fusion plasma diagnostics with mm-waves (hartfuß/fusion plasma diagnostics with mm-waves) || active...

65

3Active Diagnostics

This chapter aims at introducing the basic principles of millimeter-wave diagnos-tics based on the arrangements in which a microwave beam is passed throughthe plasma to actively probe its dielectric properties (Figures 1.7–1.9). To keepthe discussion most fundamental, the plasma dielectric properties are consid-ered sufficiently well described in the cold-plasma limit. As derived in detail invarious paragraphs of Chapter 2, the refractive index is exclusively determinedwithin this limit, by the electron density and the �B-field. Only these quantitiescan therefore be determined by evaluating the changes of the physical param-eters characterizing the probing wave when interacting with the plasma. Thediagnostic systems are called interferometry, polarimetry, and reflectometry. De-spite interferometry and polarimetry diagnostics are realized in fusion-relevantplasmas mainly in the submillimeter and the far-infrared wavelength regions,they are nevertheless introduced in this context, as the basic principles haveevolved from the microwave region. Experimental and technical details of typ-ical setups are discussed in Chapter 8, after the technical components of itsrealization have been introduced. There, we restrict the discussion, however, tothose diagnostic systems that are realized in the millimeter and submillime-ter wavelength regions and will not discuss the experimental background oflaser-based interferometer and polarimeter systems. As mentioned earlier, theprobing wave is scattered by the plasma electrons, and the analysis of the ex-tremely weak process provides information on the scattering centers. The physicsof scattering is briefly discussed at the end of this chapter. Again, the discus-sion of experimental realizations is restricted to those in the millimeter-waverange.

3.1Interferometry

The refractive index N of all types of waves described so far depends on theelectron density ne through the plasma frequency ωp with ω2

p ∝ ne. As shown inFigure 3.1, this is true for a wave passing a nonmagnetized plasma as well as for the

Fusion Plasma Diagnostics with mm-Waves: An Introduction, First Edition.Hans-Jurgen Hartfuß and Thomas Geist.© 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.

66 3 Active Diagnostics

0.50 5E+19 1E+20 1.5E+20 2E+20

Electron density ne (m−3)

NR

NX

NO

NL

0.6

0.7

0.8

0.9

1

Ref

ract

ive

inde

x N

Figure 3.1 The electron-density-dependentrefractive indices NL and NR of the left-and right-hand circular polarized charac-teristic waves as well as NO and NX ofthe ordinary and extraordinary waves. Theprobing frequency is ω = 2π × 200 GHz,

while the cyclotron frequency is chosenωc = 2π× 70 GHz corresponding to a mag-netic field of B0 = 2.5 T. The figure givesthe typical electron density range mod-ern fusion experiments are operating in,(5 − 20) × 1019m− 3.

characteristic waves passing a magnetized plasma. These are the circular polarizedL- and R-waves for a wave propagating along the �B-field direction (Equation2.86 and Equation 2.87) and the linear polarized O- and X-mode waves whenpropagating perpendicular to the �B-field direction with their �E-field either parallelor perpendicular to �B0 (Equation 2.97 and Equation 2.98). Since they all depend onωp, each of these waves can basically be used to gain information on the electrondensity. In the torus geometry of modern fusion experiments, it is experimentallyadvantageous to use a probing wave propagating perpendicular to the �B-field. Toavoid the dependency of the refractive index on the local �B-field, the O-mode is

preferably used with the refractive index given by NO =√

1 − ω2p/ω

2. The most

basic arrangement is shown in Figure 3.2. The wave is propagating perpendicular to�B with its �E-field parallel to it. According to Equation 2.97, the plasma is transparentif the probing wave frequency ω is higher than the plasma frequency ωp,

ω > ωp =√

nee2

ε0me0(3.1)

Equivalently for a given frequency ω, the electron density must not exceed a criticaldensity, the cutoff density nc, for which the plasma frequency reaches the wavefrequency, ne < nc =ω2(ε0me0/e2). Expressing the ratio ωp/ω by ne/nc, the O-moderefractive index becomes

NO =√

1 − ω2p

ω2=√

1 − ne

nc(3.2)

3.1 Interferometry 67

B

ne = constant

Figure 3.2 The figure gives a poloidal cross section of the plasma with nested circular fluxsurfaces. The probing beam path in a single-chord interferometer arrangement is aligned tocross the plasma axis and it is oriented perpendicular to the magnetic field.

Choosing the probing frequency much higher than the plasma frequency, corre-spondingly the critical density much higher than the electron density, Equation 3.2can be approximated by

NO =√

1 − ne

nc≈ 1 − 1

2

ne

nc= 1 − ω2

p

2ω2(3.3)

With sufficient accuracy, the linear dependence of the O-mode refractive index onthe electron density is obtained if the normalized quantities obey ne/nc ≤ 0.4 andωp/ω ≤ 0.6, respectively.

Probing of the electron density can most easily be conducted in an arrangementas sketched in Figure 3.3. The phase of a wave passing the plasma column iscompared with the phase of a reference wave propagating outside the plasma. Forthe reference wave, the refractive index is equal to the vacuum refractive index,NV = 1. The phase difference in units of 2π is given by the ratio of the optical pathlength difference �Lopt of probing and reference paths and the vacuum wavelengthλ corresponding to the probing frequency ω.

ω

Nv

ne(x )

No(x )

x1 x2

Φ

Figure 3.3 An O-mode wave passing the plasma suffers a phase shift � compared to areference wave in vacuum. The refractive index NO(x) is a local quantity. It is determinedby the local electron density ne(x). The total phase shift accumulation results in the line-integrated electron density.


In the case where the electron density is not constant but varies as ne(x) alongthe beam path through the plasma, the phase is given by

�(x)

2π= �Lopt

λ=

∫ x2

x1

(NV − NO(x′)) dx′

λ≈ 1

2λnc

∫ x

0ne(x′) dx′

= 4.48 × 10−16

(λ

m

)∫ x

0

(ne

(x′)

m−3

)(dx′

m

)(3.4)

The variation ne(x) has been considered smooth along the path in the senseas discussed in the frame of the WKB approximation, Equation 2.140, with thedensity gradient length being much larger than the wavelength of the probingwave. The phase shift in each infinitesimal plasma slab then adds up to the lineintegral. With NO(x) ∝ kO(x), we get � ∝ ∫ kO(x) dx identical to the WKB resultof Equation 2.138. The method described is the oldest microwave diagnostic andthe one most widely used [1–3]. The term interferometry refers to the fact thatin early experiments the phase shift was measured by means of interferometricarrangements, superimposing signal and reference waves, and by evaluating thechanges in the interference pattern when the plasma builds up. Since all phasechanges are proportional to changes in the optical path length, extending from thesignal generator through the plasma to the phase detection system, the waveguiderun must be constructed mechanically extremely stable to avoid other phase changecontributions than those introduced by the plasma. The problem is discussed inmore detail in Chapter 8.

3.1.1Single-Chord Interferometry

Measuring the phase � results in the measurement of the integral quantity∫ x2

x1ne(x′) dx′, called line-integrated density or line density for short. It is one of

the most important quantities in fusion experiments. The information contentis sufficiently high that the plasma particle content in fusion experiments iscontrolled either by keeping the line-integrated density constant in time or byvarying it during the course of the experiment in an appointed way by using theline-integrated density as the actual value within a control loop.

In toroidal fusion experiments, the path through the plasma is typically chosento lie in a poloidal plane at constant toroidal angle with the beam path, includingthe plasma axis (Figure 3.2). The measure assures that all changes of the densityprofile become noticeable in the integral quantity. The typical order of magnitudeof the total phase shift is estimated by assuming an average plasma density alongthe path of 〈ne〉 = 0.4 × 1020 m− 3, a probing beam wavelength of 2 mm, and a pathlength of 1 m, which results in a total phase of �= 36 × 2π. The large phase anglepromises high sensitivity and high-density resolution of the method. However, thelarge phase angle of many multiples of 2π may result in experimental problemswith the unambiguousness of the measurement. The problem is discussed in thecontext of experimental details of the method in Chapter 8.

3.1 Interferometry 69

3.1.2Multiple Chords

A single sightline through the plasma allows for the determination of the line-integrated density. Each slab of the plasma column along the probing beam pathcontributes to the total phase shift. Therefore, no information on the local densitycan be deduced from the integral. This becomes possible in the case where theplasma column is probed simultaneously with a number of chords, that is, with anumber of independent interferometers. Assuming the electron density constanton flux surfaces and assuming further the flux surfaces transformed to nestedcylinders of circular cross section, as given in Figure 1.6 and Figure 3.2, theelectron density is a function of the radial coordinate r alone.

The phase shift �(y) a beam is experiencing when passing along a path parallelto the x-axis in a distance y, as sketched in Figure 3.4, can be calculated from

2λnc�(y)

2π= F(y) =

∫ +∞

−∞ne

√x2 + y2 dx (3.5)

With r2 = x2 + y2 and dx = r dr/√

r2 − y2, the integral becomes

F(y) =∫ +∞

−∞ne(r)dx = 2

∫ +∞

yne(r)dx = 2

∫ a

yne(r)

r dr√r2 − y2

(3.6)

Since the plasma is bounded to the radial range r ≤ a, the upper integration limit+∞ of the integration with respect to r is replaced by the plasma radius a.The local electron density ne(r) can then be gained by Abel integral transform ofEquation 3.6 to give

ne(r) = − 1

π

∫ a

r

dF(y)

dy

dy√r2 − y2

(3.7)

a

r y

y

x

ne(r ) = constant

φ (y )

φi(yi)

B0

Figure 3.4 In case the plasma column is probed simultaneously by a number of beams,the local electron density profile ne(r) can be determined. For that, the variation of thephase shift � with distance y needs to be determined. Typically, 10–20 independent sight-lines, that is, independent interferometers, are used to determine this dependency.


This integral expression allows for the determination of the local density ne(r), thatis, the electron density profile, in the case where the quantity dF(y)/dy is knownwith sufficient accuracy. Since, according to Equation 3.5, F(y) is proportionalto the phase shift �(y), a probing beam at distance y is suffering, its variationwith distance y is determined by applying a number of parallel probing beams,measuring simultaneously �i(yi) at discrete fixed positions yi through the plasma.From the neighboring channels, defining the difference quotient ��i/�yi, thedifferential quotient d�/dy can approximately be determined. Typically, 10–20beams are used for this purpose [3, 4]. In general, for arbitrarily shaped, noncircularnested flux surfaces, the inversion needs to be done numerically [5]. Again theassumption is made that the density is constant on flux surfaces. Their geometryneeds to be known to solve the inversion problem.

3.2Polarimetry

Polarimetry makes use of the birefringence properties of a magnetized plasma. Ashas been derived in the frame of the discussion of the cold-plasma dielectric tensor,the magnetized plasma is circular birefringent, as NL = NR. It exhibits the Faradayeffect in the case where the wave is propagating along the �B-field direction (Equation2.86 and Equation 2.87), as shown in Figure 3.5 as function of the electron density.It is also linear birefringent, as NO = NX, thus exhibiting the Cotton–Moutoneffect, in the case where the wave is propagating perpendicular to the �B-field

00 5×1019 1020 1.5×1020 2×1020

Electron density ne (m−3)

NO − NX

NL − NR

0.05

0.1

0.15

0.2

0.25

Diff

eren

ce o

f ref

ract

ive

inde

x

Figure 3.5 The figure gives the differ-ence of the refractive indices NL − NR andNO − NX of the characteristic waves for par-allel and perpendicular propagations as afunction of the electron density. The dif-ferences are responsible for the Faraday

and the Cotton–Mouton effect. Calcula-tions are made for a probing frequency ofω = 2π × 200 GHz and B0 = 2.5 T. It is obvi-ous that under otherwise similar conditions,the Faraday effect is much stronger than theCotton–Mouton effect.

3.2 Polarimetry 71

(Equation 2.97 and Equation 2.98). Generally, both effects are present in the casewhere the wave is propagating at an arbitrary angle. Faraday and Cotton–Moutoneffects result in changes of the polarization state of the propagating wave. Thechanges can be used to gain information on the quantities determining the sizeof the birefringence, that is, the electron density ne and the magnetic field �B.After an elementary introduction of the Faraday and Cotton–Mouton effects inthe following two paragraphs, assuming the �B-field either purely parallel or purelyperpendicular, a more generalized description is given subsequently.

3.2.1Faraday Effect

A linear polarized wave of frequency ω is considered, propagating throughthe plasma along the z-direction, parallel to �B, entering the plasma at z = 0(Figure 3.6). The linear polarized wave polarized along the x-direction can bethought of being composed of a left-hand (L) and a right-hand (R) circular polar-ized wave of identical �E-field amplitudes E0/2. If the two wave components arepropagating through the plasma along �B in an interferometer-like arrangement,each of the two components suffers a phase shift �φL,R compared to a referencewave propagating outside the plasma.

According to Equation 3.4, the phase difference of each of the partial waves isgiven by �φL,R = (ω/c)

∫ z0 (NL,R − 1)dz′. Since the refractive index NL differs from

the index NR, the phase shifts that the two-phase components are experiencingwhen passing the same plasma slab differ from each other by ��F =�φL −�φR,

��F(z) = ω

c

∫ z

0(NL − 1) − (NR − 1)dz′ =ω

c

∫ z

0(NL − NR)dz′ =

∫ z

0(kL − kR)dz′

(3.8)

x

y

Plasma

B

αz

= +

Figure 3.6 A linear polarized wave propa-gating into z-direction parallel to the mag-netic field �B of a magnetized plasma canbe split into a right- and a left-hand circularpolarized component. The two componentsare experiencing different refractive indices

when passing the plasma, causing a phasedifference. Recombination of the two par-tial waves results again in a linear polarizedwave, but its plane of polarization is rotatedby the angle α. The phenomenon is calledFaraday effect.


