Abstract

Laser sources offer advantages over white-light sources in some phase-shifting projected fringe profilometry applications. These benefits, however, are gained at the cost of incurring speckle noise. Some basic statistics of speckle-induced phase-measurement errors are investigated based on the multiplicative noise model for image-plane speckles. First, the dependence of phase-error distribution and measurement uncertainty on speckle size and grating pitch is numerically studied, based on the Karhunen–Loève expansion method. Then an analytical expression that relates phase-error distributions to optical system parameters is derived as a direct extension of the simulation results. This expression is useful for system design and optimization. Analysis shows that phase noise caused by speckles can be modeled as additive white Gaussian noise. Optical system design and noise-reduction algorithms are also briefly discussed, based on the simulation results.

© 1999 Optical Society of America

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References

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1996 (1)

T. Matsumoto, Y. Kitagawa, T. Minemoto, “Sensitivity-variable moiré topography with a phase shift method,” Opt. Eng. (Bellingham) 35, 1754–1760 (1996).
[CrossRef]

1995 (2)

D. Donoho, “Denoising by soft thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[CrossRef]

D. Donoho, I. Johnstone, G. Kerkyacharian, D. Picard, “Wavelet shrinkage: asymptopia?” J. R. Stat. Soc. 57, 301–369 (1995).

1994 (1)

1993 (1)

X. Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

1991 (1)

1988 (1)

1985 (2)

1984 (2)

G. T. Reid, R. C. Rixon, H. I. Messer, “Absolute and comparative measurement of three-dimensional shape by phase measuring moiré topography,” Opt. Laser Technol. 16, 315–319 (1984).
[CrossRef]

V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105–3108 (1984).
[CrossRef] [PubMed]

1976 (2)

1973 (2)

F. T. S. Yu, E. Y. Wang, “Speckle reduction in holography by means of random spatial sampling,” Appl. Opt. 12, 1656–1659 (1973).
[CrossRef] [PubMed]

R. Barakat, “First-order probability densities of laser speckle patterns observed through finite-size scanning apertures,” Opt. Acta 20, 729–740 (1973).
[CrossRef]

1971 (2)

J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327–339 (1971).
[CrossRef]

J. C. Dainty, W. T. Welford, “Reduction of speckle in image plane hologram reconstruction by moving pupils,” Opt. Commun. 3, 289–294 (1971).
[CrossRef]

1970 (3)

1969 (1)

1966 (1)

D. Slepian, E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745–1759 (1966).

1965 (1)

1962 (1)

A. Lohmann, “Grating diffraction spectra as coherent light sources for two or three beam interferometry,” Opt. Acta 9, 1–12 (1962).
[CrossRef]

Allen, J. B.

Arsenault, H. H.

Barakat, R.

R. Barakat, “First-order probability densities of laser speckle patterns observed through finite-size scanning apertures,” Opt. Acta 20, 729–740 (1973).
[CrossRef]

Baribeau, R.

Bell, B. W.

B. W. Bell, “Digital heterodyne topography,” in Photomechanics and Speckle Metrology, F. Chiang, ed., Proc. SPIE814, 754–768 (1987).

Brooks, R. E.

Chiang, F. P.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 350–393.

Dainty, J. C.

J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327–339 (1971).
[CrossRef]

J. C. Dainty, W. T. Welford, “Reduction of speckle in image plane hologram reconstruction by moving pupils,” Opt. Commun. 3, 289–294 (1971).
[CrossRef]

Delves, L. M.

L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, Cambridge, UK, 1985).

Donoho, D.

D. Donoho, “Denoising by soft thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[CrossRef]

D. Donoho, I. Johnstone, G. Kerkyacharian, D. Picard, “Wavelet shrinkage: asymptopia?” J. R. Stat. Soc. 57, 301–369 (1995).

Dorsch, R. G.

Fairman, P.

B. F. Oreb, K. G. Larkin, P. Fairman, M. Ghaffari, “Moiré based optical surface profiler for the minting industry,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. SPIE1776, 48–57 (1992).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Ghaffari, M.

B. F. Oreb, K. G. Larkin, P. Fairman, M. Ghaffari, “Moiré based optical surface profiler for the minting industry,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. SPIE1776, 48–57 (1992).

Glatt, I.

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley Interscience, New York, 1990).

