Abstract

The modern schiefspiegler is an optical system consisting of two prolate spheroid mirrors coupled by conjoining the distal focus of the first spheroid with the proximal focus of the second. A ray through the proximal focus of the first spheroid must, after reflection, pass through the common focus and then, after a second reflection, through the distal focus of the second spheroid. The pseudo axis is such a ray with the property that all other meridional rays are symmetric about it. The entrance and the exit pupils are perpendicular to the pseudo axis and are located, respectively, at the first proximal focus and the second distal focus. Therefore rays through the foci are chief rays. It is shown that all chief rays are symmetric with respect to the pseudo axis and that, therefore, magnification is uniform for all object points. An expression for magnification is obtained that leads to a distortion function that depends only on magnification.

© 1998 Optical Society of America

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References

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  1. A. Kutter, Der Schiefspiegler (Biberacher Verlagsdruckerei, Biberach an der Riss, Germany, 1953).
  2. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1968), p. 48.
  3. O. N. Stavroudis, A. J. Ames, “Confocal prolate spheroids in an off-axis system,” J. Opt. Soc. Am. A 9, 2083–2088 (1992).
    [CrossRef]
  4. O. N. Stavroudis, “Schiefspiegler: an off-axis reflecting optical system,” Opt. Eng. 33, 116–124 (1994).
    [CrossRef]
  5. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Arizona, 1976).

1994

O. N. Stavroudis, “Schiefspiegler: an off-axis reflecting optical system,” Opt. Eng. 33, 116–124 (1994).
[CrossRef]

1992

Ames, A. J.

Buchroeder, R. A.

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

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1968), p. 48.

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1968), p. 48.

Kutter, A.

A. Kutter, Der Schiefspiegler (Biberacher Verlagsdruckerei, Biberach an der Riss, Germany, 1953).

Stavroudis, O. N.

O. N. Stavroudis, “Schiefspiegler: an off-axis reflecting optical system,” Opt. Eng. 33, 116–124 (1994).
[CrossRef]

O. N. Stavroudis, A. J. Ames, “Confocal prolate spheroids in an off-axis system,” J. Opt. Soc. Am. A 9, 2083–2088 (1992).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

O. N. Stavroudis, “Schiefspiegler: an off-axis reflecting optical system,” Opt. Eng. 33, 116–124 (1994).
[CrossRef]

Other

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

A. Kutter, Der Schiefspiegler (Biberacher Verlagsdruckerei, Biberach an der Riss, Germany, 1953).

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1968), p. 48.

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Figures (2)

Fig. 1
Fig. 1

Prolate Spheroid. A ray from the proximal focus is reflected by the spheroid and then passes through its distal focus. ϕ and ϕ are the angles the ray makes with the spheroid axis. ρ and ρ are the ray path lengths before and after reflection.

Fig. 2
Fig. 2

Coupled Spheroids. A is the location of the proximal focus of the first spheroid; C the distal focus of the second; B is where the distal focus of the first spheroid coincides with the proximal focus of the second; s1 and s2 are the spheroid axes; and β is the subtended angle. The thicker line is the pseudo axis; α is the angle the pseudo axis makes with the axis of the first spheroid; and α is between the pseudo axis and the axis of the second spheroid in image space. θ is the angle a chief ray makes with the pseudo axis at the first focus; θ is the angle at the last focus; and s0, s0, and s0 are unit vectors along the pseudo axis in object space, in the space between the two reflections, and in image space, respectively.

Equations (88)

