Electromagnetic propagation in periodic stratified media. I. General theory

TL;DR: In this article, a diagonalization of the unit cell translation operator is used to obtain exact solutions for the Bloch waves, the dispersion relations, and the band structure of the medium.

Abstract: The propagation of electromagnetic radiation in periodically stratified media is considered. Media of finite, semi-infinite, and infinite extent are treated. A diagonalization of the unit cell translation operator is used to obtain exact solutions for the Bloch waves, the dispersion relations, and the band structure of the medium. Some new phenomena with applications to integrated optics and laser technology are presented.

Summary

I. INTRODUCTION

• Periodic optical media and specifically stratified periodic structures play an important role in a number of applications.
• This application benefitted largely from the pioneering analysis of Abeles.
• The two papers that follow describe a general theory of electromagnetic propagation in periodic media.
• In addition to demonstrating the application of the theory to a number of familiar problems, such as reflectivity of multilayer films, the authors consider in general form a variety of some experimental situations which include Bragg waveguides, birefringence and group velocity at arbitrary directions, phase matching in nonlinear optical applications, multichannel waveguides, and optical surface waves.

II. MATRIX METHOD AND TRANSLATION OPERATOR

• For the sake of clarity in introducing the basic concepts the authors will consider first the simplest type of periodically stratified medium.
• The distribution of some typical field components can be written E(x,z)=E(x)e e-'t (3) The electric field distribution within each homogeneous layer can be expressed as a sum of an incident plane wave and a reflected plane wave.
• As a matter of fact, only one vector (or two components of two different vectors) can be chosen arbitrarily.
• The matrix in ( 9) is the unit cell translation matrix which relates the complex amplitudes of the incident plane wave a,-, and the reflected plane wave bn l in one layer of a unit cell to those of the equivalent layer in the next unit cell.

dn-to dn

• These matrices, however, possess the same trace .
• As will be shown later, the trace of the translation matrix is directly related to the band structure of the stratified periodic medium.
• As noted above, only one column vector is independent.
• The Bloch waves which result from (26) can be considered as the eigenvectors of the translation matrix with eigenvalues e'IKA given by (25) .
• The two eigenvalues in (25) are the inverse of each other, since the translation matrix is unimodular.

III. BLOCH WAVES AND BAND STRUCTURES

• The periodically stratified medium is equivalent t one-dimensional lattice which is invariant under the lattice translation.
• These are the so-called "forbidden" bands of the periodic medium.
• This completes the solution of the Bloch waves.
• The eigenvectors corresponding to the eigenvalues (25) are obtained from (24) and are uncoupled.

IV. BRAGG REFLECTOR

• Periodic perturbation in a dielectric medium has been used extensively in fabricating distributed feedback lasers"," 1 (DFB) and distributed Bragg reflection lasers 12 (DBR).
• In the case of square well alternation which corresponds to the layered medium described above, an exact solution is obtained by their matrix method.
• The geometry of the structure is sketched in From (19) the authors have the following relation:.
• The peaks of the reflectivity occur at the centers of the "forbidden" bands.
• For large N the second term in the denominator approaches zero exponentially as &-2(Nf1)KiA.

V. GUIDED WAVES

• Multilayer waveguides are becoming increasingly important in integrated optics.
• Analytic expressions for the mode dispersion relations and field distributions can be obtained by the matrix method.
• The authors are looking for guided waves propagating in the positive z direction.
• The N modes are simply due to the splitting of an N-fold degenerate band as the separations between the N identical slab waveguides are reduced from infinity.

B. Asymmetric type

• The first matrix of the matrix product in (47) accounts for the replacement of nl by na.
• The surface modes only exist when the separation is larger than a, .
• Only the confined modes in the first allowed band which corresponds to the lowest-order modes of the uncoupled individual channel waveguide are shown.
• There are exactly m +nN zero crossings in the transverse field distribution for the mnth mode with n zero crossings in each guiding channel and m zero crossings in the N-1 separation layers.

