C0.'.llMUKIC.\T!OK5
evident
that
the
curves
for
the
highly
conducting
subst.ratum
are
virtually
identical.
Thus,
if
either
the
surface
impedance
E,/H,
or
the
wa;e-tilt
ratio
H.!
H, is
mea.~ured,
the
re.-rnlt.~
can
be
intBrpreted
in
terms
of
the
plane-wave
correction
factor
Q.
However,
in
the
C&'e
of
the
high!~·
insulating
substratum,
for
this
value
of S,
there
is a
marked
departure
of
the
curves
between
fl.
and
o,
and
from
the
plane-wave
case
denoted
by
Q.
Actually,
if S is
increased
to
100,
the
curves
all
become
complet€ly
indistinguishable
fnr
both
K = 100
and
O.OL
F1:-IAJ,
HDIARK
These
result:',
which
are
a
sample
of
numerous
calculations,
illustrate
our
point
that
the
plane-wave
theory
mw<t
be
used
with
caution
for
intBrpreting
electromagnetic
data
for a st.ratified
earth
\Yith
highly
resistive
sub-4rat.a.
flEFTrnENCES
[1]
E. D. Sunde,
Earth
Conduction
Effects
in
Transmission
Systems.
Xew York: Dover. 1968.
{2]
.l.
R. \Yait.
Electromaonetic
Waves
in
Stratified
"J,frdia, 2nd
ed.
Ox-
ford. England: Pf'rgamon. 1970. ·
[3]
--,
"The
electric fields of a long current-carrying wire
on
a
strati-
fied eart.h."
J.
Geophys.
Res .. vol. 57. pp. 481-485. Dec. 1952.
[4]
J. F. Hermance and
W.
R.
Peltier. ":1.fagnetotelluric fields of a line
currPnt."
J.
Genphys.
Res.,
vol.
7.5.
pp. 33.51-3356,
.Tune
1970.
[5]
D. Shanks. "Kon-linear transformations of divergent
and
slowly
convergent sequences."
J.
,\falh.
I'hys
.. vol. 34. pp. 1-42. 1955.
[6]
A.
H. Stroud and D. Secrest.
Gaussian
Quadrature
Formulas.
Engle-
wood Cliffs. X .
.T.:
Prenticp.-Hall. 1966.
[ii
G.
Y.
I\:eller
and
F. C. Frischknecht.
Electrical
,Uethnds
in
Gen-
physfral
Prospt>ctinq.
Oxford.
England:
Pergamon,
1g66.
p_
197.
Parametric Interactions Between Alfven Waves
and
Sonic Waves
CHARLES
ELACHI
Abstract-Parametric
interactions
between
a
sonic
pump
wave
and
a
weak
Alfven
wave
are
studied.
It
is
shown
that
if
the
Alfven-
wave
velocity
FA
is
small
relative
to
the
sonic-wave
velocity
F,
there
is a
time-growing
instability
leading
to
the
increase
of
the
Alfven
wave
at
the
expense
of
the
sonic
wave.
This
phenomenon
can
be
of
importance
in
solar
and
stellar
physics.
For
"VA
large
relative
to V,,
the
interaction
is
of
the
stop-band
type.
l.
J:\TIWDL'CTIO:\
In
this
communication
we
apply
a well-developed
method
11Red
in
>'tudying elect.romagnetic
wave
propagat.ion
and
source
radiation
in
><pace-time
periodic
media
[I
J-[5]
to
inve~tigate
t.he
parametric
interaction,;
between
an
Alfven
wave
and
a sonic waYe. The$e
type.,; of
interac1imh
can
occ11r
in
the
Jaborator~·
or
in
stellar
media
such
as
the
sim's
atmosphere.
This
nonlinear
problem
can
be
linear-
ized if
·we
s11ppose
the
Alfren
wave
is
weak
relative
t-0
the
sonic
wave
which
modulates
the
pla.sma
density
in
a
wave-like
manner.
\Ve ,:how
that
a nonconYec1ive (tirne-growing·1
instability
can
occur,
leading
to
the
grmnh
of
the
Alfven
wave
at
the
expense of
the
sonic \\"ave.
The
method
u,;ed
can
he
applied
t.o
a
wide
spectrum
of
a.~trophyKical,
planetar~·,
and
laboratory
problems
involving
inter-
:l.Tanuseript received
:1.Iay
i.
