Abstract

We describe an aberration of a double-focus lens made of a uniaxial crystal with the optic axis perpendicular to the optical axis of the lens. One of the focal points of the lens is formed by ordinary rays in the crystal, and the other is formed by extraordinary rays. The extraordinary rays have astigmatism at the focal point owing to birefringence of the crystal even under the paraxial condition. The magnitude of the aberration depends on the lens thickness and the ratio of the principal refractive indices.

© 1992 Optical Society of America

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References

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  1. L. Dayson, “Common-path interferometer for testing purposes,”J. Opt. Soc. Am. 47, 386–390 (1957).
    [CrossRef]
  2. N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A dynamic zone-plate interferometer for measuring aspherical surfaces,” Opt. Commun. 54, 257–261 (1985).
    [CrossRef]
  3. K. Iwata, N. Nishikawa, “Profile measurement with a phase-shift common-path polarization interferometer,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 389–394 (1990).
    [CrossRef]
  4. A. Yariv, P. Yeh, “Electromagnetic propagation in anisotropic media,” in Optical Waves in Crystals (Wiley, New York, 1983), Chap. 4, pp. 69–120.

1985 (1)

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A dynamic zone-plate interferometer for measuring aspherical surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

1957 (1)

Dayson, L.

Honda, T.

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A dynamic zone-plate interferometer for measuring aspherical surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

Ichimura, I.

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A dynamic zone-plate interferometer for measuring aspherical surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

Iwata, K.

K. Iwata, N. Nishikawa, “Profile measurement with a phase-shift common-path polarization interferometer,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 389–394 (1990).
[CrossRef]

Nishikawa, N.

K. Iwata, N. Nishikawa, “Profile measurement with a phase-shift common-path polarization interferometer,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 389–394 (1990).
[CrossRef]

Ohyama, N.

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A dynamic zone-plate interferometer for measuring aspherical surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

Tsujiuchi, J.

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A dynamic zone-plate interferometer for measuring aspherical surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

Yamaguchi, I.

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A dynamic zone-plate interferometer for measuring aspherical surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, “Electromagnetic propagation in anisotropic media,” in Optical Waves in Crystals (Wiley, New York, 1983), Chap. 4, pp. 69–120.

Yeh, P.

A. Yariv, P. Yeh, “Electromagnetic propagation in anisotropic media,” in Optical Waves in Crystals (Wiley, New York, 1983), Chap. 4, pp. 69–120.

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

N. Ohyama, I. Yamaguchi, I. Ichimura, T. Honda, J. Tsujiuchi, “A dynamic zone-plate interferometer for measuring aspherical surfaces,” Opt. Commun. 54, 257–261 (1985).
[CrossRef]

Other (2)

K. Iwata, N. Nishikawa, “Profile measurement with a phase-shift common-path polarization interferometer,” in Laser Interferometry: Quantitative Analysis of Interferograms: Third in a Series, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 389–394 (1990).
[CrossRef]

A. Yariv, P. Yeh, “Electromagnetic propagation in anisotropic media,” in Optical Waves in Crystals (Wiley, New York, 1983), Chap. 4, pp. 69–120.

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Figures (6)

Fig. 1
Fig. 1

Wave vector of an extraordinary ray in the k space. The crystal axis (c axis) of a uniaxial crystal corresponds to the optic axis. In this figure, the axis is parallel to the X direction, which corresponds to the kx axis. The principal refractive indices are no and ne. The wave vector ke reaches the ellipse. If the vector is of an ordinary ray, it reaches the circle. Ray velocity Ve is normal to the surface of the ellipse.

Fig. 2
Fig. 2

Refraction at a boundary between an isotropic medium and a uniaxial crystal. The length of the projection of the refracted wave vectors ko, and ke on the boundary is the same as that of the incident wave vector k. The vectors Vo and Ve are the ray velocities of the ordinary and the extraordinary rays.

Fig. 3
Fig. 3

Schematic diagram of the ray trace. A ray from the point S:(xs, ys, zs) is refracted at points P:(xp, yp, zp) and Q:(xq, yq, zq) on the lens surfaces. In the crystal lens the direction of a wave vector, in general, is different from the propagation direction of the ray.

Fig. 4
Fig. 4

Astigmatism of the uniaxial crystal lens. For simplicity, the lens thickness t is not shown and the second reflection point Q is shown as corresponding to P.

Fig. 5
Fig. 5

Principal parameters of the double-focus lens. The lens may be considered as having two kinds of thickness, and the parameters depend on the image formations in the X–Z and the YZ planes.

Fig. 6
Fig. 6

Image formation of a lens with thickness t. The refractive indices of the object, the lens, and the image media are n, nL, and n′ respectively. The first and the second spherical surfaces have radii R1 and R2 of curvature. This figure shows the negative R2. S is an object point on the optical axis, and its image is formed at SL in the lens space and at S′ in the image space. The arrows pointing leftward indicate negatives, that is, s and sL′ have negative signs in this figure.

