (1967).
Clark, A. D., Zoom Lenses, American Elsevier Pub. Co., Inc., New York (1973).
Schott Optical Glass (Schott No. 3050/66), Jenaer Glaswerk, Schott and Gen, West Germany (1966).
Walsh, J. T., Photometry, Dover Pub., New York (1965).
Welford, W. T., Aberration of the Symmetrical Optical System, Academic Press, London (1974).
72
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