マイクロ波の基礎と応用
九州大学産学連携センター 間瀬 淳
1.
PLASMA DIAGNOSTICS WITH ELECTROMAGNETIC WAVES
ABSTRACT
Principles and methods of diagnosing plasmas with electromagnetic waves in a range
from microwave to visible are reviewed. In particular, interferometric and reflectometric
methods for density and density fluctuation measurement, the method of determining electron
temperature distribution from electron cyclotron emission, and the scattering method for
measuring local electron and ion temperature, and density fluctuations are described.
1.1 Electromagnetic Waves in Plasma
Electromagnetic waves in plasma are described by Maxwell’s equations, including current
density J and space charge density 

  E   B t
  H  J   0 E t
  E   0
(1.1)
,
B  0
B  0 H
and Ohm’s law
J   E
,
(1.2)
where  0 and 0 are the permittivity and the permeability in the vacuum, and [is the
conductivity tensor.
z
k
B0

y
x
-1-
Fig. 1-1. Geometry of wavenumber and magnetic field
vectors
From Eq. (1.1) we obtain the wave equation as
 E 

E 
 0 J   0 0
0 .
t 
t 
(1.3)
When we consider an electric field as
E  E0 exp i(t  k  r ) .
(1.4)
The Fourier component of Eq. (1.3) is written by
k  k  E  ( 2 c 2 ) E  0 ,
(1.5)
or using the refractive index, N  ck  , as
N  N  E  εE  0 ,
(1.6)
where c is the speed of light, and
   1    i 0
(1.7)
is the complex dielectric tensor. The property of the plasma is described by the permittivity
 0   through the conductivity []. The conductivity tensor is obtained from the equation of
motion of a single electron including a static magnetic field B0 in z-axis (Fig. 1-1), current
density given by
dv
 eE  v  B0 
,
dt
J  neev
me
(1.8)
and Ohm’s law (1.2), where me and e are the mass and the charge of an electron respectively,
and ne is the electron density of plasma.
Since the electromagnetic waves travel at phase velocity close to the speed of light in
plasmas whose electron thermal velocity vt is much smaller than c. Thus, we ignore thermal
particle motions and utilize, so called, the “cold plasma approximation”.
The dielectric tensor is obtained by
 xx  xy  xz 


    yx  yy  yz 


 zx  zy  zz 
-2-
2
2
 1   2pe ( 2  ce

)
 i  2pece  ( 2  ce
)
0


2
2

 i  2pece  ( 2  ce
)
1   2pe ( 2  ce
)
0


0
0
1   2pe  2 


,
(1.9)
where
 2pe  nee2 me 0
ce  eB me
.
Then the three components of Eq. (1.6) are
( N 2   xx ) Ex   xy E y  0
  yx Ex  ( N 2 cos 2    yy ) E y  N 2 sin  cosEz  0 ,
(1.10)
 N 2 sin  cosE y  ( N 2 sin 2    zz ) Ez  0
where is the angle between k (wave vector of the incident wave) and z-axis. In order to
have non-zero solutions of Ex, Ey, Ez in Eq. (1.10), the determinant of the matrix of
coefficients must be zero, which gives the dispersion relation of the 4th order of the refractive
index N. Then we obtain the followings:
tan   
 zz [ N 2  ( xx  i xy )][ N 2  ( xx  i xy )]
.
2
2
( N 2   zz )[ xx N 2  ( xx
  xy
)]
(1.11)
Now we consider the two cases of the propagation direction: parallel and perpendicular to
the external magnetic field.
i) Parallel propagation, k // B (  0) :
When waves propagate parallel to the external magnetic field, tan2 =0, the solutions are
N 2   xx  i xy .
(1.12)
From Eq. (1.10) it is shown that the sign “  ” corresponds to the following relationship
between x and y components of the electric field,
E y  iE x
(1.13)
As shown in Fig. 1-2, the “+” sign corresponds to the left-hand circular polarized wave, and
the “－” sign corresponds to the right-hand circular polarized wave.
-3-
Fig. 1-2 Two-types of circular polarized wave for =0.
Substituting Eq. (1.9) into Eq. (1.12), we obtain
12
  2pe
 
