TELKOM
NIKA
, Vol.16, No
.4, August 20
18, pp. 1490
~14
9
9
ISSN: 1693-6
930,
accredited First Grade by
Kemenristekdikt
i
, Decree No: 21/
E/KPT/2018
DOI
:
10.12928/TELKOMNIKA.v16i4.5191
1490
Re
cei
v
ed
No
vem
ber 1
4
, 2016; Re
vi
sed
Jan
uar
y 30, 2
018; Accepte
d
April 16, 20
18
Modelling Optical Waveguide Bends by the Method of
Lines
Ar
y
S
y
ahriar*
1
, Nabil Ra
y
h
an S
y
ahriar
2
, Jusman Sy
afiie Djamal
3
1,3
Electrical Engin
eeri
ng D
epa
rtment, F
a
cult
y of Sc
ience a
n
d
T
e
chnolo
g
y
Univers
i
t
y
al A
z
har Indo
nes
ia
,
Jakarta Indo
ne
sia
2
Mechanic
a
l E
ngi
neer
in
g Dep
a
rtmen
t, Band
ung Institute of
T
e
chnol
og
y
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: ar
y
@
uai.
a
c.i
d
A
b
st
r
a
ct
A rigorous
a
nalytic
al a
nd
semi an
alytica
l
me
t
hod
of li
nes has
be
en
used to calc
ulate th
e
transverse-
ele
c
tric field
atten
uatio
n co
efficie
n
t of gu
id
ed
mode
as it trav
el
s in w
a
ve
gui
de
ben
ds structur
e
.
Both a
ppr
oach
e
s the
n
w
e
re
c
o
mpar
ed to
g
e
t a
better
und
er
standi
ng
on
h
o
w
the atten
uati
on
beh
aves
al
o
n
g
singl
e c
u
rve w
a
veg
u
id
es w
i
th
consta
nt ra
di
us of c
u
rvatur
e.
T
he Hel
m
h
o
lt
z
Eq
uati
on i
n
p
o
lar
co
ordi
nate
w
a
s transformed i
n
to a c
u
rv
alin
ier c
oord
i
n
a
te to si
mu
late
the w
a
veg
u
id
e be
nds
usin
g
the
metho
d
of
lin
e
ana
lysis.
T
h
e
simple
a
b
sorpt
i
on bo
un
dary cond
itions
ar
e
use
d
i
n
to th
e
metho
d
of
lin
es to
de
mo
nstrate
evan
esce
nt fiel
d of the
gui
de
d
mo
de
natur
e a
s
its travels i
n
w
a
vegui
de
ben
ds structures.
T
he resu
lts sh
ow
that a reaso
n
a
b
le a
g
ree
m
ent
bet
w
een b
o
th theor
etical a
ppr
oach
e
s.
Ke
y
w
ords
: Optical w
a
vegu
id
e ben
ds, Meth
od of lin
es, T
he transverse-
el
ectric field atte
nuati
on co
effici
ent
Copy
right
©
2018 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
One of the most impo
rta
n
t optical integrat
e
d
devi
c
e
s
buildin
g blocks is wa
veguide
bend
s b
e
cau
s
e it d
e
fine
s
the overall si
ze
of in
teg
r
at
ed opti
c
s in
singl
e subtrates.
Waveg
u
ide
bend
s
are
re
quire
d in
ma
ny ba
sic opti
c
al
structu
r
e
s
, inclu
d
ing
direction
a
l
cou
p
l
ers,
mod
u
lat
o
rs,
ring resonato
r
s [1], arraye
d waveg
u
ide
filters [2],
optical del
ay line
s
[3], S-bend
attenuators [4].
and M
a
ch-Z
e
hnde
r inte
rferometers.
Ho
wever,
wa
ve
guide
ben
ds
experie
nce lo
ss a
s
the
gui
ded
mode ente
r
s the cu
rved se
ction which d
epen
ds o
n
co
nfinement fa
ctors a
nd ra
di
us of cu
rvature.
The l
o
ss
ca
n
be mi
nimized
by in
crea
sin
g
the
mod
e
confineme
n
t i.
e. by in
crea
si
ng the
refra
c
t
i
ve
index differe
nce
s
bet
wee
n
co
re an
d clad
ding laye
rs o
r
by de
crea
sing
radi
u
s
of cu
rvatures.
Incre
a
si
ng
m
ode
co
nfinem
ent will
in
cre
a
se
the
coupl
ing lo
sse
s
wh
en
waveg
u
id
e is
coupl
ed
i
n
to
the fiber opti
c
s and decreasing r
adius of
curvature w
ill increase
the overa
ll integrated opti
cs
size. In a
silica-ba
se
d
wav
eguid
e
, there
is no
rmally
o
n
ly a very
slig
ht variation i
n
refractive in
d
e
x
across th
e
cross-se
ction, t
o
allo
w lo
w l
o
ss couplin
g to
a
single
mod
e
fibre.
