[84, 105].
5.3
Features of the distribution function
It is clear that without proper characterisation of the laser lineshape information about the velocity distribution function from the contrast measurements is very uncertain. However, the correction does not affect the phase and hence insight regarding the velocity distri- bution function can still be obtained by looking for a non-linear dependence on delay in the phase measurements. The contrast measurements are still useful as the variability in the contrast due to contamination effects can indicate the magnitude of the modulations expected in the phase. For this study all measurements were made at the reference plasma condition.
Firstly, the IVDF in the azimuthal direction is considered. The phase was measured over a range of impact parameters (p = 0 cm, 1 cm, 2 cm and 2.5 cm) in the high magnetic field region (z= 47.8 cm) and was tracked over interferometric delay. The central columns of the phase measurement, where the line-of-sight is near vertical, was considered so that the measurement was only sensitive to the asymmetric phase component corresponding to the effects of the rotational flows. The phase measured at each position has been averaged over a 40×40 pixel area and these points are plotted in figure 5.19. A linear fit of φD
verses φ0 has been applied to the measurements taken at each impact parameter.
There is a clear linear dependence with the delay suggesting that in the case of the az- imuthal component of the IVDF is Maxwellian. The variability in the results is consistent with what was predicted in the case of a<2% secondary line contamination (see section 5.2.1). The contour plot shown in figure 5.19 shows that the measured phase with delay is linear for all radii including the flow shear region (r >2 cm).
The challenge is now to determine if the IVDF in the axial direction is also Maxwellian. The axial flows will be obscured by the uncertainty in the laser phase. A linear ramp in axial angle must be fitted to each phase image in order to extract the axial flow (see discussion around equation 5.1). Only measurements where there is a clear ramp in the phase have been considered. The phase was considered along the image axis of symmetry (z axis) so that the measurement was only sensitive to the symmetric component of the phase (corresponding to the axial flows). This measurement is shown in figure 5.20 for positions (a) z = 52 cm and (b) 32 cm which correspond to the locations either side of the observed axial flow reversal. The red lines here have been added as a reference only so that the images may be compared easily. Figure 5.20 (c) shows a contour plot for the symmetric component of the phase (corrected for sinχ viewing angle) for all axial positions.
The variability in these measurements is larger than what was found in the symmetric component. It is unclear why the variability is so high in these measurements however this may simply just reflect large variability in the axial flows which is also suggested by the larger error bars in figure 5.8. Unfortunately the variability in these measurements is too high to draw conclusions regarding whether the phase has a linear dependence with delay.
114 Coherence imaging of the MAGPIE argon plasma: single condition study impact par ameter (cm) (a) (b) Delay ( )rad × 105 Delay ( )rad × 105 Doppler phase 𝜙𝐷
Figure 5.19: The measured Doppler phase (asymmetric component only) at (a) p= 0 cm (◦), 1
cm (+), 2 cm (?) and 2.5 cm (×) taken at axial position, z= 47.8 cm. The solid lines are linear
fits made at each position. Figure (b) shows a contour plot of the Doppler phase over each impact
parameter atz= 47.8 cm.
Nevertheless, there are trends within each measurement that supports the insights drawn from single delay results (figure 5.8). The positive phase increase with delay at 52 cm and the negative decrease in phase with delay at 32 cm are suggestive of an axial flow reversal around 48 cm. The contour plot in figure 5.8 (c) shows that the delay measurements change smoothly with axial coordinate and delay.
These measurements illustrate that with a monochromatic and known calibration source and in the absence of spectral contamination it is be possible to determine if the phase and contrast exhibit the expected spectral characteristics associated with a non-Maxwellian velocity distribution function. Under such conditions it is possible to tomographically reconstruct the 3D velocity distribution function.
