2D Doppler backscattering using synthetic aperture microwave imaging of MAST edge plasmas

In addition to SAMI, two DBS experiments have been conducted on spherical tokamak plasmas: one focusing on the edge [22], one focusing on the core [10]. Both of these experiments steered their probing beam mechanically.

SAMI uses an array of independently phased, linearly polarised antennas (see figure 3) where the receiving antennas are placed in order to minimise the side-lobe level in the reconstructed image. The array design was optimised using a simulated annealing global optimisation algorithm which is based on the analogy of annealing in metallurgy. The optimisation method is described further in [31] and [32].

SAMI acquires data at one RF frequency at a time and can switch between frequency channels with a switching time of 300 ns during the shot in any order (16 frequencies available 10, 11, 12, 13, 14, 15, 16, 17, 18, 20.5, 22.5, 24.5 26.5, 28.5, 30.5 and 34.5 GHz) primarily probing the edge region of the MAST plasma (see figure 2).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The backscattered RF signal (${{\omega}_{\text{RF}}}$) is received by eight phase sensitive antennas simultaneously. The signal from each antenna (#1 figure 4) is split into I and Q components by a hybrid coupler (#2 figure 4). The I and Q components are then downcoverted by second harmonic mixers (#3 figure 4) where the LO signal (${{\omega}_{\text{LO}}}$) is generated, via a power splitter (#4 figure 4), by a bank of 5–17.25 GHz dielectric resonance oscillators (#5 figure 4). The IF signal is then digitised by a 14 bit 250 Mega samples per second FPGA-controlled analog to digital convertor (ADC in figure 4).

Zoom In Zoom Out Reset image size

Upper and lower side band separation is performed using software post shot as outlined in [33]. IF square waves at 10 and 12 MHz from the FPGA are up-converted to the RF frequency using a second harmonic mixer and the LO signal to form the SAMI probing signal. The $2{{\omega}_{\text{LO}}}+10$ MHz and $2{{\omega}_{\text{LO}}}+12$ MHz signals are launched simultaneouly through active probing antennas 2 and 1 in figure 3 respectively.

The SAMI array is positioned behind a vacuum window made of fused silica. The receiving antennas digitise the backscattered signal along a single polarisation; therefore O and X-mode cannot be separated at present. In section 4 we will discuss the error which this can introduce into our measurements. In future experiments SAMI will be upgraded to dual polarisation antennas which will allow for O and X-mode separation as will be discussed further in section 4.

Figure 5(a) shows a poloidal cross section of the installation of SAMI on MAST. The entire vertical extent of the plasma is visible to SAMI apart from the top ${{20}^{{}^\circ}}$ which is blocked by a poloidal field coil. Figure 5(b) shows the response of the system to a point source emitting at 16 GHz. The power received at the SAMI array is recorded as a function of  ±40° horizontal and vertical viewing angles. Beam forming has been used to focus the receiving beam at each horizontal and vertical angle in the field of view. Note that although the intensity maximum measured by SAMI is in the correct angular location, the side-lobe power can be up to 50% of the maximum. This high side-lobe level results from SAMI being limited to eight receiving antennas. Future 2D DBS systems could use a greater number of receiving antennas allowing them better directivity as will be discussed further in section 4.

Zoom In Zoom Out Reset image size

SAMI can also operate in a passive imaging mode, although this is not discussed in this paper. For SAMI passive imaging experiments and further discussion of the hardware we direct the reader to the relevant papers [31, 33, 36, 37].

The image inversion algorithm employed on the SAMI active probing data is based on the beam forming technique. Beam forming involves applying a phase shift to each of the antenna channels so that when the phase shifted signals are summed together, constructive interference occurs in a particular direction.

