Abstract

The use of laser speckle for displacement and deformation measurement has found wide applications in mechanics and metrology. Traditionally, a deterministic approach has been adopted to analyze the process. In this paper a statistical approach is utilized, resulting in a better understanding of the limitations and potentials of one-beam subjective laser-speckle interferometry (or speckle photography). We find that the amplitude modulation of the information-carrying cosine fringes is controlled by three factors: a decorrelation factor γ, a displacement factor Ω, and a displacement-gradient factor ψ. We introduce the concept of an ambiguity factor that sets a critical value for the product γΩψ. We then propose a criterion for fringe discernibility from which the upper limit of measurement is established.

© 1985 Optical Society of America

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References

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  1. J. M. Burch, J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatters,” Opt. Acta 15, 101–111 (1968).
  2. E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
    [CrossRef]
  3. R. P. Khetan, F. P. Chiang, “Strain analysis by one-beam laser speckle interferometry, 1: single aperture method,” Appl. Opt. 15, 2205–2215 (1976).
    [CrossRef] [PubMed]
  4. F. P. Chiang, “A new family of 2D and 3D experimental stress analysis techniques using laser speckles,” Solid Mech. Archives 3, 27–58 (1978).
  5. F. P. Chiang, J. Adachi, R. Anastasi, J. Beatty, “Subjective laser speckle method and its application to solid mechanics problems,” Opt. Eng. 21, 379–390 (1982).
    [CrossRef]
  6. R. K. Erf, ed., Speckle Metrology (Academic, New York, 1978).
  7. L. I. Goldfischer, “Autocorrelation function and power spectrum density of laser produced speckle patterns,”J. Opt. Soc. Am. 55, 247–253 (1965).
    [CrossRef]
  8. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 2.
    [CrossRef]
  9. J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
    [CrossRef]
  10. I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
    [CrossRef]
  11. I. Yamaguchi, “Fringe formation in speckle photography,” J. Opt. Soc. Am. A 1, 81–86 (1984).
    [CrossRef]
  12. J. B. Chen, F. P. Chiang, “Statistical analysis of whole-field filtering of specklegram and its upper limit of measurement,” J. Opt. Soc. Am. A 1, 845–849 (1984).
    [CrossRef]
  13. T. Asakara, “Surface roughness measurements,” in Speckle Metrology, R. K. Erf, ed. (Academic, New York, 1978), Chap. 3.
    [CrossRef]
  14. D. W. Li, J. B. Chen, F. P. Chiang, “Statistical analysis of one-beam subjective laser speckle interferometry,” (State University of New York, Stony Brook, N.Y., April1983).
  15. M. Pedretti, F. P. Chiang, “Effect of magnification change in laser speckle interferometry,”J. Opt. Soc. Am. 68, 1742–1748 (1978).
    [CrossRef]
  16. F. P. Chiang, D. W. Li, “Study on point-wise filtering for one-beam laser speckle interferometry,” (State University of New York, Stony Brook, N.Y., August1984).
  17. G. H. Kaufmann, “On the numerical processing of speckle photograph fringes,” Opt. Laser Technol. 12, 207–209 (1980).
    [CrossRef]
  18. G. H. Kaufman, A. E. Ennos, B. Gale, D. J. Pugh, “An electric-optical read-out system for analysis of speckle photographs,”J. Phys. E 13, 579–584 (1980).
    [CrossRef]
  19. B. Ineichen, P. Eglin, R. Dändliker, “Hybrid optical and electronic image processing for strain measurements by speckle photography,” Appl. Opt. 19, 2191–2195 (1980).
    [CrossRef] [PubMed]
  20. V. J. Parks, “The range of speckle metrology,” Exp. Mech. 20(6), 181–191 (1980).
    [CrossRef]

1984 (2)

1982 (1)

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

1981 (1)

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

1980 (4)

G. H. Kaufmann, “On the numerical processing of speckle photograph fringes,” Opt. Laser Technol. 12, 207–209 (1980).
[CrossRef]

