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Department of Mechanical and Aerospace Engineering (MAE)Faculty and Researchers Collection
1965-11
Behavior of Separation Constants for Finite
Gyromagnetic Plasmas
Bevc, Vladislav
IEEE Transactions on Antennas and Propagation, v. 13:6, 918-926
http://hdl.handle.net/10945/30361
Downloaded from NPS Archive: Calhoun
IEEE TRASWCTIONS Oh;AWTE-WAS AND PROPAGATION
VOL.
H. Norinder, “The waveform of the electric field in atmospherics,” Arkiv. Geofysik, vol. 2, pp. 161-194, June 1954.
1%’. 0. Schumann, “Uber die Oberfelder bei Ausbreitung langer
elektrischer iyellen urn die Erde und die Signale des Blitzes,‘‘
Nuovo Czmento, vol. 9, pp. 1-23, December 1952.
J. R. JVait, “On the theory of the slowtailportion of atmospheric
waveforms,” J . Geophys. Res., vol. 65, pp. 1939-1946, July 1960.
J. R. FVait, “On the propagation of ELF pulses in the earthionosphere waveguide,’’ Cunad. J . Phys., vol. 40, pp. 136CL1369,
October 1962.
j.-R.
‘Ci’ait, “Introduction to the theory of VLF propagation,”
Proc. IRE, vol. 50, pp. 1624-1647, July 1962.
J. R. Johler and L. A. Berry, “Analysis of the LF radio signal in
AP-13, KO. 6
NOTEMBER,
1965
the presence of the ambient and disturbed ionosphere” 1965,s
to be published.
[21] J. R. F$:ait and C. Froese, “Reflection of a transient electromagnetic wave a t a conducting surface,” J . Geophys. Res., vol.
60, pp. 97-103, hfarch 1955.
1221 A. G. Jean, L. J. Lange, and J. R. IVait, “Ionospheric reflection
coefficients a t VLF from sferics measurements.” Geofis. Pura
A p p l . , vol. 38,pp. 147-153, 1957.
J. R. Johler, “Radio pulse propagation by a reflection process at
the lower Ionosphere,” J . Res. X.B.S., vol. 67D, no. 5 , pp. 481-499,
September-October 1963.
Behavior of Separation Constants for
Finite Gyromagnetic Plasmas
VLADISLAV BEVC
Absfmcf-The type of the functions appearing inthe expressions
for electromagneticfields in fjnite magnetoplasmas depends only
on the properties of the medium manifestedin separation constants.
The 0-8 plane of the Brillouin diagram is partitioned into a definite
number of zones characterized by the type of possible field solutions.
The general behavior of the separation constants in the 4 plane
including the complex region based on calculations covering a wide
variation of the determining parameter is described. Effects of ion
motions are discussed, and the generality of the presented approach
is pointed out.
INTRODUCTION
F THE KUhIEROUS papers [I] concerned with
0
the
propagation
of electromagnetic
waves
through waveguides filled withanisotropicor
gyrotropic plasma columns in which ions are assumed
to be stationary and the collision and thermal effects
are neglected, the majority are confined either to the
quasi-staticapproximation [2]-[j] or to a discussion
of the problemingeneral
termswithtabulations
of
auxiliary functions [6], such as, e.g., the biradial functions [’i] and an outline of the general method of approach for obtaining the solution [6]-[12] in the form
of dispersion curves without a systematic and comprehensive quantitative presentation and classification of
numeroustypes of w-/3 diagramsand field solutions
such as are known for empty waveguides [13] and for
SIOW wavestructures[14].Only
a few quantitative
dynamic solutions representativeof general behavior of
dispersion characteristics of the W-pdiagram have been
published thus far [15]- [21], and these did not include
a
complete systematic survey
of all possible cases although
Manuscript received February 3, 1965; revised June 9 and June
15, 1963. This work was first reported in condensed form a t the 1964
International Conference on Microwaves, CircuitTheory, and Information Theory, Tokyo, Japan, and has subsequently appeared in
United States Kava1 Postgraduate School Research Report 47,
Monterey, Calif., February 1965.
