Abstract

A geometrical theory of aberrations for the vicinity of the focus of arbitrary off-axis sections of conic mirrors is derived. It is shown that an off-axis conic mirror introduces linear astigmatism in the image. However, in classical two-mirror telescopes this aberration can be eliminated by tilting the secondary parent mirror axis. It is also shown that the practical geometrical-optics performance of a classical off-axis two-mirror telescope with no linear astigmatism is equivalent to the performance of an on-axis system, proving that both systems have identical third-order coma. To demonstrate the applicability of the theory developed in a practical system, a fast (i.e., f2), compact, obstruction-free classical off-axis Cassegrain telescope is designed.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. N. Wilson, Reflecting Telescope Optics (Springer, 1996).
    [CrossRef]
  2. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (Optical Science Center, University of Arizona, Tucson, Arizona 1976).
  3. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (Optical Science Center, University of Arizona, Tucson, Arizona 1980).
  4. J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” in International Lens Design Conference, W. H. Taylor and D. T. Moore, eds., Proc. SPIE554, 76–81 (1985).
  5. P. J. Sands, “Aberration coefficients of plane symmetric systems,” J. Opt. Soc. Am. 62, 1211–1220 (1972).
    [CrossRef]
  6. J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. (Bellingham) 33, 2045–2061 (1994).
    [CrossRef]
  7. B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3292–3307 (1994).
    [CrossRef]
  8. D. J. Schroeder, Astronomical Optics (Academic, 2000), Chap. 5.
  9. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1997).
  10. S. Chang, “Geometrical theory of aberrations for classical offset reflector antennas and telescopes,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 2003).
  11. W. V.T. Rusch, A. Prata, Y. Rahmat-Samii, R. A. Shore, “Derivation and application of the equivalent paraboloid for classical offset Cassegrain and Gregorian antennas,” IEEE Trans. Antennas Propag. 38, 1141–1149 (1990).
    [CrossRef]
  12. O. N. Stavroudis, “Confocal prolate spheroids in an off-axis system,” J. Opt. Soc. Am. A 9, 2083–2088 (1992).
    [CrossRef]
  13. K. W. Brown, A. Prata, “A design procedure for classical offset dual reflector antennas with circular apertures,” IEEE Trans. Antennas Propag. 42, 1145–1153 (1994).
    [CrossRef]
  14. S. Chang, A. Prata, “The design of offset Dragonian reflector antennas with circular apertures,” IEEE Trans. Antennas Propag. 52, 12–19 (2004).
    [CrossRef]
  15. S. Chang, A. Prata, “A design procedure for classical offset inverse Cassegrain antennas with circular apertures,” in IEEE Antennas and Propagation Society Symposium Digest (Institute of Electrical and Electronics Engineers, 2001), pp. 534–537.

2004

S. Chang, A. Prata, “The design of offset Dragonian reflector antennas with circular apertures,” IEEE Trans. Antennas Propag. 52, 12–19 (2004).
[CrossRef]

1994

B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3292–3307 (1994).
[CrossRef]

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. (Bellingham) 33, 2045–2061 (1994).
[CrossRef]

K. W. Brown, A. Prata, “A design procedure for classical offset dual reflector antennas with circular apertures,” IEEE Trans. Antennas Propag. 42, 1145–1153 (1994).
[CrossRef]

1992

1990

W. V.T. Rusch, A. Prata, Y. Rahmat-Samii, R. A. Shore, “Derivation and application of the equivalent paraboloid for classical offset Cassegrain and Gregorian antennas,” IEEE Trans. Antennas Propag. 38, 1141–1149 (1990).
[CrossRef]

1972

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1997).

Brown, K. W.

K. W. Brown, A. Prata, “A design procedure for classical offset dual reflector antennas with circular apertures,” IEEE Trans. Antennas Propag. 42, 1145–1153 (1994).
[CrossRef]

Buchroeder, R. A.

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (Optical Science Center, University of Arizona, Tucson, Arizona 1976).

Chang, S.

S. Chang, A. Prata, “The design of offset Dragonian reflector antennas with circular apertures,” IEEE Trans. Antennas Propag. 52, 12–19 (2004).
[CrossRef]

S. Chang, “Geometrical theory of aberrations for classical offset reflector antennas and telescopes,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 2003).

S. Chang, A. Prata, “A design procedure for classical offset inverse Cassegrain antennas with circular apertures,” in IEEE Antennas and Propagation Society Symposium Digest (Institute of Electrical and Electronics Engineers, 2001), pp. 534–537.

