Fiber Optics Digital Transmission Fiber System

Advantages of Fiber Optics: Speed Distance Immunity to electromagnetic interference (EMI) Security

Optical Fiber Specifications

Fiber Specifications Optical Characteristics

Core Diameter (µm) 6-62.5 Attenuation @850nm (dB/km)

2.4 - 2.5

Clad Diameter (µm) 125 Attenuation @1300nm (dB/km)

0.3 - 0.7

Coating Diameter (µm) 245 Attenuation @1550nm (dB/km)

0.15 - 0.3

Numerical Aperture 0.17±.015

Zero Dispersion Wavelength (nm) 1297-1316

Group Refractive Index @1300nm 1.479

WDM/ DWDM: Upgrade fiber index profile to support more wavelength transmission

- Reducing / shifting / flattening dispersion - Reducing attenuation - Operating wavelength (1330 nm, 1550 nm) - EDFA

Basic configuration of EDFA

Developments in Photonics Photoelectric effect – Einstein E = hν - φ Laser – Townes and Schalow (1958) Laser types:

a) Solid state lasers: Ruby, Nd:YAG, Nd:glass b) Gas lasers: He-Ne, Argon ion, CO2 c) Liquid lasers: Dyes d) Semiconductor lasers

Semiconductor lasers: (injection laser or diode laser) Emitting wavelength – Material dependent

GaAs/AlGaAs -- 0.8 µm InGaAsP/InP -- 1.3 to 1.55 µm Quantum cascade long wavelength lasers -- 2 to 10 µm

Structure PN heterojuction Edge emitting Vertical cavity surface emitting

Threshold Output Power

Fiber Characteristics and Processing Structure of Fiber Core – higher index Cladding – lower index Coating Single mode Multimode Fabrication of optical waveguides: Deposition – MCVD, VAD Consolidation Drawing Coating Cable (Protection) Index control via doping SiO2

GeO2

P2O5

B2O3

Optical Fiber Spectral Attenuation Characteristics: Absorption UV – Electronic excitation IR – Vibration excitation Rayleigh Scattering λ-4 dependence Attenuation: dB/km

Fiber Bending Loss: Macrobending

Microbending

Fiber Drawing Tower:

Fiber Cable

Cross section of various types of fiber optic cable

Maxwell Equations: For charge free and perfect dielectric medium (i.e., with ρ=0 and σ=0)

where ED ε= and HB µ= Boundary conditions: Tangential component of electric field: Et is continuous Normal component of electric displacement: Dn is continuous Tangential component of magnetic field: Ht is continuous Normal component of magnetic induction: Bn is continuous

tBE∂∂−=×∆

tDH∂∂=×∆

0=×∆ D0=×∆ B

21 tt EE =

2211 nn EE εε =

21 tt HH =

2211 nn HH µµ =

Wave equation: (E-field)

Helmholtz equation: (E-field)

where nkk 0= Solution of Helmholtz equations yields the planar wave equations:

Using Maxwell equations, one obtains

The vectors E (electric field), H (magnetic field), and k (wave propagation direction) are at right angles to each other. Thus the field associated with a plane electromagnetic wave are transverse to the direction of propagation.

).(0

rktieEE −= ω

).(0

rktieHH −= ω

ωεHkE ×−=

ωµEkH ×=

012

2

22 =

∂∂−∇

tE

cE

tieE ω−~

0220

2 =+∇ EnkE

Polarization Linearly polarized light: Phase difference between Ex and Ey is nπ (n = 0, 1, ..) Circularly polarized light: Phase difference between Ex and Ey is (2n+1)π/2 (n = 0, 1, …) Elliptically polarized light: a) TE: Electric field lying perpendicular to the plane of incidence b) TM: Electric field lying in the plane of incidence Waves in metallic structures Two boundaries:

Modes: (ray model) Incidence angle determines phase of a wave propagation, giving rise to different modes. Rectangular waveguide: Simple boundary condition – metallic boundary: E=∝ E=0 -a 0 a x