VI. BRAGG REFLECTION WAVEGUIDES

• Optical dielectric waveguides with a slab configuration are capable of supporting lossless confined modes provided the index of refraction of the inner layer exceeds the indices of the two bounding media.
• The use of stratified media in dielectric waveguiding has been proposed by Ash' 9 who, however, did not consider the case of confined propagation in low index materials.
• 2 0 The new feature in this case is the form of the wave EK(x) eiKX in the stratified periodic medium (iii) where EK(x) eiK is given by (28).
• The field distribution is then even symmetric about plane a.
• If the waveguide is designed so that these conditions are satisfied for a given transverse mode, they will not be satisfied by other transverse modes, except accidentally.

VII. ELECTROMAGNETIC SURFACE WAVES

• It is the purpose of this section to investigate the electromagnetic surface waves guided by the boundary of a semi-infinite periodic multilayer dielectric medium.
• 2 3 In this paper, the band theory of the periodic dielectric medium is employed to study the surface wave with an eigenvalue in the "forbidden" band.
• The fundamental surface wave may happen to be in the zeroth or the first forbidden gap.
• This is due to the fact that when the waveguides are separated infinitely from each other the singlet state has an eigenvalue B lower than that of the infinitely degenerate state.
• In a finite system, the allowed band is not fully occupied.

