Fluid flow vorticity measurement using laser beams with orbital angular momentum

We report on the direct measurements of fluid flow vorticity using a spatially shaped beam with a superposition of Laguerre-Gaussian modes that reports on the rotational Doppler shift from microparticles intersecting the beam focus. Experiments are carried out on fluid flows with well-characterized vorticity and the experimental results are found to be in excellent agreement with the expected values even for 200 ms integration times. This method allows for localized real-time determination of vorticity in a fluid flow.

in gas (e.g. air) flows. A capability to directly measure vorticity in fluid flows in a non-intrusive time-resolved manner would greatly impact the field of fluid mechanics.
Direct non-intrusive measurement of vorticity requires a laser-based method that is sensitive to rotational motion. Translational velocities can be measured with laser Doppler velocimetry (LDV) by taking advantage of the (linear) Doppler Effect, which causes a frequency shift when objects move towards or away from a source of light.
Analogously, but much less utilized, the Rotational Doppler Effect (RDE) can be used to measure the angular velocity of a rotating object. [5][6][7][8] Measuring with RDE requires the use of Laguerre-Gaussian (LG) light beams that possess orbital angular momentum (OAM), a spatial (azimuthal) modulation of the beam phase front. The creation of beams with arbitrary orbital angular momentum l, or beams having a superposition of counter-rotating OAM (±l), requires computer controlled 2D spatial light modulators (SLM) capable of introducing complex phase designs.
The use of LG laser beams with counter-rotating OAM (±l) to determine the angular speed of rotating objects based on RDE was recently reported by Lavery et al. 6 When the illumination comprises two helically phased beams of opposite values of l, their scattering into a common detection mode gives opposite frequency shifts resulting in an intensity modulation of frequency mod 2 2 f l   , where  is the angular velocity of the rotating object. Lavery et al. 6 tested this type of setup and were able to measure the angular velocity of a spinning disk. Similar same concepts have been employed to spin and to measure the angular velocity of a microparticle trapped and spinning in an optical trap. 9,10 We present here what we believe is the first direct vorticity measurement in a fluid flow based on angular velocity measurement of micron-sized particles free flowing in the fluid using RDE and LG laser beams with OAM.
Very small particles faithfully track the fluid flow and, at steady state, they move with the local flow speed and rotate with the local angular velocity of the fluid (or half the local flow vorticity at the particle center) 11 . We demonstrate the technique in a flow field known as solid body rotation or rigid body flow field-for which the angular rotational velocity is uniform and particles carried by the flow also rotate about their center as if they were part of the rigid body. In this type of flow the vorticity is the same everywhere. We present two sets of experiments.
In the first, the signal from a group of 6 μm microparticles is integrated to obtain the average fluid rotation rate about the beam optical axis within a 100 μm illumination region, thus obtaining the spatially-averaged vorticity within that region. In the second experiment, the same information is obtained by measuring the angular velocity of a single 100 μm particle in the flow. The latter is the type of transient measurement required to determine vorticity in more complex flows fields.
The experimental setup for measurements of the local flow angular velocity and vorticity is shown in Fig. 1(a).
The 488 nm continuous wave beam from an optically pumped semiconductor laser (Genesis MX, Coherent, USA) with initially Gaussian beam profile is expanded by a telescope (L1, L2) and shaped by a two-dimensional liquid crystal on silicon spatial light modulator (LCOS-SLM, Hamamatsu, Japan). The SLM is programmed with a diffraction pattern that introduces the LG spatial modulation and diffracts the spatially shaped beam as shown in Fig.   1(b). The shaped beam possesses the orbital angular momentum corresponding to a superposition of LG ±18 modes, and its far-field intensity profile corresponds to a circular periodic structure with 36 petals (Fig 1(c)). The use of optical angular momentum in the context of the experiments presented here has been reviewed recently. 12 The beam is then focused with long focal length lens L3 and first diffraction order is selected with an aperture. Lens L4 collimates the beam, which after reflection from dichroic mirror (DM) is focused by lens L5 (60mm focal length) into the center of a rotating cylindrical container with the beam optical axis aligned along the rotation axis. The beam diameter at the focus is measured to be about 120 µm and the average power is 12 mW, an intensity that is at least one order of magnitude too weak for causing laser trapping. The container is filled with fluorescent micro- Epi-directional fluorescent light from the irradiated particles is collected with lens L6 and is focused onto a photodiode. A small diameter pinhole is set before photodiode in order to spatially filter out signal from outside of focal volume in fluid. The intensity modulated signal from the fluorescent particles is recorded at 10 kHz sampling rate and spectrally analyzed. Earlier efforts from our group to detect vorticity based on back scatter resulted in very 4 poor signal to noise ratios due to multiple sources of scatter. 13 The use of epi directional detection and the use of fluorescent particles, in combination with clean OAM shaped laser excitation, allows us to reject scattered light from the rotating surfaces of the container and guarantees the measured signal originates from within the rotating body of fluid. From the spectral peaks in Figure 2  While the experiments we have reported here represent the extension of the work of Lavery et al. 6 to the field of fluid dynamics, there are certain differences between the two as well. In the latter, the scattering signal originates from the planar surface of a spinning disk. In ours, measurements are carried out within the body of the fluid and the scattering signal is from a finite volume inside the fluid. For measurements with high spatial resolution, this scattering volume needs to be localized to a small region. In the current experiments, this was achieved by focusing the laser beam to about 100 µm diameter inside the liquid container.
We have presented the first direct and localized non-intrusive measurement of vorticity in a fluid flow using the Rotational Doppler Effect and Laguerre-Gaussian spatially modulated light beams that possess orbital angular momentum. The approach has been demonstrated in the flow field of solid body rotation where the flow vorticity is known precisely. In one experiment, measurements with a group of 6 μm microparticles is used to obtain the average fluid rotation rate about the beam optical axis within the 100 μm illumination region, and therefore, the spatially-averaged vorticity within. In another experiment, the same information is obtained by measuring the angular velocity of a single 100 μm particle in the laser beam. In both experiments, the measured results are in excellent agreement with those expected from the prescribed rotation frequencies of the rotating fluid container.
Although, the technique is demonstrated here in a simple flow where vorticity is uniform and steady, the approach holds great promise for unsteady flows with spatially varying vorticity field. We plan to explore extensions of this measurement technique to more complex flow environments.