Thomson Scattering



One of the most powerful methods of diagnosis is to use the scattering of electromagnetic radiation from the plasma. The attractiveness of this diagnostic derives from two main features. First, it is, for all practical purposes, a nonperturbing method, requiring only access of radiation to the plasma. Second, it offers the potential of determining detailed information about the distribution function of electrons and sometimes even of the ions too. These advantages are sufficient to offset the fact that the measurements are generally very difficult technically to perform. Electromagnetic wave scattering diagnostics are now widespread, especially in hot plasma experiments.

5.1 General Scattering Theory

The process of electromagnetic wave scattering by charged (elementary) particles may be thought of as follows. An incident electromagnetic wave impinges on the particle. As a result of the electric and magnetic fields of the wave, the particle is accelerated. The charged particle undergoing acceleration emits electromagnetic radiation in all directions. This emitted radiation is the scattered wave.
Consider an electron in free space, whose unperturbed orbit is given by position r(t), scattering a monochromatic input wave
Ei(r,t)  =  Ei exp  i(ki ·r − ωi t)   .
(5.1)
It can be shown [1] from basic electromagnetic theory that this electron produces a scattered field at a distant point x whose Fourier spectral component a scattered frequency ωs is
Es (ωs) = re eiks ·x
2πx
 ⌠
⌡

T′ 
 κ′ ↔
Π
 
 ·Ei ei(ωt′− k ·r′) dt′   .
(5.2)















Fig 5.1 Coordinate vectors for the scattering process.
Figure 5.1 illustrates the geometry and the following, fully relativistic, definitions should be noted:
∧i is the input wave direction unit vector, ki/ki;
∧s is the scattered wave direction unit vector, x/x ( ≈ R/R);
ks is the scattered wave-vector, ∧s ωs/c;
ω is the "scattering" frequency, ωs−ωi;
k is the "scattering" wave-vector, ks−ki;
re is the classical electron radius, 2.818 ×10−15m;
primes denote evaluation at "retarded time" t′ ≡ t −(x − ∧s ·r′)/c;
κ ≡ 1 − ∧s · β relates retarded and normal time via
dt = κ′dt′ for a particle with speed v, β ≡ v/c, and
Π↔ is a tensor that transforms input wave polarization,
∧e ≡ Ei/Ei, to scattered wave polarization:
↔
Π
 
 ·
^
e
 
 ≡ ( 1 − β2 )1/2
κ3
^
s
 
 ∧⎧
⎨
⎩ 		⎡
⎣
^
s
 
 − β ⎤
⎦ 		∧⎡
⎣
^
e
 
 − ( β  · 
^
e
 
 ) β + ( β  · 
^
e
 
 )
^
i
 
 − ( β  · 
^
i
 
 )
^
e
 
⎤
⎦ 		⎫
⎬
⎭ 		   .
(5.3)
In general, we need to add up contributions of the form Eq. (5.2) from all the electrons in the scattering region. If one could assume the phases of all contributions were completely uncorrelated then the prescription would be simple. For any such case of incoherent summation we know that the powers add. So we need simply to take the modulus squared of the electric field and sum over all electrons. The problem is, though, that we do not know a priori that the phases are purely random, because plasma is a medium supporting all kinds of collective effects in which electron positions and motions are correlated. Thus, in some cases the summation requires coherent addition, which will give a very different result.
Now, the Debye shielding effects of a plasma cause a test charge to be surrounded by shielding charges in a cloud of characteristic size approximately the Debye length, λD. The combination of the charge and its shield is referred to as a dressed particle; consideration of the interaction of radiation with the combination of charge and shield enables us to see when particle correlations are important and when they are not, as follows.
A test electron has a shielding cloud (whose total charge is +e) consisting purely of electrons, or rather, the absence of them. This is because electrons move so much faster than ions (for comparable temperatures) that the ions "can't keep up" with the electron in order to contribute to its shielding. On the other hand, a thermal test ion is slow enough to allow the other ions to participate in the Debye shielding, so it will be surrounded by a cloud of roughly −e/2 total charge of electrons and −e/2 of (absence of) ions.
Now, if the phase difference between the scattering from an electron and from electrons in its shielding cloud is large, as will be the case if k λD >> 1, then the random distribution of the electrons within the cloud will be sufficient to ensure that the scattered fields of electron and shielding cloud are incoherent. In this case, no correlation alterations to the power are necessary and the total scattered power is a simple sum of single-electron powers.
If, on the other hand, k λD << 1, the contribution from test particle and cloud will add up coherently since there is negligible phase different between them. In this case we have coherent or "collective" Thomson scattering and we are then probing the collective behavior of the plasma. Also, the scattering from a test electron and its shielding cloud approximately cancel, so that ions tend to dominate.

