Laser Doppler Imaging of Microflow

We report a pilot study with a wide-field laser Doppler detection scheme used to perform laser Doppler anemometry and imaging of particle seeded microflow. The optical field carrying the local scatterers (particles) dynamic state, as a consequence of momentum transfer at each scattering event, is analyzed in the temporal frequencies domain. The setup is based on heterodyne digital holography, which is used to map the scattered field in the object plane at a tunable frequency with a multipixel detector. We show that wide-field heterodyne laser Doppler imaging can be used for quantitative microflow diagnosis; in the presented study, maps of the first-order moment of the Doppler frequency shift are used as a quantitative and directional estimator of the Doppler signature of particles velocity.

Microfluidics has emerged as an important tool for engineering, fundamental science, biology, etc. [1]. The ability to control flow patterns at a typical scale of a micrometer through advanced microfabrication techniques [2] offers a neat and powerful guide for the development of analytical microfluidics chips. Of primary importance is the charaterization of flow patterns in microsystems [3] and much of the technological effort has focused on direct determination of flow profiles of liquids inside micro-capillaries. The size reduction indeed pre-cludes the use of external probes and therefore dismisses most of traditional macroscopic techniques for velocimetry (eg. hot-wire anemometry [4]). Particle-based flow visualization techniques (particle image velocimetry (PIV) [5], that often use fluorescent microspheres of a size of order of 100 nm in dilute suspensions at volume fraction of order of 10.5) are popular as little intrusive tools that offer high spatial resolution in the characterization of local flow properties (especially -PIV, for biological or complex fluids [6] [8] and biphasic flows [9]). Holographic configurations have been used for particle field diagnosis [10] and velocimetry [11,12]. Digital holographic PIV (HPIV) constitutes an active field of research, where off-axis HPIV appears to be a better configuration than inline HPIV in terms of signal-to-noise ratio (SNR) [13,14]. Other visualisation techniques, based on fluorescence bleaching [15], correlation spectroscopy [16], caged fluorescence [17], molecular tagging velocimetry [18], total internal reflection fluorescence [19,20], etc., are also used to investigate local flow properties.
In conventional laser Doppler velocimetry (LDV), the scattered laser light coming from a focus point onto a sample is detected by a single detector and analyzed by a spectrum analyzer. The power spectrum of coherent monochromatic light scattered by moving particles is broadened as a result of momentum transfer. The resulting broadening of light is linked to the velocity distribution of scatterers [21,22]. Doppler maps can be realized by scanning the focus point on the sample surface, which constitutes the principle of scanning Laser Doppler Imaging (LDI) [23][25], for which spatial resolution is typically low. Although LDV is a standard technique for macroflow studies, several difficulties limit its use in microflow: The focusing point of the laser beam has to be very accurately controlled, its micron-scale extension limits the accuracy of the velocity measurements, and the temporal resolution diminishes with the spatial scale of the sample [3,26].
We designed a parallel imager aimed at LDI [27], alleviating the issue of spatial scanning. Our technique involves a frequency selective heterodyne detection of the light on a multi pixel CCD detector which records digital holograms in an off axis configuration. The image is reconstructed numerically by using standard numerical holography algorithms. By sweeping the optical frequency of heterodyne detection local oscillator (reference arm), we record the spatial distribution of tunable frequency components of the object field sequentially.We demonstrate here that the technique can be employed as a tool for in-plane flow analysis in microfluidic networks and is especially wellsuited for building up Doppler maps (absolute values along with gradient measurements) on intermediate scale (∼ 1 cm) microfluidics webs.
