## 2 Theory  
When a laser shoots a flowing particle, the laser output power modulation is induced by the interaction between the resulting particle-induced back-scattered light and the laser inner cavity light. In order to describe the laser resonant behavior, here a three-mirror model is shown in Figure 1[20]. The entire system consists of two cavities, one is the laser cavity of length \(L_{c}\) between the two mirrors (M1, M2) of reflectivity r1 and r2, respectively; the other is the external cavity of length \(L_{ext}\) from the laser to the particle in the medium of refractive index \(n\). \(r_{ext}\) is the electric field amplitude ratio of the scattered light re-entering the cavity over the original laser light. The modulated laser output power \(P(t)\) in the presence of the distant target can be expressed using Equations (1) and (2), with the original laser output power \(P_{0}\), the modulation index \(m\), and the phase \(\phi\)[21]. \[P(t)=P_{0}(1+m\cos\phi) \tag{1}\]  
\[m=\frac{4r_{ext}(1-r_{2}^{2})}{r_{2}}\frac{\tau_{p}}{\tau_{l}} \tag{2}\]  
where \(\tau_{p}\) is the photon lifetime and \(\tau_{l}\) is the round-trip flight time inside the laser cavity. The instantaneous external cavity length, \(L_{ext}(t)\) is expressed by the initial length \(L_{0}\) and the instantaneous displacement \(\Delta L\) using Equation (3). \[L_{ext}(t)=L_{0}+\Delta L \tag{3}\]  
Equation (1) can be rewritten as:  
\[P(t)=P_{0}\bigg{[}1+m\cos\bigg{(}\frac{4\pi L_{ext}}{\lambda}\bigg{)}\bigg{]} \tag{4}\]  
Considering the external cavity length determines the phase \(\phi\), the displacement \(\Delta L\) can be measured with a resolution of half wavelength. \[\phi=\frac{4\pi}{\lambda}(L_{0}+\Delta L) \tag{5}\]  
Due to the well-known Doppler effect, the frequency of the laser output is shifted by \(f_{D}\):  
\[f_{D}=\frac{2V\sin\theta}{\lambda} \tag{6}\]  
where \(V\) is the particle flowing velocity and \(\theta\) is the incident angle between the laser emission axis and is perpendicular to the plane of the particle flowing direction. When an individual particle flows through the laser beam emission area from position A to position C, as shown in Figure 2, referring to the laser emission direction, the external cavity facet moves backward from A to B and then forward from B to C, so that \(\Delta L\) varies the particle diameter \(D\) (double the radius \(r\)) value during the particle passage. Hence, the particle diameter \(D\) can be retrieved from the SMI signal by counting the number of fringes \(N\), with a resolution of half-wavelength order, as shown in Equation (7). \[D=N\frac{\lambda}{2} \tag{7}\]  
Figure 1: Schematic three-mirror cavity model. Based on [22], we propose the following hypothesis: the self-mixing operates on a Gaussian intensity distribution and the self-mixing interference only occurs during the particle passage from point-in-time \(t_{0}\), so the SMI signal amplitude also varies as a Gaussian function in the time domain, as shown in Equations (8) and (9). \[P(t)=P_{0}\bigg{[}1+m\cos\bigg{(}\phi_{0}+\frac{4\pi}{\lambda}\cdot\Delta L \bigg{)}\bigg{]}\exp\bigg{(}-\frac{-(t-t_{0})^{2}}{2w^{2}}\bigg{)} \tag{8}\]  
where \(\phi_{0}\) is the initial phase. Equation (8) can be extended as:  
\[P(t)=P_{0}\cdot\exp\bigg{(}-\frac{-(t-t_{0})^{2}}{2w^{2}}\bigg{)}+m\cdot P_{0} \cos\bigg{(}\phi_{0}+\frac{4\pi}{\lambda}\cdot\Delta L\bigg{)}\cdot\exp\bigg{(} -\frac{-(t-t_{0})^{2}}{2w^{2}}\bigg{)} \tag{9}\]  
From Equation (9), it can be seen that the SMI signal induced by an individual flowing particle consists of two parts: (1) A Gaussian-shaped waveform pedestal with peak amplitude \(P_{0}\) and passage duration \(w\) (dashed line in Figure 3b) and (2) A sinusoidal oscillation of peak amplitude value \(mP_{0}\), also with Gaussian amplitude (Figure 3a,b). This part is the essential result of the interactions inside the cavity between the scattering light from the particle and the initial light containing the particle information such as the particle flowing velocity and particle size.