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「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 1/31 マイクロ波の基礎と応用 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. 電磁波伝搬の基礎方程式 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, 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) E From Eq. (1.1) we obtain the wave equation as E 0 J 0 0 0 (1.3) t t 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 c2 ) E 0 (1.5) or using the refractive index, 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 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 dv me e E v B0 (1.8) dt J neev and Ohm’s law. We ignore thermal particle motions and utilize, so called, the “cold plasma approximation”. The dielectric tensor is obtained by xx xy xz 1 2 ( 2 ce ) pe 2 i 2 ce ( 2 ce ) pe 2 0 2 yx yy yz i pece ( 2 ce ) 2 1 2 ( 2 ce ) pe 2 0 (1.9) zx zy zz 0 0 1 2 2 pe Then the three components of Eq. (1.6) are ( N 2 xx ) E x xy E y 0 (1.10) yx E x ( N 2 cos2 q yy ) E y N 2 sin q cosqE z 0 N 2 sin q cosqE y ( N 2 sin 2 q zz ) E z 0 where q 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 we obtain the followings: zz [ N 2 ( xx i xy )][N 2 ( xx i xy )] tan 2 q (1.11) ( N zz )[ xx N ( xx xy )] 2 2 2 2 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, tan2q =0, the solutions 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 iEx (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 12 2 Nl ,r 1 pe (1.14) 2 ce the subscript l and r of N denote the left-hand and right-hand circular-polarized waves. 5/31 ii) Perpendicular propagation : When waves propagate perpendicular to the magnetic field, tan2q =∞, the denomination of Eq. (1.10) has to be zero, which gives two solutions as N 2 zz or N 2 ( xx xy ) xx 2 2 (1.15) From Eq. (1.15), we obtain the following polarizations E x E y 0 and E z 0 (1.16) E x , E y 0 and E z 0 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 2 1 pe pe 2 2 Nx (1.18) 2 2 2 ce 2 pe Relativistic Effect When we include the effect of thermal electron motion, the first order of the expansion parameter N 2 (Te mec2 ) 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 propagation The dispersion relation of the left-hand and the right-hand circular-polarized waves are given by 2 2 kTe 1 1 (1.19) pe pe Nl , r 2 ( ce ) ( ce ) mec ii) Perpendicular propagation The dispersion relation of the ordinary wave (E//B0) 2 1 pe kTe 2 No 1 2 pe (1.20) 2 2 m c2 ce e The dispersion relation of the extraordinary wave (E⊥B0) [1 ( pe / ) 2 ]2 (ce / ) 2 Nx 1 ( pe / ) 2 (ce / ) 2 2 ( 2 2 ) 2 ce (7 2 4 2 ) 8ce k Te 2 4 (1.21) 1 pe pe pe ( 2 4ce ) ( 2 2 ce ) 2 2 pe 2 me c 2 1.2 Electromagnetic Wave Scattering from Plasma 1.2.1 Theory of Scattering 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 E0 exp i it ki x dv e (1.23) dt me The vector potential due to the electron motion at the position of Q is e v A(r , t ) 4 0c 2 R t t ' (1.24) t 't 1 R0 q x c where t’ is retarded time, q=R/R, and |r|=R0. The scattered wave at the receiving point is A(r , t ) E s (r , t ) (1.25) t Substituting Eq. (1.24) into (1.25), we obtain e dv Es q q (1.26) 4 0c 2 R dt t ' 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 Es (r , t ) 0 E0 q(q E0 ) dx ne ( x, t ' ) expi(it 'ki x ) r (1.28) R where r0=(e2/4pe0mec2) is the classical electron radius. The electron density in Fourier component is shown by d expi(it k x)ne (k , ) dk ne ( x, t ) 2 (1.29) (2 ) 3 We obtain d E s (r , t ) 0 E0 q(q E0 ) dx r dk R V (2 )3 2 R i expi ( i ) t 0 q x (k k ) x ne (k , ) ks=qs/c. (1.30) c c The scattered wave at the center frequency s and bandwidth Ds s Ds / 2 R Es (r , t ) 0 E0 q(q E0 ) r exp is t 0 ne (ks - ki , s i )d c (1.31) s Ds / 2 R The scattered power averaged over the observation time T is given by c 0 Nr0 2 c D s E0 1 sin 2 q s cos2 S k s ki , s i 1 2 DE s (r, t ) dt 2 Ps 0 lim 2 T T 2R 2 2 V (1.32) where 2 2 ne (k , ) S (k , ) lim (1.33) T ,V TV Ne 2 2 is the power spectral density of the density fluctuations, E0 q(q E0 ) E0 1 sin q s cos , 2 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 k i , s i (1.34) 10/31 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 I dS drdd (1.35) The radiation energy is given by j d Sdrdd (1.36) The energy difference between entering and leaving the small volume corresponds to the difference between Eqs. (1.35) and (1.36), that is, I d I d S d d I dS d d (1.37) that is, d I I j (1.38) dr When the refractive index of the plasma