Int
ern
at
i
onal
Journ
al of Ele
ctrical
an
d
Co
mput
er
En
gin
eeri
ng
(IJ
E
C
E)
Vo
l.
9
, No
.
3
,
J
un
e
201
9
,
pp. 162
0~16
29
IS
S
N: 20
88
-
8708
,
DOI: 10
.11
591/
ijece
.
v9
i
3
.
pp
1620
-
16
29
1620
Journ
al h
om
e
page
:
http:
//
ia
es
core
.c
om/
journa
ls
/i
ndex.
ph
p/IJECE
Crosst
al
k in
misa
ligned free s
pa
ce
op
tical
int
er
conn
ec
ts
:
mod
elling an
d si
mula
tion
Ned
al A
l
-
Aba
bneh
Depa
rtment
o
f
E
le
c
tri
c
al E
ngin
eering,
Jordan
Uni
ver
sit
y
of
Sci
ence
&
T
ec
hnolog
y
,
Jordan
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
J
un
11
, 2
01
8
Re
vised
Dec
2
,
201
8
Accepte
d
Dec
27
, 201
8
W
e
int
roduc
e
co
nveni
en
t
m
odel
and
an
opti
m
iz
ation
sche
m
e
to
o
pti
m
iz
e
th
e
signal
-
to
-
cro
ss
talk
ratio
(SCR
)
i
n
a
fr
ee
spa
ce
o
pti
c
al
in
te
r
connect
s
(FS
OIs
)
s
y
stem
th
at
uses
m
ic
role
nses
wit
h
finite
ci
r
cul
ar
ape
rtur
es.
In
th
is
m
odel
,
we
conside
r
bo
th
th
e
stra
y
li
gh
t
cro
ss
ta
lk
and
th
e
cro
ss
ta
lk
due
to
the
diffr
a
ct
io
n
at
the
m
icrole
ns
ape
rtur
es
to
ev
al
ua
te
th
e
SC
R.
Us
ing
c
y
l
indri
c
al
form
of
Coll
ins
diffr
a
ct
i
on
int
egr
a
l
and
t
he
La
gu
err
e
–
Ga
uss
ia
n
(LG)
beam
m
odel
,
we
der
ive
an
appr
oximate
cl
osed
form
for
m
ula
for
the
op
ti
c
a
l
fie
ld
of
a
m
ult
imode
LG
bea
m
propa
gat
ing
through
circul
ar
ap
ert
ur
ed
FSOIs
by
expa
nding
th
e
h
ard
edge
ci
rcu
lar
ape
rtur
e
func
t
i
on
of
the
m
ic
rol
ens
in
te
rm
s
of
complex
Gau
ss
ia
n
func
t
ions.
The
anal
y
s
es
i
ndic
a
te
th
at
th
e
size
of
th
e
det
e
ct
or
is a
n
importa
nt factor
to
opti
m
iz
e the SCR for
both the
ap
ert
ure
d
and
the
un
ape
rtu
red
m
isal
igne
d
FS
OIs
s
y
stem.
The
e
ffe
ct
of
h
ighe
r
o
rde
r
m
ode
of
the l
ase
r
sour
ce
on
th
e
SC
R
is
al
so
conside
r
ed.
Ke
yw
or
d
s
:
Coll
ins d
i
ffract
ion
i
nteg
ral
Lag
uerre
-
Gaus
sia
n
(
LG
) bea
m
C
om
plex
Ga
ussi
an
f
unct
ions
O
ptica
l
c
r
os
sta
lk
O
ptica
l i
nterc
onnects
Copyright
©
201
9
Instit
ut
e
o
f Ad
vanc
ed
Engi
n
ee
r
ing
and
S
cienc
e
.
Al
l
rights re
serv
ed
.
Corres
pond
in
g
Aut
h
or
:
Ned
al
Al
-
A
babneh
,
Dep
a
rtm
ent o
f El
ect
rical
En
gi
neer
i
ng,
Jo
r
da
n Un
i
ver
s
it
y
of
Scie
nce
& Tec
hnology,
I
r
bid, J
orda
n.
Em
a
il
: ned
al
k@
just.
e
du.jo
1.
INTROD
U
CTION
The
n
ee
d
for
hig
h
de
ns
it
y
and
high
sp
e
ed
interco
nnec
ts
is
de
m
and
ing
f
or
high
sp
eed
dig
it
al
app
li
cat
io
ns
.
Fr
ee
s
pace
op
ti
cal
interconn
ect
s
(F
S
OIs)
can
be
us
e
d
as
data
li
nks
with
high
a
ggreg
at
e
b
an
dwidth
i
n
di
gital
syst
e
m
s
[1
]
-
[6
]
.
T
hese
op
ti
cal
interc
onnects
syst
em
s
can
be
im
plem
ented
us
i
ng
ver
ti
cal
cavit
y sur
f
ace e
m
itti
ng
lasers
(V
CSE
Ls)
a
rr
a
y at
the tran
sm
it
te
r
side an
d p
ho
t
od
et
ect
or
s a
rr
ay
at the rece
ivin
g
side
co
nn
e
ct
ed
t
og
et
he
r
by
a
n
op
ti
cal
fr
ee
s
pa
ce
channel
co
ntainin
g
m
ic
ro
le
ns
arr
ay
to
c
ollim
at
e
the
li
gh
t
an
d
lim
it
the ligh
t s
pr
ea
ding.
Desp
it
e
us
in
g
m
ic
ro
le
ns
a
rr
a
ys,
cr
os
sta
lk
be
tween
nei
ghbori
ng
opti
cal
channels,
due
t
o
stray
li
gh
t
and
t
he
li
gh
t
diffracte
d
fro
m
har
d
a
per
t
ures
of
m
ic
ro
le
ns
,
is
sti
ll
on
e
of
the
m
ajor
lim
it
ing
facto
rs
that
determ
ine
the
per
f
orm
ance
of
the
F
SOIs
includi
ng
t
he
sign
al
-
to
-
c
ross
ta
l
k
rati
o
(S
C
R),
cha
nn
el
de
ns
it
y,
the
interco
nne
ct
s
capaci
ty
[4
]
-
[6
]
.
T
her
e
f
or
e
,
we
consi
de
r
it
on
e
of
the
m
ajo
r
iss
ue
s
to
stud
y
the
im
pact
of
cro
sstal
k
on
F
SO
I
s
syst
e
m
s
and
de
velo
p
m
e
thods
an
d
m
odel
s
to
descr
i
be
the
pro
pag
at
i
on
of
the
li
ght
in
these
syst
e
m
s.
