Abstract

We present a simple method with which to reconstruct wavefronts that are transilluminating supersonic jets through measurement of their local slopes. The wavefront slopes are measured in two orthogonal directions by use of the average displacements suffered by a random-dot pattern in its shadow cast by the wavefront in front of it. A smooth wavefront is estimated from its measured slopes by a least-squares curve fitting technique. The calculated wavefront distortion is tomographically inverted to yield the density distributions of the object. The results are comparable to that obtained by use of a phase-retrieval algorithm from axial intensity transport measurements.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. W. Faris and R. L. Byer, "Three dimensional beam deflection optical tomography of a supersonic jet," Appl. Opt. 27, 5202-5212 (1988).
    [CrossRef] [PubMed]
  2. G. W. Faris and R. L. Byer, "Beam-deflection optical tomography of a flame," Opt. Lett. 12, 155-157 (1987).
    [CrossRef] [PubMed]
  3. T. Sochacki, M. Sochacka, and C. Gomez-Reino, "Reconstruction of the axisymmetric refractive-index profiles from transverse interferograms in the presence of the immersion-air separating window," J. Opt. Soc. Am. A 7, 211-215 (1990).
    [CrossRef]
  4. B. H. Timmerman, D. W. Watt, and P. J. Bryanston-Cross, "Quantitative visualization of high-speed 3D turbulent flow structures using holographic interferometric tomography," Opt. Laser Technol. 31, 53-65 (1999).
    [CrossRef]
  5. A. Barty, K. A. Nugent, A. Roberts, and D. Peganin, "Quantitative phase microscopy," Opt. Commun. 175, 329-336 (2000).
    [CrossRef]
  6. L. McMackin, B. Masson, and N. Clark, "Hartmann wavefront sensor studies of dynamic organized structure in flowfields," AIAA J. 33, 2158-2164 (1995).
    [CrossRef]
  7. S. Olivier, V. Laude, and J.-P. Huignard, "Liquid-crystal Hartmann wave-front scanner," Appl. Opt. 39, 3838-3846 (2000).
    [CrossRef]
  8. K. Muralidhar, "Temperature field measurement in buoyancy-driven flows using interferometric tomography," Annu. Rev. Heat Transfer 12, 265-375 (2000).
  9. M. Burnett and P. J. Bryanston-Cross, "Measurement of transonic shock structures using shearography," in Laser Interfeometry III, R. J. Pryputniewicz, G. M. Brown, and W. P. Jueptner, eds., Proc. SPIE 2861, 124-135 (1995).
    [CrossRef]
  10. H. Thayyullathil, R. Langoju, R. Padmaram, R. M. Vasu, R. Kanjorodan, and L. M. Patnaik, "Three-dimensional optical tomographic imaging of supersonic jets through inversion of phase data obtained through the transport-of-intensity equation," Appl. Opt. 43, 4133-4141 (2004).
    [CrossRef]
  11. L. Venkatakrishnan and G. E. A. Meier, "Density measurement using background oriented Schlieren technique," Exp. Fluids 37, 237-247 (2004).
    [CrossRef]
  12. N. J. Lawson, "The application of particle image velocimetry to high-speed flows," Ph.D. dissertation (Loughborough University of Technology,1995).
  13. R. H. Hudgin, "Wave-front reconstruction for compensated imaging," J. Opt. Soc. Am. 67, 375-382 (1977).
    [CrossRef]
  14. D. C. Ghiglia and L. A. Romero, "Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods," J. Opt. Soc. Am. A 11, 107-117 (1994).
    [CrossRef]
  15. N. Fomin, L. Lavinskaja, W. Merzkirch, and D. Vitkin, "Speckle photography applied to statistical analysis of turbulence," Opt. Laser Technol. 31, 13-22 (1999).
    [CrossRef]
  16. R. D. Keane and R. J. Adrian, "The theory of cross-correlation analysis of particle image velocimetry images," Appl. Sci. Res. 49, 191-215 (1992).
    [CrossRef]
  17. G. W. Faris and R. L. Byer, "Beam-deflection optical tomography," Opt. Lett. 12, 72-74 (1987).
    [CrossRef] [PubMed]
  18. D. L. Fried, "Least-squares fitting a wave-front distortion estimate to an array of phase-difference measurements," J. Opt. Soc. Am. 67, 370-375 (1977).
    [CrossRef]
  19. B. R. Hunt, "Matrix formulation of the reconstruction of phase values from phase differences," J. Opt. Soc. Am. 69, 393-399 (1979).
    [CrossRef]
  20. W. H. Southwell, "Wave-front estimation from wave-front slope measurements," J. Opt. Soc. Am. 70, 998-1006 (1980).
    [CrossRef]
  21. L. A. Feldkamp, L. C. Davis, and J. W. Kress, "Practical cone-beam algorithm," J. Opt. Soc. Am. A 1, 612-619 (1984).
    [CrossRef]
  22. A. H. Andersen and A. C. Kak, "Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm," Ultrason. Imaging 6, 81-84 (1984).
    [CrossRef] [PubMed]
  23. R. Clack and M. Defrise, "Overview of reocnstruction algorithms for exact cone-beam tomography," in Mathematical Methods in Medical Imaging III, F. L. Bookstein, J. S. Duncan, N. Lange, and D. C. Wilson, eds., Proc. SPIE 2299, 230-241 (1994).
    [CrossRef]
  24. H. S. Ko and K. D. Kihm, "An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography," Exp. Fluids 27, 542-550 (1999).
    [CrossRef]
  25. I. H. Lira and C. M. Vest, "Refraction correction in holographic interferometry and tomography of transparent objects," Appl. Opt. 26, 3919-3928 (1987).
    [CrossRef] [PubMed]
  26. M. A. Anastasio and X. Pan, "Full- and minimal-scan reconstruction algorithms for fan-beam diffraction tomography," Appl. Opt. 40, 3334-3345 (2001).
    [CrossRef]
  27. A. Devaney, "Algorithms for fan-beam diffraction tomography," Ultrason. Imaging 7, 264-275 (1985).
    [CrossRef] [PubMed]
  28. R. E. Pierson, D. F. Olson, and E. Y. Chen, "Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow," Opt. Eng. 39, 838-846 (2000).
    [CrossRef]

