Abstract

Among the conceivable uses of Fresnel Zone Plates (FZP) are image formation, synthesis of holograms, coherence measurements, spectrometry, optical analog computation, and optical testing. Sometimes it is desirable to change the scale of the FZP continuously, for example to give a zoom lens effect when the FZP is used for image formation. Here we describe four ways of creating a FZP pattern as a moiré effect by superposing pairs of suitable masks. The relative position of the two masks determines the FZP scale. The theory presented here is sufficiently general to allow the synthesis of patterns other than the FZP by means of a variable moiré effect.

© 1967 Optical Society of America

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References

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  1. B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966); A. W. Lohmann, D. P. Paris, H. W. Werlich, Appl. Opt. 6, 1139 (1967).
    [CrossRef] [PubMed]
  2. A. W. Lohmann, D. P. Paris, Appl. Opt. 6,Oct. (1967).
    [PubMed]
  3. A. W. Lohmann, Optik 18, 514 (1961).
  4. A. Schuster, Phil. Mag. 48, 609 (1924); C. V. Raman, S. K. Datta, Trans. Opt. Soc. 27, 51 (1925/26); S. K. Datta, Trans. Opt. Soc. 28, 214 (1926/27); E. Lau, Optik 12, 23 (1955); G. Oster, M. Wasserman, C. Zwerling, J. Opt. Soc. Am. 54, 169 (1964).
    [CrossRef]

1967 (1)

A. W. Lohmann, D. P. Paris, Appl. Opt. 6,Oct. (1967).
[PubMed]

1966 (1)

1961 (1)

A. W. Lohmann, Optik 18, 514 (1961).

1924 (1)

A. Schuster, Phil. Mag. 48, 609 (1924); C. V. Raman, S. K. Datta, Trans. Opt. Soc. 27, 51 (1925/26); S. K. Datta, Trans. Opt. Soc. 28, 214 (1926/27); E. Lau, Optik 12, 23 (1955); G. Oster, M. Wasserman, C. Zwerling, J. Opt. Soc. Am. 54, 169 (1964).
[CrossRef]

Brown, B. R.

Lohmann, A. W.

Paris, D. P.

A. W. Lohmann, D. P. Paris, Appl. Opt. 6,Oct. (1967).
[PubMed]

Schuster, A.

A. Schuster, Phil. Mag. 48, 609 (1924); C. V. Raman, S. K. Datta, Trans. Opt. Soc. 27, 51 (1925/26); S. K. Datta, Trans. Opt. Soc. 28, 214 (1926/27); E. Lau, Optik 12, 23 (1955); G. Oster, M. Wasserman, C. Zwerling, J. Opt. Soc. Am. 54, 169 (1964).
[CrossRef]

Appl. Opt. (2)

Optik (1)

A. W. Lohmann, Optik 18, 514 (1961).

Phil. Mag. (1)

A. Schuster, Phil. Mag. 48, 609 (1924); C. V. Raman, S. K. Datta, Trans. Opt. Soc. 27, 51 (1925/26); S. K. Datta, Trans. Opt. Soc. 28, 214 (1926/27); E. Lau, Optik 12, 23 (1955); G. Oster, M. Wasserman, C. Zwerling, J. Opt. Soc. Am. 54, 169 (1964).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Cylindrical FZP: ϕ = (x/a)3 + y/b. (a) M(x,y); (b) M(+)M(−); (c) P(x).

Fig. 2
Fig. 2

Cylindrical FZP: ϕ = x2y/a3. (a) M(x,y); (b) M(+)M(−); (c) P(x).

Fig. 3
Fig. 3

Spherical FZP: ϕ = (xy2 + x3/3)/a3. (a) M(x,y); (b) M(+)M(−); (c) P(r).

Fig. 4
Fig. 4

Spherical FZP: ϕ = r2φ/a2. (a) M(r,φ); (b) M(+)M(−); (c) P(r).

Equations (13)

