1666
(23)
(24)
(25)
(26)
(27)
(28)
[29)
[30)
(31)
M.
J.
E.
Golay,
"Monochromaticity
and
noise
In
a
regenerative
electric
oacillator,"
Proc.
IRE,
vol.
48,
pp.
1473-1477,
Aug.
1960.
K.
H.
Sann,
"Phase
stability
of
oscillators,"
Proc.
IRE,
vol.
49,
pp.527-528,Feb.
1961.
L. P. Malling,
"Phase-stable
oscillators
for
space
communications,
including
the
relationship
between
the
phase
noise,
the
spectrum,
the
short-term
stability
and
the
Q
of
oscillators,"
Proc.
IRE,
vol.
50,
pp.
1656-1664,
July
1962.
L.
S.
Cutler,
"A
frequency
standard
of
exceptional
spectral
pur-
ity
and
long-term
stability,"
Hewlett-Packard
Co.,
July
1961.
W.
R.
Atkinson,
L.
Fey,
and
J.
Newman,
"Spectrum
analysis
of
extremely
low
frequency
variations
of
quartz
oscillators,"
Proc.
IEEE,
vol.
51,
p.
379,
Feb.
1963.
H.
P.
Stratemeyr,
"The
stability
of
standard-frequency
oscilla-
tors,"
The General Radio Experimenter, vol.
38,
pp.
1-16,
June
1964.
B. B.
Mandelbrot,
"Sporadic
random
functions
and
conditional
spectral
analysis:
self-similar
examples
and
limits,"
in
Proc.
5th
Berkeley
Symp.
on
Mathematical Stati8tics and Probabllity,
Berkeley,
and
Los Angeles, Univ. California Press,
1967,
pp.
155-178.
-,
"Some
noises_
with
1
ff
spectrum,
a bridge
between
direct
current
and
white
noise,"
IEEE
Trans. Inform. Theory, vol. IT-
13,
pp.
289-298,
Apr.
1967.
R.
Vessot,
L. Mueller,
and
J.
Vanier,
"The
specification
of
oscil-
lator
characteristics
from
measurements
made
In
the
frequency
PROCEEDINGS
OF
THE
IEEE,
VOL.
64,
NO. 12,
DECEMBER
1976
(32)
(33)
[34)
[35)
(36)
(37)
(38)
(39]
(40)
domain,"
Proc. IEEE, vol.
54,
pp.
199-207,
Feb.
1966.
D.
Halford,
"A
general
mechanical
model
for
lfla
spectral
den-
sity
random
noise
with
spectral
reference
to
flicker
noise
1/lfl,"
Proc. IEEE, vol.
56,
pp.
251-258,
Mar.
1968.
A. L.
McWhorter,
"a/f
noise
and
related
surface
effects
In
ger-
manium,"
M.I.T. Research
Lab
of
Electronics,
Lexington,
MA,
Tech.
Rep.
295,
May
20,
1955.
W. E.
Leavitt,
"Shipboard
satellite
communication
sets
AN/WSC-
2(XN-l)(V),
phase noise
specification
considerations,"
5430-
169A,
NRL
Prob.
ROl-36,
Aug.
2,
1972.
-,
"Shipboard
satellite
communication
sets
AN/WSC-2(XN-1)
(V),
phase
noise
specification
considerations,"
5430-59A,
NRL
Prob.
ROl-36,
Mar.
10,
1972.
"Precision
frequency
measurements,"
AN-116,
Hewlett-Packard
Co.,
July
1969.
"The
stability
of
standard
frequency
oscillators,"
General Radio
Experimenter, vol.
38,
pp.
1-16,
June
1964.
M.
A.
Caloyannides,
"A
mathematical
model
and
experimental
investigation
of
microcycle
spectral
estimates
of
semiconductor
flicker
noise,"
Ph.D.
dissertation,
California
Inst.
Technol.,
Pasa-
dena,
1972.
C. B. Searles,
R.
Ashley,
and
F.
