TELK
OMNIKA
,
V
ol.
15,
No
.
3,
September
2017,
pp
.
1461
1469
ISSN:
1693-6930,
accredited
A
b
y
DIKTI,
Decree
No:
58/DIKTI/K
ep/2013
DOI:
10.12928/telk
omnika.v15.i3.7215
1461
Numerical
Sim
ulation
of
Highl
y-Nonlinear
Disper
sion-Shifted
Fiber
Optical
P
arametric
Gain
Spectrum
with
Fiber
Loss
and
Higher
-Or
der
Disper
sion
Coefficients
K.G.T
a
y
*
,
N.
Othman
,
N.S.M.
Sha
h
,
and
N.A.
Cholan
F
aculty
of
Electr
ical
and
Electronics
Engineer
ing,
Univ
ersiti
T
un
Hussein
Onn
Mala
ysia,
86400
Batu
P
ahat,
Mala
ysia.
*
K.G.T
a
y
,
e-mail:
ta
y@uthm.edu.m
y
Abstract
The
pre
vious
study
in
v
estigated
the
fiber
par
ameters
on
the
analytical
one-pump
fiber
optical
par
a-
metr
ic
amplifier
(FOP
A)
gain
spectr
um
of
a
loss-free
highly-nonlinear
dispersion-shifted
fiber
(H
NL-DSF)
and
got
the
optim
um
results
f
or
each
par
ameter
.
Ho
w
e
v
er
,
the
FOP
A
gain
of
th
e
combination
of
all
the
optim
um
v
alues
of
the
considered
par
ameters
w
as
not
repor
ted.
Hence
,
this
paper
intends
to
in
v
estigate
the
analyti-
cal
FOP
A
gain
of
the
combination
of
all
the
opt
im
um
v
alues
of
the
considered
par
am
eters
from
the
pre
vious
study
.
Later
,
the
analytical
gain
w
as
compared
with
the
n
umer
ical
gain
from
the
f
our
th-order
Runge-K
utta
(RK4)
method
and
Matlab
b
uilt-in
function,
ode45.
Ne
xt,
the
eff
ect
of
fiber
loss
and
higher
order
dispersion
coefficients
such
as
f
our
th
and
sixth-order
dispersion
coefficients
w
ere
studied.
It
is
f
ound
that
RK4
giv
es
a
smaller
error
an
d
the
gain
reduces
while
the
bandwidth
remains
same
in
presence
of
fiber
loss
.
The
f
our
th-order
dispersion
coefficient
broadens
the
bandwidth
a
bit
while
maintaining
the
gain
and
there
are
tw
o
narro
w-band
gains
gener
ated
to
the
left
and
r
ight-side
of
the
broad-band
gain.
The
sixth-order
dispersion
coefficient
just
shifts
the
tw
o
narro
w-band
gains
to
w
ard
or
a
w
a
y
from
the
broadba
nd
gain
depending
on
the
positiv
e
or
negativ
e
signs
of
the
sixth-order
dispersion
coefficient.
K
e
yw
or
ds:
HNL-DSF
,
FOP
A,
RK4,
ode45,
higher-order
dispersion
coefficients
Cop
yright
c
2017
Univer
sitas
Ahmad
Dahlan.
All
rights
reser
ved.
1.
Intr
oduction
Optical
comm
unication
is
used
in
handling
the
e
v
er
increasing
demand
of
Inter
net
tr
affic.
The
propagation
of
an
optical
pulse
through
optical
fiber
at
long
haul
distance
will
e
xper
ience
the
deca
y
in
amplitude
because
the
nonlinear
ity
cannot
compensate
the
dispersion
due
to
the
fiber
loss
.
Hence
,
the
optical
pulse
needs
to
be
amplified.
The
commonly
used
amplifier
is
Erbium-
doped
fiber
amplifier
(EDF
A)
which
w
as
in
v
ented
in
1980.
It
amplifies
light
in
C-band
(w
a
v
elength1
r
anges
1530nm-1565nm)
where
telecomm
unication
fibers
ha
v
e
a
minim
um
loss
.
Ho
w
e
v
er
,
EDF
A
adds
noise
and
nonlinear
ity
to
the
amplified
signal.
Another
f
amily
of
the
optical
amplifier
is
Raman
amplifier
(RA)
which
uses
stim
ulated
Raman
scatter
ing
(SRS)
to
tr
ansf
er
energy
from
one
or
more
pump
sources
to
the
tr
ansmitted
optical
signal.
