1st Tranche Slides

8/2/2019 1st Tranche Slides

1/93

Dipartimento di ElettronicaInformatica e Sistemistica

G. Tartarini Advanced Electromagnetic Transmission Techniques and DevicesElectromagnetic Technologies for Link Design M

Outline of the III Module of the course

Modal propagation and transmission properties of dielectricwaveguides (Dielectric slab, Optical Fiber)

Optical transmitters, receivers and main optical devices:

fundamentals and principal characteristics.

Engineering and design aspects of present optical transmissionsystems.

(Course notes: http://elearning.ing.unibo.it/)
https://campus.cib.unibo.it/https://campus.cib.unibo.it/


2/93



Application scenarios of optical fiber links

mono-modefiber

fiber/copper

mono-modefiber

multi-modefiber

multi-modefiber

IP network

CentralOffice

Access Networks

In-Building Networks

Core Network


3/93



Basic Fiber Optic Communication System

DirectlyModulated

Transmitter(Based on LED

or LASER)

Input

ElectricalSignal(DigitalOrAnalog)

OpticalFibre

Direct DetectionReceiver

(Based on PIN orAPD)

OutputElectricalSignal(DigitalOrAnalog)


4/93



Ex.: Optical Fibre Submarine systems

Up to 9000 kmof total length

National,Transoceanicand

Intercontinentallinks

Very highcapacity,high qualityrequred.


5/93



Ex.: Optical Fibre Access Network

Typical distances1 20 km

MetropolitanAreaNetwork (MAN),Fiber to the home

(FTTH),Fiber to the Curb(FTTC)

Highcapacity,high quality

requred.


6/93



Radio coverage of:

Crowded sites

Non-LOS Areas

(Line Of Sight)

Unit (RAU)IndoorRAU

Radio BaseStation (RBS)

Outdoor Remote Antenna

OpticalFiber

Ex.: Fiber Distributed Antenna Systems


7/93



Advantages of Fiber Optic Transmission

High Bandwidth (

High Bit Rate )

Low Attenuation

Short Dimensions

No interference


8/93



Band utilized in optical communications

SonarServo-

mechanisms

Radio

Radar

Infrared

Visible Light

Ultraviolet

X RaysGamma

Rays

Cellular

Telephones

TV

1 Mm 1 km 1 m 1 mm 1 nm1 mm

1 kHz 1 MHz 1 GHz 1 THz 1015 Hz 1018 Hz(f)

(l)

Frequency Band utilized in Optical Communications


9/93



Band expressed as Dfor DlFrom ITU-T Recommendation G-692.

Nominal Frequencies in C band (1528-1561 nm):

81 carriers separated by Df=50 GHz starting from 196.1 THzIn terms ofl the range goes from 1528.77 nm to 1560.61 nm

The interval Dl is not constant. It goes from 0.389 nm to 0.405 nm

196.1 THz196.05 THz

192.1 THz

1528.77 nm 1560.61 nm


10/93



Review on dB,dBm, dB(Hz), etc.

Attenuation : P1/P2

Attenuation|Logarithmic units = 10 log10(P1/P2) (dB)

Gain/Loss: P2/P1

Gain/Loss|Logarithmic units = 10 log10(P2/P1) (dB)etc.

P1 P2Adimensional quantities dB:

Note : AttenuationFIBER=10aL/10

AttenuationFIBER|Logarithmic units =aL (dB) [a]=[dB/km]

Power Level|Logarithmic units : dB(W), dB(mW) [or dBm], etcBandwidth|Logarithmic units : dB (Hz)

Temperature|Logarithmic units : dB (K)

etc.

Non-Adimensional quantities (W, Hz ) dB(W), dB(Hz)


11/93



The Optical Transmission Channel

Electromagnetic Fundamentals to be recalled:

Plane waves propagation in homogeneous media

Continuity Conditions / Reflection Coefficients of the

electromagnetic field

Dielectric slab: analythical resolution

From the dielectric slab to the optical fiber

Transmission characteristics

Attenuation

Dispersion

Non linearities (not treated here)


12/93



The Dielectric Slab: Description

x

z

(n2)

(n1)

0...

y

y

(x=d)

(x=-d) (n2)

Waveguide invariant

in the y and z directions.

Central layer

of refraction index n1 and

of thickness 2d

Two external Layersof refraction indexes n2 < n1,

infinitely extended in the

+x andx directions.

