The experimental setup is shown in Figure 4, where a 0.3-mm-thick LNP laser with coated end mirrors (M₁: 95% transmittance at 808 nm, 99.8% reflectance at the lasing wavelength of 1,048 nm; M₂: 98% reflectance at 1,048 nm) was used.

The LNP lasing wavelength λ was 1,048 nm. The threshold pump power was 30 mW and the slope efficiency was 40%. A part of the output beam (96%) was divided into two access beams and each beam was focused on a speaker through two sets of PbMoO₄ acousto-optic modulators (AOMs), i.e., optical frequency shifters. A microscope objective lens (numerical aperture: NA = 0.25) was used to focus the beams. Here, the first AOM1 introduced the common upward frequency shift of 80 MHz for two channels. The second AOM2 and third AOM3 induced −78.65 and −78.25 MHz downward frequency shift, respectively. In short, the round-trip carrier-frequency shift, 2f_AOM, for two channels was f_c,1 = 2.7 MHz for channel 1 and f_c,2 = 3.5 MHz for channel 2, respectively. Another part (4%) of the output beam was delivered to an InGaAs photo-diode receiver (PD). The electrical signal from the photo-diode was used to measure waveforms and power spectra of modulated and demodulated outputs with a digital oscilloscope (DO) and a rf spectrum analyzer (SA).

The modulated signal was composed of two FM-type intensity modulated waves whose carrier frequencies are f_c,1 = 2.7 MHz and f_c,2 = 3.5 MHz, respectively. As a result, the observed power spectrum consists of two power spectra around 2.7 and 3.5 MHz whose frequency components show FM sidebands. An example power spectrum of the modulated signal and a magnified view around f_c,2 = 3.5 MHz are shown in Figures 5(a) and 5(b) respectively, where the common sinusoidal voltage was applied to the two speakers at modulation frequency f_m = 914 Hz. The intrinsic relaxation oscillation intensity noise is indicated by f_RO in the figure. The effect of relaxation oscillations on vibration measurements was not observed.

Figure 5(c) shows demodulated output voltages V_o(t) of two channels corresponding to Figures 5(a), each of which corresponds to a velocity variation v(t) of each speaker’s rough surface. The vibration waveform, D_p(t), can be deduced by integrating v(t) over time. In the present case, the vibration waveform D_p(t) is similar in shape to v(t), i.e., V_o(t). Due to the poor quality of the speakers, slightly irregular amplitude variations are seen. It was confirmed that the cross-talk between two speakers was absent. In short, the demodulated output waveform of one speaker was kept unchanged when we blocked an access beam to another speaker. This is because that the nonlinear wave mixing between the strong lasing field at frequency, f, and extremely weak re-injected fields in the laser, f + f_c,1 and f + f_c,2, e.g., combination-tone polarizations are not created near re-injected light frequencies under such weak feedback conditions, especially when two carrier frequencies are incommensurate like the present frequency arrangement.

A super-heterodyne method with a central frequency of 10.7 MHz was employed to demodulate the FM wave; frequencies of external oscillators were tuned to provide the 10.7 MHz central frequency to both channels and the amplifier in each channel had a maximum gain of 20 dB and a 3-dB bandwidth of 111 kHz. In Figure 5(c), the displacement D_p(t) of each speaker’s surface is also shown on the right vertical axis, where the displacement is related to the output voltage as D_p/V_o = 20 [μm/V]. To obtain such a voltage-displacement relationship, we used Hilbert transformation of the modulated output waveform and phase-sensitive detection by using LabView software on a personal computer (PC), assuming the relation D_p (t) = λ ΔΦ(t)/2π, where ΔΦ(t) is the analytic phase difference between the carrier (reference) wave and the modulated wave, as calculated from Gabor’s analytic signal [28,29]. The analytic phase Φ_A is related to analytic signal V_A and its time average <V_A> by V_A − <V_A> = R_A(t)exp[iΦ_A (t)]. Here, V_A(t) = I(t) + iI_H(t), where I(t) is the time series of the scalar intensity and I_H is its Hilbert transform. The maximum vibration amplitude V_max was also evaluated accurately from the FM modulation index β = 32 determined from ratios of FM sideband intensities to the carrier intensity of modulated signals using the relation V_max = λβ/2π yielding D_p = 5.3 μm, which parallels the vibration amplitude shown in Figure 5(c) [right vertical axis] calibrated from the phase-sensitive detection on the PC. From the systematic measurement for different voltages to the speaker, the vibration amplitude was found to vary in the relation of V_max = 60 [nm/V] at f_m = 8 kHz as shown in Figure 6. Simultaneous real-time measurements were performed successfully even if the speakers were driven at different frequencies.