Although,
Raman
amplification
can
oper
ate
at
an
y
w
a
v
elength
depending
on
the
pump
w
a
v
elength,
b
ut,
it
n
eeds
high
pump
po
w
er
and
has
prob
lems
with
doub
le
Ra
yleigh
bac
kscatter
ing
(DRB)
which
creates
noise
.
Fiber
optical
par
ametr
ic
amplifier
(FOP
A)
is
oper
ated
based
on
f
our-w
a
v
e
mixing
(FWM)
process
.
FWM
occurs
when
a
signal
light
with
angular
frequency
!
s
is
launched
in
an
optical
fiber
along
with
tw
o
strong
pumps
with
angular
frequencies
!
p
1
and
!
p
2
.
At
the
end
of
the
fiber
,
the
sig-
nal
is
amplified
and
an
idler
w
a
v
e
is
gener
ated
with
angular
frequency
!
i
through
phase-matching
condition,
!
i
=
!
p
1
+
!
p
2
!
s
.
When
the
frequencies
of
these
tw
o
pumping
w
a
v
es
are
identical,
the
”degener
ated
f
our-w
a
v
e
mixing”
(DFWM)
occurs
such
that
!
i
=
2
!
p
!
s
.
FOP
As
can
be
classified
into
one-pump
(1-P)
where
a
single
pump
w
a
v
e
is
used
in
fiber
and
into
tw
o
pump
(2-P)
Receiv
ed
March
16,
2017;
Re
vised
J
uly
4,
2017;
Accepted
J
uly
30,
2017
Evaluation Warning : The document was created with Spire.PDF for Python.
1462
ISSN:
1693-6930
in
which
tw
o
pump
w
a
v
es
are
applied
into
the
fiber
.
FOP
A
can
perf
or
m
similar
functions
compar
a-
b
le
to
e
xisting
amplifiers
on
top
of
remar
kab
le
f
eatures
that
are
not
off
ered
b
y
e
xisting
amplifiers
such
as
wide
bandwidth
[1],
adjustab
le
gain
spectr
a,
adjustab
le
center
frequency
,
w
a
v
elength
con
v
ersion,
phase
conjugation,
pulsed
oper
ation
f
or
signal
processing
and
0-dB
noise
figure
[2].
The
de
v
elopment
of
a
dispersion-shifted
fiber
(DSF)
with
z
ero-dispersion
w
a
v
elength
(ZD
W)
located
in
the
C-band
[3]
and
the
same
of
a
silica-based
highly-nonlinear
DSF
(HNL-DSF)
[4]
has
spar
k
ed
the
researches
of
FOP
A.
A
good
FOP
A
needs
to
pro
vide
broad
bandwidth
and
high
gain.
The
perf
or
mance
of
a
FOP
A
is
dependent
on
fiber
par
ameters
.
The
influence
of
fiber
par
ameters
such
as
fiber
length,
pump
po
w
er
and
dispersion
order
to
w
ards
a
host
lead-silicate
baased
binar
y
ulti-clad
microstr
uctur
e
fiber
w
as
analyz
ed
b
y
[5].
Cheng
et.
al
[6]
studied
the
eff
ect
of
pump
w
a
v
elengths
and
po
w
ers
on
dual-pump
configur
ation
of
FWM
on
highly
nonlinear
tellur
ite
fibers
with
tailored
chromatic
dispersion.
Maji
and
Chaudhur
i
[7]
in
v
estigated
fiber
par
am-
eters
such
as
fiber
length,
pump
po
w
er
and
pump
w
a
v
elength
to
w
ards
the
analytical
FOP
A
gains
of
three
ZD
W
in
the
near-z
ero
ultr
a-flat
photonic
cr
ystal
fibers
(PCF)
around
the
comm
unication
w
a
v
elength
b
y
tuning
pumping
condition.
A
study
in
[8]
concluded
that
b
y
cha
nging
temper
ature
of
the
fiber
and
tuning
the
chromatic
dispersion
profile
and
ZD
W
of
the
optimiz
ed
PCF
,
then
a
wide
gain
spectr
um
in
the
comm
unication
w
a
v
elength
can
be
achie
v
ed
.
Othman
et.al
[9]
studied
the
influence
of
fiber
par
ameters
lik
e
fiber
length,
pump
po
w
er
,
pump
w
a
v
elength
and
dispersion
slope
on
the
analytical
FOP
A
gain
of
a
single-pump
loss-free
HNL-DSF
.
