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In dual-beam optical traps, two counterpropagating, divergent laser beams emitted from opposing laser fibers trap and manipulate dielectric particles. We investigate the lensing effect that trapped particles have on the beams. Our approach makes use of the intrinsic coupling of a beam to the opposing fiber after having passed the trapped particle. We present measurements of this coupling signal for PDMS particles, as well as a model for its dependence on size and refractive index of the trapped particle. As a more complex sample, the coupling of inhomogeneous biological cells is measured and discussed. We show that the lensing effect is well captured by the simple ray optics approximation. The measurements reveal intricate details, such as the thermal lens effect of the beam propagation in a dual-beam trap. For a particle of known size, the model further allows to infer its refractive index simply from the coupling signal.
© 2015 Optical Society of America
1. Introduction
Optical traps are widely used to trap and manipulate microparticles, be it single atoms, molecules or biological cells of several micrometer size. They exert forces using the interaction of light with a dielectric material.
Dual-beam optical traps were first described by Ashkin, who experimentally demonstrated the stability of such a trap [1]. Their use has increased in the recent decade since the invention of the optical stretcher, a dual-beam laser trap specifically used for trapping and deforming biological cells [2, 3]. Force calculations for particles of several micrometers in size, like cells in suspension, are often based on ray optics [2, 4]. Approximations to full solutions of the Maxwell equations have been done [5], showing that these do not differ considerably from the ray optics estimates.
In this work, we do not focus on the action of the beam on the trapped particle, but on the inverse action of the particle on the beam. The ray optics approach suggests that a trapped particle with a sphere-like shape acts on a beam like a lens, i.e. it has a focusing effect on it due to its surface curvature and its refractive index higher than that of the surrounding medium. This effect has first been used to measure the refractive index of urchin egg cells in 1921 [6]. Moreover, elongated cells have been used not as lenses, but as beam guides [7]. Continuous refocusing of light beams by multiple serially aligned microparticles (optical binding) has also been treated [8]. We investigate the lensing effect and show that ray optics, more specifically Gaussian beam optics, is sufficient to describe it.
Our approach is to use one of the laser fibers to collect light, which has passed the trapped particle, by the respective opposite fiber. We present a model for the dependence of this fiber coupling signal on particle size and refractive index.
This coupling signal allows not only to explore the lensing effect of trapped particles, but also two further details of the beam propagation, which will be treated in order to render the beam propagation model as complete as possible. One is the thermal lens effect, which, to our knowledge, has so far been ignored with regard to dual-beam traps; next to this, the gravitational displacement of particles from the optical axis of the trap influences the coupling signal.
The refractive index of biological specimen has become of interest in biomedical research and diagnosis, for example in the determination of solid content of cells [9] or in disease research [10]. To this end, methods for live cell refractometry have been developed, which often rely on interferometry, such as tomographic phase microscopy [11] or holographic phase contrast [12]. Additional to the general interest in the refractive index of particles, the interpretation of optical stretcher measurements critically relies on knowledge about the refractive index of the cells, as the stretching and trapping forces depend on it [2].
We present a new approach to live cell refractometry in an optical stretcher without the need for additional interferometrical assays, which is based on the inherent lensing effect described here. Knowing the particle size and given that it is of near-spherical shape, its refractive index can be inferred from the fiber coupling signal. As the method is primarily conceived as a supplement to optical stretchers, where cells are also imaged via phase contrast microscopy in order to monitor their deformation, the necessity of knowing their size does not pose an important obstacle. First, this new method is validated using homogeneous, spherical PDMS beads of known refractive index. In the last part, we show the possibility and the limits of the application of this method to inhomogeneous biological cells.
2. Experimental setup
We use an optical stretcher setup as described in [2]; a sketch is shown in Fig. 1.
