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Abstract
As a compact interferometry technique, self-mixing interferometry (SMI) is a promising tool for micro particle detection in biochemical analysis and the monitoring of laser manufacturing processing, and currently SMI based micro particle detection is attracting increasing attention. However, unlike the typical displacement or vibration measurement driven by a macro target, only a small amount of literature has targeted the SMI effect induced by a single micro moving particle. In this paper, two numerical models were investigated to describe the characteristics of the signal sparked by individual particle. We compared the measurement results with the two models’ simulations in three signal characteristic aspects: the temporal waveform, frequency spectrum, and phase profile. From these results, we established that both amplitude modulation and frequency modulation effects apply under different conditions in the self-mixing process. And for the first time, we analyzed the effect of the laser illumination spot size on the particle-induced SMI signal features with two optical arrangements. When the laser beam size is larger than the particle size, the signal bursts are likely to result from frequency modulation, and vice versa. Our results can improve the capability of SMI technology in particle size discrimination and particle sorting.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Micro scale particles have played a crucial role in various domains considering the rapid advancements in nano-technology and microscale fabrication technology, such as drug carriers and imaging agents in biomedical observation [1], the important raw materials for ultra-solution 3D printing [2]. Thus, the characterization of the particle velocity, size, concentration, aggregation state, and other properties remains an extremely fundamental aspect in various fields. For example, antitumor nanoparticle aggregation significantly impacts nano drug delivery to target tumor tissues, thus nanoparticle sizing is of great concern in the design of nano drugs [3]. Another example is that the laser-induced plume is an inherent physical phenomenon in the processing of high-power fiber laser welding. The plume dynamic behaviors are closely related to the weld dynamics, so the in-situ sizing measurement of the plume particles aids in the monitoring of the manufacturing quality [4]. Therefore, a broad effort has been devoted to classification of micro/nanoparticles, such as light scattering imaging [5–9] and confocal microscopy [10]. These implementations represent outstanding performance; however, for these technologies, usually expensive devices are required, such as high-speed cameras for imaging the scattering patterns, and the whole system is often mechanically complex and bulky in size. Therefore, a simple, fast, and robust detection technology that enables accurate classification of particle size has remained elusive for practical nano medicine and diagnosis applications.
The novel heterodyne technology so-called Self-mixing interferometry (SMI) has intrinsic advantages, such as high capacity, low cost, self-alignment, and the same resolution as that of the conventional dual paths systems. Thus, it has been widely used in micrometry, velocimetry, and absolute distance measurements in traditional industrial fields [11–16]. During the last several decades, SMI technology has been applied to assess microscale particle size distribution, such as monitoring the quality of laser welding by sizing the plume particle during high power laser welding processing [17,18], or flow velocity measurement in microfluidic chips [19–23]. Based on dynamic light scattering theory, the size distribution of the particle population can be retrieved by analyzing the SMI signal frequency spectrum [24–27]. However, this method needs relatively high particle sample concentration for sufficient signal level, so refinement is indispensable. Until several years ago, SMI-based single particle detection technology had not been reported. Moreira et al. for the first time implemented a single particle detection system using SMI technology, they succeeded in observing the presence of single sub microscale particles and measuring the particle flowing velocity in a channel [28]. Zhao et al. proposed a non-dimensional parameter for particle sizing, and compared the SMI signal from living cells and polystyrene beads [29], a clear signal induced by a single particle was acquired. Unlike the continuous SMI signal from a bulk translating target, particle-induced signal represents a discrete waveform burst. To date, several works on single particle detection with this new topic have been reported, but the theoretical mechanism of the self-mixing effect subject to the single particle was poorly studied. Herbert et al. developed a comprehensive numerical model to simulate the pulsed SMI signal properties arising from the mono particle in a flowing media. Their model was based on the typical excess phase equation model and amplitude modulation theory [30]. However, it is still necessary to validate a synthetic model by other experimental works.
This article is organized as follows: first, two theoretical frames of SMI signals based on different modulation mechanisms within the context of single particle detection are presented simultaneously; after that, two synthetical time-discrete signal burst simulation models that take into account the incident angle and particle size are implemented. Second, to identify how the two models work in the realistic particle detection, a series of experiments with different size artificial polymer beads are applied using two SMI detection experimental setups. By comparing the experimental signal bursts with simulation results, we discuss how the relation-ship between the particle size and the laser spot size impact on the SMI signal characteristics, and thus obtain the working condition of each simulation model.