Geo
m
et
rical
ray
t
racin
g
m
e
tho
ds
can
be
us
e
d
to
descr
ibe
t
he
pro
pagat
io
n
of
li
ght
bea
m
s
in
un
a
pe
rtur
e
d
opti
cal
sy
stems
[
6]
and
do
not
ta
ke
into
c
onsid
erati
on
the
dif
f
racti
on
a
nd
t
he
cl
ipp
in
g
of
the
li
gh
t
du
e
t
o
fi
nite
aper
t
ur
es
.
I
n
re
al
it
y,
op
ti
cal
syst
e
m
s
hav
e
li
m
it
ed
siz
e
and
in
tur
n
finite
aper
t
ur
e.
The
r
efore,
m
et
ho
ds
t
o
de
al
with
these
a
per
t
ur
e
d
syst
em
s
are
need
e
d.
In
[1
]
a
di
ffra
ct
ion
f
or
m
al
is
m
based
on
H
uyge
ns
-
Fr
es
nel
pr
inci
ple
was
us
e
d
to
stud
y
the
cr
os
sta
lk
ef
fect
in
FSOIs
ta
king
into
acco
unt
the
m
ic
ro
le
ns
finite
aper
t
ur
e.
T
he
op
ti
cal
wa
ve
e
m
itted
from
the
VCSEL
w
as
a
plane
wa
ve
dif
fr
act
e
d
by
a
finite
ap
ertur
e
window.
In
[
7]
a
m
od
ifie
d
m
od
e
exp
a
ns
io
n
was
us
e
d
to
stud
y
the
dif
frac
ti
on
of
the
l
aser
beam
.
In
[5
]
ray
traci
ng
a
nd
dif
fr
act
io
n
pro
pa
gation
m
et
ho
ds
was
com
bin
ed
an
d
us
e
d
to
pro
pa
gate
the
li
gh
t
in
the
op
ti
cal
interco
nnect
s
yst
e
m
.
Ho
we
ve
r,
these
m
et
h
od
s
ca
n
be
us
e
d
to
deal
with
al
ign
ed
FS
OI
s
syst
e
m
s.
In
pract
ic
al
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
&
C
om
p
En
g
IS
S
N: 20
88
-
8708
Crosstalk i
n misali
gned
fr
ee s
pa
ce
opti
cal i
nt
erconnects:
m
od
el
li
ng
and si
mu
l
ation
(
Ne
dal Al
-
A
babne
h
)
1621
app
li
cat
io
ns
,
t
he
opti
cal
com
po
ne
nts
a
re
s
ubje
ct
to
m
isa
l
ignm
ent
wh
e
n
c
on
structin
g
the
F
SO
I
s
syst
em
s
[8
, 9
]
.
In
fact,
m
isa
li
gn
m
ent
of
op
ti
c
al
com
po
ne
nts
degra
des
the
pe
rfor
m
ance
of
FSOIs,
increa
s
es
the
c
ro
s
sta
lk,
a
nd
decr
ease
s
syst
em
eff
ic
ie
ncy.
Ther
e
f
or
e,
it
is
i
m
po
rtant
t
o
s
tud
y
the
c
r
os
st
al
k
in
du
ce
d
by
the
m
isa
l
ign
m
ent
of
FSOIs.
In
[10]
,
a
m
e
tho
d
c
om
bin
ing
fr
ee
-
s
pace
Ga
us
sia
n
beam
pr
op
a
ga
ti
on
the
or
y
a
nd
ge
neral
iz
ed
Coll
ins
diffracti
on
the
or
y
was
pr
e
se
nted
t
o
pro
pagat
e
the
beam
s
in
a
m
isa
li
gn
ed
F
SOIs
syst
e
m
with
finite
aper
t
ur
e
.
The
a
naly
sis
in
[
10]
assum
e
d
the
fun
dam
ental
Her
m
it
e
–
Ga
us
sia
n
m
od
el
(
HG
00
)
f
or
the
li
gh
t
bea
m
s
in
rectan
gu
la
r
c
oor
din
at
e syst
e
m
.
So
luti
ons
of
t
he
par
a
xial
Hel
m
ho
lt
z
equ
at
i
on
can
be
fou
nd
as
c
om
bin
at
i
on
s
of
He
rm
ite
–
Ga
us
sia
n
m
od
es
(w
hose
a
m
plit
ud
e
prof
il
es
are
sep
arab
le
in
x
an
d
y
us
ing
Ca
r
te
sia
n
coord
i
na
te
s)
or
sim
i
lar
ly
as
com
bin
at
ion
s
of
La
guer
re
–
Ga
us
sia
n
m
od
e
s
(
whose
a
m
plit
ud
e
pro
file
s
are
s
epar
a
ble
in
r
and
θ
us
in
gcyl
ind
ric
al
coo
r
din
at
es
)
.
Althou
gh
th
e
re
are
ot
her
po
ssible
m
od
al
de
com
po
sit
ion
s,
these
two
so
l
utions
are the
m
os
t us
efu
l
for pr
oble
m
s w
it
h
com
pact b
eam
s in
whic
h
the
opti
cal
p
owe
r
is c
onfined al
ong t
he o
ptica
l
axis.
Be
am
pr
of
il
es
w
hich
a
r
e
ci
rcu
la
rly
sym
m
e
tric
(o
r
la
sers
with
ca
vit
ie
s
that
are
cy
li
ndrical
ly
sy
mm
et
ric)
are
oft
en
best
so
lve
d
usi
ng
the
Lag
uerre
-
Ga
us
sia
n
m
o
dal
dec
om
po
sit
ion
.
T
hese
be
a
m
s
are
def
i
ned
i
n
cy
li
nd
rical
co
ordinates
an
d
can
hav
e
a
dv
a
ntage
s
ov
er
Her
m
it
e
–
Gau
ss
m
od
es
in
the
presen
ce
of
cy
li
nd
r
ic
al
sy
m
m
e
try
.
In
t
his
pa
pe
r,
a
cro
s
sta
lk
m
odel
for
a
m
isa
l
ign
ed
le
ns
base
d
FSOIs
syst
em
with
ci
rc
ular
a
per
t
ur
e
will
be
pr
e
sente
d
assum
ing
Lag
uerre
–
Ga
us
sia
n
m
od
el
of
any
order
(
LG
pm
)
fo
r
the
be
a
m
in
a
c
yl
i
ndrical
coor
din
at
e
sys
tem
.
This
m
od
el
is
de
rive
d
ba
se
d
on
the
cy
li
nd
rical
f
orm
o
f
Coll
ins
dif
fr
act
io
n
i
nt
egr
al
.
An
a
ppr
oxim
a
te
analy
ti
cal
fo
rm
ula
for
the
op
ti
cal
fiel
d
at
the
receive
r
side
is
de
rive
d
by
e
xpan
ding
th
e
m
ic
ro
le
ns
ape
r
ture
f
un
ct
i
on
i
n
te
rm
s
of
com
plex
Gau
ssian
f
un
ct
io
ns
.