5.2 Incoherent Thomson Scattering Theory

When k λD >> 1, the spectrum of the scattered power for an experiment with input polarization Ei perpendicular to the scattering plane defined by ∧s and ∧i and selecting scattered light polarized in the Ei direction can be be shown[1] to be
d2P
dΩs dωs
= r2e ⌠
⌡

V 
 < Si > d3 r ⌠
⌡ 		|
^
e
 
 · ↔
Π
 
 ·
^
e
 
 |2 κ2 f δ(k ·v − ω) d3v
= r2e ⌠
⌡

V 
 < Si > d3 r ⌠
⌡ 		⎢
⎢ 		1 −
⎛
⎝ 		1 −
^
s
 
 ·
^
i
 
⎞
⎠
( 1 − βi ) (1 − βs )
 βe2 ⎢
⎢ 		2
 
 ⎢
⎢ 		1 − βi
1 − βs
 ⎢
⎢ 		2
 
    ×( 1 − β2 ) f δ( k ·v − ω) d3 v   ,
(5.4)
where d2P/dΩs dωs is the power scattered per solid angle Ωs and per frequency interval dωs from a volume V, in which the time averaged input wave Poynting vector (power flux) is < Si > . Subscripts indicate vector components in the direction of i, s or e respectively. The terms inside the velocity space integral have a simple interpretation. The first, which is always less than or equal to 1, is the extent of depolarization of the radiation due to relativistic effects. The second, |1−βi|2/|1−βs|2, is simply ω2s/ω2i, the ratio of scattered to incident frequency squared. As such, it can be taken outside the integral, being independent of v⊥. The third term, (1−β2), can be thought of as due to the relativistic mass increase of the electron that decreases its scattering efficiency (remember re ∝ 1/me).
For very high temperatures, the higher order β terms must be included and the fully relativistic Maxwellian distribution used. The velocity integral of Eq. (5.4) can then be performed analytically only by making the simplifying approximation of treating the depolarization term as a constant that is independent of velocity. With this approximation one finds that the scattering can be evaluated [2] for a relativistic Maxwellian distribution with temperature T, and may be expressed in terms of a differential photon cross section as
ωs
r2e
 d2 σp
d ωs d Ωs
= q(T) {2K2 ( m0c2/T ) }−1 ω2r
√{1 − 2 ωr
^
i
 
 ·
^
s
 
 + ω2r }
×exp⎡
⎣ 		− m0c2
T
 √⎧
⎨
⎩ 		1 + ( ω2r − 1 )2
2 ωr ⎛
⎝ 		1 −
^
i
 
 ·
^
s
 
⎞
⎠
⎫
⎬
⎭ 		⎤
⎦ 		   .
(5.5)
where ωr ≡ ωs/ωi and q(T) is the appropriate mean value of the depolarization factor |1 − β2e(1 − ∧i ·∧s)/(1 − βi)(1 − βs)|2, slightly smaller than 1.
Although the modified Bessel function K2 may be retained in this formula, a more convenient expression is obtained by using the asymptotic approximation
[2K2( x ) ]−1 ≈ ⎡
⎣ 		x
2 π
 ⎤
⎦ 		1/2
 
 ex ⎡
⎣ 		1 + 15
8 x
 ⎤
⎦ 		   .
(5.6)
valid for x = m0c2/T >> 1. The value of q may be estimated by substituting typical thermal values of β, βt  ∼ √(T/moc2), into the depolarization factor, giving q ≈ (1 − T/m0c2)2 for 90o scattering. The resulting expression agrees with an exact numerical integration to within negligible error for practical purposes [3]. The shape of the theoretical spectra is illustrated in Fig. 5.2 for several different electron temperatures.


























