We describe a highly sensitive, real-time method of detecting small particles in a fluid flow by self-mixing laser Doppler measurement with a laser-diode-pumped, thin-slice solid-state laser with extremely high optical sensitivity. Asymmetric power spectra of the laser output modulated by the re-injected scattered light from the small particles moving in a dilute sample-flow, through a small-diameter glass pipe, were observed. The observed power spectra are shown to reflect the velocity distribution of the fluid flow, which obeys Poiseuille’s law. Quick measurements of flow rate and kinetic viscosities of water-glycerol mixtures were also performed successfully. Measurable low-concentration limits for 262-nm polystyrene latex spheres and 3-μm red blood cells in a fluid flow were below 1 and 10 ppm, respectively, in the present self-mixing laser Doppler velocimeter system.
© 2007 Optical Society of America
Sensitive and rapid detection of small particles in a fluid flow is needed for many applications, such as the production of pharmaceuticals and semiconductor equipment, conservation of water quality, and marine science. For example, integrated circuits use electrical lines as narrow as nearly 0.1 μm, and highly efficient detection of microparticles and nanometer-sized particles in washing water is required to evaluate semiconductor wafer contamination. The detection of small particles in a fluid flow has been performed extensively using a magnetic flowmeter [1, 2], ultrasound Doppler velocimeter [3,4], and laser Doppler velocimeter (LDV) [5–9]. The advantages of the LDV for a dilute sample-flow compared with conventional measuring techniques lies in the high spatial and temporal resolution and the possibility of non-contact measurement for the detection of particles 1 nm to 10 μm in size in a dilute sample. In the LDV, the velocity of moving particles is related to the Doppler shift frequency fD of scattered light by 
Here, λ is the wavelength, v is the velocity of the moving particle, vz is its velocity along the probe beam axis, and θ is the angle between the velocity vector and the probe beam axis.
The LDV system, which utilizes the self-mixing modulation effect in semiconductor lasers, has been applied to vibrometry [11–13], speckle pattern interferometry , and measurements of the “average” velocity and flow of small particles and red-blood cells [15–19]. The self-mixing LDV system with cheap optical components can be easily aligned, and can give a larger modulation depth than other techniques. We have developed several self-mixing laser-Doppler measurement systems with thin-slice solid-state lasers , including real-time vibration measurement systems [21–23] and a self-mixing particle analyzer [24, 25]. Solid-state lasers with highly doped active ions having an extremely short absorption length for laser-diode (LD) pumping are suitable for thin-slice cavity configurations, leading to an extremely short photon lifetime compared with the fluorescence lifetimes. Self-mixing LDV systems that have higher optical sensitivity get this feature from the large fluorescence-to-photon lifetime ratios inherent in thin-slice solid-state lasers, which are larger than those of self-mixing lasers with semiconductor lasers [11–19]. In our previous work, a self-mixing particle analyzer and vibrometry system, consisting of an LD-pumped Nd:GdVO4 thin-slice laser, a simple photodiode, and an rf spectrum analyzer, were shown to perform quick, accurate sizing of 262-nm polystyrene latex (PS) spheres undergoing Brownian motion in a dilute solution (0.05 parts per million (ppm)) and real-time measurement of 1-nm vibration .
In this work, we measured the power spectrum of the modulated laser output, which corresponds to the velocity distribution of light scattered from small particles in a flow of water, using the LD-pumped self-mixing Nd:GdVO4 thin-slice laser, and aqueous flowmetry for dilute sample-flows was demonstrated.