The
sl
ight variatio
n
in
index is m
o
st
helpful a
s
it
permi
t
s
the v
e
ctor wave
e
quation to
be
repla
c
e
d
by
a scala
r
eq
ua
tion
in whi
c
h t
he
electri
c
fiel
d i
s
represente
d
by on
e vector compo
n
e
n
t. This
simpl
i
fication i
s
kn
own
as the wea
k
-guid
a
n
c
e ap
proximatio
n. Ther
efore a
pre
c
ise kn
owledge o
n
be
nd wave
guid
e
s
cha
r
a
c
teri
stics be
come im
portant to de
si
gn a com
p
a
c
t integrated o
p
tical sy
stem
s.
So far, a n
u
m
ber
of efficien
t numeri
c
al
te
chni
que
s h
a
ve bee
n p
r
op
o
s
ed fo
r the
a
nalysi
s
of optical
wa
veguide
s. Th
ese in
clu
de t
he finite
differen
c
e meth
o
d
(F
DM), the
finite eleme
n
t
method
(FE
M
), the b
eam
pro
pag
ation
method
(BP
M
), an
d the
method
of lin
es
(MoL
) [5].
The
finite differen
c
e m
e
thod i
s
the olde
st nu
meri
cal
m
e
th
od for
solvin
g
partial
differential eq
uatio
ns.
It is simple t
o
pro
g
ra
m a
nd ea
sily ap
plied to
no
n-homog
eno
us refra
c
tive in
dex profile
s.
This
method sub
d
i
vides
the d
o
main
into many
su
breg
ions, in
whi
c
h the pa
rtial
derivatives
are
repla
c
e
d
by finite differen
c
e operators. A set
of linear equatio
ns a
r
e then solve
d
to obtain the
eigenvalu
e
s.
The
dra
w
b
a
ck of
the
FDM
is it offers
l
e
ss flexibility in the
mod
e
lin
g of th
e d
o
m
a
in
sin
c
e th
e
su
b
r
egio
n
i
s
normally re
ctan
g
u
lar
in
sh
ape
[6]. The finite element m
e
thod (FEM
) can
model the m
o
st intricate domain g
e
o
m
etrie
s
. In
FEM, the waveguid
e
cross sectio
n is di
vided
into surfa
c
e o
r
volume ele
m
ents an
d the field in
each element is
approxim
ated
by a polynomial.
The field
co
n
t
inuity conditi
ons
are
impo
sed
on a
ll i
n
terface
s
between the
different ele
m
ent
s.
A variational
expre
ssi
on for Maxwell’s equation
s
t
hen is
emplo
y
ed to obtai
n an eige
nvalue
matrix eq
uati
on
whi
c
h i
s
solved by
sta
ndard m
e
tho
d
s.
Thi
s
met
hod req
u
ire
s
a
mo
re com
p
le
x
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Modellin
g Op
tical Wa
ve
gui
de Bend
s by t
he Method of
Line
s (Ary Syahria
r)
1491
prog
ram
m
ing
structu
r
e
an
d is mo
re
de
mandin
g
in
b
o
th comp
uter time a
nd
m
e
mory [7]. T
h
e
beam
propag
ation meth
od
(BPM) h
a
s b
een
used to
a
nalyze
vario
u
s
two-
and
three-di
men
s
ion
a
l
optical
devices. Th
e o
r
igi
nal BPM u
s
e
d
an
FFT
alg
o
rithm
and
solved a
pa
ra
xial scala
r
wave
equatio
n. The
ba
sic ide
a
of
the BPM i
s
t
o
repres
ent t
he el
ectroma
gnetic field
b
y
a supe
rpo
s
i
t
ion
of plane wav
e
s propa
gatin
g in homog
e
nou
s media.
The advanta
ges of the BPM are that it can
be ap
plied to
a structu
r
e
with an
arbit
r
ary cr
o
s
s-section, and tha
t
both guid
e
d
and
radi
ative
wave
s are i
n
clu
ded in the analysi
s
.
Howeve
r si
nce the formulation is
derived u
nde
r the
assumptio
n
t
hat the refra
c
tive index variation in
the
tran
sverse
direction i
s
ve
ry small, the
F
FT-
BPM cannot
be appli
ed to stru
ctures
with large in
dex discontin
uities [6].
The m
e
thod
of line
s
(MoL
) h
a
s be
en
proved to
be
a
very u
s
eful to
ol for the
ana
lysis
o
f
gene
ral
wave
guide
system
s [8]. It is a
semi a
nalyt
ical method, in
whi
c
h the
wave equ
ation
is
discreti
zed
a
s
fa
r a
s
ne
cessary
in th
e tra
n
sv
e
r
se
dire
ction
an
d solved
an
alytically in t
he
longitudin
a
l dire
ction, whi
c
h re
sult
s in
less
co
mp
utational effort. An accurate
result
can
b
e
obtaine
d sin
c
e the MoL
b
ehave
s
in a
stationa
ry fashio
n and
co
nverge
nce is monotoni
c [
9
].
