Abstract

Quantitative analysis of 3-D phase objects by moire deflectometry is suggested. The method is based on measuring the deflection of rays from a collimated light beam due to gradients in the refractive index. To analyze asymmetric density fields it is necessary to obtain data of deflections at sufficient angular viewing angles about the test section and to mathematically invert the data. The moire deflectometry inversion procedure is shown to be simpler than interferometry inversion since no numerical differentiation of the measured data has to be performed. The method is demonstrated by mapping a temperature field generated above the top of two heated cylinders.

© 1984 Optical Society of America

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References

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  1. O. Kafri, “Noncoherent Method for Mapping Phase Objects,” Opt. Lett. 5, 555 (1980).
    [Crossref] [PubMed]
  2. J. Stricker, O. Kafri, “New Method for Density Gradient Measurements in Compressible Flows,” AIAA J. 20, 820 (1982).
    [Crossref]
  3. E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, “Measurements of Temperature Distribution of Flames by Moire Deflectometry,” Appl. Opt. 20, 4263 (1981).
    [Crossref] [PubMed]
  4. J. Stricker, E. Keren, O. Kafri, “Axisymmetric Density Field Measurements by Moire Deflectometry,” AIAA J. 21, 1767 (1983).
    [Crossref]
  5. M. V. Berry, D. F. Gibbs, “Interpretation of Optical Projections,” Proc. R. Soc. London Ser. A 314, 143 (1970).
    [Crossref]
  6. C. M. Vest, “Formation of Images from Projections: Radon and Abel Transformers,” J. Opt. Soc. Am. 64, 1215 (1974).
    [Crossref]
  7. R. C. Weast, Ed. CRC Handbook of Chemistry and Physics (CRC Press, Ohio, 1973).

1983 (1)

J. Stricker, E. Keren, O. Kafri, “Axisymmetric Density Field Measurements by Moire Deflectometry,” AIAA J. 21, 1767 (1983).
[Crossref]

1982 (1)

J. Stricker, O. Kafri, “New Method for Density Gradient Measurements in Compressible Flows,” AIAA J. 20, 820 (1982).
[Crossref]

1981 (1)

1980 (1)

1974 (1)

1970 (1)

M. V. Berry, D. F. Gibbs, “Interpretation of Optical Projections,” Proc. R. Soc. London Ser. A 314, 143 (1970).
[Crossref]

Bar-Ziv, E.

Berry, M. V.

M. V. Berry, D. F. Gibbs, “Interpretation of Optical Projections,” Proc. R. Soc. London Ser. A 314, 143 (1970).
[Crossref]

Gibbs, D. F.

M. V. Berry, D. F. Gibbs, “Interpretation of Optical Projections,” Proc. R. Soc. London Ser. A 314, 143 (1970).
[Crossref]

Glatt, I.

Kafri, O.

J. Stricker, E. Keren, O. Kafri, “Axisymmetric Density Field Measurements by Moire Deflectometry,” AIAA J. 21, 1767 (1983).
[Crossref]

J. Stricker, O. Kafri, “New Method for Density Gradient Measurements in Compressible Flows,” AIAA J. 20, 820 (1982).
[Crossref]

E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, “Measurements of Temperature Distribution of Flames by Moire Deflectometry,” Appl. Opt. 20, 4263 (1981).
[Crossref] [PubMed]

O. Kafri, “Noncoherent Method for Mapping Phase Objects,” Opt. Lett. 5, 555 (1980).
[Crossref] [PubMed]

Keren, E.

J. Stricker, E. Keren, O. Kafri, “Axisymmetric Density Field Measurements by Moire Deflectometry,” AIAA J. 21, 1767 (1983).
[Crossref]

E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, “Measurements of Temperature Distribution of Flames by Moire Deflectometry,” Appl. Opt. 20, 4263 (1981).
[Crossref] [PubMed]

Stricker, J.

J. Stricker, E. Keren, O. Kafri, “Axisymmetric Density Field Measurements by Moire Deflectometry,” AIAA J. 21, 1767 (1983).
[Crossref]

J. Stricker, O. Kafri, “New Method for Density Gradient Measurements in Compressible Flows,” AIAA J. 20, 820 (1982).
[Crossref]

Vest, C. M.

AIAA J. (2)

J. Stricker, O. Kafri, “New Method for Density Gradient Measurements in Compressible Flows,” AIAA J. 20, 820 (1982).
[Crossref]

J. Stricker, E. Keren, O. Kafri, “Axisymmetric Density Field Measurements by Moire Deflectometry,” AIAA J. 21, 1767 (1983).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Proc. R. Soc. London Ser. A (1)

M. V. Berry, D. F. Gibbs, “Interpretation of Optical Projections,” Proc. R. Soc. London Ser. A 314, 143 (1970).
[Crossref]

Other (1)

R. C. Weast, Ed. CRC Handbook of Chemistry and Physics (CRC Press, Ohio, 1973).

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Figures (4)

Fig. 1
Fig. 1

Schematics of the experimental setup and the coordinate system. O.R. is a typical optical ray passing through the phase object P.O. and deflected by an angle φ. θ is 90° minus the viewing angle. G1 and G2 are Ronchi rulings and S is a transparent mat screen.

Fig. 2
Fig. 2

Deflectograms of the temperature field generated by two heated cylinders shown at three viewing angles: (a) 0°, (b) 30°, and (c) 90°.

Fig. 3
Fig. 3

Temperature variation vs r for different viewing angles obtained from solving Eq. (5).

Fig. 4
Fig. 4

Temperature contour map reduced from Fig. 3.

Equations (7)

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p = p 2 sin ( α / 2 ) .
φ 1 n - [ n ( x , y , z ) y ] d x ,
N ( r , ψ ) = n ( r , ψ ) - n .
F ( y , θ ) = - - N ( r , ψ ) δ [ y - r sin ( ψ - θ ) ] d x d y ,
N ( r , ψ ) = 1 2 π 2 - π / 2 π / 2 d θ - ( F / y ) d y r sin ( ψ - θ ) - y .
F y = y [ n ( x , y , z ) - n ] d x = [ n ( x , y , z ) - n ] y d x ,
φ = α h / Δ ,

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