We describe a method for quickly and easily measuring the size of small particles in suspensions. This method uses a self-mixing laser Doppler measurement with a laser-diode-pumped, thin-slice LiNdP4O12 laser with extremely high optical sensitivity. The average size of the particles in Brownian motion is determined by a Lorentz fitting of the measured power spectrum of the modulated self-mixing laser light resulting from the motion. The dependence of the measured power spectra on particle size and concentration was quantitatively identified from the results of a systematic investigation of small polystyrene latex particles with different diameters and concentrations. The sizes and ratios of particles with different diameters mixed in water were accurately measured. An application of this self-mixing laser method for estimation of the average size of plankton in seawater showed that it is a practical method for characterizing biological species.
© 2006 Optical Society of America
The coherent nature of laser light has been utilized in dynamic light-scattering (DLS) methods  for characterizing the motion of small particles in suspension, including gasses, liquids, solids, and biological tissues [2–5]. The DLS approach to measuring diffusion broadening of scattered light from moving small particles can be used to extract useful information about particles in Brownian motion and to determine their size. One scheme for implementing the DLS method measures the fluctuation in intensity of scattered light passing through a small pinhole, which represents the beat signals of Doppler-shifted fields scattered by different particles. The particle sizes and their distribution are estimated based on analysis of a long time-series of intensity variations given by an autocorrelation function [1–4]. Another scheme measures the beat signals between a reference light field and a scattered light field using an optical interferometer, in which a frequency shifter in one arm creates a frequency-shifted field, and an unshifted reference beam in the other arm acts as a local oscillator field. In this heterodyne detection scheme, the particle size is estimated based on the broadening of the spectrum .
We have developed a method that uses a self-mixing microchip laser with extremely high optical sensitivity to estimate particle size. The laser itself is intensity-modulated by beat signals due to the interference between a lasing field and a frequency-shifted scattered field reinjected into the laser [6,7]. The self-mixing modulation effect is enhanced by the large fluorescence-to-photon lifetime ratio in thin-slice microchip solid-state lasers. As a result, the extremely weak optical feedback condition is compensated for by the extremely large lifetime ratios in self-mixing thin-slice lasers. We previously demonstrated that self-aligned, self-mixing photon correlation spectroscopy using backscattered light is applicable to quick sizing of particles in Brownian motion, directly from the power spectrum of the modulated laser intensity , without using optical interferometry or sophisticated signal processing.
We systematically measured small polystyrene (PS) latex particles having different diameters in suspension using a self-mixing laser. We used a wide range of particle concentrations as well as samples containing particles with different diameters. The dependence of the modulated signal, i.e., power spectra, on the particle size and concentration corresponded well to those of light scattering theory under Rayleigh-Debye (RD) approximation. We also used this method to capture the Brownian as well as voluntary motion of phytoplankton in seawater in real time.
2. Experimental setup of self-mixing laser measurements of small particles in suspension
The experimental setup is shown in Fig. 1; it is similar to the one we used previously [6, 7]. The laser was a laser-diode (LD) pumped, 0.3-mm-thick LiNdP4O12 (LNP) laser with a mirror coating on each end. It was operated at wavelength λ of 1048 nm. The laser’s threshold level of pump power was 30 mW and its slope efficiency was 40%. The collimated beam from the LD (wavelength λp = 808 nm) passed through a pair of anamorphic prisms and then focused on the LNP crystal by a microscope objective lens. Almost all (96%) of the output light was frequency-shifted by two acousto-optic modulators (AOMs). One modulator up-shifted the frequency, and the other modulator down-shifted the frequency. The light was focused by a microscope objective lens (numerical aperture; NA = 0.3), and was delivered to the scattering cell.
