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LIBRARY

Michigan State
University

 

This is to certify that the

thesis entitled

THE INTERACTION 0F ELECTROMAGNETIC

RADIATION WITH A BOUNDED PLASMA
presented by

Andrew RostysIaw MeInyk

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Ph2§iCS

 

Date November 13, 1967

 

0-169

ABSTRACT

THE INTERACTION OF
ELECTROMAGNETIC RADIATION
WITH A
BOUNDED.PLASMA

by Andrew Rostyslaw Melnyk

The interaction between.e1ectromagnetic.waves and
a bounded plasma is studied by extending the.Fresnel equa-
tions of reflection and transmission to include-plasma
waves with irrotational electric fields.. The preperties
of the plasma are incorporated in the dispersion rela-
tions for propagating waves. These dispersion relations
are obtained from a frequency and complex wave vector
dependent dielectric tensor §(§,w) calculated from a
linearized Boltzmann transport equation. .Results show
structure in the absorptance, reflectance, and trans-
mittance spectrum of.thin.p1asma slabs (d <.2nc/wp) due
to resonance phenomena.at frequencies where kd = nu;
(n = l, 3, 5,...). .These results suggest.that the dis-
persion relations of metallic plasmas such.as silver films

may be optically.measured.

THE.INTERACTION OF
ELECTROMAGNETIC RADIATION
WITH A
BOUNDED.PLASMA

By

Andrew Rostyslaw Melnyk

A.THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

.Department of Physics

1967

ACKNOWLEDGEMENTS

I wish to thank Professor Michael J. Harrison-for
suggesting this research problem, for his advice and
his continued encouragement throughout the course of the
work. Also I extend my thanks to Professor Truman O.
Woodruff, Professor Donald J.Montgomery, Dr. Paul
Petersen, and Mr. Don.Slanina for helpful discussions.
Finally I am grateful to the National Science Foundation
for financial support during the course of this investi-

gation.

ii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS

LIST OF
LIST OF

Chapter
I.

II.

III.

FIGURES
APPENDICES

INTRODUCTION . . . . . . . . . .

1. Fundamentals
Solid State Plasmas
Acoustic Plasma Waves
Plasma Waves .
2. Interactions of Plasmas with Electro-
magnetic Waves

THEORY OF REFLECTION AND TRANSMISSION . .

1. Equations of Transmission and Reflection
for a Medium Bounded by a Plane .
Snell's Law . . . . . . . .
Boundary Conditions
Fresnel Equations
General Equations .
2. Properties of the General Equations
Energy Conservation .
3. Equations of Transmission and Reflection
for a Slab . . . . . . . . .

DISPERSION RELATIONS FOR A COLLISIONAL
PLASMA . . . . . . . .

Theoretical Formulation .

Calculation of the Distribution Function

The Conductivity Tensor . . . . .
Evaluation of a and R

Dispersion Relat1ons

b “NH

iii

Page

ii

. viii

MMNH I-"

O‘

Table of.Contents/cont.

Chapter
IV. RESULTS . . .
1.. Plasma Slab of Infinite Thickness
2. Plasma Slab of Finite Thickness
Numerical Results . .

V. DISCUSSION AND CONCLUSION

APPENDICES
REFERENCES

iv

Page
57
57
59
59

83

89

95

Figure

4a.

4b.

4c.

5a.

5b.

5c.

6a.

6b.

6c.

LIST OF FIGURES

Wave ko~incident-on some lossy medium excites
a wave .15.: 131 + 1E2. 0 o o o o o o 0

Reflection and refraction of s and.p polarized
waves at a boundary. . . . . . . . .

Reflection and.transmission by a conducting
slab of thickness d.

Transmittance versus frequency for p-polar-
ized radiation incident at an angle of 30
degrees on plasma slabs of thicknesses d/A =
0.0305, 0.0915, 0.1545,.where 1 .is the P
vacuum.wave1ength at the plasma frequency,

Ap = ch/w . ‘The plasma 13 described by a
Fermi veloBity VF = 1.4x10 cm/sec and a col-
lision frequency corresponding to wpr = 100

Reflectance versus frequency for 30 degree
incidence . . . . . . . . . . . .

Absorptance versus frequency for 30 degree
incidence . . . . . . . . . . .

Transmittance versus frequency for 60 degree
incidence and various thicknesses . . . .

Reflectance versus frequency for 60 degree
incidence and various thicknesses . . .

Absorptance versus frequency for 60 degree
incidence and various thicknesses . . .

Transmittance versus frequency for 80 degree
incidence for various thicknesses .

Reflectance versus frequency for 80 degree
incidence for various thicknesses

Absorptance versus frequency for 80.degree
incidence for various thicknesses

Page

14

17

28

62

63

64

65

66

67

68

69

70

List of Figures/cont.

Figure

7a.

7b.

7c.

8a.

8b.

8c.

9a.

9b.

9c.

Transmittance versus frequency for slab
thickness d = 0.03051 at various angles

of incidence . . .p . . . . .

Reflectance versus frequency for slab
thickness d = 0.03051p at various angles
of incidence . . .

Absorptance versus.frequency for slab
thickness d = 0.03051 .at various angles
of incidence . . .p . . .

Transmittance versus frequency for slab
thickness d = 0.09151 at various angles
of incidence . . .p . . . . . .

Reflectance versus frequency for slab
thickness d = 0.09151 at various angles
of incidence . . .p . . . . . .

Absorptance versus frequency for slab
thickness d = 0.09151 at various angles
of incidence . . .p . . . .

Comparison of the transmittance versus fre-
quency calculated using the Fresnel equations
(11.41), dashed curve, and the genera1.equa-
tions, solid curve, for a thickness

d = 0.03051 and for 60 degree angle of
incidence . . . . . . . . . .

Comparison of the reflectance.versus.fre-
quency calculated using the Fresnel equations
(11.41), dashed.curve, and the general equa-

.tions, solid.curve, for a thickness

d = 0.03051 and for 60 degree angle of
incidence . . . . . . . .

.Comparison of the.absorptance versus fre-

quency calculated using the Fresnel equations
(11.41), dashed.curve,.and the general equa-
tions, solid curve,.for a thickness

d = 0.03051 and for 60 degree angle of

incidence P .

vi

Page

71

72

73

74

75

76

77

78

79

List of Figures/cont.

Figure

10a.

10b.

10c.

11.

Comparison of the transmittance versus fre-
quency calculated using the Fresnel equations
(11.41), dashed curve, and the general equa-
tions, solid curve, for.a.thickness d = 0.0915
and for 60 degree angle.ofincidence

Comparison of the reflectance versus fre-
quency calculated using the Fresnel equations
(11.41), dashed curve, and.the general equa-
tions, solid.curve, for a thickness d = 0.0915
and for 60 degree angle.of.incidence

.Comparison of the absorptance versus.fre-

quency calculated using the Fresnel equations
(11.41), dashed curve, and the general equa-
tions, solid curve, for a thickness d = 0.0915
and for 60 degree angle of incidence .

Coordinate system defining direction of
tensor components . . .

vii

Page

80

81

82

90

LIST OF APPENDICES

Appendix Page
A. EVALUATION OF 3 AND V . . . .. . . . . 89

viii

I. INTRODUCTION

1. Fundamentals

 

Plasma physics came into existence in 1929 when Tonks
and Langmuir1 presented their now famous theory of plasma
oscillations in an ionized gas to explain certain anomalies
in arc discharges.2 Idealizing the ionized gas as elec-
trons imbedded in a uniform positive background, they found
that a small displacement of a slab of these electrons from
their equilibrium position produces, by Coulomb interac-
tions, a restoring force which to first order is pr0por-
tional to the displacement. Thus the electrons oscil-
late in simple harmonic motion with a.characteristic
frequency mp;

(.02 = 41TN€2

p m , (1.1)

 

where N is the density, e the charge, and m the mass of
the electrons.* Noting the similarity between these os-

cillations and the oscillations of a jelly plasma, Tonks

 

* If the material in the positive background is polar-

izable, e.g. the lattice ions in a solid state plasma,

th . _ 41rNe2 .
e relat1on becomes mp - _ETfi—" where so 15 the

dielectric constant.

and Langmuir christened them ”plasma oscillations" and

named the nearly neutral part of the ionized gas a "plasma.”
Today a plasma is defined as: "the portion of-a mater-

ial body which is much larger than the shielding length
which in turn is larger than the interparticle distance

of the charged particles moving through the body."

The shielding length 1D is the distance in which the

Coulomb field of a test charge will be screened out by

the charged particles. For particles with a Maxwellian

energy distribution,

A2 .__ KT = KT/m
4TrNe2 2
OJp

 

 

(1.2)

and is called the Debye length;3 and for particles with

a Fermi-Dirac distribution,

 

V2

1;; = F (1.3)
5002
P

and is called the Fermi-Thomas length. In Eqs. 1.2 and
1.3, K is the Boltzmann constant, T is the temperature,

and VP is the Fermi velocity.

Solid State Plasmas. Until 1951, only gaseous plasmas,

 

consisting of electrons and ions were known and studied.
Besides laboratory plasmas, these included naturally
occurring gaseous plasmas such as stellar atmOSpheres,
interstellar gas clouds, the ionosphere, etc.. But the

1948 experiments of Ruthemann and Lang,4 in which they

observed that kev electrons upon passing through thin
metallic foils lost energy in discrete steps characteris-
tic of the metal, led to the discovery.of the solid state
plasma. By treating the metal as a plasma of electrons
in a positive lattice, Bohm and.PinesS explained the
discrete energy losses as the result of exciting quantized
plasma oscillations, calledplasmOns,6 each quantum
possessing energy equal to hop.
While alike in their essential features, solid state
and gaseous plasmas are dissimilar in one important
respect: stability.. Gas plasmas, because they are pro-
duced by violent means such as spark discharges, are
far from therma1.equilibrium, are.not easily contained,
and as a consequence of one or more of the many insta-
bilities tend to break up. .Thus the central problems in
gaseous plasma research are containment and control of all
the instabilities.. The solid state plasma, however, is
absolutely stable and contained by the neutralizing lat-
tice. The problem is no longer how to produce and con-
tain the plasma, but given the plasma, what to do with
it; i.e., how to throw it out of equilibrium and produce

instabilities.