Let us consider a receiving array positioned at the origin of a cartesian coordinate system with its field of view centred along the positive x axis. Let the beam be focused at a generalised point specified by $\boldsymbol{r}$ which is a distance r from the origin and is located at horizontal and vertical angles given by θ and ϕ in SAMI image coordinates respectively. The unit vector, $\widehat{\boldsymbol{r}}$, pointing towards $\boldsymbol{r}$ from the origin is given by

$$\begin{eqnarray}&&\widehat{\boldsymbol{r}}(\theta,\phi )=\left(\begin{array}{c} \cos \phi \cos \theta \\ -\cos \phi \sin \theta \\ \sin \phi \end{array}\right)\end{eqnarray} \tag{ 2 }$$

If the position of the i^th antenna in the array is given by $\boldsymbol{x}_{{\boldsymbol{i}}}$ and the array is receiving radiation with a wavelength λ, then in order to focus the array at point $\boldsymbol{r}$ the phase shift applied to the i^th antenna is given by

$$\begin{eqnarray}&&{{\psi}_{i}}(\theta,\phi )=-\frac{2\pi}{\lambda}|r\widehat{\boldsymbol{r}}-\boldsymbol{x}_{{\boldsymbol{i}}}|\end{eqnarray} \tag{ 3 }$$

It is intuitive to see that if a point source were placed at $\boldsymbol{r}$ the signal would be received at each of the antennas with a slightly different phase given by equation (3). Therefore applying this phase shift to each antenna signal before summing them together results in constructive interference in the ($\theta,\phi$) direction. Before being phase shifted and summed together, if one considers the time interval $\Delta t$, a Fourier transform is applied to each antenna channel

$$\begin{eqnarray}&&\hat{S}_{i}^{\text{A}}(\nu )={{\mathop{\int}^{}}_{\Delta t}}S_{i}^{\text{A}}(t){{\text{e}}^{2\pi j\nu t}}\text{d}t\end{eqnarray} \tag{ 4 }$$

where $S_{i}^{\text{A}}$ is the complex signal made up of the I and Q components from the i^th antenna ($S_{i}^{\text{A}}={{I}_{i}}+j{{Q}_{i}}$) and $j=\sqrt{-1}$. The complex signal from the i^th antenna in the frequency domain is denoted by $\hat{S}_{i}^{\text{A}}$. Applying a Fourier Transform to each antenna channel allows the phase shifts denoted in equation (3) to be applied in the frequency domain. This is computationally advantageous as when conducting active probing SAMI DBS experiments one is typically only interested in a small subset of the available  ±125 MHz IF spectrum; most commonly ${{\nu}_{\text{probe}}}\pm 0.2$ MHz where ${{\nu}_{\text{probe}}}$ is the active probing IF frequency. In the frequency domain only the values of interest have to be phase shifted thereby increasing the computational efficiency by a factor of 625 (relative to a time domain phase shift). Let the subset of frequency values included in the receiving beam be denoted by $\Delta \nu$, then the frequency domain synthesised beam signal is given by

$$\begin{eqnarray}&&{{\hat{S}}^{\text{B}}}(\Delta \nu,\theta,\phi )=\underset{i=1}{\overset{N}{\mathop \sum}}\,{{w}_{i}}\hat{S}_{i}^{\text{A}}(\Delta \nu ){{\text{e}}^{j{{\psi}_{i}}(\theta,\phi )}}\end{eqnarray} \tag{ 5 }$$

where w_i is a complex calibration factor correcting for amplitude and phase imbalances between antenna channel hardware and the sum is over N antennas. To measure the intensity in the (θ, ϕ) direction within the frequency range $\Delta \nu$, the magnitude of ${{\hat{S}}^{\text{B}}}$ is integrated across $\Delta \nu$:

$$\begin{eqnarray}&&I(\Delta \nu,\theta,\phi )={{\mathop{\int}^{}}_{\Delta \nu}}|{{S}^{\text{B}}}\left(\nu ;\theta,\phi \right){{|}^{2}}\text{d}\nu \end{eqnarray} \tag{ 6 }$$

By evaluating I over a range of horizontal and vertical viewing angles a 2D map of intensities, such as that shown at the figure 5(b), can be plotted.

Erroneously in equation (3) it is assumed that that the location of the source is known. However, in the far field ($r\gg |\boldsymbol{x}_{{\boldsymbol{i}}}|$) the phase is not sensitive to r and much more so to the directionality, $\widehat{\boldsymbol{r}}$. Therefore, a meticulous value of r is not necessary for accurate image reconstruction. SAMI is sufficiently far away from the plasma in MAST ($\frac{|\boldsymbol{x}_{{\boldsymbol{i}}}|}{r}\sim 0.1$) for the far field approximation to hold. In practice r was estimated using the location of the LCFS. If installed on other experiments where $|\boldsymbol{x}_{{\boldsymbol{i}}}|\sim r$, it will be critical to make as good an estimate as possible for r using density profile diagnostics and magnetic equilibria reconstructions.

