Abstract

Noting the structural similarity between a distorted lattice in a crystal and a spatially phase-modulated fringe pattern in optical metrology, we propose a crystallographic lattice heterodyne technique. The technique detects lattice distortions as spatial phase modulations, where the phase change by 2π corresponds to the displacement of atoms or lattice points by a distance equal to the lattice constant. By virtue of heterodyne detection, the technique has the potential for determining the lattice distortions to several hundredths of the lattice constant.

© 1996 Optical Society of America

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References

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  1. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]
  2. R. P. Millane, “Phase problems for periodic images: effects of support and symmetry,” J. Opt. Soc. Am. A 10, 1037–1045 (1993).
    [CrossRef]
  3. R. W. Harrison, “Phase problem in crystallography,” J. Opt. Soc. Am. A 10, 1046–1055 (1993).
    [CrossRef]
  4. M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  5. See, for example, M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview,” Indust. Metrol. 1, 79–99 (1990).
    [CrossRef]
  6. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,”J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  7. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [CrossRef] [PubMed]
  8. M. Takeda, M. Kitoh, “Spatiotemporal frequency multiplex heterodyne interferometry,” J. Opt. Soc. Am. A 9, 1607–1614 (1992).
    [CrossRef]
  9. J. M. Huntley, J. E. Field, “High resolution moire photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
    [CrossRef]
  10. See, for example, J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 273–359.
    [CrossRef]
  11. See, for example, M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 425.
  12. M. Takeda, Q. S. Ru, “Computer-based highly sensitive electron-wave interferometry,” Appl. Opt. 24, 3068–3071 (1985).
    [CrossRef] [PubMed]
  13. See, for example, A. Tonomura, “Electron holography,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1986), Vol. 23, pp. 185–220.
    [CrossRef]
  14. Y. Aharonov, D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
    [CrossRef]
  15. See, for example, F. R. N. Nabarro, Theory of Crystal Dislocations (Oxford U. Press, Oxford, 1967), p. 791.
  16. M. Pirga, M. Kujawinska, “Two-dimensional spatial-carrier phase-shifting method for analysis of complex interferogram,” in Interferometry ‘94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, ed. Proc. SPIE2340, 163–169 (1992).
    [CrossRef]
  17. See, for example, K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. 26, 349–393.
    [CrossRef]
  18. See, for example, T. R. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
    [CrossRef]
  19. M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
    [CrossRef] [PubMed]
  20. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–721 (1988).
    [CrossRef]

1994

See, for example, T. R. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

1993

1992

M. Takeda, M. Kitoh, “Spatiotemporal frequency multiplex heterodyne interferometry,” J. Opt. Soc. Am. A 9, 1607–1614 (1992).
[CrossRef]

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
[CrossRef] [PubMed]

1990

R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
[CrossRef]

See, for example, M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview,” Indust. Metrol. 1, 79–99 (1990).
[CrossRef]

1989

J. M. Huntley, J. E. Field, “High resolution moire photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
[CrossRef]

1988

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–721 (1988).
[CrossRef]

1987

1985

1983

1982

1959

Y. Aharonov, D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

Aharonov, Y.

Y. Aharonov, D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

Bohm, D.

Y. Aharonov, D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 425.

Bryanston-Cross, P. J.

See, for example, T. R. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

Creath, K.

See, for example, K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. 26, 349–393.
[CrossRef]

Field, J. E.

J. M. Huntley, J. E. Field, “High resolution moire photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
[CrossRef]

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–721 (1988).
[CrossRef]

Harrison, R. W.

Hedley, M.

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
[CrossRef] [PubMed]

Huntley, J. M.

J. M. Huntley, J. E. Field, “High resolution moire photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
[CrossRef]

Ina, H.

Judge, T. R.

See, for example, T. R. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

Kitoh, M.

Kobayashi, S.

Kujawinska, M.

M. Pirga, M. Kujawinska, “Two-dimensional spatial-carrier phase-shifting method for analysis of complex interferogram,” in Interferometry ‘94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, ed. Proc. SPIE2340, 163–169 (1992).
[CrossRef]

Millane, R. P.

Mutoh, K.

Nabarro, F. R. N.

See, for example, F. R. N. Nabarro, Theory of Crystal Dislocations (Oxford U. Press, Oxford, 1967), p. 791.

Pirga, M.

M. Pirga, M. Kujawinska, “Two-dimensional spatial-carrier phase-shifting method for analysis of complex interferogram,” in Interferometry ‘94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, ed. Proc. SPIE2340, 163–169 (1992).
[CrossRef]

Roddier, C.

Roddier, F.

Rosenfeld, D.

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
[CrossRef] [PubMed]

Ru, Q. S.

Schwider, J.

See, for example, J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 273–359.
[CrossRef]

Takeda, M.

