Abstract

Modules are conceived as building blocks in formulating optical designs with certain useful properties. This paper provides necessary and sufficient conditions for successfully combining (coupling) modules to form such a design system. Canonical ray tracing is defined, and canonical versions of aberration coefficients are derived. These are used to obtain equations for zero values of third-order-monochromatic and primary-chromatic-aberration coefficients. Their utilization in practical problems in optical design is illustrated.

© 1975 Optical Society of America

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References

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  1. O. N. Stavroudis, J. Opt. Soc. Am. 57, 741 (1967).
    [Crossref]
  2. O. N. Stavroudis, J. Opt. Soc. Am. 59, 288 (1969).
    [Crossref]
  3. F. M. Powell, thesis (University of Arizona, 1970).
  4. R. I. Mercado, thesis (University of Arizona, 1971).
  5. R. I. Mercado, J. Opt. Soc. Am. 62, 495 (1972).
    [Crossref] [PubMed]
  6. R. I. Mercado, dissertation (University of Arizona, 1973).
  7. J. Strong, Concepts of Classical Optics (Freeman, San Francisco and London, 1958).
  8. D. P. Feder, J. Opt. Soc. Am. 41, 630 (1951).
    [Crossref]

1972 (1)

1969 (1)

1967 (1)

1951 (1)

Feder, D. P.

Mercado, R. I.

R. I. Mercado, J. Opt. Soc. Am. 62, 495 (1972).
[Crossref] [PubMed]

R. I. Mercado, thesis (University of Arizona, 1971).

R. I. Mercado, dissertation (University of Arizona, 1973).

Powell, F. M.

F. M. Powell, thesis (University of Arizona, 1970).

Stavroudis, O. N.

Strong, J.

J. Strong, Concepts of Classical Optics (Freeman, San Francisco and London, 1958).

J. Opt. Soc. Am. (4)

Other (4)

R. I. Mercado, dissertation (University of Arizona, 1973).

J. Strong, Concepts of Classical Optics (Freeman, San Francisco and London, 1958).

F. M. Powell, thesis (University of Arizona, 1970).

R. I. Mercado, thesis (University of Arizona, 1971).

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Figures (3)

FIG. 1
FIG. 1

The module and its optical parameters.

FIG. 2
FIG. 2

The hard-way couple. The main foci and the first pupils coincide. Module a is in the backward orientation, module b in the forward orientation. The resulting system is afocal.

FIG. 3
FIG. 3

The easy-way couple. The two second pupils coincide.

Tables (4)

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TABLE I Critical values and category domains for three modules, (Arabic numerals replace Roman numerals of the text).

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TABLE II Canonical properties of three modules.

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TABLE III Parameters for three generated three-module designs. General data: ψ1 = −0.109159, ψ2 = −0.161674, ψ3 = 0.101765. Focal length = 30, f/8, semifield angle = 20°, indices of refraction as indicated.

Tables Icon

TABLE IV Paraxial-ray-trace data and third-order aberration coefficients for the three generated three-module designs.

Equations (62)

