1. Electron Bernstein emission diagnostic
of electron temperature profile
at W7-AS Stellarator
Inauguraldissertation
zur
Erlangung des akademischen Grades eines
doctor rerum naturalium (Dr.rer.nat.)
an der
Mathematisch-Naturwissenschaftlichen Fakult¨at
der
Ernst-Moritz-Arndt-Universit¨at Greifswald
vorgelegt von Francesco Volpe,
geboren am 19.04.1974 in Napoli (Neapel, Italien)
Greifswald, im M¨arz 2003
2. Max-Planck-Institut f¨ur Plasmaphysik, EURATOM Association
Boltzmannstraße 2, D-85748 Garching
Tel.+49/(0)89/3299.01
Author’s e-mail: frv@ipp.mpg.de
Publication Data:
Document type PhD Thesis
Author Volpe, Francesco
Corporate Author Ernst Moritz Arndt Universitaet Greifswald (DE)
Corporate Author Max-Planck-Institut fuer Plasmaphysik, Garching (DE)
TITLE Electron Bernstein emission diagnostic of electron
temperature profile at W7-AS Stellarator
PACS numbers 52.35.Hr , 52.25.Sw , 52.70.Gw , 52.35.Mw , 52.55.Hc
Publ. Year 2003
Pages 124
Figures 97
3. Abstract
Electron temperature profiles at densities above the ECE cutoff are measured at
the W7-AS stellarator by a novel diagnostic based on black body emission and
mode conversion of electron Bernstein waves (EBWs).
The EBWs, otherwise confined within the upper hybrid layers, reach the out-
side of the plasma after Bernstein-extraordinary(X)-ordinary(O) mode conver-
sion. Such O-mode polarised output is detected along a special oblique line of
sight by an antenna with gaussian optic. This has been optimised by means of
EBWs ray tracing calculations in full stellarator geometry, in order to maximise
the conversion efficiency and to minimise the Doppler broadening due to the
oblique nature of the line of sight. The obliquely detected O-mode polarization
is elliptical, but by λ/4 phase shift and proper orientation of optical axes it is
changed into a linear polarization by an elliptical waveguide. The signal is then
spectrum-analysed in a heterodyne radiometer. Finally temperature profiles are
derived from the spectra by means of the 3D ray tracing code mentioned before.
The diagnostic has been applied to time- and space-resolved measurements of
edge-localized modes, low-to-high confinement transitions and radiative collapses
at densities up to ne = 3.8 · 1020
m−3
.
Moreover, for the first time the heat wave propagation method for the determi-
nation of local heat transport coefficients has been extended beyond ECE cutoff
density, combining EBW emission measurements at the first harmonic (f =66-
78GHz) with modulated EBW heating at the second harmonic (140GHz).
9. INTRODUCTION
0.1 Motivation
A great deal of research is being undertaken worldwide with a view to generating economi-
cally significant amounts of energy from controlled nuclear fusion reactions in magnetically
confined plasmas [1].
The focus is on two confinement concepts: the Tokamak and the Stellarator. Both are
toroidal devices where intense magnetic fields force the charged particles to spiral around
the field lines and stay far from the walls of the chamber. The principal magnetic field is
toroidal and externally generated. However, in order to balance the plasma pressure with
magnetic forces, a poloidal field is also necessary. In a tokamak this is generated by a cur-
rent in the plasma. Such an internal current may either be induced by transformer action
or driven with other, non-inductive (”current drive”) methods.
In contrast, in a stellarator also the poloidal magnetic field is mainly generated by external
coils of proper shape. On one hand the axisymmetry and Ohmic heating of a tokamak are
lost, on the other the instabilities associated with the plasma current are suppressed.
In order for frequent fusion reactions to take place the plasma must be sufficiently hot and
dense. In this context, the importance of temperature measurements at high density is ap-
parent.
In tokamaks and stellarators the electron temperature Te is typically measured by [2]:
• Langmuir probes, i.e. isolated pins inserted into the plasma. By applying a voltage V
to their non-isolated tip, a current I is driven from the plasma and Te is deduced from
the I − V characteristic.
• Soft X-ray spectroscopy, taking advantage of the Te dependence of the Bremsstrahlung
continuum.
• Thomson scattering (TS), looking at the Doppler broadening of a scattered laser beam
• Electron Cyclotron Emission (ECE), i.e. emission from the electrons gyrating around
the magnetic field lines that confine the plasma. The frequency depends on the strength
of the magnetic field and hence on the position in the plasma. If the plasma is optically
thick for the examined polarization and frequency (typically a low electron cyclotron
harmonic), the intensity of the (black body) radiation is proportional to Te in the
emitting region.
Being in contact with the plasma, the Langmuir probes cannot be inserted too deep in it,
hence they provide Te information only at the plasma edge.
The Soft-X spectroscopy is a line-integrated measurement, but the local Te can be inferred
indirectly after tomography or neural network analysis of multichord observations.
1
10. 2 Introduction
Fig.0.1: Max peak density reached, steadly or transiently, in some current tokamaks (◦), stel-
larator ( ) and spherical tokamaks ( ), as a function of the magnetic field. The 2
field-pinch2experiment (RFX) and a non-fusion experiment, the magnetic reconnection
experiment (MRX), are also shown ( ). The dashed regions are not accessible to the 1st
harmonic O-mode and to the 2nd harmonic X-mode.
Only the ECE and TS diagnostics allow direct measurements of the whole Te profile.
The ECE shows the further advantage of a time resolution down to 1µs, allowing investi-
gation of fast phenomena like sawteeth oscillations, edge-localized modes (ELMs) and heat
waves from modulated electron cyclotron resonance heating (ECRH) . In contrast, the TS
time resolution is limited by the laser repetition rate, of order 10Hz. However, ECE has
optical thickness and accessibility limitations.
A typical tokamak plasma is optically thick and accessible for the second harmonic extraor-
dinary (X) mode and the first harmonic ordinary (O) mode, but the accessibility criteria,
respectively ω2
p < ω2
− ωωc and ωp < ωc, with ωp = 4πnee2/me electron plasma frequency
and ωc = eB/mec electron cyclotron frequency, can be violated in other magnetic confine-
ment devices operating at lower magnetic field B or at higher density ne, such as spherical
tokamaks3
and stellarators (fig.0.14
). In particular, current-free stellarators are not subject
to the Greenwald limit5
but only to the radiative6
one, hence for low impurity content can
reach very high densities. For instance in recent island divertor experiments at Wendelstein
7-AS (W7-AS) a stationary central density ne = 3.8·1020
m−3
was reached, in excess of three
2
a toroidal device in which the poloidal and toroidal fields are of comparable magnitude
3
tokamaks with very low ”aspect ratio” (ratio of major to minor radius), approximating to a sphere.
4
The picture refers to core density, but some plasmas can be underdense in the core and overdense at
mid-radius or at the edge. For example the H-Mode at DIII-D is characterized by a flat density profile, such
that ωp can be “low enough” at the center but too high at the low magnetic field side, i.e. the core can
be accessible but the edge not [3]. Then, not only the local temperature information is lost, but also the
possibility to drive current off-axis, e.g. in order to stabilize the ”neoclassical tearing modes” [4].
5
an empirical limit to the density of a tokamak plasma when the line-average electron density (1020
m−3
)
equals the plasma current density (in MA · m−2
) [5].
6
another operational limit, on the impurity density, that eventually can make the plasma radiate too
much and collapse. An extensive review of this and other density limits is offered in Refs.[6, 7].
12. 4 Introduction
diagnostic
port
triangular
plane
elliptical
plane
vacuum
vessel
(schematic)
divertor
modular field coils special field coilstoroidal field coils
Fig.0.3: Cut-view of W7-AS plasma, vacuum vessel and coils
the edge density profile with a local limiter [17].
The other scheme, the O-X-B-mode conversion proposed in 1973 by Preinhaelter and Kopeck´y
[18], has the advantage of being geometrically optimised, without intervention on the plasma:
it consists in launching the O-mode from outboard along a special direction making its cut-
off surface degenerate with that for the X-mode. Under this condition the O-mode converts
completely to the slow X mode. This reverses direction and couples to the B-mode at the UH
layer. The O-X-B mode conversion was observed for the first time in a cylindrical plasma at
Nagoya University in 1987 [19] and was invoked later as a possible explanation of overdense
plasma heating in the Heliotron DR [20]. Efficient O-X-B heating was clearly demonstrated
for the first time in 1996 at W7-AS stellarator [21].
Also the first spectral measurements of emission were attained at W7-AS, with a temporary
set-up based on the reversed scheme, BXO [22].
The present thesis relates the mode-converted emission to the electron temperature profile,
describes the construction and operation of the first dedicated electron Bernstein emission
(EBE) diagnostic, presents the first temperature profiles measured with this technique and
the first extension of the heat wave method to overdense plasma, by combining modulated
EBW heating and EBE diagnostic. Finally some applications to the fast high-density phe-
nomenology at W7-AS are shown.
0.2 Wendelstein 7-AS Stellarator
W7-AS is a low-shear ( 2%) five period modular stellarator with major radius R 2m
and average minor radius a 0.18m. The low shear excludes low order rational number
resonances n/m = 1/3, 1/2, ... from the confinement region; the stability is provided by a
magnetic well rather than by strong shear.
The confining magnetic field is limited to B 2.5T on the axis and is generated by a set
of nonplanar coils, 9 per period (fig. 0.3). With the superimposed toroidal field of 5 × 2
planar coils the rotational transform7
can vary within the range 0.25 ιa 0.67. The so
7
the number of turns the helical field lines make round the minor circumference of the torus for each turn
round the major circumference
13. Sec.0.2 Wendelstein 7-AS Stellarator 5
Fig.0.4: W7-AS flux surfaces in different poloidal sections along a period, for an edge rotational
transform ιa = 5/9 implying large magnetic islands at the boundary, recently utilized
with success in a new concept of divertor [26].
called “special” and “control” coils allow for variations of the toroidal mirror term and of
the boundary islands geometry, respectively.
