Abstract

The bounds on the sensitivity and resolution of moiré deflectometry caused by diffraction effects are derived from the wave equation. It is shown that Fresnel diffraction is a necessary and sufficient framework for treating these effects. Expressions are given for angular resolution, spatial resolution, and dynamic range. The proper selection of experimental parameters involves a trade-off between the angular resolution on the one hand and spatial resolution and dynamic range on the other.

© 1985 Optical Society of America

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References

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  1. O. Kafri, Opt. Lett. 5, 555 (1980); O. Kafri, A. Livnat, Appl. Opt. 20, 3098 (1981).
    [CrossRef] [PubMed]
  2. C. A. Sciamarella, D. Davis, Exp. Mech. 8, 459 (1968).
    [CrossRef]
  3. A. J. Durrelly, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).
  4. H. Talbot, Philos. Mag. 9, 401 (1836).
  5. J. M. Cowley, A. F. Moodie, Proc. R. Soc. London Ser. B 70, 486 (1957).
    [CrossRef]
  6. D. Marcuse, Light Transmission Optics (Van Nostrand, Amsterdam, 1972).
  7. A. Papoulis, Systems and Transforms with Applications to Optics (McGraw-Hill, New York, 1968).
  8. V. Mangulis, Handbook of Series for Scientists and Engineers (Academic, New York, 1965), p. 80.
  9. O. Kafri, E. Margalit, Appl. Opt. 20, 2344 (1981).
    [CrossRef] [PubMed]

1981 (1)

1980 (1)

1968 (1)

C. A. Sciamarella, D. Davis, Exp. Mech. 8, 459 (1968).
[CrossRef]

1957 (1)

J. M. Cowley, A. F. Moodie, Proc. R. Soc. London Ser. B 70, 486 (1957).
[CrossRef]

1836 (1)

H. Talbot, Philos. Mag. 9, 401 (1836).

Cowley, J. M.

J. M. Cowley, A. F. Moodie, Proc. R. Soc. London Ser. B 70, 486 (1957).
[CrossRef]

Davis, D.

C. A. Sciamarella, D. Davis, Exp. Mech. 8, 459 (1968).
[CrossRef]

Durrelly, A. J.

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

Kafri, O.

Mangulis, V.

V. Mangulis, Handbook of Series for Scientists and Engineers (Academic, New York, 1965), p. 80.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, Amsterdam, 1972).

Margalit, E.

Moodie, A. F.

J. M. Cowley, A. F. Moodie, Proc. R. Soc. London Ser. B 70, 486 (1957).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications to Optics (McGraw-Hill, New York, 1968).

Parks, V. J.

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

Sciamarella, C. A.

C. A. Sciamarella, D. Davis, Exp. Mech. 8, 459 (1968).
[CrossRef]

Talbot, H.

H. Talbot, Philos. Mag. 9, 401 (1836).

Appl. Opt. (1)

Exp. Mech. (1)

C. A. Sciamarella, D. Davis, Exp. Mech. 8, 459 (1968).
[CrossRef]

Opt. Lett. (1)

Philos. Mag. (1)

H. Talbot, Philos. Mag. 9, 401 (1836).

Proc. R. Soc. London Ser. B (1)

J. M. Cowley, A. F. Moodie, Proc. R. Soc. London Ser. B 70, 486 (1957).
[CrossRef]

Other (4)

D. Marcuse, Light Transmission Optics (Van Nostrand, Amsterdam, 1972).

A. Papoulis, Systems and Transforms with Applications to Optics (McGraw-Hill, New York, 1968).

V. Mangulis, Handbook of Series for Scientists and Engineers (Academic, New York, 1965), p. 80.

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

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Figures (4)

Fig. 1
Fig. 1

Geometry of the experiment. G1 and G2 are gratings of pitch P. G1 is parallel to the y axis on the x, y plane. A reference frame x′, y′ is attached to G2 at the variable gap z.

Fig. 2
Fig. 2

Intensity distribution versus z for the first two Fourier image periods. Right-hand scale gives the contrast ImaxImin/Imax + Imin. Dots are the results of Eq. (28). Sinusoidal curve, truncated Fourier series [Eq. (33a)]. Triangular curve, trapezoidal approximation [Eq. (33b)].

Fig. 3
Fig. 3

Configuration for measuring the focal length F of a lens. The gratings G1 and G2 are oriented at angles +θ/2 and −θ/2 relative to the y and y′ axes, respectively.

