Abstract

We present an optical method for detection of in-plane movement of a diffusing object. The technique is based on spatial filtering of the laser speckle pattern, which is produced by illumination of the object with coherent light. Two interlaced differential comb photodetector arrays act as a periodic filter to the spatial-frequency spectrum of the speckle pattern intensity. The detector produces a zero-offset, periodic output signal versus displacement that permits measurement of the movement at arbitrarily low speed. The direction of the movement can be detected with the help of the quadrature signal, which is produced by a second pair of interlaced comb photodetector arrays. When speckle size and period of the comb photodetector arrays are matched, the output signal versus displacement is quasi-sinusoidal with statistical amplitude and phase. First- and second-order statistics of the signal are investigated. First the probability density function and the autocorrelation function of the complex Fourier transform of the speckle pattern intensity are determined. Then the statistical properties of the spectrum of the filtered signal and of the signal itself are calculated. It turns out that the amplitude of the signal is Rayleigh distributed. Both the autocorrelation function of the signal and the probability density function of the measured phase difference for a given displacement are calculated. The potential accuracy of displacement measurements is analyzed. In addition, the signal quality is investigated with respect to the geometry of the detector. The theoretical results are experimentally verified.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. I. Yamaguchi and S. Komatsu, “Theory and applications of dynamic laser speckles due to in-plane object motion,” Opt. Acta 24, 705–724 (1977).
    [CrossRef]
  2. T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
    [CrossRef]
  3. B. E. A. Saleh, “Speckle correlation measurement of the velocity of a small rotating rough surface,” Appl. Opt. 14, 2344–2346 (1975).
    [CrossRef] [PubMed]
  4. J. Ohtsubo and T. Asakura, “Velocity measurement of a diffuse object by using time-varying speckles,” Opt. Quantum Electron. 8, 523–529 (1976).
    [CrossRef]
  5. N. Takai, T. Iwai, and T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossing of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
    [CrossRef]
  6. M. Naito, M. Ishigami, and A. Kobayashi, “Spatial filter and its application to industrial measurement,” presented at IMEKO-V, Versailles, May 25–30, 1970.
  7. A. Kobayashi, “Measurement and sensors,” Sens. Actuators 13, 29–41 (1988).
    [CrossRef]
  8. H. Ogiwara and H. Ukita, “A speckle pattern velocimeter using a periodical differential detector,” Jpn. J. Appl. Phys. Suppl. 14, 307–310 (1975).
    [CrossRef]
  9. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.
  10. J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. XIV, pp. 1–46.
  11. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 60–79.
  12. Hilbert transformation algorithm implemented in the MATLAB function hilbert, in Signal Processing Toolbox for Use with MATLAB (MathWorks, Natick, Mass., 1994).
  13. F. de Coulon, “Théorie et traitement des signaux,” Traité d’Electricité (Presses Polytechniques Romandes, Lausanne, 1984), Vol. VI, pp. 111–144.
  14. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 33–50.
  15. S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 24, 46–159 (1945).
    [CrossRef]
  16. A. W. Drake, Fundamentals of Applied Probability Theory (McGraw-Hill, New York, 1967), pp. 203–221.
  17. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991), pp. 86–102.
  18. S. Lowenthal and H. Arsenault, “Image formation for coherent diffuse objects: statistical properties,” J. Opt. Soc. Am. 60, 1478–1483 (1970).
    [CrossRef]
  19. R. Dändliker and F. M. Mottier, “Determination of coherence length from speckle contrast on a rough surface,” J. Appl. Math. Phys. 22, 369–380 (1971).
    [CrossRef]

1988

A. Kobayashi, “Measurement and sensors,” Sens. Actuators 13, 29–41 (1988).
[CrossRef]

1981

T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

1980

1977

I. Yamaguchi and S. Komatsu, “Theory and applications of dynamic laser speckles due to in-plane object motion,” Opt. Acta 24, 705–724 (1977).
[CrossRef]

1976

J. Ohtsubo and T. Asakura, “Velocity measurement of a diffuse object by using time-varying speckles,” Opt. Quantum Electron. 8, 523–529 (1976).
[CrossRef]

1975

H. Ogiwara and H. Ukita, “A speckle pattern velocimeter using a periodical differential detector,” Jpn. J. Appl. Phys. Suppl. 14, 307–310 (1975).
[CrossRef]

B. E. A. Saleh, “Speckle correlation measurement of the velocity of a small rotating rough surface,” Appl. Opt. 14, 2344–2346 (1975).
[CrossRef] [PubMed]

1971

R. Dändliker and F. M. Mottier, “Determination of coherence length from speckle contrast on a rough surface,” J. Appl. Math. Phys. 22, 369–380 (1971).
[CrossRef]

1970

1945

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 24, 46–159 (1945).
[CrossRef]

Arsenault, H.

