Abstract

A new type of radial shear interferometer based on the imaging properties of Gabor zone plates is described. This interferometer is stable, easily constructed, and is a useful instrument in optical testing.

© 1974 Optical Society of America

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References

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  1. P. Hariharan, D. Sen, Opt. Acta 9, 159 (1962).
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P. Hariharan, D. Sen, Opt. Acta 9, 159 (1962).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(a) Hologram formation. The numbers −1, 0, 1 indicate the three beams diffracted by the Gabor plate. (b) Reconstruction step and arrangement of the radial shear interferometer. Many unwanted diffracted beams are not represented.

Fig. 2
Fig. 2

System of coordinates.

Fig. 3
Fig. 3

The zone plate is off axis and out of focus.

Fig. 4
Fig. 4

Appearance of the interferogram. Relative to each other, the two radially sheared wavefronts have been altered: (a) slightly tilted, (b) defocused, (c) tilted and defocused.

Fig. 5
Fig. 5

Results of the optical test. The interferogram is shown to exhibit in (a) and (b) spherical aberration and in (c) astigmatism. In (b) one of the wavefronts has been tilted with respect to the other.

Equations (26)

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E = exp [ i 2 π λ ( f 0 2 + r 2 ) 1 / 2 ] ,
E = exp { i 2 π λ [ ( f 0 2 + r 2 ) 1 / 2 W ( r ¯ ) ] } ,
W ( r ¯ ) = S r 4 spherical aberration + ( C x cos θ + C y sin θ ) r 3 coma + ( A x cos 2 θ + A y sin 2 θ + 2 A x y cos θ sin θ ) r 2 astigmatism + F r 2 field curvature + ( D x cos θ + D y sin θ ) r distortion
T 1 = a 0 + 2 a 1 cos [ 2 π λ ( f 1 2 + | r ¯ 1 d ¯ 1 | 2 ) 1 / 2 ] = a 0 + a 1 exp [ i 2 π λ ( f 1 2 + | r ¯ 1 d ¯ 1 | 2 ) 1 / 2 ] + a 1 exp [ i 2 π λ ( f 1 2 + | r ¯ 1 d ¯ 1 | 2 ) 1 / 2 ] ,
E 1 = b 0 exp { i 2 π λ [ ( f 1 2 + | r ¯ 1 d ¯ 1 | 2 ) 1 / 2 ( f 0 2 + r 1 2 ) 1 / 2 + W ( r ¯ 1 ) ] } .
O P D 1 ( r ¯ 1 ) = ( f 1 2 + | r ¯ 1 d ¯ 1 | 2 ) 1 / 2 ( f 0 2 + r 1 2 ) 1 / 2 + W ( r ¯ 1 ) .
O P D 2 ( r ¯ 2 ) = ( f 2 2 + | r ¯ 2 d ¯ 2 | 2 ) 1 / 2 ( f 0 2 + r 2 2 ) 1 / 2 + W ( r ¯ 2 ) ,
O P D 1 ( y ) ( f 0 2 + d 1 2 ) 1 / 2 d 1 ( f 0 2 + d 1 2 ) 1 / 2 y + 1 2 [ ( f 0 2 + d 1 2 ) 1 / 2 d 1 2 ( f 0 2 d 1 2 ) 3 / 2 ] y 2 + 1 2 [ d 1 ( f 0 2 + d 1 2 ) 3 / 2 d 1 3 ( f 0 2 + d 1 2 ) 5 / 2 ] y 3 1 8 [ ( f 0 2 + d 1 2 ) 3 / 2 6 d 1 2 ( f 0 2 + d 1 2 ) 5 / 2 + 5 d 1 4 ( f 0 2 + d 1 2 ) 7 / 2 ] y 4 f 0 [ 1 + 1 2 ( y f 0 ) 2 1 8 ( y f 0 ) 4 ] .
O P D 1 ( y ) d 1 2 / 2 f 0 constant phase term ( d 1 / f 0 ) y tilt ( 3 / 4 ) ( d 1 2 / f 0 3 ) y 2 defocusing + ( d 1 / 2 f 0 3 ) y 3 coma + ( 15 d 1 2 / 16 f 0 5 ) y 4 spherical aberration
O P D 1 ( x ) d 1 2 / 2 f 0 constant phase term ( d 1 2 / 4 f 0 3 ) x 3 defocusing ( 3 d 1 2 / 16 f 0 5 ) x 4 spherical aberration
astigmatism = d 1 2 / ( 2 f 0 3 ) y 2 .
O P D 1 ( r ¯ 1 ) = [ ( f 0 + Δ f ) 2 + r 1 2 ] 1 / 2 ( f 0 2 + r 1 2 ) 1 / 2 ,
O P D 1 ( r ¯ 1 ) ( f 0 + Δ f ) + 1 2 ( f 0 + Δ f ) 1 r 1 2 1 8 ( f 0 + Δ f ) 3 r 1 4 [ f 0 + 1 2 f 0 1 r 1 2 1 8 f 0 3 r 1 4 ] .
O P D 1 ( r ¯ 1 ) Δ f + Δ f 2 f 0 2 r 1 2 3 Δ f 8 f 0 4 r 1 4 .
spherical aberration defocusing = 3 4 · r 1 2 f 0 2 .
spherical aberrations defocusing = 3 16 ( f * ) 2 .
O P D 1 ( r ¯ 1 ) = W ( r ¯ 1 ) ,
O P D 2 ( r ¯ 2 ) = W ( r ¯ 2 ) ,
W ( r ¯ ) = W ( r , θ ) = i , j A i j R i cos j θ .
W ( r ¯ 1 ) = i , j A i j R 1 i cos j θ
W ( r ¯ 2 ) = i , j A i j R 2 i cos j θ ,
W ( r ¯ 1 ) W ( r ¯ 2 ) = i , j A i j [ ( r 1 r 1 m ) i ( r 2 r 2 m ) i ] cos j θ = i , j A i j R 1 i [ 1 ( r 2 r 1 · r 1 m r 2 m ) i ] cos j θ .
W ( r ¯ 1 ) W ( r ¯ 2 ) = i , j B i j R 1 i cos j θ ,
S = f 2 / f 1 .
W ( r ¯ 1 ) W ( r ¯ 2 ) = i , j B i j R 1 i cos j θ = ( k + 1 2 ) λ ,
S = 1 ( d / f 1 ) = 0.7

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