nonlinear optics lab. hanyang univ. chapter 3. propagation of optical beams in fibers 3.0...
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Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Chapter 3. Propagation of Optical Beams in Fibers
3.0 Introduction
Optical fibers Optical communication - Minimal loss - Minimal spread - Minimal contamination by noise - High-data-rate
In this chapter, - Optical guided modes in fibers - Pulse spreading due to group velocity dispersion - Compensation for group velocity dispersion
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
3.1 Wave Equations in Cylindrical Coordinates
Refractive index profiles of most fibers are cylindrical symmetric Cylindrical coordinate system
The wave equation for z component of the field vectors :
022
z
z
H
Ek where,
2
2
2
2
22
2 11
zrrrr
2222 /cnk and
Since we are concerned with the propagation along the waveguide, we assume that every component of the field vector has the same z- and t-dependence of exp[i(t-z)]
)](exp[
),(
),(
),(
),(zti
r
r
t
t
H
E
rH
rE
# Solve for zz HE ,first and then expressing HHEE rr ,,, in terms of zz HE ,
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
From Maxwell’s curl equations :
tt
E
HH
E ,
zr Hr
HiEi
1
zr Hr
HiEi
)(11
rH
rrH
rEi rz
zr Er
EiHi
1
zr Er
EiHi
)(11
rE
rrE
rHi rz
zzr Hr
Er
iE
22
zz Hr
Er
iE
22
zzr Er
Hr
iH
22
zz Er
Hr
iH
22
in terms ofWe can solve for HHEE rr ,,, zz HE ,
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
022
z
z
H
Ek (3.1-1)
zz HE ,Now, let’s determine
0)(11 22
2
2
22
2
z
z
H
Ek
rrrr
The solution takes the form : )exp()( ilrH
E
z
z
where, ...,3,2,1,0l
01
2
222
2
2
r
lk
rrr
)()()( 21 hrYchrJcr ll
)()()( 21 qrKcqrIcr ll
1)
2)
:022 k
:022 k
where,
where,
,222 kh
,222 kq
ll YJ ,
ll KI ,
: Bessel functions of the 1st and 2nd kind order of l
: Modified Bessel functions of the 1st and 2nd kind of order l
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Asymptotic forms of Bessel functions :
l
l
x
lxJ
2!
1)(
...5772.0
2ln
2)(0
xxY
l
l x
lxY
2)!1(
)(
l
l
x
lxI
2!
1)(
...5772.0
2ln)(0
xxK
l
l x
lxK
2
2
)!1()(
,...3,2,1l
,...3,2,1l
1For x lx ,1For
42cos
2)(
21
lx
xxJ l
42sin
2)(
21
lx
xxYl
xl e
xxI
21
2
1)(
xl e
xxK
21
2)(
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
3.2 The Step-Index Circular Waveguide
<Index profile of a step-index circular waveguide>
1) ar(cladding region) :
The field of confined modes :1x
*
022 k
: evanescent (decay) wave
cnkn /and 202
* : virtually zero at )( br
xl exxI 2
1
)( is not proper for the solution
zltiqrCKtE lz exp)(),(r
zltiqrDKtH lz exp)(),(rar
where, 20
22
22 knq
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
2) ar(core region) : 1x
*
022 k
: finite at
cnkn /and 101 * : propagating wave
ll xxY )( is not proper for the solution
where, 220
21
2 knh
0r
zltihrBJtH lz exp)(),(r
zltihrAJtE lz exp)(),(rar
* Necessary condition for confined modes to exist :
0201 knkn
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Other field components
zltihrBJr
lihrJAh
h
iE llr
exp)()(2
zltihrJBhhrAJr
il
h
iE ll
exp)()(2
zltihrAJE lz exp)(
zltihrAJr
lihrJBh
h
iH llr
exp)()( 12
zltihrJAhhrBJr
il
h
iH ll
exp)()( 12
zltihrBJH lz exp)(
)( core 1) ar )( cladding 2) ar
zltiqrDKr
liqrKCq
q
iE llr
exp)()(
2
zltiqrKDqqrCKr
il
q
iE ll
exp)()(2
zltiqrCKE lz exp)(
zltiqrCKr
liqrKDq
q
iH llr
exp)()( 2
2
zltiqrKCqqrDKr
il
h
iH ll
exp)()( 22
zltiqrDKH lz exp)(
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Boundary condition : tangential components of field are continuous at ar
zz HHEE ,,,
0)()()()(22
qaKq
DqaKaq
ilChaJ
hBhaJ
ah
ilA llll
0)()()()(2
22
1
