Micro Particle Sizing Using Hilbert Transform Time Domain Signal Analysis Method in Self-Mixing Interferometry

The present work envisages the development of a novel and low-cost self-mixing interferometry (SMI) technology-based single particle sensing system in a microchannel chip for real time single micro-scale particle sizing. We proposed a novel theoretical framework to describe the impulse SMI signal expression in the time domain induced by a flowing particle. Using Hilbert transform, the interferometric fringe number of the impulse SMI signal was retrieved precisely for particle size discrimination. For the ease of particle sensing, a hydrodynamic focusing microfluidic channel was employed by varying the flow rate ratio between the sample stream and the sheath liquid, and the particle stream of a controllable width was formed very easily. The experimental results presented good agreement with the theoretical values, providing a 300 nm resolution for the particle sizing measurement.


Introduction
With the rapid developments in micro-nano technology, a large number of innovative micro-or nanoscale particles have been synthesized that are known to play a crucial role in the biomedical domain such as drug carriers or imaging agents in living tissues. Thus, the characterization of the particle size and other properties remains an extremely fundamental aspect.
As a low-cost, self-aligned, nonintrusive, compact, and reliable sensing approach, self-mixing interferometry (SMI) technology-based microfluidic sensing has gained much attention during the last several decades [1][2][3][4][5]. In such schemes, the laser output amplitude and frequency are modulated when a small fraction of the laser emission is scattered backward by a small particle in the microchannel coupled into the laser cavity. The characteristics of the particle can be extracted from the properties of the SMI modulated signal in both the time-domain and frequency domain.
To date, a great deal of effort has been made in the measurement of the frequency domain. The dynamic scattering (DS) method, as a typical tool for the characterization of the Brownian motion of small particles, can provide a promising assay in SMI-based particle sizing experiments. By measuring the Doppler power frequency spectrum of scattered light from a particle group, the average particle size can be retrieved. Zakian et al. presented for the first time a theoretical model to describe the frequency-shifted signal power spectrum of a laser source subjected to dynamic scattering light from the particles based on the DS method [6,7]. They demonstrated the possibility of measuring particle sizes ranging from 20 nm to 200 nm. Sudo et al. also successfully implemented particle sizing measurement using the DS method, where the particle dimension and concentration were quantitatively characterized using the Lorentz fitting of the power spectrum of the SMI signal from polystyrene particles of diameters ranging from 20 nm to 500 nm [8,9]. They also reported a forward experimental work by means of an acoustic-optical modulated self-mixing laser metrology for the accurate measurement of diffusion constants [10][11][12][13]. More recently, Wang et al. reported a fast and economic home-made signal processing algorithm for nanoparticle size measurement. With a 16 channel analog circuit, the self-mixing AC signal was transformed into DC signals, thus retrieving the size of the nanoparticles [14,15].
However, particle size evaluation techniques in the frequency domain like the DS method are often employed on particle sizes from several to hundreds of nanometers, which are smaller than the laser spot size for a longer autocorrelation time. Moreover, these techniques can only estimate the average particle size of the sample suspension in a high particle number density and low target velocities. For low density particles in the probe volume, the detection was not qualified enough due to a low backscattering efficiency. Single particle detection using the SMI technique still remains challenging, and there are very few reports on single particle detection in a SMI setup [16][17][18]. Contreras et al. presented a particle size characterization study of single polystyrene spheres with a diameter in several tens of microns in the time domain [17]. They employed a novel SMI scheme with edge-filter enhanced self-mixing interferometry (ESMI), yielding around two orders of magnitude a higher signal-to-noise ratio (SNR) in a lower feedback region even at a 10 m operation distance. Zhao et al. facilitated a robust and real time single particle detection system based on the SMI technique [19]. However, none of the works above-mentioned could measure the particle size with a considerable applied resolution of 1 µm. The existing system still cannot satisfy high throughout real time particle size analysis.
In the present work, we presented a simple and capable SMI-based single particle size characterization method by analyzing the time domain. We proposed a novel conceptual framework of the impulse laser output power signal subject to the feedback modulation from a flowing microparticle, based on the existing well-known theory. In our theoretical model, the relationship between the particle diameter and the interferometric fringe number was investigated. A customized microfluidic chip was designed as a reactor, where a stable particle sample stream was produced to make the single particle pass through the measuring area, and the hydrodynamic focusing performance was tested by varying the flow rate from each inlet. Finally, the SMI signals resulting from the mono particles of different dimensions were acquired and processed by a home-made LabVIEW algorithm involving band-pass filter and Hilbert transform. The resulting interferometric fringe numbers were demonstrated to validate our theory.

