Abstract

By apodizing the cells of a zone plate and changing the opening ratio, it is possible to shape the relative power spectrum of its foci. We describe a novel procedure that leads to an analytical formula for shaping the focus power spectrum by using apodizers expressible as the Legendre series; these act on cells of arbitrary opening ratio. Our general result is used to design zone plates that have missing foci and to discuss a synthesis procedure using apodizers with various opening ratios. Our applications can also be used for shaping the power spectrum of 1-D gratings.

© 1990 Optical Society of America

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References

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  1. P. J. Burt, Multiresolution Image Processing and Analysis, A. Rosenfeld, Ed. (Springer-Verlag, New York, 1984), Chap. 2.
  2. O. Bryngdahl, “Image Formation Using Self-Imaging Techniques,” J. Opt. Soc. Am. 63, 416–419 (1973).
    [CrossRef]
  3. B. J. Thompson, “Multiple Imaging by Diffraction Techniques,” Appl. Opt. 15, 312 (1976).
    [CrossRef] [PubMed]
  4. G. Indebetouw, “Self-Imaging Through a Fabry-Perot Interferometer,” Opt. Acta 30, 1463–1471 (1983).
    [CrossRef]
  5. J. Ojeda-Castaneda, P. Andres, E. Tepichin, “Spatial Filters for Replicating Images,” Opt. Lett. 11, 551–553 (1986).
    [CrossRef] [PubMed]
  6. A. Davila, J. E. A. Landgrave, “Simultaneous Imaging of Periodic Object Planes,” Appl. Opt. 27, 174–180 (1988).
    [CrossRef] [PubMed]
  7. P. Jaquinot, B. Roizen-Dossier, “Apodisation,” Prog. Opt. 3, 31–184 (1964).
  8. J. Ojeda-Castaneda, P. Andres, A. Diaz, “Objects that Exhibit High Focal Depth,” Opt. Lett. 11, 267–269 (1986).
    [CrossRef] [PubMed]
  9. G. Indebetouw, H. X. Bai, “Imaging with Fresnel Zone Pupil Masks: Extended Depth of Field,” Appl. Opt. 23, 4299–4302 (1984).
    [CrossRef] [PubMed]
  10. J. P. Mills, B. J. Thompson, “Effect of Aberrations and Apodization on the Performance of Coherent Optical Systems. I. The Amplitude Impulse Response,” J. Opt. Soc. Am. A 3, 694–703 (1986).
    [CrossRef]
  11. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Bessel Annular Apodizers: Imaging Characteristics,” Appl. Opt. 26, 2770–2772 (1987).
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  12. J. Ojeda-Castaneda, L. R. Berriel-Valdos, “Arbitrarily High Focal Depth with Finite Apertures,” Opt. Lett. 13, 183–185 (1988).
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  13. J. Ojeda-Castaneda, P. Andres, A. Diaz, “Annular Apodizers for Low Sensitivity to Defocus and to Spherical Aberration,” Opt. Lett. 11, 487–489 (1986).
    [CrossRef] [PubMed]
  14. J. Ojeda-Castaneda, E. Tepichin, A. Pons, “Apodization of Annular Apertures: Strehl Ratio,” Appl. Opt. 27, 5140–5145 (1988).
    [CrossRef] [PubMed]
  15. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 440.
  16. A. W. Lohmann, D. P. Paris, “Variable Fresnel Zone Plate,” Appl. Opt. 6, 1567–1570 (1967).
    [CrossRef] [PubMed]

1988

1987

1986

1984

1983

G. Indebetouw, “Self-Imaging Through a Fabry-Perot Interferometer,” Opt. Acta 30, 1463–1471 (1983).
[CrossRef]

1976

1973

1967

1964

P. Jaquinot, B. Roizen-Dossier, “Apodisation,” Prog. Opt. 3, 31–184 (1964).

Andres, P.

Bai, H. X.

Berriel-Valdos, L. R.

Bryngdahl, O.

Burt, P. J.

P. J. Burt, Multiresolution Image Processing and Analysis, A. Rosenfeld, Ed. (Springer-Verlag, New York, 1984), Chap. 2.

Davila, A.

Diaz, A.

Indebetouw, G.

G. Indebetouw, H. X. Bai, “Imaging with Fresnel Zone Pupil Masks: Extended Depth of Field,” Appl. Opt. 23, 4299–4302 (1984).
[CrossRef] [PubMed]

G. Indebetouw, “Self-Imaging Through a Fabry-Perot Interferometer,” Opt. Acta 30, 1463–1471 (1983).
[CrossRef]

Jaquinot, P.

P. Jaquinot, B. Roizen-Dossier, “Apodisation,” Prog. Opt. 3, 31–184 (1964).

Landgrave, J. E. A.

Lohmann, A. W.

Mills, J. P.

Montes, E.

Ojeda-Castaneda, J.

Paris, D. P.

Pons, A.

Roizen-Dossier, B.

P. Jaquinot, B. Roizen-Dossier, “Apodisation,” Prog. Opt. 3, 31–184 (1964).

Tepichin, E.

Thompson, B. J.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

G. Indebetouw, “Self-Imaging Through a Fabry-Perot Interferometer,” Opt. Acta 30, 1463–1471 (1983).
[CrossRef]

Opt. Lett.

