Abstract

The effects of contrast in the Fresnel zone plate on the amplitude in the focal point due to a normally incident plane wave are obtained by direct evaluation of the Fresnel–Kirchhoff diffraction integral. The change in focal-point amplitude due to smoothing the ordinary, discontinuous, “square-wave” transmission function into a continuous, “sine-wave” function is also calculated. Finally, an expression is derived for the amplitude in the point image due to a plane wave impinging on the zone plate at a small angle of incidence.

© 1966 Optical Society of America

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References

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  1. R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), p. 37.
  2. O. Myers, Am. J. Phys. 19, 359 (1951).
    [CrossRef]
  3. K. Kamiya, Sci. Light 12, 35 (1963).
  4. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 377.
  5. Reference 4, p. 369.
  6. Reference 4, p. 379.

1963 (1)

K. Kamiya, Sci. Light 12, 35 (1963).

1951 (1)

O. Myers, Am. J. Phys. 19, 359 (1951).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 377.

Kamiya, K.

K. Kamiya, Sci. Light 12, 35 (1963).

Myers, O.

O. Myers, Am. J. Phys. 19, 359 (1951).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 377.

Wood, R. W.

R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), p. 37.

Am. J. Phys. (1)

O. Myers, Am. J. Phys. 19, 359 (1951).
[CrossRef]

Sci. Light (1)

K. Kamiya, Sci. Light 12, 35 (1963).

Other (4)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 377.

Reference 4, p. 369.

Reference 4, p. 379.

R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), p. 37.

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Figures (3)

Fig. 1
Fig. 1

Transmittance as a function of radius for the discontinuous and continuous zone plate.

Fig. 2
Fig. 2

Geometry for the Fresnel–Kirchhoff diffraction integral.

Fig. 3
Fig. 3

Geometry for the case of oblique incidence.

Equations (49)

