The Reflectometry For Measuring Density Fluctuations In Magnetized Plasmas.

S. Heuraux1, F. Clairet2, T. Gerbaud2, P. Hennequin3, C. Honoré3, G. Leclert4,
R. Sabot2, F. da Silva5, A. Sirinelli2, L. Vermare2.
1 LPMIA, CNRS-Université Henri Poincaré, Nancy-1, BP 239, 54506 Vandoeuvre-lès-Nancy, France
2 Association Euratom-CEA, CEA/DSM/DRFC, CEA-Cadarache, 13108 St Paul lez Durance, France
3 LPTP, Ecole Polytechnique, 91128 Palaiseau, France
4 LPIIM, CNRS-Université de Provence,13397 Marseille cedex 20, France.
5 Associação EURATOM/IST, Centro de Fusão Nuclear, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
The understanding of turbulence in magnetized plasmas is a key point to predict the confinement in a
fusion reactor since energy and particle transport are dominated by turbulent transport. To reach the
processes associated to the turbulence, high resolution diagnostics in time, space, frequency and
wavenumber spectra are needed. Reflectometry is a versatile diagnostic using microwave, suitable for
ITER, able to provide qualitative and quantitative measurements on density fluctuations due to Magneto-
Hydro-Dynamic modes [1] and turbulence [2]. This development is completed by simulations, which
permits to evaluate and to test the possibilities, the spatial and wavenumber resolution of a new
reflectometer configuration. It can be also used to test new data processing method or to evaluate the role
of a specific physical effect on the measurement of a given parameter. After the presentation of the
reflectometry basics, we look at the state of art of different methods for fluctuation measurement with
reflectometry and associated simulations. We start with the fixed frequency reflectometry and correlation
reflectometry. We then look at the scattering methods: Doppler reflectometry and fluctuations from fast
swept reflectometry and then discuss possible improvements of the density fluctuation measurement. As
this paper contains different kinds of reflectometry, the given references should be consulted to have a
more precise idea on the subjects presented.
Refletometry Basics: Derived from the radar principle, reflectometry is first an accurate plasma density
profile diagnostic. It measures the round-trip time of a probe microwave from an origin to a reflecting
layer called cut-off layer. This cut-off layer corresponds to a radial position r where the refractive index
N(r) of the probing wave becomes zero. This position depends on the polarization of the probing wave,
two polarizations can propagate, the Ordinary-mode (O-mode) with E//B (where E is the wave electric
field, and B the plasma magnetic field), and the extraordinary polarization (X-mode) for E⊥B waves and
is generally a function of the probing frequency, local plasma density and magnetic field. The phase shift
due to round trip into the plasma integrates the refractive index change through the beam. By sweeping
the probing frequency, the time of flight is computed through the smoothed phase temporal derivative
assuming known the frequency sweep rate. The profile reconstruction needs an inversion method, an
Abel’s inversion for the ordinary mode (O-mode) [3] and Bottolier-Curtet’s method for the extraordinary
mode (X-mode)[4]. Reflectometry accessibility to the plasma is defined by the wave polarization and the
absorption of the probing wave as shown on the figure 1 in the case of TS standard plasma. Tore-Supra is
a high field (3.8 T), large size tokamak (R0=2.4 m, minor radius: a=0.72 m), with 4 reflectometers in
operation covering the range 50 to 155 GHz. Most reflectometry capabilities and limitations to provide
density profiles in various plasma conditions [12], wave-number and frequency spectra, MHD activity and
profiles of poloidal velocity measurements obtained on Tore Supra can be found in [5]. To illustrate the
TS fast swept reflectometer capabilities, one can have a look at the spatio-temporal evolution of MHD
events like double tearing.