Goodman, J. W.

J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
[CrossRef]

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

Halioua, M.

Häusler, G.

Heflinger, L. O.

Herrmann, J. M.

Johnson, W. O.

Johnstone, I.

D. Donoho, I. Johnstone, G. Kerkyacharian, D. Picard, “Wavelet shrinkage: asymptopia?” J. R. Stat. Soc. 57, 301–369 (1995).

Kafri, O.

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley Interscience, New York, 1990).

Kanwal, R. P.

R. P. Kanwal, Linear Integral Equations (Academic, New York, 1971).

Kerkyacharian, G.

D. Donoho, I. Johnstone, G. Kerkyacharian, D. Picard, “Wavelet shrinkage: asymptopia?” J. R. Stat. Soc. 57, 301–369 (1995).

Kitagawa, Y.

T. Matsumoto, Y. Kitagawa, T. Minemoto, “Sensitivity-variable moiré topography with a phase shift method,” Opt. Eng. (Bellingham) 35, 1754–1760 (1996).
[CrossRef]

Krishnamurthy, R. S.

Larkin, K. G.

B. F. Oreb, K. G. Larkin, P. Fairman, M. Ghaffari, “Moiré based optical surface profiler for the minting industry,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. SPIE1776, 48–57 (1992).

Liu, H. C.

Lohmann, A.

A. Lohmann, “Grating diffraction spectra as coherent light sources for two or three beam interferometry,” Opt. Acta 9, 1–12 (1962).
[CrossRef]

Lohmann, A. W.

Lowenthal, S.

Matsumoto, T.

T. Matsumoto, Y. Kitagawa, T. Minemoto, “Sensitivity-variable moiré topography with a phase shift method,” Opt. Eng. (Bellingham) 35, 1754–1760 (1996).
[CrossRef]

McKechnie, T. S.

T. S. McKechnie, Opt. Quantum Electron. 8, 61–67 (1976).
[CrossRef]

Meadows, D. M.

Messer, H. I.

G. T. Reid, R. C. Rixon, H. I. Messer, “Absolute and comparative measurement of three-dimensional shape by phase measuring moiré topography,” Opt. Laser Technol. 16, 315–319 (1984).
[CrossRef]

Minemoto, T.

T. Matsumoto, Y. Kitagawa, T. Minemoto, “Sensitivity-variable moiré topography with a phase shift method,” Opt. Eng. (Bellingham) 35, 1754–1760 (1996).
[CrossRef]

Mohamed, J. L.

L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, Cambridge, UK, 1985).

Oreb, B. F.

B. F. Oreb, R. G. Dorsch, “Profilometry by phase-shifted Talbot images,” Appl. Opt. 33, 7955–7962 (1994).
[CrossRef] [PubMed]

B. F. Oreb, K. G. Larkin, P. Fairman, M. Ghaffari, “Moiré based optical surface profiler for the minting industry,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. SPIE1776, 48–57 (1992).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Progress, 2nd ed. (McGraw-Hill, New York, 1984).

Paris, D. P.

Picard, D.

D. Donoho, I. Johnstone, G. Kerkyacharian, D. Picard, “Wavelet shrinkage: asymptopia?” J. R. Stat. Soc. 57, 301–369 (1995).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Reid, G. T.

G. T. Reid, R. C. Rixon, H. I. Messer, “Absolute and comparative measurement of three-dimensional shape by phase measuring moiré topography,” Opt. Laser Technol. 16, 315–319 (1984).
[CrossRef]

Rioux, M.

Rixon, R. C.

G. T. Reid, R. C. Rixon, H. I. Messer, “Absolute and comparative measurement of three-dimensional shape by phase measuring moiré topography,” Opt. Laser Technol. 16, 315–319 (1984).
[CrossRef]

Schwider, J.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. XXVIII, pp. 271–359.

Slepian, D.

D. Slepian, E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745–1759 (1966).

Sonnenblick, E.

D. Slepian, E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745–1759 (1966).

Srinivasan, V.

Su, X. Y.

X. Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Takasaki, H.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

von Bally, G.

X. Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Vukicevic, D.

X. Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Wang, E. Y.

Welford, W. T.

J. C. Dainty, W. T. Welford, “Reduction of speckle in image plane hologram reconstruction by moving pupils,” Opt. Commun. 3, 289–294 (1971).
[CrossRef]

Yu, F. T. S.