The
y
sho
w
ed
that
the
long
fiber
length,
the
high
pump
po
w
er
,
the
pump
w
a
v
elength
close
to
ZD
W
and
the
small
dispersion
slope
giv
e
a
wider
bandwidth
and
a
high
gain.
Ho
w
e
v
er
,
the
y
did
not
plot
the
combination
of
the
optim
um
cases
f
or
each
o
f
the
abo
v
e-mentioned
par
ame-
ters
and
the
y
considered
HNL-DSF
as
lossless
.
P
akarzadeh
and
Bagher
i
[10]
in
v
estigated
the
gain
spectr
um
and
sat
ur
ation
beha
vior
of
one-pump
FOP
A
in
the
presence
of
the
f
our
th-order
dispersion
coefficient.
The
y
concluded
that
when
the
pump
w
a
v
elength
is
near
to
or
e
xactly
at
ZD
W
and
when
the
diff
erence
betw
een
the
signal
and
pump
w
a
v
es
is
large
enough,
the
f
our
th-order
coefficient
4
is
impor
tant
to
be
tak
en
into
account.
Dainese
[11]
presented
a
scheme
to
optimiz
e
the
PCF
chromatic
dispersion
cur
v
e
and
higher
order
dispersion
coefficients
up
to
14
f
or
a
broadband
FOP
A.
Motiv
ated
with
the
w
or
k
of
[9]
and
the
impor
tance
to
include
higher
order
dispersion
when
the
pump
w
a
v
elength
close
to
or
e
xactly
at
ZD
W
and
when
the
diff
erence
betw
een
the
signal
and
pump
w
a
v
es
is
large
enough.
Hence
,
hence
in
this
paper
,
the
optim
um
combination
from
[9]
is
utiliz
ed
and
the
an
alytical
gain
is
compared
with
the
f
our
th-order
Runge-K
utta
(RK4)
method
and
with
Matlab
b
uilt-in
ordinar
y
dif
f
er
ential
equation
(ode)
function,
which
is
ode45.
Pr
actically
,
fiber
is
not
loss-free
f
or
which
the
fiber
loss
is
added
to
the
optim
um
combination
case
and
the
prob
lem
is
solv
e
d
b
y
using
the
RK4
method.
Lastly
,
the
higher
dispersion
coefficient
s
4
and
6
are
added
slo
wly
to
see
their
eff
ects
on
the
FOP
A
perf
or
mance
.
2.
Mathematical
Model
The
three
coupled
amp
litude
equations
f
or
pump
A
p
,
signal
A
s
and
idler
A
i
that
descr
ibe
the
propagation
of
three
inter
acting
w
a
v
es
in
1-P
FOP
A
are
giv
en
b
y
[12]
as
dA
p
dz
=
i
A
p
j
A
p
j
2
+
2(
j
A
s
j
2
+
j
A
i
j
2
)
+
2
A
s
A
i
A
p
exp
(
i
4
z
)
1
2
A
p
=
f
1
(
z
;
A
p
;
A
s
;
A
i
)
;
(1)
dA
s
dz
=
i
A
s
j
A
s
j
2
+
2(
j
A
p
j
2
+
j
A
i
j
2
)
+
A
i
A
2
p
exp
(
i
4
z
)
1
2
A
s
=
f
2
(
z
;
A
p
;
A
s
;
A
i
)
;
(2)
dA
i
dz
=
i
A
i
j
A
i
j
2
+
2(
j
A
p
j
2
+
j
A
s
j
2
)
+
A
s
A
2
p
exp
(
i
4
z
)
1
2
A
i
=
f
3
(
z
;
A
p
;
A
s
;
A
i
)
:
(3)
The
first
ter
m
of
the
r
ight-hand
side
of
the
equations
represents
the
self-phase
modulation
(SPM),
whereas
the
second
and
third
ter
ms
are
cross-phase
mod
ulation
(XPM).
The
f
our
th
ter
m
acts
as
po
w
er
tr
ansf
er
due
to
FWM
and
fiber
atten
uation
is
represented
at
the
last
ter
m.
Here
is
the
fiber
loss
,
is
the
nonlinear
par
ameter
and
represents
comple
x
conjugate
.
Meanwhile
,
4
is
the
linea
r
w
a
v
e
v
ector
mismatch
betw
een
the
inter
acting
w
a
v
es
of
pump
,
signal
TELK
OMNIKA
V
ol.
15,
No
.
