Abstract

The properties of two-conic reflecting aplanats are analyzed and discussed on the basis of third order aberration theory. Techniques for designing infinite conjugate two mirror aplanats and computing their image properties are developed. The secondary mirror alignment characteristics of Ritchey-Chrétien and aplanatic Gregorian telescopes are examined and neutral point locations defined. Design configurations corrected for a third Seidel aberration (astigmatism, image curvature, or distortion) are identified and their properties discussed. The properties of Ritchey-Chrétien and aplanatic Gregorian telescopes are compared.

© 1972 Optical Society of America

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Errata

William B. Wetherell, "General Analysis of Aplanatic Cassegrain, Gregorian, and Schwarzschild Telescopes: Erratum," Appl. Opt. 13, 242_1-242 (1974)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-13-2-242_1

References

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  1. Scientific Uses of the Large Space Telescope (National Academy of Sciences, Washington, D.C., 1969).
  2. R. E. Danielson, High Resolution Imagery with the Large Space Telescope, at the AAAS meeting, 27–29 December 1971.
  3. Technology Study for a Large Orbiting Telescope, Itek Report 70-9443-1, 15May1970, Final Report on NASA Contract NASw-1925.
  4. Large Space Telescope: Continuation of a Technology Study, Itek Report 71-9463-2, 3September1971, Final Report on NASA Contract NASw-2174.
  5. Three-Meter Telescope Study, Perkin-Elmer Report E.R. 10713, August1971, Final Report on NASA Contract NAS 5-21540.
  6. R. V. Shack, Optical Sciences Center Newsletter 3, 64, (1969).
  7. H. P. Brueggemann, Conic Mirrors (Focal Press, London, 1968).
  8. S. Rosin, Appl. Opt. 7, 1483 (1968).
    [CrossRef] [PubMed]
  9. The sign convention is that of Conrady and Kingslake.
  10. A. T. Young, Appl. Opt. 6, 1063 (1967).
    [CrossRef] [PubMed]
  11. R. Prescott, Appl. Opt. 7, 479 (1968).
    [CrossRef] [PubMed]
  12. W. Wetherell, Instrumentation in Astronomy, SPIE Seminar Proceedings (1972) Vol. 28 (in press).
  13. W. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), Sec. 10.8, p. 272.
  14. MIL-HDBK-141, Optical Design (U. S. Government Printing Office, Washington, D.C., 5October1962), Sec. 8, 9.
  15. See Ref. 12, Appendix B, for the complete equation for E0, from which Eqs. (32) and (55) are derived.
  16. D. H. Schulte, Instrumentation in Astronomy, SPIE Seminar Proceedings (1972), Vol. 28 (in press).
  17. K. Schwarzschild, Theorie Der Spiegeltelescope (Göttingen Observatory, 1905).
  18. M. Rimmer, Appl. Opt. 9, 533 (1970).
    [CrossRef] [PubMed]

1970 (1)

1969 (1)

R. V. Shack, Optical Sciences Center Newsletter 3, 64, (1969).

1968 (2)

1967 (1)

Brueggemann, H. P.

H. P. Brueggemann, Conic Mirrors (Focal Press, London, 1968).

Danielson, R. E.

R. E. Danielson, High Resolution Imagery with the Large Space Telescope, at the AAAS meeting, 27–29 December 1971.

Prescott, R.

Rimmer, M.

Rosin, S.

Schulte, D. H.

D. H. Schulte, Instrumentation in Astronomy, SPIE Seminar Proceedings (1972), Vol. 28 (in press).

Schwarzschild, K.

K. Schwarzschild, Theorie Der Spiegeltelescope (Göttingen Observatory, 1905).

Shack, R. V.

R. V. Shack, Optical Sciences Center Newsletter 3, 64, (1969).

Smith, W.

W. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), Sec. 10.8, p. 272.

Wetherell, W.

W. Wetherell, Instrumentation in Astronomy, SPIE Seminar Proceedings (1972) Vol. 28 (in press).

Young, A. T.

Appl. Opt. (4)

Optical Sciences Center Newsletter (1)

R. V. Shack, Optical Sciences Center Newsletter 3, 64, (1969).

Other (13)

H. P. Brueggemann, Conic Mirrors (Focal Press, London, 1968).

The sign convention is that of Conrady and Kingslake.

Scientific Uses of the Large Space Telescope (National Academy of Sciences, Washington, D.C., 1969).

R. E. Danielson, High Resolution Imagery with the Large Space Telescope, at the AAAS meeting, 27–29 December 1971.

Technology Study for a Large Orbiting Telescope, Itek Report 70-9443-1, 15May1970, Final Report on NASA Contract NASw-1925.