Acoustic plasma waves.. In addition to the high.frequency

 

electron-plasma oscillations, Tonks and Langmuir showed
that a two component plasma, containing two types of

mobile charge carriers or particles, has a low.frequency

mode of oscillation. In such a two component plasma, the
high frequency oscillation results from.the two charge

'species oscillating out of phase with the frequency
w2 = w2 + w2 , (1.4)

where wp- and w + are the plasma frequencies.of each

P
species, e.g., electrons and ions. .If there are more than
two species,e.g. different ions or electrons with dif-
ferent effective masses, Eq. 1.4 is extended to include
the plasma frequency of each species. Because in a gas-
eous plasma the ions are much heavier than the electrons;
i.e. wp_ >> wp+’ their motion can be neglected and the
plasma frequency is just the electroneplasma frequency,
but in general the high.frequency oscillation of a.mu1ti-
component plasma is given by a generalization of 1.4.

The lower frequency mode results from the two.charge
species oscillating in phase, and for the electron-ion
plasma is
_ w;+(k1D_)2

l + (k1D_)2

 

(1.5)

where k is the wave vector, k = 2n/1, and.1D_ is the
screening length for the electrons. Because-the.mode

1.5 exists only for wavelengths larger than the.screening
length, i.e. for (k1D)<l, 1.5 defines a band of fre-

quencies up to the ion-plasma frequency wp+°

Plasma waves. The mathematically simple.theory of Tonks

 

and Langmuir led to a single frequency of oscillation mp,
which will not propagate, but.they.pointed out this may not
be true if the thermal motion of the electrons is in-
cluded. The first attempt to calculate the.dispersion
spectrum for the.high.frequencymode was by J. J. and

8

G. P. Thomson,7 but.V1asov. is credited.with.obtaining

the correct dispersion relation
2 = 2 2
w wp[l + 3 (1Dk) . (1.6)

Equation 1.6 shows that.longitudina1 plasma-waves exist

for a band of frequencies, the.lower.1imit.being ”p and

the upper limit is determined by the requirement that

k1 < l.

D
Because the theory of longitudinal plasma waves de-
scribes small disturbances from equilibrium whereas labora-
tory plasmas are unstable, direct experimental verifica-

tion of 1.6 for gaseous plasmas is very difficult and

9

was accomplished only recently. Indirect measurements of

1.6 for solid state plasmas have been reported, for

10 studies of the variation of energy

example-Wantanabe's
loss with angle of electrons scattered by a foil, but
no known direct measurements of wavelengths in solid
state plasmas have yet been made. Based on the theory

developed here, such.an experiment will be proposed later.

2. Interactions of.Plasmas with Elegtromagnetic Waves

 

There are two basic methods for studying plasmas:
by their interaction with charged particles and by their
interaction with electromagnetic (em) fields, especially
with em waves. Although the original discovery of plasma
oscillations was.aided by the.discovery.that arc discharges

11 the

emitted radio waves at a characteristic frequency,
problem of radiation by a-plasma was neglected because the
original theory showed the.oscillations,to be longitudinal
and hence non-radiating. Interest in the.radiation prob—
lem was revivedbyShklovsky,12 Martyn,.13,and.Haef£14
who suggested that-some radio bursts from the sun origin-
ated in coronal plasma-oscillationS..Fieldig and‘others16
developed the theory-of plasma radiation, showed.that.long-
itudinal plasma waves may.couple.with transverse em

waves at density or temperature gradients or.in therres-
ence of magnetic fields, and pointed out.that-at.finite
temperatures longitudinal plasma oscillations may propa-
gate as dispersive waves.. The excitation.of.em waves

by solid state plasma.waves was first considered by

17 He noted that the energy loss method of meas-

18

Ferrell.
uring plasma frequencies was limited in accuracy and

resolution* and suggested.a more direct method. Under

 

* Because the mean free path of the incident charged
particles must exceed the thickness of the metal foil to
avoid complicated multiple scattering processes, particle

suitable.conditions.the plasmons generated by the.beam of-
charged.particles.would decay by emitting em waves, and

the-p1asma-frequency.can.be.measured directly-by detecting

19

this radiation. The predicted radiation has.been ob-

20

served by several investigators in silver foils and

also in aluminum and magnesium foils.21
The inverse of the.radiation problem is the.excita-

tion of plasma waves by.incident em waves and the.asso-

ciated reflection and.transmission of em waves.by a plasma.

Compared.to the.incoherent-scattering.of-radio.waves by

a plasma,22 and more recently parametric excitation of

plasma waves,23 the linear coherent.coupling of em waves

to plasma waves has received little.attention. On the

basis of Ferrell'si7.physica1 picture for the radiation

24 predicted that p-polarized

peak, Ferrell and Stern,
(with the electric.field.in.the plane of incidence) em
radiation obliquely incident on a thin metallic film
would be.anomalously transmitted and reflected at the

25 found that the

plasma frequency.. McAlister and Stern
transmission spectrum from a silver foil had a dip at
the-plasma.frequency.for.pepolarized.radiation but showed no
such structure for s~polarized radiation. But they ex-

plained.their.results as due to a surface plasma wave

 

energies of 10 kev or more are necessary.. But the
energy loss is only a.few electron volts, thus the quan-
tity of interest is the-difference of two large and
practically equal energies.

(surface plasmon) and not bulk plasma waves. FedOrchenko26

studied coupling between em and bulk plasma waves by

15 But his

essentially inverting Field‘s calculatiOn.
results, which were based on the hydrodynamic approxi-
mation of a plasma, were incorrect.because of an inap-
propriate boundary.condition.27
.In.what.follows we present a general-theory of linear,
coherent coupling between em waves and.waves.in a plasma.
The calculations are based on classical electromagnetic
theory and.quantum mechanical effects are essentially
neglected. In the second chapter the usual.Fresnel.equa-
tions of reflection and.transmission.are.shown.to be
inadequate for media with longitudinal polarization waves
and more general expressions are derived for slabs of
infinite and finite thickness. Because-waves.in con—
ducting media such as plasmas are inhomogeneous, corres-
ponding to complex wave.vectors,.we develop.appropriate
dispersion relations for-inhomogeneous plasma waves in
the third chapter.- The fourth.chapter.presents.numerical

results for some typical solid state-plasmas and the

last chapter discusses the theory.and results.

II. THEORY OF REFLECTION
AND.TRANSMISSION

The problem of reflection and transmission of em
waves by a plasma is.a.specia1.case of.the.pr0pagation of
em waves.across.discontinuities in.the.e1ectric.proper-
ties of matter.. To be.more.specific,-it.is.a special
case of reflection.and.transmission.by.a.conducting med-
ium, a problem treated by most.textbooks on electromag-

netic theory.28

But.these.treatments are generally in-
complete since they usually neglect waves with irrotational
electric fields and.thereby exclude.the effects of.such
waves as plasma waves.or longitudinal.opticalsphonons.

It is for this reason that the theory used by.McAlister

25 to explain their measurements considers only.

and Stern
surface plasmons. Their expressions for transmission
and reflection are.the usual Fresnel equations which are
derived for divergence free fields only, thus excluding
plasma waves or bulk plasmons.- But inltheir theory,’
the non—propagating plasma oscillations produce surface
charge densities oscillating at the plasma frequency.
Because normal electric fields of em waves.couple to

such surface charge densities their theory predicts anom-

alous behavior in the p-polarized transmission spectrum

10

due to these surface plasmons.

The exclusion of irrotational fields in the usual
derivation of.Presnel equations for conducting-media has
been obscured by an ambiguous description of the electric
field in the medium.. Thus for-obliquely.incident,
p-polarized waves the electric field-in.the-conducting med-
ium is often described.as no longer purely transverse, but
possessing longitudinal components. At first glance a
longitudinal electric field suggests an irrotational field
due to net charges..-The difficulty is that for.inhomogen-
eous waves-the.terms.”transverse" and.Vlongitudinal" are not
equivalent to "divergence free" and-"irrotationa1" re-
spectively, and.it is well.known that for-oblique incidence
the waves in.a lossy.medium are in fact inhomogeneous.

To avoid ambiguities,-the following definitions will
be used: An electric field (or the wave-associated with

it) will be-called divergence free or.emif its diver-

 

gence vanishes; i.e. if Veg = 0, and it will be called

irrotational if its cur1.vanishes; i.e. if vx§.= 0. The

 

terms longitudinal.and.transverse will be applied respect-

 

 

ively to the.components.of.the fields parallel.and normal
to the direction of.phase.pr0pagation.- Finally,.a wave

will be called inhomogeneous if its surfaces of constant

 

phase and constant amplitude do not coincide. If a plane

11

wave is represented by*

yet) 114111.01

Re[E oexp(i§r - iwt)] (11.1)

Re[§oexp(ikl°r - iwt)]exp(-k2or),

the surface of constant amplitude, a plane-in this case,

is k2 fir= constant while the plane of constant phase at

a given time is £193 =.constant. Since only harmonic plane
waves will be considered here, a.wave.is inhomogeneous when-
ever the real and imaginary parts of the wave vector K

have different directions., Thus inhomogeneous waves can

exist in any medium.

1. Equations of Transmission and.Ref1ection.for a Medium
Bounded by a Plane

Before the equations of reflection and transmission
are derived it will be instructive to rederive Snell's
law, consider.its consequences for inhomogeneous waves,

and discuss the boundary.conditions.

 

* Where it is desirable to.distinguish.between real

and complex quantities, a complex quantity.will be indi-
cated by a tilde and its real.and.imaginary parts by

the subscripts l and 2; e.g. a = a1 + 132

12

Snell's Law. If n is a unit normal to the plane interface

 

separating the two media, let 2'1 = 0 define the surface

of the interface. The existence of boundary conditions

on the fields at any point on 3.3 = 0 at any time, requires
that the space and time.variation.of all fields be the

same on 2'3 = 0. .Consequently, the phase factors in 11.1
for all the waves must be equal at 2’3 = 0 and independent
of the nature of the boundary conditions. Since the time

factors are trivially equal we require,

(gm) (1;...)