C
om
par
ed
to
ot
her
m
od
e
ls,
thi
s
m
od
el
is
m
or
e
appr
opriat
e
to
deal
with
the
ci
rc
ular
sy
m
m
et
ric
op
ti
cal
el
e
m
ents
of
the
FS
O
I
s
wh
ic
h
co
rr
es
pond
to
the
hi
gh
est
po
ssible
sym
m
etr
y
an
d
th
us
hi
gh
interc
onnec
ts
fideli
ty
[11].
Mo
reover
,
th
is
m
od
el
ta
kes
into
account
the
e
f
fect
of
highe
r
o
r
der
m
od
es
of
the
be
am
tha
t
hav
e
been
i
gnore
d
by
s
om
e
m
od
el
s.
U
sin
g
this
m
od
el
,
the
ove
rall
crosst
al
k
is
est
i
m
at
ed
includi
ng
the
st
ray
li
gh
t
cr
os
sta
lk
.
The
SCR
of
FSOIs
syst
em
is
al
s
o
evaluate
d
a
nd
op
ti
m
iz
ed.
The
analy
ses
s
how
that
the
siz
e
of
the
detect
or
a
nd
the
stray
li
ght
cr
os
sta
lk
pla
y
an
i
m
po
rtant
r
ole
to opti
m
iz
e the SCR an
d sh
ou
ld b
e
consi
der
e
d
w
he
n desi
gn
i
ng a FSO
Is
sys
tem
.
2.
OPTIC
AL
FI
EL
D
IN
A GE
NER
AL
MISALIG
NED FI
RS
T O
R
DER
OPTIC
AL S
Y
STE
M
Figure
1
is
a
si
m
ple
sche
m
ati
c
diagr
am
fo
r
a
two
-
dim
ension
al
m
isa
li
gn
ed
opti
cal
syst
em
.
P1
and
P
2
are
tw
o
re
fer
e
nce
pla
nes
for
the
al
ign
e
d
s
yst
e
m
,
and
P
1m
and
P
2m
are
the
co
rr
es
po
nd
i
ng
pl
a
nes
f
or
t
he
m
isa
li
gn
ed
syst
e
m
.
l
is
the
longit
ud
i
nal
le
ng
t
h
betwee
n
the
input
and
the
outp
ut
planes.
1
1
,
y
x
and
2
2
,
y
x
are the
r
ect
a
ngular
c
oor
din
at
e
s on
t
he
m
isa
lig
ne
d plane
s P
1m
an
d
P2
m
, r
e
sp
ect
ively
.
Figure
1.
Mi
sal
ign
e
d op
ti
cal
s
yst
e
m
Assum
ing
a
li
near
opti
cal
syst
e
m
and
usi
ng
the
par
a
xia
l
approxim
at
io
n,
t
he
ou
t
pu
t
op
ti
cal
fiel
d
)
,
(
E
2
2
2
y
x
can
be
ob
ta
i
ne
d
from
the
input
fiel
d
)
,
(
E
2
2
1
y
x
us
in
g
Coll
ins
di
ffrac
ti
on
inte
gr
al
i
n
rectan
gu
la
r
coor
din
at
e as
[
13
]
:
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8708
Int
J
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&
C
om
p
En
g,
V
ol.
9
, N
o.
3
,
June
2019
:
1620
-
1629
1622
1
1
2
2
1
1
2
2
2
2
2
1
2
1
2
1
2
1
1
1
1
2
2
2
d
)
(
)
(
2
)
(
2
e
x
p
)
,
(
E
2
)
,
(
E
dy
x
hy
gx
fy
ex
y
x
d
y
y
x
x
y
x
a
b
ik
y
x
b
ik
y
x
(1)
/
2
k
is
the
wav
e
num
ber
and
λ
is
the
wa
velen
gth.
α
,
b
,
c
,
an
d
d
are
the
tran
sfe
r
m
at
rix
el
e
m
en
ts
of
the ali
gne
d op
ti
cal
syst
em
and
e
,
f
,
g
, a
nd
h
a
re
t
he
m
isa
l
ign
m
ent p
aram
et
ers
an
d gi
ve
n as
:
x
T
x
T
s
e
2
(2)
y
T
y
T
s
f
2
(3)
x
T
T
x
T
T
d
b
s
d
b
g
)
(
2
)
(
2
(
4
)
y
T
T
y
T
T
d
b
s
d
b
h
)
(
2
)
(
2
(5)
x
s
and
y
s
are
the x
and
y m
isa
l
ignm
ents .
x
and
y
are the
angular
m
isa
li
gn
m
ents b
et
wee
n
the ali
gn
e
d
a
nd
m
isa
li
gn
ed
tra
ns
ve
rse
ax
es
.
T
,
T
,
T
,
and
T
are the
m
isa
li
gn
m
ent
el
e
m
ents d
et
erm
i
ned b
y
d
c
b
l
a
T
T
T
T
1
,
,
,
1
(6)
Using
)
c
o
s
(
x
an
d
)
s
i
n
(
y
,
(
1) in th
e
cyl
indric
al
co
or
din
at
e
syst
e
m
can
be give
n by [
12]
:
1
1
1
1
2
1
2
1
2
2
0
2
0
1
1
1
2
2
2
c
o
s
e
x
p
2
e
x
p
2
e
x
p
2
d
θ
d
)
(
θ
b
ik
b
i
k
a
b
ik
)
,
θ
(
E
πb
ik
)
,
θ
(
E
π
(7)
1
1
,
an
d
2
2
,
are t
he
cy
li
nd
rical
c
oor
din
at
es in
P
1m
a
nd P
2m
p
la
nes, res
pecti
vely
.
2
2
2
2
2
2
2
2
s
i
n
c
o
s
θ
h
θ
g
d
(8)
2
2
2
2
2
2
2
2
/2
s
i
n
/2
c
o
s
f
e
(9)
and
/2
c
o
s
/2
s
i
n
t
a
n
2
2
2
2
1
e
f
(10)
The
(
7)
gi
ves
the
op
ti
cal
fiel
d
distrib
utio
n
at
the
ou
tp
ut
plane
f
or
a
li
ne
ar
op
ti
cal
syst
e
m
in
a
cy
l
ind
rical
coor
din
at
e syst
e
m
w
it
h
sli
ght
m
isa
li
gn
m
ent.
3.
OPTIC
AL FI
EL
D
IN
A M
I
SA
LIG
NED
F
SOIs WIT
HO
UT APE
RT
U
RE
Con
si
de
r
t
he
FSOIs
syst
em
show
n
in
Fig
ur
e
2.