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 P0, the modulation index m, and the phase ϕ [21].
where τp is the photon lifetime and τl is the round-trip flight time inside the laser cavity. The instantaneous external cavity length, Lext(t) is expressed by the initial length L0 and the instantaneous displacement ∆L using Equation (3).
Equation (1) can be rewritten as: Considering the external cavity length determines the phase ϕ, the displacement ∆L can be measured with a resolution of half wavelength.
Due to the well-known Doppler effect, the frequency of the laser output is shifted by fD: where V is the particle flowing velocity and θ 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 ∆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 halfwavelength order, as shown in Equation (7).  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 φ [21].
where τ p is the photon lifetime and τ 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 ∆L using Equation (3).
Equation (1) can be rewritten as: Considering the external cavity length determines the phase φ, the displacement ∆L can be measured with a resolution of half wavelength.
Due to the well-known Doppler effect, the frequency of the laser output is shifted by f D : where V is the particle flowing velocity and θ 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 ∆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). Figure 2. Laser incidence process upon the flowing particle and external cavity length variation ∆L from point A to C.
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 t0, so the SMI signal amplitude also varies as a Gaussian function in the time domain, as shown in Equation (8) and (9).
where ϕ0 is the initial phase. Equation (8) can be extended as: 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 P0 and passage duration w (dashed line in Figure 3b) and (2) A sinusoidal oscillation of peak amplitude value mP0, also with Gaussian amplitude (Figures 3. a, 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. Considering the tiny particle size, rext is extremely low, so the SMI sensors operate in the weak feedback regime, and the m value is normally in the order of 10 -6 to 10 -4 .   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).

Experimental Setup
where φ 0 is the initial phase. Equation (8) can be extended as: 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. Considering the tiny particle size, r ext is extremely low, so the SMI sensors operate in the weak feedback regime, and the m value is normally in the order of 10 −6 to 10 −4 .
Appl. Sci. 2019, 11, x 4 of 11 Figure 2. Laser incidence process upon the flowing particle and external cavity length variation ∆L from point A to C.
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 t0, so the SMI signal amplitude also varies as a Gaussian function in the time domain, as shown in Equation (8) and (9).
where ϕ0 is the initial phase. Equation (8) can be extended as: 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 P0 and passage duration w (dashed line in Figure 3b) and (2) A sinusoidal oscillation of peak amplitude value mP0, also with Gaussian amplitude (Figures 3. a, 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. Considering the tiny particle size, rext is extremely low, so the SMI sensors operate in the weak feedback regime, and the m value is normally in the order of 10 -6 to 10 -4 .