The field is invariant in the y direction

The field propagates in the z directionzjzj (x)eH(x)eE(x,z)(x,z) ,H,E


13/93



Procedure followed to find Slab guided modes

1. Resolution ofMaxwells Equations in the three regions

Plane Wave solutions in the three regions2. Imposition of

continuity conditions of at the interfaces through either:

Coefficients GTEand GTM+ consistency condition

on the equiphase planes

Achievement of the Characteristic Equation (TE/TM, even/odd)

3. Resolution of the Characteristic Equation

Modes propagation constant

Modes amplitude behavior

Creation and Resolution of a

Homogeneous linear System

yyzzHEHE ,,,


14/93



Plane Wave solutions in the three materials: TE case

x

z

y

zjx

yy eeEzx 222 ),(E

Upper layer:evanescent plane wave

zjxjk

y

zjxjk

yy

eeE

eeEzx

x

x

1

1

1

11 ),(

E

Central layer:

uniform plane waves

zjx

yy eeEzx 2'2'2 ),(E

Lower layer:

evanescent plane wave

1xk

1xk

ib1

rb1

22jk

x

2'2jk

x

2

02

22

'2

2

2

2

2

22

0

2)(;'2,2,1,)( knkkiknk xxixi where:

In each one of the three layers, we have zxy zxy HHHEE ,


15/93



Reflection Coefficients at the interfaces: TE case

x

z

y

G

1

21

1

1

2

21

21

21

21

21

21

1

1

coscos

coscos

x

x

x

kjtg

x

x

xx

xx

ti

ti

djky

djk

y

TE

e

jk

jk

kk

kk

nn

nn

eE

eE

22 jkx

2'2jk

x

( )

( )

G 121

1

1 2

1

1... x

x

x

kjtg

djk

y

djk

y

TE eeE

eE


16/93



x

z

y

The triangular points belongingto the same equi-phase plane must be

separated in phaseby an integer multiple of 2p.

The same for the square points.

Considering e. g. points A and C, itmust be:

A

'C

'

B''

B

''C

TE

CBjn

TEABjn

Ay

mj

AyCy

e

eE

eEE

G

G

'''01

'

01

''

1

2

11

p

Consistency Condition on Equiphase planes: TE case


17/93



It must then be:

p

jm

kjtg

djn

kjtg

djn

e

e

e

e

e

x

i

x

ii

2

2

cos

2

2

2coscos

2

1

21

01

1

21

01

x

z

y

A

'C

'

B''

B

''C

i

dCB

cos

2'''

i

i

dAB

2cos

cos

2'

+d

-d

)( 'CBspan ''

)( 'ABspan

TEG

TEG

Consistency Condition on Equiphase planes: TE case


18/93



Characteristic Equation: TE case

2244

244cos

24)2cos1(cos

2

112

1

21

1

1

21

01

1

21

01

22cos

2

22coscos

2

1

21

011

21

01

pp

p

p

p

mdktgdkdmk

tgdk

mk

tgdn

mk

tgd

n

eeeee

xx

x

x

x

i

x

i

i

jmkjtgd

jnkjtgd

jnxix

ii

( )

( ) oddmdkdkd

evenmdktgdkd

xx

xx

,

,

112

112

cot

We obtain then:


19/93



u

w

p/2 p 3p/2

Curve

where v is the

Normalized Frequency:

2

2

2

10

2

2

2

1 )()(

nnd

ddkv x

Odd Modes

( )( ) )(cot)cot()()()()(

112

112

modesoddeven modes

uuwdkdkdutguwdktgdkd

xx

xx

21

22 )( dkvd x

Resolution of the Characteristic Equation: TE case

From the characteristic equation we have:

Moreover, it must be: ( ) ( ) ( ) 22222212

0

2

2

2

1 )( vwunndddkx Even Modes


20/93



Exploiting the relations:

We can solve numerically

the equation (e.g. TE case withm=0):

For w~0 it is ~n20. For w it is~n10 .

2

02

2

2

22

011

)()(

)()(

dndd

ddndkx

])()([)()()()( 220122

012

022 ddntgddndnd

w

1001 nn mw

2002 nn mw

(w)

...

