Microwave Reflection from the Region of Electron Cyclotron Resonance Heating in the L-2M Stellarator German M. BATANOV, Valentin D. BORZOSEKOV, Nikolay K. KHARCHEV, Leoni V. KOLIK, Eugeny M. KONCHEKOV, Dmitriy V. MALAKHOV, Alexaner E. PETROV, Karen A. SARKSYAN, Alexaner S. SAKHAROV, Nina N. SKVORTSOVA an Vlaimir D. STEPAKHIN Prokhorov General Physics Institute, Russian Acaemy of Sciences, Moscow 119991, Russia Receive 9 December 213 / Accepte 15 May 214 In experiments on electron cyclotron resonance ECR heating of plasma at the secon harmonic of the electron gyrofrequency in the L-2M stellarator, the effect of partial reflection of high-power gyrotron raiation from the ECR heating region locate in the center of the plasma column have been reveale. The reflection coefficient is foun to be on the orer of 1 3. The coefficient of reflection of an extraorinary wave from the secon-harmonic ECR region is calculate in the one-imensional full-wave moel. The calculate an measure values of the reflection coefficient are foun to coincie in the orer of magnitue. c 214 The Japan Society of Plasma Science an Nuclear Fusion Research Keywors: stellarator, plasma heating, microwave, gyrotron, electron-cyclotron resonance, full-wave equation DOI: 1.1585/pfr.9.342128 1. Introuction In stuying the interaction of microwave raiation with a weakly inhomogeneous high-temperature plasma in toroial magnetic confinement systems, reflection of microwaves from the electron cyclotron resonance ECR region, where the heating raiation is absorbe, is usually ignore. When the plasma is heate at the secon harmonic of the electron gyrofrequency, the absorption region is much shorter than the characteristic inhomogeneity scale lengths of the electron ensity an magnetic fiel, whereas microwaves in many cases are almost completely absorbe over a istance comparable with the raiation wavelength. Strong nonuniformity of the absorption coefficient shoul result in a partial reflection of microwaves from the region of ECR heating [1, 2]; however, this effect has not been stuie experimentally as of yet. In our experiments on ECR heating of plasma in the L- 2M stellarator see, e.g., [3], we have reveale microwave reflection from the region of the resonance at the secon harmonic of the electron gyrofrequency ω 2ω ce, where ω is the angular frequency of heating raiation an ω ce is the electron gyrofrequency by analyzing backscattere gyrotron raiation [4, 5]. 2. Experiment: Measurement Technique an Results The scheme of the experiment is shown in Fig. 1. The linearly polarize gyrotron raiation with the frequency f = ω /2π = 75 GHz splits near the plasma column author s e-mail: sakh@fpl.gpi.ru This article is base on the presentation at the 23r International Toki Conference ITC23. 342128-1 Fig. 1 Scheme of the experiment on the etection of backscattere gyrotron raiation in the L-2M stellarator: 1 gyrotron, 2 gyrotron raiation, 3 quasi-optical coupler, 4 mica splitter, 5 absorber, 6 raiation collimator, 7 etector of the gyrotron raiation power, 8 etector of scattere raiation, 9 polarization gri, 1 reference beam, 11 backscattere raiation, 12 cross section of the vacuum chamber, an 13 cross section of the plasma column. bounary into elliptically polarize extraorinary X an orinary O waves. The X-wave is absorbe in the ECR region locate in the center of the plasma column an heats plasma electrons. The O-wave passes through the plasma practically without absorption. The input raiation power was 2-35 kw. About 15% of the input power was transforme into the O-wave. The average plasma ensity was 1.7-1.8 1 13 cm 3, an the central electron temperature T e reache.7-1. kev. Figure 2 a shows signals from the etectors of the c 214 The Japan Society of Plasma Science an Nuclear Fusion Research
