Intern
ati
o
n
a
l
Jo
urn
a
l
o
f
Ad
va
nces
in Applied Sciences (IJ
A
AS)
V
o
l.
2, N
o
. 3
,
Sep
t
em
b
e
r
2013
, pp
. 16
5
~
17
0
I
S
SN
: 225
2-8
8
1
4
1
65
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJAAS
TM-Pol
arizati
o
n One-
Dimensio
nal Photonic Crystal Design
Elham Jasim Mohammad
1
, Ga
illa
n H. Abdulla
h
2
1
P
h
y
s
ics
Depar
t
m
e
nt, Col
l
ag
e of
S
c
ien
ces
,
Al-M
us
tans
iri
y
ah
Uni
v
ers
i
t
y
2
Ph
y
s
ics
Directorate, Min
i
stray
of Sc
ien
c
e &
Technolog
y
Materials Chemistr
y
Article Info
A
B
STRAC
T
Article histo
r
y:
Received J
u
1
4, 2013
Rev
i
sed
Au
g 1, 201
3
Accepted Aug 17, 2013
A theoretical
in
vestigation of o
n
e di
mensional planar
photonic cr
y
s
tal is
carried out.
Thes
e photonic
cr
y
s
tal consist of
a dielectric lay
e
r str
u
ctures with
refractiv
e index
n
1
=1.45 and n
2
= 3.45.
This
work pres
ents
a s
y
s
t
em
at
i
c
investiga
tion of
the r
e
fle
c
tion
,
forbi
dden bands
and density
of
state of
p-
polari
zat
ion. In
optic
al s
c
i
e
nc
es
,
the r
e
fractiv
e in
dex of an
optical medium is
a most fundamental quan
tity
.
Th
e refra
ctiv
e index determines the refraction
and ref
l
ection
occurr
ing at the bound
ar
y between two
media. The
propagation ang
l
e in on
e mediu
m
is take
n with
respect
to norm
a
l insid
e
th
e
first medium var
i
es between
0 an
d
π
/2
. Th
e progr
am
is writt
en
in
MATLAB
to sim
u
late
and
anal
ysis disper
sions
of electric magnetic waves in one
dimension photo
n
ic
cr
y
s
tal.
Keyword:
Pho
t
on
ic Cr
yst
a
l
Pho
t
on
ic Band G
a
p
Ref
r
action
Densi
t
y
of St
at
e
Copyright ©
201
3 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Gaillan
H.
Abdu
llah
,
Physics Direct
orate,
M
i
ni
st
ray
of
S
c
i
e
nce &
Tech
nol
ogy
M
a
t
e
ri
a
l
s C
h
em
i
s
t
r
y
,
B
a
gh
da
d, Ira
q.
Em
a
il: g
_
a
ltay
a
r@yahoo
.co
m
1.
INTRODUCTION
Ph
ot
o
n
i
c
cry
s
t
a
l
s
(PC
s
) a
r
e
st
ruct
ures
wi
t
h
pe
ri
o
d
i
cal
l
y
m
odul
at
ed di
el
ect
ri
c const
a
nt
s w
h
ose
d
i
stribu
tio
n follo
ws a p
e
riod
icity o
f
t
h
e
o
r
d
e
r of a fractio
n
o
f
th
e op
tical wav
e
leng
th
. Sin
c
e th
e first
pi
o
n
eeri
n
g
w
o
r
k
i
n
t
h
i
s
fi
el
d,
m
a
ny
new
i
n
t
e
rest
i
n
g
i
d
eas
h
a
ve
been
de
vel
ope
d
deal
i
n
g
wi
t
h
one
di
m
e
nsi
o
na
l
(1D) [1]. The sim
p
lest case of a phot
onic cry
s
tal is one di
m
e
nsional phot
onic crystal. It is a structure
built of
altern
atin
g layers of lin
ear,
un
ifo
r
m
an
d
isotro
pi
c m
a
t
e
ri
als of
t
w
o di
ffe
r
e
nt
re
fract
i
v
e i
ndi
ces
1
n
and
2
n
[2
]
.
Its sectio
n
in
x
z
pl
ane i
s
sh
ow
n o
n
Fi
g
u
re
. 1
.