Dispersion curves (w): TE case

m=0(TE0)

m=1(TE1)


21/93



The following relations areexpoited:

The plot is more readable

The curves can be referred to waveguides different from each other

Only one TE mode propagates when it is 0


22/93



Behavior of |Ey|for the modes TEm (m even, m=2k)

x

z

y

G

11

2222

2

2

1

1

11

1

21

1

yy

djkkdkj

kjtg

djk

y

yTE

EE

ee

e

eEE

xx

x

x

p

Considering for example the upper interface, we have from the characteristic equation:

In the central layer the field Ey is given by:

) ( ) zjxyzjxjkxjkyy exkEeeeEzx xx 1111 cos),( 11E


23/93



Behavior of |Ey|for the modes TEm (m even)

x

z

d-d

|Ey1|Ex. mode TE0

x

z

d-d

|Ey1|Ex. mode TE2

( ) ( )xkEexkEzx xyzj

xyy 11111 coscos),(

EIn the central layerthe expression is given by

222

0 11ppp

xkdk xx

2

3

2

3

2

311

pppp xkdk xx


24/93



Behavior of |Ey|for the modes TEm (m odd, m=2k+1)

x

z

y

G

11

22)12(2

2

2

1

1

11

1

21

1

yy

jdjkkdkj

kjtg

djk

y

yTE

EE

eee

e

eEE

xx

x

x

pp


In the central layer the field Ey is given by:

) ( ) zjxyzjxjkxjkyy exkEeeeEzx xx 1111 sin),( 11E


25/93



Behavior of |Ey|for the modes TEm (m odd)

x

z

d

-d

|Ey1|Ex. mode TE1

x

z

d-d

|Ey1|Ex. mode TE3

( ) ( )xkEexkEzx xyzj

xyy 11111 sinsin),(

EIn the central layerthe expression is given by

pppp

xkdk xx 112

pppp 2222

3 11 xkdk xx

In general |Ey|of mode TEm assumesm times the `0` value for dxd


26/93



Plane Wave solutions in the three materials: TM case

x

z

y

zjx

yy eeHzx 222 ),(H


zjxjk

y

zjxjk

yy

eeH

eeHzx

x

x

1

1

1

11 ),(

H

Central layer:uniform plane waves

zjx

yy eeHzx 2'2'2 ),(H

Lower layer:

evanescent plane wave

1xk

1xk

ib1

rb1

22jk

x

2'2jk

x

2

02

22

'2

2

2

2

2

22

0

2)(;'2,2,1,)( knkkiknk xxixi where:

In each one of the three layers, we have zxy zxy EEEHH ,


27/93



Plane Wave solutions in the three materials: TM case

x

z

y

zjxyy eeHzx

2

22 ),(H


zjxjk

y

zjxjk

yy

eeH

eeHzx

x

x

1

1

1

11 ),(

H

Central layer:uniform plane waves

zjx

yy eeHzx 2'2'2 ),(H

Lower layer:evanescent plane wave

1xk

1xk

ib1

rb1

22jk

x

2'2 jkx

2

02

22

'2

2

2

2

2

22

0

2)(;'2,2,1,)( knkkiknk xxixi

where:


28/93



Reflection Coefficients at the interfaces: TM case

x

z

y

22jk

x

2'2jk

x

( )

( )( )

G 12

12

21

1

1

/2

1

1... x

x

x

knnjtg

djk

y

djk

y

TM eeH

eH

( )( )

( )( )

( )

G

12

12

21

1

1

/2

21

2

12

21

2

12

21

2

12

21

2

12

12

12

1

1

/

/

/

/

coscos

coscos

x

x

x

knnjtg

x

x

xx

xx

ti

ti

djky

djk

y

TM

e

jknn

jknn

kknn

kknn

nn

nn

eH

eH


29/93



x

z

y

The triangular points belongingto the same equi-phase plane must be

separated in phaseby an integer multiple of 2p.

The same for the square points.

Considering e. g. points A and C, itmust be:

A

'C

'

B''

B

''C

TM

CBjn

TMABjn

Ay

mj

AyCy

e

eH

eHH

G

G

'''01

'

01

''

1

2

11

p

Consistency Condition on Equiphase planes: TM case


30/93



It must then be:

p

jm

knnjtg

djn

knnjtg

djn

e

e

e

e

e

x

i

x

ii

2

)/(2

cos

2

)/(2

2coscos

2

1

2

12

21

01

12

12

21

01

x

zy

A

'C

'

B''

B

''C

i

dCB

cos

2'''

i

i

dAB

2cos

cos

2'

+d

-d

)

(

'CB

span

''

)