M.
Palka,
"The
measurement
of
oscillator
noise
at
microwave
frequencies,"
IEEE
Trans.
Micro-
wave Theory Tech., vol. MTT-16,
pp.
753-760,
Sept.
1968.
M.
Lighthill, Introduction
to
Fourier
Analy81.r
and
Generalized
Functions.
London,
England:
Cambridge
Univ. Press,
1962,
pp.
15-57.
Active
and
Passive Periodic Structures:
Waves
•
1n
A Review
CHARLES ELACHI,
MEMBER,
IEEE
Ab.rtract-The theo:ry
and
recent
applications
of
waves in periodic
structures are reviewed.
Both
the
Floquet
and
coupled waves approach
are analyzed
in
aome detail.
The
theoretical
part
of
the
paper includes
wave propagation in
unbounded
and
bounded active
or
pusive
periodic
media, wave scattering
from
periodic boundaries, IOUice radiation
(dipole,
Cemlkov,
transition,
and
Smith-Purcell)
in
periodic media,
and
pulse traDlmiaion through a periodic slab.
The
applications
part
coven
the
recent
development
in
a variety
of
fields: distn'buted feed-
back osciDators,
mten,
mode converters, couplers, aecond-harmonic:
perators,
deftectors, modulators,
and
transducers in
the
fields
of
integrated optics
and
integrated surface acoustics.
We
also
review
the
work
on
insect
compound
eyes, mechanical. structures, ocean waves,
pulse compressions, temperature waves,
and
cholesteric liquid crystals.
Particles interaction
with
crystals is briefly reviewed,
especially
in
the
cue
of
zeolite crystals
and
supedattices. Recent advances in fabrica-
tion
techniques for very fine
gratings
are
also
CO¥ered.
Finally, specula-
tions
about
future problems
and
development in
the
field
of
waves in
periodic structures are given.
Manuscript
received
January
13,
1976;
revised
June
21,
1976.
The
submission
of
this
paper
was
encouraged
after
review
of
an
advanced
proposal.
This
paper
presents
the
results
of
one
phase
of
research
carried
out
at
the
Jet
Propulsion
Laboratory,
California
Institute
of
Technology,
under
Contract
NAS7-100,
sponsored
by
the
National
Aeronautics
and
Space
Administration.
This
work
has also
been
partly
supported
by
the
Office
of
Naval Research.
The
author
is
With
the
Jet
Propulsion
Laboratory,
California
Institute
of
Technology,
Pasadena,
CA
91103,
and
UCLA, Los Angeles, CA.
I.
INTRODUCTION
T
HE PROPAGATION
of
waves in periodically stratified
media was discussed as early
as
1887 by Lord Rayleigh
[
l],
who recognized
that
this problem was character-
ized by the
Hill
and Mathieu differential equations. Even
earlier in
the
19th
century a
number
of
scientists have investi-
gated wave propagation in lattices. Cauchy, Baden-Powell,
and Kelvin [ 2] discussed lattices
that
consist
of
identical
particles. Kelvin then proceeded
to
devise a theory
of
disper-
sion for a 2-particle lattice and a mechanical model
of
it
was
built by Vincent [ 3] . At
the
end
of
the Nineteenth century
and early Twentieth century a number
of
scientists (Vaschy,
Pupin, Campbell) used periodic networks
to
develop electric
filters.
In 1928,
Strutt
and Van der Pol
[4],
[5],
analyzed the
problem
of
an atomic grating subject
to
a periodic sinusoidal
potential, and
the
behavior
of
particles in force fields
that
are
characterized by sinusoidal and rectangular periodic variations.
In
the
same year, Bloch [ 6] generalized the results
of
Floquet
[7]
to
the use
of
partial differential equations with periodic
coefficients. The solutions, called the Bloch waves, formed
the basis
of
the theory
of
electrons in crystals, i.e.,
of
the
theory
of
solids and energy bands [ 8]
-[
12] .