The st
r
u
ct
u
r
e
has rot
a
t
i
o
nal
sym
m
e
t
r
y
aroun
d
z
axis, all
layers ex
tend
t
o
ward
s infin
i
t
y
in
x
and
y
di
r
ect
i
ons
(di
r
ect
i
o
n
y
is p
e
rp
en
dicu
lar
to sectio
n shown
on
Fig
u
re 1) and
th
e p
h
o
t
o
n
i
c crystal b
u
ilt o
f
fin
ite n
u
m
b
e
r
o
f
layers.
A pri
m
i
tiv
e cell
in
th
is case is a p
a
ir o
f
su
bsequ
e
n
t
layers with
refractiv
e in
d
i
ces
1
n
and
2
n
. Th
e nu
m
b
er of primitiv
e c
e
lls rig
h
t
fro
m
0
z
plane=
R
N
and
to th
e left=
L
N
, whe
r
e
direct
ions
right a
n
d left are i
n
acc
or
dance
wi
t
h
t
h
o
s
e o
n
Fi
g
u
re
1.
Fi
gu
re.
1
St
r
u
c
t
ure
of
o
n
e
-
di
m
e
nsi
onal
p
hot
oni
c c
r
y
s
t
a
l
(se
c
t
i
on)
[
2
]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
252
-88
14
I
J
AA
S
V
o
l. 2, No
.
3
,
Sep
t
emb
e
r :
1
6
5
–
170
16
6
Th
e primitiv
e
cell in
d
e
x
m
as depi
ct
ed
on t
h
e Fi
gu
re:
l
a
y
e
rs wi
t
h
]
,
0
[
z
co
n
s
titu
te p
r
im
it
iv
e
cell with
m
, and th
e n
e
x
t
(along
with
th
e
z
ax
is) p
a
ir h
a
s
1
m
, the first o
n
e ha
s
1
R
N
m
, the first
p
r
im
it
iv
e cell left fro
m
0
z
plane has
1
m
and t
h
e
i
nde
x
of t
h
e
l
a
st
one i
s
L
N
m
. The crystal is
su
rroun
ded b
y
a m
a
terial with
refractiv
e i
n
d
e
x
1
n
(
f
or
sim
p
lici
t
y)
[
2
].
The i
d
ea
of t
h
e desc
ri
be
d m
e
t
h
o
d
i
s
pres
ent
e
d
on
Fi
gure 2. It aim
s
to det
e
rm
ine effective reflection
from
a part of
photonic cryst
a
l on
a layer’s
bounda
ry and replace it with a m
i
rror of the sam
e
reflection. In
gene
ral
case o
f
ph
ot
oni
c cry
s
t
a
l
,
preci
se d
e
scri
pt
i
o
n o
f
den
s
i
t
y
of st
at
es (D
OS)
wi
t
h
o
u
t
ap
pr
o
x
i
m
at
i
ons
requ
ires d
i
fficult an
d
tim
e co
nsu
m
in
g
calcu
latio
n
s
.
Fi
gu
re 2 A
l
a
y
e
r of o
n
e di
m
e
nsi
o
nal
ph
ot
o
n
i
c
cry
s
t
a
l
as
a r
e
so
nat
o
r
[
2
]
.
Howev
e
r, fo
r
th
e sim
p
lest c
a
se o
f
on
e-d
i
men
s
io
n
a
l
p
hoto
n
i
c crystal it is p
o
s
sib
l
e to fo
llow the
deri
vat
i
o
n
gi
ve
n
bel
o
w i
n
t
h
e
fu
rt
he
r
part
of
t
h
e pa
pe
r.
In
o
u
r m
odel
we a
ssum
e
t
h
at
i
n
e
ach
of
l
a
y
e
rs
o
f
t
h
e
ph
ot
o
n
i
c
c
r
y
s
t
a
l
M
a
xwel
l
e
q
u
a
t
i
ons
ha
ve s
o
l
u
t
i
o
n
s
i
n
f
o
rm
of
pl
a
n
e
waves
[
2
]
:
t
i
r
k
i
e
E
E
0
(1)
wi
t
h
wave
vect
or
k
b
oun
d w
ith
an
gu
lar fr
equ
e
n
c
y
by
di
s
p
er
s
i
on
rel
a
t
i
o
n
[
2
]
:
2
2
2
2
c
n
k
.
whe
r
e
c
i
s
speed o
f
l
i
ght
i
n
v
acuum
and
n
is refractive in
dex
of th
e m
a
te
rial o
f
th
e layer.