To confirm the idea of carrier-frequency-multiplexed self-mixing laser-Doppler vibrometry using one laser and reproduce the experimental results, the numerical simulation of the model equation of a laser subjected to frequency-modulated multiple feedback lights Equations (9–12), assuming: (13) Ω D , k = Δ Ω k + β sin Ω m , k t ,where Ω_m,k = ω_m,k/κ is the normalized angular frequency of the applied voltage to the k-th target. The numerical results are shown in Figure 7 for the case of two feedback fields as demonstrated in the experiment. The photon lifetime τ_p of the present laser was estimated from the pump-dependent relaxation oscillation frequency, f_RO, using the relation f_RO = (1/2π)[(w − 1)/ττ_p], where w = P/P_th is the relative pump power and τ = 120 μs is the LNP’s fluorescence lifetime. The estimated value was τ_p = 300 ps, yielding a lifetime ratio of K = 2 × 10⁵. Other adopted parameter values, which are relevant to the present experiment, are w = 6.45, carrier frequencies Δf₁ = f_c,1 = 2.7 MHz, Δf₂ = f_c,2 = 3.5 MHz, and modulation frequencies f_m,1 = f_m,2 = 914 Hz, β = 32, delay time = 6 ns, amplitude feedback coefficient R_f = 10⁻⁵ (i.e., intensity feedback ratio of −100 dB) and ε = 10^–11.

A sample calculated power spectrum of the modulated signals and a magnified view around f_c,2 = 3.5 MHz are shown in Figures 7(a) and 7(b), respectively, where the common sinusoidal voltage was applied to the two speakers. Vibration waveforms for two channels, which were calculated from the modulated waveform by the phase-sensitive detection using Gabor’s analytic phases on PC, are shown Figure 7(c). The numerical results well reproduce the simultaneous independent measurement of vibrations of two targets shown in Figure 5. The range of acceptable levels of intensity feedback ratio, 10 log R_f², in the present system was estimated to be −100 dB from the correspondence between results of experiment and simulation.

A variety of three-dimensional (3-D) metrology systems have been developed for measuring moving objects and turbulent fluid flow [48–55]. Typical examples include 3-D laser-Doppler vibrometry systems using three oblique incident beams impinging on the target and particle image velocimetry systems. In these state-of-the-art 3-D optical metrology systems reported so far, beat signals or interference fringes between the lasing output field and the scattered optical fields from moving targets are detected using sophisticated optics and highly sensitive electronics.

On the other hand, the simplest, self-aligned 3-D nanometer-scale metrology systems with extreme optical sensitivity, without using any sophisticated optical interferometers or highly sensitive electronics, can be realized with a 3-channel self-mixing thin-slice solid-state laser metrology system.

Experimental results of simultaneous measurements of three different vibrating targets are shown. The experimental setup is shown in Figure 8, 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 1,064 nm; M₂: 98% reflectance at 1,064 nm) and three sets of TeO₂ acousto-optic modulators (AOMs) are used.

The photon lifetime τ_p of the present laser was estimated to be 235 ps from the pump-dependent relaxation oscillation frequency, f_RO, yielding a lifetime ratio of K = 4.05 × 10⁵, assuming τ = 90 μs. Here, AOM1 introduced a common upward frequency shift of 79.85 MHz for all three channels. AOM2, 3 and 4 provided 78.85-, 78.1-, and 77.6-MHz downward frequency shifts, respectively. In short, the round-trip carrier-frequency shifts 2f_AOM for three channels were f_c,1 = 2 MHz for channel-1, f_c,2 = 3.5 MHz for channel-2, and f_c,3 = 4.5 MHz for channel-3. The electrical signal from the photodiode was delivered to a software defined radio (RFSPACE Inc. SDR-14: DC-30 MHz) or three-channel frequency-modulated wave demodulation circuit (analogue FM-demodulator) having three different outputs matched to carrier frequencies f_c,1, f_c,2, and f_c,3, depending on the purpose.