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t 0 = f T 0 ( N 0 , N 1 , N 2 ; ψ ) , c 1 = C 1 ( N 0 , N 1 , N 2 ; ψ ) / f , t 1 = f T 1 ( N 0 , N 1 , N 2 ; ψ ) , c 2 = C 2 ( N 0 , N 1 , N 2 ; ψ ) / f , t ¯ = f E ( N 0 , N 1 , N 2 ; ψ ) , t ¯ = f A ( N 0 , N 1 , N 2 ; ψ ) ,
C 1 ( N 0 , N 1 , N 2 ; ψ ) = ( N 0 q ) / ( N 1 N 0 ) T 0 , T 1 ( N 0 , N 1 , N 2 ; ψ ) = N 1 ( N 0 T 0 ) / q , C 2 ( N 0 , N 1 , N 2 ; ψ ) = q / ( N 2 N 1 ) N 0 , E ( N 0 , N 1 , N 2 ; ψ ) = 2 N 2 T 0 q ( N 1 q ) / [ N 2 ( N 0 q ) ( 2 q N 1 ) + N 1 T 0 ( N 1 q ) ] , A ( N 0 , N 1 , N 2 ; ψ ) = N 0 [ T 0 ( N 1 2 N 2 ) ( N 1 q ) + N 2 N 1 ( N 0 q ) ] / 2 T 0 q ( N 1 q ) ,
T 0 = T 0 ( N 0 , N 1 , N 2 ; ψ ) = 27 N 2 2 N 1 2 ψ 3 / ( N 2 N 1 ) 2 ( N 1 N 0 ) ( 1 ψ 3 ) 2 , q = q ( N 0 , N 1 ; ψ ) = 3 N 1 2 ψ / [ ( N 1 N 0 ) ( ψ 2 + 1 ) ( 2 N 1 + N 0 ) ψ ] .
T 0 = T 0 ( N 0 , N 1 , N 2 ; ϕ ) = 27 N 2 2 N 1 2 / 2 ( N 2 N 1 ) 2 ( N 1 N 0 ) ( 1 + cos 3 ϕ ) , q = q ( N 0 , N 1 ; ϕ ) = 3 N 1 2 / [ ( 2 N 1 + N 0 ) + 2 ( N 1 N 0 ) cos ϕ ] .
T 0 ( N 2 N 1 ) 2 ( N 1 2 N 0 q ) ( N 1 q ) 2 + N 2 2 q 3 ( N 1 N 0 ) 2 = 0
y j + 1 = y j t j u j , N j + 1 u j + 1 = ( N j + 1 N j ) c j + 1 y j + 1 + N j u j , i j + 1 = c j + 1 y j + 1 u j ,
y j + 1 = y j f T j u j , N j + 1 u j + 1 = ( N j + 1 N j ) C j + 1 y j + 1 / f + N j u j , i j + 1 = C j + 1 y j + 1 / f u j ,
Y j + 1 = Y j T j U j , N j + 1 U j + 1 = ( N j + 1 N j ) C j + 1 Y j + 1 + N j U j , I j + 1 = C j + 1 Y j + 1 U j ,
Y 1 ( N 0 , N 1 , N 2 ; ψ ) = T 0 ( N 0 , N 1 , N 2 ; ψ ) / N 0 , U 1 ( N 0 , N 1 ; ψ ) = q ( N 0 , N 1 ; ψ ) / N 1 N 0 , I 1 ( N 0 , N 1 ; ψ ) = ( N 1 q ( N 0 , N 1 ; ψ ) ) / ( N 1 N 0 ) N 0 , Y 2 ( N 0 , N 1 , N 2 ; ψ ) = 1 , U 2 ( N 0 , N 1 , N 2 ; ψ ) = 0 , I 2 ( N 0 , N 1 , N 2 ; ψ ) = N 2 q ( N 0 , N 1 ; ψ ) / ( N 2 N 1 ) N 1 N 0 .
y j = Y j u 0 , u j = U j u 0 / f , i j = I j u 0 / f .
y ¯ j = Y ¯ j f u ¯ 0 , u ¯ j = U ¯ j u ¯ 0 , i ¯ j = I ¯ j u 0 .
L = N j ( u ¯ j y j u j y ¯ j ) = u 0 u ¯ 0 L ,
L = N j ( U ¯ j Y j U j Y ¯ j ) ,
p j = ( N j N j 1 ) c j / N j N j 1 = P j / f ,
P j = ( N j N j 1 ) C j / N j N j 1 ,
