Abstract

A scheme for microscopy of relatively large-size objects by using Fresnel zone plate (FZP) coded imaging (FZPCI) is digitally demonstrated. The limit on the source size in zone-plate-based microscopy comes from interference of out-of-focus multidiffraction orders of the FZP with the focused-order image. From the study of the angular spectrum of the coded image, it is shown that noise contribution from higher orders to a lower-order image can be digitally suppressed by selective propagation of spatial frequencies. Similarly, noise from aliasing and noise from lower orders to a higher-order image can be reduced by spatially limiting the coded image. To my knowledge for the first time, the results of digitally performed FZPCI-based microscopy of an object that is three times larger than the first zone of the FZP with a resolution better than 2 µm are presented and discussed.

© 2004 Optical Society of America

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References

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  1. R. H. Dicke, “Scatter hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101–L106 (1968).
    [CrossRef]
  2. H. H. Barrett, F. A. Horrigan, “Fresnel zone plate imaging of gamma rays: theory,” Appl. Opt. 12, 2686–2700 (1973).
    [CrossRef] [PubMed]
  3. H. H. Barrett, G. D. DeMeester, “Quantum noise in Fresnel zone plate imaging,” Appl. Opt. 13, 1100–1109 (1974).
    [CrossRef] [PubMed]
  4. N. M. Ceglio, L. W. Coleman, “Spatially resolved α emission from laser fusion targets,” Phys. Rev. Lett. 39, 20–24 (1977).
    [CrossRef]
  5. N. M. Ceglio, J. T. Larson, “Spatially resolved supra thermal x-ray emission from laser fusion targets,” Phys. Rev. Lett. 44, 579–582 (1980).
    [CrossRef]
  6. Be it the application of astronomy, nuclear medicine, or laser-plasma,7,8 the object to be imaged through FZPCI has been a source of one or another kind of incoherent radiation. However, the new applications that arise in the wake of digital FZPCI of this paper, such as optical tomography9 and crystallography, might involve objects that are not direct sources but are illuminated by an incoherent source placed in the vicinity.
  7. N. M. Ceglio, D. T. Attwood, E. V. George, “Zone plate coded imaging on a microscopic scale,” J. Appl. Phys. 48, 1563–1565 (1977).
    [CrossRef]
  8. N. M. Ceglio, “Zone plate coded imaging for laser produced plasma,” Appl. Phys. 48, 1566–1569 (1977).
    [CrossRef]
  9. G. S. Solanki, “Optical incoherent FZP coded 3d tomography,” manuscript available from the author, soham@cat.ernet.in.
  10. T. D. Beynon, I. Kirk, T. R. Mathews, “Gabor zone plate with binary transmittance,” Opt. Lett. 17, 544–546 (1992).
    [CrossRef] [PubMed]
  11. T. M. Cannon, E. E. Fenimore, “Coded aperture imaging: many holes make light work,” Opt. Eng. 19, 283–289 (1980).
    [CrossRef]
  12. H. D. Lueke, A. Busboom, “Binary arrays with perfect odd-periodic autocorrelation,” Appl. Opt. 36, 6612–6619 (1997).
    [CrossRef]
  13. K. Byard, “Imaging using HURA coded apertures with discrete pixel detector array,” Astron. Astrophys. 227, 634–639 (1990).
  14. M. H. Finger, T. A. Prince, “Useful classes of redundant array for imaging applications,” in Imaging in High-Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1995), p. 21.
  15. G. S. Solanki, H. C. Pant, “Computer aided zone plate coded 3d imaging: theory, software and implementation,” (2000) (available at http://www.cat.ernet.in/hpl/zpci.pdf ).
  16. G. S. Solanki, “Computer aided ZPCI diagnostic for short wavelength incoherent radiation,” Opt. Commun. (to be published).
  17. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  18. G. Gbur, E. Wolf, “Relation between computed tomography and diffraction tomography,” J. Opt. Soc. Am. A 18, 2132–2137 (2001).
    [CrossRef]
  19. Y. Yasuno, M. Nakama, Y. Sutoh, M. Itoh, M. Mori, T. Yatagai, “Optical coherence tomography by spectral interferometric joint transform correlator,” Opt. Commun. 186, 51–56 (2000).
    [CrossRef]
  20. E. A. Swanson, D. Haung, M. R. Hee, J. G. Fujimoto, “High-speed optical coherence domain reflectometry,” Opt. Lett. 17, 151–153 (1992).
    [CrossRef] [PubMed]

2001 (1)

2000 (1)

Y. Yasuno, M. Nakama, Y. Sutoh, M. Itoh, M. Mori, T. Yatagai, “Optical coherence tomography by spectral interferometric joint transform correlator,” Opt. Commun. 186, 51–56 (2000).
[CrossRef]

1997 (1)

1992 (2)

1990 (1)

K. Byard, “Imaging using HURA coded apertures with discrete pixel detector array,” Astron. Astrophys. 227, 634–639 (1990).

