Abstract

A Fresnel zone plate of variable size can be formed as a moiré pattern between a suitable pair of identical grids. The focusing power may be varied over a considerable range by relative translation of the grids if their line shapes are determined by a combination of cubic and linear functions. Elliptical and hyperbolic patterns are also obtainable. If both grids are constructed in phase reversal form, the irradiance in the diffraction image due to the zone plate is enhanced substantially. The device may then be used with photoelectric detection to monitor the straightness of a continuous linear motion. A possible application to field widening in interference spectroscopy is also suggested.

© 1977 Optical Society of America

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References

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  1. P. W. Harrison, Civ. Eng. Public Works Rev. 68, 224 (1973).
  2. B. M. New, Appl. Opt. 13, 937 (1974).
    [CrossRef] [PubMed]
  3. A. G. Bennett, Manuf. Opt. Int. 26, 88 (1973).
  4. J. M. Burch, D. C. Williams, Opt. Laser Technol. 6, 166 (1974).
    [CrossRef]
  5. G. Harburn, T. R. Welberry, R. P. Williams, Opt. Acta 22, 409 (1975).
    [CrossRef]
  6. O. Bryngdahl, J. Opt. Soc. Am. 66, 87 (1976).
    [CrossRef]
  7. O. Bryngdahl, J. Opt. Soc. Am. 64, 1287 (1974).
    [CrossRef]
  8. A. Lohmann, D. P. Paris, Appl. Opt. 6, 1567 (1967).
    [CrossRef] [PubMed]
  9. G. L. Rogers, L. C. G. Rogers, Opt. Acta 24, 15 (1977).
    [CrossRef]
  10. M. V. R. K. Murty, Appl. Opt. 3, 531 (1964).
    [CrossRef]
  11. A. G. Bennett, Manuf. Opt. Int. 26, 42 (1973).
  12. J. M. Burch, Prog. Opt. 2, 73 (1963).
    [CrossRef]
  13. O. Bryngdahl, J. Opt. Soc. Am. 65, 685 (1975).
    [CrossRef]
  14. A. E. Ennos, J. Opt. Soc. Am. 50, 14 (1960).
    [CrossRef]
  15. P. Jacquinot, Rep. Prog. Phys. 23, 267 (1960).
    [CrossRef]
  16. J. Ring, J. W. Schofield, Appl. Opt. 11, 507 (1972).
    [CrossRef] [PubMed]

1977 (1)

G. L. Rogers, L. C. G. Rogers, Opt. Acta 24, 15 (1977).
[CrossRef]

1976 (1)

1975 (2)

O. Bryngdahl, J. Opt. Soc. Am. 65, 685 (1975).
[CrossRef]

G. Harburn, T. R. Welberry, R. P. Williams, Opt. Acta 22, 409 (1975).
[CrossRef]

1974 (3)

1973 (3)

P. W. Harrison, Civ. Eng. Public Works Rev. 68, 224 (1973).

A. G. Bennett, Manuf. Opt. Int. 26, 88 (1973).

A. G. Bennett, Manuf. Opt. Int. 26, 42 (1973).

1972 (1)

1967 (1)

1964 (1)

1963 (1)

J. M. Burch, Prog. Opt. 2, 73 (1963).
[CrossRef]

1960 (2)

A. E. Ennos, J. Opt. Soc. Am. 50, 14 (1960).
[CrossRef]

P. Jacquinot, Rep. Prog. Phys. 23, 267 (1960).
[CrossRef]

Bennett, A. G.

A. G. Bennett, Manuf. Opt. Int. 26, 88 (1973).

A. G. Bennett, Manuf. Opt. Int. 26, 42 (1973).

Bryngdahl, O.

Burch, J. M.

J. M. Burch, D. C. Williams, Opt. Laser Technol. 6, 166 (1974).
[CrossRef]

J. M. Burch, Prog. Opt. 2, 73 (1963).
[CrossRef]

Ennos, A. E.

Harburn, G.

G. Harburn, T. R. Welberry, R. P. Williams, Opt. Acta 22, 409 (1975).
[CrossRef]

Harrison, P. W.

P. W. Harrison, Civ. Eng. Public Works Rev. 68, 224 (1973).

Jacquinot, P.

P. Jacquinot, Rep. Prog. Phys. 23, 267 (1960).
[CrossRef]

Lohmann, A.

Murty, M. V. R. K.

New, B. M.

Paris, D. P.

Ring, J.

Rogers, G. L.

G. L. Rogers, L. C. G. Rogers, Opt. Acta 24, 15 (1977).
[CrossRef]

Rogers, L. C. G.

G. L. Rogers, L. C. G. Rogers, Opt. Acta 24, 15 (1977).
[CrossRef]

Schofield, J. W.

Welberry, T. R.

G. Harburn, T. R. Welberry, R. P. Williams, Opt. Acta 22, 409 (1975).
[CrossRef]

Williams, D. C.