Ila-{=0 J_£o£=0 9 (11°2)

where 50 is the wave vector of the incoming wave and kj
is the wave vector of any of the possible reflected or

refracted waves. But
3; =— (1°99. - .11 X (.11 X 3:). (11.3)

so 11.2 becomes

(nxr) =kj°nx (nxr) (11.4)

IX”
0

I:
x

which by means of a vector identity can be written as

(130 x n5 1:9]. xn)°(n_x r_) = 0. (11.5)

13

Thus Snell's law, as expressed by 11.5, states that 2
and separately the real and imaginary parts of the wave
vectors E are coplanar; and the components of the wave
vectors parallel to the interface are equal.

For example, consider a wave in vacuum incident with
an angle 00 on a.semi+infinite medium in which.the wave-

vector is complex, as in Fig. l. Snell's law-requires

k0 sin 60 = k1 sin 0
(11.6)
0

k2 sin ¢

11.6 demonstrate how an obliquely incident homogeneous
wave can produceinhomogeneous waves, and-furthermore
11.6, quite graphically, showsthat.the-planes-of.con-
stant amplitude.are parallel to the interface in a lossy

medium such as a metal.28

Boundary Conditions. On the.microsc0pic-sca1e real systems

 

do not have abrupt boundaries; their electrical properties
change smoothly over some short but.finite distance.

With all charges, both free and bound, taken.explicitly

 

into account we need only introduce the total electric
vector E; the concept of a displacement vector then has
no role. For example, at a plasma-dielectric interface
the plasma extends partially into.the.dielectric and vice
versa, creating a transition region in.which-the plasma

changes gradually into the dielectric. If we construct

90

vacuum

 

14

 

medium

Fig. l. .Wave.Eo.incident on some lossy medium excites

a wave Ep= E;.+ 1E2.

15

an imaginary surface lying in this narrow transition
region then all components of the total E and E are con-
tinuous across this surface. This follows from the stand-
ard arguments which apply Maxwell's equations to pillboxes
and infinitesimal circuits passing through the imaginary
surface, provided only volume distributions of charge

and current density are present. For the case of a transi-
tion region over which the properties of the system change
smoothly but rapidly from pure plasma to pure dielectric
only volume distributions of charge and current density
can exist. Thus we shall adept the boundary.condition that
all components of E and E are continuous, keeping in mind

that we explicitly regard whatever charges are bound to

 

the dielectric as able to carry alternating currents in

the same way as the free electrons of the plasma with which
they communicate through.the electromagnetic field. The
continuity of-the.field components and Snell!s law are

thus fundamental to the solution of the problem. We
complete our idealization of the systemby-postulating that
the bulk.properties of.the-p1asma and-dielectric as they
express themselves in the.respective dispersion relations
may be extended up to the imaginary boundary surface in

the transition region. .Expressed in brief terms, our
boundary condition essentially postulates that there is

no surface charge density and that all charge.distributions

are volume distributions which can alter the.normal

16

component of the electric field only over finite distances.

Fresnel Equations. To facilitate comparison between

the Fresnel equations and the new equations incorporating
irrotational waves, we will derive the former.first. Con-
sider a plane, linearly polarized em wave incident on

a semi-infinite medium at an angle 60, as in Fig. 2. Let
g, E: and pibe a.triplet.of orthogonal.unit vectors:

2 being normal to the surface separating the two media,
and p in the plane of incidence. We assume medium 0 to
be vacuum or at worst a.dispersionless dielectric while
medium 1 is quite general. Ion, Er, and.Et are the

wave vectors of the incident, reflected, and refracted

em waves, Snell's Law.requires

10°10. =- Elna = 152

(11.7)
3.0.2 = -k

_r°E

By the superposition principle,.an arbitrary direction

of polarization can be resolved into two-cases: one with
the E-field in the plane of incidence (p-polarized) and the
other with the E-field normal to the plane of incidence
(s-polarized). For the p-polarized case let the incident,

reflected, and transmitted waves be

17

 

It!

[*1
u
o

I:

E

medium 0 medium 1

 

Fig. 2. Reflection and refraction of s and p polarized
waves-atfa.boundary.

18

mm) = (g x i—B‘olEoeXPUEVE - not)

k . .
Er(£,t) = (g x E-T)RpEoexp(1Er-r - 1wt) (11.8)
Et(£,t) = (g x Egt)TpEoexp(iEt-£ - iwt)

Note that-the fields in 11.8 satisfy 53E

fore are divergence free.

= 0 and.there-

For our harmonic waves the

magnetic fields are related to the electricby

so
flo(£,t) =
firCLt) =
§t(£’t)
where

Applying the condition

E.= (c/w)E x E. (11.9)
_$_€oE exp(iEo-_r_ ' iwt)
EeoRpEoexp(iEr-£ - iwt) (11.10)
= getTpEoexp(iEt°£ - iwt)
_ c 2
a 2 [a] _a'-k_a (11.11)
of continuity of tangential E,
nor=0 _ n x (Et)n-r=0 (11°12)

19

to 11.8, we get after some algebra,

Cko] ] - {Ck ~.
_. ,l'R - __tJ '1‘ (11.13)
I. .I p to» n.

where [5E] is the normal component of the wave vector k
n
in units of the vacuum wave vector w/c.* Similarly the

continuity of the magnetic fields 11.10 yields

1+R = T . 11.14
€0( p) 6t p ( )

Combining 11.13 and 11.14 we get the Fresnel equations of

reflection and transmission for p-polarized waves

9
I
U)

 

"U
Q
+

m

(11.15)
nor—0
w n
_T__

p a B ’

where

 

n (11.16)

 

* In some derivations (cka/w)n is written as /E; cos 0a,
with (cka/w)p = /E; sin 6a, but for complex wave vectors,
i.e. for inhomogeneous waves, the angle of refraction
becomes complex and loses its meaning, so we avoid this
notation.

20

The expression for Tp in 11.15 differs slightly from the
usual Fresnel equation because,.while the usual definition
for T is T = Et/Eo, we have defined T as
((c/w)Etht]/[(c/w)onEo]. That particular form was
chosen because it eXplicitly shows.the divergence free
property of the waves in the derivation. Our expression
for Tp.may be changed to the.usua1.Fresnel.expression by
multiplying with the factor /E;7EF .

For the s-polarized case let

E0 = EEoexp(iEo-r - iwt)
—r = ERSEOeXP(iEr'£.‘ iwt) (11.17)
Et = gTsEoexp(iEt~£ - iwt)

Upon applying the condition of continuity of tangential
E and H to 11.17 and the corresponding magnetic fields,

we find the Fresnel equations for the s-polarized case,

t]n
t1

:1

(11.18)

21

The above derivations explicitly show that the Fresnel
equations.consider only em waves with divergence free
fields and cannot be applied.to systems with so-called
"longitudinal waves" or waves with irrotational elec-

tric fields.

General Equations. To eliminate the restriction placed

 

on the Fresnel equations let us rederive.the.coefficients

assuming the most general field in the medium has irro-

tational as well as divergence free components. Let us

further assume that the irrotational and divergence free

(em) waves are non-interacting in the bulk; hence,

labeling them with subscripts 2 and t, they satisfy
51x21 = 0

(11.19)

.at.§t = 0

This assumption is satisfied by any isotropic homogeneous
medium and in the next chapter we will show that a col-
lisional but non—drifted.plasma satisfies the non-
interacting condition even for inhomogeneous waves.

For the p-polarized case, in addition to the em fields

11.8 and 11.10 let the irrotational electric field be

E£(£,t) = (5%I)LpEoexp(i§,. 5 - iwt), (11.20)

22

which has no associated magnetic field. With the addition-
a1 field 11.20 the continuity condition on tangential E

now yields
f N
[250] [1 - R'] = [EEtJ T + SEfi L (11.21)
m n p w P

where (ckl/w)p is the p component (see Fig. 2) of (c/w)E£,
and by Snell's Law is equal to (cko/w)p, hence the subscript
will be drOpped from now on.

The condition that normal E is continuous results in

[EEIPII + Rp] = (EEIPTP ‘ [E§£]an (11.22)

Solving 11.21, 11.22, and 11.14 we obtain the general

equations of reflection and.transmission

 

T = (11.23)

 

t-'
II

where

y = [%§]2[€0 - etL/Tigi] (11.24)
p n

23

Since for the s-polarized case E is always transverse to
k, irrotational waves cannot be excited, and the equations.
of reflection and transmission are the same as 11.18.

Equations similar to 11.23 have been obtained by

A. M. Fedorchenk026’27

in calculating the conversion of
transverse electromagnetic waves into longitudinal waves
at a dielectric-plasma plane interface. In Fedorchenko's
first paper26 he pointed out that the usual boundary con-
ditions specifying continuity of tangential E and H are
insufficient to solve the problem, and he added the con-
dition that the normal component of all plasma charges'
velocity vanishes at the boundary. His results were in
general incorrect, however, because this last boundary
condition applies only to a plasma-vacuum interface, since
the existence of a dielectric presupposes polarization
charges which can provide a non—vanishing value for the
normal component of current density on the plasma-
dielectric interface. But at a.plasma-vacuum boundary,
i.e. eo- 1, the normal component of the current.density
or charge motion vanishes and our results.applied.tohomo-
geneous waves coincide with Fedorchenko's 1962 results.26
Although he corrected his choice of.boundary.conditions

27 to effectively include the effect of

in the 1967 paper
polarization currents in the dielectric, his published
results still differ from ours. We ascribe this to a

misprint since he claims in the second paper that the

24

results of both papers coincide for a plasma-vacuum bound-

ary, but in fact they do not.

2.. Properties of the General Equations

Since the Fresnel equations and the.general equations
are identical for s-polarized waves, we will be concerned
only with the p-polarized case. Comparing the new equa-
tions, 11.23, with the Fresnel equations, 11.15, we see
they become identical whenever Y vanishes.. From 11.24 we
find y is proportional to three factors: the sine squared
of the angle of incidence,.(ck/w); = so sinzeo; the "trans-
verse"* conductivity pr0portiona1 to (so - at); and the
inverse of the normal component.of the irrotational wave
vector in units of the.vacuum wave vector of an.em wave.
The first factor indicates the effect of the irrotational
wave increases with the angle of incidence, which we
would expect since the normal component of the incident
electric field increases. The last factor is the.impor-
tant one. vait is real, i.e. the irrotational waves are
undamped, it is approximately.equa1 to.the.ratio.of the
phase velocity of the wave to the speed of light, which for
plasma waves is usually very small. For damped irrota-

tional waves this last factor becomes small and imaginary,

 

* The difference between transverse andlongitudinal
conductivities will become apparent in the next chapter.