W
e
can
relate
field
s
in
d
i
fferen
t layers using
con
tinu
ity con
d
ition
s
.
In
every
1
n
layer, th
e
electric field
as a sup
e
rpo
s
ition
of
co
up
led p
l
an
e
w
a
v
e
s [2
]:
]
,
[
,
,
,
,
1
,
a
m
m
z
e
b
e
a
E
t
i
z
i
x
ik
m
t
i
z
i
x
ik
m
m
j
x
j
x
(
2
)
whe
r
e
de
notes polarization T
M
(electric fiel
d i
n
pl
a
n
e
of
i
n
ci
dence
)
a
n
d
[
2
]
,
2
,
1
,
2
2
2
2
j
k
c
n
x
j
j
(3)
an
d
sim
ilar
l
y i
n
ev
er
y
2
n
layer. Co
n
tinu
ity co
n
d
ition
s
lead
to
equ
a
tio
n
relatin
g
am
p
litu
d
e
s in
co
n
s
ecu
tiv
e
p
r
im
it
iv
e cells
[2
]:
,
,
,
,
1
,
1
m
m
m
m
m
b
a
M
b
a
, whe
r
e
,
m
M
is tran
slatio
n
matrix
o
f
th
m
p
r
im
it
iv
e cell. It is
an
ob
vi
o
u
s
co
ncl
u
si
on
, t
h
at
i
f
am
pl
i
t
udes
of
pl
a
n
e
wave
s o
u
t
s
i
d
e t
h
e
ph
ot
o
n
i
c
c
r
y
s
t
a
l
are i
n
de
xed
wi
t
h
R
N
m
f
o
r
rig
h
t a
n
d
1
L
N
m
fo
r left, th
en
:
M
M
M
m
m
,
,
, whe
r
e,
D
C
B
A
M
an
d [2
]:
))
sin(
)
(
2
)
(cos(
2
1
2
2
2
2
1
2
2
1
1
2
2
2
1
b
n
n
n
n
i
b
e
A
a
i
TM
)
sin(
)
(
2
2
2
2
1
1
2
2
1
2
2
2
2
1
1
b
n
n
n
n
e
i
B
a
i
TM
(
4
)
)
sin(
)
(
2
2
1
2
2
2
2
1
2
2
1
1
2
2
1
b
n
n
n
n
e
i
C
a
i
TM
))
sin(
)
(
2
)
(cos(
2
1
2
2
2
2
1
2
2
1
1
2
2
2
1
b
n
n
n
n
i
b
e
D
a
i
TM
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
AA
S I
S
SN
:
225
2-8
8
1
4
TM-P
ol
ari
z
at
i
o
n
O
n
e
-
Di
men
s
i
o
n
a
l
P
hot
oni
c
Crysta
l
Desig
n
(Elh
am Ja
si
m Mo
hammad
)
16
7
As a
resu
lt, is si
m
p
lified
to
[2
]
:
,
0
,
0
,
,
b
a
M
b
a
N
N
N
, w
h
e
r
e,
,
0
a
and
,
0
b
= am
pl
it
udes.
Because the
bexcitation in t
h
e photonic crystal (or at
least on the
othe
r side of it) a
n
d there is
no
reflection in t
h
e i
n
finity. B
ecause
,
0
a
i
s
t
h
e am
pl
i
t
ude o
f
t
h
e
wave
i
n
ci
dent
o
n
l
a
y
e
r
bo
u
nda
ry
i
n
,
0
,
b
a
z
can be treated
as reflected wave. There
f
ore
,
we
can calculate reflection co
efficient of t
h
e whole
part
of
p
h
o
t
o
ni
c cry
s
t
a
l
refl
ec
t
i
ng
,
0
a
wav
e
, it
will b
e
[2
]:
a
i
a
i
e
a
e
b
r
1
1
,
0
,
0
(5)
2.
