1. Introduction

A class of lab-on-a-chip devices for optical analysis of particles is based on using integrated photonic waveguides for manipulating and spectroscopic fingerprinting the particles, with the goal of their identification. Possible application fields are various, and include point-of-care diagnostics and drinking water technology [1], with a focus on, for example, detection of human cells indicative of a disease and harmful bacteria. Examples in this field are labs-on-a-chip for optical trapping and Raman spectroscopy. These are on-chip versions of a laser-tweezers Raman spectroscopy (LTRS) setup [2]. LTRS unifies particle trapping with a focused laser beam and Raman spectroscopy of the trapped particle using the same concentrated light. This leads to Raman identification of the trapped particle. Prominent examples of on-chip versions are dual-fibre or dual-waveguide traps, often combined with microfluidics for particle supply. These traps are based on two opposing fibres or integrated photonics waveguides, from which counter-propagating beams emanate that define the concentrated optical field for trapping and Raman spectroscopy.

The dual-fibre trap in [3] has been used to hold and manoeuvre human cells to record their Raman signals. The trap in [4], apart from fibres for trapping and Raman generation, has fibers for collecting Raman signals. A preference is arising towards dual-waveguide devices based on microfabricated waveguides, thus avoiding delicate mounting of fibres and aiming for mass production. An important step in this direction is reported in [5,6], where the waveguides are created by local modification of the index of glass using accurate writing with a femtosecond laser beam. The resulting devices can successfully trap and stretch red blood cells (RBCs). The laser-writing technique is inherently sequential, making it particularly suitable for fast prototyping. We further mention the traps reported in [7,8], based on microfabricated Ta₂O₅ waveguides. In [7] trapping of synthetic particles and RBCs is demonstrated, with supply of particles to the trap by optical propulsion on the waveguides. In [8] Raman spectra of trapped synthetic spheres are reported, but the spectra are induced using a separate laser beam and not by the beams emitted by the waveguides. In our work [9], we have exploited the counter-propagating beams in a dual-waveguide trap built from special waveguides microfabricated with semiconductor processing techniques to their full capability, using the beams both for trapping synthetic particles and for inducing Raman signals from the trapped particles.

Recently, LTRS has been applied for studying human extracellular vesicles (EVs) [10–12]. These cell-derived particles are present in bodily fluids such as urine and have a diameter ranging from 30 to 1000 nm, the size-distribution’s maximum occurring below 200 nm. An EV comprises a lipid-bilayer membrane of the parental cell, enclosing mainly cell-derived cytoplasm with suspended DNA, RNA and proteins. Tumor-derived EVs strongly attract attention as potential biomarkers for cancer [13,14]. For cancer diagnosis such EVs may play a role similar to that of circulating tumor cells (CTCs [15],). The bio-chemical composition of a fraction of EVs may already deviate from regular in a very early stage of carcinogenesis of the organ from which the EVs derive [16], while presence of CTCs indicates full existence of organ cancer. Thus, determining EV composition using LTRS can lead to early cancer diagnosis [17]. However, only a small sub-population of EVs in bodily fluids shows a signature of carcinogenesis, if present. This circumstance asks for high throughput platforms, e.g. lab-on-a-chip devices scaled to versions with multiple parallelism [18]. Further, EVs are very small and have a low index contrast w.r.t. the suspending medium, making on-chip optical trapping of EVs challenging.

Here, from a simulation study, we present the capabilities for trapping of EVs of the dual-waveguide trap we apply in [9]. The novelty compared to [9] is that we here simulate for this type of trap the full range of optical and trapping properties, leading to qualification for the characterization of EVs. Further, our simulation results provide guidance for design and development of dual-waveguide devices for optical trapping and Raman spectroscopy of a broad range of bio-particles relevant for health and medicine.

First, the device geometry and two index models for EVs are presented, followed by a description of the simulation approach. Then, we detail the photonic ingredients of the traps, viz. the waveguide mode, the emitted on-chip beam and the energy density in the trap. Subsequently, the force curves for three trap sizes and the index models yield the trapping potentials. These in turn, via Ashkin’s criterion [19], lead to the conditions for stable EV trapping, which constitute the central results of this work. It follows that very small EVs can be trapped with the dual-waveguide traps, highlighting the clinical relevance of this lab-on-a-chip as a basic device for high throughput platforms.

2. Device geometry and optical models for EVs

The building block of the dual-waveguide trap is a waveguide with a box-shaped composite Si₃N₄−SiO₂ structure. Its cross section of 1 × 1 µm² is shown in Fig. 1(a). The walls of the square box are 50 nm thick Si₃N₄ layers. The material inside and outside the box is SiO₂. With this structure we model the slightly trapezoidal TripleX [20,21] waveguides we apply for the dual-waveguide trap in [9].

