Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
5, N
o
. 1
,
Febr
u
a
r
y
201
5,
pp
. 38
~45
I
S
SN
: 208
8-8
7
0
8
38
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
/
IJECE
Static Characterizati
on of th
e Birefringence Effect in the
Semiconductor Optical Amplifier
Using t
h
e Finite Diff
eren
ce
Meth
od
A. E
l
yam
ani
*
, A
.
Za
t
n
i
**
, H
.
Bousse
t
a
*
, A. Moumen
*
*
M.S.I.T Labo
rator
y
, Dep
a
rtmen
t
of Computer
Engineer
ing high
school of
techno
log
y
, Ibn
Zohr University
Agadir
M
o
rocco.
**
Departmen
t
of
Computer Eng
i
neering
,
h
i
gh school of
technolo
g
y
, Agadir
Moro
cco
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Oct 16, 2014
Rev
i
sed
No
v
18
, 20
14
Accepted Dec 12, 2014
Knowing the v
a
rious ph
y
s
ical mechan
isms of the semicond
uctor optical
am
plifier
(SOA) helps us
to d
e
v
e
lop
a
more
c
o
mpl
e
t
e
nume
r
ica
l
mode
l.
It
als
o
enabl
e
s
us
to s
i
m
u
late m
o
re real
is
tic
all
y
th
e s
t
ati
c
behav
i
or of the S
O
A
s
’
birefring
enc
e
ef
fect
. This
wa
y,
it al
lows
us
to s
t
ud
y
m
o
re p
r
ecis
e
l
y
the
behavior of SOA
s
, and particularly
the impact
of the amplified
spontaneous
emission (ASE)
or the pump
and probe
signals as well as the opti
c
a
l
functions based
on the non-linearity
of th
e component. In static r
e
gime, the
SOA
s
possess
a ver
y
low am
plif
ica
tion th
reshold
and
a satur
a
tio
n
power of
the gain which
mainly
dep
e
nds on the op
tica
l
power injec
t
ed in
to the act
ive
region. B
e
y
ond
the optical inpu
t power,
th
e SOA is in the saturated gain
regime which
gives it
a nonlinear
tr
ansmission behavior
. O
u
r detailed
numerical model offers a set of
equati
ons
and
an
algorithm
th
at
p
r
edic
t th
eir
behavior
. Th
e equations form a
theore
tical bas
e
from which we have cod
e
d
our m
odel in sev
e
ral f
iles
.
cpp th
a
t
th
e
Langu
age
C++
exe
c
utes
.
It
has
enabl
e
d
us, from the p
h
y
sical and g
e
o
m
etrical
par
a
meters
of the co
mponent,
to
recover
all
the r
e
lev
a
nt values
for a comprehen
s
ive stud
y
of S
O
A
s
in
stati
c
and d
y
namic r
e
gimes. In this pape
r, we p
r
opose to make a static
chara
c
t
e
riz
a
tion
of the effec
t
of the nonlin
ear polar
iz
ation
rotation
b
y
realizing
a pump-probe assemblage to
cont
rol the power
and s
t
ate o
f
polari
zat
ion
at
th
e en
tering
of
the
S
OA.
Keyword:
Birefrin
g
e
n
ce effect
Fi
ni
t
e
di
f
f
ere
n
ce m
e
t
hod
(FDM
)
Non
-
li
n
earity
Po
larisation
ro
t
a
tio
n
Sem
i
cond
uct
o
r
o
p
t
i
cal
a
m
p
lifier (SOA)
Copyright ©
201
5 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
:
A. Ely
a
m
a
ni,
M.S.I.T Labo
rato
r
y
,
D
e
p
a
r
t
men
t
of
C
o
m
p
u
t
er
En
g
i
n
eer
i
n
g h
i
gh
schoo
l of tech
no
log
y
,
I
bno
u Zohr
U
n
iv
er
sity Ag
ad
i
r
, Moro
cco
Em
a
il: ab
d
e
nb
i.elya
m
a
n
i
@g
mail.co
m
1.
INTRODUCTION
In
recen
t
years, th
ere h
a
s b
e
en
con
s
id
erab
le p
r
ogr
ess in
the ex
p
l
o
itatio
n
o
f
op
tical n
o
n
lin
earities in
SOA
s
[1
], [2
]. Mu
ch
atten
tion
h
a
s
b
een
p
a
id
to
th
e b
i
refrin
g
e
n
ce in
SOA in
a pu
m
p
-prob
e
assem
b
lag
e
. Th
e
effective refra
c
tive
indices for
tra
n
sve
r
se
-electric
(TE)
m
ode and t
r
a
n
sv
erse
-m
agnet
i
c
(TM
)
m
ode are
diffe
re
nt form
each ot
her
due
to intr
insic bi
refringence in
SOA and indu
ced bire
fri
nge
nce in SOA, thus t
h
e
pha
se cha
n
ges
i
n
SO
A
fo
r TE
an
d TM
m
ode
s. Th
e o
r
i
g
i
n
o
f
p
o
l
a
ri
zat
i
o
n-
depe
n
d
ent
gai
n
(
P
D
G
)
i
n
S
O
A
s
is
the fact that bulk active m
a
terial
h
a
s m
u
ch larg
er
TE amp
lificatio
n
th
an
TM,
wh
ich
is d
u
e to
th
e
d
i
fferent
co
nfin
em
en
t facto
r
s [3
]. In
ad
d
ition
to
th
e
cu
m
u
lativ
e effect o
f
PDG, t
h
e SOA
s
al
s
o
i
n
t
r
od
uce A
S
E
noi
se
,
whi
c
h a
ffect
s t
h
e
opt
i
cal
si
g
n
a
l
-
t
o
-
noi
se
rat
i
o
(OS
N
R
)
o
f
t
h
e
pay
l
oad
ch
a
nnel
s
.