Forbes, G. W.

Prata, A.

S. Chang, A. Prata, “The design of offset Dragonian reflector antennas with circular apertures,” IEEE Trans. Antennas Propag. 52, 12–19 (2004).
[CrossRef]

K. W. Brown, A. Prata, “A design procedure for classical offset dual reflector antennas with circular apertures,” IEEE Trans. Antennas Propag. 42, 1145–1153 (1994).
[CrossRef]

W. V.T. Rusch, A. Prata, Y. Rahmat-Samii, R. A. Shore, “Derivation and application of the equivalent paraboloid for classical offset Cassegrain and Gregorian antennas,” IEEE Trans. Antennas Propag. 38, 1141–1149 (1990).
[CrossRef]

S. Chang, A. Prata, “A design procedure for classical offset inverse Cassegrain antennas with circular apertures,” in IEEE Antennas and Propagation Society Symposium Digest (Institute of Electrical and Electronics Engineers, 2001), pp. 534–537.

Rahmat-Samii, Y.

W. V.T. Rusch, A. Prata, Y. Rahmat-Samii, R. A. Shore, “Derivation and application of the equivalent paraboloid for classical offset Cassegrain and Gregorian antennas,” IEEE Trans. Antennas Propag. 38, 1141–1149 (1990).
[CrossRef]

Rogers, J. R.

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” in International Lens Design Conference, W. H. Taylor and D. T. Moore, eds., Proc. SPIE554, 76–81 (1985).

Rusch, W. V.T.

W. V.T. Rusch, A. Prata, Y. Rahmat-Samii, R. A. Shore, “Derivation and application of the equivalent paraboloid for classical offset Cassegrain and Gregorian antennas,” IEEE Trans. Antennas Propag. 38, 1141–1149 (1990).
[CrossRef]

Sands, P. J.

Sasian, J. M.

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. (Bellingham) 33, 2045–2061 (1994).
[CrossRef]

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, 2000), Chap. 5.

Shore, R. A.

W. V.T. Rusch, A. Prata, Y. Rahmat-Samii, R. A. Shore, “Derivation and application of the equivalent paraboloid for classical offset Cassegrain and Gregorian antennas,” IEEE Trans. Antennas Propag. 38, 1141–1149 (1990).
[CrossRef]

Stavroudis, O. N.

Stone, B. D.

Thompson, K. P.

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (Optical Science Center, University of Arizona, Tucson, Arizona 1980).

Wilson, R. N.

R. N. Wilson, Reflecting Telescope Optics (Springer, 1996).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1997).

IEEE Trans. Antennas Propag.

W. V.T. Rusch, A. Prata, Y. Rahmat-Samii, R. A. Shore, “Derivation and application of the equivalent paraboloid for classical offset Cassegrain and Gregorian antennas,” IEEE Trans. Antennas Propag. 38, 1141–1149 (1990).
[CrossRef]

K. W. Brown, A. Prata, “A design procedure for classical offset dual reflector antennas with circular apertures,” IEEE Trans. Antennas Propag. 42, 1145–1153 (1994).
[CrossRef]

S. Chang, A. Prata, “The design of offset Dragonian reflector antennas with circular apertures,” IEEE Trans. Antennas Propag. 52, 12–19 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng. (Bellingham)

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. (Bellingham) 33, 2045–2061 (1994).
[CrossRef]

Other

S. Chang, A. Prata, “A design procedure for classical offset inverse Cassegrain antennas with circular apertures,” in IEEE Antennas and Propagation Society Symposium Digest (Institute of Electrical and Electronics Engineers, 2001), pp. 534–537.

R. N. Wilson, Reflecting Telescope Optics (Springer, 1996).
[CrossRef]

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (Optical Science Center, University of Arizona, Tucson, Arizona 1976).

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (Optical Science Center, University of Arizona, Tucson, Arizona 1980).

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” in International Lens Design Conference, W. H. Taylor and D. T. Moore, eds., Proc. SPIE554, 76–81 (1985).

D. J. Schroeder, Astronomical Optics (Academic, 2000), Chap. 5.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1997).

S. Chang, “Geometrical theory of aberrations for classical offset reflector antennas and telescopes,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 2003).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Representative classical two-mirror system.

Fig. 2
Fig. 2

Conic section coordinate systems.