Disco
n
tinuou
s field
s
can b
e
de
scribe
d a
c
curately
b
e
cause the inte
rface
co
nditio
n
s a
r
e in
clu
d
e
d
in the
cal
c
ul
ation. Fu
rthe
rmor
e, the
MoL i
s
relati
vely easy to
impleme
n
t
usin
g comp
u
t
er
nume
r
ical me
thods. In this
pape
r we co
mpare two dif
f
erent ap
pro
a
che
s
, nam
ely a simple
qua
si-
analytic th
eo
ry ba
sed
on
integration
of a p
hen
om
enolo
g
ical
a
b
so
rption
co
efficient, an
d
the
method of lin
es (M
oL). Bot
h
are ap
plied
to a numbe
r of different waveguid
e
ben
d curvatu
r
e
s
.
In this p
ape
r we
appli
e
d
the meth
od
of line
s
with third
orde
r ab
sorbing
b
ound
ary
con
d
ition to
analyse
wea
k
ly gui
ding
o
p
tical
wave
g
u
ide
s
b
end
s ch
aracte
risti
cs.
For the
first
approximatio
n we
have t
r
an
sform
ed t
he Helmholt
z
wave e
quat
ion in p
o
lar
co-ordinate
s
to
Carte
s
ia
n co
ordin
a
tes to
simplify the discretisati
on
of wavegui
de
stru
cture
s
. In the pro
c
e
s
s we
comp
ared th
e
re
sult
s
with a
nalytical m
e
th
ods a
s
the
co
rre
ct
referen
c
es th
at ha
s b
een
develo
p
e
d
previou
s
ly. We found that the MoL
re
sul
t
s are
in
goo
d agreeme
n
t with analytica
l
method
s. The
discre
pen
cie
s
ari
s
ed f
r
om
differents ra
di
us of
cu
rvatu
r
e u
s
e
d
in th
e cal
c
ul
ation
and the
choices
of abso
r
bin
g
boun
dari
e
s p
a
ram
e
ters.
2. Researc
h
Method
2.1. Analy
t
ic
al Approa
ch
To analyse the effect of a wavegui
de
bend,
co
nsi
d
er a bend fo
rmed by a circula
r
arc
with radiu
s
of
cu
rvature
r
a
s
sho
w
n
in
Fi
gure
1. It i
s
a
s
sumed
that
only the fu
nd
amental
mod
e
prop
agate
s
in
the guide. If the radi
us of
curvatu
r
e i
s
l
a
rge
eno
ugh
(
r
), then the propertie
s
of the mode
are effectiv
ely those of
a mode tra
v
eling in a straight guid
e
. However, a
s
r
decrea
s
e
s
, at
tenuation i
s
e
x
pected to
occur. L
e
t
P(s)
be the total p
o
we
r carried
by the mode
at
any point
s
a
l
ong the ben
d. Assumi
ng that the rate of
powe
r
loss is prop
ortion
al to the power
carrie
d by the mode at that point, we can
write:
Figure 1. Section of a curv
ed plan
ar wa
veguide
)
(
)
(
s
P
ds
s
dP
(1)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 16, No. 4, August 2018: 149
0-1499
1492
whe
r
e
is th
e attenuation
coeffici
ent. Provided
is
consta
nt, equa
tion 1 has the
solution:
s
e
P
s
P
)
0
(
)
(
(2)
Marcatili an
d
Miller
have
sho
w
n
that t
he atten
uatio
n coefficie
n
t is in
deed
con
s
tant for a fi
xed
radiu
s
, and
can be expressed a
s
[10]:
r
C
e
C
2
1
(3)
whe
r
e
C
1
a
nd
C
2
are f
unctio
n
s of the waveg
u
id
e param
eters but are in
depe
ndent o
f
r
.
Equation 3 sho
w
s that the attenuatio
n coeffi
cie
n
t incre
a
ses e
x
ponentially with decre
asin
g
bendi
ng ra
di
us; ho
weve
r, if the radiu
s
of
cu
rvature become la
rge
e
noug
h the
attenuati
on
become
s
neg
ligible. It also shows that the cha
nge of
with
r
is dominated by the value of
C
2
,
whi
c
h (fo
r
a wea
k
ly-g
uidin
g
guide
) is gi
ven by [11]:
2
2
3
2
)
2
(
2
n
n
C
eff
(4)
Her
e
n
eff
= n
eff
- n
2
, where
n
2
is the refractive index o
f
the claddin
g
.
The
C
2
val
u
e
also
provid
e
s
a m
e
thod f
o
r
cha
r
a
c
teri
sing m
ode
confineme
n
t, whi
c
h i
s
useful
whe
n
investigatin
g tech
niqu
es for redu
cing
be
nd losse
s
. Eq
uation 3 sho
w
s that
is also
affected (but less strongly)
by
the value of the coeffici
ent
C
1
, which
is defined a
s
[11]:
t
l
c
Z
C
'
1
2
1
(5)
whe
r
e the pa
rameters
Z
c
,
t
,
and
’
l
are given by:
2
2
)
2
cos(
2
2
h
h
n
Z
c
(6)
)
2
(
cos
)
sin(
2
1
2
2
h
h
h
t
(7)
and,
h
l
e
h
2
cos
2
2
'
(8)
whe
r
e
i
s
gi
ven by:
2
2
0
2
1
n
k
(9)
and
h
,
and
have their usual me
anin
g
s
[12].
