Abstract

Objects that temporally vary slowly may be superresolved by use of moving gratings. A system of this kind had been proposed three decades ago. However, it provides some distortion of the spectral response of the resolved object. In this project, an enhanced method based on Dammann gratings instead of regular gratings is suggested. The modified approach achieves results with an undistorted output and relatively high light efficiency, and it is effective for both coherent and incoherent light. Experimental results are provided for demonstrating the ability of the new approach.

© 1997 Optical Society of America

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References

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  1. M. Francon, “Amélioration de résolution d’optique,” Nouvo Climento Suppl. 9, 283–290 (1952).
  2. A. W. Lohmann, D. P. Paris, “Supperresolution for nonbirefringement objects,” Appl. Opt. 3, 1037–1043 (1964).
    [CrossRef]
  3. A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. 9, 204–206 (1960).
  4. J. W. Goodman, “Synthetic aperture optics,” Prog. Opt. 8, 1–50 (1970).
    [CrossRef]
  5. D. Mendlovic, A. W. Lohmann, “Space-bandwidth product adaptation and its applications for superresolution: fundamentals,” J. Opt. Soc. Am. A 14, 558–562 (1997).
    [CrossRef]
  6. D. Mendlovic, A. W. Lohmann, Z. Zalevsky, “Space–bandwidth product adaptation and its applications for superresolution: examples,” J. Opt. Soc. Am. A 14, 563–567 (1997).
    [CrossRef]
  7. W. Lukosz, “Optical systems with resolving powers exceeding the classical limits. II,” J. Opt. Soc. Am. 57, 932–941 (1967).
    [CrossRef]
  8. H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]
  9. H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
    [CrossRef]

1997 (2)

1977 (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

1971 (1)

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

1970 (1)

J. W. Goodman, “Synthetic aperture optics,” Prog. Opt. 8, 1–50 (1970).
[CrossRef]

1967 (1)

1964 (1)

1960 (1)

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

1952 (1)

M. Francon, “Amélioration de résolution d’optique,” Nouvo Climento Suppl. 9, 283–290 (1952).

Dammann, H.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Francon, M.

M. Francon, “Amélioration de résolution d’optique,” Nouvo Climento Suppl. 9, 283–290 (1952).

Goodman, J. W.

J. W. Goodman, “Synthetic aperture optics,” Prog. Opt. 8, 1–50 (1970).
[CrossRef]

Görtler, K.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Kartashev, A. I.

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

Klotz, E.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Lohmann, A. W.

Lukosz, W.

Mendlovic, D.

Paris, D. P.

Zalevsky, Z.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nouvo Climento Suppl. (1)

M. Francon, “Amélioration de résolution d’optique,” Nouvo Climento Suppl. 9, 283–290 (1952).

Opt. Acta (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Opt. Commun. (1)

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Opt. Spectrosc. (1)

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

Prog. Opt. (1)

J. W. Goodman, “Synthetic aperture optics,” Prog. Opt. 8, 1–50 (1970).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Optical system for obtaining an increased effective aperture by use of moving gratings.

Fig. 2
Fig. 2

Coherent (dashed lines) and optical (solid lines) transfer functions for a rectangular aperture.

Fig. 3
Fig. 3

Effective CTF obtained by use of a Dammann grating with N = 5: (a) demonstration of the replica of the CTF and (b) the summation of all those replica (effective CTF).

Fig. 4
Fig. 4

Schematic illustration of the optical setup used for the system experimental demonstration.

Fig. 5
Fig. 5

Roseta mask containing the Dammann-grating rays.

Fig. 6
Fig. 6

Output captured by the CCD camera with (a) a clear aperture, (b) a closed aperture without the rotation Roseta, and (c) a closed aperture after addition of the Roseta.

Fig. 7
Fig. 7

Grating impulse response: (a) Conventional Dammann-grating case as used in the experimental demonstration. (b) Tailored mask for achieving high-pass filtering.