3,
September
2017
:
1461
1469
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
1693-6930
1463
and
idler
as
giv
en
b
y
[12]
4
=
2
1
X
m
=1
2
m
(2
m
)!
(
4
!
)
2
m
;
(4)
where
4
!
=
!
s
!
p
is
the
frequency
detuning,
!
k
=
2
c=
k
f
or
k
2
f
p;
s
g
are
the
frequencies
of
pump
and
signal
in
ter
ms
of
w
a
v
elength,
k
.
The
dispersion
coefficients
of
odd
order
do
not
contr
ib
ute
to
the
g
ain
spectr
um
and
hence
only
e
v
en
orde
r
coefficients
are
tak
en
into
account
[12].
By
taking
m
=
1
;
2
and
3
in
e
xpression
(4),
the
linear
w
a
v
e
v
ector
mismatch
up
to
the
sixth
order
dispersion
coefficient
6
is
e
xpressed
as
4
=
2
(
4
!
)
2
+
4
12
(
4
!
)
4
+
6
360
(
4
!
)
6
:
(5)
F
or
m
=
1
;
2
and
3
,
2
;
4
and
6
are
illustr
ated
respectiv
ely
b
y
[5]
as
2
m
=
1
X
n
=2
m
n
0
(
n
2
m
)!
(
!
p
!
0
)
n
2
m
;
(6)
where
n
0
=
d
n
(
!
)
=d!
n
j
!
=
!
0
is
the
dispersion
coefficients
calculated
at
the
z
ero
dispersion
frequency
,
!
0
.
By
combining
(5)
and
(6)
f
or
m
=
1
to
3,
the
4
can
be
wr
itten
as
4
=
2
(
4
!
)
2
+
4
3
2(
!
p
!
0
)
2
+
1
4
(
4
!
)
2
(
4
!
)
2
+
6
24
(
!
p
!
0
)
4
+
1
15
(
4
!
)
4
(
4
!
)
2
:
(7)
Here
2
,
4
and
6
are
the
second,
f
our
th
and
sixth-order
dispersion
constants
.
Eq.
(7)
sho
ws
the
significance
of
4
and
6
when
pump
w
a
v
elength
approaches
to
or
precisely
at
ZD
W
and
4
!
is
large
.
If
the
fiber
loss
is
negligib
le
in
Eqs
(1)-(3),
the
analytical
par
ametr
ic
gain
is
wr
itten
as
[13]
G
=
1
+
P
p
g
sinh(
g
L
)
2
;
(8)
with
L
as
fiber
length,
P
p
as
pump
po
w
er
and
g
is
e
xpressed
as
g
=
r
(
P
p
)
2
2
2
(9)
and
the
total
phase-mismatch,
is
indicated
b
y
=
4
+
2
P
p
:
(10)
3.
Four
th-Or
der
Rung
e-K
utta
(RK4)
Method
If
the
fiber
loss
is
included,
the
Eqs
.
(1)-(3)
cannot
be
solv
ed
analytically
to
obtain
the
analytical
gain.
Hence
the
spatial
v
ar
iab
le
z
in
Eqs
.
(1)-(3)
is
made
discrete
into
n
segments
with
a
step
siz
e
h
such
that
z
0
=
0
,
z
1
=
z
0
+
h
,
z
2
=
z
1
+
h
,.
.
.
,
z
n
=
z
n
1
+
h
=
L
.
With
j
=
0
and
initial
amplitudes
f
or
pump
A
0
p
,
signal
A
0
s
and
idler
A
0
i
at
z
0
=
0
,
the
amplitudes
f
or
pump
A
1
p
,
signal
A
1
s
and
idler
A
1
i
at
ne
xt
spatial
z
1
are
giv
en
b
y
the
RK4
method
as
f
ollo
ws:
A
j
+1
p
=
A
j
p
+
k
1
+
2
k
2
+
2
k
3
+
k
4
6
;
(11)
A
j
+1
s
=
A
j
s
+
l
1
+
2
l
2
+
2
l
3
+
l
4
6
;
(12)
A
j
+1
i
=
A
j
i
+
m
1
+
2
m
2
+
2
m
3
+
m
4
6
;
f
or
j
=
0
;
1
;
::n
(13)
Numer
ical
Sim
ulation
of
Highly-Nonlinear
Dispersion-Shifted
Fiber
...
(K.G.
T
a
y)
Evaluation Warning : The document was created with Spire.PDF for Python.
