Abstract

An experimental setup for tomographic inspection of phase objects is presented. The system uses a common-path interferometer consisting of two windows in the input plane and a translating grating as its pupil. In the output, interference of the fields associated with replicated windows is achieved by a proper choice of the windows’ spacing with respect to the grating period. With a rotating object in one window and a plane wave in the second one, the phase distribution of each projection is encoded as a corresponding digital image row, which, in turn, constructs a composite interferogram over the plane of a traditional sinogram. Phase stepping of composite interferograms can be achieved by a proper translation of the grating in order to obtain the unwrapped phase distribution as the corresponding sinogram. This sinogram allows tomographic reconstruction of phase slices by standard procedures. Composite interferograms and reconstructions for some transparent samples are shown.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. L. Byer and M. Garbuny, "Pollutant detection by absorption using Mie scattering and topographic targets as retroreflectors," Appl. Opt. 12, 1496-1505 (1973).
    [CrossRef] [PubMed]
  2. R. L. Byer and L. A. Shepp, "Two-dimensional remote air-pollution monitoring via tomography," Opt. Lett. 4, 375-377 (1979).
    [CrossRef]
  3. R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, "Optical tomography for flow field diagnostics," Int. J. Heat Mass Transfer 24, 1139-1150 (1981).
    [CrossRef]
  4. L. M. Murillo-Mora, K. Okada, T. Honda, and J. Tsujiuchi, "Color conical holographic stereograms: recording and distortion compensation methods," in The Art and Science of Holography, H.J.Caulfield, ed. (SPIE, 2004), pp. 261-273.
  5. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1987).
  6. E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
    [CrossRef]
  7. C. Hauger, M. Wörz, and T. Hellmuth, "Interferometer for optical coherence tomography," Appl. Opt. 42, 3896-3902 (2003).
    [CrossRef] [PubMed]
  8. S. R. Dean, The Radon Transform and Some of Its Applications (Wiley, 1983).
  9. A. L. Auslender, G. N. Vishnyakov, and G. G. Levin, "Solution of Radon's integral equation in an optical processor," Opt. Spectrosc. 49, 518-520 (1980).
  10. C. M. Vest and P. T. Radulovic, "Measurement of three-dimensional temperature fields by holographic interferometry," in Applications of Holography and Optical Data Processing, E.Marcom, A.A.Friesem, and E.Wiener-Avnear, eds. (Pergamon, 1977), pp. 241-249.
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).
  12. A. H. Andersen, "Tomography transform and inverse in geometrical optics," J. Opt. Soc. Am. A 4, 1385-1395 (1987).
    [CrossRef]
  13. C. M. Vest, "Interferometry of strongly refracting axisymmetric phase objects," Appl. Opt. 14, 1601-1606 (1975).
    [CrossRef] [PubMed]
  14. E. W. Hansen, "Theory of circular harmonic image reconstruction," J. Opt. Soc. Am. 71, 304-308 (1981).
    [CrossRef]
  15. K. C. Tam and V. Perez-Mendez, "Tomographical imaging with limited-angle input," J. Opt. Soc. Am. 71, 582-592 (1981).
    [CrossRef]
  16. B. P. Medoff, W. R. Brody, M. Nassi, and A. Macovski, "Iterative convolution backprojection algorithms for image reconstruction from limited data," J. Opt. Soc. Am. 73, 1493-1500 (1983).
    [CrossRef]
  17. S. Gull and G. Daniel, "Image reconstruction from incomplete and noisy data," Nature 272, 686-690 (1978).
    [CrossRef]
  18. A. Mohammad-Djafari and G. Demoment, "Maximum entropy Fourier synthesis with application to diffraction tomography," Appl. Opt. 26, 1745-1754 (1987).
    [CrossRef] [PubMed]
  19. T. Liu, W. Merzkirch, and K. Oberste-Lehn, "Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density," Exp. Fluids 7, 157-163 (1989).
    [CrossRef]
  20. T. Neger, "Optical tomography of plasmas by spectral interferometry," J. Phys. D 28, 47-54 (1995).
    [CrossRef]
  21. G. Pretzler, "Single shot tomography by differential interferometry," Meas. Sci. Technol. 6, 1476-1486 (1995).
    [CrossRef]
  22. D. Dirksen and G. Von Bally, "Holographic double-exposure interferometry in the near real time with photorefractive crystals," J. Opt. Soc. Am. B 11, 1858-1863 (1994).
    [CrossRef]
  23. C. Meneses-Fabian, G. Rodríguez-Zurita, R. Rodríguez-Vera, and J. F. Vázquez-Castillo, "Optical tomography with parallel projection differences and electronic speckle pattern interferometry," Opt. Commun. 228, 201-210 (2003).
    [CrossRef]
  24. H. Hertz, "Experimental determination of 2-D flame temperature fields by interferometric tomography," Opt. Commun. 54, 131-136 (1985).
    [CrossRef]
  25. K. I. Schultz and D. L. Jaggard, "Microwave projection imaging for refractive objects: a new method," J. Opt. Soc. Am. A 4, 1773-1782 (1987).
    [CrossRef]
  26. R. Snyder and L. Hasselink, "Measurement of mixing fluid flows with optical tomography," Opt. Lett. 13, 87-89 (1988).
    [CrossRef] [PubMed]
  27. R. Snyder and L. Hasselink, "High speed optical tomography for flow visualization," Appl. Opt. 24, 4046-4051 (1985).
    [CrossRef] [PubMed]
  28. K. Widmann, G. Pretzler, J. Woisetschläger, H. Philipp, T. Neger, and H. Jäger, "Interferometric determination of the electron density in a high-pressure xenon lamp with a holographic optical element," Appl. Opt. 35, 5896-5903 (1996).
    [CrossRef] [PubMed]
  29. V. Arrizón and D. Sánchez-de-la-Llave, "Common-path interferometry with one-dimensional periodic filters," Opt. Lett. 29, 141-143 (2004).
    [CrossRef] [PubMed]
  30. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, and R. Spolaczyk, "Semiconductor wafer and technical flat planeness testing interferometer," Appl. Opt. 25, 1117-1121 (1986).
    [CrossRef] [PubMed]
  31. L. Joannes, F. Dubois, and J.-C. Legros, "Phase-shifting Schlieren: high-resolution quantitative Schlieren that uses the phase-shifting technique principle," Appl. Opt. 42, 5046-5053 (2003).
    [CrossRef] [PubMed]
  32. K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, E.Wolf, ed. (Elsevier, 1998), Vol. 16, pp. 349-393.
  33. D. W. Robinson and D. C. Williams, "Digital phase stepping speckle interferometry," Opt. Commun. 57, 26-30 (1986).
    [CrossRef]