1980 (2)

T. M. Cannon, E. E. Fenimore, “Coded aperture imaging: many holes make light work,” Opt. Eng. 19, 283–289 (1980).
[CrossRef]

N. M. Ceglio, J. T. Larson, “Spatially resolved supra thermal x-ray emission from laser fusion targets,” Phys. Rev. Lett. 44, 579–582 (1980).
[CrossRef]

1977 (3)

N. M. Ceglio, D. T. Attwood, E. V. George, “Zone plate coded imaging on a microscopic scale,” J. Appl. Phys. 48, 1563–1565 (1977).
[CrossRef]

N. M. Ceglio, “Zone plate coded imaging for laser produced plasma,” Appl. Phys. 48, 1566–1569 (1977).
[CrossRef]

N. M. Ceglio, L. W. Coleman, “Spatially resolved α emission from laser fusion targets,” Phys. Rev. Lett. 39, 20–24 (1977).
[CrossRef]

1974 (1)

1973 (1)

1968 (1)

R. H. Dicke, “Scatter hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101–L106 (1968).
[CrossRef]

Attwood, D. T.

N. M. Ceglio, D. T. Attwood, E. V. George, “Zone plate coded imaging on a microscopic scale,” J. Appl. Phys. 48, 1563–1565 (1977).
[CrossRef]

Barrett, H. H.

Beynon, T. D.

Busboom, A.

Byard, K.

K. Byard, “Imaging using HURA coded apertures with discrete pixel detector array,” Astron. Astrophys. 227, 634–639 (1990).

Cannon, T. M.

T. M. Cannon, E. E. Fenimore, “Coded aperture imaging: many holes make light work,” Opt. Eng. 19, 283–289 (1980).
[CrossRef]

Ceglio, N. M.

N. M. Ceglio, J. T. Larson, “Spatially resolved supra thermal x-ray emission from laser fusion targets,” Phys. Rev. Lett. 44, 579–582 (1980).
[CrossRef]

N. M. Ceglio, L. W. Coleman, “Spatially resolved α emission from laser fusion targets,” Phys. Rev. Lett. 39, 20–24 (1977).
[CrossRef]

N. M. Ceglio, D. T. Attwood, E. V. George, “Zone plate coded imaging on a microscopic scale,” J. Appl. Phys. 48, 1563–1565 (1977).
[CrossRef]

N. M. Ceglio, “Zone plate coded imaging for laser produced plasma,” Appl. Phys. 48, 1566–1569 (1977).
[CrossRef]

Coleman, L. W.

N. M. Ceglio, L. W. Coleman, “Spatially resolved α emission from laser fusion targets,” Phys. Rev. Lett. 39, 20–24 (1977).
[CrossRef]

DeMeester, G. D.

Dicke, R. H.

R. H. Dicke, “Scatter hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101–L106 (1968).
[CrossRef]

Fenimore, E. E.

T. M. Cannon, E. E. Fenimore, “Coded aperture imaging: many holes make light work,” Opt. Eng. 19, 283–289 (1980).
[CrossRef]

Finger, M. H.

M. H. Finger, T. A. Prince, “Useful classes of redundant array for imaging applications,” in Imaging in High-Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1995), p. 21.

Fujimoto, J. G.

Gbur, G.

George, E. V.

N. M. Ceglio, D. T. Attwood, E. V. George, “Zone plate coded imaging on a microscopic scale,” J. Appl. Phys. 48, 1563–1565 (1977).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Haung, D.

Hee, M. R.

Horrigan, F. A.

Itoh, M.

Y. Yasuno, M. Nakama, Y. Sutoh, M. Itoh, M. Mori, T. Yatagai, “Optical coherence tomography by spectral interferometric joint transform correlator,” Opt. Commun. 186, 51–56 (2000).
[CrossRef]

Kirk, I.

Larson, J. T.