The
C
1
co
efficient a
s
d
e
fin
ed in e
quatio
n 4 i
s
not a
d
i
rect fu
nctio
n
of
n
eff
, but is relate
d
to the difference bet
wee
n
the prop
agati
on con
s
t
ant
within the gui
de and the cl
addin
g
refra
c
tive
index
n
2
. Additionally, the
C
1
coeffici
en
t is
stron
g
ly
model
dep
en
dent, an
d so
ca
n be
u
s
ed
to
cal
c
ulate
the guide
shape
and other
pa
ra
mete
rs.
Th
e ab
ove form
ulation
of the
C
1
a
nd th
e
C
2
coeffici
ents
wa
s ba
sed o
n
the derivat
ion made
by
Marcatili et.al [10] and later ado
pted
by
Minford et.al
[11]. Howe
ver, two oth
e
r form
ulatio
ns have
bee
n provid
ed b
y
Lee [12] and
Marcu
s
e [13,
14]. Lee’s version of the
C
1
and the
C
2
coeffic
i
ent
s
is
as
follows
[12]:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Modellin
g Op
tical Wa
ve
gui
de Bend
s by t
he Method of
Line
s (Ary Syahria
r)
1493
h
o
e
h
h
n
k
C
)
2
(
cos
)
2
(
2
2
2
2
1
(10
)
and,
2
2
2
)
(
2
n
n
n
C
eff
(11
)
Her
e
,
k
o
,
and
h
hav
e their u
s
u
a
l
meanin
g
. In a simila
r way, Marcuse’
s version
of the
c
oeffic
i
ents
is as
follows
[14]:
)
exp(
)
(
)
2
(
2
2
2
2
2
1
2
2
1
h
k
n
n
h
C
o
(12
)
and,
2
3
2
3
2
C
(13
)
here, th
e p
r
opag
ation
co
nstant i
s
d
e
fined a
s
eff
o
n
k
. It might be
ex
pecte
d that t
hese
different form
ulation
s
wo
uld give simila
r results;
ho
we
ver, this wa
s
not found to b
e
the ca
se.
2.2. Method
of Lines
In this a
nalysis we b
egin
by con
s
id
eri
n
g the b
ehavi
our
of a g
u
id
ed mo
de a
s
i
t
travels
arou
nd a b
e
nd of co
nsta
nt curvatu
r
e.
Figur
e
2 shows a
sch
e
m
atic of the
geometry.
The
waveg
u
ide h
a
s a con
s
tant
radiu
s
of cu
rvature
r
, whi
c
h is mea
s
u
r
e
d
from the ce
ntre of the gu
ide.
The g
u
ide i
s
of width
h
, whi
c
h i
s
a
s
sume
d to b
e
much le
ss
than
r
and
i
s
centred o
n
a
comp
utationa
l windo
w of width
w
.
The core
a
nd clad
ding
refra
c
tive
indices are given
by
n
1
an
d
n
2
respec
tively.
Figure 2. Discreti
sation of
a plana
r wav
eguid
e
s b
end
s by the MoL
Assu
ming a
y
-p
ola
r
ised el
ectri
c
field, the Helm
holtz wave equati
on ca
n be written in
polar
co
-ordin
ates a
s
[15]:
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0-1499
1494
0
1
1
2
2
2
2
2
2
2
y
o
y
y
y
E
n
k
E
E
E
(14
)
To impleme
n
t
a numeri
c
a
l
solution of
equation 1
4
a modificati
on must b
e
made.
It involves ch
angin
g
the co
-ordinate
s
to
a local
co
-o
rd
inate sy
stem
that fo
llows t
he centre
of the
waveg
u
ide al
ong the pro
pagatio
n dire
ction [15].
This chan
ge
also allo
ws the po
ssi
bility of
analysi
ng the
field profile a
t
the local cro
s
s-sectio
n. By making the
sub
s
titution:
r
s
r
x
(15
)
Equation 14
may be tran
sforme
d to:
0
)
(
)
1
(
)
1
(
)
1
(
2
2
0
2
2
2
2
2
2
y
y
y
y
E
x
n
k
cx
x
E
cx
c
x
E
cx
s
E
(16
)
whe
r
e
the
co
nstant
c = 1/r
re
pre
s
e
n
ts the
waveg
u
id
e curvatu
r
e.
Note th
at it i
s
ea
sy to
se
e
ho
w
equatio
n 1
6
redu
ce
s to th
e
scala
r
wave
equatio
n for
a st
raight
wa
veguide
whe
n
c
=0. The
r
e are
some
adva
n
tage
s u
s
ing
equatio
n 16.
Firstly, t
he
comp
utationa
l win
dow ca
n be
re
stri
ct
ed
becau
se the
centre i
s
alo
n
g
the p
a
th of
the wa
ve
guid
e
. Seco
ndly, the ind
e
x profiles n
eed
not
be
altered
as th
e radi
us
of curvature cha
nge
s, in
cont
rast to the
other m
e
thod
s whi
c
h u
s
e t
he
modified in
de
x profile [15].