Our technique can be interpreted as a Doppler global (or planar) velocimetry (DGV) measurement, but on a very different temporal frequency scale than molecular absorption-based DGV [28,29]. Molecular absorption DGV relies on the absorption characteristics of iodine vapor to convert a Doppler shift to a recordable intensity. Because of the characteristics of the absorption curve of the iodine and the finite stability of laser cavities, the frequency resolution is ∼ 1 MHz (velocity resolution of ∼ 1 m/s) [30], and is therefore not suitable for microflow analysis. In our case, the measure-ment relies on a heterodyne detection extremely selective in frequency. the detection bandwidth is the reciprocal of the measurement time, hence a frequency resolution of a few Hz, compatible with microflow analysis. Our method also differentiates itself from scanning LDI and HPIV methods. Whereas scanning LDI performs sequential measurements in space and parallel measurements in the temporal frequency domain, our instrument does parallel measurements in space and sequential measurements in frequency. And unlike HPIV and its refinements (e.g. [31]), our technique does not rely on the localization of the seeding particles onto an image. Like in DGV, Doppler maps are the result of a direct selective measurement of frequency-shifted components of the scattered light. The advantages of our heterodyne holography method for wide field laser Doppler measurements are the sensitivity (heterodyne gain), the selectivity in the temporal frequency domain (coherent detection) and the large opticalétendue (product of the detector area by the detection solid angle) of multi pixel coherent detection [32,33].

Optical Setup
The experimental setup [27,34] is based on an optical interferometer sketched in Figure 1.A CW, 80mW, λ 0 = 658 nm diode (Mitsubishi ML120G21) provides the main laser beam, split by a prism into a reference (local oscillator, LO) and an object arm. The dimensionless scalar amplitudes of the fields are noted E L (incident field), E O (scattered, object field) and E LO (LO field). The object is illuminated by the laser beam with an incidence angle a. Scattered light is mixed with the reference beam and detected by a PCO PixelFly 1.3 Mpix CCD camera (1280 × 1024 square pixels, pixel size: 6.7 m, framerate ω/2π = 4 Hz, exposure time τ E = 125 ms), set at a distance d = 50 cm from the object. Two Bragg cells (acousto-optic modulators, AOM, Crystal Technology) are used to shift the LO by a tunable frequency (the difference of their driving frequencies). A 10 mm focal length lens is placed in the reference arm in order to create an off-axis (θ ≈ 1 • tilt angle) virtual point source in the object plane (see Figure 1). This configuration constitutes a lensless Fourier holography setup [35]. To perform a heterodyne detection of a frequency component of the object field, the LO field frequency ω LO is detuned by ∆ω − ω S /n with respect to the main laser beam frequency ω L . The ∆ω shift allows the LO field to match the part of the object field shifted by ∆ω. The additional shift at the n-submultiple of the camera sampling frequency provokes a modulation of the interference pattern sampled by the detector at ω S /n.
The measured power spectrum S associated to the field EO is [27]: Where A is a positive constant. The Dw spectral point is readout from a sequence of n images recorded by the camera [27]. EO is recorded in the detector plane by the frequencyshifting technique [32] which is the dynamic equivalent of the phaseshifting digital holography technique [36]. The lensless Fourier holographic setup [35] used here is less demanding in numerical calculations than the general Fresnel holography configuration. The numerical reconstruction algorithm of the image is limited to one discrete Fourier transform [37,38].
Artefacts due to the finite size of the sensor and the spatial discrete Fourier transform are negligible [39] (narrower than one pixel in the reconstructed image). We define the signal S dB represented on spectra profiles and maps: S dB (∆ω) = 10 log 10 S(∆ω) N (∆ω) whereN (∆ω) is the quantity S(∆ω)) assessed in a region of the reconstructed hologram where the object light contribution is null. It was reported to be shot-noise dominated [33].
To map the projection of velocities with the inplane component of the incident field wavevector (according to Eq. 4), we compute the first moment of the frequency shift ∆f = ∆ω/(2π): where S(∆ω) is the zeroth-order moment. These sums are calculated over the full range of measured shifts.

