2.
DIAGNOSTIC METHODS
2.1 Interferometry
2.1.1 Principle
Measurements of the refractive index are often made by interferometry. The ordinary wave that is not
necessary to consider the effect of external magnetic field and comparable to the isotropic case is used for
electron density measurement by interferometry.
The refractive index of the O-mode is given by Eq. (1.17) as
1/ 2
  2pe (r ) 

No  1 
2 

 

1/ 2
 n (r ) 
 1  e 
nc 

,
(2.1)
where nc   0me 2 / e is the “cutoff” density corresponding to the incident wave.
The
interferometry measures the phase difference  (x ) between the wave propagating in the
plasma and propagating in the outside of the plasma, which is given by.
y
 ( x)   2 (k0  k p )dy 
y1
2 y2

y1 (1  NO )dy
,
(2.2)
where k0  2 /  and k p are the wave numbers in the vacuum and plasma, respectively.
Assuming ne ≪ nc ,  (x ) is shown as the following formula.
 ( x) 
 y2
2 a
ne (r )dy 
n (r )(r 2  x 2 ) 1 / 2 rdr ,

nc y1
nc x e
rx
(2.3)
When radial profile of the density is axisymmetric, we can obtain the profiles by Abel
inversion of Eq. (2.26) as
ne (r ) 
  nc a d 2 2 1 / 2
 ( x  r ) dx , x  r
 2 r dx
(2.4)
Now we assume the plasma has parabolic distribution given by the following formula, as it is
known empirically,
-1-
Fig. 2-1 Geometry of interferometry.
  r 2 
ne r   ne 01    
  a  
(2.5)
Phase difference when the incident wave passes through the line x  0 is found by Eqs. (2.3)
and (2.5), which is proportional to ne
 0 
2  a ne 0

.
3
nc
(2.6)
2.1.2 Choice of incident wavelength
The density gradient along the diameter causes a refractive effect, when the frequency of
the incident wave becomes close to the electron plasma frequency. The value of the
refraction angleδis maximum when the incident beam propagate at the chord of x / a  0.7 as
  sin 1 ne 0 / nc   ne 0 / nc
(2.7)
Meanwhile, taking Gaussian beam theory into consideration, the beam expands along the
-2-
distance y shown in Fig. 2-1.
1/ 2

42 y 2 
d   d 02  2  2 


d 0 

(2.8)
Let us take the distance to the first collecting optics as L, and assuming d0  2L /  1 / 2 , we
obtain d  2d0 . Then, the conditions to allow measurement are led by Eq. (2.7) as
L m  L ne 0 / nc  d  2L /  1/ 2 ,
(2.9)
where nc  1.1  1013 2 ( is in the unit of cm) . On the other hand, the lower wave length
limit is determined that parasitic fringe shift  /  has no effect on measurement accuracy.
If it is 1% and below, F is fringe number due to the plasma density.
Equation (2.9) leads
range of incident wavelength as
 /   102 F
Therefore, we obtain.


4.1  108 a ne 0    1.2 1010 Lne2 0
1 / 3
(2.10)
In large devices (TFTR, JET, JT-60, DIII-D, LHD, KSTAR and so on) currently in operation
or under construction, it is shown that   100 m (far-infrared region) is the best suited
except low density region in periphery.
-3-
2.1.3 Phase Detection
An example of interferometer system and phase detector is shown in Fig. 1-4, and 1-5,
respectively
Fig. 2-2. Heterodyne interferometer using upconverter
Fig. 2-3. Quadrature-type phase detector.
-4-
2.2 Reflectometry
2.2.1
Density profile measurements
A reflectometer consists of a probing beam propagating through a plasma and a reference
beam. The microwave beam in the plasma undergoes a phase shift with respect to the
reference beam. This phase difference as a function of the probing frequency is given by
 ( )  2k
rc ( )

