Abstract

We present a system that exceeds the Rayleigh limit of resolution, by placing two fixed gratings in predetermined positions. Lukosz suggested a setup that managed to exceed the Rayleigh limit of resolution [J. Opt. Soc. Am. 56, 1463 (1966); J. Opt. Soc. Am. 57, 932 (1967)]. However, Lukosz’s technique had some drawbacks that the new suggested system attempts to resolve. Similar to Lukosz’s technique, the proposed system works without any moving elements and with no time- or wavelength-restricting conditions. It is suitable for coherent or incoherent two-dimensional imaging. However, the new system contains some important modifications. Although the system uses only two gratings, it is capable of producing superresolution without using an additional imaging lens at the output plane. The generalized Damman gratings allow for obtaining undistorted spectral restoration of information. To achieve superresolution, the input object is duplicated. The trade-off for higher resolution is a smaller field of view. Experimental results validate the theoretical analysis.

© 2000 Optical Society of America

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References

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  1. G. Toraldo Di Francia, “Resolving power and information,” J. Opt. Soc. Am. 45, 497–501 (1955).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Physik 174, 18 (1963).
  5. M. Francon, Nuovo Cimento Suppl. 9, 283–329 (1952).
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    [CrossRef]
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    [CrossRef]
  8. A. Malov, V. Morozov, I. Kompanents, Yu. Popov, “Formation of an image in a coherent synthesized-aperture system,” Sov. J. Quantum Electron. 7, 1125–1130 (1977).
    [CrossRef]
  9. A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. 9, 204–206 (1960).
  10. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “About the space bandwidth product of optical signal and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. Z. Zalevsky, D. Mendlovic, A. W. Lohmann, “Superresolution optical systems for objects with finite sizes,” Opt. Commun. 163, 79–85 (1999).
    [CrossRef]

1999 (1)

Z. Zalevsky, D. Mendlovic, A. W. Lohmann, “Superresolution optical systems for objects with finite sizes,” Opt. Commun. 163, 79–85 (1999).
[CrossRef]

1997 (4)

1996 (1)

1982 (1)

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

1977 (1)

A. Malov, V. Morozov, I. Kompanents, Yu. Popov, “Formation of an image in a coherent synthesized-aperture system,” Sov. J. Quantum Electron. 7, 1125–1130 (1977).
[CrossRef]

1969 (1)

1967 (1)

1966 (1)

1963 (1)

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Physik 174, 18 (1963).

1960 (1)

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. 9, 204–206 (1960).

1955 (1)

1952 (1)

M. Francon, Nuovo Cimento Suppl. 9, 283–329 (1952).

Bartelt, H.

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

Dorsch, R. G.

Farkas, D.

Ferreira, C.

Francon, M.

M. Francon, Nuovo Cimento Suppl. 9, 283–329 (1952).

Gartner, W.

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Physik 174, 18 (1963).

Kartashev, A. I.

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. 9, 204–206 (1960).

Kiryuschev, I.

Kompanents, I.

A. Malov, V. Morozov, I. Kompanents, Yu. Popov, “Formation of an image in a coherent synthesized-aperture system,” Sov. J. Quantum Electron. 7, 1125–1130 (1977).
[CrossRef]

Konforti, N.

Lohmann, A. W.

Lukosz, W.

Malov, A.

A. Malov, V. Morozov, I. Kompanents, Yu. Popov, “Formation of an image in a coherent synthesized-aperture system,” Sov. J. Quantum Electron. 7, 1125–1130 (1977).
[CrossRef]

Mendlovic, D.

Morozov, V.

A. Malov, V. Morozov, I. Kompanents, Yu. Popov, “Formation of an image in a coherent synthesized-aperture system,” Sov. J. Quantum Electron. 7, 1125–1130 (1977).
[CrossRef]

Popov, Yu.

A. Malov, V. Morozov, I. Kompanents, Yu. Popov, “Formation of an image in a coherent synthesized-aperture system,” Sov. J. Quantum Electron. 7, 1125–1130 (1977).
[CrossRef]

Toraldo Di Francia, G.

Zalevsky, Z.

Appl. Opt. (2)

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

Nuovo Cimento (1)

M. Francon, Nuovo Cimento Suppl. 9, 283–329 (1952).

Opt. Commun. (2)

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

Z. Zalevsky, D. Mendlovic, A. W. Lohmann, “Superresolution optical systems for objects with finite sizes,” Opt. Commun. 163, 79–85 (1999).
[CrossRef]

Opt. Spectrosc. (1)

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. 9, 204–206 (1960).

Sov. J. Quantum Electron. (1)

A. Malov, V. Morozov, I. Kompanents, Yu. Popov, “Formation of an image in a coherent synthesized-aperture system,” Sov. J. Quantum Electron. 7, 1125–1130 (1977).
[CrossRef]

Z. Physik (1)

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Physik 174, 18 (1963).

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Figures (9)

Fig. 1
Fig. 1

Optical setup for superresolution. Ob. = Object, Im. = Image, and D.G.1 and D.G.2 are the Damman gratings.

Fig. 2
Fig. 2

Experiment setup for obtaining superresolution. Att. = Attenuator.

Fig. 3
Fig. 3

Transparency function of the Damman gratings.

Fig. 4
Fig. 4

Synthetic aperture.

Fig. 5
Fig. 5

Geometrical ray’s track of the superresolution setup.

