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English Pages 100 [59] Year 1953
Boston University OpenBU
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Theses & Dissertations
Dissertations and Theses (pre1964)
1953
An application of Berek's method to triplet design. https://hdl.handle.net/2144/11303 Boston University
BOSTON UNIVERSITY GRADUATE SCHOOL
Thesis
AN APPLICATION OF BEREK'S METHOD TO TRIPLET DESIGN
by
LARKIN L. JACKSON (B.S. , University of Alabama, 1940; M.S., Stanford University, 1947.) Submitted in partial fulfillment of the requirem.ents for the degree of Master of Arts 1953
I.
INTRODUCTION  ...   
II.
NOTATIONS, DESIGN ASSUMPTIONS AND REQUIREMENTS  ... A. Notations·· ·  .. B. Design Assumptions and Requirements 
III.
IV.
DETAILED PROCEDURES A. Discussion of design variables and basic equations for their solution B. A detailed design outline for a third order solution
Page i
1 1 2
4 4
8
DISCUSSION OF DESIGN OUTLINE WITH SUMMARIZED DATA  ..  .. 
15
V.
CONCLUSIONS
42
VI.
BIBLIOGRAPHY 
vi
VII.
ABSTRACT    vii
Approved
by First Reader.
!(_Jf?,j~ ... .~·. ::~ ~~~~
Research
Prof~~f
P~ysic.s
7......... :?4..~jfl7f3
Second Reader • Assistant Professor of Physics
LIST OF ILLUSTRATIONS FIGURE 1:
Page Pz as a function of .f__and"i. p    17
FIGURE 2:
P:z. as a function of h2        19 hi
FIGURE 3:
Solution of the basic equations    21
FIGURE 4:
Summarized Best Solutions for each combination of ~ and ;£p     22
FIGURE 5:
Glass selection table    24
FIGURE 6:
Spherical Aberration (A) as a function of the first surface radius of the second element   34
FIGURE 7:
Seidel Data Calculation Sheet  36
FIGURE 8:
Axial Color Aberration Curve First Trial _______ 38
FIGURE 9:
Axial Color Aberration Curve Fourth Trial ______ 39
i
INTRODUCTION
The purpose of this project was a third order investigation, by Berek's method, of flat field photographic tripletswith an intended coverage of about 25
° halfangular
field.
This design procedure is described in ''Grundlagen
der Praktischen Optik 11 by Dr. M. Berek. The Berek procedure is essentially based on the Seidel Analysis derived 1 in 1856. The scope of this project was limited to the one purpose of investigating Berek's design procedure applied to the design of a flat field photographic triplet.
It is hoped that the procedure set forth in this thesis will serve
two purposes:
1.
A basis for comparison with other methods of 3rd order triplet design.
2.
A useful design reference for those who wish to make use of this method of design.
Wide angle triplets have, in general, suffered from an excessive curvature of field in their third order approximations.
And for that
reason, the particular design problem chosen for this investigation was that of producing a somewhat flatter field than is usually found in photographic triplets. A convenient measure of field curvature is the ratio of the curvature of the 3rd order image surface to the focal length.
1.
The usual ratio is between
The Encyclopaedia Britannica; p. 58, vol 1, 11th Ed.
ii
2 and 3, but there appears to be a possibility of a higher ratio  that is, The effect of low ratios is that when astigmatism and
a flatter field.
