2. Linear theory

We describe the interaction between the plasma column and the electric perturbation at the trap wall within the framework of the 2-D electrostatic, cold fluid drift model, whose details are found in Davidson (Reference Davidson1990). In brief, in an electrostatic approximation the electric field is related to the electric potential as $\boldsymbol {E}( \boldsymbol {r},t ) = -\boldsymbol {\nabla } \phi ( \boldsymbol {r},t)$; after averaging over the fast axial bounce and cyclotron rotation and neglecting inertial effects, the velocity of the plasma (fluid) element in the transverse plane is reduced to $\boldsymbol {v}( \boldsymbol {r},t) = -\boldsymbol {\nabla } \phi ( \boldsymbol {r},t)/B \times \hat {e}_z$, where B is the intensity of the magnetic field, directed along the longitudinal axis $\hat{e}_z$. These relations are coupled to the 2-D continuity equation $\partial _t n( \boldsymbol {r},t)+\boldsymbol {v}( \boldsymbol {r},t)\boldsymbol {\cdot }\boldsymbol {\nabla }( n( \boldsymbol {r},t) )=0$ and to the Poisson equation $\nabla ^2\phi ( \boldsymbol {r},t) = en( \boldsymbol {r},t)/\varepsilon _0$, for a distribution of electrons of charge $-e$ and density $n( \boldsymbol {r},t)$ with the boundary conditions at the trap wall $r = R_w$, and $\varepsilon_0$ the vacuum permittivity. By the aforementioned analogy, the plasma properties $\{ en/\varepsilon _0B, \boldsymbol {v}, \phi /B\}$ result isomorphic to the fluid ones $\{ \zeta, \boldsymbol {v}, \psi \}$, where $\zeta = ( \boldsymbol {\nabla } \times \boldsymbol {v})_z$ is the fluid vorticity and $\psi$ is the streamfunction defined by the relation $\boldsymbol {v} = -\boldsymbol {\nabla }\psi \times \hat {e}_z$.

A linear treatment considers small perturbations of the axisymmetric equilibrium density $n^0( r)$ and electric potential $\phi ^0( r)$, expressed as a series of azimuthally travelling waves (KH modes, also called diocotron modes in the non-neutral plasma nomenclature)

(2.1)\begin{gather} n\left(r,\vartheta,t \right) = n^0\left( r\right) + \sum_{l={-}\infty}^{+\infty} \delta n^l\left(r\right)\exp\left({\rm i}l\vartheta-{\rm i}\omega t \right), \end{gather}

(2.2)\begin{gather} \phi\left(r,\vartheta,t \right) = \phi^0\left( r\right) + \sum_{l={-}\infty}^{+\infty} \delta \phi^l\left(r\right)\exp\left({\rm i}l\vartheta-{\rm i}\omega t \right), \end{gather}

with $\omega$ the complex oscillation frequency. Plugging these into the continuity and Poisson equations we get the eigenvalue equation for the generic $l$th perturbation mode. For a uniform equilibrium density $\tilde {n}$ and radius $R_p$, i.e. $n^0( r) = \tilde {n}H(R_p - r)$ (with $H$ the Heaviside step function), in the absence of potential perturbations at the wall, the solution of the eigenvalue equation yields the linear eigenfrequency $\varOmega _l$ for each diocotron mode $l$

(2.3)\begin{equation} \varOmega_l = \omega_D\left[ l-1+\left( \frac{R_p}{R_w}\right)^{2l}\right], \end{equation}

with $\omega _D = \tilde {n}e/2\varepsilon _0B$ the fundamental diocotron frequency, that is also the rigid rotation frequency $\omega _r$ of the unperturbed circular vortex.

For our specific problem with its own boundary condition (b.c.), i.e. an electric potential perturbation at the wall $R_w$, the general solution scheme proceeds as follows:

1. (i) The b.c. is recast as a Fourier series $\delta \phi ( R_w,\vartheta,t) = \sum _{l=-\infty }^{+\infty }c_l( t)\exp ( {\rm i}l\vartheta )$.

