Vortex dynamics and applications to gaseous optical elements

Gas flow in the form of a rotation about an axis is known as a vortex. A typical vortex consists of two distinct regions: 1) an inner, or core region located near and around the principal axis, where the gas is locked into rigid body rotation (forced vortex) and 2) an outer region, where the fluid elements, while moving along a circular path, do not rotate about their own axis (free or potential vortex) [1]. The transition between these two regions depends on details of the fluid and the surroundings. Numerous models have been proposed to describe the flow characteristics as a function of the vortex parameters [2, 3, 4, 5, 6, 7].

In previous work we experimentally demonstrated [8] and simulated [9] the lensing properties of spinning gases. We showed that near the axis of rotation the gas density profile (and refractive index) has a parabolic radial dependence. Farther away from the axis the gas density deviates from a parabola, leading to aberrations. In a simple axisymmetric case it is possible to show analytically that rigid body rotation of the core has a parabolic density profile (Appendix A). By limiting the optical aperture to the core the aberrations can be substantially reduced.

In this paper we report results of direct measurements of the gas density distribution inside a vortex. Two distinct cases are described in detail: 1) short vortex with diameter of the core comparable to the length of the vortex (hereinafter referred to as the gas lens configuration), and 2) a vortex that is much longer than its core diameter. The latter configuration can potentially be operated as laser guiding structure (waveguide) if the gas is ionized (hereinafter referred to as the guiding structure configuration).

A schematic of the gas lens is depicted in Figure 1a. Compressed gas (red arrows) flows into the lens and is forced into rotation inside the lens body. Spinning gas exits the gas lens along its optical axis (blue arrows). The diameter of the exit holes defines the clear aperture of the lens. In most cases here it is larger than the core diameter of the vortex. Gas supply lines and gas lens geometry were designed to allow for the possibility of choked flow at the exit holes of the lens. In all the experiments the internal diameter of the lens body was 5 or 6 mm, the exit holes were in a range of 1.5-2 mm in diameter, and the lens thickness varied between 1 and 2 mm. The guiding structure configuration had similar dimensions except the length along the optical axis (thickness in case of a lens) that was 16 mm.

The lens had two independent gas inlets. Gas flow into each inlet was controlled and/or measured by a mass-flow-rate controller. Pressure regulators upstream of the mass flow controllers regulated the stagnation pressure $p_{0}$ of the system. A high capacity vacuum pump evacuated gas from the vacuum chamber, an adjustable valve controlled the pumping rate, and the backing pressure $p_{b}$ was monitored by a vacuum gauge. The gas lines and the optical setup are shown schematically in Figure 2a. All measurements presented in this paper were done under steady state conditions, when the gas flow rate and vacuum chamber backing pressure were constant. Nitrogen gas was used in all experiments.

The lens was secured on a holder (pink plates in Figure 1b) positioned in a vacuum chamber with optical axis aligned along a probe beam of a Mach-Zehnder interferometer using a 532 nm CW diode laser. A lens was used to image the exit plane of the gas lens onto the interferometer camera. The wavefront phase (typical example shown in Figure 2b along with lineouts in Figure 3a) was extracted from the interference patterns for different flow conditions. In each case the near-field intensity profile of the laser beam was recorded 30 cm from the lens (Figure 2c). The flow rate through the gas lens was adjusted by changing the backing pressure $p_{b}$ in the vacuum chamber, while keeping the upstream stagnation pressure $p_{0}$ fixed. For sufficiently small values of $p_{b}$ the flow rate was clamped due to a pressure discontinuity between the gas lens exit pressure $p_{e}$ and $p_{b}$ which is indicative of choking (blue dots in Figure 3d).

The phase in the central portion of Figure 3a resembles a parabola and corresponds to the core of the vortex as shown in Figure 3b. The core has the optical properties of a negative lens with a focal length inversely proportional to the curvature of the parabola. Outside of the core the phase (and the refractive index) turns over. This localized part of the refractive index variation is responsible for the ring structure observed in the near-field image in Figure 2c (see Section 3c).