Large Space Telescope: Continuation of a Technology Study, Itek Report 71-9463-2, 3September1971, Final Report on NASA Contract NASw-2174.

Three-Meter Telescope Study, Perkin-Elmer Report E.R. 10713, August1971, Final Report on NASA Contract NAS 5-21540.

W. Wetherell, Instrumentation in Astronomy, SPIE Seminar Proceedings (1972) Vol. 28 (in press).

W. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), Sec. 10.8, p. 272.

MIL-HDBK-141, Optical Design (U. S. Government Printing Office, Washington, D.C., 5October1962), Sec. 8, 9.

See Ref. 12, Appendix B, for the complete equation for E0, from which Eqs. (32) and (55) are derived.

D. H. Schulte, Instrumentation in Astronomy, SPIE Seminar Proceedings (1972), Vol. 28 (in press).

K. Schwarzschild, Theorie Der Spiegeltelescope (Göttingen Observatory, 1905).

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Figures (19)

Fig. 1
Fig. 1

Nomenclature of principal design types, the Cassegrain-type (mirrors on the same side of prime focus) and the Gregorian-type (mirrors on opposite sides of prime focus).

Fig. 2
Fig. 2

Clear aperture diameters at primary mirror, secondary mirror, exit pupil, and image.

Fig. 3
Fig. 3

Astigmatic image surfaces: T = tangential; S = sagittal; P = Petzval; I = image surface of minimum wavefront error.

Fig. 4
Fig. 4

Astigmatism constant A plotted as a function of the secondary mirror magnification m and normalized vertex back focus β = B/fp for Cassegrain-type (solid lines) and Gregorian-type (dashed lines) designs. Circles (○) indicate parameter combinations for which Dsa = Dp. Couder anastigmats are indicated by the dots (●) on the axis A = 0.

Fig. 5
Fig. 5

Sample wavefront error boundary diagram showing reduced depth of focus δ′ and mean focus shift z′ when using flat image sensor with curved field telescope.

Fig. 6
Fig. 6

Properly aligned Cassegrain-type telescope showing nomenclature for axial misalignment equations.

Fig. 7
Fig. 7

Change in the position of principal focus and the image height y′ when the mirror separation S is increased by ΔS.

Fig. 8
Fig. 8

Lateral misalignment of the secondary mirror can be represented entirely by the separation of the two optical axes at the neutral point Pω0 and the center of curvature Pcc. Axial coma is proportional to the separation Δy0ω, and the image displacement Δy′ is proportional to the separation Δycc.

Fig. 9
Fig. 9

Design constants for anastigmatic solutions showing constants for Schwarzschild concentric sphere design and its virtual image compliment (circles).

Fig. 10
Fig. 10

Anastigmatic two-mirror aplanats.

Fig. 11
Fig. 11

Design constants for flat-field solutions.

Fig. 12
Fig. 12

F/1.5 Schwarzschild of magnification m = 0.25.

Fig. 13
Fig. 13

Flat-field aplanatic Gregorian configurations.

Fig. 14
Fig. 14

Astigmatism constant A for flat-field solutions.

Fig. 15
Fig. 15

Design constants for distortion-free solutions.

Fig. 16
Fig. 16

F/1.5 distortion-free design of magnification m = 0.25.

Fig. 17
Fig. 17

Schwarzschild flat-field anastigmat.

Fig. 18
Fig. 18

Coordinate system for the derivation of the secondary mirror lateral misalignment equations.

Fig. 19
Fig. 19

Axial coma constant Kc as a function of the central obstruction diameter ratio .

Tables (2)

Tables Icon

Table I Design Parameters for Ritchey-Chrétien (RC), Conjugate Design Aplanatic Gregorian (CG), and Equal Length Aplanatic Gregorian (EG)

Tables Icon

Table II Comparison of Performance Characteristics that Vary Significantly Between Ritchey-Chrétien (RC), Conjugate Design Aplanatic Gregorian (CG), and Equal Length Aplanatic Gregorian (EG)

Equations (98)