Fig. 3
Fig. 3

OPL of an object located on the y = 0 plane.

Fig. 4
Fig. 4

Astigmatic image planes of an off-axis paraboloidal mirror (field curvatures have been neglected).

Fig. 5
Fig. 5

Effect of the aperture-stop location on a single-mirror system.

Fig. 6
Fig. 6

Classical off-axis two-mirror telescopes. (a) Cassegrain with a convex secondary mirror, (b) Gregorian, (c) Cassegrain with a concave secondary mirror, (d) inverse Cassegrain.

Fig. 7
Fig. 7

Parameters of classical off-axis two-mirror telescope.

Fig. 8
Fig. 8

Equivalent paraboloid of an off-axis Cassegrain system.

Fig. 9
Fig. 9

Side view of the case study off-axis Cassegrain system.

Fig. 10
Fig. 10

Astigmatic image surfaces of the case study telescope. (a) Positive field, (b) negative field. Generated with CODE V (Optical Research Associates, Pasadena, California).

Fig. 11
Fig. 11

Spot diagrams. (a) 1 m diameter f 2 case study telescope, (b) single 1 m diameter f 2 on-axis paraboloidal mirror. Both systems produce point images at the center of field, which are invisible in the figure. Generated with CODE V (Optical Research Associates, Pasadena, California).

Tables (1)

Tables Icon

Table 1 Telescope Data a

Equations (90)

Equations on this page are rendered with MathJax. Learn more.