When combining the two circular polarized wave components after the plasmapassage, the phase difference causes a rotation of the plane of polarization by anangle α compared to the plane of polarization of the wave when being launched.The phenomenon is called Faraday effect. To determine the angle α, the linearpolarized wave �E(z, t) is split again into its L- and R-partial wave components,�E(z, t) = �EL + �ER. Each of the circular waves can themselves be expressed by thesuperposition of two orthogonal linear polarized waves with π/2 phase shift. Inaccordance with Equation 2.89 and Equation 2.90, the two characteristic circularwaves are in complex notation with unit vectors x and y:

�EL = E0

2(x + iy)ei(kLz−ωt)

�ER = E0

2(x − iy)ei(kRz−ωt) (3.9)

At the entrance to the plasma at z = 0, we assume that the wave is linear polarizedalong the x-direction, Re{�EL + �ER} = xE0 cos ωt. After a distance z, we have fromEquation 3.9

Re{�EL + �ER} = E0

[(x cos

kL − kR

2z + y sin

kL − kR

2z

)cos

kL + kR

2z

]cos ωt

(3.10)

The angle α is determined from the ratio of the y- and the x-components:

tan α = sin((kL − kR)/2)z

cos((kL − kR)/2)z; α = kL − kR

2z (3.11)

Generalizing this result to the case of smoothly varying kL,R and NL,R, respectively,along z, one obtains the integral for α:

α(z) = 1

2��F(z) = 1

2

∫ z

0(kL(z′) − kR(z′))dz′ = 1

2

ω

c

∫ z

0(NL(z′) − NR(z′))dz′

(3.12)

As with the phase in a single-sightline interferometer, the Faraday rotation angleα is represented by a line integral along the line of sight of the probing linearpolarized wave. Instead of a single one (in the case of the interferometer), now tworefractive indices are involved, whose line-integrated difference determines α. Noexternal reference path is needed as the reference cancels in Equation 3.8. The twocircular polarized components of the linear polarized probing wave are acting asmutual references.

With the cold-plasma expressions for NL, NR of Equation 2.86 and Equation 2.87substituted, the Faraday rotation angle becomes

α = 1

2

ω

c

∫ z2

z1

⎛⎝√

1 −(

ωp

ω

)2ω

ω + ωc−√

1 −(

ωp

ω

)2ω

ω − ωc

⎞⎠dz (3.13)

To make the meaning of the integrand in Equation 3.13 more clear, the probingwave frequency ω is considered large compared to both the plasma frequency ωp

3.2 Polarimetry 73

zBp

x

r

Θ

Probingbeam

BII

⊗Bφ

Figure 3.7 The figure shows a poloidalcross section through a Tokamak plasmawith the toroidal field Bφ perpendicular theplane. The plasma current is directed out ofthe plane causing a poloidal field Bp. Thepolarization plane of a linear polarized wave

propagating into the z-direction is rotatedbecause of the Faraday effect caused by theparallel component of the poloidal field. Theparallel component depends on the positionr. It is zero for a probing beam passing theplasma center.

and the cyclotron frequency ωc, resulting in NL − NR ≈ ω2pωc/ω

3. Substituting ωp

and ωc by the electron density and the magnetic field, the latter explicitly expressedas parallel, B||, and considering as well their dependence on z, one gets for therotation angle:

α(z) = 1

2

ω

c

∫ z

0(NL(z′) − NR(z′))dz′ ≈ e3

2cε0m2e0

1

ω2

∫ z

0ne(z′)B||(z

′)dz′ (3.14)

For practical purposes with λ = 2π(c/ω), the differential rotation angle dα is givenby

dα ≈ 2.62 × 10−13

(λ

m

)2 ( ne

m−3

)(B||T

)(dz

m

)(3.15)

The angle of rotation of the polarization plane of the linear polarized wave passingthe plasma depends on the convolution of the electron density and the �B-fieldcomponent parallel to the path. In the case where one quantity is known, theline-integrated value of the other can be determined. Since the experimentalarrangement of the polarimeter is basically identical to that of an interferometer,the phase change of the linear polarized wave when passing the plasma can be usedto determine the line-integrated density as with an interferometer; the rotationof the plane of polarization can simultaneously be used to determine the line-integrated convolution of electron density with the parallel �B-field component [6].Similarly to multichannel interferometers, multichord polarimeter arrangementsallow for the determination of the local quantities ne(r) and B||(r). The method hasgained importance as a multichannel setup with a number of vertical sightlineswithin one poloidal plane of a tokamak experiment allows for the measurement ofthe local poloidal component of the total magnetic field and thus for the evaluationof the local plasma current density [7]. Figure 3.7 shows the geometry for a singlesightline.


Assuming for simplicity the main toroidal field Bφ(r) perpendicular to theprobing beam direction, the parallel �B-field component B|| determining the angleof the Faraday rotation is determined by the poloidal field Bp(r) caused by thetoroidal plasma current density j(r) of the tokamak, B||(x,r) = Bp(r)x/r. The poloidalfield Bp(r) is solely determined by the current density of the toroidal plasma current,Bp(r) = 2πμ0(1/r)

∫ r0 j(r′)r′ dr′. In the geometry of Figure 3.7 with the direction of

Bp(r) varying with the poloidal angle θ , sightlines at positive x result in rotationangles with the opposite sign compared to sightlines at negative x. The sightlinethrough the plasma center, crossing the plasma axis, experiences no Faradayrotation, as, along this sightline, the poloidal field is always perpendicular to thepropagation direction.

The Faraday rotation angle at distance x from the axis is

α(x) = const.∫ a

rmin

ne(r)dBp

dr

x√r2 − x2

dr (3.16)

The equation can be Abel inverted to give dBp/dr, from which the local currentdensity can be determined.

It should be noted that the measurement of the plasma current as described isperturbed by the linear birefringence of the plasma arising from the strong toroidalfield of the tokamak perpendicular to the probing beam (Section 3.2.2). Thus, thepolarization of the probing wave becomes also elliptical and the change of polariza-tion is no longer due to a pure Faraday effect. Owing to their different wavelengthdependencies, the perturbing Cotton–Mouton effect can be made negligibly small.It has therefore not been considered in the earlier-given introductory treatment.

In principle, the plasma current distribution across the plasma radius can becalculated on the basis of measured electron density and temperature profilesthat determine the plasma conductivity. The experimental verification, however,is possible only by measuring the Faraday rotation in a multibeam arrangement,as described. Standard magnetic diagnostics that are used to measure the plasmacurrent (Rogowski coil) determine the total net current and not the local currentdistribution.

The method has another application in the case where the parallel �B-field alongthe sightline is known, as it is the case for a sightline in the midplane of a tokamak,tangentially crossing the plasma. In the midplane, the �B-field has no componentscaused by the plasma current and is fully determined by only the toroidal mainfield. Thus, with the known parallel �B-field along the sightline, the line-integrateddensity can be calculated from the Faraday rotation angle [8]. The use of the rotationangle provided by a polarimeter instead of the phase angle from an interferometerresults in a measurement of the line-integrated electron density that is robustagainst a number of experimental problems discussed in Chapter 7 and Chapter 8.The most important advantage relies on the fact that the measured rotation angledoes not change with changes in length of the optical path outside the plasma,which is a critical issue in interferometry.

Faraday polarimetry in tokamaks has been treated in a simplified way byneglecting completely the perpendicular field components, which, however, are

3.2 Polarimetry 75

present as well. They are caused by both the toroidal field and the poloidal field.While the toroidal component is constant along the sightline in the geometry ofFigure 3.7, the perpendicular component of the poloidal field varies along thesightline. It is purely parallel only at z = 0. The perpendicular components arecausing changes in the ellipticity of the wave, which is treated in the followingsection.

3.2.2Cotton–Mouton Effect

The Faraday effect polarimeter makes use of the circular birefringence of themagnetized plasma, probing simultaneously with an L- and an R-wave. Equiva-lently, the linear birefringence can be used by probing the magnetized plasmasimultaneously with an O- and an X-wave, the characteristic waves for propagationperpendicular to the �B-field. The arrangement is analogously called Cotton–Moutonpolarimeter.

We refer to Figure 3.8 and consider a linear polarized wave, launched into positivez-direction, perpendicular to �B, which we now assume oriented in y-direction, withthe wave’s plane of polarization oriented under an angle of π/4 to the �B-field. Underthese conditions, plasma probing is accomplished simultaneously with an O-modewave with its �E-field parallel �B, �EO(z, t) = yEy0 cos(kOz − ωt), and an X-mode wave,�EX(z, t) = xEx0 cos(kXz − ωt), with the �E-field perpendicular to �B. Identical phasesare assumed when the two components are entering the plasma at z = 0.

Since NO and NX are different, a phase difference ��CM between the two linearpolarized components of the probing wave evolves when passing the plasma fromz = 0 to z along the z-axis. Equivalently to Equation 3.8, the phase difference is

x

B

y

z

X

ΔΦ

= +Plasma

Figure 3.8 An elliptically polarized wavepropagating into z-direction, perpendicu-lar to the magnetic field �B of a magnetizedplasma, can be split into its O - and X -mode components, that is, its y- and itsx-component in the geometry of the figure.Owing to the different refractive indices for

O - and X -waves, the two partial waves un-dergo different phase shifts when passingthe plasma. Combining the waves againafter the plasma transit results in a wavewhose ellipticity has changed during theplasma passage. The phenomenon is calledCotton–Mouton effect.


given by

��CM(z) = �φO − �φX = ω

c

∫ z

0(NO − 1) − (NX − 1)dz′

= ω

c

∫ z

0(NO − NX)dz′ =

∫ z

0(kO − kX)dz′

NO − NX =√

1 −(

ωp

ω

)2

−√√√√1 −

(ωp

ω

)2 ω2 − ω2p

ω2 − ω2p − ω2

c(3.17)

With the difference of the refractive indices evaluated as before for the probingfrequency ω large compared to both the plasma and cyclotron frequencies, ωp

and ωc, NO − NX ≈ ω2pω

2c/ω

4 results. Substituting the plasma and the cyclotronfrequencies by the electron density and the perpendicular magnetic field B⊥, andassuming their variation smooth along y, one obtains the integral for the phasedifference:

��CM(z) = ω

c

∫ z

0(NO(z′) − NX(z′))dz′ ≈ e4

cε0m3e0

1

ω3

∫ z

0ne(z′)B2

⊥(z′)dz′

(3.18)

With λ = 2π(c/ω), an approximate differential expression for the phase differencecan be given:

d��CM ≈ 4.89 × 10−11

(λ

m

)3 ( ne

m−3

)(B⊥T

)2

dz (3.19)

As with the Faraday polarimeter, the phase difference in the Cotton–Moutonpolarimeter is given by the convolution of the electron density and the �B-field.However, in the Cotton–Mouton polarimeter, the �B-field squared enters while itenters linearly in the Faraday polarimeter. Also, the dependence on the probingfrequency is stronger: it is proportional to ω− 3 in the Cotton–Mouton effect, butproportional to ω− 2 in the Faraday effect.

The linear superposition �EO + �EX of the two linear polarized wave componentswith their planes of polarization perpendicular to each other and with a finite phasedifference ��CM between the two results generally in an elliptically polarized wave,which can be shown as follows: with EX = Ex0cos(ωt +��CM) for the X-mode andEO = Ey0cos ωt for the O-mode, the sum in the argument of the cos-function canbe expanded to give EX = Ex0(cos ωt · cos ��CM − sin ωt · sin ��CM). Substitutingthe term cos ωt = EO/Ey0 from the O-mode component, we get(

EO

Ey0

)2

+(

EX

Ex0

)2

− 2

(EO

Ey0

)(EX

Ex0

)cos ��CM = sin2 ��CM (3.20)

The equation describes an ellipse with an angle to the x-axis (Figure 3.9), withthe angle given by

tan 2 = 2Ex0Ey0

E2x0 − E2

y0

cos ��CM (3.21)

3.2 Polarimetry 77

Ey

Ey0

Ex0 Ex

ba

Ψ

Figure 3.9 The ellipticity of an elliptically polarized wave is described by the semimajorand the semiminor axes a and b of the ellipse and the angle , the major axis is formingwith the x-axis of the coordinate system.

Aligning the ellipse with one coordinate axis, thus = 0, equivalently ��CM = π/2,transforms Equation 3.20 into the familiar equation of an ellipse. If, in addition, fieldamplitudes fulfill, Ex0 = Ey0 = E0, the equation of a circle follows, E2

X + E2O = E2

0.In the case where ��CM = 0, one gets linear polarization with =π/4, as anotherspecial case of the general form of Equation 3.20.

Summarizing, the plasma-induced phase difference ��CM in the Cotton–Mouton polarimeter changes the ellipticity of the wave when passing the plasma.The change is carrying information on the line-integrated product of density andthe square of the �B-field component perpendicular to the direction of the wave path.In Faraday polarimetry, the weaker Cotton–Mouton effect appears as perturbingside effect. However, under certain conditions, it can also be used to provide arobust line-integrated density measurement, as is discussed in Chapter 8.

3.2.3Common Generalized Description

The method applied to describe the Faraday and Cotton–Mouton effects in anelementary way, as exemplified before, was based on the decomposition of thewave launched to the plasma into its characteristic waves. Each of the two wavespropagates with its characteristic phase velocity c/N1,2, suffering neither absorptionnor refraction and in particular, it propagates without change in polarization. Thepolarization after recombining the characteristic components is determined bythe phase difference �� = (ω/c)l(N1 − N2) as developed along the path length l.To generalize the description, we follow the literature [9–11] by consideringa plane electromagnetic wave, propagating into z-direction. The �B-field is notany longer aligned with the z- or the y-direction, but now has both paralleland perpendicular components. The probing wave might be composed of twoorthogonal components, Ex(z,t) = Ex0cos(φx − ωt) and Ey(z,t) = Ey0cos(φy −ωt),respectively. The polarization can then be characterized by giving the curve theresulting �E-field is performing in the xy-plane of a Cartesian coordinate system.This curve is, in general, elliptical, as shown in Figure 3.9. Thus, the state of


polarization is characterized by the ratio of semiminor and semimajor axis b/a ofthe ellipse, the angle , the semimajor axis is forming with the x-axis, and thedirection of rotation of the �E-field vector. The parameters defining the ellipse areuniquely determined by the normalized components of the Stokes vector �s(s1, s2, s3)usually used to describe most descriptive and experimentally relevant the state ofpolarization.