is inhomogeneous and anisotropic, the equation of transfer is given by 2 d I I j Nr (1.39) N2 dr r If the plasma is in thermal equilibrium, Kirchhoff’s law is worked out; j I B (1.40) I B is the black-body radiation written by 2 h 3 1 IB Nr (1.41) 8 3c 2 exp(h / Te ) 1 In microwave region, Te , Eq. (1.41) becomes 2 2 I B Nr T 3 2 e (1.42) 8 c By use of (1.42) the solution of the transfer equation is written by I I BO 1 exp 0 (1.43) L 0 dr (1.44) 0 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 1051ne ni Z 2Te1 / 2 Gd [W m3sr-1] (1.45) Where G d , the Gaunt factor averaged over velocities, takes , 3 4Te G d (, Te ) ln 0.577 When Te 1 . The absorption coefficient becomes ei 7.0 1011ne ni Z 2 Te3 / 2 2G [m-1] (1.46) The total radiation power is obtained from integration in as P 1.6 1040 neni Z 2Te / 2 [W m3 ] ei 1 (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 1062 nenaTe3 / 2 Fd [W m3 ] (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 dP ev B0 m0 P v (1.49) dt 1 2 The value of at the angle q from the external magnetic field is obtained by e2 2 cosq // 2 2 2 c J n ( X ) J n( X ) Y 2 2 (1.50) 8 0 n 1 sin q X / 0 sin q Y n0 1 // cosq (1.51) 0 ce (1 2 )1 2 From Eq. (1.51), it is shown that has discrete line spectra with its peaks at Y=0, that is, n0 n 1, 2, 3, (1.52) 1 // cosq 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. i) Doppler broadening: Dn 2 1/ 2 nce (kTe / mec 2 )1 2 cosq (1.53) ii) Relativistic broadening: Dn 2 nce (kTe / mec ) 1/ 2 2 (1.54) It is seen that the relative importance of relativistic effect and Doppler effect is determined by the angle q. We now consider two cases 1) For the case of N cosq vte c i) n 1 2 v 2 pe te (1 2 cos q ) sin q LB 2 2 4 1 o ) 2 N o ( (O-mode) (1.55) ce c (1 cos2 q )3 0 2 2 2 2 ce v te 1 N x 1 2 cos2 q LB ( x) 2 pe (X-mode) (1.56) pe c 0 ce ii) n 2 2( n 1) 2n 2( n 1) pe vte 2 sin q 2(n 1) (1 cos2 q )no, x) (q ) LB c ( o, x ) ( n 1 (O, X-mode) (1.57) n 2 (n 1)! ce 0 where 2 pe 2 2(ce 2 ) 2 ce 2 2 N o, x 1 pe pe 2 sin q 4 2 4 cos2 q (1.58, 59) ce 2(ce 2 ) ce (sin2 q ) 2 2 2 pe ce 15/31 2) For the case of N cosq ＜vte / c (qc＜q 90 ) i) n 1 1/ 2 2 2 o 2 1 pe pe vte 2 LB 1 2 c 0 (O-mode) (1.60) ce ce 3/ 2 2 2 ce vte 4 LB x 5 2 2 1 pe 1 B z1 pe c 0 (X-mode) (1.61) 2 2 ce ii) n 2 2 n 1 / 2 2n2n 1 2 n n 1 (o) 1 pe pe vte 2n LB n2 2 c (O-mode) (1.62) 2 (n 1)! ce ce 0 n 1 / 2 pe vte 2n 1 2 2( n 1) 2 2 n 1 pe LB n n 1 c ( x) A (X-mode) (1.63) 2 (n 1)! n 2ce 2 ce 0 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 1/ 2 2 (r ) ne (r ) 1/ 2 No 1 pe 1 nc (2.1) 2 where nc 0me / e is the “cutoff” density. 2 The interferometry measures the phase difference between the waves propagating in the plasma and in the outside of the plasma, which is given by y 2 y2 ( x) 2 (k0 k p )dy (1 No )dy (2.2) y1 y1 Assuming ne ≪ nc , (x) is shown as the following formula. y2 2 a 2 1 / 2 ( x) y1 ne (r )dy x ne (r )(r x ) rdr, r x (2.3) 2 nc nc When radial profile of the density is axisymmetric, we can obtain the density profiles by Abel inversion nc a d 2 ne (r ) ( x r 2 )1 / 2 dx , x r (2.4) 2 r dx Now we assume the plasma has parabolic distribution given by the following formula, as it is known empirically, the phase difference is given by r 2 2 a ne 0 ne r ne 01 (2.5) 0 (2.6) a 3 nc 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 sin 1 ne 0 / nc ne 0 / nc (2.7) Taking Gaussian beam theory into consideration, the beam expands along the distance y. 1/ 2 2 42 y 2 d d0 2 2 (2.8) d0 Let us take the distance to the first collecting optics as L, and assuming d0 2L / 1 / 2 we obtain d 2d0 . Then, the conditions to allow measurement are L m L ne 0 / nc d 2L / 1/ 2 (2.9) 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 D / 10 2 F . Therefore we obtain 4.1 108 a ne 0D 1.2 1010 Lne 0 2 1 / 3 (2.10) 2.1.3 Phase Detection An example of interferometer system and phase detector Heterodyne interferometer using upconverter. Quadrature-type phase detector. 20/31 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 rc ( ) ( ) 2k N (r , )dr 2 within the WKB approximation. (2.11) a The refractive indexes of the O-mode and the X-mode propagations are given by 12 12 ck o pe , N ck x 1 pe x pe 2 2 2 2 No 1 (2.12) o o x x x 2 ce 2 x 2 2 2 pe 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 rp 1 2 2 2 1 pe ( ) c 2 dr (2.13) rw 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, pe c ( ) r ( pe ) d ( 2 2 )1 2 (2.14) 0 pe By separating different frequency components of the reflected wave and obtaining time-of-flight measurement for each component, the density profile can be determined. 