N. M. Ceglio, J. T. Larson, “Spatially resolved supra thermal x-ray emission from laser fusion targets,” Phys. Rev. Lett. 44, 579–582 (1980).
[CrossRef]

Lueke, H. D.

Mathews, T. R.

Mori, M.

Y. Yasuno, M. Nakama, Y. Sutoh, M. Itoh, M. Mori, T. Yatagai, “Optical coherence tomography by spectral interferometric joint transform correlator,” Opt. Commun. 186, 51–56 (2000).
[CrossRef]

Nakama, M.

Y. Yasuno, M. Nakama, Y. Sutoh, M. Itoh, M. Mori, T. Yatagai, “Optical coherence tomography by spectral interferometric joint transform correlator,” Opt. Commun. 186, 51–56 (2000).
[CrossRef]

Pant, H. C.

G. S. Solanki, H. C. Pant, “Computer aided zone plate coded 3d imaging: theory, software and implementation,” (2000) (available at http://www.cat.ernet.in/hpl/zpci.pdf ).

Prince, T. A.

M. H. Finger, T. A. Prince, “Useful classes of redundant array for imaging applications,” in Imaging in High-Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1995), p. 21.

Solanki, G. S.

G. S. Solanki, H. C. Pant, “Computer aided zone plate coded 3d imaging: theory, software and implementation,” (2000) (available at http://www.cat.ernet.in/hpl/zpci.pdf ).

G. S. Solanki, “Computer aided ZPCI diagnostic for short wavelength incoherent radiation,” Opt. Commun. (to be published).

Sutoh, Y.

Y. Yasuno, M. Nakama, Y. Sutoh, M. Itoh, M. Mori, T. Yatagai, “Optical coherence tomography by spectral interferometric joint transform correlator,” Opt. Commun. 186, 51–56 (2000).
[CrossRef]

Swanson, E. A.

Wolf, E.

Yasuno, Y.

Y. Yasuno, M. Nakama, Y. Sutoh, M. Itoh, M. Mori, T. Yatagai, “Optical coherence tomography by spectral interferometric joint transform correlator,” Opt. Commun. 186, 51–56 (2000).
[CrossRef]

Yatagai, T.

Y. Yasuno, M. Nakama, Y. Sutoh, M. Itoh, M. Mori, T. Yatagai, “Optical coherence tomography by spectral interferometric joint transform correlator,” Opt. Commun. 186, 51–56 (2000).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. (1)

N. M. Ceglio, “Zone plate coded imaging for laser produced plasma,” Appl. Phys. 48, 1566–1569 (1977).
[CrossRef]

Astron. Astrophys. (1)

K. Byard, “Imaging using HURA coded apertures with discrete pixel detector array,” Astron. Astrophys. 227, 634–639 (1990).

Astrophys. J. (1)

R. H. Dicke, “Scatter hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101–L106 (1968).
[CrossRef]

J. Appl. Phys. (1)

N. M. Ceglio, D. T. Attwood, E. V. George, “Zone plate coded imaging on a microscopic scale,” J. Appl. Phys. 48, 1563–1565 (1977).
[CrossRef]

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

Opt. Commun. (1)

Y. Yasuno, M. Nakama, Y. Sutoh, M. Itoh, M. Mori, T. Yatagai, “Optical coherence tomography by spectral interferometric joint transform correlator,” Opt. Commun. 186, 51–56 (2000).
[CrossRef]

Opt. Eng. (1)

T. M. Cannon, E. E. Fenimore, “Coded aperture imaging: many holes make light work,” Opt. Eng. 19, 283–289 (1980).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (2)

N. M. Ceglio, L. W. Coleman, “Spatially resolved α emission from laser fusion targets,” Phys. Rev. Lett. 39, 20–24 (1977).
[CrossRef]

N. M. Ceglio, J. T. Larson, “Spatially resolved supra thermal x-ray emission from laser fusion targets,” Phys. Rev. Lett. 44, 579–582 (1980).
[CrossRef]

Other (6)

Be it the application of astronomy, nuclear medicine, or laser-plasma,7,8 the object to be imaged through FZPCI has been a source of one or another kind of incoherent radiation. However, the new applications that arise in the wake of digital FZPCI of this paper, such as optical tomography9 and crystallography, might involve objects that are not direct sources but are illuminated by an incoherent source placed in the vicinity.

G. S. Solanki, “Optical incoherent FZP coded 3d tomography,” manuscript available from the author, soham@cat.ernet.in.

M. H. Finger, T. A. Prince, “Useful classes of redundant array for imaging applications,” in Imaging in High-Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1995), p. 21.