Fig. 2 a Signal of the homoyne etector before an after averaging over the time interval of 3 μs 15 reaings an the time epenences of b the phase increment of the interferometer measuring the plasma ensity along the central horizontal chor an c the average plasma ensity. quasi-optical irectional coupler recore uring one of the L-2M ischarges. The backscattere signal was recore using the homoyne etection scheme. By turning the receiving rectangular waveguie by π/2, two mutually orthogonal components of the backscattere raiation coul be recore the Y component collinear with the fiel of the incient wave an the Z component orthogonal to the fiel of the incient wave. Thus, the backscattere X- an O-waves coul be recore separately. The shape of the etecte signal shown in Fig. 2 a is a result of the interference of three signals: the reference signal, the fast oscillating signal, an the signal with a slowly varying phase. The fast oscillating signal can be attribute to gyrotron raiation backscattere from smallscale plasma fluctuations with wavenumbers k s 2k, where k is the wavenumber of the scattere wave. The signal with a slowly varying phase is quite natural to interpret as a wave reflecte from some obstacle or a quasi-steay perturbation. Hereinafter, we will call this signal quasisteay. If the quasi-optical coupler is calibrate, then it is easy to etermine the reflection backscattering coefficients of both the fast oscillating an quasi-steay components. After averaging over the fast oscillating component, the result of the interference of the reference signal an the signal with a slowly varying phase is retaine. The location of the plasma region that is the source of the quasi-steay signal can be etermine by analyzing variations in the phase of this signal. The change in the phase of the quasi-steay signal is apparently cause by a change in the average plasma ensity on the path of the incient an reflecte raiation. If gyrotron raiation is reflecte from the rear wall of the vacuum chamber, then its phase increment shoul correspon to the ouble length of the central chor of the 342128-2 cross section of the last magnetic surface. If reflection takes place insie the plasma column, then the phase increment shoul be smaller. Since the resonance at the secon harmonic of the electron gyrofrequency is locate on the magnetic axis, the phase increment of X-wave in the case of reflection from the ECR region shoul correspon to the length l of the central chor. Note that the X-wave is almost completely 99% absorbe in the ECR region. Therefore, it is har to expect that reflection of the X-wave from the rear wall of the chamber coul be etecte. Reflection from the ECR region is confirme by the calculation of the length over which the recore phase increment ϕ of the quasi-steay signal at the gyrotron raiation frequency f takes place. To calculate this length, it is necessary to have information on the change in the average plasma ensity over a given time interval. This information can be obtaine by analyzing variations in the phase of the signal from the interferometer operating on the O-wave with the frequency f 1 = ϕ 1 /2π = 141 GHz an measuring the average plasma ensity along a horizontal chor passing through the center of the plasma column. Figure 2 b shows the time evolution of the interferometer phase ϕ 1 for the same ischarge in which the backscattere signal was recore, an Fig. 2 c shows the time evolution of the average plasma ensity recovere from the interferometer signal. The change in the phase of the quasi-steay signal over a certain time interval Δt can be written as Δϕ = 2k x ΔN X x 2k x N X Δvx, 1 v where k = ω /c is the wavenumber of gyrotron raiation in vacuum, N X = 1 v 2 u/1 u v 1/2 is the refractive inex of the X-wave propagating perpenicular to the magnetic fiel, u = ω ce /ω 2, v = ω pe /ω 2, an ω pe is the plasma electron frequency. The factor 2 in formula 1 accounts for the ouble passage of raiation from the plasma bounary x = to the point of reflection x = x an back. The change in the interferometer phase Δϕ 1 is Δϕ 1 = k 1 l ΔN O x k 1 l N O v 1 Δv 1 x, 2 where N O = 1 v 1 1/2 is the refractive inex of the O- wave propagating perpenicular to the magnetic fiel, k 1 = ω 1 /c, v 1 = ω pe /ω 1 2, an integration is performe over the entire length of the central chor l = 23 cm. Taking into account that N X v = 1 u 2v + 2uv + v 2 2 1 v 2 u, 1/2 3/2 1 u v 3 N O 1 =, v 1/2 1 2 1 v 1 4
