## Abstract

It is possible to devise an experiment in which the local vorticity of a flow can be estimated by probing the fluid with Laguerre–Gauss (LG) beams, i.e., optical beams that show an azimuthal phase variation that is the origin of its characteristic nonzero orbital angular momentum. The key point is to make use of the transversal Doppler effect of the returned signal that depends only on the azimuthal component of the flow velocity along the ring-shaped observation beam. We found from a detailed analysis of the experimental method that probing the fluid with LG beams is an effective and simple sensing technique that is able to produce accurate estimates of flow vorticity.

© 2015 Optical Society of America

Vorticity describes the spinning motion of a fluid, i.e., the tendency to rotate, at every point in a flow. The interest in performing accurate and localized measurements of vorticity reflects the fact that many of the quantities that characterize the dynamics of fluids are intimately bound together in the vorticity field, being an efficient descriptor of the velocity statistics in many flow regimes. It describes the coherent structures and vortex interactions that are at the leading edge of laminar, transitional, and turbulent flows in nature [1]. The measurement of vorticity is of principal importance in research fields as diverse as biology microfluidics, complex motions in the oceanic and atmospheric boundary layers, and wake turbulence on fluid aerodynamics. However, the precise measurement of flow vorticity is difficult [2]. Here we put forward an optical sensing technique to obtain a direct measurement of vorticity in fluids using Laguerre–Gauss (LG) beams, i.e., optical beams that show an azimuthal phase variation that is the origin of their characteristic nonzero orbital angular momentum [3]. In our experimental method we make use of the transversal Doppler effect [4–6] of the returned signal, which depends only on the azimuthal component of the flow velocity along the ring-shaped observation beam to produce accurate estimates of flow vorticity.

Vorticity is defined as the curl $\mathrm{\omega \u20d7}=\nabla \u20d7\times \mathrm{U\u20d7}$ of a velocity vector field $\mathrm{U\u20d7}$. It is a measure of the amount of angular rotation of a material point about a particular position in a flow field, and it may be regarded as a measure of the local angular velocity of the fluid [1]. Typically, vorticity measurement systems are designed to probe one of the components of the vorticity vector $\mathrm{\omega \u20d7}$. For simplicity, and without loss of generality, we consider a two-dimensional flow in the $x\u2013y$ plane, where $\mathrm{U\u20d7}=(U,V,0)$ and the vorticity is $\mathrm{\omega \u20d7}=(0,0,\omega )$. The $z$-axis component is defined on the basis of velocity derivatives as $\omega =\partial V/\partial x-\partial U/\partial y$.

Several optical methods for measuring vorticity in a flow—particle image velocimetry (PIV) [7–9] and laser Doppler velocimetry (LDV) [10,11], among others—attempt to determine first the instantaneous velocity components $U$ and $V$, and then differentiate velocity data to yield the vorticity field component $\omega $. By whatever means the velocity measurements are estimated, they must be made simultaneously over several closed space locations from which spatial gradients $\partial V/\partial x$ and $\partial U/\partial y$ can be evaluated using finite difference schemes. As any real measure needs to consider the length scales $dx$ and $dy$, flow measurements are always integrated over some area and a spatial average vorticity is measured. The accuracy of the calculated vorticity depends on the spatial resolution $dx$ and $dy$ of the velocity sampling and the uncertainty error on estimates of velocity differences $dU$ and $dV$.

To overcome the shortcomings of finite difference methods for vorticity measurements, other measurement methods use imaging techniques to observe local rotation of the flow. Vorticity optical probing (VOP) uses Gaussian laser beams to illuminate the passage of probe particles embedded in the flow to obtain, by image analysis, information about their trajectories [12]. Although small probe particles suspended in a flow will react to fluctuations of rotation in the flow, allowing the vorticity of the flow to be probed directly as it moves along a streamline, it is not always possible—or even convenient—to implant probe particles into the fluid whose dynamics needs to be characterized.

Here we consider a new optical sensing technique that is not dependent upon direct velocity measurements or the use of probe particles, and directly measures the local vorticity of fluid elements. The technique we propose here is akin to laser Doppler anemometry using LG laser beams [3] to illuminate the flow and obtain, by observing the transversal Doppler effect in the reflected signal [4–6], information about vorticity (see Fig. 1).