ELACHI: WAVES IN PERIODIC STRUCTURES
1667
lo)
(b)
le\
Id)
~
~
~
%
~
~
~W///////&,
)t
WAVE
~
~
~
~
~
~
~
~
~
"""'-
~
~·~
~
~
~
-
~
~
~
~
/
~
~
/~
CHARGE
~
%
~
%
~
%
%
%
DIPOLE
(e)
(f)
(g)
lh\
~
""-....
/
-""
/
'\
~
~
-
-
--~
~
~
~
~
~
............_
............_
I
~
~
~
~
~
~
%
~
%
(I)
(j)
lk)
(I)
X(z) /
.............,~~/~~
•
(m)
(n)
(a)
WATER
WAVES
BOTTOM
(p)
(q)
(')
~I
I I I I
I/
L.!_11111/
!''''''/
~,,,,,
;.- - ---
-7
-
..
..
---
Fig.
1.
Periodic
structure
configurations
reviewed
in
this
paper.
(a)
Waves
and
particles
in
an
unbounded
periodic
medium.
(b),
(c) Wave
scattering
from
a
periodic
half-cpace.
(d)
Wave
scattering
from
a
periodic
boundary.
(e) Wave
scattering
by
a
thick
grating.
(f), (I),
(h)
Waves
in
periodic
guides
and
particles
moving
near
a
periodic
structure.
(I)
Waveguiding
and
radiation
on
a
surface
with
periodic
impedance.
(j)
Guide
with
periodic
loading.
(k)
Corrugated
fiber.
(l)
Two-
dimensional
periodic
mechanical
mesh.
(m)
Flexural
waves
in
periodically
supported
burns.
(n)
Acoustic
waves
and
flow
in a
periodic
duct.
(o)
Water waves
on
a
periodic
bottom.
(p),
(q),
(r)
Waves
and
particles
in
simple crystals, cholesteric
liquid
crystals,
and
zeolite
crystals, respectively.
Interest was also very strong in the field
of
optical multi-
layers [ 13]
-[
15] , which have many applications: filters, anti-
reflection films, beam splitters, and polarizers. The theory
of
stratified optical thin films was elegantly and considerably
investigated by Abeles [
16].
A detailed and comprehensive
review
of
the work
on
waves in periodic structures as
of
the
late 40's can be found in Brillouin's
book
Wave
Propagation
in
Periodic Structures [
17].
In the l 950's the interest in periodic structures came mainly
from the fields
of
slow wave structures and antennas. The
study
of
slow wave structures was mainly stimulated by the
development
of
microwave tubes where a periodic structure
is used
to
slow the wave, which could
then
couple
to
the
relatively slow electron beam [
18]-[22].
The structure most
often
used was the helix in the different forms: Sheath helix
(22],
tape helix
[23],
(24],
and multifilar helix
[24],
[25].
Other structures which were also studied
in
detail for slow
wave guiding and filtering were
the
tape ladder line [ 24
]-[
26]
and ridge waveguide
(24],
(27],
[28].
The investigations
of
the properties
of
traveling wave peri-
odically loaded antennas was stimulated by the successful
experimental design
of
the cigar antenna
[29],
[30] which
utilizes a modulated disk
on
rod structures. The theory
of
guiding structures with periodic modulation
of
the surface
reactance was
then
developed [
31]-[
33] . Periodic loading
of
a basically slow-wave structure produces a complex wave
1668
which continuously radiates power, but with the bulk of the
energy being bound. This permits the realization
of
a large
effective aperture
[32]-(33].
An
excellent review, with
references,
of
the work on traveling-wave antenna is given
by
Hessel [
33]
.
In the l 960's, the main emphasis in the field
of
periodic
structures was toward: 1) exact solution
of
the electromag-
netic
wave
equation
in
sinusoidally periodic and laminar
media
[34]-[38];
2)
wave
propagation in time and space-time
periodic media
(39}-[45];
and 3) localized source radiation
(dipoles, linear sources and moving charges)
[46]-[52).
