Abstract

We present an Abel-inversion algorithm to reconstruct mean and rms refractive-index profiles from spatially resolved statistical measurements of the beam-deflection angle in time-dependent, axisymmetric flows. An oscillating gas-jet diffusion flame was investigated as a test case for applying the algorithm. Experimental data were obtained across the whole field by a rainbow schlieren apparatus. Results show that simultaneous multipoint measurements are necessary to reconstruct the rms refractive index accurately.

© 1999 Optical Society of America

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References

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  1. R. J. Santoro, H. G. Semerjian, P. J. Emmerman, R. Goulard, “Optical tomography for flow field diagnostics,” Int. J. Heat Mass Transfer 24, 1139–1150 (1981).
    [CrossRef]
  2. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).
  3. R. Snyder, L. Hesselink, “Optical tomography for flow visualization of the density field around revolving helicopter rotor blade,” Appl. Opt. 23, 3650–3656 (1984).
    [CrossRef]
  4. R. Snyder, L. Hesselink, “Measurement of mixing fluids flows with optical tomography,” Opt. Lett. 13, 87–91 (1988).
    [CrossRef]
  5. G. W. Faris, R. L. Byer, “Three-dimensional beam deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202–5212 (1988).
    [CrossRef] [PubMed]
  6. E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, “Measurements of temperature distribution of flames by moiré deflectometry,” Appl. Opt. 20, 4263–4266 (1981).
    [CrossRef] [PubMed]
  7. J. Stricker, “Analysis of 3-D phase objects by moiré deflectometry,” Appl. Opt. 23, 3657–3659 (1984).
    [CrossRef]
  8. P. V. Farrell, D. L. Hofeldt, “Temperature measurement in gases using speckle photography,” Appl. Opt. 23, 1055–1059 (1984).
    [CrossRef] [PubMed]
  9. P. S. Greenberg, R. B. Klimek, D. R. Buchele, “Quantitative rainbow schlieren deflectometry,” Appl. Opt. 34, 3810–3822 (1995).
    [CrossRef] [PubMed]
  10. K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89–95 (1998).
    [CrossRef]
  11. A. K. Agrawal, N. K. Butuk, S. R. Gollahalli, D. Griffin, “Three-dimensional rainbow schlieren tomography of a temperature field in gas flows,” Appl. Opt. 37, 479–485 (1998).
    [CrossRef]
  12. L. McMackin, R. J. Hugo, R. E. Pierson, C. R. Truman, “High speed optical tomography system for imaging dynamic transparent media,” Opt. Express 1, 302–311 (1997), http://epubs.osa.org/opticsexpress .
    [CrossRef] [PubMed]
  13. Y. R. Sivathanu, J. P. Gore, “A tomographic method for the reconstruction of local probability density functions,” J. Quant. Spectrosc. Radiat. Transfer 50, 483–492 (1993).
    [CrossRef]
  14. M. R. Nyden, P. Vallikul, Y. R. Sivathanu, “Tomographic reconstruction of the moments of local probability density functions in turbulent flow fields,” J. Quant. Spectrosc. Radiat. Transfer 55, 345–356 (1996).
    [CrossRef]
  15. R. Rubinstein, P. S. Greenberg, “Rapid inversion of angular deflection data for certain axisymmetric refractive-index distributions,” Appl. Opt. 33, 1141–1144 (1994).
    [CrossRef] [PubMed]
  16. C. Dasch, “One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods,” Appl. Opt. 31, 1146–1152 (1992).
    [CrossRef] [PubMed]
  17. B. Albers, A. K. Agrawal, “Schlieren analysis of an oscillating gas-jet diffusion flame,” Combust. Flame (to be published).

1998

K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89–95 (1998).
[CrossRef]

A. K. Agrawal, N. K. Butuk, S. R. Gollahalli, D. Griffin, “Three-dimensional rainbow schlieren tomography of a temperature field in gas flows,” Appl. Opt. 37, 479–485 (1998).
[CrossRef]

1997

1996

M. R. Nyden, P. Vallikul, Y. R. Sivathanu, “Tomographic reconstruction of the moments of local probability density functions in turbulent flow fields,” J. Quant. Spectrosc. Radiat. Transfer 55, 345–356 (1996).
[CrossRef]

1995

1994

1993

Y. R. Sivathanu, J. P. Gore, “A tomographic method for the reconstruction of local probability density functions,” J. Quant. Spectrosc. Radiat. Transfer 50, 483–492 (1993).
[CrossRef]

1992

1988

1984

1981

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

E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, “Measurements of temperature distribution of flames by moiré deflectometry,” Appl. Opt. 20, 4263–4266 (1981).
[CrossRef] [PubMed]

Agrawal, A. K.