The vorticity is closely related to the flow circulation. The circulation about a closed contour in a fluid is a scalar integral quantity and measures the macroscopic rotation for a finite area of the fluid. It is defined as the line integral ${C}_{o}=\oint \mathrm{U\u20d7}\xb7\mathrm{d}\mathrm{l\u20d7}$ evaluated along the contour of the component of the velocity vector $\mathrm{U\u20d7}=(U,V)$ that is locally tangent to the contour. If circulation is considered along a closed circular loop of radius ${\rho}_{0}$ lying in the $x\u2013y$ plane, it results in ${C}_{o}={\int}_{0}^{2\pi}{U}_{\varphi}({\rho}_{0},\varphi ){\rho}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{d}\varphi $, where we make use of cylindrical coordinates $\mathrm{\rho \u20d7}=({\rho}_{0},\varphi )$ along the circular contour, and ${U}_{\varphi}=-U\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\varphi +V\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\varphi $ represents the azimuthal component of the velocity of the fluid. This component of velocity defines the local angular velocity of the flow $\mathrm{\Omega}={U}_{\varphi}({\rho}_{0},\varphi )/{\rho}_{0}$. Interestingly, for a specific circular flow area, the Stoke’s theorem ${C}_{o}={\iint}_{A}(\nabla \u20d7\times \mathrm{U\u20d7})\xb7\hat{n}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{d}A$, where $\hat{n}\equiv \hat{z}$ is the unit vector normal to the loop, allows expressing the average normal component $\omega =(\nabla \u20d7\times \mathrm{U\u20d7})\xb7\hat{n}$ of the vorticity field as the circulation ${C}_{o}$ of the velocity field around the considered fluid element divided by the loop elemental area $\pi {\rho}_{0}^{2}$. It results in a simple expression for the flow vorticity in terms of a line integral of the azimuthal components ${U}_{\varphi}$ of velocity:

Let us assume that a paraxial LG light beam propagating along the $z$ axis illuminates a system of noninteractive, independent small scatterers moving with the flow with velocity $\mathrm{U\u20d7}$ and undergoing translation relative to the scattering volume defined by the illumination beam (see Fig. 1). The incident radiation at the transverse position $\mathrm{\rho \u20d7}$ of scatterers across the beam wavefront can be written as

For an incident LG laser beam with radial mode number $p=0$, arbitrary azimuthal mode number $m>0$, and beam radius ${\omega}_{0}$, the phase $\mathrm{\Phi}(\mathrm{\rho \u20d7})$ in the transverse profile depends only on the azimuthal angle as $m\varphi $, and the intensity distribution ${|{E}_{0}(\rho )|}^{2}$ describes a central dark spot surrounded by a very narrow, bright ring whose radius of maximum intensity is ${\rho}_{0}={\omega}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{m/2}$ [3]. A moving scatterer going through the light ring will observe the azimuthal phase gradient ${\nabla \u20d7}_{\perp}\mathrm{\Phi}=m\hat{\varphi}/{\rho}_{0}$ defined by the LG beam. Consequently, and due to the transverse velocity $\mathrm{U\u20d7}$ of the scatterer, the time rate ${\nabla \u20d7}_{\perp}\mathrm{\Phi}\xb7\mathrm{U\u20d7}$ of the echo phase signal yields a frequency shift ${f}_{\perp}$ that is written as [3]

The frequency Doppler shift given by Eq. (3) depends only on the local angular velocity of the scatterer in the flow $\mathrm{\Omega}={U}_{\varphi}({\rho}_{0},\varphi )/{\rho}_{0}$ along the doughnut-shaped observation region $\rho ={\rho}_{0}$ defined by the light beam. Equation (3) allows expressing the circulation contour integral of the velocity ${U}_{\varphi}({\rho}_{0},\varphi )$ and the corresponding average vorticity $\omega $ in Eq. (1) in terms of the frequency transversal Doppler shift ${f}_{\perp}({\rho}_{0},\varphi )$ as

The line integral in Eq. (4) describes the frequency centroid $\u3008{f}_{\perp}\u3009$, the arithmetic mean or average of ${f}_{\perp}({\rho}_{0},\varphi )$ along the ring-like observation region,

The frequency centroid $\u3008{f}_{\perp}\u3009$ estimation is typically based on the spectrum of the observed signal. The return backscattered signal in time—a compound of signals with different frequency Doppler shift triggered by the multiple components of velocity ${U}_{\varphi}({\rho}_{0},\varphi )$ along the annular illumination beam—can be Fourier transformed to define its frequency spectrum. The characteristic return Doppler spectrum is a histogram of Doppler frequency components describing the spectral content of the returned signal, and it can be used to calculate the frequency centroid $\u3008{f}_{\perp}\u3009$ as the average of the frequencies present in the signal.

