chapter 3. propagation of optical beams in fibers -...
TRANSCRIPT
![Page 1: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/1.jpg)
Nonlinear Optics Lab. Hanyang 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
![Page 2: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/2.jpg)
Nonlinear Optics Lab. Hanyang 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-bz)]
)](exp[),(
),(
),(
),(zti
r
r
t
tb
H
E
rH
rE
# Solve for zz HE , first and then expressing HHEE rr ,,, in terms of
zz HE ,
![Page 3: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/3.jpg)
Nonlinear Optics Lab. Hanyang Univ.
From Maxwell’s curl equations :tt
EH
HE ,
zr Hr
HiEi
b
1
zr Hr
HiEi
b
)(11
rHrr
Hr
Ei rz
zr Er
EiHi
b
1
zr Er
EiHi
b
)(11
rErr
Er
Hi rz
zzr H
rE
r
iE
b
b
b22
zz H
rE
r
iE
b
b
b 22
zzr E
rH
r
iH
b
b
b22
zz E
rH
r
iH
b
b
b 22
in terms ofWe can solve for HHEE rr ,,,zz HE ,
![Page 4: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/4.jpg)
Nonlinear Optics Lab. Hanyang 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
rrrrb
The solution takes the form : )exp()( ilrH
E
z
z
where, ...,3,2,1,0l
01
2
222
2
2
b
r
lk
rrr
)()()( 21 hrYchrJcr ll
)()()( 21 qrKcqrIcr ll
1)
2)
:022 bk
:022 bk
where,
where,
,222 bkh
,222 kq b
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
![Page 5: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/5.jpg)
Nonlinear Optics Lab. Hanyang Univ.
Asymptotic forms of Bessel functions :
l
l
x
lxJ
2!
1)(
...5772.0
2ln
2)(0
xxY
l
lx
lxY
2)!1()(
l
l
x
lxI
2!
1)(
...5772.0
2ln)(0
xxK
l
lx
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
x
l ex
xI2
1
2
1)(
x
l ex
xK
21
2)(
![Page 6: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/6.jpg)
Nonlinear Optics Lab. Hanyang 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 bk
: evanescent (decay) wave
cnkn /and 202 b
* : virtually zero at )( br
x
l exxI 21
)(
is not proper for the solution
zltiqrCKtE lz b exp)(),(r
zltiqrDKtH lz b exp)(),(rar
where, 2
0
2
2
22 knq b
![Page 7: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/7.jpg)
Nonlinear Optics Lab. Hanyang Univ.
2) ar (core region) : 1x
*
022 bk
: finite at
cnkn /and 101 b
* : propagating wave
l
l xxY )( is not proper for the solution
where, 22
0
2
1
2 b knh
0r
zltihrBJtH lz b exp)(),(r
zltihrAJtE lz b exp)(),(rar
* Necessary condition for confined modes to exist :
0201 knkn b
![Page 8: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/8.jpg)
Nonlinear Optics Lab. Hanyang Univ.
Other field components
zltihrBJr
lihrJAh
h
iE llr b
b
b
exp)()(
2
zltihrJBhhrAJr
il
h
iE ll b
b
b
exp)()(
2
zltihrAJE lz b exp)(
zltihrAJr
lihrJBh
h
iH llr b
b
b
exp)()( 1
2
zltihrJAhhrBJr
il
h
iH ll b
b
b
exp)()( 1
2
zltihrBJH lz b exp)(
)( core 1) ar )( cladding 2) ar
zltiqrDKr
liqrKCq
q
iE llr b
b
b
exp)()(
2
zltiqrKDqqrCKr
il
q
iE ll b
b
b
exp)()(
2
zltiqrCKE lz b exp)(
zltiqrCKr
liqrKDq
q
iH llr b
b
b
exp)()( 2
2
zltiqrKCqqrDKr
il
h
iH ll b
b
b
exp)()( 2
2
zltiqrDKH lz b exp)(
![Page 9: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/9.jpg)
Nonlinear Optics Lab. Hanyang Univ.
Boundary condition : tangential components of field are continuous at ar
zz HHEE ,,,
0)()()()(22
qaK
qDqaK
aq
ilChaJ
hBhaJ
ah
ilA llll
b
b
0)()()()(2
2
2
1
qaK
aq
ilDqaK
qChaJ
ah
ilBhaJ
hA llll
b
b
0)()( qaCKhaAJ ll
0)()( qaDKhaBJ ll
(3.2-10)
![Page 10: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/10.jpg)
Nonlinear Optics Lab. Hanyang Univ.