Tonomura, A.

See, for example, A. Tonomura, “Electron holography,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1986), Vol. 23, pp. 185–220.
[CrossRef]

Werner, C. L.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–721 (1988).
[CrossRef]

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 425.

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–721 (1988).
[CrossRef]

Appl. Opt.

Indust. Metrol.

See, for example, M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview,” Indust. Metrol. 1, 79–99 (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Magn. Reson. Med.

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
[CrossRef] [PubMed]

Opt. Eng.

J. M. Huntley, J. E. Field, “High resolution moire photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
[CrossRef]

Opt. Lasers Eng.

See, for example, T. R. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

Phys. Rev.

Y. Aharonov, D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

Radio Sci.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–721 (1988).
[CrossRef]

Other

See, for example, A. Tonomura, “Electron holography,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1986), Vol. 23, pp. 185–220.
[CrossRef]

See, for example, F. R. N. Nabarro, Theory of Crystal Dislocations (Oxford U. Press, Oxford, 1967), p. 791.

M. Pirga, M. Kujawinska, “Two-dimensional spatial-carrier phase-shifting method for analysis of complex interferogram,” in Interferometry ‘94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, ed. Proc. SPIE2340, 163–169 (1992).
[CrossRef]

See, for example, K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. 26, 349–393.
[CrossRef]

See, for example, J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 273–359.
[CrossRef]

See, for example, M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 425.

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Figures (7)

Fig. 1
Fig. 1

(a) Ideal crystal with perfect periodicity. (b) Imperfect crystal with lattice distortions. (c) Displacement of a lattice point (or an atom) caused by a lattice distortion.

Fig. 2
Fig. 2

(a) Spatial-frequency spectra of the structure image of a crystal; these spectra correspond to diffraction spots in an x-ray diffraction pattern or an electron diffraction pattern observed in the focal plane of an electron microscope objective. (b) A spectrum filtered and shifted down to the spatial frequency of zero.

Fig. 3
Fig. 3

Simulated lattice image with edge dislocations.

Fig. 4
Fig. 4

Modulus of the spatial-frequency spectra of the simulated structure image in Fig. 3; these spectra correspond to diffraction spots in an x-ray diffraction pattern or an electron diffraction pattern observed in the focal plane of an electron microscope objective.

Fig. 5
Fig. 5

(a) Phase distribution ϕ10(x, y) representing the lattice distortion or the displacement of atoms in the x direction in Fig. 3; 2π phase corresponds to the distortion Δx(x, y) that is as large as the lattice constant L. (b) Phase distribution ϕ01(x, y) representing the lattice distortion or the displacement of atoms in the y direction in Fig. 3; 2π phase corresponds to the distortion Δy (x, y) that is as large as the lattice constant L.

Fig. 6
Fig. 6

Lattice fringe image of a gold crystal (courtesy of Q. S. Ru).

Fig. 7
Fig. 7

(a) Spatial-frequency spectra of the lattice fringe image of the gold crystal shown in Fig. 6. (b) Phase distribution ϕ10(x, y) representing the lattice distortion in the x direction of a gold crystal. (c) Another solution of the phase distribution ϕ10(x, y) representing the lattice distortion in the x direction of a gold crystal. In (b) and (c), 2π phase corresponds to the distortion Δx(x, y) that is as large as the lattice constant Lx.

Equations (13)

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g 0 ( x , y ) = h k G h k 0 exp [ 2 π i ( h x L x + k y L y ) ] ,
g ( x , y ) = g 0 [ x - Δ x ( x , y ) , y - Δ y ( x , y ) ]
= h k G h k 0 exp { 2 π i [ h x - Δ x ( x , y ) L x + k y - Δ y ( x , y ) L y ] }
= h k g h k ( x , y ) exp [ 2 π i ( h x L x + k y L y ) ] ,
g h k ( x , y ) = G h k 0 exp [ - i ϕ h k ( x , y ) ] ,
ϕ h k ( x , y ) = 2 π [ h Δ x ( x , y ) L x + k Δ y ( x , y ) L y ] .
G ( f x , f y ) = - g ( x , y ) exp [ - 2 π i ( f x x + f y y ) ] d x d y
= h k G h k ( f x - h / L x , f y - k / L y ) ,
log [ g h k ( x , y ) ] = log ( G h k 0 ) - i [ ϕ h k ( x , y ) + arg ( G h k 0 ) ] ,
Δ x ( x , y ) = L x ϕ 10 ( x , y ) / 2 π .
Δ y ( x , y ) = L y ϕ 01 ( x , y ) / 2 π .
Δ x ( x , y ) = ( L 2 π ) tan - 1 ( x y ) ,
Δ y ( x , y ) = ( L 2 π ) tan - 1 ( y x ) ,

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