Abstract

The use of zone plates in optical alignment and displacement measurement is discussed. A simple theory of positioning using two coherent wavefronts is proposed. A few examples, already described in the literature as well as new ones, are presented as an illustration of the theory. The application of the specific case of the off-axis cylindrical zone plate (so-called parabolic zone plate) and off-axis conical zone plate for sensing the motion along a straight line parallel and inclined to the hologram plane is established. There are also adduced results of simple experiments supporting the analysis.

© 1990 Optical Society of America

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References

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  1. G. Makosch, F. J. Schoenes, “Interferometric Method for Checking the Mask Alignment Precision in the Lithographic Process,” Appl. Opt. 23, 628–632 (1984).
    [CrossRef] [PubMed]
  2. G. T. Olaru, T. Asakura, “Moire Alignment and Positioning Using a Synthetic Fresnel Zone Plate,” Opt. Commun. 54, 327–332 (1985).
    [CrossRef]
  3. R. F. Stevens, “Zone-Plate Interferometers,” J. Mod. Opt. 35, 75–79 (1988).
    [CrossRef]
  4. V. Moreno, M. V. Perez, S. Bara, C. Gomez-Reino, “Interferometric Alignment by a Circular Zone Plate,” Proc. Soc. Photo-Opt. Instrum. Eng. 952, 367–370 (1989).
  5. T. R. Welberry, R. P. Williams, “On Certain Non-Circular Zone Plates,” Opt. Acta 23, 237–000 (1976).
    [CrossRef]
  6. M. V. Perez, C. Gomez-Reino, S. Bara, “Holographically Produced Parabolic Zone Plates,” Opt. Eng. 26, 461–466 (1987).
    [CrossRef]
  7. C. Frere, D. Leseberg, O. Bryngdahl, “Computer-Generated Holograms of Three-Dimensional Objects Composed of Line Segments,” J. Opt. Soc. Am. A 3, 726–730 (1986).
    [CrossRef]
  8. Z. Jaroszewicz, “Conical Zone Plate,” Opt. Commun. 66, 9–14 (1988).
    [CrossRef]

1989 (1)

V. Moreno, M. V. Perez, S. Bara, C. Gomez-Reino, “Interferometric Alignment by a Circular Zone Plate,” Proc. Soc. Photo-Opt. Instrum. Eng. 952, 367–370 (1989).

1988 (2)

R. F. Stevens, “Zone-Plate Interferometers,” J. Mod. Opt. 35, 75–79 (1988).
[CrossRef]

Z. Jaroszewicz, “Conical Zone Plate,” Opt. Commun. 66, 9–14 (1988).
[CrossRef]

1987 (1)

M. V. Perez, C. Gomez-Reino, S. Bara, “Holographically Produced Parabolic Zone Plates,” Opt. Eng. 26, 461–466 (1987).
[CrossRef]

1986 (1)

1985 (1)

G. T. Olaru, T. Asakura, “Moire Alignment and Positioning Using a Synthetic Fresnel Zone Plate,” Opt. Commun. 54, 327–332 (1985).
[CrossRef]

1984 (1)

1976 (1)

T. R. Welberry, R. P. Williams, “On Certain Non-Circular Zone Plates,” Opt. Acta 23, 237–000 (1976).
[CrossRef]

Asakura, T.

G. T. Olaru, T. Asakura, “Moire Alignment and Positioning Using a Synthetic Fresnel Zone Plate,” Opt. Commun. 54, 327–332 (1985).
[CrossRef]

Bara, S.

V. Moreno, M. V. Perez, S. Bara, C. Gomez-Reino, “Interferometric Alignment by a Circular Zone Plate,” Proc. Soc. Photo-Opt. Instrum. Eng. 952, 367–370 (1989).

M. V. Perez, C. Gomez-Reino, S. Bara, “Holographically Produced Parabolic Zone Plates,” Opt. Eng. 26, 461–466 (1987).
[CrossRef]

Bryngdahl, O.

Frere, C.

Gomez-Reino, C.

V. Moreno, M. V. Perez, S. Bara, C. Gomez-Reino, “Interferometric Alignment by a Circular Zone Plate,” Proc. Soc. Photo-Opt. Instrum. Eng. 952, 367–370 (1989).

M. V. Perez, C. Gomez-Reino, S. Bara, “Holographically Produced Parabolic Zone Plates,” Opt. Eng. 26, 461–466 (1987).
[CrossRef]

Jaroszewicz, Z.

Z. Jaroszewicz, “Conical Zone Plate,” Opt. Commun. 66, 9–14 (1988).
[CrossRef]

Leseberg, D.

Makosch, G.

Moreno, V.

V. Moreno, M. V. Perez, S. Bara, C. Gomez-Reino, “Interferometric Alignment by a Circular Zone Plate,” Proc. Soc. Photo-Opt. Instrum. Eng. 952, 367–370 (1989).

Olaru, G. T.

G. T. Olaru, T. Asakura, “Moire Alignment and Positioning Using a Synthetic Fresnel Zone Plate,” Opt. Commun. 54, 327–332 (1985).
[CrossRef]

Perez, M. V.

V. Moreno, M. V. Perez, S. Bara, C. Gomez-Reino, “Interferometric Alignment by a Circular Zone Plate,” Proc. Soc. Photo-Opt. Instrum. Eng. 952, 367–370 (1989).

M. V. Perez, C. Gomez-Reino, S. Bara, “Holographically Produced Parabolic Zone Plates,” Opt. Eng. 26, 461–466 (1987).
[CrossRef]

Schoenes, F. J.

Stevens, R. F.