Amplitude ratios : [from (3.2-10) with determined eigenvalue b, Report]
)(
)(
qaK
haJ
A
C
l
l
1
2222 )(
)(
)(
)(11
qaaqK
qaK
hahaJ
haJ
ahaq
li
A
B
l
l
l
l
b
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
22
2
2
1 11
)(
)(
)(
)(
)(
)(
)(
)(
khaqal
qaqaK
qaKn
hahaJ
haJn
qaqaK
qaK
hahaJ
haJ
l
l
l
l
l
l
l
l b
is to be determined for each lb
(3.2-11)
![Page 11: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/11.jpg)
Nonlinear Optics Lab. Hanyang 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)
21
2
2222
2
0
2
1
22
2
2
1
2
2
2
1
2
1
2
2
2
1 11
22)(
)(
ahaqkn
l
qaK
K
n
nn
qaK
K
n
nn
hahaJ
haJ
l
l
l
l
l
l b
By using the Bessel function relations : ,)()()( 1 xJx
lxJxJ lll
)()()( 1 xJ
x
lxJxJ lll
Rha
l
qaqaK
qaK
n
nn
hahaJ
haJ
l
l
l
l
22
1
2
2
2
11
)(
)(
2)(
)(
R
ha
l
qaqaK
qaK
n
nn
hahaJ
haJ
l
l
l
l
22
1
2
2
2
11
)(
)(
2)(
)(
21
2
2222
2
01
22
2
1
2
2
2
1 11
)(
)(
2
ahaqkn
l
qaqaK
qaK
n
nnR
l
l bwhere,
: EH modes
: HE modes
: Can be solved graphically(3.2-15)
![Page 12: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/12.jpg)
Nonlinear Optics Lab. Hanyang 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
)(
)(
)(
)(
0
2
1
1
2
2
0
1
qaKnqa
qaKn
hahaJ
haJ
![Page 13: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/13.jpg)
Nonlinear Optics Lab. Hanyang 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
22
0
2
1
2 b knh 0102& knkn b
*
222
0
2
2
2
1
2 )()()( haaknnqa
2/12
2
2
10 )(0 nnakVha
)(
)(
)0(
)0(
0
1
0
1
VVK
VK
haqaK
haK
)4
tan(1
~)1(
)1(
0
1
ha
hahahaJ
haJ
Roots of J0(ha)=0
![Page 14: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/14.jpg)
Nonlinear Optics Lab. Hanyang 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/l) for TE0m (or TM0m) waves :
212
2
2
1
0
0 2 nn
xa m
m
l
where, mx0 : mth zero of J0(x)
* Asymtotic formula for higher zeros :
)4
1(~0 mx m
![Page 15: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/15.jpg)
Nonlinear Optics Lab. Hanyang 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/l :
* Asymptotic formula for higher zero : 212
2
2
1
'1
1 2 nn
xa m
m
l
)4
1(~1 mx m
modes for'where, 1mEHmm
modesfor 1' 1mHEmm
![Page 16: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/16.jpg)
Nonlinear Optics Lab. Hanyang Univ.
The cutoff value for a/l (l>1)
212
2
2
12 nn
za lm
HE
lm
l
212
2
2
12 nn
xa lm
EH
lm
l
where, zlm is the mth root of )(1)1()( 12
2
2
1 zJn
nlzzJ ll
![Page 17: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/17.jpg)
Nonlinear Optics Lab. Hanyang Univ.
Propagation constant, b
0kn
b : (effective) mode index
)/ of valuecutoff( 0lm2 knn b#
: poorly confined
1nn# : tightly confined
# V<2.405
Only the fundamental HE11 mode
can propagate (single mode fiber)
![Page 18: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/18.jpg)
Nonlinear Optics Lab. Hanyang 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.
b hqnn ,121
<Wave equation for the Cartesian field components>
1) y-polarized waves
ztieqrBK
ztiehrAJE
il
l
il
l
yb
b
exp)(
exp)(0xE
ar
ar
(2.4-1), (3.1-2) & assume Ez<<Ey
yyx EEz
iH
b
0yH yz E
x
iH
yxz E
y
iH
y
iE
b
2
![Page 19: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/19.jpg)
Nonlinear Optics Lab. Hanyang 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 il
ly b exp)(0xE
ztieqrBKH il
lx b
b exp)( 0yH
ztieqrKeqrKBiq
H li
l
li
lz b
exp)()(2
)1(
1
)1(
1
ztieqrKeqrKBq
E li
l
li
lz bb
exp)()(2
)1(
1
)1(
1
0xE ztiehrAJE il
ly b exp)(
ztiehrJehrJAh
E li
l
li
lz bb
exp)()(2
)1(
1
)1(
1
ztiehrAJH il
lx b
b exp)( 0yH
ztiehrJehrJAih
H li
l
li
lz b
exp)()(2
)1(
1
)1(
1
Continuity condition :
)(
)(
qaK
haAJB
l
l
0201, knkn b
![Page 20: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/20.jpg)
Nonlinear Optics Lab. Hanyang Univ.
