# Method for Measuring Absolute Optical Properties of Turbid Samples in a Standard Cuvette
## 3 Results
### Development of Fit for Absolute Optical Properties  
#### 3.2.1 Optimization of Cost Space Shape  
In order to fit for the absolute optical properties \(\mu_{a}\) and \(\mu_{s}^{\prime}\) we consider the \(\chi^{2}\) function in Equation (28). This function contains the scaling parameter \(\kappa\) which balances the scale of \(\ln|\text{SR}\{\widetilde{T}\}|\) versus \(\angle\text{SR}\{\widetilde{T}\}\). The intention of such a parameter is to modify the \(\chi^{2}\) space to be as circular as possible. This circularity can be quantitatively defined by considering iso-lines in cost space and their perimeter (\(P\)) as well as area (\(A\)). A circle has the minimum  
Figure 4: Demonstration of the cancellation of coupling factors when considering the \(\ln|\text{DR}\{\widetilde{T}\}|\) and \(\angle\text{DR}\{\widetilde{T}\}\) measurements. For this simulation, random \(\widetilde{C}\)s were generated for each optode and Equations (33)–(36) implemented. All other simulation parameters are the same as Figures 1 and 2. The expected theoretical value for the measured differences is shown as a dashed line. (**a**) The two symmetric \(\ln|\text{SR}\{\widetilde{T}\}|\) measurements and the \(\ln|\text{DR}\{\widetilde{T}\}|\) measurement. (**b**) The two symmetric \(\angle\text{SR}\{\widetilde{T}\}\) measurements and the \(\angle\text{DR}\{\widetilde{T}\}\) measurement. Acronyms and Symbols: Green’s function for the complex Transmittance (\(\widetilde{T}\)), Single-Ratio of the \(\widetilde{T}\) (\(\text{SR}\{\widetilde{T}\}\)), \(\text{SR}\{\widetilde{T}\}\) amplitude (\(\text{SR}\{\widetilde{T}\}\)), natural logarithm of \(|\text{SR}\{\widetilde{T}\}|\) (\(\ln|\text{SR}\{\widetilde{T}\}|\)), \(\text{SR}\{\widetilde{T}\}\) phase (\(\angle\text{SR}\{\widetilde{T}\}\)), Dual-Ratio of the \(\widetilde{T}\) (\(\text{DR}\{\widetilde{T}\}\)), \(\text{DR}\{\widetilde{T}\}\) amplitude (\(|\text{DR}\{\widetilde{T}\}|\)), natural logarithm of \(|\text{DR}\{\widetilde{T}\}|\) (\(\ln|\text{DR}\{\widetilde{T}\}|\)), \(\text{DR}\{\widetilde{T}\}\) phase (\(\angle\text{DR}\{\widetilde{T}\}\)), and complex optical Coupling, power, and/or efficiency factor (\(\widetilde{C}\)).  
ratio of \(P\) to \(A\) of all 2-Dimensional (2D) shapes. Therefore, the dimensionless metric \(P^{2}/A\) was minimized by varying \(\kappa\) (note that \(P^{2}/A\) has a minimum theoretical value of \(4\pi\), for a circle) [33].  
The effect of the \(\kappa\) value on \(\chi^{2}\) space shape is shown in Figure 5 using the same parameters as Figures 1 and 2 (where the optical properties are the _true_ values). Figure 5b shows the optimal \(\kappa\) of \(1.2\times 10^{-3}\), which is the case where \(P^{2}/A\) was minimized. The resulting \(P^{2}/A\), for this optimal \(\kappa\) was 36 about 3 times worse than the value of \(4\pi\approx 12.6\) for a ideal circular cost space. This can be seen by how oblique the \(\chi^{2}\) iso-lines are in the \(\mu_{a}\) direction suggesting a higher relative uncertainty in \(\mu_{a}\). We investigate this further in Section 3.3. Figure 5a,c show the effect of favoring either the \(\angle\text{SR}\{\widetilde{T}\}\) or \(\ln|\text{SR}\{\widetilde{T}\}|\) term in the \(\chi^{2}\) expression (Equation (28)). In either case \(\mu_{a}\) and \(\mu_{s}^{\prime}\) are correlated and the space is spread more in \(\mu_{a}\). However, when \(\ln|\text{SR}\{\widetilde{T}\}|\) is favored (Figure 5c) \(\mu_{a}\) and \(\mu_{s}^{\prime}\) have a negative correlation while the correlation is positive when \(\angle\text{SR}\{\widetilde{T}\}\) is favored (Figure 5a).  
Note that the optimal \(\kappa=1.2\times 10^{-3}\) was found for the \(\mu_{a}\) of 0.01 mm and \(\mu_{s}^{\prime}\) of 1 mm and a different optimal \(\kappa\)s may be found elsewhere for different _true_\(\mu_{a}\) and \(\mu_{s}^{\prime}\). Despite this we have opted to utilize this one \(\kappa\) value for the rest of this work to reduce computation time, in the future a map of optimal \(\kappa\) could be found as a function of \(\mu_{a}\) and \(\mu_{s}^{\prime}\).