Abstract

In a concentric two-mirror unit magnification system, only meridional imagery suffers from higher-order field curvature and residual aberration. Here a new parameter is introduced to optimize these two problems. It is demonstrated that they are simultaneously optimized when the convex-mirror radius is exactly half of that of the concave mirror. Then the inclination of sagittal and meridional fields is balanced, and residual aberration is minimized at several times less than that of the concentric system.

© 1983 Optical Society of America

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References

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  1. A. Suzuki, Appl. Opt. 22, 3950 (1983).
    [CrossRef] [PubMed]
  2. A. Bouwers, Achievements in Optics (Elsevier, New York, 1948).
  3. A. Suzuki, in Proceedings, Forty-Second Autumn Symposium of Japan Applied Physics Society (1981), p. 96.
  4. A. Suzuki, Proc. Soc. Photo-Opt. Instrum. Eng. 275, 35 (1981).
  5. C. S. Ih, K. Yen, Appl. Opt. 19, 4196 (1980).
    [CrossRef] [PubMed]

1983 (1)

1981 (2)

A. Suzuki, in Proceedings, Forty-Second Autumn Symposium of Japan Applied Physics Society (1981), p. 96.

A. Suzuki, Proc. Soc. Photo-Opt. Instrum. Eng. 275, 35 (1981).

1980 (1)

Bouwers, A.

A. Bouwers, Achievements in Optics (Elsevier, New York, 1948).

Ih, C. S.

Suzuki, A.

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

A. Suzuki, in Proceedings, Forty-Second Autumn Symposium of Japan Applied Physics Society (1981), p. 96.

A. Suzuki, Proc. Soc. Photo-Opt. Instrum. Eng. 275, 35 (1981).

Yen, K.

Appl. Opt. (2)

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

A. Suzuki, Proc. Soc. Photo-Opt. Instrum. Eng. 275, 35 (1981).

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

A. Suzuki, in Proceedings, Forty-Second Autumn Symposium of Japan Applied Physics Society (1981), p. 96.

Other (1)

A. Bouwers, Achievements in Optics (Elsevier, New York, 1948).

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Figures (7)

Fig. 1
Fig. 1

Basic constitution and parameters of a two-mirror system.

Fig. 2
Fig. 2

Image field characteristics of an optimized system.

Fig. 3
Fig. 3

Calculation of meridional imagery in a symmetrical nonconcentric system.

Fig. 4
Fig. 4

Calculation of sagittal imagery by a projected system.

Fig. 5
Fig. 5

Residual aberration of an optimized nonconcentric system.

Fig. 6
Fig. 6

Wave aberration map at paraxial image points where astigmatism is zero: (A) concentric system (h = 100); (B) optimized system (h = 100, Δha = 0.171, so image height is 100.171). The pitch of the contour curves is 0.04λg.

Fig. 7
Fig. 7

Wave aberration map of the optimized position in the rms sense: (A) concentric system (image height is 100.40); (B) optimized system [image height 100.171, 1-μm defocus from Fig. 6(B) position]. The pitch of the contour curves is 0.02λg.

Tables (5)

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Table I Basic Constitution of a Two-Mirror System

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Table II Exact Condition for Balancing Sagittal and Meridional Image Fields

Tables Icon

Table III Projected Two-Mirror System for Calculating Sagittal Imagery

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Table IV Construction Data of a Two-Mirror System

Tables Icon

Table V Characteristics of an Optimized Nonconcentric System (Range of Good Imagery is Valid when Residual Aberration is Small)

Equations (78)