The components of the Stokes vector expressed by the orthogonal �E-field com-ponents in their original normalized definition and expressed as well by the ellipseparameters are as follows [9]:

s1 = E2x0 − E2

y0

E2x0 + E2

y0

= cos 2χ cos 2 (3.22)

s2 = 2Ex0Ey0

E2x0 + E2

y0

cos(φy − φx) = cos 2χ sin 2 (3.23)

s3 = 2Ex0Ey0

E2x0 + E2

y0

sin (φy − φx) = sin 2χ (3.24)

The angle χ is defined by the ratio of the two axes of the ellipse, tan χ =± b/a.The angle χ is positive or negative for directions of rotation of �E clockwiseand anticlockwise, respectively (−π/4 ≤ χ ≤ π/4). In the case of horizontal (+sign) and vertical (− sign) linear polarization, the Stokes vector components ares1 = ± 1, s2 = s3 = 0, while in the case of right- (+ sign) and left-hand (− sign)circular polarization, one has s3 = ± 1, s1 = s2 = 0.

The Stokes vector is defined for TEM waves. Its components are, therefore, ex-clusively expressed by the transversal components of the wave under consideration.However, as has been discussed in Section 2.6.3, the X-mode has a finite longi-tudinal �E-field component El, whose relative size compared to the perpendicularcomponent Ep is given by∣∣∣∣∣ El

Ep

∣∣∣∣∣ =∣∣∣∣DS∣∣∣∣ = XY

(1 − X − Y2)(3.25)

The ratio decreases with increasing probing frequency. With typical values forthe plasma and cyclotron frequencies of 100 and 70 GHz, respectively, the ratiois 0.1 at 200 GHz probing frequency, 0.01 at about 400 GHz, and smaller than0.001 at 1000 GHz. In the approximation with the probing frequency being largecompared to both the plasma and the cyclotron frequency, the X-mode can beassumed not significantly deviating from a TEM wave. Thus, the wave polarizationcan sufficiently well be described using the Stokes vector.

The Stokes vector components sk = s∗k/s∗0, as given in Equations 3.22–3.24, arenormalized quantities, normalized to the total intensity or power, proportional tos∗0 = E2

x0 + E2y0. The definitions are chosen such that the individual components can

experimentally be determined with a power detector and maximal two additionalpolarization-analyzing components in a direct way: the quantity s∗0 = E2

x0 + E2y0,

proportional to the total power, is measured with the detector without any additional

3.2 Polarimetry 79

analyzer. Assuming the constant of proportionality unity, the component s∗1 =E2

x0 − E2y0 is measured with the same detector but with a linear analyzer parallel to

the x-direction (Section 7.3.5). Component s∗2 = 2Ex0Ey0 cos(φy − φx) is measuredin the same way, but with the linear analyzer under 45◦ with respect to thex- and y-directions, respectively, while component s∗3 = 2Ex0Ey0 sin(φy − φx) isdetermined after the insertion of a λ/4-plate using the terminology of optics, whichshifts the phase of the y-component of the �E-field by π/2 with respect to thex-component, again with the linear analyzer under 45◦. The measured intensity isIk = (1/2)(s∗k + s∗0), k = 1, 2, 3. For nonpolarized waves, we have s∗1 = s∗2 = s∗3 = 0.

The Stokes vector is a unit vector, with its tip lying on the surface of a unitsphere (Poincare sphere). Each state of polarization is uniquely represented by apoint P on the surface, whose longitude and latitude correspond to 2 and 2χ ,respectively, as given in Figure 3.10.

As mentioned, the polarization states of the two characteristic waves for eachpropagation direction we are considering do not change when passing the plasma.They are orthogonal and lie on opposite sides of the unit sphere. A wave composedof a linear superposition of characteristic waves changes its polarization state whendeveloping the phase difference along the path.

Geometrically this corresponds to a rigid rotation of the unit sphere around thepolarization direction �sc1 of one of the characteristic waves by the angle γ . Weassume the characteristic wave with index 1 to be the slow wave, the one with thelower phase velocity, that is, the one with the larger refractive index N. This angleis identical to the phase difference discussed before, γ =�� = (ω/c)(N1 − N2)z.

To consider the changing birefringence properties of the plasma along the prop-agation path z, infinitesimal thin slabs need to be considered and the preconditionsof the WKB approximation must be fulfilled, meaning that phase changes need tobe small over the distance of a wavelength. The evolution of the polarization state

S1

S3

S2

S

2Ψ

2χ

Figure 3.10 Any polarization state of a wave can be described by the Stokes vector �s. Itis a unit vector with the components s1, s2, and s3, defining one point on the Poincaresphere, with unit radius. The point on the sphere can uniquely be defined as well by theangles 2 and 2χ . Only the upper hemisphere of the Poincare sphere is shown.


�s(z) can then generally be described by the differential equation

d�s (z)

dz= � (z) ×�s (z) (3.26)

with the absolute value of the vector � (z) containing the dielectric properties ineach slab along the propagation path z and with the direction of the polarization�sc1 of the slow characteristic wave corresponding to N1:

� (z) = ω

c[N1(z) − N2(z)] �sc1 (3.27)

If �s0 is the initial polarization, the solution of Equation 3.26 is

�s (z) = �s0 −�s0 ×∫ z

0

� (z′)dz′ (3.28)

With the three components of the vectors involved, �s0 ≡ (s10, s20, s30) and� ≡ ( 1, 2, 3), the components of the final polarization vector are explicitlygiven by

�s (z) =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

s10 − s30

∫ z

0 2

(z′)dz′ + s20

∫ z

0 3(z′)dz′

s20 − s10

∫ z

0 3(z′)dz′ + s30

∫ z

0 1(z′)dz′

s30 − s20

∫ z

0 1(z′)dz′ + s10

∫ z

0 2(z′)dz′

s

⎞⎟⎟⎟⎟⎟⎟⎟⎠

(3.29)

For the special cases of propagation purely parallel and purely perpendicular to�B, the characteristic waves have been used in Section 3.2.1 and Section 3.2.2to calculate separately in an elementary way the polarization changes due tothe Faraday and Cotton–Mouton effects. For the general case with the �B-fieldwith components (Bx,By,Bz) at an angle � to the propagation direction, thusthe combined effects acting on the wave when passing the plasma can now betreated [9]. For that, according to Equation 3.18 and Equation 3.27, the absolutevalue | � | and the direction of � (z) need to be calculated. To determine | � |, thedifference N1(z) − N2(z) has to be determined. The direction of � is identical tothe polarization �sc1 of the slow characteristic wave. The refractive indices of thecharacteristic waves N1 and N2 are given in Equation 3.21, identical to Equation2.62, Equation 2.99, and Equation 2.100. Using the notations of Equation 2.54, thecold-plasma refractive indices for arbitrary angle � are given by

N21,2 = 1 − X

1 − (Y2 sin2 �/2(1 − X)) ±√

Y4 sin4 �/4(1 − X)2 + Y2 cos2 �

(3.30)

3.2 Polarimetry 81

Identifying N1 in this equation with the plus sign in the denominator and N2 withthe minus sign, the difference, defining the size of | � |, is given by

N1 − N2 = N21 − N2

2

N1 + N2

= 1

N1 + N2

X√

Y4 sin4� + 4(1 − X)2Y2 cos2�

(1 − X)(1 − Y2 cos2�) − Y2 sin2�(3.31)

Next the components of the polarization state �sc1 as defined through the angles χ1

and 1 need to be calculated. The angle χ1 is defined by the parameters a and bof the polarization ellipse, tan χ1 = b/a. As discussed before, this ratio is given bythe absolute value of the ratio |Ex/Ey| of the orthogonal field components of theprobing wave.

From Equation 2.59, Equation 2.60, and Equation 2.61 with the notations as usedbefore (Equation 2.54), the ratio for the slow wave is given by∣∣∣∣∣Ex1

Ey1

∣∣∣∣∣ = 1

Y cos �

⎡⎣√

Y4 sin4 �

4(1 − X)2 + Y2 cos2 � − Y2 sin2 �

2(1 − X)

⎤⎦ (3.32)

The tilt angle 1 is determined by the perpendicular �B-field components Bx andBy, so that tan 1 = By/Bx (Figure 3.11). Expressing the trigonometric functions in

Equation 3.22, Equation 3.23, and Equation 3.24 by the �B-field components Bx andBy,

sin2 � = B2x + B2

y

B2, cos � = Bz

B, B = me0

eωc (3.33)

Ey

Ex

x

z

y

B⊥

B⊥

Bz

B

Θ

Ψ1

Ψ1

Figure 3.11 The geometry as used to generally describe the polarization changes. Thewave launched into the homogeneous plasma is propagating into z-direction. The �B-field lies in the xz-plane. It forms an angle � with the direction at which the wave ispropagating.


and making use of the identities, cos 2α = (1 − tan2 α)/(1 + tan2 α), sin 2α =2tan α/(1 + tan2 α), one finally gets for the Stokes vector of the slow characteristicwave

�sc1 =

⎛⎜⎜⎝

cos 2χ1 cos 21

cos 2χ1 sin 21

sin 2χ1

⎞⎟⎟⎠

= Y2sin2 �√4Y2(1 − X)2cos2 � + Y4sin4 �

⎛⎜⎜⎜⎜⎝

B2x−B2

y

B2x+B2

y

2BxBy

B2x+B2

y

2ω me0e (1 − X) Bz

B2x+B2

y

⎞⎟⎟⎟⎟⎠ (3.34)

Combining this result with Equation 3.21 and substituting parameters X , Y by thephysical quantities they are representing, the final result for the vector � is

� = 1

N1 + N2

ω2p

cω3

1

F

⎛⎜⎜⎜⎜⎜⎝

(e

me0

)2 B2x−B2

y

1−(ωp/ω)2(e

me0

)2 2BxBy

1−(ωp/ω)2

2ω(

eme0

)Bz

⎞⎟⎟⎟⎟⎟⎠

F = 1 − 1

ω2

(e

me0

)2

⎛⎜⎝ B2

x + B2y

1 −(ωp/ω

)2 + B2z

⎞⎟⎠ (3.35)

The components 1 and 2 of the vector � are describing the Cotton–Moutoneffect, while component 3 is responsible for the Faraday effect. Both effectsoccur simultaneously under the conditions of an oblique �B-field as considered.Component 2 differs from 0 only for the case of the �B-field neither aligned to thex- nor aligned to the y-direction. For clarity, the z-dependence has been suppressedin this equation. However, generally, the �B-field components as well as the plasmafrequency are functions of z. In the approximation as used before, with the probingfrequency much larger than the plasma and the cyclotron frequency, the functionF as well as N1,2 are approximated by 1, F ≈ N1,2 ≈ 1, and Equation 3.27 reduces tothe form as often used in the literature:

� = ω2p

2cω3

⎛⎜⎜⎜⎜⎜⎝

(e

me0

)2(B2

x − B2y )

(e

me0

)22BxBy

2ω(

eme0

)Bz

⎞⎟⎟⎟⎟⎟⎠ (3.36)

3.3 Reflectometry 83

The approximated vector components 1 and 3 correspond to the expressionsused in Equation 3.18 and Equation 3.14 to calculate the Cotton–Mouton phasedifference and the Faraday angle.

3.3Reflectometry

The active diagnostic methods, interferometry and polarimetry, discussed inSection 3.1 and Section 3.2, are basically operated under conditions with theprobing wave unaffected in amplitude and propagation direction when passing theplasma. This is possible as the probing wave frequency is chosen large comparedto the plasma and the cyclotron frequencies. Operation conditions are thereforeclearly distinct from any cutoff and away from resonances, resulting in wave re-flection and wave absorption, respectively. Refraction can be made negligibly smallby choosing the sightlines parallel to the density gradient. The information ofinterest on electron density and �B-field is carried by the accumulated phase change� = ∫ r2

r1k(r)dr along the path r2 − r1 through the plasma column. Owing to the

line integration of the local wave–plasma interaction in successive plasma slabs,multisightline arrangements and mathematical inversion procedures are needed togain local information on the quantities of interest. However, in the case where theprobing wave frequency ω is chosen identical to the cutoff frequency, ω =ωco(rc),at some position rc in the plasma, the wave index of refraction approaches zeroat this position, N(rc) → 0, and the wave is reflected back. The position rc canbe determined by applying a kind of RADAR technique called reflectometry, atechnique originally developed to probe the height and the electron density in theEarth’s ionosphere [12]. With a single probing sightline, reflectometry involvesgaining local information on the position of the reflecting layer by calculating theround trip time delay td from the measured phase delay � the incident wave isundergoing on its way from the plasma boundary to the cutoff position and back.The local plasma parameters determining the cutoff frequency ωco at position rc

can then be derived. The time delay is calculated from the phase change withfrequency

td = ∂�

∂ω

∣∣∣∣ω=ωco

(3.37)

with proper variation of the frequency around the cutoff frequency ωco.In a typical reflectometry geometry, as illustrated in Figure 3.12, with the wave

launching position on the outer, low-field side of the torus, in known distanced to the plasma edge at r = a, the cutoff position inside the plasma is calculatedfrom the measured time delay td as rc = d + a − (1/2)td c, with c being the speed oflight. By varying the probing frequency ω, the whole density profile can basicallybe scanned.

So far, the wave mode best suited for this kind of measurement has not beenspecified. In the geometry of Figure 3.12, the probing wave is propagating in the


Torusaxis

R0rc r = a

d

ω = ωco

ne0

ne

Figure 3.12 A wave with frequency ω

launched from a position in distanced to the plasma edge is propagatinginto the plasma, reaching cutoff at po-sition r = rc and is being reflected back.From a measurement of the round triptime delay, the wave takes from the

launch position back to the detector, thecutoff position can be calculated. The cut-off frequency ωco is determined by the localelectron density in the plasma, which there-fore can be derived. By varying the prob-ing frequency, the density profile can bescanned.

equatorial plane perpendicular to the main toroidal field. Both the linear polarizedX- and O-modes as the characteristic modes in this geometry can therefore beenvisaged.

According to Equation 2.97 and Equation 2.98, the O-mode has the cutoff at ω(O)co =

ωp, while X-mode cutoffs appear at the lower L-frequency, ω(X)co = ωL, and the higher

R-frequency, ω(X)co = ωR with ωR =

√ω2

c/4 + ω2p + ωc/2 and ωL =

√ω2

c/4 + ω2p −

ωc/2, resulting in ωR >ωp >ωL and ωR >ωc. In short, the cutoff conditions forthese three cases can be expressed by ω2

p/ω2 + Cco(ωc/ω) = 1, with Cco = 0 for

the O-mode and Cco =± 1 for the R- and L-wave cutoffs, respectively. Since theR-frequency lies below the plasma frequency, it cannot be used for reflectometryprobing. The O-mode cutoff frequency depends solely on the plasma frequency,that is, the electron density, ω2

p ∝ ne(r), while the X-mode cutoff frequency depends,

in addition, on the cyclotron frequency, that is, the �B-field, ωc ∝ B(r).The two cutoff frequencies reach their maximum value inside the plasma, which

implies that probing of the whole density profile from only one side of the torus isnot possible (Figure 3.13).

