Abstract

Effects of aperture size and shape of the photodetector and effects of the structure of the grating lines on the performance of deferred electronic heterodyne moire deflectometry are theoretically investigated. Deferred deflectometry is used for measurements of nonsteady phase objects for which it is difficult to complete the analysis of the field in real time. It has been shown that scanning of a moire fringe pattern parallel to an unshifted fringe yields periodical variations in the heterodyne phase and amplitude, which cause severe errors in the measurements. Theory indicates that these variations may be minimized by using a detector with square aperture of size ρ/p = 1.0,2.0,3.0 … or a circular detector with size ρ/p = 1.25,2.25,3.25…. To minimize errors in deflection angle measurements, the fringe inclination due to the phase object should not exceed 15° for a square detector and 13° for a ρ/p = 3.25 circular detector. Ronchi gratings with structure factor 0.5 ≲ q ≲ 0.7 are recommended.

© 1989 Optical Society of America

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References

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  1. J. Strieker, “Electronic Heterodyne Readout of Fringes in Moire Deflectometry,” Opt. Lett. 10, 247–249 (1985).
    [CrossRef]
  2. J. Strieker, “Diffraction Effects and Special Advantages in Electronic Heterodyne Moire Deflectometry,” Appl. Opt. 25, 895–902 (1986).
    [CrossRef]
  3. A. J. Decker, J. Strieker, “A Comparison of Electronic Heterodyne Moire Deflectometry and Electronic Heterodyne Holographic Interferometry for Flow Measurements,” SAE Publication No. 851896, 14–17 Oct., 1985, Long Beach Convention Center, CA.
  4. J. Strieker, “Moire Deflectometry with Deferred Electronic Heterodyne Readout,” Appl. Opt. 24, 2298–2299 (1985).
    [CrossRef]
  5. J. Strieker, “Performance of Moire Deflectometry with Deferred Electronic Heterodyne Readout,” J. Opt. Soc. Am. A, 4, 1798–1806 (1987).
    [CrossRef]
  6. A. J. Decker, J. Strieker, D. Weimer, K. E. Weiland, “Sources of Error in Heterodyne Moire Deflectometry,” Appl. Opt. 27, 1381–1383 (1988).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. E. Bar-Ziv, “Effect of Diffraction on the Moire Image. I. Theory,” J. Opt. Soc. Am. A 2, 371–379 (1985).
    [CrossRef]
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    [CrossRef]

1988 (1)

1987 (1)

1986 (1)

1985 (4)

1980 (1)

Bar-Ziv, E.

Decker, A. J.

A. J. Decker, J. Strieker, D. Weimer, K. E. Weiland, “Sources of Error in Heterodyne Moire Deflectometry,” Appl. Opt. 27, 1381–1383 (1988).
[CrossRef] [PubMed]

A. J. Decker, J. Strieker, “A Comparison of Electronic Heterodyne Moire Deflectometry and Electronic Heterodyne Holographic Interferometry for Flow Measurements,” SAE Publication No. 851896, 14–17 Oct., 1985, Long Beach Convention Center, CA.

Kafri, O.

Keren, E.

Strieker, J.

Weiland, K. E.

Weimer, D.

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Figures (10)

Fig. 1
Fig. 1

Schematic of the setup for recording the phase object: ϕ is the refraction angle; R(x) is the distance from the phase object at which the refracted ray seems to emerge.

Fig. 2
Fig. 2

Ronchi grating transmission profile.

Fig. 3
Fig. 3

Calculated heterodyne phase vs. detector position parallel to an unshifted fringe, for two circular aperture sizes: ρ/p = 3.07 and ρ/p = 7.09. Deferred mode of operation with Agfa 10E75 plate with Ronchi grating factor q = 0.7.

Fig. 4
Fig. 4

As in Fig. 3 for a detector with size ρ/p = 3.07. The three curves are for q = 0.3, 0.5, and 0.7.

Fig. 5
Fig. 5

Signal amplitude vs. q for ρ/p = 1.0: (□—□—□) square detector; (○—○—○) circular detector.

Fig. 6
Fig. 6

Peak-to-peak phase variation (for a scan parallel to an unshifted fringe) vs. detector size for various grating structure factors q. Deferred mode with a circular detector and with Agfa 10E75 photographic plate.

Fig. 7
Fig. 7

Same as Fig. 6 for a square detector.

