1. Introduction

Precise determination of particle or marker trajectories on a microscopic scale is important for many investigative applications, since it reveals the interaction with the surrounding environment, e.g. in a cell or cell constituents. The interaction with the environment can reveal important information on mechanical [1] or bio-chemical properties [2]. Especially fluorescent particles can mark specific functional parts of a cell system, whose interaction with the environment is highly relevant for the understanding of biological processes on the microscopic or even nanoscopic scale. Processes like diffusion [3, 4], directed motion [5] or anisotropic behavior can be monitored and interpreted [6, 7]. A more device-oriented application of measuring particle-environment interaction are microfluidic channels that help to obtain information on molecular diffusion [8, 9], pH analysis [10] and vascular modeling of blood flow parameters [2, 11]. It is even possible to separate and sort particles and molecules based on their diffusion coefficients [12].

Microfluidic devices have the potential to provide easily applicable clinical diagnostics and pharmaceutical screening tests [1], but they are also capable to evoke mixing processes on a microscopic scale using only a very small amount (in the order of nano-liter to micro-liter) of the needed ingredients [13].

Particle image velocimetry (PIV) and particle tracking velocimetry (PTV) are common methods for micro- and nano-fluidic analysis [7, 14]. Depending on the type of flow and the corresponding spatial and temporal resolution for resolving the fluid dynamics, adequate seeding particle densities have to be chosen. For large densities, single seeding particles cannot be identified, but a cross-correlation algorithm is applied in order to follow the 2-dimensional movement of the non-specific intensity distribution from frame to frame. PIV has a large spatial resolution which mainly depends on the interrogation window that is translated across the image. The disadvantage of this method is the lack of detecting the behavior of single-particles, e.g. when they defocus while moving along the optical axis. The free motion of seeding particles or markers in different environments e.g. in different regions of a cell plasma cannot be followed by PIV, which is a full frame cross-correlation method. In contrast to PIV, particle tracking velocimetry (PTV) is applicable [15, 16] for moderate particle densities i.e. tracking of single seeding particles is feasible. The PTV method can detect defocusing behavior and has therefore the potential of following trajectories in x, y and z coordinate. This allows a 3-dimensional evaluation of the particle trajectories and therefore offers complete spatial and temporal information on the fluid-environment interaction.

Depth (or z-) information within the depth of field (DOF) of a microscope objective is usually gained by scanning along the optical axis. This can be realized manually or with the help of a piezo-electric element [17] or an adaptive lens [18]. But any scanning method is limited in its temporal resolution because only a small z-region is observed at a time (i.e. one z-layer) and highly dynamic processes appearing between different z-layers (e.g. turbulent mixing in 3 dimensions) cannot be resolved. For the observation of such processes a scan-free wide-field approach especially for the z coordinate is required.

Possible solutions for that are common non-scanning methods as the holographic [19], 3-pinhole aperture-plate approach [20], astigmatic approach [15] or the plenoptic light-field method [21, 22]. The holographic method evaluates the phase difference between a reference and a measuring light wave. The disadvantage is that a coherent light source is needed and therefore the coherence length is limiting the achievable z-range. The 3-pinhole aperture-plate approach applies an amplitude modulation of the imaged light field by introducing a 3-pinhole plate to the setup. Each seeding particle appears as 3 light spots on the camera forming an equilateral triangle. From the expansion of the triangle, depth information can be extracted for each particle. The disadvantage of this method is the limitation of signal-to-noise ratio since it is discarding a lot of light intensity from the emitting or scattering particles. The second method introduces a cylindrical lens (static phase-modulation) to the imaging path, which results in the reproduction of the particles as elliptically shaped spots on the camera. Depending on the shape and the orientation of the ellipses, depth information can be gathered. The image processing is demanding, but the advantage is that all emitted or rather scattered light is used for information extraction on the particle depth localization.

