The schematic diagram of the experimental setup is shown in Figure 1, which can individually measure the scattered light of suspended particles [12]. The 520 nm continuous wave laser (S) is used as the illuminating light source with a 4 mm diameter beam size and a 0.7 W maximal output power. The illuminating light beam passes through the lens (L1) and the transparent window (W1) and then is focused into a small spot at the focal point. Such a convergent beam effectively enlarges the energy density of the light illuminated on the individual particles in the sample pool (SP). The suspension is stirred by a magnetic stirrer to keep the particles suspended. Particularly, the distance between the focal spot and W1 is about 30 mm, shown as the red dashed line. Once the suspended particle passes through the convergent beam in SP, it will be illuminated. Only the scattered light of the particles in the detection volume can pass through the receiving optical system which consists of the other transparent window (W2), the lens 2 (L2), and a 100-micron pinhole (P). The detection volume and the pinhole are an object–image relationship. Behind the pinhole, there is a short focal length lens (L3) to convert the scattered light to a parallel light beam before entering the photo detector (D). Note that both the illuminating beam and received light suffer from the scattering in the suspension when they respectively propagate after W1 and before W2.

In the individual measurement, if a single suspended particle passes through the scattering volume, which is determined by the focal spot and the detection volume, its scattered light contributes to the signal. When there are no particles in the scattering volume, the electronic noise, environmental light, and the scattering of water will contribute to the background, but the background is much smaller than the signal. Therefore, the signals are a series of temporal pulses. Furthermore, we can calculate the number of suspended particles in unit time, which should be linear with the concentration of the particles in the suspensions for the individual measurement. Meanwhile, the low level of the background is also an effective indicator of the individual measurement. Otherwise, if the scattering volume is too big or the concentration of the particles is too high, there is at least one particle in the scattering volume, which means that the background would be high and there would be no obvious temporal pulse.

The effective scattering cross-section ( σ s ) and the geometrical size of the scattering particle ( G s ) are related by the proportionality constant called the scattering efficiency ( Q s ) [15], as shown in Eq. 1. The scattering efficiency of the microsphere can be calculated using the Mie theory. σ s = Q s ∗ G s (1)

Moreover, the scattering coefficient ( μ s ) describes a medium containing many scattering particles at a concentration described as a volume density ( ρ s ), as shown in Eq. 2. Generally, the scattering coefficient can represent the physical properties of the suspension. For the specific diameter of microsphere particles, σ s can be considered a constant and μ s has a linear relationship with ρ s . μ s = ρ s ∗ σ s (2)

Generally, the achievement of convergent beam mainly depends on μ s and the optical path length ( D L ) that the light beam travels through the suspensions containing particles. In particular, the product of μ s and D L can be a comprehensive indicator ( β s , dimensionless), as shown in Eq. 3. β s = μ s ∗ D L (3)

To check the influence of the particle concentration on the convergence of light beam and further on the individual measurement of suspended particles, we carry out experiments on suspensions with different particle concentrations. Herein, the aquatic suspensions consisting of 3 μm polystyrene microspheres (PM) with a refractive index of 1.59 are used as samples. First, a low concentration of PM (4 × 10⁴ particles/ml) and a high concentration of PM (5 × 10⁵ particles/ml) are respectively placed in the SP and measured using the experimental setup. As shown in Figure 2A, the peak values of the pulses are much larger than the background when the concentration of PM is low, which means a high SNR. Therefore, we can easily extract the pulses of suspended particles, and each pulse presents one particle. However, when the concentration of PM is too high, as shown in Figure 2B, the background is high as almost 10 times of that in Figure 2A, so it is hard to find a pulse with high SNR. That is, the individual measurement will become hard to extract the high SNR pulse of particles in the high-concentration suspension.

Subsequently, different concentrations of PM are measured and the corresponding numbers of suspended particles detected in 3 min (NPM) are recorded, as shown in Figure 3A. Notably, when the concentration is lower than 10⁵ particles/ml, the NPM can increase linearly with the increase of concentration. However, when the concentration increases and becomes higher than 10⁵ particles/ml, the increase of NPM clearly becomes gentle. Correspondingly, the measured mean backgrounds of different concentrations of PM are shown in Figure 3B. The mean backgrounds stay at similar values when the concentration of PM is low. However, the background starts to increase when the concentration of PM approaches 10⁵ particles/ml and becomes larger than 0.18 after the concentration of PM is larger than 10⁵ particles/ml. For simplicity, we set 0.10 as the threshold (red dashed line in Figure 3B). From Figure 3, the influence of the low concentration of suspended particles on the individual measurement can be ignored, which implies that the convergence of the illuminating light beam may not be affected either. However, the mean background will increase rapidly when the concentration is higher than 10⁵ particles/ml.