Figure 5.2: Spectral shapes for relativistic Thomson scattering.
These fully relativistic expressions, though somewhat complicated, have to be used in order to obtain accurate results in high temperature plasmas. When the electron temperature is less than a few hundred eV, the non-relativistic "dipole approximation" can be used.
In the dipole approximation, these expressions become particularly simple because the quantity κ2 | Π↔ ·∧e |2 reduces to |∧s ∧(∧s ∧∧e)|2 which is independent of v. The velocity integral can then be performed trivially to obtain
d2P
d Ωs dωs
 = ⎡
⎣ 		re2 ⌠
⌡

V 
 < Si > d3 r |
^
s
 
 ∧(
^
s
 
 ∧
^
e
 
 )|2 ⎤
⎦ 		fk ⎛
⎝ 		ω
k
⎞
⎠ 		1
k
    ,
(5.7)
where fk(v) is the one-dimensional velocity distribution in the k direction:
fk(υk) ≡ ⌠
⌡ 		f ( v⊥ , υk ) d2 v⊥   .
(5.8)
(⊥ denotes perpendicular to k.) For a Maxwellian distribution this is
fk = ne ⎛
⎝ 		me
2 πTe
⎞
⎠ 		1/2

 
 exp⎛
⎝ 		− me υk2
2Te
⎞
⎠ 		  .
(5.9)
The potential power of this result is clear. The frequency spectrum (for fixed scattering geometry) is directly proportional to the velocity distribution function, giving, in principle, complete information of the electron distribution in one dimension along k.
A typical scattering spectrum is shown schematically for this case in Fig. 5.3. The spectral shape is proportional to the distribution function.


















Figure 5.3: The scattered spectrum in the dipole approximation is directly proportional to the one-dimensional velocity distribution with υk = (ωs − ωi)/k.
Unfortunately, it is rather rare for the results of a practical scattering experiment to be sufficiently accurate as to provide detailed information on the precise shape of the distribution function. Signal-to-noise limitations usually require that a specifically chosen curve shape (Maxwellian) be fitted to the spectrum obtained, thus in effect measuring Te from the width and ne from the height, providing only moments of the distribution function. Nevertheless, obtaining both the temperature and the density from a single diagnostic, with excellent spatial resolution, makes Thomson scattering very powerful.

5.3 Incoherent Thomson Scattering Experiments

The order of magnitude of incoherent scattering is determined simply by the total Thomson cross section σt = 8 πre2/3, which is a fundamental constant σt = 6.65 ×10−29 m2 (re = 2.82 ×10−15m). Thus if a beam of radiation traverses a length L of plasma of density ne, a fraction σt ne L of the incident photons will be incoherently scattered. In most laboratory plasmas this fraction is very small; for example, if ne = 1020 m−3 and L = 1 m, σneL = 6.65 ×10−9 and less than 10−8 of the photons are scattered. Of these photons, an even much smaller fraction will be detected, since one usually collects scattered radiation only from a short section of the total beam length (perhaps  ∼  1 cm) with collection optics that subtend only a rather small solid angle (perhaps 10−2 sr). The fraction of scattered photons collected is then (10−2m/1m) × (10−2sr/4π sr)  ∼ 10−5. Thus, of the input photons, only perhaps 10−13 will be collected.
This fact is the source of most of the practical difficulties involved in performing an incoherent scattering experiment. The first requirement that it forces upon us is that we must have a very intense radiation source in order to provide a detectable signal level. That is why the measurements are almost always performed with energetic pulsed lasers. Actually, of course, the number of scattered photons observed is proportional to the total incident energy (regardless of pulse length) for a given frequency. However, the noise from which the signal must be discriminated will generally increase with pulse length. Hence, high incident power as well as high energy is required.
A schematic representation of a typical incoherent Thomson scattering configuration is shown in Figure 5.4. The input laser beam is allowed to pass through the plasma, as far as possible avoiding all material obstructions.