§5.3 Features of the distribution function 115 (a) (b) z position: 52 cm z position: 32 cm (c)
Figure 5.20: Symmetric phase component (corrected for sinχviewing angle) measured over delay for axial positions (a) 52 cm and (b) 32 cm. The red reference lines are used to aid comparison of data. Figure (c) shows a contour plot of the symmetric component of the phase (corrected for
Chapter 6
Tomography of plasmas
2D coherence imaging measurements represent planar-integration of the distribution func- tion in velocity space which is then integrated along the line of sight. The interpretation so far has assumed that the ion velocity distribution function is Maxwellian and the line- integration effects are ignored. We consider here the central row of the coherence imaging measurement and apply tomographic reconstruction techniques to determine the radial profiles for the local emissivity, ion temperature and flow. The reconstructions are then compared to the line integrated approximations, which were found by applying equations 4.25, 4.27 and 4.28 to the projected brightness, contrast and phase. The comparison be- tween the reconstructed quantities and the approximations is used to identify the extent of the line-integration effects.
6.1
Tomography and the Radon transform
Tomographic reconstruction is an imaging technique used to reconstruct the internal struc- ture of an object when only line- or planar-integrated features of the object can be mea- sured. Tomographic reconstruction techniques have been used in conjunction with interfer- ometric measurements to find the 3D spatial distributions of the electron density [109,110]. The 3D refractive index fields have also been reconstructed from holographic interferom- etry [111]. Tomographic reconstruction been applied to camera images on the TORPEX plasma to find the 3D spatial distribution of the plasma emissivity [112], Doppler coherence imaging measurements taken on the DIII-D Tokamak of the doubly ionised carbon emis- sion (465 nm) have been inverted to yield radial distributions of the parallel ion flow and emissivity [52] and reconstruction of X-ray measurements have aided studies of sawtooth oscillations on the TFTR Tokomak [113]. These forms of measurements use tomography to construct a 3D spatial distribution from line-integrated measurements.
Laser Induced Fluorescence (LIF) is a technique commonly employed in plasma studies to measure features of the velocity distribution function such as bulk ion or neutral tem- peratures, flows and densities. In this technique a narrow-band laser is used to excite the plasma species over a small range of wavelengths. For a thermally broadened distribu- tion the ions and atoms will be excited when the laser frequency matches the excitation energy at their respective rest-frames. The emission intensity is measured over the fre-
118 Tomography of plasmas
quency scan to indicate the population of excited particles. Standard LIF measurements target a local position in the plasma and therefore do not require spatial tomography, how- ever, tomography over velocity-space is required to determine the 3D velocity distribution function [31].
The Radon transform, first used by Johann Radon in 1917, is a mathematical way of relating a function to its projection along a line. His work was further extended to the 3D transform where the projection is determined from the integral over a series of planes. Both types of Radon transform are relevant in this work and have already been encountered in chapter 2.1 sections 2.1.6 and 2.1.7.
x y l (r,θ) 𝑝 𝜑𝑝 ^ 𝜉 x y 𝑝 𝜑𝑝 ^ 𝜉 (a) (b) z (r,θ,φ)
Figure 6.1: Geometry showing (a) the 2D Radon transform where the function is integrated along
the line-of-sight, l to give the projection and (b) the 3D Radon transform where the function is
integrated over the 2D plane.
A 2D function f(x) viewed along a linel, where x= (x, y), is related to the 1D projection ˇf(p, ϕp) through, ˇf(p, ϕp) = Z L f(x, y) dl = Z Z R2 f(x, y)δ(p−ξ·x) dx = Z Z R2 f(x, y)δ(p−xcosϕp−ysin ϕp) dx dy (6.1)
where p and ϕp are the impact parameter and impact angle, respectively, and ξ =
(cosϕp,sinϕp) is a unit vector describing the direction of the impact parameter such
that p=ξ·x. The function f(x, y) is integrated along the line of sight and the geometry is shown in figure 6.1(a).
A 3D function f(x), where now x= (x, y, z), is viewed along the line land related to its 1D projection through
§6.2 Tomography techniques for reconstruction of the local plasma 119