Nl ,r  1  2 

   ce 

,
(1.14)
where the subscript l and r of N denote the left-hand and right-hand circular-polarized waves.
ii) Perpendicular propagation, k  B (  90) :
When waves propagate perpendicular to the magnetic field, tan=∞, the denomination of
Eq. (1.10) has to be zero, which gives two solutions as
N 2   zz
2
2
N 2  ( xx
  xy
)  xx
or
.
(1.15)
From Eq. (1.15), we obtain the following polarizations
Ex  E y  0 and Ez  0
Ex , E y  0 and Ez  0
(1.16)
.
Thus the dispersion equations of the ordinary (O-mode) and the extraordinary (X-mode)
waves are given by
1/ 2
  2 pe 
N O  1  2 

 

(1.17)
1/ 2
  2pe
 2   2pe 
N X  1  2  2
2 

    2pe  ce


(1.18)
-4-
When we include the effect of thermal electron motion, the first order of the expansion parameter
  N 2 (Te mec 2 ) is considered in the calculation. This assumption is effective when electron
temperature is less than 20 keV since is less than 0.05. Then, the dispersion relations become
followings depending on the propagation direction.
i) Parallel propagation, k // B :
The dispersion relation of the left-hand and the right-hand circular-polarized waves are
given by
Nl , r


 2pe


 1 


(



)


ce



 2pe
kT 


 e2 
1 

 (  ce ) mec 

(1.19)
ii) Perpendicular propagation, k  B :
The dispersion relation of the ordinary wave (E//B0)
  2pe 
N O  1  2 

 

2


1   pe  kTe 
  2   2 m c2 
ce
e 

(1.20)
The dispersion relation of the extraordinary wave (E⊥B0)
NX
2 2
2

[1  ( pe /  ) ]  (ce /  ) 


2
2 

 1  ( pe /  )  (ce /  ) 



2
2
2 2
2
2
2
4

  pe (   pe )  ce (7  4 pe )  8ce k Te 


1 
2
2
2
2
2 2
2
(  4ce ) (   pe  ce )
me c 



-5-
(1.21)
1.2 Electromagnetic Wave Scattering from Plasma
1.2.1 Theory of Scattering
The model of scattering by a single electron is shown in Fig. 1-3. When the electric field
of the incident wave is given by
Ei  E0 exp iit  ki  x  ,
(1.22)
the equation of motion indicates that an electron oscillates with an acceleration given by
dv
e

E0 exp i it  ki  x  .
dt
me
(1.23)
The vector potential due to the electron motion at the position of Q is
A(r , t ) 
e
v
 
40c 2  R  t  t '
,
(1.24)
1
t 't  R0  q  x 
c
where t’ is retarded time, q=R/R, and |r|=R0.
Fig. 1-3. Scattering by a single electron.
-6-
The scattered wave at the receiving point is
E s (r , t )  
A(r , t )
.
t
(1.25)
Substituting Eq. (1.24) into (1.25), we obtain
Es 

 dv 
q  q   dt  .
  t '
40c R 
e
(1.26)
2
The scattered wave shown in Eq. (1.26) is the one for a single electron. For the plasma with
many electrons, we must add each value statistically as follows:
N
ne ( x , t ' )    x  xi (t ' ) ,
(1.27)
i 1
where is the Dirac delta function. The total scattered electric field from all the electrons
with electron density ne in a volume V is then
r
E s (r , t )   0 E0  q(q  E0 ) dx ne ( x, t ' ) exp i (it 'ki  x ) ,
R
(1.28)
where r0=(e2/40mec2) is the classical electron radius.
The electron density in Fourier component is shown by

ne ( x , t ) 


dk

  ( 2 )  
3
d
exp i (it  k  x )ne (k ,  ) .
2
(1.29)
From Eqs. (1.28) and (1.29) we obtain