A more flexible and customizable way of modulating the phase of light in order to get depth information is applying a spatial light modulator, namely a liquid crystal on silicon modulator (LCoS-SLM). The reflected light is phase-only modulated for one polarization direction. With the help of an LCoS-SLM it is possible to generate a double-helix point spread function (DH-PSF) [23–25] for light emerging from a focus. Here the light focus is situated in the focal plane of a microscope objective that is directed to a particle seeded fluid flow. Each particle appears as a double-image on the camera. Within the orientation of the double-image, depth information along the optical axis is decoded. There are several kinds of double-helix generating phase masks, e.g. the here applied spiral phase mask [28, 29], fork patterns [30] and Gauss-Laguerre mode superposition masks [23]. These double-helix phase masks are optimized for Cramer-Rao lower bound (CRB) and possess superior performance than astigmatic and biplane approaches [25]. A more advanced method of PSF-engineering [26, 27] uses tetrapod phase masks resulting in a more complex image structure for each particle and a large, adjustable z-range. Its disadvantage is the much more elaborated computing cost due to the necessary maximum likelihood estimation of the image shape for each particle, instead of the simpler Gaussian mask fitting [16] that is applied for the DH-PSF [23].

In the present paper we demonstrate 3D particle tracking measurements of laminar flow fields without the need of scanning along the optical axis. We apply a DH-PSF that is generated by spiral phase mask [28]. Instead of exciting fluorescent particles with a coherent laser beam as it is known from previous work [23, 27], an incoherent light source (LED) is used in order to illuminate the fluid from the backside [29]. The light field is transferred to an LCoS-SLM that performs phase-only modulation of the imaged light field. The shadow of each particle generates one double-image on the CCD-camera. Here we focus on relatively large particles (2 µm) compared to the nanoscopic standard of 0.2 µm in literature [23–25, 27]. For counting very few photons emitted from a 0.2 µm sized particle, an EMCCD camera is needed whose frame-rate is usually limited to 20 Hz [27]. Although for larger particles spatial resolution is worse, the advantage of our method is a potentially higher temporal resolution due to a larger signal to noise ratio (SNR) i.e. higher applicable camera frame rates (up to 200 Hz). Therefore we see the potential of our method in measuring fast (or even turbulent) fluid flows, which is not possible at the time with photon-limited methods using fluorescent nano-particles [23–25].

2. DH-PSF setup and image processing

Numerous established techniques exist for a broad range of fluid flow measurements in micro-channels [7]. Here we performed 3D particle tracking velocimetry (3D-PTV) with the help of a DH-PSF (Helix-PTV). Applying a DH-PSF results in a double-image for each seeding particle. The orientation of the double-image gives information on its position along the optical axis. A reasonable advantage of this technique is gathering depth information without scanning along the optical axis (commonly known as z-stacking).

The simplest approach for evaluation double-images of seeding particles is the analysis with the help of Gaussian mask fitting [16]. In case of 2D-PTV the pixelated 2D detector image is scanned for Gaussian intensity distributions that are recognized and assigned to a particle. But for 3D-PTV each particle is represented by a double-image of two Gaussian intensity distributions. By attributing two closely located Gaussian intensity distributions to one particle and analyzing the orientation of the double-image one can extract its z-coordinate additionally to the x- and y- coordinates and therefore the full 3D information on the particle location. While translating this procedure frame by frame through all frames of the recorded video, the velocity and 3D directionality of the imaged fluid flow can be extracted.

2.1 Setup

The measurement setup for fluid flow measurements consists of five main components as it is indicated in Fig. 1, namely an LED light source (Thorlabs Green 530 nm Mounted LED, 350 mW), a micro-channel (Ibidi, 400 µm channel thickness) filled with distilled water and seeding particles (microParticles GmbH polystyrol PS-FR-Fi256, 2.21 µm diameter), a microscope objective (Nikon CFI plan S-Fluor 10x, NA = 0.5, WD = 1,2 mm), an LCoS-SLM (Holoeye PLUTO-NIR-011 Phase Only, 1920x1080 pixel, 60 Hz) and a CCD camera (Basler camera pilot piA640-210gm, 648x488 pixel, 7.4 µm pitch, 210 Hz full-frame rate). The fluid flow rate can be precisely controlled (tenth of µl/s) by a low pressure syringe pump (Cetoni) that is connected to the micro-channel. The shadow-image of the seeding particles from the micro-channel is transferred via two lens-telescopes to the LCoS-SLM. A spiral phase mask that is loaded to the LCoS-SLM creates a DH-PSF from the focal plane of the microscope objective. A polarizer selects the axis for phase-only operation of the SLM. The phase-modulated image is recorded by the CCD camera.

