1. Introduction

Over the last few decades, optical trapping has emerged as a powerful tool in the physical, biological, and engineering sciences. It is of particular interest in biology, as processes from cell motility to transcription of DNA, are driven by pico-newton forces [1–5]. The ability of optical trapping to apply molecular scale forces over a nanometer range has allowed for critical insights into the mechanical nature of fundamental bio-molecules, such as DNA, RNA and proteins, as well as bio-processes, such as transcription and replication [3, 5–11]. Despite this, the conventional optical trapping techniques are limited by throughput. By employing an optical gradient force generated by a tightly focused laser beam through a high numerical aperture objective, optical trapping instruments normally can only manipulate one object at a time. Although, methods for generating multiple optical traps, such as time-sharing a single laser beam or holographic modulation, have been reported over the years [12, 13], they require that input laser power be proportional to the number of the optical traps and thus limit the number of high stiffness traps.

Near-field optical trapping methods present a possible alternative to generate strong trapping forces while addressing the throughput limitation [14–20]. Photonic nanostructures for near-field optical trapping have been designed and demonstrated, including plasmonic optical tweezers [14, 18, 19], slot waveguides [15], whispering-gallery mode resonators [17], and photonic crystal resonators [16]. These structures have a strong gradient, from the evanescent field, which can trap a large number of particles with very low input power [14–20]. However, it has been challenging to achieve stable three-dimensional trapping that can be precisely controlled.

Recently, we have demonstrated a high-throughput, near-field nanophotonic trapping platform that achieved stable trapping with precision controllable repositioning [21, 22]. The core concept of the platform is nanophotonic standing-wave interferometry, where laser light travels through a nanophotonic waveguide, is split into two equal intensity laser beams, the two beams are guided by the waveguides and meet each other, which ultimately leads to interference of two counter-propagating laser beams and results in the formation of standing waves. The evanescent field of the antinodes of the standing wave forms an array of stable three-dimensional optical traps. We call this type of trap a nanophotonic standing-wave array trap (nSWAT). By tuning the phase difference between the two counter-propagating laser beams, the antinode locations can be precisely repositioned, and consequently, the optical trap positions can be precisely manipulated. The nSWAT device holds the capability for high-throughput precision measurements on-chip.

In this paper we present a new nSWAT design that significantly improves the stability of the trap and double the trapping force and energy for the same input laser power. For a quantitative comparison, we fabricated both the old and the new trap design on the same chip and compared their performance concurrently under the same input laser power. We implemented these devices on silicon nitride photonic platform [22, 23] that can operate at a wavelength of 1064 nm, thereby alleviate the optical absorption of water which is normally a major problem for nanophotonic trapping devices operating at 1550 nm wavelength.

2. Structure and properties of the new and old nSWAT designs

Figure 1(a) depicts the structure of the trapping device that contains both the old nSWAT and the new nSWAT design in proximity of each other and operate by the same laser power. A 50/50 beam splitter divides the input laser beam (P = B²), into two separate counter-propagating laser beams, with powers P₁ = E² and P₂ = F² (P₁ = P₂ for an ideal splitter). These beams then travel around the loop, form standing waves to create an array of traps at the old trapping region within the loop, enter the same splitter again, and eventually emerge at input arm with power P' = B'², which should be the same as P assuming no power loss, but travelling in the opposite direction. The standing wave formed by counter-propagating waves of powers P and P' should have twice the peak power of that formed by P₁ and P₂ for an ideal splitter. Thus, we have relocated the trapping region to the input arm of the nSWAT.

 

Fig. 1 nSWAT analysis and fabrication. a) A schematic of the device layout. The electric field vectors at different locations along the waveguide are indicated. Input laser beam of power P = B² is divided at the 50/50 splitter into two counter propagating waves of powers P₁ = E² and P₂ = F² and, each of which is half of P in the case of an ideal splitter. Wave of power P₂ passes through a region of the waveguide modulated by a microheater (brown) and acquires an additional phase (θ) before reaching the old trapping regions. Waves of powers P₁ and P₂ then re-enter the splitter and emerge as waves of powers P' = B'² at the input arm and O² at the output. b) An optical microscope image of the fabricated device (false colored for clarity). The device is implemented with silicon nitride waveguides over a 3.8 μm thermal oxide layer on top of a silicon wafer, with waveguide and microheater fabrication details given in [22]. The old and new trapping regions are laid side by side for direct comparison.

