# Method for Measuring Absolute Optical Properties of Turbid Samples in a Standard Cuvette
## 1 Introduction  
Samples in which the propagation of light is dominated by random scattering are considered optically diffuse. Such samples can be characterized by two absolute optical properties, the absorption coefficient (\(\mu_{a}\)) and the reduced scattering coefficient (\(\mu^{\prime}_{s}\)) [1]. The \(\mu_{a}\) represents chemical information, and its spectral measurement allows for determination of the sample's chemical constituents and concentrations. Meanwhile, the \(\mu^{\prime}_{s}\) describes the micrometer scale structure of diffuse samples. However, in many applications \(\mu^{\prime}_{s}\) is considered a confounding parameter, since measurement of \(\mu_{a}\) and chemical makeup is often the end goal. For this reason, even when diffuse sample measurement of only \(\mu_{a}\) is sought, \(\mu^{\prime}_{s}\) must also be determined since it significantly impacts the behavior of light and thus the recovered \(\mu_{a}\). Applications that seek to measure these diffuse optical properties are numerous and span many fields. For example, applications include those within biomedical research and clinical applications [1; 2; 3], of food science and quality [4; 5; 6], concerning pharmaceutical metrology [7; 8], pertaining to art and archaeology [9; 10], and within dendrology [11] to name a few. In all cases, one has two options:To make a measurement which retrieves the total attenuation coefficient (\(\mu_{t}\)) or the effective attenuation coefficient (\(\mu_{eff}\)). 2. To make a measurement that can separate both \(\mu_{a}\) and \(\mu_{s}^{\prime}\). However, only option 2 allows for careful quantitative analysis of the sample properties; since in option 1 one can only measure a coefficient, namely \(\mu_{t}\) or \(\mu_{eff}\), that couples both the \(\mu_{a}\) and \(\mu_{s}^{\prime}\) of the sample. There are few methods capable of achieving option 2. One such technique is Near-InfraRed Spectroscopy (NIRS) implemented in Frequency-Domain (FD) [12] (or Time-Domain (TD) [13]) which can recover \(\mu_{a}\) and \(\mu_{s}^{\prime}\) by using temporally modulated light. In the case of FD, photon density waves are generated by using a sinusoidally modulated source on the order of 100 MHz, and the amplitude and phase of these photon density waves are measured from the detected modulated light signal. An example of a commercially available FD instrument capable of this measurement is the ISS Imagent V2 [Champaign, IL USA] (Imagent). A second technique capable of option 2 is the integrating sphere [14; 15]. This technique measures total diffuse reflectance and total diffuse transmittance to separate \(\mu_{a}\) and \(\mu_{s}^{\prime}\). Both techniques have their strengths and weaknesses. Common implementations of FD NIRS require large sample volumes to create geometries that are effectively infinite in at least one dimensional extent making simple diffusion theory expressions valid [16]. Other methods, besides diffusion theory, exist to tackle non-simple geometries such as Monte Carlo [17] and the radiative transport equation implemented with higher order spherical harmonics [18; 19; 20]. However these methods are computationally costly compared to diffusion theory and would likely be impractical to implement on a broad range of optical wavelengths (\(\lambda\)s). Another weakness of FD regarding the number of \(\lambda\)s, is the typical implementation at only discrete \(\lambda\)s such as with the Imagent. Meanwhile, the integrating sphere requires careful calibration or a reference sample and is easily susceptible to errors induced by the measurement technique (for example, light loss causing an incorrect measurement of total reflectance and transmittance). Due to these difficulties with option 2 (and relative ease implementing option 1), most commercial spectrometers make a measurement that is based on the retrieving non-diffuse transmittance.