Asakura, T.

T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

N. Takai, T. Iwai, and T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossing of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

J. Ohtsubo and T. Asakura, “Velocity measurement of a diffuse object by using time-varying speckles,” Opt. Quantum Electron. 8, 523–529 (1976).
[CrossRef]

Dändliker, R.

R. Dändliker and F. M. Mottier, “Determination of coherence length from speckle contrast on a rough surface,” J. Appl. Math. Phys. 22, 369–380 (1971).
[CrossRef]

Iwai, T.

Kobayashi, A.

A. Kobayashi, “Measurement and sensors,” Sens. Actuators 13, 29–41 (1988).
[CrossRef]

Komatsu, S.

I. Yamaguchi and S. Komatsu, “Theory and applications of dynamic laser speckles due to in-plane object motion,” Opt. Acta 24, 705–724 (1977).
[CrossRef]

Lowenthal, S.

Mottier, F. M.

R. Dändliker and F. M. Mottier, “Determination of coherence length from speckle contrast on a rough surface,” J. Appl. Math. Phys. 22, 369–380 (1971).
[CrossRef]

Ogiwara, H.

H. Ogiwara and H. Ukita, “A speckle pattern velocimeter using a periodical differential detector,” Jpn. J. Appl. Phys. Suppl. 14, 307–310 (1975).
[CrossRef]

Ohtsubo, J.

J. Ohtsubo and T. Asakura, “Velocity measurement of a diffuse object by using time-varying speckles,” Opt. Quantum Electron. 8, 523–529 (1976).
[CrossRef]

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 24, 46–159 (1945).
[CrossRef]

Saleh, B. E. A.

Takai, N.

T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

N. Takai, T. Iwai, and T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossing of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

Ukita, H.

H. Ogiwara and H. Ukita, “A speckle pattern velocimeter using a periodical differential detector,” Jpn. J. Appl. Phys. Suppl. 14, 307–310 (1975).
[CrossRef]

Yamaguchi, I.

I. Yamaguchi and S. Komatsu, “Theory and applications of dynamic laser speckles due to in-plane object motion,” Opt. Acta 24, 705–724 (1977).
[CrossRef]

Appl. Opt.

Appl. Phys.

T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

Bell Syst. Tech. J.

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 24, 46–159 (1945).
[CrossRef]

J. Appl. Math. Phys.

R. Dändliker and F. M. Mottier, “Determination of coherence length from speckle contrast on a rough surface,” J. Appl. Math. Phys. 22, 369–380 (1971).
[CrossRef]

J. Opt. Soc. Am.

Jpn. J. Appl. Phys. Suppl.

H. Ogiwara and H. Ukita, “A speckle pattern velocimeter using a periodical differential detector,” Jpn. J. Appl. Phys. Suppl. 14, 307–310 (1975).
[CrossRef]

Opt. Acta

I. Yamaguchi and S. Komatsu, “Theory and applications of dynamic laser speckles due to in-plane object motion,” Opt. Acta 24, 705–724 (1977).
[CrossRef]

Opt. Quantum Electron.

J. Ohtsubo and T. Asakura, “Velocity measurement of a diffuse object by using time-varying speckles,” Opt. Quantum Electron. 8, 523–529 (1976).
[CrossRef]

Sens. Actuators

A. Kobayashi, “Measurement and sensors,” Sens. Actuators 13, 29–41 (1988).
[CrossRef]

Other

M. Naito, M. Ishigami, and A. Kobayashi, “Spatial filter and its application to industrial measurement,” presented at IMEKO-V, Versailles, May 25–30, 1970.

A. W. Drake, Fundamentals of Applied Probability Theory (McGraw-Hill, New York, 1967), pp. 203–221.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991), pp. 86–102.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. XIV, pp. 1–46.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 60–79.

Hilbert transformation algorithm implemented in the MATLAB function hilbert, in Signal Processing Toolbox for Use with MATLAB (MathWorks, Natick, Mass., 1994).

F. de Coulon, “Théorie et traitement des signaux,” Traité d’Electricité (Presses Polytechniques Romandes, Lausanne, 1984), Vol. VI, pp. 111–144.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 33–50.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Optical setup for detection of in-plane movement of a diffusing object with laser speckle patterns.