N (r ,  )dr 
a

(2.11)
2
within the WKB approximation, where k and  are the wave number and frequency of the
probing beam, N is the plasma refractive index, and a and rc() are the plasma and cutoff radii,
respectively. The refractive indexes of the O-mode and the X-mode propagations are given by
12
  2pe 
No 
 1  2 
o 
o 

cko
12
,
(2.12)
  2pe
 x2   2pe 
ck
N x  x  1  2  2
2 
x 
 x  x   2pe  ce


in the cold plasma approximation, where c is the speed of light,  pe and ce are the
electron plasma frequency and the electron cyclotron frequency, and subscript O and X
correspond to the ordinary mode and the extraordinary mode incident waves, respectively.
For the O-mode propagation, this integral can be analytically solved for the density profile
using an Abel inversion. For the X-mode propagation, a numerical algorithm has been
developed to invert the data.
In an ultrashort-pulse reflectometer, a very short pulse (1-100 ps) is used as a probe beam.
The pulse has a frequency component governed by the shape and duration. Each Fourier
component reflects from a different spatial location in the plasma. The time-of-flight for a
wave with frequency  from the vacuum window position rw to the reflection point at rp is
given by
2
 ( ) 
c
rp 

 2pe 
 1   2
rw 


1 2
dr .
(2.13)
-5-
In order to obtain the density profile from the time-of-flight data, the Eq. (2.13) can be
Abel inverted to obtain the position of the cutoff layer,
r ( pe ) 
c

 pe

0
 ( )
 d
( 2pe   2 )1 2
(2.14)
.
By separating different frequency components of the reflected wave and obtaining
time-of-flight measurement for each component, the density profile can be determined. The
movement of the plasma postion due to the fluctuations can be neglected during the reflection
of ultrashort pulse (ps).
2.2.2
Fluctuation measurements
Reflectometry has also been used in order to study plasma fluctuations. The instataneous
phase shift  between the local beam and the reflected beam is expressed as   0   ,
where 0 and  are the phase shift depending on the cutoff layer due to density and
density fluctuations of the plasma profile. In a simple homodyne reflectometer, the mixer
output is given by
V  El Er cos(0  )  El Er (cos 0 cos   sin 0 sin )
 El Er (cos 0   sin 0 )
(2.15)
assuming   1, where El and Er are the electric field amplitude of the local beam and the
reflected beam, respectively. The time varying component of the mixer output depends on
both amplitude and phase modulations, since Er can be time dependent due to changes in the
cutoff layer. In general, the radial fluctuations of the cutoff layer produce the phase
modulations and the poloidal (azimuthal) fluctuations cause amplitude modulations.
Therefore it is important to identify both phase and amplitude fluctuations using, such as,
heterodyne detection or quadrature type mixer.
In a simple one-dimensional model, the phase changes in the O-mode and the X-mode
propagations due to the small perturbations of the density and the magnetic field, ne and
B at the critical density layer are given by
o  2ko Ln (ne / ne )
(2.16)
,
and
-6-
x 

2k x ne ne  (ce x  2pe ) B B

(2.17)
1 Ln  (ce x  2pe ) LB
When the wavelength of the fluctuation is much longer than the spot size of the incident wave,
there is no attenuation or modulation of the reflected wave. The depth of the phase modulation
approaches the 1D geometric optics limit.
2.2.1
Reflectometer systems
Various types of reflectometer are shown in Fig. 2-4.
Fast-Sweep FM Reflectometer
AM Reflectometer
○high resolution with simple hardware
○minimal effect of density fluctuations
●phase runaway
●parastic reflections from wall and window
f
Source
f+
(t)
Short-Pulse Reflectometer
Ultrashort-Pulse Reflectometer
○measurement of real-frozen plasma
○an impulse generator
●many sorces or sweep source with
●ultrashort pulse (<10 ps) for high density
wideband switches
Source
f
Pin
Switch
plasmas
f±f/2
Pulse
Generator
Trigger
BP
Filter
Time
Delay

