Boyd T.J.M., Sanderson J.J. The Physics of Plasmas

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11.5 Stimulated Raman scattering 447

propagation of forward scattered light from the SRFS resonance to its reﬂection

density and then back to the SRBS resonance. The feedback loop established in

this way allows ampliﬁers, which are convective in the absence of feedback, to

grow temporally at those frequencies for which the two ampliﬁers are in phase.

11.5.3 Numerical solution of the SRS equations

The set of coupled equations (11.32) contains all the models that describe the

time-asymptotic state of linear SRS theory and have been solved by Barr, Boyd

and Mackwood (1994) for a plasma slab with appropriate boundary conditions.

The density proﬁle within the slab is arbitrary and this region is bounded by

semi-inﬁnite regions of homogeneous plasma. In these regions the four ﬁrst-order

equations may be solved exactly, the solutions corresponding to left and right

propagating electromagnetic and Langmuir waves. In the presence of a pump

ﬁeld the solutions are coupled and hence are neither purely electromagnetic nor

electrostatic. The set of equations (11.32) is solved in two distinct forms. Solution

as an eigenvalue problem determines the complex eigenfrequencies; in this case all

initial value terms are set equal to zero and the equations form a (mathematically)

homogeneous system. The second form of solution determines convective gain

factors at frequencies distinct from the eigenvalues and for this, initial value source

terms have to be retained.

A range of physical parameters was chosen to exploit the many features of the

model. A linear density ramp overdense to forward scattered light was used to allow

for feedback. The parameters essential for interpreting SRS characteristics and en-

abling comparisons to be made with experiment include the Landau damping of the

Langmuir wave, the collisional damping of both daughter waves, the convection of

each wave, the degree of feedback between back- and forward-scattering ampliﬁers

and the physical extent of the feedback loop.

Figure 11.7 shows the threshold contour C

= exp 2π for Raman back-scatter.

This shows two distinct regimes separated at a scattered frequency ω

= 0.63ω

The approximately horizontal section of the gain contour at lower frequencies

agrees well with the convective gain predicted by (11.45). At frequencies ω

0.65ω

the steep rise in threshold corresponds to the onset of a cut-off produced

by Landau damping. A rule of thumb is often applied for the onset of Landau

damping when kλ

= 0.3 which, for the temperature used to produce these results,

corresponds to ω

= 0.7ω

. Note that this cut-off is not predicted by the convective

damping threshold (11.28) which gives zero threshold in the absence of collisions.

Figure 11.8 plots the threshold predicted by (11.29). There is good agreement

between the Landau cut-off seen here and the result in Fig. 11.7.

The absolute character of the instability is sustained over the whole range of

emission frequencies on account of the feedback between back- and forward-

448 Aspects of inhomogeneous plasmas

Fig. 11.7. Convective gain contour showing the threshold gain contour C

= exp 2π for

SRBS from a linear density ramp underdense to forward-scattered light.

Fig. 11.8. Thresholds predicted by (11.29) for SRS from plasmas with Z = 10 (——), 30

(– – –), 50 (— · —) and 79 (······).

Raman resonances discussed above. Figure 11.9 shows absolute Raman growth

rates as a function of ω

/ω

for a Z = 10 plasma, with the corresponding thresh-

olds plotted in Fig. 11.10 which also shows thresholds for a Z = 79 plasma.

Raman scattering from quarter-critical density n

/4 shows the most rapid growth.

Figure 11.10 shows some differences between plasmas with low and high Z .For

11.5 Stimulated Raman scattering 449

Fig. 11.9. Absolute SRS growth rates versus scattered frequency for a Z = 10 plasma for

/c = 0.1(◦), 0.07 () and 0.04 (♦).

Fig. 11.10. Absolute SRS thresholds versus scattered frequency for Z = 10 () and Z =

79 (×) plasmas.

low Z (weak collisionality), thresholds increase only slowly with ω

up to the

onset of Landau cut-off. By contrast, absolute thresholds in the case of a strongly

collisional plasma rise steeply with ω

on account of the effects of damping on the

waves within the feedback loop. This proves to be the dominant effect so that no

Landau cut-off is evident for strongly collisional plasmas.