To solve eq
uation 1
6
by
the MoL, the
equatio
n is
now
discretised
usin
g the finite differen
c
e o
perato
r
, by putting:
2
1
1
2
2
)
(
2
x
E
E
E
x
E
i
i
i
y
(17
)
and:
x
E
E
x
E
i
i
y
2
1
1
(18
)
If this is done
, equation 16
can b
e
writte
n in matrix form as:
0
2
2
E
Q
ds
E
d
(19
)
whe
r
e
=
[E
1
, E
2
, E
3
,
...
...
,
E
N
]
t
is a colu
mn vector
co
ntaining discretise
d
value
s
of the field
E
y
(x),
at the p
o
ints
x
1
, x
2
, ...
. x
N
, and
Q
is a tri-dia
gon
al matrix defined
by :
2
2
2
2
2
2
2
1
0
2
2
1
2
2
2
2
2
2
2
2
2
1
2
1
2
2
)
1
(
0
0
0
..
..
..
..
0
0
)
1
(
0
0
0
0
)
1
(
0
)
1
(
0
0
..
..
..
..
)
1
(
0
)
1
(
0
0
)
1
(
0
2
1
)
1
(
2
)
1
(
0
0
..
..
..
..
0
)
1
(
)
1
(
2
)
1
(
0
0
)
1
(
)
1
(
2
)
(
1
n
cx
n
cx
n
cx
k
cx
c
cx
c
cx
c
cx
c
x
cx
cx
cx
cx
cx
cx
cx
x
Q
N
N
N
N
(20
)
E
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TELKOM
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ISSN:
1693-6
930
Modellin
g Op
tical Wa
ve
gui
de Bend
s by t
he Method of
Line
s (Ary Syahria
r)
1495
Assu
ming tha
t
there is no
back-refle
c
tio
n
, the gener
a
l
solution for
con
s
tant matrix elements h
a
s
the form:
inp
s
j
E
T
e
T
E
1
(21
)
whe
r
e
is a matrix contai
ning the eig
e
n
vectors of
arrang
ed in
column
s,
is a
diagon
al
matrix co
ntai
ning the
eig
envalue
s of
, and
inp
E
is the
input field ve
ctor. O
ne
of the mo
st
importa
nt parameters a
s
so
ciated
with th
e waveg
u
ide
is the fra
c
tion
al power that
remain
s in t
he
core at point
z
. This p
o
we
r is approxima
t
ely given by the overla
p integral:
2
)
,
(
)
0
,
(
)
(
dx
z
x
E
x
E
z
P
(22
)
whe
r
e
E(x
,
0
)
is the input field and
E(x
,
z)
is the field a
t
point
z
.
2.3. The Abs
o
rbing Bou
n
dar
y
Condition
To cal
c
ul
ate the modal fie
l
d of the cu
rved wa
veg
u
i
de, it is necessary to restrict the
extent of the co
mputatio
nal
wi
ndo
w.
On
ce a
gain
,
this is don
e usi
ng
ab
sorbin
g b
oun
dary
con
d
ition
s
. Derivation of a
ppro
p
ri
ate bo
unda
ry co
ndi
t
i
ons i
n
pola
r
co-ordinate i
s
gene
rally rather
compli
cate
d
[16]. In the cal
c
ulatio
n d
e
scrib
ed h
e
re, we h
a
ve adopte
d
a si
mpler
app
ro
ach,
applying a
straight gui
de
boun
dary co
ndition to the
curved
wav
eguid
e
equat
ion, by using
the
assumptio
n
t
hat the
radi
us of curvatu
r
e
is la
rg
e
en
ou
gh that the
m
ode in
sid
e
th
e be
nd i
s
sim
ilar
to that of straight guide.
The ab
so
rbi
n
g bou
nda
ry condition i
s
in
serted
into th
e
edge
of the
matrix com
p
o
nents
of
equatio
n 20
[16]. In this
ca
se,
we h
a
ve
use
d
the
more effective thi
r
d-ord
e
r
ab
so
rbing
bo
unda
ry
con
d
ition, wh
ere the r
adi
ca
l is approxim
ated by:
2
2
0
2
2
0
2
1
S
q
q
S
p
p
S
(23
)
Differen
c
e
s
i
n
the
ch
oice
of the
co
efficient
s,
p
and
q,
produces different f
a
milies
of
absorbi
ng bo
unda
ry
condit
i
ons. The
s
e result
i
n
diffe
rences in th
e
angle
of exa
c
t ab
so
rption
of
the incomi
ng
wave by the abso
r
bi
ng boun
dary lay
e
r. Table 1 shows a list o
f
the coefficient
values an
d a
b
so
rption
an
gles of the
a
pproxim
ation
s
that
are
m
o
st
comm
onl
y use
d
. Here
, we
have used th
e
L
type of a
pproxim
ation.