Equations (48)

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( k x 2 n o 2 + k y 2 + k z 2 n e 2 - ω 2 c 2 ) ( k x 2 + k y 2 + k z 2 n o 2 - ω 2 c 2 ) = 0 ,
V = k · ω ( k ) .
k e x = k s x + ( n e - n ) ω c e x , k e y = k s y + ( n e - n ) ω c e y , k e z = ω c n e .
V e x = c 2 ω k e x n o 2 , V e y = c 2 ω k e y n e 2 , V e z = c 2 ω k e z n e 2 .
k s = n ω c 1 [ ( x p - x s ) 2 + ( y p - y s ) 2 + ( z p - z s ) 2 ] 1 / 2 × ( x p - x s , y p - y s , z p - z s ) .
k s = n ω c ( - x p - x s z s , - y p - y s z s , 1 ) .
e ^ p = ( - x p R 1 , - y p R 1 , 1 ) .
k e x = - ω c [ ( n z s + n e - n R 1 ) x p - n z s x s ] , k e y = - ω c [ ( n z s + n e - n R 1 ) y p - n z s y s ] , k e z = ω c n e ,
V e x = - c n o 2 [ ( n z s + n e - n R 1 ) x p - n z s x s ] , V e y = - c n o 2 [ ( n z s + n e - n R 1 ) y p - n z s y s ] , V e z = c n e .
x q = [ 1 - α t n e ( n z s - n e - n R 1 ) ] x p + α t n e n z s x s ,
y q = [ 1 - t n e ( n z s - n e - n R 1 ) ] y p + t n e n z s y s ,
z q = t ,
e ^ q ( e q x , e q y , e q z ) = ( - x q R 2 , - y q R 2 , 1 ) .
k t x = k e x + ( n - n e ) ω c e q x , k t y = k e y + ( n - n e ) ω c e q y , k t z = n ω c .
k t x = - ω c { ( n z s + n e - n R 1 ) + n - n e R 2 × [ 1 - α t n e ( n z s + n e - n R 1 ) ] } x p + ω c n z s × ( 1 - α t n e n - n e R 2 ) x s , k t y = - ω c { ( n z s + n e - n R 1 ) + n - n e R 2 × [ 1 - t n e ( n z s + n e - n R 1 ) ] } y p + ω c n z s × ( 1 - t n e n - n e R 2 ) y s , k t z = ω c n .
x = k t x k t z z + x q ,
y = k t y k t z z + y q .
α t n e = 1 ( n e - n ) / R 1 + n / z s + 1 ( n - n e ) / R 2 - n / z f .
x f = - n / z s D - ( n - n e ) / R 2 [ 1 + ( α t / n e ) D ] x s ,
y f = - [ 1 - ( t / n e ) ( 1 - α ) D ] n / z s D - ( n - n e ) / R 2 [ 1 + ( α t / n e ) D ] y s + ( t / n e ) ( 1 - α ) D 2 D - ( n - n e ) / R 2 [ 1 + ( α t / n e ) D y p ,
D = - ( n z s + n e - n R 1 ) .
t n e = 1 ( n e - n ) / R 1 + n / z s + 1 ( n - n e ) / R 2 - n / z f f .
x f f = - [ 1 + ( t / n e ) ( 1 - α ) D ] n / z s D - ( n - n e ) / R 2 [ 1 + ( α t / n e ) D ] x s + t / n e ( 1 - α ) D 2 D - ( n - n e ) / R 2 [ 1 + ( α t / n e ) D ] x p ,
y f f = - n / z s D - ( n - n e ) / R 2 [ 1 + ( α t / n e ) D y s ,
f x = - n g + h - ( α t / n e ) g h ,
f x = - n n f x ,
d x = - α t n e h f x ,
d x = - α t n e g f x ,
f y = - n g + h - ( t / n e ) g h ,
f y = - n n f y ,
d y = - t n e h f y ,
d y = - t n e g f y ,
g = n e - n R 1 ,
h = n - n e R 2 .
k h = k s - ( k s · e ^ ) e ^ .
k e = k h + μ e ^ ,
e z e x , e y ,
k z k x , k y .
k h z k h x , k h y .
μ = - b + ( b 2 - a d ) 1 / 2 a ,
a = e x 2 n o 2 + e y 2 + e z 2 n e 2 , b = k h x e x n o 2 + k h y e y + k h z e z n e 2 , d = k h x 2 n o 2 + k h y 2 + k h z 2 n e 2 - ω 2 c 2 .
μ = n e ω c ,
k e x = k h x + n e ω c e x , k e y = k h y + n e ω c e y , k e z = n e ω c e z .
k h x = k s x - k s z e x e z , k h y = k s y - k s z e y e z , k h z = 0.
- n s + n L s L = n L - n R 1 ,
- n L s L + n s = n - n L R 2 .
s L - s L = t .
t n L = 1 ( n L - n ) / R 1 + n / s + 1 ( n - n L ) / R 2 - n / s .

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