The author is with the United States Kava1 Postgraduate School,
Nlonterey, Calif.
they were certainly oriented in this direction. model
The
used in these computations, with the exception of the
work by Auer et al. [ l j ] ,was a collisionless cold plasma
in which ions are assumed to be stationary. This model
was first introduced to this problem b!, Hahn [ 8 ] , and
was subsequently adopted by several other authors
who
developed the elementary theory of wave propagation
in gyrotropic plasmas [9], [lo]. Camus and Le AIezec
[18], [19], Johnson and Berett [21], and Likuski [22]
have calculated several characteristic dispersion curves
of waves with no angular variation; Bevc and Everhart
[16], [I71 and Eidson and Auld [20] have also calculated severalcasesfor
angularvariations
a = 2 1.
Hopson and Wang [ l l ] state that modes with angular
variation of the fields cannot exist. Their reasoning is
based onanerroneousassumption
forwavefunction
which had, infact,already
beendiscussed and ruled
out earlier by Suhl and Walker [lo] as being unduly
restrictive.
Even when the motion of ions, collisions, and the
thermal effects are neglected, the number of combinationsthatcanarise
for reasonablestepsin
which
parameters can be varied is considerable. The parameters to be varied are: 1) the plasma density manifested
as the plasma frequency a,,assumed uniform throughout the cross section of the waveguide, 2) the intensity
of the collimating uniform axialmagnetic field BO or
the corresponding cyclotron frequency
W H , 3) the size of
the column or the radius of the circular waveguide a ,
appearingintheconvenient
normalizing frequency
w0= c/a related to the lowest empty waveguide cutoff
frequency. For quantitative work which alone can be
suited for evaluation of experimentalresults,
it is
desirable to establisha system of classfication of modes
of propagation and to prepare a catalog of solutions.
Recently, a systematization of solutions has been proposed [23] which uses a parameter space representation
918
BEVC: SEPARATIOK CONSTAKTS FOR GYROMAGNETIC
PLAShL4S
919
similar to theAllis diagram [24] for waves in unbounded conditions, y ( u ) is a function of parameters w p ,W H and
magnetoplasma. This systematizationuses a normaliza- of w and p ; J,(ur/a) is a Bessel function of order n of
tion of characteristic parameters of the plasma column first or of second kind or of complex argument, dependup,w H , w o with respect to the frequency w a t which the ing on whether the separation constants ztl and u- are
waves propagate.
real,imaginary,
or complex,respectively
[15]- [19],
I t is believed that a more direct and,for the purposes [22],[27].
lye notein
passing that forgyrotropic
of comparing the computed dispersion curves with the
ferrites, complex values of separation constants do not
experimental results which are almost invariably given occur a t real values of frequency and propagation conin the form of w-/3 diagrams [2], [SI, [25],[26],
a stant. Likuski [22] givesatableshowingforwhich
simpler approach is one where the types of solutions are values of separation constant solutions exist in a comclassified according to certain zones in the w-/3 diagram pletely filled waveguide.
in which theyoccur. Inthis way, anyintermediate
The separation constants u 1 and 212 determining the
transformation toand from theparameterspaceis
transversevariation
of the fields aresolutions of a
avoided.
biquadratic equation first given by Hahn [8]. The deriQualitative classification of modes has already been
vation of this equation dependssolely on the properties
proposedaccording to the nature of thepropagation
of the medium and is independent of boundary condicharacteristics, viz., a s plasmaguide modes (P modes), tions. The equivalentdielectrictensor
2 forwave
Cyclotronmodes(C
modes) [2], andpredominantly
propagation in the direction
of the collimating magnetic
transverse electric (TE) and transverse magnetic (TM) field is given below.
waveguide modes [16]. In addition to this designation
and for quantitative comparisons,
i t is proposed that
the parameters, such as the ratio
of theplasmafre5 = ,-je;
j e;I1 2
O
(3)
quency w p to the normalizing frequency wo (related to
O
€33
the radius of the waveguide) and the ratio Af of the
cyclotron frequency W H to the plasma frequency wp,be For electronic plasma in which collisions and motion of
specified by subscripts and superscripts preceding the
ions are neglected, the components e i j are defined as
modedesignation letterwiththeangularandradial
follows:
variation indexesfollowingin
the manner customary
with theempty waveguidemodes.Formodeswith
angular variation, there should also be a n indication of
the sign of the angular variation number, e.g., the symbol 0.s5.0Pln
would mean a plasmaguide modewith angular
variation n= 1, radial variation number
Y =2 at the
ratio w ~ / w =
, 0.8, and the normalized plasma frequency
UP2
€33 = 1 - *
w p / o o = 5 . 0 . In using such labeling of modes, cognizance
w 2
should also be taken
of the fact that the radial variation
of cyclotron modes does not always remain the same
The biquadratic equation is then found in thefollowing
along the same dispersion curve; in suchcases two sym- form [s]-[~o]
bols may be necessary to identify the mode. This behavior of cyclotron modes will be discussed elsewhere.
