Abstract

The process of optical whole-field filtering of a double-exposure specklegram has been analyzed from a statistical point of view. It is shown that the size of the filtering aperture plays a vital role in the range of visible fringes, and this in turn determines the upper limit of measurement.

© 1984 Optical Society of America

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References

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  1. D. Casasent, D. Psaltis, “Deformation invariant space-variant optical pattern recognition,” in Progress in Optics, Vol. XVI, E. Wolf, ed. (Academic, New York, 1978).
    [Crossref]
  2. D. Casasent ed., Optical Data Processing and Applications (Springer-Verlag, Berlin, 1978).
    [Crossref]
  3. F. P. Chiang, “Techniques of optical spatial filtering applied to the processing of moiré fringes,” Exp. Mech. 6, 523–526 (1979).
  4. F. P. Chiang, “Multipurpose optical moiré processor,” Opt. Eng. 18, 456–460 (1979).
    [Crossref]
  5. R. Erf, Speckle Metrology (Academic, New York, 1978).
  6. Special issue on coherent optical techniques in experimental mechanics, Opt. Eng. 21, 377–495 (1982).
  7. F. P. Chiang, “Subjective laser speckle method and its application to solid mechanics problems,” Opt. Eng. 21, 379–390 (1982).
    [Crossref]
  8. E. Archbold, A. E. Ennos, “Displacement measurement from double exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
    [Crossref]
  9. V. J. Parks, “The range of speckle metrology,” Exp. Mech. 20, 181–191 (1980).
    [Crossref]
  10. F. P. Chiang, A. Asundi, “White light speckle method of experimental strain analysis,” Appl. Opt. 18, 409–411 (1979).
    [Crossref] [PubMed]
  11. F. P. Chiang, “Speckle method with electron microscopes,” presented at Society for Experimental Stress Analysis 1982 Spring Meeting, Oahu, Hawaii.
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1968).
  13. J. W. Goodman, in Laser Speckles and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2.
  14. R. P. Khetan, F. P. Chiang, “Strain analysis by one-beam laser speckle interferometry I: single aperture method,” Appl. Opt. 15, 2205–2215 (1976).
    [Crossref] [PubMed]
  15. G. Jaisingh, “Studies in one-beam laser speckle interferometry,” Ph.D. Dissertation (State University of New York at Stony Brook, Stony Brook, N.Y., October1979).

1982 (2)

Special issue on coherent optical techniques in experimental mechanics, Opt. Eng. 21, 377–495 (1982).

F. P. Chiang, “Subjective laser speckle method and its application to solid mechanics problems,” Opt. Eng. 21, 379–390 (1982).
[Crossref]

1980 (1)

V. J. Parks, “The range of speckle metrology,” Exp. Mech. 20, 181–191 (1980).
[Crossref]

1979 (3)

F. P. Chiang, A. Asundi, “White light speckle method of experimental strain analysis,” Appl. Opt. 18, 409–411 (1979).
[Crossref] [PubMed]

F. P. Chiang, “Techniques of optical spatial filtering applied to the processing of moiré fringes,” Exp. Mech. 6, 523–526 (1979).

F. P. Chiang, “Multipurpose optical moiré processor,” Opt. Eng. 18, 456–460 (1979).
[Crossref]

1976 (1)

1972 (1)

E. Archbold, A. E. Ennos, “Displacement measurement from double exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[Crossref]

Archbold, E.

E. Archbold, A. E. Ennos, “Displacement measurement from double exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[Crossref]

Asundi, A.

Casasent, D.

D. Casasent, D. Psaltis, “Deformation invariant space-variant optical pattern recognition,” in Progress in Optics, Vol. XVI, E. Wolf, ed. (Academic, New York, 1978).
[Crossref]

Chiang, F. P.

