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Digital holographic microscopy provides an ideal tool for 3D tracking of microspheres while simultaneously allowing a full and accurate characterization of their main physical properties such as: radius and refractive index. We demonstrate that the combination of high resolution multipoint tracking and accurate optical sizing of tracers provides an ideal tool for precise multipoint viscosity measurements. We also report a detailed evaluation of the technique’s accuracy and precision in relation to the primary sources of error.
©2011 Optical Society of America
1. Introduction
Holographic tweezers [1, 2] allow a full 3D, dynamic and contactless micromanipulation of samples in micro environments. A straightforward application of that in lab on chip technology is the use of trapped beads as a multipoint sensor of physical/chemical properties of surrounding fluid [3–10]. There are many different situations in physics and biology where monitoring fluid properties such as viscosity at many positions simultaneously can be of extreme importance [11]. For instance, cell environments can be highly inhomogeneous in time and space due to time varying concentration gradients of macromolecules around cell membranes. The basic principle of a holographic multipoint sensors is that a suspended object will experience forces that depend on fluid properties such as viscosity and temperature. The analysis of resulting trajectories [12] can be then used to extract local fluid properties. The accuracy of such light driven sensors relies clearly on the available spatial and temporal resolutions of tracking probe particles. However particle shape and size will also play a role and assuming a perfect, monodisperse nominal radius can introduce an important source of error when comparing data from different probes. An ideal tracking tool should then provide a high spatial and temporal resolution combined with simultaneous accurate particle sizing capabilities. Digital Holographic Microscopy (DHM) [13] can provide those requirements and allow multipoint viscosity measurement with a superior accuracy to other proposed tracking schemes. Although higher spatial and temporal resolutions can be achieved using quadrant photodiodes in back focal plane interferometry [6, 7, 14–17], they require additional accurate calibration and result cumbersome when tracking more than two particles. Multi particle tracking is usually performed using video microscopy [12] at the cost of a worse spatial and temporal resolution. Both back focal plane tracking and video microscopy give no direct information on particle’s size. Bead sizing is only available after a delicate calibration procedure that requires moving the probe at different distances from a fixed wall and then fitting the corresponding mobility values to Faxén formulas [18, 19]. Moreover, such a procedure relies on the assumption of a homogeneous viscosity which prevents accurate multipoint viscosity measurements in inhomogeneous media. On the other hand, DHM provides an all-optical size determination which can be obtained from one single snapshot and results in an even better accuracy [20]. In addition axial tracking is available over a much larger range [21].
DHM refers in general to optical microscopy techniques where the coherent illumination light interferes with that scattered from the sample, producing an interference pattern that is the recorded hologram. In most of the cases 3D spatial reconstruction is obtained by direct numerical back propagation [21–28] of the recorded hologram. DHM has been shown to provide a powerful tool for fast 3D particle tracking especially when using spherical probes. In that case object reconstruction can be achieved by fitting the recorded hologram to the superposition of forward propagated Mie scattering patterns. This approach [20, 28–31], which we’ll refer to as Digital Holographic Tracking (DHT), allows to track spherical colloids in a 3D space with a supposed resolution of nanometers while simultaneously measuring the mean radius a and the relative refractive index m with high but not yet well characterized accuracy.
We start by quantifying the actual precision and accuracy of DHT in relation to the primary sources of error. As a demonstration of the peculiar features of DHT we then use a system of four probe beads in blinking holographic tweezers [32] as a multipoint viscosity sensor. When the particle sizing capabilities of DHM are exploited, accuracy and precision of viscosity measurements are improved by a factor of at least four when compared to other techniques relying on position tracking alone.
2. Digital Holographic Tracking, principles and implementation
Our custom microscope system Fig. 1 is equipped with a 100x numerical aperture 1.4 oil immersion objective which can collect either the collimated light of a diode laser (λ = 0.657 μm) or the light coming out a tungsten-halogen lamp (Thorlabs OLS1-EC). The objective is coupled with a tube lens to image the focal plane on the CMOS camera (PROSILICA GC-1280) plane. The specimen’s positioning is controlled by a motorized microscope stage (Märzhäuser SCAN IM). An holographic optical trapping system consisting of a DPSS laser (Coherent Verdi 3W, λ = 532 nm) and a spatial light modulator (Holoeye LC-R 2500) is coupled to the microscope by a dichroic mirror.
Fig. 1 Schematic view of our digital holographic microscope/tweezers system. DM: dichroic mirror. Sample illumination can be switched between white light and a collimated diode laser (λ = 0.657).
