Spectrophotometry of turbid media in a cuvette

The dual-slope approach, which is based on measured quantities that are linearly dependent on source-detector separation, may be generalized by considering dual ratios of the measured FD signal. Similar to the dual slope (which is the arithmetic mean of two paired single slopes of either the logarithmic amplitude or the phase), the dual ratio also represents the arithmetic mean of either two paired logarithmic amplitude ratios or two paired phase differences. Figure 1(a) shows the geometrical configuration of a standard cuvette containing a turbid medium, and a dual-ratio configuration of two sources (1,2) and two detectors (A,B). Since the sources and detectors are on opposite sides of the cuvette, this is a configuration for transmittance measurements. In the case of frequency-domain transmittance (which can be expressed as a complex number to represent its amplitude and phase), one can express the dual ratio (DR_1AB2) as a geometric mean of two paired single ratios (SR_1AB, SR_2BA) of single-distance transmittances (T_1A, T_1B, T_2A, T_sB): 

\({\rm DR}_{1AB2}=\sqrt{{\rm SR}_{1AB}\times{\rm SR}_{2BA}}=\sqrt{\frac{\mathbf{T}_{\mathbf{1B}}}{\mathbf{T}_{\mathbf{1A}}}\times\frac{\mathbf{T}_{\mathbf{2A}}}{\mathbf{T}_{\mathbf{2B}}}}\) 

It is this dual ratio of complex transmittances that is used for spectrophotometry of turbid media in a cuvette. Figure 1(b) illustrates the cost function of dual ratio measurements for media with different absorption and reduced scattering coefficients for the geometrical arrangement of Fig. 1(a). From Fig. 1(b) it appears that absorption measurements may be affected by a greater error than scattering measurements, especially for less scattering media (the cases in the left panels of Fig. 1(b)). Furthermore, boundary conditions (which also involve the index of refraction mismatch at the cuvette walls) were found to play a key role in the measured dual ratio. These initial results indicate that optimal configurations of sources and detectors, and approaches to minimize the impact of variable boundary conditions need to be identified to maximize the accuracy of spectrophotometry of turbid media in a standard cuvette.

Fig. 1. (a) Cuvette geometry and location of sources and detectors. (b) Cost function maps representing the deviation of dual-ratio measurements of the FD transmittance from their actual values for media with different absorption and scattering properties.