Abstract

A refined Fourier-transform method of analysis of interference patterns is presented. The refinements include a method of automatic background subtraction and a way of treating the problem of heterodyning. The method proves particularly useful for analysis of long sequences of interferograms.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Z. Vučić, J. Gladić, “Growth rate of equilibrium-like-shaped single crystals of superionic conductor cuprous selenide,” J. Cryst. Growth 205, 136–152 (1999).
    [CrossRef]
  2. Z. Vučić, J. Gladić, “Shape relaxation during equilibrium-like growth of spherical cuprous selenide single crystals,” Fizika A 9, 9–26 (2000).
  3. Z. Vučić, J. Gladić, D. Lovrić, S. Mitrović, M. Milas, N. Demoli, “Mechanism of growth of equilibrium-like-shaped cuprous selenide single crystals,” Third scientific meeting of the Croatian Physical Society, Book of Abstracts, 113 (2001).
  4. G. C. Brown, R. J. Pryputniewicz, “Holographic microscope for measuring displacements of vibrating microbeams using time-averaged, electro-optic holography,” Opt. Eng. 37, 1398–1405 (1998).
    [CrossRef]
  5. 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 (1981).
    [CrossRef]
  6. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
  7. F. Gallet, S. Balibar, E. Rolley, “The roughening transition of crystal surfaces. II. Experiments on static and dynamic properties near the first roughening transition of hcp 4He,” J. Phys. (France) 48, 369–377 (1987).
    [CrossRef]
  8. J. P. Ruutu, P. J. Hakonen, A. V. Babkin, A. Yu. Parshin, G. Tvalashvili, “Growth of 4He-Crystals at mK-Temperatures,” J. Low Temp. Phys. 112, 117–164 (1998).
    [CrossRef]
  9. See for example W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Numerical Recipes,” (Cambridge UniversityU.K., 1992), Chap. 12.
  10. D. J. Bone, H.-A. Bachor, R. J. Sabdeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
    [CrossRef] [PubMed]

2000 (1)

Z. Vučić, J. Gladić, “Shape relaxation during equilibrium-like growth of spherical cuprous selenide single crystals,” Fizika A 9, 9–26 (2000).

1999 (1)

Z. Vučić, J. Gladić, “Growth rate of equilibrium-like-shaped single crystals of superionic conductor cuprous selenide,” J. Cryst. Growth 205, 136–152 (1999).
[CrossRef]

1998 (2)

G. C. Brown, R. J. Pryputniewicz, “Holographic microscope for measuring displacements of vibrating microbeams using time-averaged, electro-optic holography,” Opt. Eng. 37, 1398–1405 (1998).
[CrossRef]

J. P. Ruutu, P. J. Hakonen, A. V. Babkin, A. Yu. Parshin, G. Tvalashvili, “Growth of 4He-Crystals at mK-Temperatures,” J. Low Temp. Phys. 112, 117–164 (1998).
[CrossRef]

1987 (1)

F. Gallet, S. Balibar, E. Rolley, “The roughening transition of crystal surfaces. II. Experiments on static and dynamic properties near the first roughening transition of hcp 4He,” J. Phys. (France) 48, 369–377 (1987).
[CrossRef]

1986 (1)

1983 (1)

1981 (1)

Babkin, A. V.

J. P. Ruutu, P. J. Hakonen, A. V. Babkin, A. Yu. Parshin, G. Tvalashvili, “Growth of 4He-Crystals at mK-Temperatures,” J. Low Temp. Phys. 112, 117–164 (1998).
[CrossRef]

Bachor, H.-A.

Balibar, S.

F. Gallet, S. Balibar, E. Rolley, “The roughening transition of crystal surfaces. II. Experiments on static and dynamic properties near the first roughening transition of hcp 4He,” J. Phys. (France) 48, 369–377 (1987).
[CrossRef]

Bone, D. J.

Brown, G. C.

G. C. Brown, R. J. Pryputniewicz, “Holographic microscope for measuring displacements of vibrating microbeams using time-averaged, electro-optic holography,” Opt. Eng. 37, 1398–1405 (1998).
[CrossRef]

Demoli, N.

Z. Vučić, J. Gladić, D. Lovrić, S. Mitrović, M. Milas, N. Demoli, “Mechanism of growth of equilibrium-like-shaped cuprous selenide single crystals,” Third scientific meeting of the Croatian Physical Society, Book of Abstracts, 113 (2001).