1464
ISSN:
1693-6930
Where
k
1
=
hf
1
(
z
;
A
p
;
A
s
;
A
i
)
;
k
2
=
hf
1
z
+
h
2
;
A
p
+
k
1
2
;
A
s
+
l
1
2
;
A
i
+
m
1
2
;
k
3
=
hf
1
z
+
h
2
;
A
p
+
k
2
2
;
A
s
+
l
2
2
;
A
i
+
m
2
2
;
k
4
=
hf
1
(
z
+
h;
A
p
+
k
3
;
A
s
+
l
3
;
A
i
+
m
3
)
;
l
1
=
hf
2
(
z
;
A
p
;
A
s
;
A
i
)
;
l
2
=
hf
2
z
+
h
2
;
A
p
+
k
1
2
;
A
s
+
l
1
2
;
A
i
+
m
1
2
;
l
3
=
hf
2
z
+
h
2
;
A
p
+
k
2
2
;
A
s
+
l
2
2
;
A
i
+
m
2
2
;
l
4
=
hf
2
(
z
+
h;
A
p
+
k
3
;
A
s
+
l
3
;
A
i
+
m
3
)
;
m
1
=
hf
3
(
z
;
A
p
;
A
s
;
A
i
)
;
m
2
=
hf
3
z
+
h
2
;
A
p
+
k
1
2
;
A
s
+
l
1
2
;
A
i
+
m
1
2
;
m
3
=
hf
3
z
+
h
2
;
A
p
+
k
2
2
;
A
s
+
l
2
2
;
A
i
+
m
2
2
;
m
4
=
hf
3
(
z
+
h;
A
p
+
k
3
;
A
s
+
l
3
;
A
i
+
m
3
)
:
(14)
With
amplitudes
f
or
pump
A
1
p
,
signal
A
1
s
and
idler
A
1
i
at
spatial
z
1
,
the
amplitudes
f
or
pump
A
2
p
,
signal
A
2
s
and
idler
A
2
i
at
ne
xt
spatial
z
2
is
calculated
iter
ativ
ely
from
Eq.
(13)
.
The
process
is
iter
ated
till
z
j
=
z
n
=
L
.
At
the
end
of
the
fiber
,
let
the
output
signal
amplitude
A
n
s
=
A
s;
out
.
The
n
umer
ical
gain
G
is
calculated
as
G
numer
ical
=
10
log
(
P
s;
out
=P
s;
in
)
,
where
P
s;
out
and
P
s;
in
are
output
and
input
signal
po
w
ers
respectiv
ely
.
P
s;
out
=
j
A
s;
out
j
2
while
P
s;
in
=
j
A
0
s
j
2
.
4.
Specification
of
HNL-DSF
In
this
n
umer
ical
sim
ulation,
a
HNL
-DSF
of
OFS
compan
y
with
=
0
:
82
dB
=k
m
,
=
11
:
5
W
1
k
m
1
and
ZD
W
at
0
=
1556
:
5
nm
w
ere
used
as
the
par
ametr
ic
gain
medium
.
Then,
the
optim
um
case
of
fiber
length
L
=
500
m
,
pump
po
w
er
,
P
p
=
30
dB
m
,
pump
w
a
v
elength,
p
=
1559
and
dispersion
slope
s
=
0
:
01
from
[9]
w
ere
used
in
this
sim
ulation
with
signal
w
a
v
elength
from
1400nm
to
1700nm.
Ne
xt,
the
dispersion
D
from
the
fiber
data
sheet
v
ersus
w
a
v
elength
w
as
plotted.
Subsequently
,
the
second-order
dispersion
coefficient,
2
w
as
calculated
from
2
=
2
2
c
D
;
(15)
where
c
is
speed
of
light.
After
that,
a
second-order
deg
ree
polynomial
fit,
D
=
a
2
+
b
+
d;
where
a
,
b
and
d
are
some
coefficien
ts
w
as
perf
or
med
to
obtain
the
quadr
atic
equation
f
or
D
.
Later
,
D
w
as
diff
erentiate
twice
to
get
a
first-order
der
iv
ativ
e
,
D
and
second-order
der
iv
ativ
es
,
dD
d
in
order
to
obtain
the
f
our
th-order
and
sixth-order
dispersion
coefficients
from
the
f
ollo
wing
respectiv
e
equations:
4
=
4
(2
c
)
3
(6
D
+
6
D
+
2
dD
d
)
;
(16)
6
=
6
(2
c
)
5
(120
D
+
240
D
+
120
2
dD
d
+
20
3
d
2
D
d
2
+
4
d
3
D
d
3
)
:
(17)
The
second,
f
our
th
and
sixth-order
dispersion
coeeficients
w
ere
obtained
as
2
=
3
:
872
10
2
ps
2
=k
m
,
4
=
6
:
327
10
5
ps
4
=k
m
and
6
=
1
:
186
10
8
ps
6
=k
m
.