Fig. 1 Experimental setup. Top: Between the stretcher chamber and the lasers, 99:1 fiber splices are used to redirect 1% of the light coming from the respective opposite laser onto photo diodes. Bottom: Close-up sketch showing the dimensions of the elements in the trapping region. For reasons of clarity, just one beam (coming from the left fiber end) is shown.
It consists of two Yb-based fiber lasers (Fibotec, Meiningen, Germany) connected to single-mode laser fibers (Corning HI 1060 Specialty Fiber, Corning, New York) whose ends are embedded in a self-built SU-8 photoresist structure, aligning them opposite each other. The laser wavelength is λ0 = 1064 nm; the mode field diameter of the beam leaving the optical fiber is given by the manufacturer as ω0 = 3.1 ± 0.015 μm.
Objects are delivered to the laser trap by a self-built microfluidic system which pumps them through a square profile glass capillary (Vitrotubes 8508, Vitrocom, New Jersey, USA).
Between the trap region and the lasers, 99:1 fiber couplers (FC1064-99B, Thorlabs Inc., NJ, USA) are used to redirect 1% of the laser signals coming from the trapping region onto photo diodes (Photodiode BPW 34, Siemens, Munich, Germany). This way, the amount of laser light coupled into the respective opposite fiber, after having passed the stretcher chamber with the trapped particle, is recorded via NI-6009 USB boxes (National Instruments, Austin, Texas). We call this signal the core coupling signal, or simply coupling signal.
Leaving the fiber, the beams traverse the trapping chamber, with dimensions and refractive indices shown in Table 1.
Table 1. Dimensions of the optical elements in the trapping region
As a test system, liquid PDMS beads were used. The beads were suspended in ethanol because their refractive index difference with respect to ethanol (nEthanol = 1.3559 at λ0) is about Δn = 0.035; comparable to the refractive index difference of cells in medium, which typically is in the range of 0.01 < Δn < 0.05. This leads to focusing behavior comparable to our target system, biological cells – other typical test materials like silica or polystyrene microspheres have considerably higher refractive indices, with differences Δn > 0.1 in usual media like water or ethanol, rendering their focusing behavior different.
PDMS beads are created by mixing 5% Silicone oil (Cat. No. 378364, Sigma Aldrich, St. Louis, MO) with 95% pure ethanol and gentle manual shaking of the mixture. Beads of different sizes can readily be found and tested.
As we were unable to find exact literature values for the refractive index of PDMS at λ0, an Abbe refractometer was modified to use an infrared LED of this wavelength as sample illumination source, according to the method described in [15]. The refractive index of PDMS, using this method, was determined to be .
Section 4.2 presents coupling data of biological cells. The samples are primary cells obtained from surgery using the method of total mesometrial resection [16], yielding primary cervical cancer tissue as well as its healthy counterpart. The tissue samples were dissociated, cells were cultivated to the first passage and subsequently measured in the optical stretcher. The ethics committee of the University of Leipzig approved the research protocol (reference number 090-10-19042010) and all participants gave written informed consent according to the Declaration of Helsinki.
3. Modeling the fiber coupling
An overview of the main effects of beam propagation in the trapping region is shown in Figs. 2(a)–2(d). First, the free Gaussian beam propagation (a) is treated. However, it needs to be adjusted for the thermal lens effect which is a result of laser absorption, leading to increased beam divergence as compared to free propagation of the beam (b). Using this beam propagation model, the focusing effect of trapped particles (c) is introduced, which finally needs to be modified to include gravitational displacements of the particles from the optical axis (d).
Fig. 2 Overview of effects in light propagation in a dual-beam trap treated in this study; note that the strengths of the effects are exaggerated: (a) a beam - here coming from the left fiber - diverges as it propagates through the chamber, (b) due to laser absorption, a thermal lens emerges, increasing beam divergence, (c) a trapped particle has a lens-like, focusing effect on the beam, (d) a gravitational displacement from the optical axis diverts the beam.