x 2 + y 2 2 R z + ( 1 + K ) z 2 = 0 ,
x = x cos θ 0 z sin θ 0 + x 0 ,
y = y ,
z = x sin θ 0 + z cos θ 0 + z 0 .
( 1 + K cos 2 θ 0 ) z 2 + [ x K sin 2 θ 0 2 sin θ 0 x 0 + 2 ( 1 + K ) z 0 cos θ 0 2 R cos θ 0 ] z + ( 1 + K sin 2 θ 0 ) x 2 + 2 [ x 0 cos θ 0 R sin θ 0 + ( 1 + K ) z 0 sin θ 0 ] x + y 2 + [ x 0 2 2 R z 0 + ( 1 + K ) z 0 2 ] = 0 .
x 0 2 2 R z 0 + ( 1 + K ) z 0 2 = 0 ,
z x x = x 0 , y = 0 , z = z 0 = tan θ 0 ,
x 0 cos θ 0 R sin θ 0 + ( 1 + K ) z 0 sin θ 0 = 0
x 0 = R sin θ 0 1 + K sin 2 θ 0 ,
z 0 = R 1 + K ( 1 cos θ 0 1 + K sin 2 θ 0 ) ,
( 1 + K cos 2 θ 0 ) z 2 + ( x K sin 2 θ 0 2 R 1 + K sin 2 θ 0 ) z + ( 1 + K sin 2 θ 0 ) x 2 + y 2 = 0 .
z = a 1 x 2 + a 2 y 2 + a 3 x 3 + a 4 x y 2 + O ( x n x y n y ) ,
a 1 = ( 1 + K sin 2 θ 0 ) 3 2 2 R ,
a 2 = ( 1 + K sin 2 θ 0 ) 1 2 2 R ,
a 3 = K sin 2 θ 0 ( 1 + K sin 2 θ 0 ) 2 4 R 2 ,
a 4 = K sin 2 θ 0 ( 1 + K sin 2 θ 0 ) 4 R 2 ,
OPL = d + d ,
d = ( x x o ) 2 + ( y y o ) 2 + ( z z o ) 2 ,
d = ( x x i ) 2 + ( y y i ) 2 + ( z z i ) 2 .
x o = s sin ( θ s + θ ) ,
z o = s cos ( θ s + θ ) ,
x i = s sin ( θ s + θ ) ,
z i = s cos ( θ s + θ ) ,
d = s ( 1 μ ) 1 2 ,
μ = x s 2 sin ( θ s + θ ) x 2 s 2 [ 1 2 a 1 s cos ( θ s + θ ) ] y 2 s 2 [ 1 2 a 2 s cos ( θ s + θ ) ] + x 3 s 2 a 3 cos ( θ s + θ ) + x y 2 s 2 a 4 cos ( θ s + θ ) + O ( x n x y n y ) .
d = s x sin ( θ s + θ ) + x 2 2 cos ( θ s + θ ) [ cos ( θ s + θ ) s 2 a 1 ] + y 2 2 [ 1 s 2 a 2 cos ( θ s + θ ) ] x 3 2 cos ( θ s + θ ) { 2 a 3 sin ( θ s + θ ) s [ cos ( θ s + θ ) s 2 a 1 ] } x y 2 2 { 2 a 4 cos ( θ s + θ ) sin ( θ s + θ ) s [ 1 s 2 a 2 cos ( θ s + θ ) ] } + O ( x n x y n y ) ,
OPL = s + s + A 1 x 2 + A 1 y 2 + A 2 x 3 + A 2 x y 2 + O ( x n x y n y ) ,
A 1 = 1 2 cos ( θ s + θ ) [ cos ( θ s + θ ) ( 1 s + 1 s ) 4 a 1 ] ,
A 1 = 1 2 [ ( 1 s + 1 s ) 4 a 2 cos ( θ s + θ ) ] ,
A 2 = 1 2 cos ( θ s + θ ) { sin ( θ s + θ ) ( 1 s 1 s ) [ cos ( θ s + θ ) ( 1 s + 1 s ) 2 a 1 ] 4 a 3 } ,
A 2 = 1 2 sin ( θ s + θ ) ( 1 s 1 s ) [ ( 1 s + 1 s ) 2 a 2 cos ( θ s + θ ) ] 2 a 4 cos ( θ s + θ ) .
cos θ s ( 1 s + 1 s ) 2 ( 1 + K sin 2 θ 0 ) 3 2 R = 0 ,
cos θ s ( 1 s + 1 s ) 2 ( 1 + K sin 2 θ 0 ) 1 2 R cos 2 θ s = 0 .
K sin 2 θ 0 = sin 2 θ s .
1 s t = 2 cos 3 θ s R cos ( θ s + θ ) 1 s ,
1 s s = 2 cos θ s cos ( θ s + θ ) R 1 s ,
1 s t = 2 cos 2 θ s R 1 s + θ sin 2 θ s R + θ 2 1 + sin 2 θ s R + O ( θ 3 ) ,
1 s s = 2 cos 2 θ s R 1 s θ sin 2 θ s R θ 2 cos 2 θ s R + O ( θ 3 ) .
Δ s s t s s = s s s t s t s s = 2 sin 2 θ s R θ + 2 R θ 2 + O ( θ 3 ) .
1 s t = 1 s 0 ( 1 + θ tan θ s ) + O ( θ 2 ) ,
1 s s = 1 s 0 ( 1 θ tan θ s ) + O ( θ 2 ) ,
1 s 0 = 2 cos 2 θ s R .
A 2 = cos 3 θ s R [ sin θ s ( cos θ s θ sin θ s ) ( 1 s 1 s 0 ) + θ cos 2 θ s ( 1 s 1 R ) ] + O ( θ 2 ) ,
