Abstract

A new quantitative method for mapping of the second derivatives of a specular surface and of the refractive index of phase objects is proposed and demonstrated. Since the method is a finite fringe technique it is possible to obtain a third derivative contour map using a shearing moire technique.

© 1983 Optical Society of America

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References

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  1. O. KafriOpt. Lett. 5, 555 (1980); Phys. Bull. 33, 197 (1982).
    [CrossRef] [PubMed]
  2. O. Kafri, A. Livnat, Appl. Opt. 20, 3098 (1981).
    [CrossRef] [PubMed]
  3. O. Kafri, A. Livnat, E. Keren, Appl. Opt. 22, 650 (1983).
    [CrossRef]
  4. O. Kafri, E. Margalit, Appl. Opt. 20, 2344 (1981).
    [CrossRef] [PubMed]
  5. O. Kafri, A. Livnat, E. Keren, Appl. Opt. 21, 3884 (1982).
    [CrossRef] [PubMed]
  6. A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).
  7. E. Keren, O. Kafri, “Geometrical Approach for Mapping of Derivatives,” Comput. Phys. Commun., xx, 000 (1983). (in press)
  8. A. Assa, J. Politch, A. A. Betser, Exp. Mech. 19, 129 (1979) and references therein.
    [CrossRef]
  9. F. P. Chiang, M. Bailangadi, J. Appl. Mech. 42, 29 (1975) and references therein.
    [CrossRef]

1983 (1)

1982 (1)

1981 (2)

1980 (1)

1979 (1)

A. Assa, J. Politch, A. A. Betser, Exp. Mech. 19, 129 (1979) and references therein.
[CrossRef]

1975 (1)

F. P. Chiang, M. Bailangadi, J. Appl. Mech. 42, 29 (1975) and references therein.
[CrossRef]

Assa, A.

A. Assa, J. Politch, A. A. Betser, Exp. Mech. 19, 129 (1979) and references therein.
[CrossRef]

Bailangadi, M.

F. P. Chiang, M. Bailangadi, J. Appl. Mech. 42, 29 (1975) and references therein.
[CrossRef]

Betser, A. A.

A. Assa, J. Politch, A. A. Betser, Exp. Mech. 19, 129 (1979) and references therein.
[CrossRef]

Chiang, F. P.

F. P. Chiang, M. Bailangadi, J. Appl. Mech. 42, 29 (1975) and references therein.
[CrossRef]

Durelli, A. J.

A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

Kafri, O.

Keren, E.

O. Kafri, A. Livnat, E. Keren, Appl. Opt. 22, 650 (1983).
[CrossRef]

O. Kafri, A. Livnat, E. Keren, Appl. Opt. 21, 3884 (1982).
[CrossRef] [PubMed]

E. Keren, O. Kafri, “Geometrical Approach for Mapping of Derivatives,” Comput. Phys. Commun., xx, 000 (1983). (in press)

Livnat, A.

Margalit, E.

Parks, V. J.

A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

Politch, J.

A. Assa, J. Politch, A. A. Betser, Exp. Mech. 19, 129 (1979) and references therein.
[CrossRef]

Appl. Opt. (4)

Exp. Mech. (1)

A. Assa, J. Politch, A. A. Betser, Exp. Mech. 19, 129 (1979) and references therein.
[CrossRef]

J. Appl. Mech. (1)

F. P. Chiang, M. Bailangadi, J. Appl. Mech. 42, 29 (1975) and references therein.
[CrossRef]

Opt. Lett. (1)

Other (2)

A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

E. Keren, O. Kafri, “Geometrical Approach for Mapping of Derivatives,” Comput. Phys. Commun., xx, 000 (1983). (in press)

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic description of moire deflectometer: G1 and G2 are two transmission gratings. (b) Setup for double deflectometry. The distorted shadow is projected on a photographic plate; then the object is shifted in the y direction, the grating is rotated by an angle θ in the y-z plane, and a second exposure is taken.

Fig. 2
Fig. 2

(a) Interference of an undistorted grating with a distorted grating. This is regular deflectometry. The fringe distortion is proportional to the first height derivative. (b) Double deflectogram without shift. The lines are straight. (c) Double deflectogram without rotation. This is shearing moire deflectometry, and the contours are lines of equi second derivatives. (d) Double deflectogram. The fringe distortion is proportional to the second height derivative.

Fig. 3
Fig. 3

By shifting two identical double deflectograms (shearing double deflectometry) a contour map of equi third height derivative is obtained.

Equations (14)

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y cos ( θ / 2 ) = z sin ( θ / 2 ) + k p , for G 1 , y cos ( θ / 2 ) = - z sin ( θ / 2 ) + m p , for G 2 , k , m = 0 , ± 1 , ± 2 , ,
Z = ( k - m ) p 2 sin ( θ / 2 ) l p θ ,
[ y + ϕ ( z , y ) Δ ] cos ( θ / 2 ) = z sin ( θ / 2 ) + k p , y cos ( θ / 2 ) = - z sin ( θ / 2 ) + m p .
ϕ ( z , y ) Δ z θ + l p .
[ y + ϕ ( z , y ) Δ ] cos ( θ / 2 ) = z sin ( θ / 2 ) + k p , [ y + ϕ ( z , y ) Δ ] cos ( θ / 2 ) = - z sin ( θ / 2 ) + m p .
[ y + ϕ ( z , y ) Δ ] cos ( θ / 2 ) = z sin ( θ / 2 ) + k p , [ y + δ y + ϕ ( z , y + δ y ) Δ ] cos ( θ / 2 ) = - z sin ( θ / 2 ) + m p .
ϕ ( z , y ) - ϕ ( z , y + δ y ) δ y = z θ Δ δ y ϕ ( z ) y .
ϕ y 1 n f 2 n y 2 d x .
2 h y 2 = z θ 2 Δ δ y .
( 2 h y 2 ) incr = p 2 Δ δ y .
z = G ( y , z ) + k p
G y G ( y , z ) - G ( y - δ y , z ) d y = l p d y .
z y = 2 Δ δ y θ 3 h y 3 = l p d y ,
( 3 h y 3 ) incr = p 2 Δ δ y d y .

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