This
opti
cal
syst
e
m
consi
sts
of
V
CSELs
ar
ray
,
m
ic
ro
le
ns
arra
y, and
d
et
ect
or
s ar
ray. Th
e las
ers
arr
ay
is p
la
ced at the f
oc
us plane of th
e transm
it
te
r
m
ic
ro
le
ns.
The
m
isa
li
gn
m
ents
of
the
m
i
cro
le
ns
with
r
espect
to
the
opti
cal
axis
of
the
FSOIs
syst
e
m
are
x
s
,
0
y
s
and
0
y
x
as s
how
n
in
Fi
gure
2. The
tra
ns
fe
r
m
at
rix
el
e
m
ents o
f
the
a
li
gn
ed
FSO
Is
s
yst
e
m
are:
Evaluation Warning : The document was created with Spire.PDF for Python.
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t J
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om
p
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g
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S
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Crosstalk i
n misali
gned
fr
ee s
pa
ce
opti
cal i
nt
erconnects:
m
od
el
li
ng
and si
mu
l
ation
(
Ne
dal Al
-
A
babne
h
)
1623
1
,
/
1
,
,
/
1
d
f
c
l
b
f
l
a
t
t
(11)
and the m
isa
li
gn
m
ent p
aram
eter
s a
re:
0
,
/
1
,
0
,
/
T
t
T
T
t
T
f
f
l
(12)
Figure
2.
Mi
sal
ign
e
d
le
ns
-
bas
ed
F
SOIs
0
,
0
,
0
,
2
h
g
f
s
f
l
e
x
t
(13)
The
beam
e
m
i
t
te
d
f
ro
m
the
la
ser
s
ource
is
a
LG
beam
with
waist
ra
diu
s
0
.
I
n
this
case,
t
he
opti
cal
fiel
d
distrib
utio
n
of
a
LG
be
a
m
fo
r
any
order
in
t
he
cy
li
nd
rical
coor
din
at
e
syst
em
at
the
plane
of
the
transm
itti
ng
m
i
cro
le
ns
is
giv
e
n by [
5]
1
2
1
2
1
m
p
2
1
2
1
1
1
1
1
1
e
x
p
2
L
e
x
p
2
)
,
(
E
im
m
(14)
m
p
L
is
the
Lagu
er
r
e
po
ly
nom
ial
with
m
od
e
or
de
rs
p
an
d
m
.
Using
pro
pa
ga
ti
on
theo
ry
of
Gau
s
sia
n
bea
m
in
fr
ee
sp
ace
, th
e
beam
r
adius
at
the fro
nt s
urfac
e of the
transm
it
te
r
m
ic
ro
le
ns
is
4
0
2
2
2
0
1
1
t
f
(15)
wh
e
re
t
f
is
the
fo
cal
le
ng
t
h
of
the
transm
it
te
r
m
ic
ro
le
ns
S
ubsti
tuti
ng
(
14)
i
nt
o
(
7)
an
d
us
ing
(11
),
(
12)
,
an
d
(13),
we
ca
n o
btain the
opti
c
al
f
ie
ld at the
det
ect
or
s a
rr
ay
a
s
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S
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Int
J
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C
om
p
En
g,
V
ol.
9
, N
o.
3
,
June
2019
:
1620
-
1629
1624
4
1
2
2
2
1
2
2
2
2
t
2
2
2
2
1
2
2
2
2
2
t
2
2
2
2
2
2
2
t
2
2
1
2
1
1
2
1
2
2
2
1
2
2
2
1
2
s
i
n
/f
2
c
os
1
4
s
i
n
/f
2
c
os
e
x
p
s
i
n
/f
2
c
os
2
1
1
2
e
x
p
e
x
p
2
)
,
(
E
k
l
s
k
L
k
l
s
k
s
k
k
l
ik
im
l
ik
x
m
p
x
m
x
p
p
m
m
(
16)
In the a
bove
equati
on
k
is
l
f
l
ik
k
t
2
/
1
(
17)
The
a
bove
eq
ua
ti
on
is
cl
os
e
d
form
fo
rm
ula
for
the
opti
ca
l
fiel
d
of
t
he
LG
beam
propagati
ng
t
hro
ugh
le
ns
base
d
FSOIs
syst
e
m
with
s
li
gh
t
m
isa
li
gn
m
ent.
In
fact
(16)
prov
i
des
a
su
it
able
way
fo
r
est
i
m
at
i
n
g
the
cro
sstal
k
in
the
FSOIs
syst
em
under
c
onside
rati
on,
an
d
ca
n
be
us
e
d
to
st
udy
the
im
pact
of
beam
hig
her
order
m
od
es on the
c
ro
sstal
k
a
nd in t
urn on t
he FS
OI
s
syst
em
p
erf
orm
ance.
4.
OPTIC
AL
F
IEL
D
I
N
A
MISALIG
N
ED
FS
OIs
WITH
CIRC
ULA
R
APE
RTU
RE
US
I
NG
PROP
OSE
D MO
DEL
In
this
sect
io
n,
a
pr
op
a
gatio
n
m
od
el
of
an
LG
beam
pro
pag
at
in
g
i
n
a
ci
rcu
la
rly
a
pe
r
ture
d
F
SOIs
syst
e
m
is
der
ived
by
ex
pr
e
s
sing
the
ha
rd
aper
t
ur
e
functi
on
of
t
he
m
icr
ole
ns
into
i
n
te
rm
s
of
com
plex
Gau
s
sia
n f
un
ct
ion
s
[14,1
5]. L
et
)
(
1
A
the circ
ular
aper
t
ur
e
f
un
ct
i
on w
it
h
1
a
rad
i
us
1
1
1
1
1
0
1
)
(
a
a
A
(18)
us
in
g (7) a
nd
(
18) , we
fo
und the
fiel
d
distribu
ti
on at t
he pl
ane
of d
et
ect
or
s array
as:
1
1
1
1
2
1
2
1
2
2
1
0
2
0
1
1
1
2
2
2
)
c
os
(
e
x
p
2
e
x
p
2
e
x
p
)
(
)
,
(
E
2
)
,
(
E
d
d
b
ik
b
i
k
a
b
ik
A
b
ik
(19)
The
a
bove
i
ntegr
al
ca
n
be
s
olv
ed
by
e
xpress
ing
t
he
ci
rc
ular
ape
rtu
re
f
unc
ti
on
in
te
rm
s
of
lim
it
ed
nu
m
ber
of
com
plex
Ga
us
s
ia
n
f
unct
ion
s
[14
]
:
2
1
2
1
1
1
e
x
p
)
(
a
B
A
A
n
N
n
n
(20)
wh
e
re
n
A
and
n
B
ar
e
com
plex
coeffic
ie
nts.
