Maritime University of Szczecin
Akademia Morska w Szczecinie
2011, 26(98) pp. 47–51 2011, 26(98) s. 47–51
Modulation of weak Cotton-Mouton effect in conditions
of strong Faraday rotation
Modulacja słabego efektu Cottona-Moutona w warunkach
silnej rotacji Faradaya
Yury A. Kravtsov
1,2, Janusz Chrzanowski
21 Space Research Institute, Profsoyuznaya St. 82/34, Moscow 117997, Russia
2 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
Key words: plasma polarimetry, weakly anisotropic plasma, Cotton-Mouton effect, Faraday effect, mutual influence of the Cotton-Mouton and Faraday effects
Abstract
Evolution of polarization state is described by new mathematics, namely, by angular variables technique (AVT) which describes behavior of the angular parameters of polarization ellipse in magnetized plasma. From this point of view the polarization of electromagnetic waves in magnetized plasma is studied in conditions, when Cotton-Mouton effect is weak enough as compared with Faraday one. The method of consequent approximations is applied, which uses the ratio of Cotton-Mouton , and Faraday 3 parameters, as a small parameter of a problem and allows obtaining simple analytical expressions for ellipticity angles approximations.
Słowa kluczowe: polarymetria plazmy, słabo anizotropowa plazma, efekt Cottona-Moutona, efekt Fara-daya, wzajemne oddziaływanie efektów: Cottona-Moutona i Faradaya
Abstrakt
Przy pomocy nowego modelu matematycznego – techniki zmiennych kątowych (AVT), opisana jest ewolucja stanu polaryzacji z ukazaniem zachowania kątowych parametrów elipsy polaryzacji w namagnetyzowanej plazmie. Z tego punktu widzenia analizowany jest stan polaryzacji wiązki elektromagnetycznej w warunkach, gdy efekt Cottona-Moutona jest niewielki w porównaniu z efektem Faradaya. Zastosowana metoda kolejnych przybliżeń pozwala wykorzystać stosunek parametrów (Cotton-Mouton), oraz 3 (Faraday), jako kolejny, mały parametr, pozwalający uzyskać proste analityczne wyrażenia dla kątów opisujących stan elipsy polary-zacji.
Introduction
Competition and mutual interaction of Cotton- -Mouton and Faraday effects in magnetized plasma are of special interest for plasma polarimetry [1, 2]. According to [1, 2], Cotton-Mouton and Faraday effects are mutually independent, when plasma parameters:
0 3 3( ) ( )d W and
0 d ) ( ) ( W (1)are small enough:
1 ) (
3
W , W()1 (2)
Here, 3 and are standard parameters,
in-volved in plasma polarimetry by Segre [3, 4], is an arc length along the ray.
In this paper, the method of consequent approxi-mations is applied to solution of polarimetric prob-lems in frame of the angular variables technique (AVT). This method describes mutual interaction between the Cotton-Mouton and Faraday effects and is applicable in conditions, when Cotton-
-Mouton effect is weak enough as compared with Faraday phenomenon: 3 (3)
Because of this a small parameter: 1 3 (4)
appears in equations, used in plasma polarimetry. In distinction to frequently used in plasma pola-rimetry Stokes vector formalism (SVF) [3, 4], we apply here angular variable technique to describe behavior of angular parameters of polarization el-lipse in weakly anisotropic plasma. The equations, of AVT which were derived in [5], we intend to solve by the method of consequent approximations. We shall study here two new phenomena: modula-tion and suppression of the weak Cotton-Mouton effect by the Faraday rotation. To the author’s knowledge, these phenomena were not described in the literature so far.
The material of this paper is outlined as follows. Sect. 2 presents the basic equations of AVT, governing the evolution of angular parameters of polarization ellipse [5]. The eqs. (5) which were obtained on the basis of the quasi-isotropic approxi-mation (QIA) of the geometrical optics method, which describes properties of electromagnetic waves in weakly anisotropic media, including weakly magnetized plasma [6, 7, 8, 9, 10]. The equations for angular parameters we shall solve in Sect. 3 by the method of consequent approxima-tions. Sect. 4 reveals the phenomena of ellipticity modulation and suppression by Faraday rotation.
Basic equations of the angular variables technique
It is commonly accepted to characterize the po-larization ellipse by angular parameters and , shown at figure 1.
