Abstract

Approximate formulas are derived for the axial coma resulting from tilt and decenter of a surface and for the spherical aberration resulting from a change in its axial position. These expressions include terms that represent aberrations induced by the subsystem preceding the surface in addition to other terms that are intrinsic contributions from the misaligned surface itself. This separation of the terms gives a simple method of designing a system that is insensitive to a misalignment at a given surface. The method is illustrated by applying it to a two-mirror astronomical telescope with corrector. Two examples are given—one for tilt and the other for despace. In both examples an appreciable reduction in the sensitivity is obtained. The limitations of these solutions and the problem of simultaneous correction for two types of misalignment are examined.

© 1994 Optical Society of America

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References

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  1. D. J. Schroeder, Astronomical Optics (Academic, London, 1987), Chap. 6, pp. 102–113.
  2. W. B. Wetherell, M. P. Rimmer, “General analysis of aplanatic, Cassegrain, Gregorian, and Schwarzschild telescopes,” Appl. Opt. 11, 2817–2832 (1972).
    [CrossRef] [PubMed]
  3. G. Catalan, “Intrinsic and induced sensitivity to tilt,” Appl. Opt. 27, 22–23 (1988).
    [CrossRef] [PubMed]
  4. M. Rimmer, “Analysis of perturbed lens systems,” Appl. Opt. 9, 533–537 (1970).
    [CrossRef] [PubMed]
  5. H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, London, 1950), Chap. 3, pp. 45–47.
  6. Optical design software product of Optical Research Associates, 550 North Rosemead Boulevard, Pasadena, Calif. 91102.

1988 (1)

1972 (1)

1970 (1)

Appl. Opt. (3)

Other (3)

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, London, 1950), Chap. 3, pp. 45–47.

Optical design software product of Optical Research Associates, 550 North Rosemead Boulevard, Pasadena, Calif. 91102.

D. J. Schroeder, Astronomical Optics (Academic, London, 1987), Chap. 6, pp. 102–113.

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Figures (3)

Fig. 1
Fig. 1

Misalignment aberrations for tilt (t = 1 mrad) of the secondary mirror, decenter (e = −1.75 mm), and despace (δd = 10 mm) as a function of h2/h1 for different values of BFL (meters).

Fig. 2
Fig. 2

Axial coma as a function of the tilt angle of the secondary mirror: solid curve, the primary coma for the corrected system 3Z8, where Z8 is the corresponding Zernike coefficient; dotted curve, the value obtained from Eq. (2.1) for the corrected system; dashed curve, the value of 3Z8 for the reference system.

Fig. 3
Fig. 3

Change in spherical aberration (relative to zero displacement) as a function of axial position of the secondary mirror: solid curve, the change in primary spherical aberration for the corrected system 6Z9, where Z9 is the corresponding Zernike coefficient; dotted curve, the value obtained from Eq. (2.3) for the corrected system; dashed curve, the value of 6Z9 for the reference system.

Tables (14)

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Table 1 Reference System Data: System Construction Parameters (mm)a

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Table 2 Reference System Data: Ray Trace Dataa

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Table 3 Misalignment Wave Aberrations (in units of ⊑ = 588 nm) for Tilt, Decenter, and Despace: Comparison among Different Calculation Methods

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Table 4 Misalignment Wave Aberrations (in units of ⊑ = 588 nm) for Tilt, Decenter and Despace: Zonal Dependence of the Aberrations on the Aperture

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Table 5 Dependence of the Aberrations (in units of ⊑) on the Misalignment Parameter: Tilt and Decenter

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Table 6 Dependence of the Aberrations (in units of ⊑) on the Misalignment Parameter: Despace

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Table 7 Design Data of a Catadioptric System Corrected for Spherical Aberration, OSC and Wcta

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Table 8 Evaluations of a Telescope with Reduced Sensitivity to Tilt (Wave Aberrations in Units of ⊑): Wave Aberrations for 1-mrad Tilt and Different Values of the Aperture

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Table 9 Evaluations of a Telescope with Reduced Sensitivity to Tilt (Wave Aberrations in Units of ⊑): Ray Trace Data of the Corrected System

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Table 10 Evaluations of a Telescope with Reduced Sensitivity to Tilt (Wave Aberrations in Units of λ): rms Wave-Front Error (Wrms) for 1-mradTilt

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Table 11 Evaluations of a Telescope with Reduced Sensitivity to Tilt: Polychromatic Encircled Energy in Millimeters (Geometrical Point-Spread Function)

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Table 12 Telescope with Reduced Sensitivity to Despace: Corrected System with EFL = 49,035 mm

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Table 13 Telescope with Reduced Sensitivity to Despace: Reference System with EFL = 49 035 mm

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Table 14 Primary Spherical Aberration W40 Normalized Coma W31 and Wsd for a Two-Mirror System with Wct = Wce = 0a

Equations (35)

