Abstract

When using coherent light in projected fringe moire interferometry, a phase grating will often produce a higher contrast pattern with greater light efficiency than will an amplitude grating. This paper describes the nature and mechanics of this phenomenon and presents experimental examples using this technique.

© 1984 Optical Society of America

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References

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  1. K. G. Harding, J. S. Harris, Appl. Opt. 22, 856 (1983).
    [CrossRef] [PubMed]
  2. R. Ritter, H.-J. Meyer, Appl. Opt. 19, 1630 (1980).
    [CrossRef] [PubMed]
  3. J. P. Sihora, D. W. Taylor, “Deflection of Rotating Marine Propellers Using Projected Grid Moire Techniques,” presented at Fourth International SESA Congress, Boston (May, 1980).
  4. F. P. Chiang, “Moire Methods of Strain Analysis,” in Manual on Experimental Stress Analysis, A. S. Kobayashi, Ed. (Society for Experimental Stress Analysis, Westport, Conn., 1978), p. 58.
  5. D. Post, Opt. Eng. 21, 458 (1982).
    [CrossRef]
  6. F. P. Chiang, Proc. Soc. Photo-Opt. Instrum. Eng. 153, 113 (1978).
  7. A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).
  8. W. T. Welford, Opt. Acta 16, 371 (1969).
    [CrossRef]
  9. S. H. Rowe, J. Opt. for Am. 61, 1599 (1971).
    [CrossRef]
  10. C. Williams, O. Beckland, Optics: A Short Course for Engineers and Scientists (Wiley Interscience, New York, 1972), p. 113.
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 69.
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 401.
  13. Ref. 12, p. 403.

1983 (1)

1982 (1)

D. Post, Opt. Eng. 21, 458 (1982).
[CrossRef]

1980 (1)

1978 (1)

F. P. Chiang, Proc. Soc. Photo-Opt. Instrum. Eng. 153, 113 (1978).

1971 (1)

S. H. Rowe, J. Opt. for Am. 61, 1599 (1971).
[CrossRef]

1969 (1)

W. T. Welford, Opt. Acta 16, 371 (1969).
[CrossRef]

Beckland, O.

C. Williams, O. Beckland, Optics: A Short Course for Engineers and Scientists (Wiley Interscience, New York, 1972), p. 113.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 401.

Chiang, F. P.

F. P. Chiang, Proc. Soc. Photo-Opt. Instrum. Eng. 153, 113 (1978).

F. P. Chiang, “Moire Methods of Strain Analysis,” in Manual on Experimental Stress Analysis, A. S. Kobayashi, Ed. (Society for Experimental Stress Analysis, Westport, Conn., 1978), p. 58.

Durelli, A. J.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 69.

Harding, K. G.

Harris, J. S.

Meyer, H.-J.

Parks, V. J.

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

Post, D.

D. Post, Opt. Eng. 21, 458 (1982).
[CrossRef]

Ritter, R.

Rowe, S. H.

S. H. Rowe, J. Opt. for Am. 61, 1599 (1971).
[CrossRef]

Sihora, J. P.

J. P. Sihora, D. W. Taylor, “Deflection of Rotating Marine Propellers Using Projected Grid Moire Techniques,” presented at Fourth International SESA Congress, Boston (May, 1980).

Taylor, D. W.

J. P. Sihora, D. W. Taylor, “Deflection of Rotating Marine Propellers Using Projected Grid Moire Techniques,” presented at Fourth International SESA Congress, Boston (May, 1980).

Welford, W. T.

W. T. Welford, Opt. Acta 16, 371 (1969).
[CrossRef]

Williams, C.

C. Williams, O. Beckland, Optics: A Short Course for Engineers and Scientists (Wiley Interscience, New York, 1972), p. 113.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 401.

Appl. Opt. (2)

J. Opt. for Am. (1)

S. H. Rowe, J. Opt. for Am. 61, 1599 (1971).
[CrossRef]

Opt. Acta (1)

W. T. Welford, Opt. Acta 16, 371 (1969).
[CrossRef]

Opt. Eng. (1)

D. Post, Opt. Eng. 21, 458 (1982).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

F. P. Chiang, Proc. Soc. Photo-Opt. Instrum. Eng. 153, 113 (1978).

Other (7)

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

J. P. Sihora, D. W. Taylor, “Deflection of Rotating Marine Propellers Using Projected Grid Moire Techniques,” presented at Fourth International SESA Congress, Boston (May, 1980).

F. P. Chiang, “Moire Methods of Strain Analysis,” in Manual on Experimental Stress Analysis, A. S. Kobayashi, Ed. (Society for Experimental Stress Analysis, Westport, Conn., 1978), p. 58.

C. Williams, O. Beckland, Optics: A Short Course for Engineers and Scientists (Wiley Interscience, New York, 1972), p. 113.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 69.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 401.

Ref. 12, p. 403.

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Figures (5)

Fig. 1
Fig. 1

Simple diffraction due to sinusoidal grating.

Fig. 2
Fig. 2

System used for calculating superposition of two grating fields.

