Abstract

Inspections of moiré fringe characteristics, such as period and orientation, conventionally are done by two approaches; namely, parametric equation and Fourier analysis methods. In some cases these methods yield different results. This inconsistency is removed by revising the derivation of the indicial equation for moiré fringes by the parametric equation method.

© 2007 Optical Society of America

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References

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  1. K. Patorski and M. Kujawinska, Handbook of the Moiré Fringe Technique (Elsevier, 1993).
  2. O. Kafri and I. Glatt, The Physics of Moiré Metrology (Wiley, 1989).
  3. W. Yu, D. Yun, and L. Wang, "Talbot and Fourier moire deflectometry and its applications in engineering," Opt. Lasers Eng. 25, 163-177 (1996).
    [CrossRef]
  4. R. Torroba and A. A. Tagliaferri, "Precision small angle measurements with a digital moire technique," Opt. Commun. 149, 213-216 (1998).
    [CrossRef]
  5. A. Asundi, "Moire interferometry for deformation measurement," Opt. Lasers Eng. 11, 281-292 (1989).
    [CrossRef]
  6. O. Kafri and I. Glatt, "Moire deflectometry: a ray deflection approach to optical testing," Opt. Eng. 24, 944-960 (1985).
  7. J. C. Bhattacharya and A. K. Aggrawal, "Measurement of the focal length of a collimating lens using the Talbot effect and the moire technique," Appl. Opt. 30, 4479-4480 (1991).
    [CrossRef] [PubMed]
  8. M. Tavassoly and M. Abolhassani, "Specification of spectral line shape and multiplex dispersion by self-imaging and moiré technique," Opt. Lasers Eng. 41, 743-753 (2004).
    [CrossRef]
  9. S. Yokozeki, Y. Kusaka, and K. Patorski, "Specification of spectral line shape and multiplex dispersion by self-imaging and moiré technique," Appl. Opt. 15, 2223-2227 (1976).
    [CrossRef] [PubMed]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), p. 8.

2004 (1)

M. Tavassoly and M. Abolhassani, "Specification of spectral line shape and multiplex dispersion by self-imaging and moiré technique," Opt. Lasers Eng. 41, 743-753 (2004).
[CrossRef]

1998 (1)

R. Torroba and A. A. Tagliaferri, "Precision small angle measurements with a digital moire technique," Opt. Commun. 149, 213-216 (1998).
[CrossRef]

1996 (1)

W. Yu, D. Yun, and L. Wang, "Talbot and Fourier moire deflectometry and its applications in engineering," Opt. Lasers Eng. 25, 163-177 (1996).
[CrossRef]

1991 (1)

1989 (1)

A. Asundi, "Moire interferometry for deformation measurement," Opt. Lasers Eng. 11, 281-292 (1989).
[CrossRef]

1985 (1)

O. Kafri and I. Glatt, "Moire deflectometry: a ray deflection approach to optical testing," Opt. Eng. 24, 944-960 (1985).

1976 (1)

Appl. Opt. (2)

Opt. Commun. (1)

R. Torroba and A. A. Tagliaferri, "Precision small angle measurements with a digital moire technique," Opt. Commun. 149, 213-216 (1998).
[CrossRef]

Opt. Eng. (1)

O. Kafri and I. Glatt, "Moire deflectometry: a ray deflection approach to optical testing," Opt. Eng. 24, 944-960 (1985).

Opt. Lasers Eng. (3)

W. Yu, D. Yun, and L. Wang, "Talbot and Fourier moire deflectometry and its applications in engineering," Opt. Lasers Eng. 25, 163-177 (1996).
[CrossRef]

M. Tavassoly and M. Abolhassani, "Specification of spectral line shape and multiplex dispersion by self-imaging and moiré technique," Opt. Lasers Eng. 41, 743-753 (2004).
[CrossRef]

A. Asundi, "Moire interferometry for deformation measurement," Opt. Lasers Eng. 11, 281-292 (1989).
[CrossRef]

Other (3)

K. Patorski and M. Kujawinska, Handbook of the Moiré Fringe Technique (Elsevier, 1993).

O. Kafri and I. Glatt, The Physics of Moiré Metrology (Wiley, 1989).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), p. 8.

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Figures (2)

Fig. 1
Fig. 1

Orientation of the two straight-line gratings in the x y plane. Lines of first grating are parallel to and those of the second make a small angle θ with the y axis. They have periods d 1 and d 2 , and are indicated by h and k parameters, respectively.

Fig. 2
Fig. 2

Two families of infinite ( M , N ) lines families in the lattice point: (a) ( 1 , 1 ) line family and (b) ( 2 , 1 ) line family. The value of index p in each line family has been calculated by Eq. (13) for corresponding values of M and N .

Equations (16)

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d 1 M d 2 ; M Z .
T 1 ( x , y ) = m = a m exp ( i m 2 π d 1 x ) ,
T 2 ( x , y ) = n = b n exp [ i n 2 π d 2 ( x cos θ y sin θ ) ] ,
T M = T 1 ( x , y ) T 2 ( x , y )
= m = n = a m b n exp { i 2 π [ ( m d 1 + n cos θ d 2 ) x n sin θ d 2 y ] } .
f = ( f X 2 + f Y 2 ) 1 / 2 .
f m n = ( m 2 d 1 2 + n 2 d 2 2 + 2 m n cos θ d 1 d 2 ) 1 / 2 .
d M = 1 f m 0 n 0 = d 1 d 2 n 0 2 d 1 2 + m 0 2 d 2 2 + 2 m 0 n 0 d 1 d 2 cos θ .
d M = d 1 d 2 d 1 2 + M 2 d 2 2 2 M d 1 d 2 cos θ .
x d 1 = h ; h Z ,
x cos θ y sin θ d 2 = k ; k Z .
h k = p ; p Z .
d 1 d 2 d 1 2 + d 2 2 2 d 1 d 2 cos θ ,
M h + N k = p ; p Z ,
N x cos θ y sin θ d 2 + M x d 1 = p ; p Z .
d 1 d 2 N 2 d 1 2 + M 2 d 2 2 + 2 M N d 1 d 2 cos θ .

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