# Self-Mixing Interferometry-Based Micro Flow Cytometry System for Label-Free Cells Classification
## 2 Theory  
When the individual micro particle or cell inside the channel flows through the laser beam volume, part of the scattered light re-enters the laser cavity. The interaction between the back-scattered light and the original light inside the cavity gives rise to a modulation of the laser output power. Meanwhile, due to the Doppler effect, the output power signal presents a spectral shift in the frequency domain, which depends on the flowing cell velocity.  
In order to describe the self-mixing behavior in the presence of individual cell, we proposed a novel theoretical framework based on the well-known three-mirror model as shown in Figure 1[32]. The entire system consists of two cavities: the first cavity is the laser cavity of length \(L_{c}\) between the two cavity mirrors (M1, M2) of reflectivity \(r_{1}\) and \(r_{2}\), respectively; and the other is the external cavity of length \(L_{ext}\) from M2 to the cell; where \(r_{c}\) is the electric field amplitude ratio of the scattered light from the cell re-entering the cavity over the original laser output.  
The laser output power subjected to the self-mixing effect \(P(t)\) can be expressed as [25]:  
\[P(t)=P_{0}[1+m\cos(2\pi f_{D}\cdot t)] \tag{1}\]  
\[m=4\frac{r_{c}(1-r_{2}^{2})}{r_{2}}.\frac{\tau_{p}}{\tau_{l}} \tag{2}\]  
\(P_{0}\) is the laser initial output power without self-mixing effect, \(m\) is the modulation parameter which is critical for representing the modulation strength and the SMI signal level. Taking into consideration the cell dimension, \(r_{c}\) is extremely low, thus the SMI operates generally in the weak feedback regime [25]. \(\tau_{p}\) is the photon lifetime, \(\tau_{l}\) is the round-trip time inside the laser cavity. When the cell crosses the laser emission area with a constant velocity \(V\), a Doppler frequency shift \(f_{D}\) is induced. The absolute value of the frequency shift can be expressed in the following form [33]:  
\[f_{D}=\frac{2V\cdot\sin\theta}{\lambda} \tag{3}\]  
where \(\theta\) is the incident angle, \(\lambda\) is the laser wavelength.  
Considering a spheroid cell flowing through the laser beam over a period \(w\) as depicted in Figure 2, the amount of scattering light coupling into the cavity is at its maximum when the cell center is on the beam central axis (Figure 2b). Here, the value of the modulation index \(m\) reaches the maximum as well, resulting in the maximal amplitude of signal \(P(t)\). In order to describe the amplitude variation over the passage, we propose the following hypothesis: the laser irradiation profile obeys a Gaussian distribution, and the self-mixing interference is only generated when the cell enters the laser beam volume from point-in-time \(t_{0}\) to \(t_{0}+w\), the SMI signal amplitude then varies as a Gaussian function [34]:  
\[P(t)=P_{0}[1+m\cos(2\pi f_{D}\cdot t)]\exp\left(-\frac{-(t-t_{0})^{2}}{2w^{2} }\right) \tag{4}\]  
We can expect that the total amount of light re-entering the laser cavity will be impacted by the particle nature: for example it will be greater for a larger particle, or for a particle with a large refractive index difference with the carrying fluid. It can also be expected that the cell membrane nature, the surface roughness, or the intracellular contents influence the SMI signal parameters.  
Figure 1: The schematic of three-mirror model.