Abstract

Remote projection and viewing of moiré contours are examined analytically for a system employing separate projection and viewing optics, with specific attention paid to the practical limitations imposed by the optical systems. It is found that planar contours are possible only when the optics are telecentric (exit pupil at infinity) but that the requirement for spatial separability of the contour fringes from extraneous fringes is independent of the specific optics and is a function only of the angle separating the two optic axes. In the nontelecentric case, the contour separation near the object is unchanged from that of the telecentric case, although the contours are distorted into loweccentricity (near-circular) ellipses. Furthermore, the minimum contour spacing is directly related to the depth of focus through the resolution of the optics.

© 1983 Optical Society of America

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References

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  1. Lord Rayleigh, “On the manufacture and theory of diffractiongratings,” Philos. Mag. 47, 81–93, 193–205 (1874).
  2. T. Merton, “Nouvelles méthodes de fabrication des réseaux,” J. Phys. Radium,  13, 49–53 (1952).
    [CrossRef]
  3. D. M. Meadows, W. O. Johnson, and J. B. Allen, “Generation of surface contours by moiré patterns,” Appl. Opt. 9, 942–947 (1970).
    [CrossRef] [PubMed]
  4. M. Idesawa, T. Yatagai, and T. Soma, “Scanning moiré method and automatic measurement of 3-D shapes,” Appl. Opt. 16, 2152–2162 (1977).
    [CrossRef] [PubMed]
  5. L. Pirodda, “Shadow and projection moiré techniques for absolute or relative mapping of surface shapes,” Opt. Eng. 21, 640–649 (1982).
    [CrossRef]
  6. T. Yoshizawa and H. Tashiro, “Localization of fringes in moiré topography,” Opt. Lasers Eng. 3, 29–43 (1982).
    [CrossRef]
  7. M. Takeda, “Fringe formula for projection type moiré topography,” Opt. Lasers Eng. 3, 45–52 (1982).
    [CrossRef]

1982 (3)

L. Pirodda, “Shadow and projection moiré techniques for absolute or relative mapping of surface shapes,” Opt. Eng. 21, 640–649 (1982).
[CrossRef]

T. Yoshizawa and H. Tashiro, “Localization of fringes in moiré topography,” Opt. Lasers Eng. 3, 29–43 (1982).
[CrossRef]

M. Takeda, “Fringe formula for projection type moiré topography,” Opt. Lasers Eng. 3, 45–52 (1982).
[CrossRef]

1977 (1)

1970 (1)

1952 (1)

T. Merton, “Nouvelles méthodes de fabrication des réseaux,” J. Phys. Radium,  13, 49–53 (1952).
[CrossRef]

1874 (1)

Lord Rayleigh, “On the manufacture and theory of diffractiongratings,” Philos. Mag. 47, 81–93, 193–205 (1874).

Allen, J. B.

Idesawa, M.

Johnson, W. O.

Meadows, D. M.

Merton, T.

T. Merton, “Nouvelles méthodes de fabrication des réseaux,” J. Phys. Radium,  13, 49–53 (1952).
[CrossRef]

Pirodda, L.

L. Pirodda, “Shadow and projection moiré techniques for absolute or relative mapping of surface shapes,” Opt. Eng. 21, 640–649 (1982).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “On the manufacture and theory of diffractiongratings,” Philos. Mag. 47, 81–93, 193–205 (1874).

Soma, T.

Takeda, M.

M. Takeda, “Fringe formula for projection type moiré topography,” Opt. Lasers Eng. 3, 45–52 (1982).
[CrossRef]

Tashiro, H.

T. Yoshizawa and H. Tashiro, “Localization of fringes in moiré topography,” Opt. Lasers Eng. 3, 29–43 (1982).
[CrossRef]

Yatagai, T.

Yoshizawa, T.

T. Yoshizawa and H. Tashiro, “Localization of fringes in moiré topography,” Opt. Lasers Eng. 3, 29–43 (1982).
[CrossRef]

Appl. Opt. (2)

J. Phys. Radium (1)

T. Merton, “Nouvelles méthodes de fabrication des réseaux,” J. Phys. Radium,  13, 49–53 (1952).
[CrossRef]

Opt. Eng. (1)

L. Pirodda, “Shadow and projection moiré techniques for absolute or relative mapping of surface shapes,” Opt. Eng. 21, 640–649 (1982).
[CrossRef]

Opt. Lasers Eng. (2)

T. Yoshizawa and H. Tashiro, “Localization of fringes in moiré topography,” Opt. Lasers Eng. 3, 29–43 (1982).
[CrossRef]

M. Takeda, “Fringe formula for projection type moiré topography,” Opt. Lasers Eng. 3, 45–52 (1982).
[CrossRef]

Philos. Mag. (1)

Lord Rayleigh, “On the manufacture and theory of diffractiongratings,” Philos. Mag. 47, 81–93, 193–205 (1874).

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Figures (9)

Fig. 1
Fig. 1

Optical system for projection moiré contouring

Fig. 2
Fig. 2

Optics of fringe projection and viewing.

Fig. 3
Fig. 3

Geometric relationship between the model coordinate space (x, y, z) and the image coordinate spaces of the projection (a′, b′, c′) and observation (α′, β′, γ′) arms.

Fig. 4
Fig. 4

Fringes cast by the projection arm.

