## 2 Experimental Setup and Theoretical Analysis
### Interferometer Setup  
As mentioned before, a Mach-Zehnder interferometer was integrated into the optical trap system. The experimental setup corresponding to the interferometer used is shown in Figure 2. The interferometer light source is a KIMMON KOHA laser (Sukagawa, Japan) (model KBL-100-A) with a wavelength (\(\lambda_{B}\)) of 405 nm and a maximum power of 100 mW; in this laser, the coherence length decreases below 300 \(\upmu\)m when a power lower than 1 mW is selected [28], and this makes it possible to obtain interferograms that are free from spurious interference, and also to perform non-invasive measurements (without affecting the object under study). The laser beam passes through a beam expander system and is incident on a 50:50 beam splitter. The reflected part--the reference arm--is incident on a mirror that is attached to a piezoelectric (PZT), which is controlled by a THORLABS variable voltage source (model TDC001), in order to apply the phase-shifting technique in the interferogram demodulation, then, continuing on its path, the wavefront is redirected to a second beam splitter, also 50:50. On the other hand, the part transmitted by the beam splitter at the laser output passes through Microscope Objective 1, MO1 (Edmund brand, model 59880, of 100X, with an NA of 0.8 and a WD of 3 mm), and then, after going through the sample under study, light is captured by a second microscope objective (MO2), to be redirected via a mirror, until it crosses the second beam splitter. Here is where the beams recombine, and interference is generated. The image-capturing video system consists of a THORLABS camera (model DCC1240c) based on a CMOS sensor with a pixel size of 5.3 \(\upmu\)m and a resolution of 1280 \(\times\) 1024 pixels. The field of view generated between Microscope Objective 2 and the camera is 61.24 \(\times\) 48.99 \(\upmu\)m.  
With the system in question, it is possible to obtain interference patterns such as the one shown in Figure 2, whose capture was performed with the LabVIEWTM program, while image processing was carried out with the MATLABTM programming language. The phase difference (\(\Delta\phi\)) between the two interfered wavefronts, previously mentioned, is given by [1]  
\[\Delta\phi(x,y)=\frac{2\pi}{\lambda}\text{OPD}(x,y), \tag{1}\]  
where \(\lambda\) is the wavelength and OPD is the optical path difference. Figure 2: Scheme of a Mach–Zehnder interferometer. (MO1—Long Working Distance Microscope Objective, MO2—Oil Immersion Microscope Objective). An idealized model of an interference pattern with time-phase shifting modulation (\(I(x,y,t)\)) is described using the following:  
\[I(x,y,t)=a(x,y)+b(x,y)\text{cos}[\phi(x,y)+\omega_{0}t]. \tag{2}\]  
Here, \(a(x,y)\) is the background signal, \(b(x,y)\) is the modulation signal (or fringe contrast), \(\omega_{0}\) is a phase carrier, and, for simplicity, the phase difference term of interest will hereafter be referred to as \(\phi\). If the \(I(x,y,t)\) term is known, a system of at least three equations is necessary to determine \(\phi(x,y)\), due to the infinite possibilities in the other parameters; in practice, it is common to introduce a variation that can be a sequential increase of \(\omega_{0}t=2\pi t/N\) (also called \"phase shifting\" or \"phase step\"), \(N\) being the total number of fringe patterns [29]. It is important to note that \(t\) only indicates the instant at which the phase shifting is applied; it is not directly related to a specific time.