5.
Results
and
Anal
ysis
The
analytical
gain
of
a
lossless
HNL-DSF
w
as
calculated
b
y
using
Eqs
.(8)-(10).
Figure
.
1(a)
por
tr
a
ys
the
analytical
gain
of
the
abo
v
e
lossless
HNL-DSF
.
It
sho
ws
a
gain
spectr
um
from
1514nm
to
1607nm
with
a
peak
gain
of
43.9234dB
which
has
a
broader
bandwidth
if
compared
to
TELK
OMNIKA
V
ol.
15,
No
.
3,
September
2017
:
1461
1469
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
1693-6930
1465
[9].
Meanwhile
,
Eqs
(1)-(3)
with
=
0
dB
=k
m
w
ere
solv
ed
n
umer
ically
b
y
using
Eqs
.
(11)-(13)
of
the
RK4
method
with
a
step
siz
e
h
=
0
:
1
.
The
RK4
gain
w
as
plotted
in
Fig.
1(b).
Intuitiv
ely
,
both
Figs
.
1(a)
and
1(b)
are
indistinguishab
le
.
1400
1450
1500
1550
1600
1650
1700
0
5
10
15
20
25
30
35
40
45
Wavelength (nm)
Gain (dB)
1400
1450
1500
1550
1600
1650
1700
0
5
10
15
20
25
30
35
40
45
Wavelength (nm)
Gain (dB)
Figure
1.
(a)
The
analytical
gain
spectr
um
(b)
The
RK4
gain
spectr
um
Later
,
Eqs
.(1)-(3)
with
=
0
dB
=k
m
w
ere
again
solv
ed
n
umer
ically
b
y
using
Matlab
b
uilt-
in
ode
function,
kno
wn
as
ode45.
The
ode45
gain
w
as
plotted
in
Fig.
2(b).
Fig.
2(a)
depicts
the
analytical
gain.
Here
also
,
both
the
Figs
.
2(a)
and
2(b)
are
identical.
1400
1500
1600
1700
0
5
10
15
20
25
30
35
40
45
Wavelength (nm)
Gain (dB)
1400
1500
1600
1700
0
5
10
15
20
25
30
35
40
45
Wavelength (nm)
Gain (dB)
Figure
2.
(a)
The
analytical
gain
spectr
um
(b)
The
ode45
gain
spectr
um
T
o
chec
k
whether
both
the
analytical
gain
(Fig.
1(a))
and
RK4
gain
(Fig.
1(b))
are
abso-
lutely
similar
,
the
y
w
ere
plotted
on
the
same
g
r
aph
as
displa
y
ed
in
Fig.
3(a).
Whereas
,
both
the
analytical
gain
(Fig.
2(a))
and
ode45
gain
(Fig.
2(b))
w
ere
plotted
on
the
same
g
r
aph
as
displa
y
ed
in
Fig.
3(b)
to
chec
k
if
the
y
are
liter
ally
look-alik
e
.
It
is
noticed
that
both
the
analytical
and
RK4
gains
are
o
v
er
lapped
accur
ately
.
Similar
ly
,
both
the
analytical
and
ode45
gains
are
o
v
er
lapped
on
one
another
.
Numer
ical
Sim
ulation
of
Highly-Nonlinear
Dispersion-Shifted
Fiber
...
(K.G.
T
a
y)
Evaluation Warning : The document was created with Spire.PDF for Python.
1466
ISSN:
1693-6930
1400
1500
1600
1700
0
5
10
15
20
25
30
35
40
45
1400
1500
1600
1700
0
5
10
15
20
25
30
35
40
45
Wavelength (nm)
Gain (dB)
Figure
3.
(a)
The
o
v
er
lapped
analytical
and
RK4
gain
spectr
um
(b)
The
o
v
er
lapped
analytical
and
ode45
gain
spectr
um
Ev
en
though
g
r
aphically
both
the
analytical
and
RK4
gains
are
o
v
er
lapped
e
xactly
,
the
absolute
errors
betw
een
the
analytical
and
RK4
gains
w
ere
calculated
and
plotted
in
Fig.