Table 1. Co
efficients fo
r different third
-
o
r
der ab
so
rbi
n
g
bound
ary co
ndition
s,
after reference [4.16]
T
y
pe of
app
roxi
mation
p
0
p
2
q
2
angle of exact a
b
s
orption (
o
)
Pade’ 1.0000
-0.7500
-0.2500
0.0
Cheb
y
s
hev
L
0.9997
-0.8086
-0.3165
11.7, 31.9, 43
.5
Cheb
y
s
hev point
s
0.9965
-0.9129
-0.4725
15.0, 45.0, 75
.0
Least square (
L
2
)
0.9925
-0.9223
-0.5108
18.4, 51.3, 76
.6
Cheb
y
s
hev-P
ade
’ (C-P)
0.9903
-0.9431
-0.5556
18.4, 53.1, 81
.2
Ne
w
m
an points
1.0000
-1.0000
0.6697
0.0, 60.5, 90.
0
Cheb
y
s
hev
L
0.9565
-0.9435
0.7038
26.9, 66.6, 87
.0
q
0
= 1.000
0 for each app
rox
i
mation
3. Results a
nd Analy
s
is
To get a better und
erstan
ding on ho
w the gui
de
d mo
de evolution
durin
g its pro
pagatio
n
in
s
i
de
c
u
r
v
e
d
w
a
veg
u
i
de
s
;
w
e
ha
ve
us
ed
tw
o me
tho
d
s, i.e. a
n
a
nal
ytical theo
ry t
hat is a
s
sum
ed
to be the
ri
ght app
ro
ach and
the
method
of li
nes. In
both
cal
c
ulatio
ns paramete
r
s of a
waveg
u
ide b
end
s with
a radiu
s
50
00
m thro
ugh
an
angl
e of 4
5
o
with
different refrac
tive
index
T
Q
Q
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93-6
930
TELKOM
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Vol. 16, No. 4, August 2018: 149
0-1499
1496
cha
nge
s
hav
e be
en
used.
Figu
re
3
sho
w
s the va
riati
on of
C
2
cal
c
ulated usi
ng equatio
n
4
wi
th
wavele
ngth for differe
nt values of
n
, i.e. for different degree
s of waveguid
e
co
n
f
inement. The
s
e
results sh
ow
that
C
2
i
n
creases a
s
the
co
nfineme
n
t
be
come
s
hi
gher an
d al
so that
C
2
val
ues
gene
rally de
crea
se sl
owly
at long wavel
ength
s
.
Figure 3. Vari
ation of the param
eter
C
2
with wavelength, for different values
of
n
as p
r
edi
cted
by equation 4
Figure 4 (a
)
and (b)
sho
w
a comp
ari
s
o
n
of the
C
1
a
nd the
C
2
val
ues a
s
a fun
c
tion of
waveguide width, cal
c
ul
ated
by
Marcatili’
s
approximation
of equation 4 and 5;
Lee’s
approximatio
n of equatio
n 10 and 11
; and Marcu
s
e’
s app
roxi
mation of eq
uation 12 an
d 13
respec
tively. Here, the parameters
of
n
1
=
1
.463,
n
2
=14
58,
=1.5
25
m, with
h
varying fro
m
4
m to 7
m, have been u
s
ed.
(a)
(b)
Figure 4. A compa
r
ison of different app
roximations fo
r (a
) the
C
1
value, and (b)
the
C
2
value
Figure 4 (a)
sho
w
s that the
C
1
coeffici
e
n
ts of Lee’
s and Ma
rcuse’s expre
s
sion
are in a
good
ag
reem
ent. Ho
weve
r, Marcatili’s
equatio
n giv
e
s m
u
ch lo
wer valu
e. Thi
s
mig
h
t well
be
becau
se of
the differe
nt app
roa
c
h
e
s u
s
e
d
to
derive th
e
C
1
coeffici
ent. The M
a
rcatili
approximatio
n is obtaine
d from the compl
e
x
sol
u
tion of the eigenvalue
equation of
the
waveg
u
ide
b
end, while
bo
th Lee
and
M
a
rcuse u
s
e
a
different a
pproximation b
a
sed
on th
e lo
cal
rate of
po
we
r radi
ation f
r
o
m
the b
end.
Furthe
rmo
r
e,
for the
C
2
coefficient, Lee’
s and M
a
rcat
ili’s
approximatio
ns
give a
go
od ag
re
emen
t, while M
a
rcuse’
s
a
p
p
r
oa
ch gives app
arently
in
co
rrect
values. It
ca
n be
con
c
lu
ded th
at Le
e’s
expre
s
si
on i
s
the
m
o
st li
kely to
be
co
rrect.
This
assumptio
n
will be validated later by co
mpar
i
s
o
n
wit
h
the rigo
rou
s
method of lines.
0
0.
0
005
0
.
001
0.
0
015
0
.
002
0.
0
025
0
.
003
1
.
3
1
.3
5
1
.4
1
.
4
5
1
.
5
1
.5
5
1
.6
(
m)
C
2
(
m
-1
)
n
=
0
.
007
n
=0
.
0
0
6
n
=0
.
0
0
5
0
10000
20000
30000
40000
50000
60000
70000
80000
44
.
5
55
.
5
66
.
5
77
.
5
Wi
d
t
h (
m)
C
1
(m
-1
) .
