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Fluidic vortices generated from optical vortices in a microdroplet cavity

We harness the momentum of light resonating inside a micro-droplet cavity, to experimentally generate micro-flows within the envelope of the drop. We 3D map these optically induced flows by using fluorescent nanoparticles; which reveals circular micro-streams. The flows are parametrically studied and, as expected, exhibit an increase of rotation speed with optical power. The flow is non-circular only when we intentionally break the axial symmetry of the droplet. Besides the fundamental interest in light-flow interactions including in opto-fluidic cavities, the optically controlled flows can serve in bringing analytes into the maximum-power region of the microcavity.

the circulating WGM of a droplet. Let's also assume that most of the optical losses in this resonator are due to absorption. Momentum conservation consideration implies that this absorbed microWatt must apply azimuthal force on the liquid that is tangential to its equatorial line.
Also relevant here is that in contrast to solids, where one generally has to overcome the static friction to start motion; liquid viscosity is typically speed independent. As a result, optically induced flows do not have an optical power threshold. Thus, there is no question if light will initiate flow; the only question is what will be the speed of this flow. Surprisingly maybe, the optically induced flows in the microdroplet cavity are independent of its optical quality factor. This is because it does not matter if the microWatt of optical power will be absorbed after circulating one round-trip or 1000 round-trips in the cavity. For in both cases a similar amount of momentum will be transferred to the microdroplet.
As for mapping the flows, we do so by inserting and mixing fluorescent nanoparticles in the liquid. As explained in section 5, we took special care in verifying that the nanoparticles are probing the flows while drifting at the exact speed of the liquid molecules. Although we will focus here on droplet resonators, the transfer of momentum from light to liquid as well as the resulting flows are relevant for any optofluidic device.
Assuming only absorption optical losses in the cavity, the optical force that pushes the liquid near the air-liquid interface to circulate is where P is the input optical power, c is the speed of light and n the refractive index of the droplet liquid. The drag forces, Fs, which resist the optical forces, is estimated using the shear-flow equation: where µ is the dynamic viscosity, u is to the speed of liquid at the airliquid interface and y is the distance from the liquid-air interface to the stem at the region of motion. A is the area of the moving liquid at the liquid-air interface, and is equal to the circumference of the droplet multiplied by the width of the optical mode along the stem. As mapped in [20], the width of the mode is typically 20% of the droplet extension along the stem.
After accelerating and reaching steady state, the optical force (Eq. 1) is fully balanced by the drag force (Eq. 2). Using this relation, we can calculate the speed at the liquid-phase boundary: To give a relevant scale, a typical droplet might be 100 μm in diameter and the stem holding it might be 10 μm in diameter. The liquid can have a viscosity of 10 cSt (10 times higher than water) and a refractive index equal 1.399. Coupling 4 microwatt of light to such a droplet will induce a 9.22 μm/s flow-speed according to equation 3. This estimation in fact implies that flows exist in almost any optofluidic resonator including the ones described in .
We fabricate our droplet cavities by dipping a tapered silica fiber in silicone oils (Sigma-Aldrich #378321 or #378356 ) [20,21]. As a result of interfacial tension between the solid anchor and the liquid, droplets tend to drift toward the thicker part of the taper to maximize the solid-liquid interface. We prevent this drift and pin the droplet to the taper end by turning the solid taper-end into a small sphere, using a fiber splicer.
As one can see in Figure 1, we use a tapered fiber to evanescently couple [25][26][27] a tunable laser (970 nm) into the droplet resonator. The light transmitted through the droplet is coupled out through the other side of the taper and measured with a photodetector. A nano positioning system is used to control the distance between the coupler and the resonator. This optically-pumped droplet per-se is sufficient for optically induced flows. However, in order to film these flows, we mix fluorescent polymer nanospheres (Bang FS02F envy green) in the liquid. We film the nanoparticle's path which represents the flow streamline. Furthermore, when defocused, the microscope smears the nanoparticle's image into a larger spot. The size of the yellow fluorescent spot, together with its diameter, allows us therefore to get its position in 3 dimensions. For minimizing the optical forces on the nanoparticles probe, we use small nanoparticles with low absorption coefficient. The mean diameter of the nanospheres is 60 nm and theirs refractive index is 1.57. The nanospheres are fluorescently excited by a green laser (532nm) and emit yellow light (565nm). Using a yellow-pass filter, we film the streamlines by tracking the nanoparticles.