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The proximity of the old and the new trapping region allows for a direct comparison of the trap properties including bead displacement and trap stiffness. This also eliminates comparison uncertainties that might be introduced by variations due to fabrication and laser coupling efficiency to the waveguide, from device to device. Figure 1(b) shows an optical image of such a device, fabricated following a previously described protocol [22].

In practice, a beam splitter cannot be precisely fabricated to partition light to 50/50 due to slight variations in the waveguide dimensions and splitter gap distances during fabrication. To understand how these variations might impact the standing-wave quality and trapping stability, we have carried out the following theoretical analysis.

In the device shown in Fig. 1, the standing-wave can be formed in new and old trapping regions. First, we calculate the relationship between the standing wave position and the phase shift introduced by the microheater in both the new and old trapping regions. Second, we show that the standing-wave quality in the new trapping region is less sensitive to fabrication variations. For this discussion we assume the coherence length of the laser is much longer than any length scale in this device (which is the case for the laser we have used) and thus its effect on the standing-wave is ignored.

By introducing the laser (amplitude B) to the splitter, the output amplitudes E and F at the forward ports of the splitter can be obtained as follows:

We denote the propagation constant of the optical mode in the waveguides as β = 2πn_eff/λ, where n_eff is the effective mode index and λ is the wavelength in vacuum. The standing-waves in the new and old regions along the waveguide (z) are formed as the result of interference between B and B', and E and F respectively as follows:

The stored energy, to the first order approximation, is proportional to the squared magnitude of the electric field, which can be obtained from Eqs. (3) and (4) respectively:

These expressions show that interference fringes vary sinusoidally along the waveguide and can be shifted via the phase-shifter θ. Furthermore, the same phase shifter repositions the trap arrays in the old and new regions in an identical fashion.

The trapping force depends on the gradient of the squared magnitude of the electric field. In the case of an ideal splitter, $k=t=1/\sqrt{2}$ and I_new = 2I_old. Consequently the trap stiffness at the new region is twice of that at the old region.

When the splitter is non-ideal (k ≠ t), the quality of the standing wave may be measured by the standing wave ratio (SWR) which is ratio of the maximum to minimum energy:

Figure 2(a) shows that the SWR in the new region is much less sensitive to deviation of t from its optimum value of $1/\sqrt{2}$ and thus is more tolerant of fabrication variations.

 

Fig. 2 Standing wave quality in the old and new trapping regions. a) Standing wave ratio as a function of t. b) Power ratio of forward to backward propagating beams as a function of t. c) Scattering power normalized to that at t = 1 as a function of t.

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A related criterion to characterize device sensitivity to splitter imperfection is the power ratio (R) of the forward to backward propagating beams at each trapping region:

In the case of non-ideal splitter, the particle will experience not only a trapping force, but also a scattering force that pushes the particle in the direction of net power flow. Such a scattering force shifts the trapping center and breaks the force symmetry around in the trapping center, leading to reduced trapping stability. The scattering force depends on the net power flow which is:

Figure 2(c) shows the net power flow versus t at both regions. The scattering power is minimal for a larger range of splitter coefficient t in the new region. In fact, there is a local minimum of zero near $t=1/\sqrt{2},$making such a device much less sensitive to variations in t over a broad range.

Thus, in comparison with the old design, the new nSWAT design offers three major improvements. First, the optical trapping stiffness and force are enhanced by a factor of 2. Second, the quality of the standing wave, as characterized by the standing wave ratio parameter, is much less sensitive to an imperfect 50/50 splitter ratio and thereby is more robust against fabrication imperfections. Third, the new design provides more stable trapping against scattering force resulting from an imbalance of the counter-propagating beams due to an imperfect splitter.

In addition, we have conducted finite-element-method based 3D full-wave electromagnetic simulations to determine how the trapping force varied with the bead size [22]. For the waveguide geometry and the laser used in this work (250 nm × 550 nm silicon nitride waveguide, polystyrene beads, 1064 nm laser), the optimal bead size is 300 nm diameter [Fig. 3]. The 380 nm bead size used in the experiments is close to the optimal size, though not ideal.