G. S. Solanki, H. C. Pant, “Computer aided zone plate coded 3d imaging: theory, software and implementation,” (2000) (available at http://www.cat.ernet.in/hpl/zpci.pdf ).

G. S. Solanki, “Computer aided ZPCI diagnostic for short wavelength incoherent radiation,” Opt. Commun. (to be published).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

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Figures (10)

Fig. 1
Fig. 1

Schematic set up for construction of the coded image. Incoherent radiation from a planar source f(x, y) are recorded in the coded-image plane h(ξ, η) through an FZP, g(x, y). Diffraction effects must be avoided in an ideal zone plate camera.

Fig. 2
Fig. 2

Spatial-frequency distribution for dc and the first three converging and diverging orders in the coded image as a function of radius. The solid line along ρ axis represents the base-band extension of the source spectrum, which in turn determines the spatial extension of various orders in the coded image. The horizontal dashed–dotted lines from it interfere with different-order lines. The normal from these interference points to the r axis gives the spatial extension of the orders.

Fig. 3
Fig. 3

Schematic representation in the spatial-frequency domain of various orders of the coded aperture that contributed to the coded image. The low-pass filter, shown by the dotted lines with a stop-band at zero frequency, suppresses the background noise in the first-order reconstruction. (The spectral profiles of the individual orders need not be exactly triangular but depend on the source-spectrum profile. However, for a base-band source they may be approximated thus.)

Fig. 4
Fig. 4

First-order reconstructions (shown to right of the arrows) of a two-point object from different portions (shown at left) of the original coded image demonstrate Eq. (12). The resolution of the reconstruction (a), decoded from a full coded image, is almost two times better than that of (b). The size of the segmented coded image in (b) is one half and the area one fourth of that in (a). This shows that resolution increases almost linearly with symmetrical increase in the size of the coded image. Panel (c) demonstrates the effect of asymmetrical inclusion of higher frequencies.

Fig. 5
Fig. 5

Schematics of enhanced digital decoding. Abbreviations used: SL, spatial limiting; (shown by an aperture), PWS, plane-wave spectrum; LPF, low-pass filtering; IFFT, inverse fast Fourier transform.

Fig. 6
Fig. 6

2D object containing 51 bright pixels each of size 2 µm. The size of the object is 64 µm×64 µm, approximately three times larger than the central zone diameter of the FZP.

Fig. 7
Fig. 7

Coded image of the object of Fig. 6 at size 512 µm×512 µm.

Fig. 8
Fig. 8

Coded image of the object of Fig. 6 spatially limited to 256 µm×256 µm in size.

Fig. 9
Fig. 9

Reconstruction of the image through the enhanced digital decoding scheme to decouple out noise contribution from the higher orders and aliasing. The reconstruction is perfect, mapped pixel by pixel with the object, with a resolution better than 2 µm and without artifacts.

Fig. 10
Fig. 10

Normal reconstruction, without any enhanced image processing, would have resulted in artifacts and poor resolution owing to the large size of the object. Notice the uncorrected axial inversion of the image, which is inherent in optical reconstruction and can very easily be taken care of in digital decoding.

Equations (17)

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h(ξ, η, z2)=Cf(x, y, z1)gξ+nxm, η+nymdxdy,
C=14π(z2+z1)2,
m=z2+z1z1,n=z2z1.
h(ξ, η)=h0+m=-(odd)hp(ξ, η),
hp(ξ, η)=1ipπCf(x, y, z)×exp-ipπr2r12CircrrNdxdy
h0=12Cf(x, y, z)CircrrNdxdy;
r=[(η+ny)2+(ξ+nx)2]1/2/m,
Hp(α, β)=CpF(α, β)×Zp(α, β),
Hp(α, β)=F{hp(ξ, η)},
F(α, β)=F{f(x, y)},
Zp(α, β)=Fexp-ipπr2r12CircrrN;
Cp=Cipπforpother than zero,
=C2forp=0.
Hp(ρ)=S0CpBexp-ipπr2r12CircrrN,
ρ=pr12r,forrrN.
H(α, β, z)=H(α, β, 0)exp[-iz(k2-α2-β2)1/2],
U(x, y)=exp(jkz)jzλ×expjk2z(x2+y2)whole plane×H(ξ, η)expjk2z(ξ2+η2)×exp-jkz(xξ+yη)dξdη.

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