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m = F / F p
η = B / D p ,
β = B / f p ,
f p = D p F p .
F = 1 2 sin U ,
S = D p ( F - η ) / ( m + 1 ) = f p ( m - β ) / ( m + 1 ) .
ψ = S / f p = ( m - β ) / ( m + 1 ) .
f = F D p .
c p = - 1 2 D p F p .
c s = ( 1 - m 2 ) / 2 m D p ( F p + η ) .
α = α ( m F + η ) / m ( F p + η ) = α ( m 2 + β ) / m ( 1 + β ) .
l e p = - m F D p ( F p + η ) / ( m F + η ) = m 2 f p ( 1 + β ) / ( m 2 + β ) .
D I = θ I F D p .
D s a = ( F p + η ) D p / ( F + F p ) .
D s = D p ( F p + η + θ i F p F - η ) / ( F + F p ) .
D h p = D p [ η ( F p + η ) + θ T F p ( F 2 - η 2 ) ] F ( F p + η ) .
D e p = D p m ( F p + η ) / ( m F + η ) = D p m ( 1 + β ) / ( m 2 + β ) .
( min ) = D s / D p
( min ) = D h p / D p ,
z ( r ) = c r 2 / [ 1 + ( 1 - κ c 2 r 2 ) 1 2 ] ,
z ( r ) = c r 2 / { 1 + [ 1 - ( k + 1 ) c 2 r 2 ] 1 2 } .
κ p = - 2 ( F p + η ) / m 2 ( F - η ) ,
κ s = - [ 2 F ( m + 1 ) ( m - 1 ) 3 ( F - η ) + 4 m ( m - 1 ) 2 ] .
z s ( r ) = c s r 2 1 + ( 1 - c s 2 r 2 ) 1 2 + c s 3 ( κ s - 1 ) r 4 8 .
z s ( r ) = F r 4 / 2 D p 3 ( F p + η ) 3 ( F - η ) .
z petz = - α 2 D p F [ m 2 ( F + F p - η ) + η ] / 2 m ( F p + η ) ,
z t - s = 2 z s - p .
z I = z t - s + z petz .
z t - s = - α 2 D p F [ ( 2 m + 1 ) F + η ] 2 m ( F p + η ) .
z I = y 2 / 2 R I = α 2 F 2 D p 2 / 2 R I ,
R I = - m D p F ( F p + η ) / [ m 2 ( F + F p - η ) + η ] = - m 2 f p ( 1 + β ) / [ m 2 ( m + 1 - β ) + β ] .
R I - D p F p ( F p + η ) / ( F + F p - η ) .
E 0 = ω 2 = W 20 2 ( 1 - 2 ) 2 12 + W 20 W 02 ( 1 - 2 ) 2 12 + W 02 2 ( 1 + 4 ) 16 ,
W 02 = z t - s / 8 λ F 2 = - α 2 D p [ ( 2 m + 1 ) F + η ] 16 λ m 2 F p ( F p + η ) .
W 20 = - W 02 / 2
ω = α 2 D p ( 2 m + 1 ) F + η ( 1 + 2 + 4 ) 1 2 32 ( 6 ) 1 2 λ m 2 F p ( F p + η ) .
W 02 = - α 2 D p A / 16 λ F p ,
A = [ ( 2 m + 1 ) F + η ] / m 2 ( F p + η ) = [ ( 2 m + 1 ) m + β ] / m 2 ( 1 + β ) .
α I = 32 ( 6 ) 1 2 λ m 2 F p ( F p + η ) ω I α 2 D p ( 2 m + 1 ) F + η ( 1 + 2 + 4 ) 1 2 .
W 20 ( Δ z ) = Δ z / 8 λ F 2 .
δ = ± 16 ( 3 ) 1 2 λ ω I F 2 / ( 1 - 2 ) .
z ( α , ω I ) = z I + Δ z I ,
Δ z I = ± δ [ 1 - ( α / α I ) 4 ] 1 2 .
δ y y = α 2 ( F - η ) 4 m 2 ( F p + η ) 2 [ ( m 2 - 2 ) F + ( 3 m 2 - 2 ) η ] = α 2 ( m - β ) 4 m 2 ( 1 + β ) 2 [ m ( m 2 - 2 ) + ( 3 m 2 - 2 ) β ] .
δ y / y = α 2 ( m 2 - 2 ) / 4             ( β = 0 ) ,
m = y / y p = - l s / l s = F / F p ,
δ p = ± 16 ( 3 ) 1 2 λ ω I F p 2 / ( 1 - 2 )
δ = m 2 δ p .