3.1. Free Gaussian beam propagation
The propagation of a Gaussian beam through an optical system composed of different media is modeled using ray transfer matrices and the Gaussian beam parameter q, defined as [17]:
Here, zR is the Rayleigh length and z the distance from the beam waist along the optical axis. λ is the wavelength of the laser, ω is the beam width and R the radius of curvature of the wavefronts. One starts with q(0) = izR, the beam parameter of the beam leaving the optical fiber, and uses ray transfer matrices known from ray propagation. Here, we need only the matrices for propagation in a medium Mp and propagation through an interface of two media Mi [17]: where z is the propagation distance inside one medium and n1, n2 are the refractive indices of the respective media. The beam parameter is then transformed according to the trapping geometry [17]: The beam propagation can thus easily be modeled up to a desired position, where beam width ω and radius of curvature R are extracted from the beam parameter q. Using ω and R, the beam fields are constructed, up to the Gouy phase which can be omitted here [17]: The intensity of the coupling of one beam to the opposite fiber is given by the squared amplitude of their overlap: Here, the integral is extended over the plane perpendicular to the optical axis.For our setup, this yields a beam width of ωmiddle ≈ 8.2 μm in the middle of the chamber, where cells are trapped, and ωend ≈ 15.4 μm at the opposite fiber. The absolute coupling coefficiency without a trapped object is calculated to be Cempty ≈ 0.15. However, since the laser heats the medium, a thermal lens emerges that makes coupling depend on the laser power.
3.2. The thermal lens effect in dual-beam traps
A thermal lens (TL) leads to self-defocussing of a laser beam, caused by a refractive index distribution that arises as a consequence of the laser heating the medium [18]. The heating is strongest on the optical axis, and because dn/dT < 0 for usual media, the optical density is lower on the optical axis than off the axis – comparable to a defocussing lens, thus continuously increasing the divergence of the laser beam. In an optical stretcher setup, peak heating was measured to be about 12.5 K/W on the optical axis [19].
The coupling signal of an emtpy setup (with medium but without particle) thus does not increase linearly with laser power (cf. Fig. 3(a)). In the case of water as medium, the coupling value at 1 W per fiber is about 8.5 times higher than at 0.1 W (trapping power), thus 15% lower than expected under the assumption of linearity.
Fig. 3 (a): Due to the termal lens effect, the core coupling is sub-linear with respect to the laser power. The effect is much stronger for ethanol than for water. (b): Simulation results for beam propagation with and without thermal lens effect; for 2*1 W stretching power in water (only one beam shown). The increased beam divergence merely influences the beam width in the middle of the trap, while the coupling of the beam into the opposite fiber is significantly altered.
The effect is much stronger with ethanol as medium. This case is important here because PDMS beads in ethanol are used to verifiy the model.
The optical power DTL of a thermal lens created by a Gaussian beam is approximated as [20]:
where P is the laser power, ω the beam radius, α the absorption coefficient, κ the thermal conductivity, the refractive index change with temperature and l the length of the lens. The optical power per length and laser power is then simply dTL.This approximation is valid near the beam axis in cylindrically symmetric situations, where the exponential integral resulting from the integration of the heat equation can be expanded around r = 0. The beam shape given in the experimental situation described here is not fully cylindrically symmetric, as the beam diverges on its path through the stretcher chamber and thus, the optical power of the lens will be z-dependent. However, with two beams coming from mutually opposite sides, this dependence is partially canceled.
Optical powers per length and laser power are calculated to be for water (using values given by [21, 22]) and for ethanol (using values given by [23]). The effect of thermal lensing on the coupling is incorporated into the ray transfer matrix calculation using lens matrices [17]:
To account for the thickness of the thermal lens, which continuously acts along the whole beam path, the propagation matrix is replaced by a repetition of ’thin’ lens and propagation matrices, each 1 μm wide: The relative core coupling without a trapped object was measured at laser powers from 100 mW up to 1500 mW per laser fiber for water and ethanol. The calculated coupling using the above model is compared to these curves in Fig. 3. As the modeled curves match well to the measured values, the calculated values are used for beam propagation calculations further on.3.2.1. Thermal lensing with trapped particles
The presence of trapped particles might influence the temperature distribution and thus the thermal lensing due to their differing extinction and thermal conductivity.