r
dk
d
E s (r , t )   0 E0  q(q  E0 ) dx 
3 
R
(2 )   2
V



 R     i 
 exp i (  i ) t  0   
q  x  (k  k )  x  ne (k ,  )
c   c 



, (1.30)
where k and  are the wave number and frequency of the density fluctuations. When we
observe the scattered wave using a receiver with center frequency s and bandwidth s , we
obtain
-7-
s  s / 2
  R 
exp is  t  0  ne (ks - ki , s  i )d ,
c 

s  s / 2 
r
Es (r , t )   0 E0  q(q  E0 )
R

(1.31)
where ks=qs/c.
Therefore, the scattered power averaged over the observation time T is given by
Ps 

c 0
1
lim
2 T  T
c 0 Nr02
2R2
 E s (r , t )
2
dt
V

E02 1  sin 2  s cos 2 


S k s  ki , s  i  s
2
，
(1.32)
where
2 ne (k ,  )
S (k ,  )  lim
Ne
T ,V  TV
2
(1.33)
is the power spectral density of the density fluctuations, Ne is the mean electron density,
2


N=NeV is the total density in the volume V, E0  q(q  E0 )  E02 1  sin 2  s cos 2  ,  is the
angle between E0 and ks-ki plane. It is noted that the scattered power is observed when
following matching conditions are satisfied.
k  k s  ki
(1.34)
  s  i
-8-
1.3 Electromagnetic Wave Radiation from Plasma
1.3.1 Radiation process in plasma
The radiation process is described by the equation of transfer which includes the emission
and absorption in plasma. Let us consider the small volume dS  dr in the plasma as shown
in Fig. 1-4, where I is the intensity of radiation along a ray whose unit is watts per square
meter, per steradian, and per radian frequency. Then, the energy absorbed along the distance
dr is given by
 I dS drdd ,
(1.35)
where  is the absorption rate of radiation per unit path length. This small volume of
plasma also radiates the energy. Putting the radiation coefficient j,whose unit is watts per
unit volume, per steradian, and per radian frequency, the radiation energy is given by
j d Sdrdd
.
(1.36)
The energy difference between entering and leaving the small volume corresponds to the
difference between Eqs. (1.35) and (1.36)
I  d I d S d d  I dS d d
,
(1.37)
that is,
d I
 I  j .
dr
(1.38)
When the refractive index Nr of the plasma is inhomogeneous and anisotropic, the
equation of transfer is given by
Fig. 1-4. Radiation along a ray.
-9-
N r2
d  I 
  I  j .
dr  N r2 
(1.39)
If the plasma is in thermal equilibrium, Kirchhoff’s law is worked out;
j   I B ,
(1.40)
here I B is the black-body radiation written by
3
1
2 h
,
I B  Nr

3 2 exp( h / T )  1
8 c
e
(1.41)
where h is the Planck’s constant. In microwave region,   Te , Eq. (1.41) becomes
I B  N r2
2
Te .
8 3c 2
(1.42)
By use of (1.42) the solution of (1.39) is written by
I  I BO 1  exp   0  ,
(1.43)
L
 0    dr ,
(1.44)
0
here 0 is called as “optical thickness”. When  0 ≫1 , I equals to the intensity of black body
radiation.
1.3.2 Bremsstrahlung
In a plasma there exists electromagnetic radiation due to collisions of electrons with ions
and neutral particles since the electrons deaccelerated in the electric field. For example, the
radiation power due to the electron-ion collision is given by
dP ei  1.09 1051 ne ni Z 2Te1 / 2 Gd
- 10 -
[W  m3sr -1] ,
(1.45)
where G d , the Gaunt factor averaged over velocities, takes
G d ( , Te ) 

3   4Te 
ln 
  0.577 

    