Here, three vibrating targets, each of which consisted of a mirror attached to a piezoelectric-element, were used as shown in Figure 8. Figure 9 shows vibrating waveforms calculated by the integral of the output signals V_o(t) from the FM-demodulator which is proportional to the instantaneous velocity of the piezoelectric-element over time for each of the three channels. Here, the displacement, D_p(t) = λΔΦ/2π was estimated from the Hilbert transformation of a modulated signal and the phase-sensitive detection for each channel (i.e., carrier frequency) by using LabView software on a personal computer, as mentioned in 4.1.1. The measured vibration amplitude was confirmed to coincide with those measured with a capacitive displacement sensor [39]. The cross talk among three channels was completely eliminated by tuning the filter bandwidth of the FM-demodulator circuit around the carrier frequency.

The vibration amplitudes measured at different driving frequencies are summarized in Figure 10. The software-defined radio enabled us to perform phase-sensitive detection of modulated signals to obtain vibration amplitudes directly in the form of D_p(t) = λΔΦ/2π. The vibration amplitude was found to be independent of the driving frequencies in the range from 1 Hz to 5 kHz for vibration amplitudes larger than 30 nm. The measurable vibration amplitude in the present three-channel real-time vibrometry system was about 2 nm at 5 kHz, as indicated by open circles in Figure 10.

The present set-up can be applied to dynamic bending measurements of metal plates by impinging access beams on at least three positions of the target. If we focus three access beams on the same position of the target from three different directions, vibration test and 3-dimensional vibration analysis of small structures, housing and components can be performed.

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 micro-particles and nanometer-sized particles in washing water is required to evaluate semiconductor wafer contamination. The detection of small particles in fluid flow has been performed extensively using a magnetic flowmeter [56,57], ultrasound Doppler velocimeter [58,59], and laser Doppler velocimeter (LDV) [60–64]. The advantage 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 of 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 f_D of scattered light by [65]: (14) f D = 2 v z λ = 2 v sin   θ λ .

Here, λ is the wavelength, v is the velocity of the moving particle, v_z 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 [66–68], speckle pattern interferometry [69], and measurements of the “average” velocity and flow of small particles and red-blood cells [16,17,21–23,70,71].

Before presenting 3-D measurement of liquid flow, let us show the liquid flow in a vertical glass column measured with a single access beam using the same 3 at.% Nd-doped Nd:GdVO₄ self-mixing laser metrology system as that used in the experiment described in 4.1.3 under the single-channel configuration. The distinct feature of self-mixing thin-slice solid-state laser scheme, which is not expected in self-mixing semiconductor lasers, includes the inherent high optical sensitivity due to 2–3 orders of magnitude larger K values, much narrower line-width as well as smaller spontaneous emission noise than LDs.

The flow passage is illustrated in Figure 11. A Doppler-shift frequency can be increased with increasing the angle. Therefore, the large angle is suitable for measuring velocity distributions of liquids with high viscosities. For measuring liquid flow with lower viscosities, on the other hand, we need a narrower frequency span (i.e., high frequency resolution) for power spectrum analysis to extract the information in the vicinity of 2f_AOM. Therefore, the angle was set 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 of 100 mm long and 1.2 mm in diameter. Water-glycerol mixtures with various glycerol concentrations were used as liquid, and polystyrene latex standards (PS) with a diameter of 262 nm were added as the tracer. 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.

Figure 12 shows time-dependent power spectra of the modulated wave for PS particles in water. Before the flow, the power spectrum showed a Lorentz shape representing the Brownian motion of PS particles as will be shown in Section 5. 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 2f_AOM = 2 MHz for θ > 0 (see the corresponding Supplementary Material).

The averaged power spectra for PS particles in various glycerol-water mixtures are shown in Figure 13.

The power spectrum became narrow and the peak frequency approached 2f_AOM as the glycerol concentration increased. This approach toward 2f_AOM 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 Equation (14). The narrowing of the power spectrum with decreasing velocity of dilute sample-flow was predicted in Reference [72].

To 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 Figure 12(b–e). 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: (15) v ( r ) = Δ P 4 π η ( a 2 − r 2 ) .where 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.

Let us calculate the power spectrum of light scattered from PS particles in a dilute sample-flow in the passage. The volumetric flow rate Q/t for a dilute sample-flow in the passage in the vertical direction is related to the maximum of the velocity distribution v_max by: (16) Q t = π a 2 v avg = 1 2 π a 2 v max .

