Abstract

An analysis is made of two-mirror systems consisting of spherical reflecting surfaces. Solutions are found for those systems having zero third-order spherical aberration. It is shown that no practical solution exists for the configuration resembling the Cassegrainian telescope; there are three one-parameter families of solutions. These are given by

c1=(q1)/2t0t1=(t0f)/qc2=q/2ft0=(27f/32)sec2θq=3[1+4cos23(θ+πr)]1,
where c1 and c2 are the two curvatures; t1 the axial separation of the two reflecting surfaces; t0 the distance from a focus to the corresponding surface; and f the focal length. The free parameter is θ and r = 0, 1, −1.

© 1967 Optical Society of America

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References

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  1. D. P. Feder, J. Opt. Soc. Am. 41, 630 (1951).
    [CrossRef]
  2. F. C. Champion, Light (Blackie & Son Ltd., London, 1941), p. 41.
  3. L. E. Dickson, New First Course in the Theory of Equations (John Wiley & Sons, Inc., New York, 1939), pp. 42–45.
  4. G. A. Korn and T. M. Korn, in Mathematical Handbook for Scientists and Engineers (McGraw-Hill Book Company, New York, 1961), p. 23.

1951 (1)

Champion, F. C.

F. C. Champion, Light (Blackie & Son Ltd., London, 1941), p. 41.

Dickson, L. E.

L. E. Dickson, New First Course in the Theory of Equations (John Wiley & Sons, Inc., New York, 1939), pp. 42–45.

Feder, D. P.

Korn, G. A.

G. A. Korn and T. M. Korn, in Mathematical Handbook for Scientists and Engineers (McGraw-Hill Book Company, New York, 1961), p. 23.

Korn, T. M.

G. A. Korn and T. M. Korn, in Mathematical Handbook for Scientists and Engineers (McGraw-Hill Book Company, New York, 1961), p. 23.

J. Opt. Soc. Am. (1)

Other (3)

F. C. Champion, Light (Blackie & Son Ltd., London, 1941), p. 41.

L. E. Dickson, New First Course in the Theory of Equations (John Wiley & Sons, Inc., New York, 1939), pp. 42–45.

G. A. Korn and T. M. Korn, in Mathematical Handbook for Scientists and Engineers (McGraw-Hill Book Company, New York, 1961), p. 23.

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Figures (10)

F. 1
F. 1

Cassegrain-like and Gregory-like systems. The height of a ray is designated by y0 on the object plane, by y1 at the first mirror and by y2 at the second. In each of the above illustrations y0=0. u0, u1, and u2 represent the slopes of a ray between the object plane and the first mirror, between the two mirrors and following the second mirror, respectively.

F. 2
F. 2

Group-A solutions showing separation (T1) and the curvatures of the first and second mirrors (C1,C2) vs object distance. Focal length=1.

F. 3
F. 3

Group-A solutions showing positions of mirror vertices and centers of curvature relative to object point. Ordinate is the parameter θ. Focal length=1.

F. 4
F. 4

Group A. Third-order aberrations. Abscissa is the parameter θ. Focal length=1.

F. 5
F. 5

Group-B solutions showing separation (T1) and the curvatures of the first and second mirrors (C1,C2) vs object distance. Focal length=1.

F. 6
F. 6

Group-B solutions showing positions of mirror vertices and centers of curvature relative to object point. Ordinate is the parameter θ. Focal length=1.

F. 7
F. 7

Group-B. Third-order aberrations. Abscissa is the parameter θ. Focal length=1.

F. 8
F. 8

Group-C solutions showing separation (T1) and the curvatures of the first and second mirrors (C1,C2) vs object distance. Focal length=1.

F. 9
F. 9

Group-C solutions showing positions of mirror vertices and centers of curvature relative to object point. Ordinate is the parameter θ. Focal length=1.

F. 10
F. 10

Group C. Third-order aberrations. Abscissa is the parameter θ. Focal length=1.

Tables (5)

Tables Icon

Table I Group-A parameters.

Tables Icon

Table II Group-B parameters.

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Table III Group-C parameters.

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Table IV Axial rays, Group C, θ= 1.4, f=9.52 mm. φ and φ′ are the angles between the ray and the axis in object and image space, respectively.

Tables Icon

Table V y0 = 0.5 Group C, θ = 1.4, f=9.524 mm. φ and φ′ are the angles between the ray and the axis in object and image space, respectively.