2. (ii) The b.c. thus obtained is Laplace transformed in the time variable to get $\delta \phi ( R_w,\vartheta,s) = \sum _{l=-\infty }^{+\infty }c_l( s)\exp ( {\rm i}l\vartheta )$.

3. (iii) The eigenvalue problem is solved in the Laplace-transformed domain to obtain the perturbed potential at the plasma surface $\delta \phi ( R_p,\vartheta,s)$.

4. (iv) By Laplace inverse transforming the solution we finally get $\delta \phi ( R_p,\vartheta,t)$. Depending on the form of the b.c., this expression will manifest an eventual resonance condition yielding a secular growth of a specific mode.

The general solution of the problem is in the form

(2.4)\begin{equation} \delta \phi\left( r\right) = \begin{cases} Ar^l+Br^{{-}l} & r\in\left[0,R_p\right]\\ Cr^l+Dr^{{-}l} & r\in\left[R_p,R_w\right], \end{cases} \end{equation}

whose coefficients are found using the b.c. as well as potential finiteness at $r=0$, radial electric field discontinuity at $r=R_p$ and potential continuity at $r=R_p$. In the Laplace-transformed domain they read

(2.5) \begin{equation} \begin{cases} A = \dfrac{c_l\left( s\right)\left( {\rm i}s-l\omega_D\right)R_w^{{-}l}}{{\rm i}s-\varOmega_l}\\ B = 0\\ C = \dfrac{c_l\left( s\right)\left( {\rm i}s-\left(l+1 \right)\omega_D\right)R_w^{{-}l}}{{\rm i}s-\varOmega_l}\\ D = \dfrac{c_l\left( s\right)\omega_dR_p^{2l}R_w^{{-}l}}{{\rm i}s-\varOmega_l} \end{cases} \end{equation}

As an explicit example, let us use a dipole rotating-field b.c. for $N=4$ sectors

(2.6)\begin{equation} \delta\phi\left(r=R_w,\vartheta,t \right) = \sum_{m=0}^3 V_m\left(t\right) \left[ H\left( \vartheta-m\frac{\rm \pi}{2}\right) - H\left( \vartheta-\left( m+1\right)\frac{\rm \pi}{2}\right) \right], \end{equation}

with

(2.7)\begin{equation} V_m\left(t \right) = V_{dr}\cos\left( \omega_{dr}t+\sigma m \frac{\rm \pi}{2}\right). \end{equation}

This condition corresponds indeed to four potentials $V_m$ oscillating with amplitude $V_{dr}$ and a $\sigma m {\rm \pi}/2$ phase shift between adjacent sectors; in particular, with $\sigma = -1$ the potential is a corotating dipole, i.e. a dipole field rotating with the same orientation as $\omega _r$, while $\sigma = +1$ is a counterrotating dipole. For $\sigma = \pm 2$ we would have a purely oscillating (non-rotating) quadrupole. Let us consider $\sigma = \pm 1$. Rewriting the b.c. as a Fourier series, only some terms of the series appear, and specifically only odd ones

(2.8)\begin{equation} c_l = \frac{V_0-V_1}{\left( 4k+1\right){\rm \pi}}+\frac{V_0-V_1}{\left( 4k+3\right){\rm \pi}}-{\rm i}\left( \frac{V_0+V_1}{\left( 4k+1\right){\rm \pi}}-\frac{V_0+V_1}{\left( 4k+3\right){\rm \pi}}\right),\quad k\in \mathbb{Z}^+, \end{equation}

and with $V_m$ forms of (2.7) for $m = 0,1$ we get two different cases for $\sigma = \pm 1$