Appl. Opt. (10)

Bell Syst. Tech. J. (1)

D. Slepian, E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745–1759 (1966).

IEEE Trans. Inf. Theory (1)

D. Donoho, “Denoising by soft thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[CrossRef]

J. Opt. Soc. Am. (3)

J. R. Stat. Soc. (1)

D. Donoho, I. Johnstone, G. Kerkyacharian, D. Picard, “Wavelet shrinkage: asymptopia?” J. R. Stat. Soc. 57, 301–369 (1995).

Opt. Acta (3)

A. Lohmann, “Grating diffraction spectra as coherent light sources for two or three beam interferometry,” Opt. Acta 9, 1–12 (1962).
[CrossRef]

J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327–339 (1971).
[CrossRef]

R. Barakat, “First-order probability densities of laser speckle patterns observed through finite-size scanning apertures,” Opt. Acta 20, 729–740 (1973).
[CrossRef]

Opt. Commun. (2)

J. C. Dainty, W. T. Welford, “Reduction of speckle in image plane hologram reconstruction by moving pupils,” Opt. Commun. 3, 289–294 (1971).
[CrossRef]

X. Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Opt. Eng. (Bellingham) (1)

T. Matsumoto, Y. Kitagawa, T. Minemoto, “Sensitivity-variable moiré topography with a phase shift method,” Opt. Eng. (Bellingham) 35, 1754–1760 (1996).
[CrossRef]

Opt. Laser Technol. (1)

G. T. Reid, R. C. Rixon, H. I. Messer, “Absolute and comparative measurement of three-dimensional shape by phase measuring moiré topography,” Opt. Laser Technol. 16, 315–319 (1984).
[CrossRef]

Opt. Quantum Electron. (1)

T. S. McKechnie, Opt. Quantum Electron. 8, 61–67 (1976).
[CrossRef]

Other (11)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

R. P. Kanwal, Linear Integral Equations (Academic, New York, 1971).

L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, Cambridge, UK, 1985).

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

A. Papoulis, Probability, Random Variables, and Stochastic Progress, 2nd ed. (McGraw-Hill, New York, 1984).

B. W. Bell, “Digital heterodyne topography,” in Photomechanics and Speckle Metrology, F. Chiang, ed., Proc. SPIE814, 754–768 (1987).

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley Interscience, New York, 1990).

B. F. Oreb, K. G. Larkin, P. Fairman, M. Ghaffari, “Moiré based optical surface profiler for the minting industry,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. SPIE1776, 48–57 (1992).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 350–393.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. XXVIII, pp. 271–359.

J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic diagram of a projected fringe profile measurement system with a coherent light source. L, distance between object and pupil planes; L, distance between pupil and image planes. (b) Definitions of region R and the xoy coordinate system. Region R is the effective aperture of the detection cell in question. The origin of the xoy system is chosen to be the center of the cell, and 2Δ is the cell dimension.

Fig. 2
Fig. 2

Phase-measurement errors for Rs(Δx, Δy)=jinc[(1.22π/w)Δx]jinc[(1.22π/w)Δy], Rs(Δx, Δy)=sinc[(π/w)Δx]sinc[(π/w)Δy], and Rs(Δx, Δy)=exp{[-3(Δx2+Δy2)/2w2]}. n, p, Number of speckles in a cell and number of cells per grating period, respectively. w, Δ, Half-widths of Rs(Δx, Δy) and the detection cells, respectively. Functions jinc(x) and sinc(x) are defined in Eqs. (35a) and (35b).

Fig. 3
Fig. 3

Mean value of phase-measurement error for different p and n, where p and n are the number of cells per grating period and the number of speckles in a cell, respectively.

Fig. 4
Fig. 4

Dependence of phase error on p and n, where p and n are the number of cells per grating period and the number of speckles in a cell, respectively.

Fig. 5
Fig. 5

Phase-error probability-density functions for several values of n and p=10, where p and n are the number of cells per grating period and the number of speckles in a cell, respectively.

Fig. 6
Fig. 6

(a) Phase-error probability-density function for (a) p=10 and n=1, (b) p=10 and n=4, and (c) p=10 and n=16, where p and n are the number of cells per grating period and the number of speckles in a cell, respectively.