Equations (61)

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c 1 = ( q 1 ) / 2 t 0 t 1 = ( t 0 f ) / q c 2 = q / 2 f t 0 = ( 27 f / 32 ) sec 2 θ q = 3 [ 1 + 4 cos 2 3 ( θ + π r ) ] 1 ,
y 1 = y t u u 1 = 2 c 1 y 1 u ,
( y 1 u 1 ) = ( 1 t 0 2 c 1 ( 1 + 2 t 0 c 1 ) ) ( y 0 u 0 ) , ( y 2 u 2 ) = ( 1 t 1 2 c 2 ( 1 + 2 t 1 c 2 ) ) ( y 1 u 1 ) .
( y 2 u 2 ) = ( 1 2 c 1 t 1 t 0 + t 1 + 2 t 0 c 1 t 1 2 ( c 2 c 1 2 c 1 t 1 c 2 ) 1 + 2 t 1 c 2 2 t 0 ( c 2 c 1 2 c 1 t 1 c 2 ) ) ( y 0 u 0 ) .
1 + 2 t 1 c 2 2 t 0 ( c 2 c 1 2 c 1 t 1 c 2 ) = 0 ,
c 2 = 1 2 { ( 1 + 2 t 0 c 1 ) / [ t 0 t 1 ( 1 + 2 t 0 c 1 ) ] } .
t 0 + t 1 + 2 t 0 c 1 t 1 = y 2 / u 0 = f ,
t 1 = ( t 0 f ) / ( 1 + 2 t 0 c 1 ) .
c 2 = ( 1 + 2 t 0 c 1 ) / 2 f .
q = 1 + 2 t 0 c 1 .
{ c 1 = ( q 1 ) / 2 t 0 t 1 = ( t 0 f ) / q c 2 = q / 2 f .
1 2 ( c 2 c 1 2 c 1 t 1 c 2 ) 1 .
1 2 [ q 2 f q 1 2 t 0 2 q 1 2 t 0 t 0 f q q 2 f ] 1 = f .
( y 2 u 2 ) = ( ( f q + t 0 f ) / t 0 q f 1 / f 0 ) ( y 0 u 0 ) .
( y ¯ 2 u 2 ) = ( 1 n 2 0 1 ) ( y 2 u 2 ) ( y ¯ 0 u 0 ) = ( 1 n 0 0 1 ) ( y 0 u 0 ) .
( y ¯ 2 u 2 ) = ( 1 n 2 0 1 ) ( ( f q + t 0 f ) / t 0 q f 1 / f 0 ) ( 1 n 0 0 1 ) ( y ¯ 0 u 0 ) = ( ( f q + t 0 f ) / t 0 q n 2 / f n 0 ( f q + t 0 f ) / t 0 q n 0 n 2 / f f 1 / f n 0 / f ) ( y ¯ 0 u 0 ) .
y ¯ 2 = y ¯ 0 ,
{ ( f q + t 0 f ) / t 0 q n 2 / f = 1 n 0 ( f q + t 0 f ) / t 0 q n 0 n 2 f f = 0 .
n 2 = [ ( f t 0 ) ( q 1 ) / t 0 q ] f .
t = f ( f q + t 0 f ) / t 0 q .
y 1 = t 0 u 0 y 2 = f u 0 u 1 = q u 0 u 2 = 0 i 1 = 1 2 ( q + 1 ) u 0 i 2 = 1 2 q u 0
y ¯ 1 = ( t 0 f ) y ¯ 0 / f y ¯ 2 = ( t 0 f ) ( q 1 ) y ¯ 0 / t 0 q u ¯ 1 = [ q ( t 0 f ) + f ] y ¯ 0 / f t 0 u ¯ 2 = y ¯ 0 / f i ¯ 1 = [ f ( q 1 ) t 0 ( q + 1 ) ] y ¯ 0 / 2 f t 0 i ¯ 2 = [ f ( q 1 ) t 0 ( q + 1 ) ] y ¯ 0 / 2 f t 0 .
the Lagrange invariant I = u 0 y ¯ 0 , the Petzval contributions P 1 = 2 c 1 = ( q 1 ) / t 0 P 2 = 2 c 2 = q / f ,
S 1 = c 1 y 1 2 / I S 2 = c 2 y 2 2 / I .
D = C + 1 2 P I
E = S ¯ i i ¯ + 1 2 ( u 1 2 u 2 )
B = [ t 0 ( q 1 ) ( q + 1 ) 2 f q 3 ] u 0 3 / 8 y ¯ 0 ,
F = [ f q 2 t 0 ( q 2 1 ) ] [ f ( q 1 ) t 0 ( q + 1 ) ] u 0 2 / 8 t 0 f ,
C = [ f q t 0 ( q 1 ) ] × [ f ( q 1 ) t 0 ( q + 1 ) ] 2 u 0 y ¯ 0 / 8 t 0 2 f 2 ,