Fig. 4
Fig. 4

Effects of a discontinuity. (a) Sharp image at low sensitivity. (b) Blurred image at high sensitivity (large z). Two diffraction orders are visible, but only the first degrades the contrast. Distorted fringes in (b) result from imperfections in the gratings and the collimated beam, which show up at high sensitivities.

Equations (91)

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f ( x , 0 ) = 1 2 + 1 π n = - ( - ) n exp [ - i ( 2 n + 1 ) q x ] 2 n + 1 ,
q = 2 π / p .
f ( x , z ) = exp [ i ( π 4 - k z ) ] ( λ z ) 1 / 2 - d x f ( x , 0 ) × exp [ - i k ( x - x ) 2 2 z ] ,
k = 2 π / λ .
f ( x , z ) = exp ( - i k z ) ( 1 2 + 1 π n = - ( - ) n × exp { - i [ ( 2 n + 1 ) q x - q 2 z 2 k ( 2 n + 1 ) 2 ] } ( 2 n + 1 ) ) .
ζ q 2 z 2 π k = l ,             l = 0 , 2 , 4 , .
f ( x , z ) = exp ( - i k z ) { 1 2 + 1 π × n = - ( - ) n exp [ - i ( 2 n + 1 ) q x ] 2 n + 1 } .
f ( x , z ) = exp ( - i k z ) { 1 2 - 1 π × n = - ( - ) n exp [ - i ( 2 n + 1 ) q x ] 2 n + 1 } ,
Z l = 2 π k l q 2 = p 2 l λ             or             ζ l = l ;             l = 1 , 2 , 3 , ,
f ( x , y , 0 ) = 1 4 π 2 - d k x - d k y ψ ( k x , k y ) × exp [ - i ( k x x + k y y ) ] ,
ψ ( k x , k y ) = - d x - d y f ( x , y , 0 ) exp [ i ( k x x + k y y ) ] .
f ( x , y , z ) = 1 4 π 2 - d k x - d k y ψ ( k x , k y ) × exp [ - i ( k x x + k y y + k z z ) ] ,
k z ( k x , k y ) = ( k 2 - k x 2 - k y 2 ) 1 / 2
ψ ( k x , k y ) = 2 π ψ ( k x ) δ ( k y ) ,
ψ ( k x ) = - d x f ( x , 0 ) exp ( i k x x ) ,
f ( x , z ) = 1 2 π - d k x ψ ( k x ) × exp { - i [ k x x + z ( k 2 - k x 2 ) 1 / 2 ] } .
f ( x , 0 ) = n = - a n exp ( - i q n x )
ψ ( k x ) = 2 π n = - a n δ ( k x - q n ) .
f ( x , z ) = n = - a n exp { - i [ q n x + z ( k 2 - q 2 n 2 ) 1 / 2 ] } .
( k 2 - q 2 n 2 ) 1 / 2 k - q 2 n 2 2 k ,
f ( x , z ) exp ( - i k z ) n = - a n exp [ - i ( q n x - z q 2 n 2 2 k ) ]
I 1 ( x , z ) = f ( x , z ) f * ( x , z ) = 1 4 + 1 π n = - g n ( z ) exp [ - i ( 2 n + 1 ) q x ] + g n * ( z ) exp [ i ( 2 n + 1 ) q x ] 2 n + 1 + 1 π 2 n = - n = - g n ( z ) g m * ( z ) exp [ 2 i ( m - n ) q x ] ( 2 n + 1 ) ( 2 m + 1 ) ,
g n ( z ) ( - ) n exp { i z [ k - ( k 2 - q 2 ( 2 n + 1 ) 2 ) 1 / 2 } ( - ) n exp [ i q 2 ( 2 n + 1 ) 2 z 2 k ] .
I 2 ( x , z , ω ) = 1 2 + 1 π × n = - ( - ) n exp { - i ( 2 n + 1 ) [ q x + 2 π ω ( x , y ) ] } ( 2 n + 1 ) ,
I ( x , z , ω ) = 1 p x - p / 2 x + p / 2 I 1 ( x , z ) I 2 ( x , z ) d x ,