Also
of
importance
was
the use
of
interdigital transducers in the
field
of
integrated acoustics (
53].
In the early l 970's, new technological advances
in
the
development
of
passive and active thin-film optical waveguides
and the fabrication
of
solid state, optical, and acoustical
gratings had generated a new interest in, and
gave
a strong
impetus to, the field
of
waves . in periodic structures. The
interest was mainly stimulated by
th~
fact that new exotic
materials, with a wide range
of
properties (nonlinear, piezo-
electric, anisotropic, pyroelectric magnetoelastic, magneto-
optic, electrooptic, etc.) can now be used in different forms
(bulk, thin films, fibers, etc.) and with very fine periodicities
to
support electro-magnetic, acoustic or electron
waves.
Active materials were used
to
develop distributed feedback
lasers [
54
]-(98].
Periodic nonlinear materials were proposed
and used for parametric interaction
[99)-[
103),
and high-
mobility semiconductors were suggested
to
develop surface
or bulk sources far optical, IR, magnetic, and acoustic waves
[
104J-[107].
Work
on the properties
of
periodic structures
was also active in the fields
of
structural engineering, classical
acoustic, liquid crystals, and insect vision. In Fig. l ,
we
show
a variety
of
periodic structures that have been studied by
numerous authors and that will be reviewed in this paper.
Periodic structures are widely encountered in nature in the
form
of
crystals. They can also
be
simply generated by a
standing wave, i.e., an acoustic wave in a fluid
or
solid, or an
electromagnetic
wave
in a nonlinear or active medium. Finally,
large periodic structures can
be
developed by just simply
repeating a basic unit. These were factors in generating the
interest of scientists
to
study their characteristics. However,
two special properties made these structures
so
unique and
important:
1)
their eigenmodes consist
of
an infinite number
of space-harmonics with phase velocities varying from zero to
infinity; and 2) they can support propagating
waves
only in
well-specified propagation bands.
The uniqueness
of
the first property is that
it
allows the
periodic structure
to
support waves that have a very low phase
velocity and therefore can
be
efficiently coupled
to
relatively
slowly moving charges
or
sources.
It
also allows the coupling
of different types of waves, or similar waves in different
modes, without requiring them
to
have inherently identical
wave
vectors (in the absence
of
the periodicity). In other
words, the periodic structure has an inherent
wave
vector
(K =
21'f/A,
where A
is
the period
of
the structure) that
is
adjustable by the designer and can be used
to
conserve the
momentum (or the
wave
vector) in the coupling between any
two
waves.
The second property is commonly known
as
the distributed
feedback (Bragg reflection), which is a result
of
the cumulative
reflection from each unit cell in the structure.
As
we
shall
see
later, in certain frequency bands the propagation·
wave
vector can only be complex. This implies that a
wave
propa-
gating in the structure with a frequency in the stopband
will
~ncounter
successive reflection, i.e., "distributed feedback,"
PROCEEDINGS
OF
THE IEEE, DECEMBER 1976
and thus cannot extend far away from its source. This
is the reason for the presence
of
forbidden bands in crystals.
All
types
of
waves
exhibit the above properties when they
propagate in a periodic structure. The
wave
could
be
an
acoustic, electromagnetic, magnetoelastic, plasma, electron,
flexural, or water
wave.
The structure could have a periodic
boundary, a periodic support,
or
a periodic bulk parameter
(i.e., index
of
refraction, plasma density, electric potential,
nonlinearity constant, gain, density, etc.) The only require-
ment is that the propagation properties
of
the
wave
are some-
how related to the perturbed parameter.
In this review paper
we
will discuss analytically and physi-
cally the unique properties, review the theoretical and experi-
mental work in the last
15
years, and speculate on some
future developments in the field
of
waves in periodic media.
In Section II,
we
will
use, with no loss
of
generality, the
propagation
of
an electromagnetic wave in an unbound peri-
odically modulated medium
as
an example to derive and
explain the unique properties
of
periodic structures.