(2.9)\begin{equation} \sigma ={+}1:\quad c_l\left(t\right) = \begin{cases} \dfrac{V_{dr}\exp\left( -{\rm i}{\rm \pi}/4\right)}{{\rm \pi}\left( 4k+1\right)}{\rm e}^{+{\rm i}\omega_{dr}t} = \varepsilon {\rm e}^{+{\rm i}\omega_{dr}t} & l = 4k+1,\\ \dfrac{V_{dr}\exp\left( +{\rm i}{\rm \pi}/4\right)}{{\rm \pi}\left( 4k+3\right)}{\rm e}^{-{\rm i}\omega_{dr}t} = \varepsilon' {\rm e}^{-{\rm i}\omega_{dr}t} & l = 4k+3, \end{cases} \end{equation}

and

(2.10)\begin{equation} \sigma ={-}1:\quad c_l\left(t\right) = \begin{cases} \dfrac{V_{dr}\exp\left( -{\rm i}{\rm \pi}/4\right)}{{\rm \pi}\left( 4k+1\right)}{\rm e}^{-{\rm i}\omega_{dr}t} = \varepsilon {\rm e}^{-{\rm i}\omega_{dr}t} & l = 4k+1,\\ \dfrac{V_{dr}\exp\left( +{\rm i}{\rm \pi}/4\right)}{{\rm \pi}\left( 4k+3\right)}{\rm e}^{+{\rm i}\omega_{dr}t} = \varepsilon' {\rm e}^{+{\rm i}\omega_{dr}t} & l = 4k+3. \end{cases} \end{equation}

To observe the evolution of the perturbation on the vortex edge, we Laplace transform the coefficients

(2.11)\begin{equation} \sigma ={+}1:\quad c_l\left(t\right) = \begin{cases} \dfrac{\varepsilon}{s-{\rm i}\omega_{dr}} & l = 4k+1,\\ \dfrac{\varepsilon'}{s+{\rm i}\omega_{dr}} & l = 4k+3, \end{cases} \end{equation}

and

(2.12)\begin{equation} \sigma ={-}1:\quad c_l\left(t\right) = \begin{cases} \dfrac{\varepsilon}{s+{\rm i}\omega_{dr}} & l = 4k+1,\\ \dfrac{\varepsilon'}{s-{\rm i}\omega_{dr}} & l = 4k+3. \end{cases} \end{equation}

Using these expressions in (2.5) we can evaluate the potential (2.4) and in particular the perturbation value at the plasma edge. For the case $\sigma =+1$, via Laplace inverse transform we get $\delta \phi (R_p,\vartheta,t)$ to be

(2.13)\begin{equation} \left. \begin{array}{ll} \displaystyle\varepsilon\left( \dfrac{R_p}{R_w}^l\right) \left[ \dfrac{\varOmega_l-l\omega_D}{\varOmega_l+\omega_{dr}}\exp\left( {\rm i}l\vartheta-{\rm i}\omega_{dr}t\right)+\dfrac{l\omega_D+\omega_{dr}}{\varOmega_l+\omega_{dr}}\exp\left( {\rm i}l\vartheta+{\rm i}\omega_{dr}t\right)\right], & l \!= 4k\!+\!1,\\ \displaystyle\varepsilon'\left( \dfrac{R_p}{R_w}^l\right) \left[ \dfrac{\varOmega_l-l\omega_D}{\varOmega_l-\omega_{dr}}\exp\left( {\rm i}l\vartheta-{\rm i}\omega_{dr}t\right)+\dfrac{l\omega_D-\omega_{dr}}{\varOmega_l-\omega_{dr}}\exp\left( {\rm i}l\vartheta-{\rm i}\omega_{dr}t\right)\right], & l \!= 4k\!+\!3, \end{array} \right\} \end{equation}

where we can see that, while the coefficients $l = 4k+1$ have no poles, the $l = 4k+3$ ones yield a resonance at $\omega _{dr} = \varOmega _l$. In this case the potential perturbation can be recast as

(2.14)\begin{equation} \delta\phi\left( R_p,\vartheta,t\right) = \varepsilon^{\prime}\left( \frac{R_p}{R_w}^l\right)\left[ 1+{\rm i}\left( l\omega_D-\varOmega_l\right)t\right]\exp\left( {\rm i}l\vartheta-{\rm i}\varOmega_lt\right), \end{equation}

where a secular growth appears for a specific mode with $l = 4k+3$, where $k$ is automatically selected as the drive frequency matches one of the $l = 4k+3$-type modes. If we chose $\sigma = -1$ the roles would be flipped in (2.13) and the resonant growth would occur for $l = 4k+1$ wavenumbers while the $l = 4k+3$ ones would be stable. Similarly, the choice $\sigma = \pm 2$ would yield coefficients with $l = 4k+2$ terms only, leading to resonant secular growth of modes with such $l$ values.