DENS
ITY OF
STATE
I
N
1
D
-P
HOTO
N
I
C
CR
YST
A
L
Pho
t
on
ic cryst
a
ls h
a
v
e
b
e
en
th
e su
bj
ect
o
f
in
ten
s
e i
n
v
e
stig
atio
n
d
u
e
to th
eir ab
ility to
co
n
t
ro
l t
h
e
pr
o
p
ert
i
e
s
of
p
hot
on
s [
3
]
.
Wh
en a c
o
l
l
i
m
a
t
e
d l
i
ght
ray
of
w
a
vel
e
n
g
t
h
i
n
a
h
o
m
ogene
ous
m
e
di
um
(e.g.
air
)
reaches t
h
e s
u
rface of the
1D
photonic
cryst
a
l slab at an incidence a
ngle
inc
. Fo
r
sim
p
lic
ity
we con
s
ider the
case where t
h
e
direction of
wave
pr
op
ag
atio
n
is restricted
in
the
xy
p
l
an
e (Figu
r
e
3).
After en
try in
t
o
th
e
ph
ot
o
n
i
c
cry
s
t
a
l
,
t
h
e l
i
ght
ra
y
pr
opa
gat
e
s at
an an
gl
e
of
re
fract
i
o
n
pc
.
To c
o
m
put
e t
h
e rel
a
t
i
ons
hi
p
bet
w
een
inc
and
pc
fo
r a gi
v
e
n wa
vel
e
n
g
t
h
,
we m
a
t
c
h t
h
e fre
que
ncy
an
d
t
a
nge
nt
i
a
l
com
p
o
n
e
n
t
of t
h
e
wave
vect
o
r
for the inci
de
nt and
refracted
wave ac
ross the interface
usi
ng t
h
e following sim
p
le proc
edure.
We s
p
e
c
ify the
ang
u
lar fre
q
u
e
n
cy
f
2
(
f
i
s
t
h
e freque
ncy
o
f
l
i
g
ht
) an
d t
h
e i
n
c
i
dence a
ngl
e
inc
i
n
t
h
e
hom
oge
n
e
ou
s
m
e
di
um
. Usi
ng t
h
e rel
a
t
i
ons
hi
ps
2
2
2
y
x
r
k
k
and
)
/
(
)
tan(
x
y
inc
k
k
we
can find the wave vector in the
i
n
ci
dent
m
e
di
um
. Here
r
is th
e relativ
e
p
e
rm
i
ttiv
it
y o
f
t
h
e in
ci
d
e
n
t
m
e
d
i
u
m
, an
d
x
k
an
d
y
k
are the
com
pone
nts of the wa
ve vector pe
rpe
n
di
cular and pa
rallel, respectively, to
the interface bet
w
ee
n the
hom
oge
neo
u
s
m
e
di
um
and the p
hot
o
n
i
c
cr
y
s
t
a
l
.
(
W
e ass
u
m
e
0
z
k
fo
r sim
p
licity.) W
e
can co
m
p
u
t
e th
e
1D
p
h
o
t
o
ni
c c
r
y
s
t
a
l
di
spe
r
si
on
rel
a
t
i
o
nshi
p
,
o
r
ph
ot
o
n
i
c
ban
d
st
ruct
ure
,
)
(
k
, using
th
e t
r
an
sfer m
a
tri
x
m
e
thod, a sta
nda
rd technique
c
o
m
m
only
found i
n
the
literature.
He
re
the a
n
gular
freque
ncy
and th
e
paral
l
e
l
com
ponent
o
f
t
h
e wa
ve vect
o
r
y
k
,
are
t
h
e sam
e
as t
h
ose i
n
t
h
e i
n
ci
dent
h
o
m
ogen
e
ou
s
m
e
di
um
.
The
tran
sfer m
a
trix
tech
n
i
qu
e allo
ws
u
s
to
fi
nd
th
e p
e
rp
end
i
cu
lar co
m
p
on
en
t o
f
th
e
wave v
ecto
r
x
k
in
t
h
e
p
h
o
t
on
ic crystal. Fro
m
)
(
k
we can t
h
en
com
put
e t
h
e ph
ot
o
n
i
c
c
r
y
s
t
a
l
gr
ou
p
vel
o
ci
t
y
usi
n
g
)
/
)
(
,
/
)
(
,
/
)
(
(
)
(
z
z
x
k
g
k
k
k
k
k
k
k
. From
t
h
e com
ponent
s o
f
t
h
e gr
o
up
vel
o
ci
t
y
we can
det
e
rm
i
n
e
t
h
e angl
e o
f
re
fra
ct
i
on usi
n
g
)
/
(
tan
1
x
g
y
g
pc
[4]
.