2004

2003

1996

1995

T. Neger, "Optical tomography of plasmas by spectral interferometry," J. Phys. D 28, 47-54 (1995).
[CrossRef]

G. Pretzler, "Single shot tomography by differential interferometry," Meas. Sci. Technol. 6, 1476-1486 (1995).
[CrossRef]

1994

1989

T. Liu, W. Merzkirch, and K. Oberste-Lehn, "Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density," Exp. Fluids 7, 157-163 (1989).
[CrossRef]

1988

1987

1986

1985

R. Snyder and L. Hasselink, "High speed optical tomography for flow visualization," Appl. Opt. 24, 4046-4051 (1985).
[CrossRef] [PubMed]

H. Hertz, "Experimental determination of 2-D flame temperature fields by interferometric tomography," Opt. Commun. 54, 131-136 (1985).
[CrossRef]

1983

1981

1980

A. L. Auslender, G. N. Vishnyakov, and G. G. Levin, "Solution of Radon's integral equation in an optical processor," Opt. Spectrosc. 49, 518-520 (1980).

1979

1978

S. Gull and G. Daniel, "Image reconstruction from incomplete and noisy data," Nature 272, 686-690 (1978).
[CrossRef]

1975

1973

1969

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Andersen, A. H.

Arrizón, V.

Auslender, A. L.

A. L. Auslender, G. N. Vishnyakov, and G. G. Levin, "Solution of Radon's integral equation in an optical processor," Opt. Spectrosc. 49, 518-520 (1980).

Brody, W. R.

Burow, R.

Byer, R. L.

Creath, K.

K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, E.Wolf, ed. (Elsevier, 1998), Vol. 16, pp. 349-393.

Daniel, G.

S. Gull and G. Daniel, "Image reconstruction from incomplete and noisy data," Nature 272, 686-690 (1978).
[CrossRef]

Dean, S. R.

S. R. Dean, The Radon Transform and Some of Its Applications (Wiley, 1983).

Demoment, G.