4(a)
which
re
v
eals
that
still
there
are
some
small
errors
betw
een
the
analytical
a
nd
RK4
gains
with
the
maxim
um
error
of
0.02215.
On
the
other
hand,
the
absolute
errors
betw
een
the
analytical
and
ode45
gains
w
e
re
pictured
in
Fig.
4(b)
and
it
e
xposes
that
there
are
some
errors
betw
een
the
analytical
and
ode45
gains
with
a
maxim
um
error
of
0.9682.
Compar
ativ
ely
,
Fig
4
pro
v
es
that
the
RK4
method
giv
es
a
smaller
error
if
compared
to
ode45.
1400
1450
1500
1550
1600
1650
1700
0
0.005
0.01
0.015
0.02
0.025
Wavelength (nm)
Gain (dB)
1400
1450
1500
1550
1600
1650
1700
0
0.5
1
1.5
Wavelength (nm)
Gain (dB)
Figure
4.
(a)
Absolute
error
of
analytical
and
RK4
gain
spectr
um
(b)
Absolute
error
of
analytical
and
ode45
gain
spectr
um
Fur
ther
more
,
there
is
no
analytical
solution
if
fiber
loss
as
=
0
:
82
dB
=k
m
is
tak
en
into
consider
ation
in
Eqs
.(1)-(3).
Hence
,
based
on
the
smaller
error
giv
en
b
y
the
RK4
method
in
compar
ison
with
ode45
in
Fi
g.
4,
Eqs
.(1)-(3)
w
ere
solv
ed
n
umer
ically
b
y
the
RK4
method
up
to
the
second
order
dispersion
coefficient
as
e
xpressed
in
v
ector
mismatch
Eq.
(7).
The
FOP
A
gains
f
or
both
the
loss-free
fiber
and
lossy
fiber
with
an
atten
uation
coefficient
of
=
0
:
82
dB
=k
m
TELK
OMNIKA
V
ol.
15,
No
.
3,
September
2017
:
1461
1469
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
1693-6930
1467
are
sho
wn
in
Fig.
5.
It
is
f
ound
that
the
peak
gain
is
reduced
when
the
fiber
loss
is
tak
en
into
account,
b
ut
the
bandwidth
remains
the
same
.
This
patter
n
is
similar
with
research
w
or
k
in
[10]
that
included
fiber
loss
and
the
f
our
th-order
coefficient.
The
y
compared
their
w
or
k
with
pre
vious
research
w
or
ks
without
including
fiber
loss
and
the
f
our
th-order
coefficient
and
it
w
as
obser
v
ed
that
the
peak
gain
spectr
um
with
the
fiber
loss
w
as
reduced.
1400
1420
1440
1460
1480
1500
1520
1540
1560
1580
1600
1620
1640
1660
1680
1700
−5
0
5
10
15
20
25
30
35
40
45
Wavelength (nm)
Gain (dB)
db/km
α
=0
α
=0.82
Figure
5.
The
gain
spectr
um
up
to
second-order
dispersion
coefficient
of
a
loss-free
and
lossy
HNL-DSF
Lastly
,
to
study
the
e
ff
ect
of
higher-order
dispersion
coefficient
such
as
the
f
our
th
order
and
sixth
order
dispersion
coefficients
to
w
ard
the
perf
or
mance
of
a
lossy
HNL-DSF
FOP
A,
Eqs
.(1-
3)
w
ere
solv
ed
n
umer
ically
b
y
the
RK4
method
up
to
the
f
our
th
order
and
sixth
orde
r
dispersion
coefficients
giv
en
in
v
ector
mismatch
Eq.
(7)
one
b
y
one
.
The
FOP
A
gains
of
the
second,
f
our
th
and
sixth
order
dispersion
coefficients
are
giv
en
in
Fig.
6.
It
is
noticed
that,
the
f
our
th-order
dispersion
coefficient
broadens
the
bandwidth
a
bit
while
maintainin
g
the
gain
peak
as
repor
ted
b
y
[10].
Besides
,
th
ere
are
tw
o
narro
w-band
gains
at
the
left
and
r
ight
sides
of
the
wide-band
gain
due
to
the
eff
ect
of
the
f
our
th-order
dispersion
coefficient.
The
narro
w-band
at
the
r
ight
side
of
the
wide-band
gain
w
as
repor
ted
b
y
[11]
and
[14]
in
their
half
r
ange
FOP
A
gain
spectr
um
when
the
negativ
e
sign
of
the
f
our
th-order
dispersion
coefficient
w
as
present.
