Maritime University of Szczecin
Akademia Morska w Szczecinie
2011, 26(98) pp. 5–9 2011, 26(98) s. 5–9
Numerical modeling of polarization effects in a plasma
at the W7-X stellarator
Numeryczne modelowanie efektów polaryzacyjnych w plazmie
stellaratora W7-X
Bohdan Bieg
1, Matthias Hirsch
2, Yury A. Kravtsov
1,31 Maritime University of Szczecin, Faculty of Maritime Engineering, Department of Physics Akademia Morska w Szczecinie, Wydział Mechaniczny, Katedra Fizyki
70-500 Szczecin, ul. Wały Chrobrego 1–2
2 Max Planck Institute for Plasma Physics, D-17491 Greifswald, Wendelsteinstrasse, Germany 3 Space Research Institute, Moscow 117997, Profsoyuznaya St. 82/34, Russia
Key words: Faraday, Cotton-Mouton, plasma density Abstract
The interferometer-polarimeter in the stellarator W7-X will operate in the far infrared band in the double passage regime: the electromagnetic beam will be reflected back from a corner cube retroreflector and thus pass through the magnetized plasma twice. Evolution of the polarization state of the beam along the polarimeter sightline in the standard magnetic field configuration of W7-X is analyzed on the basis of differential equations, for traditional polarization parameters – azimuthal angle and ellipticity angle . The influence of the Faraday and the Cotton-Mouton phenomena on polarization parameters and are analyzed. Calculations are performed for different plasma densities. Finally the potential sources of inaccuracy are considered, such as the shift in the probing beam position.
Słowa kluczowe: Farady, Cotton-Mouton, gęstość plazmy Abstrakt
W interfero-polarymetrycznym systemie diagnostycznym stellaratora W7-X podczerwona wiązka pomiarowa ulega podwójnemu przejściu przez obszar namagnesowanej plazmy. Ewolucja stanu polaryzacji fali elektro-magnetycznej, podczas jej propagacji w plazmie o standardowej konfiguracji pola magnetycznego, analizo-wana jest na podstawie równań różniczkowych, opisujących zmiany tradycyjnych parametrów polaryzacyj-nych: kąta azymutu i kąta eliptyczności . Przeanalizowano wpływ zjawiska Faradaya i Cottona-Moutona na zmiany wartości parametrów i przy różnych gęstościach plazmy w stellaratorze. Zbadano również wpływ przesunięcia przestrzennego pozycji wiązki pomiarowej na obserwowane zmiany polaryzacji fali elek-tromagnetycznej.
Introduction
Polarimetry is one of the basic diagnostic at thermonuclear devices. In a tokamak, the change of the polarization state determined by the plasma density and magnetic field is often used to derive the unknown plasma currents if the density is known from the other measurements. In a stellara-tor like Wendelstein 7-X where plasma current are small and thus the magnetic field components are well known, from the measurement of Farady and
Cotton-Mouton effects the information on line average plasma density can be unfolded. For Wen-delstein 7-X a single channel far-infrared interfero-polarimeter system has been proposed. An instru-ment of this kind has already been tested at W7-AS and its capability measure the line averaged density has been demonstrated [1]. One of the main crite-rion for choosing the appropriate probing beam wavelength for W7-X is expected value of the pola-rimetric effect. In this paper, we present the estima-tion of Faraday rotaestima-tion and Cotton-Mouton phase
shift at W7-X geometry, for standard magnetic field configuration and typical plasma densities.
Geometry of the system
The poloidal cross section of the interfero-polarimeter system, against a background of the part of the W7-X vessel and plasma in standard magnetic field configuration, is illustrated in figure 1. The beam is transmitted through the equatorial port, crosses the triangular shaped plasma along the midplane on the distance of 140 cm and is reflected back from a corner cube reftroreflector (CCR) mounted to the rear vessel wall. The total path-length into the vessel is about 2L = 420 cm. As in the zeroth-order of approximation the port, the plasma and the CCR have common axis of symme-try and as a result the position of the forward and backward beams are shifted on the same distance with respect to this axis and are in the same plane.