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r n = ( n λ f ) 1 2 ,
T 0 = T max + T min ,
C 0 = ( T max - T min ) / T 0 .
T ( r ) = 1 2 T 0 [ 1 + C 0 sin ( π r 2 / λ f ) ] .
U ( P ) = - i A e i k s cos δ λ s a T ( ξ , η ) e i k g ( ξ , η ) d ξ d η ,
g ( ξ , η ) = - l ξ - m η + ( ξ 2 + η 2 ) / 2 s - ( l ξ + m η ) 2 / 2 s ,
U ( P ) = - i A e i k f λ f a T ( ξ , η ) d ξ d η = - ( i A e i k f / λ f ) ( T max a 1 + T min a 2 ) ,
g ( ξ , η ) = ( ξ 2 + η 2 ) / 2 s = r 2 / 2 s .
U ( 0 , 0 , f ) = - i A e i k f ( C + i S ) ,
C = 1 λ f 0 2 π d θ 0 r N T ( r ) ( cos π r 2 λ f ) r d r ,
S = 1 λ f 0 2 π d θ 0 r N T ( r ) ( sin π r 2 λ f ) r d r .
C = 0 v N T ( v ) cos v d v ,
S = 0 v N T ( v ) sin v d v .
C = T max 0 v 1 cos v d v + T min v 1 v 2 cos v d v + + T N v N - 1 v N cos v d v = T max [ sin v ] 0 π + T min [ sin v ] π 2 π + + T N [ sin v ] ( N - 1 ) π N π + 0 ,
S = - T max [ cos v ] 0 π - T min [ cos v ] 2 π π - - T N [ cos v ] N π ( N - 1 ) π = 2 T max - 2 T min + ± 2 T N , S = N C 0 T 0             N             even ,
S = ( N C 0 + 1 ) T 0             N             odd .
U = N C 0 T 0 A e i k f             N             even ,
U = ( N C 0 + 1 ) T 0 A e i k f             N             odd .
C = 1 2 T 0 0 v N ( 1 + C 0 sin v ) cos v d v = 1 2 T 0 [ sin v + 1 2 C 0 sin 2 v ] 0 N π = 0 ,
S = 1 2 T 0 0 v N ( 1 + C 0 sin v ) sin v d v = 1 2 T 0 [ - cos v + 1 2 C 0 ( v - sin v cos v ) ] 0 N π = 1 2 T 0 ( ± 1 + 1 2 C 0 N π + 1 ) , S = 1 4 π N C 0 T 0             N             even ,
S = ( 1 4 π N C 0 + 1 ) T 0             N             odd ,
U = 1 4 π N C 0 T 0 A e i k f             N             even ,
U = ( 1 4 π N C 0 + 1 ) T 0 e i k f             N             odd .
U = 0             N             even ,
U = T 0 A e i k f             N             odd .
U = i λ ( K 1 - K N ) 1 2 T 0 A e i k f             N             even ,
U = i λ ( K 1 + K N ) 1 2 T 0 A e i k f             N             odd ,
K N K 1 = - ( i / 2 λ ) ( 1 + cos δ ) = - i / λ .
U = 0             N             even ,
U = T 0 A e i k f             N             odd ,
U ( P ) = - i A e i k s cos φ λ s a T ( ξ , η ) e i k G ( ξ , η ) d ξ d η ,
G ( ξ , η ) = ξ sin φ - l ξ - m η + ( ξ 2 + η 2 ) / 2 s - ( l ξ + m η ) 2 / 2 s .
x = z sin φ + x cos φ ,
y = y ,
l ξ = ( z / s ) ξ sin φ + ( x / s ) ξ cos φ .
G ( ξ , η ) = ( ξ 2 + η 2 ) / 2 s - ξ 2 sin 2 φ / 2 s = r 2 ( 1 - cos 2 θ sin 2 φ ) / 2 f = ( r 2 sin 2 φ / 4 f ) [ ( 1 + cos 2 φ ) / sin 2 φ - cos 2 θ ] .
U ( P ) = - i A e i k f cos φ 2 λ f 0 r N 0 2 π e - i w cos 2 θ 2 d θ T ( r ) e i b w r d r ,
0 4 π e - i w cos α d α = 4 π J 0 ( - w ) = 4 π J 0 ( w ) .
U ( P ) = - i A e i k f 2 cos φ sin 2 φ 0 w N T ( w ) J 0 ( w ) e i b w d w .
J 0 ( w ) 1 - w 2 / 4.
U ( P ) = - i A e i k f ( 2 cos φ / sin 2 φ ) ( C + i S ) ,
C = 0 w N T ( w ) ( 1 - w 2 4 ) cos b w d w ,
S = 0 w N T ( w ) ( 1 - w 2 4 ) sin b w d w .
C = T max 0 w 1 ( 1 - w 2 4 ) cos b w d w + T min w 1 w 2 ( 1 - w 2 4 ) cos b w d w + + T N w N - 1 w N ( 1 - w 2 4 ) cos b w d w = 1 / b { T max [ F c ( w 1 ) - F c ( 0 ) ] + T min [ F c ( w 2 ) - F c ( w 1 ) ] + + T N [ F c ( w N ) - F c ( w N - 1 ) ] } ,
F c ( w ) = sin b w ( 1 + 1 / 2 b 2 - w 2 / 4 ) - ( w / 2 b 2 ) cos b w .
C = 1 / b [ C 0 T 0 j = 1 N - 1 ( - 1 ) j + 1 F c ( w j ) - T max F c ( 0 ) + T N F c ( w N ) ] .
S = 1 / b [ C 0 T 0 j = 1 N - 1 ( - 1 ) j + 1 F s ( w j ) - T max F s ( 0 ) + T N F s ( w N ) ] ,
F s ( w ) = - cos b w ( 1 + 1 / 2 b 2 - w 2 / 4 ) - ( w / 2 b 2 ) sin b w .
U ( P ) = - i A e i k f [ 2 cos φ / ( 1 + cos 2 φ ) ] × [ C 0 T 0 j = 1 N - 1 ( - 1 ) j + 1 F ( w j ) - T max F ( 0 ) + T N F ( w N ) ] .