 

Fig. 3 Maximum trapping force as a function of bead diameter. Simulations were performed using 1 W total local laser power (0.5 W power from each direction). Parameters used are: 250 nm × 550 nm silicon nitride waveguide, polystyrene beads, and 1064 nm laser, bead to waveguide surface-to-surface distance 5 nm, with other configurations detailed in [22].

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3. Experimental results and discussion

The fabricated device are characterized with a tunable laser (New Focus TLB6721P) cascaded to a laser amplifier operating at 1064 nm wavelength. By monitoring the power at the output port of the device we can find the optimal operation wavelength for the splitter, which occurs when the power at the output port of the device is minimal. The trapping device has been designed for the transverse-magnetic (TM) polarization mode of the waveguide to provide a greater optical trapping force than with the transverse-electric (TE) polarization mode [24]. Experiments were carried out with 380 nm carboxylated polystyrene beads in 2 mM Tris HCl buffer at pH7.8.

To visually assay the power inside the waveguides at both the old and new locations [Figs. 4(a) and 4(b)], we imaged the scattered laser light from the waveguide at both locations. This scattering is expected to arise from slight imperfections in the sidewalls of the waveguides introduced during fabrication and should be proportional to the laser power inside the waveguide [25]. The scattered light was imaged by a 60 × , 1.2 NA water-immersion objective (Nikon) using an InGaAs camera. To average out non-uniformity of the scattered laser intensity along the waveguide, the intensity of the scattered laser was then integrated over an 80 µm length of the waveguide at either the old or new region. Figure 4(c) shows that the average scattered laser intensity in the new trapping region is twice that of the old location, consistent with the doubling of the laser power in the new trapping region.

 

Fig. 4 Direct comparison of the power in a waveguide in the old and new trapping regions. a) Scanning electron micrograph of the waveguides in the two trapping regions. The waveguides are 550 nm in width and 250 nm in height. b) Image of scattered light from the two waveguides. The brightness of the scattered light from each waveguide is proportional to the power transmitted through the waveguide. c) Intensity plot of the waveguides. The intensity plot is generated by taking the image in (b), integrating intensity values along the z axis, and plotting the integrated intensity value along the x direction. The solid red curve is a fit to the sum of two Gaussian functions, corresponding to the scattered light at the old and new trapping region respectively. The cumulative intensity of the scattered light at each region is determined as the area under the corresponding Gaussian. d) Relation in the cumulative intensities of the scattered light in the waveguides in the new and old trapping regions. Measurements were made over a range of input laser power (40 to 160 mW). The resulting plot gives a linear fit of a slope 2.006 ± 0.016.

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We first demonstrate the same microheater can be used to position a trap array at both the old and new locations. Previously we showed that the trap may be repositioned by introducing a phase difference θ between the counter-propagating waves via the thermo-optic effect using an on-chip electric microheater [21, 22, 24] [Fig. 1(a)]. Here we show that the same microheater may also be used in the same fashion to reposition the array trap in the new trapping region, though a trapped bead move in an opposite direction in the new trapping region. Because θ increases linearly with the refractive index change, or the temperature change, of the underlying waveguide, while the temperature change is proportional to the heater power or the square of the applied voltage to the heater, the resulting trap displacement is proportional to the square of the voltage. In Fig. 5, the displacement of the trap was monitored as the voltage applied to the microheater was increased in order to translate the trapped beads. As shown, at both the old and new locations, bead displacement increases linearly with the square of the voltage with nearly identical slopes: −10.37 ± 0.03 and 10.39 ± 0.02 nm/V² respectively. The trap period λ_z was measured to be 360 ± 5 nm for both regions, using methods previously described [21, 22].

 

Fig. 5 Displacement of a trapped bead versus the square of the voltage applied to the microheater at both new and old trapping regions. The linear fit has a slope of −10.37 ± 0.03 nm/V² for the old trapping region, and 10.39 ± 0.02 nm/V² for the new trapping region. Exactly one trapping period is shown for both the new and old trapping regions.

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We show that a trap at the new trapping region can be repositioned with the same precision as that at the old region. To determine how well a trapped bead could be positioned, we trapped a bead at both the new and old regions, and then applied square-wave voltages to the microheater while measuring the positions of the trapped beads. As shown in Fig. 6, for beads trapped at the old and new regions, a 10-nm step size can be readily discerned and a 2.5-nm step size could also be resolved. This indicates the new trap region design has comparable potential for detecting molecular events that occur at the nanometer scale as the old trap region.