A fused-quartz cell was filled with water containing small particles. Changing the modulation frequencies of the two AOMs produced a shift in the optical carrier frequency, fAOM, of 2 megahertz (MHz) at the end of the round-trip. The rest of the output light (4%; 80 μW) was detected by an InGaAs photoreceiver (New Focus 1811: DC-125 MHz), and the electrical signal produced by this device was fed to an rf spectrum analyzer (Tektronix 3026: DC-3 GHz). The power spectrum was obtained by averaging 100 traces from the analyzer over 30 s at a measurement frequency span of 1 kHz, for example. The demodulated waveform detected using an FM receiver and a digital oscilloscope (Tektronix TDS 540D: DC-500 MHz) corresponded to the averaged motion of particles in the small focal volume of the laser beam, in which a long-term intensity probability distribution of the demodulated output corresponded to the averaged displacement of particles . The response of the demodulated wave was sensitive to changes in the position and angle of the scattering cell with respect to the focal plane. The scattering cell was adjusted such that the displacement probability distribution of the demodulated waveform was symmetric .
The samples were water containing PS particles purchased from STADEX.
3. Results on polystyrene latex particles in Brownian motion
3.1. Lorentz broadening of power spectrum
The self-mixing effect is produced by interference between the laser field and the Doppler-frequency-shifted field fed back from the moving particles to the laser. This leads to intensity modulation of the laser at the beat frequency between the two fields. The dynamic light-scattering by particles in Brownian motion is examined in real time to detect the frequency-modulation-driven variations in the intensity of the output laser light at beat frequencies between the lasing field and the light field scattered by the particles, fAOM + fD(t), where fAOM is 2 MHz and fD(t) is the time-dependent Doppler frequency shift of the scattered field, which is reinjected into the laser.
As given by the fluctuation-dissipation theorem, the frequency spectrum of light scattered by Brownian particles has a Lorentz profile. When each particle is moving independently and the amplitude of the light scattered by each particle is constant, the power spectral densities of the modulated signals should be fitted by the following Lorentz function, similar to the case of heterodyne detection using an interferometer .
where A is a proportionality constant related to the amplitude of the light scattered by the Brownian particles, neglecting a weak quantum noise of the laser we used; ω is the angular frequency; D is the diffusion constant; d is the diameter of the Brownian particles; η is the liquid medium’s coefficient of viscosity; and kBT is the Boltzmann factor. The k is the magnitude of the scattered wave vector
where n is the refractive index of the liquid medium and θ is the scattering angle (π here). It is obvious that the observed rf power spectrum is a good representation of the frequency spectrum of laser light scattered by independent particles in Brownian motion.
Using Eqs. (1) and (2) and computer software (Microcal Origin 5.0), we performed a least-squares, best-fitting procedure for the power spectrum of PS particles to obtain the particle size. An example power spectrum obtained from the spectrum analyzer and the fitting curve is shown in Fig. 2. It is for 115-nm-diameter PS particles in water with 0.1 wt. % concentration. The particle diameter of 113.1±0.9 nm is in good agreement with the value provided by the manufacturer, d = 115 nm. These results suggest that the average size of Brownian particles can be estimated directly from the power spectrum obtained from the rf spectral analyzer in our self-mixing laser Doppler measurement.
We could successfully transform the average motion of particles into sound from the demodulated output voltage. A pulsed-code modulation (PCM) audio file (3.03 Mbytes) linked to Fig. 2 carries the sound of averaged motion of 1 wt. %, 207-nm diameter Brownian particles in suspension. The laser beam was occluded over several seconds (no sound in playback) and then again impinged on the scattering cell.
3.2 Dependence of scattered light intensity on concentration and particle size
We measured the power spectrum for PS particles in water with various concentrations and particle sizes to clarify the relationships between the intensity of the scattered light and the characteristics of the particles, such as their size and number.
Figure 3(a) shows example power spectra obtained for 115-nm-diameter PS particles in water with concentrations ranging from 0.001 to 10 wt. %. The power spectrum for each sample follows the Lorentz function given by Eq. (1). Figure 3(b) shows the concentration dependence of the proportionality constant of the Lorentz function A obtained from Lorentz fitting of the curves in Fig. 3(a). When the concentration was increased, the proportionality constant rose linearly below 0.1 wt. % and deviated from linearity above 0.1 wt. %.