Fig. 3
Fig. 3

Photomicrograph of amplitude grating.

Fig. 4
Fig. 4

Photomicrograph of phase grating: (a) surface of phase grating; (b) pattern 60 μm behind phase grating; (c) pattern 120 μm behind phase grating.

Fig. 5
Fig. 5

Real-time moire pattern photographed from video: (a) moire pattern using amplitude grating; (b) moire pattern using phase grating.

Equations (22)

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sin 2 π f x + sin 6 π f x 3 + sin 10 π f x 5
z = sin 2 π f r ,
t ( x , y ) = exp [ i π ( 1 + m sin 2 π f r ) ] ,
sin ϕ = m λ f ,
1 F = 1 d 1 + 1 d 2 + d 3 ,
M = d 2 + d 3 d 1 .
M f 1 = f 2 .
1 2 [ 1 + cos ( 2 π f 1 ξ ) ] ,
U ( u ) = 1 i λ d 3 1 i λ d 2 1 i λ d 1 1 2 [ 1 + cos ( 2 π f 1 ξ ) ] exp [ i π λ d 1 ( x - ξ ) 2 ] d ξ exp ( - i π π F x 2 ) exp [ i π λ d 2 ( r - x ) 2 ] d x exp [ i π [ 1 + m sin ( 2 π f 2 r ) ] ] exp [ i π λ d 3 ( u - 4 ) 2 ] d r
= exp ( i π λ d 3 u 2 ) / i λ 3 d 1 d 2 d 3 1 2 [ 1 + cos ( 2 π f 1 ξ ) ] exp ( i π λ d 1 ξ 2 ) exp [ i π λ ( 1 d 2 + 1 d 3 ) r 2 ] exp [ i π m sin ( 2 π f 2 r ) ] exp ( - 2 π i λ d 3 u r ) exp [ i π λ ( 1 d 1 + 1 d 2 - 1 F ) x 2 ] exp [ - 2 π i λ d 1 ( ξ + d 1 d 2 r ) x ] d x d r d ξ ,
i λ ( 1 d 1 + 1 d 2 - 1 F ) - 1 exp [ - i π λ ( 1 d 1 + 1 d 2 - 1 F ) - 1 1 d 1 2 × ( ξ 2 + d 1 2 d 2 2 r + 2 d 1 d 2 ξ r ) ] .
U ( u ) = [ exp ( i π λ d 3 u 2 ) / λ 2 d 1 d 2 d 3 ] ( d 1 d 2 M / d 3 ) 1 2 [ 1 + cos ( 2 π f 1 ξ ) ] exp ( i π λ M F F - d 2 d 3 2 ξ 2 ) exp [ i π m sin ( 2 π f 2 r ) ] exp [ - 2 π i r λ d 3 ( u + M ξ ) ] exp { i π r 2 λ d 3 [ 1 + M F ( F - d 1 ) ] } d r d ξ ,
exp [ i π m sin ( 2 π f 2 r ) ] = g = - J g ( m π ) exp ( i 2 π g f 2 r ) ,
U ( u ) = ( M / λ 2 d 3 2 ) exp ( i π u 2 / λ d 3 ) g = - J g ( m π ) 1 2 [ 1 + cos ( 2 π f 1 ξ ) ] exp [ i π λ M F ( F - d 2 ) d 3 ] exp [ - i 2 π r ( u λ d 3 + M ξ λ d 3 - g f 2 ) ] d r d ξ .
δ [ M ξ λ d 3 - ( g f 2 - u / λ d 3 ) ] ,
U ( u ) = ( M / λ 2 d 3 2 ) exp ( i π u 2 / λ d 3 ) g = - J g ( m π ) 1 2 [ 1 + cos ( 2 π f 1 ξ ) ] exp [ i π λ M F ( F - d 2 ) d 3 ξ 2 ] δ [ M S λ d 3 - ( g f 2 - u / λ d 3 ) ]
= ( M / λ 2 d 3 2 ) ( λ d 3 M ) 2 exp ( i π u 2 / λ d 3 ) g = - J g ( m π ) 1 2 ( 1 + cos { 2 π f 1 [ ( λ d 3 g f 2 / M ) - u / M ] } ) × exp { [ i π λ d 3 ( F - d 2 ) / M F ] ( g f 2 - u / λ d 3 ) 2 } .
I - 1 = J - 1 2 ( m π ) 1 4 { 1 + cos [ 2 π f 1 ( u M + λ d 3 f 3 M ) ] } 2 ,
I 0 = J 0 2 ( m π ) 1 4 [ 1 + cos ( 2 π f 1 u / M ) ] 2 ,
I + 1 = J + 1 2 ( m π ) 1 4 { 1 + cos [ 2 π f 1 ( u M - λ d 3 f 3 M ) ] } 2
2 π f 1 λ d 3 f 2 M = π ,             d 3 = M 2 λ f 1 f 2 ,
m π = 2.4 m = 0.76 2 π λ d 3 f 1 f 2 M = π 2 d 3 = M 4 λ f 1 f 2 .

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