Experimental Setup
The experimental measurement setup is illustrated in Figure 4, which consists of three parts: the laser diode based SMI sensor, the microfluidic platform, and the signal processing algorithm. The SMI sensor was based on a commercial distributed feedback laser diode (DFB) working at a wavelength of 690 nm (Thorlabs, Newton, US), and the laser operated at a constant voltage through a DC voltage-stabilized source (Rigol, Beijing, CN). The incident angle θ between the laser emission axis and perpendicular to the plane of the channel chip was set to 4°, weighing a satisfying trade-off that allows us to maintain a good signal-to-noise ratio and a sufficient Doppler shift frequency that fitted our acquisition devices. For the laser beam shaping, we used two spherical lenses in the same mode (Thorlabs, Newton, USA) with an 8 mm focal length. The DFB laser and the lens pair were assembled within a cage system (Thorlabs, Newton, USA). Subsequently, the entire optical arrangement was mounted on a 3D translation stage that allowed the adjustment of the focus location. The DFB sensor and the optical arrangement were mounted on a home-made printed circuit band (PCB); by monitoring the embedded photodiode inside the package, the output power fluctuation of the laser diode in the presence of feedback modulation was retrieved. A home-made transimpedance amplifier was employed here to enhance the signal by transforming the photodiode current to voltage. Finally, the signal was acquired using a National Instrument acquisition card (NI-6361 USB) driven by a customized LabVIEW algorithm. The time domain signal was processed by a home-made MATLAB script for burst certification and particle sizing.
In the experimental setup, we applied the single particle sizing measurements using five different diameters polystyrene sphere (PS) aqueous dispersions (Zhongkeleiming, Beijing, CN). The dimension coefficients of variation (CV) values of the particle dimension were better than 3% for each sample. Since the mass density of PS is 1.05 g/cm 3 , similar to that of water, the particles can suspend uniformly in water after being mixed well with an ultrasonic cleaner. As shown in Figure 5, a customized hydrodynamic focusing polydimethylsiloxane (PDMS) microfluidic channel of 100 μm thickness was employed as the particle sequence generator chamber and sensing reactor. The dispersions were injected into the channel from the central inlet. Meanwhile, two deionized water fluids were injected using another pump through the other inlets on the left and right side as the sheath flow. Compressed by the double sheath fluids, the particle solution was confined within the center of the channel in the horizontal direction, so the particle group was transformed to a one-byone sequence after well particle suspension dilution [23]. This allowed for an individual particle to be characterized as they passed through the SMI sensing region at a constant velocity. The SMI sensor was based on a commercial distributed feedback laser diode (DFB) working at a wavelength of 690 nm (Thorlabs, Newton, US), and the laser operated at a constant voltage through a DC voltage-stabilized source (Rigol, Beijing, CN). The incident angle θ between the laser emission axis and perpendicular to the plane of the channel chip was set to 4 • , weighing a satisfying trade-off that allows us to maintain a good signal-to-noise ratio and a sufficient Doppler shift frequency that fitted our acquisition devices. For the laser beam shaping, we used two spherical lenses in the same mode (Thorlabs, Newton, USA) with an 8 mm focal length. The DFB laser and the lens pair were assembled within a cage system (Thorlabs, Newton, USA). Subsequently, the entire optical arrangement was mounted on a 3D translation stage that allowed the adjustment of the focus location. The DFB sensor and the optical arrangement were mounted on a home-made printed circuit band (PCB); by monitoring the embedded photodiode inside the package, the output power fluctuation of the laser diode in the presence of feedback modulation was retrieved. A home-made transimpedance amplifier was employed here to enhance the signal by transforming the photodiode current to voltage. Finally, the signal was acquired using a National Instrument acquisition card (NI-6361 USB) driven by a customized LabVIEW algorithm. The time domain signal was processed by a home-made MATLAB script for burst certification and particle sizing.
In the experimental setup, we applied the single particle sizing measurements using five different diameters polystyrene sphere (PS) aqueous dispersions (Zhongkeleiming, Beijing, CN). The dimension coefficients of variation (CV) values of the particle dimension were better than 3% for each sample. Since the mass density of PS is 1.05 g/cm 3 , similar to that of water, the particles can suspend uniformly in water after being mixed well with an ultrasonic cleaner. As shown in Figure 5, a customized hydrodynamic focusing polydimethylsiloxane (PDMS) microfluidic channel of 100 µm thickness was employed as the particle sequence generator chamber and sensing reactor. The dispersions were injected into the channel from the central inlet. Meanwhile, two deionized water fluids were injected using another pump through the other inlets on the left and right side as the sheath flow. Compressed by the double sheath fluids, the particle solution was confined within the center of the channel in the horizontal direction, so the particle group was transformed to a one-by-one sequence after well particle suspension dilution [23]. This allowed for an individual particle to be characterized as they passed through the SMI sensing region at a constant velocity. In the present work, a home-made data processing algorithm was also proposed. Hilbert transform (HT) is a useful signal processing tool for the description of the complex envelope of a carrier signal. Such a method could represent a signal in its analytical form by performing a 90° phase shift of the original signal over the orthogonal plane without modifying the amplitude, and the phase information is independent of the signal amplitude variations [24]. So far, HT has presented a good performance on SMI signals, as proven in many reports [25][26][27]. Herein, we employed the HT to unwrap the simultaneous external phase ϕ(t) and extract the fringe number correctly.
The transform HT(u(t)) was calculated as the convolution of function u(t) with an impulse response function h(t) = 1/πt.
In this work, the SMI signal P(t) was transformed into a complex analytical form and was expressed as a combination of two orthogonal parts: namely, the real part Φ(t) and the imaginary part Θ(t), respectively, and the imaginary part realizes a phase shift of roughly 90 ° compared to the original SMI signal [28]. The complex form and the phase of the signal can be denoted by the following equations.
The flow chart of signal acquisition and processing based on the work in [19] is depicted in Figure 6. First, a band-pass filter operated on the raw signal in a certain frequency range (from 1 kHz to 10 kHz) to remove the unwanted high-frequency disturbance from electronic devices and low frequency noise from the signal pedestal in Equation (8). Subsequently, considering the noise floor, a trigger of 200 mV amplitude was applied to the filtered signal. Once the signal amplitude level reached the threshold, the signal sequence in the current acquisition window was saved. Finally, the number of fringes was counted by the Hilbert transformation for single particle characterization. In the present work, a home-made data processing algorithm was also proposed. Hilbert transform (HT) is a useful signal processing tool for the description of the complex envelope of a carrier signal. Such a method could represent a signal in its analytical form by performing a 90 • phase shift of the original signal over the orthogonal plane without modifying the amplitude, and the phase information is independent of the signal amplitude variations [24]. So far, HT has presented a good performance on SMI signals, as proven in many reports [25][26][27]. Herein, we employed the HT to unwrap the simultaneous external phase φ(t) and extract the fringe number correctly.
The transform HT(u(t)) was calculated as the convolution of function u(t) with an impulse response function h(t) = 1/πt.
In this work, the SMI signal P(t) was transformed into a complex analytical form and was expressed as a combination of two orthogonal parts: namely, the real part Φ(t) and the imaginary part Θ(t), respectively, and the imaginary part realizes a phase shift of roughly 90 • compared to the original SMI signal [28]. The complex form and the phase of the signal can be denoted by the following equations.
Θ(t) = HT(P(t)) (12) The flow chart of signal acquisition and processing based on the work in [19] is depicted in Figure 6. First, a band-pass filter operated on the raw signal in a certain frequency range (from 1 kHz to 10 kHz) to remove the unwanted high-frequency disturbance from electronic devices and low frequency noise from the signal pedestal in Equation (8). Subsequently, considering the noise floor, a trigger of 200 mV amplitude was applied to the filtered signal. Once the signal amplitude level reached the threshold, the signal sequence in the current acquisition window was saved. Finally, the number of fringes was counted by the Hilbert transformation for single particle characterization.