2) x-polarized waves (similar procedure to the case y-polarized waves)
ztiehrAJE il
lx b exp)( 0yE
ztiehrAJH il
ly b
b exp)(0xH
ztiehrJehrJAh
H li
l
li
lz b
exp)()(2
)1(
1
)1(
1
ztiehrJehrJAh
iE li
l
li
lz bb
exp)()(2
)1(
1
)1(
1
In core (r<a)
In cladding (r>a)
ztieqrBKE il
lx b exp)( 0yE
ztieqrBKH il
ly b
b exp)(0xH
ztieqrKeqrKBq
H li
l
li
lz b
exp)()(2
)1(
1
)1(
1
ztieqrKeqrKBq
iE li
l
li
lz bb
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.
![Page 21: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/21.jpg)
Nonlinear Optics Lab. Hanyang Univ.
Graphical Solution for the confined modes (l=0)
22
0
2
2
2
1
2 )()()(,, haknnqaqaYhaX
)(
)(
)(
)( 11
qaK
qaKq
haJ
haJh
l
l
l
l
modes: lmLP22
1
2
0 lmlm hnk b<Possible distribution of LP11>
![Page 22: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/22.jpg)
Nonlinear Optics Lab. Hanyang Univ.
Mode cutoff value of a/l
0)(1 VJ l 212
2
2
1
212
2
2
10 2 nna
nnakV l
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
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Nonlinear Optics Lab. Hanyang 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
hrJA
S
l
l
z
b
bar
ar
)()()(2
11
222 ahJahJahJAa lll
b
)()()(2
11
222 qaKqaKqaKBa lll
b
2
0 0
a
zcore rdrdSP
2
0 azclad rdrdSP
])()()([2
11
2
222 ahJahJq
hahJAa lll
b
![Page 24: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/24.jpg)
Nonlinear Optics Lab. Hanyang Univ.
The ratio of cladding power to the total power, G2 :
G
)()(
)()(1
11
222
22ahJahJ
ahJqaha
Vpp
P
p
P
ll
l
cladcore
cladclad
![Page 25: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/25.jpg)
Nonlinear Optics Lab. Hanyang Univ.
3.4 Optical Pulse Propagation and Pulse Spreading in Fibers
One bit of information = digital pulse
Limit ability to reduce the pulse width : Group velocity dispersion
Group velocity dispersion
Considering a Single mode / Gaussian pulse, temporal envelope at z=0 (input plane of fiber) :
)]Re[exp(),(),0,,( 0
2
0 tityxutyxE
where, ),(0 yxu : transverse modal profile of the mode
Fourier transformation :
2/1
0 and
])exp()(~
)exp(),(Re[),0,,( 00 dtiftiyxutyxE
where,
212
2
4
)2exp()][exp()(
~
tFTf
![Page 26: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/26.jpg)
Nonlinear Optics Lab. Hanyang Univ.
Propagation delay factor for wave with the frequency of
),(0 yxu
0 ])(exp[: 0 zi b
Let’s take complex expression and omit the
(are not invloved in the analysis and can be restored when needed)
dztiftzE ]})()[(exp{)(~
),( 00 b
Taylor series expansion : ...2
1)()( 2
2
2
00
00
b
bbb
d
d
d
d
z
vd
d
v
ztifdztitzE
gg
2
00
1
2
1exp)(
~)](exp[),(
b
),()](exp[ 00 tzzti Eb
where, velocitygroup
11),(
0
00 gvd
d
bbb
z
vd
d
v
ztifdtz
gg
1
2
1exp)(
~),(
E
za
v
ztifd
g
exp)(~
(3.4-5): Field envelope
![Page 27: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/27.jpg)
Nonlinear Optics Lab. Hanyang Univ.
The pulse spreading is caused by the group velocity dispersion characterized by the parameter,
b
d
dv
vvd
d
d
da
g
gg
22
2
2
11
2
1
2
1
0
(3.4-3)(3.4-5) :
d
v
ztiiaztz
g 4
1exp
4
1),( 2
E
222
2
22
2
161
)(4exp
161
)(exp
41
1
za
vztazi
za
vzt
zai
gg
![Page 28: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/28.jpg)
Nonlinear Optics Lab. Hanyang Univ.
