Abstract

Effects of diffraction on the performance of electronic heterodyne readout of moire fringes are investigated. The sensitivity, accuracy, and resolution of the system are calculated, and it is shown that these features are significantly improved compared with the conventional intensity moire readout technique. The sensitivity of the system can be tripled without changing the distance between gratings. The system was evaluated experimentally by measuring the refractive-index derivatives of a weak phase object consisting of a large KD*P crystal. Effects of nonlinear fringe modulation were studied both theoretically and experimentally. It is shown that in this case the electronic phase is not linearly related to the fringe shift, and calibration of the system is necessary.

© 1986 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);O. Kafri, A. Livnat, E. Keren, “Infinite Fringe Moire Deflectrometry,” Appl. Opt. 21, 3884 (1982);Z. Karny, O. Kafri, “Refractive-Index Measurements by Moire Deflectometry,” Appl. Opt. 21, 3326 (1982).
    [CrossRef] [PubMed]
  2. E. Bar-Zim, S. Sgulim, O. Kafri, E. Keren, “Temperature Mapping in Flames by Moire Deflectometry,” Appl. Opt. 22, 698 (1983).
    [CrossRef]
  3. J. Stricker, O. Kafri, “New Method for Density Gradient Measurements in Comprehensible Flows,” AIAA J. 20, 820 (1982);J. Stricker, E. Keren, O. Kafri, “Axisymmetric Density Field Measurements by Moire Deflectometry,” AIAA J. 21, 1767 (1983).
    [CrossRef]
  4. L. Horowitz, Y. B. Band, O. Kafri, D. F. Heller, “Thermal Lensing Analysis of Alexandrite Laser Rods by Moire Deflectometry,” Appl. Opt. 23, 2229 (1984).
    [CrossRef] [PubMed]
  5. O. Kafri, A. Livnat, “Reflective Surface Analysis Using Moire Deflectometry,” Appl. Opt. 20, 3098 (1981);“Second and Third Optical Differentiation by Double Moire Deflectometry,” Appl. Opt. 22, 2115 (1983).
    [CrossRef] [PubMed]
  6. E. Keren, O. Kafri, “Diffraction Effects in Moire Deflectometry,” J. Opt. Soc. Am. A 2, 111 (1985).
    [CrossRef]
  7. E. Bar-Ziv, “Effect of Diffraction on the Moire Images. I: Theory,” J. Opt. Soc. Am. A 2, 371 (1985).
    [CrossRef]
  8. E. Bar-Ziv, S. Sgulim, D. Manor, “Effect of Diffraction on the Moire Image. II: Experiment,” J. Opt. Soc. Am. A 2, 380 (1985).
    [CrossRef]
  9. E. Bar-Ziv, “Effect of Diffraction on the Moire Image for Temperature Mapping in Flames,” Appl. Opt. 23, 4040 (1984).
    [CrossRef] [PubMed]
  10. G. S. Rau, E. Bar-Ziv, “Deflection Mapping of Flames Using the Moire Effect,” Appl. Opt. 23, 2686 (1984).
    [CrossRef] [PubMed]
  11. J. Stricker, “Electronic Heterodyne Readout of Fringes in Moire Deflectometry,” Opt. Lett. 10, 247 (1985).
    [CrossRef] [PubMed]
  12. J. Stricker, “Moire Deflectometry with Deferred Electronic Heterodyne Readout,” Appl. Opt. 24, 2298 (1985).
    [CrossRef] [PubMed]
  13. E. R. Mansen, A Table of Series and Products (Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 449.
  14. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982).
  15. O. Kafri, E. Marglith, “Double Exposure Moire Deflectometry for Removing Noise,” Appl. Opt. 20, 2344 (1981).
    [CrossRef] [PubMed]
  16. D. Weimer, “Pockels-Effect Cell for Gas-Flow Simulation,” NASA Tech. Publ. 2007 (1982).
  17. A. J. Decker, J. Stricker, “Comparison of Electronic Heterodyne Moire Deflectometry and Electronic Heterodyne Holographic Interferometry for Flow Measurements,” SAE Tech. Paper Series, Paper No. 851896. SAE Aerospace Technology Conference and Exposition14–17 Oct. 1985, Long Beach Convention Center, CA.