25

so that in media in which.the.irrotational waves are highly

damped we recover the Fresnel equations.

Eneggy Conservation. Because physical observations are

 

usually made on the reflected or transmitted intensities

or energies rather than amplitudes we next relate the coeffi-
cients 11.15 to the energies of the waves. For an electro-
magnetic system the mean energy flow is given by the real

part of the complex Poynting vector E,
s C 1R *
—='4'172’ e Exfl (11.25)

where A* indicates the complex conjugate of A. According
to the principle of energy conservation the normal com-
ponent of energy flow across the interface, given by

BfiE, must be continuous. If we construct the Poynting

vectors for the em waves 11.9,

. c 2 ck
E0 E? €°E°[*Uo]

c 2 ck
—r = 3?'€°E°|RpIZLTEr] (11.26)

C 2 2 CR .
Et EE'E°ITp| Re[e§[d;t]] exp(-21_<_t2 1)

it is a simple matter to show that the Fresnel equations
satisfy the energy principle. But, if we apply the condi-

tion

26

(11.27)

to the system with irrotational waves by substituting

11.15 into 11.26.we find

 

I (D
II
°°|
:1
m
0
f-_\
8

(2°

it
2)n,r=0. 55°]DB:[4R3(1°“°1 ], (11.28)

Ido+Bo+Y0I2

which we may identify as the energy flux flowing into the
irrotational wave (the bracketed expression being the frac-
tion of the incident energy going into irrotational waves).
The origin of 11.28 may be seen more clearly if we notice
that the total electric field in the medium is the sum

of Bt and E2, consequently the Poynting vector consists

of two terms: the first is just E1 and the second is

c *
§E Re(_E_’Q x Et) (11.29)

Designating 11.29 as E£ and substituting the expressions

for E2 and Et(11.20 and 11.10) we find

2
= .C_ E [SE] Re[e*L T*] , (11.30)
m p

h
. :3
(D

—f—£)E-_=O 8n t p p

which reduces to 11.28 with 11.15. Because irrotational
waves transport energy mechanically not electromagnetically
11.29 should not be identified as the Poynting vector of

the irrotational wave.

27

3. Equations of Reflection and.Transmission.for a Slab
Some of the most interesting and useful phenomena in
optics are produced by the interference of em waves in
thin dielectric slabs. Because we expect that irrotational
waves may exhibit similar interference properties we next
calculate the reflection and transmission equations for
a slab of material capable of supporting irrotational
waves.
_Suppose our conducting medium is bounded by two
planes; one being 373 = 0 and the other being a}: = d,
as in Fig. 3.. To the left of this slab of thickness d is
a dispersionless dielectric, so, and to the right is another
dispersionless dielectric, 82. Since no irrotational
waves can be excited for s-polarized incidence we will
pconsider.only the p-polarized case. The incident and

reflected waves in so are

E. = (s x 21531) exp(ik_o°£ - iwt)
E0 = 560 exp(iEo.£ - iwt)
, (11.31)
as = (_s_ x Egan exp(ii_<_5or_ - iwt)
E; ' 360R exp(iEfi-£ - iwt),

where the primed quantities refer to the reflected waves.
The conducting medium contains four waves, irrotational

and em waves propagating to the right and to the left,

28

 

 

 

co conducting medium 52

 

 

Fig. 3. Reflection and transmission by a conducting slab
of thickness d.

29

_ CKt ~ , _ .
Et — (g x.-E )T exp(1k_t 1 int)
Et = g ctT exp(iEt-£ — iwt)

I
E - (g'x 3%t)T' exp(iE;°£ - iwt)

 

—t

(11.32)
' o 0
at = §_ etT' exp(1E£°_r_'_ - 101t)
E2 = 2&2 L exp(iE£°£ - iwt)
I
E; = 5%£ L' exp(iE£°£ - iwt)
Finally the transmitted wave in 52 is

E2 = (_s_ x c132” exp(ik_2-r_ - iwt)

(11.33)

H2 = g 52K exp (152'3 - iwt)

Snell's law requires the p components of all wave vectors
to be equal (law of refraction) and the n components of
the primed wave vectors to be equal to the negative n
component of the corresponding unprimed wave vector (law
of reflection).. Applying the continuity condition to

E and H at the surface 3°: = 0.we obtain

(at + R) = 1:311”) - [2w - v]

30

Similarly from the boundary.conditions at the surface

n-g = d we have

 

k , ' ’ck , ' ck
[675.] [11. - T .t] + If] [1% + L M - [ 002] T
n \ p n
ck , v ck , ' _ ck
[5.] [T¢t + T ¢t] - [—62] [L¢£ - L ¢2] - [5—] T (11.35)
P n P
et[1¢t + T'¢;] = e21
where
.¢t/£ = exp(iE°Et/£d) (11.36)
¢'¢ = 1
T = K exp(iE°E2d) (11.37)

Solving 11.34 and 11.35 for R and T we find the equations

for reflection and transmission for a slab to be
(11.38)

where

I
N = (1-¢t¢£) AoDz'D0A2¢t¢£ ' (¢t‘¢£) C0B2¢t'B0C2¢£J

31

Nt = (A0+Do)¢t¢2[(¢t‘¢£)(AO'BO) + (¢£‘¢£)(A0'Co)]

(Ir.39)
M = (1‘¢t¢1)[DoD2'AoA2¢t¢l] ' (¢t'¢£)[BoBz¢f'CoC2¢£]
and
A = a - B - y
B = a - B + y
(II.40)

The subscript 2 on 11.40 indicates that so and k0 are~
replaced by 82 and.k2 in the corresponding expressions
for a, B, and y; i.e. they correspond to reversing the
positions of the dielectrics so and 22.

Although the.above expressions are complicated, cer-
tain important features are directlyevident.. Thus for
example our suspicion that irrotational waves as well as
em waves interfere is supported by the appearance of the
phase factor ¢£. As we have done for the semi-infinite
case, it will be instructive, to compare the above equa-
tions with the corresponding Fresnel equations of reflec-
tion and transmission by a slab. They are not difficult
to derive,znu1 because they may be found in the liter-

ature29 we will write them-down directly

32

 

R0 - R2¢ 2
R = t
1 " R0P2¢t2
(11.41)
. 2
(10 + 80 (l " R0)¢t

 

 

where R0 and R2 are the Fresnel reflection equations

11.15 for medium 0 and 2. We would expect 11.38 to reduce
to 11.41 if the irrotational wave is absent or highly
damped out. But if Im(k£)+ m then ¢2 + O, y + 0 and

11.39 become

Nr = (do-Bo)(az+82) ’ (00+Bo)(02'82)¢:
Nt = 4aoBo¢t (II.42)
M = (00+Bo)(02+82) ’ (Go-Bo)(d2‘82)¢t2

and we recover 11.41. Also we expect to recover the equa-
tions for the semi-infinite case if d + w. Even if the

waves are only slightly damped ¢t’ o2 + 0 as d + w, so

NT '* AOD2

Nt + 0 (11.43)

M + DODZ

33

and we find

- A0 = 00 ' Bo - Y0 = ..
R - D? a0 + Bo + Y0 Rp (11.44)

 

III. DISPERSION RELATIONS FOR
A COLLISIONAL PLASMA

1. Theoretical Formulation

 

Basically, there are two theoretical methods of study-
ing the properties of plasmas; a microscopic, kinetic treat—
ment using the Boltzmann transport equation or some equiv-
alent means30 to obtain the distribution function for the
charged particles, or a macroscopic, hydrodynamic treat-
ment using a closed set of moment equations. Together with
Maxwell's equations of electrodynamics, either of the above
treatments provides a closed description of a plasma. The
microscopic treatment has the advantage of providing a more
detailed description of plasma dynamics, but has the disad-
vantage of being mathematically more difficult. The macro-
scopic treatment, on the other hand, is relatively simple
mathematically and thus provides better physical insight into
the various phenomena, but has a narrower range of applica-
tion and neglects some important processes, for example it
cannot predict Landau damping. In the following paragraphs,
the two methods will be briefly sketched and their rela-
tionship will be illustrated.

As in other many—body problems, one is interested
only in the knowledge of macrosc0pic properties of the

plasma, such as the mean particle density, mean velocity

34

35

or temperature, conductivity tensor, etc., rather than
detailed.description.of individual particle motion. It

is sufficient, therefore,.to describe a given class of
particles by a.distribution function f(£, v, t) such that
fdavdar represents at time t the probable number.of par-
ticles with velocities between X and v.+ d1, and positions
between 3 and 1 + d£.. All macroscopic properties may be
determined from the knowledge of f, which for.each species
of particles.is governed by the Boltzmann transport

equation
' (111.1)

where ‘7; and I}; represent the gradient operators in coor-
dinate and velocity space, E is the force field at point
(3, v), and (6f/6t)co11;.represents the.time rate of
change of f due to collisions. The essential feature of
III.l.is the collision term. .Because charged-particles
interact primarily through.their electromagnetic fields,
predominantly.by Coulomb interactions, these collisions
conveniently divide into three classes.31
The first corresponds.to.collisions.with.impact.para-
meters less than the mean parameter for 90° deflection,
sometimes called the distance of closest approach. These

large angle deflections, usually described by the familiar

two-body collision.integrals of the kinetic theory of

36

gases, are negligible in most cases of interest. The
second class consists of.a succession of uncorrelated

small angle Coulomb deflections with impact parameters
greater than the distance of minimum approach but less than
the Debye length, AD. .These encounters are best treated
by the Fokker-Planck collision integral. The third class
of encounters with impact parameters greater than the

Debye length describe a plasma. With.many particles in

a sphere of radius AD; i.e° N15 > 1, N being the particle
density, macroscopic electric forces suppress density fluc-
tuations over distances greater than AD, hence the col-
lision term is replaced by a.macroscopic em force field

due to the correlated effect of many particles. Because

8 first used this fluid-like treatment of a plasma,

Vlasov
the resulting collisionless Boltzmann equation usually
bears his name.