G. H. Kaufman, A. E. Ennos, B. Gale, D. J. Pugh, “An electric-optical read-out system for analysis of speckle photographs,”J. Phys. E 13, 579–584 (1980).
[CrossRef]

V. J. Parks, “The range of speckle metrology,” Exp. Mech. 20(6), 181–191 (1980).
[CrossRef]

B. Ineichen, P. Eglin, R. Dändliker, “Hybrid optical and electronic image processing for strain measurements by speckle photography,” Appl. Opt. 19, 2191–2195 (1980).
[CrossRef] [PubMed]

1978 (2)

F. P. Chiang, “A new family of 2D and 3D experimental stress analysis techniques using laser speckles,” Solid Mech. Archives 3, 27–58 (1978).

M. Pedretti, F. P. Chiang, “Effect of magnification change in laser speckle interferometry,”J. Opt. Soc. Am. 68, 1742–1748 (1978).
[CrossRef]

1976 (1)

1972 (1)

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

1970 (1)

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[CrossRef]

1968 (1)

J. M. Burch, J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatters,” Opt. Acta 15, 101–111 (1968).

1965 (1)

Adachi, J.

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

Anastasi, R.

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

Archbold, E.

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

Asakara, T.

T. Asakara, “Surface roughness measurements,” in Speckle Metrology, R. K. Erf, ed. (Academic, New York, 1978), Chap. 3.
[CrossRef]

Beatty, J.

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

Burch, J. M.

J. M. Burch, J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatters,” Opt. Acta 15, 101–111 (1968).

Chen, J. B.

J. B. Chen, F. P. Chiang, “Statistical analysis of whole-field filtering of specklegram and its upper limit of measurement,” J. Opt. Soc. Am. A 1, 845–849 (1984).
[CrossRef]

D. W. Li, J. B. Chen, F. P. Chiang, “Statistical analysis of one-beam subjective laser speckle interferometry,” (State University of New York, Stony Brook, N.Y., April1983).

Chiang, F. P.

J. B. Chen, F. P. Chiang, “Statistical analysis of whole-field filtering of specklegram and its upper limit of measurement,” J. Opt. Soc. Am. A 1, 845–849 (1984).
[CrossRef]

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

M. Pedretti, F. P. Chiang, “Effect of magnification change in laser speckle interferometry,”J. Opt. Soc. Am. 68, 1742–1748 (1978).
[CrossRef]

F. P. Chiang, “A new family of 2D and 3D experimental stress analysis techniques using laser speckles,” Solid Mech. Archives 3, 27–58 (1978).

R. P. Khetan, F. P. Chiang, “Strain analysis by one-beam laser speckle interferometry, 1: single aperture method,” Appl. Opt. 15, 2205–2215 (1976).
[CrossRef] [PubMed]

D. W. Li, J. B. Chen, F. P. Chiang, “Statistical analysis of one-beam subjective laser speckle interferometry,” (State University of New York, Stony Brook, N.Y., April1983).

F. P. Chiang, D. W. Li, “Study on point-wise filtering for one-beam laser speckle interferometry,” (State University of New York, Stony Brook, N.Y., August1984).

Dainty, J. C.

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[CrossRef]

Dändliker, R.

Eglin, P.

Ennos, A. E.

G. H. Kaufman, A. E. Ennos, B. Gale, D. J. Pugh, “An electric-optical read-out system for analysis of speckle photographs,”J. Phys. E 13, 579–584 (1980).
[CrossRef]

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

Gale, B.

G. H. Kaufman, A. E. Ennos, B. Gale, D. J. Pugh, “An electric-optical read-out system for analysis of speckle photographs,”J. Phys. E 13, 579–584 (1980).
[CrossRef]

Goldfischer, L. I.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 2.
[CrossRef]

Ineichen, B.

Kaufman, G. H.

G. H. Kaufman, A. E. Ennos, B. Gale, D. J. Pugh, “An electric-optical read-out system for analysis of speckle photographs,”J. Phys. E 13, 579–584 (1980).
[CrossRef]

Kaufmann, G. H.

