Abstract

The diffraction patterns of the central aperture and the first 10 rings of the classical Fresnel zone plate are calculated in the principal focal plane for off-axis distances as large as the radius of the 10th ring. The zone-plate pattern is found by summation; near the axis it is compared with closed-form approximations. The integrated flux within circles as large as the 10th ring is also calculated.

© 1969 Optical Society of America

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References

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  1. F. Zernike and B. R. A. Nijboer in La Théorie des Images Optiques (Revue d’ Optique, Paris, 1949), p. 227.
  2. B. R. A. Nijboer, thesis, Groningen, 1942;Physica 10, 679 (1947).
  3. J. Boersma, Math. Comp. 16, 232 (1962).
  4. A. V. Baez, J. Opt. Soc. Am. 51, 405 (1961).
    [CrossRef]
  5. G. Möllenstedt, K. H. von Grote, and C. Jönsson in X-Ray Optics and X-Ray Microanalysis, H. H. Pattee, Ed. (Academic Press Inc., New York, 1963), p. 73.
  6. A. Boivin, Théorie et Calcule des Figures de Diffraction de Revolution (Les Presses de l’ Université Laval, Quebec, 1964), Ch. 5.
  7. Principles of Optics, M. Born and E. Wolf, Eds. (Pergamon Press, Inc., New York, 1959), p. 436.
  8. See Ref. 7, p. 397.

1962 (1)

J. Boersma, Math. Comp. 16, 232 (1962).

1961 (1)

Baez, A. V.

Boersma, J.

J. Boersma, Math. Comp. 16, 232 (1962).

Boivin, A.

A. Boivin, Théorie et Calcule des Figures de Diffraction de Revolution (Les Presses de l’ Université Laval, Quebec, 1964), Ch. 5.

Jönsson, C.

G. Möllenstedt, K. H. von Grote, and C. Jönsson in X-Ray Optics and X-Ray Microanalysis, H. H. Pattee, Ed. (Academic Press Inc., New York, 1963), p. 73.

Möllenstedt, G.

G. Möllenstedt, K. H. von Grote, and C. Jönsson in X-Ray Optics and X-Ray Microanalysis, H. H. Pattee, Ed. (Academic Press Inc., New York, 1963), p. 73.

Nijboer, B. R. A.

F. Zernike and B. R. A. Nijboer in La Théorie des Images Optiques (Revue d’ Optique, Paris, 1949), p. 227.

B. R. A. Nijboer, thesis, Groningen, 1942;Physica 10, 679 (1947).

von Grote, K. H.

G. Möllenstedt, K. H. von Grote, and C. Jönsson in X-Ray Optics and X-Ray Microanalysis, H. H. Pattee, Ed. (Academic Press Inc., New York, 1963), p. 73.

Zernike, F.

F. Zernike and B. R. A. Nijboer in La Théorie des Images Optiques (Revue d’ Optique, Paris, 1949), p. 227.

J. Opt. Soc. Am. (1)

Math. Comp. (1)

J. Boersma, Math. Comp. 16, 232 (1962).

Other (6)

G. Möllenstedt, K. H. von Grote, and C. Jönsson in X-Ray Optics and X-Ray Microanalysis, H. H. Pattee, Ed. (Academic Press Inc., New York, 1963), p. 73.

A. Boivin, Théorie et Calcule des Figures de Diffraction de Revolution (Les Presses de l’ Université Laval, Quebec, 1964), Ch. 5.

Principles of Optics, M. Born and E. Wolf, Eds. (Pergamon Press, Inc., New York, 1959), p. 436.

See Ref. 7, p. 397.

F. Zernike and B. R. A. Nijboer in La Théorie des Images Optiques (Revue d’ Optique, Paris, 1949), p. 227.

B. R. A. Nijboer, thesis, Groningen, 1942;Physica 10, 679 (1947).

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Figures (6)

F. 1
F. 1

Diffraction pattern of central aperture in principal focal plane of zone plate. Re (A0) and Im (A0) are the real and imaginary components of the amplitude as a function of ρ/r1.

