「プラズマ科学のフロンティア」 2009 年 9月 2-4 日 核融合科学研究所

マイクロ波の基礎と応用マイクロ波の基礎と応用ー計測を中心としてーー計測を中心としてー

九州大学産学連携センター　間瀬　淳

　　　　　　　　　　　内　容

1. 電磁波伝搬の基礎方程式

2. マイクロ波計測の原理と手法

3. 磁場閉じ込めプラズマにおける

　 マイクロ波計測の進展

4. マイクロ波計測の産業応用

1/31

　　　　　　　　　　

1.1 Electromagnetic Waves in Plasma Electromagnetic waves in plasma are described by Maxwell’s equations, including current 　density J and space charge density r,

(1.1)

and Ohm’s law (1.2)

From Eq. (1.1) we obtain the wave equation as (1.3)

When we consider an electric field as (1.4)

The Fourier component of Eq. (1.3) is written by (1.5)

or using the refractive index, as (1.6)

where c is the speed of light, and (1.7)

is the complex dielectric tensor

HB

B

E

EJH

BE

0

0

0

0

t

t

EJ

0000

t

EJ

tE

)(exp0 rkEE ti

0)( 22 EEkk c

0 EεENN

0i 1

1. 1. 電磁波伝搬の基礎方程式電磁波伝搬の基礎方程式

The property of the plasma is described by the permittivity through the conductivity [s]. 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

(1.8)

and Ohm’s law.We ignore thermal particle motions and utilize, so called, the “cold plasma approximation” .The dielectric tensor is obtained by

　　　 (1.9)

Then the three components of Eq. (1.6) are

(1.10)

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

v

vv

en

edt

dm

e

e

J

BE 0

zzzyzx

yzyyyx

xzxyxx

22

222222

222222

100

0)(1)(

0)()(1

pe

cepececepe

cecepecepe

i

i

0)sin(cossin

0cossin)cos(

0)(

222

222

2

zzzy

zyyyxyx

yxyxxx

ENEN

ENENE

EEN

we obtain the followings:

(1.11)

We consider two cases of propagation direction: parallel and perpendicular to the external magnetic field.

i) Parallel propagation :When waves propagate parallel to the external magnetic field, tan2=0, the solutions

　 (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, (1.13)

The “+” sign corresponds to the left-hand circular polarized wave, and the “ －” sign corresponds to the right-hand circular polarized wave. Substituting Eq. (1.9) into Eq. (1.12), we obtain

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 (1.1

4)

the subscript l and r of N denote the left-hand and right-hand circular-polarized waves.

)]()[(

)]()][([tan

2222

222

xyxxxxzz

xyxxxyxxzz

NN

iNiN

2xyxx iN

21

2

2

, 1

ce

perlN

xy iEE

ii) Perpendicular propagation :When waves propagate perpendicular to the magnetic field, tan2 =∞, the denomination of Eq. 　(1.10) has to be zero, which gives two solutions as or (1.15)

From Eq. (1.15), we obtain the following polarizations (1.16)

Thus the dispersion equations of the ordinary (O-mode) and the extraordinary (X-mode) waves are given by

(1.17)

(1.18)

zzN 2xxxyxxN )( 222

0 0 ,

0 0

and

and

zyx

zyx

EEE

EEE

2/1

2

21

pe

oN

2/1

222

22

2

2

1

cepe

pepexN

5/31

)( 22 cmTN ee

2

22

, )(1

)(1

cm

kTN

e

e

ce

pe

ce

perl

11222

2

2

2

cm

kTN

e

e

ce

pepeo

2222222

42222222

)()4(

8)47()(1

cm

Tk

e

e

cepece

cepecepepe

22

222

)/()/(1

)/(])/(1[

cepe

cepexN

When we include the effect of thermal electron motion, the first order of the expansion parameter 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

i) Parallel propagationThe dispersion relation of the left-hand and the right-hand circular-polarized waves are given by