To probe the whole profile, two reflectometry setups are needed, one probingfrom the inner and the other from the outer side of the torus. The necessarycondition for the application of the method is that the cutoff frequency is increasingwith distance to the launch position of the probing wave. Considering O-modeprobing and a central electron density of ne0 = 1020 m− 3, as given in the examplesof Figure 3.13, the cutoff frequency ω

(O)co varies from the edge to the plasma center

from ω(O)co = 0 at r = a to almost ω

(O)co = 90GHz at the plasma axis at r = 0.

To determine the whole density profile, successive variation of the probingfrequency is necessary to determine the cutoff positions from individual roundtrip time delays for a wide range of frequencies corresponding to a wide range ofdifferent electron density samples on the density profile. Since it is not possibleto vary the probing frequency over such a wide range, only parts of the profile


0−1

(a)

(b)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Normalized plasma radius

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1


2E+19

4E+19

6E+19

8E+19

1E+20

0

30

60

90

120

150

Ele

ctro

n de

nsity

ne

(m−3

)

Cut

off f

requ

ency

(G

Hz)

Cut

off f

requ

ency

(G

Hz)

0

2E+19

4E+19

6E+19

8E+19

1.2E+20

1E+20

0

25

50

75

100

150

125

Ele

ctro

n de

nsity

ne

(m−3

)

ne

ne

fR

fR

fOfc

fc

fL

Figure 3.13 The cutoff frequencies f O ofthe O-mode and f L, f R of the X-mode alonga sightline in the equatorial plane of a fusionplasma. A peaked profile of the electron den-sity has been assumed and a tokamak-like�B-field variation (a). Cutoff frequencies f O,

f L approach zero at the plasma edge. Toavoid this, X-mode probing with f R > f c canpreferably be used to probe the profile. Itsuse enables probing of even flat profiles, asshown in (b).


are accessible with O-mode reflectometry. Generally, technical reasons restrict themaximum frequency interval covered with a single reflectometer to one octave(Chapter 6 and Chapter 7), with the lowest frequency applicable at all of about20 GHz, demanding for more than one reflectometry setup to probe even from oneside of the torus.

One way out is the use of the X-mode for probing with ω(X)co = ωR. The lowest

value of ωR appears at the plasma edge, at vanishing electron density, approachingthe electron cyclotron frequency, ωR → ωc(r = a). As a result, the frequency intervalnecessary to probe the full accessible profile regime is much smaller than in the O-mode case. Assuming, for example, the �B-field dependence along the normalizedplasma radius r(n) = r/a, − 1 ≤ r(n) ≤ 1 tokamak-like as B = B0A/(A + r(n)), andassuming the value on axis B0 = 2.5 T and the aspect ratio A = R0/a = 5, the lowerlimit of the cutoff frequency ωR at the plasma outer edge is about 60 GHz. Thefrequency interval for X-mode probing then ranges from 60 to about 130 GHz. SinceωR > ωc is always valid, the probing signal inside the plasma cannot be absorbedat resonances occurring at ω = nωc, n = 1, 2, 3, . . . , which is not generally true forO-mode probing, as demonstrated in the example of Figure 3.13. However, underconditions of higher �B-field or lower density, the relation ωp < ωc becomes thenpossible everywhere, avoiding resonance absorption.

The examples in Figure 3.13 demonstrate another distinctive feature of X-modereflectometry: the possibility to probe the density beyond the density maximum oralong a density plateau. This feature is due to the fact that ωR is still increasingwith increasing �B-field along the probing beam path despite the electron density isconstant or even decreasing. This is of course only true for the geometry, as givenin Figure 3.12, probing from the low-field side. High-field side probing under theseconditions reduces the accessible profile range. Summing up, the radial rangethat can be scanned by reflectometry corresponds approximately to the gradientregion of the density profile, more accurate to that radial range in which the cutofffrequency is monotonically increasing with distance to the launch position of theprobing wave.

3.3.1Time Delay Measurement

Reflectometry allows for the determination of the electron density profile ne(r)by varying the probing frequency ω and measuring the corresponding time delaytd(ω) of the wave reflected back from the cutoff position rc to the detector. Inthe case where a continuous wave is used, the time delay is derived, according toEquation 3.37, from the phase change ∂ � when varying the wave frequency by∂ω around the cutoff frequency ωco. Since time and frequency are complementaryquantities, a frequency change �ω results in a time uncertainty of �t ≥ 2π/�ω,which translates to an uncertainty �d in the measurement of a distance d toa plane mirror of �d = (1/2)(�tc) ≥ πc/�ω. The expression defines the ultimateaccuracy; a distance measurement can be performed in this way, thus determiningthe ultimate resolution of the method. Usually, the frequency is varied at a constant


rate ∂ω/∂ t, and the delay time td is calculated from the resulting measured phasechange in time:

∂�

∂t= td

∂ω

∂t(3.38)

So far, the reflectometry method was described as a method measuring the distanceto a mirror-like reflection plane. The medium plasma was not considered at all.With plasma, the wave undergoes plasma-parameter-dependent phase changes onits way to the reflecting cutoff layer and back, which need to be considered. Inthe case where the conditions of the WKB approximation are fulfilled, that is,plasma parameter changes are small over a local wavelength, the phase variesas in an interferometer along the path from the plasma edge to some position rinside the plasma as �(r) = ∫ r

a k(r′)dr′. When probing with O-mode, the conditioncannot be fulfilled at the very plasma edge. The probing wavelength is increasingwhen approaching the edge, however, the density gradient length Ln is decreasing;thus, the ratio Ln/λ tends to zero. Figure 3.14 gives an example showing that thevery plasma edge demanding for probing frequencies below about 20–30 GHzis not treatable in this way, The more important question is how to treat theplasma layer close to the reflection position, as the validity condition of WKB,(1/k2)dk(r)/dr = W � 1, cannot be fulfilled when approaching the cutoff layer. Atcutoff k2 goes through zero and the small quantity W grows beyond all limits; thusWKB breaks down (Figure 3.15). The phase change along the envisaged path to thecutoff layer and back, however, needs to be considered, including that part whereWKB fails.

00 0.20.1 0.40.3 0.60.5 0.80.7 10.9


2E+19

4E+19

6E+19

8E+19

1E+20

0

10

20

30

40

50

Ele

ctro

n de

nsity

ne

(m−3

)

L/λ

ne

L /λ

ωp ≈ 30 GHz

Figure 3.14 The figures show for a given density profile how the ratio on gradient lengthLn and O-mode cutoff wavelength λ are varying with the local coordinate. The validity ofWKB approximation demands for Ln/λ � 1, which cannot be fulfilled at the very plasmaedge.


(a)

(b)

0 (r /a)c0.2 0.4 0.6 0.8 1

Normalized plasma radius r /a

0 0.2 0.4 0.6 0.8 1

Normalized plasma radius r /a

0

500.0E+3

1.0E+6

1.5E+6

2.0E+6

2.5E+6

3.0E+6

Squ

are

of p

ropa

gatio

n co

nsta

nt k

2

W

0

2E+19

4E+19

6E+19

8E+19

1E+20

0

0.02

0.04

0.06

0.08

WKB not valid W = (1/k2)dk /dr

0.1

Ele

ctro

n de

nsity

ne

(m−3

)

0

2E+19

4E+19

6E+19

8E+19

1.2E+20

1E+20

Ele

ctro

n de

nsity

ne

(m−3

)

ne

ncutoff

ne

k2

WKB not valid

Figure 3.15 (a) The figures showingthe situation probing the density pro-file with an O-mode at 80 GHz. The cor-responding cutoff position is located atr/a = 0.33. The quantity k2 passes almostlinearly the cutoff. (b) W = (1/k2) d k/d r � 1

is shown along the propagation path ofthe probing wave. It sharply increaseswhen approaching the cutoff position,demonstrating that WKB approximationis not valid any longer for r < 0.38 in thisexample.


3.3.2Phase Change at Cutoff

In Section 2.8.1, the Helmholtz equation has been solved for a wave propagatingthrough the plasma with the WKB approximation valid along the whole path.Figure 2.9 demonstrates the smallness of the quantity W(r) for an O-mode waveat frequency ω = 2π × 150 GHz crossing the plasma. The frequency that has beenchosen in this example is clearly above the maximum cutoff frequency alongthe path, which is about ωp = 2π× 90 GHz. By reducing the wave frequency toω = 2π × 80 GHz under otherwise identical conditions, cutoff density is reached atabout ne = 8 × 1019 m− 3 corresponding to r(n)

c ≈ 0.33 in Figure 3.15. The figuresgive the value of W(r(n)) and demonstrate that the conditions of the WKB ap-proximation are not valid when approaching the cutoff position, most obvious forr(n) < 0.35, where W(r(n)) is strongly growing.

Following the discussions in [12, 13], Equation 2.132 can be solved analytically inthe case where k2 approaches linearly the cutoff position at rc as k2(r) ∝ (r − rc). Thecondition of linear dependence seems fulfilled in the example given in Figure 3.15.As before, we assume the �B-field oriented in z-direction and the toroidally shapedplasma approximated by a straight cylinder extending along the z-coordinate. Theprobing wave propagates perpendicular to it along the x-axis, approaching thecutoff position at x = xc. According to Figure 3.16, the wave’s starting point islocated at the plasma edge at x = 0, corresponding to r = a in Figure 3.12. Equation2.132 then reads(

d2

dx2+ k2 (x)

)E(x, t) = 0 (3.39)

In case of O-mode probing, one has E ≡ Ez, and in the case of X-mode, ne-glecting for simplicity its longitudinal component, one has E ≡ Ey. Referringto Figure 3.16, the linear dependence of k2(x) on position x is assumed,given by k2 = (ω/c)2N2 = (ω/c)2(1 − x/xc), with d2/dx2(k2) � d/dx(k2). Changingthe variables by defining ξ as ξ = |d/dx(k2)|1/3

ω=ωco (x − xc) = [(ω/c)21/xc]1/3(x − xc)= −((ω/c)xc)2/3N2, Equation 3.38, called Stokes equation, then reads as(

d2

dξ 2− ξ

)E(ξ , t) = 0 (3.40)

The wave is penetrating the plasma in positive x-direction. Suppressing the timedependence, pairs of linear-independent solutions of the Stokes equation are theAiry functions Ai(ξ ) and Bi(ξ ). Since Bi(ξ ) is growing for ξ > 0, it has no physicalmeaning. In integral representation, the solution Ai(ξ ) is given by

E(ξ )

E0= Ai(ξ ) = G

π

∫ ∞

0cos(

�3

3+ ξ �

)d� (3.41)

with G a constant [14]. In Figure 3.17, the function is given in the vicinity of thecutoff position. The solution for ξ > 0, x > xc, N2 < 0 is an exponentially damped


1

xcx

N2

WKB not valid

Figure 3.16 The figures sketches the assumptions made to obtain a solution for the �E-fieldof a wave propagating beyond the validity region of WKB. A linear dependency of N2(x) isassumed when approaching the cutoff position at xc.

−0.6−10 −8 −6 −4 −2 0

ζ

−0.4

−0.2

0

0.2

0.4

0.6

Ai(ζ

)

0

0.05

0.1

0.15

0.2

0.25

0.3

Ai2 (

ζ)

Ai(ζ )

Ai2(ζ )

FWHM

Cutoff

Figure 3.17 The figure gives the Airy-integral function Ai(ζ ) representing the �E-field of awave approaching cutoff at ζ = 0. The full width at half maximum (FWHM) of the last max-imum of Ai2(ζ ) gives the range where most of the reflected power is originating from. Itswidth might be used to estimate the localization of the reflecting layer.


wave, decaying beyond the cutoff position as

E(ξ ) = E(0)1

2√

πξ− 1

4 exp(

−2

3ξ

32

)(3.42)

Equation 3.41 is an asymptotic form of Ai(ξ ) for large argument. In front of thecutoff position at ξ < 0, x < xc, the field oscillates, corresponding to a standingwave as caused by interference of the incident wave and the reflected one. Forlarge values of |ξ | – the condition |ξ | ≥ 5 seems sufficient to make the errorsmaller than 1% – the following asymptotic form of the Airy function can be usedas well:

E(ξ )

E0≈ G√

πξ− 1

4 sin(

2

3ξ

32 − π

4

)(3.43)

This expression is matched to the WKB solution at larger distances by adjusting thefactor G [12]. The first term in the argument of the sin-function can be expressedas follows:

2

3ξ

32 = 2

3

[(ωc

xc

)23N2

] 32

= 2

3

ω

cN3(x)xc = ω

c

∫ xc

x

√1 − x

xcdx =

∫ xc

xkdx

(3.44)

which means that Equation 3.43 gives a kind of WKB solution for the total phaseshift along the path x to xc. Thus, the argument is composed of the integral givenin Equation 3.44, and in addition the constant − π/4, the second term in theargument of the sin-function in Equation 3.43.

The round trip phase shift between the incident wave at x = 0 and the re-flected one arriving after reflection at x = xc back at x = 0 is then obviouslygiven by

� = 2(

2

3

ω

cxc − π

4

)= 2

ω

c

∫ xc

0

√1 − x

xcdx − π

2= 2∫ xc

0kdx − π

2(3.45)

Despite WKB is not valid at the cutoff position, the result is the same as expectedfrom the WKB geometric optics approximation, obtained with Equation 2.138,apart from the additive fixed phase of −π/2.