Again, it would just be a matter of lengthy but similar algebra to evaluate the case of the b.c. applied to $N=8$ sectors. In this case the b.c. reads

(2.15)\begin{equation} \delta\phi\left(r=R_w,\vartheta,t \right) = \sum_{m=0}^7 V_m\left(t\right) \left[ H\left( \vartheta-m\frac{\rm \pi}{4}\right) - H\left( \vartheta-\left( m+1\right)\frac{\rm \pi}{4}\right) \right], \end{equation}

with

(2.16)\begin{equation} V_m\left(t \right) = V_{dr}\cos\left( \omega_{dr}t+\sigma m \frac{\rm \pi}{4}\right), \end{equation}

where $\sigma$ can take a larger number of values, corresponding to dipole ($\sigma = \pm 1$), quadrupole ($\sigma = \pm 2$) and sextupole ($\sigma = \pm 3$) rotating fields or an oscillating octupole field ($\sigma = \pm 4$). The respective driven modes are listed in table 1.

Table 1. Summary of KH modes that can be excited with a rotating electric field. Given a number $N$ of sectors with a corresponding sector angular span ${\rm \pi} /2$ (for $N=4$) or ${\rm \pi} /4$ (for $N=8$), the suitable multipolarity and rotation orientation of the field is determined by the phase shift between adjacent sectors $\Delta \varphi$. First column indicates the modes $l$ excited with a 4-fold split electrode, second and third rows with an 8-fold split electrode. The integer $k = 0,1,2,\ldots$ finally determines the unique excited mode and is automatically chosen by setting the resonant mode frequency.

We detailed the cases $N=4,8$ as they are the most common choices in experimental set-ups, yet from this analysis we can easily extrapolate to a general case. For any even number of sectors $N$, it is possible to generate co- and counterrotating fields of multipolar order $\leq N-2$ (i.e. $|\sigma | \leq N/2-1$) and an oscillating $N$-polar field ($|\sigma | = N/2$), which enables the excitation of all modes with the exception of wavenumbers that are integer multiples of $N$. Odd sectoring $N$ makes it possible to generate only rotating fields of multipolar order $\leq N-1$; again, all modes can be excited except the integer multiples of $N$.

3. Apparatus, preparation routine and diagnostics

All experiments were performed in the ELTRAP (ELectron TRAP) device, a PMT whose electrode stack is sketched (not to scale) in figure 1. The electrodes are hollow cylindrical shells with an inner diameter of $90$ mm made out of oxygen-free high-conductivity copper. The ‘C’ electrodes are $90$ mm long, ‘S’ ones are $150$ mm long. Any pair of non-adjacent electrodes can be biased to a negative potential – typically $-100$ to $-200$ V – to create an axial electrostatic well with a maximum trapping length ${\sim }1$ m. Electrodes S2, S4 and S8 are azimuthally split into 2, 4 and 8 patches, respectively, which can be used either to impart electric field perturbations or to detect transverse collective modes of the confined particle sample (see figure 2). As one of the confinement voltages is reduced to zero, the plasma can be destructively detected as it flows out of the trap and hits either a charge collector plate (on the left) or a P43 phosphor screen (on the right) set to a positive potential high enough to produce phosphorescence by electron impact (in these experiments, $7$ kV) and thus producing an image of the axially integrated density distribution. The image is captured by a charge-coupled device (CCD) camera with a maximum resolution of $1344\times 1024$ pixels. The electrode stack is installed within a vacuum vessel; the pressure in these experiments was typically in the mid $10^{-9}$ mbar, corresponding to electron–neutral collision times of the order of some tens to $100$ ms. The whole apparatus is surrounded by a solenoid providing a uniform magnetic field of intensity $B\leq 0.2$ T directed along the trap axis.