The t
h
e
o
ret
i
cal
fo
un
dat
i
o
n
of
SO
As m
odel
i
ng was e
s
t
a
bl
i
s
hed i
n
1
9
80s
.
Si
nce t
h
e
n
, m
a
jo
r p
r
og
res
s
conce
r
ni
n
g
S
O
A m
odel
i
ng
h
a
s st
art
e
d t
o
f
o
cu
s o
n
t
h
e m
a
t
e
ri
al
gai
n
c
o
effi
ci
ent
,
s
p
ont
aneo
us em
i
ssion
rat
e
and t
h
e fract
i
o
n o
f
sp
o
n
t
a
ne
o
u
s em
i
ssi
ons cou
p
l
e
d
wi
t
h
t
h
e gui
ded
wave
s i
n
an am
pl
i
f
ier. S
o
m
e
num
eri
cal
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
S
t
a
tic Cha
r
a
c
t
e
riza
tio
n o
f
t
h
e Birefrin
g
e
n
ce
Effect in
th
e
S
e
micon
d
u
c
tor
Op
tica
l
Amp
lifier …
(
A Elyamani
)
39
sim
u
l
a
t
i
ons w
h
i
c
h use t
h
e
Fi
n
i
t
e
Di
ffere
nce
M
e
t
h
o
d
(F
DM
) [
4
]
have
bee
n
st
udi
ed i
n
t
e
nsi
v
el
y
t
o
sol
v
e c
a
rri
er
rat
e
eq
uat
i
o
n a
n
d
p
h
o
t
o
n t
r
a
v
el
i
ng
wave
eq
u
a
t
i
ons
[5]
,
[
6
]
.
In
th
is
p
a
p
e
r,
we propo
se to
mak
e
a static c
h
aract
erization of the effect of
th
e no
n
lin
ear p
o
l
arization
rot
a
t
i
o
n
by
rea
l
i
z
i
ng a
p
u
m
p
-pr
o
be assem
b
l
a
ge t
o
c
ont
rol
the power and
state of polariz
a
tion at t
h
e e
n
teri
ng
of t
h
e S
O
A
.
T
h
ere
f
o
r
e, t
h
i
s
p
a
per i
s
dedi
cat
ed t
o
t
h
e st
u
d
y
of m
odi
fi
cat
i
o
ns t
o
m
odel
s
u
gge
st
ed
by
C
o
nnel
l
y
t
o
t
a
ke i
n
t
o
ac
cou
n
t
t
h
e ef
fec
t
of bi
re
fri
nge
nce i
n
t
h
e S
O
A. T
h
e ex
pre
s
si
ons
of t
h
e c
o
upl
e
d
m
ode eq
uat
i
o
n
s
m
u
st
be
m
odi
f
i
ed. We have
devel
ope
d
a
m
o
re det
a
i
l
e
d m
odel
base
d
on
t
h
e eq
uat
i
o
ns
of t
h
e c
o
u
p
l
e
d
m
odes
whi
c
h de
pe
nd
on t
h
e p
o
l
a
ri
za
t
i
on an
d
use t
h
e fi
ni
t
e
di
ffe
re
nce m
e
t
hod (F
DM
). T
h
i
s
m
odel
t
r
eat
s sepa
rat
e
l
y
t
h
e TE
com
p
o
n
ent
an
d TM
c
o
m
pone
nt
of t
h
e
opt
i
cal
fi
el
d, i
n
t
r
od
uces
d
i
ffere
nt
c
o
n
f
i
n
em
ent
fact
o
r
s
fo
r t
h
e
two
po
larization
states an
d
tak
e
s in
to
accoun
t th
e ph
en
om
eno
n
o
f
t
h
e T
E
and TM
m
odes ene
r
gy
co
upl
i
n
g.
The res
u
l
t
s
we
have
obt
ai
ne
d
pr
ovi
de an i
n
st
ruct
i
v
e i
n
si
g
h
t
i
n
t
o
S
O
A
.
These re
sul
t
s
a
r
e al
so be
nefi
c
i
al
for
devi
ce desi
gn
and
o
p
t
i
m
i
zat
ion
.
2.
THEORY OF SOA
2.
1.
T
h
e
o
reti
c
al
M
o
del
We dec
o
m
pos
e t
h
e i
n
com
i
n
g
arbi
t
r
a
r
i
l
y
pol
ari
zed el
ect
r
i
c fi
el
d i
n
a
TE com
pone
n
t
and TM
com
pone
nt
as il
l
u
st
rat
e
d i
n
fi
gu
re 1. T
h
ese t
w
o p
o
l
a
ri
zat
i
o
n di
rect
i
o
ns ar
e al
ong t
h
e p
r
i
n
ci
pal
axes (
x,
y
) that
di
ag
onal
i
ze t
h
e
wa
ve
pr
o
p
agat
i
on i
n
t
h
e
S
O
A
.
Fi
gu
re
1.