 

Fig. 1 Optical setup for shadow imaging of the seeding particles. For setup alignment a laser beam (532 nm) transmitted through a single-mode fiber (SMF) was used. For fluid flow measurements light of a green LED (532 nm) was focused to the micro-channel (MC). A microscope objective (MO) (10x, NA = 0.3, 40x, NA = 0.65) images the shadows of the particles followed by a telescope consisting of lenses (L) f₁ = 5 cm, f₂ = 5 cm, a mirror (M) and a second telescope built of f₃ = 3 cm, f₄ = 10cm, an iris (I), and a polarizer (P). The LCoS Holoeye Pluto (SLM) is loaded with a spiral phase mask (SPM) and the image is focused to a Basler pilot camera (CCD) with a lens f₅ = 6 cm.

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The change in orientation angle $d\psi$ with distance change $dz$ along the optical axis can be expressed for a spiral phase mask in the following way [28]:

For calibration of the setup alignment, green Laser light (Cobolt MDPL, 532 nm, 30 mW) was coupled into a single-mode fiber. The bright transmission spot from the single-mode fiber output facet (Thorlabs 460-HP, core diameter 3.5 µm, NA = 0.13) was suitable for calibrating the setup. As it is shown in Fig. 2 the fiber facet was driven through the focal plane of the microscope objective in the range of $\pm$150 µm. The double-image of the fiber facet exhibits a rotation of its orientation by $\pm$ 40°. For optimal contrast a spiral phase mask with N = 10 was selected. A rest of unmodulated light is still observable in the center of the double-image up to 38° due to the limited performance of the polarization filter and the fill factor of the SLM (93%). The error bars indicate an angle estimation error of $\pm$2.5° which results in an error for depth localization of $\pm$6 µm.

 

Fig. 2 Calibration measurement for optimizing the setup. The image of a laser spot from a single mode fiber (532 nm) is converted to a double-image by a applying a SPM of ∆𝑙 = 2 and N = 10. With the help of the bright single spot emission the alignment and quality of the setup can be assured. Error bars for angle determination is given as vertical stripes. Up to $\pm$38° there is a rest of unmodulated laser light in the center of the double-image. The error bars indicate an angle estimation error of $\pm$2.5° which results in an error for depth localization of $\pm$6 µm.

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2.2 Image processing and particle tracking software

In order to find the best value N of the spiral phase mask for imaging 2 µm sized particles with the DH-PSF, a series of N from 1 to 5 was recorded for static particles sticking onto the micro-channel wall in water. As it can be seen from Fig. 3(a), N = 2 gives the best results in terms of contrast and recognizability of a Gaussian intensity distribution. Although for increasing N a larger separation of the parts within the double-image can be advantageous, here it leads to the unsymmetrical deformation and degradation of the double-image quality. The double-image recognition is demonstrated in Fig. 3(b). Two red circles indicate the detected center of the Gaussian intensity distribution and a red line connects the parts of the double-image to assign it to one particle. The original image has been inverted so that the shadows turn into bright spots. For some data sets it is necessary to apply contrast enhancement and background subtraction. The image was cross-correlated with a Gaussian intensity distribution mask I(x,y) as it is written in Eq. (2) [16] and the center of the spots (x_0, y₀) have been determined.

 

Fig. 3 a) Static micro-particles (2 µm in diameter) in a micro-channel for increasing N. With increasing N, the rotation sensitivity $d\psi /dz$ changes and the double-image separation increases while the overall contrast is reduced (see rainbow color coded images below the original ones). b) Inverted image for one particle applying spiral phase mask of N = 2. Two Gaussian intensity distributions are recognized and attributed to one particle c) The total angle rotation for N = 2 is 65° which leads to detectable z-range of 85 µm. The slope of the calibration curve is (0.8 ± 0.017) °/µm.