 

Fig. 1 (a) Cross section of the composite SiO₂−Si₃N₄ waveguide, the building block of the dual-waveguide trap. The material inside and outside the square box with Si₃N₄ walls is SiO₂. (b) Three-dimensional impression of the dual-waveguide trap. The waveguide facets, with separation w, are part of the walls of the fluidic channel. The origin of the coordinate system is the trap centre, where in the figure an EV is located with a core-shell structure.

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The geometry of the dual-waveguide trap built from a pair of such waveguides is depicted in Fig. 1(b). The waveguides are opposed and guide light to a fluidic channel, used for supply of EVs. The waveguide facets, which emit a narrow light beam, are part of the walls of the fluidic channel. The counter-propagating beams created in this way define a strongly confined optical field and thus form an optical trap. The gap width w between the waveguide facets is 5, 10 or 15 µm, equal to the widths in the devices leading to the results in [9]. The trap centre is the origin of the coordinate-system (see Fig. 1(b)). The x-direction is the longitudinal direction and the y- and z-directions are the transverse directions.

EVs are spherical particles [22], with a diameter $d_{\text{EV}}$. We represent their optical properties by two models for the refractive index $n$. The first model assumes a homogeneous EV index, for which we take $n_{\mathrm{hom}}=1.37$ [23]. In the second model we apply the realistic core-shell structure of EVs, using $n_{\text{core}}=1.37$and $n_{\text{shell}}=1.46$ [24], and taking a shell thickness of 5 nm. The homogeneous index can be accurate enough for big vesicles since these have a small surface-to-volume ratio. The core-shell model will be more appropriate for small vesicles since for these the shell contribution to the total optical force is expected to be rather high.

3. Simulation approach

We derive the electromagnetic fields in the traps with the 3D finite-difference time-domain (FDTD) method, using Lumerical FDTD solutions [25]. The first goal is to obtain the optical forces acting on an EV, as a function of its position on the three axes defined in Fig. 1(a), and for the 10 or 15 µm traps on other axes parallel to the y-axis. This is done by simulating traps containing an EV and empty traps. The fields with the EV in the trap are needed for Lumerical’s force calculations. The fields for the empty trap are used to independently calculate the forces for comparison with Lumerical’s method, as discussed below.

The TE mode obtained from Lumerical’s mode solver is excited in each waveguide, for wavelength ${\lambda }_0=785\text{nm}$. This is our preferential wavelength [9], in view of generating Raman signals from the trapped particle. We use planar sources, placed at 2 µm from the waveguide facet, that oscillate in phase and emit a short Gaussian pulse of carrier frequency $f=c/{\lambda }_0$, the frequency corresponding to the mode’s wavelength. The time traces of the electromagnetic fields at each mesh point are Fourier transformed to obtain the spatial dependence of the complex amplitudes of the harmonic fields for ${\lambda }_0$. This procedure yields the beams emitted from the facets, which according to the TE polarization have $\overrightarrow{E}$//$\widehat{z}$.

The mesh size must be small enough to resolve the fields, on the scale of i) the wavelength in the highest index material, ii) the smallest structure of the device and iii) the details of the EV structure. The simulation time, however, is inversely proportional to the mesh size to the fourth power, implying a practical limit. To fulfill the mesh-size requirements, while keeping simulation time within reasonable bounds, we subdivide the total simulation domain into three subdomains of different mesh size. The total simulation domain is 17 and 11 µm in the y- and z-direction, respectively. The length in the y-direction exceeds that in the x-direction for the TE polarization used, in view of the side lobes emitted away from the beam axis (see Section 4). In the x-direction the simulation domain is $w+5.4$ µm. At the domain’s boundaries we use 2 µm thick perfectly matched absorbing layers.

The first subdomain is a rectangular block of cross section 2 × 2 µm², enclosing the waveguides. It has mesh size ${\Delta }_1=10\text{nm}$, imposed by the Si₃N₄ waveguide walls. The block length equals the total domain’s length in the x-direction. The second subdomain is a cube of edge $d_{\text{EV}}+10{\Delta }_2$ surrounding the EV. For the homogeneous index model, we choose ${\Delta }_2=d_{\text{EV}}/20$, with an upper limit of 10 nm. For the core-shell model we take ${\Delta }_2=2\text{nm}$, in view of the thin EV shell. The third subdomain is the remainder of the total domain, for which we take ${\Delta }_3={\lambda }_{\text{0}}/(14n_{\text{SiO2}})$. In simulating the empty trap, we use ${\Delta }_4=10\text{nm}$ in the first and second subdomain and ${\Delta }_3={\lambda }_{\text{0}}/(14n_{\text{SiO2}})$ in the third subdomain.

