## 2 Materials and Methods
### Setup  
A microfluidic flowmetry system is built in this work for numerical simulation and experimental validation, the setup is illustrated in Figure 2a. One 1 mm \(\times\) 1 mm polydimethylsiloxane (PDMS) channel and two 96 \(\upmu\)m \(\times\) 96 \(\upmu\)m dry film (DF) photoresist channels realized the microfluidic reactors, and one of the DF channels was built with an additional gold coating film on the rear interface to demonstrate the channel reflectivity influence of the OFI signal. The OFI sensor is based on a 670 nm VCSEL (Vertical-Cavity Surface-Emitting Laser)at a constant bias current of 3.5 mA through a custom-designed laser driver. The incident angle \(\theta\) between the laser emission axis \(Z^{\prime}\) and perpendicular to the plane of the channel chip is set to \(15^{\circ}\) as this value has experimentally proven the best signal level. A doublet lens pair (MAP103030B, Thorlabs Co., Ltd, Newton, NJ, USA) constituted by two lenses of the same focal length of 30 mm was used for laser beam focalization, resulting in a 4.6 \(\upmu\)m (1/e\({}^{2}\)) laser beam waist in the micro-channel. Polystyrene (PS) microspheres (MicroParticles, GmbH, Berlin, Germany) of two different diameters (552 nm and 4.96 \(\upmu\)m) were used as the tracing particles. The PS solutions were pumped into the channel by a syringe pressing pump (PHD 22/2000 Harvard Apparatus, Holliston, MA, USA). The OFI signal was achieved through the monitor photodiode embedded in the VCSEL package. After amplification by a custom-made transimpedance amplifier, the OFI signal was acquired using a National Instrument acquisition card (NI USB 6251, National Instruments, Austin, TX, USA) at a sampling frequency of 400 kHz. Processing of the signal consists in averaging of 30 consecutive Fast Fourier Transform (FFT) of \(2^{15}\) samples. All the data acquisition procedures and processing were automated by a custom-made LabVIEW algorithm.  
Figure 1: Incidence and scattering vectors when the laser beam was shot at the particle.  
To implement our theoretical model, multiple numerical calculation tools were applied. Firstly, the entire optical part of the OFI system and the microfluidic channel chip were modeled using the commercial ray tracing software Zemax to simulate the laser beam propagation. All parameters such as channel geometrical sizes, materials, lens film coatings, particle characteristics were chosen to be in a realistic agreement with the experimental parameters in the 3D model. To calculate the OFI signal in different particle positions, a cuboid mesh was built inside the channel, and the fine 3D grid intervals were set to be \(\text{dx}=\text{dy}=0.5~{}\upmu\text{m}\), \(\text{dz}=1~{}\upmu\text{m}\). A PS particle was placed at each node and then the ray tracing processing was repeated at all the mesh node coordinates. By the Monte Carlo statistical approach, the individual particle scattering angular distribution and the feedback light power re-entering-cavity coupling efficiency \(r_{ext}\) were computed. Secondly, the flowing velocities at each node were calculated through simulation with computational fluid dynamics software Fluent. Finally, a Matlab algorithm has been designed that calculated the feedback modulation contributions to \(P_{f}\) and the associated Doppler frequency shift \(f_{d}\) as expected from Equations (3) and (7) and that simulates the Doppler frequency spectrum signal. The values of the modeling parameters are listed in Table 1.