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Single particle tracking provides information on the particle trajectories. A common way is frame-by-frame tracking of the particles. Here the temporal resolution is mainly limited by the applicable camera frame rate that depends on the required exposure time for one image. A trade-off between the necessary SNR for reliable particle identification and the expected flow velocity is important.

3. Measurement results of DH-PSF shadow imaging

As a proof of concept experiment distilled water with 2 µm sized seeding particles has been injected into a 400 µm thick micro-channel. Without inducing a flow, free motion of the particles is observable [Fig. 4]. Four seeding particles have been tracked in a volume of about 20x20x50 µm. Figure 4(a) and Visualization 1 show the video recording (contrast enhanced and background subtracted). After inversion of the images all four particles are detectable by cross-correlation with the Gaussian intensity-distributed mask. Particle coordinates are extracted from 500 video-frames as it is depicted in Fig. 4(c) as semi-transparent black balls. (the temporal resolution is 20 ms). The XY-projection [Fig. 4(c), red dots] exhibits a directed motion in XY-plane in form of closed trajectories for each particle. In Z-coordinate [Fig. 4(c), green dots: XZ-projection] the scattering of the particle location is about $\pm$3 µm.

 

Fig. 4 a) Free motion of 2 µm sized particles in water within a micro-channel (see Visualization 1). b) Four different particles are identified. Particles 1, 2, 3 and 4 are located in a depth of Z = 81 µm, 58 µm, 61 µm and 49 µm, respectively. c) A weak directed drift from the left to the right side in the XY-Plane is observed (red dots). The total measurement range along the optical axis Z is 55 µm.

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In terms of fluid velocimetry, flow profiles are of interest in order to understand the particle interaction with its environment. Our intent is to measure only one set of video data in order to extract full flow-field information $\overrightarrow{v}(\overrightarrow{r},t)$with $\overrightarrow{r}(x,y,z)$ from only one recorded video, i.e. all three velocity components within a volume FOV x DOF that is defined by the field of view (FOV) and the DOF of the microscope objective. For comparison with the scan-free approach, a reference measurement was taken. By scanning along the optical axis, a parabolic flow profile was measured within a 400 µm thick micro-channel as it is shown in Fig. 6 (data is plotted as black rectangles). The measurement was done by performing one PTV evaluation for each z-layer within the thickness of the micro channel. This kind of procedure is time-consuming, yields large amount of partially redundant video data and requires PTV evaluation for each z layer.

 

Fig. 5 a) Measurement data of a laminar flow seeded with 2 µm sized particles in a 400 µm thick micro-channel (see Visualization 2) b) Coordinate labeling with respect to the micro-channel. c) Particle trajectories have been identified with the DH-PSF shadow-imaging method. The seeding particles keep their z-position (green dots: XZ projection) while flowing from the right to the left side (red dots: XY projection). The spatial resolution is 2 µm in X, Y and Z-coordinate. The measurement volume had a size of 40x40x40 µm.

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Fig. 6 Parabolic flow profile measurements in a 400 µm thick micro-channel. Black data points have been captured by scanning PTV measurements i.e. one PTV evaluation per z-layer. The data has been fitted by a parabola within the error bars. Orange data points have been extracted from only one PTV evaluation of double-helix PSF measurements. The evaluable axial range for z-localization is here 55 µm (compare also calibration curve in Fig. 3).

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A small region of interest of DH-PSF video data of a laminar micro-channel flow is presented by several video-frames in Fig. 5(a) (see also Visualization 2, Fig. 5(a)). Compare also the XYZ axis with respect to the micro-channel dimension in Fig. 5(b). The 3D-PTV evaluation gives the measured laminar flow field that is plotted in terms of trajectories in Fig. 5(c). 3D localization data is projected to the XY (red dots) and XZ (green dots) plane, respectively. The trajectories can be identified by eye and are indicated by blue solid lines. It can be seen that the seeding particles follow a unidirectional laminar flow of the water. There are closed trajectories in XY plane. But also in XZ projection one can discriminate different particle trajectories holding their directionality in one z layer.