We obtain the time-averaged force $<\overrightarrow{F}>$ acting on an EV using Lumerical’s volumetric technique, by calculating the volume integral of the time-averaged Lorentz force density for a cube enclosing the EV. The edge of the integration cube is $d_{\text{EV}}+7{\Delta }_2$, some smaller than the cube’s edge that defines the sub-meshing. The Lorentz force density $<\overrightarrow{f}>$ is calculated at each mesh point inside the cube and on its faces using:

To calculate the forces for the smaller EVs with an independent approach, enabling comparison with Lumerical’s method applied to EVs with an assumed homogenous index, we use the expression for the so-called time-averaged gradient force acting on a homogeneous dielectric particle, here an EV [26]:

The gradient force of Eq. (2) inherently is a conservative force. Consequently, in the approximation of negligible scattering force, a unique value of the trapping potential can be assigned at any point by integrating the force along a suitable path. We anticipate that the forces of the dual-waveguide trap obtained with the volumetric technique are rather close to the gradient force and thus can be considered as conservative. Therefore, we derive the position dependent optical potential by integrating the force obtained with the volumetric technique. For example, the transverse potential $U_y(x_0,y_0,0)$ in the point $y_0$ for an axis parallel to the y-direction through the point $(x_0,z=0)$ ($x_0,y_0\ge 0$, the negative branches follow from symmetry) is given by:

Calculating the potential for three axes, yields the main information on the potential wells. For stable trapping of an EV the potential well should be deep enough. For this we use Ashkin’s stability criterion, according to which trapping in a potential well of depth $U_0$ is stable for $U_0/kT\ge 10$ [19]. Since the well depth is proportional to the optical power supplied to the trap by the waveguides, the criterion leads to a condition for the optical power.

4. Waveguide mode, emitted beam, and energy density in the trap

We have calculated the TE mode of the waveguide of Fig. 1(a) using Lumerical’s mode solver for $\lambda =785$nm. At this wavelength the waveguide supports a single mode per polarization. We use refractive indices of 2.00, 1.45 and 1.33 for Si₃N₄, SiO₂ and water, respectively. The mode profile is shown in Fig. 2(a), where the colors represent the intensity of the light. As can be seen, for this polarization the electric field concentrates in the upper and lower Si₃N₄ walls of the box. Further, the field is relatively low in the interior of the box (SiO₂), quite contrary to the case of usual homogeneous ridge waveguides. The electric field drops abruptly upon entering the vertical walls of the box from inside or outside, satisfying the boundary condition for the normal component for the dielectric displacement.

 

Fig. 2 (a) Profile of the TE mode of the waveguide in Fig. 1(a), for a wavelength of 785 nm. The colors indicate intensity, red (blue) meaning high (low), and do not relate to the scale bar to the right. (b) Energy density of the optical field in the xy-plane, both in the waveguide and the water, obtained from a FDTD simulation. The horizontal dashed line is the beam axis. Vertical dashed lines indicate the facet position and are the transverse lines for which in (d) beam profiles are plotted. The color bar defines the energy-density values. (c) Longitudinal energy-density profile on the beam axis. (d) Transverse beam profiles on the axes in (b).

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FDTD simulation leads to the energy-density distribution in the xy-plane in Fig. 2(b), both in the waveguide and the water, using a source in the waveguide emitting 1 W of power towards the facet. At the facet the guided mode is converted into radiation modes that enter the water present for $x\ge 0$. The oscillatory pattern inside the waveguide and its walls represents a standing wave resulting from interference of the forward propagating mode and its part reflected back from the facet. Near the facet, the radiation modes take the form of two coherent beamlets emitted by the upper and lower waveguide walls. These beamlets merge, yielding a main beam (further called beam) coaxial with the waveguide axis, which co-exists with two side lobes directed away from the axis. The beam reaches its energy-density maximum on the axis at 1.2 µm from the facet, whereafter it decays smoothly and simultaneously broadens. The maximum and the decay of the beam are quantified by the longitudinal energy-density profile plotted in Fig. 2(c). The transverse beam profiles in Fig. 2(d) further quantify the beam, in particular its spreading with increasing distance to the facet. The transverse profiles show shoulders at distances of 1.2 and 2.5 µm that arise from the aforementioned side lobes. We have compared the beam in Fig. 2(c) with the one we report in [29] for a Si₃N₄ solid core waveguide of the same cross section. We find that the decay length of the present beam is about three times longer and the beam spreading correspondingly weaker, meaning that the composite Si₃N₄−SiO₂ waveguide produces a tighter beam and thus is a better waveguide to build a strong optical trap.

For three traps built from the waveguide of Fig. 1(a) with facet separations of 5, 10 and 15 μm we have simulated the optical fields resulting from the coherent counter-propagating beams. In Fig. 3 we show time averaged energy-density distributions in xy-plane of the traps. A total power of 1W is supplied to the traps, each waveguides contributing 0.5 W. We omit the distribution in the xz-plane, since it portrays the same type of behavior.