2. Theory
2.1 Excess phase equation under self-mixing effect
When a laser beam shoots at a moving object, part of the reflected light from the object goes back into the laser cavity and “self-mixes” with the initial free-running laser field. This coherently interaction induces modulations in amplitude and frequency (AM and FM) of the cavity field, with a cosinusoidal and sinusoidal driving term for AM and FM, respectively [16].
The schematic diagram for the SMI system is shown in Fig. 1. M1 and M2 are the laser diode cavity mirrors, and r1 and r2 are the reflectivity values of each mirror. rext is the reflectivity of the target. Lc is the laser inner cavity length, and the external cavity length Lext is the distance between the laser output mirror M2 and the target.
The coherent interaction between the reflected light and the free-running laser field gives rise to a simultaneous periodic amplitude and frequency modulation in the laser output power. [15,31,32]. The laser output power subjected to the self-mixing effect P(t) can be expressed from [22] as follows:
P0 is the initial output power without feedback distribution and m is the modulation parameter, which is proportional to the re-injection light coupling efficiency into the laser cavity, whose value can be expressed by Eq. (2). τp is the photon lifetime, and τl is the round-trip flight time inside the laser cavity.In Eq. (1) the critical item is the instantaneous phase ϕ(t) under the feedback effect. Normally, the ϕ(t) value can be calculated by the well-known excess phase equation for the steady-state response of a laser diode experiencing optical feedback [30,33]. The feedback light produces an excess phase onto the laser wave as given by Eq. (3):
where ϕ0(t) is the external round-trip phase of the status of the free-running laser without the feedback effect. C denotes the feedback parameter, indicating the “feedback” strength. α is the linewidth enhancement factor, which can be considered as a constant. In self-mixing interference context as shown in Fig. 1, ϕ0 can be calculated easily as the phase delay resulting from the round trip over the external cavity length Lext [34].In a weak feedback level, i.e., for C < 1, Eq. (3) yields a single solution of the ϕ value versus ϕ0 with a 2π period [34]. Therefore, the output power P(t) profile over the time interval can be derived easily as well using Eq. (1). For example, in Fig. 2 the SMI signal P(t) is plotted for C = 0.1 with a spherical object vibration trace. The whole distance is 8 µm. This typical AM theoretical frame has been widely used in many metrology applications, such as vibration and displacement measurement [13,16]. When C > 1, multiple solutions will exist for ϕ, meaning that the phase of the laser can wrap before reaching the bounds (Fig. 2). However, by considering the context of the microscale particle detection, the C value is always smaller than 1 [22,23], so in this paper we only consider the weak feedback region (C < 1).
2.2 AM-SMI model in context of a passage of a single microsphere particle
From the above theory, the variation of external cavity length ΔLext can be calculated with respect to the phase variation Δϕ value by Eq. (4). When a micro spherical particle passes through the laser illumination area as shown in Fig. 3, the external cavity facet (rext) is the particle curved surface rather than a solid bulk plane in Fig. 1. When referring to the laser axis (the red lines in Fig. 3), if we consider the laser beam spot size is so small as to be negligible, the external cavity length Lext varies during the passage of the particle, resulting in a small phase variation Δϕ. When an interference fringe appears in P(t) signal trace, a 2π phase variation occurs. Consequently, the particle diameter D can be retrieved from the SMI signal phase variation Δϕ.
Another issue should be noted: the SMI effect, when subject to a moving particle, is in a time-discrete form. Only when the particle moves into the spatially limited measurement area is the reflected or scattered feedback light sufficient to spark the SMI effect. The value of m is varying during the particle passage. Here, we assume the modulation index m value varies as Gaussian profile during the passage period τ. At the time point t0, the particle center approaches the laser focal position, and the amount of feedback light reaches the maximum, thus evoking the maximal amplitude of the SMI signal [35].
The simulation conditions are listed in Table 1.