Figure 5.4: Typical configuration for an incoherent Thomson scattering experiment.
Naturally, it must pass through a vacuum window at its entrance (and possibly exit) to the plasma chamber. At these points unwanted scattering of the laser beam occurs, even from the most perfect windows, whose intensity can far exceed the plasma scattering. This is the second important restriction we face arising from the small magnitude of the Thomson cross section: the need to avoid detection of this "parasitic radiation," usually called stray light. The baffles indicated in the figure are often used to reduce the stray light. Also, removing the windows and other optics far back into the ports is another way to reduce stray light, and the purpose of a viewing dump, when present, is primarily to reduce the effect of multiple scattering of the stray light from vacuum surfaces finally entering into the collection angle. In other words, the viewing dump provides a black background against which to view the scattered light.
Despite these types of precautions the stray light may often still exceed the Thomson scattered light. Fortunately, it can be discriminated against by virtue of the fact that the stray light appears precisely at the input frequency, whereas the Thomson scattered spectrum is broadened out from ωi by the Doppler effect, which is our main interest. Thus, provided we avoid the frequency ωi, we can avoid the stray light. Various filters and high rejection spectral techniques exist for this purpose.
Another very important source of noise is in light emitted from the plasma due to bremsstrahlung or particularly line radiation. This is often a very serious problem.
A numerical example discussed by Sheffield [4] is that of scattering of ruby laser (λ = 694.3 nm) light from a plasma with parameters Te = 500 eV, ne = 2.5 ×1019 m−3, L = 0.7 cm, Ωs = 2.3 ×10−2 sr, scattering angle θ = 90o, and detector quantum efficiency Q = 0.025. The number of scattered photons detected when the laser pulse energy is W joules is QNs = 2.7 ×102W. For a Q-switched laser pulse of duration 25 ns, a bremsstrahlung estimate of background light suggests that less than one noise photon should be detected. On paper then, the signal to noise with a 6 J pulse looks very good. However, it was found in practice that the background plasma light was about 500 times more intense than expected. The source of this extra light was identified as primarily line radiation, against which no special precautions were taken. This experience serves as a cautionary tale against relying too heavily upon low background light calculations based upon bremsstrahlung alone.
























Figure 5.5: Schematic illustration of the configuration used for multiple point Thomson scattering (TVTS) (after Bretz et al. [5]).
A more modern example illustrating the information to be gained from incoherent scattering is depicted in Fig. 5.5, which shows the so called TV Thomson scattering (TVTS) system originally developed at Princeton University by Bretz et al. [5]. It gains its name from the intensifier tube in the detection system that uses principles similar to a television to receive a two-dimensional image from the detection spectrometer. The optics are arranged so that one axis of this image represents different wavelengths while the other receives scattered light from different spatial positions in the plasma. Thus, a large number of scattered spectra are measured simultaneously, providing good spatially resolved measurements of plasma temperature and density as illustrated in Fig. 5.6.




























Figure 5.6: Typical results from the TVTS system on the Princeton large torus.
The ruby laser has dominated incoherent plasma scattering experiments since its availability first made them possible in the 1960s. It still offers a well proven way of meeting the requirements of incoherent scattering, primarily high power and energy with good beam quality (low divergence, etc.) at a wavelength (694.3 nm) where sensitive detectors are available. However, recent detector developments have made possible the use of neodymium lasers (λ = 1.06 μm) with avalance photodiodes for scattering experiments. The major advantage that such lasers have, particularly when yttrium aluminum garnet (YAG) is used as the solid state laser medium, is the ability to fire repetitively at up to  ∼  100 Hz. This can then enable the time evolution of electron temperature and density to be followed in long plasma pulses typical of modern fusion experiments, as had been demonstrated, for example, by Rohr et al. [6]. An incidental advantage also observed is that the plasma light due to impurity line radiation is often less troublesome near 1 μm because there are fewer lines there.

5.4 Coherent Scattering

When scattering is performed with k λD < 1, correlation between the electrons must be taken into account, and a coherent sum of scattered electric fields calculated. This can be done quite readily [1] and yields
d2P
dωs dΩs
 = r2ePi
2πA
 | ↔
Π
 
 ·
^
e
 
 |2 ne VS (k, ω)   ,
(5.10)
where
S(k, ω) ≡ ( 2 π)8
neTV
 |Ne ( k, ω)|2   .
(5.11)
and
Ne ( k, ω) ≡ 1
( 2 π)4
 ⌠
⌡