 

Fig. 3 Time averaged energy density in the yx-plane for dual-waveguide traps with a facet separation of 5, 10 and 15 μm. The total power supplied to each trap is 1 W. The 5 μm trap has one global hot region centred between the facets. The 10 and 15 μm traps have two global hot regions, each near a facets. The color bar defines the energy-density values.

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A common characteristic of the traps is the interference pattern between the facets, which has a period of 306 nm. The EV size distribution is such that a small EV can be locally trapped at a single maximum of the interference pattern, here further called hot spot. But also multiple small EVs may be trapped at one hot spot, similar to trapping of multiple EVs with a laser trap [10]. A hot spot is narrower in the longitudinal than in a transverse direction. Therefore, the confined Brownian motion of an EV has wider excursions in a transverse than in the longitudinal direction.

The three traps, however, also show an important difference concerning the position of the regions where the interference maxima are strongest, i.e. the so-called global hot regions. For the 5 μm trap a single global hot region exists (see Fig. 3), symmetrically placed aroundthe center. The 10 and 15 μm traps, on the contrary, have two global hot regions near the facets. This difference results from the longitudinal profile of the interfering beams shown in Fig. 2(c), in particular their decay and the position of their maximum. These are such that the central global hot region of the 5 μm gap is split into two global hot regions for the wider gaps. We note that the trap built from Si₃N₄ solid core waveguides [29] already shows two global hot regions for facets 5 μm apart. This agrees with the weaker beam decay in the present traps.

5. Force characteristics of the traps

We have calculated the longitudinal force curve $F_x(x)$ and the transverse force curves $F_y(y)$ and $F_z(z)$ in the traps. For up to about two dozen diameters of homogeneous EVs in the range 50−1000 nm and for 6 diameters of core-shell EVs in the range 50−450 nm such curves have been obtained, of which a fraction is shown in Fig. 4. Forces acting on homogeneous EV have also been calculated using Eq. (2), giving the gradient force. Gradient-force data are included in Fig. 4(a), 4(b) and 4(c) for $d_{\text{EV}}$ = 100 nm (black dots), but such data have been obtained for other diameters as well. We concentrate on the 5 and 15 μm traps, since qualitatively results for the 10 μm trap are very similar to those of the 15 μm trap.

 

Fig. 4 Force curves for the 5 and 15 µm traps, calculated using the volumetric technique. The black dots have been obtained using Eq. (2). The legends in (a) define EV size and EV index model (homogeneous vs. core-shell) and apply to each panel. (a) Transverse force $F_y(y)$ for the 5 µm trap. (b) Transverse force $F_z(z)$ for the 5 µm trap. (c) Longitudinal force $F_x(x)$ for the 5 µm trap. (d) Longitudinal force $F_x(x)$ for the 15 µm trap, with part of the x-axis omitted to obtain the same horizontal scale as in (a-c). The curves in (a-d) are piecewise cubic spline interpolations between the data points. $F_y(y)$ and $F_z(z)$ curves have 15 data points. $F_x(x)$ curves have 33 data points per µm.

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In Figs. 4(a), 4(b) and 4(c) curves $F_y(y)$, $F_z(z)$, and $F_x(x)$ for the 5 μm trap are shown, respectively, for diameters as indicated and in Fig. 4(a) and 4(b) for both index models. The unit of force is pN/W, i.e. the force is normalized to the power delivered to the trap. In Fig. 4(d) $F_x(x)$ is shown for the 15 μm trap, restricted to homogeneous EVs of the same diameters as in the other sub-figures. In view of symmetry, the plots are limited to the positive half axes. Transverse force curves for the 15 μm trap are omitted, since qualitatively these are similar to those of the 5 μm trap in Fig. 4(a). In condensed form, the properties of the three traps are quantitatively reflected by trap stiffnesses in Table 1 below and by stability curves in Section 7.

Table 1. Normalized trap stiffnesses ${\kappa }_x$ and ${\kappa }_y$ of the strongest trapping sites for the three traps for diameters of core-shell EVs as indicated. The quantities $d_{\text{EV}}$, w, ${\kappa }_{x(y)}$ denote EV diameter, facet separation of the trap, and stiffnesses for the x(y)-direction, respectively.

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_{$F_y(y)$}for the 5 µm trap is linear near the origin and has a negative slope. This reflects the restoring character of a Hookean force pulling back the EV to the hot spot at the origin, a behavior described by $F_y(y)=-{\kappa }_{y}y$, with ${\kappa }_y$ the normalized stiffness for the y-direction. Normalized stiffnesses for the other directions are defined accordingly. With increasing distance $|F_y(y)|$ reaches an absolute maximum, followed by a local maximum. The absolute maximum relates to the maximum gradient of the beam in Fig. 2(b). The local maximum relates to the side lobes in that figure. For $d_{\text{EV}}$ = 1000 nm, the local maximum is less pronounced, since the EV for this size overlaps with both the beam and the side lobe, making their individual contributions less distinct. This behavior of $F_y(y)$ applies to both index models, but for small diameters the force on core-shell EVs (curves shown for $d_{\text{EV}}$ = 100, 150 nm) is considerably higher than that for homogeneous EVs. For $d_{\text{EV}}$ = 50 nm, $F_y$ on average is 110% higher for the core-shell EV than for the homogeneous EV, while for $d_{\text{EV}}$ = 250 nm the percentage still is 25%. For $d_{\text{EV}}$ = 500 nm the forces for the two models are virtually the same. Apparently, for the smaller diameters the shell, though only 5 nm thick, contributes significantly to the force and thus cannot be omitted in the EV model.