As a plane monochromatic wave impinges on a dielectric sphere, the scattered light interferes with the incident beam resulting in a fringe pattern. A magnified image of the interference pattern is projected on a CMOS camera by the microscope objective. Comparing the experimental hologram with the predictions given by Lorenz-Mie theory [33] we can extract particle’s 3D position together with its radius and relative refractive index. Let us assume the incident light is approximately a linearly polarized plane wave E i(r) = exp(jk r · ẑ)x̂, where the spatial vector position is denoted as r, ẑ is the unit vector in the direction of beam propagation and x̂ is the unit vector in the direction of beam’s polarization. The light field scattered by a dielectric sphere particle is Ei f(r – r p, a, m), where f is an analytically known function of the spatial position relative to the particle’s center r p, as well as of the particle’s radius and refractive index relative to the medium. The theoretical light intensity, normalized by |Ei|2, is
where the parameter α is used to adjust the scattering efficiency relative to the ideal case (α = 1).In practice, recorded holograms always suffer from a mainly multiplicative “background” noise due to scattering from unwanted objects inevitably found within the sample environment and in the whole optical path outside of the sample chamber. Such a background noise can be partially corrected by dividing the recorded hologram by a previously recorded empty field of view image [20]. The resulting pattern provides the experimental normalized intensity Iexp. We fit the obtained holograms Iexp to Ith using a Levenberg-Marquardt nonlinear least-squares minimization algorithm with the five free parameters: r p = (xp, yp, zp), a and m. A considerable speed up in chi squared evaluations is achieved exploiting the computational power of a CUDA based Graphic Processing Unit (NVidia GTX260). The computation of Ith is very expensive since it involves the evaluation of special functions (spherical Bessel functions and Legendre functions) which are expressed as recursive series and hence not suited for parallelization. However, special functions are only to be evaluated as a function of radial distance. To speed up the computation, we tabulate them once on a 1D array an then use linear interpolations. Our software can analyze 23 particle’s hologram per second (i.e. real time analysis). When the radius and the relative refractive index are previously calibrated, we can reduce the fitting parameters with a speed up of 50% (i.e. 35 fps). We account for laser intensity variations introducing a multiplicative factor β which can be inferred by minimizing the whole error :
where the sum is performed over all pixels.However, small variations in the empty field of view hologram can occur over time and dividing by a previously recorded background could result in significant errors at the required levels of accuracy. For example, we report in Fig. 2 a typical the time correlation function c(t) of recorded empty field background fluctuations:
where bi(t) is the intensity on the ith camera pixel normalized by the full array average, N is the total number of pixels and 〈·〉 represents time averages. At the beginning, the correlation function suddenly decreases of about 20% due to the intrinsic sensor noise of the camera. Then a slower decay is observed on a few minutes time scale.Fig. 2 The correlation function c(t) of background fluctuations is plotted as a function of time. It vanishes in approximately two minutes.
It is also worth noting that when approaching a precision level of about one percent other, usually negligible, imperfections in the actual hologram recording, such as camera distortions and discretization, may affect performance. For example, one would expect to find an α parameter that is smaller than the ideal value of 1 since Mie theory doesn’t account for the effect of bead nanoscale roughness or absorption. On the contrary we noticed that a value greater than 1 by a few tens of percent is usually needed to get the best fit. We believe this is an artifact due to distortions in the CMOS. In particular, we found that the response function of our camera is given by a straight line shifted by a constant offset:
where I is the light intensity on the camera, Ts is the exposure time, Ig is the pixel gray level and b and c are two positive parameters depending on the camera properties. Such parameters can be experimentally evaluated by imaging with different shutters a beam with fixed intensity. In our case we found a value for c of about 38 over a dynamic range of 255. Taking into account the camera’s response and neglecting quadratic scattering terms, the recorded normalized intensity Ig will be: which explains why the measured value of α′ is larger than the true α. If recorded images are correctly linearized we always obtain α values smaller than one.Previously reported precisions of DHT [20, 28, 29] are of few nanometers for the Cartesian coordinates (x, y and z), and a one part per thousand error for the particle radius a and relative refractive index (m). Although precision can be easily measured experimentally, accuracy is less easily accessible due to the lack of an independent, high resolution, determination of bead size, and refractive index. Regarding probe position, one could think of tracking a nanopositioned bead to get a direct estimate of accuracies. Unfortunately, accurate nanopositioning is usually achieved by having a bead that is stuck on a fixed support, like a coverslip, rigidly connected to a piezo stage. However DHT heavily relies on the assumption of a spherical and homogeneous bead that is embedded in an optically homogeneous medium. The presence of a nearby glass surface would produce a hologram that wouldn’t agree with Mie theory. For this reason we now proceed to evaluate accuracy and precision using synthetic “experimental” holograms obtained as follows:
- Compute a set of theoretical normalized intensities Ith of a Brownian moving particle.