F. 2
F. 2

Diffraction pattern of first ring in principal focal plane of zone plate. Re(A1) and Im(A1) are the real and imaginary-components of the amplitude as a function of ρ/r1. The geometrical projection of the ring is marked by r2 and r3.

F. 3
F. 3

Diffraction by zone plate consisting of central aperture and one ring. Re(S1) and Im(S1) are the real and imaginary components of the amplitude. The dashed lines are the base terms in the near-axis approximations [Eqs. (22) and (23)]. The dotted line is the quadratic approximation of the imaginary base term in Eq. (32).

F. 4
F. 4

Diffraction by zone plate with, five rings. Re(S5) and Im(S5) are the real and imaginary components of the amplitude. The dotted line is the extension of the approximation represented by Eq. (22). The dashed line is the base term in Eq. (22).

F. 5
F. 5

Diffraction by zone plate with 10 rings. Re(S10) and Im(S10) are the real and imaginary components of the amplitude. The dashed line is the real base term in Eq. (22). Note the similarity between Im (S10) and Im (S5) in Fig. 4.

F. 6
F. 6

Fraction of transmitted flux (EN/ET) falling within a circle of radius ρ0 for various values of N, the number of rings in the zone plate. The crosses indicate ρ0 values equal to the outer radius. A cross section of the rings is shown at the top.

Tables (4)

Tables Icon

Table I Real base term Re ( A 0 ) 3 J 1 [ y ( 3 2 ) 1 2 ] / [ y ( 3 2 ) 1 2 ] and imaginary base term Im ( A 0 ) ( i / π ) J 0 [ y ( 3 2 ) 1 2 ] for near-axis calculation of zone-plate diffracted-light amplitude.

Tables Icon

Table II Real and imaginary components of amplitude and irradiance for zone plate with 10 rings. ρ/r1 = 0.00909241 u.

Tables Icon

Table III Fraction of transmitted flux EN/ET falling within circle of radius ρ0.

Tables Icon

Table IV Flux in central disk (EC) as fraction of transmitted flux (ET) and incident flux (EZ).

Equations (34)