Full text

Electromagnetic
propagation
in
periodic
stratified
media.
I. General
theory*
Pochi
Yeh,
Amnon
Yariv,
and
Chi-Shain
Hong
California
Institute
of
Technology,
Pasadena,
California
91125
(Received
8 November
1976)
The
propagation
of
electromagnetic
radiation
in periodically
stratified
media
is
considered.
Media
of finite,
semi-infinite,
and infinite
extent
are treated.
A diagonalization
of the
unit
cell translation
operator
is used
to
obtain
exact
solutions
for
the Bloch
waves,
the dispersion
relations,
and
the band
structure
of the
medium.
Some
new
phenomena
with
applications
to
integrated
optics
and laser
technology
are
presented.
I.
INTRODUCTION
Periodic
optical
media
and
specifically
stratified
peri-
odic
structures
play
an important
role
in
a number
of
applications.
These
include
multilayer
high-reflectance
coatings
for
both
high
reflection
and
antireflection.
This
application
benefitted
largely
from
the
pioneering
analysis
of
Abeles.
1 Other
proposals
involve
the use
of
these
structures
for
phase
matching
in
nonlinear
optical
applications2-
4
and
for obtaining
optical
bire-
fringence
in
stratified
media
composed
of
isotropic
or
cubic
materials.
5,6
Recent
developments
in the
crystal-growing
field,
es-
peciallyin
molecularbeam
technology,
7
make
is
pos-
sible
to
grow
multilayer
media
with
well-controlled
periodicities
and with
layer
thicknesses
down
to
10 A.
We
may
thus
well
consider
the
periodic
optical
struc-
ture
as
a new
optical
medium
to
take
its
place
along
with
that
of,
say,
homogeneous
isotropic
and
aniso-
tropic
materials.
Before
proceeding
with
the
many
applications
envisaged
for periodic
media
we
need
to
understand
precisely
and
in detail
the
nature
of electro-
magnetic
wave
propagation
in
these
media.
Although
a
number
of
special
cases
have
been
analyzed,
a general
theory
is
not available.
To illustrate
this
situation
we
may
point
out,
as one
example,
that
the
present
state
of
the
theory
does
not
answer
questions
such
as
that
of
the
direction
of
group
and
energy
velocities
of
waves
in
periodic
media
or
even
that of
the
birefringence
at
arbitrary
angles
of
incidence.
The
two
papers
that
follow
describe
a general
theory
of
electromagnetic
propagation
in
periodic
media.
The
theoretical
approach
is
general,
so
that
many
situations
considered
previously
will
be
shown
to be
special
cases
of our
formalism.
The
theory
has
a
strong
formal
similarity
to
the quantum
theory
of electrons
in
crys-
tals
and
thus
makes
heavy
use
of
the
concepts
of
Bloch
modes,
forbidden
gaps,
evanescent
waves,
and surface
waves.
In
addition
to
demonstrating
the
application
of the
theory
to a
number
of
familiar
problems,
such
as
re-
flectivity
of
multilayer
films,
we consider
in
general
form
a variety
of
some
experimental
situations
which
include
Bragg
waveguides,
birefringence
and group
velocity
at arbitrary
directions,
phase
matching
in
nonlinear
optical
applications,
multichannel
waveguides,
and
optical
surface
waves.
We
also
consider
the
im-
portant
problem
of propagation
and
reflection
in
media
with
periodic
gain
and loss
alternation
which
is relevant
to x-ray
laser
oscillation
in artificially
layered
media,
8
423
J. Opt.
Soc. Am.,
Vol.
67,
No. 4,
April
1977
II.
MATRIX
METHOD
AND
TRANSLATION
OPERATOR
For
the sake
of
clarity
in
introducing
the basic
con-
cepts
we
will
consider
first
the simplest
type
of
peri-
odically
stratified
medium.
The
extension
to
the
more
general
case
is
presented
in
Appendix
A. The
strati-
fied
medium
treated
in
what
follows
consists
of
alter-
nating
layers
of different
indices
of refraction.
The
index
of refraction
profile
is
given
by
n2,
O<x<b,
n(x)=
n:
b<x<A
with
n(x
+A) =n(x)),
(1)
(2)
where
the
x axis
is normal
to the
interfaces
and
A
is
the period.
The
geometry
of the
structure
is
sketched
in Fig.
1.
The
distribution
of
some
typical
field
com-
ponents
can
be
written
E(x,z)=E(x)e
e-'t
(3)
The
electric
field
distribution
within
each
homogeneous
layer
can
be
expressed
as
a sum
of
an
incident
plane
wave
and
a reflected
plane
wave.
The
complex
ampli-
tudes
of
these
two
waves
constitute
the
components
of
a
column
vector.
The
electric
field
in
the a
layer
of
the
nth
unit
cell
can
thus
be
represented
by
a column
vector
nb(a)
*
As a
result,
the electric
field
distribution
in the
same
b
ni
-A-
-a-
FIG.
1. Portion
of a
typical
periodic
stratified
medium.
Copyright
©
1977
by the
Optical
Society
of
America
423
.I-