curvature are corrected by higher order aberrations at a field angle near the margin of the field, the imagery at the lower angles is afflicted with significant astigmatism, 'which deteriorates image quality in the intermediate field.
The tangential and sagittal image curves usually have .
+
s: '7
A,
+
2 7. 49
+ 4. + 7
r;
A ·z'..
= +95. sr;
A3 ;::: ~
~z.
+ 2.1. 
z 6 .7 g
~£
2.
+ I§. 7 6
lj b.
For the whole lens:
A= A1
+~JAz. r(I..~ 1JA 3
Berek, p 12.2..
the derived equation for the whole lens:
A== 43.~3 + 27,49
4.47 r 4. Cfg ri.Cfl
II 'Z.
tl
r1
+ 6,14 ~3
Yi'2.
Coma a.
For the individual elemen t:
B~ :.(~~
\ 0.1.. T 2 '1'1.,{ Yt~+l/ 0·)~ _L (Yt.: t I Jp· u"" uA. rt Y\.t I
YlA. \) r.L
1..
the derived equations for the individual elements:
B, :::
I 0' +9
r:l I.J"l :=.
B3;:: b.
B~
 3. 2. 3
b .Cf Cf £.
Yl b ,I 3
1
gq 
r'Z
3. 3s
For the whole lens:
B,
t::?
Berek, Page 122
,~~Bt ~~[33 + d, (HA2+~)[d.~~d._]A3 the derived equation for the whole lens:
B~
17.
z '8 
"
Astigmatism a.
'3. 2. 3
For the whole lens:
c =.zr + z.J 1 h ;
.
+ :4Q±
VI\
o. Cf 2 ,_3 r,
T
y'1..
t0.43 0.87 3
J
Berek, Page 122
rei h~
r.: z_.
r~"L
/'vt )'"~ h~h +dz
B? + :z (~~) ~~ .. __::.. B3 i d. (:. Az. +~~ d
k )1..
~ 1
I
L n 1
the derived equation for the whole lens:
C :: .... ),44+ ~ _ y'l..
D.;? t 0· 010
t3
t;. L
I
o. 43
r;z. 3
2
A3
32
17.
Setting up equations and substitution for B and C: An attempt was made to solve the bending equations simultaneously,
but this was ·quickly abandoned when it became evident that terms up to tenth power would be involved  to say nothing of square root terms liberally sprinkled throughout.
Then a synthetic division method was
tried with considerably more success. Numerical value;s given under design requirements were substitued for B and C; but no substitution was made for A.
The reason will appear
under step 18. 18, 19, 20 and 21.
Solution of the equations:
Since the expression for C ha.s only two unknown5in it, we may immediately obtain numerical values from it by assuming a numerical value for one of the radii and solving for the other. Since the central element has almost twice as much power as the other two elements, it was considered probably that it would have the steepest curves of any of the elements: and therefore it was deemed best to solve the equation for r and s ub stitute reasonable values of r 2 •
3
These values were varied on both
sides of the equiconcave radii, determined from solution of the paraxial lens makers equation: P
= n1 IV~
An inspection of the curve reveals that the focal point is /
changing rapidly between the wavelengths fpr F and G light. not be too good for color photography.
This would
However, it would be satisfactory
for black and white photography with a very light yellow filter  ideally, a filter with a cut off around the F wavelength.
This would prevent the out of
FIGURE 7 SEIDEL DATA CALCULATION
36
SHEET
Form .·
£FL 1.0
•
z
I
R
6
t .333 0
t3 .432 .145
 .390 0
I 1.617
1.617 I
I I. 72
I . 72 I.
I 1.745
I 745 I
t .813 . 501
!_. 346 1.467
tl.467
t 813 3. 224 0
 . 013 1.990
2 .381 2.887
5.160 tl.l72
.3 . 2 2 5
2.003
5.268
:t3.988
1/n'
I o618
. 618 I
1.000 .581
0 761
. 761 I. 995
2.887 t .396
•·
1/n'u
.761
I. 234
2.490
u
vn/R n/u
.
n/R n/u
Q
1/n
1/nu
1/nU l/t1't.r• L\
I
ul{ /uJi.1
( U~)~=h
.
1/ h 2 Q
E:
I
.
~ ~
T
h'l Q2
.
. .
ftmc 1/rV /R