In what follows, we monitor the optically induced flows while changing the optical input power, the resonator size and the viscosity of the liquid.
As one can see in figure 2 (violet), increasing the input power from 0 to 4 µW, for a 10cst-viscosity droplet with a diameter of 100 μm and a refractive index of 1.399 reveals a linear increase of the velocity from rest to 12.5 µm/s. Such a linear behavior is expected from equation 3.
We now repeat this experiment while changing the diameter of the droplet to 200 µm. As one can see in figure 2 (green) speed goes down with resonator size. The flow velocity was calculated by averaging three measurements and the error bar of the velocity is the standard deviation.
Using equation 3, we can calculate that a rise of a droplet radius from 100 µm to 200 µm with the same 10cSt viscosity, and assuming that the mode area in the bigger droplet is twice as large as the mode area in the smaller droplet, will reduce the speed by 2 (while the experimental speed reduction is 1.97). We then repeat the experiment measuring the speed as a function of an input power, but this time at a higher viscosity which slows down the flow. As expected from equation 3, increasing the viscosity indeed reduced the speed; yet, the exact amount of speed reduction is less that what is predicted by equation 3. One explanation for that might be that the high viscosity liquid has more scattering losses when compared to the low viscosity liquid.
As one can see in figure 4 a, moving the particle toward the camera results in blurring. By calibrating this blur diameter, we can estimate the distance of the particle from the focal plan where its image has the smallest diameter. In order to get a lookup table of the distance to the microscope as a function of spot size, we take a coverslip, put on a fluorescent nanoparticle and captured images with difference focus by controlling the distance with a XYZ piezo stage. This measured relation between blurring and distance is presented in Fig 4 a. We merge 18 frames from a movie of the flowing nanoparticle and present them in figure 4b. As one can see (Fig 4b), the position and bluer of the 18 spots reveal the 3D position of the nano particle for each of the 18 measurements. Using this method, we now map the nanoparticle trajectory in 3D. As one can see in Figure 5, the streams are almost circular. In Fig. 6 we present instabilities observed in a droplet with a diameter of 250μm where due to the gravity, there is a fall in the droplet symmetry and the flow is non circular. The anchor of this droplet is 28 μm away from the center and the deviation of the droplet from a circular shape is 3%. To conclude our measurements, we were examining resonators with axial symmetry as well as resonators which are non axially symmetric which were accordingly hosting circular and non-circular flows. The flow speed increases with power and decreases with resonator size and with viscosity.
As the nanoparticle functions here as a probe, it is important to show that the nanoparticle drifts with flow. In what follows, we will show that the nanoparticle does not move with respect to the flow. We will start by calculating the force that light applies on the nanoparticle.
Using the Rayleigh scattering equation, one can calculate the cross section area for scattering, sca  , using [28]: Throughout the experiment the nanospheres are suspended in the silicon oil droplet, the particles within the optical mode volume will experience the strongest scattering forces. The intensity within the mode volume is: where F is the optical finesse, P is the source power and m A is the mode area in a direction perpendicular to propagation.
Taking the most pesimistic assumption that all scattering is in the backward direction we get the strongest force that light can apply on the nanoparticle: where c is the light speed. Now if the particle is moving in respect to the liquid, it will experience a drag force: where v is the particle speed relative to the surrounding medium,  is the dynamic viscosity. At steady state, the drag force (Eq.7) is equal to the scattering force (Eq.6), which gives us the nanoparticle-liquid relative speed: From Eq. 8, we find out that in our experimental conditions, the fastest nanoparticle-liquid speed is 0.142 µm /s which is102 times slower than the flow speeds measured on the liquid. One can therefore say that the nanoparticle motion represent the flow motion with an accuracy better than 1%. An additional source of error might be the Brownian motion of the nano particle. We were therefore performing a control-group experiment where the nanoparticles were monitored while the pump was off. The nanoparticle drift during the time of a typical measurement (such as in Figure 4-6) was 3.1 µm, which is less than the thickness of the line in figures 5-6 We experimentally map streamlines in an optofluidic whispering-gallery resonator, and reveal that optically induced vortices are abundant in the fluidic whispering-gallery embodiment. Our work is relevant to optofluidic devices including microcavities that part of their optical mode overlaps with liquids.