('AB

span

TMG

TMG

Consistency Condition on Equiphase planes: TM case


31/93



Characteristic Equation: TM case

22

)/(44

2)/(

44cos

24)2cos1(cos

2

11

2

1

22

1

2

12

21

1

1

2

12

2101

1

21

01

2

2

cos

222cos

cos

2

1

2101

1

2101

pp

p

p

p

mdktgdkn

ndm

knntgdk

mknn

tgdn

mk

tgd

n

eeeee

xx

x

x

x

i

x

i

i

jmk

jtgd

jn

k

jtgd

jn

xix

i

i

( )

( ) oddmdkdknnd

evenmdktgdknnd

xx

xx

,

,

11

2

122

11

2

122

cot)/(

)/(

We obtain then:


32/93



u

w

p/2 p 3p/2

( ) ( )( ) ( ) )(cot)/(),(cot

)()/(),(2

12

2

12

oddoddeveneven

TMuunnwTEuuwTMutgunnwTEutguw

Resolution of TE and TM Characteristic Equations

From the characteristic equation we have:

Moreover, it must be:

TE0

TM0

TM1TE1TM2TE2

TM3

TE3

The graphic solution gives thevalues of

dwdkux 21

,

22 uvw

of the guided modes for a given valueof the normalized frequency:

2

2

2

1

2

2

2

10 )/( nndcnndv w


33/93



Exploiting again the relations:

We can solve numericallythe equation to find (w)(e.g. TMcase withm=0):

and add the TMdispersion curves to the TE ones

2

02

2

2

22011

)()(

)()(

dndd

ddndkx

( ) ])()([)()(/)()(

22

01

22

01

2

12

2

02

2

ddntgddnnndnd

w

10

01

n

n

mw

20

02

n

n

mw

...

Dispersion curves (w): TE and TM case

(TE0)

(TE1)

(TM0)

(TM1)


34/93



The following relations areexpoited:

The plot is more readableThe curves can be referred to waveguides different from each other

Only one TE and one TMmode propagate when it is 0


35/93



Behavior of |Hy|for the modes TMm (m even, m=2k)

x

z

y

G

11

2222

)/(2

2

1

1

11

12

12

21

1

yy

djkkdkj

knnjtg

djk

y

yTM

HH

ee

e

eHH

xx

x

x

p


In the central layer the field Hy is given by:

) ( ) zjxyzjxjkxjkyy exkHeeeHzx xx 1111 cos),( 11H


37/93



Behavior of |Hy|for the modes TMm (m odd, m=2k+1)

x

z

y

G

11

22)12(2

)/(2

2

1

1

11

12

12

21

1

yy

jdjkkdkj

knnjtg

djk

y

yTM

HH

eee

e

eHH

xx

x

x

pp


In the central layer the field Hy is given by:

) ( ) zjxyzjxjkxjkyy exkHeeeHzx xx 1111 sin),( 11H

B h i f | | f h d TM ( dd)


38/93

Dipartimento di ElettronicaInformatica e Sistemistica G. Tartarini Advanced Electromagnetic Transmission Techniques and Devices

Electromagnetic Technologies for Link Design M

Behavior of |Hy|for the modes TMm (m odd)

x

z

d

-d

|Hy1|Ex. mode TM1

x

z

d-d

|Hy1|Ex. mode TM3

( ) ( )xkHexkHzx xyzj

xyy 11111 sinsin),(

HIn the central layerthe expression is given by

pppp

xkdk xx 112

pppp 2222

3 11 xkdk xx

In general |Hy|of mode TMm assumesm times the `0` value for dxd


39/93

Dipartimento di ElettronicaInformatica e Sistemistica G. Tartarini Advanced Electromagnetic Transmission Techniques and DevicesElectromagnetic Technologies for Link Design M

Guided Modes and Radiation Modes22

01

2

1 )( nkx2

2

2

22

02

2

2 )( nkx

2

01 )( n

2

02 )( n

0

0

0

Planewavespropagatingalong z

Planewavespropagatingalong xin thecentral

layer

Plane

wavespropagatingalong xin theouterlayers

Planewavesattenuating

along z

Planewavesattenuating along xin the outer layers

GuidedModes

Radiation

modespropagatingalong z

Radiationmodesattenuating

along z

The set of guided and radiation modes forms a Complete Set of Solutionsof MaxwellsEquations for the dielectric slab Any field of the slab can be expressed as a linearcombination of these modes