Table 1. Simulation Parameters
The simulated SMI signals induced by the microsphere series in different diameters are shown in Fig. 4. The figures show that the fringe number values are 3, 7, 12, and 15 for 2, 5, 8, and 12 µm particle diameters, respectively, consistent with Eq. (6). All the temporal SMI signal bursts exhibit a symmetrical structure; so the symmetry axis occurs exactly at t = t0, and the laser axis passes through the sphere center of each particle (B point in Fig. 3).
Fig. 4. Simulated Gaussian shaped AM-SMI signal in different particle sizes. (a) 2 µm, (b) 5 µm, (c) 8 µm, (d) 12 µm.
If we ignore the laser beam spot size, the translation time period τ only depends on the velocity V and the transmission distance (here it is approximately the particle diameter D in Fig. 2). For the given particle size D, the τ value is inversely proportional to V. The fringe number does not change with different velocities; the phase variation during the particle flight Δϕ still depends on the particle size rather than the velocity.
Due to the spherical surface, neither the external cavity length variation nor the phase variation change linearly, as shown in Fig. 4. When the beam reaches the particle edge region (A or C in Fig. 2), ΔLext or Δϕ changes much faster than at the center (B in Fig. 3). Consequently, the intervals between the adjacent fringes increase from the particle edge (A or C) to the particle center (B), exhibiting the nonlinear phase variation. Moreover, a prominent and broadening fringe in each figure can be explained with the higher reflectivity and slower phase variation in the center.
Using the Hilbert Transform, we also calculate the phase in the simulation signal [36]. For example, Fig. 5(a) shows the phase profile for a 5 µm micro sphere, the red solid curve is the phase variation value calculated by Eq. (4) without considering the phase flipping due to the external cavity length changing from increasing to decreasing. The black point-dashed line denotes the sphere center position, where the phase deviation approaches the maximal value. The re-constructed phase curve is hemi-spherical shaped as well, consistent with the external cavity length variation due to the particle surface. The frequency domain characteristics are also investigated in the model: the frequency spectrum of a 5 µm micro sphere via MATLAB’s FFT algorithm as shown in Fig. 5(b). From the figure, we can see that there are many harmonic peaks in the spectrum.
Fig. 5. (a) Phase profile calculated by Hilbert transform in the AM-SMI model during a 5 µm particle flight through the detection volume. The blue dashed line is the flipped phase curve; the red solid line is the unflipped phase curve; the black point dashed line denotes the center position of the sphere. (b) Frequency spectrum of a translating particle in the AM-SMI model.
Similar to what was done in Ref. [30], we also consider another scenario where the particle passes the beam axis with an incident angle θ (see Fig. 3(a)). In this case, the particle velocity has a component in the laser incidence direction. The shape of the external cavity length trace is skewed by the addition of the displacement from the particle translation along the laser axis. The simulations with 0, 8, 20 deg incident angle using AM model as shown in Fig. 6. We can note that with the same particle size (12 µm), the signal burst with an incident angle (Fig. 6(b)) is different from the one without an incident angle (Fig. 4(d)). This can be explained by the unsymmetrical external cavity length trace shown in Fig. 6(a). The shape of the burst becomes asymmetric and some fringes are merged (in the dashed line frame) due to the distortion of the amplitude modulation index m profile. However, the phase profile in Fig. 6(c) and the frequency spectrum in Fig. 6(d) are almost identical to the one without an incident angle.
Fig. 6. (a) SMI external cavity length variation curve versus time in different incident angles of 12 µm PS bead. (b) AM modulation based simulated 12 µm particle SMI signal burst in 8 deg incident angle. (c) The Phase profile. (d) Frequency spectrum.
In the AM model, we assume that the laser beam scans the particle surface, and the external cavity length change in micro scale leads to the SMI signal phase oscillation and signal amplitude modulation waveform.
2.3 FM-SMI model in context of single microsphere particle passage
Apart from the amplitude modulation, due to the well-known Doppler Effect, we can assume that when the particle passes through the laser detection region, the feedback light continuously contributes a Doppler frequency modulation on the laser output power. Similar to velocimetry, this scheme is more directly interpreted as the result of the heterodyne measurement [19,37,38].