F. P. Chiang, “Subjective laser speckle method and its application to solid mechanics problems,” Opt. Eng. 21, 379–390 (1982).
[Crossref]

F. P. Chiang, A. Asundi, “White light speckle method of experimental strain analysis,” Appl. Opt. 18, 409–411 (1979).
[Crossref] [PubMed]

F. P. Chiang, “Techniques of optical spatial filtering applied to the processing of moiré fringes,” Exp. Mech. 6, 523–526 (1979).

F. P. Chiang, “Multipurpose optical moiré processor,” Opt. Eng. 18, 456–460 (1979).
[Crossref]

R. P. Khetan, F. P. Chiang, “Strain analysis by one-beam laser speckle interferometry I: single aperture method,” Appl. Opt. 15, 2205–2215 (1976).
[Crossref] [PubMed]

F. P. Chiang, “Speckle method with electron microscopes,” presented at Society for Experimental Stress Analysis 1982 Spring Meeting, Oahu, Hawaii.

Ennos, A. E.

E. Archbold, A. E. Ennos, “Displacement measurement from double exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[Crossref]

Erf, R.

R. Erf, Speckle Metrology (Academic, New York, 1978).

Goodman, J. W.

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

J. W. Goodman, in Laser Speckles and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2.

Jaisingh, G.

G. Jaisingh, “Studies in one-beam laser speckle interferometry,” Ph.D. Dissertation (State University of New York at Stony Brook, Stony Brook, N.Y., October1979).

Khetan, R. P.

Parks, V. J.

V. J. Parks, “The range of speckle metrology,” Exp. Mech. 20, 181–191 (1980).
[Crossref]

Psaltis, D.

D. Casasent, D. Psaltis, “Deformation invariant space-variant optical pattern recognition,” in Progress in Optics, Vol. XVI, E. Wolf, ed. (Academic, New York, 1978).
[Crossref]

Appl. Opt. (2)

Exp. Mech. (2)

F. P. Chiang, “Techniques of optical spatial filtering applied to the processing of moiré fringes,” Exp. Mech. 6, 523–526 (1979).

V. J. Parks, “The range of speckle metrology,” Exp. Mech. 20, 181–191 (1980).
[Crossref]

Opt. Acta (1)

E. Archbold, A. E. Ennos, “Displacement measurement from double exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[Crossref]

Opt. Eng. (3)

Special issue on coherent optical techniques in experimental mechanics, Opt. Eng. 21, 377–495 (1982).

F. P. Chiang, “Subjective laser speckle method and its application to solid mechanics problems,” Opt. Eng. 21, 379–390 (1982).
[Crossref]

F. P. Chiang, “Multipurpose optical moiré processor,” Opt. Eng. 18, 456–460 (1979).
[Crossref]

Other (7)

R. Erf, Speckle Metrology (Academic, New York, 1978).

D. Casasent, D. Psaltis, “Deformation invariant space-variant optical pattern recognition,” in Progress in Optics, Vol. XVI, E. Wolf, ed. (Academic, New York, 1978).
[Crossref]

D. Casasent ed., Optical Data Processing and Applications (Springer-Verlag, Berlin, 1978).
[Crossref]

G. Jaisingh, “Studies in one-beam laser speckle interferometry,” Ph.D. Dissertation (State University of New York at Stony Brook, Stony Brook, N.Y., October1979).

F. P. Chiang, “Speckle method with electron microscopes,” presented at Society for Experimental Stress Analysis 1982 Spring Meeting, Oahu, Hawaii.

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

J. W. Goodman, in Laser Speckles and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2.

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Figures (4)

Fig. 1
Fig. 1

Optical arrangement for whole-field spatial filtering of a specklegram.

Fig. 2
Fig. 2

Schematic of Airy disks created by filtering aperture and the areas of integration for Eq. (5).

Fig. 3
Fig. 3

Displacement fringe patterns of a disk under in-plane rotation as obtained by three filtering apertures: (a) Df = 6 mm, (b) Df = 3 mm, and (c) Df = 1.5 mm. Fringe sensitivity P = 20 μm.

