Microwave polarization plasma diagnostic

Head of group: Prof. dr hab. Yu. Kravtsov, Institute of Physics, Maritime University of Szczecin

Members of group: Dr. G. Żegliński, Institute of Electronics and Informatics,

Szczecin University of Technology.Dr. J. Zaleśny, Institute of Physics, Szczecin University of Technology.

Msc. P. Berczyński, Institute of Physics, Szczecin University of Technology

Potential European partners:

• ENEA (the Italian organization for nuclear energy and fusion research), Frascati, Ital

• Intention List with ENEA is in preparation:• Dr. Francesco Orsitto • Prof. S. Segre       • Dr. Vincenzo Zanza • Dr. P. Buratti• JET: Prof. Jerzy Brzozowski• IPP, Greifswald, Germany, Prof. H. J. Hartfuss• Dr. M. Henderson• CRPP, Lozanna Dr.H. Weisen

The main purpose of the presentation isto suggest efficient theory of microwave

polarization diagnostics of the magnetized plasma.

1) The proposed theory is developed on the basis of quasi-isotropic approximation (QIA) of geometrical optics.

2) QIA describes linear interaction of normal modes in inhomogeneous magnetized plasma and embraces both Faraday effect (quasi-longitudinal propagation) and Cotton-Mouton phenomenon (quasi-transverse propagation).

3) In combination with paraxial complex geometrical optics, updated recently in Szczecin, the method under consideration reduces description of both diffraction and polarization evolution of the electromagnetic beams to the solution of ordinary differential equations, which can be readily solved numerically.

4) This method looks quite promising for data interpretation on microwave sounding of inhomogeneous magnetized plasma. The method can be helpful for solution of problems, connected with microwave energy delivery into hot inhomogeneous plasma.

Quasi-isotropic approximation (QIA).

Quasi-isotropic approximation (QIA) of the geometrical optics method, was suggested by Kravtsov (1969) for description of electromagnetic waves polarization in weakly anisotropic media. The QIA is based on supposition that all the components of the anisotropy tensor

mnmnmn 0

are small as compared with an electric permittivity of the isotropic (non-magnetized) plasma 0

mn is the tensor of electrical permittivity, mn is a unit tensor

QIA involves small parameter 1||max mnanisotr

characterizing degree of anisotropy

According to Kravtsov (1969); Fuki, Kravtsov, Naida (1997); Kravtsov (1990, 2005) solution of the Maxwell equations in the lowest approximation in weakly anisotropic media can be presented as

)exp( ikAΓE Γ is a polarization vector, which is orthogonal to the leading ray,

as in isotropic medium.

If e1 and e2 together with the tangent unit vector l form an orthogonal trihedral, corresponding to parallel transport transfer, then the components of the polarization vector 2211 eeΓ obey the equations

)(21

)(21

2221212/1

02

2121112/1

01

ikds

d

ikdsd

Here s is an arc length along the ray

For high frequency electromagnetic waves in a magnetized plasma these equations take a form:

22/1121

2

222/1111

sin)1(2

cos

cossin)1(2

uviGGds

d

GuviGdsd

2022 )/(,/4 mceHumNev e Standard plasma parameters:

Ne is the electron concentration, H0 is a static magnetic field,

is an angle between magnetic vector H0 and the ray tangent l

2/12/1)1(21 vuvkG Parameter

For weak magnetic field, u<<1, the main role in above equation belong to the Faraday effect, which is linear in magnetic field and parameter . Polarization of electromagnetic wave is determined by Faraday rotation of the

polarization plane at angle

is an angle between magnetic vector and the ray tangent l

H

E

E

EE

dlNH e cos~ 0

0H 2/1u

0H

Faraday diagnostic is the most spread in plasma physics, both in laboratory and in ionosphere conditions but disposes with two shortcomings:

1) Faraday method determines integral of electron concentration (with weighting multiplier ) and doesn’t allow to extract local values of Ne

cos

2) Faraday approach becomes ineffective, when the ray is orthogonal to the magnetic field in some point: in this case quasi-longitudinal Faraday approximation fails.