The exact Floquet approach and also the approximate but
simple coupled modes approach are analyzed. In Section III,
we
include the effect
of
boundaries.
In
Section IV,
we
analyze
the case
of
periodic boundaries. The cases
of
sources and
transients are discussed in Sections V and
VI.
Active periodic media and their applications in a wide range
of
fields are studied in detail in Section VII. Section VIII
addresses the recent applications
of
passive periodic struc-
tures. The wide field
of
electrons in crystals is briefly re-
viewed in Section IX. The fabrication techniques are pre-
sented in Section X and speculations for future development
are in Section
XI.
This paper does not address the work on
waves
in space-time and temporal periodic structures which
require in itself a special review paper.
Throughout this paper, an exp
(-iwt)
time dependence
is
assumed.
II. WAVES IN AN UNBOUND PERIODIC MEDIUM
The wave equation in a symmetrically periodic medium can
be reduced to a differential equation
of
the form (Appen-
dix
A)
(1)
(
211')
00
f(z)
= f z + - = L
On
cos
(nKz)
K n-o
(2)
where on are related
to
the Fourier coefficients
of
the peri-
odicity function and to the wavenumber k =
211'/)..,
A
is
the
wavelength
of
the propagating
wave
and
A=
21f/K
is the period
of
the medium. o
0
is also related to the transverse
wave
vector
whenever it exists. The solutions
of
(
l)
are the
Hill
functions
of
which Mathieu's functions are a special
case
(when
On=
0
for n
-:/=
0, 1). The
Hill
equation also applies for
f<.z)
odd. The
general solution can be written in a Floquet form
I/;= exp
(iKz)A(z)
(3)
where A (z)
is
a periodic function, and K is termed the charac-
teristic exponent and is a single valued function
of
the an's.
The periodic function
A(z)
may next be expanded in a Fourier
series. The solution can then be written
as
n•+co
I/I=
L
An
exp (i(K + nK)z].
(4)
-
ELACHI: WAVES IN PERIODIC STRUCTURES
The different components
of
iJ;
are called
the
space harmonics
of
the propagating wave, in analogy
to
the familiar time
harmonic expansion for an arbitrary function in time. The
values
of
Kn
= K +
nK
represent physically the propagation
wave numbers
of
these space harmonic contributions
to
the
total field. The space harmonics do
not
exist independently.
They are portions
of
a total solution.
Introducing (4) into the wave equation, one obtains
~[-
(K
+
nK)
2
An+!~
am(An-m
+
An+m)]
· exp [i(K +
nK)z)
=
0.
(5)
Since the above relation must hold for any value
of
z,
then
( S) reduces
to
an infinite set
of
homogeneous equations
-2(K
+nK)
2
An
+Lam(An-m
+An+m)
m
= 0, n = 0, ± 1,
...
(6)
which can be written in a matrix form
llMll · IAl=O.
(7)
The solution will be nontrivial
if
det
I
IMI
I=
o.
(8)
This
is
the dispersion relation which gives the value
of
K
as
a
function
of
the
an
's. The solution
of
(7) would
then
give
the
relative values
of
the
space harmonics, i.e.,
An/A
0
• The value
of
Ao
itself is determined from the boundary
or
source condi-
tions. An analytical solution for (8) was given by Hill, which
relates the free space wave
number
k
to
the characteristic
exponent K
[35].
In the simple case where
an
= 0 for n
=I=
0, 1, (6) reduces
to
where
a
0
-
(K
+
nK)
2
Dn =
2--'"-----
a1
(9)
(10)
Applying an iterative process on (9) one obtains
the
continued
fractions
[34]
~--_lj
_ ___U _
_J_J_,,,
(11)
An-l
- Dn
~
1Dn+2
An
~
1 1
-A--=
- Dn -
rn:=-:
-
n+l
which when combined yield
_LJ_,.,
rn,;:;·
D =
_u
-
___J__J
- ___U _
...
