Abstract

The characteristics of a two-mirror unit magnification system are investigated. This system consists of a concave mirror and a convex mirror and is optimized off-axis. In a concentric system, sagittal imagery has no aberration, whereas meridional imagery has higher-order field curvature and residual aberration. It is shown that the inclination is proportional to 1/Fe3 and the residual aberration to 1/Fe7 The limits of this system are also discussed.

© 1983 Optical Society of America

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References

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  1. A. Offner, U.S. Patent3,748,015.
  2. A. Suzuki, U.S. Patent4,167,677.
  3. See, for example, A. Offner, Opt. Eng. 14, 130 (1975).
    [CrossRef]
  4. I. Kano, Proc. Soc. Photo-Opt. Instrum. Eng. 273, 48 (1979).
  5. C. G. Wynne, Optical Instruments and Techniques 1969 (Oriel Press, Newcastle upon Tyne, 1970).
  6. A. P. Grammatin, Sov. J. Opt. Technol. 38, 210 (1971).
  7. D. P. Feder, J. Opt. Soc. Am. 41, 630 (1951).
    [CrossRef]
  8. Y. Matsui, Lens Sekkei-Hou (Kyouritsu, Tokyo, 1972).
  9. A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1960).
  10. R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).
  11. A. Suzuki, in Proceedings, Forty-First Autumn Symposium of Japan Applied Physics Society (1980), p. 43.
  12. A. Suzuki, Appl. Opt. 22, 3950 (1983).
    [CrossRef] [PubMed]

1983 (1)

1980 (1)

A. Suzuki, in Proceedings, Forty-First Autumn Symposium of Japan Applied Physics Society (1980), p. 43.

1979 (1)

I. Kano, Proc. Soc. Photo-Opt. Instrum. Eng. 273, 48 (1979).

1975 (1)

See, for example, A. Offner, Opt. Eng. 14, 130 (1975).
[CrossRef]

1971 (1)

A. P. Grammatin, Sov. J. Opt. Technol. 38, 210 (1971).

1951 (1)

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1960).

Feder, D. P.

Grammatin, A. P.

A. P. Grammatin, Sov. J. Opt. Technol. 38, 210 (1971).

Kano, I.

I. Kano, Proc. Soc. Photo-Opt. Instrum. Eng. 273, 48 (1979).

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).

Matsui, Y.

Y. Matsui, Lens Sekkei-Hou (Kyouritsu, Tokyo, 1972).

Offner, A.

See, for example, A. Offner, Opt. Eng. 14, 130 (1975).
[CrossRef]

A. Offner, U.S. Patent3,748,015.

Suzuki, A.

A. Suzuki, Appl. Opt. 22, 3950 (1983).
[CrossRef] [PubMed]

A. Suzuki, in Proceedings, Forty-First Autumn Symposium of Japan Applied Physics Society (1980), p. 43.

A. Suzuki, U.S. Patent4,167,677.

Wynne, C. G.

C. G. Wynne, Optical Instruments and Techniques 1969 (Oriel Press, Newcastle upon Tyne, 1970).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

See, for example, A. Offner, Opt. Eng. 14, 130 (1975).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

I. Kano, Proc. Soc. Photo-Opt. Instrum. Eng. 273, 48 (1979).

Proceedings, Forty-First Autumn Symposium of Japan Applied Physics Society (1)

A. Suzuki, in Proceedings, Forty-First Autumn Symposium of Japan Applied Physics Society (1980), p. 43.

Sov. J. Opt. Technol. (1)

A. P. Grammatin, Sov. J. Opt. Technol. 38, 210 (1971).

Other (6)

C. G. Wynne, Optical Instruments and Techniques 1969 (Oriel Press, Newcastle upon Tyne, 1970).

A. Offner, U.S. Patent3,748,015.

A. Suzuki, U.S. Patent4,167,677.

Y. Matsui, Lens Sekkei-Hou (Kyouritsu, Tokyo, 1972).

A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1960).

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).

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Figures (9)

Fig. 1
Fig. 1

Basic constitution of the two-mirror system.

Fig. 2
Fig. 2

Behavior of the telecentric principal ray.

Fig. 3
Fig. 3

Basic parameters for calculating image fields.

Fig. 4
Fig. 4

Basic parameters of the two-mirror system.

Fig. 5
Fig. 5

Sagittal imagery of the concentric system. Object point A and image point B are on the common axis PP′. By rotational symmetry, sagittal imagery has no aberrations.

Fig. 6
Fig. 6

Image fields of the concentric system at image height h.

Fig. 7
Fig. 7

Inclination of meridional image field in the concentric system. (Sagittal image field has no inclination.)

Fig. 8
Fig. 8

Calculation of meridional residual aberration.

Fig. 9
Fig. 9

Residual aberration of meridional imagery at h = 100.

Tables (2)

Tables Icon

Table I Basic Constitution of a Two-Mirror System

Tables Icon

Table II Characteristics of Concentric System (Range of Good Imagery is Valid when the Residual Aberration is Small)

Equations (56)