P ˆ = P 1 + P 2 = [ N 2 ( N 0 q ) + T 0 q ] / N 2 N 1 N 0 T 0 .
p = P ˆ / f .
s j = N j 1 ( N j N j 1 ) y j ( u j i j ) / 2 N j L = S j ( u 0 / f u ¯ 0 ) ,
S j = N j 1 ( N j N j 1 ) Y j ( U j I j ) / 2 N j .
S 1 = T 0 ( N 1 2 N 0 q ) / 2 N 1 2 N 0 , S 2 = q / 2 N 0 .
s ¯ j = N j 1 ( N j N j 1 ) y ¯ j ( u ¯ j i ¯ j ) / 2 N j L = S ¯ j ( f u ¯ 0 / u 0 ) ,
S ¯ 1 = N 0 ( E T 0 ) [ T 0 ( N 1 2 N 0 q ) N 0 E ( N 0 q ) ] / 2 N 1 2 T 0 E 2 , S ¯ 2 = A [ N 0 ( N 2 2 N 1 2 ) N 2 q A ] / 2 N 2 3 N 0 .
B j = s j i j 2 = B ˆ j ( u 0 3 / f 3 u ¯ 0 ) , B ˆ j = S j I j 2
F j = s j i j i ¯ j = F ˆ j ( u 0 2 / f 2 ) , F ˆ j = S j I j I ¯ j
C j = s j i ¯ j 2 = C ˆ j ( u 0 u ¯ 0 / f ) , C ˆ j = S j I ¯ j 2 .
F ˆ = F ˆ 1 + F ˆ 2 = N 2 q 2 [ N 2 ( N 0 q ) T 0 ( N 1 q ) ] 2 T 0 N 1 N 0 2 ( N 2 N 1 ) 2 ( N 1 q ) , F = F ˆ ( u 0 2 / f 2 ) .
D j = C j + 1 2 L p j = ( C ˆ j + 1 2 P j ) ( u 0 u ¯ 0 / f ) ,
D = 1 2 P ˆ ( u 0 u ¯ 0 / f ) ,
D ˆ = 1 2 P , D = D ˆ ( u 0 u ¯ 0 / f ) .
E j = s ¯ j i j i ¯ j + 1 2 ( u ¯ j 2 u ¯ j 1 2 ) = E ˆ j u ¯ 0 2 ,
E ˆ j = S ¯ j I j I ¯ j + 1 2 ( U ¯ j 2 U ¯ j 1 2 ) .
E = E ˆ u ¯ 0 2 ,
E ˆ = S ¯ 1 I 1 I ¯ 1 + S ¯ 2 I 2 I ¯ 2 + 1 2 ( 1 / N 2 2 1 / E 2 ) .
axial color a j = y j i j N j 1 ( R j 1 R j ) = a ˆ j ( u 0 2 / f ) , lateral color b j = y j i ¯ j N j 1 ( R j 1 R j ) = b ˆ j ( u 0 u ¯ 0 ) ,
a ˆ j = Y j I j N j 1 ( R j 1 R j ) , b ˆ j = Y j I ¯ j N j 1 ( R j 1 R j ) .
a ˆ = a ˆ 1 + a ˆ 2 , b ˆ = b ˆ 1 + b ˆ 2 .
a = a ˆ ( u 0 2 / f ) , b = b ˆ ( u 0 u ¯ 0 ) .
t 0 = f a A ( N 2 , N 1 , N 0 ; ψ a ) , c 1 = C 2 ( N 2 , N 1 , N 0 ; ψ a ) / f a , t 1 = f a T 1 ( N 2 , N 1 , N 0 ; ψ a ) , c 2 = C 1 ( N 2 , N 1 , N 0 ; ψ a ) / f a , t 2 = f a T 0 ( N 2 , N 1 , N 0 ; ψ a ) + f b T 0 ( N 2 , N 3 , N 4 ; ψ b ) , c 3 = C 1 ( N 2 , N 3 , N 4 ; ψ b ) / f b , t 3 = f b T 1 ( N 2 , N 3 , N 4 ; ψ b ) , c 4 = C 2 ( N 2 , N 3 , N 4 ; ψ b ) / f b , t 4 = f b A ( N 2 , N 3 , N 4 ; ψ b ) .
f a E ( N 2 , N 1 , N 0 ; ψ a ) + f b E ( N 2 , N 3 , N 4 ; ψ b ) = 0 .
f a T 1 ( N 2 , N 1 , N 0 ; ψ a ) > 0 , f b T 1 ( N 2 , N 3 , N 4 ; ψ b ) > 0 , f b T 0 ( N 2 , N 1 , N 0 ; ψ a ) + f b T 0 ( N 2 , N 3 , N 4 ; ψ b ) > 0 .