G~ €11
e33]
9
r
'4 BASISFOR
OF
THE SE'STEMATIZATION
FIELDSOLUTIOKS
1
1'
+ L{[(!)2
+ [
€122
-
2)!(
y:([
-
€11]2}
+
(:)b
The first step toward a systematization of field solutions is to determine what type
of functions occur in cer- ...(€,,2
-€
4
tain regions of the w-/3 plane. I t is well known [SI-[IO]
€11
that in a circularcylindricalwaveguide
filled with a
cold collisionless plasma of uniform density collimated
The symbols denote thefollowing quantities:
by an axial magnetic field, the radial variation of the
U = uc/wpa
longitudinal high-frequency electric and magnetic fields
/3 =propagation constant appearing in expj ( w t -pz)
E, and Hz, from which transverse field components can
w = frequency
- of the wave
be derived, are given by the following expressions:
u p= v'qpO/~O=electron plasma frequency
Cdn(zw/a)
CJn(z42v/a)
w H =q&
electron cyclotron frequency
E, =
(1)
a
=radius
of the waveguide
2412 - 2422
c =velocity of light.
Cly(ul)Jn(ulr/4
C2y(uz)Jn(u2r/a)
H, =
*
(2) I t is possible to normalize all quantities in
(3)-(5) in
2$12 - u 2 2
such a way that, so long as no boundary conditions are
The constants Cl and C2 are determined by boundary introduced, the only parameter that needs to be varied
[(y
+
+
IEEE TRANSACI'IONS ON ANTENNAS AND PROPAGATION
920
NOVEMBER
is the ratio L M = W H / W ~ . Equation (5) is not restricted
stants are complex is given by the following expression:
only to electronic plasma.I t is quite generally applicable
to media composed of various species when appropriate
components of the equivalent dielectric tensor areused.
This will be illustrated in the discussion of the effect of
ion motion given a t the conclusion of this paper. The
equationsderivedsubsequently,
however, arenot so
general;theyare
validonlyformediacomposed
of
species with no drift velocity, as i t will become readily
apparent.
The combination of signs m u s t be such that both w and
The explicit solution of the biquadratic equation for p are positive. For a collisionless cold uniform electronic
U12and U.2 is
plasma with stationary ions, this expression reduces to
[MI, [ I ~ I
u
1
2
2 = (1
(!2)2
+
w
o<-<1
;(E) [ + 2)
$1
* +(E)' ( .):
U P
1-
Y
L\w/
€11
- €33
(Eu
-
\w/
The positive sign in frontof the square root
in the previous equation is consistently used for the calculation of
the value of U1and the negative sign for the calculation
of the value of U2.
Evidently, five different types of expressions for the
field solutions of (1) and (2) canoccurdependingon
whether the values of U are real, imaginary,or complex
conjugates, viz.,
1) U12, UZ2 bothpositive;
2) VI2positive, UZ2negative;
3) U12 negative, U22 positive;
4) UI2, U22 both negative;
5) U12, U22complex conjugates.
4-
dl
-
(r)2].
(8)
WP
The region just defined will be referred to as complex
region. I t isshown on Figs. 2 and 3 tobeanovalshapedclosed
curve, beginning and ending a tt h e
origin, touching the line w / w p= 1 a t & / w p = 1, and the
line
UP
Forimaginaryvalues
of U1 and UZ, the modified
Bessel function I,((ur/a) replaces the J,(ur/a) in (1)
and (2). When the behavior of U1 and lJ2 in the 0-8
plane is known, i t is possible to tell what type of function appears in theexpression for the fields. Thus, if in a
region of the w-/3 plane, U1 is found to be imaginary and
U2 to be real, it can concluded
be
t h a t , subject to certain
conditions, the radial variation of the field is determined
by theoscilIatory part J,(uzr/a)of the solution sincethe
contribution due to the part with
I,(ulr/a) may, because of its exponential character, become negligible or
unessential. Such is the case with slow wave solutions,
the propagation curves of which may, under very restrictive conditions, be computed with the use of quasistatic
approximation.