when  Te  1. The absorption coefficient becomes
 ei  7.0 1011 ne ni Z 2 Te3 / 2  2G
[m-1]
(1.46)
from the Kirchhoff’s law, where ni is the ion density and Z is the atomic number, and Te is in
the uint of K. The total radiation power is obtained from integration in as
Pei  1.6  1040 ne ni Z 2Te1 / 2 [W  m 3 ] .
(1.47)
Meanwhile for low-temperature weakly-ionized plasma, the radiation power occurs due to the
collision between electron and neutral particles, and is given by
dP ea  3.9 1062 nenaTe3 / 2 Fd
[W  m3 ] ,
(1.48)
where n a represents the density of neutral particles, F is approximately value of 1.
1.3.3 Cyclotron emission
A plasma in an external magnetic field radiates as a result of acceleration of electrons in
their orbital motions around the magnetic field lines. This emission is called as electron
cyclotron emission. The cyclotron emission power is also calculated from the integration of
the coefficient of self emission over the distribution function. The equation of motion in the
magnetic field is
dP
 ev  B0 
dt
P
m0
1  2
,
(1.49)
v
- 11 -
where   v / c , m0 is the electron rest mass. The value of  at the angle  from the
external magnetic field is obtained by
 
2

e2 2    cos    //  2
2 2
 
 Y  .
J
(
X
)


J
(
X
)

n

n
8 2  0c n 1 sin  

(1.50)
Here
X   / 0   sin 
Y  n0   1   // cos   ,
(1.51)
0  ce (1   2 )1 2
 //  v// c and   v c are the components of parallel and perpendicular to the external
magnetic field,  2   //2   2 ,  Y  is the delta function, J n  x  is n th order Bessel
function, and J n' x   dJ n x  / dX .
From Eq. (1.52), it is shown that  has discrete line spectra with its peaks at Y=0, that is,

n0
1   // cos
n  1,
2, 3,     .
(1.52)
The total emission power is obtained by the integration of Eq. (1.50) over the distribution
function.
The spectrum of electron cyclotron emission exhibits the broadening due to the physical
processes in plasmas. There are several possible mechanisms for the broadening. Those are
i) Doppler broadening:
n  2 1/ 2 nce (kTe / mec 2 )1 2 cos 
(1.53)
ii) Relativistic broadening:
n  2 1 / 2 nce (kTe / mec 2 )
(1.54)
It is seen that the relative importance of relativistic effect and Doppler effect is determined
by the angle . The optical thickness is also calculated depending on the value of . We now
consider two cases
- 12 -
1) For the case of N cos  vte c
i) n  1
1(o)
  pe 

  2 N o 
 ce 
2
2
 v te  (1  2 cos 2  ) 2 sin 4  L


 B
2 3
 c 
0
(1  cos  )


  2pe 
1( x)   2 N x 1  2 
  
ce 

2
 ce 


  pe 


2 v 2
 te 


 c 


L
cos 2   B
0
(O-mode)
(1.55)
(X-mode)
(1.56)
ii) n  2

( o, x )
n
2
 2n2(n 1)   pe   vte 


 n 1
2 (n  1)!  ce   c 

2( n 1)

sin  2(n1) (1  cos2  )n(o, x) ( ) LB
0
(O, X-mode)
(1.57)
are obtained. Here N o2,x is the refractive index propagating in the direction  from the
magnetic field, and is given by
2
No2, x
2
2(ce
  2pe )
  pe 

 1  
2
2
2
2
 ce  2(ce   pe )  ce (sin    )
(1.58)
2
2
 ce
  2pe 
2
4

  sin   4
cos 2 
 2

ce


(1.59)
2) For the case of N cos ＜vte / c (c＜  90 )
ⅰ） n  1
1/ 2
  2pe 
1o    2 1  2 
  
ce 

2
  pe   vte 2 L B


    c  
ce
0


  2pe 

x
2
1  5 2 1  2 
 2 
ce 

3/ 2
(O-mode)
(1.60)
2
  ce   vte 4 LB

B z1  
  pe   c  0


- 13 -
(X-mode)
(1.61)
ⅱ） n  2
 n(o)

 2n 2n 1 
2n 1(n  1)! 
1
 2pe 
2 
n 2ce


 2pe 
 n
 n( x)  n 1
A1  2 2 
2 (n  1)!  n ce 
2 2( n 1)
where A and
n 1 / 2
n 1 / 2
2
  pe   vte 2n


    c 
 ce 
LB
0
  pe   vte 2n 1 LB

 


0
ce

  c 
(O-mode) (1.62)
2
(X-mode) (1.63)
B z1  depend on n and  pe ce , and close to 1 for not so high density.
- 14 -