f b = f a E a / E b ,
t 0 = f a A a , c 1 = C 2 a / f a , t 1 = f a T 1 a , c 2 = C 1 a / f a , t 2 = f a ( T 0 a T 0 b E a / E b ) , c 3 = E b C 1 b / E a f a , t 3 = f a T 1 b E a / E b , c 4 = E b C 2 b / E a f a , t 4 = f a A b E a / E b .
A a = A ( N 2 , N 1 , N 0 ; ψ a ) , T 1 b = T 1 ( N 2 , N 3 , N 4 ; ψ b ) , etc .
T 1 a ( T 0 a T 0 b E a / E b ) > 0 , T 1 a T 1 b E a / E b < 0 .
t 0 = f b T 0 ( N 2 , N 3 , N 4 ; ψ b ) , c 1 = C 1 ( N 2 , N 3 , N 4 ; ψ b ) / f b , t 1 = f b T 1 ( N 2 , N 3 , N 4 ; ψ b ) , c 2 = C 2 ( N 2 , N 3 , N 4 ; ψ b ) / f b , t 2 = f b A ( N 2 , N 3 , N 4 ; ψ b ) + f c A ( N 6 , N 5 , N 4 ; ψ c ) , c 3 = C 2 ( N 6 , N 5 , N 4 ; ψ c ) / f c , t 3 = f c T 1 ( N 6 , N 5 , N 4 ; ψ c ) , c 4 = C 1 ( N 6 , N 5 , N 4 ; ψ c ) / f c , t 4 = f c T 0 ( N 6 , N 5 , N 4 ; ψ c ) .
f b T 1 ( N 2 , N 3 , N 4 ; ψ b ) > 0 , f c T 1 ( N 6 , N 5 , N 4 ; ψ c ) > 0 , f b A ( N 2 , N 3 , N 4 ; ψ b ) + f c A ( N 6 , N 5 , N 4 ; ψ c ) > 0 .
I : A T 1 > 0 ; II : A T 1 < 0 .
A : T 0 T 1 > 0 and E T 1 > 0 B : T 0 T 1 > 0 and E T 1 < 0 C : T 0 T 1 < 0 and E T 1 > 0 D : T 0 T 1 < 0 and E T 1 < 0 .
c 1 = C 2 a / f a , t 0 = f a A a , t 1 = f a T 1 a , c 2 = C 1 a / f a , t 2 = f a ( T 0 a T 0 b E a / E b ) , c 3 = E b C 1 b / E a f a , t 3 = f a T 1 b E a / E b , c 4 = E b C 2 b / E a f a , t 4 = f a E a A b / E b + f c A c , c 5 = C 2 c / f c , t 5 = f c T 1 c , c 6 = C 1 c / f c , t 6 = f c T 0 c .
T 1 a ( T 0 a T 0 b E a / E b ) > 0 , T 1 a T 1 b E a / E b < 0 , f a E a T 1 b / E b < 0 , f c T 1 c > 0 , f a E a A b / E b + f c A c > 0 .
F = f c E b / E a .
f c = F E a / E b .
y 0 = F / 2 b .
u ¯ 0 = tan β .
F T = F ˆ a y 0 2 / f a 2 F ˆ b N 2 2 u 2 2 + F ˆ c y 4 2 / f c 2 .
f a 2 = F 2 ( F ˆ b F ˆ a ) / F ˆ c .
P T = P a / f a + P b / f b + P c / f c = 0
f a = F ( P a E a / E b P b ) / P c .
a T = a ˆ a y 0 2 / f a 2 + a ˆ b f b N 2 2 u 2 2 + a ˆ c y 4 2 / f c .
f a = F ( a ˆ a E b / E a a ˆ b ) / a ˆ c .
E T = ( E ˆ a E ˆ b + E ˆ c ) u ¯ 0 2 ,
b T = b ˆ a N 0 y 0 u ¯ 0 b ˆ b N 2 u 2 y ¯ 2 + b ˆ c N 4 y 4 u ¯ 4 ,