Space–time correlation properties of dynamic laser speckle in the near diffraction field of a longitudinally moving diffuse object

Marek Kowalczyk; Eusebio Bernabeu

doi:10.1364/JOSAA.6.000758

Journal of the Optical Society of America A
Vol. 6,
Issue 6,
pp. 758-764
(1989)
•https://doi.org/10.1364/JOSAA.6.000758

Space–time correlation properties of dynamic laser speckle in the near diffraction field of a longitudinally moving diffuse object

Marek Kowalczyk and Eusebio Bernabeu

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Marek Kowalczyk and Eusebio Bernabeu, "Space–time correlation properties of dynamic laser speckle in the near diffraction field of a longitudinally moving diffuse object," J. Opt. Soc. Am. A 6, 758-764 (1989)

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Abstract

The space–time autocorrelation function of detected speckle-intensity fluctuation is derived for a quite general case in which both the diffuse object and the photodetector move along the axis of an illuminating Gaussian beam. Correlation properties of some particular cases of this general arrangement are then analyzed. The considerations presented here are limited to the near diffraction field in which the speckle complex amplitude is assumed to obey circular Gaussian statistics. New time-stationary processes are revealed, and the possibility of their application in speckle velocimetry is discussed.

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Values of ϑ z₀, and R₀ If Not Arbitrary	Schematic Arrangement	Space–Time Autocorrelation	Time Autocorrelation	Stationarity	Stationarity Conditions	Velocimetry Formula
ΔI(x; t)	Fig. 1	Γ_Δ_I (x₁, x₂; t₁, t₂) [Eq. (19)]	Γ_Δ_I(x;t₁, t₂) [Eq. (20)]	−
ΔI′(x; t),ϑ = 0, z₀ = 0	Fig. 1	Γ_Δ_I_′ (x₁, x₂;t₁, t₂)^a	Γ_Δ_I_′ (x; t₁, t₂)	−
			γ_Δ_I_′ (0; t₂ − t₁) [Eq. (25)]	+	x = 0, R₀ ≫ z_j	$\begin{array}{l} ∣ v ∣ = 4 π^{2} {w_{0}}^{2} Δ f / λ \\ = 2 π^{2} {w_{0}}^{2} N_{0} / \sqrt{3} λ \end{array}$
ΔI″(x; t), ϑ = 2, R₀ = z₀	Fig. 2	Γ_Δ_I_″(x₁, x₂; t₁, t₂)^a	Γ_Δ_I_″(x; t₁, t₂) [Eq. (28)]	−
$Δ {I_{s}}^{″} (0; t) = \frac{Δ I^{″} (0; t)}{π {w_{0}}^{2} / z^{2}}$	Fig. 2		γ_{ΔI_s″}(0; t₂ − t₁) [Eq. (31)]	+	z_j > a	$∣ v ∣ = π^{2} {w_{0}}^{2} N_{0} / \sqrt{3} λ$
ΔI‴(x; t), ϑ = 1	Fig. 3	Γ_Δ_I_‴(\|x₂ − x₁\|; t₁, t₂) [Eq. (43)]	γ_Δ_I_‴(t₂ − t₁) [Eq. (34)]	+	Unconditional	$\begin{array}{l} ∣ v ∣ = 4 π^{2} {w_{0}}^{2} Δ f / λ \\ = 2 π^{2} {w_{0}}^{2} N_{0} / \sqrt{3} λ \end{array}$
ΔI_r‴(x; t)	Fig. 4			+	Unconditional	$\begin{array}{l} ∣ v_{r} ∣ = \| \frac{d (z / 2)}{d t} \| \\ = 2 π^{2} {w_{0}}^{2} Δ f / λ \\ = π^{2} {w_{0}}^{2} N_{0} / \sqrt{3} λ \end{array}$
ΔI_i‴(t), ϑ = 1	Figs. 3 and 4^b		Γ_{ΔI_i} (t₁, t₂) Eq. (45)	−

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Journal of the Optical Society of America A