The additive phase advance is yet unimportant in the evaluation of reflectometrymeasurements. No absolute phase measurement needs to be conducted, as allinformation on the cutoff layer position is contained in the differential phasechange, ∂�/∂ω|ω=ωco

.In summary, Equation 3.44 corresponds to the well-known WKB solution,

summing up all phase contributions along the path to the cutoff position. Theround trip phase delay written in the variables as used in the previous sectionsis then given by �(rc) = 2

∫ rc

a k(r)dr − π/2. The corresponding time delay can becalculated from Equation 3.37 and Equation 3.42. For O-mode reflectometry with

kO(z) = (1/c)√

ω2 − ω2p(z), it is given by


td(rc) = ∂�

∂ω= 2

∂

∂ω

∫ rc

a

1

c

√ω2 − ω2

p(r)dr = 2

c

∫ rc

a

ω√ω2 − ω2

p

dr (3.46)

The integrand in Equation 3.46 corresponds to the inverse group velocity of theO-mode wave,

∫ rc

a∂

∂ωkO(r)dr = ∫ rc

a 1/vg(r)dr. The equation does not include thedistance d from the wave launch position to the plasma edge as sketched inFigure 3.13. Inclusion results in an additional phase term �d = 2(ω/c)d with thecorresponding time delay td = 2(d/c).

3.3.3Profile Reconstruction

From Equation 3.46, the density profile can be reconstructed by Abel inversion:

rc(ωp) = r0 + c

π

∫ ωp

ω0

td(ω)√ω2

p − ω2dω

r0 = rc(ω0); td(ω) = ∂�

∂ω

∣∣∣∣ωco

(3.47)

with ω0 being the lowest probing frequency possible and with r0 the correspondingprofile position.

Thus, a frequency scan covering ω0 ≤ ω ≤ωp needs to be conducted. Since themeasurement cannot be performed, starting with ω0 = 0, the initialization mustbe done with measurements from other diagnostics systems or, if not available, byextrapolating the density profile from r0 to the very edge at r = a with reasonableassumptions. The inversion procedure cannot be conducted analytically in the caseof X-mode reflectometry. In this case, the group velocity (∂ k/∂ω)− 1 is an explicitfunction of position, as it depends on the local �B-field. Thus, the inversion must beconducted iteratively. The numerical procedure described in Ref. [15] is basicallyapplicable to O- as well as X-mode reflectometry; it is thus sketched in the following.

Position r0 might be the known position corresponding to the lowest frequencyω0, and ωi, 1 ≤ i ≤ n, be the frequencies where the time delay tdi has been measured,tdi = ∂�/∂ω|ω=ωi

, with ri being the corresponding cutoff positions. Defining φi =∑ij=1 tdi(ωj − ωj−1) and Ai,j = (1/2)(ωi/c)(Ni,j − Ni,j−1) with 1 ≤ j ≤ i and Ni,j the

refractive index for frequency ω =ωi and position r = rj, and Ni,i = 0, the integral

of Equation 3.45 can be approximated by the sum φi =∑ij=1 Ai,j(rj − rj−1), which

corresponds to

⎛⎜⎜⎜⎜⎝

�1

�2

�3

. . .

�n

⎞⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎝

A1,1 0 0 . . .

A2,1 − A2,2 A2,2 0 . . .

A3,1 − A3,2 A3,2 − A3,3 A3,3 . . .

. . . . . . . . . . . .

An,1 − An,2 An,2 − An,3 An,3 − An,4 . . .

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

r1

r2

r3

. . .

rn

⎞⎟⎟⎟⎟⎠ (3.48)


From this relation, the positions ri can be calculated step by step:

r1 = �1

A1,1

r2 = [r1(A2,2 − A2,1) + �2]

A2,2

r3 = [r1(A3,2 − A3,1) + r2(A3,3 − A3,2) + �3]

A3,3

. . .

rn = [r1(An,2 − An,1) + r2(An,3 − An,2) . . . + rn−1(An,n − An,n−1) + �n]

An,n

(3.49)

With the known positions ri and the corresponding cutoff frequencies ωi, theplasma parameters at the positions can be calculated.

3.3.4Localization of Reflecting Layer

The Airy function solution (Figure 3.17) for the �E-field in the vicinity of the cutoffposition, ξ = 0, shows the difference to the reflection at a mirror with the �E-fieldexactly zero at this position. The wave’s �E-field is decaying from the positionof the maximum of the last lobe at ξ =− 1.02, still clearly differing from zeroat ξ = 0. To gain a measure for the localization finally determining the localresolution of the method, the width of the square of the Airy function is beingidentified with the minimum width of the reflecting layer. This is the regionfrom which most of the reflected wave power is originating. The positions of halfmaximum of Ai2(ξ ) around its maximum at ξ = − 1.02 are at ξ =− 0.092 and atξ =− 1.722. The resulting full width half maximum (FWHM) �ξFWHM of Ai2(ξ )is then �ξFWHM ≈ 1.6. Using this value as the layer thickness and the minimumerror in localization, respectively, one obtains in physical coordinates consideringthe ξ -definition, ξ = |d/dx(k2)|1/3

ω=ωco (x − xc),

�x ≈ 1.6

[(ω/c)2d/dx(N2)|x=xc]

13

(3.50)

In the O-mode case, we have d/dx(N2)|ω=ωco= 1/Ln = (1/ne)dne/dx; thus the

width �x can be expressed by the density gradient length Ln at the cutoff position

�x ≈ 1.6

[(ω/c)21/Ln]13

= 1.6Ln

[(ω/c)Ln]23

(3.51)

The value depends on the gradient length of the density profile under considerationand is smaller for steeper density profiles. Figure 3.18 demonstrates this featureby giving examples. It must be pointed out that the �x-value, as defined inEquation 3.50, represents the limiting spatial resolution a reflectometry probing


Δr

Δr

ne

ne

(a)

(b)

0 0.2 0.4 0.6 0.8 1

Normalized plasma radius (r /a)

0 0.2 0.4 0.6 0.8 1


Nor

mal

ized

spa

tial r

esol

utio

n Δr

0

2E+19

4E+19

6E+19

8E+19

1E+20

0

0.01

0.02

0.03

0.04

0.05

Nor

mal

ized

spa

tial r

esol

utio

n Δr

0

0.01

0.02

0.03

0.04

0.06

0.05

Ele

ctro

n de

nsity

ne

(m−3

)

0

2E+19

4E+19

6E+19

8E+19

1E+20

1.2E+20

Ele

ctro

n de

nsity

ne

(m−3

)

Figure 3.18 The figures give the quan-tity defined as minimum localizationerror and reflecting layer thickness, respec-tively, for a peaked (a) and a broad (b)electron density profile under condition of

probing with an O-mode wave. The quan-tity is normalized to the minor plasmaradius a. In regions of steeper gradient ofthe density profile, better localization isexpected.

measurement might achieve. It should be mentioned, however, that the quantityas defined with Equation 3.50 is not mandatory.

In the literature are equivalent expressions in use, identifying the length[(ω/c)2dN2/dx|x=xc

]−1/3, with �x being smaller by the factor 1.6. This lengthis the characteristic length describing the exponential decay of the Airy functionat and beyond the cutoff position. In addition to what has been discussed so far,other effects are contributing to the reflected wave as well. They are discussed inthe literature [16] and will not be treated in more detail here.


3.3.5Relativistic Corrections

As discussed in Section 2.7.3 in the context of relativistic effects modifyingthe elements of the dielectric tensor, the cutoff density increases with electrontemperature (Figure 2.10). This dependence needs to be necessarily considered fortemperatures above about kBTe ≥ 5 − 10 keV. Since reflectometry probing aims atdetermining the location of the cutoff density, relatively large errors arise in thecase where the relativistic mass increase is not included properly. With the masscorrection term, as introduced in Chapter 2, me = me0

√1 + 5(kBTe/me0c2), the

refractive index for O- and for X-modes are given by Equation 2.129 and Equation2.130. Using these expressions, the increase of the cutoff density �nc = n(rel)

c − nc

over the cold-plasma value nc can be calculated. With Cco = 0 for the O-mode andCco = 1 for the higher X-mode (R-wave), one has

�nc

nc=√

1 + 5(kBTe/me0c2) − 1

1 − Cco(ωc/ω)(3.52)

The increase is higher for the X-mode than for the O-mode and is highest inthe case where the probing frequency is close to the cyclotron frequency. Forgiven density, the corresponding cutoff frequencies are lowered. Figure 3.19 showsthe examples of how the cutoff frequencies are affected as function of plasmaradius for typical broad density and peaked temperature profiles. As obvious, theevaluation of reflectometry data at high electron temperatures requires necessarilythe knowledge of the temperature profile for proper reconstruction of the densityprofile.

The different temperature dependence of O- and X-mode cutoffs might be usedwith advantage in next-generation, high-temperature burning plasma experiments(BPXs) to reconstruct iteratively both the density and the temperature profileby applying reflectometry simultaneously in O- and in X-mode as proposed in[17]. Two independent reflectometers probing the same plasma column thenneed to be operated. The two systems generate two independent phase data sets�O(ω(O)), �X(ω(X)), which are obtained by simultaneous reflection from the O- andthe X-mode cutoff layers. In the first step, the density profile is inverted using theO-mode data �O(ω(O)), assuming Te ≡ 0. Then using the X-mode data togetherwith the density profile as obtained in the first step, an estimate of the temperatureprofile is derived from the X-mode data set �X(ω(X)). The resultant Te-profile isthen used together with the O-mode phase data to obtain in a next step the seconditeration of the density profile and so forth until the derived profiles becomeconstant and errors are negligible.

3.3.6Influence of Density Fluctuations

The treatment of the reflectometry method to probe the plasma electron densityprofile as introduced in the previous sections was based on the assumption of a


1 keV

1 keV

50 keV

50 keV

Te(r /a)

(a)

−1 −0.5 0 0.5 1

Te(r /a)


(b)

−1 −0.5 0 0.5 1Normalized plasma radius (r /a)

Nor

mal

ized

ele

ctro

n te

mpe

ratu

re

0

20

40

60

80

100

0

0.2

0.4

0.6

0.8

1

Nor

mal

ized

ele

ctro

n te

mpe

ratu

re

0

0.2

0.4

0.6

0.8

1

O-m

ode

cuto

ff fr

eque

ncy

(GH

z)

50

75

100

125

150

O-m

ode

cuto

ff fr

eque

ncy

(GH

z)

Figure 3.19 The cutoff frequencies for O-(a) and X-modes (b) with the maximumcentral electron temperature as parameter:kBTe = 1, 5, 10, 20, 50 keV. The peaked tem-perature profile is included to illustrate theradius dependence. A broad density profile

has been used, as can be concluded fromthe O-mode cutoff frequency radius depen-dence. The relativistic effect needs to benecessarily considered at central tempera-ture above about kBTe = 5 keV. It is strongerin X-mode.


smooth time-independent density profile. However, in fusion plasmas, turbulenceplays an important role, affecting, in particular, heat and particle transport in amost significant way. Fluctuations of density and temperature of local magneticand electric fields and, in particular, correlated fluctuations of both density andtemperature with the fluctuating fields give rise to enhanced radial particle andenergy transport. The relative magnitude of fluctuations is of the order 0.1% inthe plasma center and 1% at its boundary. The fluctuations are characterizedby spatial scales, that is, by their wave number spectra in radial and poloidaldirections, S(kr) and S(k�), respectively, by their correlation lengths, and bytheir typical time scales, that is, their frequency spectra S(ωf). While the latterextend from the kilohertz into the megahertz range of frequencies far below theprobing wave frequency regimes, their typical fluctuation wavelengths λf = 2π/kf

cover a very wide range from much larger down to much smaller than thelocal wavelength of the probing wave at position x, λ(x) = 2π/k(x) = λ0/N(x).Reflectometry measurements are affected by the fluctuating electron density alongthe probing path. The random irregularities in density δne(x,t) give rise to phasefluctuations δ�(t) complicating the phase measurements as conducted for profilemeasurements in various ways. On the other hand, however, the phase fluctuationinformation from reflectometry can beneficially be used to determine the frequencyand the wavenumber spectra of the density fluctuations as well as size and motionof turbulent density irregularities in the plasma, enabling their characterizationand largely increasing the diagnostic potential of reflectometry. However, we willnot discuss the complex and comprehensive role of reflectometry as a fluctuationdiagnostic. A detailed mathematical description is challenging and beyond the scopeof this book; nevertheless, the most important aspects should briefly be introduced,as density fluctuations affect in various ways also density profile measurements,with implications also on the experimental realization (Section 8.3).

In the previous sections, the total round trip phase delay has been calculatedone-dimensionally. In a simple model, also the phase fluctuations caused by densityfluctuations will be treated in this way. However, it is not self-evident that a one-dimensional expression can as well be used to describe the phase of a wave reflectedfrom density irregularities [18]. The observation of large-amplitude fluctuations ofthe reflected wave suggests that multidimensional effects play an important roleand need to be taken into account to model the observations. Multidimensionalmeans density disturbances along the propagation direction of the probing wavebut also perpendicular to it influence the reflected signal. The reflected wave canthen no longer be considered as plane as the waves reflected from the rippledcutoff layer propagates into different directions as sketched in Figure 3.20. Thereflectometry receiver then detects a superposition of waves, that is, the cumu-lative result of varying random contributions from various directions reflected atvarious positions. Because the different components have different amplitudesand phases, strong fluctuations in amplitude and phase of the detected signalarise, much complicating the profile measurements. We are assuming for a firstguess nevertheless the plasma fluctuations parallel to the propagation directionof the probing wave and so small in amplitude that a one-dimensional model


Cutoff

Rippled surface

y

x

Figure 3.20 Reflection of the probing wave at the cutoff position in the plasma differsfrom that at a plane mirror. Density fluctuations generate a rippled surface, which reflectspartial waves into different directions, causing strong phase fluctuations when combined atthe detector.

can be used. Thus, the fluctuations in permittivity are varying only along thedirection of wave propagation. The electron density along the path x is givenby ne(x) = ne(x) + δne(x), with ne(x) representing the smooth density profile andδne(x) the small disturbance superimposed, δne(x) � ne(x). To evaluate their in-fluence in the phase measurement, the perturbations are considered constantin time, that is, temporally frozen. The situation can experimentally be realizedby conducting fast probing frequency scans ∂ω/∂ t, as discussed in Section 8.3.With xc the cutoff position, the round trip phase of the reflected wave of fixedfrequency ω is given by � = 2

∫ xc

0 k(x)dx − π/2. The density perturbations causelocal variations in k(x) with amplitude δk(x) = (∂ k/∂ ne)δne(x).