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M ( x ) = { 1 ; if ( x / d ) - N q / 2 0 ; otherwise } N = 0 , ± 1 , ± 2 , , M ( x ) = - + ( n ) A n exp [ 2 π i n ( x / d ) ] ; A n = sin ( n π q ) / n π .
M ( x , y ) = { 1 ; if ϕ ( x , y ) - N q / 2 0 ; otherwise } N = 0 , ± 1 , ± 2 , , M ( x , y ) = Σ ( n ) A n exp [ 2 π i n ϕ ( x , y ) ] ; A n = sin ( n π q ) / n π .
ϕ ( x , y ) = ( x 2 + y 2 ) / R 1 2 ; q = 1 2 ; M ( r ) = { 1 ; if r 2 / R 1 2 - N 1 4 0 ; otherwise } N = 0 , 1 , 2 , .
M ( x , y ) = Σ A n exp [ 2 π i n ϕ ( x , y ) ] ; ϕ ( x , y ) = ( x 2 + y 2 ) / R 1 2 ; P ( x ) = M ( x + Δ x / 2 , y ) M ( x - Δ x / 2 , y ) d y = Σ Σ A n A m exp [ 2 π i { } ] d y ; { } = n ϕ ( + ) + m ϕ ( - ) = 1 2 ( n + m ) { ϕ ( + ) + ϕ ( - ) } + 1 2 ( n - m ) { ϕ ( + ) - ϕ ( - ) } ; 1 2 { ϕ ( + ) + ϕ ( - ) } = { x 2 + ( Δ x / 2 ) 2 + y 2 } / R 1 2 ; 1 2 { ϕ ( + ) - ϕ ( - ) } = x Δ x / R 1 2 .
( 1 / h ) - h / 2 + h / 2 exp [ 2 π i ( n + m ) y 2 / R 1 2 ] d y = { 1 ; if n + m = 0 0 ; if n + m 0.
P ( x ) = Σ A n A - n exp [ 2 π i 2 n x Δ x / R 1 2 ] .
M 1 = 1 ; if r 2 / R 1 2 - N 1 4 ; q 1 = 1 2 ; M 2 = 1 ; if r 2 / R 1 2 - N 1 6 ; q 2 = 1 3 .
B n = A n ( 1 ) A - n ( 2 ) ; B n = sin ( n π / 2 ) sin ( n π / 3 ) / ( n π ) 2 ; B 1 = ( 3 ) 1 2 / ( 2 π 2 ) ; B 2 = B 3 = B 4 = 0 ; B 5 = - B 1 / 25.
ϕ 1 ( x , y ) = N ; ϕ 2 ( x , y ) = N ( N = 0 , ± 1 , ± 2 , ) .
1.             C y l i n d r i c a l F Z P ; ± Δ x / 2 ; : .. d y ϕ ( x , y ) = ( x / a ) 3 + y / b ; M 1 ( x , y ) = 1 ; if ϕ ( x , y ) - N q 1 / 2 ; M 2 ( x , y ) = 1 ; if ϕ ( x , y ) - N q 2 / 2 ; P ( x ) = M 1 ( x + Δ x / 2 , y ) M 2 ( x - Δ x / 2 , y ) d y = Σ B n exp [ 2 π i n ( x / x 1 ) 2 ] ; B n = A n ( 1 ) A - n ( 2 ) exp [ 4 n π i ( Δ x / 2 a ) 3 ] ; 1 / x 1 2 = 3 Δ x / a 3 .
2 .             C y l i n d r i c a l F Z P ; ± Δ y / 2 ; d y ϕ ( x , y ) = x 2 y / a 3 ; M 1 , 2 ( x , y ) = 1 ; if ϕ ( x , y ) - N q 1 , 2 / 2 ; P ( x ) = M 1 ( x , y + Δ y / 2 ) M 2 ( x , y - Δ y / 2 ) d y = Σ B n exp [ 2 π i n ( x / x 1 ) 2 ] ; B n = A n ( 1 ) A - n ( 2 ) ; 1 / x 1 2 = Δ y / a 3 ; 1 / f = Δ y 2 λ / a 3 .
3 .             S p h e r i c a l F Z P ; ± Δ x / 2 ; d φ ϕ ( x , y ) = ( x y 2 + x 3 / 3 ) / a 3 ; M 1 , 2 ( x , y ) = 1 ; if ϕ ( x , y ) - N q 1 , 2 / 2 ; P ( r ) = M 1 ( x + Δ x / 2 , y ) M 2 ( x - Δ x / 2 , y ) d φ ; x = r cos φ ; y = r sin φ ; P ( r ) = Σ B n exp [ 2 π i n ( r / R 1 ) 2 ] ; B n = A n ( 1 ) A - n ( 2 ) exp [ 2 π i n ( Δ x / a ) 3 / 12 ] ; 1 / R 1 2 = Δ x / a 3 ; 1 / f = Δ x 2 λ / a 3 .
4 .             S p h e r i c a l F Z P ; + Δ φ / 2 ; d φ ϕ ( r , φ ) = r 2 φ / a 2 ; M 1 , 2 ( r , φ ) = 1 ; if ϕ ( r , φ ) - N q 1 , 2 / 2 ; N = 0 , 1 , 2 , ; 0 φ < 2 π ; P ( r ) = M 1 ( r , φ + Δ φ / 2 ) M 2 ( r , φ - Δ φ / 2 ) d φ = Σ B n exp [ 2 π i n ( r / R 1 ) 2 ] ; 1 / R 1 2 = Δ φ / a 2 ; 1 / f = Δ φ 2 λ / a 2 .

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