Fi
gu
re
3
Il
l
u
st
r
a
t
i
on
of
h
o
w
a
1D
p
h
o
t
o
ni
c cr
y
s
t
a
l
m
i
ght
be
use
d
t
o
di
s
p
ers
e
l
i
ght
of
di
ffe
r
e
nt
wa
vel
e
n
g
t
h
s [
4
]
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
252
-88
14
I
J
AA
S
V
o
l. 2, No
.
3
,
Sep
t
emb
e
r :
1
6
5
–
170
16
8
In
recent years, high
quality factor, sm
all-
m
ode
volum
e cavities based
on
phot
onic c
r
ystals have
attracted
sign
ifican
t atten
tio
n
,
b
ecau
s
e of their ab
ility
to
m
o
d
i
fy th
e d
e
n
s
ity o
f
op
tical states stro
ng
l
y
. An
increase i
n
DOS of t
h
e lasing
m
ode causes
signi
ficant e
nha
ncem
ent
of t
h
e
sp
o
n
t
a
ne
ous
e
m
i
ssi
on rat
e
[
5
]
.
The
co
n
c
ep
ts
o
f
d
e
n
s
ity o
f
state hav
e
p
r
ov
en essen
tial in
th
e
stu
d
y
o
f
electr
o
mag
n
e
tic w
a
v
e
p
r
op
agatio
n th
rough
peri
odi
c st
ruct
ures
. T
h
e
D
O
S
i
s
us
ual
l
y
de
fi
ned
as
[6]
:
m
BZ
m
BZ
k
d
k
A
N
2
))
(
(
1
)
(
(6)
whe
r
e the i
n
tegral is take
n
over t
h
e
th
m
ban
d
a
nd
BZ
A
is the area
of t
h
e Brilloui
n zone
(BZ). E
q
.
(6) can
also
b
e
written
as
[6
]:
m
EFS
BZ
ds
d
dk
A
N
m
1
)
(
(7)
whe
r
e t
h
e i
n
t
e
gral
i
s
t
a
ke
n al
on
g t
h
e
th
m
equi
f
r
e
que
ncy
s
u
r
f
ac
es (EF
S
) at
fre
que
ncy
and
d
dk
g
1
i
s
t
h
e i
nverse
gr
o
up vel
o
ci
t
y
. The D
O
S w
a
s fi
rst
used i
n
u
nde
rst
a
n
d
i
n
g t
h
e
m
odi
fi
ca
t
i
on of s
p
o
n
t
a
neo
u
s
e
m
issio
n
in
pho
ton
i
c crystals. Th
e DOS
p
l
ays an
i
m
p
o
r
tant ro
le in
lig
h
t
trapp
i
ng
fo
r so
l
a
r
cel
l
s
and i
n
m
ode
co
nfin
em
en
t in
pho
ton
i
c cry
s
tal stru
ctures. Th
e
d
e
fi
n
ition
in
equ
a
tio
n
(6) su
gg
ests t
h
e typ
i
cal m
e
t
h
od
b
y
whi
c
h t
h
e DO
S i
s
co
m
put
ed
:
usi
ng t
h
e f
u
l
l
band st
r
u
ct
u
r
e and bi
n
n
i
n
g
by
freq
u
ency
t
o
appr
o
x
i
m
at
e t
h
e
integral. T
h
e freque
ncy binning m
e
thod ca
n be im
proved
i
f
the group ve
locities are
also available [2], [6].
Pho
t
on
ic cryst
a
ls are st
rong
ly wav
e
leng
th
sen
s
itiv
e,
h
i
gh in
d
e
x
con
t
rast
d
i
electric m
a
te
rials. Th
is sen
s
itiv
ity
ori
g
i
n
at
es f
r
o
m
t
h
e di
sper
si
ve p
r
ope
rt
i
e
s o
f
t
h
e
ph
ot
o
n
i
c
crystal structure as a resu
lt
o
f
th
e wav
e
leng
th
scale
ref
r
act
i
v
e i
nde
x m
odul
at
i
on
wi
t
h
cert
a
i
n
cry
s
t
a
l
sym
m
e
try
[7]
.
