Abstract

Surface features are measured by the analysis of the diffracted field intensity distribution generated by a scanning Gaussian beam. The measured data are shown to represent the amplitude of the Gabor expansion coefficients of the complex function embossed by the surface onto the reflected wave front. The surface is reconstructed by use of a custom-designed algorithm based on generalized projections, with a resolution exceeding the classical diffraction limit.

© 1997 Optical Society of America

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References

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  1. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. Part 3 93, 429–457 (1946).
  2. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image representation,” Optik (Stuttgart) 35, 237–246 (1972).
  3. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  4. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  5. A. Levi, H. Stark, “Image restoration by the method of generalized projections with applications to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  6. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  7. A. Papoulis, “A new algorithm in spectral analysis and band limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [CrossRef]
  8. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  9. T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” SIAM J. Optimiz. (to be published).
  10. M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–101 (1980).
  11. P. D. Einziger, “Gabor’s expansion of an aperture field in exponential elementary beams,” Electron. Lett. 24, 665–666 (1988).
    [CrossRef]
  12. M. Zibulski, Y. Y. Zeevi, “Oversampling in the Gabor scheme,” IEEE Trans. Signal Process. 41, 2679–2687 (1993).
    [CrossRef]
  13. I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
    [CrossRef]
  14. I. Daubechies, Ten Lectures on Wavelets, Vol. 61 of Conference Board of the Mathematical Sciences–National Science Foundation Regional Conference Series in Applied Mathematics Lecture Series (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
  15. J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–220 (1990).
    [CrossRef]

1993 (1)

M. Zibulski, Y. Y. Zeevi, “Oversampling in the Gabor scheme,” IEEE Trans. Signal Process. 41, 2679–2687 (1993).
[CrossRef]

1990 (2)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–220 (1990).
[CrossRef]

1988 (1)

P. D. Einziger, “Gabor’s expansion of an aperture field in exponential elementary beams,” Electron. Lett. 24, 665–666 (1988).
[CrossRef]

1984 (1)

1982 (1)

1980 (1)

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–101 (1980).

1978 (2)

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef] [PubMed]

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and band limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1974 (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image representation,” Optik (Stuttgart) 35, 237–246 (1972).

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. Part 3 93, 429–457 (1946).

Bastiaans, M. J.

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–101 (1980).

Cohen, N.

T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” SIAM J. Optimiz. (to be published).

Daubechies, I.

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

I. Daubechies, Ten Lectures on Wavelets, Vol. 61 of Conference Board of the Mathematical Sciences–National Science Foundation Regional Conference Series in Applied Mathematics Lecture Series (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Einziger, P. D.

P. D. Einziger, “Gabor’s expansion of an aperture field in exponential elementary beams,” Electron. Lett. 24, 665–666 (1988).
[CrossRef]

Fienup, J. R.

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. Part 3 93, 429–457 (1946).

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image representation,” Optik (Stuttgart) 35, 237–246 (1972).

Kotzer, T.

T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” SIAM J. Optimiz. (to be published).

Levi, A.

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Raz, S.

J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–220 (1990).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image representation,” Optik (Stuttgart) 35, 237–246 (1972).

Shamir, J.

T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” SIAM J. Optimiz. (to be published).

Stark, H.

Wexler, J.

J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–220 (1990).
[CrossRef]

Youla, D. C.

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

Zeevi, Y. Y.

M. Zibulski, Y. Y. Zeevi, “Oversampling in the Gabor scheme,” IEEE Trans. Signal Process. 41, 2679–2687 (1993).
[CrossRef]

Zibulski, M.

M. Zibulski, Y. Y. Zeevi, “Oversampling in the Gabor scheme,” IEEE Trans. Signal Process. 41, 2679–2687 (1993).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

P. D. Einziger, “Gabor’s expansion of an aperture field in exponential elementary beams,” Electron. Lett. 24, 665–666 (1988).
[CrossRef]

IEEE Trans. Circuits Syst. (2)

A. Papoulis, “A new algorithm in spectral analysis and band limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

IEEE Trans. Inf. Theory (1)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

IEEE Trans. Signal Process. (1)

M. Zibulski, Y. Y. Zeevi, “Oversampling in the Gabor scheme,” IEEE Trans. Signal Process. 41, 2679–2687 (1993).
[CrossRef]

J. Inst. Electr. Eng. Part 3 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. Part 3 93, 429–457 (1946).

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

Opt. Acta (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (2)

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–101 (1980).

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image representation,” Optik (Stuttgart) 35, 237–246 (1972).

Signal Process. (1)

J. Wexler, S. Raz, “Discrete Gabor expansions,” Signal Process. 21, 207–220 (1990).
[CrossRef]

Other (2)

I. Daubechies, Ten Lectures on Wavelets, Vol. 61 of Conference Board of the Mathematical Sciences–National Science Foundation Regional Conference Series in Applied Mathematics Lecture Series (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” SIAM J. Optimiz. (to be published).

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Figures (5)

Fig. 1
Fig. 1

Electro-optical setup used to measure the sliding-window spectrum along a profile on the investigated surface.

Fig. 2
Fig. 2

Computer simulation of the reconstruction of a surface profile by use of the generalized projection algorithm: (a) the original surface profile, (b) reconstruction from the lowest eight frequencies, and (c) reconstruction from an extrapolated spectrum.

Fig. 3
Fig. 3

Experimentally measured spectrum of a perfectly reflective step. The horizontal axis represents the position of the Gaussian beam relative to the surface, whereas the vertical axis gives the position of the detector in the detection plane in arbitrary units.

Fig. 4
Fig. 4

Extrapolated spectrum generated by the algorithm by use of only the lowest eight frequencies of the measured spectrum as input.

Fig. 5
Fig. 5

Reconstruction of the surface profile from an experimentally measured spectrum. The solid curve represents the reconstruction from the originally measured spectrum, whereas the dashed curve represents the reconstruction from the extrapolated spectrum. The horizontal axis gives the position in micrometers; the vertical axis gives the height of the profile in radians.

Equations (22)

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STFTfxα, β=fxγx-αexp-j2πβxdx.
Cmn=fxγx-maexp-j2πnbxdx.
γmnx=γx-maexp-j2πnbx,
gmnx=gx-maexp-j2πnbx.
fx=m,nCmngx-maexp-j2πnbx.
ux, y=γx, yfx-mxa, y-mya.
u1x1, y1-- γx, yfx-mxa,y-myaexp-j2πxx1+yy1/λf1dxdy.
u2nxb, nyb--γx, yfx-mxa, y-myaexp-j2πbxnx+yny/λf1dxdy.
u2nxb-γxfx-mxaexp-j2πxnxb/λf1dx,
u2nbexp-j2πmnab/λf1-fxγx+maexp-j2πxnb/λf1dx.
C-mn2-fxγx+maexp-j2πxnb/λf1dx2.
C1=fx:fx=1,x.
C1=fx:fx1,x.
C2=fx:fxγmnxdx=Cmn, m, nI1.
P1fx=expj argfx.
P1fx=fxif fx<1expj argfxif fx>1.
d1f1, f2=f1-f2=f1x-f2x2dx12.
d2f1, f2=m,nCmn1-Cmn22,
Cmn1,2=f1,2xγxdx.
P2fx=m,nI1C˜mngmnx+m,nN1Cmngmnx,
fkx=Pkfx, k=1, 2.
fi+1x=f1x+f2x2-.

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