Δ z p = - m 2 Δ S ,
ω δ = ( 1 - 2 ) Δ z p / 16 ( 3 ) 1 2 λ F 2 .
ω δ = ( 1 - 2 ) Δ S / 16 ( 3 ) 1 2 λ F p 2 .
Δ y a = ( m α + α ) Δ S = α Δ S [ m 2 ( 2 + β ) + β ] / m ( 1 + β ) .
Δ I . S . = ( y + Δ y a ) / y .
Δ y r a = α Δ S ( 1 - m 2 ) / 4 ( 1 + β ) ,
W 40 = ( m + 1 ) [ 2 m ( m - 1 ) ( F - η ) + 1 ] Δ S 256 λ F 3 F p ( F - η ) .
E 0 = ω 2 = W 20 2 ( 1 - 2 ) 2 + W 20 W 40 6 ( 1 - 2 ) ( 1 - 4 ) + W 40 2 45 ( 4 - 2 - 6 4 - 6 + 4 8 ) ,
W 20 = - W 40 ( 1 - 4 ) / ( 1 - 2 ) .
δ z = - ( m + 1 ) [ 2 m ( m - 1 ) ( F - η ) + 1 ] ( 1 - 4 ) Δ S 16 F p F ( F - η ) ( 1 - 2 ) ,
Δ z = Δ z p + δ z .
ω δ τ = W 40 ( 1 - 2 ) 2 / 6 ( 5 ) 1 2 .
L ω 0 / R s = ( m + 1 ) / [ ( m + 1 ) - ( κ s - 1 ) ( m - 1 ) ] .
L ω 0 = ( 1 - m ) ( F p + η ) ( F - η ) D p m 2 ( F - η ) + ( F p + η ) .
l ω 0 = D p F p ( F p + η ) m 2 ( F - η ) + ( F p + η ) .
ω c = σ d Δ y ω 0 ,
σ d = 0.0037 λ F p 3 [ 1 + ( F p + η ) m 2 ( F - η ) ] = 0.0037 λ F p 3 [ 1 + ( 1 + β ) m 2 ( m - β ) ] .
σ d 0.0037 / λ F p 3             m 1.
Δ y 1 = ( 1 - m ) Δ y c c .
β = - ( 2 m 2 + m ) .
ψ = 2 m ,
S = 2 f .
κ p = ( 2 m - 1 ) / m 3 ,
κ s = ( 2 m - 1 ) 2 / ( 1 - m ) 3 ,
R I = D p F ( 2 m - 1 ) / ( 2 m 2 - 1 ) ,
Δ y / y = α 2 m ( 2 - 3 m ) / ( 1 - 2 m ) 2 .
β = m 2 / ( m - 1 ) .
ψ = m / ( 1 - m 2 ) .
κ p = 2 ( m 2 + m - 1 ) / m 2 ,
κ s = 2 / ( 1 - m ) .
A = ( 2 m 2 - 1 ) / m ( m 2 + m - 1 ) ,
Δ y / y = - α 2 [ ( 4 m - 1 ) ( m 2 - 1 ) + 1 ] / 4 ( m - 1 ) 2 ( m 2 + m - 1 ) .
β = m ( 2 - m 2 ) / ( 3 m 2 - 2 ) .
ψ = 4 m ( m - 1 ) / ( 3 m 2 - 2 ) .
δ W = - 2 δ r · g ( i · g ) ,
i = [ - ( x / d ) , - ( y / d ) , 1 - ( ρ 2 / 2 d 2 ) ] ,
g = [ - c 3 x + 1 2 ( 1 - κ s ) c s 3 ρ 2 x - c s y + 1 2 ( 1 - κ s ) c s 3 ρ 2 y , 1 - 1 2 c s 2 ρ 2 ] ,
( i · g ) = 1 - ( ρ 2 / 2 d 2 ) [ 1 - c s d ] 2 .
δ r = ( 0 , Δ y , 0 ) .
δ W d = 2 c s Δ y { y - [ ( 1 d - c s ) 2 + ( 1 - κ ) c s 2 ] ρ 2 y 2 } .
Δ y d = δ W d ( tilt ) + ( d / y ) = 2 c s Δ y ( S + B ) .
Δ y d = ( 1 - m ) Δ y .
W c d = - Δ y 32 λ F p 3 [ 1 + ( F p + η ) m 2 ( F - η ) ] .
δ r = ( 0 , 1 2 ϕ c p 2 , - ϕ y ) .
δ W ϕ = 2 ϕ { y + [ c s 2 - 1 2 ( 1 d - c s ) 2 ] y p 2 } .
Δ y ϕ = 2 ϕ D p ( F p + η ) m / ( m + 1 ) .
Δ y ϕ = ( 1 - m ) Δ y c c .
W c ϕ = - ϕ D p ( m - 1 ) ( F p + η ) 32 λ F p 3 m 2 ,
ω c = - K c Δ y ω 0 λ F p 3 [ 1 + ( F p + η ) m 2 ( F - η ) ] ,
K c = 1 32 [ 1 + 2 + 4 + 6 8 - ( 1 + 2 + 4 ) 2 9 ( 1 + 2 ) ] 1 2 ,

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