For the O-mode with ∂k/∂ne = −(ω/c)(1/2nc

√1 − ne/nc), the phase fluctuations

are then given by

δ�O(x) = −ω

c

∫ xc

0

δne(x)

nc

√1 − ne/nc

dx (3.53)

Owing to the presence of the denominator in the integrand, approaching zero atcutoff, strong weighting of the phase fluctuations at the cutoff position is expected.In the case where the wavelength of the perturbation is large, the resulting phasechange δ�O can also be considered as caused by the change in reflection locationδx, which is related to the density perturbation by δne = (∂ ne/∂ x)δx. Thus, inO-mode we have

δ�O = 2k0δxO = 2k0

(∂ne

∂x

)−1

δne = 2k0Ln

δne

ne(3.54)

with the gradient length Ln calculated at the cutoff position. In the case of X-modeprobing, the �B-field gradient length LB needs to be considered in addition, resultingin

δ�X = 2k0δxX = 2k01

(1/Ln + (ωcω0/ω2p)1/LB)

δne

ne(3.55)

These one-dimensional estimates can only be used in the case of large fluctuationwavelengths λf � λ0, equivalently, kf � k0.


In the next step of sophistication, the role of position and wavenumber onthe magnitude of the phase response δ� needs to be studied by solving theStokes equation with the density perturbation included. It has been conductednumerically, for example, in Ref. [19], replacing the electron density ne in the termk2 = (ω/c)2[1 − (e2/ω2ε0me0)ne] of Equation 3.39 by ne = ne + δne. The perturbationis assumed time-independent, localized at position x0 near the cutoff positionwith a width �f and a characteristic wave-number kf. The density disturbance isassumed with Gaussian width and amplitude δne0 as given by

δne(x) = δne0e−(x−x0)2

�2f sin kf (x − x0) (3.56)

It turns out that the main contribution to phase fluctuations comes from theposition xB, at which the wave number kf of the fluctuations matches the wavenum-ber of the probing wave,

kf = 2k(xB) = 2N(xB)k0 (3.57)

Under this Bragg resonance condition, the path difference of waves backscatteredfrom successive density maxima of the periodic density perturbation is a multipleof half the wavelength of the probing wave, their coherent superposition thusmaximizing the phase response. The position of maximum δ� moves away fromthe cutoff position xc when the fluctuation wavelength approaches the probingwavelength λf → λ0. Since in real plasmas, the density perturbations are coveringa certain range of wavenumbers characterized by the k-spectrum S(k), the Braggresonance condition always selects a certain k-range, generating the largest phasefluctuations.

Although phase fluctuation’s response to density perturbations is largest nearthe cutoff position, Bragg back scattering is nevertheless present everywhere alongthe path of the probing beam from the plasma edge to the cutoff position. Theincident wave will, therefore, be affected or, in a fluctuation diagnostics point ofview, will probe the large-scale fluctuations (i.e., small kf) always near its cutofflayer, however, simultaneously small-scale density fluctuations correspond to largerkf ≈ 2k0 near the plasma edge, with k0 the vacuum wavenumber of the probingwave. In this way, a range of wave numbers contribute to the phase fluctuationsas the probing beam is basically sensitive to the k-range as defined by k0 ≥ kf ≥ kco.The smallest k at cutoff may be estimated via the reciprocal characteristic Airylength by kco ≈ |k2

0d/dx(N2)|1/3.So far the density perturbations along the probing direction are assumed time

independent. However, they are strongly time dependent, with frequency spectraextending into the megahertz range of frequencies. This means that at fixed probingfrequency, broadband phase fluctuations appear, whose interpretation and spatialallocation are not straightforward. The one-dimensional modeling seems sufficientin the case of long fluctuation wavelengths and small amplitudes [20]. However,interference effects caused by the generally two-dimensional structure of the pertur-bations play an important role, demanding for full-wave two-dimensional modelingto interpret the phase fluctuations in terms of localized density fluctuations, in


particular, under the conditions of broadband, short-wavelength turbulence [21].In density profile measurements, the amplitude and phase fluctuations of thereflected wave mask the phase variation needed to determine the round trip timedelay. One way out is to scan the density profile in such a short time that the densityfluctuations appear time-independent (frozen). Subsequent averaging of a numberof density profiles obtained in this way then results in smooth profiles.

In the case where the density perturbations are moving with velocity �vf witha velocity component parallel to the �k-vector of the probing wave, Doppler shiftof the reflected signal’s frequency of �ω = �k · �vf arises. The interpretation of thefrequency-shift �ω then depends on the observation geometry of the reflectometrysystem. In the case of the perpendicular probing geometry discussed so far, suitedfor density profile measurements, the frequency-shift causes an artificial shift incutoff layer position. For dedicated systems (Section 8.3.7), however, with theobservation geometry tilted in such a way that the probing �k-vector has poloidal ortoroidal components, the frequency-shift can be interpreted in terms of perturbationvelocity �vf along the observation direction [22].

3.4Scattering

In the previous chapters, we considered the influence of the plasma dielectric prop-erties on the propagation of a monochromatic plane wave through the magnetizedplasma. The diagnostic potential to determine certain plasma parameters from theplasma’s impact on phase and polarization of the probing wave was identified,forming the basis of interferometry, polarimetry, and reflectometry. The plasmawas treated as a continuum with the smoothed ensemble-averaged electron densityas the most important plasma parameter, which, together with the local magneticfield, determines exclusively the interaction with the electromagnetic wave at fre-quencies comparable to the electron plasma frequency. At electron temperatureskBTe higher than about 3–5 keV, relativistic effects need to be taken into account,introducing the electron temperature as an additional parameter influencing wavepropagation at fusion-relevant temperatures. Average quantities characterizing theensemble of electrons in a volume element determine the wave propagation. Inthe treatment conducted so far, wave interaction with individual plasma particleswas assumed so small that momentum and energy transfer to the electrons arenegligible. However, although acceleration of the plasma electrons by the electricfield component of the wave was considered in the derivation of the dielectrictensor (Equation 2.29, Equations 2.65–2.67), the important consequence of thisacceleration still needs to be discussed. Accelerated charge carriers are sources ofelectromagnetic radiation. Thus, the electrons become emitters of electromagneticwaves themselves. The process is called Thomson scattering, treated in detail, forexample, in Refs. [23–25]. The re-emitted radiation is very weak – the reason whyit could be neglected so far – but it has a high diagnostic potential as shown in thefollowing section.

3.4 Scattering 101

3.4.1Single-Particle Thomson Scattering

To calculate the radiation caused by the acceleration of a single plasma electronin the field of a plane wave, we recall the well-known way to treat the problem:starting with the Lienard–Wiechert scalar �LW(�r, t) and vector potential �ALW(�r, t) ofa moving, accelerated point charge, and applying the relation �E = −(∂ �ALW/∂t) −�∇�LW, the electric field �Es at position P(R,t) at time t in distance R to the origin isdetermined.

Given the geometry of Figure 3.21 and Figure 3.22, the distance between theaccelerated charge and the observer at position P is R′(t′) = |�R − �r0(t′)|. The timet′ is the retarded time at the electron’s position, t′ = t − R′(t′)/c. For large distanceR, we approximate R ≈ R′. The retardation then becomes t′ ∼= t − |R − s · �r0/c|,where s is the unit vector from the electron to the observer at P, now assumedconstant in time.

With the normalized electron velocity �β = �ve/c, its acceleration �β = (1/c)d�ve/dt′,(1 − �β)dt′ = dt, the 1/R-dependent radiation part of the electric field in the wavezone (R � λ), is given by [26]

�Es(R, t) = −e

4πε0

⎡⎣ s ×

((s − �β

)× �β)

c(1 − s · �β)3R

⎤⎦

ret

(3.58)

The expression needs to be evaluated at the retarded time t′, the time the electronis emitting.

If no static �E- and �B-fields are assumed, the acceleration �β is exclusively causedby the fields �Ei and �Bi = i × �Ei/c of the incident plane wave passing the plasma.The �E-field at the electron position is given by �Ei(�r, t′) = �Ei0 cos(�ki�r0 − ωit

′). Thewave propagates into the direction i = �ki/|�ki|, linear polarized with �E-field directionei = �Ei0/|�Ei0|.

Considering explicitly also the relativistic mass increase, the acceleration is

�β = 1

c

d�ve

dt= −e

c

√1 − β2

me0[�Ei + �β × (i × �Ei) − �β( �β · �Ei)] (3.59)

The acceleration is inversely proportional to the mass me0. This is the reason whyacceleration of plasma ions in the context of Thomson scattering can completely

Origin

r0(t ′)

e (t ′)

R ′(t ′)

R(t )

P(R,t )

s

Figure 3.21 Geometry to calculate the electric field of an accelerated electron at position�r0(t′) generated at P(R,t) in distance R. The unit vector pointing from the electron to theobserver is s.


r0(t ′)

EsEi

ki,

ks,

i

Θ

φs

P (R,t )

dΩs

Figure 3.22 Scattering geometry used to calculate the scattered power within solid angled s. The incident wave propagates along direction i. The scattered power is observed alongdirection s under an angle �. The scattered wave’s �E-field is measured with the polarizationtilted by an angle φ with respect to the incident wave’s polarization.

be neglected compared to that of the electrons. Substituting this expression intoEquation 3.58, the resulting scattered field �Es(R, t) becomes

�Es(R, t) = e2

4πε0me0c2

√1 − β2

⎡⎢⎣s ×((

s − �β)×(�Ei + �β ×

(i × �Ei

)− �β(

�β ·�Ei

)))(

1 − s · �β)3

R

⎤⎥⎦

ret

(3.60)

The acceleration by the magnetic field component of the wave is smaller by a factorβ compared to the acceleration by the electric field. It is therefore neglected in thecase of small β. To outline the physical process, only the nonrelativistic case isdiscussed in the following. In the limit, β → 0, Equation 3.60 reduces to

�Es(R, t) = e2

4πε0me0c2R

[s ×(

s × �Ei0

)]ret

cos(�ki�r(t′)− ωit

′)

= re

R

∣∣∣�Ei0

∣∣∣ cos(�ki�r(t′)− ωit

′) [

s × (s × ei

)]ret (3.61)

with the classical electron radius re = e2/4πε0me0c2 = 2.818 × 10− 15 m introduced.The �E-field of the scattered wave is pointing into the direction given by the doublecross product s × (s × ei). The amplitude is extremely small, smaller than theincident one by at least the ratio re/R.

3.4.2Doppler Shift

An underlying constant velocity of the accelerated electron has not been consideredso far. In the case where the electron is moving with velocity �ve, as sketched inFigure 3.23, the electron experiences the Doppler shifted frequency ωi − �ki · �ve.The frequency of the wave scattered into the direction of the observer is onceagain Doppler shifted, contributing with another term, ωs = ωi − �ve · �ki + �ve · �ks.Defining the scattering wave vector �k = �ks − �ki, the twice Doppler shifted frequencyis ωs = ωi + �k · �ve, which might, in terms of the unit vectors of incident and scattered

3.4 Scattering 103

waves and the velocity �ve as well, be expressed as ωs = ωi(c − i · �ve)/(c − s · �ve). Thedifferential scattered power per solid angle dPs/d s, most important for theevaluation of experiments, can now be expressed. It is convenient to give it in termsof the time-averaged Poynting flux, generally defined as

⟨�S⟩t= 1

μ0〈�E × �B〉t =

�k|�k|

1

2cε0|E|2 (3.62)

With the relation dPs/d s = R2〈�Ss〉t · s, with Ss = |�Ss|, one then obtains

dPs

d s= R2 1

2cε0|Es|2 = 1

2cε0 r2

e |Ei0|2|s × (s × ei)|2 = r2e 〈Si〉tL(�s, �) (3.63)

The equation gives the scattered power per unit solid angle out of the scatteringvolume expressed in terms of the power flux �Si into it. The function L(�,�)considers the scattering geometry.

Although still concerned with the scattering from a single electron, we aimedat the scattering from an ensemble of electrons within a certain volume Vs withcross section As. The incident Poynting flux 〈Si〉t is then identical to the inputbeam power per area, 〈Si〉t = Pi/As. According to Figure 3.22, the function L(�s,φ)as introduced in Equation 3.63 is given by L(�s, φ) = |s × (s × ei)|2 = 1 − |s · e|2 =1 − sin2 �scos2 φ, with φ being the angle between the �E-field directions of thescattered wave �Es and the incident wave �Ei. For φ = 0, the function L(�s,φ) hasthe classical doughnut-shaped dipole radiation pattern with no radiation into thedirection of acceleration and with its maximum perpendicular to it.

Integration over the full solid angle gives the total power emitted by an acceler-

ated electron, known as the Larmor formula, P = (e2/6πε0c)| �β|2. If considering inaddition the Doppler frequency shift by multiplying with a δ-function, the differ-ential scattered power per unit solid angle and per unit frequency from a singleelectron becomes

d2Ps

d sdωs= r2

e 〈Si〉L(�s, φ)δ(�k · �ve − (ωs − ωi)) (3.64)

ki, ω i

ks, ωsk

e− Ve

Θ

Figure 3.23 The scattering geometry defining the scattering vector �k = �ks − �ki. The incidentwave frequency that is scattered by the electron moving with velocity �ve undergoes twice a

Doppler shift, resulting in ωs = ωi − �k · �ve.