According to [5], the angular variables and satisfy the equations:
2 2 sin 2 1 2 tan 2 2 cos 2 1 3 (5) where: 2 / , sin cos 2 0 0 || 2 0 || 0 3 XY k XY k (6) are plasma parameters, involved in plasma polari-metry by Segre [3, 4]. The parameters and are expressed through the standard plasma parame-ters X and Y, defined as [12, 13]:
Fig. 1. Angular parameters and characterizing orientation and ellipticity of the polarization ellipse
Rys. 1. Parametry kątowe i , opisujące orientację i nachyle-nie elipsy polaryzacji
2 2 2 2 π 4 m N e X p e , c mc eB Y 0 (7)
(in the book [13] they were denoted as v and u). In the eqs. (7) Ne is electron density, B0 means
static magnetic field, is an angle between the ray
and the magnetic vector B0, and is an angle
between the unit vector e1 of the coordinate system
and the transverse component of the magnetic field
B0, as presented at figure 2.
Fig. 2. Orientation of the static magnetic field B0 in frame of the coordinate system (e1, e2, l)
Rys. 2. Kierunek statycznego pola magnetycznego B0 we współrzędnych (e1, e2, l)
Keeping terminology, used in [3], we shall treat
as shear angle. For the electromagnetic waves || || 0 B 0 B 0 B 1 e 2 e
of millimeter and sub-millimeter bands, used in plasma polarimetry, parallel transport coordinate system [14, 15] practically coincides with the Cartesian one.
The first term 3 in eq. (5) is linear in magnetic
field B0 and corresponds to Faraday effect, while
parameter is quadratic in B0 and corresponds to
Cotton-Mouton effect [3, 4].
Introducing plasma parameters 1 = cos2
and [3, 4], we may present eq. (5) also in the form:
2 cos 2 sin 2 1 2 tan 2 sin 2 cos 2 1 2 1 2 1 3 (8)where parameters 1 and 2 are involved instead of
. Another form of the equation (5) implies an
explicit introduction of the small parameter (4). Using dimensionless variable along the ray:
0 3 3( ) ( )d ) ( W Z (9)We may transform eq. (5) to the form:
2 2 sin 2 1 d d 2 tan 2 2 cos 1 2 1 d d Z Z (10)In what follows, we prefer to use for analysis the equation (5), considering, when necessary, dimensional parameter as a small parameter of
a problem.
Solution by the method of consequent approximations
Let (0) and (0) are the initial values of angular variables. Presenting differential eqs. (5) in the integral form, we have:
0 0 d 2 2 sin 2 1 ) 0 ( ) ( d 2 tan 2 2 cos 2 1 2 1 ) 0 ( ) ( W (11) In view of smallness of Cotton-Mouton parame-ter in respect to the Faraday parameter 3,we can apply the method of consequent approxi-mations to eqs. (11), assuming that in the zero-th approximation we deal with the wave of linear polarization, obeying pure Faraday effect:
0 ) 0 ( ) ( ), ( 2 1 ) 0 ( ) ( 3 0 0 0
W (12)Subsequent approximations n and n are
sup-posed to satisfy the equations:
0 1 0 1 d 2 2 sin 2 1 0 d 2 tan ) ( 2 ) ( 2 cos 2 1 2 1 0 n n n n n W (13) Starting with the zero-th approximation (12), in the first iteration we have:
0 0 1 0 1 d 2 2 sin 2 1 0 2 1 0 W (14) whereas the second iteration provides (15):
0 1 1 2 0 0 0 0 1 0 1 0 1 2 d 2 2 sin 2 1 0 d d 2 2 sin 2 2 cos 2 1 ) ( d 2 tan 2 2 cos 2 1 (15)In what follows we consider the case of linear initial polarization, when initial ellipticity angle
(0) equals zero:
(0) = 0 (16)
Besides, we shall into account that the shear angle f in toroidal plasma is small enough [16].
In these conditions we arrive to the following equa-tions of the first order:
0 0 1 0 1 d 2 sin 2 1 2 1 0 W (17)Analogously, the equations of the second order (15) will take a form:
0 1 1 2 0 0 0 0 0 0 1 0 0 2 d 2 sin 2 1 d d 2 sin 2 cos 2 1 d 2 tan 2 cos 2 1 (18)Phenomenon of ellipticity modulation by Faraday rotation
In the first order of the method of subsequent approximations, joint action of the Cotton-Mouton and Faraday effects in frame of the first order ap-proximation is described by the eqs. (17). Accord-ing to eq. (17), the azimuthal angle , correspond-ing to the Faraday rotation = (1/2)W3(), does
not depend on the Cotton-Mouton effect at all. On the contrary, the ellipticity angle is undergone to the essential influence of Faraday rotation. As we shall see below, Faraday rotation is responsible for periodic modulation of ellipticity and for phenome-non of ellipticity suppression.