In the laser cavity, the FM-SMI signal is generated by combining the initial laser waveform at frequency f0 and the Doppler-shifted echo wave from the particle at frequency of f0 ±fD. As a result, the SMI signal can be viewed as having the form of a typical heterodyne signal representing the differential frequency fD in the frequency spectrum. The frequency shift fD value is proportional to the scalar product of the velocity vector $\vec{V}$ and the vector difference between the incidence and scattering vectors $\vec{k}\; $[35]. The expression is shown in Eq. (4), and the positive or negative characteristic depends on the particle translation direction.
The system has an extremely high initial optic frequency (∼1014 Hz). In practical optoelectronic experiments, the bandwidths of an oscilloscope and opto-electronic detectors cannot reach such high values (no more than 109 Hz normally). Thus the ultra-high frequency region signal (within the dashed line frame in Fig. 7 will vanish in the electronic circuit, only the heterodyne difference-frequency signal part in the low frequency region can be observed in the figure center.
Similar to the previous AM-SMI model, the FM-SMI heterodyne signal also indicates a Gaussian shaped envelop of modulation index m value, so the signal expression can be written as the following equation. When t = t0, the laser beam shoots perpendicularly onto the particle or cell center, and the particle center passes the light axis.
In Fig. 8, the simulated Gaussian shaped FM-SMI signals for 2 µm, 5 µm, 8 µm, and 12 µm are depicted with the same parameters in Tablal.1. In the figures, the fringe series, unlike the AM-SMI model, are arranged uniformly, and the interval is constant.
Fig. 8. Simulated SMI signals of microspheres in different sizes: (a) 2 µm, (b) 5 µm, (c) 8 µm, (d) 12 µm.
In FM-SMI model, the phase variation Δϕ can be calculated as the product of Doppler frequency and the transition time period.
Considering the particle velocity V and the incidence angle θ is constant during the short signal period, so according to Eq. (10), Δϕ is linearly related to τ. The phase profile in the FM model is expected to have a linear trend versus time as shown in Fig. 9.
Since the frequency spectrum is the most interesting property of the signal burst in the FM model, we try to derive the frequency model based on the well-known Fourier Transform. We ignore the DC part in Eq. (9), and apply a Fourier transform on the FM-SMI signal burst, yielding:
To simplify the derivation, we set t0 = 0, yielding:
Now, F(ω) can be derived as follows:
Finally, we obtain Eq. (13) as a function of fD:
In Eq. (14) the single particle induced SMI signal also represents a Gaussian form in the frequency spectrum. Here, we consider only the positive part of the spectrum, yielding:
We have simplified the frequency spectrum F(f) in Eq. (15). The central frequency points to the Doppler frequency fD, and the width of the frequency spectrum Wf value is inversely proportional to the transition period τ. By using Eq. (11), at a given velocity we find that Wf decreases with the particle size D. ${W_f} \propto \frac{1}{{\pi \cdot \tau }}$
We applied Fourier transform on the simulated signal bursts to extract the frequency spectra in different particle sizes are shown in Fig. 10(a). Well-defined Gaussian-shaped frequency peaks always can be found at the same frequency exactly corresponding to the Doppler frequency shift fd by Eq. (8). The full width at half maximum (FWHM) value curve versus different particle sizes is depicted in Fig. 10(b), which fits a function that is inversely proportional.
Fig. 10. (a) Typical signal frequency spectra of microsphere in different sizes. (b) The FWHM of the signal spectrum as a function of particle size (the blue line is the inverse proportional fitting result).
3. Experimental study
3.1 System description
The SMI-based particle detection system is shown in Fig. 11. A commercial 1310 nm distributed feedback (DFB) laser diode (Allwave Lasers Device, Inc.) was employed as the laser source and the heterodyne sensor. In choosing this laser diode we considered the benefits of its ultranarrow linewidth low power consuming, and single frequency output. A homemade polydimethylsiloxane (PDMS) hydrodynamic focusing channel was used [29], and the incident angle between the laser axis and the normal of the channel was set to be 8 degree. Ultrapure water was injected simultaneously as the sheath liquid. We set the pumping flow rate ratio of the sheath/sample/sheath fluid at 20-20-20 µL/min, and a stable laminar fluid was induced in the channel, with a 40 µm width core liquid carrying the scatterers. Through an aspherical collimating lens (C240TME-C, Thorlabs, Inc.) and a 20× microscope objective (Daheng, Inc.), the laser output beam was focused tightly and shot perpendicularly onto the microfluidic chip. After adjustment of the optical arrangement position, the laser focal spot was located exactly at the channel center, and the focal spot size was measured by a beam profiler (BP209-IR, Thorlabs, Inc.) to be approximately 20 µm.