The concentration of PM in the suspension affects the individual measurement in a complex manner. There is a threshold concentration (about 10⁵ particles/ml). When the concentration of PM is larger than this threshold, the scattering of suspended particles in the optical path of the illuminating beam will destroy the individual measurement in some way. The destructiveness can be indicated by the linear relationship between the NPM and the concentration of PM, and the mean background. Since the convergence of the illuminating beam plays a vital role in the individual measurement, one can imagine that the convergence of the illuminating beam may be no longer valid for the cases with a concentration of PM larger than the threshold concentration.

When there is more than one particle in the light beam, we will get interference and speckle image on any plane. However, for the experimental setup in Figure 1, only the light scattered by the particle in the scattering volume can be received, which may be interference by the very close particles in the light beam. If there is more than one particle in the scattering volume on average, the individual particle measurement is not valid, and the signals would be like those in Figure 2B. Therefore, the particle concentration should be seriously considered to ensure that the light beam statistically illuminates one particle.

To investigate how the convergence of the illuminating beam is affected, we use the Monte Carlo simulation to mimic the light propagation and scattering in the suspensions. The Monte Carlo simulation has been widely used to study the behavior of photons as they propagate through complex media [16, 17], and its principles are clearly described elsewhere [18]. Herein, Monte Carlo simulation is used to simulate the effects of different factors on the convergent beam, including size and concentration of suspended particles, scattering coefficient of the suspension, and optical path length of the illuminating beam before the focal spot in the suspension.

The geometric model of the convergent beam is set in Figure 4A. In accordance with the experiment setup, the radius of the lens (r _in ) is set as 2.0 mm. The wavelength of the incident light is 520 nm. The total number of photons is 10⁹ with a normal distribution in the incident area. The incident photons would statistically undergo propagation and scattering. Some of them will eventually pass through the optical path in suspension and finally arrive at the receiving plane, and the area distribution of the collected photons can be evaluated, and the full width at half-maximum (FWHM) of the distribution can also be obtained. In this work, we use FWHM to reflect the size of the convergent beam and further estimate the situation of the convergence of the light beam after the particle scattering in the suspension. The refractive indexes of the medium and suspended particles are respectively 1.33 and 1.59.

If the FWHM of the convergent beam is low, the convergent beam statistically illuminates one particle, as shown in Figure 4B. A larger FWHM means that the convergent beam would have the opportunity to illuminate more particles with less energy density, which leads to a low SNR, as shown in Figure 4C. According to the signals in Figure 2, a large FWHM will generally disable the individual measurement of suspended particles.

The framework of the Monte Carlo simulation used in this research is illustrated in Figure 4D [16]. The multilayered light scattering and transport algorithms follow the public code [17]. First, we need to set the mentioned necessary simulation parameters, including r _in , μ _s , σ_s, D _L , the number of photons and their wavelength, and the refractive indexes of the medium and suspended particles. Herein, the focal distance and the thickness of the medium are set as the same values which can represent the optical path length, i.e., D _L . The scattering functions of spheres for every angle are calculated and stored in the memory as a database to reduce the computation time. Then, the photon is initialized, propagates, and scatters until it is outside the medium. The program will output the photon information once all photons are calculated. Finally, we can get the Gaussian distribution of the collected photons and further obtain the FWHM of the distribution. The phase changes due to the propagation and scattering of each photon are also traced and considered after arriving at the receiving plane and superposing with other photons, which includes the interference effect between particles.

We have known that particle scattering will redirect the light [15]. Different concentrations of spherical particles in different sizes are simulated, as shown in Figure 5A. The concentration of particles largely affects the FWHM, which changes significantly with the particle sizes. However, each of them has a turning point of concentration before which FWHM stays low and grows gently, but after that, FWHM rapidly increases with concentration. The turning point is related to the particle sizes, and it is smaller when the particle size is bigger. Especially, the turning point of 3 μm diameter particles is at about 10⁵ particles/ml, which is quite consistent with the experiment result in Figure 3.

The scattering coefficient μ _s is determined by the concentration and diameter of a sphere as shown in Eq. 2, so we can combine these two factors by using μ s . Then, we redraw these results as shown in Figure 5B. The curves in Figure 5A are combined by μ _s as the argument, and they almost share the same turning point, μ s 0 = 1.7 cm⁻¹. Before μ s 0 , FWHM is almost unchanged and relatively close to 0, which means that particles in the suspension basically have no effects on the convergent beam.

Subsequently, we simulate the changes of FWHM in different D _L and μ _s of the suspension, as shown in Figure 5C. It is notable that FWHM almost stays low and grows negligibly for a small μ _s but increases very fast for a large μ _s . However, the turning points μ s 0 are smaller for larger D _L . From these results, we can use the dimensionless quantity, β _s , to comprehensively consider their influences on the FWHM. Then, we redraw the curves with β _s of different D _L and show them in Figure 5D. There is a turning point at about β s 0 = 5.0 before which FWHM will be almost not affected.

Till now, we can judge whether the convergence of the illuminating beam can be maintained by the use of β s 0 . We know that the convergence of the illumination beam will determine whether the individual measurement can work in the linearity range for the suspensions. Moreover, according to Eq. 3, μ _s and D _L are equivalent to determining the dimensionless quantity β _s . These simulation results encourage us to manipulate D _L to ensure the convergence of the illuminating beam and then facilitate the individual measurement of suspended particles.