The macrosc0pic treatment begins with.the hydrody-
namic equations of.motion,.which presents a problem since
they can be an infinite set of coupled equations. For the
sake of illustration only the equations of motion in the
low temperature.approximation (LTA) will be.considered.
Let N be the particle density, E.the force.on the par-
ticles of mass m, g = Ny_the net-particle current, and
a the pressure tensor resulting from.the particle motion.

Then the hydrodynamic equation of motion of the particles

is

<5 —1 -lV.
fig- aEN m g . (111.2)

In the LTA some assumption must be made about the pres-

sure tensor, for example
p = va2 (111.3)

The above equations along with.the continuity equation

'3? N + Vt.) = 0. (111.4)

and Maxwell's equations determine the system of charged
particles.

To show how the microsc0pic and macroscopic treat-
ments are related, we start with the following moment

integrals

N(£,t) = Jf(£,v,t)d3v a

F 3
9131.0 =J1f(1.x.t)d v, (111.5)
£(£,t) = mvvf(£,v,t)d3v.

 

Obviously N, g, and glare respectively the mean particle
density, current, and pressure. Note that 111.3 is just
the trace of the expression for a, 111.5. If we now take

the velocity integral of the Boltzmann equation 111.1,

38

we arrive at the continuity Eq. 111.4. Similarly multi-
plying the Boltzmann equations by the velocity v and
integrating we arrive at the hydrodynamic Eq. 111.2. In
performing the integrals we have assumed that the particle
number and momentum.or-current are each conserved under
collisions so that the collision terms vanish. We have
also assumed that f is well behaved in velocity space and
its surface integrals vanish. Higher moment equations
are formed by integrating.the Boltzmann equation with
additional velocity factors; e.g. the next moment equation
gives the equation of motion of the pressure tensor 3
in terms of the heat flow tensor 3, etc.. .Each higher
moment recovers more information about the system that is
completely contained in f.. This infinite set of.coup1ed
equations must be truncated and.some approximation given
to the last term.

Because the macroscopic approach.excludes certain
detailed information such as the effect of collisions due

to other mechanisms, e.g. scattering by lattice, we will

use the microscopic approach.

2. .Calculation of the Distribution Function

 

Let us consider a homogeneous, isotropic, and un-
bounded gas of charged particles,.whose net-charge is
zero. This gas.may contain different species of charged
particles; each species being characterized by a set of

parameters including mass, charge, temperature, Fermi

39

energy, etc.. 1n.addition to any external.force fields,

E in 111.1 contains the interaction between all the charged
particles, while the.collision term contains all other
scattering mechanisms such as impurities, phonons, etc..
Suppose the system is disturbed from equilibrium such that
the force field is harmonic in space and time, i.e. propor-
tional to exp(ik-r - iwt). Or, to put it in another
equally valid way, since any disturbance in an unbounded
steady state system can be Fourier decomposed into har-
monic components, we choose one. Then the differential

operators in space and time.become algebraic:

§?.= -iw , (Z.= 15 . (111.6)

With no static external electric or magnetic.fields the

force field is

§,= -e(§ + %.g x g) (111.7)
Although the following calculations will be made for
electrons, the results are quite general and can.be applied
to any "free".charged.particles by changing.the para-
meters.. 111.7 may be rewritten in terms of the B field

only,

 

)1‘.E (111.8)

40

by the use of Maxwellls equation

- - .1 Q.
vrxg- Gag, (111.9)

where l in 111.8 represents the unit identity.tensor.
We will treat the collision term by the relaxation time

ansatz

of

t I -Y[f(£’1’t) - f3(£919t)] 9 (111.10)
coll.

where y is the collision frequency, f is the distribution
function before scattering, and f5 is the distribution
function after scattering, i.e. the local equilibrium
function. The total distribution function for the.e1ec-

trons is

3N0 r2m 3’2 [ [EF'EU'I
fo = -—————' —— l + exp _FT_ . (111.11)
811er [hz]

where No is the average density, a the energy per particle,
5F the Fermi energy, and RT the thermal energy. Because

the Fermi energy is.a function of density, at T = 0 being

0

_ h .2 an
.61: - '2? (311 N) , . (111.12)

and because the density.varies in space and time,

N(_r_,t) = N0 + Nlexp(ik°r - iwt) (111.13)

41

the local equilibrium distribution also varies. Assuming
the density variations are small compared with the equili-
brium value (No >> N1), we can make a linear expansion of

fS about £0,
f = £0 + [is-3'] [gfi-F] N1(£,t). (111.14)

From now on quantities subscripted with.l.are small com-
pared to the equilibrium value (subscript O) and their
spaceetime dependence is exp(ik-r - iwt), which.will not
be written explicitly.

Before proceeding any further something.must be
said about the temperature of the electrons.and their equil-
ibrium distribution. In general 111.11.is-difficult to
handle except in the limit.of very low.or very high temp-
eratures. But.because.the.Fermi.energy of electrons at
room temperature is so much greaterthan the thermal
energy, 111.11 may be approximated to be at-a temperature
of absolute zero. On the other hand,charged carriers in
a semiconductor or gas plasma have high temperatures and
111.11 may be approximated by the Boltzmann-distribution.
Because we are interested in a low temperature solid
state plasma we shall use the zero temperature approxi-

mation. Thus 111.14 may be rewritten as

0
. 8f 2 8F
= I- 4— ——I—- I

42
and 111.1.becomes

6 , -_ e -, 11$ 12- , _
E f + XVf " fil- E1 [1.11—- + (l-T)L] Vf - -Y[f'fs](111.16)

If we assume 111.16 has a solution linear in harmonic

quantities.

(111.17)

1+:
A
[’1
I<
fl
V
H
H')
o
+
H";
H
(D
N
'U
A
He
I7?
IN
I
He
8
ff
v

we find

 

0
= éfo 1112.1. ._ _2_ E: 5f. yN.
f1 e[XE_]VTTETTE:i 3 0[6€ ]y-iw+ikov (111.18)

3. The Conductiviterensor

The conductivity tensor may be best described as the
linear response function of a system relating the com-
ponents of an electric field to the components of a cur-
rent density produced-by this field. The.most general
linear relation possible-between.two.vector-quantities
that vary in space and time is

£(r,t) = Idar'det'g(r,t,r',t')°§(r',t'), (111.19)
i.e. the current density g at the position 3 and time t
depends on the electric field B at all points in space

and all times through some tensorial relation g. But for

43

a homogeneous causally related system Q is a function of

relative position and time only,and its Fourier eXpansion is
(111.20)

Since we are interested in fields with harmonic space time

dependence,
§1(r',t') = §(k,w)exp(ik°r' - iwt') (111.21)
111.19 may be reduced to

1113.0 = g1l<_.w)°§1(1.t) (III-22)
by the use of the orthogonality property of the exponen-
tial function.

The current density gand the distribution function

f are related by the moment equation

__ 3
i —.§qijd Vixifi, (111.23)
where qi is the charge and vi.is the velocity of the
ith species of charged particles. The current for the elec-
trons whose distribution function we have.just.calcu-

lated is

44
J (£:t) = -6Jd3vxf1(1.t) (111.24)

Only f1 contributes to the current since f0 is.symmetric
in v. Because fl is a function of E1 and N1, 111.24 has
two parts: The conductivity current which is proportional
to E1; and the diffusion current, which is due to the
variation of the local equilibrium density and.is pro-

portional to N1.

 

g1(£,t) = g'(k,w)°§1 — ew§(k,w)N1 (111.25)
where

g'(k,w) = e2[d3v[-%§£];TI%:¥ETV (111.26)

3 = [davé 3;. %[-§:o]y-iw%i£'l (111.27)

To reduce 111.25 to the form 111.22 we define the tensor

3 as

3(149) = 3(§.w)£ . (111.28)

so that the equation of continuity

1°11 + ewN1 = 0 (111.29)

45

may be rewritten as

3’11 = -ewBN1- (111.30)
Substituting 111.30 into 111.25, the generalized conduc-
tivity tensor becomes

1

5115.4) = [,1 .- gcmm)’ 9315.4). (111.31)

where 5-1 represents the inverse of tensor A. Note, if
more than one charged species is present 3 and 2' are

'
replaced by £31 and 121.

Evaluation of g' and 3. .At absolute zero.the Fermi-Dirac

 

distribution function 111.11 becomes a step function in
energy or velocity which permits us to reduce velocity

integrations to solid angle integrations by the.identity

Jd3vG(X)['%§£] = Z;i§£; JdQG(vF), (111.32)
P

Since 111.32 is a good approximation for.temperatures
less than the Fermi temperature we apply it to our metal-

lic plasma and obtain

2
w
_ 3 1
2' --z% IE'VTTB l . (111.33)
VF 1

46

The integrals

I'-]

= I 11.31“ (111.35)

1+1a-§

and

- - “d9
y_ - (151:1: , (111.36)

where r is a unit radial vector in spherical coordinates

and

I”
III
1:
<

(111.37)

are evaluated in Appendix A.

So far in this chapter, k_has been a general vector,
but now we will choose our coordinate system byrequiring
the real part of k, i.e. the direction of phase propaga-
tion, to be in the positive 2 direction. Since, as was
demonstrated in Chapter 11, for inhomogeneous waves the
imaginary part of R has a different direction, we choose
the x direction so that the most general k lies in the
x-z plane. In other words k_and a possess only x and
2 components.

In this coordinate system we find.(Appendix A) for

A
homogeneous waves with a = azg,

IIQ

where

3w

Ot - 8n y-iw

3w2

2 n y-iw

and R has only the z-z

where a represents the

and for the homogeneous-wave is just a

Defining ofi as

47

 

 

 

Gt 0 0
= 0 0t 0
0 O oi
2
-£— [1+3 arctan a - l]
a2 a

1_ 1 _ arctan a
2 a

in

component

= 1_. 1 _ arctan a
1w a 9

scalar magnitude of a;

20

111.38 takes the simple form

(111.