G. H. Kaufmann, “On the numerical processing of speckle photograph fringes,” Opt. Laser Technol. 12, 207–209 (1980).
[CrossRef]

Khetan, R. P.

Li, D. W.

D. W. Li, J. B. Chen, F. P. Chiang, “Statistical analysis of one-beam subjective laser speckle interferometry,” (State University of New York, Stony Brook, N.Y., April1983).

F. P. Chiang, D. W. Li, “Study on point-wise filtering for one-beam laser speckle interferometry,” (State University of New York, Stony Brook, N.Y., August1984).

Parks, V. J.

V. J. Parks, “The range of speckle metrology,” Exp. Mech. 20(6), 181–191 (1980).
[CrossRef]

Pedretti, M.

Pugh, D. J.

G. H. Kaufman, A. E. Ennos, B. Gale, D. J. Pugh, “An electric-optical read-out system for analysis of speckle photographs,”J. Phys. E 13, 579–584 (1980).
[CrossRef]

Tokarski, J. M. J.

J. M. Burch, J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatters,” Opt. Acta 15, 101–111 (1968).

Yamaguchi, I.

I. Yamaguchi, “Fringe formation in speckle photography,” J. Opt. Soc. Am. A 1, 81–86 (1984).
[CrossRef]

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

Appl. Opt. (2)

Exp. Mech. (1)

V. J. Parks, “The range of speckle metrology,” Exp. Mech. 20(6), 181–191 (1980).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Phys. E (1)

G. H. Kaufman, A. E. Ennos, B. Gale, D. J. Pugh, “An electric-optical read-out system for analysis of speckle photographs,”J. Phys. E 13, 579–584 (1980).
[CrossRef]

Opt. Acta (4)

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[CrossRef]

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

J. M. Burch, J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatters,” Opt. Acta 15, 101–111 (1968).

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

Opt. Eng. (1)

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

Opt. Laser Technol. (1)

G. H. Kaufmann, “On the numerical processing of speckle photograph fringes,” Opt. Laser Technol. 12, 207–209 (1980).
[CrossRef]

Solid Mech. Archives (1)

F. P. Chiang, “A new family of 2D and 3D experimental stress analysis techniques using laser speckles,” Solid Mech. Archives 3, 27–58 (1978).

Other (5)

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

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 2.
[CrossRef]

T. Asakara, “Surface roughness measurements,” in Speckle Metrology, R. K. Erf, ed. (Academic, New York, 1978), Chap. 3.
[CrossRef]

D. W. Li, J. B. Chen, F. P. Chiang, “Statistical analysis of one-beam subjective laser speckle interferometry,” (State University of New York, Stony Brook, N.Y., April1983).

F. P. Chiang, D. W. Li, “Study on point-wise filtering for one-beam laser speckle interferometry,” (State University of New York, Stony Brook, N.Y., August1984).

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Figures (9)

Fig. 1
Fig. 1

Optical arrangement for the recording process.

Fig. 2
Fig. 2

Pointwise-filtering arrangement.

Fig. 3
Fig. 3

Relation between the illuminated area S and the fringe-contributing area S1.

Fig. 4
Fig. 4

Geometry for calculating Ω: rx is the radius of the illuminated area and |ds0| is the speckle displacement in the area.

Fig. 5
Fig. 5

Displacement factor Ω as a function of |ds0|/2rx.

Fig. 6
Fig. 6

Strain factor ψ for a circular aperture and negligible displacement.

Fig. 7
Fig. 7

Optical arrangement for whole-field filtering.

Fig. 8
Fig. 8

Combined effect of factors Ω and ψ.

Fig. 9
Fig. 9

Geometry for calculating FS ratio for whole-field filtering.