450 Aspects of inhomogeneous plasmas

11.6 Radiation from Langmuir waves

In this section we consider a laser–plasma interaction in the presence of a density

gradient so steep that WKBJ theory cannot be relied on to provide insights into the

physics. The interaction in question is one in which Langmuir waves are coupled to

the radiation ﬁeld. Although longitudinal and transverse modes are always coupled

in inhomogeneous plasmas, the coupling is generally weak for plasmas in thermal

equilibrium. However, in plasmas in which a non-thermal spectrum of Langmuir

waves is excited, steep density gradients may lead to signiﬁcant radiation from the

plasma over a narrow band around the plasma frequency.

Radiation by Langmuir waves ﬁrst attracted attention in attempts at interpreting

the characteristics of certain types of emission from the Sun, in particular type III

solar radio noise. One of the striking characteristics of type III emission spectra is a

drift in frequency over time, predominantly from high to low frequencies. Not only

is the bandwidth narrow but in many instances a second harmonic band is observed.

These characteristics along with the sudden onset of the emission, are consistent

with a conjecture put forward by Wild (1950) to interpret type III emission. The

sequence of events starts with bursts of energetic electrons produced in solar ﬂares

injected into the chromosphere, travelling outwards through the corona exciting

Langmuir waves in the coronal plasma. The sudden onset of type III bursts is

consistent with a threshold set by the collisional damping of Langmuir waves

in the colder and denser chromospheric plasma. Likewise the bandwidth of the

emission should reﬂect the narrow bandwidth of the Langmuir waves determined

by Landau damping. Wild’s original conjecture has since been supported by the

direct observation of both electron beams and the Langmuir waves they generate,

from satellite observations of type III bursts (see Lin et al. (1986)). The appearance

of a second harmonic in some, though by no means all, recorded type III spectra

is another signature pointing to Langmuir waves as the source of the emission. By

and large more is known about the excitation phase in which suprathermal levels of

Langmuir waves are generated, than the coupling phase, in which electrostatic en-

ergy is converted into radiation, with the generation of both a fundamental plasma

line and its second harmonic.

In a very different corner of parameter space, high intensity laser interactions

with dense plasmas afford another example of Langmuir waves coupling to the

radiation ﬁeld. These interactions lead to jets of energetic electrons which can

excite suprathermal levels of Langmuir waves in the superdense plasma which

in turn may radiate in the very steep density gradients present. The fact that this

radiation is generated at the plasma frequency which, for sufﬁciently overdense

plasmas, is far above that of the incident light suggests that plasma emission may,

under suitable conditions, have potential as a source of XUV light.

11.6 Radiation from Langmuir waves 451

Fig. 11.11. Normalized frequency spectrum showing laser harmonics in reﬂection and

plasma line emission for a plasma with n/n

= 30 and a

= 0.5; m = ω/ω

The problem of plasma emission was revisited by Boyd and Ondarza-

Rovira (2000) in PIC simulations of the interaction of moderately intense laser

light with slab plasmas of density up to 200 times critical density. Figure 11.11

shows the spectrum of back-reﬂected light from a plasma of density n/n

= 30

ﬁrst observed by Ondarza-Rovira (1996). In this spectrum the plasma line at

ω/ω

= (n/n

)

1/2

 5.5 appears as a dominant feature against the background

of harmonics of the incident laser light, comparable in intensity to the third laser

harmonic. Moreover, the plasma line is in reality a broad feature in the spectrum

reﬂecting the fact that in practice there will be some plasma emission at normalized

frequencies up to (n/n

)

1/2

. However, at lower densities the intensity is corre-

spondingly lower and is swamped by low harmonics of the incident laser light.