The extinction of PDMS at 1064 nm is αPDMS < 0.3dB/cm [24], lower than that of water or ethanol [23, 21], and its thermal conductivity κPDMS = 0.18K/(Wm) similar to that of ethanol κeth = 0.167K/(Wm) [25].
In the case of biological cells, the absorption depends strongly on the cell type and exact values are typically unknown; however the thermal lensing in water is very weak at trap powers of P = 100mW (see Fig. 3(a)).
Within the realm of this study, the thermal lens effect is assumed as unperturbed by the trapped particles.
3.3. The focusing effect of trapped particles
An easy way to calculate - and also to imagine - the coupling is by following the idea presented in [26]: The coupling of one beam to the opposite fiber field is equivalent to the coupling of that beam propagated halfway through the chamber to the opposite beam propagated halfway back, i.e. the coupling of the two fields in the middle of the trap, instead of at the end. Mathematically this is done by introducing a unity matrix 1 = T+zmiddle T−zmiddle, with two inverse propagators T, into Eq. (5), which is thus transformed to
This allows to analytically calculate the beam fields up to the middle of the trap, and to insert a (numeric) phase mask there:
where Q is the phase mask modeling the phase change due to the increased optical path length; i.e. in the case of a homogeneous sphere of refractive index difference Δn and radius R: Due to the symmetry of the situation, the two beam fields have identical amplitudes, but inversed phases. The coupling efficiency depends on the matching of the phases only - a beam field overlapped with itself would lead to a coupling of 100%. The focusing effect can also be thought of as an increased matching of the wavefronts of the two beams caused by the phase object, as represented in Fig. 4(a).Fig. 4 (a): The coupling can be thought of as the overlap of the left and right beam in the middle of the trap (see text). A phase mask between the two beams increases this overlap by making the wavefronts match better. (b): Calculated core coupling over particle radius for different refractive index differences. The dotted line shows the core coupling curve for the core/shell model with Δnshell = 0.03, Δncore = 0.005 and , which has a pronouncedly higher tail, allowing for discrimination between simple and more complex architectures of the trapped particles.
Treating trapped particles as phase masks is possible under two conditions.
First, their absorption and side scattering must be negligible in order to leave the amplitudes of the wave fields unchanged, i.e. they must not be amplitude objects but phase objects. Here, absorption can be ruled out: on the way through the chamber, about 0.1% of the light intensity is absorbed by the water content alone [21], already leading to significant heating [19]; any absorption significantly lowering the amplitude would immediately lead to boiling water. Next, side scattering is several orders of magnitude lower than forward scattering for cells in suspension [27], thus not significantly altering field amplitudes.
Second, light diffraction must be negligible; the phase mask approximation requires that wavefronts are changed according to the optical path only, and not due to diffraction at the entrance ’pupil’ of the particle. Diffraction starts to widen a beam from the Rayleigh range zR on; in other words, because the coupling is computed directly behind the cell, it is enough that particles be considerably smaller than their Rayleigh range to fulfill this condition:
This condition is fulfilled for the particles presented in this study. Typically, cells in suspension – with some exceptions as, e.g., erythrocytes – match this criterion.Coupling values first increase with particle size up to a peak size Rpeak ≈ 7 μm, before they decrease again, as shown in Fig. 4(b), with the exact peak position depending on the refractive index. Using lensmaker’s equation for thin lenses, setting object and image distance to z = 100 μm and relative refractive index to Δn = 0.04, maximal focusing is estimated to occur for a symmetric biconvex lens with a radius of curvature of
An ideal lens of this size would image one fiber end onto the other - the strong optical power in spite of the low refractive index difference is a result of the small radius of curvature. However, the coupling maximum is not found here because particles of this size miss important parts of the beam, which has a width of ω ≈ 8 μm in the trap region. From R > ω on, particles cover virtually the whole beam width and coupling is expected to decrease as the optical power of the particles decreases. From a ray optics perspective, a value of Rfocus < Rpeak < ω is thus expected.As it is not very convenient to measure absolute coupling powers - for instance, the quality of the fiber splice is not exactly known - all fiber coupling values presented from now on are taken relative to the empty chamber coupling value Cempty for a laser power of 100 mW. As mentioned above, Cempty ≈ 0.15, thus a beam focusing by a trapped particle leads to relative coupling values C in the range 1 < C ≲ 6.