1985 (5)

1984 (3)

1983 (1)

1982 (2)

J. Stricker, O. Kafri, “New Method for Density Gradient Measurements in Comprehensible Flows,” AIAA J. 20, 820 (1982);J. Stricker, E. Keren, O. Kafri, “Axisymmetric Density Field Measurements by Moire Deflectometry,” AIAA J. 21, 1767 (1983).
[CrossRef]

D. Weimer, “Pockels-Effect Cell for Gas-Flow Simulation,” NASA Tech. Publ. 2007 (1982).

1981 (2)

1980 (1)

Band, Y. B.

Bar-Zim, E.

Bar-Ziv, E.

Decker, A. J.

A. J. Decker, J. Stricker, “Comparison of Electronic Heterodyne Moire Deflectometry and Electronic Heterodyne Holographic Interferometry for Flow Measurements,” SAE Tech. Paper Series, Paper No. 851896. SAE Aerospace Technology Conference and Exposition14–17 Oct. 1985, Long Beach Convention Center, CA.

Heller, D. F.

Horowitz, L.

Kafri, O.

Keren, E.

Livnat, A.

Manor, D.

Mansen, E. R.

E. R. Mansen, A Table of Series and Products (Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 449.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982).

Marglith, E.

Rau, G. S.

Sgulim, S.

Stricker, J.

J. Stricker, “Moire Deflectometry with Deferred Electronic Heterodyne Readout,” Appl. Opt. 24, 2298 (1985).
[CrossRef] [PubMed]

J. Stricker, “Electronic Heterodyne Readout of Fringes in Moire Deflectometry,” Opt. Lett. 10, 247 (1985).
[CrossRef] [PubMed]

J. Stricker, O. Kafri, “New Method for Density Gradient Measurements in Comprehensible Flows,” AIAA J. 20, 820 (1982);J. Stricker, E. Keren, O. Kafri, “Axisymmetric Density Field Measurements by Moire Deflectometry,” AIAA J. 21, 1767 (1983).
[CrossRef]

A. J. Decker, J. Stricker, “Comparison of Electronic Heterodyne Moire Deflectometry and Electronic Heterodyne Holographic Interferometry for Flow Measurements,” SAE Tech. Paper Series, Paper No. 851896. SAE Aerospace Technology Conference and Exposition14–17 Oct. 1985, Long Beach Convention Center, CA.

Weimer, D.

D. Weimer, “Pockels-Effect Cell for Gas-Flow Simulation,” NASA Tech. Publ. 2007 (1982).

AIAA J. (1)

J. Stricker, O. Kafri, “New Method for Density Gradient Measurements in Comprehensible Flows,” AIAA J. 20, 820 (1982);J. Stricker, E. Keren, O. Kafri, “Axisymmetric Density Field Measurements by Moire Deflectometry,” AIAA J. 21, 1767 (1983).
[CrossRef]

Appl. Opt. (7)

J. Opt. Soc. Am. A (3)

NASA Tech. Publ. (1)

D. Weimer, “Pockels-Effect Cell for Gas-Flow Simulation,” NASA Tech. Publ. 2007 (1982).

Opt. Lett. (2)

Other (3)

A. J. Decker, J. Stricker, “Comparison of Electronic Heterodyne Moire Deflectometry and Electronic Heterodyne Holographic Interferometry for Flow Measurements,” SAE Tech. Paper Series, Paper No. 851896. SAE Aerospace Technology Conference and Exposition14–17 Oct. 1985, Long Beach Convention Center, CA.

E. R. Mansen, A Table of Series and Products (Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 449.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982).

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic of a moire deflectometer with electronic heterodyne readout. R.S., T.S., and G.S. are the reference, test, and gating signals, respectively. S is a mat screen, G1 and G2 are the Ronchi gratings. φ is the deflection angle, (b) Moire fringe pattern formed by two Ronchi gratings.

Fig. 2
Fig. 2

Block diagram of the phase meter.

Fig. 3
Fig. 3

Calculated σ, electronic phase readout, Δψ, the phase shift between test signal and reference signal, for A/p = 5. The four curves correspond to four different positions of the reference detector.

Fig. 4
Fig. 4

Spatial resolution determination.

Fig. 5
Fig. 5

Effect of δφ, refraction angle variation along x, on the spatial resolution.

Fig. 6
Fig. 6

Flow simulator—crystal of KD*P.