M
a
r
c
a
t
ili
Le
e
Ma
rc
u
s
e
0
500
1000
1500
2000
2500
44
.
555
.
5
66
.
5
77
.
5
Wi
d
t
h
(
m)
C
2
(m
-1
) .
M
a
r
cat
i
l
i
Le
e
Ma
r
c
u
s
e
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Modellin
g Op
tical Wa
ve
gui
de Bend
s by t
he Method of
Line
s (Ary Syahria
r)
1497
To illust
rate t
he way in wh
ich the
guid
e
d
mode
evolves in
a curve
d
wave
guid
e
, we first
comp
are the
input
and
o
u
tput mod
e
shape
s
obtai
n
ed afte
r trav
elling
rou
nd
a be
nd
of ra
diu
s
5000
m thro
ugh a
n
a
ngle
of 45
o
. Fig
u
re 5
(a) sho
w
s re
sults obtai
ned fo
r a
gui
de of
co
re in
dex
1.464,
claddi
ng ind
e
x 1.4
58 an
d
width
5
m at a wavelength
of
1.525
m, w
h
ile Fig
u
re
5
(b
)
sho
w
s re
sult
s for a si
mila
r but less st
rongly co
nfini
ng guid
e
whi
c
h ha
s a
core index of 1.463.
The cal
c
ul
ations h
a
ve bee
n done by u
s
i
ng the MOL
schem
e.
(a)
(b)
Figure 5. Input and output
field distrib
u
tion of
the fund
amental mo
d
e
after travelli
ng aro
und a
waveg
u
ide b
end of radi
us
5000
m thro
ugh an a
ngle
of 45
0
. (a)
n
= 0.006, (b)
n
= 0.005
Figure 5 de
monst
r
ate
s
that the outpu
t field
profile of the mode gene
rally extend
s into
the claddi
ng
and its pea
k i
s
red
u
ced,
so
that it
is gra
dually r
adi
ating power. Th
e amount of the
power lo
ss d
epen
ds o
n
the degree of confineme
n
t.
For exampl
e, in Figure 5
(
a
)
, the output field
extends i
n
to t
he cl
addi
ng o
n
ly to a very l
i
mit
ed extent, and th
e inp
u
t and the
out
put field
sha
p
e
s
are ve
ry simil
a
r. Ho
weve
r,
in Figu
re 5
(
b
)
the
output fi
eld extend
s
much fu
rthe
r
into the cl
add
ing
due to the re
ductio
n
in co
nfinement. T
he deg
re
e of asymmetry
also in
crea
se
s con
s
ide
r
abl
y as
the confin
em
ent is red
u
ce
d.
We no
w use
the result
s of the method of li
nes ca
lculatio
n to estimate an e
ffective
attenuation
coefficient al
o
ng a
uniformly cu
rv
ed
waveg
u
ide.
This
ca
n b
e
done
by u
s
ing
equatio
n 1,
whe
r
e th
e
ds
s
dP
)
(
value
s
a
r
e fou
nd by eval
ua
ting the diffe
rence in th
e i
n
tegrate
d
optical po
we
r acro
ss the
mode in the com
putat
ional area b
e
twee
n two
adjacent axial
prop
agatio
n steps. Figu
re
6 sho
w
s the
attenuation
coe
fficient fou
nd in this
wa
y as a functio
n
of
bendi
ng angl
e
, for the parameters
n=
0.005, 0.006,
a
nd 0
.
007,
n
2
=1.458,
=1.5
25
m,
h
=5
m, and
r
=5
000
m.
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
-2
0
-
1
0
0
1
0
2
0
L
a
t
e
r
a
l
pos
i
t
i
on
x (
m)
Transverse Field amp
litude (
a
r
b
. unit)
i
nput
f
i
e
l
d
out
put
f
i
e
l
d
n
=0.
006
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-2
0
-
1
0
0
1
0
2
0
L
a
t
e
r
a
l
po
s
i
t
i
o
n
x (
m)
Transverse field amplitude (arb.
unit)
in
p
u
t
f
i
e
l
d
ou
t
p
u
t
f
i
e
l
d
n
=0
.
0
05
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93-6
930
TELKOM
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Vol. 16, No. 4, August 2018: 149
0-1499
1498
Figure 6. Attenuation
coeffi
cient a
s
a fun
c
tion of the b
endin
g
angle
, for a different
degree
s of co
nfinement
In each
ca
se,
the value of
is not con
s
tant, but rise
s
grad
ua
lly fro
m
zero at the start of
the ben
d a
n
d
settle
s
to
a stea
dy-state value
only
after
some
rapi
d fluctu
ations. In
ea
rly
analyses, the
fluctuation
s
have bee
n a
s
so
ciated
with
transitio
n lo
ss [17-18]. Ho
wever, re
cently
it
wa
s d
e
mon
s
t
r
ated
that the
y
are
me
rely
a math
emat
ical a
r
tefact
wh
ich i
s
i
nhe
ren
t
in nu
meri
ca
l
modellin
g of bend
s u
s
ing
beam p
r
opa
g
a
tion method
s, and the ste
ady-state val
ue is an a
c
cu
rate
estimate of th
e attenuation
coeffici
ent after the
mod
e
has
settled to
its final lateral positio
n.