The spectrum in Fig. 11.11 shows that as the harmonic line spectrum weakens

beyond m = 5 the broad plasma line becomes increasingly dominant. There is

no clear indication of a second harmonic of the plasma line at 2ω

 11ω

due in part to the fact that this is a weak feature for the parameters chosen and

in part to the dominance of the broad feature that appears in the spectrum on

the blue side of the plasma line centred about ω  9ω

with intensity about an

order of magnitude below that of the plasma line. This unexpected spectral detail

turns out to be a robust feature in the simulations. Figure 11.12 reproduces the

reﬂected spectrum from a plasma of density n/n

= 200, again showing a feature

at ω  1.5ω

with intensity about an order of magnitude weaker than the plasma

line. In addition a second harmonic of the plasma line appears in the spectrum with

a weak third harmonic, shifted slightly to the blue. Surprisingly, additional lines

452 Aspects of inhomogeneous plasmas

Fig. 11.12. Reﬂected harmonic spectrum for n/n

= 200 and a

= 0.5.

Fig. 11.13. Electron trajectories for n/n

= 100 and a

= 0.5.

appear at approximately fourth and ﬁfth harmonics of the plasma line. The PIC

simulations showed clear evidence of electrons of relativistic energy from direct

forward acceleration by the v × B force at high intensities.

Figure 11.13 shows electron trajectories with electron velocities spread over a

wide band and moreover, evidence of beam-like behaviour, with electrons pene-

trating into the dense plasma. Across this region of penetration, Langmuir waves

excited by these electrons are detected. The simulations show strongly driven

Langmuir waves at the front edge of the plasma slab and this localization of the

11.7 Effects in bounded plasmas 453

Langmuir spectrum is important for the subsequent coupling of the plasma waves

to the radiation ﬁeld by means of the localized density gradient in the (perturbed)

peak density region.

Plasma emission from laser-produced plasmas has been observed by Teubner

et al. (1997) who detected a plasma line and a second harmonic although in fact

the harmonic appeared at a frequency a little over 1.6 times the fundamental. They

attribute the plasma line to surface emission and interpret the harmonic as due

to radiation from Langmuir plasmons in the interior where the plasma density is

assumed to be lower. On these grounds they expect the harmonic line to be more

intense than the fundamental. However, in the simulations by Boyd and Ondarza-

Rovira the opposite is the case, with the second harmonic typically between two

and four orders of magnitude weaker than the fundamental. If one recalls that the

line at ∼ 1.5 ω

is between two and three orders of magnitude more intense than

the second harmonic it is possible that it is this feature which has been detected

by Teubner et al. given that the relative frequencies of the lines they observe are

approximately 1:1.6.

Other aspects of the emission have been reported by Lichters, Meyer-ter-Vehn

and Pukhov (1998) in simulations in which second and third harmonics of the

plasma line were seen but not the fundamental. Spectra were recorded after the

laser pulse had ﬁnished so that no laser harmonics are present. The absence of any

effect in the spectrum in the region of the feature found by Boyd and Ondarza-

Rovira suggests that the presence of the plasma line is critical for its appearance.

11.7 Effects in bounded plasmas

Boundedness and inhomogeneity in a sense go hand in hand since a bounded

plasma is always inhomogeneous to some degree. Any attempt to classify modes

as was done in Chapter 6 would be to miss an essential point about bounded, inho-

mogeneous plasmas where the boundary and the structure of the density proﬁle are

integral in determining wave characteristics. Not only do plasma boundaries mod-

ify the dispersion characteristics of modes within the plasma (the ‘bulk’ modes) but

distinct modes may propagate along the surface of the plasma (‘surface’ waves) in

addition to bulk modes.

11.7.1 Plasma sheaths

We look next at the inhomogeneous layer that is formed where plasma is in contact

with a material boundary. Close to any wall or at any surface of contact with

plasma, a sheath is formed across which electron and ion densities no longer

balance. The reason for this stems from the very much greater mobility of electrons

454 Aspects of inhomogeneous plasmas

over ions (see Section 8.2.1). The net ﬂux of electrons means that any surface in

contact with a plasma rapidly acquires a negative charge. The potential resulting

from this then acts to reduce the electron ﬂux and enhance the ﬂux of positive ions

to the surface until a stage is reached where the two balance and the net current

to the surface disappears. In this steady state the surface potential is known as the

ﬂoating potential.