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f = r 1 2 / λ ,
a = ( 2 i p ) 0 1 t d t exp ( i p t 2 ) J 0 ( q t ) ,
a = 2 i ( π p ) 1 2 exp ( 1 2 p i ) j = 0 ( 2 j + 1 ) ( i ) j × J j + 1 2 ( 1 2 p ) J 2 j + 1 ( q ) / q .
p 2 n + 1 = ( 2 n + 1 ) π
q 2 n + 1 = y ( 2 n + 1 ) 1 2 with y = 2 π ρ / r 1 .
a 2 n + 1 ( y ) = 2 π ( 1 ) n j = 0 ( 2 j + 1 ) ( i ) j × J j + 1 2 [ ( n + 1 2 ) π ] J 2 j + 1 [ y ( 2 n + 1 ) 1 2 ] / y .
a 2 n + 1 = 4 J 1 [ y ( 2 n + 1 ) 1 2 ] / [ y ( 2 n + 1 ) 1 2 ] ( 24 i / π ) × ( 2 n + 1 ) 1 J 3 [ y ( 2 n + 1 ) 1 2 ] / [ y ( 2 n + 1 ) 1 2 ] ,
p 2 n = 2 n π
q 2 n = y ( 2 n ) 1 2
a 2 n = 2 π ( 1 ) n + 1 j = 1 2 j ( 1 ) j 1 J j + 1 2 ( n π ) × J 2 j + 1 [ y ( 2 n ) 1 2 ] / y .
a 2 n = 4 J 3 [ y ( 2 n ) 1 2 ] / [ y ( 2 n ) 1 2 ] ( 12 π i / n 2 ) J 5 [ y ( 2 n ) 1 2 ] / y ( 2 n ) 1 2 .
A n = a 2 n + 1 a 2 n .
A n = π 1 2 1 2 d x exp ( i π x ) J 0 [ y ( 2 n + 1 2 + x ) 1 2 ] .
A n = π m = 0 ( C m / m ! ) [ y / 2 ( 2 n + 1 2 ) 1 2 ] m × J m [ y ( 2 n + 1 2 ) 1 2 ]
C m = 1 2 1 2 x m exp ( i π x ) d x .
Re ( A n ) = 2 J 0 [ y ( 2 n + 1 ) 1 2 ] + y 2 ( π 2 8 ) / ( 16 π 2 ) × J 2 [ y ( 2 n + 1 2 ) 1 2 ] / ( 2 n + 1 2 )
Im ( A n ) = i y / [ 2 π ( 2 n + 1 2 ) 1 2 ] × { 2 J 1 [ y ( 2 n + 1 2 ) 1 2 ] + y 2 ( π 2 8 ) / ( 16 π 2 ) J 3 [ y ( 2 n + 1 2 ) 1 2 ] / ( 2 n + 1 2 ) } .
C m = ( m + 1 ) i π C m 1 ( m even ) .
d Im ( A n ) / d y = ( i y / 2 π ) Re ( A n ) .
S N A 0 + 1 2 N + 1 2 A n d n = A 0 + ( π / 2 ) m = 0 ( y / 2 ) m 1 ( C m / m ! ) × { ( 2 N + 3 2 ) 1 2 ( m 1 ) J m 1 [ y ( 2 N 3 2 ) 1 2 ] ( 3 2 ) 1 2 ( m 1 ) J m 1 [ y ( 3 2 ) 1 2 ] } .
d Im ( S N ) / d y = ( i y / 2 π ) Re ( S N ) ,
Re ( S N ) = ( 2 N + 3 2 ) 2 J 1 [ y ( 2 N + 3 2 ) 1 2 ] / [ y ( 2 N + 3 2 ) 1 2 ] 3 J 1 [ y ( 3 2 ) 1 2 ] / [ y ( 3 2 ) 1 2 ] + Re ( A 0 ) ,
Im ( S N ) = ( i / π ) J 0 [ y ( 2 N + 3 2 ) 1 2 ] ( i / π ) J 0 [ y ( 3 2 ) 1 2 ] + Im ( A 0 ) ,
y = [ 2 π ( 21 ) 1 2 / 504 ] u .
Re ( S N ) = 2 ( N + 1 ) and Im ( S N ) = 0 .
( 2 N + 3 2 ) π r 1 2 / λ f = ( 2 N + 3 2 ) π ,
E N ( ρ 0 ) = 2 π 0 ρ 0 d ρ ρ I N ( ρ ) .
E Z = ( 2 N + 1 ) π r 1 2 ,
E T = ( N + 1 ) π r 1 2 .
E N ( ρ 0 ) = ( π / 4 ) ( Δ ρ ) 2 I N ( 0 ) + 2 π m = 1 M m ( Δ ρ ) 2 I N ( m Δ ρ ) ,
Re ( S N ) ( 2 N + 3 2 ) { 2 J 1 [ y ( 2 N + 3 2 ) 1 2 ] / [ y ( 2 N + 3 2 ) 1 2 ] } + 1 2 .
Im ( S N ) ( i / π ) { 1 J 0 [ y ( 2 N + 3 2 ) 1 2 ] + y 2 / 8 } .
( E C / E T ) Re = ( 2 N + 3 2 ) / [ π 2 ( N + 1 ) ] × { 1 J 0 2 ( x 0 ) + [ 1 J 0 ( x 0 ) ] / ( 2 N + 3 2 ) + x 0 2 / [ 8 ( 2 N + 3 2 ) 2 ] } .
ρ 0 = 1 2 r 1