Hydrodynamic Focusing of Dye and Particle Streams
First, the hydrodynamic focusing process and laminar flow stability inside the microfluidic chip were observed using an optical microscope. In order to distinguish the boundaries and the widths of the fluids more clearly, 0.1 %wt methylene blue solution was injected into the channel as an indicator. Figure 7 shows the different blue-dyed stream widths of (a) 286 μm, (b) 183 μm, (c) 104 μm, and (d) 51 μm produced by different pumping flow-rate ratios of the analyte/sheath fluid. These photographs indicated well-defined core fluids in the chamber. As the flow rate ratio decreased, the blue inner stream was constricted immediately. All of the streams maintained a constant and straight form over a two-hour period, proving the focusing stability of our channel. Finally, considering the maximum PS dimension (<10 μm), we determined the flow rates of the sample and sheath to be 1 and 25 μL/min, respectively, for a 20 μm core fluid width. The maximum Doppler frequency induced by the flowing particle was calculated to be 4.8 kHz by Equation (6).

Single Particle Sizing Measurement
Hundreds of monodispersed signal bursts in different particle sizes were captured and processed by our system as mentioned in the previous section. The sampling frequency and acquisition data number were 500 kHz and 2 17 , respectively. The data processing results of an 8 μm diameter PS particle are shown in Figure 8. In Figure 8a, the blue and red curves represent the raw SMI signal and the filtered signal in the time domain, respectively. It can be observed that the bandpass filter effectively eliminated the electronic noise, and the impulse waveform evoked from the flowing particle was distinguished easily. Though the fringes could be observed, there was still some signal perturbation ambiguity in the waveform edge. Thus, after the filter, the phase characterization

Hydrodynamic Focusing of Dye and Particle Streams
First, the hydrodynamic focusing process and laminar flow stability inside the microfluidic chip were observed using an optical microscope. In order to distinguish the boundaries and the widths of the fluids more clearly, 0.1 %wt methylene blue solution was injected into the channel as an indicator. Figure 7 shows the different blue-dyed stream widths of (a) 286 µm, (b) 183 µm, (c) 104 µm, and (d) 51 µm produced by different pumping flow-rate ratios of the analyte/sheath fluid. These photographs indicated well-defined core fluids in the chamber. As the flow rate ratio decreased, the blue inner stream was constricted immediately. All of the streams maintained a constant and straight form over a two-hour period, proving the focusing stability of our channel. Finally, considering the maximum PS dimension (<10 µm), we determined the flow rates of the sample and sheath to be 1 and 25 µL/min, respectively, for a 20 µm core fluid width. The maximum Doppler frequency induced by the flowing particle was calculated to be 4.8 kHz by Equation (6).