From above (fig. 2.23) W7-AS looks like five toroidally linked magnetic mirrors with the
mirror ends in the corners, where the field line curvature is higher. In this way the degrading
effect on confinement of the “bad” curvature is compensated by the beneficial increase of
the magnetic field strength. Being a multiple helicity, l = 2/l = 3 stellarator, within a
semiperiod the flux surface cross-sections rotate and change gradually from a triangular
shape (l = 3) in the weak curvature region at toroidal angle ϕ = 00
to an elliptical shape
(l = 2) in the pentagon corner at ϕ = 360
(fig. 0.4). In the elliptical section B has a
“tokamak-like” dependence on the major radius R, like 1/R, while in the triangular section
B is characterized by a more complicated, non-monotonic dependence (fig. 2.2).
For the following it is convenient to label the flux surfaces with the average minor radius or
effective radius, reff . This is defined as the radius of a circle whose area equals half the sum
of the ϕ = 00
and ϕ = 360
cross-sectional areas of the considered magnetic surface.
The W7-AS magnetic configuration is the result of equilibrium computations at minimal
geodesic curvature of magnetic field lines [23, 24]. As a result, both the radial transport
and the Pfirsch-Schl¨uter currents are reduced8
. From these points of view W7-AS can be
considered an “advanced stellarator” (AS) on the way to the fully optimized one, W7-
X, presently under construction in Greifswald [25], meeting at the same time all the 7
optimization criteria discussed in [24].
Fig. 2.23 summarizes the diagnostics W7-AS is equipped with, and the heating systems
are also sketched, viz. the electron cyclotron resonance heating (ECRH) with 1 × 70 and
4 × 140GHz gyrotrons (total power 2.4MW), the neutral beam injection (NBI, 4MW) and
the ion cyclotron heating (ICRH).
The plasma was formerly faced by a limiter, recently replaced by an island divertor which has
8
Recall that a stellarator magnetic field is uniquely determined by external coils. For this reason any
parallel current in the plasma, as the Pfirsch-Schl¨uter and the bootstrap current, is undesirable and should
be minimised.
14. 6 Introduction
enhanced the machine performance in terms of quasi-stationarity, confinement and impurity
control [26]. Other important results addressed at W7-AS in the frame of neoclassical and
anomalous transport studies, improved confinement regimes, heating systems and diagnostics
are overviewed in ref.[27, 132].
0.3 Thesis outline
The motivation to use mode-converted electron Bernstein waves (EBWs) for temperature
diagnostics of dense laboratory plasmas has been highlighted in Sec.1 of the present intro-
duction. Then a description followed of W7-AS stellarator, the magnetic confinement device
where the experiments were carried out.
Chapter 1 begins with an intuitive introduction to EBWs and to the propagation mecha-
nism. Sec. 1.2 shows how the wave equation in weakly inhomogeneous plasmas leads to ray
tracing equations. Afterwards the Bernstein mode is more rigorously defined as the third
eigenmode of the hot dielectric tensor along with the ordinary (O) and extraordinary (X)
mode (Sec. 1.3). The possibility to excite EBWs in the overdense plasma core through O-X
and X-B mode conversions is clarified in Sec. 1.4 and 1.5. The time-reversed scheme BXO
permits detection of electron Bernstein emission (EBE). Since it reaches the blackbody level
and is well localized in nonuniformly magnetised high-temperature plasmas (Sec. 1.6), EBE
can be used for local measurements of temperature, provided the emission regions are com-
puted as functions of the frequency by precise ray tracing in the complex magnetic topology
of interest. A code developed with this aim is described in appendix A.
The code was first applied to identify which line of sight yields maximum transmissivity at
the cutoff and minimum Doppler broadening and shift (Sec. 2.1). A Gaussian optic antenna
was designed for observation along that line and mounted in-vessel (Sec. 2.2). Owing to the
oblique view the detected O-polarization is elliptical, but is readily linearized by a broadband
phase shifter. A part of the diagnostic development consisted in comparing possible shifters
and finding which best met the diagnostic requirements. The results given in Sec. 2.3 led
to the installation of an elliptical waveguide. Chapter 2 continues with a description of the
waveguide transmission line (Sec. 2.4), of the ECE radiometer and of its adaptation to EBE
(Sec. 2.5). The chapter ends with a paragraph on the amplification and data acquisition
system.
Apart from preparatory measurements (radiometer calibration and polarizer regulation,
Secs. 3.1- 3.2), Chapter 3 is devoted to proof of principle of the new experimental tech-
nique presented in this thesis. Aiming to be a local measurement of temperature, at first the
emission was shown to be actually localized. Results are reported in Sec. 3.3. The following
section faces the problem of incomplete transmission through the cutoff layer, making some
non-EBE radiation reach the antenna. The origin of such ”stray radiation” is discussed and
a method for its subtraction from the spectrum is demonstrated.
Sec. 3.5 illustrates the conversion of the EBE spectrum into the temperature profile, after
mapping frequencies in emitting layer positions with the aid of the ray tracing code. The
profiles agree well with Thomson Scattering.
Sec. 3.6 deals with diagnostic operational limits set by mode conversions.
The combination of overdense plasma access and good temporal resolution makes EBE
unique compared with other Te profile diagnostics such as ECE and Thomson Scattering.
15. Sec.0.3 Thesis outline 7
It is proposed to exploit this capability in investigating high density phenomena such as
confinement transitions, edge-localized modes and radiative collapses. Exemplary measure-
ments are shown in Secs. 3.7, 3.8 and 3.10 respectively. Moreover, for the first time it has
been possible to detect heat waves generated by modulated EBW heating (Sec. 3.9), that
will help the study of transport in overdense plasmas.
Future work is listed in Appendix D in conjunction with a diagnostic proposal for W7-X.
16. 1. EMISSION, PROPAGATION AND MODE
CONVERSION OF ELECTRON BERNSTEIN
WAVES
1.1 Electron Bernstein waves
Electron Bernstein waves are electrostatic waves propagating across the magnetic field in a
hot plasma. They are sustained by electron cyclotron motion and can be thought of as fronts
of electron rarefaction and compression perpendicular to the magnetic field with wavelength
λ of the order of electron Larmor radius ρ:
λ ρ (1.1)
where clearly a finite ρ should be considered. In a maxwellian plasma this implies also that
the temperature be finite. In fact in a cold magnetised plasma the electrons are tight to the
field lines and account for charge separation (and spacecharge waves) only along the field.
On the other hand at high enough temperatures the electron gyroradius ρ = mvT c/(eB)
(where vT is the electron thermal velocity and the other notations are standard) becomes
comparable with the wavelength λ. If additionally the wave frequency ω is of the order of
the cyclotron frequency ωc, the electrons can move in phase with the local wave field and
“participate” in wave motion, in the following sense: by moving in response to the wave
field, the electrons alter the local concentration of electric charge, giving their own contri-
bution to the electric field. To be constructive, this contribution must clearly have proper
frequency, wavelength and phase. We are already supposing the electrons to oscillate at
nearly the same frequency as the wave. Moreover, suppose ρ = λ/4. As a consequence the
phase is approximately locked in half a revolution, whereas in the other half the electron
moves against the wave (fig. 1.1b). Hence in its frame an electron experiences alternatively
a static and a rapidly varying electric field. The varying field averages to zero, because
within the considered semiperiod the wavefront travels a distance λ/2 forth and the electron
2ρ = λ/2 back, i.e. in total the electron samples one period of the moving periodic potential
(fig. 1.1d). Only in the other half of the revolution, when the electron “surfs” the moving
potential (fig. 1.1c), there can be a net energy exchange between the wave and the particle.
In particular, much like the Landau damping, an electron gains (loses) energy when it moves
slightly slower (faster) than the potential. Summing over many electrons the final result is
the “phase organization” or ”phase bunching” depicted in fig. 1.2: the gyrophases are no
more random as they were before the impinging of the wave, but organized depending on
the gyrocenter positions in such a way that electrons form propagating planes of net charge
accumulation/rarefaction.
8
17. Sec.1.1 Electron Bernstein waves 9
B
k
Plasma ref.frame Wave ref.frame
a)
b)
c)
d)
wavefronts
electron
k
Fig.1.1: a,b) Half-cycles of electrons gyrating in a magnetic field and immersed in a electrostatic
wave (solid wavefront: E 0, dashed: E 0)
c,d) Mechanical analogues in a reference frame moving with the potential
The discussion so far applies to resonating electrons (ω = ωc, λ = 4ρ) but coherent motion
is possible more generally at cyclotron harmonics ω nωc. Once more, the important point
is the phase velocity ω/k to equal the electron velocity. Namely the resonance condition
applies to the ratio, not to ω and k separately, which leads to some degree of freedom: elec-
trons revolving with frequency ωc and radius ρ can “surf” (and thus sustain) even harmonic
disturbances characterized by ω = nωc and λ = 4ρ/n, as follows from a generalization of
fig 1.1.
For the sake of simplicity, electrons in fig. 1.2 gyrate all with the same radius. Now let the
kE
- - - -+ + + + + +
B
e
kE
- - - -+ + + + + +
EBWsMagn.plasma:
λ ρ
ω ω
Unmagn.plasma: plasma waves
planes of constant n
c
λ λ
ω ωp
D
Fig.1.2: Electron gyration can create charge accumulation and rarefaction just like Langmuir
plasma waves do in unmagnetized plasmas.
18. 10 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
velocities (and therefore the gyroradii) be distributed according to a maxwellian of thermal
velocity vT . In an electrostatic wave propagating at high phase velocity ω/k vT , the elec-
trons sample the wave in all its phases during a revolution, even in the “active part” sketched
in fig. 1.1a. The forces acting on the electrons cancel and coherent motion is impossible.
On the contrary when ω/k 3vT the wave can interact with the bulk of the maxwellian,
and the interaction is more and more efficient (namely there are more and more resonating
electrons) for lower and lower phase velocity ω/k.
Substituting ω nωc the following inequality is found
E
B
B
B=0a)
b)
c)
Fig.1.3: Electron orbit in a)
Langmuir plasma wave,
b) upper hybrid oscilla-
tion, c) EBW
for the nth harmonic, which generalizes the 1.1: kρ
n/3.
A complementary way to look at the EBWs starts from
common plasma waves that propagate in unmagnetized
plasmas and obey Langmuir dispersion ω2
= ω2
p + 3k2
v2
T
[28]. When a static magnetic field is superimposed onto
the oscillating electric field associated with the plasma
wave, the electron orbits become ellipses. These are termed
upper hybrid oscillations and have ω = ωUH = (ω2
p +
ω2
c )1/2
. On increasing further the magnetic field, the Lorentz
force dominates over the electrostatic one and the orbits
take a circular form (fig. 1.3). By virtue of the differ-
ent restoring force, by virtue of the different relevant fre-
quency (ωc instead of ωp) and of the different condition
to obey (ω ωc rather than ω ωp), EBWs can propagate even where ω ωp (overdense
plasma), provided ω ωc still applies. The trick is that for ωp ωc the electrons describe
orbits of radius ρ larger than the Debye radius, hence can export outside the Debye sphere
the information about a space charge disturbance generated inside.