The same video data from Fig. 5 was evaluated in a larger FOV by calculating the velocity magnitude |$\overrightarrow{v}(\overrightarrow{r},t)$| of the particle motion. Since the fluid flow is directed within the XY-plane and the particles hold their trajectory within a z-layer (compare Fig. 5(c)), the velocity magnitude |$\overrightarrow{v}(\overrightarrow{r},t)$| = $\sqrt{v_x²+v_y²}$ was determined by for each frame (px/frame). Together with the video frame-rate (50 Hz) and the spatial calibration (1 µm corresponds to 4 px) one gets the velocity amplitude in µm/s as it is shown in Fig. 6. Note that for the here measured flow speed, the frame rate of 50 Hz was sufficient. But in principle SNR would allow to apply the maximal frame-rate of the CCD-camera (200 Hz) for resolving much faster fluid flows.

The measurement results of Helix-PTV are compared to the full parabolic flow profile measured by scanning PTV from 50 to 350 µm (black data points in Fig. 6), where the z-layer at 200 µm is the center of the micro-channel thickness. The parabolic flow profile is characterized by a velocity range from 150 µm/s down to 35 µm/s. Note that the borders of the parabola (wall of the micro-channel) down to zero velocity are not accessible due to very low density of seeding particles in this region. Orange data points in Fig. 6 show the results applying Helix-PTV. The data points reproduce part of the scanning reference measurement in a z-range of 55 µm. The precision of Helix-PTV for velocity measurement is comparable to the one of scanning PTV technique as it is indicated by the orange error bars.

The results show, that it is possible to gather velocity information of the fluid flow within a quarter of the whole flow profile in one shot with high spatial resolution. Here not only the amount of instantaneous information in space but also in time is increased significantly compared to the scanning method. Especially flow changes that occur within the expected time for scanning are in principle not resolvable with the common scanning PTV approach. Furthermore Helix-PTV has the potential to measure even turbulent flows where trajectories are crossing in space. The here introduced scan-free Helix-PTV technique paves the way for studying complex flow structures and mixing processes [13] on a microscopic scale.

4. Discussion and summary

In order to provide a useful test-bed for demonstrating the capability of Helix-PTV in terms of SNR and measurement speed we have chosen a micro-channel with backside-illumination by an incoherent LED light source. Seeding particles that are added to the fluid, generate shadows which are imaged with a DH-PSF resulting in double-images on the CCD camera. The orientation of the double-images is evaluated and gives depth information of the particles.

We presented scan-free Helix-PTV measurement data of a laminar fluid flow in a 400 µm thick micro-channel. 2 µm sized seeding particles have been monitored as double-images in order to extract 3-dimensional information on their trajectories and velocities. The measurement volume had a size of 40x40x40 µm. A spatial resolution of 2 µm and a temporal resolution of 20 ms was achieved (note that 5 ms is achievable by applying full camera frame-rate). The accessible z-range had a value of 55 µm. Using phase-masks that generate a more complex PSF, would allow for designing the optimum trade-off between measurable z-range and localization error [26, 27].

The here applied type of phase mask (spiral phase mask) in connection with an LCoS-SLM offers a customizable trade-off between resolvable flow velocity and spatial resolution. We see the potential of our method in measuring fast (and even turbulent) fluid flows which is currently not possible with photon-limited methods. When an optimal phase mask is found, replacing the SLM by a fabricated phase mask (DOE) [31] would increase the modulation-efficiency and therefore SNR. This would result in larger detection speed for the measurement of potentially faster velocity-profile changes and even turbulent flows [15]. By identifying single particles with Helix-PTV and enumerating them, it is in principle possible to resolve turbulent flow structures in xyz-direction where scanning PTV is not capable of. Even a control for trapping of functionalized types of particles could be implemented [32].

Usually a micro-channel is considered to have an undistorted optical access. But refractive index mismatch between the particle, the solvent and the micro-channel material can cause significant aberrations that increase the measurement error for particle localization [34–36]. Here the advantage of an adjustable phase mask that an LCoS-SLM offers is, that additionally to the DH-PSF generation, aberration correction can be performed [35, 37].