By eliminating
(Cn)
V mn
q
the
matrix
equation
b,
,n X
. I '\
.
.tb
n A nA-b
(n-l)-th
unit
cell n-th
unit
cell
FIG.
2.
Plane
wave
amplitudes
associated
with
the
nth unit
cell and
its
neighboring
cells.
layer
can
be written
. A (n-2)A.
E(x,
z) =
WOO
eikox(x-nA)
+
bra)
e-ikogx(x-nA))eifB
(4)
with
k,,x
={[(co/c)ncg]'
_ B2p
/ 2, a=
1,2
(5)
The
column
vectors
are
not independent
of each
other.
They
are
related
through
the
continuity
conditions
at
the
interfaces.
As a matter
of fact,
only
one
vector
(or
two
components
of
two
different
vectors)
can
be
chosen
arbitrarily.
In
the
case
of TE
modes
(E
vector
in
y-z
plane)
imposing
continuity
of E
and
BE/Bx
at
the inter-
face (see
Fig. 2)
leads to
an- 1+
bn
=
ie-
ik2xA
c +
e k 2xA
dn
ikl,(ai
- bn-
1
) =ik
2
. (e
k2xA cn
-e
k2sxA d
0
),
e-
ik2xa
cn + e i
k2xa dn
= e i
klxa an
+ e iklxa
bn
(6)
ik
2
x(e-ik2xa
-
e'd2xndn)
= ik(e
-eix
ban)
The four
equations
in (6)
can be
rewritten
as the
follow-
ing two
matrix equations:
/1 1
A tn-1\
\1 -1
\bn-1J
ek fk
2
xA
=
kM. e-k
2
xA
k,,
e
k2X
\
- k2.x
eik2x)
I
d
and
according
to (5)
can be viewed
as
functions
of PRO
The
matrix
in (9)
is the unit
cell translation
matrix
which
relates
the complex
amplitudes
of the
incident
plane wave
a,-, and
the reflected
plane
wave
bn l in one
layer of
a unit
cell to those
of the
equivalent
layer
in
the next
unit cell.
Because
of the
fact that
this matrix
relates
the fields
of two equivalent
layers with
the same
index
of refraction,
it
is unimodular,
i. e.,
AD
-BC =1
(14)
It is important
to note that
the matrix
which
relates
(
(Cn
dn-to
dn
is different
from
the matrix
in (9).
These
matrices,
however,
possess
the same
trace
(Appendix
A).
As
will be
shown
later, the
trace of
the translation
matrix
is
directly related
to
the band
structure
of the strati-
fied
periodic
medium.
The
matrix
elements
(A, B,
C, D) for
TM waves
(H
vector in
yz vector
plane)
are slightly
different
from
those
of the
TE waves.
They
are given
by
ATM =e iklx
[cosk2ab
-1i(
, + n2
sink2xb],
(15)
(7)
(e- k2xa
efk2xa \
(cn'
e
ik2
_ -e
2x2,
dn/
BTM
= ei
kj.,a ['
i ( kr
_- Ik)
sink2,b]
CTM=e
kn ka(
- A)
sink2ki
b]
,
DTM =eiklxa
[cosk2xb
+ ii(EII
+ EtL)
sink2xbl
e-
fi kxa
ki. e-ilxa
k
2
x'
where
we define
a, aa"),
bn a
W)X
Cn E-
an)2 dn
= b(2)
.
424
J. Opt. Soc.
Am., Vol.
67, No. 4,
April 1977
(8)
As noted
above,
only one
column
vector
is independent.
We can
choose
it, as an
example,
as the column
vector
of
the nl layer
in the
zeroth
unit cell.
The remaining
column
vectors
of the equivalent
layers are
given
as
(19)
Yeh et al.
424
0
Ctnln
bn-A,
>b (n-l)A (,,-
(bn- l
(C D)
bn)
is
obtained.
The
matrix
elements
are
A =
e-e lx,
[ cosk2,b
-i (
k2 +
kj) sink
2
xb]
B e
2klxf i
[~
2i(kk
-kkli)
sink2xb]
C =e- i
klxa [
i (M-
- )
s ink2bb]
'
D =e
ikla [cosk2xb
+
2
z lkk
1
i kIk)
sink2xJ]
(n-
A
I
L
(9)
(10)
(11)
(12)
(13)
(16)
(17)
(18)
Le
i k1"a
an
- kax e ik,
bn
k2x
(a.)
= Byn
(ao).
bn
D
bo
Y
..\\\\\\\\\\\\\\1
- A

