The biological tissue can be characterized by its optical properties, such as the absorption coefficient $\mu a$ and the reduced scattering coefficient $\mu s$’. When diffusing photons scatter from moving scatterers, they experience phase shifts and cause the speckle fluctuation at the detector side. The motion information below the tissue surface is carried in the electric field of diffuse light $E(r\u2192,\tau )$ and can be extracted from the electric field autocorrelation function, which is defined as $G1(r\u2192,\tau )=\u2329E(r\u2192,t)E*(r\u2192,t+\tau )\u232a$. It has been shown that $G1(r\u2192,\tau )$ satisfies a correlation diffusion equation, i.e.,^{1}^{,}^{3}^{,}^{4}^{,}^{37}Display Formula
$[\u221213\mu s\u2032\u22072+\mu a+13\alpha \mu s\u2032k02\u2329\Delta r2(\tau )\u232a]G1(r\u2192,\tau )=S(r\u2192),$(1)
where $k0$ is the wavenumber of the light in a medium, $\alpha $ is the fraction of photon scattering events from moving scatterers out of total scatterers, and $\u2329\Delta r2(\tau )\u232a$ represents the mean square displacement of the moving scatterers, and is commonly described using two different models, namely the Brownian motion and random flow model in biological tissues. The Brownian motion model considers the motion of scatterers as diffusive motion and $\u2329\Delta r2(\tau )\u232a=6DB\tau $,^{38}^{,}^{39} where $DB$ is the effective diffusion coefficient of the scatterers. Meanwhile, the random flow model assumes the random ballistic motion of scatterers and $\u2329\Delta r2(\tau )\u232a=\u2329V2\u232a\tau 2$ (Refs. ^{40} and ^{41}), where $\u2329V2\u232a$ represents the mean square velocity of the scatterers. Experimentally, the diffuse light electric field autocorrelation function is usually derived from the measured normalized intensity autocorrelation $g2(r\u2192,\tau )=\u2329I(r\u2192,t)I(r\u2192,t+\tau )\u232a/\u2329I\u232a2$ by using the Siegert relation:^{4}^{,}^{13}^{,}^{42}Display Formula$g2(r\u2192,\tau )=1+\beta |G1(r\u2192,\tau )|2\u2329I(r\u2192,t)\u232a2,$(2)
where $I(r\u2192,t)$ is the detected diffusing light intensity at position $r$ and time $t$, the angle bracket $\u2329\u2026\u232a$ denotes an ensemble average, and $\beta $ here is a numerical factor related to the detector geometry, number of detected speckles, and other experimental parameters. The assumption for applying the Siegert relation to obtain the electric field autocorrelation function is that the system is ergodic, which means the time average is equal to the ensemble average. The analytic solution of the correlation diffusion equation for a point light source upon a semi-infinite medium is^{4}^{,}^{43}Display Formula$G1(\rho ,\tau )=3\mu s\u20324\pi (e\u2212kDr1r1\u2212e\u2212kDr2r2),$(3)
where $kD=3\mu s\u2032\mu a+6\mu s\u20322k02\alpha DB\tau $, $r1=\rho 2+z02$ ($r2=\rho 2+(z0+2zb)2$) are the distance between source (image source) and detector, respectively, in which $\rho $ is the source detector separation, $z0=1/\mu s\u2032$, $zb=2(1\u2212Reff)/3\mu s\u2032(1+Reff)$ and $Reff$ represents the effective reflectance. Although physically the motion of blood cells may be considered as random flow, the Brownian motion model showed better fitting in the majority of cases, ranging from brain^{4}^{,}^{7}^{–}^{9}^{,}^{11}^{–}^{14}^{,}^{17} to muscle^{20}^{,}^{23} and tumor models.^{27}^{,}^{29}^{,}^{30}^{,}^{32} Therefore, the measurements of $G1(r\u2192,\tau )$ can be fitted to yield a blood flow index ($BFI=\alpha DB$) to parameterize the relative blood flow.