 

Fig. 6 Trap reposition resolution of at the new and old trapping regions. The positions of trapped beads at both trapping regions were measured as the traps were repositioned in a square-wave fashion by the microheater, in either 10 nm (a) or 2.5 nm (b) steps. The black and red curves on the bottom plots show the measured bead displacement at the old and new trapping regions, respectively, with solid curves as the fitted bead displacement.

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To directly compare the trap stiffness at the old and new regions, we trapped polystyrene beads and determined the bead positions along the waveguide (z) as a function of time were by video tracking. The trap stiffness was determined by both the variance and power spectrum methods that have been established for optical trapping measurements [Figs. 7(a) and 7(b)] [21, 22, 26–28]. In the variance method, the bead position distribution was fit to a Gaussian function. The trap stiffness was calculated from the variance of the bead position (σ²) using the equipartition theorem: k_z = (k_BT)/σ². In the power spectrum method, the stiffness was determined from the Lorentz power spectrum of the bead position: $P\left(f\right)=\frac{k_{B}T}{2{\pi }^2\gamma \left(f^2+f_c^2\right)},$ where f is the frequency, k_BT is the thermal energy, and γ is the viscous drag coefficient for the bead. The trap stiffness was obtained from the corner frequency f_c: k_z = 2πγf_c. Using these methods, we determined trap stiffness at different laser powers. As shown in Fig. 7(c), the trap stiffness ratio of the new to old trapping region was found to be 2.11 ± 0.03 for the variance method and 2.18 ± 0.04 for the power spectrum method, in agreement with the expected trap stiffness doubling at the new region.

 

Fig. 7 Trap stiffness versus laser power in the waveguides. a) An example of trap stiffness measurement using the variance method. A 380-nm polystyrene bead was trapped on the nSWAT in the old trapping region under an estimated 7.2 mW laser power (sum of the powers in both directions) at the trapping region. Local laser power intensities are estimated based on measured loss in each section of the optical path, as detailed in [22]. The bead position distribution (grey) was fit to a Gaussian function (black) to determine the variance of the bead position: σ² = 516 nm², yielding trap stiffness: k_z = 0.0079 pN/nm. b) Trap stiffness measurement using the power spectrum method. For the same set of data as in a), the power spectrum of the bead position was fit to the Lorentzian function (black) (see text). The fit yielded f_c = 103 Hz, and γ = 9.6 × 10⁻⁶ pN/(nm•Hz), giving the trap stiffness, k_z = 0.0062 pN/nm. c) Trap stiffness versus laser power. At each laser power in the old trapping region, the trap stiffness was determined from both the variance method (open square) and the power spectrum method (open circle) at both the old and new trapping regions. Fitting these results to lines, solid and dashed respectively, gives stiffness 1.04 pN/nm per watt for the variance method and 0.95 pN/nm per watt for the power spectrum method at the old design, and 2.19 pN/nm per watt for the variance method and 2.07 pN/nm per watt for the power spectrum method at the new design. The error bars are standard errors of the mean.

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To directly compare the maximum trapping force at the old and new regions, we trapped a bead in the old or new region, and then used the viscous drag to generate a force on the bead by moving the trap at a constant velocity (v_trap). We have previously shown that the bead motion may be described by a damped-forced oscillator at low Reynold’s number [22], and the equation of motion is given by: $\gamma \frac{\text{d}z}{\text{d}t}=F_{\text{max}}\sin \left[\frac{2\pi }{{\lambda }_z}\left(z-{\nu }_{\text{trap}}t\right)\right].$ The bead speed $\left({\nu }_{\text{tr}ap}=\frac{\text{d}z}{\text{d}t}\right),$ and hence of viscous drag force on the bead $\left(F=\gamma \frac{\text{d}z}{\text{d}t}\right),$ increases linearly with the trap speed until the trap speed reaches a critical value: v_trap = v_critical, beyond which the bead speed monotonically decreases: $v_{\text{bead}}=v_{\text{trap}}-\sqrt{v_{\text{trap}}^2-v_{\text{critical}}^2}.$ The maximum trapping force F_max is determined by the critical trap speed: F_max = γv_critical. Figure 8 shows the measured bead velocity versus trap velocity for a bead trapped in either the old or new trapping region. For each data set, the fit yielded a critical speed and therefore a maximum force. The ratio of the maximum force of the new to old region was 1.80 ± 0.27, in reasonable agreement with the expected force doubling at the new region.