Here, v_avg is the average of the velocity distribution, where v_avg = (1/2)v_max. The velocity distribution calculated for PS in water is shown in Figure 14(a). The theoretical study indicates that the power spectrum of light scattered from a rotating ground glass or a particle moving uniformly is a Gaussian spectrum whose width depends on the transient time of a single irregularity or a particle across the incident light beam [72]. Therefore, the observed power spectrum for PS particles in dilute sample-flow can be interpreted as the summation of Gaussian spectra with continuously changing width and peak frequency as: (17) I ( ω ) = ∑ i A i w i exp { − [ ω − 2 π ( 2 f AOM + f D , i ) ] 2 w i 2 } .where subscript i is an element of the velocity in the passage. Empirical parameters A and w are the amplitude and width of the Gaussian spectrum relating to the velocity of the particle, respectively. f_D is the Doppler shift frequency of scattered light given by Equation (14). Figure 14(b) shows the calculated power spectrum for each PS particle moving in a dilute sample-flow together with the experimental power spectrum.

Here, the fitting procedure is as follows:

Thirty Gaussian functions are set within the observed power spectrum, whose center frequencies f_Di are equally spaced, to describe the velocity distributions.

The thin grey curve 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.

From the excellent correspondence between the experimental power spectra and the theoretical velocity distributions discussed so far, it is concluded 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.

Now, let us evaluate the maximum flow velocities and kinetic viscosities from experimental power spectra. The peak frequency of the observed power spectrum, (f_max)_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, (f_max)_V=Vmax, as shown in Figure 13(b). The relationship between these two 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 (f_max)_appear is proportional to (f_max)_V = _Vmax as (f_max)_V = _Vmax = 1.0783 × (f_max)_appear − 156518.

Figure 16(a) shows the relationships between (v_max)_meas against (v_max)_cal for PS particles in glycerol-water mixtures. Here, (v_max)_cal was calculated from (f_max)_V = _Vmax and (v_max)_meas was obtained from the measured volumetric flow rate.

There is good agreement between (v_max)_cal and (v_max)_meas. The kinetic viscosity of the dilute sample was calculated for PS in glycerol-water mixtures from Equation (15) and the result is shown in Figure 16(b). The deviation from the calculated values in the low velocity (i.e., high viscosity) region is considered to arise from the substantial contribution of Brownian motions of particles. Another systematic determination of the maximum flow velocity along the central axis will be discussed in Section 4.2.5. As shown in this section, the flow velocity as well as the kinetic viscosity for a dilute sample-flow in the passage are measured quickly and directly from the power spectrum of a modulated self-mixing laser intensity.

Applying the liquid flow measurement, we can detect an extremely small amount of small particles in a fluid flow. 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 17(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, while 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. An example result for dilute samples including another size particle, i.e., red blood cells of sheep with a diameter of 3 μm in water, is shown in Figure 17(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 depended on a particle size since the intensity of scattered light depended on the particle size. According to the investigation of Brownian particles, which will be discussed in chapter 5, the lower bound of the concentration detected by the present self-mixing flowmetry is estimated < 1 ppm for particles of 40–500 nm diameters.

The experimental setup for 3-D measurement and the spatial configuration of the three access beams on the liquid flow are shown in Figure 18. A glass pipe with a 2 mm inner diameter was set slightly tilted with respect to the optical stage (θ = 10.00° and ψ = 7.25° in Figure 18) and the velocity of water flow containing PS particles (110-nm diameter, 1 wt% concentration) was controlled precisely by the syringe pump. The water flow rate was controlled to give laminar flow whose radial velocity distribution obeyed Poiseuille’s law with maximum velocity at the center of the glass pipe. The maximum flow velocity along the central axis was estimated to be 76.32 mm/s assuming Equation (16), where a volumetric flow rate was Q/t = 479.53 mm³/s. In the present experiment using the same laser as Figure 8, the direction of the f_c,3 beam was set along the Z-axis, while the f_c,1 and f_c,2 beams were in the YZ- and XZ-planes, respectively. The crossed beam-focus of the three access beams was adjusted to the central axis of the glass pipe.

Observed power spectra around the three carrier frequencies of the laser output, which were modulated by feedback scattered fields from moving PS particles in water flow, are shown in Figure 19. Peculiar asymmetric power spectra, reflecting a radial velocity distribution obeying Poiseuille’s law, which can be fitted by Equation (17) are seen. Indeed, the observed power spectra were found to be fitted remarkably well by the summation of Gaussian spectra given by Equation (17), as depicted by the fitting curves in Figure 19(a). The parameter w_i, which is proportional to v_i, was assumed to be w_i = 574,100v_i as given in 4.2.2.

