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).
    [Crossref] [PubMed]
  12. J. Ojeda-Castaneda, L. R. Berriel-Valdos, “Arbitrarily High Focal Depth with Finite Apertures,” Opt. Lett. 13, 183–185 (1988).
    [Crossref] [PubMed]
  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 (3)

1987 (1)

1986 (4)

1984 (1)

1983 (1)

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

1976 (1)

1973 (1)

1967 (1)

1964 (1)

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. (6)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

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

Opt. Lett. (4)

Prog. Opt. (1)

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

Other (2)

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

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

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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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