I ( x , z , ω ) = 1 4 + 2 π 2 n = 0 cos [ k z - z ( k 2 - q 2 ( 2 n + 1 ) 2 ) 1 / 2 ] cos [ 2 π ( 2 n + 1 ) ω ( x , y ) ] ( 2 n + 1 ) 2 .
I ( x , z , ω ) = 1 4 + 2 π 2 n = 0 cos [ π ζ ( 2 n + 1 ) 2 ] cos [ 2 π ( 2 n + 1 ) ω ( x , y ) ] ( 2 n + 1 ) 2 .
C ( ζ ) I max - I min I max + I min = | 8 π 2 n = 0 cos π ζ ( 2 n + 1 ) 2 ( 2 n + 1 ) 2 | .
A ( ζ ) 1 2 - 1 π 0 ζ d t - d n sin [ π t ( 2 n + 1 ) 2 ] = 1 2 - 1 π ( ζ 2 ) 1 / 2 .
ζ - l min < ¼ ,             l = 0 , 1 , 2 , .
A ( z ) = 1 4 + 2 π 2 n = 0 n max cos { k z - z [ k 2 - q 2 ( 2 n + 1 ) 2 ] 1 / 2 } ( 2 n + 1 ) 2 ,
z l = l π k - ( k 2 - q 2 ) 1 / 2 .
I 1 ( x , y , ζ ) 1 4 + 2 π 2 cos ( π ζ ) cos 2 π ω ( x , y ) ,
I 2 ( x , y , ζ ) 1 2 { I 0 [ ω ( x , y ) + ζ 2 ] + I 0 [ ω ( x , y ) - ζ 2 ] } ,
I 0 ( ψ ) = 1 4 + 2 π 2 n = 0 cos [ 2 π ( 2 n + 1 ) ω ] ( 2 n + 1 ) 2 .
ψ ( k x ) = 2 n = - a n sin [ ( k x - q n ) d ] k x - q n .
f ( x , z ) = 1 π n = - a n - d k x sin [ ( k x - q n ) d ] k x - q n × exp { - i [ k x x + z ( k 2 - k x 2 ) 1 / 2 ] } .
f ( x , z ) = n = - a n ( exp { - i [ q n x + z ( k 2 - q 2 n 2 ) ] } U ( d - x ) + z λ 1 / 2 exp ( - 3 π i 4 ) { exp [ - i ( q n d + k r + ) ] r + 1 / 2 [ k ( x - d ) - q n r + ] - exp [ i ( q n d - k r - ) ] r - 1 / 2 [ ( x + d ) - q n r - ] } ) ,
r ± 2 = z 2 + ( x d ) 2 .
d = ( N + ¼ ) p .
f ( x , z ) = χ ( x , z , d ) + z 2 ( 2 π k ) 1 / 2 × exp ( - 3 π i 4 { exp [ - i ( k r + + π γ + 2 ) ] r + 1 / 2 cos [ π γ + 2 ( x - d ) ] - exp [ - i ( k r - + π γ - 2 ) ] r - 1 / 2 cos [ π γ - 2 ( x + d ) ] } ) ,
γ ± k ( x d ) q r ± .
n = - 1 ( 2 n + 1 ) ( γ - 2 n - 1 ) = n = - 1 γ 2 - ( 2 n + 1 ) 2 = - π 2 γ tan ( π γ 2 ) .
W ( r , γ ) z exp [ - i ( 3 π 4 + k r + π γ 2 ) ] 2 ( 2 π k r ) 1 / 2 ( x - d ) cos ( π γ 2 ) = z exp [ - i ( 3 π 4 + k r + π γ 2 ) ] 2 π 1 / 2 r 3 / 2 q γ cos ( π γ 2 ) ,
W ( r , 2 j + 1 ) = z exp [ - i ( 3 π 4 - k r ) ] λ 1 / 2 r 3 / 2 q ( 2 j + 1 ) × [ 1 2 + i ( 2 j + 1 ) π n j 1 ( 2 n + 1 ) ( 2 j - 2 n ) ] = z exp [ - i ( 3 π 4 - k r ) ] λ 1 / 2 r 3 / 2 q ( 2 j + 1 ) [ 1 2 - i π ( 2 j + 1 ) ] ,
f ( x j , z ) = χ ( x j , z , d ) - 1 π ( 2 j + 1 ) × exp ( - i { π ( j + ½ ) + q 2 r k ( 2 j + 1 ) 2 + z [ k 2 - q 2 ( 2 j + 1 ) 2 ] 1 / 2 } ) + W ( r , 2 j + 1 ) .
f ( x j z ) = χ ( x j , z , d ) - 1 π ( 2 j + 1 ) × exp { - i [ π ( j + ½ ) + k z + π ζ ( 2 j + 1 ) 2 ] } + exp [ - i ( 3 π 4 + k z ) ] 2 π ζ 1 / 2 ( 2 j + 1 ) [ 1 - 2 i π ( 2 j + 1 ) ] .
W ( z , 2 j ) = exp [ - i ( 3 π 4 + k z ) ] 4 π j ζ 1 / 2 .