Fig. 7
Fig. 7

Calibration curves σ against test detector position. y-position of the reference detection is different in curves a and b.

Fig. 8
Fig. 8

x-derivative of refractive index for flow simulator.

Equations (31)

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p = p 2 sin ( θ / 2 ) p θ .
Δ = l p 2 λ l = 1 , 2 , 3 , .
φ x ( r ) = θ δ h y ( r ) Δ = δ h y ( r ) p p Δ .
φ x ( r ) = 1 n s z 0 Z f n ( x , y , z ) x d z ,
I ( y , Δ * ) = 1 4 + 2 π 2 n = 0 cos [ π ( 2 n + 1 ) 2 Δ * ] ( 2 n + 1 ) 2 × cos [ 2 π ( 2 n + 1 ) ( χ p + y θ p ) ] ,
ψ ( y , t ) = 2 π ( χ p + y θ p + V p t ) ,
I R ( y , Δ * , t ) = 1 4 + 2 π 2 n = 0 cos [ π ( 2 n + 1 ) 2 Δ * ] ( 2 n + 1 ) 2 × cos [ 2 π ( 2 n + 1 ) ( χ p + y θ p + Ω 2 π t ) ] ,
Ω = 2 π V p .
I R 1 ( y , Δ * , t ) = 1 4 + 2 π 2 cos ( π Δ * ) cos [ 2 π ( χ p + y θ p + Ω 2 π t ) ] ,
I T ( r , R , Δ * , t , φ x ) = [ R ( x ) Δ + R ( x ) ] × ( 1 4 + 2 π 2 n = 0 cos { π ( 2 n + 1 ) 2 Δ * ( R ( x ) Δ + R ( x ) ) } ( 2 n + 1 ) 2 × cos { 2 π ( 2 n + 1 ) × [ χ p + y θ p + φ x ( r ) Δ p + Ω 2 π t ] } ) ,
I T 1 ( r , R , Δ * , t , φ x ) = [ R ( x ) Δ + R ( x ) ] ( 1 4 + 2 π 2 × cos { π Δ * [ R ( x ) Δ + R ( x ) ] } cos { 2 π [ χ p + y θ p + φ x ( r ) Δ p + Ω 2 π t ] } ) .
Δ ψ ( r ) = 2 π φ x ( r ) Δ p .
Δ ψ 1 ( r ) = 2 π θ p ( y T y R ) ,
Δ ψ 2 ( r ) = 2 π θ p ( y T y R ) + 2 π φ x ( r ) Δ p ,
I R = C R cos [ 2 π ( ψ 0 + A p cos ω t ) ] ,
I T = C T cos [ 2 π ( ψ 0 + Δ ψ 2 π + A p cos ω t ) ] ,
ψ 0 = χ p + y R θ p , Δ ψ = 2 π ( y T y R p + φ x Δ p ) , C R = 2 π 2 cos ( π Δ * ) , C T = R Δ + R 2 π 2 cos [ π Δ * ( R Δ + R ) ] .
σ = tan 1 V B V A .
I A = C R C T 2 { cos ( Δ ψ ) + cos [ 2 π ( 2 ψ 0 + Δ ψ 2 π + 2 A p cos ω t ) ] } ,
I B = C R C T 2 { sin ( Δ ψ ) sin [ 2 π ( 2 ψ 0 + Δ ψ 2 π + 2 A p cos ω t ) ] } .
σ = tan 1 { sin ( Δ ψ ) J 0 ( 4 π A p ) sin [ 2 π ( 2 ψ 0 + Δ ψ 2 π ) ] cos ( Δ ψ ) + J 0 ( 4 π A p ) cos [ 2 π ( 2 ψ 0 + Δ ψ 2 π ) ] } .
σ = 3 Δ ψ
σ φ x = 3 ( 2 π Δ p ) ,
Δ x p = ρ * + 2 Δ * ,
Δ x Δ α λ 4 π .
( Δ α ) min = λ * 4 π ( ρ * + 2 Δ * ) ,
( Δ φ ) min = λ * 360 Δ * ,
Δ x Δ φ λ 360 ( 2 + ρ * Δ * ) λ 180 .
δ ( Δ x ) = ( φ x x Δ x ) Δ .
φ x x < 1 Δ ( ρ * + 2 Δ * ) .
2.3 × 10 3 ( φ / X ) max 2.1 × 10 2 mm 1 .

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