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W c t = t ( x h ) Δ ( n ) + 1 2 t h ( n i ) 2 Δ ( 1 / n ) ,
W c e = e c ( x h ) Δ ( n ) + 1 2 e c h ( n i ) 2 Δ ( 1 / n ) 1 2 e h 3 c 3 Q Δ ( n ) ,
W s d = δ d 2 Δ n [ ( n sin I ) 2 ( n i ) 2 n n ( x 2 h 2 ) c 2 ] δ d 2 ( h c ) 4 Q Δ ( n ) 1 8 δ d Δ [ n ( h 2 c 2 i 2 ) 2 ] ,
W c t = t [ 2 ( x h ) + h i 2 ] ,
W c e = e c [ 2 ( x h ) + h i 2 h 3 c 2 Q ] ,
W s d = δ d [ h 4 c 4 Q + 1 4 ( h 2 c 2 i 2 ) 2 + ( x 2 h 2 ) c 2 + ( sin 2 I i 2 ) ] .
W c = ( x u + x l 2 X a ) h 3 E F L ,
W s = x u h 4 E F L .
δ ( OPD ) = ( δ r · g ) ( n cos I n cos I ) ,
c [ x 2 + y 2 ( 1 + Q ) z 2 ] 2 z = 0 ,
g x = c x { ( c x ) 2 + [ 1 c z ( 1 + Q ) ] 2 } 1 / 2 = c x [ 1 1 2 c 2 x 2 + c z ( 1 + Q ) ] + O ( 4 ) .
z = 1 2 c x 2 + O ( 4 ) .
g x = c x 1 2 c 3 x 3 Q + O ( 5 ) , g z = 1 c z ( 1 + Q ) { ( c x ) 2 + [ 1 c z ( 1 + Q ) ] 2 } 1 / 2 = 1 { 1 + c 2 x 2 / [ 1 c z ( 1 + Q ) ] 2 } 1 / 2 .
g z = 1 1 2 c 2 x 2 + ( 3 8 1 + Q 2 ) c 4 x 4 + O ( 6 ) .
n cos I n cos I = n n 1 2 ( n sin 2 I n sin 2 I ) 1 8 ( n sin 4 I n sin 4 I ) .
( δ r · g ) = e g x .
δ ( OPD ) = e ( c x 1 2 Q c 3 x 3 ) × [ ( n n ) 1 2 ( n sin 2 I n sin 2 I ) + O ( 4 ) ] = e c x ( n n ) + 1 2 e c x ( n sin I ) 2 ( 1 n 1 n ) 1 2 e c 3 x 3 Q ( n n ) .
δ W = δ ( OPD ) = e c h ( n n ) e c ( x h ) ( n n ) + 1 2 e c h ( n i ) 2 ( 1 n 1 n ) 1 2 e c 3 h 3 ( n n ) ,
W c e = e c ( x h ) ( n n ) + 1 2 e c ( n i ) 2 ( 1 n 1 n ) 1 2 e c 3 h 3 Q ( n n ) .
δ W OSC = e c h ( n n ) W tilt ,
W tilt = [ e c h ( n n ) h ] x ,
δ W OSC = W tilt h x W tilt .
OSC = x h 1 ,
δ W OSC e c h ( n n ) OSC .
δ r = ( o , o , δ d ) ,
( δ r · g ) = δ d g z ,
δ ( OPD ) = δ d [ 1 1 2 c 2 x 2 + c 4 x 4 ( 3 8 1 + Q 2 ) ] × [ ( n n ) 1 2 ( n sin 2 I n sin 2 I ) 1 8 ( n sin 4 I n sin 4 I ) ] .
δ W = δ d ( n n ) δ ( OPD ) , δ W = δ d [ 1 2 c 2 x 2 ( n n ) 1 2 ( n sin 2 I n sin 2 I ) + c 4 x 4 ( 3 8 1 + Q 2 ) ( n n ) + 1 4 c 2 x 2 ( n sin 2 I n sin 2 I ) 1 8 ( n sin 4 I n sin 4 I ) ] .
δ W = δ d { 1 2 c 2 x 2 ( n n ) 1 2 ( n sin I ) 2 ( 1 n 1 n ) 1 2 Q c 4 x 4 ( n n ) 1 8 Δ [ n ( c 2 x 2 sin 2 I ) 2 ] } ,
δ W = δ d { 1 2 c 2 h 2 ( n n ) 1 2 ( n i ) 2 ( 1 n 1 n ) + ( n n ) [ ( n sin I ) 2 ( n i ) 2 n n c 2 ( x 2 h 2 ) ] 1 2 Q c 4 h 4 ( n n ) 1 8 Δ [ n ( h 2 c 2 i 2 ) 2 ] } .
W s d = δ d { ( n n ) [ ( n sin I ) 2 ( n i ) 2 n n c 2 ( x 2 h 2 ) ] 1 2 Q c 4 x 4 ( n n ) 1 8 Δ [ n ( h 2 c 2 i 2 ) 2 ] } .
OHC = 2 z u 2 R 1 ,
δ d [ 1 2 c 2 h 2 ( n n ) + 1 2 ( n i ) 2 ( 1 n 1 n ) ] = { δ d [ 1 2 c h 2 ( n n ) + 1 2 ( n i ) 2 ( 1 n 1 n ) ] 2 z u 2 R } × 1 1 + OHC .
δ W s d W 20 + δ d [ 1 2 c 2 h 2 ( n n ) + 1 2 ( n i ) 2 ( 1 n 1 n ) ] OHC .
δ W s d = δ d ( c 2 h 2 + i 2 ) OHC .

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