Most optical particle-sizing methods, such as angular static light scattering and dynamic light scattering, are performed using dilute media. It is difficult to estimate the particle size in the high concentration range because of interactions between particles and multiple scatterings. Typically, the effect of particle interactions is significant when the solid volume fraction of particles is greater than 5% or the average interparticle spacing is smaller than the wavelength of the light. As a result, the particle diameter estimated from experiments can deviate from the true diameter [9,10]. Furthermore, the light scattered by the particles is affected by the multiple scattering above some concentration level, and this upper concentration limit varies with particle size and instrument design . The theoretical assessment of multiple scattering is a rather complex task requiring additional information, such as the scattering geometry, the intensity distribution of the incident laser beam, and the sample dimensions .
In our experimental results, it is apparent that the linear increase in the proportionality constant (i.e., power spectral density) with an increasing concentration below 0.1 wt. % was caused by the increase in the number of PS particles because the total scattered-light intensity was the linear superposition of the intensity of the light scattered by each particle. The deviation from the linear dependence of the proportionality constant above 0.1 wt. % was supposedly caused by the multiple scatterings and particle interactions. In spite of the deviations from the theoretical power spectral density at high concentrations, the particle sizing was fairly accurate over a wide range of concentrations, i.e., 0.001–1 wt. %.
Next, let us discuss the particle-size dependence of the intensity of the scattered light. Figure 4(a) shows example power spectra for PS particles with diameters of 115, 262, and 474 nm in water with 0.05 wt. % concentration. The power spectrum obtained from the experiment for each sample was in good agreement with the Lorentz function. Figure 4(b) shows the PS-size-dependent proportionality constant for PS particles with diameters of 20, 115, 204, 262, and 474 nm in water with concentrations of 0.05 and 0.1 wt. %. The proportionality constant was at its maximum at around 262 nm.
In general, the scattering of light by particles suspended in a medium can be categorized into two types, i.e., Rayleigh scattering and Mie scattering. Rayleigh scattering occurs when the particles are much smaller than the light wavelength, where the intensity of the scattered light is angularly isotropic. Mie scattering occurs when the particles are larger than the wavelength, where a scattered light intensity indicates a strong angular dependence. In the marginal particle-size region, the intensity of light scattered by a spherical particle is expressed well by the following relationships over a wide particle-size region under the modified RD approximation 
where V is the volume of a particle, N is number of particles in a unit volume, and P(θ) is the form factor. In RD approximation, the intensity of the scattered light depends not only on the scattering angle and wavelength but also on the particle size. In Fig 4(b), the solid and dotted lines indicate the particle-size dependence of the intensity of scattered light calculated from the RD approximation and Rayleigh theory, respectively. In the limit of Rayleigh scattering, I ∝ V2N ∝ d3 at a fixed concentration. The experimental results for particle-size dependence of the proportionality constant, i.e., the power spectral density, are in good agreement with the theoretical predictions.
We thus conclude that the power spectral densities of modulated signals from a self-mixing laser with extreme optical sensitivity well represent the particle-size and concentration dependencies of the intensity of the scattered light, which obey the well-established light scattering theory.
3.3 Measured power spectrum of mixed polystyrene particles in suspension
We tested 0.05 wt. % suspensions containing polystyrene latex particles with two different diameters—115nm (A) and 474nm (B). We mixed A and B with various fractions. Figure 5(a) shows the power spectrum when A:B was 1:1 together with those of A and B alone.
For a two-component sample, the power spectra of the modulated laser output intensity can be expressed by the superposition of two Lorentz functions. We calculate the particle size for the two-component mixture by summing two Lorentz functions 
where Γi,mix and Ai,mix are fitting parameters (i = 1,2). The diameters of the distributed PS particles, estimated using the least-squares, best-fitting curve of Eq. (4), were in good agreement with the actual diameters. Figure 5(b) shows the proportionality constants, A115,mix (i.e., A1,mix) and A474,mix (i.e., A2,mix), as a function of the ratio of the number of PS particles estimated to be in the concentration: (N115/N474)real. The value of A115,mix increased and that of A474,mix decreased with increasing (N115/N474)real.