Fig. 7
Fig. 7

(a) Probability-density functions obtained from simulations, calculations from Eq. (49), and curve fitting for (a) p=10 and n=9, (b) p=10 and n=16, and (c) p=10 and n=25, where p and n are the number of cells per grating period and the number of speckles in a cell, respectively.

Fig. 8
Fig. 8

Definitions of regions R1 and R2, which represent two detection cells centered at (ξ1, η1) and (ξ2, η2) in the image plane.

Tables (1)

Tables Icon

Table 1 Phase Errors Obtained from Simulations and Eq. (50) for Square and Circular Aperturesa

Equations (75)

Equations on this page are rendered with MathJax. Learn more.

|f(ξ, η)|2=B(ξ, η)+M(ξ, η)cos2πdξ+φ(ξ, η)+HOH,
i(ξ, η)=-+-+f(ξ, η)r(ξ, η)×p(ξ, η)h(ξ-ξ, η-η)dξdη,
i(ξ, η)=f(ξ, η)r(ξ, η)-+-+p(ξ, η)×h(ξ-ξ, η-η)dξdη=f(ξ, η)r(ξ, η)s(ξ, η),
s(ξ, η)=-+-+p(ξ, η)h(ξ-ξ, η-η)dξdη.
s(ξ, η)=sr(ξ, η)+jsi(ξ, η),
I(ξ, η)=|i(ξ, η)|2=B(ξ, η)+M(ξ, η)×cos2πdξ+φ(ξ, η)×|r(ξ, η)|2|s(ξ, η)|2.
I(ξ, η)=B(ξ, η)+M(ξ, η)×cos2πdξ+φ(ξ, η)×|r(ξ, η)|2S(ξ, η).
I¯=r024Δ2-ΔΔ-ΔΔB0+M0 cos2πdx+φh+φ0×S(x, y)dxdy,
φ=tan-1I4¯-I2¯I1¯-I3¯,
Ii¯=r024Δ2-ΔΔ-ΔΔB0+M0 cos2πdx+φh+φ0+(i-1)π2S(x, y)dxdy
I4¯-I2¯=r02M02Δ2sin(φ0+φh)-ΔΔ-ΔΔ cos2πdx×S(x, y)dxdy+cos(φ0+φh)-ΔΔ-ΔΔ sin2πdx×S(x, y)dxdy,
I1¯-I3¯=r02M02Δ2cos(φ0+φh)-ΔΔ-ΔΔ cos2πdx×S(x, y)dxdy-sin(φ0+φh)-ΔΔ-ΔΔ sin2πdx×S(x, y)dxdy.
φ=φ0+φh+Δφ,
Δφ=tan-1-ΔΔ-ΔΔ sin2πdxS(x, y)dxdy-ΔΔ-ΔΔ cos2πdxS(x, y)dxdy.
s(x, y)=n=0anΦn(x, y),
-ΔΔ-ΔΔΦm(x, y)Φn*(x, y)dxdy=δ[m-n].
an=-ΔΔ-ΔΔs(x, y)Φn(x, y)dxdy.
E(aman*)=υn2δ[m-n],
-ΔΔ-ΔΔRs(x-x, y-y)Φn(x, y)dxdy
=υn2Φn(x, y),
Rs(x-x, y-y)Es(x, y)s*(x, y).
Rs(x-x, y-y)=F[Λ(α, β)],
Rs(x-x, y-y)=Rx(x-x)Ry(y-y),
Φn(x, y)=ψk(x)ϕl(y),
-ΔΔRx(x-x)ψk(x)dx=χkψk(x),
-ΔΔRy(y-y)ϕl(y)dy=κlϕl(y).
s(x, y)=k,l=0cklψk(x)ϕl(y),(x, y)R,
ckl=-ΔΔ-ΔΔs(x, y)ψk(x)ϕl(y)dxdy,