3.3.1. The core/shell phase mask model
Treating phase masks numerically, they can be shaped in an arbitrary way in order to model phase objects more complex than homogeneous spheres. In section 4.2, we will present coupling measurements of biological cells, which are optically inhomogeneous due to their composition of cytosol, organelles and the nucleus.
In order to include inhomogeneities in a simple way, we introduce a core of higher optical density into the phase mask. Thus, the phase mask represents an inhomogeneous, concentric core/shell architecture, with the radius and refractive index increment of the core as two additional free parameters. The according phase mask is thus composed as
Results for this model are also shown in Fig. 4(b). Note that the core/shell coupling curves have a distinctly different shape than the simple coupling curves, with their tail markedly lifted; the homogeneous model is unable to predict such curves.3.4. Particle displacements
Displacements of the object’s center-of-mass from the optical axis can be caused by positional fluctuations or by residual flow of the medium, but most importantly by gravitation. The displacement Δy is given by balancing gradient force and gravitational force. The model used for calculating the gradient force is taken from [4]. For PDMS in ethanol, an excess density of ΔρPDMS = 190 kg/m3 is used, while for cells in medium a typical value of ΔρCells = 60 kg/m3 is assumed [28].
Results for gradient forces and resulting displacements of PDMS beads are shown in Fig. 5(a). The gradient force increases with particle size R until the particle covers the whole beam width, i.e. R ≈ ω ≈ 8 μm, however gravity increases as Fgrav ∼ R3. Therefore, object displacements become noticeable from R ≳ ω on, and can get up to 1 μm. For biological cells, the displacement is usually smaller due to their lower mass density.
Fig. 5 (a): Equilibrium gravitational displacement of trapped particles with an excess density of Δρ = 190 kg/m3 at a trapping power of 2 * 0.1 W. The inset shows the corresponding gradient (trapping) force exerted by the lasers. Because the trapping force decreases with particle size for R ≳ 9 μm, displacements become important for particles bigger than that. (b): Overview of modeled core coupling under the influence of the different effects (cf. Fig. 2) discussed in this study. Thermal lens effect for 2 * 0.1 W trapping power in ethanol. Gravitational displacements taken from Fig. 5(a). While the thermal lens and the displacements slightly influence height and position of the peak and the tail, the coupling peak is distinctly wider in the core/shell phase mask model.
The influence of the particle displacement on the fiber coupling is simply treated by translating the phase mask accordingly. Results are shown in Fig. 5(b).
4. Experimental results
4.1. PDMS beads
Results for the coupling signal of PDMS beads are shown in Fig. 6(a). The radii of beads were extracted from phase contrast images using an edge detection algorithm as described in [3]. Trapped beads have radii between 3 and 14 μm.
Fig. 6 (a): Dependence of the fiber coupling of PDMS beads on bead size. Crosses are experimental values. The error bars indicate the bead size uncertainty u(R) = 0.2 μm; this is estimated by the effect that repeated trapping of the same bead under slightly changed microscope focus has on the detected contour. The red line shows the prediction of the presented model, using a value of Δn = 0.0335 for the refractive index. The green dashed line shows the prediction of the simple model without thermal lens effect and gravity displacement correction, which is insufficient. The inset shows an example of a PDMS bead with its contour masked red as given by the edge detection. Scale bar is 10 μm. (b): The fiber coupling signal during a cell stretch process in an optical stretcher. After the laser switches to stretch power, the coupling signal decreases because thermal equilibration is not completely reached; further, the deformation of the cell increases the signal due to the increasing curvature of its surfaces. After the laser switches back to trapping power, the relaxations of the thermal lens and of the cell deformation overlap again.