(1.19)

ii) Perpendicular propagationThe dispersion relation of the ordinary wave (E//B0)

(1.20)

The dispersion relation of the extraordinary wave (E⊥B0)

(1.21)

Relativistic Effect

1.2 Electromagnetic Wave Scattering from Plasma

1.2.1 Theory of Scattering

When the electric field of the incident wave is given by

(1.22)

the equation of motion indicates that an electron oscillates with an acceleration given by

(1.23)

The vector potential due to the electron motion at the position of Q is

(1.24)

where t’ is retarded time, q=R/R, and |r|=R0.

xq

rA

0

20

1'

'4),(

Rc

tt

ttRc

et

v

xkE iie

tim

e

dt

d exp0v

xkEE iii ti exp0

t

tts

),(),(

rArE

'2

04 ts dt

d

Rc

e

v

qqE

N

iie ttn

1

)'()',( xxx

)'(exp)',( )(),( 000 xkxxEqqErE iies titndR

rt

),()(exp2)2(

),(3

kxk

kx eie nti

ddtn

The scattered wave at the receiving point is

Substituting Eq. (1.24) into (1.25), we obtain

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:

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

where r0=(e2/4pe0mec2) is the classical electron radius.

The electron density in Fourier component is shown by

(1.25)

(1.26)

(1.27)

(1.28)

(1.29)

We obtain

),()()(exp

2)2()(),(

0

3000

kxkkxq

kEqqErE

ei

i

Vs

ncc

Rti

dddx

R

rt

The scattered wave at the center frequency s and bandwidth s

dn

c

Rti

R

rt isisess

ss

ss

),( exp )(),( 02/

2/00

0

k-kEqqErE

e

e

VT N

n

TVS

2

,

),(2lim),(

kk

2220

200 cossin1)( sE EqqE

ks=qs/c.

where

is the power spectral density of the density fluctuations, , is the angle between E0 and ks-ki plane.

It is noted that the scattered power is observed when following matching conditions are satisfied.

The scattered power averaged over the observation time T is given by

isis , kkk

2, cossin1

2

c ),(

1lim

2222

02

20020 s

isissV

sT

s SER

Nrdttr

T

cP

kkE

(1.30)

(1.31)

(1.32)

(1.33)

(1.34)

ddrddSI

dSdrddj

dddSIddSdIdI

jI

dr

Id

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.

The energy absorbed along the distance is given by

The radiation energy is given by

The energy difference between entering and leaving the small volume corresponds to the difference between Eqs. (1.35) and (1.36), that is,

that is,

When the refractive index of the plasma is inhomogeneous and anisotropic, the equation of transfer is given by

jI

N

I

dr

dN

rr

2

2

(1.36)

(1.35)

(1.37)

(1.38)

(1.39)

10/31

BIj

1)/exp(

1

8 23

32

erB Thc

hNI

In microwave region, , Eq. (1.41) becomes eT

erB Tc

NI

23

22

8

0exp1 BOII

L

dr00

0 is called as “optical thickness”.

When , I equals to the intensity of black body radiation10 ≫

If the plasma is in thermal equilibrium, Kirchhoff’s law is worked out;

BI is the black-body radiation written by

By use of (1.42) the solution of the transfer equation is written by

(1.40)

(1.41)

(1.42)

(1.43)

(1.44)

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

]srm[W 1009.1 -132/1251 deieei GTZnndP

,

Where , the Gaunt factor averaged over velocities, takes dG

577.0

4ln

3),(

e

edT

TG

When . The absorption coefficient becomes 1eT

][m 100.7 -122/3211 GTZnn eieei

]m[W 106.1 32/1240 eieei TZnnP

The total radiation power is obtained from integration in as

]m[W 109.3 32/362 FdTnndP eaeea

Meanwhile for low-temperature weakly-ionized plasma, the radiation power occurs due to the collision between electron and neutral particles, and is given by

(1.45)

(1.46)

(1.47)

(1.48)

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

YXJXJe

nnnc

1

2222

//2

22)()(

sin

cos

08

2120

//0

0

)1(

cos1

sin/

ce

nY

X

From Eq. (1.51), it is shown that has discrete line spectra with its peaks at Y=0, that is,

,3 ,2 ,1cos1 //

0 nn

The total emission power is obtained by the integration of Eq. (1.50) over the distribution function.