Indeed,
it
appears
that
the
plasmaguide modes described by Trivelpiece and Gould
[2] are confined to a particular part of the w-/3 plane
for the reasons very similar to those outlined above.
PARTITION
OF THE w-8
PLANE AXD
at
w
-= d ( 1
*P
+ M2)/(2 +
M2).
The complex region becomes narrower as the value of
:ki increases. Since on the boundary curve of (7) or (S),
the discriminant vanishes, the valuesof separation constants on i t become identical. I t has been stated [27]
that for modes of propagation, the separation constants
cannot be equal since it would appear from expressions
of (1) and (2) that longitudinalfieldswouldthenbe
indeterminate.Calculations[16],
[22] show,however,
that dispersion curves can traverse the complex region,
and that longitudinal field components for values of w
and /3 lying on the boundary of the complex region are
finite. One can readily show by meansof a limiting process that the longitudinalcomponent of the electric
field E, of (1) becomes in this instance
ZOWEBOUXDARIES
As already mentioned, the separation constants can
assume complex conjugate values [IS] for certain real
values of w and /3. From (6), i t is apparent that the
boundary of the regioninwhich
the separation con-
A similar expression is found for the longitudinal magnetic field Hz.
BEVC: SEPARATION CONSTANTS FOR GI'ROXAGNETIC PLAShlAS
1965
The value of the separation constants a t the origin
depends on the path of the 0-0 diagram along which
the origin is approached; the separation constants are
multivalued a t t h e origin.
I t is further of interest to determine whether, and in
which points on the boundary of the complex region,
the valueof separation constants becomes equalto zero.
Clearly, U = U 1 = U2vanishes a t
(2
I .o
a8
I
a707
0.6
a4
0.2
w
PC
-- 1and
- = 1.
UP
UP
The other points are found by setting the
expression for
L7z in (6) equal to zero, and solving the resulting equation in (w/wP)
(ty-
921
M2
2
+ M2)
- 0.
Fig. 1. Ordinates W L and w e of the points on the boundary of the
complex region in which the values of the separation constants
Cl 1 - CT2 vanish in dependence of the ratio of the cyclotron frequency to plasma frequency M.
W
WP
2 .o
1.6
Factoring this equation, we obtain
I .e
I.o
.[(3
-2):"(
-
(t)I;+
0.8
=
0.
(11)
0.4
Theroots of thisequationarerealforpositivereal
values of X . The roots of interest can be conveniently
plotted in dependence of the parameter X , with the use
of the following relation
=
fM
u:>o
I
0.4
0.e
1.2
I
I
I
1.6
2.0
2.4
Fig. 2. Zones of the w-j3 plane in which the functions of l?l2 and
U2* are of the same or of opposite sign. General behavior for
values of M < 1.
This relationship is shown graphically on Fig. 1. From
Figs. 2 and 3, it is apparent that one point in
which
both Ul and UP vanish lies on the left boundary of the
complex region, and the other such point on the right
hand side of the boundary. The coordinates
of these two
points \\-ill be designated as PL,W L and PR, PA, respectively.
Furthermore, we recall that either I;; or I;? can
vanish on curves defined by the equations [16], [IS]
Thisisevidentfrom
(5). Forthecase
of electronic
plasma under consideration, (13) reduces to
0.4
0.8
u:>o.
1.2
Uf>O
1.6
2.0
2.4
P C
WP
These two curves lie outside of the complex region, but
3. Zones in the w-j3 plane in which the functions U 1 and Uz
they touch its boundarya t the points where the values Fig.are
of the same or of omosite sign. General behavior for values
of separationconstantsvanish.This
is also shown on
of J[>I.
-
I
IEEE TRANSACTIONS
ON
929
BhWEhTNAS AND PROPAGATION
NOVEMBER
Figs. 2 and 3. We note that these two curves are identi- BEHAVIOR
OF THE SEPXR~TION
CONSTANTS OUTSIDE
cal with the propagation curves for the ordinary and
OF THE COMPLEX REGIOW
extraordinary ray [28] in the direction of the magnetic
The behavior of the separation constants in the w-@
field in an unboundedmagnetoplasma.