Prog. Opt.

P. Jaquinot, B. Roizen-Dossier, “Apodisation,” Prog. Opt. 3, 31–184 (1964).

Other

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 440.

P. J. Burt, Multiresolution Image Processing and Analysis, A. Rosenfeld, Ed. (Springer-Verlag, New York, 1984), Chap. 2.

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Figures (9)

Fig. 1
Fig. 1

Generation of apodizing functions with the same transmittance profile but different opening ratio.

Fig. 2
Fig. 2

Traditional method of focus elimination by changing the opening ratio of rectangular cells.

Fig. 3
Fig. 3

Focus elimination by changing the opening ratio of cells apodized with the first-order Legendre polynomial, as in Fig. 1.

Fig. 4
Fig. 4

Amplitude transmittance: (a) dotted line, the zero-order Legendre polynomial; (b) dashed line, the second-order Legendre polynomial; (c) solid line, the combination of (a) and (b) as in Eq. (22).

Fig. 5
Fig. 5

Focal power spectrum of the apodized zone plate in Fig. 4.

Fig. 6
Fig. 6

Pyramidlike apodizer (solid line) generated by adding with various weights three rectangular functions (discontinuous lines) of different opening ratio.

Fig. 7
Fig. 7

Focal power spectrum of pyramidlike apodized zone plate in Fig. 6.

Fig. 8
Fig. 8

Continuous apodizing function (solid line) generated by adding with different weights three functions (discontinuous lines) like that of Fig. 4 but with a different opening ratio.

Fig. 9
Fig. 9

Focal power spectrum of the continuous apodizer in Fig. 8.

Equations (28)

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F ( x , s ) = m = - C m ( s ) exp ( i 2 π x m / d ) ,
C m ( s ) = ( 1 / d ) - s d / 2 s d / 2 F ( x , s ) exp ( - i 2 π x m / d ) d x .
exp ( - i x y ) = n = 0 ( - i ) n ( 2 n + 1 ) P n ( x ) j n ( y ) ,
exp ( - i 2 π x m / d ) = n = 0 ( - i ) n ( 2 n + 1 ) P n ( 2 x / s d ) j n ( m π s ) .
C m ( s ) = n = 0 ( - i ) n ( 2 n + 1 ) [ ( 1 / d ) - s d / 2 s d / 2 F ( x , s ) P n ( 2 x / s d ) d x ] j n ( m π s ) ,
t = 2 x / s d ,             G ( t ) = F ( x , s ) ,
C m ( s ) = ( s / 2 ) n = 0 ( - i ) n ( 2 n + 1 ) j n ( m π s ) [ - 1 1 G ( t ) P n ( t ) d t ] .
F ( x , s 1 ) = F ( s 2 x / s 1 , s 2 ) .
G ( t ) = q = 0 a q P q ( t ) .
C m ( s ) = s q = 0 ( - i ) q a q j q ( π m s ) .
C m ( s ) = s j 0 ( π m s ) = s sin ( π m s ) / ( π m s ) .
H ( r 2 , s ) = m = - h m ( s ) exp [ i 2 π m ( r / R ) 2 ] ,
h m ( s ) = ( 1 / R 2 ) 0 s R 2 H ( r 2 , s ) exp [ - i 2 π m ( r / R ) 2 ] d ( r 2 ) .
x = r 2 - s R 2 / 2 , J ( x , s ) = H ( r 2 , s ) , d = R 2 .
h m ( s ) = ( 1 / d ) - s d / 2 s d / 2 J ( x , s ) exp ( - i 2 π m x / d ) d x ,
h m ( s ) = s q = 0 ( - i ) q a q j q ( π m s ) .
H ( r 2 ) = J ( x = r 2 - s R 2 / 2 ) = q = 0 a q P q ( t = 2 r 2 / s R 2 - 1 ) .
a q = δ 0 q ,             G ( t ) = P 0 ( t ) = 1 ,
h m ( s ) = s j 0 ( π m s ) = sin ( π m s ) / π m .
h m ( s ) = ( - i s ) j 1 ( π m s ) = ( - i s ) [ sin ( π m s ) / ( π m s ) 2 - cos ( π m s ) / ( π m s ) ] .
π m s = π ( 1.43 )             or             π m s = π ( 2.47 ) .
G ( t ) = ( 2 / 3 ) P 0 ( t ) - ( 2 / 3 ) P 2 ( t ) = 1 - t 2 .
h m ( s ) 2 = 4 s 2 [ sin ( π m s ) / ( π m s ) 3 - cos ( π m s ) / ( π m s ) 2 ] 2 ,
x = ( s d / 2 ) t ,             J ( x , s ) = G ( t ) .
f ( x ) = k = 1 K e k J ( x , s k ) .
G ( t ) = f [ ( d / 2 ) t ] = k = 1 K e k J [ ( d / 2 ) t , s k ] .
x = ( s d / 2 ) t ,             J ( x , s ) = k = 1 K e k J ( x / s , s k ) k = 1 K e k J ( x , s k s ) ,
h m ( s ) = k = 1 K e k h m ( s k s ) .

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