TECHNICAL AND SCIENTIFIC SERVICES IN GEOMETRICAL AND PHYSICAL OPTICS

• Document Type: CREST
• Collection: NGA Records (Formerly NIMA)
• Document Number (FOIA) /ESDN (CREST): CIA-RDP79B00873A001800010007-8
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 41
• Document Creation Date: December 28, 2016
• Document Release Date: September 13, 2012
• Sequence Number: 7
• Publication Date: June 15, 1967
• Content Type: MISC

Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
STAT
Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declaslsified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
TECHNICAL AND SCIENTIFIC SERVICES
IN
GEOMETRICAL AND PHYSICAL OPTICS
Submitted by
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
STAT
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
?
TECHNICAL AND SCIENTIFIC SERVICES
IN
GEOMETRICAL AND PHYSICAL OPTICS
STAT
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
It is requested that the data set forth
herein shall not be disclosed outside the Govern-
ment or be duplicated, used or disclosed in whole
or in part for any purpose other than to evaluate
the proposal; provided, that if a contract is
awarded to this offeror as a result of or in
connection with the submission of such data, the
Government shall have the right to duplicate, use,
or disclose this data to the extent provided in
the contract. This restriction does not limit the
Government's right to use information contained in
such data if it is obtained from another source.
?
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
TABLE OF CONTENTS
.Page
1 TECHNICAL PROGRAM .............??? 1
BACKGROUND AND EXPERIENCE ........ 4
COST DATA ........................ 17
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
TECHNICAL PROGRAM
Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
~} -
?
i
This technical proposal presents a program
designed to supply, on a time and materials basis over
a period of one year, certain technical and scientific
services of a staff augmentation nature. The services
proposed will lie within the general area of reconnais-
sance-intelligence technology and will include but not
be limited to the following disciplinary areas:
1,. Transfer function theory.
2. Wave and geometrical aberration theory.
3. Statistical properties of photographic
grain.
4. Evaluation of lens designs.
5. Measurements of optical system performance.
6. Evaluation of Viewers.
The purpose of the proposed program will be to
supply technical services at various levels as back-up
to the customer's staff. Work undertaken may be independent
investigation of problems of customer interest or may be
formulated as a continuation of lines of study initiated
by the customer.
The work to be pursued under this program will be
carried out under mutually agreeable work plans. In
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
?
addition to experimentation and analysis performed at
facility, consulting, advisory STAT
and/or other services will be supplied to the customer at
his facility and upon his request. It is proposed that
certain lines of investigation be established within thirty
(30) days after contract, such investigative lines being
based upon current problems of customer interest. Such
fields of investigation might include:
1. Relations between areal images formed
by lens systems and the resulting photo-
? graphic image for various films.
2. Evaluation of existing optical systems and
viewers.
3. Assistance in fabrication of unusual photo-
graphic targets and gratings.
The work proposed will be performed under the direct
supervision ot~
as Chief Investigator.
President, who will act STAT
will be assisted by STAT
(respectively STAT
Chief Consultant and Director of Research). Other staff
members will assist as required. A total of 1138 hours
as proposed with engineering and scientific services to
be supplied in the following categories:
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
Consultant:
Computer Services
Project Manager
Senior Scientist
Scientific Investigator
Staff Engineer
Draftsman
Technician
STAT
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
?
?
?
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
I
is well equipped from
?
?
both facility and personnel standpoints to conduct ex-
perimental and theoretical investigation in all subject
related fields.
The material contained in the following pages
is descriptive of Company experience, staff, and facility.
STAT
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
BACKGROUND AND EXPERIENCE
We have assembled a group of craftsmen, engineers,
and physicists whose skill in optics and related fields
is unsurpassed.
Our capability encompasses the optical problem
from they mathematical theory of lens design and systems
analysis, through the modern test facilities necessary
for the evaluation of advanced optical hardware.
Our contracts have ranged in value from a few
hundred dollars for brief technical consulting tasks,
to over one million dollars for the production of .high
precision optical assemblies. Our projects most often
include design and development, as well as manufacture.
We hope that the following synopsis of represen-
tative contracts will serve to .indicate our range of
interest and experience.