The dependence of a fitting parameter of A_i on the Doppler-shift frequency is shown in Figure 19(b). It is obvious from this Figure that A_i-values peak at different frequencies for the three access beams. This may imply that the scattered light intensity was maximum at these peaks, i.e., Doppler-shift frequencies, for the three access beams impinging on the liquid flow from different directions because A_i is proportional to the intensity of light scattered from particles back into the laser. These peak frequencies were expected to coincide with the Doppler-shift frequencies derived from the maximum flow velocity along the central axis of the glass pipe, assuming the impinging angles of the three access beams, since the focal position of the three access beams was adjusted to the center of the glass pipe where the effective plane-wave scattering occurred as the LDV experiment using a rotating cylinder experiment tells us [41].

Therefore, the velocity components of water flow in the center of the pipe along three orthogonal axes, X, Y, and Z, can be calculated from the three peak frequencies in Figure 19(b), and the corresponding velocity vector, i.e., flow direction, can be determined by the trigonometric method. Results are summarized in Table I.

The experimental velocity vector, i.e., flow direction, coincides with that expected for the experimental setup within 6% accuracy.

In order to capture such overall dynamics of many Brownian particles, the dynamic light scattering method with a self-mixing thin-slice laser was employed. In this scheme, the laser itself is intensity-modulated by beat signals between the lasing field and a frequency-shifted scattered field through the interference of the two fields. The distinct advantage of our self-mixing scheme over any other method [73–75] is that we can demodulate the laser intensity fluctuations through a simple frequency-modulated (FM) wave demodulation circuit [29,37,39] and identify in real time the overall dynamics of many Brownian particles with high time resolutions.

The experimental setup is the same as Figure 20, where the laser-diode-pumped 0.3-mm-thick LiNdP₄O₁₂ (LNP) laser was employed in this experiment as shown in Figure 24(a). Example power spectra of modulated intensities obtained for 107 nm, 207 nm, and 458 nm particles (concentrations: 1 wt %) are shown in the right of Figure 24(b), where the average spectrum of 100 power spectra is shown. Within the effective scattering volume (i.e., imaging field), V_s = πw₀²f_D (w₀: spot size, f_D = λ/2(NA)² = 8.5 μm: focal depth), a sufficient number of particles existed, i.e., N ∼2.1 × 10³, 2.3 × 10⁴, 1.7 × 10⁵ for 458, 207 and 107 nm particles, respectively, assuming a measured spot size of w₀ = 20 μm. Here, the relatively thin effective scattering region along the laser axis with respect to the beam diameter was formed as depicted on the left of Figure 24(b). Measured power spectra are well fitted by Lorentz function given by Equation (18). Next, let us investigate the net motion of an ensemble of Brownian particles, i.e., motion of an effective medium at the center of mass of particles (called a “virtual particle” hereafter) within the imaging field by demodulating the laser output intensity. The FM-wave demodulation circuit has a maximum amplifier gain of 20 dB and a 3-dB bandwidth of 111 kHz. The demodulated output voltage from the FM demodulator is proportional to the velocity of a single virtual particle along the laser axis, v_x(t), in the form of V_o(t) = C (2v_x(t)/λ).

Here, the slope, C, is determined by the voltage versus frequency relationship around the carrier frequency of the FM demodulator we used, which depends on the amplifier gain of the circuit. Temporal variations in calibrated velocities for different particles and the corresponding displacements, x ( t ) = ∫ 0 t v x ( t ) dt, are shown in Figures 25(a) and 25(b), respectively, where the observation time interval was 10 μs, i.e., a sampling rate of 100 kHz which is lower than 3-dB bandwidth of 111 kHz of the FM-demodulator.

Since the virtual particle is moving at the velocity given by the vector sum instantaneous velocities of many individual particles, its velocity v_x(t) is considered to be much smaller than those of individual particles. Indeed, the root mean-square velocity of a virtual particle, which is calculated from the long-term velocity variation shown in Figure 25(a), was v rms * = < v x ( t ) 2 > = 110, 46, 21 μm/s for 107, 207, and 458 nm particles, respectively, i.e., three-orders of magnitude smaller than those of real particles: v rms = k B T / m = 7.97, 2.98, and 0.91 cm/s.