qaKaq
ilDqaK
qChaJ
ah
ilBhaJ
hA llll
0)()( qaCKhaAJ ll
0)()( qaDKhaBJ ll
(3.2-10)
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Amplitude ratios : [from (3.2-10) with determined eigenvalue Report]
)(
)(
qaK
haJ
A
C
l
l1
2222 )(
)(
)(
)(11
qaaqK
qaK
hahaJ
haJ
ahaq
li
A
B
l
l
l
l
A
B
qaK
haJ
A
D
l
l
)(
)(
: the relative amount of Ez and Hz in a mode
Condition for nontrivial solution to exist : (Report)
2
0
222
222
21 11
)(
)(
)(
)(
)(
)(
)(
)(
khaqa
lqaqaK
qaKn
hahaJ
haJn
qaqaK
qaK
hahaJ
haJ
l
l
l
l
l
l
l
l
is to be determined for each l
(3.2-11)
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Mode characteristics and Cutoff conditions
(3.2-11) is quadratic in )(/)( hahaJhaJ ll Two classes in solutions can be obtained, and designated as the EH and HE modes. (Hybrid modes)
(3.2-11) 2
1
2
2222
2
021
222
21
22
21
21
22
21 11
22)(
)(
ahaqkn
l
qaK
K
n
nn
qaK
K
n
nn
hahaJ
haJ
l
l
l
l
l
l
By using the Bessel function relations : ,)()()( 1 xJx
lxJxJ lll
)()()( 1 xJx
lxJxJ lll
R
ha
l
qaqaK
qaK
n
nn
hahaJ
haJ
l
l
l
l22
1
22
211
)(
)(
2)(
)(
R
ha
l
qaqaK
qaK
n
nn
hahaJ
haJ
l
l
l
l22
1
22
211
)(
)(
2)(
)(
21
2
2222
2
01
22
21
22
21 11
)(
)(
2
ahaqkn
l
qaqaK
qaK
n
nnR
l
l where,
: EH modes
: HE modes
: Can be solved graphically(3.2-15)
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Special case (l=0)
1) HE modes
)(
)(
)(
)(
0
1
0
1
qaqaK
qaK
hahaJ
haJ
)()(,)()( 111'0 xJxJxKxK (3.2-15b) &
From (3.2-10), 0CA (Report)
Therefore, from (3.2-6)~(3.2-9), nonvanishing components are EHH zr ,, (TE modes)
)()(,)()( 111'0 xJxJxKxK (3.2-15a) &
From (3.2-10), 0DB (Report)
Therefore, from (3.2-6)~(3.2-9), nonvanishing components are HEE zr ,, (TM modes)
2) EH modes
)(
)(
)(
)(
021
122
0
1
qaKnqa
qaKn
hahaJ
haJ
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Graphical Solution for the confined TE modes (l=0)
)(
)(
)(
)(
0
1
0
1
qaqaK
qaK
hahaJ
haJ
2
1
)0(
)0(
0
1 haJ
J
)ln()(
2~
)(
)(222222
0
1
ahVahVVhaqaK
VhaK
q should be real to achieve the exponential decay of the field in the cladding
220
21
2 knh 0102& knkn
*
2220
22
21
2 )()()( haaknnqa
2/122
210 )(0 nnakVha
)(
)(
)0(
)0(
0
1
0
1
VVK
VK
haqaK
haK
)4
tan(1
~)1(
)1(
0
1
hahahahaJ
haJ
Roots of J0(ha)=0
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
* If the max value of ha, V is smaller than the first root of J0(x), 2.405 => no TE mode
* Cutoff value (a/) for TE0m (or TM0m) waves :
2122
21
0
0 2 nn
xa m
m
where, mx0: mth zero of J0(x)
* Asymtotic formula for higher zeros :
)4
1(~0 mx m
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Special case (l=1)
<EH modes> <HE modes>
* HE mode does not have a cutoff.* All other HE1m, EH1m modes have cutoff value of a/* Asymptotic formula for higher zero : 212
221
'1
1 2 nn
xa m
m
)4
1(~1 mx m
modes for'where, 1mEHmmmodesfor 1' 1mHEmm
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
The cutoff value for a/ (l>1)
2122
212 nn
za lm
HE
lm
2122
212 nn
xa lm
EH
lm
where, zlm is the mth root of )(1)1()( 122
21 zJ
n
nlzzJ ll
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Propagation constant,
0kn
: (effective) mode index
)/ of valuecutoff( 0lm2 knn #: poorly confined
1nn# : tightly confined
# V<2.405 Only the fundamental HE11 mode can propagate (single mode fiber)
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
3.3 Linearly Polarized Modes
The exact expression for the hybrid modes (EHlm, HElm) are very complicated.If we assume n1-n2<<1 (reasonable in most fibers) a good approximation of the field components and mode condition can be obtained. (D. Gloge, 1971) Cartesian components of the field vectors may be used.