Th O ti l Fib Cl ifi ti


40/93


The Optical Fiber: Classification

SilicaFibres

PlasticFibres

Single Mode

Multimode

Multimode

Graded

Index

StepIndex

StepIndex

Graded

Index

StepIndex

Standard

DispersionShifted

P

ERFORM

ANCES+

COSTS

Long DistanceConnections

In-Building,Local AreaConnections

HomeConnections

S O


41/93


Core: SiO2+ GeO210 mm (Single Mode),50, 62.5 mm (Multi Mode),n1~ 1.443

Cladding: SiO2125 mm

n2~ 1.44

Primary Coating

400 mm Secondary Coating1 mm

Silica Optical Fibres

Plastic Optical Fibers


42/93


Core:Polymethylmethacrylate (PMMA),Perfluorinate (CYTOP), 62.5 mm 980 mmn1~ 1.49

Cladding): Other Polymericmaterial250 mm 1 mmn

2~ 1.41

External coating~2 mm

Plastic Optical Fibers

P d t fi d O ti l Fib G id d M d


43/93


n1>n2

r

fn2

n1

a1. Resolution ofMaxwells Equations in the three regions

Solutions in terms of Bessel functions in the three

regions

2. Imposition of

continuity conditions of at the

interfacer = a

Homogeneous Linear System to be solved

Achievement of the Characteristic Equation (TE/TM,

even/odd)

3. Resolution of the Characteristic Equation

Modes propagation constant

Modes amplitude behavior

HEHE

,,, zz

)(w

Procedure to find Optical Fiber Guided Modes

S l i P d f M ll E ti


44/93


zj

zzrr

zj

zj

zzrr

zj

eiHiHiHer,z)(r

eiEiEiEer,z)(r

ff

ff

)(),(,

)

(),(,

H

E

H

E

Evaluation of Ez, Hz:

0)(0)(

22

0

2

220

2

ziz

ziz

HnH

EnE

Evaluation ofEt, Ht:

)()(

1

)

()(

1

22

0

22

0

ztzizt

i

rrt

ztzzt

i

rrt

EijHjkn

iHiH

HijEjkniEiE

w

wm

H

E

r

fn2

n1

(2.1)

(2.2)

Solving Procedure of Maxwells Equations

(i=1 for core

i=2 for cladding)

Expressions of HE


45/93


Where:

Jm ( ) = Bessel Function of First Kind of order m

Km ( ) = Modified Bessel Function of Second Kind of order m

Solving (2.1) (2.2) in cyilindrical coordinates yields:

r< a

r> a

r< a

r> a

2

02

22

2

22

01

2

1

n

nkt

Expressions of

r

fn2

n1

zzHE,

zj

m

zj

tmz

zj

m

zj

tm

z

emDmDr)(K

emCmCr)(kJ,z)(r

emBmBr)(K

emAmAr)(kJ,z)(r

f

f

)sin()cos()sin()cos(,

)sin()cos(

)sin()cos(,

212

211

212

211

H

E

E i f th Ch t i ti E ti


46/93


Imposing the Continuity Condition the Characteristic Equation is obtained

Introducing the approximation D = (n12 - n22 ) / (2 n12 )


47/93


Review on Bessel Functions of First Kind

)(0 xJ

)(1 xJ

Zeros of J0(x):2.40485.52018.6537

Zeros of J1(x):3.83177.015610.1735

Review on Modified Bessel Functions of Second Kind


48/93


Review on Modified Bessel Functions of Second Kind

)(0 xK

)(1 xK

All Km(x)tend tofor x0 and fallexponentiallyfor increasing x.

The ratio Km(x)/ Km+1(x)

tend to 1 for increasing x

Both ratiosx*Km(x)/ Km+1(x)andx*Km+1(x)/ Km(x)are equal to 0

for x=0 andtend to x for increasing x

LP d ith 0 fi di k


49/93


2 4 6 8 10

))((

))((

)( 21

20

2

1

2

12

1

2

akvK

akvK

akvt

t

t

)(

)(

10

111

akJ

akJak

t

tt

LP01 LP02

LP03

kt1a

LPmn modes with m=0: finding kt1

LP d ith 1 fi di k


50/93


2 4 6 8 10

2 2

0 12 2

1 2 2

1 1

( ( ) )

( ) ( ( ) )

t

t

t

K v k a

v k a K v k a

0 11

1 1

( )

( )

tt

t

J k ak a

J k a

LP11LP12

LP13

kt1a

LPmn modes with m=1: finding kt1

O ti l Fib N li d Di i C


51/93


2

2

2

10 nnav

2

2

2

0

2

1

2

0

2

2

2

0

2

nn

nb

Optical Fiber: Normalized Dispersion Curves

Behaviors similar

to the ones of the

dielectric slab

B h i f M d A lit d


52/93


Linearly Polarized modes which are obtained utilizing the condition D


53/93


Modes LP0n are two times degenerate: they can haveEt =Ex orEt =Ey

x

y

LP01LP02

Modes LPmn are 4 times degenerate:

They can haveEt =Ex orEt =Ey

They can have azimuthal dependence cos(mf) or sen(mf)

LP11

f

x

y

Degeneration of LP modes

Attenuation in Optical Fibers


54/93


Attenuation in Optical Fibers

Source of

Information

Fiber Optic

Cable

Electrical

Signal

Destination

Optical

Source

Optical

Detector

Optical

Receiver

Attenuation in the Optical Fibers


55/93


Intrinsic absorption

Extrinsic Absorption

Linear Diffusion

Attenuation in the Optical Fibers

Material: SiO2, SiO2+GeO2, SiO2+F,Atl~0.1 mm: electronic transitionsAt l~9. mm: photon interactionswith material molecular vibrations

Rayleigh Scattering: a part of the optical power is transferred from the propagationModes to other modes, including radiation modes, at the same frequency.It is due to slight inhomogeneities of the crystal over distances shorter than 1 mm

(Due to impurities)Interactions with bond vibrations ofOH-ion

Attenuation in SiO2 Fibers (SMF and MMF)


56/93


Attenuation in SiO2 Fibers (SMF and MMF)

5

4

3

2

1

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

a(dB/km)

l (mm)

Rayleigh Scattering

OH- ionAbsorption

(This peak canbe reduced if

the fabricationprocedure isimproved)

Infraredabsorption

UltravioletAbsorption

Classical Transmission Windows


57/93


Classical Transmission Windows

1 2 3

5

4

3

2

1

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

a(dB/km)

l (mm)

I windowl~ 0.8mm

II window

l~ 1.3mm III windowl~ 1.55mm

~ 0.2dB/km

~ 0.4dB/km

Recent Definition of new Transmission Windows


58/93


Recent Definition of new Transmission Windows

I window

0.8mm


59/93


Attenuation in Plastic Optical Fibers

500

400

300

200

100

0.45 0.5 0.55 0.6 0.65 0.7

a(dB/km)

l (mm)

250

200

150

100

50

0.6 0.8 1.0 1.41.2

a(dB/km)

l (mm)

Step-Index PMMA Graded-Index in Perfluorinated Polymer

Attenuation in Optical Fibres: Conclusions


60/93


SiO2 Fibre

Attenuation in Optical Fibres: Conclusions

Utilizing appropriate optical transmission bands (III window, II window, etc),the attenuation is very low (a< 0.5 dB/km).

Long-distance links (unrepeateredspans of tens of km) Metropolitan networks

In-Building networks

The high attenuation (at present a> 30 dB/km) is one of the limiting factors forPOF links, which at present do not exceed tens or a very few hundreds of meters

of length.

Home networks Connections inside cars, trains, ships, etc

Plastic Fibre

Coupling Losses AC1 due to Numerical Aperture:1 N i l A t f th Fib


61/93


1. Numerical Aperture of the Fibre

( )2

2

2

1sin nnNA MFFibre

Incindent rays which remain inside the AcceptanceCone of semi-aperture MF are guided inside thefibre

SMF : NAFibra~ 0.15MMF : NAFibra~ 0.30PMMF : NAFibra~ 0.50

Coupling Losses AC1 due to Numerical Aperture:2 N i l A t f th S


62/93



( )MSSourceNA sin

2

101 log10Fibre

SourceC

SourceFibre

NANAA

NANAIf

For a fixed NASourcemultimodal fibres have lower values of AC1 withrespect to single mode ones

2. Numerical Aperture of the Source

Coupling Losses AC2due to Source Emitting Area:S E i l t Di t


63/93



Source Fibre DCoreDSource

2

102 log10Fibre

SourceCSourceCore

D

DADDIf

SMF : DCore~ 10 mmMMF : Dcore~ 50-62.5 mmPMMF : Dcore~ 62.5-980 mm

For a fixed DSourcemultimodal fibersexhibit lower AC2values with respect tothe single mode ones

Source Equivalent Diameter

Definitions referred to a pulse signal


64/93



x(t)

t

t

Rms puse width

(rms = root mean square)

Average pulse arrival time

( )

1/ 2

2

1 1( ) , ( )

( ) ( )

t tx t dt t t x t dt

x t dt x t dt

Definitions referred to a pulse signal

Dispersion in Optical Fibres


65/93



Dispersion in Optical Fibres

Source of

Information

Fiber Optic

cable

Electrical

Signal

Destination

Optical

Source

Optical

Detector

Optical

Receiver

Types of Dispersion in Optical Fibres


66/93



Types of Dispersion in Optical Fibres

Exists only in the MMF.