T′ 
 ⌠
⌡

V 
 Ne ( r′ , t′) ei(ωt′− k ·r′) dt′d3 r′  .
(5.12).
is the Fourier transform of the total density fluctuation level, including the discreteness of the particles. Also V is the scattering volume, A is its cross-section, Pi is the input power, and T is the duration of the scattering.
In a uniform, unmagnetized plasma, S(k, ω), the scattering form factor, is
S(k, ω) = 2 π
kne
 ⎡
⎣ 		⎢
⎢ 		1 − χe
1 + χe + χi
 ⎢
⎢ 		2
 
 fek ⎛
⎝ 		ω
k
⎞
⎠ 		   +   ⎢
⎢ 		χe
1 + χe + χi
 ⎢
⎢ 		2
 
∑
i 
 Zi2 fik ⎛
⎝ 		ω
k
⎞
⎠ 		⎤
⎦ 		   ,
(5.13)
where χe,i is the plasma susceptibility due to electrons or ions, and fk is the particle distribution function in the direction of k.
In a magnetized plasma, on the other hand, this expression is modified in two important ways. The first is that χe,i are the susceptibilities for a magnetized plasma. The second is that fk must be replaced by a sum over cyclotron harmonics [4,7]. A species j requires this substitution unless Ωj << ω. If Ωj >> ω, then the substitution is simply:
fjk ⎛
⎝ 		ω
k
⎞
⎠ 		→ k
k||
 fj|| ⎛
⎝ 		ω
k||
⎞
⎠
(5.14)
where f|| is the particle distribution in the direction parallel to B. This is because the particle species is effectively constrained to move only in the parallel direction.
The attractive feature of collective scattering is that it gives an opportunity to obtain a rather direct measurement of f(υ) for the ions, when the ion term in equation (5.13) is dominant. The shape of the ion feature is affected by any resonances, where the dielectric constant (1 + χe + χi) is zero, but by judicious choice of scattering geometry these can sometimes be avoided.
There are numerous practical difficulties involved with detecting the ion feature. Many of these are common to incoherent scattering too but some arise specifically because of the requirements of coherent scattering. In particular the requirement k λD \lsapprox 1 enforces longer scattering wavelength (smaller k-scattering). This can only be achieved by decreasing the scattering angle or increasing the incident radiation wavelength. The former approach has limits, since the scattered radiation must be separated from the input beam. As a result only rather dense plasmas (typically ne \grapprox 1022 m−3) are suitable for detection of the ion feature using visible radiation.
For more typical plasmas it is essential to use longer wavelength radiation such as that obtained from a CO2 laser (10.6 μm) or specially developed far infrared lasers (λ approximately a few hundred micrometers). There have been one or two experiments on Tokamaks [8,9] and more elsewhere [10] that have observed the thermal bulk ion feature. However, the technical difficulties are so great that there seems little prospect of this becoming a useful routine diagnostic.
A recently proposed application is to diagnose fusion-produced alpha particles using their collective scattering feature. Although this seems at least as difficult as observing the thermal bulk ion feature (the signal is smaller, though the spectrum is broader) there seem not to be many alternatives for the vitally important measurement of fusion-product behavior inside the plasma. Therefore, experiments are under serious consideration [11].
In addition to the density fluctuations that arise in an otherwise uniform plasma owing to the discreteness of the particles, most laboratory plasmas experience density fluctuations caused by various types of instability within the plasma. Their frequencies may extend from very low frequency up to the characteristic frequencies of the plasma (ωp, etc.). Considerable interest focuses upon the ability to diagnose these fluctuations because, in the case of "naturally" occurring fluctuations, they can be responsible for enhanced transport, while in the case of deliberately excited waves, such as those launched for heating purposes, internal detection allows the wave dynamics to be investigated directly.
Generally, the fluctuation levels encountered far exceed the thermal levels; so by judicious choice of incident frequency, the detection problems can be made considerably easier than they are for thermal scattering.
The equations governing this process are again simply those we have had before [Eqs. (5.10 - 5.14)] except that now we need not consider the Klimontovich density, Ne but instead can ignore the discreteness of the particles and use the smoothed density ne:
Es (ωs) = re ei k ·x
2 πx
 ↔
Π
 
 ·Ei ⌠
⌡

T 
 ⌠
⌡

V 
 ne e−i(k ·r − ωt) dt  d3 r  .
(5.15)
When, as is often the case, we wish to characterize a density fluctuation spectrum ne(k, ω) that is broad in k and ω, that is, a rather turbulent spectrum, the frequency spectrum may be obtained by appropriate frequency analysis of the scattered waves. The k spectrum, on the other hand, is most easily obtained by varying the scattering angle so that k = 2kisinθ/2 scans an appropriate domain. Sometimes it is convenient to observe scattering simultaneously at various different scattering angles so as to obtain reasonably complete k information simultaneously. Figure 5.7 shows an example of such a setup and Figure 5.8 shows some typical k and ω spectra.


