hqnn ,121
<Wave equation for the Cartesian field components>
1) y-polarized waves
ztieqrBK
ztiehrAJE
ill
ill
y
exp)(
exp)(0xE
ar ar
(2.4-1), (3.1-2) & assume Ez<<Ey
yyx EEz
iH
0yH yz Ex
iH
yxz E
y
iH
y
iE
2
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Expressions for the field components in core (r<a)
After tedious calculations, (3.3-6)~(3.3-17), … (x, y)
Expressions for the field components in cladding (r>a)
ztieqrBKE illy exp)(0xE
ztieqrBKH illx
exp)( 0yH
ztieqrKeqrKBiq
H lil
lilz
exp)()(2
)1(1
)1(1
ztieqrKeqrKBq
E lil
lilz
exp)()(2
)1(1
)1(1
0xE ztiehrAJE illy exp)(
ztiehrJehrJAh
E lil
lilz
exp)()(2
)1(1
)1(1
ztiehrAJH illx
exp)( 0yH
ztiehrJehrJAih
H lil
lilz
exp)()(2
)1(1
)1(1
Continuity condition :
)(
)(
qaK
haAJB
l
l
0201, knkn
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
2) x-polarized waves (similar procedure to the case y-polarized waves)
ztiehrAJE illx exp)( 0yE
ztiehrAJH illy
exp)(0xH
ztiehrJehrJAh
H lil
lilz
exp)()(2
)1(1
)1(1
ztiehrJehrJAh
iE lil
lilz
exp)()(2
)1(1
)1(1
In core (r<a)
In cladding (r>a)
ztieqrBKE illx exp)( 0yE
ztieqrBKH illy
exp)(0xH
ztieqrKeqrKBq
H lil
lilz
exp)()(2
)1(1
)1(1
ztieqrKeqrKBq
iE lil
lilz
exp)()(2
)1(1
)1(1
Continuity condition Mode condition :
)(
)(
)(
)( 11
qaK
qaKq
haJ
haJh
l
l
l
l
)(
)(
)(
)( 11
qaK
qaKq
haJ
haJh
l
l
l
l
and/or simpler than (3.2-11)
: This results also can be obtained from the y-polarized wave solution. x- and y-modes are degenerated.
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Graphical Solution for the confined modes (l=0)
220
22
21
2 )()()(,, haknnqaqaYhaX
)(
)(
)(
)( 11
qaK
qaKq
haJ
haJh
l
l
l
l
modes: lmLP221
20 lmlm hnk
<Possible distribution of LP11>
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Mode cutoff value of a/
0)(1 VJ l 2122
21
2122
210 2 nn
annakV
where,0: q (3.3-27)
Ex) l=0, cutoffno:)LP(0at0)()( 0111 VVJVJ
)LP(832.3at0)( 021 VVJ
Ref : Table 3-1Cutoff value of V for some low-order LP
Asymptotic formula for higher modes :
22
3)(
lmLPV lm
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Power flow and power density
The time-averaged Poynting vector along the waveguide :
**Re2
1xyyxz HEHES
(3.3-18), (2.3-19)
)(
2
)(2
22
22
hrKB
hrJAS
l
l
z
ar
ar
)()()(2
11222 ahJahJahJAa lll
)()()(2 11
222 qaKqaKqaKBa lll
2
0 0
a
zcore rdrdSP
2
0 a zclad rdrdSP
])()()([2
11
2
222 ahJahJq
hahJAa lll
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
The ratio of cladding power to the total power, 2 :
)()(
)()(1
11
222
22 ahJahJ
ahJqaha
Vpp
P
p
P
ll
l
cladcore
cladclad
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
3.4 Optical Pulse Propagation and Pulse Spreading in Fibers
One bit of information = digital pulseLimit ability to reduce the pulse width : Group velocity dispersion
Group velocity dispersionConsidering a Single mode / Gaussian pulse, temporal envelope at z=0 (input plane of fiber) :
)]Re[exp(),(),0,,( 02
0 tityxutyxE
where, ),(0 yxu : transverse modal profile of the mode
Fourier transformation :
2/10 and
])exp()(~
)exp(),(Re[),0,,( 00 dtiftiyxutyxE
where, 212
2
4
)2exp()][exp()(
~
tFTf
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Propagation delay factor for wave with the frequency of
),(0 yxu
0 ])(exp[: 0 zi Let’s take complex expression and omit the
(are not invloved in the analysis and can be restored when needed)
dztiftzE ]})()[(exp{)(~
),( 00
Taylor series expansion : ...2
1)()( 2
2
2
00
00
d
d
d
d
z
vd
d
v
ztifdztitzE
gg
200
1
2
1exp)(