2004

2001

2000

R. E. Pierson, D. F. Olson, and E. Y. Chen, "Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow," Opt. Eng. 39, 838-846 (2000).
[CrossRef]

A. Barty, K. A. Nugent, A. Roberts, and D. Peganin, "Quantitative phase microscopy," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

S. Olivier, V. Laude, and J.-P. Huignard, "Liquid-crystal Hartmann wave-front scanner," Appl. Opt. 39, 3838-3846 (2000).
[CrossRef]

K. Muralidhar, "Temperature field measurement in buoyancy-driven flows using interferometric tomography," Annu. Rev. Heat Transfer 12, 265-375 (2000).

1999

B. H. Timmerman, D. W. Watt, and P. J. Bryanston-Cross, "Quantitative visualization of high-speed 3D turbulent flow structures using holographic interferometric tomography," Opt. Laser Technol. 31, 53-65 (1999).
[CrossRef]

N. Fomin, L. Lavinskaja, W. Merzkirch, and D. Vitkin, "Speckle photography applied to statistical analysis of turbulence," Opt. Laser Technol. 31, 13-22 (1999).
[CrossRef]

H. S. Ko and K. D. Kihm, "An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography," Exp. Fluids 27, 542-550 (1999).
[CrossRef]

1995

M. Burnett and P. J. Bryanston-Cross, "Measurement of transonic shock structures using shearography," in Laser Interfeometry III, R. J. Pryputniewicz, G. M. Brown, and W. P. Jueptner, eds., Proc. SPIE 2861, 124-135 (1995).
[CrossRef]

L. McMackin, B. Masson, and N. Clark, "Hartmann wavefront sensor studies of dynamic organized structure in flowfields," AIAA J. 33, 2158-2164 (1995).
[CrossRef]

1994

D. C. Ghiglia and L. A. Romero, "Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods," J. Opt. Soc. Am. A 11, 107-117 (1994).
[CrossRef]

R. Clack and M. Defrise, "Overview of reocnstruction algorithms for exact cone-beam tomography," in Mathematical Methods in Medical Imaging III, F. L. Bookstein, J. S. Duncan, N. Lange, and D. C. Wilson, eds., Proc. SPIE 2299, 230-241 (1994).
[CrossRef]

1992

R. D. Keane and R. J. Adrian, "The theory of cross-correlation analysis of particle image velocimetry images," Appl. Sci. Res. 49, 191-215 (1992).
[CrossRef]

1990

1988

1987

1985

A. Devaney, "Algorithms for fan-beam diffraction tomography," Ultrason. Imaging 7, 264-275 (1985).
[CrossRef] [PubMed]

1984

L. A. Feldkamp, L. C. Davis, and J. W. Kress, "Practical cone-beam algorithm," J. Opt. Soc. Am. A 1, 612-619 (1984).
[CrossRef]

A. H. Andersen and A. C. Kak, "Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm," Ultrason. Imaging 6, 81-84 (1984).
[CrossRef] [PubMed]

1980

1979

1977

Adrian, R. J.