The be
havi
or o
f
a p
hot
on
wi
t
h
a cert
a
i
n
fre
que
ncy
will de
pend
on the propag
ati
on
direction wit
h
in the
phot
oni
c
crystals. The
m
odulation
of the
refractiv
e ind
e
x
will cau
se that certain
en
erg
i
es and
d
i
r
ectio
n
s
are forb
i
d
den
fo
r
ph
o
t
o
n
s
. A reg
i
on
of energ
i
es
wh
ere th
e pho
to
n
i
c crystal d
o
es no
t allo
w
ph
o
t
o
n
s to
p
r
o
p
a
gat
e
re
gar
d
l
e
s
s
of
t
h
ei
r
di
rec
t
i
on a
nd
p
o
l
a
ri
zat
i
o
n
is called
a co
mp
lete ph
o
t
on
ic
b
a
nd
g
a
p
(cPB
G)
[8
].
3.
R
E
SU
LTS AN
D ANA
LY
SIS
We
procee
d wi
th the calc
u
lation of
fields for TE m
odes. M
A
TL
AB is a
great and easy tool to
use t
o
si
m
u
late o
p
tical electro
n
i
cs.
All th
e resu
lts b
e
lo
w are
go
t aft
e
r fo
llowing
these step
s:
1.
Calculate the reflectance
func
tion.
2.
Im
pl
em
ent
a
ti
on
of
di
s
p
ersi
on
rel
a
t
i
o
n
o
f
el
e
c
t
r
om
agnet
i
c
wave
s i
n
1
D
p
hot
oni
c c
r
y
s
t
a
l
.
3.
Fo
un
d t
h
e
no
r
m
al
i
zed fre
que
ncy
.
4.
Found
t
h
e
ray angle with res
p
ect
to norm
al
inside 1
st
m
e
di
um
vari
es bet
w
een
0
-
2
/
and i
n
si
de t
h
e
2
nd
m
e
di
um
usi
ng
Snel
l
l
a
w
(
45
.
1
1
n
and
45
.
3
2
n
).
T
h
en
t
r
a
n
sf
orm
i
n
ci
dence
angl
e i
n
deg
r
ee
s.
5.
Sel
ect
poi
nt
s w
h
i
c
h bel
o
n
g
t
o
t
h
e
f
o
r
b
i
d
de
n b
a
nd
s.
6.
In
orde
r to com
pute the DOS we
found the
inve
rse
gr
oup
velocity at each sam
p
le
poi
nt
and the
n
the
DOS
can be di
rect
l
y
obt
ai
ne
d usi
n
g
equat
i
o
n (
7
).
Fi
gu
re 4 i
s
ab
out
t
h
e re
fl
ect
ance f
unct
i
o
n
vers
us wa
vel
e
ngt
h wi
t
h
ave
r
age m
ean val
u
es=0.
8
05
5,
m
e
di
an=0.
8
58
4 an
d st
an
dar
d
devi
at
i
o
n (ST
D
)=
0.
26
7
3
. T
h
e
m
a
gni
t
ude
o
f
refl
ect
ance=
1
wi
t
h
i
n
t
h
e ra
n
g
e o
f
i
n
fra
re
d. Fi
g
u
r
e 5 sh
o
w
s f
o
r
b
i
dde
n ba
n
d
s p
-
pol
a
r
i
zat
i
on, i
t
expl
ai
n t
h
e rel
a
t
i
ons
hi
p b
e
t
w
een i
n
ci
de
nce
angl
es
vers
us
n
o
rm
al
ized f
r
e
que
ncy
.
The i
n
ci
de
nce
an
gl
e i
s
an
i
m
po
rt
ant
param
e
t
e
r w
h
i
c
h e
f
f
e
c
t
s t
h
e wi
dt
h
o
f
ban
d
g
a
ps. Th
e m
e
a
n
= 0.572
, m
e
d
i
an
= 0
.
60
57
,
m
o
d
e
= 0
.