(III

(III

(III

(III

(III

38)

.39)

.40)

.41)

.42)

.43)

48

0t 0 0
g = 0 0t 0 (111.44)
0 0 02

 

 

The tensor 3 for the inhomogeneous case, as evaluated in

Appendix A, may be put in the following form:

I7?
[w

R = [1 - EIE£§E_E] (111.45)
- 0 1“) 8

Similarly, the non-zero components of g' may be written as

k 2
X 1
xx 0t + E-E on

0
II

(111.46)

where 02' and 0t are given by 111.39 and 111.40 and

 

2
3 “p 1 3+a
' = ' - = —I- _ '-
ou o£ 0t 86 V715 a2 [3 a arctan a] (111.47)

After some simple but tedious algebra, 111.45 and 111.46

may be combined to give the components of g

 

Oyy - 0t
k2 (111.48)
= x
022 02 cp,
k
0 = o = kxkz 0
x2 zx E55 0 ’
where 0t and 02 are given by 111.39 and 111.43 and
cp = ou' + Rog (111.49)
so that
02 = OH + ot (111.50)
4. Dispersion Relations

 

A dispersion relation, in the sense that we employ
the term, gives the wave vector k, which in general may
be complex, as a function of the frequency m, which we
assume to be real. Because of our sign convention, a
positive imaginary part of k corresponds to a wave growing
in space, while a negative value corresponds to a wave

attenuating in space.

50

The dispersion relation is derived by requiring the

fields of the wave to satisfy Maxwell's equations

-11 -léEl
VXEI“CJ3_ C 1'.
(111.51)
_ _l 5&1
‘7X E1 - c 8?

Assuming g and B are related by 111.22, we solve for g
((C/leg-Ll. - (C/w)21<_ 5 - ;(l<_.w)]°§1 = 0 (111.52)

where

4fli

 

3 (111.53)

is the dielectric tensor. For a non-trivial solution to

111.52 we demand that the secular equation

||(C/w)2£f§l - (c/w)2£ 5 - gll = 0 (111.54)

be satisfied, which gives us the desired dispersion rela-
tions.

Using the conductivity tensor we have just calcu-
lated and 111.53 we may write down the components of the

dielectric tensor for an inhomogeneous wave:

 

 

 

 

 

 

 

 

k2
_ x
EXX _ at + k2 EU
Syy = at
(111.55)
Exz Ezx k2 9
k2
e = e - x e ,
22 2. kg 11
where
n wz. 1+a?
= _ . 123 3 J -
at 1 Z w(w+iy;) 23? aj arctan aj 1 (111.56)
J
n 8;) 3 arctan aJ
Z me+1y.) a7? 1 - a.
2 =1-1 3 J .441 (11157)
2 n y. arctan a. '
1 + ii —l 1 - J ]
. w a.
J J
cu = 62 - st . (111.58)

At this point we have generalized our results to an n-
component plasma and that is the reason for the summa-
tion in above expressions.

For homogeneous waves, i.e. for kx = 0, the secular
equation yields three solutions; two degenerate diSper-

sion relations

52
(c/w)2_lgt°kt = et(§t,w) (111.59)

associated with electric fields polarized in the x and y
direction; i.e. transverse waves, and the dispersion re-

lation
€£(££,w) = 0 (111.60)

associated with the electric field in the z-direction;
i.e. a longitudinal wave. These results are well known,
but it should be pointed out that usually no distinction
is made between the longitudinal dielectric function a,
and the transverse dielectric function at.
For inhomogeneous waves, however, we would expect
the dispersion relation for y polarized waves to be 111.59,
but the x and z polarized waves to be mixed. Indeed

the y-polarized solution factors out and we are left with

[(6/6)2k§ - EXX][(c/6)Zk; - 622]
(111.61)
-[(c/w)2kxkz + exz][(c/w)2kzkx + 62x] = 0.

But-the above equation may be simplified with 111.58 into

[(C/w)2k2 - at] a, = o , (111.62)

53

which shows the remaining two waves to be independent with
"transverse" and."longitudinal" like dispersion relations.
But they are not transverse and longitudinally polarized
waves as in the homogeneous case. To see how the fields
are related to the wave vectors let us solve 111.52

explicitly for the fields:

II
CD

[(C/w)2k: - ExxJEx - [Cc/w)2kxkz + EszEz
(111.63)

ll
0

[(c/w)2kzkx + an Ex - [(6/6)21<fc - EZZJEZ

Substituting the "transverse" dispersion relation 111.59

into either Eq. 111.63 we find

or (111.64)

59..”

Similarly the "longitudinal" dispersion relation 111.60

substituted into 111.63 gives

x z z x
or (111.65)
£2 x E1 = 0

In other words, the dispersion relation 111.59 corresponds

54

to irrotational wave fields or polarization waves. Thus
in Chapter 11, Eq. 11.11 which defines e is just the dis-
persion relation for the em waves. Furthermore, we have
proved that in a homogeneous, isotrOpic plasma with col-
lisions, the wave fields can be separated into two inde-
pendent parts, one irrotational and the other divergence
free. In general this is not true, for example in a
drifted.p1asma where a static external electric field is
present isotropy is.destroyed, and the wave fields

can no longer be separated in this manner.

Discussion.of Dispersion Relations. We will now examine

 

some of the general properties of the waves in a plasma
as given by the dispersion relations 111.59 and 111.60.
From the definition.of a (Eq. 111.37).and the fact that

c >> VF, the absolute value of a is always less than 1
for the em wave. .Thus we may approximate at by expanding

111.56 as a power series in a. Since
arctan a = Z (-l) fHTT_ (111.66)

we find

(-1)n332n

2
w -
= - 1 I
et 1 E w1w+iyi 1 + % (2n+3)(2n+1) (111°67)

 

Or to zero and first order in a2 we have

55

6t = 1 ‘ WEE-{7T (111.68)
wz k-hv2
.- _ l Z;__§;.
St - 1 EXPO-TEE? 1 + 5 . (111.69)

(Wiv)2

For the sake of simplicity we wrote 111.68 and 111.69 for
a one component plasma, but the results may be easily
generalized to a multi—component plasma. Solving 111.69
explicitly for kt,we have

(02

 

. 1 _
[EEtJ‘ = “(9+1Y5 , . (111.70)
w + 1 VF 2mm ‘
1 '5' 2;" TLM 3

Since (VF/C)2 is about 10.5 or less, Eq. 111.70 indicates

that the commonly used expression 111.68, for ET , is
good for frequencies w/wp > 10-2.

We next examine the behavior of the irrotational
polarization wave in a one component plasma. For fre-
quencies near mp, |a| < 1 provided y < w, and $2, expanded

to first order in a2, is

'2 2 2
.w [1 + 3 kLVF 2]
= 1 - 919*115 kivng (“+1Y5 . (111.71)
- -1 .
1 1w 3iw+1yi‘

Note, that to zero order in a2, a, and e are equal, but

I:
because this approximation neglects all except the infinite

wavelength dispersion of the irrotational, for our problem

we cannot equate 82’

56

with €t° For a collisionless plasma

(y = 0), the dispersion relation becomes

[slur

(0

As for the em wave, k2

w<w,
P

real for w > w
P

2 2
a2.) ML] 4] (111.72)
VF w
P
is primarily imaginary for
and vanishes near mp The pri-

mary difference is that k2 is greater than kt by approx-

imately %— or inversely the em wavelengths are always larger

F

than the irrotational wavelength by the factor %—.

This
F

fact, we shall later see, has important consequences.

IV. RESULTS

1. Plasma Slab of Infinite Thickness

 

Taking the dispersion relations calculated in the last
chapter,we shall evaluate the expressions for transmission
and reflection of em waves by a plasma. Since the size
of the effect due to the irrotationa1.p1asma wave depends
on the ratios of y to a and B (Eqs. 11.16 and 11.24) we
shall consider.approximate.values of these quantities and
determine if the.effect is measurable. Neglecting the

collision frequency in 111.68 and 111.72 we find

 

/cos2 6 - x:2 (1V.1)

ID
ll

 

Y = /3/5 sin2 e[:§],

x2 x - l

where x = w/wp, e is the angle.of incidence, and the

first medium is chosen to be vacuum, so so = 1. Since

the fraction of the energy flowing into the irrotational

xvave (Eq. 11.29) has a maximum when 8 = [Sgt] vanishes,
‘ n

the largest effect of the irrotational wave should occur

1mhen the em wave is refracted into a surface wave along

57

58

the interface. This condition is achieved in IV.1 when

cos 6 = x‘1 = (mp/w), (1v.2)

so that

a = Sin2 6 cos 6 ,

(1v.3)
- /3/5 (VF/C) sin 6 cos3 9.

.4
I

Since there is no transmitted em wave.the condition for
all the energy of the incident.wave to go into the long-

itudinal plasma wave.is a = y,or

tan a sec 6

/3/5 (VF/C)
(IV.4)

w/wp

sec 6

For typical solid state plasmas with a Fermi velocity of
about 108cm/sec,this critical angle is only a few min-
utes, i.e. almost norma1.incidence, and.the.frequency
exceeds the plasma.frequency by.a few tenths of a per-
cent. But.because the usual Brewster angle, corres-
ponding to a = B, is also a few minutes near the plasma
frequency, the dip in the reflection due to the longi-

tudinal plasma wave would be.indistinguishable from the

59

dip due to the usual Brewster condition.

2.. Plasma Slabs of Finite.Thickness

Although the experimental conditions for observing
irrotational.polarization.waves in the.ref1ection spec-
trum of.a very thick plasma slab appear difficult to rea-
lize, they may be.met for thin slabs. Since the problem
lies with the fact.that.at the frequency and.ang1e at which
the polarization.wave.has a.measurable.effect, the reflec-
tion changes drastically.because.the plasma becomes trans-
parent.to em waves, let us.consider-p1asma.slabs or films
thin enough to be transparent.to.these em.waves. In
other words,-we can-exploit the fact that the wavelengths
of the em waves.in.plasmas exceed the wavelengths of the
polarization waves.by.the.factor.c/VF, by choosing plasma
slabs thin compared to the wavelength of the em-wave;
.ReCngt)d << n, but.thick compared to the wavelength
of the polarization.wave; Re(n-k1)d >.n. Such a slab
would be almost entirely-transparent.to em waves per-
mitting the effects of the.irrotational waves to be

observable.