R. D. Keane and R. J. Adrian, "The theory of cross-correlation analysis of particle image velocimetry images," Appl. Sci. Res. 49, 191-215 (1992).
[CrossRef]

Anastasio, M. A.

Andersen, A. H.

A. H. Andersen and A. C. Kak, "Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm," Ultrason. Imaging 6, 81-84 (1984).
[CrossRef] [PubMed]

Barty, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Peganin, "Quantitative phase microscopy," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Bryanston-Cross, P. J.

B. H. Timmerman, D. W. Watt, and P. J. Bryanston-Cross, "Quantitative visualization of high-speed 3D turbulent flow structures using holographic interferometric tomography," Opt. Laser Technol. 31, 53-65 (1999).
[CrossRef]

M. Burnett and P. J. Bryanston-Cross, "Measurement of transonic shock structures using shearography," in Laser Interfeometry III, R. J. Pryputniewicz, G. M. Brown, and W. P. Jueptner, eds., Proc. SPIE 2861, 124-135 (1995).
[CrossRef]

Burnett, M.

M. Burnett and P. J. Bryanston-Cross, "Measurement of transonic shock structures using shearography," in Laser Interfeometry III, R. J. Pryputniewicz, G. M. Brown, and W. P. Jueptner, eds., Proc. SPIE 2861, 124-135 (1995).
[CrossRef]

Byer, R. L.

Chen, E. Y.

R. E. Pierson, D. F. Olson, and E. Y. Chen, "Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow," Opt. Eng. 39, 838-846 (2000).
[CrossRef]

Clack, R.

R. Clack and M. Defrise, "Overview of reocnstruction algorithms for exact cone-beam tomography," in Mathematical Methods in Medical Imaging III, F. L. Bookstein, J. S. Duncan, N. Lange, and D. C. Wilson, eds., Proc. SPIE 2299, 230-241 (1994).
[CrossRef]

Clark, N.

L. McMackin, B. Masson, and N. Clark, "Hartmann wavefront sensor studies of dynamic organized structure in flowfields," AIAA J. 33, 2158-2164 (1995).
[CrossRef]

Davis, L. C.

Defrise, M.

R. Clack and M. Defrise, "Overview of reocnstruction algorithms for exact cone-beam tomography," in Mathematical Methods in Medical Imaging III, F. L. Bookstein, J. S. Duncan, N. Lange, and D. C. Wilson, eds., Proc. SPIE 2299, 230-241 (1994).
[CrossRef]

Devaney, A.

A. Devaney, "Algorithms for fan-beam diffraction tomography," Ultrason. Imaging 7, 264-275 (1985).
[CrossRef] [PubMed]

Faris, G. W.

Feldkamp, L. A.

Fomin, N.

N. Fomin, L. Lavinskaja, W. Merzkirch, and D. Vitkin, "Speckle photography applied to statistical analysis of turbulence," Opt. Laser Technol. 31, 13-22 (1999).
[CrossRef]

Fried, D. L.

Ghiglia, D. C.

Gomez-Reino, C.

Hudgin, R. H.

Huignard, J.-P.

Hunt, B. R.

Kak, A. C.

A. H. Andersen and A. C. Kak, "Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm," Ultrason. Imaging 6, 81-84 (1984).
[CrossRef] [PubMed]

Kanjorodan, R.

Keane, R. D.

R. D. Keane and R. J. Adrian, "The theory of cross-correlation analysis of particle image velocimetry images," Appl. Sci. Res. 49, 191-215 (1992).
[CrossRef]

Kihm, K. D.

H. S. Ko and K. D. Kihm, "An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography," Exp. Fluids 27, 542-550 (1999).
[CrossRef]

Ko, H. S.

H. S. Ko and K. D. Kihm, "An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography," Exp. Fluids 27, 542-550 (1999).
[CrossRef]

Kress, J. W.

Langoju, R.

Laude, V.

Lavinskaja, L.

N. Fomin, L. Lavinskaja, W. Merzkirch, and D. Vitkin, "Speckle photography applied to statistical analysis of turbulence," Opt. Laser Technol. 31, 13-22 (1999).
[CrossRef]

Lawson, N. J.

N. J. Lawson, "The application of particle image velocimetry to high-speed flows," Ph.D. dissertation (Loughborough University of Technology,1995).

Lira, I. H.

Masson, B.