To study these phenomena, it is convenient to use a variable Z() = W3() as integration variable
in eq. (17) instead of the arc length . This variable, which has been already used in eq. (10), plays a role of dimensionless distance along the ray.
The variable Z allows presenting ellipticity angle
1, eq. (17), in the form:
Z
Z Z Z Z Z
sin2 0 d 2 1 0 3 1 (19)An integrand in eq. (19) oscillates in variable Z with a period 2, so that positive and negative half-periods of the sinusoidal function conceal each other. Therefore the result of integration at Z >> 1 is determined by small section of the unit length: Z ~ 1. Therefore the local maximum of ellipticity angle, which should be small, as assumed above, can be estimated as:
1 ~ | | 3 max 1 (20)According to eq. (20), the local maximum of ellipticity is suppressed with the rate 3 of Faraday
rotation, the ratio (20) is playing a role of suppres-sion factor. The suppressuppres-sion factor:
3 (21)shows how maximum ellipticity angle |
1|max decreases with Faraday parameter 3().This qualitative analysis of the phenomena of ellipticity modulation and suppression can be illu-strated by the example of homogeneous plasma with:
const
3
, const (22)
In this case integral (19) can be calculated expli-citly:
cos2 0 cos2 0
2 1 d 0 2 sin 2 1 3 0 3 1
Z Z Z Z Z (23)or, after simple transformations:
2 sin 2 ) 0 ( 2 sin ) ( 3 3 3 1 (24)
Here, functions sin(2(0) + 3/2) and
sin(3/2) describe modulation, whereas factor
/3 corresponds to suppression of ellipticity by
Faraday rotation. Period of modulation, which equals to, is determined by the rate of Faraday rota-tion 3.
For (0) = /4 we can rewrite last relation in form:
3 3 1 2 sin 1 ) ( (25)According to eq. (23), local maximum of ellip-ticity can be presented in the following form:
|| || 2 3 max 1 cos 2 sin ) ( ) ( | | Y (26)
This value does not exceed a unit under condi-tion cos (Y/2)sin2, which corresponds to so
called “quasi-longitudinal” propagation [13]. Mag-netic parameter Y, which equals to the ratio of the electron cyclotron frequency c = eB0/mc to the
sounding frequency , Y = e/, is small enough in
the sub-millimeter and far-infrared frequency bands. It is worth noticing, for instance, that at
B0 = 1 T and = 1012 magnetic parameter Y is
about 10–3. Therefore, suppression factor (21) as
a rule is sufficiently small under polarimetric mea-surements.
Inverse influence of Cotton-Mouton effect on Faraday rotation
Such an influence can be revealed in the second order of the method of consequent approximations, eq. (18). Let us consider inverse influence of weak Cotton-Mouton effect of Faraday rotation for a homogeneous plasma. In the second approxima-tion:
2 3 3 3 2 1 2 2sin 2 1 2 1 (27) According to eq. (27), azimuthal angle expe-riences variations, quadratic in small parameter = ()/3() and with doubled frequency of
Faraday rotation. These features will be illustrated below by the results of numerical modeling.
Conclusions
This paper analyzes properties of electromagne-tic waves, experiencing influence of moderate Faraday effect and weak Cotton-Mouton effect. Analysis of electromagnetic wave’s polarization is performed on the basis of angular variables tech-nique (AVT), which deals with the equations for angular parameters of polarization ellipse. Equation for angular parameters are solved by the method of consequent approximations under assumption that
Cotton-Mouton effect is weak enough, as compared with the Faraday effect might be of moderate value.
It is shown that ellipticity of electromagnetic wave experiences modulation and suppression under action of Faraday rotation. In turn, it is dis-covered that weak Cotton-Mouton effect influences on Faraday phenomenon.
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This work, supported by the European Com-munities under the contract of Association between EURATOM and IPPLM (project P-12), was carried out within the framework of the European Fusion Development Agreement. This work is supported also by Polish Ministry of Science and High Educa-tion (grant Nr 202 249535).
Recenzent: prof. dr hab. inż. Janusz Kwaśniewski Akademia Górniczo-Hutnicza w Krakowie