Figure 5.7: An example of a scattering system design for simultaneous measurement of S(k) at various k values (after Park et al. [12]).
Although CO2 laser radiation has been extensively used for the purposes of density fluctuation measurements, its wavelength (10.6 μm) tends sometimes to be rather smaller than desirable. As a result, very small scattering angles are required, which usually prevent one from obtaining spatial resolution along the incident beam. Figure 5.9 illustrates this point. In this respect longer wavelength lasers in the submillimeter spectral region prove more satisfactory, although their technology is less well developed. Microwave sources have also been used extensively. Their main drawback is that the frequency tends to be so low that the beam suffers from considerable refraction by the plasma. Also, diffraction limits the minimum beam size obtainable. From the theoretical point of view, the treatment we have outlined presupposes ωi >> ωp, which may not be well satisfied for microwaves. Moreover, fluctuation wave numbers greater than 2ki are not obtainable so relevant parts of the k spectrum may not be accessible with low ki microwaves.






















Figure 5.8: Typical fluctuation spectra obtainable from a collective scattering experiment (after Semet et al. [13]).






















Figure 5.9: Scattering with shorter wavelength radiation gives poorer spatial resolution along the beam than longer wavelength (for the same k).
More often that not coherent detection techniques are used together with a continuous, rather than pulsed, source. A particular reason for using heterodyne, rather than homodyne, detection is that it enables the direction of propagation, for a given ks, to be determined. Waves give rise to either positive or negative frequency shift, ω, depending on their propagation direction. Heterodyne detection, often achieved with a local oscillator a few MHz or so from the scattering input frequency, can determine the sign of the shift. This has been significant recently in indicating the presence of fluctuations travelling in the ion diamagnetic drift direction on tokamaks, as Fig 5.10 illustrates. These observations may prove to be crucial in determining the underlying causes of turbulent transport.





























(a) ―ne = 1.8×1019 m−3    (b) ―ne = 7.5×1019 m−3
Fig 5.10 Frequency spectra for two discharge conditions in the TEXT tokamak at two scattering wave-vectors. Negative frequency corresponds to the electron diamagnetic drift direction. (Brower et al, 1988 [14]).

References

[1]
I.H. Hutchinson Principles of Plasma Diagnostics, Cambridge University Press, New York, (1987).
[2]
V.A. Zhuravlev and G.D. Petrov, Soviet J. of Plasma Phys., 5, 3 (1979).
[3]
A.C. Selden Culham Laboratory Report CLM R220 (1982).
[4]
J.Sheffield Plasma Scattering of Electromagnetic Radiation, Academic, New York, (1975).
[5]
N. Bretz, et al, Appl. Opt. 17, 192 (1978).
[6]
H. Rohr et al, Nucl. Fusion 22, 1099 (1982).
[7]
e.g. T.P. Hughes and S.R.P. Smith, Nucl. Fusion 21, 1043 (1988).
[8]
P. Woskoboinikow et al, 11th Europ. Conf. on Controlled Fusion and Plasma Phys., Europhysics Conference Abstracts, Vol 7D, Part II, p81 (1983).
[9]
R. Behn, et al, Proc. 10th Int. Conf. on IR and MM waves, Lake Buena Vista, FL, p143 (1985).
[10]
W. Kasparek and E. Holtzhauer, Phys.Rev. A 27, 1737 (1983).
[11]
e.g. A.E. Costley et al, JET report R(88)08 (1988).
[12]
H. Park et al, Rev. Sci Instr. 53, 1535 (1982).
[13]
A. Semet et al, Rev. Sci. Instr. 45, 445 (1980).
[14]
D.L. Brower, W.A. Peebles, S.K. Kim, and N.C. Luhmann, Rev. Sci. Instr. 59, 1559 (1988).


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