Figure 1: Cut-offs positions in Tore Supra versus probing frequency in continuous lines for fcux upper fclx
lower X-mode branch and fpe for O-mode, dashed line cyclotron frequencies fce and upper-hybrid
frequency fuh computed with a magnetic field of 3.8 T and central density 7 1019 m-3. The V &W bands
cover the frequency range 50-110 GHz and the D band corresponds to 105-155 GHz, in these conditions
the reflectometers can be always used to probe the TS plasma. Only the V band can be used to probe the
plasma with the O-mode polarization on TS.
Fixed Fluctuations measurements: The simplest method to obtain information on the density fluctuation
is to measure the phase fluctuation of a reflectometer working at fixed frequency. In this case the main
contribution on the phase fluctuation comes from large spatial scale density fluctuation in the vicinity of
the cut-off layer corresponding to radial wave number smaller than Airy’s wavenumber [5]. However a
low contribution comes from Bragg backscattering all along the beam path when the wave-numbers
matches the Bragg rule kf= -2ki(r) (ki(r) is the local incident wave-vector and kf is the density fluctuation
wave-vector). To recover the absolute value of density fluctuations, a formula linking phase variation and
density fluctuations obtained under the Born approximation is used with a constant ad hoc wave number
(1 cm-1). Simulations with different broadband turbulence are required to determine the value of the main
wavenumber used as input in the previous formula to reach the absolute value of the density fluctuation
amplitude and, if possible, the experimental wavenumber spectrum should be used in the simulations.
Correlation reflectometry: Correlation of a dual fixed frequency reflectometer is another tool to reach the
turbulence characteristics. But the interpretation of the measured correlation length is complicated by the
non-linear effects [6-7], except in the plasma center where the amplitude of the density fluctuation is low
enough to consider that the response is linear. Simulations of correlation reflectometry have confirmed
that the non-linear effects modify drastically the correlation length measurements [8]. The correlation
length is shortened more and more as the density fluctuation amplitude increases.
Doppler reflectometry: The Doppler reflectometry taking advantages from reflectometry and Bragg
backscattering is also used to provide information on the poloidal wave number spectrum and poloidal
velocity of the density fluctuations [9]. This technique is based on the possibility of separately detect the
field backscattered on fluctuations along the beam path from the field reflected at the cut-off layer
(standard reflectometry), by launching the probing beam in oblique incidence with respect to the cut off
layer. The localized swelling of the incident field at the cut-off, amplifies the scattering process with
selected wave-vector via the Bragg rule and allows localization of the scattering process near the cut-off
layer. At the cut-off, the component of the probing beam wave-vector perpendicular to the refractive
index layer vanishes: in O-mode, kir and thus kf,r=0, and the tilt angle of the beam with respect to
density/magnetic surfaces determines the local wave-number: in slab geometry, ki = ko sinθ, where ko is
the vacuum wave-number and θ is the angle of the probing beam. To improve and increase the
information extracted from the Doppler reflectometer, the simulations have helped us in a significant way.
For example, it has been simulated to know that it is possible to localise thin velocity shear layers
associated to internal transport barriers, to determine the velocity range in the shear layer and to give a
qualitative velocity profile [10] and further simulations are dedicated to see if it is possible to recover the
absolute value of the density fluctuation from the backscattered signal.
Fluctuation from fast swept reflectometer: Simulations have shown that it is possible to measure the
density fluctuation profile from fast frequency sweep reflectometer in both polarizations O and X-modes.
The principle has been described in [2] and used an analytical formula for O-mode. The method follows a
procedure, which consists first to determine phase fluctuations versus position, then compute the Fourier
transform after the applied inverse relations to recover the wavenumber spectrum of the density
fluctuation and then obtain the amplitude of the density fluctuation using Parceval’s theorem. For the Xmode
a transfer function is needed due to the fact that the Bragg backscattering condition is a complex
function of the density and magnetic field. For each shot and for all the windows used, this transfer
function is computed using the experimental magnetic field and the density profile, data processing can be
found in [11] and has been applied on TS date as shown on figure 2 (right). The simulation has been used
to define the domain of validity. An evaluation of the maximal value extractable of the density fluctuation
amplitude has be deduced for this data processing method. It can be applied on density fluctuation level
up to 30 % with a relative accuracy better than 20 %. However the spectrum is more sensitive to the nonlinear
processes and is distorted as soon as the density fluctuation amplitude reaches 1% for the X-mode
and 3 % for the O-mode for standard TS parameters. This fact defines the plasma zone where we can
expect to use this method without having distorted spectrum by non-linearity that is to say in the centre of
the plasma on a restricted wavenumber range defined by the window size. The simulations of the TS Xmode
reflectometer have also permitted to show that this method is able to separate different spectra in a
plasma with the standard TS density and magnetic field profile with a constant density fluctuation level
(0.5%) as shown on figure 2 (left).
Statistical properties of the density of the density fluctuations, for example, the probability density
function could also be obtained with this method. From simulations it should be possible to establish a
method, taking into account the nonlinear effects at high density fluctuation level able to extract the radial
wavenumber spectrum from the phase fluctuation spectrum.

Figure 2: (left) Simulation of wavenumber spectrum reconstruction with a continuous transition from
power law (Kolmogorov type) at the edge to Gaussian spectrum shape at the centre, (right) experimental
density fluctuation profiles obtained during linear ramp of current from 0.4 to 1 MA over 14s for the
discharge TS32398.
By looking at the high-speed imaging [15], we see that the sweeping time of the reflectometers used is
of the order of the event shown on the movies. The sweeping time 20 μs is also greater than the
correlation time around 10 μs [16], which is just valid over the window used to reconstruct the density
fluctuations profile. This last point restricts the possibility of accessing to large wavenumbers up to the
Bragg limit (kf < 2ko, ko vacuum wavenumber of the probing wave). Reducing by a factor of five the
sweeping time 4 μs, it becomes possible to probe single event as blobs, to extend the probed wavenumber
range and to study the turbulent wave front generated by heating pulse.
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[10] F. da Silva, S. Heuraux, M. Manso Nuclear Fusion to be published.
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[12] F. Clairet, L. Colas, S. Heuraux, G. Lombard Plasma Phys Cont Fusion (2004) 46, 1567.
[13] P. Maget, F. Imbeaux, G. Huymans F. Clairet etal Nuclear Fusion (2005) 45, 69.
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