Equations (77)

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t ( x ) = I 1 ( x ) + I 2 ( x ) ,
R c ( x 1 , x 2 ) = I 1 ( x 1 ) I 2 ( x 2 ) ,
I 1 ( x 1 ) I 2 ( x 2 ) = I 1 ( x 1 ) I 2 ( x 2 ) + A 1 ( x 1 ) A 2 * ( x 2 ) 2 ,
A 1 ( x 1 ) A 2 * ( x 2 ) 2 = A 1 * ( x 1 ) A 2 ( x 2 ) 2 = | P ( r ) P * ( r - d r ) exp ( j ϕ ) × exp [ - j k r q + Δ q · ( Δ x - d s ) ] d σ r | 2 ,
d r = d p ( ξ 0 ) - p [ m 0 ( ξ ) · d ( ξ ) ] ξ 0 - [ r · d ( ξ ) ] ξ 0 - d z ( ξ 0 ) p r + d z ( ξ 0 ) p ξ 0 ,
d s = - M ( d p + ξ d z p ) - ( Δ q q - M d z 1 p ) [ d p - p ( m 0 · d ) ] ξ 0 ,
ϕ = k { 1 q + Δ q ( Δ q q - M d z 1 p ) [ d z r 2 p + r · ( r · d ) ξ 0 ] + r 2 d z 2 p 2 } ,
I 1 2 = I 2 2 = P ( r ) 2 d σ r 2 .
I 1 ( x 1 ) I 2 ( x 2 ) = I 1 ( x 1 ) I 2 ( x 1 + Δ x ) = | P ( r ) 2 d σ r 2 + P ( r ) P * ( r - d r ) exp ( j ϕ ) exp [ - j k r q + Δ q · ( Δ x - d s ) ] d σ r | 2 ,
I 1 ( x ) I 1 ( x + Δ x ) = [ P ( r ) 2 d σ r ] 2 + | | P ( r ) | 2 exp [ j k r q + Δ q · Δ x ] d σ r | 2 ,
A 1 A 1 * max 2 = P ( r ) 2 d σ r 2 .
A 1 A 2 * max 2 = P ( r ) P * ( r - d r ) exp ( j ϕ ) d σ r 2 .
γ = P ( r ) P * ( r - d r ) exp ( j ϕ ) d σ r 2 / P ( r ) 2 d σ r 2 .
I 1 ( x ) I 2 ( x + Δ x ) = c ( 0 ) + γ c ( Δ x - d s ) ,
c ( x ) = | | P ( r ) | 2 exp ( - j k r · x q + Δ q ) d σ r | 2 .
A ( u ) = H ( x ) t ( x ) exp ( - j 2 π u · x λ f 1 ) d σ x ,
H ( x ) = { H 0 ( x ) in the illuminated area s 0 elsewhere .
A ( u ) = A 1 ( u ) + A 2 ( u ) ,
A i ( u ) = s H 0 ( x ) I i ( x ) exp [ - j 2 π x · u λ f 1 ] d σ x ,             i = 1 , 2 , .
I ( u ) = A ( u ) 2 = A 1 A 1 * + A 2 A 2 * + A 1 A 2 * + A 1 * A 2 .
I v ( u ) = I ( u ) = A 1 A 1 * + A 2 A 2 * + A 1 A 2 * + A 1 * A 2 .
A i A l * = s H 0 ( x 1 ) H 0 * ( x 2 ) I i ( x 1 ) I l ( x 2 ) × exp [ - j 2 π ( x 1 - x 2 ) · u λ f 1 ] d σ x 1 d σ x 2 ,             i , l = 1 , 2 ,
A 1 A 1 * = s H 0 ( x 1 ) H 0 * ( x 2 ) [ c ( 0 ) + c ( x 2 - x 1 ) ] × exp [ j 2 π ( x 2 - x 1 ) · u λ f 1 ] d σ x 1 d σ x 2 , = c ( 0 ) | s H 0 ( x ) exp [ j 2 π x · u λ f 1 ] d σ x | 2 + s H 0 ( x 1 ) { s H 0 * ( x 2 ) c ( x 2 - x 1 ) × exp [ j 2 π ( x 2 - x 1 ) · u λ f 1 d σ x 2 ] } d σ x 1 .
