Abstract

Cassegrainian-type reflecting objectives are useful as in-line imaging devices for fast spectrometer systems. They are inherently free from chromatic aberrations and may be corrected for spherical aberration. A particularly simple design, consisting of two mirrors with oppositely equal curvatures, is possible when unit magnification is desired. This system has zero third-order spherical aberration, and all other aberrations are small also. Its main disadvantage is the loss due to the central obstruction of the beam, amounting to about 28% exclusive of reflection losses.

© 1974 Optical Society of America

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References

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  1. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), pp. 367–368.
  2. D. P. Feder, J. Opt. Soc. Am. 41, 630 (1951).
    [CrossRef]

1951 (1)

Feder, D. P.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), pp. 367–368.

J. Opt. Soc. Am. (1)

Other (1)

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), pp. 367–368.

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Figures (4)

Fig. 1
Fig. 1

(a) Marginal-ray path. (b) Principal-ray path for aperture stop at concave mirror.

Fig. 2
Fig. 2

Transverse spherical aberration Δy vs marginal-ray slope u (r = 400 mm, m = 2.4).

Fig. 3
Fig. 3

Dependence of Seidel coefficients F, C, D, and E on partial magnification m (r = 400 mm, u1 = −7°, h1 = 5 mm, aperture stop at concave mirror).

Fig. 4
Fig. 4

Loss L due to central obstruction vs partial magnification m.

Tables (2)

Tables Icon

Table I Seidel Aberration Coefficients of Reflecting Objective (r = 400 mm, m = 2.25, u1 = −7°, h1 = 5 mm) for Three Selected Positions of Aperture Stop

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Table II Design Data fo Final System

Equations (50)

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m 1 = u / u 1 , m 2 = u / u 2 .
M = u 1 / u 2 = m 2 / m 1 ,
Δ spher = h 1 ( B 1 + B 2 ) / u 1 ,
B 1 = S 1 i 1 2 , B 2 = S 2 i 2 2 ,
S 1 = a 1 ( u i 1 ) / h 1 ,
S 2 = + a 2 ( u 2 i 2 ) / h 2 .
1 a 1 + 1 a 1 = 1 a 1 ( 1 m 1 ) = 2 r 1 ,
1 a 2 + 1 a 2 = 1 a 2 ( 1 + m 2 ) = 2 r 2 .
i 1 = 1 2 ( u 1 + u ) = 1 2 ( 1 + m 1 ) u 1 ,
i 2 = 1 2 ( u + u 2 ) = 1 2 ( 1 m 2 ) u 2 .
S 1 = 1 4 h 1 ( m 1 1 ) 2 r 1 u 1 ,
S 1 = 1 4 h 2 ( m 2 + 1 ) 2 r 2 u 2 ,
B 1 = 1 16 h 1 ( m 1 2 1 ) r 1 u 1 3 ,
B 2 = 1 16 h 2 ( m 2 2 1 ) r 2 u 2 3 .
r 1 r 2 = h 1 u 2 3 ( m 2 2 1 ) 2 h 2 u 1 3 ( m 1 2 1 ) 2 = m 1 4 ( m 2 2 1 ) 2 m 2 4 ( m 1 2 1 ) 2 ,
m 1 = m 2 m ,
r 1 = r 2 r .
a 1 = 1 2 ( m 1 ) r , a 2 = 1 2 ( m + 1 ) r ,
d = r / m .
P 1 = 2 / r 1 , P 1 = 2 / r 2 ,
I = h 1 u 1 ,
ϕ 1 = i 1 j 1 ( υ j 1 ) y 1 / I ,
ϕ 2 = i 2 j 2 ( υ 2 j 2 ) y 2 / I .
F 1 = S 1 i 1 j 1 ,
F 2 = S 2 i 2 j 2 ;
C 1 = S 1 j 1 2 ,
C 2 = S 2 j 2 2 ;
D 1 = C 1 + 1 2 P 1 I ,
D 2 = C 2 + 1 2 P 2 I ;
E 1 = ϕ 1 + 1 2 ( υ 2 υ 1 2 ) ,
E 2 = ϕ 2 + 1 2 ( υ 2 2 υ 2 ) .
Δ coma = h 2 F = effective size of coma patch ,
Δ tan = 2 h 2 ( 2 C + D ) Δ sag = 2 h 2 D } major and minor axes of astigmatic patch , in paraxial image plane ,
Δ dist = h 2 E = difference of distorted and ideal image heights .
υ 1 = h 1 a 1 d = 2 h 1 ( m + 2 ) r m ( m + 1 ) ,
υ = υ 2 = h 2 a 2 = 2 h 1 r ( m + 1 ) ,
j 1 = 1 2 ( υ 1 + υ ) = 2 h 1 r m ,
j 2 = υ ,
y 1 = υ 1 d = 2 h 1 m ( m + 1 ) ,
y 2 = 0 ,
d = d r 1 r 1 + 2 d = r m + 2
F = ( m 2 1 ) u 1 2 4 m ,
C = D = ( 2 m 1 ) h 1 u 1 m 2 r ,
E = 4 ( 2 m + 1 ) h 1 2 m 3 ( m + 1 ) r 2 .
A 1 = 2 a 1 u 1 , A 2 = 2 a 2 u 2 ,
u 2 = A 1 2 ( a 2 d ) = a 1 u 1 a 2 d .
2 ( a 1 d ) u 1 < A 2 < 2 a 2 u 2 = [ ( 2 a 1 a 2 ) / ( a 2 d ) ] u 1 .
L = ( u 2 / u 2 ) 2 = [ a 1 / ( a 2 d ) ] 2 ,
L = m ( m 1 ) m ( m + 1 ) 2 ,
T = ( 1 L ) ( 1 L ) ( 1 R ) 2 ,

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