A Plant and Facilities list, and. appropriate
resumes are also included.
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
STAT
I ASTRONOMICAL SYSTEMS AND TELESCOPES
1. Design study and fabrication of two complete sets
of reflecting optics for the N.A.S.A. Goddard Ex-
perimental Package Orbiting Astronimical Observa-
tory, each set consisting of one Kanigen coated
beryllium primary mirror 30 inches in diameter,
one fused silica secondary mirror 13 inches in
diameter, one Kanigen coated beryllium spectro-
meter mirror 23 inches in diameter, and one
Kanigen coated beryllium grating blank 10 inches
in diameter, all of which are aspheric, (on the
order of 100a except for the grating blank) and
accurate to 1/4a.
2. Design, (mechanical) and fabrication of tempera-
. ture insensitive 18 inch aperture 1/4a accuracy
catadioptric telescope. '
?
3. Design and fabrication of 24 inch aperture six
element Fresnel prism for use with fast Schmidt
type optical systems. The assembly was made
from astronomical objective quality glass and
has resolution accuracies on the; order of two
arc seconds. '
4. Design and fabrication of 24 inch aperture Casse-
grainian Telescope assembly for use in tracking
operations.
5. Development and fabrication of reflecting optical
components for astronomical research in the X
radiation region.
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
STAT
II PHOTOGRAPHIC RECONNAISSANCE,
COPYING EQUIPMENT, AND VIEWERS
?
1. Design, development, and fabrication of a variety of
long focal length high acuity aerial reconnaissance
lenses, both in prototype and production quantities.
These lenses are, in general, diffraction limited
and contain multiple aspheric surfaces.
2. Design and manufacture of very high aperture proto-
type lenses requiring advances in the state of the
art of lens design, and maximum skill in optical
fabrication techniques.
3. Design and fabrication of two 400 k/mm near ultra-
violet high acuity copying lenses for use in con-
tinuous printing. These unusual 12 inch focal
length f/2.5 lenses are presently in use in a U.S.
Air Force installation.
4. Design and optical fabrication of a wide field,
high magnification, high resolution film viewer
using new techniques.
5. Design and fabrication of large aperture f/0.88
Schmidt system engineered as integral part of
television projection system.
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
STAT
III OPTICAL TEST AND CALIBRATION EQUIPMENT
?
1. Design and fabrication of diffraction limited
24 inch aperture 240 inch focal length off-axis
catadioptric collimator for use in testing
large aperture telescopes on the Atlantic
Missile Range.
2. Fabrication of high precision 38 inch aperture
380 inch focal length fused silica Cassegrainian
optical collimator system.
3. Design and fabrication of 5 inch aperture star
field simulator with 3 stars having 5 arc second
size and collimation accuracy, independent
magnitude variation, and continuous angular
adjustment.
4. Research and development in the field of optical
system analysis techniques, including aspects
relating to photographic film and the photo-
graphic process. '
5. Quantity production, by photography, of high
resolution test targets having spatial fre-
quencies up to 1000 k/mm.
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
STAT
IV COHERENT OPTICAL SYSTEMS
(prototype and production)
1. Design and fabrication of multi-element optical
systems involving the use of ultra-clean 1/8a
accuracy cylindrical surfaces.
2. Design and fabrication of diffraction limited
optical assemblies for 6328 ~ light including
highly aspheric f/1 components.
?
?
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
Q
Next 3 Page(s) In Document Denied
STAT
Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
- 1'+ -
?
?
The company occupies a custom designed and constructed 39,000 square
foot building on a four acre plot in the Bedford-Burlington research
area. Activities in the building include design, research, develop-
ment and manufacturing; testing areas include a thermally controlled
fifteen by seventy-five foot room containing, among other equipment,
our master collimator and lens bench mounted on a thirty foot long
vibration isolated concrete pad.
Design facilities include the Royal-McBee RPC-4000 computer, standard
computer accessories and Monroe desk calculators.