1973: revised :I.fay 29. 1973. This work
was supportf'd by XASA under
Contract
).!ASi-100
and
by
the
l:.S.
Air Force Office of
SC'ience
Research
under
Grant
AFOSR-68-1400.
The
author
is
with
the
Jet
Propulsion Laboratory, California
Institute
of Technology, Pasadena, Calif. 91103.
______________.
Ale
VEN
\\'AVE
907
K,
i!,
SONIC
PUMP
Fig.
1.
Geometry of interaction. Dashed lines represent crests of sonic
wave.
Both
sonic
and
A1fven
waves
propagate
parallel
to
n1agnetic
field.
actions
between
electromagnetic,
acoustic,
hydromagnetic,
aud
space-charge
wave:'
in
fixed
or
moving
pla.sma.s [G],
[I].
The
geometry
considered
is
shown
in
Fig.
1.
The
sonic
pump
wave
(dashed
line.~
repre.<;ent
the
wave
cres1>'
·,
rnodnla1e;;
the
pla.-;ma
density
po
in
a
wave-like
manner
po(z,n
=
po[l
+
,if(J\.z
-
rn1]
where
f(~I
is a
normalized
periodic
function
m=+oo
f(O
=
L
a,.1ei·11~
m=-OC
K
and
Sl/27r
are
the
wavenumber
and
frequency
of
the
sonic
pump,
V, =
<;J/
K is
the
sonic
velocity,
and
'7
< 1
C•>1Tesp•)nd"
to
the
strength
of
the
pump
and
is
supposed
tn
he
small
enough
so
we
can
neglect
terms
of
the
second
order
(or
hig;her i
in
7J.
The
apprnpriate
equations
governing
the
Alfven
wave
behaYior
arc
[8]
[
a
a]
B,
p
0
(z,tl
- +
Uiz,t1
- v
+-
X
\7
X b
at
az
-tr.
0
il
I
ab
- -
\7
X
(v
X
Bo)
at
·
()
!2)
U = ril'
..
f1Kz
-
Wie,
where
U is
the
particle,!
velocity
a
..
,-<ociated
with
t.he
sonic
wave;
b
and
v
are
the
magnetic
field
and
partides'
velocity
associated
with
the
Alfven
wave,
and
we
took
in1o c·onsideratiun
that
1his
wave
is
transverse
and
does
not
generate
change
in 1he
pla$ma
density.
Bo = R
0
e,
is a stati<· magnetic· field. Bec-anse
of
the
space-time
periodicity,
the
wave
soluiion
Cllihists of
a11
infinite
number
of
space
and
time
harmonic-<
iFlnquet
form
i
n=+o:o
v el
2:
'l'n
exp
(
iK.,~Z
-
i(J),J
l
n=+ro
b e,
~
bn
exp
(iKnZ
-
iw"t
I
Kn
= K +
nK,
Wn
= W
+nil
where
e, is
the
transverse
unit
vector,
v,.
and
/,"
c•Jfrespond
to
the
generated
harmonics,
w/2.,,.
is
the
principal
Alfven
freqnency,
and
908
IEEE
TRANSACTIONS
ON
ANTENNAS
AND
PROPAGATIOJI.",
NOVEMBER
1973
Fig.
2.
(b)
Brillouin
diagram_:
brok<>n !inf>--?/ =
O:
solid
line-71
¢ 0.
•:ai
v.! > v,.
(bi
\.'_4
< ''··
K is a
wavenumber
which
will
be
determined
from
the
disper~ion
equation.
Heplacing
the
quantitie."
in
(1)
and
(2)
by
their
Floquet
form,
equating
the
terms
with
the
same
frequency,
and
solving
the
resulting
equation,.:, we
obtain
where
bn
= - Kn
BoVn
Wn
i=+co
DnYn
+
1)
L a,._;V; = 0
i=-00
Kn Kn
TT
,1
2
-
Wn
Vs
Dn
= 1
___
_..:;.·
___
_
Wn Wn
-
Knl
..
s.
1'_
4
= B
0
/ (
4rrp
0
)
1
i2
=
Alfven
velocity.
(3)
The
solution
of
the
system
of
equations
13) gives
the
relative
amplit.ude.~
rn/Yo,
and
the
nontriviality
condition
(system's
deter-
minant
=
0)
gives
the
dispersion
relation
which
determines
the
Brillouin
diagram.