L. McMackin, B. Masson, and N. Clark, "Hartmann wavefront sensor studies of dynamic organized structure in flowfields," AIAA J. 33, 2158-2164 (1995).
[CrossRef]

McMackin, L.

L. McMackin, B. Masson, and N. Clark, "Hartmann wavefront sensor studies of dynamic organized structure in flowfields," AIAA J. 33, 2158-2164 (1995).
[CrossRef]

Meier, G. E. A.

L. Venkatakrishnan and G. E. A. Meier, "Density measurement using background oriented Schlieren technique," Exp. Fluids 37, 237-247 (2004).
[CrossRef]

Merzkirch, W.

N. Fomin, L. Lavinskaja, W. Merzkirch, and D. Vitkin, "Speckle photography applied to statistical analysis of turbulence," Opt. Laser Technol. 31, 13-22 (1999).
[CrossRef]

Muralidhar, K.

K. Muralidhar, "Temperature field measurement in buoyancy-driven flows using interferometric tomography," Annu. Rev. Heat Transfer 12, 265-375 (2000).

Nugent, K. A.

A. Barty, K. A. Nugent, A. Roberts, and D. Peganin, "Quantitative phase microscopy," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Olivier, S.

Olson, D. F.

R. E. Pierson, D. F. Olson, and E. Y. Chen, "Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow," Opt. Eng. 39, 838-846 (2000).
[CrossRef]

Padmaram, R.

Pan, X.

Patnaik, L. M.

Peganin, D.

A. Barty, K. A. Nugent, A. Roberts, and D. Peganin, "Quantitative phase microscopy," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Pierson, R. E.

R. E. Pierson, D. F. Olson, and E. Y. Chen, "Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow," Opt. Eng. 39, 838-846 (2000).
[CrossRef]

Roberts, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Peganin, "Quantitative phase microscopy," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Romero, L. A.

Sochacka, M.

Sochacki, T.

Southwell, W. H.

Thayyullathil, H.

Timmerman, B. H.

B. H. Timmerman, D. W. Watt, and P. J. Bryanston-Cross, "Quantitative visualization of high-speed 3D turbulent flow structures using holographic interferometric tomography," Opt. Laser Technol. 31, 53-65 (1999).
[CrossRef]

Vasu, R. M.

Venkatakrishnan, L.

L. Venkatakrishnan and G. E. A. Meier, "Density measurement using background oriented Schlieren technique," Exp. Fluids 37, 237-247 (2004).
[CrossRef]

Vest, C. M.

Vitkin, D.

N. Fomin, L. Lavinskaja, W. Merzkirch, and D. Vitkin, "Speckle photography applied to statistical analysis of turbulence," Opt. Laser Technol. 31, 13-22 (1999).
[CrossRef]

Watt, D. W.

B. H. Timmerman, D. W. Watt, and P. J. Bryanston-Cross, "Quantitative visualization of high-speed 3D turbulent flow structures using holographic interferometric tomography," Opt. Laser Technol. 31, 53-65 (1999).
[CrossRef]

AIAA J.

L. McMackin, B. Masson, and N. Clark, "Hartmann wavefront sensor studies of dynamic organized structure in flowfields," AIAA J. 33, 2158-2164 (1995).
[CrossRef]

Annu. Rev. Heat Transfer

K. Muralidhar, "Temperature field measurement in buoyancy-driven flows using interferometric tomography," Annu. Rev. Heat Transfer 12, 265-375 (2000).

Appl. Opt.

Appl. Sci. Res.

R. D. Keane and R. J. Adrian, "The theory of cross-correlation analysis of particle image velocimetry images," Appl. Sci. Res. 49, 191-215 (1992).
[CrossRef]

Exp. Fluids

L. Venkatakrishnan and G. E. A. Meier, "Density measurement using background oriented Schlieren technique," Exp. Fluids 37, 237-247 (2004).
[CrossRef]

H. S. Ko and K. D. Kihm, "An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography," Exp. Fluids 27, 542-550 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

A. Barty, K. A. Nugent, A. Roberts, and D. Peganin, "Quantitative phase microscopy," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Opt. Eng.

R. E. Pierson, D. F. Olson, and E. Y. Chen, "Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow," Opt. Eng. 39, 838-846 (2000).
[CrossRef]

Opt. Laser Technol.

B. H. Timmerman, D. W. Watt, and P. J. Bryanston-Cross, "Quantitative visualization of high-speed 3D turbulent flow structures using holographic interferometric tomography," Opt. Laser Technol. 31, 53-65 (1999).
[CrossRef]

N. Fomin, L. Lavinskaja, W. Merzkirch, and D. Vitkin, "Speckle photography applied to statistical analysis of turbulence," Opt. Laser Technol. 31, 13-22 (1999).
[CrossRef]

Opt. Lett.