A
SO
A a
n
d
t
h
e t
w
o
pol
a
r
i
zat
i
on
di
r
ect
i
ons T
E
a
n
d
TM
sc
hem
a
The t
o
tal electric field is
defi
ned
by [1]:
(1
)
Wi
t
h
:
e
x
p
∅
e
x
p
∅
Whe
r
eas
is th
e in
pu
t po
larizatio
n
ang
l
e,
∅
and
∅
are in
itial p
h
a
se of inp
u
t
sig
n
a
l for TE
m
o
d
e
and
TM m
ode, respecti
v
ely. In our cal
culation, in left
(input) a
nd
righ
t
(out
put
)
facets of SOA have powe
r
reflectiv
ities
R
,
and
R
, respectiv
ely.
Th
e sign
al wav
e
g
e
ts
p
a
rtia
lly tran
sm
i
tte
d
an
d reflected fro
m
th
e two
facets of t
h
e a
m
plifier. Due t
o
the
input si
gnal, the
spatially varying c
o
m
ponent
of the field i
n
the
S
O
A can
be de
com
pose
d
i
n
t
o
f
o
r
w
ar
d
and
bac
k
war
d
pr
o
p
agat
i
n
g
E
+
, E
-
, an
d the trav
eling
-
wav
i
ng
equ
a
tion
for TE
m
o
d
e
an
d TM
m
o
d
e
ar
e [1
],
[7
],
[8
] an
d [9
]:
,
,
,
,
,
,
,
,
,
,
,
,
,
∗
,
(2
)
,
,
1
2
,
,
,
,
,
,
,
1
2
,
,
,
,
∗
,
Th
ose s
p
ont
a
n
eou
s
em
i
ssi
on
ph
ot
o
n
rat
e
s a
r
e o
b
ser
v
e
d
i
n
t
h
e
fol
l
o
wi
n
g
e
quat
i
o
n:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
5, No
. 1, Feb
r
uar
y
20
1
5
:
3
8
– 45
40
,
,
,
,
,
,
,
,
,
,
,
,
(3
)
,
,
,
,
,
,
,
,
,
,
,
,
Whe
r
eas
√
1
,
,
and
,
are the coe
fficients
of c
o
upling m
odes
TE-TM a
nd T
M
-TE
respectively.
,
,
,
,
,
,
(4)
Whe
r
eas
i
s
t
h
e c
o
upl
i
n
g
c
onst
a
nt
[1
0]
,
t
h
e st
a
r
re
prese
n
t
s
t
h
e c
o
m
p
l
e
x c
o
nj
u
g
at
e.
The
ot
her
coefficients a
r
e
de
fine
d in [5], [7].
The
de
vel
o
ped
m
odel
base
d
o
n
t
h
e
ass
u
m
p
t
i
on
o
f
q
u
asi
-
st
a
t
i
onary
i
s
sum
m
ed u
p
i
n
a s
e
veral
se
ct
i
o
n
division
of the
com
pone
nt
gain re
gion s
o
as to ta
ke
into account t
h
e
non unif
orm
distribution
of c
a
rrier
d
e
nsity an
d refractiv
e ind
e
x
.
Th
e carrier
d
e
nsity in
sectio
n
i
in
sid
e
th
e
SOA
o
b
e
ys t
h
e
rate equ
a
tio
n [7
]:
,
,
2
,
,
2
,
(5
)
Whe
r
eas
I
is th
e am
p
lifier b
i
as cu
rren
t,
d
and
w
are
the active
re
gion thic
kne
ss
and
width,
respectively.
Th
e reco
m
b
in
atio
n
rate
term
i
s
gi
ve
n
by
:
(6
)
Whe
r
eas
1
,
and
are
rec
o
m
b
ination c
o
efficients.
,
,
,
,
,
and
,
are
defi
ned
by
e
q
uat
i
o
ns
[7]
,
[
9
]
:
,
∑
,
,
,
,
(7
)
,
∑
,
,
,
,
(8
)
,
∑
,
,
,
,
(9
)
,
∑
,
,
,
,
(1
0)
SOA
geom
etrical and m
a
teria
l
param
e
ters us
ed in
the
steady-state
m
odel a
r
e
give
n in tabl
e I.
2.
2.
P
o
l
a
ri
z
a
t
i
on-
D
epe
nden
t
G
a
i
n
(P
DG)
The
de
pende
n
ce of SOA to
pola
r
ization is
cha
r
act
erized by
m
easuring the
polarization-de
pende
n
t
gai
n
(
P
D
G
)
.
I
n
pract
i
ce t
h
i
s
consi
s
t
s
of m
e
asuri
ng t
h
e gai
n
of t
h
e
devi
ce i
n
al
l
possi
bl
e cases of
pol
a
r
i
zat
i
o
n
si
gnal
s
.
To
si
m
p
li
fy
ou
r st
u
d
i
e
s,
we h
a
ve
sim
p
l
y
sou
ght
t
w
o
ort
h
o
g
ona
l
axes
of l
i
n
ea
r
pol
a
r
i
zat
i
on
(
TE a
n
d
TM axes). The
gain hasn’t, therefor
e
,
bee
n
measured
but for the
polari
ze
d signals along these two a
x
e
s
. The
pol
a
r
i
zat
i
o
n
-
de
pen
d
e
n
t
gai
n
c
a
n
be cal
c
u
l
a
t
e
d
vi
a t
h
e
f
o
l
l
o
wi
n
g
fo
rm
ul
a:
|
|
(1
1)
Two cases of
measuring the
PDG can the
n
be presen
t
e
d:
con
f
i
g
urat
i
o
ns i
n
sel
f-sat
urat
i
o
n an
d cr
oss-
satu
ration
.
