Abstract

Within the framework of ABCD matrix theory an exact analytical expression is derived for the space–time-lagged photocurrent covariance that is valid for arbitrary (complex) ABCD optical systems, i.e., systems that include Gaussian-shaped apertures and partially developed speckle. General expressions are derived for the mean spot size and both the mean speckle size and the temporal coherence length. Additionally, a general description of both speckle boiling and speckle translation in an arbitrary observation plane is given. Included in the analysis is the effect of a finite wave-front-curvature radius for the Gaussian-shaped laser beam illumination of the target. The effects of diffraction and wave-front-curvature radius are discussed for both imaging systems and a Fourier transform system. It is shown that, whereas diffraction affects the speckle dynamics in both cases, a finite wave-front curvature affects only the speckle dynamics in the Fourier transform system. Further, the effects of finite detector apertures are considered, in which the effects of speckle averaging are included and discussed. In contrast to previous work, the obtained analytical results are expressed in a relatively compact form yet fully contain all diffraction effects and apply to an arbitrary ABCD optical system.

© 1998 Optical Society of America

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References

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  1. T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
    [CrossRef]
  2. H. T. Yura, S. G. Hanson, “Laser-time-of flight velocimetry: analytical solution to the optical system based on ABCD matrices,” J. Opt. Soc. Am. A 10, 1918–1924 (1993).
    [CrossRef]
  3. H. T. Yura, S. G. Hanson, L. Lading, “Laser Doppler velocimetry: analytical solution to the optical system including the effects of partial coherence of the target,” J. Opt. Soc. Am. A 12, 2040–2047 (1995).
    [CrossRef]
  4. B. Rose, H. Imam, S. G. Hanson, H. T. Yura, “Effects of target structure on the performance of laser time-of-flight velocimeter systems,” Appl. Opt. 36, 518–533 (1997).
    [CrossRef]
  5. T. Yoshimura, “Statistical properties of dynamic speckle,” J. Opt. Soc. Am. A 3, 1032–1054 (1986).
    [CrossRef]
  6. In this regard, included in Ref. 5 is an excellent review of the literature before 1986.
  7. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.
  8. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.
  9. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  10. B. Rose, H. Imam, S. G. Hanson, H. T. Yura, R. S. Hansen, “Laser-speckle angular-displacement sensor: theoretical and experimental study,” Appl. Opt. 37, 2119–2129 (1998).
    [CrossRef]
  11. For simplicity in notation we omit, in the following, the multiplicative factor exp[iω0t], where ω0 is the angular frequency of the incident light.
  12. S. Wolfram, mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).
  13. H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD systems,” J. Opt. Soc. Am. A 10, 316–323 (1993).
    [CrossRef]

1998

1997

1995

1993

1987

1986

1981

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

Asakura, T.

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

Goodman, J. W.

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

Grum, T. P.

Hansen, R. S.

Hanson, S. G.

Imam, H.

Lading, L.

Rose, B.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.

Takai, N.

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

Wolfram, S.

S. Wolfram, mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).

Yoshimura, T.

Yura, H. T.

Appl. Opt.

Appl. Phys.

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

J. Opt. Soc. Am. A

Other

In this regard, included in Ref. 5 is an excellent review of the literature before 1986.

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

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.

For simplicity in notation we omit, in the following, the multiplicative factor exp[iω0t], where ω0 is the angular frequency of the incident light.

S. Wolfram, mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).

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Equations (48)

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C(p1, p2; τ)=i(p1, t)i(p2, t+τ)-i(p1, t)i(p2, t+τ),
i(t)=αdpW(p)I(p, t),
i(p, t)αSI(p, t),
I(p, t)=|U(p, t)|2,
U(p, t)=drU0(r, t)G(r, p),
G(r, p)=-ik2πBexp-ik2B(Ar2-2r·p+Dp2),
U0(r, t)=Ui(r, t)ψ(r, t),
Bψ(r1, r2)=4πk21πrc2exp-(r1-r2)2rc2,
rc=rhσϕ,
ψ(r, t+τ)=ψ(r-vτ, t).
C(p1, p2; τ)=|U(p1, t)U*(p2, t+τ)|2=dr1dr2Ui(r1, t)Ui*(r2, t+τ)  ×G(r1, p1)G*(r2, p2)Bψ(r1, r2τ)2,
r2τ=r2+vτ.
Ui(r)=2Pπrs21/2 exp-r21rs2+ik2R,
C(p1, p2; τ)=C0 exp-2 p12+p22ω2|γ(Δp; τ)|2,
C0=8PαSπrs2k2ρ022,
|γ(Δp; τ)|=exp-Ai-2Brkrs22(vτ)2+Δp-Ar+2Bikrs2vτ2ρ02,
ρ0=8|B|2k2rs2+4kIm(BA*)+rc2|A|2+4|B|2k2rs4+4 Im(BA*)krs21/2,
ω=ρ0|B|22[Im(BD*)Im(BA*)-(Im B)2]+4|B|2krs2Im(BD*)+rc2I1/2,
I=k Im(DB*)|A2|+4|B2|k2rs4+4 Im(BA*)krs2-k Im(AB)-2rs2(Bi2-Br2),
|γ(0; τ)|=exp(-τ2/τc2),
τc=ρ0|v|1(Kb2+Kt2)1/2=ρ0|v||A2|+4|B2|k2rs4+4 Im(BA*)krs2-1/2,
Kt=Ar+2Bikrs2,
Kb=Ai-2Brkrs2.
|γ(Δp; τ)|=exp-(τ-τd)2τc2exp[-(βΔp/ρ0)2],
τd=Δp·v|v|2KtKb2+Kt2=Kt(τc/ρ0)2(Δp·v),
β=Kb(Kb2+Kt2)1/2.
Ar+2Bikrs2Ai-2Brkrs2(speckleboiling).
Ai-2Brkrs2=0(speckletranslation).
vs=Ktv=Ar+2Bikrs2v.
W(p)=exp-2p2σ2,
C¯(Δp; τ)=dp1W(p1)dp2W(p2)C(p1, p2; τ).
C¯(Δp; τ)=C¯0 exp-2Δp-2Ktω2vτ(ρ02+2ω2)2ρ02/(1+ρ02/2ω2)+2σ2-2(vτ)2Kb2ρ02+Kt2(ρ02+2ω2),
C¯0=C0 11+σ2ω21+σ2ω2+2σ2ρ02.
i=αdpW(p)I(p)=C01+σ2/ω2exp-Δp22ω2.
C¯(p1, p2; τ)=C¯0 exp-2(Δp-Ktvτ)2ρ02+2σ2-2Kb2(vτ)2ρ02,
C¯0=C0[1/(1+2σ2/ρ02)].
M=i2-i2i2=C¯(p, p; 0)i2.
M=1+σ2/ω21+σ2/ω2+2σ2/ρ02.
M=1/N,
N=1+2σ2/ρ02.
Δp=Ar+2Bikrs2vτ.
Δpvτ=-1+4f2k2rs2σa2=-N/(N-1),
N=1+krsσa2 f2.
C(k)=j=knijij-k-ijij-k,
Ci(Δpx)=i(px)i(px+Δpx)-i(px)i(px+Δpx),
|γ(Δpx)|=exp-Ai-2Brkrs22(R0θ)2+Δpx-Ar+2Bikrs2R0θ2ρ02.
Δpx=Ar+2Bikrs2R0θ.
px0=2θf+R0θfR-4fz1k2rs2σa2,

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