308
4 an
d
STD
=
0
.
2
586
. Th
e th
ick
n
e
sses and
the in
dex
cont
rast
of t
h
e
ph
ot
o
n
i
c
cry
s
t
a
l
det
e
rm
i
n
at
e
m
a
ny
of i
t
s
op
t
i
cal
prope
rt
i
e
s. Pl
ay
i
ng o
n
t
h
ese t
w
o pa
ram
e
t
e
rs
,
we ca
n
obt
ai
n
fre
que
ncy
ran
g
e
s f
o
r
w
h
i
c
h l
i
ght
pr
o
p
ag
at
i
o
n i
s
f
o
rbi
dde
n i
n
t
h
e
m
a
t
e
ri
al
and
ot
hers
ra
n
g
es
fo
r
whic
h light can propagate. These freque
ncy ranges a
r
e
also scale depende
n
t. Reducing the size of the
ele
m
en
tary cel
l o
f
th
e p
e
ri
o
d
i
c lattice
sh
ifts th
e who
l
e
frequ
en
cy rang
e to h
i
g
h
e
r v
a
l
u
es. Fig
u
re 6
rep
r
esen
ts
t
h
e rel
a
t
i
o
ns
hi
p
bet
w
ee
n t
h
e
no
rm
al
i
zed fre
que
nci
e
s
ver
s
u
s
de
nsi
t
y
o
f
st
a
t
e fo
r M
p
o
l
a
ri
zat
i
on.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
AA
S I
S
SN
:
225
2-8
8
1
4
TM-P
ol
ari
z
at
i
o
n
O
n
e
-
Di
men
s
i
o
n
a
l
P
hot
oni
c
Crysta
l
Desig
n
(Elh
am Ja
si
m Mo
hammad
)
16
9
0.
1
0.
1
5
0.
2
0.
2
5
0.
3
0.
3
5
0.
4
0.
4
5
0.
5
0
0.
5
1
1.
5
2
2.
5
3
x 1
0
5
N
o
rm
a
l
i
z
ed
F
r
eq
ue
nc
y
D
ens
i
t
y
O
f
S
t
a
t
es
(
a
.
u
.
)
0
10
20
30
40
50
60
70
80
90
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
I
n
c
i
denc
e A
ngl
e (
deg)
N
o
r
m
al
i
z
ed F
r
eque
ns
y
Fi
gu
re
4 R
e
fl
ect
ance v
e
rs
us t
h
e
wavel
e
ngt
h.
The
re
flectanc
e
is extrem
ely high
ove
r t
h
e range
.
1460
825
nm
Fi
gu
re 5 Inci
de
nce
a
ngl
e ve
rs
us no
rm
al
i
zed
fre
que
ncy
.
It
s
h
o
w
s
t
h
e fo
r
b
i
dde
n ba
nd
s
f
o
r
p
-
p
o
l
a
ri
zat
i
o
n.
N
o
t
t
h
at
, t
h
e l
a
rge
r
t
h
e di
ffe
rence
bet
w
ee
n t
h
e
t
w
o i
n
di
ces t
h
e
w
i
der t
h
e
ban
d
g
a
ps
bec
o
m
e
.
Fi
gu
re 6
De
ns
i
t
y
of
st
at
e ve
rsus
no
rm
al
i
z
ed fre
q
u
ency
.
M
i
nim
u
m
,
m
a
xi
m
u
m
DOS are bet
w
ee
n
=0
.0
137
to infin
ity an
d th
e med
i
am
=5
.8
1
.
4.
CO
NCL
USI
O
N
We ca
n c
onst
r
uct
ve
ry
com
p
act
opt
i
cal
de
v
i
ces wi
t
h
d
e
sired
o
p
tical prop
erties su
ch as lasers, ligh
t
e
m
it
tin
g
d
i
od
es, filters, waveg
u
i
d
e
s,
h
o
l
ey
fib
e
rs, an
d
ph
o
t
o
n
i
c in
teg
r
ated
circu
its by d
e
sig
n
i
n
g
sp
ecific
p
h
o
t
on
ic crystal stru
ctures and in
tro
d
u
c
ing
some d
e
fects i
n
a PC,.
