# Impact of Fat Content and Lactose Presence on Refractive Index in Different Types of Cow Milk
## 2 Materials and Methods
### Methodology  
Once the experimental setup is prepared, the following methodology is pursued:  
1. The polarizer is adjusted to 90\({}^{\circ}\) to select \(p\)-polarization. The polarization is significant because the technique offers two types of linear polarization, and the choice of this can be substantial. In this manuscript, \(p\)-polarization was chosen. 2. Data acquisition is essential because the system will rotate to change the angle of incidence on the sample; for this reason, it is necessary to consider some data in the program that performs the measurements. * Initial angle: 20\({}^{\circ}\)
* Final angle: 75\({}^{\circ}\)
* Angular pitch: 0.1 degree. The angular step in which the motors move. * Integration parameter: 0.003. It is the amount by which the spectrometer averages each measurement, that is, every 0.1 degree. * Multiplier: 2. This parameter regulates the motor of the goniometer associated with the spectrometer, ensuring it moves at twice the speed when dealing with specular reflection compared to the goniometer bearing the sample. 3. A measurement of reference is necessary; this calibration curve is considered without the sample to obtain the light spectrum values for each angle of incidence for a prism-air interface. This measurement will be carried out whenever a sample has been used to observe that our system continues to be well calibrated and that the cleaning of the prism, due to the sample, was adequate. 4. The post-processing of the data obtained from each measurement is analyzed to obtain the reflectance curve as a function of the angle of incidence and wavelength. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Component** & **Model** & **Features** \\ \hline High-intensity fiber light source with & & High-Output of 150 W Bulbs with 3200 K \\ collimation package & & Wavelength range: 400–1600 nm \\  & & Adjustment: 0–100\% \\ Glan-Thompson Polarizer with rotation & & Calcite polarizer 10 \(\times\) 10 mm; extinction \\ mount, set at 90 deg & & ratio 100,000:1; wavelength range \\  & & 350 nm–2.3 μm \\ Motorized Precision Rotation Stage with & & Resolution: 2 arcsec \\ D.C. Servo Motor Driver (Goniometers) & & Bidirectional Repeatability: \(\pm\)0.1\({}^{\circ}\) \\  & & Max. Rotation Velocity: 25 deg/s \\ Spectrometer & & Wavelength range: 200–1000 nm \\  & & Spectral accuracy: \(<\)2 nm @ 633 nm \\ Prism & BK7 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optical components of the experimental setup. 5. All data are completed at this point, and different reflectance curves can be analyzed. One of them is calculated as a function of the angle of incidence at a fixed wavelength or vice versa. The purpose of this is to compare measurements with theory in a simpler way; for example, theoretical graphs of an interface prism-air can be generated using [PERSON]'s composite reflectance formula for \(p\)-polarization [4], or if a complex sample is carried out, Equation (1) can be used. \[r_{p123}=\frac{r_{p12}+r_{p23}\times e^{2 ih\sqrt{n_{l}^{2}-n_{i}^{2}\ sen^{2} \theta_{i}}}}{1+r_{p12}\times r_{p23}\times e^{2 ih\sqrt{n_{l}^{2}-n_{i}^{2}\ sen^{2} \theta_{i}}}}. \tag{1}\]  
To compare with the experiment, Equation (1) should be considered as the optical reflectance [4], which is defined in Equation (2). \[R_{p123}=\big{|}r_{p123}\big{|}^{2}. \tag{2}\]  
where \(n_{l}\) is the refractive index of the incidence medium, which, in our case, will be the BK7-prism, \(n_{t}\) is the refractive index of the transmitted medium, and \(\theta_{i}\) is the angle of incidence. \(r_{p12}\) is the Fresnel's coefficient of the first interface, prism-sample, and \(r_{p23}\), for the second interface, glass-air, and \(h\) is the thickness of the sample, which was measured using the formula \(h=V/A\), where \(V\) is the measured volume of the sample, and \(A\) is the area of the coverslip.