As can be seen in Fig. 4(b), $F_z(z)$behaves very much the same as $F_y(y)$, the differences being that $|F_z(z)|$ is some smaller than $|F_y(y)|$, accompanied by a slightly less steep initial slope, that the maximum of $|F_z(z)|$occurs at a slightly larger distance and that a local minimum of $|F_z(z)|$ is absent. These differences can be traced back to a less tight beam in the xz-plane and absence of side lobes in that plane. $F_y(y)$ and $F_z(z)$ being so close, we will further refer to $F_y(y)$ as the transverse force.

_{$F_x(x)$} is an oscillatory function, as exemplified in Figs. 4(b) and 4(c) for the 5 and 15 µm traps. The oscillation period equals the period of the fields in Fig. 3, implying that the origin is interference. As a function of EV size, a hot spot of the interference pattern attracts or repels the EV, depending on whether the local force-curve slope is negative or a positive. In Figs. 4(b) and 4(c) attraction occurs for 100, 150 and 950 nm, while repulsion occurs for 450 and 1000 nm. This sign reversal of the force agrees with [30], which reports that the sign in a standing wave trap depends periodically on particle size and that the particle’s equilibrium position alternates between interference maxima and minima. Further [30], particle sizes exceeding the interference-pattern period exist for which the net optical force is zero as a result of complete cancellation of attractive forces of neighbouring hot spots. For the 1000 nm EV this cancellation almost occurs, giving weak oscillations in Figs. 4(b) and 4(c). EVs clearly smaller than the interference-pattern period are attracted by only a single hot spot, giving an amplitude of $F_x(x)$ considerably higher than the maximum of $|F_y(y)|$. This results from the spot’s stronger intensity gradient in the x-direction than in the y-direction. This difference in force strength for the two directions can clearly be seen from the curves for 100 and 150 nm EVs in Figs. 4(a) and 4(c).

The global hot region of the 5 µm trap is centered at $x=0$. Therefore, for the smaller EVs the force oscillations in Fig. 4(b) are weaker towards the facets. The hot spots near the origin thus are more preferential trapping sites, and one may expect the type of hopping motion we report in [9] towards the center of the global hot region. For the 1000 nm EV the situation is different, since for this size the less intense hot spots near the facet lead to increasingly weaker cancellation near the facet. Preferential trapping is still at the origin, in view of the force direction imposed by the background, on which the oscillations are superimposed.

For the 15 µm trap the global hot regions are located near the facets. For most of the diameters in Fig. 4(c) this leads to stronger oscillations with increasing distance from the center and thus to stronger local traps near the facets. For the 1000 nm EV the changed position of the global hot region leads to weaker oscillations towards the facet, followed by a few strong oscillations as a result of entering the relatively “cold” on-axis region very close to the facet (see Fig. 3).

We now compare the forces obtained using the volumetric technique and the gradient formula Eq. (2). In Figs. 4(a) and 4(b) it is seen that the results for $d_{\text{EV}}$ = 100 nm agree very well. The relative root-mean-square deviation between the forces according to the models amounts to 7% for $F_y(y)$ and 11% for $F_x(x)$. For $d_{\text{EV}}$ = 50 nm (curve not shown) these differences are 2.9% and 2.5%, respectively. These differences indicate that the scattering force in this size range is only a small fraction of the total force. This results from the counter-propagation of the beams of low divergence. For larger EVs the difference increases, indicating break down of the dipole approximation, on which the gradient formula is based.

In Table 1 the values of the normalized trap stiffnesses ${\kappa }_x$ and ${\kappa }_y$ for the traps are compiled for 50, 100 and 150 nm core-shell EVs and for the strongest trapping sites, i.e. for the most intense hot spots of the respective global hot regions. The forces for the y- and z-direction being very comparable, ${\kappa }_y$ is taken as representative for ${\kappa }_z$. The Table shows that both ${\kappa }_x$ and ${\kappa }_y$ decrease with increasing facet separation. This arises from the decreasing maximum constructive interference of the beams. Further, all ${\kappa }_x$ strongly exceed the related ${\kappa }_y$ owing to the much stronger gradients in the x-direction. Finally, the systematic increase of ${\kappa }_x$ and ${\kappa }_y$ with EV size for each trap agrees with the diameter dependence of the power required to trap core-shell EVs (Section 7).