The
convergence
of
the
Floquet
sum
(3 l
can
be
determined
by
appl}ing
the
Poincare
convergence
theorem
[l].
For
a
sinusoidal
modulation
the
sum
would
converge
if I
lim"~"'
Dn
I > 1.
This
condition
is
always
sati~fied,
and
therefore
the
sum
converges.
Exact
:-;olutions
would
require
extensive
numerical
calculation,
but
the
limit
11
« 1
can
be
studied
analytically
and
gives a
good
idea
of
the
general
solution.
For
'7
= 0,
(3)
reduces
t-0
D,. = 0=:}wn = ±YAKn· (4)
The
corresponding
Brillouin
diagram
is a
serie.~
of lines of slope
±VA
centered
at
On
=
-nK,
-n..<1 (see
dashed
lines
in
Fig.
2
·,,
but
only
the
harmonic
n = 0
has
any
physical
meaning.
For
'7
p
O,
strong
interactions
occur
near
the
intersection
points
because
of
the
phase-matched
coupling
between
the
harmonic.'3.
In
the
limit
17
« 1,
Taylor
series
development
could
be
applied
t-0
analytically
deter-
mine
the
behavior
near
the
interaction
regions.
Let
u"
consider
the
two
harmonics
n = 0
and
n =
-1
(the
fol-
l<nYing
is
valid
for
any
interaction
region
between
two
successive
harmonics".
For
'7
= 0,
the
corresponding
Brillouin
diagrams
inter-
sect
at
Ko
= K./2 + Q/2VA
and
w
0
= rl/2 +
KV.d2.
For
'7
p
0,
but
small,
the
solution
near
the
interaction
region
can
be
written
as
K =
K'
(I
+
17r
i
and
w = w
0
I1 +
'78)
and
all
harmonics
other
than
n = 0
and
-1
can
be
neglected.
Therefore,
(3)
reduce"
to
(for
f
(~)
= cos
~.
i.e.,
0±1
=
~
and
a'"
= 0,
otherwise)
'7'
J)oD_, =
J.
(5)
Heplacing
Kand
w
by
their
Taylor
series
expansion,
(5)
reduces
to
v"
- v [v" - v
]'
r2-s2=
A~
A_
V,
+ l'A
4V_4
(6)
The
resulting
conclusions
are
as follows.
1)
For
Y.,
<FA
(Fig.
2(a)),
(6)
corresponds
t-0
a st-0p-band
interaction
where
w is complex
(sis
real
and
r is
imaginary),
and
the
generated
harmonics
can
be
relatively
large
near
the
first
interaction
region.
2)
For
l',
>
Y_,
(Fig. 2
(bl),
(6)
corresponds
to
an
unstable
time-
growing
interaction
"·here
w is
complex
(sis
imaginary
and
r is
real).
This
type
of
instability
means
that
if
the
magnetopla.-;ma
supports
a
strong
sonic
wave,
a
small
Alfven
disturbance
would
grow
by
extracting
power
from
the
sonic
wave
until
some
nonlinear
processes
stop
ib
growth.
The
rnte
of
growth
for a
first-order
interaction
is
determined
by
r
lm
"']
V, -
i·.4
[v,
-l·.!Jv'
max
---;:;;- =
'7
41·
l'
' V
L
.1
-~I
A
and
the
unstable
Alfven frequencies
correspond
t-0
the
intersection
points
between
the
different
space
harmonics.
The
intersection
bet'IYeen
the
nth
and
mth
harmonic
give."'
1'_4Kn
l
()
[
,
-+w=~
(n
-m)~-ln+m)
r
j'
r,
Wm=
-l
A.Km
where
n,m =
0,±1,·
·
·.
III.
Co;:-.;n,rs1ox
The
well-developed
techniques
for t.he
study
of
electromagnetic
waves
in
space-time
periodic
media
were
applied
for
the
study
of
the
interaction
between
sonic
and
Alfven
wave.~.
The
results
of t.his
communication
can
be
applied
to
many
physical
phenomena,
es-
pecially
for
the
generation
of Alfven
waves
in
the
atmosphere
of
the
Slln.
Strong
:'lHnHI
\\·ave.'l
can
exist in t.he
sun's
chromosphere,
and
the
sound
vehieity,
which
i~
usually
of
the
order
of
10
km/s,
can
exceed
the
Alfven
velocity,
'd1irh
changes
from
=0.2
km/s
in
the
phot.o-
sphere
to
1000
km/s
in
the
chromosphere
[0].