Analysis of SAMI active probing data differs from that used on conventional DBS data. Limited to eight receiving antennas only a partial suppression of signal from outside the chosen probing direction is possible. As mentioned in section 1, the back-scattering efficiency is strongly dependent on ${{K}_{\bot}}$. Partial suppression results in the dominant low ${{K}_{\bot}}$ signal affecting all received spectra. This can make the spectra of received signals difficult to interpret.

Figures 6(a) and (b) show spectra at 300 ms into MAST shot 27 969 when the 16 GHz beam was focused at the points where the values of blue minus red and red minus blue-shifted power where at their greatest respectively. The red and blue extrema were observed at ($-{{8}^{{}^\circ}}$, $-{{12}^{{}^\circ}}$) and (0°, $-{{28}^{{}^\circ}}$) in image coordinates respectively. On both of these figures there is a large un-shifted power spike at the active probing IF frequency (12 MHz). This is due to reflections off the window and only partially suppressed normal incidence reflections off the plasma.

Zoom In Zoom Out Reset image size

No Doppler peak is visible in the spectra as obtained during single horn DBS experiments [2, 4–10, 22] due to imperfect side-lobe suppression. In addition, SAMI cannot separate O and X-mode polarisations at present which will increase the number of different ${{K}_{\bot}}$s being sampled, thereby delocalising the scattering location. However, the directional weighting imposed by the phased array does allow a red-blue power imbalance to be measured allowing information to be attained which will be discussed further in section 3.4.

Due to low ${{K}_{\bot}}$ dominance, interference between multiple backscattered signals and both O and X-mode polarisations being present, a quantitative explanation of the observed spectra will require a full-wave treatment; such a study is planned using the cold-plasma full-wave code EMIT-3D [38]. Such a study will aid in turbulent velocity and k-spectra measurements through better understanding of observed spectra

The range of ${{K}_{\bot}}$ values which will affect SAMI spectra can be estimated using an analytic formula [4] which takes into account the curvature of the beam and cutoff layer. For a Gaussian probing beam, where the amplitude profile is $E(x)\propto \exp \left(-{{x}^{2}}/{{w}^{2}}\right)$ and w is the 1/e beam radius, the range of ${{K}_{\bot}}$ present in the backscattered spectrum is given by

$$\begin{eqnarray}&&\Delta {{K}_{\bot}}=\frac{2\sqrt{2}}{w}{{\left[1+{{\left(\frac{{{w}^{2}}{{k}_{0}}}{\rho}\right)}^{2}}\right]}^{\frac{1}{2}}}\end{eqnarray} \tag{ 7 }$$

where k₀ is the wave vector of the probing beam in vacuum and ρ is the effective curvature radius within the spot $1/\rho =1/{{R}_{\text{plasma}}}+1/{{R}_{\text{beam}}}$. In the Rayleigh region of the beam, ${{R}_{\text{beam}}}\to \infty$ so $\rho \to {{R}_{\text{plasma}}}$.

The typical radius of curvature for the cutoff layer in a MAST plasma is ${{R}_{\text{plasma}}}\sim 1.3$ m. Taking into account the widths of central maxima produces a $\Delta {{K}_{\bot}}$ range of 0.7–2.4 cm⁻¹ for 10–34.5 GHz probing.

Figures 7(a) and (b) show the distribution of ${{k}_{\parallel}}$ and ${{k}_{\bot}}$ values respectively as a function of probing orientation that are accessible at 16 GHz, 230 ms into MAST shot 27 969 calculated using the beam-tracing code TORBEAM [39]. Doppler backscattering is most efficient when the incident beam is aligned perpendicular to the magnetic field at the scattering location (along ${{k}_{\parallel}}=0$ in figure 7(a)).