1.2 Ray tracing: eikonal ansatz and WKB approximation
In a space- and time-varying medium the displacement vector D at given position x and
time t depends on electric field E at different x and t [29]. A tokamak or stellarator
plasma, however, is stationary within a microwave timeperiod, therefore we can ignore the
nonlocality-in-time and Fourier transform the wave equation with respect to time:
× ( × E) =
ω2
c2
E +
4πiω
c2
j (1.2)
Without losing generality, one can cast:
E(x, ω) = a(x, ω) eiS(x,ω)
(1.3)
k
def
= S (1.4)
After some algebra the field eq. 1.2 takes the form of the amplitude equation:
( + ik) × (( + ik) × a) =
ω2
c2
a +
4πiω
c2
e−iS
j, (1.5)
19. Sec.1.2 Ray tracing: eikonal ansatz and WKB approximation 11
In a weakly inhomogeneous medium, characterized by an inhomogeneity scale L λ, it is
reasonable to seek solutions to the Maxwell eq. 1.2 which slightly deviate from a plane wave.
The amplitude a will be a slow (and with slow derivative) spatial function compared to the
phase S:
|ik × ( × a)| |ik × (ik × a)| (1.6a)
| × ( × a)| |ik × (ik × a)| (1.6b)
Provided also the response of the plasma in the wave field to be slow1
, in the sense:
| × (ik × a)| |ik × (ik × a)| (1.6c)
the left hand side (LHS) of eq. 1.5 approximates to ik×(ik×a). To be consistent, the right
hand side (RHS) must keep only zeroth order in ik
, i.e. the local Ohm’s law must be used2
:
ji(x) = σij(k, x)aj(x)eiS(x)
(1.7)
where σij is the conductivity tensor. Ohm’s law applies to small electric field amplitudes,
when the induced current is linear in a. This restriction is well satisfied in passive plasma
diagnostics with microwaves [30].
In addition, it is necessary to introduce a current jext, which is not a response of the medium
to E but accounts for spontaneous emission (of EBWs, in our case), which in turn will be
discussed in Sec. 1.6. Then, eq. 1.5 rewrites:
ik × (ik × a) =
ω2
c2
a +
4πiω
c2
e−iS
jext (1.8)
where ij = δ ij + 4πiσij/ω is the local dielectric tensor and δ ij the Kronecker symbol.
In terms of the refractive index N = c
ω
k, on abbreviating
Λij = ij − NiNj + N2
δ ij (1.9)
and splitting hermitian and antihermitian parts, Λh
= (Λ + Λ†
)/2 and Λa
= (Λ − Λ†
)/2i,
eq. 1.8 becomes:
Λh
a = −iΛa
a +
4πi
ω
e−iS
jext (1.10)
The former term on the RHS describes absorption and stimulated emission, the latter spon-
taneous emission [31]. Far from emitting and absorbing locations the equation reduces to
Λh
a = 0, (1.11)
and describes non-dissipative wave propagation. Its solvability condition is:
D
def
= det Λh
= 0
1
The conductivity can vary rapidly even in a slow varying plasma, near a resonance, thus violating the
geometrical optics limit (M.Brambilla, private comm.)
2
It can be demonstrated that in a strongly non-uniform medium j depends also on remote positions x
[29].
20. 12 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
or, to emphasize the dependences of D and ω,
D(k, ω(k, x, t), x, t) = 0 (1.12)
If D vanishes, also its derivatives do vanish and, after manipulations known as method of
characteristics [32, 33], the following hamiltonian ray tracing equations are found:
dx
dt
=
∂ω
∂k
(1.13a)
dk
dt
= −
∂ω
∂x
(1.13b)
The passages from eq. 1.2 (in the unknown E) to eq. 1.5 (in the unknown a) and finally to
eq. 1.8 (in a, supposed to vary slowly over a wavelength), can be so summarized:
eikonal
ansatz
−→ + ik
WKB
−→ ik
Remarkably, the eikonal ansatz is only a formal rearrangement without approximations. The
Wentzel-Kramer-Brillouin (WKB) approximation consists in neglecting the medium-part
with respect to the wave-part k of the + ik operator.
In other words eq. 1.5 is quite general, while eq. 1.8 holds when the conductivity σ varies
slowly on the scale of λ. In general σ is a function of the properties of the medium (this is
why was called medium-part). In hot maxwellian plasmas, in particular, σ will be shown
to depend on density, temperature and magnetic field. These three quantities are therefore
required to vary on scales Ln, LT , LB λ in order for the WKB approximation to apply.
This condition is necessary but not sufficient 1
: the condition of staying far from resonances
should be added. Furthermore spontaneous-emitting layers must also be far enough, in order
to remove jext from 1.8.
Notice, finally, that a non-dissipative medium was assumed in writing eq. 1.11. This is
equivalent to stating that also the dissipative scales (related to the imaginary part of k,
hence to Λa
) are much longer than λ, so that it takes many periods for the wave to damp.
1.3 Ordinary, extraordinary and Bernstein mode
The electrostatic electron cyclotron waves, previously studied by Gross, Stepanov and oth-
ers3
are better known as electron Bernstein waves in honour of Ira B.Bernstein and of his
fundamental article of 1958 [35]. By the way, here the approach of [36, 37] in deriving the
dispersion relation is preferred, because it highlights the occurrence of the Bernstein mode as
third cyclotron mode in hot plasmas, along with the ordinary and extraordinary one which
exist in cold plasmas also.
The wave equation 1.11 can be viewed as null eigenvalue problem of matrix Λij = ij −
NiNj + N2
δ ij, where the dielectric tensor in a cold plasma magnetized along the z axis,
3
The history is surveyed at page 275 of Akhiezer’s book [34]
21. Sec.1.3 Ordinary, extraordinary and Bernstein mode 13
at frequencies ω high for ion motion, is given by [38]:
=
1 −
ω2
pe
ω2−ω2
ce
−iωce
ω
ω2
pe
ω2−ω2
ce
0
iωce
ω
ω2
pe
ω2−ω2
ce
1 −
ω2
pe
ω2−ω2
ce
0
0 0 1 −
ω2
pe
ω2
=
S −iD 0
iD S 0
0 0 P
, (1.14)
Due to cylindrical symmetry Ny = 0 can be set without losing generality. The solvability
condition det Λ = 0 is of type AN4
+BN2
+C = 0, sometimes referred to as Booker quartic.
For propagation along the magnetic field (Nx = 0) this splits in:
N2
= S − D N2
= S + D P = 0 (1.15)
corresponding to two electromagnetic (e.m.) solutions, left and right circular polarised, and
one electrostatic (e.s.) solution, the Landau plasma oscillations.
Across the field (Nz = 0) there is no e.s. solution4
but only two e.m. eigenmodes of refractive
index:
N2
= (S2
− D2
)/S N2
= P (1.16)
termed extraordinary (X) and ordinary (O) mode because a charge moving in their field is
respectively influenced or not by the static ambient magnetic field.
An alternative way to write the dispersion eq. 1.16,
c2
k2
⊥ = (ω2
− ω2
L)(ω2
− ω2
R)/(ω2
− ω2
UH),
points out immediately the 0 of the left and right cutoffs (k⊥ → 0):
ωL, R = [(ωce/2)2
+ ω2
pe]1/2
ωce/2
and of the so-called upper hybrid (UH) resonance (k⊥ → ∞) at:
ωUH = (ω2
ce + ω2
pe)1/2
Depending on the phase velocity ω/k one distinguishes fast (F) and slow (S) X-modes.
With a broadly accepted language abuse, we will call ordinary and extraordinary also the
eigenmodes for oblique propagation5
.
It is more convenient for the remainder to express density and magnetic field in dimensionless
form:
X =
ω2
pe
ω2
Y =
ωce
ω
(1.17)
Let us introduce now a finite electron temperature Te, hence a finite electron thermal velocity,
β =
vT
c
=
2
c
KBTe
me
1/2
, (1.18)
4
The physical reason was anticipated in Sec. 1.1: for vanishing gyroradius the electrons are tight to the
field lines, hence do not account for charge separation ⊥ B.
5
even though charged particle motion in both oblique propagating modes is influenced by B
22. 14 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
Fig.1.4:
Dispersion relation of ordinary (O), fast
(F) and slow (S) extraordinary (X)
waves propagating across a magnetic
field Y = 0.9. The dashed line corre-
sponds to propagation in vacuum, ω =
ck, for reference.
and a finite Larmor radius (FLR) parameter,
µ =
1
2
k2
⊥ρ2
=
1
2
N⊥β
Y
2
. (1.19)
Finally express the distance from −nth cyclotron harmonic in Doppler shift unity:
ζn =
ω + nωce
|kz|vT
=
1 + nY
|Nz|β
. (1.20)
Ignoring the (slow) ion motion, the hot plasma dielectric tensor can be written as [38]:
= I + Xζ0
+∞
n=−∞
n2
µ
˜InZn
−in˜InZn
n2
µ
˜In − 2µ˜In Zn
−n 2
µ
˜In(1 + ζnZn) −i
√
2µ˜In(1 + ζnZn) 2ζn
˜In(1 + ζnZn)
(1.21)
where I denotes the identity tensor, the components 12 = − 21, 13 = 31 and 23 = − 32
are omitted and two abbreviations are used:
˜In = e−µ
In(µ) Zn = Z(ζn),
In being the nth order modified Bessel function and Z the plasma dispersion function[39]:
Z(ζ) =
1
√
π
∞
−∞
e−ξ2
dξ
ξ − ζ
, Im(ζ) 0 (1.22)
From the power series expansions of ˜In and Zn for small µ and large ζn it is easy to recognize
that the hot tensor 1.21 reduces to eq. 1.14 when β → 0 (or, equivalently, µ → 0 and
ζn → ∞).