The power spectra of the velocity fluctuations shown in Figure 25(a) indicate Lorentz-type of functions just like a single Brownian particle, which exhibits random motion known as the Ornstein-Uhlenbeck (OU) process resulting from the following standard Langevin equation [82]: (21) m * dv x ( t ) dt = − γ * v x ( t ) + F thermal ( t )where F thermal ( t ) = 2 k B T γ * ξ ( t ) is a random thermal force with < ξ (t) >=0 and < ξ(t)ξ(t′) >= δ(t – t′). Here, m* and γ* are the mass of a virtual particle and friction coefficient of a virtual liquid, respectively.

Figure 26(a) shows power spectra of the temporal evolutions of the displacement, x(t), which is given by S(ω) ∝ 2D* / ω² (1+ω²τ_m *²) from Equation (21), where τ_m* = m*/γ* = ρ* a*²/18η* (ρ*: density) is called the persistence time, and characterizes the time for which a given velocity is “remembered” by the system and D* is the diffusion constant of the virtual particle. Note that the roll-off frequencies from the f⁻²-type random walk toward asymptotic f⁻⁴ behavior, which are indicated by arrows, appear at frequencies that are 5–6 orders of magnitude lower than those for real particles. Here, fitting curves are indicated by solid lines. Because these roll-off frequencies are much lower than the 3-dB bandwidth of 111 kHz, the observed behavior is considered to be generic in the net motion, being free from measurement limitations of the instrument. The roll-off frequency was measured to decrease with decreasing the particle concentrations and the net motion approached pure random walk.

Figure 26(b) shows the mean-square displacement, <x(τ)²> = <[x(t + τ) − x(t)]²>, where τ is a variable time delay. The experimental values (open circles) were fitted surprisingly well by the following equation derived from Equation (21) [81,82]: (22) < x ( τ ) 2 > = 2 D * [ τ − τ m * { 1 − exp ( − τ / τ m * ) } ] ,as indicated in Figure 26(b), where fitting curves for virtual particles and theoretical results for real particles are also shown by solid and dashed lines, respectively.

Relevant diffusion coefficients D*, which respectively coincide with the theoretical values for real particles D, were found to be attained for virtual particles. This is reasonable because the power spectra of modulated signals of 12.8-ms-long frames for accurate sizing of real particles shown in Figure 24(b) should hold for virtual particles for observation time scale shorter than the frame length in Figure 26(b). On the other hand, the persistence times of virtual particles, τ_m* = 0.43, 0.78, and 1.67 ms, obtained from curve fittings are 5–6 orders of magnitude longer than those of real particles with 107, 207 and 458 nm diameters: τ_m = 0.71, 2.67, and 13.1 ns, respectively. These τ_m*-values also coincided well with those obtained from the curve fitting shown in Figure 26(a), and the roll-off toward the pure random walk appeared at τ_r = 1/f_r, which are indicated by arrows in Figure 26(b). It is obvious that the high-velocity components were substantially suppressed, as shown in Figure 26(a), and a drastic reduction in the mean-square displacement compared with purely diffusive motion arose on the observation time scale shorter than τ_r in Figure 26(b).

The mean-square displacement was calculated by using long-term time series (10 seconds) of demodulated signals measured at a time resolution of 200 μs, in which the gradient was decreased gradually as τ increases similar to Figure 26(b) and the pure Brownian random walk <x(τ)²> = 2Dτ was confirmed to appear in the regime of τ > 10 ms.

Such extremely long persistence times cannot be explained in terms of collective diffusion due to hydrodynamic interactions among particles [81]. Also, the non-diffusive motion of a Brownian particle featuring elongate persistence times due to the liquid's inertia, which is expressed by Equation (3) of Reference [83], indicated deviations from the <x(τ)²> = 2Dτ relation around τ = 0.1, 0.2 and 0.5 μs for 107, 207 and 458 nm particles, respectively. Consequently, the observed behavior is considered to be inherent in the overall dynamic of Brownian particles.

The similar overall dynamics, which obeys Equation (21), have always appeared for Brownian particles with different diameters (i.e., 115, 262 and 474 nm). The theoretical treatment is anticipated for explaining the OU process with elongate persistence times and the link between single-particle and overall dynamic, including the effects of number of particles moving within the imaging field and observation time intervals on the net motion.

