# Dual-Slope Diffuse Reflectance Instrument for Calibration-Free Broadband Spectroscopy
## 2 Materials and Methods
### Analysis  
In Section 2.1, two techniques were described: Multi-Distance Frequency-Domain Near-Infrared Spectroscopy (MD FD-NIRS; Section 2.1.1) and Dual-Slope Continuous-Wave broadband Diffuse Reflectance Spectroscopy (DS CW-bDRS; Section 2.1.2). MD yields the spatial dependence of reflectance across multiple distances, while DS yields two symmetric spatial dependencies of reflectance. Additionally, FD-NIRS yields complex reflectance (at a few wavelengths), while CW-bDRS yields CW reflectance (at many wavelengths). However, regardless of these differences, the same basic analysis was used to find absolute optical properties (with the process repeated for each separate wavelength). In the following, we first describe the analysis of complex reflectance across multiple distances (MD FD-NIRS). Then, we explain how the method is modified to handle DS data and what additional assumptions are needed to use CW data (DS CW-bDRS). Considering a semi-infinite medium with extrapolated-boundary conditions, the complex reflectance (\(\widetilde{R}\)), defined as the net optical flux exiting the tissue per unit source power, may be written as [2; 39]:  
\[\widetilde{R}=\frac{1}{4\pi}\Big{(}\widetilde{C_{1}}e^{-i\widetilde{\
u_{f}} \widetilde{\
u_{1}}\tau_{1}}+\widetilde{C_{2}}e^{-i\widetilde{\
u_{2}} \widetilde{\
u_{f}}\tau_{2}}\Big{)}, \tag{1}\]  
Figure 3: Expected spectral features of near-infrared dyes in water, provided by the manufacturer (QCR Solutions, Palm City, FL, USA). [37] (**a**) Normalized Absorbance (A) versus wavelength (\(\lambda\)) of NIR746A dye (N7). (**b**) Normalized A of NIR869A dye (N8). \[\widetilde{C_{1}}=z_{0}\left(\frac{1+\widetilde{\mu_{eff}}r_{1}}{r_{1}^{3}}\right) \qquad\widetilde{C_{2}}=-z_{0}^{{}^{\prime}}\left(\frac{1+\widetilde{\mu_{ eff}}r_{2}}{r_{2}^{3}}\right), \tag{2}\]  
\[r_{1}=\sqrt{\rho^{2}+z_{0}^{2}}\qquad r_{2}=\sqrt{\rho^{2}+z_{0}^{{}^{\prime}2}}, \tag{3}\]  
where \(z_{0}\) is the depth of the real isotropic source (\(z_{0}=1/\mu_{s}^{{}^{\prime}}\)), \(z_{0}^{{}^{\prime}}\) is the height of the imaginary isotropic source (\(z_{0}^{{}^{\prime}}=-z_{0}+2z_{b}\)), \(\rho\) is the source-detector distance on the surface of the semi-infinite medium, and \(\widetilde{\mu_{eff}}\) is the complex effective attenuation coefficient. The extrapolated boundary height (\(z_{b}=2A/(3(u_{s}^{{}^{\prime}}+\mu_{a}))\)) and the complex effective attenuation coefficient (\(\widetilde{\mu_{eff}}=\sqrt{3(\mu_{s}^{{}^{\prime}}+\mu_{a})(\mu_{a}-\omega n _{im}i/c)}\) ) are both expressed in terms of the absorption coefficient (\(\mu_{a}\)), reduced scattering coefficient (\(\mu_{s}^{{}^{\prime}}\)). The remaining parameters are the angular modulation frequency (\(\omega\)), the reflection parameter (\(A\)), the index of refraction inside the medium (\(n_{in}\); assumed to be 1.4, typical for biological tissue [2]), and the speed of light in vacuum (\(c\)). Additionally, \(\omega=2\pi f_{mod}\), where \(f_{mod}\) is the modulation frequency. Finally, \(A\) is a function of \(n_{in}\) and the index of refraction outside the medium (\(n_{out}\); assumed to be 1.0) [39]. To yield a linear relationship, Equation (1) can be rewritten as:  
\[ln\left[\frac{4\pi\widetilde{R}}{\widetilde{C_{1}}+\widetilde{C_{2}}e^{i \widetilde{\mu_{eff}}(r_{1}-r_{2})}}\right]=-\widetilde{\mu_{eff}}r_{1}, \tag{4}\]  
which shows a linear dependence of a modified complex reflectance (\(\widetilde{y}=ln[4\pi\widetilde{R}/(\widetilde{C_{1}}+\widetilde{C_{2}}e^{i \widetilde{\mu_{eff}}(r_{1}-r_{2})})]\)) on \(r_{1}\) (i.e., \(\widetilde{y}=-\widetilde{\mu_{eff}}r_{1}\)). The linear complex slope (\(-\widetilde{\mu_{eff}}\)) in Equation (4) is used to calculate the desired optical properties (\(\mu_{a}\) and \(\mu_{s}^{{}^{\prime}}\)). However, \(\widetilde{y}\) is dependent on \(\mu_{a}\) and \(\mu_{s}^{{}^{\prime}}\), and \(r_{1}\) is dependent on \(\mu_{s}^{{}^{\prime}}\).