We use numerical simulations and experiments with selected engineered flows to demonstrate the viability of the proposed method. When a set of independent scatterers, moving with velocity $\mathrm{U\u20d7}$, passes the ring-like observation region given by Eq. (2), it generates a burst of optical echoes that contributes to the received optical signal. We apply a superposition model for the scattering process that directly gives the complex amplitude of the return signal as the sum of the fields scattered by all the scatterers illuminated by the LG beam (see Supplement 1 for details). The use of a realistic signal model illustrates the dependence of the results on the different experimental parameters and allows addressing the problem of vorticity estimation under the supposition of both additive (receiver) and multiplicative (speckle) noise, those producing great return signal variability. We assume that the Doppler measurement system uses heterodyne detection—the most straightforward to set up experimentally—where the scattered light is coherently mixed on the receiver with a more intense reference beam, which acts as a local oscillator [13].

In order to proceed with the numerical experiments, we simulate the signal returns by direct implementation of the superposition model (Eq. (S1) in Supplement 1). Figure 2 shows the result of our numerical experiments on two different flow patterns. The technique is tested in a steady laminar flow [Fig. 2(a)], in which the flow vorticity is known, and in a complex flow around a circular cylinder immersed in a uniform flow [Fig. 2(b)]. The use of realistic numerical experiments illustrates the dependence and the effects of several flow and illumination parameters on the performance of the probing technique. It allows choosing the best measurement parameters and addressing the optimization problem of vorticity estimation. In these experiments, we consider an incident LG laser beam with radial mode number $p=0$, azimuthal mode number $m=10$, and beam radius ${\omega}_{0}=45\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$. The illuminating beam phase changes from zero to $2\pi $ ten times around the azimuth and the intensity distribution shows a bright ring of radius ${\rho}_{0}=100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$.

Even though any value of the mode number $m$ would produce similar results on the measurement of flow vorticity, using a large $m$ is desirable as it makes the frequency Doppler shift ${f}_{\perp}$ given by Eq. (3) higher. As the frequency centroid estimation $\u3008{f}_{\perp}\u3009$ on the spectral measurements is enhanced when higher frequencies are present, using increasingly larger values of $m$ seems desirable. However, optical beams that show larger azimuthal phase variation (large value of $m$) are usually difficult to implement in practical experimental systems that make use of spatial light modulators (SLMs), since smaller spatial features should be controlled. In the experiments shown here we used spatial modes with different values of $m$. In most cases a value of $m=10$ turned out to be an optimum choice.

Figure 2(a) shows the measurement of vorticity in a laminar pipe flow. In the numerical experiment, the fluid is flowing along the longitudinal $y$ axis through a closed channel of radius $R=2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$. The transversal velocity $\mathrm{U\u20d7}=(0,V)$ of the flow describes a parabolic profile of velocities along the transversal $x$ axis that varies from zero at the channel ends to a maximum of ${V}_{0}=4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$ along the center of the channel. The parabolic profile of velocities $V={V}_{0}[1-{(x/R)}^{2}]$ gives the linear vorticity profile $\omega =2\text{\hspace{0.17em}}\text{\hspace{0.17em}}{V}_{0}/{R}^{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x$. The measurements with LG beams reproduce very closely these expected vorticity values.

In a different numerical experiment, Fig. 2(b) shows vorticity in a complex flow created by the unsteady separation of fluid around a cylindrical object located upstream (not shown in the graph). We estimate the velocity field $\mathrm{U\u20d7}$ using a numerical tool for flow simulation. From the numerical velocity field $\mathrm{U\u20d7}=(U,V)$ we calculate the expected $z$ component of vorticity as $\omega =\partial V/\partial x-\partial U/\partial y$. The flows on opposite sides of the cylindrical object interact in an extended region and produce a regular circulation pattern. The energy of the vortices is ultimately expended by viscosity as they move farther downstream, and the regular pattern disappears. The velocity field is pictured in the right graph with a set of streamlines that are tangent to the flow velocity vector. The color scale in the same graph gives an idea of the vorticity magnitude. The left plot compares a measure of flow vorticity with LG beams and the corresponding theoretical expectations. In the simulation, the measurement is realized across the flow, downstream from the cylindrical object.