A partial inventory of major laboratory and fabrication equipment
includes:
1 Abrasive Surface Grinder
3 Bridgeport Millers
1 Cincinnati Miller
1 .Grob Band Saw
1 Kalamazoo Metal Saw
1 Porter Cable Belt Sander
1 Prentice Lathe
1 Sigma Precision Lathe
1 South Bend Precision Lathe
POLISHING
2 16" Strasbaugh Polishers (8 spindles)
4 18" Strasbauch Polishers (16 spindles)
2' 16" Polishers
7 12" Hand Finishing Spindles
1 40" Hand Finishing Spindle
1 Conic Section Generator
4 50" Triple Motion Polishers
(two with aspherizing units)
4 10" Elgin Polishers (16 spindles)
4 26" Loh Double Arm Polishers (two with aspherizing units)
4 6" Cylindrical Polishers
ROUGH GRINDING
16"
16"
34"
STAT
STAT
STAT
STAT
Diamond Glass Cutting Saw STAT
5 Sp
Edger and Grinder STAT
Sphere Generator
Sphere Generator
indle Edging Machine STAT
1 Fe er Diamon Saw
1 16" Lensmaster Rou h Hand Surfacer
1 30" Rough Hand Surfacer
1 24" Strasbaugh Generating and Edging Machine
Diameters are nominal: larger sizes can often be accommodated.
STAT
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
?
STAT
1 Spectra Physics Model 131 Gas Laser
1 Abbe Refractometer
1 Beckman DU Spectrophotometer
2 Cadillac Hei ht Ga es
1 10 foot, .001" Accuracy, Optical Bench STAT
1 Radius Measuring Bench
1 38" Aperture 1/20a Optical Flat
1 144" f/8 Parabolic Collimator
2 Aspheric Surface Measuring Devices
1 20" Aperture Test Sphere
1 arry ontro s Vi ration Isolation System on Air Mounts
30' x 4 1/2' x 2'
3 Precision Rotary Tables
1 Nikon Collimator 1 2 second
1 16" Aperture 22' Radius 1/20a Test Sphere STAT
1 Edcraft Clean Bench
3 Gaertner Laboratory Optical Benches
1 Graf Apsco Binocular Microscope
1 Hilger & Watts Angle Dekkor
1 Perkin-Elmer Infrared Recording Spectrophotometer
4 R.C.A. Air Gages - 10,000:1 amplification
1 Electronic Profile Measuring Gage
PHOTOGRAPHIC LABORATORY
Jo ce-Loebl, Mark IIIC, Microdensitometer
Automatic Processing Equipment
Emulsion Coating Equipment
B&L Micrographic Camera
Visible and UV Light Sources
Vacuum Operating Contact Printer
Modulation Transfer Test Equipment
Microdensitometer
Precision Reduction Camera 1:10
Precision Reduction Camera 1:10
Precision Reduction Camera 1:50
Precision Reduction Camera 1:100
?
Kuhlmann Model GM 1/1 Pantograph
Large Separated Dust Free Area with Special Ventilating and
Cleaning Facilities
1 Edwards High Vacuum Unit with 8" Diameter Bell Jar
1 Bendix Balzer Vacuum Unit with 20" Diameter Bell Jar
STAT
STAT
STAT
Declassified in Part - Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
The Role of Eikonal and 1Natrix I~lethods in
Contrast Transfer Calculations
W. Brouwer, E. L. O'Neill, and A. Walther
a reprint from Applied Optics
volume 2, number 12, December 1963
Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
.,u,s liqu?
Reprinted rom AP~ClI~ OPTICS, Vol. 2, page 1239, December 1963
Copyright 1963 by the Optical Society of America and reprinted by permission of the copyright owner
The Role of Eikon~al and Matrix Methods in
Contrast Transfer Calculations
,-
W. Brouwer, E. L. O'Neill, and A. Walther
The notion that the optical contrast transfer function is a useful tool for assessing the performance of
image-forming instruments has been accepted generally for some time and is now well established. This
paper discusses one method of making the transition from ray-trace data to the evaluation of this im-
portant function. First, the light distribution in the point image is rigorously derived in terms of an
integral over angular coordinates involving the eikonal function about a reference surface at infinity.