The
va
lues
of
th
ese
coeffic
ie
nts
f
or
10
N
are
ta
bu
la
te
d
in
[14].
Substi
t
uting
(
20)
into
(
19)
a
nd
perform
ing
the inte
gr
at
io
n,
we ob
ta
in
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
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om
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g
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S
N: 20
88
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8708
Crosstalk i
n misali
gned
fr
ee s
pa
ce
opti
cal i
nt
erconnects:
m
od
el
li
ng
and si
mu
l
ation
(
Ne
dal Al
-
A
babne
h
)
1625
2
2
1
2
2
1
2
1
2
2
2
2
2
t
2
2
2
2
1
2
1
2
2
2
2
2
t
2
2
2
2
1
2
1
1
1
2
1
2
1
2
2
2
2
t
2
2
1
2
2
2
1
2
2
2
2
2
s
i
n
/f
2
c
os
1
4
s
i
n
/f
2
c
os
e
x
p
1
1
s
i
n
/f
2
c
os
2
2
e
x
p
e
x
p
2
)
,
(
E
a
B
k
l
l
s
k
L
a
B
k
l
s
k
a
B
k
a
B
k
A
s
l
ik
im
l
ik
n
x
m
p
n
x
p
n
N
n
p
m
n
n
m
x
m
(
21)
The
(21
)
is
the
m
ajo
r
c
on
tri
buti
on
in
t
his
pa
per
an
d
repres
ents
a
n
a
pprox
i
m
at
e
analy
ti
ca
l
relat
ion
f
or
the
op
ti
cal
fiel
d
of
the
LG
beam
pr
op
a
gat
ing
thr
ough
th
e
ci
rcu
la
rly
aper
ture
d
m
isa
l
i
gn
e
d
FS
OI
s
s
yst
e
m
sh
ow
n
in
Fi
gure
2.
N
ote
that
fo
r
infi
nite
ap
ertur
e
siz
e,
(21
)
beco
m
es
identic
al
to
(16)
der
ive
d
in
previo
us
sect
ion
.
I
n
fac
t,
this
relat
ion
pro
vid
es
a
s
ui
ta
ble
way
f
or
est
i
m
at
ing
th
e
cr
os
sta
lk
i
n
a
sli
gh
tl
y
m
isalign
ed
FSOIs syste
m
t
akin
g
int
o
acc
ount t
he
e
ff
ect
of f
i
nite ape
rtu
r
e of the
m
ic
ro
le
ns
.
5.
EST
IMA
TI
N
G CRO
S
STA
LK
-
TO
-
SIG
N
AL RATIO
Fo
r
the
sa
ke
of
est
im
at
ing
the
SCR
w
e
c
on
si
der
Fig
ur
e
2.
At
the
tra
ns
m
itter
m
ic
ro
le
ns
es
ar
ray,
the
ov
e
rlap
be
tween
the
int
end
e
d
m
ic
ro
le
ns
ape
rture
an
d
stray
li
gh
t
from
neigh
bor
ing
la
sers
cau
ses
a
cro
sstal
k
noise
.
The
stray
crosst
al
k
c
an
be
s
een
as
the
por
ti
on
of
the
transm
itted
powe
r
from
the
inten
de
d
so
urce
t
hat
pas
ses
thr
ough
nei
ghbori
ng
m
ic
ro
le
ns
es
an
d
rea
ch
oth
er
detect
or
s
as
s
how
n
i
n
Fi
gure
2.
T
he
stray
cro
sstal
k
powe
r
ca
n
be
f
ound
as
the
total
po
wer
receiv
ed
by
al
l
neig
hbor
i
ng
de
te
ct
or
s
f
r
om
the
li
gh
t
c
om
ing
thr
ough
neig
hbori
ng
m
ic
ro
le
ns
es
ass
um
ing
on
ly
the
inten
de
d
source
is
on.
I
n
this
case,
we
can
est
i
m
a
te
the
stray
cr
os
sta
lk
powe
r by
1
1
1
2
1
1
1
1
1
1
2
1
1
1
12
11
1
2
1
)
,
(
E
4
)
,
(
E
4
d
d
d
d
P
P
P
c
c
c
(22)
11
c
P
an
d
12
c
P
are
the
cro
sstal
k
no
ise
s
recei
ved
by
t
he
inte
nd
e
d
det
ect
or
from
the
neig
hbor
a
nd
next
neig
hbor
la
ser
s
resp
ect
ively
.
1
and
2
are
the
areas
co
ver
e
d
by
neig
hbor
a
nd
ne
xt
neig
hbor
tra
ns
m
itted
m
ic
ro
le
nes,
res
pecti
vely
.
O
n
t
he
oth
er
ha
nd,
at
the
detect
ors
plane
,
the
ove
rlap
betwee
n
t
he
inte
nd
e
d
de
te
ct
or
aper
t
ur
e
an
d
th
e
diffracte
d
li
ght
beam
s
fr
om
oth
e
r
VCSELs
through
ot
her
transm
itted
m
i
cro
le
nes
al
so
c
auses
a
cro
sstal
k
noi
se.
This
cr
os
st
al
k
is
cal
le
d
diffracti
on
in
duc
ed
crosst
al
k.
T
he
total
diffrac
ti
on
induce
d
cr
os
sta
lk
powe
r
recei
ve
d
by
the
inte
nded
detect
or
c
an
be
f
ound
a
s
the
power
re
cei
ved
by
al
l
neig
hbori
ng
de
te
ct
or
s
from
the
li
gh
t
com
ing
thr
ough
inte
nd
e
d
m
ic
ro
le
ns
powe
r
assum
ing
only
the
inte
nded
so
urce
is
on.
I
n
this
case,
we
ca
n
es
tim
a
te
the d
if
frac
ti
on
in
duce
d cros
sta
l
k powe
r by
2
2
2
2
2
2
2
2
2
2
2
2
2
2
22
21
2
4
3
)
,
(
E
4
)
,
(
E
4
d
d
d
d
P
P
P
c
c
c
(23)
21
c
P
and
22
c
P
are
the
cro
sstal
k
noise
s
received
by
th
e
detect
or
f
ro
m
the
neighb
or
and
nex
t
neig
hbor
detect
ors
resp
ect
ively
.
3
and
4
are
the
areas
co
ver
e
d
by
neig
hbor
and
nex
t
neig
hbor
detect
or
s
,
r
especti
vely
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8708
Int
J
Elec
&
C
om
p
En
g,
V
ol.
9
, N
o.
3
,
June
2019
:
1620
-
1629
1626
The
total
po
we
r
of
the
c
ro
ssta
lk
noise
is
the
su
m
of
1
c
P
an
d
2
c
P
.