Sheath dynamics is one of the classical problems of plasma physics having been

identiﬁed and largely resolved by Langmuir. It requires the solution of Poisson’s

equation along with the dynamical equations for both ions and electrons with

suitably chosen boundary conditions. The plasma parameters are strongly inho-

mogeneous across the sheath and inhomogeneity, added to the non-linearity of the

governing equations, means that a general solution can only be found numerically.

Exceptionally, in plane geometry an approximate analytical solution is possible.

The key approximation lies in the distinct assumptions made about electron and

ion dynamics. For the electron ﬂuid, the pressure gradient is dominant over the

momentum term while for the ion ﬂuid the reverse is the case, i.e. the ions are cold.

With these assumptions we may integrate the respective equations of motion for

the two ﬂuids. For the electrons

(x) = n

exp[eV(x)/k

] (11.48)

where V (x) is the electrostatic potential and we have made use of the boundary

condition V (x →∞) = 0andn

(x →∞) = n

. Integrating the ion momentum

equation and the continuity equation gives, for a hydrogen plasma,

(x) = n



1 −

2eV(x)



−1/2

(11.49)

where u

= u

(x →∞). The spatial variation of electron and ion densities across

the sheath is illustrated in Fig 11.14. If we now make use of (11.48) and (11.49) in

Poisson’s equation we ﬁnd

V (x)



exp



eV(x)



−



1 −

2eV(x)



−1/2



(11.50)

This non-linear equation is the plasma sheath equation in plane geometry. Be-

fore integrating (11.50) it helps to rewrite it in terms of dimensionless variables

φ =−

M =

)

1/2

ξ =

11.7 Effects in bounded plasmas 455

Fig. 11.14. Sheath formation at a plasma boundary.

With these deﬁnitions (11.50) reads

dξ



1 +

2φ



−1/2

− e

−φ

(11.51)

Multiplying by dφ/dξ allows us to integrate (11.51) once to give



dφ

dξ



= M





1 +

2φ



1/2

− 1



+ (e

−φ

− 1) (11.52)

Note that (11.52) effectively determines φ(ξ) in terms of the ‘Mach number’ M.

A second integration can only be done numerically unless we make an additional

approximation. If we conﬁne our attention to the plasma side of the sheath where

φ is small we may approximate (11.52) by



dφ

dξ





1 −



(11.53)

which has a monotonic solution provided M

> 1. In other words





1/2

(11.54)

456 Aspects of inhomogeneous plasmas

a result due to Bohm (1949) and known as the Bohm sheath criterion. A sheath

forms at a material boundary provided the streaming velocity of ions entering the

region close to a wall exceeds the ion acoustic velocity c

If we return to (11.51) and approximate conditions within the sheath proper by

supposing that we may neglect the electron contribution (this would only be valid

for a wall or probe surface at a high enough negative potential, |φ|1) then

(11.51) reduces to

dξ



(2φ)

1/2

(11.55)

Integrating (11.55) twice across the sheath, assuming dφ/dξ = 0 at the edge,

conﬁrms that the sheath is typically a few Debye lengths thick.

11.7.2 Langmuir probe characteristics

This outline of sheath properties enables us to understand and interpret Langmuir

probe characteristics. Langmuir probes provide reliable electron temperature and

density diagnostics in relatively cool, low-density plasmas. The probe itself is a

small metal electrode – cylindrical, spherical or in the shape of a disk – inserted

into the plasma. The sheath that envelops the probe shields the plasma from the

probe potential.

The essence of the Langmuir probe technique is to monitor the current to the

probe as the probe voltage changes. Assuming that current is positive when it ﬂows

out from the probe, ion current drawn to the probe will be negative. The probe

characteristic is shown in Fig. 11.15 as a current–voltage plot. For a potential more

negative than the ﬂoating potential V

the electron contribution to the currrent drops

off until the probe draws only ion current given by the Bohm value, i.e.

eu

A =



πm



1/2

(11.56)

Here n

denotes the plasma ion density at the edge of the sheath and A the surface

area of the probe. If we choose the sheath edge to be the point at which u

≡

2(2k

/πm

)

1/2

and with a potential V

−k

/2e relative to the plasma then

 n

so that

= n

exp(eV

)  0.6n

The Bohm ion saturation current I

is therefore

 0.24n





1/2

(11.57)