2. Experimental setup
The experimental procedure is shown in Fig. 1, where a 3 at.% Nd-doped 1-mm-thick Nd:GdVO4 laser with coated end mirrors (M1: 95% transmittance at 808 nm, 99.8% reflectance at the lasing wavelength of λ = 1063 nm; M2: 98% reflectance at 1063 nm) was used as a self-mixing laser. A collimated beam from the laser diode operating at 808 nm was passed through a pair of anamorphic prisms and focused onto a laser crystal. Part of the laser beam (4%) was sent to an InGaAs photoreceiver (New Focus 1811: DC-125 MHz) connected to a spectrum analyzer (Tektronics TDS540D: DC-3GHz) and/or AD converter (RFSPACE SDR-14). The main beam (96%) was passed through two TeO2 acousto-optic modulators (AOMs) (central frequency: 80 MHz) and delivered to a scattering cell through a microscope objective lens with a numerical aperture of 0.25. The beam diameter at the focal plane was estimated to be 100 μm. The lasing threshold was Pth = 32 mW, and the slope efficiency was 24%. By tuning the up- and down-shift frequencies of the AOMs to +80 and -79 MHz, we set the roundtrip frequency shift (i.e., carrier frequency) 2fAOM to 2 MHz, and the short-term stability of the carrier frequency was Δf = 10 Hz. A dilute sample containing 10 wt% PS standard spheres with a diameter of 262 nm purchased from Estapor was diluted to various concentrations. Power spectra of the laser output modulated by the re-injected scattered light from the PS particles moving in the dilute sample-flow were measured because the velocity of PS particles in a dilute sample-flow can be obtained from these spectra.
3. Experimental results of sample-flow
3.1 Dilute sample-flow through a vertical glass column
The flow passage in the first experiment is illustrated in Fig. 2. A Doppler-shift frequency can be increased with increasing the angle. Therefore, the large angle is suitable for measuring velocity distributions for liquids with high viscosities. For measuring liquid flow with lower viscosities, on the other hand, we need a wider frequency span (i.e., lower frequency resolution) for power spectrum analysis and in particular the important information around 2f AOM is lost. Therefore, we set the angle to 10° to measure liquids having various viscosities with reasonable resolutions under the same scattering scheme. In the scattering cell, 3.7 cm3 of fluid flowed from the reservoir at the top down through a vertical glass column 100 mm long and 1.2 mm in diameter. Water-glycerol mixtures with various glycerol concentrations were used as liquid, and PS standards with a diameter of 262 nm were added. Here, the flow velocity becomes slower by adding glycerol, because the viscosity of the glycerol is much higher than that of the water. The concentration of PS particles used in the first experiment was 0.05 wt% in all mixtures.
As an example of the results for the dilute sample-flow, Fig. 3 shows time-dependent power spectra of the modulated wave for PS particles in water. In the stationary state of the dilute sample, the power spectrum had showed a Lorentz shape representing the Brownian motion of PS particles . When the dilute sample began to flow in the passage, a power spectrum with an asymmetric shape appeared on the higher frequency side with respect to the carrier frequency 2fAOM = 2 MHz.
To better explain the observed power spectra, let us characterize the measured power spectra during the time period when stationary flow (i.e., constant peak frequency shift) was established, as shown in Figs. 3(b)–3(e). The averaged power spectra for PS particles in various glycerol-water mixtures are shown in Fig. 4. The power spectrum became narrow and the peak frequency approached 2fAOM as the glycerol concentration increased. These results indicate that this approach toward 2fAOM resulted from a decrease in the velocity of PS particles moving in the vertical direction in the fluid flow as the glycerol concentration was increased. In other words, the velocity of the dilute sample-flow containing PS particles decreased as the viscosity of the water-glycerol mixture increased, and the decrease in velocity of PS particles led to a decrease in the Doppler frequency-shift of the modulated wave, as represented by Eq. (1). The narrowing of the power spectrum with decreasing velocity of dilute sample-flow was predicted in Ref. .
3.2 Dilute sample dropped from a vertical glass column
To clarify the physical mechanism of the asymmetric power spectrum observed for a dilute sample-flow in the passage, we allowed a dilute sample to drop vertically from the passage and measured the power spectrum using a self-mixing LDV system. The flow passage used in the second experiment is illustrated in Fig. 5. The probe light impinged on the dilute dropped sample (hereinafter, sample-drop) at 10°, and the distance of the probing position from the passage, d, was changed systematically. As an example of the results for the dilute samples-drop, Fig. 6 shows time-dependent power spectra for 0.05 wt% PS in water. When the dilute sample began to drop, the Gaussian spectrum was observed in a frequency range higher than 2fAOM. The probing position dependence of the averaged power spectra for 0.05 wt% PS in water is shown in Fig. 7. At d > 0 mm, the power spectrum observed for the dilute sample-drop had two clear peaks. The two peak frequencies approached each other as d was increased. At d = 2.50 mm, the two peaks merged, and the power spectrum became a single Gaussian spectrum. These results indicate that the power spectrum of the light scattered from the dropped particles at a location far from the pipe was a Gaussian spectrum. On the other hand, the power spectrum of the light scattered from moving particles in the pipe deviated from a Gaussian spectrum because the velocity of these particles had a distribution caused by the restrictions imposed by the glass wall.