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P=ρ(sin ϕ sin σ,sin ϕ cos σ, cos ϕ),
ρ=r1- cos ϕ.
N=1K(sin ϕ sin σ, sin ϕ cos σ, cos ϕ-),
K2=1+2-2 cos ϕ.
P=ρ(ξ, η, ζ),
ρ=r1-ζ.
N=1K(ξ, η, ζ-),
K2=1+2-2ζ.
cos i=S·N=1-ζK.
S=-S+2 cos iN.
ξ=1-2K2ξ,
η=1-2K2η,
ζ=1K2[(1+2)ζ-2].
ρ=ρ K21-2.
ξ1=ξ,
η1=η cos α+ζ sin α,
ζ1=-η sin α+ζ cos α.
P=ρ(ξ1, η1, ζ1),
ρ1=r11-1ζ1,
N=1K1(ξ1, η1, ζ1-1),
K12=1+12-21ζ1.
ρ1=ρ1 K121-12,
ξ1=1-12K12ξ,
η1=1-12K12(η cos α+ζ sin α),
ζ1=1K12[(1+12)(-η sin α+ζ cos α)-21].
ξ2=ξ1,
η2=η1 cos β+ζ1 sin β,
ζ2=-η1 sin β+ζ1 cos β.
ξ2=1-12K12ξ,
η2=1K12[-21 sin β+Aη+Bζ],
ζ2=-1K12[21 cos β+Cη+Dζ],
A=(1-12)cos β cos α-(1+12)sin β sin α,
B=(1-12)cos β sin α+(1+12)sin β cos α,
C=(1-12)sin β cos α+(1+12)cos β sin α,
D=(1-12)sin β sin α-(1+12)cos β cos α.
K12K22=(1+22)K12+22(21 cos β+Cη+Dζ)=(1+12)(1+22)+412 cos β+2η[1(1+22)sin α+2C]-2ζ[1(1+22)cos α-2D].
ρ2=r21-2ζ2=r2K22K22+2(21 cos β+Cη+Dζ),
ρ2=ρ2 K221-22.
ξ2=(1-22)(1-12)K12K22ξ,
η2=1-22K12K22(-21 sin β+Aη+Bζ),
ζ2=-1K12K22[(1+22)(21 cos β+Cη+Dζ)+22K12]=-1K12K22(2P+Qη-Rζ),
P=2(1+12)+1(1+22)cos β
Q=412 sin α+(1+22)C,
R=412 cos α-(1+22)D.
K¯12K¯22 sin α=(1-22)(-21 sin β+B),
K¯12K¯22 cos α=-2P+R,
K¯12K¯22=(1+12)(1+22)+412 cos β-2[1(1+22)cos α-2D].
sin(α+θ)=1-22K12K22(-21 sin β+A sin θ+B cos θ),
cos(α+θ)=-1K12K22(2P+Q sin θ-R cos θ).
K12K22=(1+12)(1+22)+412 cos β+2[1(1+22)sin α+2C]sin θ-2[1(1+22)cos α-2D]cos θ.
sin θ=sin(α+θ)cos α-cos(α+θ)sin α,
sin θ=1K12K22K¯12K¯22{2(BP-1R sin β)×(1-cos θ)+[AR+BQ-2(AP+1Q sin β)]}.
BP-1R sin β
=0=[(1-12)cos β sin α+(1+12)sin βcos α]×[2(1+12)+1(1+22)cos β]-1 sin β{412 cos α-(1+22)[(1-12)×sin β sin α-(1+12)cos β cos α]}
=(1-12){[1(1+22)+2(1+12)cos β]×sin α+2(1-12)sin β cos α}.
1(1+22)sin α+2C=0,
tan α=-2(1-12)sin β1(1+22)+2(1+12)cos β.
22(1-12)2 sin2 β+[1(1+22)+2(1+12)cos β]2=22(1-12)2(1-cos2 β)+12(1+22)2+212(1+12)(1+22)cos β+22(1+12)cos2 β=12(1+22)2+22(1-12)2+212(1+12)(1+22)cos β+22[(1+12)2-(1-12)2]cos2 β=(1+1222)(12+22)+212[(1+1222)+(12+22)]cos β+41222 cos2 β=(1+1222+212 cos β)(12+22+212 cos β).
p2=1+1222+212 cos β,
q2=12+22+212 cos β,
sin α=-2(1-12)sin β/pq,
cos α=[1(1+22)+2(1+12)cos β]/pq.
p2+q2=(1+12)(1+22)+412 cos β,
p2-q2=(1-12)(1-22).
K12K22=p2+q2-2pq cos θ,
K¯12K¯22=(p-q)2.
A=(1-12)[2(1+12)+1(1+22)cos β]/pq,
B=1(p2+q2)sin β/pq,
C=1(1-12)(1+22)sin β/pq,
D=-[2(1-12)2+1(p2+q2)cos β]/pq.
sin α=1(1-22)sin β/pq,
cos α=[2(1+2)+1(1+22)cos β]/pq.
P=pq cos α,
Q=(p2-q2)sin α,
R=(p2+q2)cos α.
ξ2=p2-q2p2+q2-2pqζξ,
η2=1p2+q2-2pqζ×{[ζ(p2+q2)-2pq]×cos α+η(p2-q2)sin α},
ζ2=1p2+q2-2pqζ×{[ζ(p2+q2)-2pq]×sin α-η(p2-q2)cos α},
ξ*=p2-q2p2+q2-2pqζξ,
η*=p2-q2p2+q2-2pqζη,
ζ*=ζ(p2+q2)-2pqp2+q2-2pqζ.
tan θ=(p2-q2)sin θ(p2+q2)cos θ-2pq=M(θ)tan θ,
M(θ)=(p2-q2)cos θ(p2+q2)cos θ-2pq.
M(0)=M=p+qp-q,
D(θ)=M(θ)-M(0)=p+qp-q2pq(1-cos θ)(p2+q2)cos θ-2pq.
D¯(θ)=2pq(1-cos θ)(p2+q2)cos θ-2pq.
p=M+1M-1q.
D¯(θ)=(M2-1)(1-cos θ)(M2+1)cos θ-(M2-1),

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