- Record a time series of true background holograms Ibg with an empty field of view.
- Obtain time series of synthetic noisy Mie interference patterns Isim = IthIbg.
- Emulate the camera discretization to 256 gray levels.
We can now fit our synthetic dataset made of 4000 simulated holograms based on about a 8 minute long time series of experimental backgrounds, which we also used to work out correlation function c(t) shown in Fig. 2. Resulting accuracies, shown in Table 1, are computed as the standard deviation of the absolute errors. We performed this calculation for three different cases in order to discriminate between different sources of error.
Table 1. Estimated Accuracies and Precisions of Position, Radius and Relative Refractive Index Measurements*
Case A: We skip steps ii) and iii) imposing Isim = Ith so that the fitted interference pattern is noiseless but discretized to 256 levels. This case shows that the camera discretization is sufficient to set the accuracy limit to values that (except for m) are much higher than the floating point precision which is expected for a noiseless image.
Case B: We compute the normalized holograms dividing Isim by a background recorded few seconds earlier. This means that the background image is still very correlated and thus we obtain a low noise normalized hologram. That case gives an estimate of the best accuracy we can achieve in a real experiment.
Case C: We compute the normalized holograms dividing Isim by a background recorded eight minutes earlier. In that case the background is already in the low correlated regime (see Fig. 2).
In order to estimate the precision of the system, we repeated the same procedure starting off the same theoretical Mie pattern (ideally fixed bead). The standard deviations of fitted parameters are reported in Table 1, where we skipped the case (a), since it’s meaningless. Furthermore, we also checked if there are any significant correlations between the adjustable parameters while tracking the Brownian motion of single actual 2.07 μm diameter silica bead. No mutual dependence between parameters appears, except for the radius and the relative refractive index for which a weak anti-correlation is observed (the slope of the regression line is −0.07 μm −1).
The above analysis shows that tracking performance is mainly deteriorated by the background noise. A partially coherent source helps reducing the artifacts that are visible when using a fully coherent illumination [34]. In addition, carefully cleaning sample and optics can also substantially reduce background inhomogeneities.
3. Multipoint viscosity measurements
It has been suggested that, using holographic optical tweezers, multiple beads can be trapped in target points to map viscosity [18]. However, the behaviour of a bead inside a fluid is significantly influenced by its dimensions and by hydrodynamic interactions with other bodies. For example, due to the short working distance of high numerical aperture objectives, we usually observe particles that are few tens μm distant from the cover slip wall. In such a case, the diffusivity D of a spherical bead due to Brownian motion is dependent on the radius a and on the distance h from the wall according to Faxén correction formulas [35–37]:
We can use for example the measured value of D to get viscosity if we know particle size a and height h. A relative error on a will produce a relative error on η of δa/a, while an uncertainty on h contributes a relative error on η of about (9/16)(a/h)(δh/h).For instance, for a 1 μm radius bead floating at about 10 μm from a solid wall, if we are interested in getting a viscosity value with a one percent accuracy then we should know particle height within 1 μm and particle size within 10 nm (1%). Highly uniform microspheres are commercially available with a typical size variation that is at best about 5%, therefore much bigger than what DHT can provide. Bead radius uncertainty is then a major source of systematic error on tweezers based viscosity measurements [7, 18].
As a demonstration of DHT capabilities for multipoint viscosity measurements, we test our technique in an homogeneous medium (distilled water) whose rheology properties are well known, and evaluate the agreement of the local measured viscosities to the expected value. We holographically trapped four 2.07 μm silica beads. We place them on the vertices of a 8 μm × 8 μm square that lies on a plane parallel to the objective focal plane and ten microns above the cover slip wall. Using a chopper we periodically release them from the traps and record their free Brownian motion for 62.5 ms, before traps are switched on again and particles are reset to the same starting positions. Acquisition framerate is set to 160 Hz and 8 frames per release are processed. Particles displacements in the z directions are 260 nm at most. Therefore we can safely neglect any variation of Faxén correction Eq. (6). Working with freely diffusing tracers has the advantage that hydrodynamic interactions on single particle diffusivities can be easily neglected. In particular it is found [32] that, due to hydrodynamic interaction with neighboring beads, single particle diffusivities are only affected by a correction which is at most of order (a/r)4, with a particle radius and r interparticle distance. In our geometry (r/a ∼ 8) that hydrodynamic correction is only expected to affect diffusivities, and hence viscosities, on the level of a few part over a thousand (8−4). Another advantage of blinking tweezers for viscosity measurement is that data acquisition is only performed in the absence of laser light. Even in the presence of a small local heating due to trapping beams, that will diffuse away in a few microseconds once trapping light is switched off.