A 1 A 1 * = c ( o ) | s H 0 ( x ) exp [ - j 2 π x · u λ f 1 ] d σ x | 2 + s H 0 ( x 1 ) 2 d σ x 1 δ c ( Δ x ) exp [ j 2 π Δ x · u λ f 1 ] d σ Δ x ,
A 1 A 2 * = s H 0 ( x 1 ) H 0 * ( x 2 ) [ c ( o ) + γ c ( x 2 - x 1 - d s ) ] × exp [ - j 2 π ( x 1 - x 2 ) · u λ f 1 ] d σ x 1 d σ x 2 = c ( o ) | s H 0 ( x 1 ) exp [ - j 2 π x 1 · u λ f 1 ] d σ x 1 | 2 + s H 0 ( x 1 ) { H 0 * ( x 2 ) γ c ( x 2 - x 1 - d s ) × exp [ j 2 π ( x 2 - x 1 ) · u λ f 2 ] d σ x 2 } d σ x 1 .
A 1 A 2 * = c ( o ) | s H 0 ( x 1 ) exp [ - 2 π x 1 · u λ f 1 ] d σ x | 2 + γ s H 0 ( x 1 ) H 0 * ( x 1 + d s ) ( s c ( x 2 - x 1 - d s ) × exp { j 2 π [ ( x 2 - x 1 ) · u λ f 1 ] } d σ x 2 ) d σ x 1 = c ( o ) | s H 0 ( x 1 ) exp [ - j 2 π x · u λ f 1 ] d σ x 1 | 2 + γ s H 0 ( x 1 ) H 0 * ( x 1 + d s ) exp [ j 2 π d s · u λ f 1 ] d σ x 1 × δ c ( Δ x ) exp [ j 2 π Δ x · u λ f 1 ] d σ Δ x .
s H 0 ( x 1 ) H 0 * ( x 1 + d s ) exp [ j 2 π d s · u λ f 1 ] d σ x 1 = s 1 H 0 ( x 1 ) H 0 * ( x 1 + d s ) exp [ j 2 π d s · u λ f 1 ] d σ x 1 .
d s = d s 0 + [ d s I x I | 0 x I + d s I x I I | 0 x I I ] e ^ I + [ d s I I x I | 0 x I + d s I I x I I | 0 x I I ] e ^ I I ,
s 1 H 0 ( x ) H 0 * ( x + d s ) exp [ j 2 π d s · u λ f 1 ] d σ x = exp [ j 2 π d s 0 · u λ f 1 ] s 1 H 0 ( x ) H 0 * ( x + d s ) × exp [ j 2 π u · x λ f 1 ] d σ x ,
= I e ^ I + I I e ^ I I ,
( I I I ) = [ d s I x I d s I I x I d s I x I I d s I I x I I ] | at 0 ( cos α sin α ) ,
α = tan - 1 u I I u I .
I 0 ( u ) = c ( o ) | s H 0 ( x ) exp [ - j 2 π x · u λ f 1 ] d σ x | 2 ,
I h ( u ) = [ s H 0 ( x ) 2 d σ x ] δ c ( x ) exp [ j 2 π x · u λ f 1 ] d σ x ,
A 1 A 1 * = I 0 ( u ) + I h ( u )
A 1 A 2 * = I 0 ( u ) + γ I h ( u ) exp [ j 2 π d s 0 · u λ f 1 ] × s 1 H 0 ( x ) H 0 * ( x + d s ) exp [ j 2 π u · x λ f 1 ] d σ x s H ( x ) 2 d σ x .
I v ( u ) = 4 I 0 ( u ) + 2 I h ( u ) { 1 + γ cos [ 2 π d s 0 · u λ f 1 ] × s 1 H 0 ( x ) H 0 * ( x + d s ) exp [ j 2 π u · x λ f 1 ] d σ x s H ( x ) 2 d σ x } .
Ω = s 1 / s
ψ = s 1 H 0 ( x ) H 0 * ( x + d s ) exp [ j 2 π u · x λ f 1 ] d σ x / s 1 s H 0 ( x ) 2 d σ x / s ,