+ -1..J -
__u
n Dn+l
IDn:;.2
~
Dn-1
~
(12)
(13)
This expression is another form
of
the dispersion relation.
The above continued fractions can be shown [ 7]
to
converge
if
IDn
I;;;;.:
2 for n
;;;;.:
N where N is a finite integer. An inspec-
tion
of
Dn shows
that
this condition is usually satisfied.
The relation between
K,
k, and
the
parameters
of
the
pertur-
bation can be illustrated in the form
of
a stability diagram,
which
is
customary in
the
study
of
Mathieu's equation. Fig.
2 shows
the
stability diagram for the case where
an
= 0 for
n
=I=
0, 1. The unshaded areas are the so-called "stable regions"
6
N
""
;,._
0
N
4
2
4
4o
/K2
0
6
1669
10
Fig. 2. Stability diagram for
the
case
of
Mathieu's equation. The
continuous lines correspond
to
the
boundaries separating the regions
of
complex and real solution. The dashed lines correspond
to
a fixed
solution for
K.
Line 1 corresponds
to
the
case
of
f"lxed
K and a
1
(i.e., perturbation magnitude and signal frequency) while a
0
varies;
i.e., transverse wave vector. Line 2 corresponds
to
the
case
of
a wave
incident
on
a periodic
half
space where
the
perturbation and
the
incidence angle are
nxed.
A change in these
two
parameters would
lead
to
a change in
the
slope
of
the
line. Line 3 corresponds
to
the
case
of
a wave in a modulated guide, where
the
transverse wave vec-
tor
is fixed and
the
perturbation is fixed [
34).
-2 - I
4
4a
K2
0
I
I
I
I
I
I I
I I
I I
I
I
I
I I
I I
I
12.
7
I I
I I
/2.4
I
7
I
I
I
I
I
10
Fig. 3. Stability diagram for the case
of
TM
waves in a sinusoidally
periodic dielectric with a relative perturbation
11
= 0.4
of
the
di-
electric constant. Shaded regions correspond
to
the regions where
the solution for K is complex [ 35 I.
wherein the solution for K is pure real. The term "stable"
refers
to
the fact
that
the corresponding solutions are bounded
for any value
of
z. Outside the stable regions, K
is
complex
and its value is
± K =
mK/2
+ ia,
m = 0, 1, 2,
...
(14)
where
(mK/2)
is the absolute value
of
K at the boundaries
of
the appropriate regions and a is strictly real. These regions
are referred
to
as
"unstable regions" because one
of
the values
of
K yields a solution which is
not
bounded at infinity. How-
ever, in actual physical situations, this solution is eliminated
by
the
radiation condition.
To illustrate,
let
us consider the case
of
an electromagnetic
wave in an unbounded sinusoidally periodic medium where
the modulation coefficient is
f'/
and the average dielectric
constant is e
0
e,. This specifies a line across
the
diagram (line
2, Fig. 2) which is the locus
of
the solution K
as
a function
of
the frequency
w/2rr
or
the unperturbed wavenumber k.
We
observe
that
at
low frequencies the solution for K is
real.
As
k increases, a certain value is reached where K is equal
1670
PROCEEDINGS
OF
THE
IEEE, DECEMBER
1976
(a)
2W/A
= L
10
(b)
2W/ A = 1 2
"I
2W/A
= 9
10
81
Fig.
4.
Regions
of
real
(unshaded)
and
complex
(shaded)
solution
for
the
case
of
a rectangular
periodicity
for
different
values
of
2w/A
(9).
to
K/2. After that, real
(")
stays constant while a increases
from 0
to
amax
then
decreases
to
0. This corresponds
to
the
first stopband region.
For
higher frequencies, the solution
crosses the next propagating region until " =
K,
where 1t crosses
into the second stopband, and so on.
We
remark
that
the
width
of
the
stopband and
the
value
of
amax increase with
17.
For
17
small, the stopbands are centered at k
Ve,.
