Abstract

Theoretical expressions for the modulation depth of moiré signals under incoherent illumination are derived. Consequently, the modulation depth of a moiré signal is related to the following factors: the geometric shape and the size of the light source; the diffraction effect of the grating, which relates to the number of lines in the grating; the line and the space ratio; the grating pair gap; the geometric shape and the size of the receiving window; etc. In addition, the influence of the grating pair on the period and the inclination of moiré fringes under noncollimated illumination are discussed, and the changes in the moiré signal modulation depth under noncollimated illumination with that under collimated illumination are made. Finally, some experimental results are given to verify the theoretical expressions. This research is useful for the actual design of grating sensors.

© 1998 Optical Society of America

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References

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  1. Y. Nakano, K. Murata, “Talbot interferometry for measuring the small tilt angle variation of an object surface,” Appl. Opt. 25, 2475–2477 (1986).
    [CrossRef] [PubMed]
  2. E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, “Temperature mapping in flames by moiré deflectometry,” Appl. Opt. 22, 698–705 (1983).
    [CrossRef] [PubMed]
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1986 (1)

1983 (1)

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Figures (9)

Fig. 1
Fig. 1

Noncollimated illumination.

Fig. 2
Fig. 2

Influence of gap space on the moiré fringe period under noncollimated illumination: (1) ϖ = 0.16 mm, (2) ϖ = 0.125 mm, (3) ϖ = 0.04 mm, (4) ϖ = 0.02 mm.

Fig. 3
Fig. 3

Influence of gap space on the inclination of the moiré fringes under noncollimated illumination: (1) ϖ = 0.16 mm,(2) ϖ = 0.125 mm, (3) ϖ = 0.04 mm, (4) ϖ = 0.02 mm.

Fig. 4
Fig. 4

Curves of the modulation depth of the moiré signal with the gap space under noncollimated and collimated illumination: I, II, under collimated light; I′, II′, under noncollimated light; I, I′, ϖ = 0.16 mm; II, II′, ϖ = 0.04 mm.

Fig. 5
Fig. 5

Schematic of the experimental setup: 1, 2, micropositioners; 3, photodetector; S, regulated power supply; V, digital voltmeter.

Fig. 6
Fig. 6

Gratings (a) of ϖ = 0.125 mm illuminated by HG-413, (b) of ϖ = 0.04 mm illuminated by HG-413, (c) of ϖ = 0.125 mm illuminated by HG-504, (d) of ϖ = 0.04 mm illuminated by HG-504.

Fig. 7
Fig. 7

Schematic diagram of the experimental setup.

Fig. 8
Fig. 8

Change in the period and the inclination of moiré fringes under noncollimated illumination as R 2 is changed.

Fig. 9
Fig. 9

(a) Curve of the period and the R 2 of moiré fringes: ——, theoretical value; ○, experimental value. (b) Curve of the inclination and the R2 of moiré fringes: —·—·, theoretical value; △, experimental value.

Tables (1)

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Table 1 Theoretical and Experimental Values of ξ and Δφ for Different Values of R2

Equations (37)