(
AD)
(an)
=eiKA
(()
G n bn
(24)
2
I7
7r
27
37r
4,r
57
6,r
$(inunitsof
I )
FIG.
3.
TE
waves
(E
perpendicular
to
the
direction
of period-
icity)
band
structure
in the
w-:
plane.
The
dark
zones
are
the
allowed
bands.
By
using
(14),
the
above
equation
can
be
simplified
to
times
any
arbitrary
constant.
The
Bloch
waves
which
result
from
(26)
can
be
considered
as
the
eigenvectors
of
the
translation
matrix
with
eigenvalues
e'IKA
given
by
(25).
The
two
eigenvalues
in
(25)
are
the
inverse
of
each
other,
since
the
translation
matrix
is
unimodular.
Equation
(25)
gives
the
dispersion
relation
between
w,
,1,
and
K
for
the
Bloch
wave
function
(20)
K(/3,
w) =
(1/A)
cos-'
[2 (A
+D)1
.
(27)
(an)
D
-B~n
/ao\.
tbnJ
-C
A
J
b,
The
column
vector
for
the
n
2
layer
can
always
be
ob
tained
by
using
Eq.
(8);
more
generally,
we
can
sp(
the
field
uniquely
by
specifying
any
ai
and
b,.
III.
BLOCH
WAVES
AND
BAND
STRUCTURES
The
periodically
stratified
medium
is
equivalent
t
one-dimensional
lattice
which
is
invariant
under
the
lattice
translation.
The
lattice
translation
operator
is defined
by
Tx
=x
+ 1A
,
where
I
is an
integer;
it
fo
lows
that
TE(x)
=E(T-
1
x)
=E(x
-ZA)
.
(21)
The
ABCD
matrix
derived
in
Sec.
II
is
a
representa-
tion
of
the
unit
cell
translation
operator.
According
to
the
Floquet
theorem,
a wave
propagating
in
a
periodic
medium
is
of
the
form
9
EK(X,
z)
=EK(X)
eixeis
,
where
EK
(x)
is periodic
with
a
period,
A,
i. e,
EK(x+A)=EK(X)
.
(22)
The
subscript
K indicates
that
the
function
E,(x)
de-
pends
on
K.
The
constant
K
is
known
as
the
Bloch
wave
number.
The
problem
at
hand
is
thus
that
of
determining
K and
EK(x).
In
terms
of our
column
vector
representation,
and
from
(4),
the
periodic
condition
(22)
for
the
Bloch
wave
is
simply
(23)
(
n)
=eJKA
( an)
It
follows
from
(9)
and
(23)
that
the
column
vector
of
the
Bloch
wave
satisfies
the
following
eigenvalue
prob-
lem:
425
J.
Opt.
Soc.
Am.,
Vol.
67,
No.
4,
April
1977
Regimes
where
I a(A
+D)
I <
1
correspond
to
real
K
and
thus
to
propagating
Bloch
waves,
when
I
2 (A
+D)I
>
1,
K
=
m7r/A
+
iKi
and
has
an
imaginary
part
Ki
so
that
the
Bloch
wave
is
evanescent.
These
are
the
so-called
"forbidden"
bands
of
the
periodic
medium.
The
band
edges
are
the
regimes
where
I
(A
+ D)
I
= 1.
According
to
(4)
and
(23)
the
final
result
for
the
Bloch
wave
in
the
nl
layer
of
the
nth
unit
cell
is
E_(x)eiKx
(aoeik1x(x-nA)
+ boe-ik
x(x-nA))
e-iK(x-nA)I
eiKx
(28)
where
ao
and
bo
are
given
by
Eq.
(26).
This
completes
the
solution
of
the
Bloch
waves.
The
band
structure
for
a
typical
stratified
periodic
medium
as
obtained
from
(27)
is
shown
in
Figs.
3
and
4 for
TE
and
TM
waves,
respectively.
It
is
interesting
to
notice
that
the
TM
"forbidden"
bands
shrink
to
zero
when
13
=
(co/c)n
2
sin9B
with
OB
as
the
Brewster
angle,
since
at
this
angle
the
incident
and
reflected
waves
are
w
2,
37r
47r
57T
67
,e
( in units
of
I
)
FIG.
4.
TM
waves
(H
perpendicular
to
the
direction
of
peri-
odicity)
band
structure
in
the
w)-fl
plane.
The
dashed
line
is
P
= (w/c)n
2
sinh.
The
dark
zones
are
the
allowed
bands.
Yeh
et
al.
425
uic
(ao
B
\bQ)
\e
iKA
AJ
4ic
13
r
.
-
3
(26)
The
phase
factor
e-KA
is
thus
the
eigenvalue
of
the
translation
matrix
(A,
B,
C,
D)
and
is
given
by
e
=K
2
(A
+D)
±{
[2
(A
+D)]
-1
}
(25)
The
eigenvectors
corresponding
to
the
eigenvalues
(25)
are
obtained
from
(24)
and
are