3.4.3Incoherent Scattering

We are now considering an ensemble of Ne electrons in the scattering volumeVs with average electron density ne. If there are no correlated motions of theplasma electrons, the phases of the scattered waves originating from differentelectrons are uncorrelated and the total scattered power from the assembly ofelectrons within the volume Vs is given by the sum of the scattered power fromthe single electrons, resulting in Ne times the differential scattered power as givenin Equation 3.64. This is the presupposition of the process of incoherent Thomsonscattering. We take it for the moment as valid and discuss the conditions inmore detail in Section 3.4.5. The frequencies of the scattered waves originatingfrom single electrons of the assembly depend on the individual electron’s velocitycomponent along the scattering vector �k. Given a normalized velocity distributionfunction f (�ve), the power level in the frequency interval dωs at ωs is given by thenumber of particles Nk = Nefk(vk)dvk within the dvk velocity interval at vk, in whichthe electrons have a velocity component along �k resulting in the Doppler shiftedscattering frequency ωs = ωi + �k · �ve. The spectrum of the total scattered radiationis then determined by the velocity distribution function f k(vk) of the electronsalong the scattering vector �k. Assuming the electron assembly within the scatteringvolume in thermodynamic equilibrium, this one-dimensional distribution functionis obtained by integrating the Maxwell–Boltzmann distribution over the twoperpendicular velocity components with temperature Te,

fk(vk) =∫

f (v⊥, vk)d2v⊥ = ne

√me0

2πkBTee− me0vk

2

2kBTe (3.65)

Expressing the velocity component vk by ω/k = (ωs −ωi)/k, dvk = (1/k)dωs, weobtain

d2P

d dωs= r2

e 〈Si〉tVsL(�s, φ)ne1

kfk

(ωk

)

= r2e 〈Si〉tVsL(�s, φ)ne

1

k

√me0

2πkBTee− me0ω2

2kBTek2 = r2e 〈Si〉tVsL(�s, φ)S(k, ω)

(3.66)

The function S(k,ω) introduced is called the scattering form factor. Since thefrequency ω in the argument of S(k,ω) corresponds to the Doppler shift of theincident frequency ωi to the frequency ωs of the scattered wave, the form factorS(k,ω) directly reflects the velocity distribution f k(vk) along the scattering vector �k.Measuring the scattering spectrum around the incident frequency ωi, therefore,allows for the determination of the electron temperature Te by fitting the measureddata to a Gaussian.

Experiments are conducted by sending a well-collimated beam through theplasma, observing the scattered radiation under an angle �s with another well-defined detection beam. Their intersecting common volume defines the scatteringvolume Vs. The scattering volume is a function of the scattering angle �s, being

3.4 Scattering 105

smallest with the incident beam and the beam of the collection optics perpendicularto each other, �s = 90◦.

The Thomson scattering arrangement has high diagnostic potential. It measuresthe electron temperature from the distribution function and, in the case where thescattering system is absolutely calibrated with respect to power, it measures, inaddition to the temperature, the local electron density ne within Vs. Density andtemperature profile measurements are possible by the simultaneous observationof the scattering spectrum at a number of positions along the primary beam paththrough the plasma. Weakening along that path is negligible as well as multiplescattering, as the scattering cross section is very small. Defining the differentialcross section of a single electron by dσ/d s = Ps/〈Si〉d s = r2

e sin2 �s, the totalcross section is obtained by integrating over the full solid angle to obtain theso-called Thomson scattering cross section:

σT = r2e 2π

∫ π

0sin3 �s d�s = 8π

3r2

e = 6.652 × 10−29 m2 (3.67)

Adding up this cross section Ne times to consider all electrons within the scatteringvolume Vs = lsAs with length ls and cross section As allows for an estimate of thetotal scattered power related to the input power. Using Ps = Neσ T〈Si〉, the ratiobecomes Ps/Pi = (Vs/As)ne (8π/3)r2

e = nelsσT. With ne = 1020 m− 3 and ls, a fewmillimeters, Ps/Pi ≈ 10− 11 results. Approximating the observation solid angle withd s ≈ 10− 2sr, the observable scattering power is about 10− 13 of the input power.

Note that with Thomson scattering the electron energy distribution functionis probed neither along the primary beam nor along the observation beam, butalong the scattering vector �k, whose orientation is determined by the scatteringgeometry, �k = �ks − �ki, as shown in Figure 3.23. The value of �k is given by k =√

k2s + k2

i − 2kskicos�s. Since momentum transfer from the wave to the scatteringelectron (Compton effect) is completely negligible in the microwave and opticalrange of frequencies, the absolute value of the incident vector |�ki| does not changein the scattering process so that |�ks| = |�ki|. Thus, the value of the scattering vectorbecomes

|�k| = 2|�ki| sin�s

2(3.68)

It shows that the size of the scattering vector can be adjusted by the choiceof the wavelength of the probing wave, λi = 2π/ki, as well as by the scatteringgeometry. The value of the scattering vector needs to be known to determine thetemperature from the scattering spectrum, but most importantly, it determinesthe presupposition made while deriving the scattering spectrum. It was assumedthat the scattered power from individual electrons could be added up incoherently.This is valid in the case where electron motions are uncorrelated. However,considering the cloud of electrons moving with the ions, the electron motions arecorrelated. The characteristic size of this cloud is the Debye length λD. The ions aredynamically dressed with a cloud of electrons moving along with them, but witha random distribution of the fast mobile electrons within this cloud. It dependson the size of the product of kλD whether or not the presupposition is valid. In


the case where kλD � 1, the phase difference of scattered waves from individualelectrons within the cloud is large enough and random that the presuppositionsof incoherent scattering can be assumed valid. Since, according to Equation 3.68,|�k| is of the order of |�ki|, the condition demands for the wavelength of the probingwave small compared to the Debye length, λi � λD. With this condition, theincident wave is probing the granularity within the Debye sphere in the sense thatneighboring electrons within the sphere are generating random, sufficiently largephase differences of the scattered waves. Since, for typical fusion plasmas, theDebye length is of the order of several 10− 5m, the condition kλD � 1 demands forprobing wavelengths in the optical or near-infrared range. The condition cannotbe fulfilled with millimeter-waves. If, on the other hand, kλD ≤ 1, correspondingto λi ≥ λD, the incident wave is probing an ensemble of many electronssimultaneously. The scattering fields from individual electrons then need to beadded up coherently. The process is called coherent Thomson scattering (CTS). Itis discussed in Section 3.4.6.

3.4.4Relativistic Incoherent Scattering Spectrum

The nonrelativistic description conducted so far, which resulted in a Gaussianscattering spectrum Figure 3.24 (a), representing the velocity distribution of theelectrons within the scattering volume, is valid only up to temperature kBTe ≈ 1 keV.To describe Thomson scattering at fusion-relevant temperatures, Equation 3.60needs to be used together with a relativistic Maxwell distribution function. Sincethe fully relativistic results can only be obtained by numerical integration, ananalytical approximation of the scattering spectrum as derived in Ref. [27] is givenhere. The approximation expresses the relativistic spectrum by the nonrelativisticone, multiplied with a function containing temperature-dependent corrections. Theapproximation can be used up to about kBTe ≈ 30 keV. With α = (1/2)me0c2/kBTe,k2 = 2k2

i for scattering at � = 90◦, and κ = (λs − λi)/λi = (ωi/ωs) − 1, one gets forthe nonrelativistic scattering form factor function as defined through Equation 3.66Snr(κ) = √

α/2π exp(−(α/2)κ2). The relativistic form factor is approximated bymultiplying with velocity-dependent corrections:

Sr(κ) = Snr(κ){

1 − 7

2κ + 1

2ακ3 + 39

32α+ 29

8κ2 − 7

4ακ4 + 1

8α2κ6

}(3.69)

The two functions Snr and Sr are plotted in Figure 3.24 for various temperatures inthe range 1 ≤ kBTe ≤ 30 keV. The relativistic ones are exhibiting a clear asymmetryof the scattering spectrum and a large blue shift of their maxima at electrontemperatures of fusion-relevant plasmas. This strong deviation from a Gaussianshape is due to the fact that relativistic electrons in the rest frame do not emit witha spatial radiation pattern as a classical dipole, but with a pattern that is enhancedin forward direction.

The higher emission into forward direction results in an enhanced Dopplershift to higher frequencies, in this way introducing the increasing asymmetry of

3.4 Scattering 107

0

1 keV

10 keV

3 keV

30 keV

0.2(a)

(b)

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8ω i /ωs

0.4 0.6 0.8 1 1.2 1.4 1.6ω i /ωs

1

2

3

4

5

6

7S

pect

ral s

hape

0

1

2

3

4

5

6

7

Spe

ctra

l sha

pe

1 keV

3 keV

10 keV

30 keV

Figure 3.24 Scattering spectra as obtained for temperatures kBTe = 1, 3, 10, 30 keV with rel-ativistic corrections (b) and without (a). With increasing temperature, the spectra exhibit ablue shift and become strongly asymmetric, deviating more and more from the Gaussianshape as obtained from nonrelativistic calculations.

the scattering spectrum with temperature. More details of incoherent Thomsonscattering are not given here because, as was pointed out before, the condi-tions cannot be met with millimeter-waves. To make use of the high diagnosticpotential of incoherent Thomson scattering, scattering arrangements in the op-tical and near-infrared spectral ranges with powerful laser sources need to beset up [23].


3.4.5Role of Density Fluctuations

Under the conditions of incoherent scattering, the total power per solid angleinto the direction of the observer was calculated by just summing up the powercontribution from each of the Ne = neVs individual electrons within the scatteringvolume Vs, with the density ne assumed constant. The result obtained is correct.However, the correct argumentation demands for the existence of electron densityfluctuations, not considered so far. This can be realized in the following way [23].We consider the scattering volume divided into C cells, each cell containingM electrons on average. The total number of electrons is Ne = neVs = CM.The momentary number of electrons in cell j at position �rj is Mj because ofdensity fluctuations deviating from the average value by δMj = Mj − M. The

scattering field from this cell has phase φj = �k • �rj and amplitude EsC. The totalscattering field composed of all contributions from the individual cells is thengiven by Es = EsC

∑Cj=1(M + δMj)e

−iφj . Because of the large number of cells, toeach phase of a scattered wave originating from one cell, a wave with the oppositephase could be found, cancelling the first. Therefore, all terms containing Mvanish, Es = EsC

∑Nj=1 Me−iφj = 0. The remaining part of the scattering field is

Es = EsC

∑Cj=1 δMje

−iφj . It is determined by the individual deviations δMj of thenumber of electrons within the cells, that is, the degree of inhomogeneity of theplasma and the distribution of scattering centers, respectively. The scattered poweris given by

Ps ∝ E2s =⟨

E2sC

C∑n,m=1

δMnδMme−i(φm−φn)

⟩(3.70)

In the case where no correlations exist, 〈δMnδMm〉 = 0, the only nonzero terms arethose with m = n to give⟨

E2sC

C∑n,m=1

δMnδMme−i(φm−φn)

⟩= CE2

sC〈δM2〉 (3.71)

Thus the scattered power is determined by 〈δM2〉 = 〈(M − M)2〉. In countingPoisson statistics, describing the natural fluctuations in the granularity of theplasma, the mean square deviation equals M. The resulting total scattered powerfrom C cells is then proportional to Ne = CM. This is exactly the ad hoc resultobtained before in the incoherent scattering case, kλD � 1, stating that the totalscattered power is given by the sum of the power from individual scattering centers,but the arguments leading to it are completely different.

3.4.6Coherent Scattering

If kλD ≤ 1, the case of the so-called cooperative or coherent Thomson scatteringis reached. Now correlations exist, 〈δMnδMm〉 = 0, and the degree of coherency

3.4 Scattering 109

between the fields of the various electrons needs to be considered in more detail.The way the form factor S(�k, ω) is being calculated in the coherent scatteringcase is sketched later by following Refs. [23, 24]. Only those correlations areconsidered, which are always present because of basic plasma properties, thatis, the perturbation of the background of particles due to the presence of oneparticle at a given position, described by the picture of dressed particles. Inparticular, each ion is dragging a screening cloud of electrons (Debye shielding),causing ion-induced perturbations on the otherwise homogeneous electron densitydistribution. No instability-driven density turbulence is considered, which mightstrongly enhance correlated motions and then dominate the scattering spectrum.Since density fluctuations are the necessary ingredient, the general treatment startsby calculating the density fluctuations for a given plasma state. On this basis,the function S(�k, ω) can be determined via the ensemble average of the Fouriertransform δne(�k, ω) of the density fluctuations from

S(�k, ω) = limT→∞V→∞

1

VT

〈|δne(�k, ω)|2〉n0e

(3.72)

The time T in the equation enters with the Fourier transform. We assume aprobability distribution function Fα(�r, �v, t) =∑Nα

j=1 δ[�r − �rj(t)]δ[�v − �vj(t)] for speciesα, which gives the number of particles with velocity �v at �r and t per unitvolume. The index represents electrons and ions with charge Z. The densityis obtained by summing over velocities, nα(�r, t) =∑+∞

v=−∞ dvFα(�r, �v, t). The den-sity has the mean value n0α(�r, t) = Nα/Vs. It is locally fluctuating around thisvalue by δnα = nα−n0α. It is this quantity that determines the scattered powerfrom the plasma. The function δFα = Fα(�r, �v, t) − F0α describes the deviationsfrom the mean value; its Fourier–Laplace transform is given by δFα(�k, �v, ω) =∫∞

0 e−iωt dt∫ +∞−∞ δFα(�r, �v, t)ei�k�r d�r. Thus, the fluctuating electron density can be

calculated from δnα(�k, ω) =∑v

d�vδFα(�k, �v, ω), with the result [23]

δne(�k, ω) = −i

⎡⎣(1 − χe

ε

) N∑j=1

ei�k�rj

ω − �k • �vj

+ Zχe

ε

N/Z∑l=1

ei�k�rl

ω − �k • �vl

⎤⎦ (3.73)

Here, ε(�k, ω) = 1 + χe(�k, ω) + χi(�k, ω) is the dielectric function, and χ e and χ i arethe susceptibility contributions from the electrons and the ions with charge Zseparately. Generally, they are given by

χα(�k, ω) = Zαe2

mα0ε0

∫ +∞

−∞

1

k2

�k · ∂ fαk/∂�vω − �k · �v

d�v (3.74)

The function f αk(vk) is, as mentioned before, the one-dimensional velocity distri-bution function of the electrons, α = e, and the ions, α = i. It is connected with thedistribution function Fα by F0α = (n0α/Zα)fα(�v).


3.4.7Electron and Ion Feature

The three terms of Equation 3.73 describe the electron density fluctuations ascaused by the electrons themselves; this is the first sum including pre-factors, andelectron density fluctuations caused by ions with charge Z, the second.

Since χ e/ε ≈ (kλD)− 2, terms with this pre-factor in Equation 3.73 vanish in theincoherent case, as kλD � 1. The remaining first sum describes the phases ofthe individually moving electrons, unaffected by the motion and presence of theothers. The second and the third term are becoming important in the coherentcase, kλD ≤ 1.

