Abstract

A technique to improve the sensitivity of in-plane deformation measurements on transparent and reflective specimens under tension or compression is described. The principle of this technique is that of moiré fringe multiplication. Experiments with 20-cycle/mm gratings yielded results having eighteen times the sensitivity of the conventional moiré technique. One fringe shift of each moiré fringe corresponded to an in-plane deformation of 2.7 μm.

© 1973 Optical Society of America

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References

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  1. J. Guild, The Interference Systems of Crossed Diffraction Gratings (Oxford U. P., London, 1956).
  2. A. E. Ennos, J. Sci. Instrum. 1, 731 (1968).
    [CrossRef]
  3. P. M. Boone, Opt. Technol. 2, 94 (1970).
    [CrossRef]
  4. P. Boone, R. Verbiest, Optica Acta 16, 555 (1969).
    [CrossRef]
  5. C. A. Sciammarella, N. Lurowist, J. Appl. Mech. 34, 425 (1967).
    [CrossRef]
  6. D. Post, Appl. Opt. 6, 1938 (1967).
    [CrossRef] [PubMed]
  7. D. Post, Exptl. Mech. 8, 63 (1968).
    [CrossRef]
  8. C. A. Sciammarella, Exptl. Mech. 9, 179 (1969).
    [CrossRef]
  9. D. Post, Appl. Opt. 10, 901 (1971).
    [CrossRef] [PubMed]
  10. D. Post, T. F. MacLaughlin, Exptl. Mech. 11, 408 (1971).
  11. K. Matsumoto, M. Takashima, J. Opt. Soc. Am. 60, 30 (1970).
    [CrossRef]

1971 (2)

D. Post, T. F. MacLaughlin, Exptl. Mech. 11, 408 (1971).

D. Post, Appl. Opt. 10, 901 (1971).
[CrossRef] [PubMed]

1970 (2)

1969 (2)

P. Boone, R. Verbiest, Optica Acta 16, 555 (1969).
[CrossRef]

C. A. Sciammarella, Exptl. Mech. 9, 179 (1969).
[CrossRef]

1968 (2)

A. E. Ennos, J. Sci. Instrum. 1, 731 (1968).
[CrossRef]

D. Post, Exptl. Mech. 8, 63 (1968).
[CrossRef]

1967 (2)

C. A. Sciammarella, N. Lurowist, J. Appl. Mech. 34, 425 (1967).
[CrossRef]

D. Post, Appl. Opt. 6, 1938 (1967).
[CrossRef] [PubMed]

Boone, P.

P. Boone, R. Verbiest, Optica Acta 16, 555 (1969).
[CrossRef]

Boone, P. M.

P. M. Boone, Opt. Technol. 2, 94 (1970).
[CrossRef]

Ennos, A. E.

A. E. Ennos, J. Sci. Instrum. 1, 731 (1968).
[CrossRef]

Guild, J.

J. Guild, The Interference Systems of Crossed Diffraction Gratings (Oxford U. P., London, 1956).

Lurowist, N.

C. A. Sciammarella, N. Lurowist, J. Appl. Mech. 34, 425 (1967).
[CrossRef]

MacLaughlin, T. F.

D. Post, T. F. MacLaughlin, Exptl. Mech. 11, 408 (1971).

Matsumoto, K.

Post, D.

D. Post, Appl. Opt. 10, 901 (1971).
[CrossRef] [PubMed]

D. Post, T. F. MacLaughlin, Exptl. Mech. 11, 408 (1971).

D. Post, Exptl. Mech. 8, 63 (1968).
[CrossRef]

D. Post, Appl. Opt. 6, 1938 (1967).
[CrossRef] [PubMed]

Sciammarella, C. A.

C. A. Sciammarella, Exptl. Mech. 9, 179 (1969).
[CrossRef]

C. A. Sciammarella, N. Lurowist, J. Appl. Mech. 34, 425 (1967).
[CrossRef]

Takashima, M.

Verbiest, R.

P. Boone, R. Verbiest, Optica Acta 16, 555 (1969).
[CrossRef]

Appl. Opt. (2)

Exptl. Mech. (3)

D. Post, Exptl. Mech. 8, 63 (1968).
[CrossRef]

C. A. Sciammarella, Exptl. Mech. 9, 179 (1969).
[CrossRef]

D. Post, T. F. MacLaughlin, Exptl. Mech. 11, 408 (1971).

J. Appl. Mech. (1)

C. A. Sciammarella, N. Lurowist, J. Appl. Mech. 34, 425 (1967).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Sci. Instrum. (1)

A. E. Ennos, J. Sci. Instrum. 1, 731 (1968).
[CrossRef]

Opt. Technol. (1)

P. M. Boone, Opt. Technol. 2, 94 (1970).
[CrossRef]

Optica Acta (1)

P. Boone, R. Verbiest, Optica Acta 16, 555 (1969).
[CrossRef]

Other (1)

J. Guild, The Interference Systems of Crossed Diffraction Gratings (Oxford U. P., London, 1956).

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Figures (10)

Fig. 1
Fig. 1

Optical system for transparent specimen.