The physical meaning of the new variables is the allowance for thermal effects, namely for
23. Sec.1.3 Ordinary, extraordinary and Bernstein mode 15
Doppler (ζn) and FLR (µ) effects. As outlined in Sec. 1.1, the waves we are looking for are
based on FLR effects, hence for a first understanding the (nonrelativistic) Doppler effects
can be neglected by restricting to orthogonal propagation (Nz = 0). The simplification will
be removed in time for interpretation of measurements, which involve oblique propagating
EBWs.
For Nz = 0 the quantity ζn diverges, 1 + ζnZn approximates to −1/2ζ2
n and the matrix 1.9
takes the form:
Λ =
Λxx Λxy 0
−Λxy Λyy 0
0 0 Λzz
(1.23)
An eigenvector is immediate from Λzz = 0 and has only z component, i.e. E is parallel to
the externally produced B0. This is a generalization of the foreseen ordinary mode, with
dispersion relation N2
= zz.
Two other eigenmodes, both extraordinary (E ⊥ B0), satisfy the equation:
Λxx Λxy
−Λxy Λyy
= 0 (1.24)
or, equivalently, fixing Ny = 0 without lack of generality,
N2
= ( xx yy + 2
xy)/ xx (1.25)
In contrast to the cold counterpart, eq. 1.16,
z
y
x
hot plasma
z
y
x
cold plasma
BB
O
X
X
O
k
k
Fig.1.5: Relative orientation of O, X and B mode
polarizations with respect to k and B for
perpendicular propagation in cold and hot
plasma, when FLR are taken into account.
yy = xx and xx, yy, xy are functions of
the unknown N2
. Furthermore, while the
cold equation was algebraic, the present
one is transcendental because µ appears as
exponent. Third and most important, the
sum over Bessel functions brings a large
number of new possible ω roots at fixed
N, i.e. a large number of new waves (ω, k)
termed electron Bernstein waves6
or cy-
clotron harmonic waves. These are the
harmonics of the so called Bernstein (B)
mode.
The fact that the dispersion relation 1.25
describes both the extraordinary modes,
namely the preexisting X and the FLR-
induced B-mode, is the key point for their
coupling. For its importance, eq. 1.25 will be solved numerically in a dedicated section, 1.5.
6
In the same way also the ion Bernstein waves are found, when ion thermal motion is included in eq. 1.21
instead of considering ions as a fixed neutralizing background.
24. 16 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
v
interaction withX-mode to be included
f(v)
Fig.1.6: EBW dispersion relations for perpendicular propagation and different values of ω2
p/ω2
c ,
adapted from [40].
However, under large µ approximations 7 8
the two modes decouple and a separated analytic
expression can be derived for the dispersion of the B-mode. It follows in fact from the
asymptotic behaviour [38]
˜In =
1
√
2πµ
(1 −
4n2
− 1
8µ
+ . . . ) (µ 1) (1.26)
that the boxed term in tensor 1.21 dominates at small wavelengths, i.e. at large gyroradii.
7
The opposite limiting case, of small µ, is not interesting: owing to perpendicular propagation it is already
ζn → ∞, which in conjunction with µ → 0 would coincide with the cold limit.
8
Different authors decouple by different limits the transverse (X) and longitudinal (B) part of eq. 1.25.
Bernstein adopts c2
k2
ω2
c + ω2
p [35], Krall c2
k2
ω2
p [36]. Here, with Timofeev [37], we let µ → ∞, which
implies c2
k2
ω2
c . For frequencies of our interest, ωp ωc ≈ ω, all the three are electrostatic limits. The
basic idea is common: find an ordering making the off-diagonal components of eq. 1.21 negligible. Stix [38]
reverses the order: he uses the e.s.approximation from the beginning, later warning and demonstrating that
it applies only when N2
|Λij| for any i and j.
25. Sec.1.3 Ordinary, extraordinary and Bernstein mode 17
Hence the dispersion eq. 1.24 reduces to Λyy = 0 (correspondingly E has only the y com-
ponent) and Λxx = xx = 0 (only x component). By projecting on k, i.e. along the x
axis, the two modes appear to be respectively transverse and longitudinal. The former can
be identified with the old X-mode. The latter is new (the cold dispersion relation had no
e.s. solutions) and is the B-mode. Its dispersion relation xx = 0 can be written as:
µ = X
∞
n=−∞
n2 ˜In(µ)
1 + nY
(1.27)
where also the approximation Zn −1/ζn has been used. Notations apart, this is the
original equation worked out by Bernstein [35] and reported among others in Stix’s book
[38]. The solutions are plotted in fig. 1.6 for different values of ω2
p/ω2
c , a quantity expressing
how overdense the considered plasma is. It should be emphasized again that eq. 1.27 and
fig. 1.6 are valid only for large values of kρ. The rough threshold kρ = n/3 established in
Sec. 1.1 is depicted in the present picture as a “guide for eyes”. On the left of the dashed
line, there are no more electrons the e.s.wave can interact with (and be sustained by!) and
the dispersion modifies by connecting to the e.m. X-mode. In other words the dispersion
plot 1.6 is expected to match somehow the cold dispersion plot 1.4 in the small gyroradius
limit, on the left of the dashed line (see Sec. 1.5).
In the same spirit, two solid lines cut regions of fig. 1.6
-1.5 -1.0 -0.5 0 0.5 1.
1
2
3
4
w/wc
k D
2 2
l
Fig.1.7: Oblique EBW dispersion re-
lation at θ = 80◦ from the
magnetic field, for k⊥vT /ωc =
1, adapted from [38], showing
harmonic coupling (solid line)
with respect to the perpen-
dicular case (fig. 1.6). How-
ever, intense absorption at
cyclotron harmonics can also
be recognized from the imagi-
nary part of the dispersion re-
lation (dashed).
where EBWs interact mainly with the bulk (|v|
vT ), with mid-velocity (vT |v| 2vT ) or tails
(2vT |v| 3vT ) of the maxwellian velocity distribu-
tion. These demarcations give a flavour of where the
dispersion relation may be altered by non-maxwellian
features.
Incidentally, mind that also the dielectric tensor 1.21
later used in ray tracing calculations is founded on
the assumption of maxwellian plasma. We will come
back on this point before interpreting the experimen-
tal results (Chap.3), demonstrating the assumption
to be valid for the discharges under examination.
From fig. 1.6 EBWs may be recognized to be mostly
backward waves, namely they have opposite phase
(ω/k) and group (∂ω/∂k) velocity. In 3D calcula-
tions the vectors vph = ω/k and vg = ∂ω/∂k will be
shown to be even not aligned, in general.
Second, orthogonal EBWs exist only between cyclotron
harmonics, where their emission and absorption take
place. This property does not hold for oblique waves,
whose harmonics can couple [38] (fig. 1.7). Math-
ematically, such harmonic coupling originates from
the dispersion function Zn, which does no longer ap-
proximate to −1/ζn: the insertion of its exact ex-
pression 1.22 in eq. 1.27 removes kρ divergences at multiple values of ωc. Intuitively, the
26. 18 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
“Langmuir-like” component ( B) of EBWs does not interact with the electron cyclotron
motion (⊥ B), suffers only the Landau damping and in principle can pass through cyclotron
harmonics. This is no longer true for the huge optical thickness computed in Sec. 1.6 for
the oblique-viewing experiments which this thesis is concerned with. As a consequence the
same inequality as for perpendicular EBWs applies:
nωc ω (n + 1)ωc, (1.28)
with n integer. This condition must be verified everywhere between emission and mode-con-
version into slow X. The condition may be refined after a glance to forbidden frequency bands
in fig. 1.6. These bands, sometimes called Gross gaps after E.P.Gross, depend in a compli-
cated way on ω2
p/ω2
c , therefore it is difficult to write down a simple and general inequality.
However, the present thesis deals only with 1st and 2nd harmonics, when, irrespectively of
the ω2
p/ω2
c value, the condition 1.28 is replaced by:
ωc ω min(2ωc, ωUH) (1.29)
2ωc ω min(3ωc, ωUH) (1.30)
The plasma frequency enters in these relations trough ωUH, setting a lower density limit for
EBWs. Namely 1st and 2nd harmonic must be excited/detected within the upper hybrid
layers. On the other hand, as it is evident from the dispersion plot 1.6 where arbitrary large
values of ω2
p/ω2
c are retained9
, there is no upper density limit for EBWs.
1.4 O-SX mode conversion
A p-polarised10
ray of light travelling in a transparent medium, is totally transmitted in a
second medium if it strikes the boundary at the Brewster angle arctan(N2/N1), with N1 and
N2 refractive indexes of first and second medium. However a good transmissivity [41],
T = 1 − R = 1 − tan2
(θi − θt)/ tan2
(θi + θt),
is achieved for any incidence angle θi sufficiently close to the Brewster one: after making the
transmission angle θt explicit from Snell’s law, the angular dependence of fig. 1.8 is found.
It is also all day life in long-distance wireless communication that 3 to 30 MHz radio waves
reach the receiver after one or more reflections between ground and ionosphere, provided
the incidence angle exceeds a critical angle (fig. 1.9). Partial reflection occurs also in the
opposite limit of small angle, i.e. vertical incidence (the ”ionograms”11
are based on this
principle [43]).
Similar situations, at a different range of frequencies, take place in tokamak and stellarator
plasmas. Reflectometry, for instance, is the tokamak counterpart of a ionogram; the O-SX
conversion we are going to discuss is just full transmission at Brewster incidence, accompa-
nied, like in the ionosphere [43], by a continuos change of polarization from O to SX.
9
Values ω2
p/ω2
c 1 are also possible, provided ω ωUH (see fig.1.3 of ref.[10]).
10
with E in the plane of incidence
11
Measurements of the delay between a radio pulse and its ”echo”. Since a unique relationship exists
between the sounding frequency and the reflecting ionization density, by sweeping the frequency the profile
of ionization degree vs. altitude is obtained.
27. Sec.1.4 O-SX mode conversion 19
Brewster angle
arctan (N /N )2 1
Critical or “total
internal reflection”
angle, arcsin (N /N )2 1
Fig.1.8: Transmissivity of a p-polarised wave
from a medium with N1 = 1 into a
medium with N2 = 0.2, as a function
of the incidence angle.
RT
Ionosphere
Earth
Fig.1.9: Long-distance transmission of radio
waves. For excessive antenna eleva-
tion, i.e. too small incidence angle, the
ray penetrates the ionosphere [42].
The idea behind is that, when propagating in the critical direction, the O and SX cutoff
surfaces coincide. The polarizations also coalesce, locally. Such a mode degeneracy permits
waves to approach the cutoff as O-waves and go ahead with a different name, “slow X”.