Studies of the longer, guiding channel vortices were made with the same experimental apparatus and with the same set of diagnostics. In this case the gas density inside the vortex was reduced, to decrease the accumulated phase shift and to make it comparable to that obtained in the shorter gas lens structure. Since the long vortex was designed for plasma-based laser-guiding applications, it was important to estimate the absolute gas density inside the vortex channel. Interferometry, however, can provide only the information about the relative density [10]. To get the absolute gas density we inserted a thin probe into the gas vortex as shown in Figure 1a (green tube). The probe consists of a small thermocouple attached to an open end of a 0.7 mm diameter tube. The tube was connected to a manometer. The probe was positioned radially at the edge of the clear aperture and longitudinally near the mid-plane of the vortex. The end of the tube was oriented perpendicularly to the gas flow streamlines, providing static pressure and temperature measurements. Pressure and temperature at known locations inside the vortex provided the gas density value from the ideal gas equation and were used for calibration of the interferometry data (Section 4).

The major tool for the experimental data analysis is an open source, in-house-developed, Python Zernike polynomials library ’zernike’ [11]. The library has tools for expansion of an arbitrary wavefront into an orthogonal set of Zernike polynomials as well as reconstruction of the wavefront from known Zernike coefficients. Such a decomposition of the optical phase surface provides a general means to characterize the optical properties of the gas lens, including the focal length and aberrations [9].

In contrast to short gas lenses, where the absolute value of the gas density might seem unimportant, in laser guiding applications it is critical to know the plasma (gas) density and to be able to adjust it for particular experimental requirements [16]. Fluid dynamics simulations of the gas lens [9] indicate that the radial gas density profiles are almost the same along the optical axis inside the lens. The gas density drops quickly as soon as gas exits the lens. Using these assumptions on the gas density distribution, we can retrieve the actual density values from interferometric measurements. If we ignore small contributions from the vortex structure outlets, the density is actually given by the average density measured by means of interferometry. This is especially true for the long vortices we propose to use as laser guiding structures.

The radial gas density profile in the vortex is proportional to the measured phase $\Phi(r)$

where $n_{0}$ is the refractive index of the gas at a specific number density $N_{g0}$, $k$ is the laser wavenumber, and $L$ is the length of the vortex [10]. The relative phase was extracted from the measurements of the optical interference pattern generated by the group velocity delay between a reference laser pulse and a laser pulse propagating through the vortex. The final phase change $\Phi(r)$ was calculated by subtracting this relative phase from the reference (flat) one, acquired in a separate measurement.

Since the Mach-Zehnder interferometer is highly susceptible to shot-to shot instabilities Equation 2 required an additional calibration. The gas density calibration was done by direct determination of the gas density at one fixed location. Using the probe described in Section 2, we measured gas temperature and static pressure at the edge of the clear aperture of the guiding tube (a small shadow was visible on a side of interferometry image). We assumed that small tube of the probe did not disturb the flow (we detected no effect on interference image). For known pressure and temperature the gas number density can then be calculated from the ideal gas equation:

where $P_{1}$ and $T_{1}$ are measured by the probe pressure and temperature, $N_{A}$ is Avogadro number, and $R_{g}$ is gas constant.

The probe was installed in a 16 mm long guiding structure and the gas density was recorded for different flow rates. Figure 5 shows readings of the probe used in calibration of the gas density profiles. The density curves in Figure 5 are radial lineouts of stationary vortices at corresponding flow rates. The vortices demonstrate good stability, but retrieval of the phase was hampered by fluctuations around central point and the far wings, where interferometry readings were noisy. As expected, for lower absolute gas densities the relative depth of the density depression decreases.

While the vortex lens can find applications in both negative (neutral gas) and positive (plasma) configurations [8], laser guiding is possible only in a plasma channel with a density minimum on axis. As shown above, the gas density in the vortex is already optimized for laser guiding and it is important to ionize the gas in a ’soft’ manner, without significant changes to the density profile. Laser induced ionization and simple capacitive discharges modify the density by shock waves that move gas or plasma [17]. A similar setup involving bulk, soft ionization is under consideration at the AWAKE project at CERN [18]. A promising approach is to use a high power helicon discharge, capable of generation of a meter-long cylinder of high density plasma [19]. Ionization of the flowing gas is not well understood and in case of rotational motion might lead to new interesting physics, due to the unusual thermal properties of spinning gases [20, 21].

The neutral gas lenses have small useful aperture (the core) and long focal length, leading to effective f-numbers in the range of hundreds. Ionization, in addition to flipping the sign of the focal length, will shorten the focal length by an order of magnitude or more [8] and reduce the f-number to useful values. Such a plasma lens can be used for enhanced focusing of high intensity lasers [22] and for coupling of intense laser beams into guiding channels.