The
m
easu
r
em
en
t o
f
th
e
PDG in
th
e self-satu
r
ation
en
ables u
s
to
inqu
ire
o
n
t
h
e in
trin
sic
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
S
t
a
tic Cha
r
a
c
t
e
riza
tio
n o
f
t
h
e Birefrin
g
e
n
ce
Effect in
th
e
S
e
micon
d
u
c
tor
Op
tica
l
Amp
lifier …
(
A Elyamani
)
41
d
e
p
e
nd
en
ce of th
e SOA co
mp
on
en
t wh
ich
is a v
e
ry im
portant m
easure. The m
easures
of the
PDG i
n
the
co
nf
igu
r
ation
pu
mp
-p
rob
e
allow
stud
ying
t
h
e
o
p
tical gates
which
operate i
n
cross-m
o
dulation.
(a)
(b
)
Fi
gu
re
2.
Sc
he
m
a
t
i
c
vi
ew o
f
a t
y
pi
cal
SO
A
st
ruct
ure t
o
m
e
asure
the
static gain and t
h
e PDG:
(a)
con
f
i
g
urat
i
o
n
s
e
l
f-sat
u
r
at
i
o
n;
(b
) c
o
n
f
i
g
urat
i
o
n
cr
oss
-
sat
u
ra
t
i
on
(p
um
p-p
r
o
b
e)
.
Figure 2 sc
he
matizes
the principle of these two m
easures. In eac
h
m
easure, the PDG of t
h
e de
vice is
evaluate
d by c
o
m
p
aring t
h
e
curves
of
static gain
obtaine
d
wit
h
a c
onti
n
uous
pum
p,
according to the two
ext
r
em
e confi
g
urat
i
o
ns
of
p
o
l
a
ri
zat
i
on
(TE a
n
d
TM
).
The greater the
diffe
r
ence
bet
w
een
the two
obtained T
E
and TM c
u
rves
is, the
hi
ghe
r t
h
e PDG.
Table I. SOA
geom
etrical and m
a
terial para
m
e
ters [1]
,
[
7
]
.
Sy
m
b
o
l
Par
a
m
e
t
e
r
s
Valu
e
an
d
u
n
it
y
M
o
lar
fr
action of Ar
senide in the active r
e
gion.
0.892
L
Cavity
length
1000
d
Active r
e
gion thickness
0.2
W
Centr
a
l active
r
e
gion width
2
I
Bias cur
r
e
nt
260
m
A
K
g
Bandgap shr
i
nka
g
e
coefficient
0.9
10
,
/
E
quivalent r
e
fr
active index at zer
o car
r
i
er density
.
3.22
,
Differential of equivalent refractive
i
n
d
e
x
with
resp
ect
to
carrie
r
d
e
n
s
ity
.
1
.4
4
,
1
.2
0
1
0
,
Optical conf
ine
m
e
n
t f
actor
0.3
,
0.25
I
nput couplin
g loss
.
3.0
Output couplin
g lo
ss.
3.0
Input facet reflecti
v
ity
51
0
Ouput facet reflectivity
51
0
Car
r
i
er
independent abso
rption loss
coef
f
i
cient
6200
Car
r
i
er
dependent absor
p
tion loss coe
fficient
7500
1
0
Linear
radiative r
e
co
m
b
ination coef
f
i
cient
11
0
Bi
m
o
l
ecular radiat
ive
reco
m
b
ination coef
f
i
cient
2.5
10
Auger reco
m
b
inati
on coeff
i
cient
3.0
10
Bandgap ener
gy
quadr
atic coefficient
1.35
Bandgap ener
gy
quadr
atic coefficient
0
.775
Bandgap ener
gy
quadr
atic coefficient
0.149
Ef
f
i
ctive
m
a
ss of
electron in the CB
4.10
1
0
Ef
f
i
ctive
m
a
ss of
heavy hole in the V
B
4.19
1
0
Effictive
m
a
ss of light hole in the VB
5.06
1
0
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
5, No
. 1, Feb
r
uar
y
20
1
5
:
3
8
– 45
42
3.
RESULTS
A
N
D
DI
SC
USI
O
N
The m
easure
o
f
st
at
i
c
gai
n
co
nsi
s
t
s
o
f
m
easuri
ng t
h
e am
pl
i
f
i
cat
i
on
of a
s
i
gnal
as i
t
pass
es t
h
r
o
ug
h a
SOA d
e
p
e
nd
ing
on
th
e power o
f
the in
pu
t sig
n
a
l. Th
er
e a
r
e two types of
m
easures: self-gai
n saturation and
cross
-
gai
n
sat
u
rat
i
on or p
u
m
p
-p
ro
be.
T
h
e fi
r
s
t
m
easure
in
vo
lv
es inj
ecting a sin
g
l
e sign
al
in
th
e SOA (fig
u
re
.2
(a)
)
w
h
ereas
t
h
e sec
o
nd
o
n
e
i
n
v
o
l
v
e
s
t
w
o
si
gnal
s
cal
l
e
d
pum
p an
d
p
r
o
b
e
.
3.
1.