Then, the ray-trace procedure is. developed in the language of refraction and translation matrices culmin-
ating in matrix elements which are simply related to the eikonal coefficients of wave optics. Finally,
the numerical evaluation of the contrast transfer function in amplitude and phase from these eikonal
coefficients is presented, and the paper ends with an example showing the off-axis transfer function for
line structures oriented at various azimuths. All calculations are carried out to fifth order in the eikonal
coefficients, and emphasis is placed on the usefulness of this approach on relatively slow, low-capacity
computing machines.
1. Introduction
The application of Fourier techniques to the theory
? of image formation has been studied extensively in
the preceding decade.
The theory that was developed has been accepted
generally as a useful tool in the analysis of optical
systems. It centers on two concepts: point-spread
function and frequency-transfer function, one being
the Fourier transform of the other. Each of these
functions can, in principle, be determined when the
geometrical aberrations of the lens are known.l-a
In practical problems of lens design the ability to
evaluate the transfer function numerically would be a
great asset to the lens designer. The authors have
the impression that one step in the procedure of
computing this function is not well known to many
workers in the field : the conversion of ray-trace data
into wavefront deviations. Following Luneberg4 and
Wolfs we shall treat this problem in a way which is at
once simpler and more rigorous, by using eikonal
functions instead of wavefront shapes. This approach
has the additional advantage that it shows in a unique
way the transition from wave optics to geometrical
optics.
?
W. Brouwer and A. Walther are with Diffraction Limited,
Inc., Bedford, Massachusetts. E. L. O'Neill is with the Physics
Department, Boston University, Boston, Massachusetts,
Received 5 August 1963.
In the ray-tracing calculations a system of two by
two matrices as introduced by Smiths and Brouwer'
will be used to great advantage: the relation between
the matrix elements and the eikonal functions will be
shown to be very simple, and easier to apply than the
usual ray intercepts. This leads to a way of calculat-
ing the transfer function that is well suited to rela-
tively slow computers with a rather small memory
capacity.
In Sec. II we derive an expression for the point-
image amplitude distribution using the eikonal function
to describe the aberrations and perform a Fourier
transformation over the angular coordinates of a
reference surface at infinity. In Sec. III we relate the
matrix elements determined from geometrical lens
design calculations to the eikonal function. Finally,
in Sec. IV, starting with the coefficients in the eikonal
expansion, we show several examples of off-axis transfer
functions using the numerical integration scheme of
Hopkins.s
I1. Transition from Geometrical Wave Optics
A. Notation
We shall have occasion to use four planes associated
with a rotationally symmetrical lens [Fig. 1(a)~.
These planes are perpendicular to the axis of the lens
and are, respectively, the object plane (coordinates
x anal y), the entrance pupil plane (coordinates xl and
yl), the exit pupil plane (coordinates x~' and yt'), and
Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
x' x'
Y Y, Y~
(a)
pupil about which we shall say more later, and where
~ ,~gnstants and the finite area of integration have
been absorbed into F(vx,vv). Now by virtue of the fact
that we are treating the optical system as a filter of ?
spatial frequencies there exists a further Fourier
transformation between s(x',y') and D(vz,vv) in the
form
/ J s(x', y') exp [-2~ri(vzx' -f- vvy')l~'dy'
J J s(x', y') dx'dy'
Entrance Image
Pupil '
Fig. 1. (a) Coordinate systems. (b) Definition\of angle-angle
eikonal.
the image plane (coordinates x' arld y'). The four
axes marked with x lie in one plane, and so do the
axes marked with y. The x and y axes are mutually
perpendicular and intersect in the axis of the lens,
marked by z, zl, zl', or z' depending on which of the
four planes is used as a reference. The refractive
index in the object space is denoted by n; in the image
space it is n'. Since we consider only axially sym-
metric lenses and, unless indicated otherwise, an object
point will be understood to lie on the x axis, we are
allowed to refer to the x-z plane as the meridional
plane.