Dirksen, D.

Dubois, F.

Elssner, K.-E.

Emmerman, P. J.

R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, "Optical tomography for flow field diagnostics," Int. J. Heat Mass Transfer 24, 1139-1150 (1981).
[CrossRef]

Garbuny, M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

Goulard, R.

R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, "Optical tomography for flow field diagnostics," Int. J. Heat Mass Transfer 24, 1139-1150 (1981).
[CrossRef]

Grzanna, J.

Gull, S.

S. Gull and G. Daniel, "Image reconstruction from incomplete and noisy data," Nature 272, 686-690 (1978).
[CrossRef]

Hansen, E. W.

Hasselink, L.

Hauger, C.

Hellmuth, T.

Hertz, H.

H. Hertz, "Experimental determination of 2-D flame temperature fields by interferometric tomography," Opt. Commun. 54, 131-136 (1985).
[CrossRef]

Honda, T.

L. M. Murillo-Mora, K. Okada, T. Honda, and J. Tsujiuchi, "Color conical holographic stereograms: recording and distortion compensation methods," in The Art and Science of Holography, H.J.Caulfield, ed. (SPIE, 2004), pp. 261-273.

Jäger, H.

Jaggard, D. L.

Joannes, L.

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1987).

Legros, J.-C.

Levin, G. G.

A. L. Auslender, G. N. Vishnyakov, and G. G. Levin, "Solution of Radon's integral equation in an optical processor," Opt. Spectrosc. 49, 518-520 (1980).

Liu, T.

T. Liu, W. Merzkirch, and K. Oberste-Lehn, "Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density," Exp. Fluids 7, 157-163 (1989).
[CrossRef]

Macovski, A.

Medoff, B. P.

Meneses-Fabian, C.

C. Meneses-Fabian, G. Rodríguez-Zurita, R. Rodríguez-Vera, and J. F. Vázquez-Castillo, "Optical tomography with parallel projection differences and electronic speckle pattern interferometry," Opt. Commun. 228, 201-210 (2003).
[CrossRef]

Merzkirch, W.

T. Liu, W. Merzkirch, and K. Oberste-Lehn, "Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density," Exp. Fluids 7, 157-163 (1989).
[CrossRef]

Mohammad-Djafari, A.

Murillo-Mora, L. M.

L. M. Murillo-Mora, K. Okada, T. Honda, and J. Tsujiuchi, "Color conical holographic stereograms: recording and distortion compensation methods," in The Art and Science of Holography, H.J.Caulfield, ed. (SPIE, 2004), pp. 261-273.

Nassi, M.

Neger, T.

Oberste-Lehn, K.

T. Liu, W. Merzkirch, and K. Oberste-Lehn, "Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density," Exp. Fluids 7, 157-163 (1989).
[CrossRef]

Okada, K.

L. M. Murillo-Mora, K. Okada, T. Honda, and J. Tsujiuchi, "Color conical holographic stereograms: recording and distortion compensation methods," in The Art and Science of Holography, H.J.Caulfield, ed. (SPIE, 2004), pp. 261-273.

Perez-Mendez, V.

Philipp, H.

Pretzler, G.

Radulovic, P. T.

C. M. Vest and P. T. Radulovic, "Measurement of three-dimensional temperature fields by holographic interferometry," in Applications of Holography and Optical Data Processing, E.Marcom, A.A.Friesem, and E.Wiener-Avnear, eds. (Pergamon, 1977), pp. 241-249.

Robinson, D. W.

D. W. Robinson and D. C. Williams, "Digital phase stepping speckle interferometry," Opt. Commun. 57, 26-30 (1986).
[CrossRef]

Rodríguez-Vera, R.

C. Meneses-Fabian, G. Rodríguez-Zurita, R. Rodríguez-Vera, and J. F. Vázquez-Castillo, "Optical tomography with parallel projection differences and electronic speckle pattern interferometry," Opt. Commun. 228, 201-210 (2003).
[CrossRef]

Rodríguez-Zurita, G.

C. Meneses-Fabian, G. Rodríguez-Zurita, R. Rodríguez-Vera, and J. F. Vázquez-Castillo, "Optical tomography with parallel projection differences and electronic speckle pattern interferometry," Opt. Commun. 228, 201-210 (2003).
[CrossRef]

Sánchez-de-la-Llave, D.

Santoro, R. J.