Sel
f
-G
ai
n
Sa
tur
a
ti
on
The m
easure of self-gain sat
u
ration
of the SOA ta
ke
s into
account the ASE noi
se
. Indee
d
, in the case
o
f
a l
o
w op
tical in
j
ection
,
t
h
e
ASE
po
wer con
s
titu
tes a con
s
id
erab
le add
itiv
e
n
o
i
se
d
i
sturb
i
ng
th
e
d
e
tect
io
n
o
f
t
h
e am
pl
i
f
i
e
d s
i
gnal
.
It
i
s
nece
ssary
t
o
m
easure t
h
e
gai
n
by
r
e
m
ovi
n
g
t
h
e A
S
E c
ont
ri
but
i
o
n.
The sha
p
e
of t
h
e above curve (figure
3)
ha
s two zo
nes:
a
zone
whe
r
e t
h
e gai
n
i
s
con
s
t
a
nt
, cal
l
e
d
sm
a
ll sig
n
a
l reg
i
m
e
, an
d
an
area wh
ere th
e
g
a
in
falls lin
early, called
saturatio
n
reg
i
m
e
.
Wh
ile th
e first
zon
e
allo
ws to
m
eas
u
r
e th
e sm
all
s
i
g
n
a
l g
a
in
G
0
, the second one
m
easures that of
t
h
e sat
u
rat
i
on
po
wer at
-3
dB
(P
sat, in
).
It
sho
u
l
d
al
so
be n
o
t
e
d t
h
at
t
h
e hi
ghe
r t
h
e
bi
as
cu
rre
nt
of
SOA i
s
, t
h
e m
o
re available carriers for
electron-hole recom
b
ination
th
ere are,
p
a
rticu
l
arly for stim
u
l
a
t
ed
e
m
iss
i
o
n
,
and
thu
s
,
th
e h
i
gh
er th
e s
m
all
sig
n
a
l
g
a
in
is.
In
add
itio
n
,
t
h
e
m
o
re th
e cu
rren
t is, th
e m
o
re im
p
o
r
tan
t
th
e ASE. Thu
s
, th
e SOA saturation
p
o
wer is low
[5
] (Th
e
ASE particip
ates in
t
h
e
g
a
in
satu
rati
o
n
).
-
4
5
-
3
0
-
1
5
0
1
5
0
1
0
2
0
3
0
P
s
a
t
,
i
n
=
-
8
,
5
d
B
m
G
0
-
3
G
a
i
n
(
d
B
)
I
n
p
u
t
p
o
w
e
r
(
d
B
m
)
P
o
l
a
r
i
s
a
t
i
o
n
T
E
G
0
Fi
gu
re 3.
Static g
a
in of t
h
e SOA in
sel
f-sat
uratio
n
1
4
0
0
1
4
2
5
1
4
5
0
1
4
7
5
1
5
0
0
1
5
2
5
1
5
5
0
1
5
7
5
1
6
0
0
-
1
0
-
5
0
5
1
0
1
5
2
0
2
5
3
0
P
u
m
p
i
n
T
E
P
u
m
p
i
n
T
M
G
a
i
n
f
i
b
e
r
-
t
o
-
f
i
b
e
r
(
d
B
)
W
a
v
e
l
e
n
g
t
h
(
n
m
)
(a)
1
4
0
0
1
4
2
5
1
4
5
0
1
4
7
5
1
5
0
0
1
5
2
5
1
5
5
0
1
5
7
5
1
6
0
0
-
6
0
-
5
0
-
4
0
-
3
0
-
2
0
-
1
0
P
u
m
p
i
n
T
E
P
u
m
p
i
n
T
M
A
S
E
p
o
w
e
r
(
d
B
m
)
W
a
v
e
l
e
n
g
t
h
(
n
m
)
(b
)
Fi
gu
re
4.
Gai
n
and
A
S
E s
p
ect
ra acc
or
di
n
g
t
o
t
h
e TE
an
d
T
M
pol
a
r
i
zat
i
o
n
f
o
r
Pi
n =
-
2
0
dB
m
.
Fi
gu
re 4
pre
s
ent
s
t
h
e gai
n
and
ASE s
p
ec
t
r
a so
lve
d
for the TE and
TM polarizati
o
n.
W
e
note
effectively a trans
p
are
n
cy and pea
k
wa
vele
ngt
h differ
e
n
c
e
as well as a
diffe
rence
between m
a
xim
a
. This
mean
s th
at th
e SOA used
h
a
s so
m
e
in
trin
sic
pol
a
r
i
zat
i
on d
e
pen
d
e
n
ce. Ac
cor
d
i
n
g
t
o
t
h
e SO
A
s
ge
om
et
ry
and
th
e g
a
in
and
ASE sp
ectra, th
is ten
d
e
n
c
y comes fro
m
th
e
elastics co
n
t
rain
ts app
lied
to
th
e SOA wh
ich
h
a
ve
larg
ely fav
o
red th
e TE tran
sitio
n
s
.
We t
h
en studied the
PDG
of the
SOA i
n
self-sat
u
r
atio
n.
For
th
is, we h
a
v
e
u
s
ed
th
e figu
re
schem
a
t
i
zed above
(
f
i
g
ure
2
(
a
)).