The
reflection coefficient
for a
pla
n
e
wave inci
dent
up
o
n
a peri
od
i
c
dielectric structure
was
anal
y
zed. The
researc
h
pres
e
n
t
e
d
i
n
t
h
i
s
pa
per f
o
cuse
s on pl
ana
r
o
n
e di
m
e
nsi
o
nal
ph
ot
o
n
i
c
st
r
u
ct
u
r
es
c
onsi
s
t
s
o
f
altern
atin
g
layers o
f
m
a
te
rial with
d
i
fferen
t d
i
electric
co
nstan
t
s (
45
.
1
1
n
and
45
.
3
2
n
).
It
has bee
n
sh
own
th
at th
e larg
er th
e
d
i
fferen
ce
b
e
tween
th
e two
in
dices the wide
r the ba
nd ga
ps
becom
e
. Also, as the
wi
dt
h
o
f
t
h
e ai
r l
a
y
e
rs bec
o
m
e
sm
al
l
e
r i
n
com
p
ari
s
on t
o
t
h
e wi
dt
h
o
f
t
h
e
di
el
ect
ri
c l
a
y
e
rs, t
h
e wi
dt
h
of
ba
n
d
gap
s
w
oul
d de
crease. T
h
e an
gl
e of i
n
ci
de
nc
e of t
h
e l
i
g
ht
wave i
s
al
so a
not
her
fact
or
whi
c
h ef
fect
s t
h
e wi
dt
h
of
ba
n
d
gaps
.
DO
S i
s
a
n
i
m
port
a
nt
fact
or i
n
l
i
ght
t
r
a
p
pi
n
g
i
n
p
h
o
t
o
ni
c cry
s
t
a
l
st
ruct
u
r
es.
400
600
80
0
1000
1200
1400
1600
0
0.
2
0.
4
0.
6
0.
8
1
1.
2
W
a
v
e
l
engt
h (nm
)
R
e
f
l
ec
t
anc
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
252
-88
14
I
J
AA
S
V
o
l. 2, No
.
3
,
Sep
t
emb
e
r :
1
6
5
–
170
17
0
REFERE
NC
ES
[1]
A. H. Araf
a. "
E
l
ectrom
a
gne
tic
W
a
ves
Propagation Characteri
stics in Supercond
ucting Phot
on
ic Cr
y
s
ta
ls",
Chap
t
e
r-
4.
[2]
R. Adam
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BIOGRAP
HI
ES
OF AUTH
ORS
E
l
ha
m Ja
si
m Moha
mma
d,
wa
s
born in Ir
aq,
s
h
e re
ce
ived h
e
r P
h
.D. d
e
gree
in
Optoele
c
troni
cs
P
h
y
s
ics
S
c
ienc
e
from
Al-M
us
tans
iri
y
ah Univers
i
t
y
, h
e
r M
.
S
.
degree in Im
age P
r
oces
s
,
P
h
y
s
ics
S
c
ienc
e from
Al-M
us
tans
iri
y
ah
Univers
i
t
y
. S
h
e
rece
ived B
.
S
.
de
gree in
P
h
y
s
i
cal
S
c
ienc
e from
Al-M
us
tans
iri
y
a
h
Univers
i
t
y
.
S
h
e works
as
a Uni
v
ers
i
t
y
P
r
offes
s
o
r in th
e Dep
a
rtm
e
nt of P
h
ys
ic
s
Science from Al-Mustansiriy
ah
University
, Bag
hdad, I
r
aq.
Gaill
an H. Abdu
llah
,
was born
in
Bagdad
,
Ir
aq; h
e
re
ceiv
e
d B
.
S. d
e
gree
in Ph
ysic
a
l
Scien
c
e
from
Baghdad Univ
er
sity
, M.S. degr
ee in
Laser and
Op
toelectron
i
cs (
A
lrasheed Co
llege/University
of
Techno
log
y
-Bag
hdad) and his
Ph.D. degree in
Laser and Optoelectronics technolog
y
from
Techno
log
y
University
/Ir
a
q-Bagdad. He
joined
to the
Laser
and Optoel
ec
tr
onic c
e
nt
er in
Ministr
y
of Science
and Technolog
y
and carried
out
research in Thin
Films design s
y
stem and
optical d
e
sign.
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