B. Summary of Fourier Optics
In order to establish our notation and for reference
purposes we shall give a brief summary of the basic
concepts in Fourier optics. In what follows, F(xi',yl')
describes the complex scalar disturbance over the exit
pupil plane, a(x',y') the complex amplitude distribu-
tion inthe image of a point source, s(x',y') the intensity-
spread function, and finally D(v2jvv) the frequency
response or transfer function for the system. By a
direct application of Huygen's principle together with
approximations that are quite valid for most optical
systems it is not difficult to show that a(x',y') and
F(xl',yl') are Fourier transform pairs in the form
a(x', y') = J J F(v=, vv) eXP [2~ri(vzx' -E- vvy')1 dvxdvv,
where vz and v? are reduced coordinates in the exit
1240 APPLIED OPTICS /Vol. 2, No. 12 /December 1963
where the denominator-has been introduced for nor-
malization purposes (ID(0)~ = 1). Capitalizing both
on the fact that s(x',y') _ ~a(x',y')I z and the con-
volution theorem for the transform of a product, we
end up with a direct relation between the disturbance
over the exit pupil and the frequency response in terms
of the well-known integral
+f~f
J J F(?=, ~~)F*(r~z - vs, l~v - vv)d?=da?
J J I F(~s, rev) ~ Zdksd?v
These relations together with their physical interpreta-
tions have been fully described in the literature, and the
uninitiated reader is invited to consult the references
for further details of the Fourier approach (e.g., refs.
2 and 3). As it stands, the phase portion of F(?x,
uv) describes a surface of constant phase about a ref-
erence sphere passing through the center of the exit
pupil whose center is an appropriate point in the
image plane. Unfortunately, we have not found this a
convenient reference surface in our attempts to bridge
the gap between wave optics and geometrical optics.
This transition is an important point. We wish to
emphasize that the shape of the wavefront can be
determined to any accuracy desired by means of the
laws of geometrical optics. Consequently, any ap-
proximation in the translation of these geometrical
data to data used in the work on diffraction involves
an unnecessary waste of available information.
Methods have been suggested to remove the wave-
front deformation function from the exponent of the
diffraction integral in an artificial manner so as to use
quasi-geometrical methods in passing from the exit
pupil to the image plane. These techniques (e.g.,
spot diagrams) must be treated with utmost care;
in this paper we shall not -take recourse to these
approximations.
?
?
Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
C. Wavefront Deviations
Our first aim shall be to establish a reasonably
exact derivation foi? Eq. (1), which will automatically
.lead to the proper definition of vx and vv. We shall,
as is usual in this field, restrict ourselves to the Huygens-
Fresnel diffraction theory. There are several un-
desirable features in the usual derivation of Eq. (1).
We mention:
(1) The shape of the Wavefront depends on where
one choses its location.
(2) Some authors measure the deviation of the
Wavefront along the rays. Others measure along the
radii of the reference sphere.
(3) 1'he relations between the coordinates that
define a i?ay geometrically and the coordinates suited
to diffraction calculations are very complicated.
(4) The diffraction integral takes only approxi-
matelythe form of a Fourier integral.
The approach taken to this problem in the pres-
ent paper, found to be due to Luneberg,4 and also
treated by Wolfs avoids these problems.
In Fig. 2, let P(x,y) be an object point of Which an
optical system creates a diffraction image in the
(not necessarily Gaussian) image plane (x',y'). Let
P'(xo' -~ ~',yo' + ~') be a point in which ~~ie wish to
evaluate the amplitude of the diffracted light; ~' and
~' are coordinates in the diffraction pattern, measured
in the image plane with respect to a reference point
? (xo',yo') which conventionally is chosen as the inter-
section point of the principal ray with the image
plane.
When the light has traversed the optical system,
the pencil of light is completely determined by a
Wavefront E. The line AQ represents a ray in the
image space.
The Wavefront being a surface of equal phase, the
amplitude in the point P' is proportional to
a(P') = J F(A) exp 1 2~ W IdQ,
x
in which T~ = AP' and F(A) is the amplitude distribu-
tion over the Wavefront E. A discussion- of this ampli-
tude distribution in the Wavefront is outside the scope
of this paper.
Let the direction of rays in the image space be given
in terms of their. optical direction cosines (L',M',N')
which are defined as the geometrical direction cosines
multiplied with the refractive index of the image space.