We extracted the monitoring photodiode current at the rear facet of the laser package. The optical current signal was converted to a voltage signal and amplified by a homemade trans-impedance amplifier circuit. Afterward, the SMI voltage signal was digitized using a fast speed data acquisition card (NI-6361 USB, NI, Inc.) working at a 2 × 106 sampling frequency, and the number of signal acquisition windows was 1214 samples. Both the signal acquisition and saving operation were automated by a home-made LabVIEW routine on a PC.
3.2 Experimental results
First, SMI signal sequences, which are subject to particles in different sizes, were extracted using the above-mentioned system. The SMI signal waveforms induced by polystyrene (PS) beads of varying sizes are illustrated in Fig. 12. One can note that well-defined fringe sequence can be observed in the entire range of particle sizes from 2 µm to 12 µm, and that the interference fringe number increases as the particle diameter increases. From the figures we find that the measured SMI burst signals for a single particle are quite consistent with the FM model, the fringe array is arranged uniformly, and the fringe intervals are almost kept constant without a dominating central fringe.
Fig. 12. Signal burst of the PS particles with different diameters in the SMI measurements (a) 2 µm, (b) 5 µm, (c) 8 µm, (d) 12 µm.
The frequency spectra of the PS particles of varying diameters were also illustrated as shown in Fig. 13. At the given flow rate, the spectral peak was defined to be 11 kHz for different particle sizes, and as shown in Fig. 14(a), the spectral width Wf decreases with the particle size, the same as the simulations show in Fig. 10(b). The measured Wf values were in good agreement with the FM model simulation, except that in 12 µm particle case, the vale was smaller than the simulated one, it can be explained by the limited frequency resolution of the FFT algorithm.
Fig. 13. Frequency spectra of PS particles in different diameters: (a) 2 µm, (b) 5 µm, (c) 8 µm, (d) 12 µm.
Fig. 14. (a) The FWHM of the frequency spectrum as a function of particle size. The blue marks with error bars denote the measured mean values and the standard deviation, the black line denotes the simulation results. (b) Phase-time curve in 12 µm PS particle; the red dashed line is the linear fitting.
Furthermore, we calculated the signal burst phase by the Hilbert Transform [29]; the signal phase-time profile for a 5 µm PS particle was a good linear function in the signal duration as shown in Fig. 14(b), which was also in good agreement with the FM model simulation. All the temporal waveform, frequency spectrum and phase profile behaviors unambiguously disclose that FM model applicability in this experimental scheme.
3.3 Experimental study by confocal system
To gain deeper insight into the influence of the laser spot size on the SMI signal features, a new confocal microscope scheme was developed. The schematic of the second system is depicted in Fig. 15(a). In the new system, the same laser diode source was used, and after the first collimation and focalization with the lens pair (C240TMD-C and C330TME-C, Thorlabs, Inc.) we placed an 800 µm diameter pinhole at the first focal point as a spatial beam filter. After the pinhole, there was an aspherical lens (C240TME-C, Thorlabs, Inc.) that collimates the filtered beam again. An objective lens (power 50 ◊) focused the beam into the microchannel. The laser beam spot size was measured to be 2 µm, which was much smaller than that of the former SMI setup. Figure 15(b) illustrates that only when the central point of the particle o is in the focal plane (the dashed line), the scattered light (red line) can be coupled efficiently into the laser cavity aperture, arising considerable SMI signal.
Fig. 15. The confocal SMI system. (a) The schematic graph of the system; (b) The confocal optical beam path.
We reused the PS particles of different diameters as the scatterers to extract the SMI signal using the secondary confocal system and plotted the burst signals in Fig. 16. In contrast to the signals retrieved by the first common doublet-lenses system, no interference fringe sequences were found in a PS size range of 2–8 µm. The new SMI signals were spike-like forms with much smaller illumination area rather than the former Gaussian-shaped sinusoidal waveform proposed in Fig. 12.
