# Method for Measuring Absolute Optical Properties of Turbid Samples in a Standard Cuvette
## 2 Methods
### Types of Measurement  
The signal obtained between a single temporally modulated source and a single detector recovers the Green's function for the complex Transmittance (\(\widetilde{T}\)) with FD NIRS. \(\widetilde{T}\) is a complex number to represent the amplitude and phase of the transmitted photon density waves modulated on the order of \(100\,\mathrm{MHz}\). These signals are named: \(\widetilde{T}_{1\mathbf{A}}\), \(\widetilde{T}_{1\mathbf{B}}\), \(\widetilde{T}_{2\mathbf{A}}\), and \(\widetilde{T}_{2\mathbf{B}}\); where the first subscript indicates the source and the second the detector. The short \(\rho\) measurements (\(\rho=14.9\,\mathrm{mm}\)) are \(\widetilde{T}_{1\mathbf{A}}\) and \(\widetilde{T}_{2\mathbf{B}}\) while the long \(\rho\) measurements (\(\rho=25.1\,\mathrm{mm}\)) are \(\widetilde{T}_{2\mathbf{A}}\) and \(\widetilde{T}_{1\mathbf{B}}\). From these \(\widetilde{T}\) measurements, ratios between the short and long \(\rho\) measurements may be obtained. Therefore we introduce the Single-Ratio of the \(\widetilde{T}\)s (SR\(\{\widetilde{T}\}\)s) for the geometry in Figure 1 as follows:  
\[\text{SR}\{\widetilde{T}\}_{1\mathbf{B}}=\frac{\widetilde{T}_{1\mathbf{B}}}{ \widetilde{T}_{1\mathbf{A}}} \tag{1}\]  
\[\text{SR}\{\widetilde{T}\}_{2\mathbf{B}A}=\frac{\widetilde{T}_{2\mathbf{A}}}{ \widetilde{T}_{2\mathbf{B}}} \tag{2}\]and the Dual-Ratio of the \(\widetilde{T}\) (DR\(\{\widetilde{T}\}\)) as the geometric mean of the two symmetric SR\(\{\widetilde{T}\}\)s.  
\[\text{DR}\{\widetilde{T}\}_{\texttt{1 AB2}}=\sqrt{\text{SR}\{\widetilde{T}\}_{ \texttt{1 AB}}\times\text{SR}\{\widetilde{T}\}_{\texttt{2 BA}}}=\sqrt{\frac{ \widetilde{T}_{\texttt{1B}}\widetilde{T}_{\texttt{2A}}}{\widetilde{T}_{\texttt {1A}}\widetilde{T}_{\texttt{2B}}}} \tag{3}\]  
This forms a similar type of measurement to DS/SC but replacing the concept of slope with that of ratio (The word slope in Dual-Slope (DS) is historical [29; 30] as no slopes are actually considered in this work). We acknowledge that the notation used here is verbose since, that even for DR\(\{\widetilde{T}\}\) we utilize subscripts to show all optodes used. However, for this work we have opted to use this notation to distinguish explicitly, the origin of the measurements. This is helpful in observing the differences in SR\(\{\widetilde{T}\}\)s, particularly in Section 3.1.2. In other work we opt to utilize a simpler notation with numbered subscripts [30]. We can further expand the expression for SR\(\{\widetilde{T}\}\) to consider SR\(\{\widetilde{T}\}\) amplitude (\(|\text{SR}\{\widetilde{T}\}|\)) and SR\(\{\widetilde{T}\}\) phase (\(\angle\text{SR}\{\widetilde{T}\}\)). For example, with 1 AB we have:  
\[|\text{SR}\{\widetilde{T}\}|_{\texttt{1 AB}}=\frac{|\widetilde{T}_{\texttt{1B} }|}{|\widetilde{T}_{\texttt{1A}}|} \tag{4}\]  
\[\angle\text{SR}\{\widetilde{T}\}_{\texttt{1 AB}}=\angle\widetilde{T}_{\texttt{ 1B}}-\angle\widetilde{T}_{\texttt{1A}} \tag{5}\]  
and we introduce the final ratio type, the natural logarithm of \(|\text{SR}\{\widetilde{T}\}|\) (\(\text{In}\,|\text{SR}\{\widetilde{T}\}|\)):  
\[\text{In}\,|\text{SR}\{\widetilde{T}\}|_{\texttt{1 AB}}=\text{In}\,|\widetilde {T}_{\texttt{1B}}|-\text{In}\,|\widetilde{T}_{\texttt{1A}}| \tag{6}\]  
The motivation for utilizing the natural logarithm in this way is that it partly linearizes typical expressions for diffuse optical measurements (\(\widetilde{T}\) in this case) as a function of \(\rho\)[31]. Additionally it shows a symmetry between \(\ln|\text{SR}\{\widetilde{T}\}|\) and \(\angle\text{SR}\{\widetilde{T}\}\) as they are both differences. This work focuses on utilizing these \(\ln|\text{SR}\{\widetilde{T}\}|\) and \(\angle\text{SR}\{\widetilde{T}\}\) ratio types in the development of the proposed method.