Single Particle Sizing Measurement
Hundreds of monodispersed signal bursts in different particle sizes were captured and processed by our system as mentioned in the previous section. The sampling frequency and acquisition data number were 500 kHz and 2 17 , respectively. The data processing results of an 8 μm diameter PS particle are shown in Figure 8. In Figure 8a, the blue and red curves represent the raw SMI signal and the filtered signal in the time domain, respectively. It can be observed that the bandpass filter effectively eliminated the electronic noise, and the impulse waveform evoked from the flowing particle was distinguished easily. Though the fringes could be observed, there was still some signal perturbation ambiguity in the waveform edge. Thus, after the filter, the phase characterization

Single Particle Sizing Measurement
Hundreds of monodispersed signal bursts in different particle sizes were captured and processed by our system as mentioned in the previous section. The sampling frequency and acquisition data number were 500 kHz and 2 17 , respectively. The data processing results of an 8 µm diameter PS particle are shown in Figure 8. In Figure 8a, the blue and red curves represent the raw SMI signal and the filtered signal in the time domain, respectively. It can be observed that the band-pass filter effectively eliminated the electronic noise, and the impulse waveform evoked from the flowing particle was distinguished easily. Though the fringes could be observed, there was still some signal perturbation ambiguity in the waveform edge. Thus, after the filter, the phase characterization via Hilbert transform was also employed to extract the fringe number precisely, as shown in Figure 8b, and the fringe number ambiguity was further improved. We processed 30 signals in the range of 5 μm and 10 μm particle diameters. The distribution of the fringe numbers of the PS particle is shown in Figure 9. It should be emphasized that though the particles were spatially restricted in a small area (20 μm width) inside the hydrodynamic focusing channel center, Doppler frequency shift deviations still existed from 3 kHz to 4.8 kHz. However, according to a given size particle, the fringe number did not vary dramatically at different Doppler frequencies. The fringe number was the eventual characteristic of the given particle. The filtered SMI signal waveforms of different sizes PS samples are illustrated in Figure 10. Welldefined fringe patterns were observed from each particle size, and the waveform performance reassembled that of the hypothesis above-mentioned in Equation (9) and Figure 3. It was noted that although the 200 nm PS particle was smaller than half of the laser wavelength (350 nm), one integrated waveform could still be presented in the time domain in Figure 10a. Another point worth highlighting is that the signal frequency was not constant during the passage period, and the upchirp phenomena were observed in the signal bursts, particularly in the bigger particles as shown in Figure 10d. The intervals of the fringes were reduced due to the increase in the frequency. This phenomenon can be explained by the fact that due to the incident angle between the particle flowing direction and the laser axis, the scattering light direction and the Doppler frequency changed during acquisition. We processed 30 signals in the range of 5 µm and 10 µm particle diameters. The distribution of the fringe numbers of the PS particle is shown in Figure 9. It should be emphasized that though the particles were spatially restricted in a small area (20 µm width) inside the hydrodynamic focusing channel center, Doppler frequency shift deviations still existed from 3 kHz to 4.8 kHz. However, according to a given size particle, the fringe number did not vary dramatically at different Doppler frequencies. The fringe number was the eventual characteristic of the given particle. via Hilbert transform was also employed to extract the fringe number precisely, as shown in Figure  8b, and the fringe number ambiguity was further improved.
(a) (b) We processed 30 signals in the range of 5 μm and 10 μm particle diameters. The distribution of the fringe numbers of the PS particle is shown in Figure 9. It should be emphasized that though the particles were spatially restricted in a small area (20 μm width) inside the hydrodynamic focusing channel center, Doppler frequency shift deviations still existed from 3 kHz to 4.8 kHz. However, according to a given size particle, the fringe number did not vary dramatically at different Doppler frequencies. The fringe number was the eventual characteristic of the given particle. The filtered SMI signal waveforms of different sizes PS samples are illustrated in Figure 10. Welldefined fringe patterns were observed from each particle size, and the waveform performance reassembled that of the hypothesis above-mentioned in Equation (9) and Figure 3. It was noted that although the 200 nm PS particle was smaller than half of the laser wavelength (350 nm), one integrated waveform could still be presented in the time domain in Figure 10a. Another point worth highlighting is that the signal frequency was not constant during the passage period, and the upchirp phenomena were observed in the signal bursts, particularly in the bigger particles as shown in Figure 10d. The intervals of the fringes were reduced due to the increase in the frequency. This phenomenon can be explained by the fact that due to the incident angle between the particle flowing direction and the laser axis, the scattering light direction and the Doppler frequency changed during acquisition. The filtered SMI signal waveforms of different sizes PS samples are illustrated in Figure 10. Well-defined fringe patterns were observed from each particle size, and the waveform performance reassembled that of the hypothesis above-mentioned in Equation (9) and Figure 3. It was noted that although the 200 nm PS particle was smaller than half of the laser wavelength (350 nm), one integrated waveform could still be presented in the time domain in Figure 10a. Another point worth highlighting is that the signal frequency was not constant during the passage period, and the up-chirp phenomena were observed in the signal bursts, particularly in the bigger particles as shown in Figure 10d. The intervals of the fringes were reduced due to the increase in the frequency. This phenomenon can be explained by the fact that due to the incident angle between the particle flowing direction and the laser axis, the scattering light direction and the Doppler frequency changed during acquisition. The particle characterization of each particle size was repeated around 100 times, the fringe number trend as a function of the PS particle diameter is depicted in Figure 11. This particle diameter range corresponds to typical human cell dimensions in capillaries. It was observed that the results were in good agreement with the theoretical results calculated by Equation (7), increasing linearly with the particle diameter. The minimal available particle size discrimination by this system was around a half wavelength (~300 nm), and a considerable detection size range from 200 nm to 10 μm was approached, which can be qualified in clinical red blood cell detection. Figure 11. The fringe number of the SMI signal burst as a function of particle diameter. The error bars represent the standard deviation values. The black dashed line denotes the calculation result from Equation (7).