1 s 0 = cos θ s ( cos θ s sin 2 θ s K ) R .
A 2 = cos 5 θ s R 2 θ + O ( θ 2 ) .
( W + a ) tan ψ = ( ρ 0 + a ) tan θ b ,
a = ( W + a ) tan ψ tan γ ,
tan θ b = ( W ρ 0 ) tan ψ 1 ( 1 W ρ 0 ) tan γ tan ψ .
θ b = ψ W ρ 0 + ψ 2 W ρ 0 ( 1 W ρ 0 ) tan ψ s + O ( ψ 3 ) .
θ = ψ θ b = ψ ( 1 W ρ 0 ) ψ 2 W ρ 0 ( 1 W ρ 0 ) tan ψ s + O ( ψ 3 ) .
2 θ s + ( A B C θ b ) + ( B A C + θ a ) = π .
ρ 0 θ b = C D ¯ sin ( π 2 + ψ s ) + O ( ψ ) ,
ρ 0 θ a = C D ¯ sin ( π 2 ψ s ) + O ( ψ ) ,
θ a = θ b ρ 0 ρ 0 + O ( ψ 2 ) .
θ s = ψ s + W 2 ( 1 ρ 0 1 ρ 0 ) ψ + O ( ψ 2 ) .
W c = W ( 1 + ψ tan ψ s ) + O ( ψ 2 ) .
1 s 0 = 1 ρ 0 ψ W ρ 0 2 tan ψ s + O ( ψ 2 ) .
Δ s s t s s = 2 ( 1 W ρ 0 ) sin 2 ψ s R ψ + 2 R ( 1 W ρ 0 ) [ ( 1 W ρ 0 ) + W ( 1 ρ 0 1 ρ 0 ) cos 2 ψ s 2 W ρ 0 sin 2 ψ s ] ψ 2 ( 1 s t 1 s s ) + O ( ψ 3 ) ,
A 2 = cos 3 ψ s R { sin ψ s cos ψ s ( 1 s 1 ρ 0 ) + [ ( 1 W ρ 0 ) cos 2 ψ s ( 1 s 1 R ) + W ρ 0 2 sin 2 ψ s ] ψ } + O ( ψ 2 ) ,
Δ s 2 s s 2 s t 2 = 2 ω 1 2 sin 2 i 2 R 2 ( 1 s t 2 1 s s 2 ) + O ( ω 2 ) ,
s t 2 = W c s t 1 ,
s s 2 = W c s s 1 ,
1 s t 2 = 1 2 ω [ 1 2 2 ( tan i 1 + tan i 2 ) + 1 2 tan i 2 ] + O ( ω 2 ) ,
1 s s 2 = 1 2 + ω [ 1 2 2 ( tan i 1 tan i 2 ) 1 2 tan i 2 ] + O ( ω 2 ) .
Δ s s t 2 s s 2 = 2 ω 1 2 2 ( 1 R 1 sin 2 i 1 2 R 2 sin 2 i 2 ) + O ( ω 2 ) ,
1 = R 1 2 cos 2 i 1 ,
1 R 1 sin 2 i 1 = 2 R 2 sin 2 i 2 .
tan β 2 = ( e 1 e + 1 ) 2 tan β θ 0 2 ,
tan β 2 = ( e + 1 e 1 ) 2 tan β θ 0 2 .
TA x f = 0 Δ x f ,
AA x f = TA x f f .
ATC = 0 f x f ( A 2 p x p 3 + A 2 s x s 3 ) x a = ( D 2 ) ,
x s x f ( 1 ) δ + 1 1 cos i 2 ,
x p x s 1 cos i 2 2 cos i 1 .
ATC = ( 1 ) δ + 1 3 4 D 2 0 f [ A 2 p 1 cos 3 i 1 1 2 A 2 s 1 cos 3 i 2 ( 2 1 ) 2 ] .
A 2 p = ω cos 5 i 1 R 1 2 + O ( ω 2 ) .
A 2 s = ω cos 5 i 2 R 2 ( 1 2 2 ) ( tan i 1 tan i 2 2 R 2 + 1 ) + O ( ω 2 ) .
2 R 2 = tan i 1 sin 2 i 2 ,
tan i 1 = 0 + 2 0 tan i 2 ,
cos 2 i 2 R 2 = cos 2 i 1 R 1 ( 1 2 + 1 0 ) ,
1 R 2 = 1 2 ( 1 0 + 1 0 + 2 ) + 1 R 1 1 2 ( 0 0 + 2 ) .
A 2 s = 1 4 ω cos 3 i 2 1 0 2 2 ( 1 2 R 1 1 0 2 2 2 ) + O ( ω 2 ) .
f = ( 1 ) δ + 1 0 1 2 .
ATC = 3 ω 16 ( f D ) 2 + O ( ω 2 ) .
1 s t 2 1 s s 2 = 2 ω 1 2 sin 2 i 2 R 2 + ω 2 2 R 2 [ ( 1 2 ) 2 ( 1 2 1 0 ) 1 2 cos 2 i 2 + 1 2 2 tan i 2 sin 2 i 2 ] ( 1 s t 2 1 s s 2 ) + O ( ω 3 ) ,
1 s t 2 1 s s 2 = s t 1 s s 1 ( W c s t 1 ) ( W c s s 1 ) .
s t 1 = R 1 2 cos 2 i 1 ( 1 ω tan i 1 1 2 ω 2 ) + O ( ω 3 ) ,
s s 1 = R 1 2 cos 2 i 1 [ 1 + ω tan i 1 + ω 2 ( 1 2 + tan 2 i 1 ) ] + O ( ω 3 ) .
1 s t 2 1 s s 2 = 1 2 2 [ 2 ω tan i 1 + ω 2 ( 1 4 2 tan i 1 tan i 2 + tan 2 i 1 ) ] + O ( ω 3 ) .
Δ s = ω 2 f 0 [ f ( 1 tan i 1 tan i 2 ) ( 1 ) δ ] + O ( ω 3 ) .

Metrics