The optical characteristics of each design are crucial to the trapping capabilities of the platform. Optical mode simulations were performed in Lumerical FDE for both the liquid channel and solid core ridge waveguide using 532 nm as the wavelength. The zero order E field intensity mode profile distributions for both the liquid channel and ridge waveguide are shown in
Figure 4a,b. The simulations are approximate and, for simplicity, do not take into account the exact meniscus shape. The gradient force is possible due to the optical power distribution in the liquid channel. This distribution creates the electric field gradient which enables particle focusing in the center of the channel where the optical intensity is the highest. In order for the optofluidic focusing section of the gradient force design to be effective, it needs to be long enough to give the gradient force enough time to pull particles to the center of the channel under typical fluid velocities.
Managing the optical loss in the various sections of the optofluidic platform is a critical parameter for high trapping efficiency. These devices do not utilize reflecting layers like those found in anti-resonant reflective optical waveguide (ARROW) structures, so the optical loss in the liquid channel is expected to be high since the aqueous solutions have a lower refractive index than the surrounding silicon dioxide walls. The average liquid channel optical loss using top view scattering was measured at 6.61 cm
−1 while the ridge waveguide was 0.50 cm
−1 [
22]. Due to the high optical loss, this platform can only be used in proximity applications where optofluidic sections are kept short. The optical transmission directly correlates to the optical scattering force available to influence particles. The transmission for the orthogonal force design from chip edge to the protrusion cavity is calculated to be 19%. The transmission from chip edge to protrusion cavity for the gradient force design is 17.9%. Transmission values were calculated based on measured values for waveguide loss and coupling coefficients between the fiber and chip as well as between the ridge waveguide and liquid channel [
22,
24,
25].
3.1. Particle Manipulation in the Orthogonal Force Design
The orthogonal force design introduces radiation pressure from the ridge waveguide orthogonal to the liquid channel. This design brings the particles as close to the ridge waveguide as possible. It can be used in applications that require an array of multiple ridge waveguides and trapping cavities where the trapping efficiency of an individual cavity is not the main priority. The orthogonal force design also allows for selectively sorting particles based on their size or refractive index. The forces acting on a particle in the optofluidic region of the orthogonal design is provided as an illustration in
Figure 5. The fluid simulations were done in Ansys Fluent and are in the laminar flow regime.
The optical scattering force is the trapping mechanism for the orthogonal force design. It can be determined using
where
Q is a dimensionless variable relating the momentum transfer efficiency,
a is the radius of the particle,
n is the index of refraction of the liquid,
c is the speed of light, and
I is the intensity of the optical power incident on the particle. This equation can be used to calculate the minimum optical power required to capture particles for various sizes and flow velocities as show in
Figure 6. For the minimum power required to trap a particle, we consider the worst-case scenario where a particle is on the side wall furthest from the protrusion cavity. A particle against the wall will be moving at its slowest rate as the velocity is not constant along the channel. Our calculation uses an average fluid velocity in the x direction to account for different possible flow rates as the particle transitions towards the cavity. If the velocity induced on the particle by the scattering force can match the velocity of the fluid flow, the particle will be trapped. The velocity produced by the scattering force is found by considering the matching drag force on the particle (which is velocity dependent). Because of the square geometry formed by the 10 μm waveguide and 10 μm channel width under consideration near the protrusion cavity entrance; a particle would travel the same distance in the y direction as in the x to be caught. We are assuming a terminal velocity due to the short span of the acceleration. These calculations are based on an approximation where the power is equally distributed vertically. The drag force equation holds due to laminar flow inside the microfluidic channel.
Minimum trapping power was calculated for a range of particle diameters and flow velocities as shown in
Figure 6. High velocity particles and smaller particle diameters require more optical power for trapping. If the minimum radiation pressure is not induced on the particle, it will pass through the optofluidic region without being contained in the protrusion cavity.
Experiments were performed using 400 mW of optical power from a laser source, 72 mW at the optofluidic trapping site due to the 19% transmission efficiency shown previously, 532 nm wavelength, and 1 μm diameter particles. The trapping efficiency can be found in the Results section. The intersecting lines and dot in
Figure 6b demonstrates the power, particle size, and flow velocity expected by our experimental values. A video showing particles being trapped for these conditions can be found in
Video S1. The fluid flow for these videos was approximately 100 μm/s. Given that we are operating very close to the minimum optical power necessary to trap particles, we do not expect all of them to trap as they flow past the protrusion cavity. The fact that the ridge waveguide does not line up vertically with the center of the liquid microchannel, thus causing uneven illumination, also leads us to expect incomplete trapping.
3.2. Particle Manipulation in the Gradient Force Design
The gradient force design was developed as an efficient optical trapping system. The beam propagation and liquid flow are both in the same direction and act in concert to push particles toward the protrusion cavity. The gradient force causes the particles to be centered in the channel and helps prevent particles from escaping.
The gradient force can be analyzed using Equation (2) below
where
is the index of refraction of the particle,
is the radius of the particle,
c is the speed of light,
is the ratio of the index of refraction of the particle to the index of the medium and
is the optical intensity [
26]. Equations (1) and (2) hold for homogeneous spheres corresponding to the microbeads used in our experimentation. The gradient force is plotted for various particle diameters versus the channel position in
Figure 7, utilizing the electric field intensity as shown in
Figure 4. The gradient force increases from the wall of the channel towards the center. Particles are attracted to the region where the optical intensity is the strongest. This enables particles to be pulled from anywhere in the channel to the center. As shown in Equation (2), the gradient force is directly proportional to the gradient of optical intensity. Higher intensity allows for faster and stronger gradient force channel centering.