-ΔΔψm(x)ψn*(x)dx=δ[m-n],
-ΔΔϕk(y)ϕl*(y)dy=δ[k-l],
E(cklcmn*)=χkκnδ[k-l, m-n].
Rs(x-x, y-y)=sinπw(x-x)sinπw(y-y)πw2(x-x)(y-y),
sr(x, y)=m,n=0cmnψm(x)ϕn(y),
si(x, y)=k,l=0dklψk(x)ϕl(y).
s(x, y)=m,n=0cmnψm(x)ϕn(y)+jk,l=0dklψk(x)ϕl(y).
S(x, y)
=|s(x, y)|2
=m,n=0m,n=0cmncmnψm(x)ψm(x)ϕn(y)ϕn(y)
+k,l=0k,l=0dkldklψk(x)ψk(x)ϕl(y)ϕl(y).
Δφ=tan-1n=0m=0m=0(cmncmn+dmndmn)-ΔΔ sin2πdxψm(x)ψm(x)dxl=0k=0k=0(cklckl+dkldkl)-ΔΔ cos2πdxψk(x)ψk(x)dx.
nΔw,
pd2Δ,
sincπwΔxsincπwΔy,exp-32w2(Δx2+Δy2),
jinc1.22πwΔxjinc1.22πwΔy
sinc(x)sin xx,
jinc(x)2J1(x)x,
3σphase=0.328p-0.992n0.386n+0.362n+0.267
(p10),
S(x, y)¯E[S(x, y)]S¯,
ΔS(x, y)S(x, y)-S¯;
ΔΘ1-ΔΔ-ΔΔ sin2πdxΔS(x, y)dxdy,
Γ2S¯-ΔΔ-ΔΔ cos2πdxdxdy,
ΔΘ2-ΔΔ-ΔΔ cos2πdxΔS(x, y)dxdy.
Δϕ=tan-1ΔΘ1Γ2+ΔΘ2.
σ1=-ΔΔ-ΔΔ-ΔΔ-ΔΔ sin2πdxsin2πdx×RΔS(x-x, y-y)dxdxdydy1/2,
σ2=-ΔΔ-ΔΔ-ΔΔ-ΔΔ cos2πdxcos2πdx×RΔS(x-x, y-y)dxdxdydy1/2.
ρ=1σ1σ2-ΔΔ-ΔΔ-ΔΔ-ΔΔ sin2πdxcos2πdx×RΔS(x-x, y-y)dxdxdydy.
fΔφ(Δφ)=k2(tan2 Δφ+1)π(k22 tan2 Δφ+1)×exp-k1221+πk122(k22 tan2 Δφ+1)1/2×expk122(k22 tan2 Δφ+1)×erfk1[2(k22 tan2 Δφ+1]1/2,
erf(x)2π0x exp(-z2)dz,
k1=Γ2/σ2,
k2=σ2/σ1.
fΔφ(Δφ)=k2(tan2 Δφ+1)(k22 tan2 Δφ+1)k122π(k22 tan2 Δφ+1)1/2×erfk1[2(k22 tan2 Δφ+1)]1/2×exp-k12k22 tan2 Δφ2(k22 tan2 Δφ+1).
σphase=12πfΔφ(0)=1k1k2×erf(k1/2).
S(x, y)=-+P(λ)Sλ(λ; x, y)dλ,
S¯total=S¯λ-+P(λ)dλ.
ρ(λ1, λ2; Δx, Δy)=Ψ(λ1, λ2)T(λ1, λ2; Δx, Δy).
Ψ(λ1, λ2)=exp-8π2ra21λ1-1λ22,
T(λ1, λ2; Δx, Δy)=-+-+Λ(λ1x0-L sin θ, λ1y0)Λ*(λ2x0-L sin θ, λ2y0)exp[i(x0Δx+y0Δy)]dx0dy0-+-+|Λ(λ1x0, λ1y0)|2dx0dy0-+-+|Λ(λ2x0, λ2y0)|2dx0dy01/2.
RΔS(Δx, Δy)
=S¯total2--P(λ1)P(λ2)|ρ(λ1, λ2; Δx, Δy)|2dλ1dλ2-P(λ)dλ2.
Δφ(ξi, ηi)=tan-1Θ1(ξi, ηi)Θ2(ξi, ηi),
Θ1(ξi, ηi)-ΔΔΔΔ sin2πdξ×S(ξ-ξi, η-ηi)dξdη,
Θ2(ξi, ηi)-ΔΔΔΔ cos2πdξ×S(ξ-ξi, η-ηi)dξdη,
2w=1.22(1+Mimage)Fλ,
n=Δ0.61(1+Mimage)Fλ2.

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