R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, "Optical tomography for flow field diagnostics," Int. J. Heat Mass Transfer 24, 1139-1150 (1981).
[CrossRef]

Schultz, K. I.

Schwider, J.

Semerjian, H. G.

R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, "Optical tomography for flow field diagnostics," Int. J. Heat Mass Transfer 24, 1139-1150 (1981).
[CrossRef]

Shepp, L. A.

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1987).

Snyder, R.

Spolaczyk, R.

Tam, K. C.

Tsujiuchi, J.

L. M. Murillo-Mora, K. Okada, T. Honda, and J. Tsujiuchi, "Color conical holographic stereograms: recording and distortion compensation methods," in The Art and Science of Holography, H.J.Caulfield, ed. (SPIE, 2004), pp. 261-273.

Vázquez-Castillo, J. F.

C. Meneses-Fabian, G. Rodríguez-Zurita, R. Rodríguez-Vera, and J. F. Vázquez-Castillo, "Optical tomography with parallel projection differences and electronic speckle pattern interferometry," Opt. Commun. 228, 201-210 (2003).
[CrossRef]

Vest, C. M.

C. M. Vest, "Interferometry of strongly refracting axisymmetric phase objects," Appl. Opt. 14, 1601-1606 (1975).
[CrossRef] [PubMed]

C. M. Vest and P. T. Radulovic, "Measurement of three-dimensional temperature fields by holographic interferometry," in Applications of Holography and Optical Data Processing, E.Marcom, A.A.Friesem, and E.Wiener-Avnear, eds. (Pergamon, 1977), pp. 241-249.

Vishnyakov, G. N.

A. L. Auslender, G. N. Vishnyakov, and G. G. Levin, "Solution of Radon's integral equation in an optical processor," Opt. Spectrosc. 49, 518-520 (1980).

Von Bally, G.

Widmann, K.

Williams, D. C.

D. W. Robinson and D. C. Williams, "Digital phase stepping speckle interferometry," Opt. Commun. 57, 26-30 (1986).
[CrossRef]

Woisetschläger, J.

Wolf, E.

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Wörz, M.

Appl. Opt.

Exp. Fluids

T. Liu, W. Merzkirch, and K. Oberste-Lehn, "Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density," Exp. Fluids 7, 157-163 (1989).
[CrossRef]

Int. J. Heat Mass Transfer

R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, "Optical tomography for flow field diagnostics," Int. J. Heat Mass Transfer 24, 1139-1150 (1981).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. D

T. Neger, "Optical tomography of plasmas by spectral interferometry," J. Phys. D 28, 47-54 (1995).
[CrossRef]

Meas. Sci. Technol.

G. Pretzler, "Single shot tomography by differential interferometry," Meas. Sci. Technol. 6, 1476-1486 (1995).
[CrossRef]

Nature

S. Gull and G. Daniel, "Image reconstruction from incomplete and noisy data," Nature 272, 686-690 (1978).
[CrossRef]

Opt. Commun.

C. Meneses-Fabian, G. Rodríguez-Zurita, R. Rodríguez-Vera, and J. F. Vázquez-Castillo, "Optical tomography with parallel projection differences and electronic speckle pattern interferometry," Opt. Commun. 228, 201-210 (2003).
[CrossRef]

H. Hertz, "Experimental determination of 2-D flame temperature fields by interferometric tomography," Opt. Commun. 54, 131-136 (1985).
[CrossRef]

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

D. W. Robinson and D. C. Williams, "Digital phase stepping speckle interferometry," Opt. Commun. 57, 26-30 (1986).
[CrossRef]

Opt. Lett.

Opt. Spectrosc.

A. L. Auslender, G. N. Vishnyakov, and G. G. Levin, "Solution of Radon's integral equation in an optical processor," Opt. Spectrosc. 49, 518-520 (1980).

Other

C. M. Vest and P. T. Radulovic, "Measurement of three-dimensional temperature fields by holographic interferometry," in Applications of Holography and Optical Data Processing, E.Marcom, A.A.Friesem, and E.Wiener-Avnear, eds. (Pergamon, 1977), pp. 241-249.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

S. R. Dean, The Radon Transform and Some of Its Applications (Wiley, 1983).

L. M. Murillo-Mora, K. Okada, T. Honda, and J. Tsujiuchi, "Color conical holographic stereograms: recording and distortion compensation methods," in The Art and Science of Holography, H.J.Caulfield, ed. (SPIE, 2004), pp. 261-273.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1987).