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
S
t
a
tic Cha
r
a
c
t
e
riza
tio
n o
f
t
h
e Birefrin
g
e
n
ce
Effect in
th
e
S
e
micon
d
u
c
tor
Op
tica
l
Amp
lifier …
(
A Elyamani
)
43
The
resul
t
obt
a
i
ned
(Fi
g
.5
) s
h
o
w
s t
h
at
t
h
e
SO
A p
r
ese
n
t
s
a PD
G i
n
sel
f
-
s
at
urat
i
o
n
of t
h
e o
r
der
of
1
d
B
in
sm
all si
g
n
a
l
reg
i
m
e
. Th
e PDG is subsequ
e
n
tly in
creasin
g
l
y low
wh
en
en
tering in
satu
ration
reg
i
me
.
Thi
s
i
s
ex
pl
ai
n
e
d by
t
h
e
di
f
f
e
r
ence i
n
gai
n
due t
o
p
r
efe
rre
d t
r
a
n
si
t
i
ons T
E
or
TM
bet
w
een co
n
duct
i
o
n
an
d
val
e
nce
ban
d
s.
Expe
ri
enci
ng
l
o
w p
u
m
p
i
ng,
t
h
e ban
d
s a
r
e
not
f
u
l
l
,
so t
h
ere i
s
a di
f
f
e
r
ence i
n
di
st
ri
but
i
o
n
bet
w
ee
n t
h
e ca
rri
er TE a
nd T
M
. At
hi
g
h
p
u
m
pi
ng, al
l
t
h
e
ban
d
s are sat
u
r
a
t
e
d, t
h
e di
st
ri
but
i
o
n i
s
eq
ual
and t
h
e
anisotropy sat
u
rates. T
h
ese
curves a
r
e t
h
us cha
r
acter
i
zed
by
sm
al
l
si
gn
al
gai
n
s
TE a
n
d TM
di
f
f
ere
n
t
fr
om
each
othe
r a
n
d saturation
power
TE a
n
d T
M
differe
n
t from
one anothe
r as well.
Thes
e
differe
n
ces a
r
e m
o
re
p
r
ecisely no
ted b
e
low:
8.5
6
Th
e SOA u
s
ed
in
th
is
st
u
d
y
is in
trin
sically
polarizatio
n
-
d
e
p
e
n
d
e
n
t
.
-
6
0
-
5
0
-
4
0
-
3
0
-
2
0
-
1
0
0
1
0
2
0
0
5
1
0
1
5
2
0
2
5
3
0
P
u
m
p
i
n
T
M
P
u
m
p
i
n
T
E
G
a
i
n
-
f
i
b
e
r
-
t
o
-
f
i
b
e
r
(
d
B
)
I
n
p
u
t
p
o
w
e
r
(
d
B
m
)
P
D
G
=
1
d
B
Fi
gu
re 5.
The
PD
G of SO
A
i
n
sel
f
-sat
u
r
at
i
o
n.
3.
2.
Cr
oss
-
G
a
i
n
Sa
tur
a
ti
on
Thi
s
m
easure consi
s
t
s
of i
n
jec
t
i
ng t
w
o co
nt
i
n
uo
us si
g
n
al
s i
n
t
h
e SO
A, o
n
e
of fi
xe
d p
o
we
r
,
t
h
e p
r
o
b
e,
wh
ose
gai
n
i
s
m
easured
de
pe
ndi
ng
o
n
t
h
e
p
o
we
r o
f
t
h
e
ot
h
e
r si
g
n
al
, t
h
e
p
u
m
p
. Fi
gu
re
2 (b
) sh
o
w
s t
h
e s
c
hem
e
of
assem
b
l
a
ge
of
suc
h
a
cha
r
a
c
t
e
ri
zat
i
on.
-
5
0
-
4
0
-
3
0
-
2
0
-
1
0
0
1
0
2
0
0
5
1
0
1
5
2
0
2
5
3
0
P
r
o
b
e
g
a
i
n
(
d
B
)
P
u
m
p
p
o
w
e
r
i
n
j
e
c
t
e
d
(
d
B
m
)
P
r
o
b
e
a
t
-
2
0
d
B
m
P
r
o
b
e
a
t
-
1
5
d
B
m
P
r
o
b
e
a
t
-
1
0
d
B
m
P
r
o
b
e
a
t
-
5
d
B
m
S
o
n
d
e
à
0
d
B
m
Fi
gu
re 6.
St
at
i
c
gai
n
i
n
p
u
m
p
-pr
o
be
f
o
r di
ffe
rent
p
o
we
rs of
pr
o
b
es. The
p
u
m
p
and pr
o
b
e are
TE
p
o
l
a
ri
z
e
d
An exam
ple of c
u
rves
obtained w
ith
the
s
a
m
e
SOA
b
u
t
fo
r
diffe
re
nt p
r
o
b
e
p
o
we
rs is
re
po
rted i
n
figure
6. T
h
e c
u
rves
present a
n
ide
n
ti
cal
fo
rm
to
th
at o
f
self-g
ain
satu
rati
on
. In
d
e
ed
,
b
e
it
trav
ersed
b
y
on
e or
m
o
re si
gnal
s
, t
h
e SO
A has t
w
o
ope
rat
i
n
g r
e
gi
m
e
s: t
h
e
sm
al
l
-
si
gnal
gai
n
re
gi
m
e
and t
h
e sat
u
rat
i
on r
e
gi
m
e
.
