## 1 Introduction  
Diffuse correlation spectroscopy (DCS) is a well-established optical technique capable of sensing blood flow in biological tissue using multiply scattered light and has widely been applied to quantify tissue perfusion in vivo [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]. The light source used in DCS is a long coherence length laser, which is delivered to a medium of interest through a large-core optical fiber while the back-scattered (reflected) intensity is collected using a small-core (single mode) fiber, placed at a fixed distance (the source-detector separation SDS) from the center of the source fiber. The detected signal is transmitted to a photon counter that in turn is electronically coupled to a hardware correlator board [14; 15]. Experimental DCS measurements yield the normalized intensity autocorrelation function, \(g_{2}(\tau)\) measured for a given fiber geometry, which in turn is quantified by fitting via an analytical expression to extract flow-dependent coefficients that characterize the dynamically scattered photon transport [15; 16; 17]. Theoretically, solutions to the unnormalized field correlation \(G_{1}(\tau)\) are found by considering it to be governed by the modified photon diffusion equation  
\[\left[D\
abla^{2}-\mu_{a}-\frac{1}{3}\mu_{b}^{\prime}k_{0}^{2}\langle\Delta r^ {2}(\tau)\rangle\right]G_{1}(r,\tau)=S(r) \tag{1}\]As discussed previously [15; 16; 17], \(D\) in Equation (1) is the photon diffusion coefficient of the medium (determined from its absorption and reduced scattering coefficients \(\mu_{a}\), \(\mu_{s}^{\prime}\) as \(D=[3(\mu_{a}+\mu_{s}^{\prime})]^{-1}\)), \(\langle\Delta r^{2}(\tau)\rangle\) is a correlation-time (\(\tau\)) dependent term and represents the ensemble-averaged, mean-squared displacement of moving scatterers in time \(\tau\). \(k_{0}\) is the wave-vector associated with the laser source and \(S(r)\) describes the spatial geometry of the source to be modeled [15]. Two models are commonly used for \(\langle\Delta r^{2}(\tau)\rangle\): the Brownian flow model (where \(\langle\Delta r^{2}(\tau)\rangle=6D_{B}\tau\)) with Brownian flow coefficient \(D_{B}\); the random flow model (where \(\langle\Delta r^{2}(\tau)\rangle=V^{2}\tau^{2}\)) where \(V\) is the average volumetric flow-speed of the scattering particles [15; 16; 18; 19; 20]. In a semi-infinite turbid medium, for a collimated pencil-beam input, closed form analytical solutions to \(G_{1}(\tau)\) in cylindrical geometry is given by  
\[G_{1}(\tau)=\frac{1}{4\pi D}\bigg{[}\frac{\exp(-K(\tau)\cdot r_{1})}{r_{1}}- \frac{\exp(-K(\tau)\cdot r_{2})}{r_{2}}\bigg{]} \tag{2}\]  
Here, \(K(\tau)\) is dependent on the medium's absorption, scattering and correlation-transport properties as \(K(\tau)=3\mu_{a}\mu_{s}^{\prime}+\mu_{s}^{\prime 2}k_{0}^{2}\Delta r^{2}(\tau)\). \(r_{1}\) and \(r_{2}\) are given as \(r_{1}^{2}=\rho^{2}+z_{0}^{2}\) and \(r_{2}^{2}=\rho^{2}+(z_{0}+z_{b})^{2}\), where \(\rho\) is the SDS (i.e., the distance from the detection point on the surface of the semi-infinite medium and the collimated input source). \(z_{0}\) and \(z_{b}\) are dependent on the medium's optical properties (via the boundary conditions specified) as \(z_{0}=1/\mu_{s}^{\prime}\) and \(z_{b}=2 AD\) and \(A\) is obtained from the relative refractive index between the medium and detector, as discussed previously [17; 19; 21]. To match theoretical predictions to experimental measurements the unnormalized field correlation \(G_{1}(\tau)\) is transformed into the normalized intensity correlation via the Siegert relation, which gives that \(g_{2}(\tau)=1+\beta[g_{1}(\tau)]^{2}\), where \(g_{1}(\tau)=G_{1}(\tau)/G_{1}(0)\), and \(\beta\) is a scalar constant that depends on detection geometry [15; 16]. Equation (2) is derived from the formalism of photon diffusion theory and as such would be expected to carry constraints on its validity (dictating geometry of measurements and the choice of laser wavelengths) [22; 23; 24; 25; 26].