However, few studies were focused on the theoretical analysis of intrinsic properties of two-layered diffusion model in SFDI for optical property estimations. As an emerging optical measuring technique, SFDI is capable of noncontact and wide-field mapping of \(\mu_{a}\) and \(\mu_{s}\)' on a pixel-by-pixel basis, which is absent in other techniques using point light source (e.g., spatially-resolved, time-resolved, and integrating sphere). SFDI can be used for estimating optical properties of homogeneous tissues, as well as layered samples, by using appropriate light transfer models (e.g., diffusion approximation and Monte Carlo). Recently, [PERSON] et al. (2018) developed a two-layer look-up-table inversion algorithm for extracting \(\mu_{a}\) and \(\mu_{s}\)'of the bottom layer, in which Monte Carlo simulations were conducted natively in the spatial-frequency domain [9]. The results showed that optical property extractions of the bottom (tumor) layer were determined to be within 20 and 11% of the true values for \(\mu_{a}\) and \(\mu_{s}\)', respectively. Several other studies have also been reported for estimating \(\mu_{a}\) and \(\mu_{s}\)' of layered tissues using the SFDI technique [8; 9; 10; 11; 12]. Some researchers estimated the four optical properties (\(\mu_{a1}\), \(\mu_{s1}\)', \(\mu_{a2}\) and \(\mu_{s2}\)') simultaneously with the known top-layer thickness (all-at-once method) [8], while others estimated \(\mu_{a}\) and \({\mu_{s}}^{\prime}\) of one (often the top) layer first, followed by estimating \(\mu_{a}\) and \(\mu_{s}\)\({}^{\prime}\) of the other layer (stepwise method) [12]. Our previous study demonstrated that the efficacy and accuracy of the stepwise method for estimating optical properties of two-layered samples were superior to that for the all-at-once method, under the constraining conditions for the top-layer thickness between 0.2 and 2.0 mm [13]. In the paper of our recent research, the stepwise method with frequency optimization was employed for measuring the optical properties of apple peel and flesh tissues, and the results showed that there were still relatively large error values (22.8%) for estimating  
pattern, diffuse reflectance at the surface can be yielded by applying the partial-current boundary condition [16]:  
\[R_{d}(f_{x})=\frac{3 Aa^{\prime}}{\left(\mu_{eff}^{\prime}/\mu_{tr}+1\right)\left( \mu_{eff}^{\prime}/\mu_{tr}+3A\right)} \tag{1}\]  
where \(A=\frac{1-R_{eff}}{2\left(1+R_{eff}\right)}\) is proportionality constant, \(R_{eff}\approx 0.0636n+0.668+0.71/n-1.44/n^{2}\) is the effective reflection coefficient, in which \(n\) is the refractive index of the medium, \(a^{\prime}=\mu_{s}^{\prime}/\mu_{tr}\) is the reduced albedo, \(\mu_{tr}=\mu_{a}+\mu_{s}^{\prime}\) is the transport coefficient, \(\mu_{a}\) and \(\mu_{s}^{\prime}\) are absorption coefficient and reduced scattering coefficient, respectively, \(\mu_{eff}^{\prime}=\left(3\mu_{a}\mu_{tr}+\left(2\pi f_{x}\right)^{2}\right)^{ 1/2}\) is the scalar attenuation coefficient, and \(f_{x}\) is the spatial frequency. For a two-layered turbid medium (Figure 1), light within the medium decays exponentially, such that the light source term is different in each layer with the bottom layer being assumed to be infinitely thick. By applying appropriate boundary conditions to the diffusion model, diffuse reflectance at the surface can be expressed using Equation (2) [17]:  
\[R_{d}(f_{x})=A\cdot\frac{\mu_{s1}^{\prime}}{\mu_{eff1}^{\prime}}\cdot\frac{A_{ 1}+A_{2}}{A_{3}}x \tag{2}\]  
where \(A_{1}\), \(A_{2}\) and \(A_{3}\) are constants determined by the boundary conditions, and subscript 1 of \(\mu_{s1}^{\prime}\) and \(\mu_{eff1}^{\prime}\) refers to the top layer.