Let OS be the normal drawn from the origin in the
(x',y') plane onto the ray AQ. Then the optical path
length PAS considered as a function E(x,y,L',M') of
x,y,L', and 111' is known as the (point-angle) mixed
? eikonal of the system.9? io
When this function is known, the coordinates (x',y')
of the intersection point of a ray with the image
plane are given by:
c~E _ -x, aE - -y,
aL' ' c~M'
and the direction cosines of the rays in the object
space:
c~E _ -L ~E _ -M.
ax cry
Let RP' be the normal from P' drawn onto the
ray AQ. Then we can write for the path length
AP' in Eq. (4)
AP' = E(L', M') - PA + SR + (AP' - AR), (6)
in which the dependence of E on x and y is omitted
because we assume the object point to be fixed. The
path- PA is constant and may accordingly be dropped.
The location of the Wavefront is irrelevant, as long as
we do not choose it "too close to the image plane."
A great simplification is obtained if we make full use
of this freedom and move the Wavefront out to infinity.
In that case the term (AP' - AR) in (6) reduces to
zero and the line AP' becomes parallel to the ray AQ.
For the projection SR of the line OP' onto the ray
AQ we can write:
SR = L'(xo' -?- ~') ~- M'(ya' -I- n');
consequently, we have:
W = E(L,' M') -I- L'xo' -{- M'yo' -I- ~'L' -F- n'M'.
A point on the ~vavefront is now no longer specified by
linear coordinates; it must be specified as a direction.
So the diffraction integral (1) reduces to
a(I") = J f F(I.'M') exp 2~2 [Wo(L',D2') ~- ~'L'
E
Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
in which
~o(L,'M') = L' (L,'M') -I- L'xo -I- M'1/o, (8)
a function which is open to easy numerical evaluation,
as will be shown subsequently.
Comparing Eqs. (7) and (8) with Eq. (1) we observe
first of all that the rather vague concept of wavefront
deformation has been replaced by awell-defined eikonal
function. We also notice that (7) represents a Fourier
transformation, provided that we use as variables in
the frequency domain:
vl = L'/~,
vv = M'/~.
Every pair of values for vz and v? defines uniquely a
point in the exit pupil. The integration over the exit
pupil coordinates is now replaced by an integration over
direction cosines in the image space. These direction
cosines are a natural product of the ray trace, and, con-
sequently, one does not need the linear exit pupil
coordinates at all. Luneberg4 has shown that the
limiting procedure of moving the wavefront to in-
finity is an essential step in the electromagnetic theory
of image formation (see also Wolfb).
The amplitude function F(L',M') is still unknown
and must be determined by external means. It may
often be assumed to be constant. (See, however,
refs. 11 and 12, in which an object point is assumed that
radiates uniformly in all directions.)
Equations (7) and (8) form an ideal bridge between
geometrical optics and wave optics. Equation (7)
shows that the diffraction integral may be considered as
a Fourier transform, even for wide apertures and large
field angles. Neither an artificially tipped image
plane nor a troublesome reference sphere need be
introduced. Equation (7) shows a close relationship
between the eikonal functions and diffraction theory.
This relationship becomes even more apparent when we
apply the
method of
stationary phase
to Eq.
(7).
For large
aberrations,
the direction
(L',M')
con-
tributing most to the amplitude in the point (xo' -F'
i;',yo'~-~') is found by requiring that the exponent be
stationary with respect to L' and M'. This yields:
conventional ray-tracing procedure it is not feasible to
determine the relation between the heights of inter-
section in the image plane and the pupil coordinates in a
closed form. We must sample this function numeri-
cally, and, if we wish, then determine intermediate
values by interpolation. For the eikonal function we
have to apply the same technique, for the same reason.
This could be done by path-length computations along
the rays traced; however, to attain the desired
accuracy a double word-length computation has to be
used. With moderately small computers this becomes
time-consuming. One can, however, simplify these
computations considerably by applying the inter-
polation not to the eikonal function itself but to its
first derivatives with respect to the rotationally
invariant variables. In a power series development
of the eikonal function a term of a degree n in the
linear variables leads to terms of a degree n - 2 in
the power series of the above-defined first derivative,
whence the greater accuracy obtainable. Further-
more, these first derivatives are directly related to
matrix elements .(obtained from a ray trace), that we
shall define presently.
