ASPHERIC CALCULATION

• Document Type: CREST
• Collection: NGA Records (Formerly NIMA)
• Document Number (FOIA) /ESDN (CREST): CIA-RDP79B00873A001600030002-3
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 4
• Document Creation Date: December 28, 2016
• Document Release Date: August 16, 2012
• Sequence Number: 2
• Publication Date: January 8, 1965
• Content Type: MEMO

Declassified in Part - Sanitized Copy Approved for Release 2012/08/16: CIA-RDP79B00873AO01600030002-3
IA
Aspheric Calculation
c
it Technical Memorandum
w
8 3anua t
ry.
: STAT
Abstract: A technique for eliminating high order positive spherical aberration
is presented. The idea of positive high order spherical is defined, the aspheri.c
equation is discussed, and from the differential rates of changes of aspherics
coefficients versus actual ray deviation a matrix is formed by predetermined
ray intercepts.
Reducing image curvature very often introduces high order positive spherical
aberration. In Figure 1 a doublet is shown where the front positive el.emeni: has
undercorrected spherical. If the negative rear element were not present, the
marginal rays of the first element would focus in front of the paraxial focal
plane. As it i.s,?these rays strike the negative lens with higher angles of
incidence than they would otherwise if undercorrected spherical were not present.
The deep curvature of the negative lens in this case reduces image curvati:re,
which is desirable, but it overcorrects the marginal rays, which is undesirable.
-- paraxial focal. plane
marginal ray focus front lens
Figure 1
I .
Either the positive or negative lens can be aspherized to recorrect these marginal
rays. The assumption is the shapes and glass indexes of-the elements cannot be
varied to reduce the overcorrected spherical. "Rolling off" a surface on the
positive lens or negative lens will introduce power into the marginal rays and
bring them to focus at the paraxial focal plane, or any other point in front (jr
behind. It can be seen that the spherical aberration must be overcorrected if
the aspherizing is to be easy. Correcting undercorrected marginal spherical
means adding neative power to the outer rim which is a much more difficult
operation.
The desired shape of the rolled off sphere can be arrived at by adjusting the
coefficients of the aspheric equation. This equation is an even powered polynomial-`'
Written in a form having a spherical. expression and deformation terms. This form
facilitates starting with a spherical. surface equation and adding power curves to
it which leave the basic sphere unaltered while changing it above a determined ordinate.
Declassified in Part - Sanitized Copy Approved for Release 2012/08/16: CIA-RDP79B00873AO01600030002-3
 Declassified in Part - Sanitized Copy Approved for Release 2012/08/16: CIA-RDP79B00873AO01600030002-3
cy
x 1+ Y 1 - c y
Equation (1) is the expression for a sphere where the center of curvature is located
a distance equal to the radius. It is also made single-valued so that any value of
the ordinate (y), only one value for x is obtained which is the one closest to the
(1)
without upsetting the computation.
origin. Furthermore, the radius, entered
Figure 2
Figure 2 illustrates the geometry. Now if power curves of the type
n
(2) y f. kx
or to be consistent, written as x a function of y
(3) xi - I/ki yn
are added to the expression of a sphere, equation (1), the spherical curve in
Figure 2 would be modified according to be magnitude of k and n. That is, if k
and n are large, equation (2) has the curve flat through small ordinates and
whipping away sharply as y grows large, as shown in Figure 3.
~... .tom
?4 r
Figure 3
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 Declassified in Part - Sanitized Copy Approved for Release 2012/08/16: CIA-RDP79B00873A001600030002-3
By adjusting k and choosing an appropriate n, a curve can be arrived at which
when added to a circle curve alters the circle significantly above a certain
height. For any n the value of x at y w 1 is always 1/k. For a curve with
small x, or sag, contribution, k would have to be large. The magnitude of n
determines how rapidly sag increases above y 81. It is easy to see that
raising y to a large n when y < I produces small. sag and large sag when y 1.
To effect a "roll off" on a spherical surface a deformation curve must have
negligible sag contribution below a certain ordinate and a predetermined amount
above it. Figure 4 illustrates adding such a curve to a sphere. The deformation
itself can be the sum of several power curves.
-.-` Resulting Aspheric
Deformation curve .._.__......_.,. '_.... __...~. ,.-?,~ d,.
Sphere-.. '..
Ordinate A1\ Zriirh A-l- m_I-_ r? _
Figure 4
The method for calculating the deformation curve is a practical. approach to a
solution, easy to understand, a,nci accu?a-e. A surface is chosen for spherizing
and it is known from the ray trace'I,p,fotrs exactly at what height deformation is
to begin. Figure 5 is a plot of on-~axps ,spherical aberration where lateral
ray intercept height on an image plane is plotted against fraction of entrance
pupil height.
Figure 5
Fraction Entrance Pupil
Declassified in Part - Sanitized Copy Approved for Release 2012/08/16: CIA-RDP79B00873A001600030002-3
 Declassified in Part - Sanitized Copy Approved for Release 2012/08/16: CIA-RDP79B00873AO01600030002-3
The equation for a general surface of revolution is in.the form of equation (4).
(ii)
or
jr- 1 c 2y
The three ray intercepts 1, 2, 3, in Figure 5 are to be brought into the dotted
portion of the curve. A small change is made in coefficient b and the three
rays 1, 2, 3 traced. The assumption is that for small changes in the coefficients
b, c, d, there is a proportional change in the ray intercepts. In other words,
the region of Linearity is assumed. Analytically,
b 4!.hl
k1 Gh1
(S) likewise
b - k2 X112
b - k3 . esh3
Similarly, differential changes are recorded for c and d
(6)
;ac
,(1 ,Uh1
ac
2 1.h2
C~c
;(3 /Sh3
(7)
d
m1. h1
Atd
m2 112
.: .d m3 Ah3
A change in
Putting the
(8)
6111T is
equations into form, a 3 x 3 matrix is obtained.
1
/+ b l + Qc a d 1;, ! hI.T
+ ay4- i by6 + cy8 + dy10
1r l
(9)
(10)
t~h3T
for ray 3.
&b 1+ Qc 1+ Ad 1 - 6h2T
k2 If 2 m2
Lib 1_ 6c + 4,d 1 . 6bh3
If the changes needed are large the calculations go outside the region of
linearity and the exact changes 0 hjT, Gah2T, and Ah3T are not obtained.
It is then necessary to repeat the process, solving f(-.?r A b, Ac, and & d,
each time until solution is reached.
desired for ray 1, 4h2T for ray 2, and
Declassified in Part - Sanitized Copy Approved for Release 2012/08/16: CIA-RDP79B00873AO01600030002-3