Fig. 4
Fig. 4

Displacement fringe patterns of a disk under in-plane rotation as obtained by three filtering apertures: (a) Df = 6 mm, (b) Df = 3 mm, and (c) Df = 1.5 mm. Fringe sensitivity P = 10 μm.

Equations (23)

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t ( x , y ) = f ( x , y ) + f [ x - d x ( x , y ) , y - d y ( x , y ) ] ,
H ( ξ , η ) = { 1 for [ ( ξ - ξ 0 ) 2 + ( η - η 0 ) 2 ] 1 / 2 D f / 2 0 otherwise ,
g ( x , y ) = { f ( x , y ) + f [ x - d x ( x , y ) , y - d y ( x , y ) ] } * h ( x , y ) ,
h ( x , y ) = - H ( ξ , η ) exp [ j 2 π λ f ( x ξ + y η ) ] d ξ d η = exp [ j 2 π λ f ( ξ 0 x + η 0 y ) ] F T { circ ( ξ 2 + η 2 ) 1 / 2 } .
h ( x , y ) = { exp [ j 2 π λ f ( ξ 0 x + η 0 y ) ] for ( x 2 + y 2 ) 2 0.61 λ f D f 0 otherwise .
g ( x , y ) = s f ( x , y ) exp [ - j 2 π λ f ( ξ 0 x + η 0 y ) ] d x d y + s f [ x - d x ( x , y ) , y - d y ( x , y ) ] × exp [ - j 2 π λ f ( ξ 0 x + η 0 y ) ] d x d y ,
s f ( x , y ) exp ( - j 2 π λ f { ξ 0 [ x + d x ( x , y ) ] + η 0 [ y + d y ( x , y ) } ) d x d y ,
g ( x , y ) = { 1 + exp [ - j 2 π λ f ( ξ 0 d x + η 0 d y ) ] } g 0 ( x , y ) + g 1 ( x , y ) + g 2 ( x , y ) exp [ - j 2 π λ f ( ξ 0 d x + η 0 d y ) ] ,
g 0 ( x , y ) = s 0 f ( x , y ) exp [ - j 2 π λ f ( ξ 0 x + η 0 y ) ] d x d y , g 1 ( x , y ) = s 1 f ( x , y ) exp [ - j 2 π λ f ( ξ 0 x + η 0 y ) ] d x d y , g 2 ( x , y ) = s 2 f ( x , y ) exp [ - j 2 π λ f ( ξ 0 x + η 0 y ) ] d x d y .
p r ^ i [ g k ( r ) , g k ( i ) ] = 1 2 π σ k 2 exp { - [ g k ( r ) ] 2 + [ g k ( i ) ] 2 2 σ k 2 } , σ k 2 = s k f ( x , y ) 2 d x d y ,             k = 0 , 1 , 2 ,
σ k 2 = c s k ,             k = 0 , 1 , 2 ,
g k ( x , y ) 2 = 2 σ k 2 = 2 c s k ,             k = 0 , 1 , 2.
I ( x , y ) = g ( x , y ) 2 ,
I ( x , y ) = | { 1 + exp [ - j 2 π λ f ( ξ 0 d x + η 0 d y ) ] } × g 0 ( x , y ) | 2 + g 1 ( x , y ) 2 + g 2 ( x , y ) 2 + cross terms .
I ( x , y ) = 2 c { 2 [ 1 + cos 2 π λ f ( ξ 0 d x + η 0 d y ) ] s 0 + s 1 + s 2 } .
cos 2 π λ f ( ξ 0 d x + η 0 d y ) = - 1 ,
cos 2 π λ f ( ξ 0 d x + η 0 d y ) = 1.
r · d = { N λ f for bright fringes ( N + ½ ) λ f for dark fringes ,
p = λ f r = λ f ( ξ 0 2 + η 0 2 ) 1 / 2 .
v = I max - I min I max + I min = 2 s 0 2 s 0 + s 1 + s 2 .
D s = 1.22 λ f D f .
d max = 0.8 D s = 0.98 λ f D f ,
N max = ± 0.98 r D f .

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