If we use the definition of factor a,
# Pulse duration t at z (FWHM)
2
2
0
0
2ln81)(
ttt
aLL
initial pulse width
# |aL|>>t0 (large distance) :0
)2ln8(~)(
tt
aLL
0
2
2ln4)(
tt
L
d
dv
vL
g
g
Practical Expression :
2
2
0
2
0
2ln21)(
t
l
tt
DL
cL
where,
2
2
2
2/
b
l
l
d
dc
L
ddTD a
c2
4
l
T : pulse transmission time through length L of the fiber
![Page 29: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/29.jpg)
Nonlinear Optics Lab. Hanyang Univ.
Group velocity dispersion
1) Material dispersion : n() depends on
Waveguide dispersion : blm depends on (& geometry of fiber)
cnnnkn lmlmlm
b ),,( 210
i)
1
)(
b
b
d
d
d
dv lm
lm
lmg: modal dispersion
ii) Single mode fiber,
c
nnn
n
nn
n
n
cd
d
vg
b 2
2
1
1
1
material dispersion waveguide dispersion
(3.4-18)
![Page 30: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/30.jpg)
Nonlinear Optics Lab. Hanyang Univ.
From the uniform dielectric perturbation theory,
2
22
2
11
2
2 nnc
b GG
where, : Fractions of power flowing in the core and cladding 21, GG
G
n
n
n
n 11
1
G
n
n
n
n 22
2
(3.4-18)
c
nnn
n
nn
n
n
cd
d
v wg
G
G
b 222
111
1
![Page 31: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/31.jpg)
Nonlinear Optics Lab. Hanyang Univ.
In weakly guiding fiber : n1~n2
m
nnn
21
c
nnn
cd
d
v wmg
b1
c
nnn
c wm
ll
l
Group velocity dispersion :
wm
nn
cD
2
2
2
2
ll
l
ex) GeO2-doped silica : m3.1at02
2
ll
m
n
# depends on core diameter, n1, n2 control the waveguide shapew
n
2
2
l
![Page 32: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/32.jpg)
Nonlinear Optics Lab. Hanyang Univ.
Group velocity dispersion & dispersion-flattened and dispersion-shifted fibers
![Page 33: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/33.jpg)
Nonlinear Optics Lab. Hanyang 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
b
(3.4-6)
where,
b
d
dv
vd
da
g
g
22
2
2
1
2
1
Total optical phase :
222
2
00161
)(4),(
za
vztazzttz
g
b
Optical frequency :
)(
1618),(),(
2220 gvzt
za
aztz
ttz
0d
dvg
![Page 34: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/34.jpg)
Nonlinear Optics Lab. Hanyang 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
t : envelop impulse response
of a fiber of length z
![Page 35: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/35.jpg)
Nonlinear Optics Lab. Hanyang Univ.
Compensation for pulse broadening
1) By optical fiber with opposite dispersion
)(~
)i 1 f
)exp()(~
)(~
)ii 2
1112 Liaff
)exp()(~
)(~
)iii 2
2223 Liaff
2
22111 )(exp)(~
LaLaif (a1L=-a2L)
![Page 36: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/36.jpg)
Nonlinear Optics Lab. Hanyang Univ.
2) By phase conjugation
dtiftitf 00 exp~
)exp()(conjugatortoInput
dtif 0
* exp~
conjugatorfromOutput
)(~
)i f
)exp()(~
)(~
)ii 2
1112 Liaff
)exp()(~
)(~
)(~
)iii 2
11
*
1
*
23 Liafff
2
2211
*
1 )(exp)(~
LaLaif
)exp()(~
)(~
)iv 2
2234 Liaff
(a1L=a2L)
![Page 37: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/37.jpg)
Nonlinear Optics Lab. Hanyang Univ.
<Experimental setup> <Eye diagram>
Where are (b) and (c) ??
Refer to the text
![Page 38: Chapter 3. Propagation of Optical Beams in Fibers - Hanyangoptics.hanyang.ac.kr/~choh/degree/nonlinear_optics_2009...Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Propagation of Optical](https://reader035.vdocuments.site/reader035/viewer/2022071415/6111cc2f7ffe4b54f7561a46/html5/thumbnails/38.jpg)
Nonlinear Optics Lab. Hanyang 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-distance
optical communication