The measured mean square displacement (MSD) in the x coordinate is fitted by:
where 〈·〉 represents averaging over both time origin and the ensemble of recorded trajectories. The second term is an effect due to the finite exposure time Ts of the camera (2.5 ms in the present case) that affects the measure of the MSD [38]. Measured MSDs and their best fit to Eq. (7) are shown in Fig. 3. In Fig. 4 a recorded hologram and its corresponding best fit are compared. Black circles are the normalized radial intensities showing an excellent agreement to the theoretical best fit reported as a solid line.Fig. 3 Mean square displacements of all particles along the x direction averaged over 10000 trajectories. Experimental data (squares and circles) are fitted to Eq. (7) (solid lines). The size of error bars is smaller than symbols size.
Fig. 4 Experimental (solid circles) and best fit (solid line) normalized radial intensities. Experimental and theoretical holograms are shown as insets.
The viscosity measurements for the four particles are plotted in figure Fig. 5 as a function of the time interval over which the MSDs are averaged. Left and right panel refer respectively to viscosity determinations using measured or nominal probe radius (1.04 ± 0.10 μm, Bangs Laboratories Inc.). It is evident that assuming a uniform probe size results in artificial local viscosity inhomogeneities which are much reduced when the single probe sizing capabilities of DHT are exploited. Moreover, we found that probes radii are systematically smaller than the nominal one (see Table 2), so that even when averaging over different probes the measured viscosity will always be affected by a systematic error. The mean accuracy on viscosity is 2.7%, that is about four times better than the mean accuracy (10.5%) resulting from assuming in Eq. (6) the nominal values for the beads’ radii instead of the measured ones. Such an accuracy of about 10% is comparable with that one obtained in former viscosity measurements performed with HOT [18] and back focal plane interferometry [6,7]. Another advantage of DHT is that any probe contamination, such as stuck small particle, can be easily spotted when the probe particle hologram doesn’t match the theoretical scattering pattern of a perfect Mie sphere. Hence, low accurate viscosity measurements obtained from those probes can be rejected.
Fig. 5 Viscosity measured at the vertices of a 8μm × 8μm square tracking for a time t the free Brownian motion of four 2.07μm diameter silica beads (shown in inset). In plot (a), viscosity is inferred using the measured radii of particles, while in plot (b) the nominal radius is adopted. The dotted line shows the predicted viscosity for water at 23° C. Both accuracy and precision of viscosity measurement improve when the radius of the tracked probes can be measured.
Table 2. Radius, Relative Refractive Index and Measured Viscosity for the Four Particles*
When comparing position accuracies, back focal plane interferometry provides better performances allowing for subnanometer accuracies while in our implementation of DHT the estimated accuracy is limited by background noise to a few nanometers. However, when one is interested in measuring viscosity through diffusion coefficients, the required accuracy on position tracking has to be small when compared to the largest root mean squared displacement. For example in our case Brownian motion is tracked up to a root mean squared displacement of about 140 nm (see Fig. 3). In those conditions a tracking accuracy of even 10 nm would be enough for 1% viscosity determinations, always provided one has simultaneous access to precise particle size. A low background noise is essential to get the higher tracking accuracy that is necessary when, relying on the ever increasing CMOS framerates, fast viscosity changes have to be detected.
4. Conclusions
Digital Holographic Tracking provides an ideal tool for multipoint sensing with holographic tweezers, simultaneously allowing precise, individual probe sizing. Estimated tracking accuracy is better than standard video microscopy but still worse than the subnanometer capability of back focal plane interferometry. Nevertheless, DHT allows to track multiple probes over a much higher axial range while probe size is readily available together with position and relative refractive index from fitting of a single snapshot. This is in contrast to other tracking techniques, where probe size is not directly available but has to be deduced from multiple acquisitions taken at different distances from a fixed wall. We demonstrate that using reliable probe sizes allows to reach a mean viscosity accuracy of 2.7% that is four times smaller than what is obtained using nominal bead size.
Acknowledgments
This work was supported by IIT-SEED BACTMOBIL project and MIUR-FIRB project RBFR08WDBE.
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