Flannery, B. P.

See for example W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Numerical Recipes,” (Cambridge UniversityU.K., 1992), Chap. 12.

Gallet, F.

F. Gallet, S. Balibar, E. Rolley, “The roughening transition of crystal surfaces. II. Experiments on static and dynamic properties near the first roughening transition of hcp 4He,” J. Phys. (France) 48, 369–377 (1987).
[CrossRef]

Gladic, J.

Z. Vučić, J. Gladić, “Shape relaxation during equilibrium-like growth of spherical cuprous selenide single crystals,” Fizika A 9, 9–26 (2000).

Z. Vučić, J. Gladić, “Growth rate of equilibrium-like-shaped single crystals of superionic conductor cuprous selenide,” J. Cryst. Growth 205, 136–152 (1999).
[CrossRef]

Z. Vučić, J. Gladić, D. Lovrić, S. Mitrović, M. Milas, N. Demoli, “Mechanism of growth of equilibrium-like-shaped cuprous selenide single crystals,” Third scientific meeting of the Croatian Physical Society, Book of Abstracts, 113 (2001).

Hakonen, P. J.

J. P. Ruutu, P. J. Hakonen, A. V. Babkin, A. Yu. Parshin, G. Tvalashvili, “Growth of 4He-Crystals at mK-Temperatures,” J. Low Temp. Phys. 112, 117–164 (1998).
[CrossRef]

Ina, H.

Kobayashi, S.

Lovric, D.

Z. Vučić, J. Gladić, D. Lovrić, S. Mitrović, M. Milas, N. Demoli, “Mechanism of growth of equilibrium-like-shaped cuprous selenide single crystals,” Third scientific meeting of the Croatian Physical Society, Book of Abstracts, 113 (2001).

Macy, W. W.

Milas, M.

Z. Vučić, J. Gladić, D. Lovrić, S. Mitrović, M. Milas, N. Demoli, “Mechanism of growth of equilibrium-like-shaped cuprous selenide single crystals,” Third scientific meeting of the Croatian Physical Society, Book of Abstracts, 113 (2001).

Mitrovic, S.

Z. Vučić, J. Gladić, D. Lovrić, S. Mitrović, M. Milas, N. Demoli, “Mechanism of growth of equilibrium-like-shaped cuprous selenide single crystals,” Third scientific meeting of the Croatian Physical Society, Book of Abstracts, 113 (2001).

Parshin, A. Yu.

J. P. Ruutu, P. J. Hakonen, A. V. Babkin, A. Yu. Parshin, G. Tvalashvili, “Growth of 4He-Crystals at mK-Temperatures,” J. Low Temp. Phys. 112, 117–164 (1998).
[CrossRef]

Press, W. H.

See for example W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Numerical Recipes,” (Cambridge UniversityU.K., 1992), Chap. 12.

Pryputniewicz, R. J.

G. C. Brown, R. J. Pryputniewicz, “Holographic microscope for measuring displacements of vibrating microbeams using time-averaged, electro-optic holography,” Opt. Eng. 37, 1398–1405 (1998).
[CrossRef]

Rolley, E.

F. Gallet, S. Balibar, E. Rolley, “The roughening transition of crystal surfaces. II. Experiments on static and dynamic properties near the first roughening transition of hcp 4He,” J. Phys. (France) 48, 369–377 (1987).
[CrossRef]

Ruutu, J. P.

J. P. Ruutu, P. J. Hakonen, A. V. Babkin, A. Yu. Parshin, G. Tvalashvili, “Growth of 4He-Crystals at mK-Temperatures,” J. Low Temp. Phys. 112, 117–164 (1998).
[CrossRef]

Sabdeman, R. J.

Takeda, M.

Teukolsky, S. A.

See for example W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Numerical Recipes,” (Cambridge UniversityU.K., 1992), Chap. 12.

Tvalashvili, G.

J. P. Ruutu, P. J. Hakonen, A. V. Babkin, A. Yu. Parshin, G. Tvalashvili, “Growth of 4He-Crystals at mK-Temperatures,” J. Low Temp. Phys. 112, 117–164 (1998).
[CrossRef]

Vetterling, W. T.

See for example W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Numerical Recipes,” (Cambridge UniversityU.K., 1992), Chap. 12.

Vucic, Z.