Generally we observed only a negligible change in the coupling signal for translations of particles along the optical axis, which is also backed up by the model. Additionally, the coupling values presented throughout the manuscript are found to be symmetric with respect to the two fiber diodes.
Qualitatively, the size dependence of the coupling signal of the PDMS beads shows the expected curve shape as presented in the modeling section, for example in Fig. 5, peaking for a bead radius around Rpeak ≈ 7 μm.
This measured size dependence was then quantitatively compared to the predictions made by the model. Input values for the model are the geometry and refractive indices of the trapping region and the mode field diameter ω0 of the optical fibers (all given in section 2); further, the strength of the thermal lens effect and gravitational displacements as described in sections 3.2 and 3.4.
The simple (homogeneous) phase mask model was used, with the refractive index difference of the particle as the free parameter. In Fig. 6, the difference is ΔnPDMS = 0.0335.
The modeled coupling curve agrees very well to the measured values in the range from 2 to 9 μm, with slight deviations occurring for bigger particles (see Fig. 6). The predictions of the coupling model without the thermal lens effect and particle displacement corrections are also shown; their agreement to the experimental data is distinctly worse as the peak is too high while the tail is too low, indicating the necessity of these corrections.
The curve can be divided into three regions. For radii up to 5 μm, the fraction of unscattered light makes up more than 50% of the beam power, dominating the coupling. For radii between 5 μm and 9 μm, particle size and refractive index govern the curve shape. Finally, the gravitational displacement comes into play for radii bigger than 9 μm.
The remaining deviation for bigger particles in the corrected model may have several reasons. First, the calculation of gravitational displacements may become inaccurate for these particle sizes. Second, the influence of particles on the thermal lens is not modeled, with bigger particles possibly stronger influencing the beam. Finally, the phase mask models the particles as infinitely thin; this approximation also becomes less valid for bigger particles. Within the realm of this study, it could not be evaluated which of these factors is the dominating source of the deviation.
There are several sources of error and uncertainty. The major factor leading to uncertainty of the refractive index is the uncertainty of the mode field diameter of the optical fiber. Varying this parameter in the range given by the manufacturer, ω0 = 3.1 ± 0.15 μm, leads to variations of the best refractive index fit parameter of about 0.003. Another source of uncertainty is the bead size detection, which suffers from a possibly suboptimal microscope focus or inexact edge detection, allowing for an uncertainty of the bead size of about 0.25 μm. However, assuming that this value varies from bead to bead, these uncertainties should have a tendency to cancel each other. Varying the strength of the thermal lens effect in the possible range also leads to deviations which are small compared to the effect of the uncertainty of the mode field diameter.
Therefore, the resulting uncertainty of the measurement is given by this range:
As stated in sec. 2, we were unable to find a literature value to compare this value to, but instead we measured the refractive index using an Abbe IR-refractometer [15], leading to an absolute value of .In order to compare the two values, the refractive index of the medium, nEthanol = 1.3548 ± 0.0004 [14] must be added to the result given in Eq. (18), thus yielding a value of , measured using the new method presented here – in good agreement to the existing value.
4.2. Biological cells
4.2.1. Focusing effect during the trap and stretch phases
The dual-beam trap in our setup is mainly used as an optical stretcher, i.e. a device not only to trap cells, but also to deform them using higher laser powers (typically Ptrap = 100 mW, Pstretch ≈ 1000 mW per fiber).