The components of magnetic field B in the
vicinity of the central polarimeter sightline exhibit highly symmetric configuration. Figure 2 shows these components in the standard W7-X coordinate
system (poloidal-horizontal: Bx, toroidal: By and
poloidal-vertical: Bz) for sightlines that are shifted
by ±0, 1.8, 3 and 5 cm vertically and horizontally. For a given position on the SL the components By
and Bz are constant in the plasma region, regardless
on the beam position. Bx reveal more complicated
behavior – its value depends not only on the posi-a) 0.0 0.5 1.0 1.5 2.0-0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08
B
x[T]
z=5 z=3 z=1.8 z=0 z=-1.8 z=-3 z=-5Position on SL [m]
-1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5 0,0 0,5 1,0 1,5 2,0y=0
B
xB
zPlasma
B
[T]
Position on SL [m]
B
y y = 0 Position on SL [m] Position on SL [m] Bx [T] 0.08 0.06 0.04 0.02 0.00 –0.02 –0.04 –0.06 –0.08 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 B [T] 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2 .0 0.0 0.5 1.0 1.5 2.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5z=0
B
xB
zPlasma
B
[T]
Position on SL [m]
B
y 0,0 0,5 1,0 1,5 2,0-0,08 -0,06 -0,04 -0,02 0,00 0,02 0,04 0,06 0,08B
x[T]
y=5 y=3 y=1.8 y=0 y=-1.8 y=-3 y=-5Position on SL [m]
b) z = 0 Bx [T] 0.08 0.06 0.04 0.02 0.00 –0.02 –0.04 –0.06 –0.08 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 B [T] 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2 .0 Position on SL [m] Position on SL [m]Fig. 2. Magnetic field components in the vicinity of the polarimeter sightline in the poloidal (a) and toroidal plane (b) for different shifts of the sightline
Rys. 2. Wartość składowych indukcji pola magnetycznego w otoczeniu osi optycznej polarymetru: a) w płaszczyźnie poloidalnej, b) w płaszczyźnie toroidalnej
Fig. 1. Poloidal cross-section of W7-X vessel, showing the polarimeter sightline and the standard magnetic field configu-ration
Rys. 1. Poloidalny przekrój przez komorę stellaratora W7-X, z zaznaczonym biegiem wiązki pomiarowej i plazmą w stan-dardowej konfiguracji pola magnetycznego
tion on the SL, but also on the SL position in the space. Nevertheless always the symmetry
Bx (x,y,z) = –Bx (x,–y,–z) is preserved.
Evolution equations for angular parameters of polarization ellipse
The equations for angular parameters and , derived in [2], could be written in the form:
2 cos 2 sin d d 2 tan 2 sin 2 cos d d 2 1 2 1 2 1 2 1 3 2 1 (1)Here is an arc length along the ray path and 1, 2, 3 are polarimetric parameters, involved in [3]:
2 2
3 1 1C Ne Bx By y x e B B N C1 3 2 2 z eB N C3 2 3 (2) 2 1 11 2 1 21 12.4610 cmOe 2.4610 mT C 1 13 1 17 3 5.2610 Oe 5.2610 T Cwith x and y the direction perpendicular to the beam, z in the beam direction (respectively paral-lel O the z, y and x axis in W7-X coordinate sys-tem). It is worth noting that when components of vector are comparable, as long as azimuthal angle remain close to 45 or 0 and ellipticity angle to 0 (sin2 1 or cos2 1 and tan2 << 1), the pola-rization changes could be described as a pure Fara-day and pure Cotton-Mouton effect:
d or d d 2 2 1 1 2 1 3 2 1 (3) Otherwise the Faraday and Cotton-Mouton ef-fects combine nonlinearly and accurate determina-tion of plasma density value becomes very compli-cated [2, 3].Results of numerical modeling
Figure 3 shows the behavior of vector com-ponents as a function of the coordinate along the beam path on the forward (f) and backward (b) propagation for = 195 m, Ne = 11020 cm–3 and
two sample beam positions: shifted by 2 cm verti-cally and horizontally.
All three components are in the same order of magnitude, in spite of the fact that Bz Bx,By.
It can be also seen that the variation of 1 and 2 along the path are independent on the initial beam position and that 1() = 1(2L – ), 2() = – 2(2L – ), 3() = 3(2L – ). Such a high sym-metry determine the evolution of the angular para-meters. Figure 4 shows the change of and from initial state (45,0) in case, when Faraday and Cot-ton-Mouton effects are independent (Ne = 11020
cm–3) and when there is a nonlinear interaction between both effects (Ne = 51020 cm–3). The jump
in and at = 210 cm is caused by reflection from CCR and additional jump at = 420 cm restore the initial coordinate system.