I t shouldbe
plane outside of the complex region is best represented
pointed out that the separation constants U1or Lr2 do by contour diagrams of. Figs. 4-7. For stationary multinot vanish along the entire length of these curves as is speciesplasmas, thecontourcurvesare
givenby
a
evident from Figs. 2 and 3. This is of importance in the simple biquadratic equation in p obtained from ( 5 ) in
partitioning of the w-fl plane by zone boundaries but which the value of U is set constant, viz.,
has not always been recognized in previous treatments When a component of the plasma has a drift velocity,
of thisproblem.Camusand
LeMezec [18] show on the propagation constant enters the expressions for the
their diagrams the curves of (14) as having no point in components of the equivalent dielectric tensor eiiwith the
common with the boundaryof the complex region, and, consequence t h a t a solution can no longer be obtained
in the same form as given previously. No simple rule
thus, their partition of the w-fl plane is not quite corappears to exist for determiningwhetherthecurves
rect.Thispartition
is not discussedin theirearlier
paper, however [19]. Johnson and Berett [21] present a obtained from (16) are contour curves of constant U1
partition of the w-/3 plane for the value of the ratio or U2. This question can always be resolved, however, by
calculating, with the given value of w/wp and the values
1M< 1, but the detailon their diagrams is insufficient to
shorn the determinationof the sign of the separation con- of &/wp obtained from (16), the desired function U1 or
stants in all zonesof the w-/3 plane. To determine whether U2, as the case may be, using the rule concerning the
Ul or U2vanishes on a certain segment of these curves, selection of sign stated for ( 6 ) and determining whether
rulessimilar to those given by Booker [29] must be the calculated value of Ul or U2is equal to the value of
U originally set in (16).
applied. The results can be summarized as follows:
u1=0
W
PC
o<-<1
-=1
(JP
*P
W
WR
WP
CJP
0<-<-
u2=0
0
WP
BC
-=
//
W l 2
/I-I
-
1
Fig. 4.
Contourcurvesfor constant real and imaginary values of
U I in the w-8 plane at the value of M=0.8.
W
U P
Fig. 6.
Contourcurvesfor constant real and imaginaryvalues of
Ulin the a-8 plane at the value of M= 1.6.
w
U P
2.8
2.8
2.4
2.4
2.0
2.0
1.6
I .6
I.2
1.2
0.8
0.8
0.4
0.4
0
0
0
I
I
0.4
0.8
I
1.2
I
1.6
I
I
2.0
2.8
I
I
2.4
PC
U P
Fig. 5. Contour curves for constant real and imaginary values of
U2 in the W-8plane a t the value of "0.8.
Fig. 7.
Contour curves for constant real and imaginary values of
LT2in the 0 - p plane a t the value of M = 1.6.
Typical diagrams of contour curves on which Ul and is the only parameter which determines the appearance
been verified
U2are positive real or imaginary constants are
shown on and typeof the contour diagrams. This has
for a wide
Figs &7. I t is seen that the only essential difference be- by aIgebraicandnumericalcalculations
tween the cases where the value of M< l and X > l is range of values of M. Within the two groups for 1 4 4 < 1
foundwith thecontour diagram of U2. We wish t o and AT> 1, there is yet some variation in the sequence
emphasize that,albeitthe
presenteddiagrams
are of special frequencies, but this variation only changes
shown for two particular numerical values of M only, the relative position and sequenceof special frequencies
and does not modify the nature of the contour diagram
viz., M = O . 8 and 1.6, thesediagramsrepresentthe
or thepartitioning of the w-fl plane. The following
generalbehavior
of theseparationconstantsforall
cases
values of M. Also, i t is now apparent that the ratio &
l are distinguished :
0
NOVE3BER
B E E TlXNSACTTOiYS ON ANTENNAS Ah?) PROPAGATION
934
< M < 1/42
2
WP
1/42<M<1
o<-<--+
WP
R
d7-
-+1<-<<<1<<"2+1
2
W
WP
L
1<M<3/2
2
"0
M>3/2
M
o< --+
2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
BC
UP
M2
/--+I
v
4
WR WL
<-<-<
WP
1<M<d&f2+1.
(17)
WP
Fig. 8. Contourcurves for constantvalues of the modulus p of the
functiontheU=p/d
in
complex region of the W-0plane at the
value of ~ = 0 . 8 7
BEHAVIOR
OF THE SEPARATION
CONSTANTS
WITHIN THE COMPLEX REGION
In the complex region, the separation constants can
be convenientlyrepresentedbytheirmodulus
p and
argument 9, viz.,
C 1 = p&
and
U z = p/-+
-
Since tables of Bessel functions of complex arguments
[30] tabulate the arguments in polar form, these tables
can be readily
used withthe previous representation.The
modulus p and the argument 4 can be readily found
from (6). Equations of contourcurvesforconstant
modulusandconstantargumentcanthenbeagain
/3c
found for stationary species as solutions of biquadratic
UP
equations. The calculation is straightforward and need
Fig.