In a cell stretching experiment, when the cell is stably trapped, the laser switches to stretch power for a period of typically 2 s, before it switches back to trapping power in order to observe the relaxation behavior. An example of the coupling signal of such a trap-and-stretch-process is shown in Fig. 6(b).
The coupling signal generally follows the laser pattern. However, during the stretch phase it decreases because thermal effects continue to evolve on a timescale of about 1 s [19]. At the same time, the deformation of the cell has a tendency to increase the coupling due to the increasing surface curvature; this tendency only manifests itself after the thermal effects have reached equilibrium. Thus, at the end of the stretch phase, the signal increases.
It would be desirable to use the increasing focusing as a measure for the cellular deformation, as it is done in [26], where lower laser powers are used for deformation than in our experiments. However, this is obstructed here by the thermal lens, which has an effect of similar magnitude on the coupling value as the deformation itself; consequently, this idea is not further pursued here.
Nonetheless, the coupling signal of cells during the trap phase can be used for refractometry in the same way as for the PDMS beads, as is shown in the next section.
4.2.2. Cell populations
Coupling data of two exemplary measurements of biological cells are shown in Fig. 7(a). Both samples are primary cells obtained from surgery (see section 2). The first and obvious difference to PDMS beads is that they exhibit a big variation in their coupling efficiencies at a given particle size.
Fig. 7 (a): Core coupling of 2 primary cell samples. Sample 1 is typical of many primary and cell lines measured. The core/shell model describes the averaged curve shape, while the homogeneous model cannot (not shown here). Note that in sample 2, some cells (marked by the ellipse) have distinctly lower coupling values. The inset shows a phase contrast sample image of a cell trapped in the stretcher, with the detected edge; scale bar is 10 μm. (b): Fitted refractive index (shell) values (relative to the medium), assuming a constant core value of Δncore = 0.007. In this measurement, there seems to be a subpopulation of possibly apoptotic cells in sample 2 that has a distinctly lower refractive index.
Examing the data a little closer, we first want to focus on sample 1, which is typical also for many cell lines measured in our lab (data not shown here). The moving average of the coupling signals peaks at an object size of around 8 μm, i.e. at bigger sizes compared to the PDMS peak at around 7 μm. The model predicts that with a peak shift to bigger object sizes, peak coupling values also increase (see Fig. 4(b)). However, peak coupling values are lower than in the PDMS sample, while the tail of the distribution is higher.
Thus, the homogeneous model does not match the data well, unless the cell’s refractive index strongly increases with cell size – which is not to be expected [12]. The shifted peak combined with the lifted tail rather indicates inhomogeneity, as it happens in the complex core/shell phase mask introduced in section 3.3.1, which will be used to model the coupling curves of biological cells.
By introducing a core with a radius of 0.8 times the cell radius, with a refractive index higher by an additional increment of Δcore = 0.007, one arrives at a coupling curve matching well the moving average of sample 1 (see Fig. 7(a)).
It is tempting to associate the core in our model with the cell nucleus and the shell with the cytosol, thus mimicking a basic feature of cellular structure; however, studies on nuclei are in disagreement whether its refractive index is higher or lower than that of the cytosol [29]. So far, the core/shell model is only a first attempt to model inhomogeneity, which is impossible to be further specified by our data.
Assuming the aforementioned core/shell architecture of Rcore = 0.8 Rshell and Δcore = 0.007, we use the coupling signal of each individual cell to assign a refractive index Δnshell to it. This is done by choosing a start value for Δnshell, which is iteratively changed until the modeled coupling value matches the measured value with sufficient accuracy. This way, we create refractive index distributions of cell populations.
These distributions should be taken with a grain of salt, as one cannot ensure the applicability of the core/shell architecture to the individual cell.
Compared to sample 1, the cells of sample 2 exhibit a different coupling behavior, with a big fraction of cells having distinctly lower coupling values (marked by the ellipse in Fig. 7(a)). Consequently, these cells are assigned a lower refractive index by our method. A histogram comparing the refractive index distributions (relative to the medium) of the two cell samples is shown in Fig. 7(b). This representation now hints at a new insight: sample 2 is composed of two subpopulations whose refractive indices cluster around two values; one close to the refractive index of sample 1, and the other value close to zero.