The value of at the angle from the external magnetic field is obtained by

vBvP

2

00

1

me

td

dP (1.49)

(1.50)

(1.51)

(1.52)

The spectrum of electron cyclotron emission exhibits the broadening due to the physical processes in plasmas. There are several possible mechanisms for the broadening. i) Doppler broadening: cos)/(2 2122/1 cmkTn eecen

)/(2 22/1 cmkTn eecen ii) Relativistic broadening:

It is seen that the relative importance of relativistic effect and Doppler effect is determined by the angle . We now consider two cases

)(sin)(2

)(21

2222

2222,

cepece

pece

ce

pexoN

2

2

2

2242 cos4sin

ce

pece

cN tevcos1) For the case of

2 ii) n

0

),(2)1(2

)1(22

1

)1(22),( )()cos1( sin

)!1(2

Bxo

nn

nte

ce

pen

nxo L

cn

nn

v ((O, X-mode)

1 i) n

where

(1.53)

(1.54)

032

42222

2)(1

)cos1(

sin)cos21(

Bte

ce

peo

o L

cN

v

0

2

222

2

22)(

1 cos1

Bte

pe

ce

ce

pex

x L

cN

v

(O-mode)

(X-mode)

(1.55)

(1.56)

(1.57)

(1.58, 59)

1 i) n

0

222/1

2

22

1 1

Bte

ce

pe

ce

peo L

c

v

0

42

1

2/3

2

22

12

125

Bte

pe

ce

ce

pex L

czB

v

2 ii) n

0

222/1

22

2

1

122)( 1

)!1(2

Bn

te

ce

pen

ce

pen

no

nL

cnn

n

v

0

1222/1

22

2

1

)1(22)( 1

!)1(2

Bn

te

ce

pen

ce

pen

nx

nL

cnA

n

n

v

)90( /cos ＜＜ cte cN v2) For the case of

(O-mode)

(X-mode)

(O-mode)

(X-mode)

(1.60)

(1.61)

(1.62)

(1.63)

15/31

内　容内　容

　　　　　　　　　　　1. 電磁波伝搬の基礎方程式

2. マイクロ波計測の原理と手法

3. 磁場閉じ込めプラズマにおける　 マイクロ波計測の進展

4. マイクロ波計測の産業応用

2.1 Interferometry

2.1.1 Principle

Measurements of refractive index are often made by O-mode interferometry given by

where is the “cutoff” density.The interferometry measures the phase difference between the waves propagating in the plasma and in the outside of the plasma, which is given by

Assuming (x) is shown as the following formula.

When radial profile of the density is axisymmetric, we can obtain the density profiles by Abel inversion

emn ec /20

ce nn ≪

xrrdrxrrnn

dyrnn

xa

x ey

y ce

c ,))((

2)()( 2/1222

1

rxdxrxdx

dnrn

a

rc

e

,)()( 2/1222

2/12/1

2

2)(

1)(

1

c

epeo n

rnrN

(2.1)

2

1

2

1)1(

2)()( 0

y

y oy

y p dyNdykkx (2.2)

(2.3)

(2.4)

Now we assume the plasma has parabolic distribution given by the following formula, as it is known empirically, the phase difference is given by

2

10a

rnrn ee

c

en

na 0

3

20

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

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 . Therefore we obtain

Taking Gaussian beam theory into consideration, the beam expands along the distance y.

cece nnnn /0/0sin 1

2/1

20

2

2

220

4

d

ydd

Let us take the distance to the first collecting optics as L, and assuming we obtain . Then, the conditions to allow measurement are

2/10 /2 Ld

02dd

2/1/2/0 LdnnLL cem

F210/

3/12108 0102.10101.4

ee Lnna

7.0/ ax

(2.6)(2.5)

(2.7)

(2.8)

(2.9)

(2.10)

2.1.3 Phase Detection

An example of interferometer system and phase detector

Heterodyne interferometer using upconverter. 　 Quadrature-type phase detector.