Fig. 2
Fig. 2

In-plane deformation of a specimen. The points of intersection of the rulings before and after deformation are marked with white circles and black circles, respectively. d(x,y) is the shift of the rulings at the point P(x,y). In Figure, d(x,y) should read d(x,y).

Fig. 3
Fig. 3

Moiré fringe pattern.

Fig. 4
Fig. 4

Out-of-plane deformation of a grating after loading. w(x,y) is the amount of the deformation at point P(x,y).

Fig. 5
Fig. 5

Optical system for reflective specimen suitable for low order diffracted waves.

Fig. 6
Fig. 6

Optical system for reflective specimen suitable for high order diffracted waves.

Fig. 7
Fig. 7

Moiré patterns obtained with mismatch. The optical system used is shown in Fig. 1 and the diffraction by the rulings parallel to the y axis is revealed. (a) Moiré pattern with 0th and 1st-order diffracted waves from a deformed grating that produces the same sensitivity as that of conventional moiré techniques. (b) Moiré pattern with ±5th-order diffracted waves that produces ten times greater sensitivity. (c) Moiré pattern of eighteen times greater sensitivity with ±9th-order diffracted waves.

Fig. 8
Fig. 8

Moiré patterns obtained without mismatch and misalignment. The optical system used is shown in Fig. 1, and the diffraction by the rulings parallel to the x axis of the same crossed gratings as that used in Fig. 7 is revealed. For (a), (b), and (c) the diffracted waves brought into interference are the same as those in Fig. 7.

Fig. 9
Fig. 9

Reflective specimen. The arrow shows the loading direction.

Fig. 10
Fig. 10

Moiré pattern with mismatch for the reflective specimen shown in Fig. 9. (a) Moiré pattern with 0th and 1st-order diffracted waves. (b) Moiré pattern of six times greater sensitivity with ±3rd-order diffracted waves. (c) Moiré pattern of ten times greater sensitivity with ±5th-order diffracted waves.

Equations (19)

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T x ( x , y ) = m = - b x m exp { 2 π i m p [ x + u ( x , y ) ] } ,
b x m = 1 p - p / 2 p / 2 T x ( x , y ) exp ( - 2 π i m p ) d x .
V ill = exp ( 2 π i / λ ) ( x sin α 1 + y sin β 1 ) + exp ( 2 π i / λ ) ( x sin α 2 + y sin β 2 ) ,
V diff = V ill T x ( x , y ) = m = - b x m exp 2 π i p [ ( m + p λ sin α 1 ) x + m u ( x , y ) + p λ y sin β 1 ] + m = - b x m exp 2 π i p [ ( m + p λ sin α 2 ) x + m u ( x , y ) + p λ y sin β 2 ] ,
α 1 = sin - 1 [ - ( N / p ) λ ] , α 2 = sin - 1 [ - ( M / p ) λ ] , β 1 = 0 , β 2 = 0 ,
I ( x , y ) = b x N exp 2 π i [ x sin θ x λ + y sin θ y λ + N u ( x , y ) p ] + b x M exp 2 π i [ M u ( x , y ) p ] | 2 = b 2 x N + b 2 x M + 2 b x N b x M cos [ 2 π λ ( x sin θ x + y sin θ y ) + 2 π N - M p u ( x , y ) ] ,
I ( x , y ) = b x 1 2 + b x 0 2 + 2 b x 1 b x 0 cos [ 2 π λ ( x sin θ x + y sin θ y ) + 2 π p u ( x , y ) ] .
I ( x , y ) = b x N 2 + b x - N 2 + 2 b x N b x - N cos [ 2 π λ ( x sin θ x + y sin θ y ) + 2 π p 2 N u ( x , y ) ] ,
x sin θ x λ + n sin θ n λ + 2 N p u ( x , y ) = N x ,
φ = - cot - 1 ( sin θ x / sin θ y ) ,
τ = - ( λ / sin θ x ) cos φ .
x = X cos φ + Y sin φ , y = - X sin φ + Y cos φ ,
X - ( 2 N / p ) u ( x , y ) τ = - N x τ .
X = - N x τ ,
t ( X , Y ) = ( 2 N / p ) u ( x , y ) τ .
u ( x , y ) = ( p / 2 N ) [ t ( X , Y ) / τ ] .
Δ φ m ( x , y ) = w ( x , y ) ( n - cos θ m ) ,
Δ φ m ( x , y ) = w ( x , y ) ( 1 + cos θ m ) .
θ m = sin - 1 ( m λ / p ) ,

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