The quickest way to see that starts with putting the cold determinantal equation met at the
beginning of previous section, det Λh
= 0, in the form of the well-known Appleton-Hartree
equation [38]:
N2
⊥ + N2
z = 1 −
2X(1 − X)
2(1 − X) − Y 2 sin2
θ ± Γ
(1.31)
Γ = [Y 4
sin4
θ + 4(1 − X)2
Y 2
cos2
θ]1/2
Unlike special solutions 1.15 and 1.16, now the wave-vector forms an arbitrary angle θ with
the static magnetic field. The upper sign at denominator refers to O-mode and the lower to
X-mode.
The O and X branches cross -i.e., the modes coalesce- at X = 0 and X = 1.
The former cross-point is mere degeneracy at low density, when both O and FX tend to
vacuum propagation (fig. 1.4).
In the latter cross-point, O and SX dispersion connect and form one “O-SX” curve, which
partly develops in the evanescence half-plane N2
⊥ 0 (fig. 1.10a). Evanescent values are
avoided only when Γ vanishes i.e. only if -at the same time- Y sin θ = 0 and (1−X)Y cos θ =
0, which in a magnetoplasma happens only if θ vanishes at the X = 1 layer, i.e. only if vph
arrives parallel to B at the cutoff. In that case eq. 1.31 returns:
N2
z = N2
z,opt =
Y |X=1
Y |X=1 + 1
, (1.32)
where Y |X=1 is Y evaluated at the X = 1 layer.
Imposing θ to vanish at X = 1 is like imposing the turning point to lie at the cutoff. This
is done by elevating the antenna until the ray meets the cutoff (transition from ray 2 to ray
3 in fig. 1.11) but before it becomes perpendicular (ray 1).
Under these conditions, the O-wave can excite a wave beyond the cutoff, with the same k
and polarization, but belonging to the SX branch. This continues undisturbed.
28. 20 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
Fig.1.10: a) Appleton-Hartree dispersion relation for optimal and slightly non-optimal oblique
detection in Y = 0.9.
b) Like a) but in the variables of fig. 1.4 and only for optimal angle.
Note, however, that it does no longer make sense to call (extra)ordinary the waves at the
turning point: in that point the wave-vector is parallel to B and the eigenpolarizations are
circular (see eq. 1.15). In particular the wave under consideration is left-handed. This can
be deduced from the Clemmow-Mullaly-Allis (CMA) diagram [38, 39]. It is also derived in
a different way here, further down.
From the second row of the cold wave equation, with Ny = 0 -because finite values of Ny
will be shown to be detrimental for the O-SX conversion-,
1 − X
1−Y 2 − N2
z −iY X
1−Y 2 NxNz
iY X
1−Y 2 1 − X
1−Y 2 − N2
x − N2
z 0
NxNz 0 1 − X − N2
x
Ex
Ey
Ez
= 0, (1.33)
the following polarization at the cutoff (Nx = 0) is found:
i
Ex
Ey
=
1
Y
+
(N2
z − 1)(1 − Y 2
)
XY
(1.34)
Solving eq. 1.31 or 1.33 for X at Nx = 0, the familiar X = 1 cutoff and the cutoffs
X = (1 − N2
z )(1 ± Y )
are deduced, where the minus sign corresponds to lower density, i.e. to the right or FX cutoff
(see also fig. 0.2 and 1.4). The plus sign corresponds in turn to the left-handed (i.e. SX) or
to the O-mode cutoff, depending on N2
z /N2
z,opt and according to fig. 1.10a. The results are
plotted in fig. 1.13a. In different units, they illustrate of how far the cutoff layers are from
each other. The O-mode can exist only below the continuous curve, the SX only above the
dashed line.
29. Sec.1.4 O-SX mode conversion 21
Fig.1.11:
1) a quasi-perpendicular O-wave is reflected at the
cutoff; 2) a quasi-parallel one turns without meet-
ing the cutoff; 3) for an “optimal” inclination in
between, the turning point coincides with the cut-
off. Locally the wave is L-polarised and degener-
ate with another solution of the Appleton disper-
sion relation for Nz = Nz,opt, which belongs to the
SX branch and has exactly the same wavevector
and polarization.
B
O−mode cutoff
1
2
x
z
n
∆
L
L
L
SX
O O
3
Fig.1.12: Rigorous version of ray 3 of sketch 1.11, when the decoupling of phase and group
velocity is taken into account: a) A detail of plot 1.10b extended to k⊥ 0 shows
how ∂ω/∂k⊥ remains finite at the cutoff k⊥ = 0. b) Consequent ray tracing for ray 3.
The arrows represent the wave-normals at different positions. They also give the vph
direction, which strongly differs from vg (tangent to the ray) . A piece of ray after the
SX turning point is also shown.
These X curves lead -through eq. 1.34- to the polarization plot 1.13b, where both O and SX
mode appear to be left-polarised12
(iEx/Ey = 1) at N2
z = N2
z,opt.
As a result, at the turning point the wave is sub-
O
L
at the cutoff
at the antenna magn. field
SX
ray
beyond the cutoff
Fig.1.14: Continuous change of pola-
rization from O to SX, pass-
ing through a degenerate left-
circular (L) state.
ject only to the left-cutoff ωL
13
and immune to the
ordinary one.
We remark that in the slab geometry under con-
sideration Nz is conserved. As a consequence, and
efforting the degeneracy in k apparent from fig. 1.10
and that in polarization apparent from fig. 1.13b,
the launch of an ordinary wave with N=
( 1 − N2
z,opt, 0, Nz,opt) results in excitation, at the
X = 1 layer, of an SX wave having N = (0, 0, Nz,opt).
In contrast to perpendicular SX waves, which then
travel immediately back toward lower density, ob-
lique SX waves penetrate deeper. Actually in contrast to fig. 1.4, fig. 1.10b hints that
frequencies slightly below the cutoff are admitted in the oblique case, therefore densities
moderately higher than that at the cutoff are also admitted, in fig. 1.10a.
This is confirmed by the SX turning point density computed from eq. 1.31 for ∂ω/∂Nx = 0
12
Controversial definitions of handedness may be found in literature. Here we follow that of [30].
13
lying deeper in the plasma, as a consequence of ωL ωp
30. 22 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
Fig.1.13: a) cutoff density and b) polarization at the cutoff, both as functions of N2
z , showing
coalescence of O and SX mode at N2
z = Y
1+Y 0.47 in uniform field Y = 0.9.
The density corresponding to the SX turning point (t.p.) is also plotted in a).
and also plotted in fig. 1.13a.
The replacement of graph 1.4 with 1.10b allows now to move from O to SX branch chang-
ing in a continuous way wavevector and -since everything in eq. 1.34 varies continuously-
polarization (fig. 1.14). Care was taken not to access regions of evanescence, so that the
wave remains undamped. There being no dissipation, energy conservation writes simply
R + T = 1, with R and T energy flux fractions associated with the reflected O-mode and
the transmitted SX-mode. To definitely believe in finite transmissivity, it remains only to
solve the wave equation 1.2 with the initial condition of an O-wave incident with optimal
Nz. At least a non-vanishing left-handed component of the wave-field, E+ = 0 is expected
beyond the cutoff, but surprisingly the transmission will be found to be complete: T = 1.
Consider a plasma magnetized along z and plane-stratified along x. Suppose the inhomogene-
ity is weak enough to make use of the local dielectric tensor. The plasma be invariant under
y and z translations. Fourier-transform in those directions and handle the x-inhomogeneity
with the eikonal representation:
E(x, ω) = E0(x, ω) eiS(x,ω)
kx
def
= dS/dx
Then observe that ky = 0 can only degrade the conversion efficiency because N2
y 0 implies
evanescence of the x-component (thus dissipation) in order to guarantee N2
⊥ = 0 at the
conversion point (see fig. 1.10a). Setting ky = 0 and replacing
kzx −→ x
the wave eq. 1.2 writes in components:
iN2
z dxEz = (S − N2
z )Ex − iD Ey (1.35a)
−N2
z d2
xEy = iD Ex + (S − N2
z )Ey (1.35b)
−d2
xEz + idxEx = PEz/N2
z (1.35c)
where Stix notation is used for the cold dielectric tensor 1.14. and the subscript has been
dropped from E0. Moreover, for brevity the operator
dx = d/dx + ikx
31. Sec.1.4 O-SX mode conversion 23
has been introduced. Perturbing the refractive index of eq. 1.32,
N2
z = N2
z,opt + δ N2
z ,
and assuming a linear density profile X = 1 + x/Ln give:
−S + N2
z =
Y + x/Ln
1 − Y 2
+ δ N2
z (1.36a)
D =
Y + Y x/Ln
1 − Y 2
, (1.36b)
Hence, if N2
z is optimal and Y 1, which is the case of EBE at W7-AS (0.94 Y 1),
then −S + N2
z = D and eq. 1.35bb rewrites:
−N2
z,optd2
xEy = iDEx − DEy (1.37)
At this point, it is convenient to define E± = Ex ± iEy and to impose the initial condition
of incoming O-mode, or identically the final condition of SX-polarization: according to
fig. 1.13b, for optimal incidence both O and SX have E+ = 0 in an interval of the cutoff. As
a consequence, eq. 1.37 becomes:
d2
xE− = 0
i.e. the left-handed component decouples from the set 1.35. The real part of the equation is:
d2
E−
dx2
= k2
x(x) (1.38)
In the vicinity of a cutoff (a zero of k2
x(x)), we can retain only the leading order in the Taylor
expansion
k2
x(x) = β2
(x − x0) + γ2
(x − x0)2
+ O[(x − x0)3
]. (1.39)
It follows from fig. 1.10a that for slightly non-optimal incidence there are two zeros, that the
leading terms are linear and eq 1.38 splits in two Airy’s equations:
d2
E−
dx2
+ β2
(x − xO)E− = 0
d2
E−
dx2
+ β2
(xSX − x)E− = 0 (1.40)
where xO and xSX are the O and SX-mode cutoff positions. The same slopes β2
are assumed.
The most general solutions of eqs. 1.40 are linear combinations of Airy functions Ai and Bi,
but far from cutoffs the WKB approximation applies, yielding [39]:
E−(x) =
A1
kx(x)
exp i
x
kx(x )dx +
A2
kx(x)
exp −i
x
kx(x )dx ,
where A1 and A2 are the amplitudes of forward (incident) and backward (reflected) wave.