The mean refractive index increment of sample 1, according to this method, is
With the index of the medium nmedium = 1.325, an absolute value of ntotal = 1.357 results.5. Conclusion
Starting from the simple ray-optics picture of optically trapped particles acting like lenses, we have investigated this effect in a dual-beam optical trap. It has been shown for the stretching forces that full solutions to Maxwell equations do not differ considerably from ray optics estimates [5], thus suggesting the validity of ray optics approximations for this type of trapping experiment. Consequently, in our case a simple approach using Gaussian beams and phase masks turned out to be sufficient to describe the measured coupling signal over a wide range of particle radii (3 – 14 μm).
The simplicity of the model in turn allowed to discriminate and to grasp the influence of a range of effects on the beam propagation and interaction. Besides the lensing effect, the thermal lens effect was discussed, which has so far mostly been ignored in dual-beam traps. Further on, the gravitational displacement of the particles in the trap turned out to have a small, but measurable influence on the fiber coupling signal. These displacements were estimated using force predictions from a study by Imbert and Roosen [4], also using ray optics.
Having understood these effects, we succeeded in predicting the particle size dependence of the fiber coupling for the case of homogeneous PDMS beads. The coupling signal can thus be used as a means to measure the refractive index of (homogeneous) microparticles.
The situation turned out to be more difficult for inhomogeneous particles, like suspended biological cells. We show that in spite of the simplicity of our approach, the inhomogeneity leaves a trace in the coupling curves. The phase mask approach allows for simple introduction of more complex internal particle architecture. Assuming a core/shell structure – albeit it cannot definitely be assigned to the cellular nucleus/cytosol structure – allows the coupling behavior of a cell population to be described. This also allows to estimate the refractive index of single cells measured in an optical stretcher.
We want to stress that it is important not to draw any conclusion from the fact that one cell sample is from cancerous tissue, while the other is not. The anomaly present in sample 2 has happened in a fraction of primary cell measurements in our lab; so far, we do not know about its significance. In this context, the two samples just serve to show that strong differences between their optical properties can be detected.
While the core/shell structure works in this context, it leaves some questions open. It is not possible to determine a correct value for the core diameter; more generally spoken, the model parameters are underdetermined by the experiment. Completely different models of inhomogeneities, such as randomly distributed substructures within the particles, would also be conceivable. Thus, a follow-up study addressing the validity of the refractive index values for different cell types, the biological relevance of the core/shell structure, and its potential use as a cell-marker is highly interesting.
Nevertheless, the presented example of primary cell samples highlights the importance of monitoring cellular optical properties in a dual-beam trap. The strongly different refractive index distributions of the two cell samples point to the strength of this method, which lies in the fact that especially when working with primary samples, it is impossible to rely on literature values of optical properties. Instead, live monitoring is required for deeper insight into cellular properties and further to calculate meaningful optical forces. In conclusion, we presented an inherent, simple and effective addition to an optical stretcher system to assess single cell optical properties.
Acknowledgments
The project was funded by the Leipzig graduate school of Natural Sciences – Building with Molecules and Nano-objects “BuildMoNa” of the Universität Leipzig; by the state of Saxony ( “Initiative des Sächsischen Landtags zur Förderung von Forschungsprojekten im Bereich Biotechnologie und Lebenswissenschaft 2013/14”); and by the Deutsche Forschungsgemeinschaft (KA 1116/9-1).
We thank Prof. Dr. Dr. Michael Höckel and Prof. Dr. med. Lars Christian Horn of the Universitätsklinikum Leipzig for supplying primary cell samples. We further thank Moritz Kreysing for fruitful discussions and Bernd Kohlstrunk for technical support.
References and links
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