We can see from figure 7(b) that many values of ${{K}_{\bot}}$ (via equation (1)) can be measured simultaneously using a 2D DBS device although unlike other steerable monostatic DBS systems in its current configuration SAMI has been unable to measure K-spectra. This is due to the incomplete directional separation provided by the phased array. Therefore, additional antennas would result in a greatly reduced side-lobe level.

Zoom In Zoom Out Reset image size

SAMI is a proof of principle diagnostic and DBS experiments have never been attempted using a phased array previously. Limited directional weighting can result in difficult interpretation of spectra as discussed in section 3.1. However, conducting 2D DBS allows flexibility in the alignment of the probing beam. Figure 8 shows the results from a beam that was aligned with time averaged maximum blue shift during MAST shot 28 100 found at $-{{20}^{{}^\circ}}$ on the horizontal and 8° on the vertical. The active probing beam was launched 10 MHz above $2{{\omega}_{\text{LO}}}$ and the data was digitised on four RF frequency channels: 14, 15, 16 and 17 GHz with a switching time of 200 μs. It is not possible to resolve a discrete Doppler peak in the received spectrum as discussed in section 3.1. The spectra will be a convolution across many probing beam orientations with a directional weighting applied by the phased array. SAMI can still investigate the turbulent velocities of the plasma by looking at the centre of mass of this spectrum. However, this only provides a qualitative comparison with previously observed trends. The centre of mass spectrum will give a gross under estimate of the turbulent velocity at any one point. Forward modelling simulation work is required in order to make a quantitative turbulent velocity comparison to previous charge exchange recombination spectroscopy (CXRS), beam emission spectroscopy (BES) and conventional DBS experiments.

Zoom In Zoom Out Reset image size

Figure 8(a) shows the 40 ms moving average centre of mass of the turbulence velocity across $10\pm 0.2$ MHz in the IF with the central unshifted peak ($10\pm 0.01$ MHz) notched as a function of time. During these measurements the receiving beam was focused in the ($-{{20}^{{}^\circ}}$, 8°) direction. The turbulent velocity was calculated from the observed Doppler shift using the value of ${{k}_{\bot}}$ at the scattering location as calculated by TORBEAM (figure 8(b)).

A gradual increase in the centre of mass turbulence velocity is observed from 70 ms onwards following 2.5 MW of NBI power being applied by one of MAST's on axis, co-injected, NBI sources (figure 8(c)). This rise in observed Doppler shift results from spin-up caused by momentum injection from the NBI system and has been observed during numerous other DBS experiments, for example [2, 10]. Once the plasma enters H-mode at 215 ms (indicated by a decrease in ${{D}_{\alpha}}$, figure 8(d)) there is an abrupt change in the sign of the observed turbulence velocity. A sharp change in the turbulent velocity coinciding with the onset of H-mode has also been measured during previous DBS experiments [2, 10] and results from the edge turbulence velocity being dominated by toroidal fluid velocity during L-mode and diamagnetic velocity in H-mode. The steep pressure gradient that forms in the edge region during H-mode results in a sharp increase of the diamagnetic velocity [2].

Figure 9(a) shows the 5 ms moving average of the total Doppler shifted power. The power plotted is the summed power across the IF $10\pm 0.2$ MHz where the central unshifted frequency peak is notched ($10\pm 0.01$ MHz) and the average background noise level is subtracted. The Doppler power steadily increases after the NBI is applied (70–215 ms). This is likely to be caused by an increase in the density (figure 9(f)) and the scattering location moving closer to the SAMI array (figure 9(b)) and/or an increase in the turbulence amplitude due to an increasing electron temperature gradient at the scattering location (figure 9(d)); the electron density gradient stays notably constant during this period (figure 9(d)). Temperature and electron density gradients are calculated from Thomson scattering data.

Zoom In Zoom Out Reset image size

There is a sharp drop in power as the plasma enters H-mode at 215 ms despite the scattering location not changing significantly (figure 9(b)). This decrease is caused by the suppression of turbulence in the edge region. The ramp up in power during NBI injection and drop in power as the plasma enters H-mode has been observed in other DBS experiments [3]. In figure 9(a) microwave bursts are observed after the plasma enters H-mode as each Edge Localised Mode (ELM) coincides with microwave emission up to four orders of magnitude above thermal [37]. The average background emission is subtracted but this will only nullify the ELM emission if the burst is evenly distributed across the IF.