K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, E.Wolf, ed. (Elsevier, 1998), Vol. 16, pp. 349-393.

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 (7)

Fig. 1
Fig. 1

Common-path interferometer consisting of a 4 f telecentric coherent imaging system with two windows (L, F) in the object space ( x ) and a Ronchi ruling in the frequency plane ( u ) . In the image plane ( x ) , the diffraction orders + 1 , 0 of one window can overlap with the diffraction orders 0, 1 of the other one, provided that the focal length f, the wavelength λ, and the period of the grating d are related to the windows’ separation x 0 , with x 0 = λ f d . In that case, interference patterns result within a region such as the ones encircled with dashed curves.

Fig. 2
Fig. 2

Relationship between the width (ms) and the amplitude (V) of the pulse applied to the actuator that drives the grating and corresponding fringe shifts (fringe number). The pulse width varies between 1 and 120 ms, as indicated in the vertical axis (left). The several pulse amplitude values are indicated in the inset with data symbols related to voltage levels between 4 and 9 V. The horizontal axis shows the number of fringes passing by the center of an interference pattern (from an auxiliary Michelson interferometer, see text) when a pulse with a constant voltage value is applied. The straight lines fit data for each voltage level.

Fig. 3
Fig. 3

Experimental tomographic setup using a grating interferometer. The object plane t ( x , y ) consists of two windows of equal sides a and b with separation between the centers of x 0 [inset (a)]. A Ronchi ruling acts as the spatial filter F ( u , v ) . t f ( x , y ) is the amplitude distribution in the image plane. In front of one of the windows is the sample O immersed in the liquid gate and appended to a stepping motor. f, focal lengths of transforming lenses; M, one mirror of an auxiliary Michelson interferometer; DC, actuator driving the grating; CCD, camera; PC, personal computer. The arrows indicate data flow and control from the PC.

Fig. 4
Fig. 4

Composite interferograms in the plane of a conventional sinogram from projections of two glass tubes of different diameters and four grating positions, showing phase shifting as a whole. All images are data arrays of 400 × 400   pixels .

Fig. 5
Fig. 5

Composite interferograms in the plane of a conventional sinogram from projections of a microscope slide. (a)–(d) Constructed interferograms from phase-shifted projections, corresponding to phase shifts of 0°, 90°, 180°, and 270°, respectively; (e) the resulting wrapped phase; (f) the resulting unwrapped phase; and (g) and (h) two plots of the reconstructed object obtained from (f). All images are data arrays of 400 × 400   pixels .

Fig. 6
Fig. 6

Transparent samples for experimental tomographic inspection [one piece from a microscope slice (see Fig. 5) and three glass blocks]. Relevant dimensions are given in millimeters.

Fig. 7
Fig. 7

Four phase-shifted composite interferograms in the plane of the sinogram from projections of the three glass blocks shown in Fig. 6. All images are data arrays of 400 × 400   pixels .

Equations (7)

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

F ( μ , ζ ) = rect ( μ a w ) n = δ ( μ μ 0 n x 0 ) ,
F ̃ ( x , y ) = a w x 0 n = exp ( i 2 π n μ 0 x 0 ) sinc ( n a w x 0 ) δ ( x n x 0 , y ) .
t ( x , y ) = w ( x + x 0 2 , y ) + w ( x x 0 2 , y ) exp [ i ϕ ( x x 0 2 , y ) ] ,
t 0 f ( x , y ) = a w x 0 { w ( x + x 0 2 , y ) + w ( x x 0 2 , y ) exp [ i ϕ ( x x 0 2 , y ) ] } ,
t + 1 f ( x , y ) = a w x 0 exp ( i 2 π μ 0 x 0 ) sinc ( a w x 0 ) { w ( x x 0 2 , y ) + w ( x 3 x 0 2 , y ) exp [ i ϕ ( x 3 x 0 2 , y ) ] } .
I 0 , + 1 ( x , y ) = 1 + sinc 2 ( a w x 0 ) + 2 sinc ( a w x 0 ) cos [ ϕ ( x , y ) + 2 π μ 0 x 0 ] ,
V 0 , + 1 ( a w ) = 2 sinc ( a w x 0 ) 1 + sinc 2 ( a w x 0 ) .

Metrics