The key point of this strategy is that we use a small tube or a similar container to contain the suspension and leave only distilled water in the sampling pool. Then, we change the location of the tube along the illuminating beam to shorten D _L , and a series of experiments are carried out for different D _L to investigate the effect of this strategy. In practice, 5 × 10⁵ particles/mL PM suspension is contained in a K9 glass bottle (GB), as shown in Figure 6A. GB is gradually moved away from W1, and the NPM is correspondingly recorded for each location. The experiment is repeated three times, and the result is shown in Figure 6B. GB keeps moving away from W1, and the NPM increases. When D _L is less than 4 mm, β _s would be less than 5 and the convergence of the illuminating beam is well kept, which ensures the individual measurement and can achieve the maximal NPM. When D _L becomes larger than 6, the convergent beam’s FWHM is enlarged and the pulses originating from the individual particle are weakened and the background becomes larger so that the pulses with enough SNR become less, and then the NPM decays.

Recall that it is hard to extract pulses with high SNR using the original experimental setup where D _L is 30 mm. For the individual measurement using GB where D _L is 4 mm, the NPM is much larger because the mean background is less than 0.1. This can be explained by the dramatic increase of FWHM of the illuminating beam after β _s is larger than 5 in Figure 5D. Notably, the shortened D _L can obviously promote the achievement of the individual measurement and the probing of the maximal NPM by ensuring the convergence of the illuminating beam. A future design may be practical if we could use a small-diameter tube to let the suspension flow through the scattering volume.

Furthermore, we also consider the emerging demand for the in situ detection of the suspended particle in aquatic environments [19–21]. Therefore, we propose the shielding strategy that the propagation path between the W1 and focus point is sealed and filled with distilled water, and the illuminating light and scattered light pass by a K9 glass window (W3), as shown in Figure 7. Now, compared with the original experimental setup in Figure 1, D _L has been reduced from 30 to 8 mm.

Then, the concentration experiments of PM are carried out using the new setup, and the results are collected and shown in Figure 8A. The linearity range of NPM covers the particle concentration to 2 × 10⁵ particles/ml, whose maximum is almost 3 times larger than that in the original experimental setup. Moreover, as the second indicator of the individual measurement, for the new setup in Figure 7, the mean background stays at a low level, less than 0.1, even at a much higher concentration of PM than that for the original setup. Also, the increase rate of mean background becomes gentler with concentration than those in Figure 3B with the original setup. Similar to Figure 3B, the background grows sharply after the turning point of the concentration [10⁶ in Figure 8B], which can be explained by the FWHM’s sharp growth in Figure 5 after the turning point. In addition, we can find that NPM linearly increases with the concentration of PM when the mean background is low.

We notice that there is a slight difference in the slopes in Figures 3A, 8A, which may originate from different stirring speeds. In the experiment, the stirring speed has little effect on the convergence of the illuminating beam but can affect the movement velocity of suspended particles. Therefore, the accuracy control of the stirring speed can generally decrease the errors of the results. Herein, we use a continuously adjustable magnetic stirring apparatus with the range of 300–1,800 rpm (Topolino, IKA Works GmbH & Co. KG, Germany) to rotate the magnetic rotor. However, the magnetic stirring apparatus cannot show the specific speed. We first find a fixed location of the rotary knob to ensure that the magnetic rotor can stably rotate, and then we adjust the rotary knob to the fixed location for each experiment. It may lead to a slight difference in the stirring speed. Moreover, the Brownian motion of suspended particles and the fluctuations can be ignored because the Brownian motion is much slower than the stirring speed.

Then, we calculate β _s based on the concentration experiments using Eq. 3, as shown in Figure 9. Impressively, NPM will linearly increase with the increase of β _s when β _s is less than 5, which is quite consistent with the simulation result. After that, the NPM loses the linearity with β _s . Therefore, β _s is an effective indicator to comprehensively demonstrate the achievement of the individual measurement and it can further reveal the convergence of the illuminating beam in the suspension.

As a dimensionless quantity, β _s combines the physical properties of the suspension and the optical path length of the illuminating beam. According to the physical explanation of the scattering in turbid media, β _s describes the logarithmic attenuation of the intensity suffered from the scattering in the suspensions [15]. In statistics, β _s usually can be used to determine the multiple scattering times with the mean free path [18]. From the experimental results and simulation results, we can see that if the scattering occurs less than five times, the convergence of the illuminating beam can be kept effectively. Otherwise, the convergence of the illuminating beam will seriously suffer from particle scattering. These conclusions will help the optical design of the particle probers in the suspension.

For the natural water ecosystem, the suspended particles not only include the spherical particles but also the non-spherical particles. Also, the suspended particles in natural water may have diverse sizes, morphologies, materials, and pigments. The convergence of the illuminating beam in natural water is worth to investigating in the future. However, the investigation method used in this work would be beneficial and the conclusions may be suitable or at least instructive for natural water.