Microfluidic devices with velocity gradients
In this study, we designed a fluidic network made using standard microfabrication techniques [2]: a circuit is patterned onto a silicon substrate using micro-photolithography (spincoating of a photoresist exposed to UVs through a mask and developped to obtain hard patterns resolved at ≈ 1µm), molded into a transparent elastomer polydimethylsiloxane (PDMS) and stuck on a glass substrate. We thus obtain closed circuits for liquid transport of typical dimensions 10 50 5000 µm. The actual devices we use are special (Figure 2) as they generate a flow via evaporation in a dead end microchannel [40], with a flow field which varies in space. Details for fabrication of these 2-layer systems (one layer for liquid transport, one layer for gas transport for evaporation) are given elsewhere [41] and the main feature we shall retain is the linear dependence of mean velocity in a channel as a function of position (while by design [40], these  [40]. This devices is made of PDMS on glass and has two layers: one for the fluid separated from the other for the gas removal by a thin PDMS membrane (e ∼ 10µm). Typical dimensions are L 0 ∼ 1cm, h ∼ 20µm. b. The permeation-induced flow is stationnary, and varies linearly in space between 0 and a typical upper bound v max ≈ 50µ/s (longest channel) which actually depends on L 0 . devices are used to concentrate a solute near the dead end of the channels, an effect we shall also evidence here). Such a heterogeneous flow field is a specificity of permeation-induced flows [42], and we will exploit it for testing the construction of velocity maps.
The microsystems are filled with a solution of pure water seeded with 1.0 m diameter latex spheres (with carboxylate stabilizing surface groups, by Molecular Probes). Evaporation induces a permanent flow that drives the liquid from the reservoir towards the dead end of the channel, with a velocity slowing down linearly from ≈ 50µm/s down to 0 (see Figure 2). This result holds for the mean velocity; two effects alter this simple view: the flow field actually shows a parabolic profile (Poiseuille flow) and the particles undergo Brownian dynamics.

Wide field Laser Doppler mapping of microfluidic webs
A 2 × 10 −4 volumic concentration suspension of latex beads is perfused through the microfluidic web. This concentration of latex corresponds to 3.82 × 10 −4 particles per cubic micron, which was the lowest seeding for which those particles would lead to an acceptable SNR, in our configuration. A Mie algorithm 1 was used to assess its optical parameters : the resultant mean free path is l s = 1.5mm, and the anisotropy coefficient is g = 0.93. This suspension of latex beads was perfused in the microfluidic device, composed of 6 channels of h = 23µmw = 100µm cross section, which optical thickness (defined as the thickness of the channel in mean free path units) h×µs ∼ 10−2 is much smaller than unity. The incidence angle a of the laser beam (see Figure 1) is ∼ 45 • . In this experiment, Doppler shift maps were measured in the -64 Hz → +64 Hz range (spacing interval : 2 Hz). S(∆ω) was calculated with a n image demodulation and averaged over a m image sequence. Here : m = 16, n = 8. The total measurement time of the spectral cube took ∼ 8 minutes and 30 seconds (= 16 images × 65 frequency shifts × 1/4 second exposure time). Nevertheless, in these conditions, the measurement of one complete spectral map at an arbitrary frequency shift ∆f took only 16 1/4 = 4 seconds.
Three S dB maps, measured at ∆f = 0, -22 and -44 Hz are represented on Figures 3a to c. The whole microfluidic web appears to respond at each arbitrary ∆f frequency. Such maps are difficult to interpret, since they represent a signal which combines both scatterers density and Doppler signature. Additionally, these results reveal an averaged response over the height of the channel. The zeroth-order moment map of the frequency shift, represented on Figure 3d, displays a signal allegedly proportional to the moving particles concentration  [43] (under the approximation of a small scatterers densities), which shows the expected increase in density at the end of the channels, in region (vi).