~)](exp[),(
),()](exp[ 00 tzzti E
where, velocitygroup
11),(
0
00 gvd
d
z
vd
d
v
ztifdtz
gg
1
2
1exp)(
~),(
E
za
v
ztifd
g
exp)(~
(3.4-5): Field envelope
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
The pulse spreading is caused by the group velocity dispersion characterized by the parameter,
d
dv
vvd
d
d
da g
gg22
2
2
11
2
1
2
1
0
(3.4-3)(3.4-5) :
d
v
ztiiaztz
g 4
1exp
4
1),( 2E
222
2
22
2
161
)(4exp
161
)(exp
41
1
za
vztazi
za
vzt
zaigg
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
If we use the definition of factor a,
# Pulse duration at z (FWHM)
2
20
0
2ln81)(
aL
L
initial pulse width
# |aL|>>0 (large distance) :0
)2ln8(~)(
aL
L
02
2ln4)(
L
d
dv
vL g
g
Practical Expression :
2
20
2
0
2ln21)(
DL
cL
where,
2
2
2
2/
d
dc
L
ddTD a
c2
4
T: pulse transmission time through length L of the fiber
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Group velocity dispersion
1) Material dispersion : n() depends on Waveguide dispersion : lm depends on (& geometry of fiber)
cnnnkn lmlmlm
),,( 210
i) 1
)(
d
d
d
dv lm
lmlmg : modal dispersion
ii) Single mode fiber,
c
nnn
n
nn
n
n
cd
d
vg
2
2
1
1
1
material dispersion waveguide dispersion
(3.4-18)
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
From the uniform dielectric perturbation theory,
222
211
22 nn
c
where, : Fractions of power flowing in the core and cladding 21,
n
n
n
n 11
1
n
n
n
n 22
2
(3.4-18)
c
nnn
n
nn
n
n
cd
d
v wg
22
211
1
1
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
In weakly guiding fiber : n1~n2
m
nnn
21
c
nnn
cd
d
v wmg
1
c
nnn
c wm
Group velocity dispersion :
wm
nn
cD
2
2
2
2
ex) GeO2-doped silica : m3.1at02
2
m
n
# depends on core diameter, n1, n2 control the waveguide shapew
n
2
2
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Group velocity dispersion & dispersion-flattened and dispersion-shifted fibers
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Frequency chirping
: modification of the optical frequency due to the dispersion
,161
)(4exp
161
)(exp
41
1),(
222
2
0022
2
za
vztazizti
za
vzt
zaitzE gg
(3.4-6)
where,
d
dv
vd
da g
g22
2
2
1
2
1
Total optical phase :
222
2
00161
)(4),(
za
vztazzttz g
Optical frequency :
)(161
8),(),(2220 gvzt
za
aztz
ttz
0d
dvg
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
3.5 Compensation for Group Velocity Dispersion
(3.4-5)
dti
v
ziiazftz
g
)exp(exp)(~
),( 2E
2exp)(~
,
iazf
v
ztzFT
g
E
Fiber transfer function
By convolution theorem, (1.6-2),
tdttaz
itf
zitz
2
4exp)(
4
1),(
E
2
4exp
4
1)( t
az
i
zit
: envelop impulse response
of a fiber of length z
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
Compensation for pulse broadening
1) By optical fiber with opposite dispersion
)(~
)i 1 f
)exp()(~
)(~
)ii 21112 Liaff
)exp()(~
)(~
)iii 22223 Liaff
222111 )(exp)(
~ LaLaif (a1L=-a2L)
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
2) By phase conjugation
dtiftitf 00 exp~
)exp()(conjugatortoInput
dtif 0* exp
~conjugatorfromOutput
)(~
)i f
)exp()(~
)(~
)ii 21112 Liaff
)exp()(~
)(~
)(~
)iii 211
*1
*23 Liafff
22211
*1 )(exp)(
~ LaLaif
)exp()(~
)(~
)iv 22234 Liaff
(a1L=a2L)
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
<Experimental setup> <Eye diagram>
Where are (b) and (c) ?? Refer to the text
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
3.7 Attenuation in Silica Fibers
Recently, 400 Mb/s, 100 km @ 1.55 m
Residual OH contamination of the glass
1.55 m is favored for long-distanceoptical communication