In an ultrashort-pulse reflectometer, a very short pulse is used as a probe beam. The time-of-flight for a wave with frequency from the vacuum window position rw to the reflection point at rp is given by

In order to obtain the density profile from the time-of-flight data, the Eq. (2.43) can be Abel inverted to obtain the position of the cutoff layer,

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 given by

)(

2),(2)(

cr

a

drrNk

21

222

22

2

221

2

2

1 ,1

cepex

pex

x

pe

x

xx

o

pe

o

oo

ckN

ckN

within the WKB approximation.

The refractive indexes of the O-mode and the X-mode propagations are given by

drc

p

w

r

r

pe

21

2

2

12

)(

dc

rpe

pepe

02122 )(

)()(

By separating different frequency components of the reflected wave and obtaining time-of-flight measurement for each component, the density profile can be determined.

(2.11)

(2.12)

(2.13)

(2.14)

20/31

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

)sin(cos

)sinsincos(cos)cos(

00

000

rl

rlrl

EE

EEEEV

)/(2 eenoo nnLk

Bpexcen

pexceeexx

LL

BBnnk

)(1

)(22

2

In a simple homodyne reflectometer, the mixer output is given by

The time varying component of the mixer output depends on both amplitude and phase modulations. In general, the radial fluctuations of the cutoff layer produce the phase modulations and the poloidal (azimuthal) fluctuations cause amplitude modulations. 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, at the critical density layer are given by

(2.15)

(2.16)

(2.17)

● Measure the group delay, or the return phase, as a function of frequency

● Deduce the distance to the cutoff as a function of cutoff density　 － A simple inversion procedure can be used for O-mode radiation

Reflectometer utilizes r

eflected wave from the

cutoff layer of plasmas.

We can obtain reflected waves from each cutoff-layer corresponding to each radial position by injecting an incident wave with wide frequency region.

fp 1

2e2n

e(r)

me

0

1 2

ddt

()ddt

, ()2c 1

pe2

2

r

ant

rc

1 2

drLs L

rc

p()

w()

rcr

ant c

p()

pe2 2

1 2

0

pe

d

Detector

Source

Cutoff layer

Plasma

(t)

rc

rant

Ls

Lr

ReflectometryReflectometry －－ PrinciplePrinciple

Various Types of ReflectometryVarious Types of Reflectometry

Source

(t)

f

f+

Sourcef, f± fmmod.

f

Phase Detection

fm fm+

Sourcef±f/2Pin

Switchf

Time Delay

Trigger

Pulse Generator

BP Filter

Digitizer and/or TAC

Fast-Sweep FM RefletometerFast-Sweep FM Refletometer ○high resolution with simple hardware ●phase runaway

AM ReflectometerAM Reflectometer ○minimal effect of density fluctuations ●parastic reflections from wall and window

Short Pulse ReflectometerShort Pulse Reflectometer ○measurement of real-frozen plasma ●many sorces or sweep source with wideband switches

Ultrashort Pulse ReflectometerUltrashort Pulse Reflectometer ○an impulse generator ●ultrashort pulse (<10 ps) for high density plasmas

2.3 Thomson Scattering

2.3.1. Collective scattering

By using microwave as an incident wave, scattering parameter is usually larger than unity, so called collective scattering. Most laboratory plasmas have density fluctuations caused by various types of instability. 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 qs is written by

),(),( kk SVnpP Tseisss where pi is the power density of the incident wave, Vs is the scattering volumn, T is the cross section

of Thomson scattering, and S(k,w) is the power spectral density of the density fluctuation given by 2~

),(2

1)(

e

kse n

nVndSkS

k

where 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 fro

m the integration of k as.

kn~

ddSntrn ee kk

),(

2

1),(~

42

For the thermal fluctuations, S(k)~1, however, S(k)>>1 for the non-thermal fluctuations. Assuming 32 1010/~ ee nn , ne=1019 m-3, and Vs=10-5 m3, S(k)=108-1010.