0
(tlh) / hJ._ h"
ht'Q2..,
.5 81 I
I . 573
 ~ . ~88
.3~~
 . 864 .212
~1.384
.651
~ .'77 .
3. 833
t 1.155
. 573 I
...
.. 212 1.266
I
.: I~ 478
ITJ.~·LL
..
I ·· . ~  . 790

··
.69:!_~ f' ~ " 47f 2.515
1.~ .i'If·: 2·~~} 
.499 
.310
. 291 4.480  . 864 .647
....
:~397 
T.525
.

  f 
±
~
0
___! __ ' . 691 _ . 4 u____ .546 ' . 62 1.__  ~  . I !   ..
_o_
L
_ ____
o_ _ j __ .~
· ;:;r   
~224
0
.224
.1 73
ttl0.400
4.011
6326· 3.626 
1i. 915 ~ ~~ :4s __e
8
2.454
1.231
t2. 471 I . 234  ...003
.
o
t

.490 . . 490
t:i49 ~tl:877 :t.o1 2
. 499
A
p
.2?.L
t .3 10
c . 761

520 . 5.724
'5.75Kif5.ol_a ; 
i
I '

D
 .618
:t . 618


!
·.124
.044
o
1.096
...
I
·= ~_43
I
 [  ~ + r
'    ·   : . ...   ·   '
I
:==~~~~~~~~L I
I
~   r  
I
i

B.=  0 '. 200 C=tO . 060 Pr= 4.964 1. 201 P=
 .255 1 3 ._o53_._:___2:4:7 3L:_:_Q.am
l
  .. 
··  
.3'39_rsA6l__f~= 0 .889 ____
t . 474 t2 . 818  1 ,1.193 I 256

i
 2 . 732 1U16L t ._6~6.j ~ . £0_9_ .
t .997
I
I
T(C+P) •
  ~

  = ·: · . · I. 387 . 418
I
~
  ..,    :    I o· . o~ _: . 15~L t o ··· j ._145 : .9__ T
t/n hk Xhk
E+S
I I 2.003
1.157 2.698 2.698 I+ .790
LOI?
 ' ~~ L
I
I I 3. 225
h2 h2Q
rB
5
 . 420 . 155
n•
t'A
4
126 . 15 0
n
~
3
:310 0
t

RF
SURFACE
TERM
=fI
I
NOTE: THIS SHEET IS FOR THE FINAL THIN LENS SOLUTION AFTER AXIAL COLOR HAS BEEN CORRECTED
! ;
I
I
I
37
of focus blue light from fuzzing up the image. 26.
Correction (swinging) of the axial color curve:
If, as for the color photography case mentioned above, the basic axial color curve is not satisfactory, we can attempt a correction by changing the glass in one or more elements, keeping the element powers constant and keeping the ratio between first and second radii constant. Since these characteristics of the lens remain constant, the thirdorder aberrations remain practically unaffected.
That is, spherical aberration,
coma, and astigmatism and distortion,to a high degree of approximation; depend only on the ratio between radii, powers and spacingnone of which change.
Unless the index nD of the substituted glass happens to be the same
as the one it replaces, the Petzval sum will change slightly, but this is usually negligible. Four color curves were obtainedthree besides the basic combination.
Curves for the basic combination of glasses and for the combination
used in the fourth trial are shown in Figures 8 and 9.
A different glass was
tried in each element, but the change was too great when element 2 was changed and not enough when element 3 was changed.
The effect of a glass
change on axial color seems to be related to the power of the element and .the change in dispersion of the glass.
The results of the determinations
are shown here in tabular form: Element changed
Old Glass
New Glass
no change 1
DBC1
DBC ... 2
Element Power
ilvVnew ... )fold
1. 99
0
1. 99
+3.9
Axial Color (~ v6) CG' inclusive 0.0038 0. 002l(ChoseiJ
38
FIGURE 8 AXIAL COLOR ABERRATION CURVE FIRST TRIAL
::I
E
0
~ f()
ow

lO
w ::I
E
0 0
w
w
I'! 000 lO
•
r< ::I
E
0 0
lO
LLW