Fig. 16. Signal burst of the PS particles in different diameters retrieved by confocal system: (a) 2 µm, (b) 5 µm, (c) 8 µm, (d) 12 µm.
Notably, when the particle diameter was 12 µm, 6 times of the laser spot size, the signal burst in Fig. 16 (d) was significantly different from the ones arising out of the smaller particles shown in Fig. 16 (a–c). The fringe sequence appeared again, but the fringe intervals were not uniform as they were in the former setup, and the array was asymmetric. This signal appearance resembles the AM-SMI model with a slight incident angle as shown in Fig. 6 (b). A sufficient quantity of signal bursts from the same particle size and experimental condition were captured, and all the results showed similar phenomena disclosing that the experimental results were consistent with the AM model simulation.
Moreover, both the phase and the spectral profile showed coherence with the AM simulation results, thus effectively proving that when the particle size is much bigger than laser spot, the single particle signal burst is subjected to amplitude modulation resulting from the external cavity variation due to the particle surface. The phase profile measured from the confocal system is presented in Fig. 17(a); the curved profile is different from the straight one in the typical setup in Fig. 12(b), but more like the one in the AM model in Fig. 6(c). Additionally, in the frequency spectrum no clear and visible Doppler frequency peak at 11 kHz can be found over the entire range of particle sizes (for example, 12 µm PS bead as shown in Fig. 9(b)), which was in good agreement with the AM-SMI model.
Fig. 17. 12 µm PS particle confocal SMI signal properties: (a) The phase profile (b) The frequency spectrum.
Figure 17 (a) shows that when the laser beam is smaller than the particle, the measured phase variation value from 12 µm particle was only around 37 rad (from 14360 to 14397 rad), much less than the simulation value (around 90 rad) in Fig. 6(c). To explain this phenomenon, let’s revisit the particle reflection scenario in Fig. 18. In case of very small laser spot, due to the system numerical aperture and incident beam convergence, only a very spatial-limited region at the particle center contributed the SMI effect like Fig. 18(a). Moreover, according to the smaller particles, the correlation between the particle and laser incident beam is too weak to give raise to interference, so only a spike can appear in Fig. 16 (a-c). However, if the laser spot size in bigger than the particle like the former setup in our work, the particle is merged inside the illumination area during the trip in Fig. 18(b), continuously contributes frequency modulation effect, whereas external cavity length variation is negligible, so the signal properties is dominated by FM model theory in this case.
Fig. 18. Schematic diagram of reflected light power from a microscale particle (a) The beam size is smaller than the particle size (b) The beam size is larger than the particle size.
4. Conclusions
To obtain a deeper insight into the self-mixing effect mechanism subject to a single particle, we investigated the characteristics of the single particle SMI signal theoretically and experimentally. Based on the typical self-mixing theoretical frames, we developed two theoretical SMI signal models based on amplitude modulation (AM) and frequency modulation (FM), respectively. Then, we developed two SMI based experimental devices to retrieve the single particle SMI signal. By conducting a comparative study between the self-mixing simulation models and the experimental results, we found, for the first time, that both AM and FM models affect the single particle detection scheme. It is noted that the beam spot size is an essential parameter influencing the multiple properties of the micro particle induced SMI signal, including temporal waveform, the frequency spectrum, and phase profile. When the laser spot size is bigger than the particle size, the SMI oscillation can be viewed as a heterodyne scheme invoked by the Doppler frequency effect, and all the measured signal characteristics were satisfied with the FM model. When the laser spot size is much smaller than the particle size, the signal characteristics are more consistent with the AM model. The SMI signal phase variation is mainly due to the external cavity length changing caused by the particle surface.
On the other hand, we also conclude that the measured phase profile could be another promising tool for SMI based particle detection apart from temporal waveform and frequency spectrum, considering the phase variation strongly depends on the particle size. In this work, we have established that SMI can offer an adaptable single micro/nanoparticle sensing and particle size classification, especially suitable for low-concentration and valuable samples, which is useful for laser processing monitoring and chemical analysis.
Funding
National Natural Science Foundation of China (61905005); General Program of Science and Technology Development Project of Beijing Municipal Education Commission (KM202110005004).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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