Each mode of the fiber carries with a different group velocity vgaportion of the modulating signal

Exists both in SMF and in MMF.

Each portion of the signal spectrum (wave packet) travels with adifferent vg.

Exists in SMF, can be neglected in MMF.

The signal is divided between the two degenerate x-polarized and

y-polarized LP01 modes. These modes ideally should have thesame vg, but in practice exhibit different vgs.

IntermodalDispersion

Chromatic

Dispersion

PolarizationModeDispersion

Dispersion in few words


67/93



Dispersion in few words

The signal and/or its spectrum, exhibit an undesired subdivision in different portionsWhich travel with different group velocities vg. The complete denomination of thephenomenon is:

Group Velocity Dispersion (GVD)

Limitations at the Bit Rate of the link in case of digital transmission

Limitations at the Pass Band of the link in case of analog transmission

Consequences

Summarizing:

Intermodal Dispersion


68/93



p

u= kt1a

Normalized frequency v high

u ~ 2.4048, 3.8317,5.1356,,That is, kt1,mn is independent from v from w

For modeLP01 it is kt1,01= 2.4048/(a)For modeLP11 it is kt1,11= 3.8317/(a)

For modeLP21 it is kt1,21= 5.1356/(a)

2 2

0 12 2

12 2

1 1

( ( ) )( )

( ( ) )

t

t

t

K v k av k a

K v k a

kt1,01a =2.4048

kt1,11a = 3.8317

kt1,21a = 5.1356

Computation of vg as dw/d


69/93



( )

( ) ( )

mng

mnmnmnt

mnt

senn

c

d

dv

senc

nn

n

k

n

d

d

kn

w

m

mw

m

w

mw

1

1

2

01

2

10

2

2

,

2

10

2

,

2

10

2

cos11

n10

kt1,mn

n1

n2

n2

mn

For the fundamental mode it is vg = (c/n1) [1- (p/2a)2/(n1k0)

2]1/2 ~ c/n1

For the mode at cut-off it is sen mn =n2/n1vg~ (c n2/n12

)

For a span of lengthL the maximum difference between the arrival times is:

Dt=Ln1/c(n1/n2-1)~Ln1D/c

p g

Computation of vg through geometrical optics


70/93



The same result can be obtained through geometrical optics:

DtAB= n1LiD/cDt= n1 {SLi}D/c=n1LD/c

Li

n1

n2

c

A

B

p g g g p

Pulse spreading and Bit rate


71/93



Estimation of the transmittable bit rate:

Dt TB that is Br


72/93



n(r)

(n2)

B

With the optimum value ofa (a ~ 2) it is:

Dt~(n1LD/c) (D/2)

B L=BrL~ (Gbit/sec) Km

a

n( r )=n1[1-2D(r/a)

a]1/2 if ra

A

Graded Index Multimode Fiber

Product Band x Distance


73/93



Product Band x Distance

The effect of intermodal dispersion is represented by theProduct Band x Distance(~ Bit Rate x Distance)

Typical values:

Plastic Fibres

Step-Index Graded-Index

Silica Fibres ~800-1500 MHz Km~tens of MHz Km

~200-300 MHz Km~10-20 MHz Km

Chromatic Dispersion: main causes


74/93



Waveguide Dispersion

The behavior ofis non linear

with wdue to the presence of the

two asymptotes=n2k0 and=n1k0

Material Dispersion

The fact that n1=n1(w) and n2=n2(w),

introduces a further cause of non lineardependence ofwith w.

w

1001 nkn mw

2002 nkn mw

(w)

...