It
is
obser
v
ed
that
the
positiv
e
sign
of
the
sixt
h
order
dispersion
coefficient
just
mo
v
es
the
tw
o
narro
w-band
gain
to
w
ard
the
wide-
band
gain
while
the
peak
gain
is
unchanged
and
its
broadband
gain
is
o
v
er
lapped
e
xactly
with
the
broadband
gain
from
the
f
our
th-order
dispersion
coefficient.
1400
1420
1440
1460
1480
1500
1520
1540
1560
1580
1600
1620
1640
1660
1680
1700
−5
0
5
10
15
20
25
30
35
40
45
Wavelength (nm)
Gain (dB)
β
2
β
4
β
6
Figure
6.
The
gain
spectr
um
up
to
second,
f
our
th
and
positiv
e
sixth-order
dispersion
coefficients
of
a
lossy
HNL-DSF
If
the
sixth-order
dispersion
coefficient
tak
es
negativ
e
sign
as
seen
in
Fig.
7,
the
tw
o
Numer
ical
Sim
ulation
of
Highly-Nonlinear
Dispersion-Shifted
Fiber
...
(K.G.
T
a
y)
Evaluation Warning : The document was created with Spire.PDF for Python.
1468
ISSN:
1693-6930
narro
w-band
gains
will
mo
v
e
a
w
a
y
from
the
wide-band
gai
n
in
compar
ison
to
the
positiv
e
sign
of
the
sixth-order
dispersion
coefficient.
The
broadband
gain
of
the
negativ
e
sign
of
the
sixth
order
dispersion
coefficient
is
o
v
er
lapped
e
xactly
with
the
broadband
gain
from
the
f
our
th-order
dispersion
coefficient
too
.
1400
1420
1440
1460
1480
1500
1520
1540
1560
1580
1600
1620
1640
1660
1680
1700
−5
0
5
10
15
20
25
30
35
40
45
Wavelength (nm)
Gain (dB)
β
2
β
4
β
6
Figure
7.
The
gain
spectr
um
up
to
second,
f
our
th
and
negativ
e
sixth
-order
dispersion
coefficient
of
a
lossy
HNL-DSF
.
6.
Conc
lusion
This
paper
has
sim
ulated
the
analytical
gain
spectr
um
of
the
combination
of
all
the
opti-
m
um
v
alues
of
a
lossless
HNL-DS
F
par
ameters
[9]
and
compared
with
the
RK4
method
as
w
ell
as
Matlab
b
uilt
-in
function
ode45.
It
can
be
concluded
that
the
RK4
method
giv
es
a
smaller
er-
ror
if
compared
with
ode45.
In
pr
actical,
fiber
is
lossy
and
it
is
impor
tant
to
include
higher
order
dispersion
coefficients
if
the
pump
w
a
v
elength
is
close
to
or
e
xactly
at
ZD
W
and
when
the
diff
er-
ence
betw
een
the
signal
and
pump
w
a
v
es
is
large
enough.
Hence
,
fiber
loss
and
higher-order
dispersion
coefficients
w
ere
added
slo
wly
to
the
sim
u
lation
and
w
ere
solv
ed
n
umer
ically
b
y
the
RK4
method.
It
can
be
concluded
that
fiber
loss
damps
the
peak
gain
of
the
FOP
A
while
the
bandwidth
is
unchanged.
On
top
of
that,
the
f
our
th-order
dispersion
coefficient
broadens
the
bandwidth
slightly
and
gener
ates
a
narro
w-band
gain
at
the
left
and
r
ight
sides
of
the
broadband
gain.
Ho
w
e
v
er
,
the
sixth
order
dispersion
coefficient
just
mo
v
es
the
left
and
r
ight
narro
w-band
gains
to
w
ard
or
a
w
a
y
the
broadband
gain
depending
on
its
positiv
e
and
negativ
e
signs
.
Ac
kno
wledg
ement
This
w
or
k
w
as
par
tly
suppor
ted
b
y
ORICC
UTHM
and
the
Ministr
y
of
Higher
Education
Mala
ysia
u
nder
Research
Accultur
ation
Collabor
ativ
e
Eff
or
t
(RA
CE)
V
ot
1509,
registr
ar
of
Univ
er-
siti
T
un
Hussein
Onn
Mala
ysia
and
ORICC
Univ
ersiti
T
un
Hussein
Onn
Mala
ysia.
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