J. M. Burch, D. C. Williams, Opt. Laser Technol. 6, 166 (1974).
[CrossRef]

Williams, R. P.

G. Harburn, T. R. Welberry, R. P. Williams, Opt. Acta 22, 409 (1975).
[CrossRef]

Appl. Opt. (4)

Civ. Eng. Public Works Rev. (1)

P. W. Harrison, Civ. Eng. Public Works Rev. 68, 224 (1973).

J. Opt. Soc. Am. (4)

Manuf. Opt. Int. (2)

A. G. Bennett, Manuf. Opt. Int. 26, 42 (1973).

A. G. Bennett, Manuf. Opt. Int. 26, 88 (1973).

Opt. Acta (2)

G. Harburn, T. R. Welberry, R. P. Williams, Opt. Acta 22, 409 (1975).
[CrossRef]

G. L. Rogers, L. C. G. Rogers, Opt. Acta 24, 15 (1977).
[CrossRef]

Opt. Laser Technol. (1)

J. M. Burch, D. C. Williams, Opt. Laser Technol. 6, 166 (1974).
[CrossRef]

Prog. Opt. (1)

J. M. Burch, Prog. Opt. 2, 73 (1963).
[CrossRef]

Rep. Prog. Phys. (1)

P. Jacquinot, Rep. Prog. Phys. 23, 267 (1960).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Grid corresponding to Table I: vertical axis of symmetry is off center.

Fig. 2
Fig. 2

Four circular fringes: grid edges are coincident on horizontal axis of symmetry.

Fig. 3
Fig. 3

Maximum number of circular fringes.

Fig. 4
Fig. 4

Four circular fringes: power ten times greater than in Fig. 2.

Fig. 5
Fig. 5

Elliptical fringes: vertical displacement less than horizontal displacement.

Fig. 6
Fig. 6

One-dimensional zone plate: vertical and horizontal displacements equal.

Fig. 7
Fig. 7

Rectangular hyperbolas: vertical displacement only.

Fig. 8
Fig. 8

Illustrating symbols used in the text: the grid displacement corresponds to Fig. 3.

Equations (32)

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ϕ ( x , y ) = a 2 ( x 3 + y 3 ) = n .
ϕ ( x , y ) = a ( x 3 + 3 x y 2 ) = n .
ϕ ( x + u , y ) = a [ ( x + u ) 3 + 3 ( x + u ) y 2 ] = n a ;
ϕ ( x - u , y ) = a [ ( x - u ) 3 + 3 ( x - u ) y 2 ] = n b .
Δ ϕ ( x , y ) = 2 a u [ 3 ( x 2 + y 2 ) + u 2 ] = m ,
r = [ m 6 a u ] 1 / 2 .
F = 2 m λ / r 2 = 12 a u λ ,
ϕ ( x , y / 3 ) = a x ( x 2 + y 2 ) .
ϕ ( x , y ) = a [ x 3 + x ( 3 y 2 + b ) ] = n ;
Δ ϕ ( x , y ) = 2 a u [ 3 ( x 2 + y 2 ) + u 2 + b ] = m .
r + = [ m 6 a ( u - v ) ] 1 / 2 ;
r - = [ m 6 a ( u + v ) ] 1 / 2 .
x 1 = r 0 + u 0 ;
x 2 = r 0 - u 0 .
r = x 1 - u
m = 6 a u ( x 1 - u ) 2 .
m = 8 9 a x 1 3 ,
u = 1 3 x 1 .
d / d x [ ϕ ( x + u , 0 ) ] = a [ 3 ( x + u ) 2 + b ] ;
d / d x [ ϕ ( x - u , 0 ) ] = a [ 3 ( x + u ) 2 + b ]
d ϕ ¯ d x = a [ 3 ( x 2 + u 2 ) + b ] .
d / d x [ Δ ϕ ( x , 0 ) ] = 12 a u x ,
g = | 1 4 ( x u + u x ) + b 12 u x | .
b > 3 x 1 ( x 1 - 2 u ) .
s = 2 F 0 [ 1 - ( 1 - F 0 F ) 1 / 2 ] ,
Δ ϕ = ϕ ( x + α y + u , y - α x ) - ϕ ( x - α y - u , y + α x ) .
Δ ϕ α = 2 α a y [ 3 ( y 2 - x 2 + u 2 ) + b ] .
δ y = - y Δ ϕ α / 2 y 2 δ ϕ = - a 6 u [ 3 ( 3 y 2 - x 2 + u 2 ) + b ] .
t = p 2 / λ ,
E Z E L = 1 π 2 sin 2 ( π w ) ,
E M E L = ( E Z E L ) 2 = 1 π 4 sin 4 ( π w ) ,
l = m λ ( f / r ) 2 = 6 a u λ f 2 ,

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