a) 195μm;Ne11020m3;r0[020]
b) 195μm;N 11020m3;r0[002]
e
Fig. 3. Profiles of vector components along the beam path for two different initial beam positions: a) r0 = [0 2 0], b) r0 = [0 0 2]. The retroreflector is located at = 2.1 m Rys. 3. Przebieg zmian wartości wektora wzdłuż biegu promienia dla dwóch różnych położeń początkowych wiązki pomiarowej: a) r0 = [0 2 0], b) r0 = [0 0 2]. Retroreflector znajduje się w pozycji = 2,1 m
In the lower density case, the change of the azi-muthal angle after the first passage throw the plasma is almost completely cancelled after the second passage – the Faraday effect vanishes in the zeroth order of approximation. In opposite, ellip-ticity angle after the second passage is twice as
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 -0,0015 -0,0010 -0,0005 0,0000 0,0005 0,0010 0,0015
[m
-1]
[m]
1f 2f 3f 1b 2b 3b 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 -0,0015 -0,0010 -0,0005 0,0000 0,0005 0,0010 0,0015
[m
-1]
[m]
1f 2f 3f 1b 2b 3b [m] [m] [m –1 ] [m –1 ]large as after the first plasma crossing – the Cotton-Mouton effect is doubled. As on the whole path sin 1, the line integrated plasma density value could be calculated from the approximate relation:
L y z e B B C N 2 0 2 2 3 1 d 2 (4)The calculations carried out for different initial beam positions revealed that final and value does not change significantly with the beam shift. a) 195μm;N 11020m3;r0[002] e b) 195μm;N 51020m3;r0[002] e
Fig. 4. Change of and along the beam path for two plasma densities: a) Ne = 11020 m–3, b) Ne = 51020 m–3
Rys. 4. Zmiany polaryzacji fali elektromagnetycznej podczas przejścia przez plazmę o gęstości: a) Ne = 11020 m–3, b) Ne = 51020 m–3
In the higher density case, when both effects in-teract, symmetries described previously are broken: the final value is different from the initial one and is not doubled. During the beam propagation the polarization angles become large and relations (3) are not fulfilled – the is no longer a linear measure of N and to determine its value the com-e
plete equations system (1) has to be solved. What’s more, as in this case the value depend on the
beam position (Fig. 5), some additional sources of inaccuracy are expected: sensitivity to the beam displacement and the beam depolarization as a re-sult of switch from the “narrow beam” approxima-tion to finite beam diameter.
195μm; 5 10 m; , 45 ,0 0 0 3 20 Ne
Fig. 5. The dependence of the value on the beam position Rys. 5. Zależność zmian kąta eliptyczności od położenia wiąz-ki
Another source of inaccuracy is the shift bet-ween the CCR axis and the central sightline, which may appear as a result of long term thermal drift. As it effect mainly the Faraday rotation angle, even in the lower density case (Fig. 6), the residual Fara-day component can be regarded as a measure of such a displacement. ] 2 0 0 [ ; m 10 1 ; μm 195 20 3 0 N r e
Fig. 6. The dependence of the value on the shift between the CCR axis and the central sightline
Fig. 6. Zależność zmian kąta azymutu od względnego przesu-nięcia osi symetrii retroreflektora i osi optycznej polarymetru 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 -47 -46 -45 44 45 46 47 -10 -8 -6 -4 -2 0 2 4 6 8 10
[deg]
[m]
[deg]
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 -50 -40 -30 30 40 50 -40 -30 -20 -10 0 10 20 30 40 [deg]
[m]
[deg]
[m] [ de g] [ de g] [m] [ de g] [ de g] -1,0 -0,5 0 0,5 -1,5 1,0 -1,0 -0,5 0,0 0,5 1,0 -1,0 -0,5 0,0 0,5 1,0 y [cm] z [cm ] [deg] [deg] y [cm] z [c m] -25,0 -27,5 -30,0 -32,5 -35,0 -37,5 -22,5 -4 -2 0 2 4 -4 -2 0 2 4 [deg] y [cm] z [cm ] y [cm] z [c m] [deg]The polarization changes depend also on the plasma density profile and the effect of refraction of the beam in the plasma region. To minimize the error of the measurement, the optimization consi-dered all factors has to be performed, based on the initial polarization state, the beam position and wavelength – but it goes beyond the scope of pre-sented work.
Conclusions
The proposed interfero-polarimeter system for W7-X is characterized by highly symmetric mag-netic field configuration in the vicinity of the sightline. As a result for densities up to Ne 1020
m–3, the Faraday effect vanishes although in the plasma region the parallel magnetic field compo-nent is different from zero and the shift of ellipticity angle caused by pure Cotton-Mouton effect is a linear measure of the line integrated plasma den-sity value. Azimuthal angle change different from zero ( 0) signify that the sightline and the CCR axis are shifted vertically. The strength of interac-tion between Faraday and Cotton-Mouton effects at high density plasma region could be minimized through proper choice of the beam position and initial polarization state.
References
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2. GUENTHER K.: Approximate method to extract the pure Faraday and Cotton–Mouton effects from polarimetry measurements in a tokamak. Plasma Phys. Control. Fusion, 2004, 46, 1423–1441.
3. SEGRE S.E.,ZANZA V.: Derivation of the pure Faraday and Cotton–Mouton effects when polarimetric effects in a to-kamak are large. Plasma Phys. Control. Fusion, 2006, 48, 339–351.
Others:
4. CZYZ Z.H.,BIEG B.,KRAVTSOV YU.A.: Complex polariza-tion angle: Relapolariza-tion to tradipolariza-tional polarizapolariza-tion parameters and application to microwave plasma polarimetry. Phys. Let. A, 2007, 368, 101–107.
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Acknowledgements
This work, supported by the European Commu-nity under the contract of Association between EURATOM and IPPLM (project P-12), was carried out within the framework of the European Fusion Development Agreement.
Recenzent: prof. dr hab. inż. Janusz Kwaśniewski Akademia Górniczo-Hutnicza w Krakowie