R. F. Stevens, “Zone-Plate Interferometers,” J. Mod. Opt. 35, 75–79 (1988).
[CrossRef]

Welberry, T. R.

T. R. Welberry, R. P. Williams, “On Certain Non-Circular Zone Plates,” Opt. Acta 23, 237–000 (1976).
[CrossRef]

Williams, R. P.

T. R. Welberry, R. P. Williams, “On Certain Non-Circular Zone Plates,” Opt. Acta 23, 237–000 (1976).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

R. F. Stevens, “Zone-Plate Interferometers,” J. Mod. Opt. 35, 75–79 (1988).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

T. R. Welberry, R. P. Williams, “On Certain Non-Circular Zone Plates,” Opt. Acta 23, 237–000 (1976).
[CrossRef]

Opt. Commun. (2)

Z. Jaroszewicz, “Conical Zone Plate,” Opt. Commun. 66, 9–14 (1988).
[CrossRef]

G. T. Olaru, T. Asakura, “Moire Alignment and Positioning Using a Synthetic Fresnel Zone Plate,” Opt. Commun. 54, 327–332 (1985).
[CrossRef]

Opt. Eng. (1)

M. V. Perez, C. Gomez-Reino, S. Bara, “Holographically Produced Parabolic Zone Plates,” Opt. Eng. 26, 461–466 (1987).
[CrossRef]

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

V. Moreno, M. V. Perez, S. Bara, C. Gomez-Reino, “Interferometric Alignment by a Circular Zone Plate,” Proc. Soc. Photo-Opt. Instrum. Eng. 952, 367–370 (1989).

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Figures (3)

Fig. 1
Fig. 1

Hologram recording and reconstruction: P1, hologram recording plane; P2, hologram plane during reconstruction; diffracted beams, A 1 = U 2 t 1 * , A 2 = U 2 + U 1 t 1 * , A 3 = U 1 + U 2 t 1 , A 4 = U 1 t 1.

Fig. 2
Fig. 2

Interference pattern for the PZP: (a) in-plane displacement and (b) out-of-plane displacement.

Fig. 3
Fig. 3

Interference pattern for the off-axis CZP: (a) in-plane displacement and (b) out-of-plane displacement.

Equations (27)

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U i ( r ) exp [ i k Φ i ( r ) ] for i = 1 , 2.
t ( r ) 1 + ɛ t 1 ( r ) + ɛ t 1 * ( r ) ,
U i ( r + Δ r ) exp { i k [ Φ i ( r ) + Φ i ( r ) Δ r ] } for i = 1 , 2.
[ U 1 ( r + Δ r ) + U 2 ( r + Δ r ) ] t ( r ) exp [ i k Φ 1 ( r ) ] { exp [ i k Φ 1 ( r ) Δ r ] + ɛ exp [ i k Φ 2 ( r ) Δ r ] } + exp [ i k Φ 2 ( r ) ] { exp [ i k Φ 2 ( r ) Δ r ] + ɛ exp [ i k Φ 1 ( r ) Δ r ] } + remaining terms .
[ Φ 1 ( r ) - Φ 2 ( r ) ] Δ r = s λ ,
Φ 1 ( r ) = sin α x + cos α z ,             Φ 2 ( r ) = - sin α x + cos α z ,
Δ x = s λ \ 2 sin α .
Φ 1 ( r ) = ( x 2 + y 2 ) / 2 z + z ,             Φ 2 ( r ) = z .
[ x - ( Δ x / Δ z ) z ] 2 + [ y - ( Δ y / Δ z ) z ] 2 = ( z 2 / Δ z 2 ) [ Δ x 2 + Δ y 2 - 2 s λ Δ z ] .
x · Δ x z + y · Δ y z - s λ = 0.
Φ 1 ( r ) = x 2 / 2 z + z ,             Φ 2 ( r ) = sin α y + cos α z .
[ x - ( Δ x / Δ z ) z ] 2 = ( z 2 / Δ z 2 ) [ Δ x 2 - 2 Δ y Δ y sin α + 2 ( 1 - cos α ) Δ z 2 - 2 s λ Δ z ] ,
2 D 1 = 2 z 2 λ / Δ z ,
2 D 1 - ( 2 z / Δ z ) { 2 ( 1 - cos α ) Δ z 2 - 2 λ Δ z trunc [ ( 1 - cos α ) Δ z / z ] } 1 / 2 ,
( x / z ) Δ x + sin α Δ y = s λ ,
Δ y = s λ / sin α .
Δ z = s λ \ ( 1 - cos α ) .
Φ 1 ( r ) = [ x 2 / 2 ( d + z cos α 1 - y sin α 1 ) ] - y sin α 1 + z cos α 1 , Φ 2 ( r ) = y sin α 2 + z cos α 2 ,
x d + z cos α 1 - y sin α 1 Δ x + [ x 2 sin α 1 2 ( d + z cos α 1 - y sin α 1 ) 2 - sin α 1 - sin α 2 ] Δ y - [ x 2 cos α 1 2 ( d + z cos α 1 - y sin α 1 ) 2 + cos α 1 - cos α 2 ] Δ z = s λ .
y = d / sin α 1 - x Δ x / s λ sin α 1 .
D = λ ( d - y sin α 1 ) / Δ x .
Δ x = λ sin α 1 / tan ξ .
y = d / sin α 1 ± x sin α 1 2 s λ Δ i A i - 2 B i / A i ,
Δ i = 2 λ sin 2 α 1 / ( A i tan 2 ξ + 2 B i sin 2 α 1 ) .
Δ i = λ / [ A i ( D 1 2 / 2 d 2 ) + B i ] ,
Δ z = Δ y tan α 1 ,
Δ l = s λ / sin ( α 1 + α 2 ) ,

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