# Method for Measuring Absolute Optical Properties of Turbid Samples in a Standard Cuvette
## 3 Results
### Investigation of Difference Measurements
#### 3.1.2 Optode Coupling and Auto-Calibration  
As has been stated above including in (Again we note that the Dual-Ratio of the \(\widetilde{T}\) (DR\(\{\widetilde{T}\}\)) and a Single-Ratio of the \(\widetilde{T}\) (SR\(\{\widetilde{T}\}\)) are the same when not considering optode coupling (shown in Section 3.1.2)), SR\(\{\widetilde{T}\}\) & DR\(\{\widetilde{T}\}\) (as well as \(|\text{SR}\{\widetilde{T}\}|\) & \(|\text{DR}\{\widetilde{T}\}|\), \(\angle\text{SR}\{\widetilde{T}\}\) & \(\angle\text{DR}\{\widetilde{T}\}\), and \(\ln|\text{SR}\{\widetilde{T}\}|\) & \(\ln|\text{DR}\{\widetilde{T}\}|\)) are equivalent when optode coupling is not considered. For this reason, other sections of this manuscript are not careful to distinguish between them as theoretical calculations are being carried out and coupling is not a consideration. However, in this section we show the effect of optode coupling and the auto-calibration of the DR\(\{\widetilde{T}\}\) which is inherited/inspired by the SC method [24]. To do this, first we define a complex optical Coupling, power, and/or efficiency factor (\(\widetilde{C}\)) for each optode: \(\widetilde{C}_{1}\), \(\widetilde{C}_{2}\), \(\widetilde{C}_{\mathbf{A}}\), and \(\widetilde{C}_{\mathbf{B}}\). Physically, \(|\widetilde{C}|\) represents a multiplicative factor (attenuation or amplification) on the amplitude of \(\widetilde{T}\) and \(\angle\widetilde{C}\) represents a phase shift on the phase of \(\widetilde{T}\). \(\widetilde{C}\)s applied to sources (number subscripts) have units of mW since their amplitude also includes source power; while \(\widetilde{C}\)s for detectors (letter subscripts) are unitless. Therefore, adding _coup_ subscripts to our measurements when they are confounded by coupling (opposed to the theoretical value without the _coup_ subscript) we have the following signals considering coupling:  
\[\widetilde{T}_{\mathbf{1A},coup} =\widetilde{C}_{1}\widetilde{C}_{\mathbf{A}}\widetilde{T}_{ \mathbf{1A}} \tag{33}\] \[\widetilde{T}_{\mathbf{1B},coup} =\widetilde{C}_{1}\widetilde{C}_{\mathbf{B}}\widetilde{T}_{ \mathbf{1B}}\] (34) \[\widetilde{T}_{\mathbf{2A},coup} =\widetilde{C}_{2}\widetilde{C}_{\mathbf{A}}\widetilde{T}_{ \mathbf{2A}}\] (35) \[\widetilde{T}_{\mathbf{2B},coup} =\widetilde{C}_{2}\widetilde{C}_{\mathbf{B}}\widetilde{T}_{ \mathbf{2B}} \tag{36}\]Now, let us revisit Equations (1)-(3) but with optode coupling considered:  
\[\text{SR}\{\widetilde{T}\}_{\texttt{1 AB},cup}=\frac{\widetilde{C}_{1}\widetilde{C }_{\texttt{B}}\widetilde{T}_{\texttt{1B}}}{\widetilde{C}_{1}\widetilde{C}_{ \texttt{A}}\widetilde{T}_{\texttt{1A}}}=\frac{\widetilde{C}_{\texttt{B}} \widetilde{T}_{\texttt{1B}}}{\widetilde{C}_{\texttt{A}}\widetilde{T}_{\texttt{1 A}}}=\frac{\widetilde{C}_{\texttt{B}}}{\widetilde{C}_{\texttt{A}}}\text{SR}\{ \widetilde{T}\}_{\texttt{1 AB}} \tag{37}\]  
\[\text{SR}\{\widetilde{T}\}_{\texttt{2 BA},cup}=\frac{\widetilde{C}_{\texttt{2}} \widetilde{C}_{\texttt{A}}\widetilde{T}_{\texttt{2A}}}{\widetilde{C}_{\texttt{ 2}}\widetilde{C}_{\texttt{B}}\widetilde{T}_{\texttt{2B}}}=\frac{\widetilde{C}_ {\texttt{A}}\widetilde{T}_{\texttt{2A}}}{\widetilde{C}_{\texttt{B}}\widetilde{ T}_{\texttt{2B}}}=\frac{\widetilde{C}_{\texttt{A}}}{\widetilde{C}_{\texttt{B}} \widetilde{T}_{\texttt{2B}}}=\frac{\widetilde{C}_{\texttt{A}}}{\widetilde{C}_{ \texttt{B}}}\text{SR}\{\widetilde{T}\}_{\texttt{2 BA}} \tag{38}\]  
\[\text{DR}\{\widetilde{T}\}_{\texttt{1 AB2},cup}=\sqrt{\frac{\widetilde{C}_{ \texttt{B}}\widetilde{T}_{\texttt{1B}}\widetilde{C}_{\texttt{A}}\widetilde{T}_{ \texttt{2A}}}{\widetilde{C}_{\texttt{A}}\widetilde{T}_{\texttt{1A}}\widetilde{C} _{\texttt{B}}\widetilde{T}_{\texttt{2B}}}}=\sqrt{\frac{\widetilde{T}_{\texttt{ 1B}}\widetilde{T}_{\texttt{2A}}}{\widetilde{T}_{\texttt{1A}}\widetilde{T}_{ \texttt{2B}}}}=\text{DR}\{\widetilde{T}\}_{\texttt{1 AB2}} \tag{39}\]  
showing that the measured \(\text{DR}\{\widetilde{T}\}\) is the same as the theoretical values regardless of the various optode couplings \(\widetilde{C}\).