D = { [ f q t 0 ( q 1 ) ] [ f ( q 1 ) t 0 ( q + 1 ) ] 2 + 4 t 0 f [ f ( q 1 ) t 0 q ] } u 0 y ¯ 0 / 8 t 0 2 f 2 ,
E = ( q 1 ) ( t 0 f ) 2 [ t 0 2 ( q + 1 ) 2 f 2 ( q 1 ) 2 ] y ¯ 0 2 / 8 f 3 t 0 3 .
t 0 ( q 1 ) ( q + 1 ) 2 f q 3 = 0 ,
( t 0 f ) q 3 + t 0 q 2 t 0 q t 0 = 0 .
q = x 1 3 t 0 / ( t 0 f ) ,
( t 0 f ) x 3 t 0 ( 4 t 0 3 f ) 3 ( t 0 f ) x t 0 27 ( t 0 f ) 2 × ( 16 t 0 2 45 t 0 f + 27 f 2 ) = 0 ,
x = 1 3 y / ( t 0 f ) ,
y 3 3 t 0 ( 4 t 0 3 f ) y t 0 ( 16 t 0 2 45 t 0 f + 27 f 2 ) = 0 .
y = u ( z + 1 / z ) ,
u 3 ( z 3 + 1 / z 3 ) + 3 u ( z + 1 / z ) [ u 2 t 0 ( 4 t 0 3 f ) ] t 0 ( 16 t 0 2 45 t 0 f + 27 f 2 ) = 0 .
u 2 = t 0 ( 4 t 0 3 f ) ,
u 3 z 6 t 0 ( 16 t 0 2 45 t 0 f + 27 f 2 ) z 3 + u 3 = 0 ,
z ± 3 = ( 2 u 3 ) 1 [ t 0 ( 16 t 0 2 45 t 0 f + 27 f 2 ) ± H ] ,
H 2 = t 0 2 ( 16 t 0 2 45 t 0 f + 27 f 2 ) 2 4 u 6 = 27 t 0 2 f ( t 0 f ) 2 ( 27 f 32 t 0 ) .
z ± 3 = t 0 / 2 u 3 { ( 16 t 0 2 45 t 0 f + 27 f 2 ± 3 ( t 0 f ) [ 3 f ( 27 f 32 t 0 ) ] 1 2 } .
27 k 2 f = 27 f 32 t 0 ,
t 0 = 27 / 32 ( 1 k 2 ) f .
u = 9 f / 16 [ ( 1 k 2 ) ( 1 9 k 2 ) ] 1 2
z ± 3 = ( 1 k 2 ) [ 27 k 4 + 36 k 2 + 1 ) ± 2 k ( 27 k 2 + 5 ) ] ( 1 k 2 ) 3 2 ( 1 9 k 2 ) 3 2 .
z ± 3 = ( 1 k 2 ) ( 1 ± k ) ( 1 ± 3 k ) 3 / ( 1 k 2 ) 3 2 ( 1 9 k 2 ) 3 2 ,
z ± 3 = ( 1 k 1 ± k ) 1 2 ( 1 3 k 1 ± 3 k ) 3 2 .
z ± ( r ) = ( 1 k 1 ± k ) 1 6 ( 1 3 k 1 ± 3 k ) 1 2 d r ( r = 0 , 1 , 1 ) ,
q = 3 ( t 0 f ) 1 [ u ( z + 1 / z ) t 0 ] .
q r = [ 3 / ( 27 k 2 + 5 ) ] × { 2 ( 1 k 2 ) 1 3 [ ( 1 k ) 1 3 ) ( 1 3 k ) d r + ( 1 + k ) 1 3 ( 1 + 3 k ) d r ] 3 ( 1 k 2 ) } .
q r = 3 / { 2 [ ( 1 + k 1 k ) 1 3 d r + ( 1 k 1 + k ) 1 3 d r ] + 1 } .
q 0 = 3 { 1 + 4 cosh 1 3 log [ ( 1 + k ) / ( 1 k ) ] } 1 .
q 0 = 3 { 1 4 cosh 1 3 log [ ( k + 1 ) / ( k 1 ) ] } 1 .
t 1 / f = ( 5 + 27 k 2 ) / 32 q , t 0 / f = 27 ( 1 k 2 ) / 32 ,
q r = 3 { 1 + 2 [ ( 1 + i m 1 i m ) 1 3 d r + ( 1 i m 1 + i m ) 1 3 d r ] } 1 r = 0 , 1 , 1 .
q r = 3 { 1 + 2 [ ( exp 2 i θ ) 1 3 exp ( 2 π r i / 3 ) + ( exp 2 i θ ) 1 3 exp ( 2 π r i / 3 ) ] } 1 = 3 ( 1 + 2 { exp [ ( 2 i / 3 ) ( θ + π r ) ] + exp [ ( 2 i / 3 ) ( θ + π r ) ] } ) 1 = 3 [ 1 + 4 cos 2 3 ( θ + π r ) ] 1 .
t 0 = 27 / 32 ( 1 + m 2 ) f = ( 27 f / 32 ) sec 2 θ .
t 0 = 278.166 mm , t 1 = 232.237 mm , c 1 = 0.0038768 / mm , c 2 = 0.0607296 / mm .