9.
Contour
curves
for
constant
values of the argument 9 of the
not be given here.
A set of typical contour curves is
function L:=p/&in the complex region of the 0-8 plane at the
shownin Figs. 8 and 9. The contour curves for convalue of X =0.8.
stant argument exhibit essentially the same behavior
for all values of M ; there is, however, some variance
in appearance of the contour curves for constant values
of the modulus p when the value of AT is greater or
smaller than unity. This variation
is not very significant.
In allcases, the contour curves for the modulus surround the points in which the argument is not defined
(where the value of U is zero), and the contour curves
In these expressions, the subscriptse and i denote quanforconstantargument
4 emanatefromandconnect
tities
belonging to electrons and ions, respectively.
these points as well as the origin.
For ions with only one positive electronic charge, the
following relation holds
THEEFFECT
OF I O N M O T I O K
T h e effect of ion motion on the behaviorof separation
constants is revealed when the ion conductivity tensor
is added to the equivalent dielectric tensor of (3). The
components of the resulting equivalent dielectric tensor
~ i for
j
this two species medium are now a s follows:
1965
BEVC: SEPARATIOX CONSTAhTS FOR GYRONAGhrETIC P ” A S
925
RELATIONSHIP
OF SEPARATION
CONSTAKTS
TO THE PROPAGATION
CONSTANTS
OF HOMOGENEOUS
PLANE
WAVES
Fig. 10. Zone boundaries in the ws plane for a case including ion
motion. Logarithmic scale.
T h e forementionedcalculations
proceedfrom
the
premise that transversevariation of fieldsexists for
waves propagating in the direction
of the collimating
magnetic field. Not all values of separation constants
can satisfy actual boundary conditions when they are
imposed. In absence of boundary conditions, the separationconstants
discussedhere
pertaintocylindrical
waves. A circularly cylindrical wave, as is well known,
can be synthetized from homogenous plane waves
whose
normals form a circular cone of aperture angle a with
the direction of the axis [ 3 3 ] . In the case under consideration, we have
1
+x
J,(Kr sin CY)
=e x p j [ k ( a ) rsin CY cos 4 124]d4 (21)
2T
quencies, the representation of Fig. 10 makes use of the
logarithmic scale for both w and 0. As is apparent from
the diagram, there are now two complex regions: one
where k ( a ) is the propagation constant of a homogeneous plane wave with a normal forming the angle a with
the direction of the magnetic field, and which is given
by the Appleton-Hartree formula [31]
COMPLEX
REGION
&-e
U P
similar to that of cases where ion motion is neglected
but no longer extending all the way to zero frequency,
and another, much narrower
region in the vicinity of
thefrequencygivenby
u j w p e = p M . Moreover,both
curves of (13) now have two branches, the singularities
now occur a t
s_,
+
Moreover, i t is evident that
When the expressionsof the previous equations are substituted in (22), one obtains ( 6 ) , the biquadratic equa*pe
%e
tion for U.In this way, there follows also a direct exrespectively. There are nom four points in which the planation of the form of the fields given by (1) and (2).
separation constants vanish, three of which have their
Clemow andMulaly [32] havegiven a systematic
ordinates determined as positive roots of the follo~ving: classification of all possible types of wave surfaces given
by (22).
Correspondence
between
thediagrams
of
Clemow and Nulaly andof this paper is given by (22)(24).
ACKNOWLEDGMENT
The research leave and theuse of computer facility at
the United States Xaval Postgraduate School is grateT h e method j u s t outlined’ for the electronic plasma can
fully acknowledged.
be readily applied to this twospecies plasma, and a systematization can be achieved in the same way. We obREFERENCES
serve thatthe effects of ionsbecome manifestonly
[l] H. Unz, “Propagation in arbitrarily magnetized ferrites between
two conducting parallel planes,” Ohio State Research Foundawhen the operating frequency approaches the value of
tion, Columbus, Rept. 1021-1022, AF 33 (616)-6782, November
the ion cyclotron frequencyw ~ i = p : M Wand
~ ~ that, due to
1962. Quotes over 300 articles on related topics.
the small values of the electron to ion mass ratio found [ 2 ] A. W. Trivelpiece and R. W. Gould, “Space charge waves in
cylindrical plasma columns,” J . AppZ. Plzys., vol. 30, pp. 1784in nature, the diagrams of Figs. 2-9 for the electronic
1793, Kovember 1959.