Table 1 Position of the reflection region an reflection coefficient of the X-wave as functions of time. 3. Discussion of the Results We performe numerical calculations of the coefficient of reflection from the ECR region for the conitions corresponing to the experiments on ECR plasma heating at the secon harmonic of the electron gyrofrequency in the L-2M stellarator [3]. Let us consier a plane X-wave propagating along the x axis an polarize in the x, y plane perpenicular to the external magnetic fiel B xe z. The propagation of such a wave near the resonance ω = 2ω ce is escribe by the following set of one-imensional fullwave equations [1, 6]: f E x x x + i x f E y x + SE x ide y =, 8 i f E x + 1 + f E y + ide x + SE y x x x x =. 9 for the conitions of our experiment u.25, v.25, v 1.7, we can approximately write N X Δϕ 2k x v Δ v 2.2k x Δ v, 5 N O Δϕ 1 k 1 l Δ v 1.52k 1 l Δ v 1 v 1.28k l Δ v, 6 where the upper bar stans for the averaging over the central chor. Thus, for the known values of Δϕ an Δϕ 1, the istance x from the plasma bounary to the point of reflection can be foun by the formula x l.125 Δϕ Δϕ 1. 7 The Table 1 presents the values of x /l calculate by formula 7 an the measure values of the power reflection coefficient R 2 X in the quasi-steay component of the backscattere X-wave for several time intervals uring the L-2M ischarge illustrate in Fig. 1. It can be seen from the table that the x positions etermine from the measure change in the phase of the quasisteay component of the backscattere X-wave lie close to the center of the plasma column. The average values of R 2 X an x /l with allowance for their root-mean-square eviations from their average values over the time interval 52-56 ms are 1.8 ±.8 1 3 an.52 ±.6, respectively. Here, S = k 2 1 u v 1 u, D = v u k2 1 u, f = v 2u F ω 2ω ce 7/2, βω where β = T e /m e c 2 an F 7/2 x is the Dnestrovskii function [7], which accounts for the thermal effects near the resonance at the secon harmonic of the electron gyrofrequency in weakly relativistic β 1 Maxwellian plasma, ReF 7/2 ξ 1 for ξ 1, ξ for ξ> ImF 7/2 ξ = 8 15 i π ξ 5/2 e ξ for ξ<. Set of equations 8 an 9 possesses the energy integral Here, 1 P + W =. 11 x P = c2 Im E E y y 8πω x + E x iey Ex + ie y f x 12 is the power flux ensity where the first an secon terms in parentheses correspon to the Poynting flux an the kinetic power flux, respectively an W = c2 8πω x Ex + ie y 2 Im f > 13 is the power ensity absorbe by the plasma. Since set of equations 8 an 9 consists of two secon-orer ifferential equations, it escribes two types of waves. In a uniform plasma, the spatial epenence of the electrical fiels of these waves can be represente as expikx, where the wavenumber k satisfies the following biquaratic ispersion relation: 342128-3 fk 4 S + 2 S + D f k 2 + S 2 D 2 =. 14 Far from the resonance ω 2ω ce /ω β, f, the solutions to ispersion relation 14 correspon to the leftan right-propagating X-waves, k X ± S 2 D 2 /S, 15 as well as Bernstein B moes at the secon harmonic of the electron gyrofrequency, k B ± S/ f, k B k X. 16
Fig. 3 Calculate profiles of E y heavy soli line an E x light soli line near the resonance ω = 2ω ce. The ashe line shows ReE y as a function of x at a certain instant of time. The fiels are normalize to the amplitue E of the incient wave in vacuum. The B-moes are propagating at ω 2ω ce > Imk B = an evanescent at ω 2ω ce < Imk B Rek B. In a nonuniform magnetic fiel, the X- an B-moes turn out to be couple. The coupling is the strongest in the ECR region, where the ispersive properties of the meium vary most rapily, an graually weakens with increasing istance from the resonance region. As a result, a fraction of the X-wave energy can transform in the resonance region into the energy of the B-moe propagating towar the lower magnetic fiel. Set of equations 8 an 9 was solve for the conitions close to the experiment on ECR plasma heating in the L-2M stellarator. It was assume that the X-wave with the frequency f = 75 GHz λ =.4 cm was incient from the right from the low-fiel sie an propagate along the x axis towar negative values of x towar the major axis of the vacuum chamber. The magnetic fiel was irecte along the z axis. The resonance point ω = 2ω ce was at x =. The graient of the magnetic fiel along the x axis was assume to be constant an equal to its value on the minor