Standing waves: Reflection by metallic mirror: y

02202

2

2

2

=+∂∂+

∂∂ Enk

yE

xE

Incoming wave: )( ykxk yxAe + Reflected wave: )( ykxk xxBe − x Applying boundary condition ay y = 0,

TE wave: E-field normal to the plane of incidence TM wave: H field normal to the field of incidence

02202

2

=+∂∂ Enk

xE

ikxikx BeAeE −+~kxkx sincos~ βα +

==

=,...2,sin,...,cos

0

23

20

ππ

ππ

kakxEkakxE

E

( ) BABAe xikx −=⇒=+ 0

Multi-dimensional problems Wave propagating between two parallel metallic plates: x a z -a

A different interpretation on wave propagating between two parallel metallic plates: x k0n z θ A wave bounces between two metallic boundaries. Effective component along z: βθ =cos0nk Effective component along x: θsin0 nk The wave must be periodic after bouncing one complete cycle, i.e.,

πθ nank =×2sin0 ank

m

02sin πθ =⇒

Hence ( )2222

0akm

eff nn π−= ⇒ Same result !

02202

2

2

2

=+∂∂+

∂∂ Enk

zE

xE

znikzi effeeE 0~ =β

( ) 022202

2

=−+∂∂ Ennk

xE

eff

( ) ,...3,2,22/1220 πππ=− annk eff

( )22

220ak

meff nn π−=⇒

Wave propagating along a rectangular metallic waveguide 2-D cross section of a rectangular metallic waveguide: y

02202

2

2

2

2

2

=+∂∂+

∂∂+

∂∂ Enk

zE

yE

xE

znikzi effeeE 0~ =β b -a a x -b

By separation of variables of the above partial differential equation,

with yx EEE = and ( )2220

22effyx nnkkk −=+

Normal incidence on one dimensional dielectric boundary: (ε = n2)

0)(2202

2

=+∂∂ Exnk

xE

n1 n2

x=0 x :0<x xnikxnik BeAeE 1010 −+= :0>x xnikxnik DeCeE 2020 −+=

Assume the wave comes from left. Therefore, .0=D Eǁ continuous CBA =+⇒ Hǁ continuous CnkBAnk 2010 )( =−⇒ C

nnBA

1

2=−⇒

⇒ Ann

nC21

12+

=

AnnnnB

21

21

+−=

022

2

=+∂∂

xx Ekx

E

022

2

=+∂∂

yy Eky

E

( ) 022202

2

2

2

=−∂∂+

∂∂ Ennk

yE

xE

eff

Poynting vector:

⋅

=

ncEA 0

2

2ε

Flux coming in: 201

2Acn

reflected: 2

21

21201

2

+−⋅

nnnnAcn

transmitted: 2

21

21201

)(4

2 nnn

Acn

+⋅

Energy conservation: Incoming flux = flux reflected + flux transmitted. http://www.cyber.rdg.ac.uk/ISP/infrared/technical_data/substrate_optical_properties/page01.htm Dielectric interface: y incident beam reflected A B θin n1 x n2 θr C refracted

Incident: )(sin1010 ysocxniknki ininAerAe ⋅−⋅=⋅ θθ

Reflected: )cos(sin10 yxnik ininBe θθ +

Refracted: )cos(sin20 yxnik rrCe θθ − TE wave: E ǁ Z Boundary condition 1: Eǁ continuous at y = 0

Snell’s law: rin nn θθ sinsin 21 = 2

1

sinsin

nn

in

r =⇒

θθ

( ) ee xnikxnik rin CBA θθ sinsin 2010 =+⇒

if n1 > n2, inr θθ <

Boundary condition 2: Hǁ continuous

( )E

nknkkji

Ek x

000cossin 1010 θθ −=× is continuous.