In the vicinity of orthogonality point between the ray and magnetic field, where

2/ and 0cos Faraday rotation gives up its place to Cotton-Mouton effect, quadratic in magnetic field.

Having measured intensities and after transition through the orthogonality point one can

I1 I2

I2/I1 (Ne)loc

rightIleftI

find local electron concentration Ne in the orthogonality point:

Coutton-Mouton effect strong interaction between left-handed and right-handed circular polarizations.

To describe such an interaction in the framework of QIA, one can linearize the right-hand part of QIA equation in parameter s, counted from orthogonality point:

1sin,/cos s

characterizes the relative curvature of the ray and magnetic field. Involving the amplitudes of circular waves

2/)( 122,1 i

one can derive the following system of linearized equations for 2,1

212/1

2

22/1

11

2//

2//

iipdd

ipidd

2)1( vGup is a single parameter of interaction2/1

0 ]2/[ kvs is dimensionless parameter along the ray

The lineralized QIA equation admit the exact solution for the intensities

21right ||I and

22left ||I

If the initial (before interaction) values are:

1|)(|)(I 21right 0|)(|)(I 2

2left then after

interaction

)4/exp(|)(|)(I 21right p

)4/exp(1|)(|)(I 22left p

The value 1)4/exp()4/exp(

)4/exp(1)(I)(I

right

left

p

pp

represent the coefficient of the right-polarized wave transformation into the left-polarized one.

Having measured intensities rightI and leftI after transition through the orthogonality point, one can determine local plasma parameter p from relation

)1ln(4

p

and thereby to find local electron concentration Ne in the orthogonality point.

For the ionosphere plasma diagnostics this method was suggested by Kravtsov, Naida

(1976) and for laboratory plasma by Kravtsov, Naida (2000, 2001).

Another method for polarization description in a magnetized plasma was suggested by Segre (2001) . His method deals with Stokes vector-parameter S, which obeys the equation

SMS

dsd

with vector M, characterizing parameters of the magnetized plasma. It can be shown that above Segre equation and QIA equations are equivalent to each other in a wide area of parameters. In particular, in the area of QIA applicability Segre equation stems from QIA equations.

Like traditional geometrical optics, CGO starts with the presentation of the wave field in the form of Debye’a expansion in inverse powers of wave number:

00

0

rexprrm

mm ik

ikAu

where is eikonal and A is amplitude. We consider here 2D scalar problem (Cartesian coordinates are x and y) and restrict ourselves only by the main term in Debye’a expansion for the amplitude , that is we take . Substituting Debye’a expansion into wave equation:

0AA 0

AA

0rrr 20 uku

one has the eikonal equation for eikonal

:

2

Complex geometrical optics (CGO)1.Basic equations of Complex Geometrical Optics (CGO)

and transport equation for 0AA

0)( 2 Adiv

All the values are complex-valued in framework of CGO.

2. The ray-based and eikonal-based forms of CGO

Complex geometrical optics (CGO) is known to have two equivalent forms:the ray-based and eikonal-based ones. Yu.A.Kravtsov. Geometrical Optics in Engineering Physics. Alpha Science International, Harrow, UK, 2005.The ray-based form deals with the complex trajectories,

rppr

21,

dd

dd

Surprising feature of CGO is its ability to describe Gaussian beam diffraction.It has been demonstrated analytically (for homogeneous medium)

in frame of the ray-based form still 38 years ago in the paper: Yu.A. Kravtsov, 1967.

)(p is the ray “momentum” and d relates to the elementary arc length d by

/dd

.