The
total
sig
nal
powe
r
is
the
powe
r
recei
ve
d
by
the
inten
de
d
detect
or
f
rom
the
li
gh
t
com
ing
from
the
intende
d
s
our
ce
throu
gh
t
he
intende
d
m
ic
r
olens
.
Ba
sed on t
his,
we
est
im
at
e th
e sig
nal po
wer
as:
2
2
2
2
2
2
2
)
,
(
E
d
d
P
s
(24)
is
the
area
cov
ere
d
by
inten
ded
detect
or.
Af
te
r
est
im
ati
ng
the
sig
nal
and
the
c
ro
sst
al
k
for
the
FS
O
I
s
syst
e
m
,
the
SCR
cou
l
d
be
determ
ined.
I
n
our
m
od
el
,
no
de
te
ct
or
pream
pli
fier
is
us
ed
an
d
the
c
r
os
sta
lk
nois
e
resu
lt
in
g
f
ro
m
stray
li
gh
t
at
transm
itted
m
i
cro
le
nes,
li
ght
diffracti
on
f
rom
aper
ture
s,
a
nd
m
isa
li
gn
m
e
nt
are
include
d. Ba
se
d on (
22), (
23), an
d
(24), the
S
CR
r
at
io ca
n b
e d
et
erm
ined
a
s
2
1
c
c
s
P
P
P
SC
R
(25)
6.
NUMER
IC
A
L RES
ULTS
AND DIS
C
USSION
The
f
ollow
i
ng
pa
ram
et
ers
are
us
e
d
in
th
e
sim
ulatio
ns
:
f
or
the
l
ase
r
sourc
e
the
w
avelen
gth
is
0.850
µm
an
d
t
he
waist
ra
dius
of
eac
h
outp
ut
beam
is
3
µm
;
f
or
the
t
ran
sm
it
te
r
m
ic
ro
le
ns
the
f
ocal
le
ng
t
h
a
nd
the
diam
et
er
are
72
0
µm
and
25
0
µm
,
resp
ect
ively
.
Th
e
la
sers
ar
ray
is
placed
i
n
th
e
fo
c
us
plane
of
t
he
m
ic
ro
le
ns
a
rr
a
ys.
The
interc
onnecti
on
le
ngt
h
of
is
2.5
m
m
and
the
i
ntercon
necti
on
s
sp
aci
ng
is
25
0
µm
.
The
SCR
will
be
use
d
as
t
he
perf
or
m
ance
m
easur
e.
In
the
si
m
ulatio
ns
,
the
ef
fect
of
finite
ape
rtur
e
,
m
isa
li
gn
m
ent,
high
orde
r
m
od
es,
a
nd
stray
cro
sstal
k
on
th
e
SCR
will
be
e
m
ph
asi
zed
.
The
sim
ulati
o
n
s
ar
e
perform
ed
us
ing
t
he
ge
neral
iz
ed
Coll
ins
diffracti
on
t
heory
[
16
]
a
ssu
m
ing
a
lin
ear
F
SOIs
s
yst
e
m
.
To
a
pproxim
ate
the
ape
rtu
re
functi
on
we
us
e
the
sam
e
coeffic
ie
nts
us
ed
in
[14]
wit
h
10
N
.
Figure
3
sh
ows
the
a
pp
roxim
at
ed
aper
ture
f
unct
i
on
with
rad
i
us
mm
a
1
2
5
.
0
1
.
It
is
cl
ear
th
at
10
c
om
plex
Ga
us
sia
n
functi
ons
are
su
ff
ic
ie
nt
to
com
po
se
the
har
d
a
per
t
ure
fu
ncti
on.
Sim
ula
ti
on
resu
l
ts
in
the
fo
ll
ow
i
ng
su
bse
ct
ions,
6.1
a
nd
6.3
,
ass
um
e
on
ly
the
f
undam
ental
LG
m
od
e,
L
G
00
,
f
or
the
la
se
r
s
ource.
T
he
im
pact
of
VCSELs
’ hig
he
r
m
od
es
will
b
e c
onsidere
d
i
n
the
s
ub
sect
io
n 6.3.
Figure
3.
A
ppr
ox
im
at
ed
aper
t
ur
e
fun
ct
io
n wi
th r
a
diu
s
and
N=10
6
.
1.
S
C
R
f
or ali
gn
ed
FS
OIs
sy
s
tem
Figure
4
s
how
s
the
SCR
ve
rs
us
detect
or
ra
di
us
f
or
the
ap
er
ture
d
a
nd
the
unpe
rtu
rb
e
d
FS
OI
s
syst
em
s
with
a
n
interc
onnecti
on
le
ng
t
h
of
2.5
m
m
.
I
t
is
cl
ear
that
the
SCR
=
is
25
for
the
a
per
t
ured
FS
O
Is
a
nd
i
t
is
23
for
the
un
a
pert
ur
e
d
on
e
.
F
ur
therm
or
e,
t
he
SCR
dro
ps
qu
ic
kly
as
the
de
te
ct
or
rad
i
us
decr
ease
s.
M
oreo
ver,
op
ti
m
u
m
detect
or
r
adi
us
ca
n
be
ob
ta
ine
d
a
s sh
ow
n
in
Fig
ur
e
4.
F
or
e
xam
ple,
w
hen
the r
e
qu
i
red
SCR
is above
10
t
he
co
rr
es
pondin
g
detect
or
ra
diu
s
ranges
from
5
µ
m
to
77
µm
.
Fo
r
m
axim
u
m
SCR
,
t
he
opti
m
u
m
de
te
ct
or
rad
ii
f
or
t
he
a
pe
rtur
e
d
a
nd
the
unpe
rt
urbe
d
s
yst
e
m
s
are
22
µ
m
and
2
3
µm
,
res
pecti
vely
.
The
e
xistence of
s
uc
h
op
ti
m
u
m
can
be
ex
plaine
d
by
consi
der
i
ng
the
desig
n
form
ula
us
ed
to
m
od
e
l
the
cro
sstal
k.
In
this
pape
r
,
we
c
on
si
der
t
he
dif
fr
act
io
n
in
du
ce
d
c
ross
ta
lk
a
nd
t
he
stray
li
gh
t
cr
os
sta
lk
.
The
di
ffract
io
n
in
duced
cr
osst
al
k
mm
1
2
5
.
0
1
a
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
&
C
om
p
En
g
IS
S
N: 20
88
-
8708
Crosstalk i
n misali
gned
fr
ee s
pa
ce
opti
cal i
nt
erconnects:
m
od
el
li
ng
and si
mu
l
ation
(
Ne
dal Al
-
A
babne
h
)
1627
dep
e
nds
on
th
e
detect
or
radi
us
wh
il
e
the
stray
cr
os
sta
lk
de
pends
on
t
he
m
ic
ro
le
ns
rad
i
us
.