[3]S. S. Jha and G. S. Kino, = A variational technique for deterplasmain
which ionmotion
is neglected aresubmining the propagating modes of a finite plasma,” J. Electronics
stantially correct.
a d Control (GB),vol. 14, pp. 167-185, February 1963.
FOL.
IEEE TRANSACTIONS OIi ANTENN.ZS A N D PROPAGATION
AP-15, NO. 6
NOVEMBER,
1965
National D’Etudes des Telecommunications, Issy-les-Moulineaux, France, Etude 674 P. D. T., October 1962.
-, “Propagation des ondes dans un guide rempli de plasma
en presence d’un champ magnetique,” ElectromagneticTheory
a?td Antennas, Proceedings of a Symposium. liew York: Macmillan, 1963, pt. I , pp. 323-347.
B. A. Auld and J. C. Eidson, “Coupling of electromagnetic and
quasi-static modes in plasma loaded waveguides,” I. Appl.
Phys., vol. 34, pp. 478481, March 1963.
P. 0. Berrett and C. C. Johnson, “1Taves in a circular plasma
column with general impedance boundary conditions,” IEEE
Trans. on Xuclear Science, vol. SS-11, pp. 3b-40, January 1964.
R. Likuski, “Free and driven modes in anisotropic plasma guides
and resonators,” University of Illinois Elec. Engrg. Research
Lab., Urbana,Rept. ALTDR-64-157, AF 33(657)-10224, July
1964. Likusky uses the notation Z = u / a .
1’. L. Granatstein andS. P. Schlesinger, “Electromagnetic waves
in parameter space for finite magnetoplasmas,” J . Appl. Pkys.,
vol. 35, pp. 28G-2855, October 1964.
See for instance: Allis et al., Wanes in Anisotropic Plasnta.
Cambridge, Mass.: M.I.T. Press, 1963.
R. N. Carlile, “-4 backward-wave surface mode in a plasma
waveguide,” J . Appl. Phys., vol. 35, p ~ 1384-1391,
May 1964.
.
J. H. Malmberg and C.B. 1%’harton, Collisionless damping of
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B.ifTieder, “Microwave propagation in an overdense bounded
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Stratton, Electrmmgnetic Theory, New York: IVIcGraw-Hill,
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H. G. Booker, “Some general properties of the formulae of the
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L. J. Briggs, Director.
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wave normal,” in The Pl~ysicsof tlze Ioltosphere-Report
of the
1954 Cambridge Conference, pp. 340-350.
J. A. Stratton [28] p. 361.
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H. Suhl and L. R. 1t:alker, “Topics in guided wave propagation
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pp. 579-659, May 1954. Notation used in this
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x=(u/a)’/(~~-d).
J. E. Hopson and C. C. Wang, “Electromagnetic wave propagatlon In gyromagnetic plasmas,” Sperry Gyroscope Co., Great
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Propagation of Electromagnetic Waves
Through Inhomogeneous Slabs
Abslracf-ih approximation technique is considered for computing transmission and reflection coefficients for plane wavespropagating through s t r a s e d slabs. The approximation uses a mapping of the
given problem onto a similar, but exactly solvable problem. The
approximate solution is then expressed in terms of the solution of the
exactly solvable problem. As an illustration of the technique, the
reflection and transmission coefficients of a transition layer arecomputed. Agreement between the exact and approximate values of the
coefficients is good.
INTRODUCTIOK
S A RESULT O F the intensification of space re-
A
search, the problem of maintaining radio communications with space vehicles during their reentry into the earth’s atmosphere has
received much
attention. A refractive environment, such as a plasma
sheath, may be created over the radiating structure of
the vehicle by its passage through the atmosphere. The
formation of such a sheathaboutthe
vehicle may
severely alter and,in some cases,disrupt radio transmisManuscript received February 15, 1965; revised June 4, 1965.
The research reported i n this paper was sponsored by Air Force sion to and from thevehicle. I t is of interest, therefore,
Cambridge Research Laboratories under Contract A F lO(604)-8386. to gain insightintotheradiowavetransmissionand
The author is with the Antenna Department, Hughes Aircraft
reflection properties of such sheaths.
Company, Culver City, Calif.
986