Proc. SPIE

R. Clack and M. Defrise, "Overview of reocnstruction algorithms for exact cone-beam tomography," in Mathematical Methods in Medical Imaging III, F. L. Bookstein, J. S. Duncan, N. Lange, and D. C. Wilson, eds., Proc. SPIE 2299, 230-241 (1994).
[CrossRef]

M. Burnett and P. J. Bryanston-Cross, "Measurement of transonic shock structures using shearography," in Laser Interfeometry III, R. J. Pryputniewicz, G. M. Brown, and W. P. Jueptner, eds., Proc. SPIE 2861, 124-135 (1995).
[CrossRef]

Ultrason. Imaging

A. H. Andersen and A. C. Kak, "Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm," Ultrason. Imaging 6, 81-84 (1984).
[CrossRef] [PubMed]

A. Devaney, "Algorithms for fan-beam diffraction tomography," Ultrason. Imaging 7, 264-275 (1985).
[CrossRef] [PubMed]

Other

N. J. Lawson, "The application of particle image velocimetry to high-speed flows," Ph.D. dissertation (Loughborough University of Technology,1995).

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 (9)

Fig. 1
Fig. 1

Illustration of shadow casting by a distorted wavefront. The distorted wavefront is approximated locally by tilted plane waves that cast shadows that are displaced compared to the shadows cast by the undistorted wave: W, distorted wavefront; L, linear approximation of W; D, dot pattern; Δz, distance between the dot pattern and the plane where the shadow is imaged; θ, angle that the normal to the distorted wavefront makes with the undistorted wavefront; d, displacement of the shadow with respect to its position if the wavefront were not distorted.

Fig. 2
Fig. 2

(Color online) Experimental setup used for recording shadows cast by the distorted beam: BE, beam expander; L 1, collimation lens; O, nozzle and flow; L 2, auxiliary lens used for imaging plane II onto the CCD camera. The value of Δz used in the experiment is 1 cm. The origin of coordinates is on the axis of the nozzle at its exit plane.

Fig. 3
Fig. 3

Geometry of the cone-beam algorithm that we employed. The source vertex moves about the object on a circle.

Fig. 4
Fig. 4

(Color online) Phase difference calculated from the shadow patterns obtained with parallel-beam illumination.

Fig. 5
Fig. 5

(Color online) Phase difference calculated from the shadow patterns obtained with cone-beam illumination.

Fig. 6
Fig. 6

Typical x–z cross sections through the reconstructed density distributions in the parallel-beam case: (a) contour plot; (b) cross-sectional plot through the center of the image in (a) at y = 0.5 cm. (c) and (d) same as (a) and (b), respectively, at y = 4.5 cm.

Fig. 7
Fig. 7

Typical x–y cross sections through the reconstructed density distributions in the parallel-beam case: (a) at z = −0.3 cm, (b) at z = 0.0 cm, (c) at z = 0.3 cm.

Fig. 8
Fig. 8

Typical x–z cross sections through the reconstructed density distributions in the cone-beam case: (a) contour plot; (b) cross-sectional plot through the center of the image in (a) at y = 0.5 cm. (c) and (d) Same as (a) and (b), respectively, at y = 4.5 cm.

Fig. 9
Fig. 9

Typical x–y cross sections through the reconstructed density distributions in the cone-beam case: (a) at z = −0.3 cm, (b) at z = 0.0 cm, (c) at z = 0.3 cm.

Equations (7)

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

ϵ = i = 0 M 2 j = 0 N 1 ( W i + 1 , j W i , j Θ i , j x ) 2 + i = 0 M 1 j = 1 N 2 ( W i , j + 1 W i , j Θ i , j y ) 2 .
W i + 1 , j + W i 1 , j + W i , j + 1 + W i , j 1 4 W i , j = Θ i , j x Θ i 1 , j x + Θ ij y Θ i , j 1 y .
2 W x 2 + 2 W y 2 = ρ ( x , y ) .
W ^ i , j = ρ ^ i j 2 [ cos ( π i / M ) + cos ( π j / N ) 2 ] .
n ( r ) = 1 4 π 2 S O 2 ( S O + r   ⋅   x ^ ϕ ) 2 P ˜ ( ϕ , a , b ) d ϕ ,
( n 1 ) ρ = K ( λ ) ,
K ( λ ) = 2.2244 × 10 4 [ 1 + ( 6.7132 × 10 2 λ ) 2 ] .

Metrics