A ray-trace procedure consists in following a ray
through an optical system. There are fundamentally
two steps involved. At each refracting surface the
ray changes direction, and in going from one surface
to the next the intersection point of the ray with this
next surface has to be found. Let us first describe the
refraction of a ray at the ith surface of the system.
The coordinates of the point of intersection of the ray
with the surface in the space before refraction are
denoted by x,, y~ and by xt' and yi' when considered
in the space after refraction. The optical direction
cosines in these two spaces are denoted by Li, Mt and
Lt' and Mi'. It has been shown that the refraction
can be written in the form (see ref. 7)
CL,-~ - rl -A` II Lt 1 = R;rL:
J
n;' cos~o;' - n; cosy;
A; _
r;
n and n' are the refractive indices of the media in
front and behind the surface, ~l and ~o~' are the angles
of incidence and refraction, and rt is the subnormal of
the refracting surface for the ray considered. For a
spherical surface this is the radius itself.
Now, with a similar notation, we can write for the
translation to the next surface:
?
?
?
Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
Introducing (13) and (5) into Eq. (12) yields
CD aE + C ~- aEl x -}- I D aE + aE\L' = 0.
aul auz \ auz 21ua
Since these relations are valid for all values of
x and L' each coefficient should be zero; yielding:
?
Fig. 3. Rotation of coordinate axes for the computation of the
contrast transfer function.
CL,+~~ - rl O~rLti,~ - TrLti,
M;+~ L 0 M;'1 (M;'1
y:+~ ~ CT.' 1JLy:' J - TyLya' J
t~' is the distance measured along the ray between the
points xs', yi' and x~+~, y~+~. It should be noted that
the ray between object point and point of incidence
on the first surface can be described by a matrix of the
form T and will be denoted by To. In the same way in
the image space we have a matrix Tn giving the co-
ordinates of the ray in the image plane in terms of
the coordinates of the ray at the last surface k. When
all matrices R~ and T{ are computed the ray coordinates
in the image plane in terms of the coordinates- of the
ray in the object plane can be found by
L'1 rLl
x' J = TkRkT~IRk-i...TzRzTiR,To Cx J,
and similarly for M' and y' as a function of M and y.
Since both T and R are 2 X 2 matrices, the result of the
matrix multiplications will be a 2 X 2 matrix of the
form:
Cx''~ _ L D CA..ILx] and L/''~ - [ D CA1 L y1
J J
(12)
Note that the final "x" and "y" matrices are the same;
this is due to the rotational symmetry.
Since the optical systems considered here have an
axis of symmetry, we can achieve a simplification by
introducing the following rotationally invariant
quantities
uz = L'x ~- M'y,
aE _ A
au, B'
aE _ 1
auz B'
aE D
a~=B'
Numerical values for these partial derivatives of the
eikonal function are thus easily obtained from ray
traces. A simple integration will then give the func-
tion E.
There are many ways in which this calculation can
be performed. In the following a procedure is worked
out using truncated power series up to and including
the sixth order in the aperture variables to find E as a
function of u2 and u3, and thus as a function of the co-
ordinates L' and M` that we wish to use in the work on
diffraction.
Since we are interested only in the function E for
one object point, ul is a constant and E is a function
of uz and u3 only. Let us assume the form
E(uz, ua) = E:~;E;;uz`uai
Eoo + E,ouz + Eoiua
+ Ezouzz + Euuzua + Eozusz
+ Eao'uza + E2luzzua + Eizuzuaz. + Eoausa
From this it follows that
aE 1
auz = -B = E,o + 2Ezouz + Euua + 3Eaauzz + 2Eziuzu9
aE __ D
= Eoi
au3 B
All coefficients appearing in the desired function E
appear in the functions (I6) except Eoo. However,
Eoo is a constant and therefore not needed in the com-
putations on diffraction. The coefficients Etj in (16)
can now be found from the values of the matrix ele-
ments Band Dfor five or more rays. We use a least-
December 1963 /Vol. 2, No. 12 /APPLIED OPTICS 1243
Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
square method where the number of rays used is
determined by the desired fit.