The Doppler shift for one scattering event by a particle in translation with velocity v is [44] : where q = k s − k i is the momentum transfer, k i is the incident wave vector, ks the scattered wave vector and v the scatterer instant velocity. In the chosen optical configuration (Figure 1), k s v = 0. Hence the Doppler shift in a region where the average speed of the particles is v : At the entrance of the longest canal (canal (1) in Figure 5a) the particles translation speed is v ∼ 50µm.s −1 . According to the linear behaviour [40] of  In diluted samples (in very good approximation the ones which optical thickness is much lower than 1), the root mean square (RMS) Doppler shift of light scattered by a particle undergoing brownian motion is ω 2 B 1/2 = q 2 D B whereD B is the spatial diffusion coefficient of the brownian particle; in the particular case of a spherical particle of radius r, its expression is : where k B is the Boltzmann constant and T the absolute temperature. For our latex beads in suspension in water (of viscosity η = 10 −3 Pa.s), at room temperature (k B T = 4.0 × 10 −21 J), we have D B = 4.2 × 10 −13 m 2 s −1 . The average value of q 2 , for a scattering angle (k i , k s ) = β is [45]: which gives, in the chosen optical configuration ( Figure 1): where α ′ is the incidence angle of light in water of refractive index n = 1.33, and λ = λ 0 /n is the optical wavelength in water. The numerical value of ω 2 B 1/2 is ω 2 B 1/2 /(2π) = 30.9 Hz. This value of the RMS brownian motion-induced Doppler shift is compatible with the width observed, in Figure 4vi, where the fluid velocity is nearly zero. In other regions : (i) to (v), the observed spectrum corresponds to the convolution of the brownian spectrum with the spectrum associated to the distribution of velocities in the flow (Poiseuille flow). The narrow peak, observed on the spectra near zero frequency, is a parasitic signal not related to the flow (light scattered by the PDMS substrate...). Figure 5 represents the first moment of the Doppler shift with respect to the measured Doppler linewidths on each pixel, mapped and plotted versus the longitudinal position along two microfluidic channels. Figure 5b and 5c show a linear decrease of δf with the position along channels (1) and (2), highlighted in Figure 5a. This corresponds to the expected linear decrease of the average velocity along the channels.

Conclusion
We have presented a wide field laser Doppler measurement in a microfluidic device, and demonstrated that the velocity resolution is compatible with microflow diagnosis. In the reported pilot study, the setup, based on off-axis heterodyne holography, performs a parallel measurement of the power spectrum of the light scattered by a microfluidic device on a CCD camera. The scheme is different from conventional laser Doppler time-domain measurements : each frequency point of the spectrum is acquired at a time, by tuning the LO frequency at the desired shift. Since the spectrum is processed optically, by an interferometric scheme between scattered light and a detuned local oscillator, the span of the measurable spectrum is not limited by the detector bandwidth. Furthermore, the optical configuration allows to measure algebraic Doppler shifts since the hologram records the optical field in quadrature, rather than its intensity.
In this paper, the stress was put on the potential of the wide field laser Doppler instrument in terms of frequency (and hence velocity) diagnosis : high frequency resolution, high SNR, high dynamic range of restituable frequencies on the same image, and directional discrimination. The scope of the presented study was to provide measurements in large scale microfluidic devices into which large velocity gradients can be created at the same time.
The spatial resolution of the digital holographybased apparatus is about the extent of 1 pixel on the reconstructed image [37]. Much better resolution (at the expense of the field of view) can be obtained by using an holographic microscopy configuration, which enables theoretically sub-micronic lateral resolution [46] (up to the diffraction limit). Nevertheless, time-averaged measurements of rarefied seeds at low spatial scales, might become noisy and reduce the spatial resolution of velocity maps.
The frequency resolution is given by the heterodyne bandwidth of the detection (the inverse of the total acquisition time of the image sequence). Although measured spectra are stained by the Doppler linewidth contribution of Brownian movement, the setup has an intrinsic ability to map small velocities (up to ∼ 10 micron per second, but the actual limit should be even lower, thanks to the heterodyne selectivity). Since these (and lower) flow velocities are commonly encountered in