 

Fig. 8 Bead velocity along the waveguide versus trap velocity. Two trapped beads, one in the old trapping region and another in the new trapping region, were monitored as the traps were moved at a constant speed. Because the two beads in the two regions moved in opposite directions under microheater modulation, the positive trap speed data at the old trapping region were acquired simultaneously with the negative trap speed data at the new trapping region. Measurements were performed at 7.2 mW laser power in the old trapping region. The red squares are measured bead velocity in the new trapping region, while the black dots are measured bead velocity in the old trapping region. For each region, data in each direction were fitted by a piecewise function based on the theoretical predications [22]. For the new (old) trapping region, the critical velocity from the fit was 69.16 μm/s (38.02 μm/s) in the positive direction, 68.22 μm/s (38.45 μm/s) in the negative direction, averaging to be 68.69 μm/s (38.24 μm/s). Therefore the maximum trapping force (F_max = γv_critical) was found to be 0.625 pN and 0.347 pN for the new and old region, respectively.

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In addition, data in Fig. 8 show a high degree of anti-symmetry at both the old and new trapping regions, suggesting am optimized beam splitter as well as low waveguide power loss. Therefore, for a given laser power entering the splitter, the maximum force is estimated to be 60 pN/W in the old trapping region and is 120 pN/W in the new trapping region.

Finally, we demonstrate the robustness of trap stability of the new design to fabrication imperfections. We characterize trap stability using the input laser power at which a trapped bead can no longer be confined within a single trapping center and instead periodically escapes to a neighboring trap. Below this escape laser power, the bead will undergo one-dimensional Brownian motion that is biased in the direction of net power flow along the waveguide. A near perfect standing wave will correspond to a low escape laser power.

Figure 9(a) show the bead behavior at both the old and new trapping regions as the input laser power was steadily decreased with time. We measured the escape input laser powers for both the old and the new trapping regions, and then examined the ratio of the power for the old to the new region. This ratio was then determined for devices that covered a range of gap distance (defined as the edge-to-edge distance between the two waveguides in a beam splitter) [Fig. 9(b)].

 

Fig. 9 Escape power measurement. a) Displacements of trapped beads in the new and old regions as laser power is steadily decreased. Measurements were formed using a device with a gap distance of 620 nm. Different colors correspond to different trapped beads in a given region. From these traces, we determined that the escape power ratio of the old to the new region was 2.30 ± 0.09 for this device. b) Bead escape power ratio of the old to the new trapping region versus edge-to-edge splitter gap distance. At a 560 nm gap distance, the splitter is closest to 50/50 splitting ($t=k=1/\sqrt{2}$).

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Figure 9(b) shows that this ratio reaches a minimum of ~2 for devices with a splitter gap distance of 560 nm, suggesting that this gap distance provides the most stable trapping in the new region compared with that of the old region. This gap distance coincides with that of devices that supported power doubling and trap stiffness doubling in the new region [Figs. 4, 7 and 8]. This is also consistent with the standing wave being nearly perfect (i.e., $t=k=1/\sqrt{2}$) when the power at the new region is twice of that of the old region under the same input power. Indeed, all measurements in Figs. 4-8 were performed using devices with 560 nm gap. As the gap distance deviated from the optimal distance (i.e., $t<1/\sqrt{2}$ or $t>1/\sqrt{2}$), the power ratio increased. This increase demonstrates that when the gap distance is imperfect, the quality of the standing wave deteriorates more severely in the old region than in the new region. Thus the new region is more robust to variations in fabrication and supports enhanced trap stability. In addition, the method outlined in Fig. 9 also serves as method for assaying the quality standing waves in nSWAT devices.

4. Conclusion

In conclusion, we show that the new nSWAT devices maintain the trap manipulation precision of the old design, but offer twice of the trap stiffness and force as the old devices, as well as robustness against fabrication imperfections. These improvements will allow the nSWAT to be better integrated with various microfluidics and lab-on-a-chip devices, where the laser power budget is limited. We envision this as another step towards tweezers-on-a-chip, which enables mobile routine optical-trapping experiments in various biological laboratories.