Nonlinear Optics Lab. Hanyang Univ. Chapter 3. Propagation of Optical Beams in Fibers 3.0 Introduction Optical fibers Optical communication - Minimal

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


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 ,


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 ,


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


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)(


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


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


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)(


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)


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)


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)


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


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


* 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


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


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


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)


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


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


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.


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>


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


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


The ratio of cladding power to the total power, 2 :

)()(

)()(1

11

222

22 ahJahJ

ahJqaha

Vpp

P

p

P

ll

l

cladcore

cladclad


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


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


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


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


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)


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


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


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

(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


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


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)


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)


<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-distanceoptical communication

Documents

Nonlinear Optics Lab. Hanyang Univ. Chapter 3. Propagation of Optical Beams in Fibers 3.0 Introduction Optical fibers Optical communication - Minimal