2.2.2 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 In a simple homodyne reflectometer, the mixer output is given by V El Er cos( 0 ) El Er (cos0 cos sin 0 sin ) (2.15) El Er (cos0 sin 0 ) 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 o 2ko Ln (ne / ne ) (2.16) x 2k x ne ne (cex 2 ) B B pe (2.17) 1 Ln (cex 2 ) LB pe Reflectometry－Principle Cutoff layer Reflectometer utilizes Ls reflected wave from the Source cutoff layer of plasmas. Lr 12 1 e2ne (r) fp m Detector Plasma 2 e 0 rc (t) rant ● Measure the group delay, or the return phase, as a function of frequency 1 2 d () d , () 2 r c 2 Ls Lr c p() w () dt dt c 1 pe 2 dr r ant ● Deduce the distance to the cutoff as a function of cutoff density －A simple inversion procedure can be used for O-mode radiation c pe p( ) d rc rant 1 2 0 2 2 pe We can obtain reflected waves from each cutoff-layer corresponding to each radial position by injecting an incident wave with wide frequency region. Various Types of Reflectometry Fast-Sweep FM Refletometer 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 plasmas f Pin f±Df/2 Source Pulse Switch Generator Trigger BP Filter Time Delay Digitizer and/or TAC 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 P (ks ,s ) pi neVsT S (k ,) s (2.18) 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 1 nk S (k ) 2 S (k , )d neVs ne (2.19) ~ where nk is the amplitude of density fluctuations with wave number k. The wave number spectrum can be obtained by changing the scattering angle qs. The density fluctuation level is then determined from the integration of k as. 4 ~ (r , t ) 2 1 n S (k , )dkd ne e 2 (2.19) For the thermal fluctuations, S(k)~1, however, S(k)>>1 for the non-thermal fluctuations. Assuming ne / ne 102 103 ne=1019 m-3, and Vs=10-5 m3, S(k)=108-1010. ~ , 25/31 Microwave Scattering Apparatus Frequency spectra for various scattering angles Wavenumber spectrum Dispersion curve of ion-wave turbulence Far-Infrared Laser Scattering Measurable wavenumber is 3＜k＜50 cm-1. Resolution is Δk＜3 cm-1. Wavenumber and frequency spectra Apparatus Far-Forward Scattering Frequency spectra for various values of end-plate bias. Detector array Fluctuation level vs. ambipolar field Dispersion relations for various values of end-plate bias 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 1 e n I n ( ) I B 0 n (2.21) 1 ree where re is the wall reflectivity (1> re > 0.9). 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 , In in optically thin case becomes I B0 I n ( ) n (2.22) 1 re 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, R BT B0 (2.23) Rx Therefore ce also varies accordingly. ECE appears resonantly in width Dxn Dn d nce / dx1 (2.24) with centering on x=x() which corresponds to nce 30/31 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 B0 1 I n ( ) n e (Te ) n (2.25) 1 re 1 re 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, M n I n I ( ) k Te (2.26) I n ( )m0 c 2 n 1 2en 2n 1 1 Mn 2n 3 (2.27) n 12n 1 2n Similarly, observing the ECE at the optically thin O-mode and X-mode waves, we obtain the electron temperature as kTe mec 2 I n0) ( ) / I nx) ( ( (2.28) 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 radiometer Conventional 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. 110-196 GHz ECE 96 channels IF system with MIC technology Heterodyne Radiometer 31/31 iv) Fourier-transform spectroscopy When electric filed of incident wave on interferometer is given by E(t ) i Ei cosit Interferometer Detector The electric field entering a detector is written by Scanning Mirror En (t ) i cosit cosit Di E (2.31) i 2 Grid Grid Rdaition from Therefore, if we take mean square of En Plasma 1 T 2 1 2 Ei2 En (t )dt Ei T 0 cos Di (2.32) Fixed Mirror i 4 i 4 Di i , and X / c Monitor Detector The second term of right-hand side of Eq. (2.28) is proportional to the auto-correlation function 1 T 1 E (t )E (t )dt Ei2 cos i T 0 RE ( ) (2.33) 2 i According to Wiener-Khinchine theorem, In is eventually obtained by the Fourier transform of RE() I ( ) 4 RE ( ) cos d (2.34) 0 In this method, the frequency resolution Df is determined from maximum m X m / c as Df 1 / m

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posted: | 4/18/2013 |

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