Z. Vučić, J. Gladić, “Shape relaxation during equilibrium-like growth of spherical cuprous selenide single crystals,” Fizika A 9, 9–26 (2000).

Z. Vučić, J. Gladić, “Growth rate of equilibrium-like-shaped single crystals of superionic conductor cuprous selenide,” J. Cryst. Growth 205, 136–152 (1999).
[CrossRef]

Z. Vučić, J. Gladić, D. Lovrić, S. Mitrović, M. Milas, N. Demoli, “Mechanism of growth of equilibrium-like-shaped cuprous selenide single crystals,” Third scientific meeting of the Croatian Physical Society, Book of Abstracts, 113 (2001).

Appl. Opt. (2)

Fizika A (1)

Z. Vučić, J. Gladić, “Shape relaxation during equilibrium-like growth of spherical cuprous selenide single crystals,” Fizika A 9, 9–26 (2000).

J. Cryst. Growth (1)

Z. Vučić, J. Gladić, “Growth rate of equilibrium-like-shaped single crystals of superionic conductor cuprous selenide,” J. Cryst. Growth 205, 136–152 (1999).
[CrossRef]

J. Low Temp. Phys. (1)

J. P. Ruutu, P. J. Hakonen, A. V. Babkin, A. Yu. Parshin, G. Tvalashvili, “Growth of 4He-Crystals at mK-Temperatures,” J. Low Temp. Phys. 112, 117–164 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. (France) (1)

F. Gallet, S. Balibar, E. Rolley, “The roughening transition of crystal surfaces. II. Experiments on static and dynamic properties near the first roughening transition of hcp 4He,” J. Phys. (France) 48, 369–377 (1987).
[CrossRef]

Opt. Eng. (1)

G. C. Brown, R. J. Pryputniewicz, “Holographic microscope for measuring displacements of vibrating microbeams using time-averaged, electro-optic holography,” Opt. Eng. 37, 1398–1405 (1998).
[CrossRef]

Other (2)

Z. Vučić, J. Gladić, D. Lovrić, S. Mitrović, M. Milas, N. Demoli, “Mechanism of growth of equilibrium-like-shaped cuprous selenide single crystals,” Third scientific meeting of the Croatian Physical Society, Book of Abstracts, 113 (2001).

See for example W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Numerical Recipes,” (Cambridge UniversityU.K., 1992), Chap. 12.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

(a) Snapshot of a growing cuprous selenide single crystal. The bright areas correspond to atomically flat surfaces, whose outward growth has been monitored by interferometry. (b) Typical frame (320 × 240 pixels) recorded during the growth. The interferogram with fringe pattern (64 × 64 pixels) has been extracted from each frame for further analysis.

Fig. 2
Fig. 2

Ratio of intensities of the first-order peak to the zero-order peak as a function of the index of the interferogram obtained by summing and normalizing successive interferograms. n = 82 concludes the first sequence, which defines the first background. The 83rd interferogram has been taken as the first one in the second sequence terminating at n = 208, etc.

Fig. 3
Fig. 3

Same frame as in Fig. 1(b), but with the background subtracted as described in the text.

Fig. 4
Fig. 4

Components q x (lower panel) and q y (upper panel) of the spatial-carrier frequency plotted versus the index of the interferograms recorded during the facet growth, and determined as described in the text.

Fig. 5
Fig. 5

Fourier transform of an interferogram defined by Eqs. (1) and (6) with parameters as given in text: (a) without the subtraction of the background; (b) with the background subtracted. The point with coordinates (in pixels) x = 33 and y = 33 represents the q = 0 point of the inverse space.

Fig. 6
Fig. 6

Filled circles: chosen value of the averaged phase field difference between consecutive interferograms. Dotted curve with open diamonds: calculated values of the averaged phase field difference without taking account of noninteger and varying values of spatial-carrier frequency. Solid curve and open squares: calculated values of the averaged phase field difference with all of the components of the spatial-carrier frequency removed as described in the text.

Equations (6)

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

gr=ar+brcos2πq·r+Φr,
Δh¯=ΔΦ¯·λ4π·1cosθ2,
|gqx, qy|=m,n |gm,n| sinπqxΔqx-mπqxΔqx-m·sinπqyΔqy-nπqyΔqy-n,
ΔGr=expiδ·r,
Gr=G0rexpiδ·r.
ar=n0ρ-b02y3-4y+22x2-2xbr=1-m0y2-2y+1×1-m04x2-4x+1.

Metrics