We compare the stiffnesses in Table 1 to those in [31], which reports trapping experiments on synthetic core-shell EVs using a focussed laser beam. For EV diameters in the range 50-150 nm lateral stifnesses in the range 1-7 fN/nm/W are reported [31]. Our ${\kappa }_x$values in Table 1 clearly exceed these laser-trap values, while our ${\kappa }_y$ values are smaller. Since for a laser trap the axial stifness is several times smaller than its lateral stiffness, it follows that the important difference and advantage of the dual-waveguide trap for trapping of these nanometer-scale EVs is the up to ten times higher stiffness in the x-direction. This results from the interference of the beams.

6. Trapping potentials

We have integrated the force curves to obtain the trapping potentials, as prescribed in Section 3. From the forces in Fig. 4, a single potential well is expected for the transverse directions and a periodic potential for the longitudinal direction. Indeed, such behavior is seen in Fig. 5, which shows the potentials $U_y(y)/kT$[at $(x,z)=(0,0)$] and $U_x(x)/kT$ [at $(y,z)=(0,0)$] derived from the force curves in Figs. 4(a), 4(c), and 4(d) for a power of 100 mW supplied to the trap, again using half axes. The potentials are normalized to $kT$ (T = 293 K) to easily connect to the criterion for stable trapping $U_0/kT\ge 10$ [19].

 

Fig. 5 Normalized trapping potentials for the 5 and 15 µm traps for 100 mW of optical power supplied to the traps, calculated from the force curves in Fig. 4. The legends in (a) define EV sizes and the EV index model (homogeneous versus core-shell) and apply to each panel. (a) transverse potential $U_y(y)/kT$ for the 5 µm. (b). Longitudinal potential $U_x(x)/kT$ for the 5 µm trap. (c) Longitudinal potential $U_x(x)/kT$ for the 15 µm trap, with part of the x-axis omitted to obtain the same horizontal scale as in (a) and (b).

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In the plot of $U_y(y)/kT$ in Fig. 5(a) it is seen that the 5 µm trap cannot stably trap 100 and 150 nm homogeneous EVs in the y-direction at 100 mW. The same holds for the 100 nm core-shell EV. The 150 nm core-shell EV, however, is almost stably trapped, implying that above 150 nm core-shell EVs are all stably trapped at this power level. The potentials for homogeneous EVs of $d_{\text{EV}}$ = 450, 950 and 1000 nm each indicate stable trapping.

_{$U_x(x)/kT$}of the 5 and 15 µm traps shows periodic oscillations of a strength and phase depending on EV size, superimposed on a background potential that depends on EV size as well. Background potential with superimposed oscillations are called global potential. For the smaller EVs the individual wells of the periodic potential are not deep enough for stable trapping at 100 mW. Thermally activated hopping between wells will occur. For $d_{\text{EV}}$ = 450, 950 nm the individual wells are deep enough for stable trapping, while for these diameters the background potential is more apparent. For $d_{\text{EV}}$ = 1000 nm the oscillations are virtually absent, in agreement with the force-cancellation in Fig. 4. Considering the smoother global potentials in more detail and using symmetry w.r.t. x = 0, one recognises a single potential well centred at the origin for the 5 µm trap and a double-well potential for the 15 µm trap (from the potential levels at either side of the cut in Fig. 5(c), one easily sees that the single-sided curves indeed define a double-well potential). These different potential shapes agree with the respective global hot regions for the three traps.

In calculating the potentials we have applied Eqs. (3) and (4), thus stretching the range where the optical force to a good approximation is found to be conservative (up to $d_{\text{EV}}$ = 100 nm, see Section 5) to $d_{\text{EV}}$ = 1000 nm. To estimate the error we make in this, we have calculated for $d_{\text{EV}}$ = 1000 nm and the 5 and 15 µm traps the line integral of the optical force along different escape paths from the trap, starting at the origin. The maximum difference we find for the paths amounts to 10%. We believe this is acceptable, also since this is a worst case scenario: with decreasing EV size the relative contribution of the scattering force becomes smaller, i.e. the approximation of conservativity becomes better.

7. Trapping stability

In line with Ashkin’s criterion, the minimum power $P_{\min }$ for stable trapping of an EV is defined by the demand that the depth of the normalized potential well in the three directions equals 10. We first make this demand explicit for the three potential wells obtained.

The transverse potential well is the well $U_y/kT$defined by curves as in Fig. 5(a). We take the transverse axis through the x-position of the most intense hot spot, which for the 5, 10 and 15 µm traps is at x = 0, ± 3.6 and ± 6.3 µm, respectively. In Fig. 6(a) $U_y/kT$ is plotted for stable trapping in the 5 µm trap. We define the maximum excursion during confined Brownian motion as the distance from y = 0 to either of the knee points of the curve (indicated by arrows in Fig. 6(a)), which is located closer to y = 0 than the classical turning points at $U_y/kT$ = 10. So, we avoid the strong influence of the very flat wings of the well. We thus get a single-sided excursion of 1.0 µm. The overall picture for $U_z/kT$ is similar.