Equation 3.73 allows us to calculate the spectral density function S(�k, ω). However,this is very cumbersome, demanding for the careful discussion of a number ofimportant aspects, as can only be given in adequate depth in the specializedliterature [23, 24]. It cannot be repeated here. With Equation 3.74 and the one-dimensional Maxwell distribution along �k for species α, the spectral densityfunction becomes

S(�k, ω) = 2π

k

∣∣∣1 − χe

ε

∣∣∣2fe0

(ωk

)+ Z

2π

k

∣∣∣χe

ε

∣∣∣2fi0

(ωk

)(3.75)

The equation explicitly shows that the first term depends exclusively on the electrondynamics, and the second one on the ion dynamics of ions with charge Z. In thecase where more than one ion population exists, which is the case if the plasma iscontaminated by impurities or, most importantly in future BPXs, by the presenceof fusion-generated high energetic α-particles, the various populations need to berepresented by specific additional terms that consider the charge, density, andvelocity distribution characterizing the individual populations. The spectral densityfunction is generally, nevertheless, of the same form, allowing for clear separationof the electron and the ion terms:

S(�k, ω) = Se(�k, ω) +∑

i

Si(�k, ω) (3.76)

With a Maxwellian velocity distribution function with temperature Tα of species α,

fαk(v) = nα

(mα0

2πkBTα

) 12

emα0v2

2kBTα (3.77)

the susceptibilities χα can be calculated using Equation 3.74 to obtain

χα =(

1

kλD

)2 (ZαnαTe

neTα

)w(xα) (3.78)

The complex function w(xα) of variable xα = (1/√

2)ω/kvα with vα = √kBTα/mα0

is related to the plasma dispersion function. It is approximately given by

w(x) ≈ 1 − 2xe−x2∫ x

0ey2

dy + i√

πxe−x2(3.79)

The real part is plotted in Figure 3.25. Since χ e is inversely proportional to (kλD)2,the second and the third term in Equation 3.75 vanish in the case of kλD � 1. The

3.4 Scattering 111

−0.50 1 2 3 4

x

0

0.5

1w

(x

)

Figure 3.25 The shape of the real part of the function w(x).

first term thus corresponds to the incoherent Thomson scattering case, reproducingthe result of Equation 3.65.

In the case of kλD ≤ 1, the other terms are nonvanishing and are then contributingas well. They might even become the dominant ones, carrying in particularinformation with high diagnostic potential. In particular, in the case of large valuesof χ e, the factor |1 −χ e/(1 +χ e +χ i)|2 in the first electron term vanishes, and onlythe ion term remains. This term depends on the ion susceptibility, the ion charge,and the ion velocity distribution function. This means that although scattering iscaused by the acceleration of the electrons, scattering under collective conditionsreflects indeed the motion of the ions. The dragged electrons surrounding the ionsprovide the information.

Because of the clear separation of the electron and the ion terms in Equation 3.75,an approximation was introduced [28, 29] by substituting χ e ≈ (kλD)− 2 in the ionterm and neglecting completely the ion susceptibility in the electron term, χ i ≈ 0.With (Equation 3.77), Equation 3.75 then becomes

S(�k, ω) =√

2π

ve�e(xe) +

√2π

viZ

(1

1 + (kλD

)2)2

�i(xi) (3.80)

with the shape function �α(x) = e−x2/|1 + ρα

2w(x)|2 and species-dependent pa-rameters

ρ2e =(

1

kλD

)2

, ρ2i = Z

Te

Ti

(1

1 + (kλD

)2)

(3.81)

The ion term of S(�k, ω) is plotted in Figure 3.26 for Te = T i and for two values ofthe parameter kλD. In the case where Z and Te are known, the ion temperaturecan basically be deduced by simulating and fitting measured spectra.

The electron term of S(�k, ω) is given in Figure 3.27. It appears in a much higherfrequency range, being clearly separated from the ion term. While identical with aGaussian for kλD � 1, it exhibits completely different shape for kλD ≈ 1.


0.0E+00

(a)0 0.002 0.004 0.006 0.008 0.01

ω /ω i

2.0E−07

4.0E−07

6.0E−07

8.0E−07

1.0E−06

1.2E−06

1.4E−06

Si(k

,ω)

0.0E+00

(b)0 0.002 0.004 0.006 0.008 0.01

ω /ω i

5.0E−06

1.0E−05

1.5E−05

2.0E−05

2.5E−05

Si(k

,ω)

k λD = 1

k λD = 1

k λD = 0.4

k λD = 0.4

Te/ Ti = 5

Figure 3.26 In (a) the ion term ofthe form factor function S(�k, ω) forkBT i = kBTe = 2 keV and two values of theparameter kλD in the vicinity of kλD ≈ 1.The figure gives the Doppler shifted fre-quency ω =ωs −ωi of the scattered wavenormalized to the incident frequency cho-sen, ωi = 2π× 100 GHz. Since no relativistic

corrections are considered, the spectra aresymmetric and only one side needs to beplotted. The scattered spectra are narrowwith FWHM of about 1%. In (b), the ratio ofelectron to ion temperature has been raisedto 5 to demonstrate the trend, despite theapproximation fails in case the ratio extends1. A pure H-plasma is considered.

As mentioned before, although scattering is caused by the electrons, collectivescattering is probing the ion motion as well. The ion feature in the scatteringspectrum appears at much lower scattering frequencies than the electron featurebecause of the ion’s much lower velocity, as shown in Figures 3.26–3.28. Theelectron and ion features are well separated on the abscissa scale, which is givenby the ratio of Doppler shifted frequency and scattering vector, ω/k. Assumingk ≈ 3 × 103 m corresponding to a probing wavelength of 3 mm, the characteristic

3.4 Scattering 113

0.0E+000 0.1 0.2 0.3 0.4 0.5

ω /ω i

2.0E−08

4.0E−08

6.0E−08

8.0E−08

1.0E−07

1.2E−07

1.4E−07S

e(k,

ω)

k λD >> 1

k λD = 1 k λD = 0.4

Figure 3.27 The electron term of the formfactor function S(�k, ω) for kBTe = 2 keVand two values of the parameter kλDfulfilling the condition of coherent scat-tering, kλD ≤ 1. The spectra are widercompared to the ion spectra by about

a factor 10–20. In addition, the electronterm is plotted for the condition of in-coherent scattering, kλD � 1. Under thiscondition, the ion term vanishes and theelectron term approaches the Gaussianshape.

electron and ion features appear at Doppler frequencies of several tens of gigahertzand 1 GHz, respectively.

This means that even at large electron temperatures, the ion feature appearsclose to the probing wave frequency.

So far only the spectral density function and the way it is derived have beensketched. In the coherent scattering case, the differential scattered power has ofcourse the same form as given in Equation 3.66 in the previous section. Only theform factor S(�k, ω) needs to be replaced by the ones given in Equation 3.75 andEquation 3.80, respectively. The relativistic treatment is beyond the scope of thisintroductory text. The reader is referred to the special literature [23, 30].

3.4.8Summarizing Comments

A few comments should be made concerning the effect of the static magneticfield �B0 in a scattering experiment conducted at Tokamaks and Stellarators. Thestatic field introduces a gyration motion of the scattering electron with frequencyωc = (e/me0)B0 and gyro-radius ρe = v⊥/ωc. The gyro-motion of the scatteringelectron introduces a periodic time retardation at frequency ωce, which leads tothe occurrence of harmonics of the cyclotron frequency in the scattering spectrum


1.0E−111.0E+05 1.0E+06 1.0E+07 1.0E+08 1.0E+09

log(ω /k)

1.0E−10

1.0E−09

1.0E−08

1.0E−07

1.0E−06

log(

Se,

i(k,ω

)Ion term

Electron term

Figure 3.28 The figure combines in alogarithmic plot the electron and theion term as given in Figure 3.26 andFigure 3.27 for the coherent scatteringcondition kλD = 0.8. It demonstrates

the clear separation of the electron andthe ion feature (note different scales com-pared to Figure 3.26 and Figure 3.27).The form factor S(�k, ω) is the sum ofthe two.

(see also Chapter 4). Depending on the scattering geometry, the scattering powerspectrum has maxima at frequency ω =ωs − ωi = lωc and is periodically reducedin between the peaks. Since the measurement of this structure is very difficult,on the one hand, and laborious, on the other hand, it has not gained diagnosticimportance; so it is not further discussed here.

Summarizing, plasma scattering of electromagnetic radiation is caused by the ac-celeration of the plasma electrons in the fields of the incident wave. The accelerationof the ions is negligible. The phases of the re-radiated waves of individual electronsdepend on the accelerating field at their position. The randomness of electronpositions and the density fluctuations connected with it are necessary to avoid thatindividual phases do not cancel out. The electron density fluctuations are caused bythe electrons themselves and are caused as well by the plasma ions surrounded bytheir electron cloud. The two phenomena lead to clearly separated features in thescattering spectra. The extent to which the ion dynamic determines the spectrumdepends on the size of kλD. The role of ions is negligible for incoherent scatteringcharacterized by kλD � 1. Under conditions of kλD ≤ 1, the scattering spectrumallows for the determination of the ion velocity distribution along the scatteringvector �k. This case can advantageously be realized with millimeter-waves. In thissense, the ion velocity distribution can basically be measured with coherent Thom-son scattering. Since the scattering cross section is very small, powerful microwave

References 115

sources delivering up to 1 MW of power need to be used to make the scatteringradiation detectable.

Exercises

3.1 What is the idea of the reference line in interferometry?3.2 Consider a density profile ne(r) = ne0[1 − (r/a)4]2 with ne0 = 1020 m− 3, a = 1 m.

What is the phase shift a central probing beam is suffering for three differentwavelengths λ1 = 1 mm, λ2 = 337 μm, λ3 = 10.7 μm.

3.3 As an estimate of the order of magnitude, consider the density profile ofExercise 3.2. The linear polarized probing beam with λ = 0.1 mm, orientedvertically, is passing the horizontal midplane at r = a/2. The �B-field is paralleland constant along the path, B|| = 0.1 T. Calculate the rotation angle due tothe Faraday effect.

3.4 Calculate the angle , as defined with Equation 3.21, caused by the Cotton–Mouton effect. The beam is linear polarized, has wavelength λ = 0.4 mm, andis passing the plasma perpendicular to the �B-field, in the horizontal midplane.Its plane of polarization is oriented 45◦ to the �B-field, B = 4 T constant alongthe path.

3.5 Verify the identities given in Equation 3.22, Equation 3.23, and Equation 3.24connecting the Stokes vector components with the angles χ and .

3.6 With the density profile of Exercise 3.2, calculate the total round trip timedelay td for a wave in O-mode polarization launched in distance d = 2 m tothe plasma edge. The wave is cutoff at density ne = 4 × 1019 m− 3.

3.7 Calculate the scattering angle ranges allowing for coherent scattering forincident waves with wavelengths λ1 = 1 mm, λ2 = 337 μm, λ3 = 10.7 μm.

References

1. Heald, M.A. and Wharton, C.B.(1965) Plasma Diagnostics with Mi-crowaves, John Wiley & Sons, Inc.,New York.

2. Veron, D. (1979) Submillimeter inter-ferometry of high-density plasmas, inInfrared and Millimeter Waves (ed. K.J.Button), New York, Academic Press.

3. Donne, A.J.H. (1995) Rev. Sci. Instrum.,66, 3407.

4. Geist, T., Wuersching, E., and Hartfuss,H.J. (1997) Rev. Sci. Instrum., 68, 1162.

5. Koponen, J.P.T. and Dumbrajs, O. (1997)Rev. Sci. Instrum., 68, 4038.

6. Dodel, G. and Kunz, W. (1978) InfraredPhys., 18, 773.

7. Soltwisch, H. (1986) Rev. Sci. Instrum.,57, 1939.

8. Jobes, F.C. and Mansfield, D.K. (1992)Rev. Sci. Instrum., 63, 5156.

9. Segre, S.E. (1978) Plasma Phys., 20, 295.10. Segre, S.E. (1995) Phys. Plasmas, 2, 2908.11. Segre, S.E. (1996) Phys. Plasmas, 3, 1182.12. Budden, K.G. (1985) The Propagation

of Radio Waves, Cambridge UniversityPress, Cambridge.

13. Ginzburg, V.L. (1964) The Propagationof Electromagnetic Waves in Plasmas,Pergamon Press, Oxford.

14. Abramowitz, M. and Stegun, I. (1972)Handbook of Mathematical Functions, 9thedn, Dover Publications, New York.

15. Mazzucato, E. (1998) Rev. Sci. Instrum.,69, 2201.

16. Hutchinson, I.H. (1992) Plasma Phys.Controlled Fusion, 34, 1225.


17. Zeng, L., Peebles, W.A., Doyle, E.J.,Rhodes, T.L., and Wang, G. (2007)Plasma Phys. Controlled Fusion, 49,1277.

18. Nazikian, R., Kramer, G.J., and Valeo, E.(2001) Phys. Plasmas, 8, 1840.

19. Bretz, N. (1992) Phys. Fluids, B4, 2414.20. Gusakov, E., Heureaux, S., Popov, A.,

and Schubert, M. (2012) Plasma Phys.Controlled Fusion, 54, 045008.

21. Conway, G.D., Kurzan, B., Scott, B.,Holzhauer, E., and Kaufmann, M. (2002)Plasma Phys. Controlled Fusion, 44, 451.

22. Hirsch, M., Holzhauer, E., Baldzuhn, J.,Kurzan, B., and Scott, B. (2001) PlasmaPhys. Controlled Fusion, 43, 1641.

23. Froula, D.H., Glenzer, S.H., Luhmann,N.C. Jr.,, and Sheffield, J. (2011) PlasmaScattering of Electromagnetic Radiation,2nd edn, Elsevier, Amsterdam.

24. Hutchinson, I.H. (2002) Principles ofPlasma Diagnostics, 2nd edn, CambridgeUniversity Press, Cambridge.

25. Bindslev, H. (1993) Plasma Phys. Con-trolled Fusion, 35, 1615.

26. Jackson, J.D. (1975) Classical Electrody-namics, 2nd edn, John Wiley & Sons,Inc., New York.

27. Matoba, T., Itagaki, T., Yamauchi, T.,and Funahashi, A. (1979) Jpn. J. Appl.Phys., 6, 1127.

28. Salpeter, E.E. (1960) Phys. Rev., 120,1528.

29. Salpeter, E.E. (1961) Phys. Rev., 122,1663.

30. Bindslev, H. (1992) On the theory ofThomson scattering and reflectome-try in a relativistic magnetized plasma.PhD thesis, Riso-R-663. Riso NationalLaboratory, Roskilde.

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