Fig. 8
Fig. 8

Signal amplitude vs. detector size: (○—○—○) circular detector; (□—□—□) square detector, q = 0.5. The amplitude shown is the maximum amplitude value over one grating pitch.

Fig. 9
Fig. 9

Relative amplitude variation (Amax-Amin)/Ā vs. detector size for various grating structure factors q. Amax, Amix, and Ā are the maximum, minimum, and average amplitudes over a distance of one grating pitch. Deferred mode with a circular detector and with Agfa 10E75 photographic plate.

Fig. 10
Fig. 10

Same as Fig. 9 for a square detector.

Tables (1)

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Table 1 Calculated δψ Various Fringe Inclination Angles and for Different Square and Circular Detector Sizesa,b

Equations (16)

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p = p 2 sin ( θ / 2 ) p θ
ϕ x ( r ) = θ δ h y ( r ) Δ = p δ h y ( r ) p Δ .
ϕ x ( r ) = 1 n f z o z f n ( r ) x d z
ϕ x ( r ) = G n f z o z f ρ ( r ) x d z ,
n 1 = G ρ ,
I ( y , Δ * , t ) = 2 π 2 cos ( π Δ * ) cos [ 2 π ( x p + y θ p + Ω 2 π t ) ] ,
Δ ψ ( r ) = 2 π ϕ x ( r ) Δ p .
G ( x ) = q + 2 π n = 0 sin ( n q π ) n cos ( 2 π n x / p ) ,
I = u { q 2 + 4 q π n = 0 sin ( n q π ) n cos ( π n 2 u Δ * ) cos ( 2 π n ζ u ) + 4 π 2 k , ι = 0 sin ( k q π ) k sin ( ι q π ) ι × cos ( 2 π k ζ u ) cos ( 2 π ι ζ u ) cos [ π u Δ * ( k 2 ι 2 ) ] } ,
| T | 2 = a 0 + a 1 I .
M ( r , Δ , R , ϕ ) = detector aperture | T | 2 G d ζ ,
G ( ζ , y ) = q + 2 π n = 0 sin ( n q π ) n cos [ 2 π n ( ζ + χ / p + y θ / p ) ]
χ = χ o + V t
S ( t ) = ( a 0 + a 1 q 2 u ) 2 α 1 π 3 θ sin ( π ρ ) sin ( π ρ θ ) cos [ 2 π ( χ o + x + y θ + Ω 2 π t ) ] + 4 a 1 q u π 4 θ α 1 sin ( π ρ θ ) k = 1 α k β k { sin [ π ( 1 k u ) ρ ] 1 k u cos [ 2 π ( 1 k u ) x + 2 π ( χ o + y θ + Ω 2 π t ) ] + sin [ π ( 1 + k u ) ρ ] 1 + k u cos [ 2 π ( 1 + k u ) x + 2 π ( χ o + y θ + Ω 2 π t ) ] } + 2 a 1 u π 5 θ α 1 sin ( π ρ θ ) × k = 1 ι = 1 α k α ι β k ι { sin [ π ( 1 j u ) ρ ] 1 j u cos [ 2 π ( 1 j u ) x + 2 π ( χ o + y θ + Ω 2 π t ) ] + sin [ π ( 1 + j u ) ρ 1 + j u cos [ 2 π ( 1 + j u ) x + 2 π ( χ o + y θ + Ω 2 π t ) ] + same two terms in which j + replaces j } ,
C ( t ) = ( a 0 + a 1 u q 2 ) 2 α 1 π 2 θ x ρ / 2 x + ρ / 2 sin ( 2 π θ ρ 2 ( x ζ ) 2 ) cos [ 2 π ( χ o + ζ + y θ + Ω 2 π t ) ] d ζ + 8 a 1 u q π 3 θ α 1 k = 1 α k β k x ρ / 2 x + ρ / 2 cos ( 2 π k ζ u ) sin ( 2 π θ ρ 2 ( x ζ ) 2 ) cos [ 2 π ( x o + ζ + y θ + Ω 2 π t ) ] d ζ + 8 a 1 u π 4 θ α 1 k = 1 ι = 1 α k α ι β k ι x ρ / 2 x + ρ / 2 cos ( 2 π k ζ u ) cos ( 2 π ι ζ u ) sin ( 2 π θ ρ 2 ( x ζ ) 2 ) × cos [ 2 π ( χ o + ζ + y θ + Ω 2 π t ) ] d ζ .
α = Δ R θ .

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