Digitizer and/or TAC
Fig. 2-4. Reflectometer systems
-7-
d   ( ) d
dt
dt
c 
2 
 ()  2c  1 pe2 
 
r 
r
1 2
Lr
dr  Ls 
c   p ( )  w( )
ant
 pe
 p ()
c
rc  rant   
d
1 2
2
2


0  pe  


-8-
2.3 Thomson Scattering
2.3.1. Collective scattering
By using microwave as an incident wave, scattering parameter   (k D ) 1 is usually
larger than unity. Then the scattering feature corresponds to so called collective scattering.
Most laboratory plasmas and magnetically confined plasmas have density fluctuations caused
by various types of instability within the plasma. These fluctuations generally have
wavelengths exceeding the Debye length and the fluctuation levels encountered far exceed the
thermal levels.
The scattered power per steradian and per radian frequency at the scattering angle s is
written by
Ps (k s , s )  pi neVs T S ( k ,  ) ,
(2.18)
where k=ks-ki, =s-i, pi is the power density of the incident wave, ne is the mean electron
density, Vs is the scattering volumn, T is the cross section of the Thomson scattering, and
S(k,) is the power spectral density of the density fluctuation given by
1
S (k ) 
2

2
n~k
 S (k , )d  neVs ne
,
(2.19)

where n~k is the amplitude of density fluctuations with wave number k. The wave number
spectrum can be obtained by changing the scattering angle s. The density fluctuation level is
then determined from the integration of k as.
4
 1 
n~e (r , t ) 2    ne  S (k ,  )dkd
 2 
(2.20)
For the thermal fluctuations, S(k)~1, however, S(k)>>1 for the non-thermal fluctuations.
Assuming n~e / ne  102  103 , ne=1019 m-3, and Vs=10-5 m3, S(k)=108-1010.
-9-
2.3.2. Microwave scattering
- 10 -
2.3.3. Far-Infrared scattering
- 11 -
2.3.4. Far-forward scattering
- 12 -
2.4 Electron Cyclotron Emission
2.4.1 Determination of electron temperature
In 1.3.1 it is shown that the intensity emitted by the plasma is given by Eq. (1.43). In
actual experiment, the plasma is produced in a metal chamber. If we consider the effect of the
reflection from the metal wall, it is known that the radiation intensity is modified as
I n ( )  I B0
1  e n
1  ree n
,
(2.21)
where re is the wall reflectivity (1> re > 0.9). Therefore, when 1  re ≪ n , In becomes nearly
equal to the black body radiation, then we call as “plasma is optically thick”. On the other
hand, when  n ≪1  re , Eq. (2.21) becomes
I B0
n
1  re
I n ( ) 
(2.22)
and “plasma is optically thin”.
Let us consider a tokamak plasma case and the coordinate system as shown in Fig2-5,
where B0 is the magnetic field intensity at the plasma center, R is the major radius. It is
know that the troidal magnetic field, BT is a function of x as
BT  B0
R
Rx
(2.23)
Therefore nce also varies accordingly. Elecron cyclotron emission (ECE) appears
resonantly in width
xn  n d nce  / dx 1
(2.24)
with centering on x  x  which corresponds to   nce . When the plasma is optically
thick, the radiation power becomes proportional to its local electron temperature.
On the other hand, when plasma is optically thin  ≪1  re ,
I n ( ) 
I B0
1

n e (Te ) n
1  re
1  re
(2.25)
The radiation power is proportional to both ne and Te profiles. Therefore, when Te profile
is obtained by different methods, we can determine ne profile, and vise visa.
- 13 -
Fig. 2-5. Electron cyclotron emission from tokamak plasma.
Furthermore observing the ECE at the optically thin n and n+1 th harmonics, we can
determine the electron temperature using the following formulas,
k Te 
M n I n  I ( )
(2.26)
I n ( )m0 c 2
2en 2n 1 
1