Matching of the near-field Airy solution with the far-field WKB solution in the intermediate
range where both the approximations apply, excludes the diverging function Bi from the
solutions of 1.40, which finally will be, apart from factors:
E− ∼ Ai[−β2/3
(x − xO)] E− ∼ Ai[−β2/3
(xSX − x)]
As fig. 1.15b illustrates, the O (SX) wave evanesces exponentially after (before) its cut-
off. Physically, this is because the polarization currents (the plasma response) grow up to
complete cancelation of the displacement current ∂E/∂t.
32. 24 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
-10 -5 0 5 10
x(a.u.)
0.2
0.4
0.6
k2
x(a.u.)
xO xSX
Fig.1.15a: Profile of k2
x vs. x. For non-optimal
N2
z it is linear around the two zeros.
For optimal N2
z it has one zero and
is locally a parabola.
-10 -5 5 10
-0.5
-0.25
0.25
0.5
0.75
1
Fig.1.15b: Airy functions describing O and SX
modes near the (distinct) cutoffs.
-10 -5 5 10
-0.5
-0.25
0.25
0.5
0.75
1
-10 -5 5 10
-0.6
-0.4
-0.2
0.2
0.4
0.6
Fig.1.15c: Even and odd solution of eq. 1.41
showing merging of O and SX waves
when cutoffs overlap.
O and SX waves appear distinct in fig. 1.15b,
but it is intuitive that when cutoffs are only
few decay-lengths apart, an O wave can excite
an SX wave on the other side, and viceversa.
In particular, for optimal N2
z the cutoffs co-
incide (xO = xSX) and the leading term in
Taylor expansion is quadratic. Eq. 1.38 takes
then the form of a parabolic cylinder equa-
tion:
d2
E−
dx2
+ γ2
x2
E− = 0 (1.41)
the cutoff having been placed in the origin
(xO = 0) for simplicity.
The solution is tabulated in [44] as linear com-
bination, with weights determined by bound-
ary conditions, of the following odd and even
functions:
Eeven
− = sign(x)× |x|J−1/4(γx2
/2) (1.42a)
Eodd
− = |x|J1/4(γx2
/2) (1.42b)
that in turn involve 1st kind Bessel functions
of fractional order.
The results are summarized in figures 1.15.
In the first, two schematic profiles of k2
x vs. x
are drawn after graph 1.10.
In the former case, of non-optimal N2
z , the
profile has two zeros. Correspondingly O and
SX modes have distinct solutions (fig. 1.15b).
In the latter case, of optimal N2
z , the k2
x pro-
file is tangent to the abscissae, and there is
one solution as a whole, breaking from one
side into another (fig. 1.15c).
A tendency to long wavelength around x = 0
(a remembrance of reflectometric, Airy func-
tion behaviour) can be recognized in the graph.
Actually λ → ∞ in that point (and only in
that point). This is clear because that point
is a cutoff, defined by kx → 0.
Remarkably, far from cutoff the waves of fig.
1.15c return sinusoidal with the same wave-
length and amplitude as before.
The treatment from eq. 1.40 to fig. 1.15b is
valid for a distance between cutoffs which is
infinite compared to wavelength.
33. Sec.1.4 O-SX mode conversion 25
Fig.1.16: Fig. 1.13a replotted with x instead of X on the vertical axis, illustrating the beneficial
effect of density gradient on “angular freedom”. A linear profile X = 1 + x/Ln is
assumed, with two different length-scales Ln. In the shaded ranges of N2
z the O and SX
cutoffs are close enough (say within λ/10) to allow appreciable O-SX tunneling.
Eq. 1.41 and fig. 1.15c, on the other hand, concern the limit of zero cutoff separation.
The generalization to intermediate cutoff distance, i.e. to k2
x = γ2
x2
+ β2
x + α2
, is straight-
forward if the Whittaker functions are used. These are tabulated in [44, 45]. Complications
arise as soon as the simplifications of coefficients 1.36 and the assumption of pure left-handed
polarisation are removed. In general, in fact, eqs. 1.35 are coupled. Moreover, they are fur-
ther complicated by including Ny = 0.
Elimination of variables leads to a 4th order differential equation [18] whose solution goes
beyond the scope of this thesis.
Here it suffices to point out the partial14
analogy with the problem of tunnelling through a
potential barrier in quantum mechanics, with −k2
x playing the role of the potential and E−
of the wave function. Semiclassical computing of ”scattering coefficients” [46] was adapted
to calculations of reflectivity and transmissivity, but by virtue of minor differences in the
approach, different workers found slightly different solutions [18, 47, 48]. Among them the
Mjølus formula [48]
T = exp −πk0Ln Y/2 2(1 + Y )(Nz,opt − Nz)2
+ N2
y (1.43)
is in best agreement with a numerical study [49].
The transmissivity diminishes like a gaussian as Nz and Ny depart from optimal values. The
width of such a gaussian bell increases inversely with magnetic field Y , vacuum wavenumber
k0 and local density length-scale Ln = n/(∂n/∂x).
A big tolerance towards Nz and Ny is of course experimentally desirable15
, but since Y and
k0 are both fixed by B and by the selected EC harmonic, as a matter of fact the ”angular
freedom” (the greatest angle at which a ray can depart from optimal direction without being
totally reflected at the cutoff) depends only on density gradient. A qualitative explanation
of its benefits follows.
Eq. 1.43 can be reviewed as exp(−d/λ0), where λ0 is the free space wavelength in perpendic-
ular direction and d the thickness of the barrier, viz. the distance between O and SX cutoff.
14
there are also several distinct differences, like the grade of the equation
15
to lower the requirements on antenna directivity and precision in matching the optimal direction, and also
to improve the flexibility with respect to plasma parameters, that affect position, orientation and grooveness
of the cutoff layer, thus inducing an error on optimal Nz and Ny.
34. 26 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
Hence a unique relationship exists between T and d, which in turn is proportional to Ln.
In fig. 1.16 d is plotted against N2
z for a steep and a mild density gradient.
It can be recognized that d, and thus T , take the same values in a wider range in the picture
on the left, for strong gradient. Conversely, when Ln is large the O-SX conversion is efficient
only in a narrow N2
z range, i.e. in a narrow cone around the best direction (picture on the
right).
1.5 SX-B mode conversion
The just discussed O-SX conversion is not of much interest for what concerns the modestly
enhanced penetration depth of SX compared to O-mode: what truly matters is rather the
coupling to SX-waves, which in turn naturally prolong into electron Bernstein waves, unaf-
fected by upper density limits.
The connection with the electrostatic B-mode is favoured by the fact that the SX-mode also
possess a partial electrostatic nature. More generally all the extraordinary waves, even for
perpendicular propagation, have a longitudinal component, as pointed out in the book of
F.Chen [50]. Such an electrostatic character is stressed by oblique propagation in presence
of a gradient ne ⊥ B and of FLR effects (allowing spacecharge separation ⊥ B): the
electron motion in the field of the oblique e.m. X-wave has a component along ne. As
a consequence, a ”packet” of electrons of a certain density moves forth (back) along ne,
exploring denser (less dense) regions. In this way, the electrons themselves perturb the local
ne, causing a longitudinal spacecharge wave i.e., ultimately, an e.s. wave [51].
Light rays turn toward regions of higher refractive index and turn away from lower refractive
index. In this sense, rays are attracted by resonances and repelled by cutoffs [52]. In fact
the cold SX mode considered in previous section, although launched toward the L-cutoff,
deviates toward the UH resonance, as fig. 1.12a documented.
In getting closer to the UH resonance, just owing to definition of resonance, the perpendicular
component of the wavevector rapidly increases. Then the electric field begins to lie parallel
to the wave vector, or the wave turns to have an electrostatic nature16
.
As k⊥ increases the wave slows down in the sense of the phase velocity component ω/k⊥, and
in the long term damps collisionally [38]. This is true in general for waves in plasmas but
dramatically for e.s. waves, since sustained by charge motion. Random collisions with ions
disturb in fact the ordered electron motion in the wave field: after a collision an electron has
zero velocity, then is accelerated by the wave and stopped again by another collision. The
average electron velocity at a given time may be derived from supposing the time between
two collisions to follow a Poisson’s distribution and appears to be reproduced by a fictitious
viscous term in the equation of motion [53]. This viscous damping of electron oscillations
results in damping of the e.s.wave with time-dependence E ∼ e−iωt−νt
. The ion-electron
collision frequency,
ν =
ln Λ
4
√
2π
ω4
p
nev3
T
, (1.44)
where ln Λ ≈ 15 is the Coulomb logarithm [50], is typically several orders of magnitude
smaller than EBE-relevant ω (see fig. 1.17). As a consequence, it takes many periods 2π/ω
16
See ref.[38] and footnote 8 of this thesis.
35. Sec.1.5 SX-B mode conversion 27
Fig.1.17: Ion-electron collision frequency as a
function of edge-relevant ne and Te.
Fig.1.18: Contours of electron Larmor radius
ρ (in µm) for values of B and Te rel-
evant for W7-AS at high density.
for the wave to damp collisionally, but a cold slow-X wave takes an infinite time to reach
the UH resonance, which is enough for being damped.
If on the other hand vT = 0, then, owing to the resonance of k⊥, the thermal frequency k⊥vT
exceeds the collisional frequency earlier than significant damping has occurred.
In brief, kinetic effects (proportional to k⊥vT , diverging) are to be included in dielectric
tensor more urgently than collisional corrections (proportional to ν, fixed).
Equivalently to present frequency viewpoint, the need for hot theory is addressed by the fact
that at the UHR the decreasing phase velocity approaches the electron thermal velocity, or
that sooner or later the decreasing wavelength becomes comparable with the Larmor radius
(fig. 1.18).
At so short wavelengths also EBWs propagate and it makes sense to study their coupling
with SX mode. The underlying physics is FLR effects, making UH oscillations acquire finite
group velocity ⊥ B. The consequent novelty vs. cold plasmas is that the ray bends such that
it is no more absorbed at the UH layer because it deviates earlier. Afterwards it connects to
the nth Bernstein harmonic, with n defined by nωc ωUH (n + 1)ωc.
A thermal correction ∼ N6
to Booker quartic mentioned at page 13 suffices to display this
behaviour [14, 54]17
, although restricted to perpendicular propagation.
