Zoom lenses and computer algebra

A. Walther

doi:10.1364/JOSAA.16.000198

Journal of the Optical Society of America A
Vol. 16,
Issue 1,
pp. 198-204
(1999)
•https://doi.org/10.1364/JOSAA.16.000198

Zoom lenses and computer algebra

A. Walther

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A. Walther, "Zoom lenses and computer algebra," J. Opt. Soc. Am. A 16, 198-204 (1999)

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Abstract

I show how computer algebra can be of material help in determining the potential quality of various zoom lens arrangements. One fortuitous result of this work is a proof that, when the object is kept at infinity, it is possible in principle to design zoom lenses for which all third-, fifth-, and seventh-order aberrations are corrected over a continuous range of zoom settings limited only by the restriction that the lens groups may not run into each other.

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Equations (41)

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i	Type	$P_{i}$	$Q_{i}$
1	$a^{2}$	$P_{1}$	$Q_{1}$
2	ab	1	-1/4α
3	ac	$4 Q_{4} α^{2}$	$- 4 Q_{4}$
4	$b^{2}$	$- 4 Q_{4} α^{2}$	$Q_{4}$
5	bc	1/4	0
6	$c^{2}$	$(1 - α - 2 Q_{1} α) / 2 α$	$Q_{6}$

i	Type	Group 1	Group 2	Group 3a
1	$a^{2}$	$P_{1}$	$Q_{1}$	$P_{6}$
2	ab	$- α f$	$f / 4$	$P_{5}$
3	ac	$2 α^{2} Q_{4}$	$- 4 Q_{4}$	$P_{3}$
4	$b^{2}$	$- α^{2} Q_{4} / 2$	$Q_{4}$	$P_{4}$
5	bc	$- α f / 4$	$Q_{2}$ a	$P_{2}$
6	$c^{2}$	$(α - 1) f / 2 - Q_{1}$	$Q_{1}$ a	$P_{1}$
7	$a^{3}$	$P_{7}$	$Q_{7}$	$P_{16}$
8	$a^{2} b$	$- 3 α f / 2$	$7 f / 32$	$P_{15}$
9	$a^{2} c$	$4 α^{2} Q_{4}$	$- 4 Q_{10}$	$P_{12}$
10	${ab}^{2}$	$- α^{2} Q_{4}$	$Q_{10}$	$P_{14}$
11	abc	$- α (f^{2} - 8 α^{2} Q_{4}^{2}) / 4 f$	$(- 128 Q_{4}^{2} + f^{2}) / 16 f$	$P_{11}$
12	${ac}^{2}$	$α^{2} (4 Q_{10} - Q_{4})$	$Q_{9}$ a	$P_{9}$
13	$b^{3}$	$- α^{3} Q_{4}^{2} / 2 f$	$2 Q_{4}^{2} / f$	$P_{13}$
14	$b^{2} c$	$α^{2} (- Q_{10} + Q_{4} / 4)$	$Q_{10}$ a	$P_{10}$
15	${bc}^{2}$	$- 7 α f / 32$	$Q_{8}$ a	$P_{8}$
16	$c^{3}$	$(α - 1) f / 2 - Q_{7}$	$Q_{7}$ a	$P_{7}$

i	Type	Group 1	Group 2	Group 3
1	$a^{2}$	20	-20	95
2	ab	-200	25/2	-50
3	ac	640	-80	640
4	$b^{2}$	-160	20	-160
5	bc	-50	25/2	-200
6	$c^{2}$	95	-20	20
7	$a^{3}$	-20	20	55
8	$a^{2} b$	-300	175/16	-175/4
9	$a^{2} c$	1280	-80	960
10	${ab}^{2}$	-320	20	-240
11	abc	974	-487/8	974
12	${ac}^{2}$	960	-80	1280
13	$b^{3}$	-256	16	-256
14	$b^{2} c$	-240	20	-320
15	${bc}^{2}$	-175/4	175/16	-300
16	$c^{3}$	55	20	-20

$x_{pup}$	$x_{im, 1}$	$x_{im, 2}$
-1.0	-0.110645	-0.011453
-0.8	-0.081387	-0.008401
-0.6	-0.056109	-0.005776
-0.4	-0.034383	-0.003530
-0.2	-0.015807	-0.001619
0.0	0.000000	0.000000
0.2	0.013421	0.001368
0.4	0.024863	0.002528
0.6	0.034798	0.003533
0.8	0.043814	0.004445
1.0	0.052683	0.005347

i	Type	$P_{i}$	$Q_{i}$
1	$a^{2}$	$P_{1}$	$Q_{1}$
2	ab	1	-1/4α
3	ac	$4 Q_{4} α^{2}$	$- 4 Q_{4}$
4	$b^{2}$	$- 4 Q_{4} α^{2}$	$Q_{4}$
5	bc	1/4	0
6	$c^{2}$	$(1 - α - 2 Q_{1} α) / 2 α$	$Q_{6}$

Abstract

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Tables (4)

Equations (41)

Journal of the Optical Society of America A