Understanding the properties of the gradient force is important in determining the optical trapping characteristics of the optofluidic manipulation chip. The effects of the gradient force on the motion of a particle can be described by the differential equation
where
is the mass of the particle and
is the second derivative of the position in the x direction in the channel in terms of time (directions shown in
Figure 8a). Solving this differential equation gives insight into the trapping characteristics of this device. The first condition considered is a worst-case scenario in which a particle is at the entrance of the optofluidic region on a side wall. For a fluid moving at a given speed, it is desirable to know how far up the channel in the z direction the particle would travel before becoming susceptible to the gradient force and centralizing as shown in
Figure 8a. MATLAB was used to solve the differential equation numerically using the ode45 solver function with initial conditions for particle position and velocity.
Figure 8b demonstrates the optical power required for centralization with corresponding travel distances based on solutions to Equation (3) for a series of spherical particle sizes (assumed to be spheres) with an index of refraction of 1.6 according to Thermo fisher. Fluid velocity was assumed to be 100 mm/s and water was used as the solution (
n = 1.33). As shown in the figure, higher optical powers are required to center smaller particles. It is important that the particles move to the center of the channel before reaching the channel bend before the protrusion to ensure that the particles are captured in the cavity. The intersecting lines and dot in
Figure 8b demonstrate the power and corresponding distance expected given the experimental parameters discussed in the Results section. Video observations confirm our centering distance predictions in
Figure 8b.
The next important consideration is what happens to a particle which has been centered in the optofluidic channel and then encounters the bend near the end of the channel, where it becomes susceptible to the fluid flow moving orthogonally to the trapping direction. This is the critical junction where trapping in the protrusion cavity will either take place or the particle will be swept away by the fluid flow. Knowing the forces acting on the particle and velocity components at this point is important to ensure that enough optical force is present to capture the particle. These forces are illustrated in
Figure 9. The gradient and scattering force components are the trapping mechanisms in this case.
If the particle does not overcome the channel flow and enter the protrusion cavity, it will not become trapped.
Figure 10a demonstrates trapped and un-trapped trajectories. The trapped trajectory is at the minimum power threshold showing that at the bend, the particle will be pulled in the direction of fluid flow, causing the curve in the particles path. In this scenario, the particle will barely enter the protrusion cavity where it will no longer be susceptible to the fluid flow and become centered through the gradient force as demonstrated in
Video S2. The same methods explained previously were used to solve Equation (3) for this scenario. The solutions were used to determine the minimum power required to ensure trapping for given particle diameters and for various flow velocities as shown in
Figure 10b. Faster flow rates and smaller particle diameters require higher power for particle trapping.
Experiments were performed by coupling 400 mW of optical power onto the chip with ~72 mW at the optofluidic trapping site due to the 17.9% transmission calculation shown previously, 532 nm wavelength, and 1 mm diameter particles and the findings can be found in the Results section. The intersecting lines and dot illustrate where the experimental parameters would appear on the graph. A video showing particles being trapped for these conditions can be found in
Video S2. The fluid flow for these videos was approximately 100 mm/s. Given that our experimental conditions are safely above the calculated optical power necessary to trap particles, we expect nearly all of them to trap as they flow past the protrusion cavity.
Solving the differential equation both at the start of the channel and at the corner proves the necessity of the gradient force as it first pushes particles to the center of the channel and then to pull them back near the turn. Solutions to Equation (3) shown in the simulation results in
Figure 8b and
Figure 10b makes it evident that without the minimum power from the gradient force, the particles will not become trapped. The scattering force alone will not be enough to trap particles, especially if they were near the right wall close to the corner.
The trapping efficiency for both designs presented here was experimentally tested by edge coupling light into the chip. 400 mW of optical power from a 532 nm laser source was coupled through an optical fiber and into the ridge waveguides on chip. Polystyrene microbeads with a diameter of 1 μm were used as the particles in the experiment. Metallic cylindrical reservoirs were attached to the microfluidic channels at the both the inlet and the outlet. A fluid solution containing microbeads with a concentration of 3.8 × 10
7 beads/mL was introduced into the inlet reservoir and the fluid was gravity fed into the system. This created an average fluid flow velocity of approximately 100 μm/s. The experiment counted the total number of beads collected at the protrusion cavity versus those that escaped. Examples of trapping events are shown in
Supplemental Videos S1 and S2. Trapped and escaped particles were counted versus time and the results from two separately tested orthogonal devices and three separately tested gradient devices are superimposed and shown in
Figure 11. Similar trends held for other tests made on twelve separate orthogonal chips resulting in efficiencies within 8% and three separate gradient chips resulting in efficiencies within 3% compared to those shown in
Figure 11.
Because we are demonstrating a flow-through device capable of discriminating particles in the fluid stream,
Figure 11 demonstrates what happens as particles move through over time. The performance does not significantly change during the course of the experiment due to accumulation of particles in the trapping region.
The number of particles captured divided by the total number of particles entering the optofluidic system was used to calculate the trapping efficiency. For the orthogonal design, the measured optical trapping efficiency was 80% and the gradient force design had a trapping efficiency of 98%. These results show that our models match closely with the experimental data. The orthogonal design experimental point did not have much margin for error as shown in
Figure 6b, causing some of the particles to pass through the optofluidic region without capture. The margin of error for the experimental point is much higher in the gradient force design, shown in
Figure 10b. This results in a more reliable trapping efficiency.