/2 -
2,T
K (in units
of-)
A
FIG.
5.
Dispersion
relation
between
w
and K
when
? = 0
(nor-
mal incidence).
Dotted
curves
are
the
imaginary
part
of K
in
arbitrary
scales.
uncoupled.
The
dispersion
relation
X vs
K for
the
spe-
cial
case
Hi=0,
i.e.,
normal
incidence,
is
shown
in
Fig. 5.
IV.
BRAGG
REFLECTOR
Periodic
perturbation
in
a dielectric
medium
has
been
used
extensively
in fabricating
distributed
feed-
back
lasers","
1
(DFB)
and
distributed
Bragg
reflection
lasers
12
(DBR).
Corrugation
over
the
guiding
layer
is
the
usual
way
of providing
periodic
perturbation.
The
optical
fields
are
determined
by
using
the
coupled-
mode
theory'
3
which
in
a truncated
(finite
number
of
terms)
form
is a
very
good
approximation
as
long
as
the
perturbation
is small.
In the
case
of
square
well
alternation
which
corresponds
to the
layered
medium
described
above,
an
exact
solution
is obtained
by
our
matrix
method.
Consider
a
periodically
stratified
medium
with
N unit
cells.
The
geometry
of
the
structure
is sketched
in
Fig.
6.
The coefficient
of
reflection
is given
by
rN = (bo/ao)bo
*
(29)
From
(19)
we have
the
following
relation:
The
reflectivity
is obtained
by taking
the
absolute
square
of rN,
IrN
12 =/IC12NA
| X|=I Cl
I'+
(sinKIls/inNKA
)2
(34)
We
have
in (34),
the
first
published
analytic
expression
of the
reflectivity
of a
multilayer
reflector.
The term
I Cl
2 is
directly
related
to the
reflectivity
of a
single
unit cell
by
IrJI2=
IC12/(
C12+1)
|
C1|2=
| al |2/(1
_
I rl1
2) -
or
(35)
(36)
The
I rl
2 for
a typical
Bragg
reflector
is usually
much
less
than
one.
As a
result,
I C l
2
is roughly
equal
to
Ir,1
2
.
The second
term
in
the denominator
of
(34) is
a
fast
varying
function
of
K, or
alternatively,
of A3
and a.
Therefore,
it
dominates
the structure
of
the reflectivity
spectrum.
Between
any
two "forbidden"
bands
there
are exactly
N - 1 nodes
where
the
reflectivity
vanishes.
The
peaks
of the
reflectivity
occur
at the
centers
of
the
"forbidden"
bands.
There
are
exactly
N -
2 side
lobes
which
are
all
under
the envelope
I Cl 2/E[l
Cl
2
+ (sinKA)
2
1. At
the band
edges,
KA
=m7T
and the
re-
flectivity
is given
by
IrgI2=
IC1
2
/[IC1
2
+(1/N)
2
]
In
the
"forbidden"
gap
KA
is
a complex
number
KA=
mr +
iKiA.
The
reflectivity
formula
of (34)
becomes
2
IC1
2
I
n X=I
C1
2
+
(sinhKA/sinhNKfA)'
(37)
(38)
(39)
For
large
N the
second
term
in the
denominator
ap-
proaches
zero
exponentially
as
&-2(Nf1)KiA.
It follows
that
the reflectivity
in the
forbidden
gap is
near unity
for
a
Bragg
reflector
with
a substantial
number
of
periods.
TE
and TM
waves
have
different
band
structures
and
different
reflectivities.
For TM
waves
incident
at
the Brewster
angle
there
is no
reflected
wave.
This
is due
to the
vanishing
of
the
dynamical
factor
I
C12
/ao
)( A
B
N
)aN\
kbV
C
D bN)
(30)
The Nth
power
of an
unimodular
matrix
can be
simpli-
fied
by
the
following
matrix
identity'
4
(see
Appendix
B):
B
AUN.,-
UNv-2
(A
D)
( CUNY1
where
UN = sin(N+
1)KA/sinKA
,
BUN.-1
DUN.l- UNv2
(31)
(32)
with
K given
by Eq.
(27).
The
coefficient
of reflection
is immediately
obtained
from
(29),
(30), and
(31) as
rN
=CUN-1/(
AUN,-1-UN.-
2
) -
(33)
FIG.
6. Geometry
of a typical
N-period
Bragg
reflector.
426
J. Opt.
Soc.
Am.,
Vol.
67,
No.
4,
April
1977
Yeh
et al.
426

