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

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


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 ,


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 ,


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


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)(


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


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


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)(


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)


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)


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)


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


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


* 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


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


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


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)


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


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


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.


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>


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


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


The ratio of cladding power to the total power, G2 :

G

)()(

)()(1

11

222

22ahJahJ

ahJqaha

Vpp

P

p

P

ll

l

cladcore

cladclad


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


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


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


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


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)


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


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


Group velocity dispersion & dispersion-flattened and dispersion-shifted fibers


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


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


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)


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)


<Experimental setup> <Eye diagram>

Where are (b) and (c) ??

Refer to the text


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

Documents

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