Let us finally examine physical parameters of virtual particles. Provided that the friction constant should be the same for real and virtual particles to insure the same diffusion coefficient for both particles, virtual particle's diameter, a*, and virtual liquid's viscosity, η*, can be given by the following relations: a* = (ρτ_m * / ρ*τ_m)^1/3 a = (τ_m * / ρ_s *τ_m)^1/3 a, η* = (ρτ_m * / ρ*τ_m)^–1/3 η = (τ_m * / ρ_s *τ_m)^–1/3 η, where ρ* and ρ_s* are density and specific gravity with respect to water of a virtual particle. The experimentally determined persistence times of virtual particles and the resultant normalized diameters, a_v* ≡ a*ρ_s* ^1/3, and virtual suspension's viscosities, η_v* ≡ η*ρ_s*^−1/3 for 107-, 207-, 458-nm particles are shown in Figure 27. Physical parameters of virtual turbid media are found resemble those expected in low-density, large-diameter particles (e.g., aerosol of several tens of microns) moving in gas.

From the application viewpoint, the present self-mixing laser dynamic light scattering method provides a promising for measuring diffusion coefficients with high accuracy from demodulated signals measured at appropriate time intervals without using a high resolution wide-band spectrum analyzer. Promising applications could include in-situ measurement of cohesion and segregation process of colloidal particles over times.

The 3-D motion of a virtual particle was captured by using the 3-channel self-mixing Nd:GdVO₄ laser flowmetry scheme mentioned in 4.2.5, where three access beams were impinged into the scattering cell containing PS particles in water. The output voltage from each channel of the 3-channel FM-demodulator is proportional to the instantaneous velocity of a single virtual particle within the limited field of vision along the axis of each access beam. The displacement of a virtual particle along each direction, which is calculated by integrating the velocity over time, i.e., D(t) = ∫ (v(t) –v̄)dt, is shown in Figure 28.

Using the trigonometric method, we reconstructed the three-dimensional movement of the virtual particle in the crossed beam-focus. The resultant trajectory of the virtual particle is shown in Figure 29. The reconstructed trajectory indicates a random walk in three dimensions, which is characteristic of the nature of Brownian motion. It should be noted that the movement along the Z-axis, which is the axis of access beam f_c,3, was smaller than the movement within the XY-plane. This peculiar nature of the motion of a virtual Brownian particle captured by the present 3-D self-mixing laser dynamic light scattering scheme is interpreted as follows: When the region of crossed beam-focus was adjusted within the scattering cell such that effective plane-wave scattering occurred for accurate sizing, as shown in Figure 24, the effective scattering volume is approximately given by V_s = πw²f_d, where w is the focused beam spot size and f_d = λ/2(N.A.)² is the focal depth of the objective lens. In the present case, the estimated focal depth, f_d = 8.5 μm, was much shorter than the focused beam diameter of 36 μm. Consequently, the movement of the virtual particle along the Z-axis is considered to be limited by the narrow field of vision of particles seen by laser beams as shown in Figure 29(c).

If we apply the present set-up to anisotropic media, e.g., liquid crystals, we expect to capture quantitatively different dynamic behaviors of molecules along three orthogonal directions.

The first experimental setup is shown in Figure 41, where the simplest self-mixing LDV experiment constructed by (a), (c) and (b), (c) was carried out by using different solid-state laser media. A 5 at.% Yb-doped 2-mm-thick Yb:YAG ceramic sample consisting of randomly distributed single-crystalline grains, whose average size was about 3.20 mm, was used and the end surfaces of this sample were coated with mirrors, M₁ (99.8% reflectance at 1,049 nm and 95% transmittance at 970 nm) and M₂ (98% reflectance at 1,049 nm). Pump light from an LD with a fiber pigtail, whose core diameter was 100 μm, operating at a wavelength of 970 nm, was impinged directly on M₁, as shown in Figure 41(a). The lasing wavelength was λ = 1,049 nm and linearly polarized TEM₀₀-mode operations were obtained. The threshold pump power was P_th = 650mW and the slope efficiency was 50%.

For comparison, a 3 at.% Nd-doped 1-mm-thick Nd:GdVO₄ single crystal sample with mirrors, M₁ (99.8% reflectance at 1,064 nm and 95% transmittance at 808 nm) and M₂ (98% reflectance at 1,064 nm), was also used. A collimated elliptical beam from the LD beam operating at 808 nm was transformed into a circular beam using an anamorphic prism pair. The beam was then focused onto M₁ by a microscope objective lens of numerical aperture (NA) = 0.25, as shown in Figure 41(b). The lasing wavelength was 1,064 nm and linearly polarized TEM₀₀-mode operations were obtained. The threshold pump power was 32 mW and the slope efficiency was 24%.