Using the above, and also noting that B⊥ is continuous, we have

Solving for B

Special Cases: 2

21

210

+−=⇒=

nnnnRinθ

190 =⇒= Rinθ

Consider innn θ2

22

21 sin1−

It can be imaginary if 1sin222

21 >inn

n θ

1

2sinnn

cr =θ or 1

21sinnn

cr−=θ critical angle

It happens when 21 nn > total internal reflection.

rin CnABnCBA

θθ coscos)( 21 −=−=+⇒

ABinr

inrnnnn

θθθθ

coscoscoscos

12

12+−=⇒

Ainin

n

n

ininn

n

nn

nn

θθ

θθ

cossin1

cossin1

12

22

21

2

12

22

21

2

+−

−−

=

If we define 220 cos irnk r =θ 110 cos knk in =θ

then AkirkirB

12

12

+−=

2

22

21

22

2122 A

rkrk

AB =++

⋅=⇒

and TEieAB φ2⋅= where

1

21tankr

TE−−=φ Goos-Hanchen phase shift

For TM polarization,

tEn

tDH

∂∂=

∂∂=×∇ 2

CBA =+

221

222

21 cos

cosnC

nn

nB

nA

in

r ⋅=−θθ

−=⇒ −

221

2121tan

nknr

TMφ

Planar waveguide: 2n d 1n θ

0n

20 nn = : symmetric waveguide 20 nn ≠ : asymmetric waveguide

( ) πφφθ mdnk 2222cos 101210 =++⋅

TE: 1

01

1

2110 tantancos

kr

krmdnk −− ++= πθ

Comparison between TE and TM: Snell’s law same

crθ same ( 90=crθ in Snell’s law) Reflection, crin θθ < different

For TE: AnnnnB

inr

inr

θθθθ

coscoscoscos

12

12

+−=

For TM: Annnn

Binr

inr

θθθθ

coscoscoscos

21

21

+−

=

inrnn θθ coscos21 >⇒< 0coscos 12 =− inr nn θθ impossible 0coscos 21 =− inr nn θθ possible using inr nn θθ sinsin 12 =

inr θπθ −=2

1

21tannn

in−=θ Brewster angle ( Bθ )

1

R 0 Bθ

2π inθ

Summary of equations for dielectric interface Reflection coefficient:

TE:

2

12

22

21

2

12

22

21

22

cossin1

cossin1

+−

−−

inin

inin

nnnn

nnnn

AB

θθ

θθ

TM: 21 nn ↔ Critical angle:

1

21sinnn

cr−=θ

Brewster angle:

1

21tannn

B−=θ

Goos-Hanchen phase shift:

TE: in

in

nnnn

θ

θφ

cos

1sintan

1

222

21

21

−−= −

TM: 21 nn ↔ Characteristic equation for modes:

πφφθ mdnk in =++ 101210 cos

Numerical examples:

5.1=glassn 1=airn

R @ 0=inθ : 04.05.25.0 22

12

12 =

=

+−=

nnnnR

crθ (glass-air): 8.415.1

1sin 1 == −crθ

Bθ (glass-air): 7.335.1

1tan 1 == −Bθ

(air-glass): 3.5615.1tan 1 == −

Bθ

TEφ @ 80 :

5.7680cos5.1

180sin15.11

tan

22

1 −=⋅

−

⋅−= −φ

TMφ @ 80 :

9.8380cos1

180sin15.15.1

tan

22

1 −=⋅

−

⋅−= −φ

Characteristic equation for modes can only be solved numerically by a computer program.

crin θθ ≥ Do loop 001.0+inθ Equation satisfied

Effective mode index ( effn ) 1n z θ

10nk = θcos10nk

θsin10nk ∴ Propagation along z has the following form: znike ⋅θsin10

010 sin knnk eff=≡ θβ where θsin1nneff =

90<<θθcr 12 nnn eff <<⇒

effn is, in general, a function of dk0 , 1n , 2n A typical “dispersion” curve: effn 1n

2n

0λd

to obtain: cut-off number of modes

Extension of the basic idea Multi-layered waveguide: 1=n 1.50 1.48 1.47 Two possibilities:

I) 3n

1n 1θ 1d

2n 2θ 2d

4n

πφφθθ mnkdnkd =+++ 241322021101 coscos where ijφ is the phase shift in reflected beam at the thij interface

II) πφφθ mnkd =++ 12131101 cos

Alternative solution to the planar waveguide x 0n

2d+ symmetric

z 1n waveguide

2d−

0n Helmholtz equation

0)(2202

2

2

2

=+∂∂+

∂∂ Exnk

zE

xE

zkinzi effeeE 0~ =β

( ) 022202

2

=−+∴ Ennkdx

Edeff

1nnn eff <<

⇒ sinusoidal solution for 2dx ≤

exponential solution for 2dx >

=

+

−−

xd

dx

Be

xkABe

E

2

1

2

2

2

cosγ

γ

or

−

+

−−

xd

dx

Be

xkABe

2

1

2

2

2

sinγ

γ

Boundary conditions: symmetric mode asymmetric mode TE: ||E BkA d =21cos BkA d =21sin

∂∂

xEH || BkAk d

2211 sin γ−=− BkAk d2211 cos γ−=

1

221tan

kk d γ

= 1

221cot

kk d γ

−=

πγ mk kd += −

1

2121 tan πγ

++= −

21

tan1

2121 mk

kd

⇒×2 πγ mdk k += −1

211 tan2 where innkk θcos101 ≡

πθ γ mdkn kin +=∴ −1

2101 tan2cos same as before!!

Boundary conditions (contd.): symmetric mode asymmetric mode

TM: ||H BkA d

2cos BkA d2sin

∂∂

xH

nE 2||

1 BnkkAn d 2

1222

1 sin γ−=− − BnkAkn d 2

022

2

1 cos γ−=

20

212

2tankn

nk d γ= 20

212

2cotkn

nk d γ−=

πγ mkn

nk d +

=⇒ −

20

2121

2 tan ( )πγ21

20

2121

2 tan ++

=⇒ − m

knnk d

What are the additional information ?

1. Field pattern ( )

22dxe −−γ

exp

kxcos kxsin

exp exp

2. Overlap dxEd

d∫−

2

2

2

3. Solution for non-uniform index distribution

)(xn varying in space gradually rather than abruptly 3LiNbO

iT

Acceptance angle Air Fiber

1 2n

crθ 1n maxθ φ

1

21sinnn

cr−=θ

crθφ −= 90

1sin

sin 1max n=φ

θ

crn θθ cossin 1max =⇒

Since 1

2sinnn

cr =θ

22

21

121

22 11cos nn

nnn

cr −=−=∴ θ

NAnn ≡−= 22

21maxsinθ

Example: ,48.11 =n 47.12 =n 17.0=⇒ NA 89.9max =θ

Cylindrical coordinate system r φ z φ

z Helmholtz equation will look like

2

2

2

2

22

22 11

zrrrr ∂∂+

∂∂+

∂∂+

∂∂=∇

φ

Maxwell equations:

⇒

∂∂−=×∇

tBE

( ) ze

zr

rz

HjEr

rErr

Hjr

EEj

HjEjEr

ωµφ

ωµβ

ωµβφ

φ

φ

φ

−=∂∂−

∂∂

=∂∂+

−=+∂∂

11

1

⇒

∂∂=×∇

tDH

( ) zr

zr

rz

EjHr

rHrr

Ejr

HHj

EjHjHr

εωφ

εωβ

εωβφ

φ

φ

φ

=∂∂−

∂∂

−=∂∂+

=+∂∂

11

1

nc=

εµ1

effnc=

βω

Only zE and zH are essential because

∂∂+

∂∂−=

∂∂−

∂∂−=

∂∂−

∂∂−=

∂∂+

∂∂−=

rEH

rqjH

Err

HqjH

rHE

rqjE

Hrr

EqjE

zz

zzr

zz

zzr

ωεφ

β

φωεβ

µωφ

β

φµωβ

φ

φ

2

2

2

2

where 2220

2 β−≡ nkq Helmholtz equations:

011 22

2

22

2

=+∂∂+

∂∂+

∂∂

zzzz EqE

rrE

rrE

φ

011 22

2

22

2

=+∂∂

+∂∂

+∂∂

zzzz Hq

Hrr

Hrr

Hφ

Solution by separation of variables: )()()()( 4321 tFzFFrAFEz φ= where )(

43 ~)()( ztjetFzF βω − Since the field is periodic, πφφ 2+= υφφ jeF =∴ )(2 , where υ is an integer

0112

221

21

2

=

−++ Fq

drdF

rdrFd

γυ : Bessel function

)()()( ztjjz eeurAJarE βωυφ

υ−=<

)()()( ztjjz eewrCKarE βωυφ

υ−=>

with 221

20

2 β−= nku 1n : core index 2

220

22 nkw −= β 2n : cladding index Likewise,

)()()( ztjjz eeurBJarH βωυφ

υ−=<

)()()( ztjjz eewrDKarH βωυφ

υ−=>

Boundary conditions at ar = :

0)()()()(

0)()(

'2

'221

21

=

−−

−−

=−

=−=−

wawKDwaKa

jCjuauJBuaJa

jAu

jEE

waCKuaAJEE zz

υυυυφφ

υυ

ωµυβω

ωµωβ

0)()()()(

0)()(

'22

'1221

21

=

−−

+−=−

=−=−

wawKCwaKa

jDjuauJAuaJa

jBu

jHH

waDKuaBJHH zz

υυυυφφ

υυ

ωευβω

ωευβ

To have non-zero A , B , C , D :

( )( )2

22

222

20

21

20

2'2

2'1

'2

'2

11

0

)()()()(

)(0)(0

)()()()(

0)(0)(

+

=ℵ+ℑℵ+ℑ⇒

=

−−−

−

wuanknk

waKaw

waKwj

uaJau

uaJuj

waKuaJ

waKw

jwaKaw

uaJu

juaJau

waKuaJ

βυ

βυωεβυωε

ωµβυωµβυ

υυυυ

υυυυ

υυ

υυυυ

υυ

where )()('

uauJuaJ

υ

υυ ≡ℑ and

)()('

wawKwaK

υ

υυ ≡ℵ

General behavior of effective mode index: (Figure 2-18, textbook) effn 1n 2n 1 2 3 4 5 6

( ) 212

221

2 nnaV −=λπ

The cut-off of higher-order modes: 405.2=V

Modes are in group: 11HE

01TE , 01TM , 21HE 11EH , 31HE , 12HE

o Modified index term: ( ) 2

2220 r

nnk effυ−

1n 0=υ 2n r 0≠υ

r

0)0(0==

≠υυ rJ : ⇒ Field is zero at the center of the guide. When 121 nnn <<− 22

22

1 β≈≈ kk

+±=ℵ+ℑ 22

11wua

υυυ

Categorized modes: 0

)()(

)()(

0

1

0

1 =+wawK

waKuauJ

uaJ TE, TM modes

0)()(

)()( 11 =+ ++

wawKwaK

uauJuaJ

υ

υ

υ

υ EH modes

0)()(

)()( 11 =− −−

wawKwaK

uauJuaJ

υ

υ

υ

υ HE modes

Order of Bessel function involved:

→−→+→

=HEEH

TMTEj

11

,1

υυ degenerate group

Modes in each degenerate group has similar effn versus V behavior.

j : jJ ⇒ the order of Bessel function TE , TM involve 1J mHE ,υ involve 1−υJ (special case: 1=υ )

mEH ,υ involve 1+υJ The characteristic equation has m solutions for each υ (depending on waveguide parameters, e.g., λ , a , NA .) Modes: mTE ,υ 0=zE

mTM ,υ 0=zH mEH ,υ zz HE > mHE ,υ zz EH >

Bessel function of the first kind:

Bessel function of the second kind:

Modified Bessel function of the first kind:

Modified Bessel function of the second kind:

Symmetry of fiber waveguide Helmholtz equation: 022

02 =+∇ EnkE :E zE

zH zji eerEE βυφ −⋅= )(

( ) 012

22202

2

=

−−+∂∂+

∂∂ E

rnnk

rE

rrE

effυ

Bessel functions: υJ sinusoidal for ar <

υK exponential for ar >

1, −⇒ υυ JHE m υφje m-th solution 1, +⇒ υυ JEH m υφje m-th solution

1,

, JTMTE

m

m⇒

υ

υ υφje m-th solution

Analysis of E-field pattern 011 JHE ⇒ a r rE 0 π2 φ

φE 0 π2 φ

Modes in the fiber

01LP mode distribution

MFD core diameter Mode field diameter of a single-mode fiber: Not all light travels through the core of the fiber, but is distributed through both the core and the cladding. The “mode field” is the distribution of light through the core and cladding of a particular fiber. Mode-Field Diameter (MFD) defines the size of the power distribution. MFD is important in determining the bending loss.

Degenerate modes: modes with similar β vs. V (dispersion curve)

11HE 0J 01LP 210101 ,, HETMTE 1J 11LP

3111, HEEH 2J 21LP 12HE 0J 02LP

4121, HEEH 3J 31LP Notes:

a) mlLP , corresponds to linearly polarized mode b) Subscript l corresponds to the order of Bessel function (e.g., 331 JLP → )

c) mm HELP 10 = d) mmm TMHELP 021 ,= e) )2(, ,1,1, ≥= −+ lEHHELP mlmlml

Fundamental mode of the fiber is 11HE mode. Depending on polarization of the laser, various groups of degenerate modes are launched into the fiber.

Graded index fiber

( )

>∆−

<<

∆−=

arn

ararnnr

21

21

21

021

1

1

α

21

22

21

2nnn −=∆

1n is the index at the fiber core axis. 2n is the cladding index α is approximately 2, therefore, the index profile within the core is parabolic.

Dispersion Pulse spreading and loss of peak power along an optical fiber:

Electric field of a wave propagating:

( )rtierEtrE ⋅−∧

= βω0),( where n

0

2λπβ =

Phase velocity: c

nf

nnfvp

11202

0

==== λπβω

λπ

Group velocity: 11 −−

=

==dkdc

dkd

ddk

ddvg

ββωβ

ω

Delay time: Ldd

cdkd

cL

vL

gg λ

βπλβτ

2

2

−===

Spreading in delay time: ( ) ( )δλλτ

δτdd g=

( )

+−= 2

222

2 λβλ

λβλδλ

π dd

dd

cL

Dispersion: λτ

dd

LD g1=

What makes β ω dependent ?

-- dispersion. Material dispersion:

)(2 λλπβ n=

−=− λλλτ

ddnn

cL

matg)(

2

2 )(1λλλ

dnd

cDmat −= [ps/(km.nm)]

Material dispersion characteristics:

Waveguide dispersion: Normalized parameters V , b

22

21

222

2

nnn

b k

−−

=β

11

21 <<−=∆n

nn

)1(2

21

2

+∆≅⇒

−−

≅⇒

bknnnn

b k

β

β

( ) ∆≅−= 22

2/122

21 kannnkaV

∆

=⇒

22anVk

∆+=− dkkbdnn

cL

wg)(

22τ

∆+=∴ − dVVbdnn

cL

wg)(

22τ

2

22 )(

dVVbdV

cnDw λ

−=

Dispersion characteristics of single-mode optical fiber:

Fiber Dispersion Characteristics: Zero dispersion region 3.1 mµ Chromatic dispersion in standard SMF (single mode fiber)

15 ps/km-nm @1.55 µm Chromatic dispersion in dispersion-shifted SMF <3 ps/km-nm @1.55 µm Polarization mode dispersion : 0.3 ps/km Polarization maintaining fiber Dispersion Control: DSF (Dispersion shifted fiber) – upgrade existing 1.3 µm fiber systems for 1.55 µm operation DFF (Dispersion flattened fiber) – by suitable adjustment of fiber

core/ cladding index profile DCF (Dispersion compensating fiber) – using fiber grating