These
curve
s
a
r
e,
howe
ve
r,
differe
nt acc
ording t
o
t
h
e
po
wer
o
f
th
e
probe sign
al. Th
is
resu
lts
fro
m
th
e fact
that the probe
signal m
a
y a
l
so, accord
ing to its power, saturates the
SO
A, a
nd thus participates
in the
d
e
pop
u
l
ation
of th
e co
ndu
ction
b
a
nd
, lead
ing
to
a sm
all si
g
n
a
l
g
a
in
g
e
n
e
rally lo
wer. In
ad
d
ition
less
po
rters,
in
th
is case,
p
a
rticip
ate in
amp
lified
spo
n
t
aneo
us em
i
ssion (ASE), which l
eads to a
n
i
n
crease of the sat
u
ration
p
o
wer o
f
th
e
SOA. No
te also
th
at th
e higher the power
probe is, the lowe
r
the slope of t
h
e curve is and thus
l
e
ss effi
ci
e
n
t
c
o
m
p
ressi
o
n
ga
i
n
by
a
pum
p
pul
se
i
s
.
Fi
nal
l
y
, f
o
r
hi
gh
p
o
w
er
p
u
m
p
, t
h
e
cu
rves
m
eet
sho
w
i
n
g
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I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
5, No
. 1, Feb
r
uar
y
20
1
5
:
3
8
– 45
44
th
at th
e g
a
in
satu
ration
d
e
pen
d
s in
trin
sically o
n
th
e
m
ili
eu rega
rdl
e
ss
of t
h
e si
g
n
al
s po
wers
whi
c
h t
h
e
y
trav
erse it.
Sub
s
equ
e
n
tly, we will stud
y th
e po
larizatio
n-d
e
p
e
nd
en
t g
a
in
(PDG) o
f
SOA
fo
r
d
i
fferen
t
p
r
obe
pol
a
r
i
zat
i
ons
.
Thi
s
st
u
d
y
i
s
b
a
sed
on
si
m
i
l
a
r m
easures
to
t
h
o
s
e ach
iev
e
d
p
r
ev
iou
s
ly bu
t
th
is ti
m
e
aro
und
th
e
p
r
ob
e co
n
tinuou
s
p
o
l
ar
ized
i
n
eith
er
TE or
TM.
Fi
gu
re 7 s
h
ow
s t
h
e resul
t
s
o
b
t
a
i
n
ed
fo
r di
f
f
ere
n
t
co
nfi
g
u
r
at
i
ons o
f
p
o
l
a
r
i
zat
i
on of t
h
e pr
o
b
e at
t
h
e
en
tering
o
f
the SOA. Acco
rd
ing
to th
is
fig
u
re, it is no
ted
th
at t
h
e SOA is relativ
ely in
sen
s
itiv
e
to
t
h
e
p
o
l
arizatio
n
wh
en th
e
po
larizatio
n
o
f
th
e
prob
e is TM
(f
ig
.7 (a)) co
m
p
ared
to
th
e
TE
po
larizatio
n
(fig.7 (b
)).
-
5
0
-
4
0
-
3
0
-
2
0
-
1
0
0
1
0
1
0
1
2
1
4
1
6
1
8
2
0
2
2
2
4
P
r
o
b
e
g
a
i
n
(
d
B
)
P
u
m
p
p
o
w
e
r
i
n
j
e
c
t
e
d
(
d
B
m
)
P
u
m
p
i
n
T
M
P
u
m
p
i
n
T
E
P
r
o
b
e
i
n
T
M
(a)
-
5
0
-
4
0
-
3
0
-
2
0
-
1
0
0
1
0
1
0
1
2
1
4
1
6
1
8
2
0
2
2
2
4
P
u
m
p
i
n
T
E
P
u
m
p
i
n
T
M
P
r
o
b
e
i
n
T
E
P
r
o
b
e
g
a
i
n
(
d
B
)
P
u
m
p
p
o
w
e
r
i
n
j
e
c
t
e
d
(
d
B
m
)
(b
)
Fi
gu
re
7.
M
eas
ure
o
f
t
h
e
P
D
G
o
f
S
O
A
i
n
di
f
f
e
rent
p
o
l
a
ri
zatio
n configu
r
ation
s
p
r
ob
e.
(a) :
Prob
e in
TM
;
(b) :
Pro
b
e
i
n
T
E
. T
h
e
po
we
r
pr
ob
e i
s
-
2
0
dB
m
The curves s
h
ow that there
isn'
t a
large variati
on co
ncer
ni
n
g
t
h
e sm
al
l si
gnal
gai
n
a
nd t
h
e gai
n
com
p
ressi
o
n
.
B
y
cont
rast
, t
h
e sat
u
rat
i
on
po
we
r o
f
t
h
e
gai
n
i
s
t
h
e
on
l
y
param
e
t
e
r
whi
c
h va
ri
es f
r
om
one
co
nf
igu
r
ation of
p
o
l
ar
ization
of
th
e pu
m
p
t
o
a
not
her
.
T
h
i
s
i
s
m
o
re preci
sel
y
n
o
t
e
d i
n
t
h
e e
quat
i
o
ns
bel
o
w
:
Case
where
the probe is in TM
Case
where
the probe is in TE
1
pump
in
0
pum
pin
5
pum
pin
0
p
ump
in
Knowing
th
at u
s
ing
th
e SOA
s
as no
nl
i
n
ear
opt
i
cal
gat
e
s
m
u
st
pr
esent a strong sm
all s
i
gnal gai
n
, a
low
power
of s
a
turation as we
ll as a
m
o
re efficient possi
ble com
p
ression with a very stee
p slope of gai
n
. For
t
h
i
s
, t
h
e
co
n
f
i
g
urat
i
o
n
o
f
pol
a
r
i
zat
i
on
o
f
bot
h t
h
e
pr
obe
a
n
d
pum
p i
n
TE i
s
t
h
e m
o
st
sui
t
abl
e
f
o
r t
h
e
u
s
e o
f
ou
r
com
pone
nt
as opt
i
cal
gat
e
. T
h
e ot
he
r cases
of co
nfi
g
u
r
at
i
on ca
n be use
d
t
o
im
pro
v
e t
h
e basi
c f
unct
i
on
of
SOA (t
h
e
am
p
lificatio
n
)
.