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d = R 2 cos θ = R 2 2 R 2 h 2 ,
L = R d = R ( 1 1 2 cos θ ) .
sin θ = h R sin ϕ = 1 2 F e ,
τ ν 1
ξ ν = E ν Q ν , ξ ν = E ν Q ν + 1 .
ξ ν = ξ ν ,
G ν = ξ ν ξ ν = 2 ξ ν .
h y ν = 1 ξ ν ( h y ν 1 ξ ν 1 τ ν 1 U y ν ) , U y ν + 1 = U y ν = U y ν + 2 h y ν r ν ,
h z ν = h z ν 1 τ ν 1 U z ν U z ν + 1 = U z ν = U z ν + h z ν G ν r ν ,
ξ 1 = ξ 1 = cos θ ( 1 tan θ cos 2 θ d α ) , ξ 2 = ξ 2 = cos 2 θ ( 1 2 tan 2 θ cos 2 θ d α ) , ξ 3 = ξ 3 = cos θ ( 1 tan θ cos 2 θ d α ) ,
G 1 = G 3 = 2 cos θ ( 1 tan θ cos 2 θ d α ) , G 2 = 2 cos 2 θ ( 1 2 tan 2 θ cos 2 θ d α ) .
O A = OA + A A = h + Δ y ,
h z 1 = 1 , U z 1 = 1 R cos θ ( 1 + tan θ d α ) .
h z 3 = 1 cos 2 θ cos θ k + cos 2 θ cos 2 θ ( tan 2 θ tan θ ) kd α , U z 4 = 1 R cos θ ( 1 + tan θ d α 2 cos 2 θ tan θ kd α ) .
S ks = h z 3 U z 4 = R cos θ ( 1 cos 2 θ cos 2 θ k tan θ d α + 2 sin 2 θ kd α ) .
x s = R cos θ R sin θ d α ( 1 ) S ks = R cos 2 θ cos θ k 2 R cos θ sin θ kd α .
d x s dy = d x s d α / dy d α = 2 tan 2 θ k .
h y 1 = 1 , U y 1 = 1 R ( 1 + 2 tan θ sin 2 θ d α ) .
h y 3 = 1 1 cos 2 θ k + 2 tan θ ( 1 cos 2 θ 1 ) d α 1 cos 2 θ ( tan 2 θ + 2 sin 2 θ 3 tan θ ) kd α , U y 4 = 1 R [ 1 2 tan 2 θ k + 2 tan θ sin 2 θ ( 2 cos 2 θ 1 ) d α 2 cos 2 θ ( tan 2 θ sin 2 θ + 3 sin 2 θ 3 tan θ ) kd α ] .
S km = h y 3 U y 4 = R cos θ [ 1 cos 2 θ cos 2 θ k tan θ cos θ cos 3 θ d α + 2 ( sin 2 θ + 4 tan 3 θ sin 2 θ ) kd α 2 cos 2 θ cos 2 θ tan 2 θ k 2 ] .
x m = R cos 2 θ cos θ k + R 2 cos 2 θ cos θ tan 2 θ k 2 4 R sin 3 θ d α 2 R cos θ ( sin 2 θ + 4 tan 3 θ sin 2 θ ) kd α .
d x m dy = 4 sin 3 θ cos θ cos 2 θ ( 2 tan 2 θ + 8 tan 3 θ sin 2 θ cos 2 θ ) k .
x s = 0 , d x s dy = 0 ,
x m = 4 R sin 3 θ d α , d x m dy = 4 sin 3 θ cos θ cos 2 θ .
x m = 0
sin θ = p sin ϕ = p 1 2 F e ,
d x m dy = 4 sin 3 θ cos θ cos 2 θ = 1 2 1 p 2 4 F e 2 ( 1 p 2 2 F e 2 ) p 3 F e 3 .
d x m = 4 sin 3 θ cos θ cos 2 θ dy m λ F e 2 ,
S = 2 dy = m 2 λ F e 2 cos θ cos 2 θ sin 3 θ = 4 m p 3 λ F e 5 1 p 2 4 F e 2 ( 1 p 2 2 F e 2 ) .
S = λ F e 2 cos θ cos 2 θ sin 3 θ .
sin θ 1 = h R cos ϕ .
d θ 1 = ½ sin θ sin 2 ϕ , cos θ 1 = cos θ + ½ sin 2 θ sin 2 ϕ .
O C 1 = R 2 2 R 2 h 2 cos 2 ϕ = R 2 cos θ R 4 tan 2 θ cos θ sin 2 ϕ .
C 1 C 0 = R 4 tan 2 θ cos θ sin 2 ϕ ,
C C 0 = C 1 C 0 tan 2 θ 1 = R 2 tan 2 θ sin θ cos 2 θ sin 2 ϕ .
d η = sin 3 θ cos θ cos 2 θ sin 2 ϕ .
y h cos ϕ cos ( ϕ 2 d η ) = tan ( ϕ 2 d η ) [ x + h cos ϕ sin ( ϕ 2 d η ) ] .
y 0 = h cos ϕ cos ( ϕ 2 d η ) = h ( 1 2 tan ϕ d η ) .
y 0 = h 2 sin 4 θ cos θ cos 2 θ sin 3 ϕ R ;
Δ t = 2 sin 4 θ cos θ cos 2 θ sin 3 ϕ R
W m = 0 ϕ Δ t cos ϕ d ϕ = sin 4 θ 2 cos θ cos 2 θ sin 4 ϕ R
= sin 3 θ 2 cos θ cos 2 θ sin 4 ϕ h .
W m = sin 7 θ 2 cos θ cos 2 θ h .
W m = 1 2 1 1 4 F e 2 ( 1 1 4 F e 2 ) 1 ( 2 F e ) 7 h .
W d = Δ d 8 F e 2 = Δ d 2 sin 2 ϕ .
Δ d = sin 3 θ cos θ cos 2 θ sin 2 ϕ h .
Δ h = sin 2 ϕ 4 h ;
R ( 1 1 2 cos Θ )
R 2 cos Θ ( 1 k )
R ( 1 1 2 cos Θ )
R ( 1 1 2 cos Θ )
R 2 cos Θ
R ( 1 1 2 cos Θ )
sin 3 Θ sin 4 ϕ 2 cos Θ cos 2 Θ h
sin 3 Θ 2 cos Θ cos 2 Θ sin 2 ϕ
λ F e 2 2 cos Θ cos 2 Θ sin 3 Θ

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