Fig. 2
Fig. 2

Typical zero-offset, quasi-sinusoidal output signal with spatial period Λ=1/p0.

Fig. 3
Fig. 3

Possible photodetector geometry for spatial filtering with sharp peaks at spatial frequencies px=±1/Λ in the transfer function. The photodetector is composed of two comb photodetector arrays s1 and s2 connected differentially. Each array has N fingers of width w and length ly. The overall detector length is lx=NΛ. A second pair of comb photodetector arrays s1q and s2q can be interlaced to detect the direction of the movement.

Fig. 4
Fig. 4

(a) Transfer function |sˆ(px, 0)|2 of the photodetector, which is shown in Fig. 3, for the case of w=Λ/4 and lx=5Λ. (b) Power spectral density GI(px, 0) of the speckle pattern intensity in the image plane in the case of a circular (solid curve) and a square (dashed curve) lens aperture. The cut-off spatial frequency pm is related to the speckle size Δxs through pm=4/πΔxs (circular lens aperture). (c) Power spectral density Gi(px, 0) of the zero-offset, quasi-sinusoidal output signal.

Fig. 5
Fig. 5

Typical measured signals i1(x) and i2(x) in phase opposition for N=5 fingers. The zero-offset, quasi-sinusoidal output signal i(x) is calculated through i=i1-i2. The amplitude u(x) is the envelope function of the output signal i(x).

Fig. 6
Fig. 6

Power spectral density Gi(px, 0) of the experimentally measured output signal i(x) in Fig. 5. The spectral bandwidth Δpx is indicated for comparison with Fig. 4(a).

Fig. 7
Fig. 7

Histogram of the amplitude u of the output signal i, obtained from 32,000 measured samples for N=3, 5, 11, and 25 fingers. The best-fitting Rayleigh distributions are shown as solid curves.

Fig. 8
Fig. 8

Histogram of the phase difference Δϕ for displacement Δx=Λ/4 and Λ/2 of the output signal i, obtained from 32,000 measured samples for N=5. The best fitting probability density functions are shown as solid and dotted curves.

Fig. 9
Fig. 9

Autocorrelation function Ci and correlation factor μm of the measured output signal i. The theoretical correlation factor μ is shown as dashed curve. Fitting the probability density function to the histograms of the phase difference Δϕ yields the values μfit indicated by circles.

Fig. 10
Fig. 10

Relative average error of the output signal i versus the number of fingers N.

Fig. 11
Fig. 11

Experimental investigation of the signal contrast Γ for comb photodetector arrays with N=3, 5, 7, 11, 25, and 41 fingers (ly/Λ=25, w/Λ=0.25, Λ=100 µm). The analytically calculated values and the values obtained from numerical integration are shown as solid and dotted curves, respectively.

Equations (53)