4.
CO
NCL
USI
O
N
W
i
t
h
t
h
e
ai
d
o
f
a n
u
m
e
ri
cal
algo
ri
t
h
m
based
on
t
h
e
fi
ni
t
e
di
ffe
rence
m
e
t
hod (
F
DM
),
we
h
a
ve st
udi
e
d
th
e static ch
aracteristics o
f
t
h
e b
i
refri
n
g
e
n
ce
effect in th
e SOA.
In
o
u
r
re
searc
h
, we
have
al
so
i
n
cl
ude
d
i
n
t
h
i
s
m
e
t
hod t
h
e
e
vol
ut
i
o
n
e
quat
i
on
o
f
t
h
e ca
rri
e
r
densi
t
y
t
o
t
a
ke i
n
t
o
acc
o
unt
a
n
y
cha
n
g
e
i
n
t
h
e bi
as c
u
r
r
ent
a
nd t
h
e
di
ffe
re
nt
carri
e
r
s’
recom
b
i
n
at
i
ons
. To
dem
onst
r
at
e
th
e cap
ab
ility o
f
th
e m
o
d
e
l, so
m
e
si
m
u
lat
i
o
n
resu
lts
o
f
SOA
h
a
v
e
b
e
en
presen
ted.
As
we
ha
ve se
en, the c
h
a
r
acteristic of the
ASE
power enab
les to
d
e
termin
e th
e m
a
x
i
m
u
m
o
f
b
i
as
current that ca
n support a
SOA. The ASE
spectra indica
te the wave
leng
th
s for wh
ich th
e SOA is ad
ap
ted
(
a
ro
und
th
e gain
p
eak)
.
Th
e characterizat
ions of static
gain show
th
e s
m
all
sig
n
a
l g
a
in
as well as th
e
satu
ration
p
o
wer an
d th
e sl
o
p
e of th
e g
a
i
n
cu
rv
e.
The m
easure
of
p
o
l
a
ri
zat
i
o
n de
pe
nde
nt
gai
n
(P
DG
) o
f
t
h
e S
O
A i
n
sel
f-sat
u
r
at
i
o
n an
d c
r
os
s-
sat
u
rat
i
o
n s
h
o
w
s t
h
at
t
h
e c
o
m
pone
nt
i
s
i
n
t
r
i
n
si
cal
l
y
de
pen
d
i
n
g
o
n
t
h
e p
o
l
a
ri
zat
i
o
n
.
O
n
ce t
h
e
o
p
t
i
m
u
m
o
p
e
rating
p
a
rameters an
d
the asso
ciated
ch
aracteristic
qu
an
tities are determin
ed
, th
e step
fo
r sp
ecifyin
g
whet
her
a S
O
A i
s
a
g
o
o
d
candi
dat
e
f
o
r
t
h
e ap
pl
i
cat
i
o
ns i
n
o
p
t
i
cal
si
gnal
pr
ocess
i
ng i
s
t
h
e
dy
nam
i
c
characte
r
ization
of the c
o
m
p
onent.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
S
t
a
tic Cha
r
a
c
t
e
riza
tio
n o
f
t
h
e Birefrin
g
e
n
ce
Effect in
th
e
S
e
micon
d
u
c
tor
Op
tica
l
Amp
lifier …
(
A Elyamani
)
45
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BIOGRAP
HI
ES OF
AUTH
ORS
First aut
h
or’s
Ph
ot
o (
3
x
4
cm
)
Abdenbi. ELYA
M
ANI
was bor
n in Kenitra, M
o
rocco, on Janu
ar
y
1
st
, 1984. H
e
received th
e
M
S
c degree
in
e
l
ec
tric
al
and
ele
c
troni
cs
s
y
s
t
em
engine
ering fro
m
facult
y
of s
c
i
e
nces
Univ
ers
i
t
y
Ibnou Zohr in 2009. He is currently
pr
epar
ing hi
s PhD at the centre of doctoral studies (Ibnou
Zohr CED).
His
res
ear
ch
inter
e
s
t
s
includ
e d
e
s
i
g
n
, ch
ara
c
ter
i
z
a
ti
on, m
odelling and optimizatio
n
of optoelectronic components
and
fi
bre optic
communications s
y
stems.
Abdelkarim
. ZA
TNI was
educat
ed at th
e T
e
le
co
m
Bretagne Uni
v
ers
i
t
y
F
r
anc
e
;
He obtain
e
d a
PhD at the National School of
Engineers of Br
es
t France in 19
94. He has b
een
teaching for 20
y
e
ars. He is currently
a Professor and the Head
of computer science dep
a
rtment in Ibnou Zohr
University
at H
i
gher School of
technolog
y
Ag
ad
ir, Morocco.
He conducts his research and
tea
c
hes
com
pute
r
s
c
ien
c
e
and
te
l
ecom
m
unication
s
.
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