Numerical Results.. Defining the transmittance, T, and

 

reflectance, R, in the usual way; as the fraction of

energy transmitted and reflected.by.the.entire slab:

g . s,
T "' Egg, R no , (IV.5)

60

we calculate them by taking the absolute value squared

of the expressions 11.38:

= mm |2

T
0
WI2 IMI2

 

(IV.6)

Nt’ Nr’ and M are_given.by 11.39, and-for a plasma slab

bounded by.a vacuum on both-sides they simplify to:

2
II

t 4a[(1 - ¢§)¢,v + (1 - ¢§)¢t8]

Z
I

r - (1 - 0.02)2AD . (0t - ¢,)ZCB (1v 7)

I:
n

(1 '-' ¢t¢3la) [D2 ' A2¢t¢pl ' l

- (0t - 0,)[02¢t - C20,]

These expressions have been computed numerically for fre-
quencies between 0.6 mp and 1.5 wp in steps of 0.005 mp,
for three slab thicknesses, and four angles of

incidence. The values of slab thickness chosen were

d/Ap . 0.0305, 0.0915, and 0.1525, where AP is the wave-

length of an em wave in vacuum at the plasma.frequency

and is defined by

A = ———-. (IV.8)
P

Angles of incidence of 10, 30, 60, and 80.degrees were

61

used in the computation.

The exact, collision dependent, dispersion relations
(Eqs. 111.56 through 111.60) were solved numerically for
a one component plasma described by a.Fermi.velocity
VF = 1.4 x 108cm/sec. and a constant collision frequency
equal to lO'ZwPprr= 100). The results were plotted
directly by the.computer,.along with the-absorptance A,

defined as

.A=1-R-T, . (1v.9)

and are presented in Figs. 4 through 10.

TRANSMITTANCE

Fig.

l.

0.

0.

0.

0

4a.

0

9

.8

7

6

.5

.4

.0
0.

62

 

 

 

 

 

 

 

~ [VF = 1.4x108cm/sec
wpt = 100
,_ e=10°
AP = ch/w
d1 = 0.03051
1 d2 = 0.09151p
d3 = 0.15251

P

 

 

 

 

 

 

1.0 1.1 1.2 1.3 1.4
1000
(/p)

Transmittance versus frequency for p-polarized
radiation incident at an angle of 30 degrees on
plasma slabs of thicknesses d/A = 0.0305, 0.0915,
0.1545, where 1 is the vacuum Bavelength at the
plasma frequency, A = ch/ . The plasma is
described by a Form? velocigg VF = 1.4x108 cm/sec
and a collision frequency corresponding to pr=100.

6 0.7 0.8 0.9

Fig.

REFLECTANCE

0

4b.

0

63

 

 

 

 

 

 

 

 

 

 

 

 

dence and various thicknesses.

 

 

.9 -
8 r
Infinite
(j/thickness
_ d1
I L I 1 __
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.
0000
(/p)
Reflectance versus frequency for 30 degree inci-

4

ABSORPTANCE

64

 

 

 

 

 

 

 

 

1.0
0.9 r
0.8 -
0.7 r
0.6 .
0.5 _
0.4 _
0.3 , .1,
d2
0.2 k \
T d3
0'1 / k
\x
0.0 1, i 1;; 1 , , V§~A~--..

 

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4
(w/wp)

4c. Absorptance versus frequency for 30 degree inci-

dence, and various thicknesses.

Fig.

TRANSMITTANCE

O
.5

5a.

0
O‘

O
01

0.3

0.1

0.0

65

 

 

0.

6

d3

0

.9

d1

1.0

(w/wp)

1

1.1

1.

2

‘ 1.

3

l.

Transmittance versus frequency for 60 degree inci—

dence and various thicknesses.

 

Fig.

REFLECTANCE

5b.

66

 

 

 

 

 

 

° 0 1 J n L A l a
0.6 0.7 0.8 0.9 1.0 1.1 1.2 ‘ 1.3
(w/wp)

Reflectance versus frequency for 60 degree inci-

dence and various thicknesses.

.4

Fig.

ABSORPTANCE

5c.

67

 

 

 

 

 

 

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.
(w/wp)
Absorptance versus frequency for 60 degree inci-

dence and various thicknesses.

Fig.

TRANSMITTANCE

6a.

68

 

d2

 

 

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 l.
(w/wp)
Transmittance versus frequency for 80 degree inci-

dence and various thicknesses.

4

69

 

 

 

 

1.0
d3
0.9
d2
0.8 ‘
0.7 - ‘11
0.6 -
[L]
L)
2
F5
L20 5 -
LL}
A
LL.
LL]
“0.4 _
0.3 g
0.2 _
0.1 _
0‘0 4 l l I l n 4*
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
(w/wp)

Fig. 6b. Reflectance versus frequency for 60 degree inci-

dence and various thicknesses.

ABSORPTANCE

Fig. 6c.

70

 

 

 

 

 

r.
P d1 2 d3
1 I __*-..——M“§w
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.

1000
(/p)
Absorptance versus frequency for 80 degree inci-

dence and various thicknesses.

Fig.

TRANSMITTANCE

7a.

71

 

 

 

 

 

 

 

 

 

 

 

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
(w/wp)
Transmittance versus frequency for slab thickness

d = 0.03051p at various angles of incidence.

1.

4

72

 

800

600

 

30°

REFLECTANCB
<3
U'l

 

 

 

 

 

 

 

 

 

 

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
(w/wp)
Fig. 7b. Reflectance versus frequency for slab thickness

d = 0.0305).p at various angles of inCidence.

73

 

 

 

 

 

1.0
0.9 F
0.8 *
0.7 '
- 0.6 -
m
D
Z
5
9,05-
0!.
O
U)
m
“ 0.4 . 30°
0 3 . 10°
600
0.2 _ 80°
0.1_ \
j ‘ J /
0.0 JL - _ A

 

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
w
(w/ p)
Fig. 7c. Absorptance versus frequency for slab thickness

d = 0.03051? at various angles of incidence.

 

Fig.

TRANSMITTANCE

8a.

.91 100

74

 

 

 

 

 

 

 

 

 

 

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4
(w/wp)
Transmittance versus frequency for slab thickness

d = 0.0915).p at various angles of incidence.

Fig.

REFLECTANCE

8b.

0.

Reflectance versus frequency for slab thicknesses

d

75

 

 

800

60°

 

 

 

 

 

6

0.09151
P

.8 0.9 1.0 1.1 1.211.
(w/wp)

at various angles of incidence.

.4

76

 

 

 

 

1.0
0.9 '
0.8 ’
0.7 t
m 006 b
L1
2
:5
n. 0 5 .
g; 10°
a:
5%
0.4 .
0.3 L
0.2 b
0.1 _
0.0

 

 

 

 

 

0.6 0.7 0.8 0-9 1.0 1.1, 1.2 1.3 1.
(w/wp)
8c. Absorptance versus frequency for slab thickness

d = 0.0915).p at various angles of incidence.

Fig.

TRANSMITTANCE

9a.

0

1.

77

 

0

.5 s \ l

.3, ’

 

l ‘ l L l J l

 

 

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
(w/wp)

Comparison of the transmittance versus frequency
calculated using the Fresnel equations (11.41),
dashed curve, and the general equations, solid
curve, for a thickness d = 0.0305).p and for 60
degree angle of incidence.

.4

REFLECTANCE

78

 

1.0

0.9 ’

 

000 ,. I l J J

 

 

0.6 0.7 0.3 0.9 1.0 1.1 1.2 1.3
00
(mp)

9b. Comparison of the reflectance versus frequency
calculated using the Fresnel equations (11.41),
dashed curve, and the general equations, solid
curve, for thickness d 0.0305).p and for 60

degree angle of incidence.

Fig.

ABSORPTANCE

9c.

79

 

 

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
(w/wp)

Comparison of the absorptance versus frequency
calculated using the Fresnel equations (11.41),
dashed curve, and the general equations, solid
curve, for thickness d = 0.0305).p and for 60
degree angle of incidence.

 

Fig.

TRANSMITTANCE

10a.

80

 

 

 

 

.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1
(w/wp)

Comparison of the transmittance versus frequency
calculated using the Fresnel equations (11.41),
dashed curve, and the general equations, solid
curve, for thickness d = 0.09151p and for 60
degree angle of incidence.

.4

Fig.

REFLECTANCE

10b.

 

81

 

 

I J L L 1 1 I

 

.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
(w/wp)

Comparison of the reflectance versus frequency

calculated using the Fresnel equations (11.41),

dashed curve and the general equations, solid

curve, for thickness d = 0.0915).p and for 60
degree incidence.

1.

 

 

Fig.

1.0

0.9

0.8

O
O‘

ABSORPTANCE
O
L.

O
4:.

10¢.

82

 

 

 

 

(w/wp)

Comparison of the absorptance versus frequency
calculated using the Fresnel equation (11.41),
dashed curve, and the general equations, solid
d = 0.0915).p and for 60

curve, for thickness

degree incidence.

1.0 1.1 1.2 1.3 1.

4

V. DISCUSSION AND.CONCLUSION

The maxima and.minima structure.in.the.reflectance,
transmittance, and.absorptance.spectra appearing in
Figs. 4 through 10 arises from spatial resonances of long-
itudinal plasma waves.of.finite wavelength propagating in
a slab.of.finite.thickness., The source and the condition
for the.occurrence of this structure become apparent if
we approximate the.equations of.ref1ection.and transmis-
sion IV.7 under the.condition that the wavelength of the
em wave.is larger than the slab.thickness.. Then

0t = exp (in-ktd) is nearly unity, and

Nr 2 [AD - BC|(1 - 0,)2

= ‘4YBC1.' ¢£)2
Nt 2 408(1 - 4,2) (Vol)
M = (D. - 42¢, - 32 -.C¢,](1 - ¢,)

4e[a(1 + 0,).+ ycl - ¢,))(1 - ¢,)

Neglecting damping, so that the.waves.are homogeneous and

a and y (Eqs.-11.16 and 11.24) are real, we find R and T

83

84

 

 

(Eqs. 1V.6):
R=liL1 ' C05 ”)1
A
(v.2)
.T.=.a2(1 + cos p)
A
where
A = 02(1 + cos 0) +.y2(l - cos 0) . (v.3)
and
1» =-Re(g‘,l_<_£)d (v.4)

Clearly, even if y << d, the reflectance (transmittance)

will have relative maxima (minima) whenever

w = nn; n = l, 3, 5,... (v.5)

i.e., whenever the slab thickness equals.an odd number of
half wavelengths.of.the.longitudina1.wave. The resonance
condition can be.visua1ized.as.a standing polarization wave,
with the volume charge.density developed.in the region

of the boundary coupling the.incident.em.wave to the

polarization wave.