Figure 1. Sketch of the ELTRAP electrode stack. Hollow cylinders C0 to SH are aligned to form a cylindrical trapping volume with the longitudinal axis set along the direction of a uniform magnetic field $B$. Diagnostic tools are placed at the ends of the stack: on the left, a collector plate connected to a digital oscilloscope, on the right, a phosphor screen biased to a potential $V_{ph} \geq 4$ kV yielding images at plasma ejection that can be captured by a CCD camera. Azimuthally split electrodes S2, S4 and S8 can be used for electric excitations or non-destructive, induced-current diagnostics.

Figure 2. Sketch and naming convention of sectored electrodes. Here, the S2, S4 and S8 electrodes are azimuthally segmented and labelled as depicted, looking from the phosphor screen into the trap.

The electron plasma is produced in situ by means of a ‘sourceless’ technique, namely a very low-power radio-frequency (RF) discharge initiated by a stochastic heating of free electrons in the neutral background. If a RF potential of amplitude 1–10 V and frequency 1–20 MHz is applied to any electrode within the trapping region, a free electron bouncing across the trap length can stochastically increase its energy to some ten electronvolts by interacting with the RF field, thus reaching energies above the first ionization threshold for light gases (the residual atmospheric gas in the vacuum chamber; no additional gas injection is performed). Over times of the order of some seconds, i.e. over many collision times, an electron plasma accumulates in the trapping region, and can reach a steady-state configuration with a production–loss balance. Further details about the RF breeding of the electron plasma can be found in Paroli et al. (Reference Paroli, De Luca, Maero, Pozzoli and Romé2010), Maero et al. (Reference Maero, Chen, Pozzoli and Romé2015) and Maero (Reference Maero2017).

The electron plasma column may often lie off axis and thus entertain a bulk rotation ($l=1$ diocotron mode). After the ionizing RF drive is turned off, the plasma is brought to the trap axis by an active feedback damping (Fine Reference Fine1988) of the offset using two opposite sectors of the S4 electrode: the amplified, filtered and phase-tuned induced-current signal picked up by a sector is sent to the opposite one and the retarding electric field slows down the rotation, thus reducing its amplitude (which is proportional to the mode frequency). Feedback damping is also required as some of the positive ions generated in the plasma-breeding stage, although not efficiently trapped, have a finite residence time in the trapping region and a destructive ion-induced instability of the $l=1$ mode (Fajans Reference Fajans1993; Kabantsev & Driscoll Reference Kabantsev and Driscoll2007; Maero Reference Maero2017) would occur otherwise. A few seconds of feedback damping are enough to let the residual ions leave while the ionization drops as the electrons cool down. The result is a centred electron plasma column (fluid vortex) with a monotonically decreasing radial profile. Depending on the control parameters, we achieve a range of radial density profiles, and albeit there is no perfectly flat radial profile, the decrease is acceptably smooth and monotonic, so that such vortices are stable against diocotron instabilities; furthermore, they have a well-defined edge with a sharp fall-off region, which is another characteristic featured in the theoretical model. Vortices of this kind will be the initial state for the diocotron mode excitation.

All the experiments reported here were performed with the following set of parameters: magnetic field intensity $0.13$ T; confinement potential $-160$ V on C1 and C7; RF plasma-breeding drive in the range 5–8 V, 3–10 MHz on C2; rotating-field drive on S8; diocotron mode pickups on S2T ($l=3$ mode detection) and S4L+S4R (quadrupolar configuration for $l=2$ mode). Figure 3 shows the two classes of initial states, achieved with slightly different parameters in a sequence of separate measurement sessions that were necessary to the systematic investigation presented in the next section (we shall refer to them as profiles A and B, respectively). Each radial density profile (black line) is obtained as a $\theta$-average of the CCD image, denoised and remapped onto a polar grid and then averaged over several shots ($25$ on the left, $10$ on the right). The shot-to-shot repeatability is very good: as repeatability indices, both total charge and dispersion of the plasma radius exhibit fluctuations of $1\,\%-2\,\%$. The initial axisymmetric vortex states displayed radii $R_p\simeq 0.5-0.55 R_w$ and core densities of low- to mid $10^{12}$ m$^{-3}$, yielding rotation frequencies (also shown in figure 3, blue lines) of some ten kHz; as a consequence, the experiments can be considered collision free as at least thousands of rotation periods are guaranteed before collisional diffusion can be observed.