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

i(x, y)=-+-+dξdηI(ξ, η)s(ξ-x, η-y)=I(x, y)s(x, y),
Gi(px, py)=GI(px, py)|sˆ(px, py)|2,
s1(x, y)=k=-+δ(x-kΛ)rect(x/w)×rect(x/lx)×rect(y/ly),k  
s2(x, y)=s1(x-Λ/2, y).
|sˆ1(px, py)|2=(p0lxw)2×k=-+δ(px-kp0) sinc2(wpx)sinc2(lxpx)×ly2sinc2(lypy),
|sˆ(px, py)|2=|sˆ1-sˆ2|2=(2p0lxw)2k=-+δ[px-(2k+1)p0]×sinc2(wpx)sinc2(lxpx)×ly2sinc2(lypy).
(2k+1)p0+12Δpx[2(k+1)+1]p0-12Δpx,
Nmin=3.
Δxs=4πλdID.
GI(px, py)=E2(I)δ(px, py)+8pm2π2×cos-1ρpm-ρpm1-ρpm2,
pm=D/λdI=4/πΔxs.
Gi(px, py)=GI(px, py)(2p0lxw)2×[δ(px-p0)+δ(px+p0)]×sinc2(wpx)sinc2(lxpx)×ly2sinc2(lypy),
i(x)=u(x)cos[2πp0x+ϕ(x)],
I(x)=d2xd2xρ(x)ρ*(x)h(x-x)h*(x-x),
Iˆ(p)=|Iˆ(p)|exp[jφ(p)]=d2xI(x)exp(-j2πpx),
pdf(|Iˆ(p)|)=|Iˆ(p)|κ2(p)exp-|Iˆ(p)|22κ2(p)|Iˆ(p)|00otherwise,
pdf(φ)=12π-πφ<π0otherwise.
η(p)=π/2|Chˆ(p)|σ,
χ2(p)=(2-π/2)|Chˆ(p)|2σ2.
CIˆ(Δp)=E[Iˆ(p)Iˆ*(p-Δp)].
CIˆ(Δp)=Chˆ(p)Chˆ(Δp-p)δ(p)δ(Δp-p)+C|hˆ|2(p)δ(Δp).
i(x)=FT-1{Iˆ(px)sˆ(px)}.
u(x)cos(2πp0x+ϕ(x))=FT-1{|Iˆ(px)|exp[jφ(px)]×[δ(px-p0)+δ(px+p0)]sinc(lxpx)},
pdf(u)=uτ2exp-u22τ2u00otherwise.
Ci(Δx)=μ(Δx)cos(2πp0Δx),
μ(Δx)=FT-1{sinc2(lxpx)}=1-|Δx|lx|Δx|lx0otherwise
pdf(Δϕ)=1-μ22π(1-β2)3/2β sin-1 β+π2β+1-β2,
nzc=213pmax3-pmin3pmax-pmin1/2.
nzc=2Λ1+(1/2N)231/22Λ1+124N2,
E2(i1)=-+-+dpxdpyGi1(px, py)δ(px, py).
E2(i1)=E2(I)(Nwly)2,
σi2=-+-+dpxdpyGi(px, py)[1-δ(px, py)].
σi2E2(I)Nlyw2Λ sinc2(w/Λ).
Γ2=σi24E2(i1)14ΛNlysinc2(w/Λ),
Iˆ(p)=d2xd2xρ(x)ρ*(x)[hˆ(p)exp(-j2πpx)hˆ*(-p)exp(-j2πpx)].
Iˆ(p)=d2phˆ(p)hˆ*(p-p)Iˆ(p)×d2xd2xρ(x)ρ*(x)×exp[-j2πp(x-x)]exp(-j2πpx).
Iˆ(p)=d2phˆ(p)hˆ*(p-p)Γ(p, p).
Γ(p, p)=m,n exp[j(φm-φn)]exp[-j2πp(xm-xn)]×exp(-j2πpxn),
Γ(p, p)=m=nan2 exp(-j2πpxn)+mnaman exp[j(φm-φn)]×exp[-j2πp(xm-xn)]×exp(-j2πpxn)
=Ka2δ(p)+α(p, p).
pdf(|α|2)=12σ2exp-|α|22σ2|α|200otherwise,
CIˆ(Δp)=Ed2pd2xd2x×hˆ(p)hˆ*(p-p)ρ(x)ρ*(x)×exp[-j2πp(x-x)]×exp(-j2πpx)d2pd2xd2x×hˆ*(p)hˆ(p-p+Δp)ρ*(x)ρ(x)×exp[j2πp(x-x)]×exp[j2π(p-Δp)x].
CIˆ(Δp)=d2pd2pd2xd2xd2xd2x×Hˆ(p, p, p, Δp)E[ρ(x)ρ*(x)×ρ*(x)ρ(x)]exp(-jψ),
Hˆ(p, p, p, Δp)=hˆ(p)hˆ*(p-p)hˆ*(p)×hˆ(p-p+Δp)
Ψ=2π[px+p(x-x)-(p-Δp)x-p(x-x)].
E[ρ(x)ρ*(x)ρ*(x)ρ(x)]=δ(x-x)δ(x-x)+δ(x-x)δ(x-x).
CIˆ(Δp)=d2pd2pd2xd2xHˆ(p, p, p, Δp)×exp(-j2πpx)exp[j2π(p-Δp)x]+d2pd2pd2xd2xHˆ(p, p, p, Δp)×exp(-j2πΔpx)×exp[-j2π(p-p)(x-x)].
CIˆ(Δp)=d2pd2pHˆ(p, p, p, Δp)δ(p)δ(Δp-p)+d2pd2pHˆ(p, p, p, Δp)δ(Δp)×δ(p-p).
GI(px(y))=E2[Ix(y)]δ[px(y)]+1pmTpx(y)pm,
σi2=σix2×σiy2=-+dpxGi(px)[1-δ(px)]×-+dpyGi(py)[1-δ(py)].
σix22Gi(p0)δpx2GI(p0)(2p0lxw)2 sinc2(wp0)1lx,
σix22E2(Ix)Nw2 sinc2(w/Λ),
σiy2Gi(0)[1-δ(py)]δpy12E2(Iy)lyΛ,

Metrics