85

Superimposed in these relative maxima (minima) is
one large maximum (minimum) due to the em wave being
refracted into a surface.wave. This effect increases with
angle of incidence as seen in Figs. 7 and 8.

It is interesting.to note that our eXpressions for the
reflectance and transmittance V.2, reduce to the corres-

25

ponding equations of McAlister.and.Stern» near the plasma

frequency. Near mp,.w becomes small so

 

 

 

 

 

 

 

¢£ = l + 10
k (v.6)
c
= 1 +'i[ 8] 2nd ’
w n
and the expressions for.R.and T become
. i 2
R 7a - 170
.an sin2 0 2
= 1 cos 6(1'6)
A
(v.7)
2 a 2
T 2d - 1yw
- 32 2
- A

 

 

where

 

_ g . 20d sin2 6
A - 25 1 1 cos e(l-e) (V.8)

86

Eqs. V.7 differ from the corresponding expressions of
McAlister and Stern (Eqs. (3) and (4) in their paper) by
the factor 1 --e 2 (wp/w)2, which approaches unity near
the.plasma frequency.

Our results indicate that the wavelength of the
longitudinal highfrequency plasma wave in a metallic
plasmasuch as an alkaline or noble metal may be meas-
ured by an.optical.experiment,.provided the metallic foil
is quite.thin.. Silver,.for example, whose conduction elec-
trons have a Fermi velocity of 1.4 x 108.cm/sec, has a
plasma frequency of 5.8 x 1015.secfl,-corresponding to a
plasma wavelength AP of 3280 Angstrom units in thickness.
For comparison, the screening length for the silVer plasma,
AP = vF/wp,is approximately 50 Angstrom units. Thus the
spectrum of the thinnest foi1,-which provides the most
resolved resonance peaks, may not be applicable since the
wavelengths are approaching the limiting screening length.
0n the other hand such an.experiment.may provide informa-
tion.concerning-the critical cut off wavelength. Another
experimental difficulty may be the collision frequency.
Although the value pr = 100 is well within.the value
opt = 233 obtained from mobility measurements in silver,
these measurements were made in large, pure samples and
may not apply to evaporated films.

A calculation was also made.of the reflectance and

transmittance by.the acoustic wave in a.two component

87

plasma. But because the.phase velocity of.the acoustic
wave is much smaller than that of thehighfrequency

mode no measurable effects.were obtained. Furthermore,
the acoustic waves are strongly Landau damped-even without
collisions. .Thus.experiments, such as those.of McWhorter

2 to excite acoustic plasma waves by electro-

and May,3
magnetic radiation seem very unlikely. .The situation
may be improved by applying static electric fields to

the plasma.to amplify.the.acoustic wave.3:5

But as a
result of the anisotropy produced.by-such a-field.the irro-
tationa1-and divergence-free waves.are.no.longer inde-
pendent, so that.the.present.theory would no longer be
applicable.

Using the method-described in Chapter.11, we have
also calculated the.case of longitudinal plasma waves
incident on a boundary and emitting em waves. This

d15 and our equation for the

problem was treated by Fiel
ratio of the em wave.electric field to the plasma wave
electric field agreed with the corresponding equation

in Field's paper. This is.not surprising, since Field
also describes the real boundary as having a finite
transition region of the order of the screening length
and he uses the continuity of the normal component of

the electric field for a boundary condition. Our results,

however, are more general since we consider a dielectric

plasma boundary.and inhomogeneous waves,.which are more

88

realistic for a lossy system such as a plasma.

In summary, we.have studied the linear.coupling be-
tween longitudinal polarization waves.and.transverse elec-
tromagnetic waves introduced.by.the.existence of plane
boundaries separating.dielectric media or vacuum from
conducting media capable of supporting polarization waves.
We have investigated this coupling by extending the usual
Fresnel equations to include the effects of irrotational
fields. Because.the.theory.was developed in terms of.
the plasma dispersion.relations a means has-been suggested
for studying experimentally.the wavelength.dependence of
plasma waves. We have also shown that the dispersion
relations for inhomogeneous waves in.the plasma are still
separable into electromagnetic-like and polarization
waves. Finally, absorptance, reflectance,.and transmit-
tance spectra were numerically calculated and the re-
sults indicate that the dispersion relations for plasma

waves in-thin metallic.films.may be optically measurable.

APPENDIX A.
EVALUATION OF I AND K

In the spherical.coordinate system, Fig. 11, the

 

 

 

integrals
I = [1"‘1‘Lfla. (A-l)
d9?
V = [1713—1 U”)
become
2n #1 A A
.1 = J',d¢J’ . , ‘1‘?” . (11.3)
0 41'1 + lazu + 1ax/l-u2 cos 4
2n f1 A
' " dur
X = [\d0J _ 94' (A.4)
O -1 1 + 1azu + 1ax/l-u5 cos ¢
For homogeneous waves, ax = 0 and A.3 and A.4 take on a

simple form which is easily evaluated:

 

 

 

 

l+a2
T = T = 2" zarctan a - 1
xx yy a 2 a2 2
z (A.5)
‘ 4 arctan a
T = n 1 _ z
z

and

 

arctan a
v = L" [1 - 2] (A6)
2

90

 

 

 

Fig. 11. Coordinate system defining direction of tensor
components.

91

all other components being zero.
To evaluate A.3 and A.4 for the general inhomogeneous

wave case we need the value of the integral

 

n
_ dx
I - J v + w cos x (A'7)

The indefinite integral of A.7 has the following values34

 

2 _ 2 ‘% . W + V COS X 2 2
(v w ) arccos[v + w cos x > w
-l
v tan (x/Z) v = w
-1
-v cot (x/2) v = -w

1 O
- w + + -v
(w2_v2) 6 1n v cos x w Sln x w2 > v2
v + w cos x

Therefore the integral A.7 is defined as long as v2 is

unequal to w2

I = -—--. v2 7‘ w2 (A.8)
V ‘W

From A.8 we find

20
I do = 20
0 v+w cos ¢ /;T:;2

 

 

 

2n

‘ cosodg, = 21. 1 _ v

v+w cos 4; w W
0 - _
20 2

'cos ¢d¢ _ 21 v2 _

J v+w cos ¢ 7 w2 [ v -w V] (A-9)
O ,
Zn

. n
Sin ¢ cos ¢d¢. = 0
v+w cos ¢
0
271
2

Sin d - £_ - “FT? 2

0 v+w cos¢ ' W2 [v v w 1

We now extend the integrals A.9 to include complex v and
w and state without proof that A.9 exists as long as the
absolute value of U does not vanish, where

= 1 + a 2 + Ziazu - (ax2 + a 2)u (A.10)

X Z

The value of the first three integrals of the form

+1
1n = undu
_i/U_

can be expressed as

2 arctan a

10: a

 

93

 

a
1. = -21—-z- [1 “eta“ 3] (A.11)
a2 a
Zaz-a2 ZaZ-a2(l+a2)
13 = ——£——§— - z X arctan a
a“ as
where
a2 = 373 = a; + a: (A.12)

With A.9 and A.11 the desired integrals may now be eval-

 

 

 

 

 

uated:
a2(l+a2) - 2a2 a2 - 2a2
T = 2n 2 x arctan a - z x
XX a5 an
a a 2
T = T = 20 x z [ - 3+3 arctan a]
X; zx a“ a
(A.13)
2
T = £1 lié. arctan a - l
YY a2 a
Zaz-a2 2a2-a2(1+a2)
Tzz = 2n ——5——§—-- z x arctan a
a“ a5

T T
xy yx yz zy

and

with the condition

H-l #-
:4

ll
H-b
:1

that

94

711'

9)

a)

I.

a
[1 -
a2

laI7‘1

 

arctan a
a

 

arctan a
a

(A.14)

(A.15)

10.

11.
12.
13.
14.
15.

REFERENCES

L. Tanks and I. Langmuir, Phys. Rev. 33, 195, 990
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the problem of plasma oscillations was first considered
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use: to

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95

16.

17.
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21.

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26.

27.

28.

96

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R. A. Ferre11,.Phys. Rev. 111, 1214 (1958).

A comprehensive.review.of the energy loss measurements
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In addition, Ferrell suggested that such.an eXperiment
would constitute decisive evidence that the energy loss
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controversy on this point. Ginzburg and Frank

(V. L. Ginzburg and I. M. Frank,.zh..eksperim. i teor.
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is emitted. V. P..Silin and E..P. Fetisov, Phys.

Rev. Letters 1 374 (1961) and R. H. Ritchie and

H. B. Eldridge, Phys. Rev. 126, 1935 (1962) extended
their calculations and.showed that for very thin
metallic foils this.approach also.predicts a radia-
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W..Steinmann,.Phys..Rev. Letters,.l§ 470 (1960)

H. Boersch, C. Radeloff, and.G. Sauerbrey, Phys. Rev.
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B. T. Arkawa,.A. L. Frank,.and R. D. Birkhoff, Phys.
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For example,.Stratton.J.,."Blectromagnetic Theory,"
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chapter on the.e1ectromagnetic theory of light in the
"Handbuch der Physik," Vol. XX, pp. 197-253, Springer,
1928. The section on metal optics generally follows

29.

30.

31.

32.

33.
34.

97

the work of C. Pfeiffer, "Beitrage zur Kentnisse der
Metallreflexion," dissertation, Giessen, 1912.

For example, 0. S. Heavens, "Optical PrOperties of
Thin Solid Films," p. 56. Academic Press, New York,
1955; or J. Stratton, ref. #28, p. 51. As with 11.15,
T in 11.41 differs by the factor /€2/€2 from the
Fresnel equations in the literature because of the
way T was defined.

For example, f may also be obtained by the path.in-
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Phys. Soc. (London).A238, 344 (1957); A65, 458 (1962).

I. B. Bernstein and S. K. Trehan, Nuc. Fusion 1,
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A. L. McWhorter and W. G. May, 1.B.M.J. of Res. Dev.
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M. J. Harrison, J. Phys. Chem..Solids 23, 1079 (1962).

G. Petit Bois, Table of Indefinite Integrals, Dover,
1961, New York, pp. 121-22.

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Lm