We calculated the number of particles of each particle diameter for the two-component mixture from the ratio of the Ai values for single and two-component systems
where the subscripts indicate the single-component (single) and two-component (mix) systems. Ni,single and Ai,single were calculated from single-component samples, as shown in Figure 5(a), by curve-fitting using the single Lorentz function, Eq. (1). Figure 6 plots N115/N474 calculated using Eq. (5), (N115/N474)exp, against (N115/N474)real. The value of (N115/N474)exp agrees well with (N115/N474)real, indicating that the diameters of particles in mixed suspensions and the ratio of their numbers can be obtained from the power spectra of the modulated laser output intensity. It should be possible to attain the diameters and number ratio for general multiple-component systems by the curve-fitting of the power spectra of the self-mixing laser output.
We carried out a histogram analysis for mixed particles in suspension directly from measured power spectra by the singular value decomposition (SVD) method , not from autocorrelation functions calculated by the Fourier transformation of power spectra as demonstrated previously . We found that a reliable particle size distribution is attained when particle sizes are well separated. An example result for the sample containing 115 nm and 474 nm particles is shown in Fig. 7. As for mixed samples containing a larger distribution of sizes, whose differences are less than a factor of 2 (e.g., 100 and 200 nm), however, a histogram peaked at the average size and we could not distinguish peaks in the histogram, similar to state-of-the-art DLS systems. The main cause was found to be attributed to the systematic thermal noise of a spectrum analyzer superimposed on the power spectra from which we perform particle sizing. This problem could be overcome by the long-term data accumulation and averaging of power spectra or the use of a real-time spectrum analyzer with higher frequency resolution and lower thermal noise.
4. Application of self-mixing laser to biological species measurement
We used this self-mixing DLS method with high optical sensitivity and quick particle sizing to characterize the motion of two types of phytoplankton, Nannochloropsis oculata and Tetraselmis tetrathele. Nannochloropsis is spherical with a diameter of about 2.7 μm (measured using a Coulter counter); it cannot move voluntarily in seawater. Tetraselmis is also spherical and has a diameter of about 7.0 μm; it can move quickly using flagellum, i.e., voluntary motion, in seawater. A scattering cell was filled with seawater containing one of these two types of plankton.
Figures 8(a) and 8(b) show time-dependent power spectra for Nannochloropsis and Tetraselmis, respectively. The power spectrum observed for non self-mobile Nannochloropsis maintains a Lorentz shape in time, implying Brownian motion. On the contrary, the power spectrum for self-mobile Tetraselmis changes in time: Broad frequency components resulting from self-motion of part of plankton are superimposed occasionally on a Lorentz spectrum, a in Fig. 8(b), corresponding to Brownian motion, resulting in shoulders on both sides of the carrier frequency as indicated by arrows in b–d in Fig. 8(b).
To give more insight into observed power spectra, let us characterize measured average power spectra in a statistical way. From curve-fitting of the averaged power spectrum for non-self-mobile Nannochloropsis shown in Fig. 9(a), the average diameter was estimated to be 2.2 μm, which is close to the diameter measured by a Coulter counter. The averaged power spectrum for self-mobile Tetraselmis is shown in Fig. 9(b), where the spectrum is decomposed into two components by a curve-fitting, assuming the superposition of a Lorentz (i.e., Brownian motion) and a Gaussian (self-motion) spectrum. The measured power spectrum is well-reproduced by such a curve-fitting as shown by the red curve, in which corresponding Lorentz and Gaussian components are also depicted by blue and green curves, respectively. From the using the relation fD=2 vz/λ, we can estimate a distribution of the velocity along the laser axis vz of Brownian or self-motion. An estimated speed is given in the upper horizontal axis.