Therefore,
it
is
pos.-ible
that
in
some
regions of
the
sun
"s
atmosphere
i"where
l',
>
1'.
1
'
time-gnm·ing
Alfven
\\·avE'o'
are
generated
at
the
expense
of -:onic \Yitve".
H1-:F1·:1n:xn:s
[l]
E.
:';,
Cassedy.
and
A.
A. Olin€'r,
"'Dispersion
relations
in
time-space
periodic
media-Part
I:
Stable
interactions."'
Proc.
IEEE,
vol.
51.
pt.
I.
pp.
1342-1359,
1963.
[2)
--.
"Dispersion
relations
in
time-space
periodic
media-Part
II:
rnstable
interactions."
Proc.
IEEE,
vol.
55,
pp.
1154-1167,
1967.
·'
COMMUNICATIONS
[3]
J.
Aslrne,
"Electromagnetic
wave
propagation
through
a
bounded
time-space
periodic
cold
plasma,"
Int.
J.
Electron ..
vol.
30,
p.
201.
1971.
[4]
C.
Elachi,
"Electromagnetic
wave
propagation
and
source
radiation
in
space-time
periodic
media,"
Caltech
Antenna
Lab.,
Pasadena,
Calif.,
Rep.
61,
1971.
[51
--.
"Dipole
antenna
in
space-time
periodic
media,"
IEEE
Trans.
Anlenna.s
Propa(lai
.. vol.
AP-20.
pp.
280-287.
May
1972.
[61
--,
"Mult!stream
interactions
in
space-time
periodic
media."
Caltech
Antenna
Lab.,
Pasadena,
Calif
.•
Rep.
59.
1971.
[7]
--.
"Cerenkov
and
transition
radiation
in
space-time
periodic
media,"
J.
Appl.
Phys
..
Sept.
1972.
1
81
J.
D.
Jackson.
Cla.ssical Electrodynamics.
~ew
York:
Wiley,
1967.
9]
H.
Zirin,
The
Solar Atmosphere. '"Waltham,
>iass.:
Blaisdell,
1966.
An Application of One-Dimensional Inverse-Scattering
Theory for Inhomogeneous Regions
A.
K.
JORDAN
AND
II. N.
KRITIKOS
Abstract-One-dimensional
inverse-scattering
theory
is
applied
to
the
study of
the
reflection of electromagnetic waves
from
an
inhomo-
geneous
region
having
a refractive
index
n(x)
=
[1-
(1/k2)q(x)
Jll2
where
k =
211"
/X,
and
Xis
the
free-space wavelength.
The
exact
refrac-
tive
index
profile
is
obtained
that
will produce a reflection coefficient
in
which
the
frequency dependence is described
by
the
Butterworth
approximation.
I.
lNTRODUC'l'IO~
Inver,;e
scattering
theory is concerned with
the
reconstruction
of
the
physical characteristics
of
an unknown
target
from
the
knowledge of
the
scattering
data,
i.e., the incident
and
scattered
electromagnetic waves. Analogously,
the
:;ynthesis problem is con-
cerned
with
the design of a physical system
in
which
the
scattering
characteristics are prescribed to have a given functional form.
As
with
direct. ocatt.ering theory,
there
are several approaches
to
inve!'lle
scattering
theory, e.g., physical optics
or
Fourier
trans-
form techniques
[I],
geumetrical optics techniqlles
[2],
extended
boundary
condition
methods
[3],
aud
"exact"
or
spect.ral
methods
[4].
A ;;ynthe>'is procedure when
the
reflection coefficient
i><.
pre-
scribed to
be
a rational function of frequency
has
been developed
by
Kay
[5]
from
the
ha.~ic
mathematical theory of Gelfand
and
Levitan
[4],
[6].
Such a
synthe:<i,,;
technique
i:<
also of intere:;;t in
the
dC8ign
of nonuniform tran,;mission lines
The
purpose of t.he
pre.~ent
communicat.ion
i:<
to
give
an
applica-
tion of l'pectral one-dimensional inverse ,;catt.ering theory ;vhen
the
reflection coefficient is specified
to
be a Butterwort.h function
of
frequency
[9].
The
physical model comiidered is usefnl in
the
study
of
the
scattering of millimeter waves
by
semiconductor surfaces
[10].