Fig. 5
Fig. 5

Planar sum and difference fringes of the telecentric configuration.

Fig. 6
Fig. 6

Locations of the hyperbolic sum fringes with varying fringe order (N).

Fig. 7
Fig. 7

Approximation to the hyperbolic sum fringes in the immediate neighborhood of the model.

Fig. 8
Fig. 8

Variation of the major and minor axes of the elliptical difference fringes as functions of the fringe-order parameter μ.

Fig. 9
Fig. 9

Location of the elliptical difference fringes in the neighborhood of the model.

Equations (66)

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Δ z = p tan θ + tan ϕ ,
I p ( a , b , c ) = I s T p ( a , b , c ) ,
I p ( x , y , z ) = I s T p ( x , y , z ) ,
I p ( a , b , c ) I p ( x , y , z )
R ( x , y , z ) = S ( x , y , z ) I p ( x , y , z ) ,
R ( x , y , z ) R ( α , β , γ )
I o ( α , β , γ ) = T o ( α , β , γ ) R ( α , β , γ ) .
R ( α , β , γ ) = S ( α , β , γ ) I P ( α , β , γ ) .
I o ( α , β , γ ) = I s S ( α , β , γ ) T o ( α , β , γ ) T p ( α , β , γ ) .
T o = T p = 1 .
T ( α , β , γ ) = T o ( α , β , γ ) T p ( α , β , γ ) .
I o ( x , y , z ) = I s S ( x , y , z ) T ( x , y , z ) ,
T ( x , y , z ) = T o ( x , y , z ) T p ( x , y , z ) .
1 L + 1 L = 1 F ,
m = L / L .
P = m p .
1 L ap 1 L ex = 1 F ,
T p ( a , b , c ) = 1 2 [ 1 + cos ( 2 π b p ) ] ,
T p ( a , b , c ) = 1 2 { 1 + cos [ 2 π p L p ( a , b ) ] } , | a | ,
L p ( a , b ) = ( L + L ex ) b L + L ex + a
D s = 2.44 λ ( f # ) ,
= 2.44 λ ( f # ) 2 ,
p D s
a = z cos θ y sin θ , b = + y cos θ z sin θ .
T p ( x , y , z ) = 1 2 { 1 + cos [ 2 π p L p ( y , z ) ] } ,
L p ( y , z ) = ( L + L e x ) ( y cos θ z sin θ ) L + L e x z cos θ y sin θ .
| a | .
T p ( x , y , z ) = 1 2 { 1 + cos [ 2 π p L p ( y , z ) ] } ,
L p ( y , z ) = d p ( y cos θ z sin θ ) d p z cos θ y sin θ .
T 0 ( x , y , z ) = 1 2 { 1 + cos [ 2 π o L o ( y , z ) ] } ,
L o ( y , z ) = d o ( y cos θ + z sin θ ) d o z cos θ + y sin θ ,
T ( x , y , z ) = 1 4 [ 1 + cos ( 2 π p L p ) ] [ 1 + cos ( 2 π o L o ) ] ,
T ( x , y , z ) = 1 4 [ 1 + cos ( 2 π p L p ) + cos ( 2 π o L o ) + cos ( 2 π p L p ) cos ( 2 π o L o ) ] .
T ( x , y , z ) = 1 4 { 1 + cos ( 2 π p L p ) + cos ( 2 π o L o ) + 1 2 cos [ 2 π ( L p p + L o o ) ] + 1 2 cos [ 2 π ( L p p L o o ) ] } .
cos [ 2 π f ( x , y , z ) ]
f ( x , y , z ) = N ,
L p p = N ,
L o o = N ,
L p p + L o o = N ,
L p p L o o = N ,
d p = d o = d , o = p ,
y cos θ z sin θ d z cos θ y sin θ ± y cos θ + z sin θ d z cos θ + y sin θ = N p d ,
d = .
( y cos θ z sin θ ) ± ( y cos θ + z sin θ ) = N p ,
Δ y = p 2 cos θ ,
Δ z = p 2 cos θ .
sin θ cos θ ,
2 y ( d cos θ z ) = N p d [ ( d z cos θ ) 2 y 2 sin 2 θ ] .
y = 0 , z = d cos θ .
z = d cos θ + y tan θ , z = d cos θ y tan θ .
y = N p d ( d 2 cos θ z ) .
Δ y = p 2 cos θ .
y = 0 , z = d 2 cos θ ,
2 sin θ [ y 2 cos θ z ( d z cos θ ) ] = N p d [ ( d z cos θ ) 2 ( y sin θ ) 2 ] ,
( z z o ) 2 A 2 + y 2 B 2 = 1 ,
z o = d 2 cos θ ( 1 μ cot θ 1 μ 2 cot θ ) ,
A 2 = ( d 2 cos θ ) 2 ( 1 μ 2 cot θ ) 2 ,
B 2 = ( d 2 cos θ ) 2 ( 1 μ 2 cot θ ) ( 1 + μ 2 tan θ ) ,
μ = N p d .
μ = 2 cot θ , μ = + 2 tan θ .
tan θ cot θ < μ < 2 tan θ .
A 2 = B 2 , μ = 0 .
Δ z = p 2 sin θ .
N t = 2 Δ z cos θ ,
N t = 2 ( f # ) ( D s p ) sin 2 θ .
N t 2 ( f # ) sin 2 θ .