SAMI has observed trends in turbulence velocity and DBS power which have been observed during previous experiments as demonstrated in figures 8 and 9. Though SAMI has some notable limitations in its current form, the results presented here are encouraging for the future feasibility of 2D DBS systems.

2D DBS experiments using a phased array could potentially provide a way of measuring the magnetic pitch angle profile with high temporal and spatial resolution.

Zoom In Zoom Out Reset image size

To form figure 10 the SAMI array was focused, using beam forming, onto each point in an equally spaced 21 by 21 grid spanning  ±40° in the horizontal and vertical viewing directions. This was done using data acquired 300 ms into MAST shot 27 969 whilst actively probing the plasma at 16 GHz. For each grid point the spectra of the received beam is analysed.

The red and blue-shifted power was calculated by summing the amplitudes of the signals between 11.8–11.99 MHz and 12.01–12.2 MHz respectively in the IF after subtracting the background passive emission. The total Doppler shifted power was taken as the sum of the red and blue-shifted power. In figure 10 the difference between the blue and red-shifted power is plotted. Net positive and negative regions show where more blue and red-shifted power is present respectively. The colour bar is normalised by dividing the red-blue difference by the total Doppler shifted power. The two regions which show the most red-blue power imbalance are marked with black crosses; the spectra observed at these locations are shown in figures 6(a) and (b). The black dashed straight line connects the points of maximum blue and red-shifted power imbalance. The magnetic field lines, as calculated from EFIT and Thomson scattering, are over-plotted in grey. Optimum backscattering occurs when the probing beam is perpendicular to the magnetic field lines. Therefore, the orientation of the red and blue maxima allow a pitch angle measurement to be made. SAMI is the first DBS system to make a pitch angle measurement. This new capability is a direct result of simultaneous 2D imaging made possible by the use of a phased array.

Figures 11(a) and (b) show the magnetic pitch angle as measured by SAMI (green line) and EFIT (dashed red line) as a function of time for MAST shot numbers 27 969 (fixed frequency 16 GHz) and 28 856 (fixed frequency 10 GHz). A fine grid of 161 by 161 was used within the  ±40° viewing aperture for greater accuracy. The SAMI pitch angle time evolution was calculated using an 8 ms sliding data window. The EFIT pitch angle was calculated at the scattering location of a ray launched along a path directly in-between the locations of maxima and minima in power deficit at each moment in time as calculated by TORBEAM.

Zoom In Zoom Out Reset image size

The reliability of the SAMI pitch angle measurements relies on there being sufficient backscattered power. Figure 11(c) shows the normalised to peak value backscattered power during MAST shots 27 969 and 28 856 where the background passive emission has been subtracted. One can see that the large departure from EFIT by the SAMI pitch at 10 ms during shot 27 969 coincides with a sharp drop in backscattered power. It is also apparent that departures between SAMI and EFIT during shot 28 856 occur between 60–130 ms and 330–350 ms when the DBS power is comparatively low. With a larger database of appropriate data it should be possible derive a relationship between the reliability of the data and back-scattered power.

It is worth noting that the optimum configuration for SAMI to make pitch angle measurements was not known when the data for these shots was taken. Nevertheless, despite being a proof of principle, first of its kind diagnostic, SAMI has made pitch angle measurements which agree well with EFIT considering SAMI's present limitations. The amount of uncertainty in the SAMI and EFIT pitch measurements is left the subject of a future publication. The evolutionary trend in the pitch, as measured by SAMI and EFIT, is in agreement throughout both shots. The fluctuation level in the SAMI pitch angle measurements during shot 28 856 is noticeably higher than during 27 969. This is expected as, being a low frequency 10 GHz shot, more of the probing beam is backscattered in the scrape off layer where the density fluctuation level is high. Though the radial scattering locations are similar in both shots (figure 11(d)) the pitch evolution varies due to different plasma current temporal profiles. It is also apparent from figure 11(d) that scattering takes place in the edge region consistent with figure 2.

As will be discussed further in section 4 there are numerous ways that SAMI can be upgraded and reconfigured to improve the accuracy of the pitch angle measurements.