Equations (29)

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U0XGX-Vt=U0X mAm exp2iπmv0X-Vt=J0X, t,
J˜0ν, t= J0X, texp-2πiνX=m Am  U0Xexp2πiXmν0-ν-mν0VtdX=m Am Ũ0ν-mν0exp-2πimν0Vt.
P˜0ν=rectνΔν.
JX, t=P˜0νJ˜0ν, texp2πiνXdν.
UX, t=JX, t GX-Vt=mAm exp2πimν0X-Vt×nAn P˜0νU˜0ν-nν0×exp-2πinν0Vtexp2πiνXdν=n,mAnAm P˜0νŨ0ν-nν0exp2πiX×ν+mν0-n+mν0Vtdν.
IX, t=UX, tUX, t*=n,mn,m AnAmAn*Am*P˜0νŨ0ν-nν0×exp2πiXν+mν0-n+mν0VtdνP˜0*νŨ0*ν-nν0×exp-2πiXν+mν0-n+mν0Vtdν,
IX, t=n,mn,mAnAmAn*Am*[P˜0ν1P˜0*ν2×Ũ0ν1-nν0Ũ0*ν2-nν0exp2πiX×ν1+mν0-ν2-mν0-m+n-n-mν0Vtdν1dν2.
τ=1ν0V=dV.
1τ-τ2τ2exp-2πiν0Vtn+m-n-mdt=1n+m-n-m=00n+m-n-m0.
IX=1τ-τ2τ2IX, tdt=for alln+m-n-m-=0AmAnAm*An*P˜0μ1+nν0×P˜0*μ2+nν0Ũ0μ1Ũ0*μ2×exp2πiXμ1-μ2dμ1dμ2.
IX=for alln+m-n-m=0AmAm*nAnP˜0μ1+nν0×Ũ0μ1exp2πiXμ1dμ1×nAn*P˜0*μ2+nν0Ũ0*μ2×exp-2πiXμ2dμ2.
IX=for alln+m-n-m=0AmAm*nAnP˜0μ1+nν0×Ũ0μ1exp2πiXμ1dμ1×nAnP˜0μ2+nν0Ũ0μ2×exp2πiXμ2dμ2.
IX=mAm2nAnP˜0μ+nν0Ũ0μ×exp2πiXμdμ2.
F˜μ=nAnP˜0μ+nν0.
ν0=Δν,
An=1nN21n>N2,
1τ-τ2τ2U0X1; tU0*X2; tdt=U0X1U0*X2δX1-X2.
Ũ0μ1=U0X1; texp-2πiX1μ1dX1, Ũ0*μ2=U0*X2; texp2πiX2μ2dX2,
IX=1τ-τ2τ2IX, tdt=n,mn,mAnAmAn*Am*P˜0μ1+nν0×P˜0*μ2+nν0exp2πiXμ1exp-2πiXμ2×exp2πiν0Xn+m-n-m×U0X1; texp-2πiX1μ1dX1×U0*X2; texp2πiX2μ2dX2exp-2πi×m+n-n-mν0Vtdtdμ1dμ2.
IX=n,mn,mAnAmAn*Am*P˜0μ1+nν0×P˜0*μ2+nν0×exp2πiXμ1exp-2πiXμ2×exp2πiν0Xn+m-n-m×exp-2πiX1μ1exp2πiX2μ2dX1dX2dμ1dμ2×U0X1; tU0*X2; texp-2πi×m+n-n-mν0Vtdt.
1τ-τ2τ2U0X1; tU0*X2; texp-2πim+n-m-nν0Vtdt=U0X1U0*X2δX1-X2 m+n-m-n=00 m+n-m-n0.
IX=for allm+n-m-n=0AnAmAn*Am*P˜0μ1+nν0×P˜0*μ2+nν0×exp2πiXμ1exp-2πiXμ2×exp2πiX2μ2U0*X2×U0X1exp-2πiX1μ1×δX1-X2dX1dX2dμ1dμ2,
IX=for allm+n-m-n=0AnAmAn*Am*P˜0μ1+nν0×P˜0*μ2+nν0×exp2πiXμ1exp-2πiXμ2×U0X22 exp-2πiX2×μ1-μ2dX2dμ1dμ2.
Ĩ0ν=U0X2 exp-2πiXνdX.
IX=for allm+n-m-n=0AnAmAn*Am*P˜0μ1+nν0×P˜0*μ2+nν0Ĩ0μ1-μ2×exp2πiXμ1-μ2dμ1dμ2.
IX=for allm+n-m-n=0AmAm*Ĩ0μ1-μ2exp2πiX×μ1-μ2nAnP˜0μ1+nν0×nAnP˜0μ2+nν0*dμ1Dμ2,
IX=mAm2Ĩ0μ1-μ2exp2πiX×μ1-μ2F˜μ1F˜*μ2dμ1dμ2,
IX=mAm2Ĩ0z1exp2πiXz1×F˜z2+z12F˜*z2-z12dz2dz1.
S˜0z1=F˜z2+z12F˜*z2-z12dz2

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