Mn 
2n 
 3
2 n 1 
2n

n  1
n 1
(2.27)
Similarly, observing the ECE at the optically thin O-mode and X-mode waves, we obtain the
electron temperature as
kTe  mec 2 I n(0) ( ) / I n( x ) 
(2.28)
Let us consider the case that non-thermal electrons (relativistic electrons or runaway
electrons) and thermal electrons (bulk electron) coexist in a plasma. When we detect the ECE
from outside of torus (low-field side), the radiation temperature is given by
Tr  T1(1  e 1 )e   Te (1  e  )
(2.29)
- 14 -
where  1 is the optical thickness for non-thermal electrons. In case we detect it from inside
of torus (high-field side),
Tr  T1(1  e1 )  Te (1  e )e1
(2.30)
is given.
2.4.2 ECE radiometry
There are several types of diagnostic systems for ECE measurements, such as , i)
Heterodyne radiometer, ii) Fourier-transform spectrometer, iii) Grating polychromator, iv)
Fabry-Perot interferometer, and v) Multuchannel mesh filter
Two representative methods will be described
i) Heterodyne radiometer
Conventional heterodyne technique is often used for the second harmonic of the ECE.
This technique has very good frequency resolution. In the initial stage this had not been used
to monitor the entire 2ce spectrum, however, wideband mixers having almost full band
responsibility hav been developed, and most of the spectrum can be covered by a few mixers.
Figure 2-6 shows an example of frequency down conversion section of an ECE heterodyne
radiometer. As an IF section we have prepared 8 channel filter bankand amplifiers fabricated
by microwave integrated circuit technology. This radiometer has 96 channels in 110-196 GHz
range. .
Fig. 2-6. Down conversion section of ECE heterodayne radiometer.
- 15 -
iv) Fourier-transform spectroscopy
When electric filed of incident wave on interferometer is given by E (t )  i Ei cos it ,
the electric field En entering a detector is written by
E
En (t )   i cos it  cosit  i 
2
i
(2.31)
Therefore, if we take mean square of En (t ) in time period T when  i can be regarded
constant, we obtain the following
Ei2
1 T 2
1 2
E
(
t
)
dt

E

 4 i  4 cos i ,
n
T 0
i
i
(2.32)
where i  i , and   X / c . The second term of right-hand side of Eq. (2.32) is
proportional to auto-correlation function RE ( ) , i.e.
RE ( ) 
1 T
1
E (t )E (t   )dt   Ei2 cos i

0
T
2 i
(2.33)
According to Wiener-Khinchine theorem, spectral density I ( ) is eventually obtained by
Fourier transform of RE ( ) , such as.

I ( )  4 RE ( ) cos  d 
(2.34)
0
In this method, the frequency resolution  f is determined from maximum  m  X m / c , as
f  1/  m .
- 16 -
Table 1. Microwave Sources
Type of Oscillator
Gunn
Frequency
280GHz
150-100GHz
90-60GHz
300GHz
IMPATT
70GHz
HTO & Multiplier
Application to Plasma
Output Power Turning Bandwidth
10mW
50-100mW
1GHz(E..T.)
100-200mW
30GHz(M.T.)
2mW
500MHz(E..T.)
5GHz(M.T.)
1W
150-60GHz
5-10mW
375GHz
1W
70-50GHz
400-40GHz
10-100mW
110W, 0.6ms
150-60GHz
200-400kW
Narrowband reflectormetry,
ECE
Multichannel interferometry
Narrowband reflectometry
10-30GHz
Fast-sweep reflectometry
Collective scattering from waves
BWO
Gyrotron
50-75GHz
75-400GHz
Broadband reflectometry
Collective scattering from waves
Collective scattering from ion
thermal fluctuatuions
E..T.: Electronic tuning
M.T.:Mechanical turning
Table 2. Comparison of Microwave Detectors
Type
Wavelength [mm]Response time Video NEP [W/Hz1/2]
Remarks
InSb
0.1-3
1-5ms
～10-13
Pyroelectric
＜1
1 ms
10 ms
～10
-9
～10
Room temperature
Ge:Ga
0.01-0.1
1ms
～10-13
Liquid helium cooled
Schottky
Barrier
Dioder
＞0.07
＜1ns
～10-10
Room temperature
Response time depends on
IF
Liquid helium cooled
-7
- 17 -