Conclusions
In summary, a novel micro scale particle sensing system based on self-mixing interferometry was developed in the present work. By interferometric fringe analysis along with the Hilbert transformation, our system facilitated a fast and precise particle size characterization. A series of particle sizing experiments using our system were applied herewith in an applicable particle The particle characterization of each particle size was repeated around 100 times, the fringe number trend as a function of the PS particle diameter is depicted in Figure 11. This particle diameter range corresponds to typical human cell dimensions in capillaries. It was observed that the results were in good agreement with the theoretical results calculated by Equation (7), increasing linearly with the particle diameter. The minimal available particle size discrimination by this system was around a half wavelength (~300 nm), and a considerable detection size range from 200 nm to 10 µm was approached, which can be qualified in clinical red blood cell detection. The particle characterization of each particle size was repeated around 100 times, the fringe number trend as a function of the PS particle diameter is depicted in Figure 11. This particle diameter range corresponds to typical human cell dimensions in capillaries. It was observed that the results were in good agreement with the theoretical results calculated by Equation (7), increasing linearly with the particle diameter. The minimal available particle size discrimination by this system was around a half wavelength (~300 nm), and a considerable detection size range from 200 nm to 10 μm was approached, which can be qualified in clinical red blood cell detection. Figure 11. The fringe number of the SMI signal burst as a function of particle diameter. The error bars represent the standard deviation values. The black dashed line denotes the calculation result from Equation (7).

Conclusions
In summary, a novel micro scale particle sensing system based on self-mixing interferometry was developed in the present work. By interferometric fringe analysis along with the Hilbert transformation, our system facilitated a fast and precise particle size characterization. A series of particle sizing experiments using our system were applied herewith in an applicable particle

Conclusions
In summary, a novel micro scale particle sensing system based on self-mixing interferometry was developed in the present work. By interferometric fringe analysis along with the Hilbert transformation, our system facilitated a fast and precise particle size characterization. A series of particle sizing experiments using our system were applied herewith in an applicable particle diameter range. The particle sizing results presented good consistency with the theoretical results, thereby successfully proving the efficiency of our method.
We believe that the developed particle sizing approach could be a promising alternative for both chemical analysis and clinical cell or gene characterization. Furthermore, considering the outstanding compactness of the SMI technique, this method can be the first step toward the embedded synthetic particle sensing scheme integrated into Lab-on-a-Chip devices.