Eikonal-based form deals directly with complex eikonal and is able to describe Gaussian beam diffraction for arbitrary smoothly inhomogeneous media,

as it will be discussed below:

3. Paraxial CGO: Gaussian beam diffraction in 2D inhomogeneous media.

3.1. Eikonal equation in curvilinear coordinatesFor the description of narrow beams propagation and diffraction in inhomogeneous medium it is convenient to go over to 2D curvilinear frame of reference( , )associated with the central ray

r

22

21h

As a result, in curvilinear coordinates the eikonal equation takes the form

h is Lame` coefficient: Kh c 1 ,where cc r

c

Krr

rr

21

and K is a ray curvature:

3.2. Solution of the eikonal equation in paraxial approximation. Reduction of diffraction problem to ODE

In frame of paraxial approximation one can expand permittivity νrr c )(in a Taylor series in small deviation from the central ray:

c

crr

rrrr

...22

2

Then, within paraxial approximation the eikonal equation takes a following form:

2

rr

2

2

22

22

2 r43r

2rr

c

ch

It is worth seeking a solution of the eikonal equation as a sum of two terms: cwhere c

is the eikonal calculated along the central ray:

cc

dd

is the eikonal calculated along the central ray:

and is a small deviation from c

.

In frame of paraxial approximation deviation is to be presented in the quadratic form, like Babich [1968]:

221 B

Here B is a complex parameter changing along the central ray. Eikonal equation leads to following ordinary differential equation of Riccati-type

for parameter B:

2BddB

,where

crr

2

2

2 rr4

3r21

[P.Berczynski,Yu.A.Kravtsov,. Physics Letters A, 2004, 331(3-4), 265-268]

iIRB

)(

)( ccR

2

0

2wk

I

The transport equation takes in paraxial approximation the following simple form:

0~21~

ABA

where AA c 4/1~ .It admits the following solution:

dBAA

21exp~~

0

This solution leads to the relation:

wwAA 0

0~~

which corresponds to the energy flux conservation through the 2D GB cross-section.

4. Numerical example: Gaussian beam reflection from a plasma layer with: yy („linear layer”).

For such case: 20

220

22

4cos4

sin12

For the following parameters: 1

10001

60

100 w

00

The Riccati equation for parameter B has the following numerical solution:

20

10

wk

The beam width:

0 1.52 3.05 4.57 6.10

2.25

4.5

6.75

9

40 10k

The beam first experiences diffraction widening, then focusing and diffraction widening again.

40

10

/

k

Phase front curvature:

40 10k

Phase front curvature approaches zero three times:

in 00 k in accordance with initial condition 00 in 4

0 107.2 k , where GB width possess maximum

in 40 106.4 k , in „beam waist”

5. CGO of 2D nonlinear media.

Let us consider for example Kerr type nonlinear medium, described by permittivity:

)1( 20 uNL

For 2D Gaussian beam, one has:

2

20

0 expww

wAu

Parameter in Riccati equation in considered case takes the following form:

30

ww or 2/30

)0(I

Ik

where }Im{BI and 2

002 uNL

For the following parameters:

mVA

Vmw NL

602

214

00 10,102.1,1,10

the Riccati equation for complex parameter B has the following numerical solution:

30 10k

Beam width: wk0

0 0.5 1 1.5 2 2.50

20

40

60

80

Phase front curvature:

30 10k

0/ k

1. Generalization of CGO for weakly anisotropic plasma: Hybrid of QIA and CGO, combining polarization and diffraction process. (Jointly with ENEA.)

2. Comparative analysis of QIA equations (Szczecin) and Segre equations (Frascati) (Jointly with ENEA.)

3. Joint (Szczecin and Frascati) analysis of JET experimental data. (Jointly with ENEA.)

Main problems to realize with Euratom partners:

4. Generalization of CGO for high power Gaussian beams: self-focusing of Gaussian beams in nonlinear plasma.

5. Elaboration of advanced numerical methods for solution of the inverse problem of polarization diagnostic: extraction of the electron density from polarization data, when contribution of the Cotton-Mouton effect is comparable with that of Faraday effect.