For
a
giv
e
n
m
ic
ro
le
ns
ra
diu
s,
the
stray
c
ro
sstal
k
is
const
ant
and
ov
e
r
wh
el
m
s
the
diffracti
on
c
ro
s
s
ta
lk
at
s
m
all
detect
or
siz
e.
In
t
his
c
a
se,
the
decr
eas
e
of
t
he
SCR
a
s
the
detect
or
siz
e
decr
eases
is
du
e
t
o
the
de
crease
in
t
he
sign
al
powe
r.
F
or
la
r
ge
detect
or
siz
e,
the
diffracti
on
c
r
os
sta
lk
is
do
m
inant
an
d
in
creases
with
the
detect
or
siz
e.
The
balance
be
tween
th
e
sig
na
l
power,
the
di
ff
racti
on
c
ro
s
s
ta
lk,
an
d
the
st
ray
crosst
al
k
e
xp
la
in
s
the
e
xistence
of opti
m
u
m
d
et
ect
or
size.
To
s
how
t
he
im
pact
of
cha
ngin
g
interc
onne
ct
s
den
sit
y
on
the
desig
n
par
am
et
ers
of
the
FS
OIs
,
we
hav
e
pl
otted
in
Fig
ure
5
t
he
SCR
ve
rsus
detect
or
rad
i
us
dif
fer
e
nt
i
nterco
nnect
s
s
pac
ing
.
It
is
note
d
from
the
F
ig
ure
that
the
SCR
decre
ases
as
inte
rc
onnects
de
ns
it
y
increases
.
F
or
a
n
i
nterc
onnects
s
pacin
g
of
25
0
m
m
,
26
0mm
,
and
27
0
m
m
t
he
c
orres
ponding
opti
m
u
m
values
of
t
he
S
CR
are
23,
39,
an
d
67,
resp
e
ct
ively
.
More
ov
e
r,
the
cor
respo
nd
i
ng
opti
m
u
m
detect
or
ra
dii
are
22
µm
,
16
µm
,
and
13
µm
,
resp
e
ct
ively
.
The
opti
m
u
m
value
of
the
de
te
ct
or
ra
diu
s
de
creases
as
int
ercon
nects
de
nsi
ty
decr
eases.
Ba
sed
on
the
above
si
m
ulati
on
s
an
d
an
al
yse
s,
it
app
ea
rs
t
hat
th
e
detect
or
siz
e
play
s
an
im
po
rta
nt
r
ole
an
d
strongly
a
ff
ec
ts
the
SCR
o
f
the
FSOIs
syst
em
.
Figure
4.
SCR
ver
s
us
detect
or
r
a
diu
s
f
or
t
he
aper
t
ur
e
d
a
nd t
he unpe
rturbe
d FS
OI
s
syst
em
Figure
5
.
SCR
ver
s
us
detect
or
r
a
diu
s
f
or
t
he
aper
t
ur
e
d
F
SO
Is
s
with i
nterc
onnects s
pacin
g
as a
par
am
et
er
6
.
2
.
S
C
R
f
or
mi
sa
li
gn
ed
FS
OIs s
ys
te
m
Figure
6
s
hows
the
SCR
ver
s
us
detec
tor
ra
diu
s
for
the
aper
tu
re
d
FS
OI
s
syst
e
m
with
an
interco
nnect
io
n
le
ngth
of
2.5
m
m
un
der
dif
fer
e
nt
m
isa
l
ignm
ents
for
t
he
m
ic
ro
le
ns
i
n
x
-
directi
on.
T
he
SCR
decr
ease
s
with
m
isa
l
ign
m
ent.
The
SCR
is
15
.
5
f
or
5
µm
m
isa
li
gn
m
ent,
wh
il
e
it
decr
ea
ses
to
10.5
when
the
m
isa
li
gn
m
ent
i
s
10
µm
.
More
ov
e
r,
t
he
opti
m
u
m
detect
or
rad
i
us
dec
reas
es
with
inc
reas
ing
the
m
isa
li
gn
m
ent.
This
ca
n
be
e
xp
la
ine
d
by
t
he
ai
d
of
us
in
g
Fig
ur
e
7.
It
is
cl
e
ar
t
hat
th
e
m
isalign
m
en
t
strongly
a
ff
e
ct
s
th
e
norm
al
iz
ed
cross
ta
lk
powe
r
plo
tt
ed
as
in
Fig
ur
e
8.
The
refore
,
opti
m
u
m
de
sign
will
choos
e
sm
a
ll
detect
or
siz
e
to m
ini
m
iz
e the crosstal
k
a
nd
in
tu
r
n
im
pr
ove the SCR.
6
.
3
.
Ef
fect
of
VCSEL
s’ hi
gher
m
od
es
on
SCR
In
Fig
ur
e
8,
w
e
ha
ve
plo
tt
ed
the
SCR
ve
rs
us
detect
or
ra
di
us
for
the
ape
r
ture
d
FS
O
Is
s
yst
e
m
in
the
pr
ese
nce
of
hi
gh
orde
r
m
od
es
f
or
the
la
ser
source
.
We
assum
ed
the
pr
ese
nce
of
LG00,
L
G
00
+LG
01
,
LG
00
+LG
01
+L
G
10
,
res
pecti
ve
ly
.
The
fr
a
ct
io
nal
coe
ff
ic
ie
nt
s
in
th
e
tw
o
m
od
es
case
for
LG
00
,
L
G
01
are
0.9
and
0.1
,
res
pec
ti
vely
,
wh
il
e
t
hey
are
0.4,
0.55,
a
nd
0.0
5
f
or
LG
00
,
LG
01,
and
LG
10
res
pe
ct
ively
,
for
th
e
three
m
od
es
[5
]
,
[17
]
.
W
e
can
see
from
Figu
re
8
that
the
SCR
,
in
the
pr
ese
nc
e
of
only
the
LG
00
m
od
e,
is
higher
than
that
w
hen
two
or
t
hr
ee
m
od
es
are
pr
e
sent.
F
or
exa
m
ple,
the
op
ti
m
u
m
SCR
fo
r
the
LG
00
m
ode
is
23
.
Howe
ver,
the
op
ti
m
u
m
SCR
fo
r
t
he
LG
00
+LG
01
m
od
es
and
LG
00
+L
G
01
+LG
10
m
odes
ha
ve
dro
pp
e
d
t
o
appr
ox
im
at
ely
14
a
nd
2.7
,
r
especti
vely
.
T
he
c
orres
ponding
opt
im
u
m
detect
or
ra
dii
f
or
sin
gle
m
od
e,
t
w
o
m
od
es,
an
d
th
r
ee
m
od
es
are
22
µm
,
28
µm
,
and
42
µm
,
res
pecti
vely
.