 

Fig. 6 (a)-(c) Potential wells for the 5 µm trap in case of stable trapping under conditions as indicated above the panels. (a) The transverse well $U_y/kT$. The indicated knee points limit the confined Brownian motion, as discussed in the text. (b) The deepest local well of $U_x/kT$. (c) The global well $U_x/kT$. (d)-(f) Stability curves for the three wells as mentioned in the panels. The legends in (d) define the trap sizes and the EV index model (homogeneous versus core-shell) and apply to each of the stability curves in (d), (e) and (f).

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The deepest local well of the longitudinal potential is, depending on EV size, the deepest local well of $U_x/kT$ at the most intense hot spot or in between the two most intense hot spots (see Fig. 5(b) for these situations). In Fig. 6(b) $U_x/kT$ is plotted for stable trapping in the central local well of the 5 µm trap. Local well depths near the centre being close, this case also represents stable trapping near the center. The single-sided maximum excursion is about 150 nm (i.e., half of the modulation period). Each of the local trapping wells near the center can be filled with one or more EVs. Access to each of these traps can be provided by translating the chip under the microscope objective for Raman spectroscopy or by shifting the interference pattern by applying a phase difference to the beams forming the trap using on-chip heaters.

The global longitudinal potential well $U_x/kT$ is defined as the background potential with superimposed oscillations. EV trapping in the global potential is only fundamentally different from trapping in an individual well if the potential’s oscillatory part is weak compared to the background. If not, the EV can still spend considerable time in a single local well or EV motion is dominated by hopping between such wells. With some arbitrariness, we translate this to the additional demand that the ratio of the deepest local well to the global well depth of 10 is smaller than 0.6.

For the 5 µm trap we find $P_{\min }$ by making the well depth equal to 10 for classical turning points at x = ± 1.9 µm, where the largest EV does just not touch the facet. This also defines the maximum excursion of 1.9 µm. The demand of a weak enough oscillatory potential leads to the condition $d_{\text{EV}}\ge$300 nm. In Fig. 6(c) by $U_x/kT$is plotted for stable trapping in the global well of the 5 µm trap.

For the 10 and 15 µm traps $P_{\min }$ is chosen such that the barrier height between the global wells of the double-well potential amounts to 10. For these traps the minima of the individual wells coincide with aforementioned positions of the most intense hot spots. Taking half of the maximum double-sided excursion for these asymmetric wells, we get the maximum single- sided excursions 1.8 (3.1) µm for the 10 (15) µm trap. The demand of a weak enough oscillatory potential leads to the condition $d_{\text{EV}}\ge$300 (400) nm for the 5 (10 and 15) µm trap, while for the 10 µm traps the range 500−650 nm is excluded.

Applying the demand of stable trapping to the three wells leads to the stability curves in Figs. 6(d)-6(f) for the traps. A stability curve is a plot of $P_{\min }$ versus $d_{\text{EV}}$. Points on and above a stability curve define stable trapping, while for points below a curve an EV cannot be stably trapped. Figures 6(d) and 6(e) are for both index models. Figure 6(f) is for the homogeneous index.

Figure 6(d) for $U_y/kT$ shows that $P_{\min }$ at fixed $d_{\text{EV}}$ increases with increasing trap size, the curves for the 10 and 15 µm traps being close. This agrees with the decreasing energy density of the most intense hot spot with increasing trap size in Fig. 3. For small homogeneous EVs with $d_{\text{EV}}\le 150$nm the curves are straight lines of a slope very close to −3. Thus, we approximately have $P_{\min }\propto 1/d_{\text{EV}}^3$, in agreement with Eq. (2) and therefore with a dominating gradient force. For core-shell EVs with $d_{\text{EV}}<150$nm, we find $P_{\min }\propto d_{\text{EV}}^{-2.56}$, while for each trap the core-shell curve lies below the curve for the homogeneous EVs. These properties of the core-shell curves agree with the stronger force acting on core-shell EVs than on homogeneous EVs. Beyond this range, the curves of core-shell EVs asymptotically approach those of homogenous EVs. For 450 nm the effect of the core can still be seen in Fig. 6(d).

In Fig. 6(e), for small $d_{\text{EV}}$, the curves for the deepest local well of $U_x/kT$ show similar behavior as in Fig. 6(d). Different from Fig. 6(d), however, and very prominent are the strong and broad peaks at $d_{\text{EV}}\approx$420, 700 and 1000 nm. For these diameters the oscillatory potential’s amplitude is close to zero due to the cancellation effect. As a result, high powers are needed to create deep enough wells for stable trapping.