x < d - ( q r / k ) .
A ( k x , k y ) exp { i [ γ ( k x , k y ) - k x x - k y y - k z z ] } .
f ( x , y , 0 ) = A ( k x , k y ) exp { i [ γ ( k x , k y ) - k x x - k y ] } n = - a n exp ( - i q n x ) .
ψ ( k x , k y ) = A e i γ δ ( k y - k y ) n = - a n δ ( k x - k x - q n ) .
f ( x , y , z ) = A exp [ i ( γ - k x x - k y y ) ] n = - a n × exp { - i [ q n x + z ( k z 2 - 2 k x q n - q 2 n 2 ) 1 / 2 ] } .
k x , k y k z k .
( k z 2 - 2 k x q n - q 2 n 2 ) 1 / 2 k z - k x q n k z - q 2 n 2 2 k ,
f ( x , y , z ) = A exp [ i ( γ - k x x - k y y - k z z ) ] × n = - a n exp { - i [ q n ( x - k x k z z ) - ζ n 2 ] } .
f ( x , y , z ) = ( - ) l exp [ - i ( k x x + k y y + k z z ) ] f ( x - k x k z z , y - k y k z z , 0 ) ,             ζ = l = 1 , 2 , 3 .
I 1 ( x , y ) = n = - a n exp [ - i n q ( x - y tan θ / 2 ] ,
I 2 ( x , y ) = n = - a n exp [ - i n q ( x + y tan θ / 2 ) ] .
I ( x , y ) = n = - a n 2 cos ( q n y θ ) .
f ( x , y , 0 ) = F S exp [ - i ( γ 0 + k x 2 + y 2 2 S ) ] × n = - a n exp [ - i q n ( x - y θ 2 ) ] ,
ψ ( k x , k y ) = 2 π F i k exp ( - i γ 0 ) × n = - a n exp { i S / 2 k [ ( k x - n q ) 2 + ( k y + n q θ 2 ) 2 ] } .
f ( x , y , z ) = F S + z exp { ( - i γ 0 - i k ) [ z + x 2 + y 2 2 ( z + S ) ] } × n = - a n exp { - i [ q S n z + S ( x - y θ 2 ) + ( q S n ) 2 2 k ( z + S ) - S ( q n ) 2 2 k ] } .
q 2 S 2 2 k ( z + S ) - q 2 S 2 k = π l ,             l = 0 , 1 , 2 ,
1 S + 1 z = 1 z l = q 2 2 π k l ,
I ( x , y ) = ( F S + z ) 2 n = - a n 2 cos [ q n x z + y θ ( S + z 2 ) z + S ] .
d x d y = - θ ( S z + 1 2 ) - θ S z ,
φ ( x , y ) tan φ ( x , y ) = x / z + S .
I ( x , y ) = n = - a n 2 cos [ q n ( z φ ( x , y ) + y θ ) ] .
F ( x , y ) = ½ 2 α x 2 ,
α ( x , y ) = s 0 n ( x , y , z ) d s 0 n ( x , y , z ) d z ,
h ( x , y ) y - y 0 = z θ φ ( x , y ) ,
h p = 1 2 π = z φ min P ,
p = p / θ .
φ min = 1 / q z .
1 z ( x , y ) = q 2 2 π k l - 1 F ( x , y ) ,             l = 1 , 2 , 3 .
z < F min 4 l = 1 4 l | δ 2 α δ x 2 | max .
φ ( x , y ) = δ α ( x , y ) / δ x ,
z < 1 4 l | φ x | max a 4 l φ max ,
z < p 2 ( F min λ ) 1 / 2 p 2 ( a λ φ max ) 1 / 2 ,
φ min = 1 π ( λ F min ) 1 / 2 .
φ min = 1 π ( λ φ max a ) 1 / 2 .
φ min φ max λ / π 2 a ,
φ min λ / a ,
φ ( x , y ) x < 1 z
γ = k ( x - x c ) / q z ,
1 π exp [ - i ( k z + π 2 - π ζ ) ] ,
x - x c = ± q z k .
Δ x = q z / k .
Δ x Δ φ = 1 k = λ 2 π .
Δ x Δ k = 1.

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