Figure 3. Typical density and rotation profiles of the initial axisymmetric vortex. (a) Profile type A. (b) Profile type B. Black line: radial density profile. Blue line: rotation frequency calculated from the density profile.

4. Selective mode excitation

After the plasma column is prepared in the initial state as described in § 3, the rotating-field drive is applied for a set time on the patches of electrode S8, after which the plasma is released onto the phosphor screen by grounding electrode C7. The phosphor screen image will be the primary source of information on the vortex deformation in this work; the data analysis routine implemented to extract the azimuthal mode decomposition from the image proceeds as follows. We can define the total $l$th mode coefficient or amplitude $A_l$ as

(4.1)\begin{equation} A_l = 2{\rm \pi}\int_{0}^{R_w} \left| \delta n^{l} \left( r\right) \right|r\,{\rm d}r; \end{equation}

hence, the image is denoised and remapped onto a polar grid; the density profile $n(r,\vartheta )$ is decomposed into a Fourier series to obtain the azimuthal perturbation coefficients for each (discrete) radial position $\delta n^{l}(r)$; each coefficient is integrated (summed) over the trap cross-section as in (4.1). The mode amplitudes of the unperturbed initial density profile, stemming from small residual asymmetries, are subtracted; they will anyway be one to two orders of magnitude smaller than the resonant deformation values. In the following, the total mode amplitude $A_l$ is always shown as normalized to the amplitude $A_0$, i.e. the total fluid circulation or plasma line density. We remark that this procedure is in general more accurate than others such as the analysis of the contour deformation, used for instance in Bettega et al. (Reference Bettega, Paroli, Pozzoli and Romé2009), as it yields an integral information that includes the density perturbation over the whole cross-section of the 2-D domain. This is crucial, especially when the vorticity profile does not exhibit a steep fall off: in such a case, the identification of the vortex edge can be difficult and arbitrary and the density perturbation can extend over a significant thickness of the vortex outermost region.

4.1. Mode excitation features: $l=3$ mode analysis

In this and the next subsection we consider the excitation of the $l=3$ mode as a paradigmatic example to analyse the main features of the phenomenon under study. The following § 4.3 will extend the discussion to the whole range of modes we observed. In the vicinity of the resonance, the selected mode grows to its maximum amplitude within some hundred microseconds, corresponding to some tens of mode periods. The approximate frequency location of the resonance can be found from the calculation of the rotation frequency in the vortex core (see figure 3) and the equation for the linear mode frequency. The dependence of the maximum amplitude reached by the excited mode on the drive amplitude is shown in figure 4(a). Here, the $l=3$ mode is induced in a vortex with the profile type A by means of a counterrotating dipole drive (see table 1, left column) in an $N=4$ electrode configuration. This drive configuration is obtained by grouping the S8 sectors into four pairs of adjacent patches. The excitation is applied for a time period of $300\,\mathrm {\mu }\mathrm {s}$, at a rotation frequency of $132$ kHz, which is found experimentally to be the approximate resonance frequency value. The diagram shows a linear dependence of the mode amplitude $A_3/A_0$ vs drive amplitude up to a saturation region above $1.5$ V. Further increase of either the amplitude or the time duration of the excitation does not result in a further growth of the mode, but induces filamentation from the nonlinear structures, i.e. the vertices of the deformed vortex, as seen in figure 4(b): here, the contour plot of the transverse density distribution image is shown for a vortex perturbed with a $1.7$ V drive; the nascent filamentation has created lobes (‘cat's eye’ structures) starting from the tips of the triangle. Once the mode reaches saturation, damping through filamentation progresses and the excited mode will eventually collapse catastrophically, initiating a cascade to lower modes towards the final axisymmetric state. As stated in the introduction, the analysis of the decay stage is among our future goals and will not be discussed here; for the reader's convenience, we just mention that a variety of diocotron mode decay mechanisms has already been observed in the past (Driscoll & Fine Reference Driscoll and Fine1990; Mitchell & Driscoll Reference Mitchell and Driscoll1994; Kabantsev et al. Reference Kabantsev, Chim, O'Neil and Driscoll2014). Notice also that mode saturation is reached already within a small-perturbation regime, as the plasma electrostatic potential at the edge of the core ($r\simeq R_w/2$) is around $20$ V, where a $1.5$ V dipole drive at the trap wall drops to half its value, so that the perturbation-to-plasma potential ratio is less than $5\times 10^{-2}$.