A. K. Agrawal, N. K. Butuk, S. R. Gollahalli, D. Griffin, “Three-dimensional rainbow schlieren tomography of a temperature field in gas flows,” Appl. Opt. 37, 479–485 (1998).
[CrossRef]

K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89–95 (1998).
[CrossRef]

B. Albers, A. K. Agrawal, “Schlieren analysis of an oscillating gas-jet diffusion flame,” Combust. Flame (to be published).

Al-Ammar, K.

K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89–95 (1998).
[CrossRef]

Albers, B.

B. Albers, A. K. Agrawal, “Schlieren analysis of an oscillating gas-jet diffusion flame,” Combust. Flame (to be published).

Bar-Ziv, E.

Buchele, D. R.

Butuk, N. K.

Byer, R. L.

Dasch, C.

Emmerman, P. J.

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

Faris, G. W.

Farrell, P. V.

Glatt, I.

Gollahalli, S. R.

K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89–95 (1998).
[CrossRef]

A. K. Agrawal, N. K. Butuk, S. R. Gollahalli, D. Griffin, “Three-dimensional rainbow schlieren tomography of a temperature field in gas flows,” Appl. Opt. 37, 479–485 (1998).
[CrossRef]

Gore, J. P.

Y. R. Sivathanu, J. P. Gore, “A tomographic method for the reconstruction of local probability density functions,” J. Quant. Spectrosc. Radiat. Transfer 50, 483–492 (1993).
[CrossRef]

Goulard, R.

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

Greenberg, P. S.

Griffin, D.

A. K. Agrawal, N. K. Butuk, S. R. Gollahalli, D. Griffin, “Three-dimensional rainbow schlieren tomography of a temperature field in gas flows,” Appl. Opt. 37, 479–485 (1998).
[CrossRef]

K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89–95 (1998).
[CrossRef]

Hesselink, L.

Hofeldt, D. L.

Hugo, R. J.

Kafri, O.

Keren, E.

Klimek, R. B.

McMackin, L.

Nyden, M. R.

M. R. Nyden, P. Vallikul, Y. R. Sivathanu, “Tomographic reconstruction of the moments of local probability density functions in turbulent flow fields,” J. Quant. Spectrosc. Radiat. Transfer 55, 345–356 (1996).
[CrossRef]

Pierson, R. E.

Rubinstein, R.

Santoro, R. J.

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

Semerjian, H. G.

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

Sivathanu, Y. R.

M. R. Nyden, P. Vallikul, Y. R. Sivathanu, “Tomographic reconstruction of the moments of local probability density functions in turbulent flow fields,” J. Quant. Spectrosc. Radiat. Transfer 55, 345–356 (1996).
[CrossRef]

Y. R. Sivathanu, J. P. Gore, “A tomographic method for the reconstruction of local probability density functions,” J. Quant. Spectrosc. Radiat. Transfer 50, 483–492 (1993).
[CrossRef]

Snyder, R.

Stricker, J.

Truman, C. R.

Vallikul, P.

M. R. Nyden, P. Vallikul, Y. R. Sivathanu, “Tomographic reconstruction of the moments of local probability density functions in turbulent flow fields,” J. Quant. Spectrosc. Radiat. Transfer 55, 345–356 (1996).
[CrossRef]

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

Appl. Opt.

E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, “Measurements of temperature distribution of flames by moiré deflectometry,” Appl. Opt. 20, 4263–4266 (1981).
[CrossRef] [PubMed]

P. V. Farrell, D. L. Hofeldt, “Temperature measurement in gases using speckle photography,” Appl. Opt. 23, 1055–1059 (1984).
[CrossRef] [PubMed]

R. Snyder, L. Hesselink, “Optical tomography for flow visualization of the density field around revolving helicopter rotor blade,” Appl. Opt. 23, 3650–3656 (1984).
[CrossRef]

J. Stricker, “Analysis of 3-D phase objects by moiré deflectometry,” Appl. Opt. 23, 3657–3659 (1984).
[CrossRef]

G. W. Faris, R. L. Byer, “Three-dimensional beam deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202–5212 (1988).
[CrossRef] [PubMed]