(2.18)

(2.19)

(2.19)

Wavenumber spectrum

Apparatus Frequency spectra forvarious scattering angles

Dispersioncurve ofion-wave turbulence

Microwave ScatteringMicrowave Scattering25/31

Wavenumber and frequency spectra

Apparatus

Measurable wavenumber is 3 ＜ k ＜ 50 cm-1.Resolution is Δk ＜ 3 cm-1.

Far-Infrared Laser ScatteringFar-Infrared Laser Scattering

Dispersion relations for various values of end-plate bias

Frequency spectra for various values ofend-plate bias.

Fluctuation level vs. ambipolar field

Detector array

Far-Forward ScatteringFar-Forward Scattering

2.4 Electromagnetic Wave Radiation from Plasma

2.4.1 Determination of electron temperature

In experiments, 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

where re is the wall reflectivity (1> re > 0.9). When 　　　　 , In becomes nearly equal to

the black body radiation, then we call as “plasma is optically thick”. On the other hand, when , In in optically thin case becomes

ner ≪1

en r1≪

Let us consider a tokamak plasma, where B0 is the magnetic field intensity at the plasma center, R is the major radius. It is know that the toroidal magnetic field is a function of x as,

Therefore ce also varies accordingly. ECE appears resonantly in width

n

n

er

eII

eBn

1

1)( 0 (2.21)

ne

Bn r

II

1)( 0

(2.22)

xR

RBBT

0 (2.23)

1/ dxndx cenn (2.24)

cen with centering on x=x() which corresponds to

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 er≪

nee

ee

Bn Tn

rr

II )(

1

1

1)( 0

The radiation power is proportional to both ne and Te profiles. Therefore, when Te is obtained by different methods, we can determine ne profile, and vise visa.

Furthermore observing the ECE at the optically thin n and n+1 th harmonics, we can determine the electron temperature using the following formulas,

20)(

)(

cmI

IIMTk

n

nne

1

12

123

2

12

1

2

n

n

n

n nn

n

enM

Similarly, observing the ECE at the optically thin O-mode and X-mode waves, we obtain the electron temperature as

)()0(2 /)( xnnee IIcmkT

(2.25)

(2.26)

(2.27)

(2.28)

30/31

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) Multichannel mesh filter

i) Heterodyne radiometerConventional heterodyne technique is often used for 2ce ECE. This technique has good frequency resolution. In the initial stage this could not be used to monitor the entire 2ce spectrum, however, wideband mixers having almost full band responsibility have been developed, and most of the spectrum can be covered by a few mixers.

96 channels IF systemwith MIC technology

110-196 GHzECE

Heterodyne Radiometer

tEtE ii i cos)(

iiii

in tt

EtE coscos

2)(

ii

ii

i

Tn

EEdttE

T cos

44

1)(

1 22

02

cXii / and,

ii

iT

E EdttEtET

R cos2

1)()(

1)( 2

0

dRI E cos)(4)(0

In this method, the frequency resolution is determined from maximum as f cX mm /mf /1

iv) Fourier-transform spectroscopy

When electric filed of incident wave on interferometer is given by

The electric field entering a detector is written by

Therefore, if we take mean square of En

The second term of right-hand side of Eq. (2.28) is proportional to the auto-correlation function

According to Wiener-Khinchine theorem, In is eventually obtained by the Fourier transform of RE()

InterferometerDetector

Scanning Mirror

Monitor 　 Detector

Rdaition from Plasma

Grid　　　 Grid

Fixed 　 Mirror

(2.31)

(2.32)

(2.33)

(2.34)

31/31