Figure 4. Growth of the excited mode vs drive amplitude. (a) The normalized amplitude of mode $l=3$ is shown for a $300\,\mathrm {\mu }\mathrm {s}$ counterrotating dipole excitation of profile A at a frequency of $132$ kHz. On the right, a contour plot of the axially integrated transverse density distribution for a $1.7$ V drive shows the beginning of mode damping in the form of filamentation creating cat's eye structures. The square box has the size of the trap inner diameter.

We can draw complementary information from the electrostatic signals induced by the transverse collective motions on proper electrode sectors. With this non-destructive and non-perturbative diagnostics, the long-term vortex evolution can be tracked by maintaining the rotating drive well beyond the selected mode saturation. Currently, our electronics, combined with the weak signals induced by high-order diocotron modes, limits our detection to modes up to $l=3$. As mentioned in § 3, a ${\rm \pi}$-span S2 sector (dipole configuration) can pick up modes $l=1,3$ and two opposite ${\rm \pi} /2$ S4 sectors (quadrupole configuration) can detect mode $l=2$. We recorded a $25$ ms $l=3$ excitation of a vortex very similar to profile B, albeit with a $20\,\%$ lower density level; the resonance at $1$ V drive amplitude was found at $54$ kHz. The dipole and quadrupole time signals (red and blue lines, respectively) are shown in the two upper left panels of figure 5; on the right, the frequency regions of interest of their spectrograms are shown, from which the mode contributions can be extracted so that the mode power can be plotted in the bottom panel. First, the signals prove that the vortex deformation seen by means of the optical diagnostics is indeed a KH rotating perturbation and not a static structure. Second, they confirm the growth and damping dynamics described before: mode $3$ reaches saturation within approximately $2$ ms and collapses, hence the fast insurgence of a mode 2 that is then also damped away within $15$ ms. Despite the resolution limitations of the spectrograms imposed by the fast dynamics, it can be seen that the damping of both modes is accompanied by a rising mode frequency, a sign of the nonlinear regime. The constant-frequency line in the dipole signal (also resulting in the higher plateau of the mode power line, bottom panel) is the track of the set frequency drive active along the whole vortex evolution.

Figure 5. Electrostatic signals detected during the vortex excitation. (a,c) Signals detected by ${\rm \pi}$- and quadrupole ${\rm \pi} /2$ angular span pickups. (b,d) Insets of the corresponding spectrograms, showing the presence of an $l=3$ wave decaying into an $l=2$ wave. (e) Mode power extracted from the spectrogram, showing the cascade from the parent mode $3$ to the daughter mode $2$.

4.2. Resonance curve analysis

We also systematically studied the behaviour of the resonance curve against the drive amplitude. We take again as an example the $l=3$ mode, induced this time by the application of a $400\,\mathrm {\mu }\mathrm {s}$ corotating sextupole drive (see table 1, middle column) in an $N=8$ sector configuration on a vortex with profile B. A resonance curve is obtained by repeating the vortex preparation, mode excitation and image acquisition cycle for a set of excitation frequencies around the resonance; iterating the resonance scan for different drive amplitudes we obtained the mode amplitude patterns shown in figure 6(a). Each data point in the diagram is the average over $5$ values, with a maximum relative error $\delta A_3/A_3$ below $10^{-1}$; the error bars are not displayed for image clarity. Here, as the forcing amplitude increases, the deformation grows into the nonlinear regime, highlighted not only by the increase in the peak value of the normalized mode amplitude $A_3/A_0$, but also by the peak shift and the broadening of the resonance curve. The frequency downshift of the resonance peak for increasing drive (and consequently, mode) amplitude is visible in figure 6(b), despite the fact that our sampling resolution limits the accuracy of data points to an uncertainty of ${\pm }0.5$ kHz, and it is an apparent sign that, above a drive amplitude of $500$ mV, the oscillator is already in the nonlinear territory, where the constant mode frequency expression obtained for the linear regime no longer holds. Another nonlinearity flag is the significant progressive broadening of the resonance curve, measured by the full width at half-maximum normalized to the peak frequency and reported in figure 6(c).