(DFB/DBR)

Fiber index profiles Step index:

Depressed-clad index:

Triangular Index:

55.1 mµ DSF

Intermodal dispersion (in multimode fiber):

Generalized refractive index profile of fiber:

[ ]

≥∆−≤<∆−

=arn

ararnrn

q

),21(0,)/(21

)( 21

212

where q represents the exponent of the power law profile. ⇒∞=q a step index profile ⇒= 2q a parabolic index profile

Pulse dispersion in step index fiber:

1

maxnc

LT = , 2

minnc

LT =

( )21 nncLT −=∆⇒

)(121 nn

cLT −=∆

⇒

skm /103

01.05×

≅

30≅ ns/km very serious !! In parabolic index fiber )2( =q :

Lc

nLT 21

2∆=∆

Pulse dispersion is 25.0≈ ns/km In optimum profile ( 98.1=q ), pulse dispersion is 06.0≈ ns/km

Graded index multimode fiber:

Pulse Propagation and Broadening Dispersion in time: E pulse duration τ time

Optical cycle 0

21ωπ==

fT

power

2Eα time Optical cycle 1410−≈ sec Pulse duration 1210−≈ sec In frequency domain: (power spectral density) )(ωS ω∆ 0ω ω

Gaussian pulse (closed form representation): titeyxutyxE 0

2

),(),0,,( 0ωα +−=

Fourier transform of the time-dependent function:

Time domain: 2te α−

Frequency domain: α

πα4

2

41 Ω−e

Finite pulse width ⇔ additional spectral content The signal is no longer a single frequency signal at 0ω . There is a spectral

width at frequency Ω+0ω . Since the propagation constant, i.e., 0

2λπn is

frequency-dependent as: ( ) m+Ω+Ω+=Ω+ 22

2

00

0021)(

ωω ωβ

ωββωβ

dd

dd

We have ),( tzE propagating as a broadening pulse. Pulse broadening due to material dispersion:

−= 2

2

λλ

dnd

cD

Pulse broadening due to waveguide dispersion:

−=

2

2

λλ

dnd

cD eff

gvd

d 1

0

=ωω

β , gv : group velocity

ωω

β

ω ddv

va

dd g

g22

2

21

21

0

−== ⇒ group velocity dispersion

Pulse width: aLaLL0

2

20

02ln82ln81)(

ττττ ≅

+=

The shorter the initial pulses, the faster it broadens. Because, the broader is the spectral content. Equivalent situation: shorter focal length longer focal length Time – space Frequency – angle

“uncertainty principle” Fiber Nonlinearity Generation of new frequency components Intermixing of optical signals Raman, Stimulated Raman Scattering – molecular vibration Brillouin Scattering – Sound wave Four-Wave Mixing in DWDM Systems

Chirping: 2

2

λdnd

A systematic change in frequency during the pulse propagation – an effect of group velocity dispersion. (also refer to Fig. 3-12, text book) If chirping is systematic, we can use it to compress the pulse – optical pulse compression. We can also make giant pulses by amplifying chirped pulse and compress it to a very short duration. Techniques of pulse compression:

1) Fiber with opposite dispersion 2) Compression by grating

Polarization maintaining fiber: An optical fiber in which the polarization of lightwave launched into the fiber is maintained during propagation with little or no cross-coupling of optical power between the polarization modes. If a fiber is perfectly symmetric, it will not preserve the polarization, since Y

effXeff nn = .

Because of built-in stress, twist, etc., Y

effXeff nn ≠

( ) fXeff

Yeff Bknnk 00 ≡−≡β , where fB is called birefringence.

Cross-sections of polarization-maintaining optical fibers range from elliptical to rectangular. Such fibers possess internal birefringence.

βπ2=pL : beat length

810−≈fB for non-birefringent fiber 310−≈ for birefringent fiber.