In the case of very short f-number plasma lenses, the paraxial approximation breaks down, and higher order terms must be added to the radial density function to correct for spherical aberration. Ideally one would like to match the profile of the ideal plasma lens [23], which accounts for both radial and longitudinal density variations. A further dimension of control is likely needed to approach this ideal, e.g., additional inlets or outlets designed to influence the longitudinal component of the flow directly.

The parabolic gas density profile demonstrated in the guiding structures can reach the low $10^{18}$ $cm^{-3}$ gas densities (Figure 5). For practical laser wakefield acceleration application one requires lower densities, approaching $10^{17}$ $cm^{-3}$, over a one-meter long guiding plasma channel [24]. The lower gas densities require redesign of the vortex structure, while meter-scale length might be achieved by stacking multiple short stages together.

In summary, we have demonstrated robust operation of gas lens at different flow parameters. The useful aperture, corresponding to the vortex core was measured and characterized in terms of Zernike polynomials. The focal length of the gas lens can be easily adjusted by varying the flow rate. The shortest focal length appeared at the choking condition. We generated a 16 mm long guiding structure, measured its absolute gas density distribution, and demonstrated the possibility of adjusting this density in range of $10^{18}$ $cm^{-3}$.

Consider the flow in the straight pipe conveying the gas to the two outlets. The fluid equations describing the gas dynamics are expressible in the form [25]

where $d/dt=\partial/\partial t+\mathbf{v}\cdot\nabla$, $\mathbf{v}$ is the velocity, $\rho$ is the density, $p$ is the pressure, and $\nu$ and $\zeta$ are the kinematic viscosity and the second viscosity, respectively.

Next employ the definition of vorticity $\boldsymbol{\omega}\equiv\nabla\times\mathbf{v}$ and take the curl of both sides of Eq. A2 to obtain

Assuming $\nabla\cdot\mathbf{v}=0$ along with azimuthal symmetry $(\partial/\partial\varphi=0)$, one can expand

where $(r,\varphi,z)$ are the cylindrical coordinates and $\alpha$ is a constant. The streamlines obtained from Eqs. A4 and A5 are consistent with those derived from the numerical results in [9] near the midplane ($z\ll$ pipe length) and close to the axis $(r\to 0)$. Note also that axial flow velocity $\mathrm{v}_{z}\approx\alpha z$ is outward on either side of the midplane $z=0$.

Another consequence of the assumption $\nabla\cdot\mathbf{v}=0$ is that Eq. A3 simplifies to

The last term in Eq. A6 is referred to as the baroclinic term. Hereinafter it is assumed that the flow is isentropic, ie, $p/\rho^{\gamma}=C$ where $\gamma$ is the ratio of specific heats and $C$ is the constant associated with the Poisson adiabatic. This is the limit of barotropic flow characterized by $\nabla\rho\times\nabla p=0$. By symmetry one may assume that $\boldsymbol{\omega}=\omega(r)\hat{e}_{z}$, in which case Eq. A6 integrates to

where $r_{B}=2\sqrt{\nu/\alpha}$, and then

is the azimuthal velocity of the flow and $\Gamma$ is a constant (proportional to the circulation).

Equations A4, A5 and A7 constitute the Burgers vortex [3] with the following well-known characteristics.

i) The circulation (a macroscopic measure of vorticity) along a closed contour of radius $R$ and normal to the flow is

ii) The angular velocity about the $z$ axis is

In the core region $r\ll r_{B}$ the motion approximates that of a rigid rotor with angular frequency $\Gamma/2\pi r^{2}_{B}$ . On the other hand, far away from the core $r\gg r_{B}$ the rigid rotor motion is dissipated away.

The density variation in the core region (where viscosity is unimportant) may be obtained as follows. Substituting the velocity components given in Eqs. A4, A5 and the top line in Eq. A8 into Eq. A2 one obtains for steady flow

In the limit of a strong vortex $\Gamma\gg 4\pi\nu$ (ie, relatively large azimuthally velocity) and making use of $p/\rho^{\gamma}=C$, Eq. A9 integrates to

The $const$ is determined by employing $p_{0}/\rho_{0}=C$ at the stagnation point $(\mathbf{v}=0,\;r=z=0),$ whence

where $c_{s}=\sqrt{\gamma p_{0}/\rho_{0}}$ is the speed of sound. Near the axis the density has a quadratic dependence on radius

provided $r\ll min\left[r_{B},\;c_{s}/\left(\Gamma/2\pi r^{2}_{B}\right)\right]$.

This work was sponsored by Office of Naval Research (ONR) (N0001417WX01788) and NRL internal program.