Figure 6. Resonance curve behaviour vs drive amplitude for an $l=3$ mode (profile B). (a) Resonance curves for the normalized mode amplitude $A_3/A_0$ at different drive amplitudes. (b) Frequency position of the resonance peak vs drive amplitude. The uncertainty is dominated by the $1$ kHz resolution of the resonance curve sampling. (c) Full width at half-maximum $\delta \omega$, normalized to the peak frequency, vs the drive amplitude. The uncertainty is once again dominated by the sampling resolution and does not exceed $\pm 1.5\times 10^{-2}$.

4.3. Excitation of modes $3$ to $7$

The features discussed for the excitation of mode $l=3$ in §§ 4.1 and 4.2 are common to higher-order modes. We successfully excited all modes up to $l=7$, using in some cases more than one field configuration. As a summary, let us view figure 7, where resonance curves are reported for all these modes. The curves were obtained for a vortex with profile B; all modes were excited with an $N=8$ electrode sector configuration and the proper multipolar field (see table 1, middle and right column) at an amplitude of $1.5$ V for consistency, and adjusting the drive duration in order to reach saturation at resonance ($400\,\mathrm {\mu }\mathrm {s}$ for modes $3$ to $5$, $1$ ms for modes $6$ and $7$). Each data point is the average over $5$ shots; the uncertainties are not reported in the diagram as the absolute error is of the order of $10^{-4}-10^{-3}$. All resonances are well defined; all curves display a contrast (peak-to-tail ratio) $\geq 20$ and the peak frequencies are evenly spaced, with $\nu _{l+1}-\nu _{l} = 32-33$ kHz. The phosphor screen images at the resonance peaks, displayed in figure 8, show that all modes have reached nonlinear saturation, indicated by the lobe structures generated by filamentation around the deformed vortex core, visible to the naked eye. The understandably very weak mode $7$ makes an exception to these general considerations and it is hardly recognized by visual inspection, yet an identifiable resonance emerges from the image analysis.

Figure 7. Resonant curves for different mode excitations (profile B). All modes were induced by suitable rotating drives in an $N=8$ sector configuration (see table 1) at $1.5$ V amplitude.

Figure 8. Vortex deformations at respective resonant frequencies. Axially integrated transverse density distributions are shown for mode excitations from $l=3$ to $7$, at the mode resonant frequencies $67$, $100$, $133$, $165$ and $193$ kHz, respectively, according to the resonance curves displayed in figure 7. The square boxes have the size of the trap inner diameter.

A proof of the perfect selectivity of this mode driving scheme is given by examining all the $A_l$ coefficients. Figure 9 reports coefficients $2$ to $10$ for each excitation geometry, at the respective resonances (e.g. corresponding to the density maps in figure 8). The dominant mode (data point circled in black) is always the one that was intentionally driven, while all other coefficients are at least one order of magnitude lower. There are some exceptions, e.g. $A_6$ and $A_9$ in the $l=3$ excitation, or $A_8$ in the $l=4$ excitation: these are not higher wavemodes but harmonics of the $l=3,4$ dominant modes. We can conclude that spurious deformation components are minimized by this method, while the dominant mode is maximized (saturated).

Figure 9. Deformation mode coefficients for excitation of modes $l=3$ to $7$. Each colour represents the deformation amplitudes $A_2$ to $A_{10}$ for the excitation of a single mode at the respective resonant frequency, corresponding to the peaks in figure 7. The intended and dominant coefficient is circled in black. Dashed lines are drawn only as a guide for the reader.