The feasibility of the proposed method to measure flow vorticity is also verified through the experiments (see Fig. 3). Using the insight realized by numerical experiments into the problem of vorticity estimation, the operation parameters of the test experimental system were established as described in Supplement 1.

A heterodyne receiver based on a modified Mach–Zehnder interferometer was used for experiments, as shown in Fig. 3(a). In the experimental setup, a collimated Gaussian beam is divided by a polarized beam splitter into a reference beam and a probe beam. A phase shifter is used to frequency shift the reference beam by 1 KHz. The probe beam acquires the desired phase profile after impinging onto a computer-controlled SLM. This structured light is filtered and made to shine onto the target flow. Light reflected by particle flows is made to interfere with the reference beam using a beam splitter, and the interference signal is captured using balanced photodetection.

In order to emulate different types of flows, we use a digital micromirror device (DMD). A DMD is an array of individually controlled micromirrors that can be switched on and off to define specific spatial and temporal reflection patterns. By controlling which specific mirrors are in the on or off states, and the timing between these states, we can emulate different types of physical trajectories and velocities of reflecting particles moving with a flow. At each position where the particle would be located, light is reflected back to the detector, while no light is reflected elsewhere. This is equivalent to having a two-dimensional flow in a transverse plane. This system is very convenient to demonstrate in the laboratory the feasibility of the scheme put forward here. It allows emulating different types of flows with good control of the experimentally relevant parameters, such as the velocity profile (Visualization 1 shows one of the flows implemented in the DMD).

In flows over stationary flat plates, there is a gradient of velocity as the fluid moves away from the plate, and the fluid tends to move in layers with successively higher speed. In Figs. 3(b) and 3(c), we test the DMD-based experimental setup with two bidimensional laminar boundary layer flows characterized by parabolic and linear velocity profiles, respectively. In the experiments, the fluid is flowing along the longitudinal $y$ axis and the transversal velocity $\mathrm{U\u20d7}=(0,V)$ has a maximum of ${V}_{0}\approx 25\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$ at a distance $R\approx 6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ from the stationary layer. As a parabolic profile of velocities gives a linear vorticity profile, a linear profile $V={V}_{0}(x/R)$ gives a constant vorticity profile $\omega ={V}_{0}/R$.

Experimental measurements show the expected linear vorticity profile over the parabolic profile of velocities [Fig. 3(b)] and a constant vorticity profile over a linear velocity profile [Fig. 3(c)]. In both cases, there are small differences between theoretical and experimental, as all measurements are subject to some uncertainty due to the limited accuracy of the measurements in the DMD-generated particle flows and the concurrent limitations to dynamic speckle reduction. Basically, as speckle noise on the spectral measurements degrades the quality of the spectral data, speckle needs to be reduced, usually by spectral accumulation. However, the number of independent unsmoothed sample spectra that can be measured on DMD-generated particle flows is limited. This lack of adequate spectral accumulation is the main source of uncertainty in our vorticity measurements. Additionally, there is some uncertainty in the speed velocity of the DMD-generated particle flows due to the discrete nature of the micromirror array that also affects the accuracy of our results.

In any case, both experiments show that the vorticity profiles extracted from the measurements using the least-squares approach in a regression analysis (blue, solid lines) are well into the uncertainty limits to the theoretical expectations as defined by the DMD-limited accuracy (red, dashed lines).

In conclusion, the problem of measuring vorticity in a flow has been confronted. We propose an optical technique that uses LG beams, characterized by ring-like intensity distributions and azimuthal phase variations, to sense rotation at every point in a flow. We develop the theoretical background behind the modeling of optical measurement of vorticity in a flow, identifying the required assumptions and input beam parameters. The spectral properties of the return signal, and the spectrum centroid integral in particular, are fundamental to interpretation of experiments used in flow vorticity monitoring. By using numerical simulations and laboratory experiments, we assess the feasibility of the sensing technique and identify the accuracy of vorticity measurements from return signals affected by target speckle and receiver noise.

## Funding

Government of Spain, Ministerio de Ciencia e Innovación (MICINN) (TEC 2012-34799); Institució Catalana de Recerca i Estudis Avançats (ICREA).

## Acknowledgment

C.R.G. would like to thank V. Rodríguez-Fajardo for useful help to emulate the experimental velocity profiles.

See Supplement 1 for supporting content.

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