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x s = R cos 2 θ cos θ k 2 R cos θ sin 2 θ k d α ,
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 ) k d α .
Δ A S h = x m x s = R 2 cos 2 θ cos θ tan 2 θ k 2 .
d x s x y = 2 tan 2 θ k , d x m d y = 4 sin 3 θ cos θ cos 2 θ ( 2 tan 2 θ + 8 tan 3 θ sin 2 θ cos 2 θ ) k .
2 tan 2 θ 8 tan 3 θ sin 2 θ cos 2 θ .
tan Θ = 4 sin 3 θ cos θ cos 2 θ .
2 tan 2 θ k = 4 sin 3 θ cos θ cos 2 θ ( 2 tan 2 θ + 8 tan 3 θ sin 2 θ cos 2 θ ) k .
k i = sin 2 θ cos 2 θ 2 ( sin 4 θ + cos 4 θ ) .
cos θ ( 1 k i ) 1 = ( 1 cos θ ) 2 ( 3 cos 3 θ + 2 cos 2 θ 2 cos θ 2 ) 2 ( sin 4 θ + cos 4 θ ) .
k a = 1 1 cos θ .
h z 1 = 1 U z 1 = 1 R cos θ ( 1 a k + a 2 k 2 ) .
h z 3 = 1 cos 2 θ cos 2 θ k 2 a k + cos 2 θ cos 2 θ a k 2 + 2 a 2 k 2 , U z 4 = 1 R cos θ ( 1 a k + a 2 k 2 ) .
S k z = h z 3 U z 4 = R cos θ [ 1 ( cos 2 θ cos 2 θ + a ) k ] .
h y 1 = 1 , U y 1 = 1 R ( 1 a k + a 2 k 2 ) .
S k y = R cos θ [ 1 ( cos 2 θ cos 2 θ + a ) k 2 ( 2 sin 2 θ cos 2 θ a 2 + 2 tan 2 θ a + cos 2 θ cos 2 θ tan 2 θ ) k 2 ] .
S k y 0 = R cos θ [ 1 ( cos 2 θ cos 2 θ + a ) k ] .
S k z = S k y 0 .
S 1 = S k z = S k y 0 ,
a = cos 2 θ 2 cos 2 θ .
Δ A S k = S k y S k y 0 = 2 R cos θ ( 2 sin 2 θ cos 2 θ a 2 + 2 tan 2 θ a + cos 2 θ cos 2 θ tan 2 θ ) k 2 .
d d a Δ A S k = 4 R cos θ k 2 ( 2 sin 2 θ cos 2 θ a + tan 2 θ ) .
a = cos 2 θ 2 cos 2 θ .
Δ A S k = R cos 2 θ cos θ tan 2 θ k 2 .
Δ h 0 = Δ A S k tan Θ = cos 2 2 θ 4 cos 2 θ sin θ k 2 R = 1 tan 2 2 θ k 2 h .
x 0 = cos 2 θ 2 cos θ ( k k 3 ) R = 1 tan 2 θ ( k k 3 ) h .
Δ h a = ( 1 cos θ ) 2 tan 2 2 θ cos 2 θ h ,
x a = cos 2 θ ( 1 cos θ ) ( 2 cos θ 1 ) 2 cos 4 θ sin θ h ,
Δ A S a = 2 tan 2 θ ( 1 cos θ ) 2 tan 2 θ h .
Δ h = sin 2 ϕ 4 h ,
Δ h Δ h a = 4 cos 4 θ cos 2 2 θ .
O 1 O 2 = R 2 cos θ k 1 k , O 1 O 3 = cos 2 θ 2 cos θ k R ,
L 1 = R ( sin θ cos ϕ + k cos 2 θ 2 cos θ sin ϕ ) ,
sin θ 1 = sin ( θ + d θ 1 ) = L 1 R , d θ 1 = ½ tan θ sin 2 ϕ + k 2 cos 2 θ cos 2 θ sin ϕ .
L 3 = R [ sin θ cos ϕ + k ( cos 2 θ 2 cos θ sin ϕ + sin 3 θ cos 2 θ cos ϕ sin 2 ϕ sin 2 2 θ 2 cos θ cos 2 θ sin 3 ϕ ) k 2 cos 2 θ sin 2 θ cos 3 θ sin ϕ ] ,