I v ( u ) = 4 I 0 ( u ) + 2 I h ( u ) [ 1 + ν d cos ( 2 π d s 0 · u λ f 1 ) ] ,
ν d = γ ψ Ω .
I h ( u ) = [ H ( x ) 2 d σ x ] P ( r ) 2 × | p ( r + λ 0 ( q + Δ q ) λ f 1 u ) | 2 d σ r ,
s f = λ f 1 d s 0 .
Ω = 2 ( θ - sin θ cos θ ) π ,
θ = cos - 1 d s 0 2 r x
ψ = s 1 exp [ j 2 π u · x λ f 1 ] d σ x / s 1 .
ψ = 2 J 1 ( ρ ) / ρ ,
ρ = 2 π u r x / λ f 1 ,
F 1 ( u ) = t 0 ( x ) exp [ - j 2 π λ f 1 x · u ] d σ x ,
F 2 ( u ) = F 1 ( u ) a ( u - u 0 ) ,
a ( u ) = { 1 u r u 0 elsewhere .
A ( η ) = F 2 ( u ) exp [ - j 2 π λ f u · η ] d σ u ,
A ( x ) = F 2 ( u ) exp [ j 2 π λ f 1 u · x ] d σ u .
A ( x ) = { F 1 ( u ) exp [ j 2 π λ f 1 u · x ] d σ u } * { exp [ j 2 π λ f 1 u 0 · x ] a ( u ) exp [ j 2 π λ f 1 x · u ] d σ u } .
F 1 ( u ) exp [ j 2 π λ f 1 u · x ] d σ u = t 0 ( x ) exp [ - j 2 π λ f 1 u · x ] exp [ j 2 π λ f 1 x · u ] d σ x d σ u = b t 0 ( x ) ,
H ( x ) = a ( u ) exp [ j 2 π λ f 1 ( x · u ) ] d σ u .
A ( x ) = t 0 ( x ) * { exp [ j 2 π λ f 1 ( u 0 · x ) ] H ( x ) } = exp [ - j 2 π λ f 1 u 0 · x ] t 0 ( x ) H ( x - x ) × exp [ j 2 π λ f 1 u 0 · x ] d σ x .
A ( x ) = t 0 ( x ) H ( x - x ) exp [ j 2 π λ f 1 u 0 · x ] d σ x .
H ( x ) = J 1 ( 2 π x r u / λ f 1 ) x r u / λ f 1
r x = 0.61 λ f 1 / r u .
I ( x ) = { 2 [ δ c ( x ) exp ( j 2 π x · u 0 λ f 1 ) d σ x ] [ H ( x ) 2 d σ x ] } × { 1 + γ cos [ 2 π d s ( x ) · u 0 λ f 1 ] × s 1 H ( x ) H * [ x + d s ( x ) ] exp [ j 2 π u 0 · x λ f 1 ] d σ x s H ( x ) 2 d σ x } .
I ( x ) = I 0 { 1 + ν d cos [ 2 π d s ( x ) · u 0 λ f 1 ] } ,
d s ( x ) u 0 = { N λ f 1 for bright fringes ( N + ½ ) λ f 1 for dark fringes ,
p eq = λ f 1 / u 0 .
Q = ( ν d ) critic .
Q = Q ( s f / s s ) = Q ( f f / f s ) ,
γ Ω ψ - Q ( s f / s s ) > 0.
γ Ω ψ - Q ( s f / s s ) = 0.
S s = 1.2 λ f 1 / l d ,
S f / S s = l d / 1.2 d s .
Δ ( d s · u 0 ) = λ f 1 .
T A T B = S f / M ,
Δ ( d s · u 0 ) / T A T B = λ f 2 / S f .
Δ ( d s · u 0 ) T A T B = Grad ( d s · u 0 ) = [ ( d s I x I cos α + d s I I x I sin α ) 2 + ( d s I x I cos α + d s I I x I I sin α ) 2 ] 1 / 2 u 0 = u 0 .
S f = λ f 2 / u 0 .
S f / S s = l d / 1.2 u 0 .
0.1 Q ( S f / S s ) < 1

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