=
mK/2
which correspond
to
the well-known Bragg condition.
The above stability diagram corresponds
to
the
most simple
case. When an
-:I=
0 for all values
of
n,
the
diagram can exhibit
some interesting behavior. Yeh
et
al.
(35) studied the case
of
TM
electromagnetic waves (see Appendix A) where
the
wave
equation reduces
to
(l)
with all
an
-:I=
0. The resulting stability
diagram is shown in Fig. 3.
It
is interesting
to
remark the
crossing
of
region boundaries which lead
to
special points
where
the
stopband vanishes for nonzero perturbation. This
means
that
if
the
medium parameters and the transverse wave
vectors are adequately chosen,
the
wave would have a real
wavevector (i.e., no stopband) even
if
the Bragg condition
is
satisfied. To illustrate, for
the
case shown in Fig. 3,
if
-yK/2
is equal
to
0.3 where
-y
2
= p
2
+ q
2
,
then
the
solution would
cross from the first passband
to
the
second passband with
no
stopband in between.
Another type
of
periodicity which has been studied ex-
tensively is the rectangular periodicity
(9),
[
10),
[
17),
(36),
[ 109)
{
o,
S(z) =
1,
0,
S(z
+A)
= S(z).
for 0
<z
<w
forw
<z<A-
w
for
A-
w<z<A
(15)
Kronig and Penney [ 10] considered the case
of
a delta
function potential. Strutt [ 109] considered
the
case where
the width
of
the
hill
and the well are equal, i.e., w = A/4.
Brillouin [ 17] and Allen
[9]
considered the general case
where w
-:I=
A/4. The above work was mainly related to the
study
of
the
motion
of
an electron in a crystal with a periodic
potential. However,
the
results are directly applicable
to
the
general case
of
waves propagating
in
infinite periodic medium
in
the
direction
of
the periodicity. Lewis and Hessel [ 36]
extended
the
previous work
to
the
case in which electromag-
netic wave propagation also occurs in a direction transverse
to
the
periodicity. In the case
of
electronic waves,
the
dis-
persion relation
(8)
reduces
to
[
w
2
u
2
+
(w
-
A/2)
2
v
2
]
cos
(2u)
cos
(2v)-
A sin
(2u)
sin
(2v)
( - 2w)
wuv
where
u =
(A/2-
wXB
1
- B
2
)
112
v=
wBY2.
=cos
(27r"/K)
(16)
In Fig. 4
we
show a typical stability diagram for
the
case
of
a square potential. A unique feature is
the
presence
of
cross-
ing points similar
to
what happens in the case
of
the
TM
waves in a sinusoidal periodicity (crossing points do
not
exist
in
the
case
of
Kronig-Penney delta potential model nor in
the case
of
TE waves in a sinusoidal potential). Allen
[9]
did
derive
the
location
of
the
crossing points
to
be
B
1
= (nA/4w)
2
B
2
=(nA/4w)
2
-[mA/4(~-w)]2
(17)
where n and m are integers with n
-:I=
0 and B
2
-:I=
0.
Aside from
its general mathematical interest, the distribution
of
these
crossing points is connected
to
the
problem
of
surface states
in
the
study
of
crystals,
as
pointed
out
by
Shockley [
110].
Another area
of
investigation was
the
propagation
of
electro-
magnetic waves in a periodic plasma which was studied by
Casey
et
al.
[38].
In
the
case
of
a TE wave in a sinusoidally
periodic plasma,
the
expression
of
Dn (9) becomes
D =
~
rl
-
k2
-
(K
+
nK)2]
n
11~
k;
(18)
where
kp
=
Wp/c
is
the
plasma wavenumber, and
Wp
is
the
plasma angular frequency, c
is
the speed
of
light in vacuum,
and
17
is
the
modulation coefficient. The corresponding sta-
bility diagram which
is
shown in Fig. 2 is still valid except the
axes now are
y =
211(;)2
x = 4(k
2
-
k~)/K
2
(19)