Fig. 6
Fig. 6

Part of the Damman grating as it seen by the CCD.

Fig. 7
Fig. 7

Objects used for the experiment. (a) 1-D object. (b) 2-D object.

Fig. 8
Fig. 8

Output plane without use of Damman gratings in (a) and (b) for the objects of Figs. 7(a) and 7(b), respectively.

Fig. 9
Fig. 9

Superresolved output results obtained with Damman gratings. A complete reconstruction of the duplicated input object of Figs. 7(a) and 7(b) was obtained in (a) and (b), respectively.

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

u0x, z=0+=m Am exp2πixmν0,
u0x, z0-=m Am exp2πixmν0+z0λ×1-λ2mν021/2.
u0x, z=z0=- ũ0νexp2πixνdν.
u0x, z0+=m Am- ũ0νexp2πixν+mν0+z0λ1-λ2mν021/2dν.
u0x=ux * l=-L/2l=L/2 δx-lδx,
ũ0ν=ũνexp-2πimδxν.
u0x, z0+=m Am- ũνexp2πixν+mν0+z0λ×1-λ2mν021/2-mδxνdν.
u0x, 0+=m Am- ũνexp2πixν+mν0+z0λ ϕ1-mδxνdν,
ϕ1=1-λ2mν021/2-1-λ2ν+mν021/2.
μ=λFν+mν0.
u0μ, 2F-=m AmũμλF-mν0exp2πi z0λ ϕ2×exp-2πimδxμλF-mν0,
ϕ2=1-λ2mν021/2-1-λ2μ/λF21/2,
1-a21/21-a2/2.
u0μ, 2F+=m AmũμλF-mν0exp2πi z0λ ϕ2×rectμΔμexp-2πimδxμλF-mν0.
u0x, 4F-=m Am- ũμλF-mν0exp2πixμλF+z0λ ϕ2rectμΔμexp-2πimδxμλF-mν0dμ,
u0x, 4F+=nm AmBn- ũμλF-mν0×exp2πixμλF+xnν1+z0λ ϕ2-mδxμλF-mν0rectμΔμdμ,
u0x, 4F+z0=nm AmBn- ũμλF-mν0×exp2πixμλF+nν1x+z0λ ϕ2+z0λ ϕ3-mδxμλF-mν0×rectμΔμdμ,
ϕ3=1-λ2μλF+nν121/2.
u0x, 4F+z0=nm AmBn- ũνrectν+mν0Δμ/λF×exp2πixν+mν0+nν1+ϕtdν,
ϕt=z0λϕ2+ϕ3-mδxν=z0λ1-λ2mν021/2-1-λ2μλF21/2+1-λ2μλF+nν121/2-mδxνz0λ1-λ2mν022-1-λ2μλF22+1-λ2μλF+nν122-mδxν=z0λ-z0λ2n2ν12+m2ν02-z0λnν1ν+mν0-mδxν=z0λ-z0λ2mν0+nν12-z0λnν1ν-mδxν.
mν0+nν1=0.
ϕt=z0λ-z0λnν1ν-mδxν0.
u0x, 4F+z0=nm AmBnK - ũνrectν+mν0Δμ/λF×exp2πixνexp-2πiz0nν1λν+mδxνdν,
u0x, 4F+z0=nm AmBnK - ũνrect×ν+mν0Δμ/λFexp2πixν×exp-2πiz0λnν1+mν0νdν.
nν1+mν0=0,
xn,m=z0λnν1+mν0
ν0=ν1.
m+n=0.
Δxδx=λz0ν0.
ν0=Δμ/λF
u0x, 4F+z0=nm AmBnK - ũνrect× ν+mν0Δμ/λFexp2πixνdν.
u0x, y, z=0+=mxmy AmxAmy×exp2πixmx+ymyν0,
u0x, y, z0-=mxmy AmxAmy exp2πixmx+ymyν0+z0λ1-λ2mxν021/2+1-λ2myν021/2.
u0x, y, z=z0=-- ũνx, νy× exp2πixνx+yνydνxdνy.
u0x, y, z0+=mxmy AmxAmy-- ũνx, νy×exp2πixνx+mxν0+yνy+myν0+z0λ1-λ2mxν021/2+1-λ2myν021/2dνxdνy.
u0x, y=ux, y * lx=-L/2lx=L/2ly=-L/2ly=L/2 δx-lxδxδy-lyδy.
lx=mx;  ly=my.
ν0=ν1,
nx+mx=0;  ny+my=0,
u0x, y, 4F+z0=nxnymxmy AmxAmyBnxBnyK×-- ũνx, νyrectνx+mxν0Δμ/λF×rectνy+myν0Δμ/λF×exp2πixνx+yνydνxdνy,
xnx,mx, yny,my=z0λν0nx+mx, z0λ0ν0ny+my.
Δx, Δyλz0ν0.
Psx, 4F+z0=nm AmBnK×- rectν+mν0Δμ/λFexp2πixνdν.
Fx, 4F+z0=|Psx, 4F+z0|2.
I0x= I0xδx-xdx.
IBx= I0xFx-xdx,
F˜ν= Fxexp-2πiνxdx.
F˜ν= P˜sν+ν2P˜s*ν-ν2dν,
P˜sν=rectνΔνSR,
F˜ν=trianνΔνSR,

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