U 4 = ϕ 2 sin 3 θ cos θ cos 2 θ sin 2 ϕ 2 k sin θ cos 3 θ sin 2 ϕ .
y = tan U 4 x + L 3 cos U 4 .
y = h R 2 sin 4 θ cos θ cos 2 θ ( 1 + cos 2 θ + cos 4 θ sin 2 θ cos 2 θ k ) sin 3 ϕ k 2 R cos 2 θ cos θ tan 2 θ sin ϕ .
Δ t = R 2 sin 4 θ cos θ cos 2 θ ( 1 + cos 2 θ + cos 4 θ sin 2 θ cos 2 θ k ) sin 3 ϕ k 2 R cos 2 θ cos θ tan 2 θ sin ϕ .
k m = sin 2 θ cos 2 θ cos 4 θ + cos 2 θ ,
k a k m = ( 1 cos θ ) 2 ( 4 cos 3 θ + 3 cos 2 θ cos θ 1 ) cos θ ( cos 4 θ + cos 2 θ ) .
k a k m .
sin 2 θ 2 sin ϕ
d s = cos 2 θ 2 cos θ R .
S 1 s = R 2 cos θ ( 2 cos 2 θ k cos 2 θ ) .
S 2 s = R 2 cos θ 1 1 k ( 2 cos 2 θ k cos 2 θ ) .
R s = R cos θ 1 2 k ( 2 cos 2 θ k cos 2 θ ) ;
x 2 a E 2 + z 2 b E 2 = 1 .
a E = R 4 cos θ 2 k 1 k ( 2 cos 2 θ k cos 2 θ ) , b E = R 2 cos θ 1 1 k ( 2 cos 2 θ k cos 2 θ ) .
W E ( z ) = R cos θ z 2 2 R cos θ 1 2 k ( 2 cos 2 θ k cos 2 θ ) ( 2 k ) ( 1 k ) z 4 2 R 3 cos 3 θ ( 2 cos 2 θ k cos 2 θ ) 3 .
W c ( z ) = R cos θ z 2 2 R cos θ 1 2 k ( 2 cos 2 θ k cos 2 θ ) ( 2 k ) 3 z 4 8 R 3 cos 3 θ ( 2 cos 2 θ k cos 2 θ ) 3 .
Δ W = W C W E = ( 2 k ) k 2 8 R 3 cos 3 θ ( 2 cos 3 θ k cos 2 θ ) 3 z 4 .
W s = 4 Δ W = ( 2 k ) k 2 2 R 3 cos 3 θ ( 2 cos 2 θ k cos 2 θ ) 3 z 4 .
z = R 2 cos θ ( 2 cos 2 θ k cos 2 θ ) sin ϕ .
W s = h 16 sin 2 θ ( 2 cos 2 θ k cos 2 θ ) ( 2 k ) k 2 sin 4 ϕ .
W s h 8 cos θ sin θ k 2 sin 4 ϕ .
W s = h 32 cos θ sin 7 θ ,
W m W s = 16 cos 2 θ cos 2 θ .
R ( 1 1 2 cos Θ )
R 2 cos Θ ( 1 k )
R ( 1 1 2 cos Θ )
1 2 cos Θ ( 1 k 1 )
1 2 cos Θ ( 1 k 1 )
R cos Θ 2 cos 2 Θ k cos 2 Θ 2 k
cos 2 Θ 2 cos Θ R
R 2 cos Θ ( 1 k )
cos 2 Θ 2 cos Θ R
R cos Θ 2 cos 2 Θ k cos 2 Θ 2 k
R ( 1 1 2 cos Θ )
R 2
R ( 1 1 2 cos Θ )
R { 1 + cos 2 Θ 2 cos 4 Θ ( 1 cos Θ ) ( 2 cos Θ 1 ) }
R { 1 + cos 2 Θ 2 cos 4 Θ ( 1 cos Θ ) ( 2 cos Θ 1 ) }
h { 1 + ( 1 cos Θ ) 2 tan 2 2 Θ cos 2 Θ }
h { 1 + ( 1 cos Θ ) 2 tan 2 2 Θ cos 2 Θ }
2 sin 3 Θ cos Θ cos 2 Θ
W s = h 4 ( 1 cos Θ ) 2 sin 2 Θ sin 4 ϕ
sin Θ cos 2 Θ 2 cos 5 Θ ( 1 cos Θ ) 2
2 λ Fe 2 cos Θ cos 2 Θ sin 3 Θ

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