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I 21 X 2 ,   Y 2 = - +   I 0 x ˜ 0 ,   y ˜ 0   | h ˜ x 2 - x ˜ 0 ,   y 2 - y ˜ 0 | 2 d x ˜ 0 d y ˜ 0 ,
x ˜ 0 = - R 2 R 1   x 0 ,   y ˜ 0 = - R 2 R 1   y 0 .
h ˜ x 2 - x ˜ 0 ,   y 2 - y ˜ 0 = F G 1 x 1 ,   y 1 × exp ikM 2 R 2 x 1 2 + y 1 2 .
U x = x 2 - x ˜ 0 / λ R 2 = x 2 / λ R 2 + x 0 / λ R 1 ,
U y = y 2 - y ˜ 0 / λ R 2 = y 2 / λ R 2 + y 0 / λ R 1 .
G 1 x 1 = n = - +   C n exp i 2 n π x 1 / ϖ 1 ,
I 21 r = I 0 n = - + n = - +   C n C n exp - iR 2 λ π M ϖ 1 2 n 2 - n 2 × exp i   2 π n - n x 2 M ϖ 1 × sinc sR 2 n - n MR 1 ϖ 1 sinc tR 2 n - n M ϖ 1 R 1 ,
I 21 c = I 0 n = - + n = - +   C n C n exp - iR 2 λ π M ϖ 1 2 n 2 - n 2 × exp i   2 π n - n x 2 M ϖ 1 2   J 1 n - n Y n - n Y ,
G 2 x 2 ,   y 2 = d = - +   B d exp i 2 π d ϖ 2 x 2 cos   θ 2 + y 2 sin   θ 2 .
I 22 r = I 21 r x 2 ,   y 2 G 2 x 2 ,   y 2 = I 0 n = - + n = - + d = - + sinc sR 2 n - n MR 1 ϖ 1 × sinc tR 2 n - n M ϖ 1 R 1 exp - iR 2 λ π M ϖ 1 2 n 2 - n 2 × C n C n B l exp i 2 π n - n cos   θ 1 M ϖ 1 - cos   θ 2 ϖ 2 x 2 + sin   θ 1 M ϖ 1 - sin   θ 2 ϖ 2 y 2 ,
I 2 nr = C 0 2 B 0 + 2 C 0 n = 1   C n B - n sinc snR 2 R 1 + R 2 ϖ 1 × sinc tnR 2 R 1 + R 2 ϖ 1 cos λ π R 1 R 2 n 2 R 1 + R 2 ϖ 1 2 × exp i 2 π n cos   θ 1 M ϖ 1 - cos   θ 2 ϖ 2 x 2 + sin   θ 2 M ϖ 1 - sin   θ 2 ϖ 2 y 2 .
M nr = 2   n = 1 cos λ π R 1 R 2 n 2 R 1 + R 2 ϖ 1 2 sinc nR 2 s R 1 + R 2 ϖ 1 × sinc nR 2 t R 1 + R 2 ϖ 1 sinc n α sinc n β .
M nrr = 2   n = 1 cos λ π R 1 R 2 n 2 R 1 + R 2 ϖ 1 2 sinc nR 2 s R 1 + R 2 ϖ 1 × sinc nR 2 t R 1 + R 2 ϖ 1 sinc n α sinc n β × sinc n γ sinc n δ ,
M nrc = 2   n = 1 cos λ π R 1 R 2 n 2 R 1 + R 2 ϖ 1 2 × sinc nR 2 s R 1 + R 2 ϖ 1 sinc nR 2 t R 1 + R 2 ϖ 1 × sinc n α sinc n β 2   J 1 2 n π r 1 / p 2 n π r 1 / p .
M ncr = 2   n = 1 cos λ π R 1 R 2 n 2 R 1 + R 2 ϖ 1 2 2   J 1 nY nY × sinc n α sinc n β sinc n γ sinc n δ ,
M ncc = 2   n = 1 cos λ π R 1 R 2 n 2 R 1 + R 2 ϖ 1 2 2   J 1 nY nY × sinc n α sinc n β 2   J 1 2 n π r 1 / p 2 n π r 1 / p .
I 2 r = C 0 2 B 0 + 2 C 0 n = 1   C n B - n sinc nsR 2 ϖ 1 f sinc ntR 2 ϖ 1 f × cos λ π R 2 n 2 ϖ 1 2 exp i 2 π n cos   θ 1 ϖ 1 - cos   θ 2 ϖ 2 x 2 + sin   θ 1 ϖ 1 - sin   θ 2 ϖ 2 y 2 .
M nrr = 2   n = 1 cos λ π R 2 n 2 ϖ 1 2 sinc nR 2 s ϖ 1 f sinc nR 2 t ϖ 1 f × sinc n α sinc n β sinc n γ sinc n δ ,
M nrc = 2   n = 1 cos λ π R 2 n 2 ϖ 1 2 sinc nR 2 s ϖ 1 f sinc nR 2 t ϖ 1 f × sinc n α sinc n β 2 J 1 2 nr 1 π p / 2 nr 1 π p ,
M ncr = 2   n = 1 cos λ π R 1 R 2 n 2 ϖ 1 2 2   J 1 nX nX × sinc n α sinc n β sinc n γ sinc n δ ,
M ncc = 2   n = 1 cos λ π R 2 n 2 ϖ 1 2 2   J 1 nX nX × sinc n α sinc n β 2   J 1 2 n π r 1 / p 2 n π r 1 / p ,
X = 2 2 π R 2 r ϖ 1 f .
sinc nR 2 s ϖ 1 f sinc nR 2 t ϖ 1 f
2   J 1 nX nX
2   J 1 2 n π r 1 / p 2 n π r 1 / p
M = 2   n = 1 sin   c n α sin   c n β sin   c n γ sin   c n δ .
F = R 1 cos   θ 1 R 1 + R 2 ϖ 1 - cos   θ 2 ϖ 2 2 + R 1 sin   θ 1 R 1 + R 2 ϖ 1 - sin   θ 2 ϖ 2 2 1 / 2 .
P = ϖ 2   R 1 R 1 + R 2 1 - cos θ 1 - θ 2 + R 2 R 1 + R 2 2 - 1 / 2 .
P 0 = ω 2 1 - cos θ 1 - θ 2 - 1 / 2 ,
ξ = 2 1 - cos θ 1 - θ 2 1 / 2 2   R 1 R 1 + R 2 × 1 - cos θ 1 - θ 2 + R 2 R 1 + R 2 2 - 1 / 2 .
ξ = P / P 0 = θ θ 2 + R 2 / R 1 2 - 1 / 2 .
φ = arctan R 1 sin   θ 1 - R 1 + R 2 sin   θ 2 R 1 cos   θ 1 - R 1 + R 2 cos   θ 2 .
Δ φ = φ - φ 0 arctan R 2 / R 1 tan - 1 θ / 2 2 + R 2 / R 1 ,
M 1 rr = 2   cos λ π R 1 R 2 R 1 + R 2 ϖ 1 2 sinc R 2 s R 1 + R 2 ϖ 1 × sinc α sinc β sinc γ ,
M 1 rr = 2   cos λ π R 2 ϖ 1 2 sinc R 2 s ϖ 1 f × sinc α sinc β sinc γ .
γ = h / P = h P θ 2 + R 2 / R 1 2 1 / 2 θ , γ =   h / P 0 .
M = I max - I min I max + I min

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