IL
DEFINITION
OF
THE
PROBLEM
The
physical model, which ii; considered here, is
the
scattering
of
electromagnetic waves from an inhomogeneous dispersive medium.
An example of such a region
i,;
an
intrinsic semiconductor whose
charge carrier density
;\'
(x)
is
a function of
the
space dimension x
[11].
The
reflection coefficient b (k) is
a..,"Snmed
to
be
known for all
(normalized)
freqnencie.~,
0
:':,'.
k < <X>, and
t-0
be a pre.scribed func-
tion of frequency. Thn,; for
the
pnrpose;; of
this
brief conummiea-
tion,
the
experiment.al and data-reduction problems of obtaining
the
functional form for
ll(k
'1
a.re
excluded.
:\Januscript.
reet'ived
Sf'ptmnhcr
13,
1071:
revised
April
O,
1073.
A.
K.
Jordan
was
with
th<'
).foorc
Sc-hool
of
Electrical
Engineering,
l'niversity
of
Penns\
!vania,
Phi!adf'lphia.
Pa.
19174,
He
is
now
with
the
8p~ce
8ystems
Dlvisinn.
:-lava.I
Researc-h
Laboratory.
Washington.
D.
C.
~0390.
H.
X.
Kritikos
is
with
Ill<'
:\foore
Hchool
of
Electrical
Engineering.
Univ<'rsity
of
PennsJ·lvania,
Philadelphia.
Pa.
19104.
Fig.
1.
u
lk,xl
=
,ikx
0
µ
0
,e:(k,x
l
-ulk,x]
x=O
Geometry
for
scattering
by
inhomogeneous
semiconductor
region.
A plane
wave
of
unit
amplit,ude
ua
e'"",
is normally incident
upon
an
inhomogeneous and dispersive region int.he
right
half-plane,
0
:':,'.
x <
ro,
as shown in Fig.
1.
A reflected wave b (k
)e-ikz,
is pro-
duced, where
b(k)
is
the
complex reflection coefficient.
The
factor
(,-;,,,,•
is suppressed
throughout
this
dixc1L~sion.
In
the
right
half-plane
the
wave function, u
(k,x)
satis:fies foe differential
equation
(fl
u(kx)
+
k2r1l(kx)1t(kx)
= 0
dx2
' ' · ' ·
where
n(k,x)
is
the
index of refraction of
the
region,
and
n
2
(k,x)
1
1 -
-q(x)
k2
(1)
(2)
where
q(x)
is
the
scattering potential. Di,;persion relations of
this
type
in practice
can
be
observed in intrinsic semiconductors where
g(x)
where e
0
is
the
charge, m
0
i;.;
the effective mass of
the
charge carrier,
Eo
is
the
permittivity
of free space,
and
N
(x)
is
t.he
charge
canier
density profile.
The
di8persion relation
!2)
can be obtained
by
con-
sidering
the
sca.t.tering of electromagnetic waves from a solid-state
pl!llima
[12].
The
problem, which is
con~idered
here,
is
to
determine
N
(x)
given
that
11
(k
),
or I b
(k)
1
2
,
i;;
a rational function of
frequency
that
exhibit,~
bandpass characterh;tics.
III.
INVERSE
SCATTElllNG
THEORY
Gelfand
and
Levitan
have
examined
[6]
the one-dimensior1al
wave
equation
tJ?.
-u(kx)
+k2u
1
kx)
dx2
' . ' '
q(x
)u
(k,x
),
x
2.
0
(3)
which is obtained from
(1)
with
the
index of refraction of (2).
The
solution of (3) can be represented
a<
[5],
[6]
u
(k,x)
u
1
(k,x)
+
J'
K(x,~)u
1
(k,~)
d~
-y
(
4)
where u
1
(k,x)
u
0
(k,x)
+
b(k)e-i"",
x
:':,'.
0.
By
considering
the
spectral properties of the
boundary
valne
problem, Gelfand
and
Levitan have
~hown
that
the
kernel function
K
(x,y)
can be uniquely determined from
the
solut.ion of
t.he
integral
equation
B(x,y)
+
K(x,y)
+ r
K(x
1
(1B(~
+
y)
d~
= 0
(5)
-y
where B (x) is
the
Cauchy principal value of
t.he
inverue Fourier
transformation of
the
t•eflection coefficient.
The
kernel function ha-;
the
properties
J((x,-x)
0
(6)