We transformed the Brownian motion of Nannochloropsis as well as the motion of Tetraselmis into sound in real time by using real-time demodulation of the self-mixing laser output signals. The observed time-dependent power spectra and sounds for the Nannochloropis and the Tetraselmis are demonstrated in the movies linked to Fig. 8.
The peculiar dynamic behavior in the voluntarily motion of Tetraselmis plankton can be observed by the real-time measurement of power spectra using a self-mixing laser with extremely high optical sensitivity. We are now investigating the mechanism of the observed voluntarily dynamics and conducting systematic measurements for various types of plankton with different sizes and shapes.
We have quantitatively characterized particles with diameters of 20–500 nm and Brownian motion in water by using the power spectra of thin-slice, self-mixing, laser output intensities detected with a conventional photodiode without using any sophisticated optics or signal-processing electronics. These particles were quickly sized for a concentration range of 0.001– 1 wt. %. Systematic examination of single- and two-component polystyrene latex suspensions showed that the power spectra of self-mixing laser intensities well represent the scattered-light intensities from Brownian particles, as derived from RD light-scattering theory. In other words, the sizes of different types of particles in suspension and their number ratio can be accurately determined from curve-fitting of the observed power spectra. We captured the motion of phytoplankton in seawater as an example of a promising application of a self-mixing laser for the characterization of biological species.
We thank Dr. K. Tsutsui of Otsuka Electronics Co. for providing the diluted polystyrene latex samples purchased from STADEX.
References and links
1. B. J. Berne and R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics (Dover, NY, 2000).
2. H. Z. Cummins, N. Knable, and Y. Yeh, “Observation of diffusion broadening of Rayleight scattered light,” Phys. Rev. Lett. 12, 150–153 (1964). [CrossRef]
4. D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion with highly scattering media,” Opt. Lett. 23, 319–321 (1998). [CrossRef]
5. M. Harris, G. N. Pearson, C. A. Hill, and J. M. Vaughan, “The fractal character of Gaussian-Lorentzian light,” Opt. Commun. 116, 15–19 (1995). [CrossRef]
6. K. Otsuka, K. Abe, J.-Y. Ko, and T.-S. Lim, “Real-time nanometer-vibration measurement with a self-mixing microchip solid-state laser,” Opt. Lett. 27, 1339–1341 (2002). [CrossRef]
7. K. Abe, K. Otsuka, and J.-Y. Ko, “Self-mixing laser Doppler vibrometry with high optical sensitivity: application to real-time sound reproduction,” New J. Phys. 5, 8.1–8.9 (2003). [CrossRef]
8. K. Otsuka, K. Abe, N. Sano, S. Sudo, and J.-Y. Ko, “Two-channel self-mixing laser Doppler measurement with carrier-frequency-division multiplexing,” Appl. Opt. 44, 1709–1714 (2005). [CrossRef] [PubMed]
9. Z. Sun, C. D. Tomlin, and E. M. Sevick-Muraca, “Approach for particle sizing in dense polydisperse colloidal suspension using multiple scattered light,” Langmuir 17, 6142–6147 (2001). [CrossRef]
10. L. B. Aberle and W. Staude, “Three-dimensional cross correlation technique: influence of multiply scattered light in the Rayleigh-Gans regime,” Phys. Chem. Chem. Phys. 1, 3917–3921 (1999). [CrossRef]
11. R. Xu, Particle Characterization: Light Scattering Methods (Kluwer, London, 2000).
12. L. B. Aberle, P. Hülstede, S. Wiegand, W. Schöer, and W. Staude, “Effective suppression of multiply scattered light in static and dynamic light scattering,” Appl. Opt. 37, 6511–6524 (1998). [CrossRef]
13. K. A. Stacey, Light-scattering in Physical Chemistry (Buttler Worth Scientific Publications, London, 1956).
14. M. Bertero, P. Boccacci, and E.R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. Royal Soc. London A 383, 15–29 (1982). [CrossRef]