It is interesting to note that in this approximation
only five meridional rays are needed. The object is
situated on the x axis, and so we have for meridional
rays:
ux = L'x,
ua = 1 sLz,
and thus
D
B = L''a -~- (Eux)L' -~- (Eoz -}- Ez,xz)L'z -I- (E,zx)L'3 + a/+EoaL'9.
Five meridional rays, using the first equation gives:
L''io, Ezo, (1/2L'u -~ 3E3o xz), Ez1, and E,z. Four of the
same rays, and the second equation gives: Eo,, El,
(Eoz -[- Ezl xz), Eoa? Simple inspection shows that all
coefficients can then be computed. When the object is
at infinity the same procedures can be used with the
help of the angle-angle eikonal as shown in the
appendix.
IV. Calculation of the Transfer Function
Having shown the relation between the information
supplied by the ray trace (the matrix elements) and the
coefficients Eta of the eikonal function we now proceed
to calculate the transfer function. In doing so it
proves convenient to define normalized variables over
which we carry out the numerical integration. Letting
L?L' and M?~' represent the maximum* direction,
cosines as seenfrom the image field point we now define
Ro = L'/L',~ and 'Yo = M'/M?,' .
Further, we define normalized polar coordinates
ao = p cosh; yo = p sink such that 0 < p < 1 and we
see that aside from scale factors which can be absorbed
into the coefficients, uz can be replaced by (3o and
ua by pz. We now can write the basic integral for the
evaluation of the transfer function in terms of these
normalized variables as:
/ J 1'(So, ?'o)r' *(~o - s, yo - t) daodYo
D(st)_ -m
J J I I''o(Ro, 'Yo) I z dRodYo
where s and t are line frequencies normalized such that
they run from 0 to 1, and where for our purposes we
take
* Actually Lm' and M?,' are symmetrized forma of the direction
cosines that limiting rays make with the z', y' axes as seen from
an off-axis field point.
r'(~o, Yo) = ex C2~riE ~ z x < 1
P ~ (~o, Yo , ~o -~- 'Yo
Finally, for ease in computation it turns out to be
convenient to perform a translation and rotation of
axes to ~3, y centered at the common area of the dis-
placed circles* such that (3 points along the direction
of the normal (?) of the line structure in the object
plane (see Fig. 3). With these changes of variables
our basic integral can be cast into the form:
J J FIQ+2,YIF*I~-2,YIdRdy
J J I F(R, 'v) ~ zdQdv
At this point we adopt the numerical integration scheme
of Hopkins and write first
D(s, ~) = aJ J exp [iksV(~, 'v; ~G)1 dRdv,
where a = f f d/3dy is the normalization constant
and V(a, y;,~) is given by:
V(R, ?; ~G) = sLE1 R + 2, Y, ~G~ - EI R - 2, y, ~G/J
aE i s Za3E ? 1 s 4a5E
_ ~/3 + 3~ 2) aka -I- 5i1 2 t ~~5.
1Vow as Hopkins$ has demonstrated (see also Marchand
and Phillips13) this integral can be approximated by a
'double summation taken over all the elementary cells
(ezj ev) that fall within the common area in the form
--//~~ 1 ainX sinY
D(S, 'Y) = N ~'mFin2sZ X Y , (21)
where N= a/(4ezev) is the number of rectangles of area
4eze~ that fall within the full area a and where
x = EzkSa~',
Y = eyksaa'
each of which is evaluated at the center of the ele-
mentary cells. Finally, to complete this description
we note that we can also write D(s, ,~) in the form
* In actual practice as one gets off-axis the region of integration
is the common area of two displaced ellipses. Further, the
effect of vignetting must betaken into consideration.
?
?
?
Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
 Declassified in Part -Sanitized Copy Approved for Release 2012/09/14 :CIA-RDP79B00873A001800010007-8
?
?
where
and where
B(S, 'G) =tan-'Dr/Da,
1 sinX sinY
1 sinX sin Y.
Dr(s, ~G) = NEmEn sinZ X Y
It is to be noted in this approach that X, Y, and Z
depend upon V((3,y; ,~) which in turn depends upon
E(/3, a; +~). We have found it convenient to express
E((3, ?; ?) in the form
k+l