For our scheme, however, it is more convenient to generalize to oblique view by considering
the determinantal equation det Λh
= 0 for arbitrary N and hot dielectric tensor 1.21. This
allows to visualize both O-SX and SX-B conversions in a comprehensive way: a numerical
search of the roots of det Λh
in the (ω, k) plane gives the results of fig. 1.19.
Note that the B-mode propagates also in the overdense part of the plasma, where X 1.
The high perpendicular refractive index in that plot reflects the fact that EBWs, since
propelled by cyclotron motion, are characterized by k⊥ρ ≈ n (Sec. 1.1 and 1.3).
The amplification of N⊥ is thus inverse proportional to the wavelength decrease from about
4mm (75GHz) to Larmor radii of fig. 1.18. Its order of magnitude can be estimated at:
N⊥ =
ck⊥
ω
≈
nc
ωρ
=
ncωc
ωvT
≈
1
βth
17
to include higher orders, one can plug ij from eq. 1.21 in eq. 1.25
36. 28 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
Fig.1.19: Hot plasma dispersion relation for oblique propagation at optimal angle, showing B-
SX-O mode conversion (points). Te = 500eV , Y = 0.9. Cold plasma dispersion relation
also plotted for comparison (line).
where βth is the thermal velocity normalized to the speed of light (see also argument 1.19 of
the Bessel functions involved in hot dielectric tensor 1.21).
Large N⊥ may drive large values of N as a mere geometrical consequence. The parallel
component does not take part in wave-particle energy exchange (based on cyclotron motion,
perpendicular) and is conserved in a plasma slab invariant under parallel translations. In
realistic configurations, however, the direction of B changes along the ray, therefore the
parallel component of N also changes. Its differential increment is dN = N⊥ dΘ when B
rotates of a polar angle dΘ with respect to N (an azimutal rotation has no influence, of
course). As a result the effect of toroidal curvature on an equatorial ray is dN = N⊥ dϕ,
which is quite a large effect: it means ∆N = 0.1N⊥ after the ray has spanned 0.1 radians
(less than 6◦
) of toroidal angle.
Add now a sheared poloidal component, i.e. let B make an angle ϑ with the equatorial plane,
and let this angle change. Call Ψ the angle of N with the vertical plane containing B. Then
N = N cos ϑ cos Ψ and, since dϕ = −dΨ, it will be
dN = −N
Bϑ
B
cos Ψ dϑ + N
Bϕ
B
sin Ψ dϕ (1.45a)
−N
Bϑ
B
dϑ + N⊥
Bϕ
B
dϕ (1.45b)
The approximation at second row applies to small ϑ. Both expressions regard only rays in
the equatorial plane.
To conclude, let us consider a possible degradation mechanism for SX-B coupling. For the
diagnostic we can exclude nonlinear effects occurring in EBW heating [21]. If collisions are
37. Sec.1.6 EBW emissivity and optical thickness 29
neglected, the only losses for SX-B conversion (with B propagating toward the overdense
plasma core) can be ascribed to partial SX-FX conversion (with FX propagating out of the
plasma). The SX mode can exist only inboard of the UH resonance, the FX only outboard of
the R cutoff. If these layers are not too far, some power of the SX-mode can tunnel through
to the FX-mode with efficiency [13, 43]
TSX−FX = exp −πk0LnY 2
(ωUH/ωc − 1)/X (1.46)
and leave the plasma. The overall O-SX-B efficiency will then be TO−SX · (1 − TSX−FX).
The formula for TSX−FX reminds of that for TO−SX (eq. 1.43) but unfortunately, in contrast
to that, it cannot be controlled by adjusting the direction of launch or detection: rather it
is determined by plasma properties at the edge and more precisely at the UH layer, where
all the quantities of eq. 1.46 are evaluated. We will come back to this point in setting the
operational limits for the EBE diagnostic (Sec. 3.6).
1.6 EBW emissivity and optical thickness
Emissivity and optical thickness are well-known concepts in traditional O and X-mode ECE,
since they legitimate, where applicable, the black-body approximation and assess the theo-
retical limit to the spatial resolution of the diagnostic. In this paragraph the same quantities
are computed and discussed for the oblique-propagating B-mode.
A simple energy balance between emission and absorption in the plasma yields the radiative
transfer equation [55]:
N2
r
d
ds
Iω
N2
r
= jω − αωIω
describing the spatial evolution of the intensity per unit frequency Iω of a beam of e.m. waves,
in terms of emissivity jω and absorption coefficient αω. These are defined respectively as
intensity emitted and fractional intensity absorbed per unit frequency and unit length.
The (de)focusing action of the plasma as a refractive medium can also affect the intensity.
This is taken into account through the ray refractive index Nr, featuring a plasma anisotropy
correction [55, 56] to the refractive index N considered so far. In fact it can be shown that
in an anisotropic plasma the wave vector (parallel to the phase velocity) and the energy
propagation direction (aligned with the group velocity) are generally distinct.
The coordinate along the ray, s, is usually adimensionalized and renormalized to the absorp-
tion lengthscale α−1
ω :
dτ = −αω ds
and is renamed optical thickness. After this substitution, the integration of the transfer
equation over a -generally curvilinear- ray path of extremes A and B provides:
Iω(A)
N2
r (A)
e−τ(A)
−
Iω(B)
N2
r (B)
e−τ(B)
=
τ(B)
τ(A)
1
N2
r
jω
αω
e−τ
dτ
In particular if B is the “origin” of the ray in the plasma, i.e. Iω(B) = 0, and A is an
external antenna, in vacuum, where neither dispersion nor absorption take place (N2
r (A) = 1,
τ(A) = 0),
Iω(A) =
τ(B)
0
1
N2
r
jω
αω
e−τ
dτ (1.47)
38. 30 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
This is how the spectrum Iω collected at the antenna is related to above listed ingredients:
refraction (through N2
r ), emission (jω) and absorption (αω).
Though in general integrated along the ray, the formula is of interest for local diagnostic
when the mentioned processes resonate in a small segment BC (see fig. 1.20a) and are
negligible elsewhere. This is the case of ECE of low harmonic O and X-mode in non-
uniformly magnetized plasmas: emission and absorption take place only where the resonance
condition is fulfilled, i.e. in a thin (Doppler and relativistically broadened) layer. Accounting
for relativistic mass increase and consequent gyrofrequency decrease with the Lorentz factor
γ, the resonance condition for the nth harmonic writes18
|ω − nωc/γ| 3k vT
The electron cyclotron emissivity jω vanishes outside this ω range. If this is smaller than
the overall ωc variation in the plasma, i.e. if the magnetic field is “non-uniform enough”, the
emitting layer is small compared to plasma size, only a small portion of the ray contributes
to integral 1.47, and profile measurements become possible.
The localization improves further when reabsorption is taken into account, since only emis-
sion from outer emitting regions is not reabsorbed and reaches the antenna. Inner regions on
the other hand do not contribute because emission and absorption cancel each other. The
self-absorption is quantified by the exponent and denominator in eq. 1.47: large values of τ
imply small e−τ
/αω, therefore small contributions to the integral i.e. intense reabsorption.
This is one of the reasons to require the plasma to be locally19
perfectly absorbing or thick
or opaque, namely to satisfy:
AB
dτ =
BA
αω ds 1 (1.48)
Like in solar physics, most observed intensity will originate in a small segment CD where
CD
dτ ≈ 1 [55]. The higher τ (the strong reabsorption) makes the “back-lighting” subpor-
tion BD invisible to an observer A (fig. 1.20a).
Under these conditions (large τ and sufficient magnetic non-uniformity) ECE of underdense
plasmas is a local measurement, in the sense that it probes locally the quantity 1
N2
r
jω/αω in
integral 1.47. Before asserting it is a local temperature diagnostic, however, the additional
criterion of local thermodynamic equilibrium (LTE) must be met. In this case Kirchhoff’s
law holds, relating emission and absorption in terms of black-body spectrum [55, 56]:
1
N2
r
jω
αω
Kirchh.
= B0(ω, T)
Planck
=
ω3
8π3c2
(e ω/KBT
− 1)−1 Jeans ω2
8π3c2
KBT (1.49)
where Planck’s formula and Rayleigh-Jeans approximation ( ω KBT) are used. Note
that in a fusion plasma ω 1meV and KBT 10eV. The plasma anisotropy correction N2
r
is also included [55]. After substituting in eq. 1.47,
Iω =
ω2
KB
8π3c2
τ(B)
0
Te−τ
dτ (1.50)
18
The factor 3 is 4 to cut reasonably somewhere the Doppler broadening gaussian bell.
19
only in emitting region BC (fig. 1.20a). In CA the plasma is required to be optically thin, i.e. transparent,
in order to let the (local) temperature information unperturbed until it reaches the antenna.
39. Sec.1.6 EBW emissivity and optical thickness 31
The derivation so far was quite general: up to eq. 1.47 only an energy balance and Snell’s
law were invoked, afterwards only black-body arguments and Kirchhoff’s law, which is a
consequence of the second principle of thermodynamics. Cyclotron resonance broadening
mechanisms, independent on the mode and on its e.m. or e.s. nature, were also quoted.
In conclusion the result 1.50 applies to B-mode also, provided the optical thickness condi-
tion 1.48 is verified. To check this, the EBW absorption coefficient αω is needed. αω can be
viewed as an intensity decay rate per unit length, corresponding to a field decay rate αω/2:
E ∼ ei(k·x−ωt)−αωs/2
This is like generalizing k → k+iαωs/2, with the unit vector s = vg/|vg| pointing in the ray
direction. To justify imaginary wavevectors, also the dispersion tensor must be generalized
by including the antihermitian part.
The real k solves the wave equation, here left-dotted with a,
a∗
· Λh
(k, ω) a
def
= Λh
aa(k, ω) = 0; (1.51)
the complex k, on the other hand, solves the following equation including absorption:
Λh
aa(k + iαωs/2, ω) + iΛa
aa(k + iαωs/2, ω) = 0
Taylor-expanding until first order in αω/2k 1 and comparing with eq. 1.51 give:
αω = −
2Λa
aa
s · ∂Λh
aa/∂k
(1.52)
At high N (fig. 1.19) the pure e.s. mode approximation applies 8. Pure e.s. modes are purely
longitudinal, characterized by a k, thus Λaa can be replaced by aa viz. kk:
αω,r = −
s
sx
2 a
kk
sx∂ h
kk/∂Nx + sz∂ h
kk/∂Nz
(1.53)
The multiplicative factor s/sx has been introduced to redefine in radial units (in x direc-
tion, corresponding to the radial direction in a toroidal device) the absorption coefficient
previously defined as decay rate per unit length in the propagation direction s.