4. Analysis of experimental results
The theoretical study indicates that the power spectrum of light scattered from a rotating ground glass moving uniformly is a Gaussian spectrum whose width depends on the transient time for scattering across the incident light beam . Using this result, the results of the power spectrum observed for the dilute sample-drop and -flow can be interpreted by superposition of the Gaussian spectrum. It is assumed that the velocity of the PS particles in the vertical direction agrees with the velocity of the medium, and the Brownian motion of PS particles is neglected. With these assumptions, the power spectrum of the light scattered from each PS particle in the light path should have a Gaussian spectrum with a uniform shape, and the observed power spectrum will be the summation of these Gaussian spectra.
Let us perform a curve-fitting procedure for the power spectrum of scattered light for the dilute sample-drop firstly to identify the experimental power spectra for the sample-flow inside the pipe. Results for the dilute sample-drop suggest that the velocity distribution of the PS particles in the light path is very small, i.e., v(r) = v (constant). Therefore, the power spectrum observed for a dilute sample-drop at a distant from the passage is a single Gaussian spectrum given by
where fD is the peak frequency shift given by Eq. (1). Empirical parameter A relates to the amplitude of the Gaussian spectrum, and w is related to the amplitude and width of the Gaussian spectrum. The observed power spectrum is in good agreement with the fitting curve, as shown in Fig. 6. The velocity of vertically falling PS particles v is calculated from fD using Eq. (1). The parameter w, which is proportional to v, was determined to be w = 574100 v by applying the least-squares method to the measured power spectrum.
On the other hand, for the results of the dilute sample-flow within the pipe, the fluid flow can be treated as a laminar flow and the velocity distribution of fluid flow can be expressed using Poiseuille’s law. The Poiseuille equation gives the velocity distribution of the fluid flow as
Here, ΔP is the pressure difference between the two ends of the passage, η is the kinetic viscosity, a is the radius of the passage, and r is the distance from the center of the passage. This equation indicates that the flow velocity must be 0 at the wall of the passage (r = a) and maximum at its center (r = 0). Thus, the velocity of a PS particle near the wall of passage is low. Consequently, the broad Gaussian spectrum caused by the PS particles moving uniformly near the wall becomes narrow, and the Doppler shift-frequency is small. In contrast, the velocity of PS particles around the center of the passage is high. The Gaussian spectrum is broad, and the Doppler shift-frequency is large. Therefore, the observed power spectrum is the summation of Gaussian spectra whose linewidth and peak frequency change continuously.
We tried to calculate the power spectrum of light scattered from PS particles in a dilute sample-flow in the passage. We measured the volumetric flow rate Q/t for a dilute sample-flow in the passage in the vertical direction to obtain the maximum of the velocity distribution vmax. The volumetric flow rate is related to this by
Here, vavg is the average of the velocity distribution, where vavg=1/2vmax. The velocity distribution calculated for PS in water is shown in Fig. 8(a). It has already been noted in discussing the results of a dilute sample-drop that the width of the Gaussian spectrum and Doppler shift-frequency are proportional to the velocity of the scattered particles. We assumed that this relation can be applied to the results of the dilute sample-flow. Under this assumption, the observed power spectrum for PS particles in dilute sample-flow in the passage must be interpreted as the summation of Gaussian spectra with continuously changing width and peak frequency as
Here, subscript i is an element of the velocity in the passage. Figure 8(b) shows the calculated power spectrum for each PS particle moving in a dilute sample-flow in the passage. The Gaussian spectrum broad and the peak frequency, fmax (=2fAOM + fD), is shifted to a higher frequency range for each element with increasing r. The red line indicates the sum of the Gaussian spectra for all elements. The calculated power spectrum is in good agreement with the power spectrum observed for a dilute sample-flow.
5. Evaluations of flow velocities and kinetic viscosities
From the excellent correspondence between the experimental power spectra and the theoretical velocity distributions discussed so far, we conclude that the present self-mixing laser can measure velocity distributions of PS particles in a dilute sample-flow in a passage that obeys Poiseuille’s law. Here, we evaluate the maximum flow velocities and kinetic viscosities from experimental power spectra.