In Fig. 6(f) the main features are the residues of the peaks in Fig. 6(e). The weakness of these features results from the dominance of the background potential in its interplay with the oscillatory potential. Further comparison of Fig. 6(f) with Fig. 6(e) shows that for each trap the $P_{\min }-d_{\text{EV}}$ curve for trapping in the global well lies below the curve for trapping in the local well. Thus, when increasing the power at fixed $d_{\text{EV}}$, trapping is first is realized in a global well, followed by trapping in a local well when the power hits the stability curve for local trapping. Finally, by considering the various global potential wells in detail, it is seen that for the 5 µm trap the effect of the oscillations is smallest, as their amplitude is negligible at the classical turning points. For the 10 and 15 µm traps the features are stronger due to relatively strong oscillations present at the barrier of the double-well potential. The curve of the 5 µm trap is below those of the other traps. This relates to energy densities in Fig. (3).

In Figs. 6(d)-6(f) $P_{\min }$ covers the range from 1 mW at 1000 nm to 10 W at 50 nm (neglecting the peak values around 700 and 1000 nm). For Raman spectroscopy limited EV excursions in the trap are prefered, suggesting trapping in a local well of $U_x/kT$. 300 mW can be supported by the glued butt coupling of a fiber to the waveguide facet at the chip edge and by the waveguide itself [32]. The maximum coupling efficiency between fiber and waveguide is 89% [32] and the estimated loss of the Y-junction used [9] to split the entrance waveguide into two waveguides routed to the trapping region is −1.0 dB [32]. We thus arrive at a power of 210 mW available for trapping, which is in the range for Raman spectroscopy on single EVs [10,12]. Figures 6(d) and 6(e) indicate that with the 5 µm trap core-shell EVs of many diameters outside the peaks, down to the smallest diameter of 115 nm, can be trapped for $P_{\min }=210$ mW. In 10 and 15 µm traps this $P_{\min }$ enables stable trapping of a core-shell EV as small as 155 and 180 nm, respectively. With these smallest EVs that can be trapped, the largest part of the EV size range is covered. We thus argue that the size range that is adressed with our integrated photonics lab-on-a-chip device is clinically relevant, also in view of the results in [33], where it is demonstrated that already big EVs of a diameter of 1 µm can be used to diagnose prostate cancer.

8. Conclusion and outlook

We have presented the capabilities of dual-waveguide traps based on box-shaped composite SiO₂-Si₃N₄ waveguides for optical trapping of extracellular vesicles (EVs), using FDTD simulations and optical force calculations. EVs are very tiny spherical, cell-derived particles of diameter in the range 30-1000 nm and have a low index contrast w.r.t. water and therefore are hard to trap. EVs attract strong attention as potential markers for cancer.

Owing to both the small size and special structure of the waveguides, the counter-propagating beams are narrow and weakly divergent, properties favorable for a tight optical trap. For facet spacings of 5, 10 and 15 μm the optical field between the facets shows clear interference maxima or hot spots, leading to an oscillatory force acting on an EV as a function of its position and thus to multiple trapping sites. For certain EV sizes the forces exerted by several hot spots cancel and only a global longitudinal force remains.

By calculating forces using Lumerical’s volumetric technique for EV diameters in the range 50−1000 nm using two refractive index distributions, viz. homogeneous and that of the realistic core-shell structure, we find that the shell, though only 5 nm thick, contributes considerably to the force up to a diameter of 450 nm. By comparing these forces with forces obtained using the gradient formula, we conclude that the gradient force clearly dominates the total force up to 100 nm EV diameter. This results from counter-propagation of the beams. Longitudinal trap stiffnesses derived from the force curves are up to ten times higher than the lateral stiffness of a laser-beam trap, a quality resulting from the interference.

The trapping potentials derived from the force curves lead to the so-called stability curves (i.e., curves of the minimum optical power required for stable EV trapping as a function of EV size). These curves indicate that EVs of a wide diameter range can be stably trapped with the attainable trapping power of 210 mW, down to a smallest size of 115 nm for the 5 μm trap. This capability in combination with the power level sufficient for Raman spectroscopy of the trapped EV, qualify this dual-waveguide trap as a promising lab-on-a-chip device with clinical relevance. Finally, we note that the drag force acting on an EV in a trap under fluidic flow can be incorporated in our procedure, leading to modified stability curves. Based on these, the smallest EV that can be stably trapped in the flow situation can again be chosen by setting the optical power, allowing all smaller EVs to be collected downstream. This defines the principle of a size filter for EVs and can be generalized to other bio-particles, such as proteins, cell fragments, and so forth.

As a short outlook, we mention that a multiple-waveguide geometry is a logical extension of the geometry presented here. Multiple waveguides lead to stronger light concentration and thus a stronger trap, since an interference pattern is created with beams coming from many directions. We have a fabrication run underway for dual-, quad-, and octo-waveguide traps tailored for experimental trapping and Raman studies of EVs.