The main difference between eq. 1.53 and similar expressions for e.m. modes is that kk ∝
N2
x xx + 2NxNz xz + N2
z zz i.e., owing to large Nx, the xx component dominates, whereas
O-mode absorption is governed by zz and X-mode by yy (Sec. 1.3).
The foregoing evaluation of αω,r splits in two cases [56]:
N vT /c ⇒ quasi − perp. (relativistic),
N vT /c ⇒ oblique (Doppler broadened),
In the former, the resonance is mainly broadened by relativistic mass increase. By using the
Dnestrovskii weakly relativistic dielectric tensor it can be demonstrated that [57]:
αω,r = −
23/2
π1/6
k0 −
XY
β5
ezn
[Ei(−zn) + iπ]
1/3
(1.54)
40. 32 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
Fig.1.20a: (adapted from [58])
2nd harmonic X-mode emissivity
(a.u.) and reabsorption-corrected
emissivity (a.u.) as functions of
the B curvature radius. A per-
pendicular line of sight in the
plasma elliptical cross-section is
considered. The non-relativistic or
“cold” resonance is located at R0.
Owing to reabsorption, only emis-
sion from the shaded curve reaches
an observer A at the low field side.
Fig.1.20b:
Relativistic absorption coeffi-
cient α and optical thickness τ
for 1st harmonic B-mode per-
pendicularly propagating in a
plasma of double density and
half temperature compared to
fig. 1.20a. Most observed inten-
sity originates in the very thin
layer ( 0.1mm) where τ 1.
206 207 208 209 210 211
R( )cm
100
200
300
400
500
alpha()cm-1
,tau
τ
( )cm-1α
R0
ne0=1020
m-3
Te0=500eV
f=70GHz
where Ei is the exponential integral function [44], zn = (1 − nY )/β2
should not be confused
with ζn (eq. 1.20) and, as usual, k0 = ω/c is the vacuum wavenumber. In addition, unlike the
original work that was more general, here N = 0 i.e. exactly normal incidence is considered
in order to isolate the relativistic broadening of 1st harmonic B-mode and compare it with
2nd harmonic X-mode (fig. 1.20a).
Substituting profiles of density, magnetic field and temperature yields the relativistic absorp-
tion profile of fig. 1.20b and, after integration, the τ profile. Since the latter is very steep and
criterion 1.48 is met, it can be concluded that perpendicular EBE is a local diagnostic. The
emissivity jω is omitted from graph 1.20b because simply related to αω by a multiplication
(eq. 1.49), with a small distortion because the factor is temperature-dependent but does
not change much on the short scales of that graph. The reabsorption-corrected emissivity,
defined by
ˆjω(R) = jω(R) exp[−
R
Rmax
αω(R ) dR ], (1.55)
where Rmax is the major radius of the first point out of the plasma, is also omitted because
it has a full width at half maximum ∆R1/2 0.1mm. For ECE-X2 at the same density and
41. Sec.1.6 EBW emissivity and optical thickness 33
temperature ∆R1/2 =1.5mm [58].
For oblique view, Doppler broadening dominates and the non-relativistic tensor 1.21 suffices
to determine:
N2
kk = N2
x xx + 2NxNz xz + N2
z zz = 1 + 2
X
Nzβ3
n
˜InZn[1 + β(1 − nY )]
Launch or detection in the (B, ne) plane (Ny = 0) was assumed for simplicity. At low
enough temperature, the quantity in square brackets approximates to unity. Then, close to
the first harmonic, the equation rewrites:
N2
kk 1 + 2
X
Nzβ3
˜I−1Z−1
After some algebra it follows from eq. 1.53 that:
αω,r = −2
s
sx
ω
c
Im(Z)
Re(Z)
sx
˜I
∂ ˜I
∂kx
+
sz
˜R
∂ ˜R
∂kz
−
Nx + Nz
N2
1
˜I ˜R
+ 2
−1
where subscripts −1 are dropped from ˜I and Z and for brevity ˜R
def
= X
Nzβ3 Re(Z−1). Since
near the EC resonance Re(Z−1) → 0, one can approximate:
αω,r = 2
s
s⊥
N2
N + N⊥
ω
c
˜IX
N β3
Im(Z) (1.56)
with the imaginary part of the dispersion function giving the peculiar gaussian form to
the Doppler-broadened absorption profile. Approximating N N⊥ N , the absorption
coefficient for obliquely propagating 1st harmonic B-mode finally writes:
αB1
ω,r = 2
√
π
s
s⊥
N
N
k0
˜IX
β3
e−ζ2
−1 (1.57)
coinciding for large gyroradii, when ˜I 1/
√
2πµ, with the limit of ref.[57].
For comparison, for oblique 2nd harmonic X-mode emission in a tenuous plasma [56]:
αX2
ω =
√
2π
8
G
N
k0
Xβ
Y
e−ζ2
−2 (1.58)
where N is the Appleton-Hartree refractive index 1.31 and G is a geometrical factor:
G =
sin2
θ
| cos θ|
1 + cos2
θ +
sin4
θ + 8 cos2
θ
√
sin4
θ + 16 cos2 θ
The more favourable scaling of eq. 1.57 with N (high) and β (low), ultimately a conse-
quence of EBW electrostatic nature, yields more rapid absorption and therefore, according
to eq. 1.55, more localised emission (figs. 1.21).
42. 34 Emission, propagation and mode conversion of electron Bernstein waves Chap.1
Fig.1.21a:
Absorption coefficient α,
optical thickness τ and
reabsorption-corrected emis-
sivity ˆj for oblique 2nd
harmonic X-mode. Doppler
broadening now prevails on
the relativistic one (compare
with fig. 1.20a).
204 206 208 210 212 214 216
R(cm)
0.0
0.2
0.4
0.6
α(cm-1
),j^(a.u.)
α
0.0
0.5
1.0
1.5
2.0
2.5
τ
τ
j^
f=140GHz
Te0=1KeV
ne0=5.
1019
m-3
θ=46.50
Fig.1.21b:
like fig. 1.21a but for 1st
harmonic B-mode in dou-
ble density, half tempera-
ture plasma. Notice different
units on vertical axis.
204 206 208 210 212 214 216
R(cm)
0.0
0.2
0.4
0.6
0.8
α(106
cm-1
),j^(a.u.)
α
0
1
2
3
τ/106
τ j^
f=70GHz
Te0=500eV
ne0=1020
m-3
θ=46.50
Fig. 1.21b refers to observation from outboard of the torus. A similar picture, reflected
around the cold resonance position, holds when the ray comes from the inner side of the
torus, i.e either if the antenna is placed on the high field side (HFS), or when the ray is
reflected by the HFS upper hybrid layer (BXBXO mode conversion, see chapter 3).
Notice, in fig. 1.21b, that an EBW never arrives at the ”cold” resonance because, before, it
damps on fast electrons belonging to the tail of the maxwellian distribution function. The
theory must be revised to include non-maxwellian features, but some qualitative statement
can already be made: a larger (smaller) population of fast electrons shifts the emitting layer
outward (inward). The contrary holds for HFS launch/detection.
For a maxwellian plasma the frequency shift is proportional to the width of the maxwellian
itself: ∆f ≈ 3k vT /2π, i.e. ∆f/f ≈ 3N βT . The corresponding radial shift in a tokamak-like
magnetic field decreasing like 1/R is:
∆R/R ≈ 3N βT
Used the other way round, this relation provides a coarse diagnostic of N when βT and the
deposition or emission position are known from other diagnostics.
43. 2. EXPERIMENTAL SET-UP
In the previous chapter it was shown that EBWs are thermally emitted in overdense tokamaks
and stellarators and mode-convert to e.m. waves able to leave the plasma along a special
direction. The present chapter deals with the construction of a diagnostic to collect and
spectrum-analyze those waves. The setup is similar to standard ECE radiometry, but also
shows some distinct differences. In fact, the unusually oblique line of sight (Sec. 2.1) affects
the antenna design (Sec. 2.2) and addresses the need for a broadband control of polarization
ellipticity, finally attained with an elliptical waveguide (Sec. 2.3). The setup is completed by
a transmission line, a heterodyne radiometer and a data acquisition system (Secs. 2.4- 2.6).
2.1 Line of sight
In Sec. 1.4, a geometrical criterion to penetrate the O-mode cutoff layer and extract tem-
perature informations from the overdense plasma core was provided. For this purpose the
refractive index must have components
Ny = 0 N2
z = N2
z,opt =
Y |X=1
Y |X=1 + 1
. (2.1)
in the coordinate system with ˆz B and ˆx ne. The optimal direction is determined by
both the absolute value of the magnetic field at the cutoff position, Y |X=1, and by its local1
direction, through Ny = 0.
As a consequence, due to spatial variation of B on the cutoff layer, in a tokamak the optimal
view angles depend on the poloidal position of the antenna, and in a stellarator they also vary
with the toroidal position. Antennas in different positions collect rays of different trajectories,
emitted in different locations and with different N . The consequent possibility to control
N by choosing the antenna position finds applications in both current-drive [60, 131] and
diagnostic. In particular, for diagnostic, N has to be as low as possible.
A finite N in the emission region, in fact, broads the resonance.
f
HFSLFS
Fig.2.1: Inhomogeneous
Doppler shift for
finite N
The high optical thickness delocalizes the emission to the out-
skirts of the broadened resonance layer, any internal emission be-
ing reabsorbed (fig. 1.21b). Spatial Doppler broadening results
thus in a temperature-dependent frequency up-shift. Originat-
ing from the hottest part of the plasma, the center of the EBE
spectrum is shifted more than the edges, colder (fig. 2.1).
This spectrum deformation is undesired for two reasons. First, it
is asymmetric and makes the spatial resolution on the high field
side (HFS) worse than on the low field side (LFS). Second, the deformation is temperature
dependent. Although ray tracing calculations are necessary in any case to transform spectra
1
Sec. 1.4 concerned non-sheared magnetic fields. The effect of shear is to rotate the plane of optimal
polarisation (at the antenna) so that locally, at the cutoff, Ny = 0 [59].
35
