Abstract

The purpose of this research is the quantitative investigation of decorrelation-induced phase errors in speckle interferometry. Measurements in speckle interferometry are inherently affected by decorrelation, i.e., by alterations of the speckle fields during measurement. Likewise, the random phases carrying the interferometric information change during decorrelation. Image plane and pupil plane decorrelation are considered for both smooth and speckle reference wave interferometers. Since the decorrelation effect depends on the aperture and the pixel size, the calculations include not only the case of speckles being well resolved by the camera but also the case of unresolved speckles. Different standard deviations of the phase error are obtained from the probability density of the pixel modulation and the phase before and after decorrelation. Most cases (apart from pupil plane decorrelation in speckle reference wave setups) appear to obey exactly the same phase error statistics. In particular, the number of speckles per pixel does not affect the phase error distribution over the whole image. The only important parameters determining the decorrelation-induced phase errors are the amount of decorrelation and the pixel modulation.

© 1997 Optical Society of America

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References

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  1. D. W. Robinsonand, G. T. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics Publishing, Bristol, UK, 1993).
  2. W. Jüptner, W. Osten, eds., Fringe’93: Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns (Akademie Verlag, Berlin, 1993).
  3. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Spring–Verlag, Berlin, 1984).
  4. M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
    [CrossRef]
  5. E. Ochoa, J. W. Goodman, “Statistical properties of ray directions in a monochromatic speckle pattern,” J. Opt. Soc. Am. A 73, 943–949 (1983).
    [CrossRef]
  6. S. Donati, G. Martini, “Speckle-pattern intensity and phase: second-order conditional statistics,” J. Opt. Soc. Am. 69, 1690–1694 (1979).
    [CrossRef]
  7. C. K. Hong, H. S. Ryu, H. C. Lim, “Least-squares fitting of the phase map obtained in phase-shifting electronic speckle pattern interferometry,” Opt. Lett. 20, 931–933 (1995).
    [CrossRef] [PubMed]
  8. J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” in Interferometry VII: Techniques and Applications, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2544, 246–257 (1995).
    [CrossRef]
  9. K. Creath, “Phase-shifting speckle interferometry,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. SPIE556, 337–346, (1985).
    [CrossRef]
  10. X. Colonna de Lega, P. Jacquot, “Interferometric deformation measurement using object-induced dynamic phase shifting,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 169–179 (1996).
    [CrossRef]

1996 (1)

M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
[CrossRef]

1995 (1)

1983 (1)

E. Ochoa, J. W. Goodman, “Statistical properties of ray directions in a monochromatic speckle pattern,” J. Opt. Soc. Am. A 73, 943–949 (1983).
[CrossRef]

1979 (1)

Colonna de Lega, X.

X. Colonna de Lega, P. Jacquot, “Interferometric deformation measurement using object-induced dynamic phase shifting,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 169–179 (1996).
[CrossRef]

Creath, K.

K. Creath, “Phase-shifting speckle interferometry,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. SPIE556, 337–346, (1985).
[CrossRef]

Donati, S.

Goodman, J. W.

E. Ochoa, J. W. Goodman, “Statistical properties of ray directions in a monochromatic speckle pattern,” J. Opt. Soc. Am. A 73, 943–949 (1983).
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Spring–Verlag, Berlin, 1984).

Hong, C. K.

Huntley, J. M.

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” in Interferometry VII: Techniques and Applications, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2544, 246–257 (1995).
[CrossRef]

Jacquot, P.

X. Colonna de Lega, P. Jacquot, “Interferometric deformation measurement using object-induced dynamic phase shifting,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 169–179 (1996).
[CrossRef]

Lehmann, M.

M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
[CrossRef]

Lim, H. C.

Martini, G.

Ochoa, E.

E. Ochoa, J. W. Goodman, “Statistical properties of ray directions in a monochromatic speckle pattern,” J. Opt. Soc. Am. A 73, 943–949 (1983).
[CrossRef]

Ryu, H. S.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

E. Ochoa, J. W. Goodman, “Statistical properties of ray directions in a monochromatic speckle pattern,” J. Opt. Soc. Am. A 73, 943–949 (1983).
[CrossRef]

Opt. Commun. (1)

M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
[CrossRef]

Opt. Lett. (1)

Other (6)

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” in Interferometry VII: Techniques and Applications, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2544, 246–257 (1995).
[CrossRef]

K. Creath, “Phase-shifting speckle interferometry,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. SPIE556, 337–346, (1985).
[CrossRef]

X. Colonna de Lega, P. Jacquot, “Interferometric deformation measurement using object-induced dynamic phase shifting,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 169–179 (1996).
[CrossRef]

D. W. Robinsonand, G. T. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics Publishing, Bristol, UK, 1993).

W. Jüptner, W. Osten, eds., Fringe’93: Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns (Akademie Verlag, Berlin, 1993).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Spring–Verlag, Berlin, 1984).

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Figures (5)

Fig. 1
Fig. 1

Different regions of the speckle field contribute to the pixel intensity before and after decorrelation: Before decorrelation, the pixel records the contributions of subareas A and B; after a displacement Δx of the speckle field, the contribution of B is replaced by that of B′.

Fig. 2
Fig. 2

Conditional probability density of the decorrelation-induced phase error Δφ [Eq. (22)], conditioned on given values of IM, IM′ and decorrelation δ [Eq. (23)]. The phase error is more likely to be small for smaller decorrelations but also for higher initial and final pixel modulations.