The peak of the power spectrum observed for dilute sample-flow in the passage is an apparent peak. The frequency of this peak, (fmax)appear, does not agree with the peak frequency of the power spectrum for a particle moving with maximum velocity at the center of the passage, (fmax) V=Vmax, as shown in Fig. 8(b). The relationship between the two peak frequencies was investigated analytically for dilute sample-flows with various velocities to obtain the flow velocity quickly and directly from the observed power spectrum. It was found that (fmax) appear is proportional to (fmax) V=Vmax as (fmax) V=Vmax = 1.0783* (fmax) appear - 156518. For the experimental results of PS particles in glycerol-water mixtures, (fmax) appear was obtained from Fig. 4, and (fmax) V=Vmax and vmax were calculated from (fmax) appear. Figure 9(a) shows the relationships between (vmax))meas against (vmax)cal for PS particles in glycerol-water mixtures. Here, (vmax) cal was calculated from (fmax) V=Vmax and (vmax)meas was obtained from the measured volumetric flow rate. There is good agreement between (vmax) cal and (vmax) meas.
Furthermore, ΔP in Eq. (3) is defined as ΔP = ρV/πa 2, where ρ is the density of a dilute sample and V is its average volume. The kinetic viscosity of the dilute sample was calculated for PS in glycerol-water mixtures from Eq. (3), as shown in Fig. 9(b). The PS concentration of a dilute sample was very small, and the kinetic viscosity was in good agreement with that of the glycerol-water mixtures.
Therefore, the flow velocity for a dilute sample-flow in the passage is obtained quickly and directly from the power spectrum observed by the present self-mixing laser, and its kinetic viscosity can be calculated by obtaining the pressure difference between the two ends of the passage.
6. Measurable concentration of small particles in flow
Finally, let us examine the concentrations that can be measured with the present self-mixing LDV system for the dilute sample-flow. In the experiment, the flow passage was set such that the sample-flow occurred through a horizontal column of a 1.2-mm diameter pipe. In this flow scheme, the flow velocity in the passage in the horizontal direction is smaller than that in the vertical direction and the time taken for the dilute sample to pass through a unit volume is long. As a result, we can measure the average power spectrum for a long period of time. Figure 10(a) shows average power spectra for 262-nm diameter PS in water as the concentration was decreased. The amplitude of the power spectrum decreased monotonically with decreasing PS concentration and the power spectral profiles were unchanged. Here, 262-nm PS particles could be detected at 0.01 parts per million (ppm) for a dilute sample-flow. Example results for dilute samples including another size particle, i.e., red blood cells of sheep with a diameter of 3 μm in water, are shown in Fig. 10(b). The red blood cells in the dilute sample-flow were detected at a concentration of 6 ppm. Note that power spectral profiles for red blood cells were different from those for 262-nm PS, in which the effect of the Brownian motion of particles near the wall was different for larger particles. The measurable low concentration limit differed with particle size since the intensity of scattered light depended on the particle size. In particular, Mie scattering occurs when the particles are larger than the wavelength, in which the scattered light intensity strongly depends on particle size, number of particle, and scatter angle . We have already reported that this scattered light intensity dependence on particle size is described by the Rayleigh-Debye (RD) equation, and the measurable lower concentration bound of the self-mixing particle sizing can be estimated by the RD equation . According to this investigation, the lower bound of the self-mixing flowmetry is estimated to be 1 ppm for particles of 40 – 500 nm diameters.
We measured power spectra of a laser’s output modulated by re-injected scattered light from small particles in a fluid flow by using a self-mixing LDV system with an LD-pumped Nd:GdVO4 thin-slice laser. The power spectrum for the small particles in the dilute sample-drop had a Gaussian spectrum. On the other hand, for a dilute sample-flow, the small particles in the light path had a velocity distribution that obeyed Poiseuille’s law, which led to an asymmetrically shaped power spectrum that deviated from the Gaussian spectrum. We successfully demonstrated quick and accurate measurements of velocity distributions of small particles in flows within a small-diameter pipe. We were able to measure the kinetic viscosities of liquid, whereas the state-of-the art LDV system can measure only averaged velocity and flow rate. In addition, highly sensitive detection of an extremely small quantity of particles in flows was achieved in the ppm concentration range.
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