axis of the stellarator chamber, B /x /B L 1.116 cm 1. The plasma ensity an temperature were assume to be constant an equal to n e = 1.75 1 13 cm 3 v.25 an T e = 1 kev, respectively. The problem was solve numerically by the sweep metho [8] with a spatial step of Δx =.4 cm. The bounary conition at the right bounary correspone to the superposition of the incient X-wave with a given amplitue, the reflecte X-wave, an the outgoing right-propagating B-wave. The bounary conition at the left bounary correspone to the transmitte leftpropagating X-wave. The heavy soli line in Fig. 3 shows the amplitue of the transverse component of the electric fiel E y. The ashe line in Fig. 3 shows ReE y as a function of x at a certain instant of time. The light soli line shows the amplitue of the longituinal components of the electric fiel E x. It is seen that the X-wave experiences strong amping in the resonance region over a length comparable with its wavelength. The coefficient of 342128-4 power transmission through the resonance region is T 2 =.59 1 2, which is very close to its value calculate for the same parameters in the geometrical optics approximation [9] T 2 =.57 1 2. In the region x >, the profile of the amplitue of the transverse fiel E y exhibits spatial oscillations with the perio equal to one-half of the wavelength of the X-wave, which correspons to the presence of a wave reflecte from the resonance region. The power reflection coefficient of the X-wave in this case is R 2 X 1.56 1 3. Thus, the result of numerical calculation of the reflection coefficient of the X-wave from the set of full-wave equations 8 an 9 turns out to be close to the values obtaine in the aboveescribe experiment. To be assure that spurious reflection from the bounaries of the computation region was negligibly small, we performe a test calculation in which the magnetic fiel was reuce by 1%, so that there was no longer a resonance at the secon harmonic of the electron gyrofrequency in the computation region i.e., the entire computation region was in the zone of the weak magnetic fiel. In this case, the power reflection coefficient of the X-wave was at a level of 1 15. Small-scale oscillations observe in the profile of the amplitue of the longituinal fiel E x see Fig. 3 are a result of the interference between the longituinal electric fiels of the X-wave an the B-wave propagating towar the region of the weak magnetic fiel. In this case, the coefficient of X-B transformation in terms of the power is R 2 B =.12 1 3. 4. Conclusions In experiments on ECR heating of plasma at the secon harmonic of the electron gyrofrequency in the L-2M stellarator, the effect of partial reflection of high-power gyrotron raiation from the ECR heating region locate in the center of the plasma column has been reveale. The reflection coefficient is foun to be on the orer of 1 3, which agrees with the results of full-wave numerical calculations. To conclue, we note that microwave reflection from the ECR heating region can contribute to the lowfrequency moulation of the gyrotron power cause by the reflecte wave. Since the reflection coefficient epens strongly on the electron temperature, this effect can, in principle, be use to estimate the electron temperature in the ECR region. Acknowlegments This work was supporte by the Russian Founation for Basic Research, project nos. 11-8-1129a an 12-2- 31522. [1] A.V. Zvonkov, Sov. J. Plasma Phys. 9, 319 1983. [2] A.D. Piliya an V.I. Feorov, in Reviews of Plasma Physics, E. by B. B. Kaomtsev Consultants Bureau, New York,
1987 Vol. 13, p. 335. [3] N.K. Kharchev, G.M. Batanov, M.S. Berezhetskii et al., Plasma Fusion Res. 6, 242142 211. [4] G.M. Batanov, V.D. Borzosekov, L.M. Kovrizhnykh et al., Plasma Phys. Rep. 39, 444 213. [5] G.M. Batanov, V.D. Borzosekov, L.V. Kolik et al., Plasma Phys. Rep. 39, 882 213. [6] D.C. McDonal, R.A. Cairns an C.N. Lashmore-Davies, AIP Conf. Proc. 43, 351 1997. [7] D.G. Swanson, Plasma Waves IOP Publishing, Bristol, 23. [8] D.E. Potter, Computational Physics Wiley, New York, 1973. [9] V.V. Alikaev, A.G. Litvak, E.V. Suvorov an A.A. Fraiman, in High-Frequency Plasma Heating, E. by A. G. Litvak AIP, New York, 1991 p. 1. 342128-5