Fig. 3
Fig. 3

Conditional and unconditional standard deviation of the decorrelation-induced phase error Δφ as a function of the decorrelation δ [Eqs. (24) and (26)]. The conditional standard deviation (broken curves) depends on the product of the initial and final pixel modulation [see Eq. (23)] it is shown for pixels having initial and final modulations of 0.25, 0.5, 1, 2, and 4 times the mean pixel modulation <IM>. For relatively low pixel modulations, the phase error is considerable already for small decorrelations. For total decorrelation, the phase error is uniformly distributed over (-π, π), leading to the standard deviation π/3. The number of speckles per pixel, or the camera resolution, does not affect the phase error distribution.

Fig. 4
Fig. 4

Before decorrelation, the lens aperture records the contributions of subareas A and B. After a displacement ΔD of the speckle field in the pupil plane, the contribution of B is replaced by that of B′.

Fig. 5
Fig. 5

Standard deviation of the decorrelation-induced phase error Δφ as a function of the decorrelation δ. The solid curve represents the theoretical standard deviation of the usual phase error, which is limited to the interval (-π, π) [Eq. (26)]. This curve applies to the situation in which two different object states are compared. The broken curves show the standard deviation of the phase error if the phase evolution is monitored during the decorrelation process and unwrapped along the temporal axis (typically for a series of consecutive images). In this case the accumulated phase errors can be considerably larger than π (yet be relatively small compared to the unwrapped phase). The simulation shows the results assuming that m = 10, 50, or 100 speckles per pixel, respectively.

Equations (39)

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I=A1+A2A1+A2*=I1+I2+2I1I2 cosφ1-φ2=I0+IM cos φ.
IM=IM expiφ=IMr+iIMi=2I1I2 expiφ1-φ2=2 A1A2*.
Ax, y=A+xAx+yAy,
A=A+ΔxAx.
pI, φx=1Iexp-IIIπI1/2×2λzπDexp-2λzπD2I φx2I.
σφxI=I2I1/2πD2λz.
σΔφΔx, I=Δx2πDλzI2I1/2
σΔφΔx, IM=ΔxIMIMDλzπ2.
σΔφΔx, IM=ΔxDλzπ2.
δ=Δx/Lx
IM=IMA+IMB.
IM=IMA+IMB.
pIM, IM=--pAIM-IMBpBIMB×pBIM-IM+IMBdIMBrdIMBi
pIMr, IMi=m4πIIRexp-m4IIRIMr2+IMi2,
mA=1-δm, mB=δm
pAIMAr, IMAi=m4πIIR1-δ×exp-m4IIR1-δIMAr2+IMAi2,
pBIMBr, IMBi=m4πIIRδ×exp-m4IIRδIMBr2+IMBi2.
pIM, IM, Δφ=m2IMIM8πI2IR2 δ2-δ×exp-mIM2+IM2-21-δIMIM cos Δφ4IIRδ2-δ.
IM=IM=πIIRm1/2,
pIM, IM, Δφ=π8IM4IMIMδ2-δ×exp-πIM2+IM2-21-δIMIM cos Δφ4IM2 δ2-δ.
μ=1-δ
pΔφ|IM,IM=a expb cos Δφ -π<Δφ<π.
b=π21-δδ2-δIMIMIM2.
σΔφIM, IM=π23+4I0bn=1-1nn2Inb1/2,
σΔφIM, IMconst.b b  1σΔφIM, IMIMIMIM.
σΔφ=π23-π arcsin1-δ+arcsin21-δ-12n=11-δ2nn21/2.
δ=1-2πarccosΔDD-ΔDD1-ΔDD21/2.
IMx, y=2 AR* AAx, y+ABx, y, IMx, y=2 AR*AAx, y+ABx, y,
pAr, Ai, Ar, Ai=exp-Ar2+Ai2+Ar2+Ai2-21-δArAr+AiAiIδ2-δπI2 δ2-δ.
IM=1/m k=1mIMk,
pIMK, IMk=exp-IMKr2+IMki2+IMkr2+IMki2-21-δIMkrIMkr+IMkiIMki4πIIRδ2-δ4πIIR2 δ2-δ.
JMkr=1/2IMkr+IMkr, JMki=1/2IMki+IMki, ΔJMkr=IMkr-IMkr, ΔJMki=IMki-IMki,
pJMkr, JMki, ΔJMkr, ΔJMki=exp-JMkr2+JMki22IIR2-δ2πIIR2-δ×exp-ΔJMkr2+ΔJMki28IIRδ8πIIRδ,
pJMr, JMi, ΔJMr, ΔJMi=m24πIIR2 δ2-δ×exp-mJMr2+JMi22IIR2-δ-mΔJMr2+ΔJMi28IIRδ,
pIM, IM=m24πIIR2δ2-δexp-mIMr2+IMi2+IMr2+IMi2-21-δIMrIMr+IMiIMi4IIRδ2-δ.
δimage plane and δpupil plane by 1-δtot=1-δimage plane1-δpupil plane.
σΔφΔx, I1, I2=Δx2πDλzII1+I22I1I21/2,
σΔφΔx, I0, IM=Δx IMIM2DλzI0I01/2,
IM=π/m½ I.

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