8 =
85'
0
° 9 z75';
in1
1
0
8= 65'-
L
A
WAAAPA
10
1-0 z55 f f
0
1.0
~15
A
I~
L
A
i~
0
r
L
o
3T
1.0
- 400
0
:1
7S
W
1.0
o 30'
O
A
| 20°
] a= 10°
8 = 0°
ol
A
A
L0 l2
00
ir/2
or
w (in units
ofA)
FIG.
7. TE waves
reflectivity
spectrum
reflector at
various angles
of incidence.
of a
15-period
Bragg
at
that angle.
The reflectivity
for some
typical
Bragg
reflectors
as
a function
of frequency
and
angle of
incidence
are shown
in
Figs. 7 and 8.
V.
GUIDED
WAVES
Multilayer
waveguides
are becoming
increasingly
im-
portant
in
integrated
optics.
The two-channel
dielec-
tric
waveguides
have
been studied
extensively
in the
theory of
branching
waveguides'
5
"
6
which is
used in
fabricating
mode
selectors,
switches,
and
directional
couplers
in integrated
optics 1
7
The analytic
treatment
for the
general
N-channel
waveguide,
however,
suffers
from the
serious
difficulty
of solving
an
eigenvalue
problem
involving
a 4N x
4N matrix,
and has
relied
heavily
on numerical
techniques.
In the
present
analysis
we employ
the
matrix method
described
in Sec.
H that
involves
only the
manipulation
of 2x 2
matrices.
Of particular
interest
is
the periodic
multichannel
dielectric
waveguide
(PMDW)
which
con-
AAAAAAA
n2,
mA'x'mA+b
n(x,z)=
(m=0,
1,2, ...
, N-1),
(n, otherwise,
(40)
with
nl<n
2
The
geometry
of
the waveguide
is sketched
in Fig.
9.
1.0
10
8 a=55
A
~
A
AAI
1.0
0 AAA
AJA
8t=45°
0
0
85
21A
37A
1.0
8-40°
0
1.0
30°
0
A
1.0
0=0
80 20'
b
A
A
8-
A
A
0A
A
n0 un/2
o
w (in
units ofCA
FIG.
8. TM
waves
reflectivity
spectrum
reflector
at various
angles
of incidence.
427
J. Opt. Soc.
Am., Vol.
67, No. 4, April
1977
of a
15-period
Bragg
Yeh
et al.
427
sists of
a stack
of dielectric
layers
of alternating
in-
dices
of refraction.
Analytic
expressions
for the
mode
dispersion
relations
and field
distributions
can
be ob-
tained
by the
matrix
method.
We
are looking
for
guided
waves
propagating
in the
positive
z
direction.
Two
important
periodic
multi-
channel
waveguides
will be
considered
in the
following.
A.
Symmetric
type
Consider
the simplest
kind
of
symmetric
PMDW
with
the index
of refraction
given by