T
he
op
ti
m
u
m
detect
or
rad
i
us
inc
re
ases
with
num
ber
of
m
od
es.
This
can
be
e
xp
la
in
ed
base
d
on
th
e
fact
that
the
sp
rea
d
of
the
li
gh
t
increa
ses
with
nu
m
be
r of m
od
es a
nd the
d
et
ect
or
rad
i
us
s
houl
d
inc
rease t
o gathe
r
m
or
e l
igh
t
for
la
r
ger
SCR
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8708
Int
J
Elec
&
C
om
p
En
g,
V
ol.
9
, N
o.
3
,
June
2019
:
1620
-
1629
1628
Figure
6
.
SCR
ver
s
us
detect
or
r
a
diu
s
f
or
t
he
aper
t
ur
e
d
F
SOIs syst
em
w
it
h
x
-
m
isa
li
gn
m
en
t
as a
par
am
et
er
Figure
7.
N
orm
al
iz
ed
cro
sst
al
k
ve
rs
us
dete
ct
or
rad
i
us
for
the
m
isa
l
ign
ed
ap
e
rtu
red
FSOIs
Figure
8
.
SCR
ver
s
us
detect
or
r
a
diu
s
f
or
t
he a
per
t
ur
e
d
F
SOIs syst
em
w
it
h
diff
e
re
nt laser
m
od
es
7.
CONCL
US
I
O
N
We
ha
ve
a
naly
zed
a
sli
gh
tl
y
m
isa
li
gn
ed
f
re
e
sp
ace
opti
cal
interco
nnect
s
syst
e
m
e
m
plo
yin
g
ar
rays
of
la
ser
sour
ces,
m
ic
ro
le
ns
es,
a
nd
detect
ors.
A
no
vel
dif
fr
a
ct
ion
m
od
el
ha
s
bee
n
pr
opose
d
us
in
g
cy
li
nd
rica
l
form
of
Coll
in
s
dif
fr
act
io
n
in
te
gr
al
.
S
om
e
a
ppr
ox
i
m
at
e
cl
o
sed
f
orm
fo
rm
ulas
f
or
t
he
li
ght
opti
cal
fiel
d
at
the
detect
or
pla
ne
wer
e
de
rive
d.
Using
t
he
pr
opose
d
dif
fr
act
i
on
m
od
el
an
d
the
de
rive
d
for
m
ulas,
the
SCR
has
been
est
i
m
at
ed
an
d
use
d
a
s
th
e
perf
or
m
ance
m
easur
e.
N
ume
rical
si
m
ulatio
ns
ha
ve
s
how
n
that
m
isa
li
gn
m
ent,
stray
li
gh
t,
m
icr
ole
ns
fi
nite
ap
ertur
e
,
a
nd
la
s
er
hi
gh
e
r
order
m
od
es
a
ff
ect
t
he
f
ree
s
pace
opti
cal
interco
nnect
s
syst
e
m
and
hav
e
stron
g
influe
nces
on
the
sign
al
-
to
-
c
r
os
sta
lk
rati
o,
an
d
it
is
i
m
po
rtant
to
consi
der
them
in
real
app
li
cat
io
ns
.
Mo
re
ov
e
r,
our
analy
ses
ha
ve
sh
ow
n
that
opt
i
m
u
m
SCR
ca
n
be
obta
ined
us
in
g
the
detec
tor
siz
e
as opti
m
iz
at
ion
p
a
ram
et
er.
REFERE
NCE
S
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a
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c
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In
t J
Elec
&
C
om
p
En
g
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88
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8708
Crosstalk i
n misali
gned
fr
ee s
pa
ce
opti
cal i
nt
erconnects:
m
od
el
li
ng
and si
mu
l
ation
(
Ne
dal Al
-
A
babne
h
)
1629
[7]
N.
S.
Petrov
ic
´
and
A.
D.
Rakic´
,
“
Modeli
ng
d
iffr
action
and
i
m
agi
ng
of
l
ase
r
b
ea
m
s
b
y
the
m
ode
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expa
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m
et
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”
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F.
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n
,
W
.
L.
Hendri
ck,
P.
J.
Marc
hand
,
and
S.
C.
Esener,
“
Misali
gnm
ent
tol
e
ran
c
e
an
alys
is
of
fre
e
-
spac
e
opti
c
al
int
er
conn
ec
ts v
ia sta
t
isti
c
a
l
m
et
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App
l
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La
cro
ix
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uf
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“
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i
m
ent
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eri
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al
an
aly
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of
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isal
ignmen
t
tol
er
anc
es
in
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e
e
-
spac
e
opt
ic
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enhua
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X
iuj
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n
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i,
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Kong
“
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isal
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gned
fre
e
-
spac
e
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c
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”
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”
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s
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err
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“
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le
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of
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al
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a
y
s
,
i
n
Progress
in
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pti
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”
ed.
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olf
(North
-
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Am
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am,
1988
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[14]
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“
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ea
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n
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rm
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rix
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”
J. Opt.
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le
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r
D.
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ia
n
L
.
Maje
ws
ki,
“
Anal
y
sis
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le
nsl
ess
fre
e
-
spac
e
optical
in
te
r
conne
c
t
s
base
d
on
m
ult
i
-
t
ran
sverse
m
ode
ver
tica
l
-
c
avi
t
y
-
surfac
e
-
emit
ti
ng
la
sers
,
”
Opti
cs
Comm
unic
ati
o
ns
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27
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.
BIOGR
AP
H
Y
O
F
AU
TH
OR
Pr
of.
Nedal
Al
-
Ab
ab
neh
rec
ei
v
ed
his
B.
Sc.
and
M.Sc.
degr
ee
s
i
n
El
ec
tr
ical
Eng
ine
er
ing
fro
m
Jordan
Univer
sit
y
of
Scie
n
ce
an
d
Te
chnol
og
y
in
1993
and
1996
respe
ct
iv
ely
.
He
rec
e
ive
d
his
Doctor
al
of
En
gine
er
ing
degr
e
e
in
El
e
ct
ri
ca
l
engi
ne
eri
ng
fro
m
Univer
sit
y
of
Mass
ac
husett
s
-
Lowe
ll
(US
A)
in
2004.
He
is
cur
ren
t
l
y
ful
l
profe
ss
or
with
the
dep
art
m
ent
of
elec
tr
ical
engi
ne
eri
ng
in
Jorda
n
Univer
sit
y
of
Scie
nc
e
an
d
Te
chnol
og
y
.
His
rese
arc
h
intere
sts
are
in
fre
e
spac
e
op
ti
c
al i
nt
erc
onne
ct
s
and
d
istri
bute
d
de
tection s
y
st
ems
.
Evaluation Warning : The document was created with Spire.PDF for Python.