C. Dasch, “One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods,” Appl. Opt. 31, 1146–1152 (1992).
[CrossRef] [PubMed]

R. Rubinstein, P. S. Greenberg, “Rapid inversion of angular deflection data for certain axisymmetric refractive-index distributions,” Appl. Opt. 33, 1141–1144 (1994).
[CrossRef] [PubMed]

P. S. Greenberg, R. B. Klimek, D. R. Buchele, “Quantitative rainbow schlieren deflectometry,” Appl. Opt. 34, 3810–3822 (1995).
[CrossRef] [PubMed]

A. K. Agrawal, N. K. Butuk, S. R. Gollahalli, D. Griffin, “Three-dimensional rainbow schlieren tomography of a temperature field in gas flows,” Appl. Opt. 37, 479–485 (1998).
[CrossRef]

Exp. Fluids

K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89–95 (1998).
[CrossRef]

Int. J. Heat Mass Transfer

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

J. Quant. Spectrosc. Radiat. Transfer

Y. R. Sivathanu, J. P. Gore, “A tomographic method for the reconstruction of local probability density functions,” J. Quant. Spectrosc. Radiat. Transfer 50, 483–492 (1993).
[CrossRef]

M. R. Nyden, P. Vallikul, Y. R. Sivathanu, “Tomographic reconstruction of the moments of local probability density functions in turbulent flow fields,” J. Quant. Spectrosc. Radiat. Transfer 55, 345–356 (1996).
[CrossRef]

Opt. Express

Opt. Lett.

Other

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

B. Albers, A. K. Agrawal, “Schlieren analysis of an oscillating gas-jet diffusion flame,” Combust. Flame (to be published).

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Figures (5)

Fig. 1
Fig. 1

Profiles of the mean and the rms refractive-index difference. Analytical profiles were obtained from Eqs. (21) and (22) with a = 0.1. Reconstructed profiles for different sampling intervals were obtained from Eqs. (9) and (17).

Fig. 2
Fig. 2

Profiles of the mean and the rms beam-deflection angle at z/ d = 10. The data were taken from a sequence of 200 schlieren field images of an oscillating jet flame taken in 3.33 s.

Fig. 3
Fig. 3

Reconstructed profiles of the mean and the rms deflection angle at z/ d = 10 in the oscillating jet flame. Profiles at a 2-pixel resolution were obtained by skipping alternate data points.

Fig. 4
Fig. 4

Correlated and uncorrelated rms refractive-index difference profiles at z/ d = 10 in the oscillating jet flame. These profiles were reconstructed from Eqs. (17) and (18).

Fig. 5
Fig. 5

Correlated and uncorrelated rms refractive-index difference profiles at z/ d = 30 in the oscillating jet flame. These profiles were reconstructed from Eqs. (17) and (18).

Equations (23)

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δr=-1πr ydyy2-r21/2.
δi=δri=-1πj=iNJ01j+j+1-jl× dlj+l2-i21/2,
δi=j=0NJ Dijj,
Dij=0 j<i,=1π Ai,j j=i,=1πAi,j-Bi,j-1 j>i,
Ai,j=j+12-i21/2-j2-i21/2-j+1lnj+1+j+12-i21/2j+j2-i21/2,
Bi,j=j+12-i21/2-j2-i21/2-j lnj+1+j+12-i21/2j+j2-i21/2.
δitn=j=iNJ Dijjtn,
δi¯=1NTn=1NT δitn,
δi¯=j=iNJ Dijj¯,
j¯=1NTn=1NT jtn
j¯=l=1NB lPjlΔ.
δitn=δitn-δi¯=j=iNJ Dijjtn,
δi2¯=1NTn=1NTδitn-δi¯2.
δi2¯=1NTn=1NTj=iNJ Dijjtnk=iNJ Dikktn.
δi2¯=1NTn=1NTj=iNJk=iNJ DijDikjtnktn
δi2¯=j=iNJk=iNJ DijDikjk¯
jk¯=1NTn=1NT jtnktn.
δi2¯=j=iNJ Dij21+Rijj2¯,
Rij=k=i,jkNJDikjk¯/Dijj2¯.
δi2¯=j=iNJ Dij2j2¯.
y, t=1+a sin 2πty exp-y2.
δ¯r=-12πexp-r2,
δrmsr=δ2¯=a22πexp-r2.

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