A comp
ari
s
o
n
of the atten
uation
coefficients
p
r
edi
cte
d
by simpl
e
theory a
nd th
e MoL
(at
large axial di
stan
ce) i
s
sh
own in Fig
u
re
8. In the analytical ap
proximations, the
C
1
and th
e
C
2
coeffici
ents n
eede
d to find the
valu
es have b
e
e
n
cal
c
ulate
d
by using e
a
ch of the t
h
ree
approximatio
ns.
Figure 8. A compa
r
ison of the attenuatio
n coeffi
ci
ents
obtaine
d from
the MoL and
from the thre
e
different anal
ytical expre
s
sions
Figure 8 d
e
m
onstrates that
good
agree
ment is
obtai
ned b
e
twe
en
the analytical
form for
the lo
ss coef
ficient b
a
sed
on
Lee’
s
expre
ssi
on
and
the M
oL
cal
c
ulatio
n. A
slight differe
nce,
however, occurs
at a smal
l
radius
. In
contrast, Marcatili’s formul
ation predi
cts
a very l
o
w value
of
wh
en
compa
r
ed to
Lee’
s a
pproximation a
n
d
the Mo
L
cal
c
ulatio
n, while
Marcu
s
e’
s
formulatio
n predict
s much highe
r value
s
.
The
re
sidual
discre
pan
cie
s
between th
e
predi
ction
s
of Lee’
s the
o
ry
and th
e Mo
L
may be
explained
a
s
follows. In
the M
oL, the
cal
c
ul
ation
result
s a
r
e
hi
ghly de
pen
d
ent on
a
pro
per
choi
ce of ab
sorbin
g bou
nd
ary con
d
ition
at the
edge o
f
the comput
ational win
d
o
w
s, so that un
-
suitabl
e cond
itions give
rise to sig
n
ifica
n
t re
fle
c
tion
back into th
e
com
putation
a
l win
d
o
w
an
d
hen
ce lo
wer
appa
rent lo
ss. A similar effect al
so app
e
a
rs in a
nothe
r numeri
c
al
scheme [19].
0
0
.
00
005
0.
0
001
0
.
00
015
0.
0
002
0
.
00
025
0.
0
003
0
.
00
035
0.
0
004
0
.
00
045
0
0
.
1
0.
2
0
.
3
0.
4
0
.
5
0.
6
0
.
7
0.
8
=s/
r
(
m
-1
) .
n
=0.
005
n
=0.
006
n
=0.
007
0
0.
0
0
5
0.
0
1
0.
0
1
5
0.
0
2
0.
0
2
5
100
0
200
0
300
0
400
0
500
0
6
000
R
a
d
i
us
of
c
u
r
v
a
t
u
r
e
(
m)
(
m
-1
) .
N
u
m
e
ri
c
a
l
(M
o
L
)
A
n
a
l
y
t
i
c
(
M
a
r
c
a
t
ili'
s
)
A
n
al
y
t
i
c
(
L
ee
'
s
)
A
n
al
y
t
i
c
(
M
a
r
cu
s
e
'
s
)
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TELKOM
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ISSN:
1693-6
930
Modellin
g Op
tical Wa
ve
gui
de Bend
s by t
he Method of
Line
s (Ary Syahria
r)
1499
4. Conclusio
n
We h
a
ve inv
e
stigate
d
different a
nalytic approx
imati
ons to th
e lo
cal lo
ss coefficient in
waveg
u
ide
b
end
s b
a
sed
on
con
s
tant
radiu
s
of
curv
ature,
and
h
a
ve un
cove
red di
sa
gree
ment
betwe
en
sev
e
ral p
r
evio
usly publis
hed analytical expre
ssi
on
s.
T
o
verify
the accuracy of
the
attenuation
coefficient o
n
several pu
b
lish a
nalytica
l
expre
ssi
on,
we h
a
ve u
s
ed th
e be
am
prop
agatio
n algorith
m
ba
sed on
th
e method of
li
nes i
n
pol
ar co
-ordinate.
We
have f
ound
rea
s
on
able a
g
ree
m
ent wit
h
the analytic app
roximati
on to the local loss
coeffi
cient ba
sed
on
Lee’
s ap
proa
ch. Thi
s
agre
e
ment mi
ght
be u
s
ed
to
e
x
tend the
cal
c
ulatio
n of lo
ss in
waveg
u
i
de
bend
s
stru
ctu
r
e in m
odelli
n
g
co
ntinuo
usl
y
-varyi
ng S-b
end
s wavegu
ides
usi
ng
ca
scade
d secti
on
method.
Re
sidual
disa
gre
e
ment i
s
a
s
crib
ed
ma
inl
y
to the m
o
derate
pe
rformance of th
e
absorbi
ng bo
unda
ry con
d
itions u
s
e
d
to lim
it the range
of the calcula
t
ion wind
ow.
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ces
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Chu, HA Ha
us
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o
resi, JP Lain
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h
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e
chno
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ulz, T
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