From the pump-dependent relaxation oscillation frequency, as mentioned in 4.1.2, the photon lifetime of the Yb:YAG laser was τ_p = 176 ps, assuming a fluorescence lifetime of τ = 900 μs, yielding the lifetime ratio of K = 5.11 × 10⁶. While, τ_p = 235 ps and K = 3:83 × 10⁵ as for the Nd:GdVO₄ laser, assuming τ = 90 μs. Therefore, the K-value of the Yb:YAG ceramic laser is about 13.3 times larger than that of the Nd:GdVO₄ laser, mainly owing to its longer fluorescence lifetime, and an increased optical sensitivity of the Yb:YAG ceramic laser is expected.

To confirm such an expectation and evaluate the increased optical sensitivity quantitatively, a simple laser-Doppler velocimetry experiment without AOM frequency shifters was carried out as shown in Figure 41(c), where the laser output beam was impinged on a rotating cylinder through a lens. Here, 4% of a laser beam split by a glass plate was used for monitoring and the remainder (96%) was sent to a target. In this scheme, the laser is intensity-modulated at f_D = 2v_z/λ (v_z moving speed along the laser axis of a target). Both lasers were operated under the same optical feedback conditions using a common target through the same optical path. The laser output was detected by an InGaAs photoreceiver (New Focus 1811: DC-125 MHz) and analyzed by a spectrum analyzer (Advantest R3131A: 9 kHz-3 GHz) or a software-defined radio (RF-Space SDR-14: DC-30 MHz).

Typical power spectra of modulated lasers are shown in Figure 42(a) (frequency resolution: 8.138 kHz) for different rotation velocities of the turntable, in which pump powers (i.e., w = P/P_th values) for both lasers were adjusted such that both show the same relaxation oscillation frequency to ensure a common frequency response. It is clear that the power spectral density of the Yb:YAG laser is about 177 times larger (i.e., 22.4 dB) than that of the Nd:GdVO₄, as was expected from the 13.3-times-larger K value (i.e., amplitude modulation index) under the same optical feedback ratio, since the power spectral density is proportional to the square of the Fourier component of the amplitude-modulated laser signal. Note that peaks 1, 2, and 3 in power spectra for the Yb:YAG laser denote relaxation oscillation sidebands, f_D ± f_RO, and peak 4 indicates the second-harmonic component, 2f_D.

Numerical simulations were carried out using the model equation for lasers subjected to frequency-shifted optical feedback including spontaneous emission noise, Equations (9–12), to reproduce experimental results and estimate the optical feedback ratio. Numerical results are shown in Figure 42(b), assuming w = 1.840 for Yb:YAG, w = 1.112 for Nd:GdVO₄ and relevant K-values for both lasers as determined experimentally in 7.1.1. The common parameters are m = 10⁻⁶ and f_m = f_D = 1.188 MHz, where R_f is the amplitude feedback ratio. Here, Gaussian white noise with zero mean was introduced as spontaneous emission to reproduce experimental noise spectra exhibiting f_RO and weak 2f_RO components. Gaussian-type frequency broadening of scattered light fields from a rotating rough surface, which results in the broadened f_D components in the experiments [Figure 42(a)], was neglected in simulations for brevity. The numerical results agree well with the experimental results, i.e., an increase in power spectral density of 24 dB at f_D for Yb:YAG, and an intensity feedback ratio, 10 logR_f², was estimated to be −120 dB. The lowest limit of intensity optical feedback ratio for the successful measurement, which is restricted by the spontaneous emission noise, is estimated to be −154 dB from the noise level in the present experiment.

A 2-mm-thick, 20 at.% Yb-doped single crystalline thin-slice Yb:YAG laser was also used instead of Yb:YAG ceramic and the similar enhancement of optical sensitivity was obtained. The threshold pump power was 500 mW and the slope efficiency was 43 %. A power spectrum in LDV experiment with the single-crystalline Yb:YAG laser is shown in Figure 43, in comparison with that of Nd:GdVO₄ laser. The enhancement of 23 dB was attained. The experimentally determined key parameters are summarized in Table 2.