1001 nkn mw

2002 nkn mw...

w...

p

Effects of Chromatic Dispersion


75/93



Reduction of peak value andincrease of pulse duration

Limitation to the signal bit-rate

Digital Modulating Signal Analog Modulating Signal

Distortion of the signalLimitation of the pass band Numerical evaluation of distortion

Effects of Chromatic Dispersion

The harmonic components (WavePackets) which constitute the spectrum of the

Modulating signal arrive to destination with different delays

The Cromatic Dispersion is presnt also in the MMF but it can be oftenneglected with respect to intermodal dispersion

Chromatic Dispersion expressed in

kmnmpsec


76/93

Dipartimento di Elettronica

Informatica e SistemisticaG. Tartarini Advanced Electromagnetic Transmission Techniques and Devices


C o at c spe s o e p essed

D

kmnm

psecl

t

ll

d

d

d

vgd

Dg

1

Enlargement of a pulse for one km of fiber length for 1 THz of bandwidth

of the modulated signal.

tgt

g=v

g-1

(n

sec/m)

4.8784.875

DDl

Minimum of tgZero of DDl

kmnm

Chromatic Dispersion expressed in

kmTHzpsec


77/93




D

kmTHz

psecw

t

ww

d

d

d

vgd

Dg

1

Enlargement of a pulse for one km of fiber length for 1 THz of bandwidth

of the modulated signal.

tgt

g=v

g-1

(n

sec/m)

4.878

4.875

DDw Minimum of tg

Zero of DDw

p p kmTHz

Material Dispersion in an homogeneous medium


78/93




Normalized group delay:

Group velocity: vg=1/tg

Dispersion:

~

2

2

2

3

2

2

2

2

2 2...)2(2

1

)2(

1

lp

l

p

pw d

nd

cdf

nd

fdf

dn

cdf

d

D D

2

2

2

3

2

2

2

2

...)2(1

2

1

l

l

p d

nd

cdf

ndf

df

dn

cdf

dD f D

2

2

2

1

l

l

lpl

d

nd

cdf

d

d

dD D

p g

df

dg

pt

)2(

1~

Behavior of n(l)for pure silica


79/93




32

0

5

22

0

42

0

33

02

2

0100

)()()(

)(

l

C

l

C

l

CCCCn

lll

lll

where:

C0=1.4508554

C1=-3.1268e-3 mm-2C2=-3.81e-5 mm

-3

C3=3.027e-3 mm2

C4=-7.79e-5 mm4

C5=1.8e-6 mm6

l=.035 mm2

(l0 in mm)

( ) p

Behaviors of and for pure silicalddn/22 / ldnd


80/93




p

For l~1.3 mm it is (Zero Material Dispersion)

The behaviors are similar for doped silica

02

2

ld

nd

Ex. Of Material Dispersion: the RainBow


81/93


Informatica e Sistemistica

G. Tartarini Advanced Electromagnetic Transmission Techniques and Devices


p

b

180-2(90-2b+a)=4b-2aa

b

b

b

a

(180-4b)/2=

=90-2b

Sun at horizon horizontal raysRays all possible angles -90


82/93





0 20 40 60 80 1000

10

20

30

40

50

Values of a around 60 degrees reflection angles very close to each other Intensity maximum.

50 55 60 65 737

38

39

40

41

42

43

Angle a (degrees)

Angle4b-2a(degrees)

The angle of the maximum varies with l: Red: 42.3 degrees, Violet: 40.4 degrees

Standard Fibres ITU-T G.652 e G.653


83/93





G.652 : Standard Single Mode Fiber

Most used type of SMF Dispersion is zero for l~ 1.3mm

G.653 : Dispersion Shifted Fiber

More expensive, less used Dispersion is zero for l~ 1.55mm(III wndow C Band)

Chromatic Dispersion: Wave Packet


84/93





( )

( )

D

DD

ft

ft

fE

ttx

p

p

w

sin

cos)0,(

0

0

D

D

D

Ld

dtf

Ld

dtf

fE

LtLtx

0

0

0

sin

cos),(

0

0

w

w

w

w

p

w

p

w

)0,(tx

t

),( Ltx

t

( ) LfjLfH exp),(

fD

f

0f0f

2/0

E

)0,()0,( fXfX

LfLfXArg )(),(

f

0f

0f

f

0f0f

2/0E

),( LfX( )F( )1-F

Chromatic Dispersion: Finite Bandwidth Signals


85/93





( ) LfjLfH

exp

),(

)0,(tx

t

t0 ),( Ltx...

t0 +Dt

t

)0,()0,( fXfX

0f0f f

Lf

LfXArg)(

),(

),( LfX0f

0f

f

f

( )F( )

1-

F

Chromatic Dispersion: Gaussian Pulse for z=0


86/93





( )tfeEtx Tt

02

0 2cos)0,(2

0

2

p

T0= rms pulse width

t

)0,(tx

0E

088.0 ET0

Chromatic Dispersion: Gaussian Pulse for z>0


87/93





)2cos(

)2(

1),(

00

))2(

1(2

)2

1(

2

2

2

2

0

0

0

2

2

2

20

2

ztfe

zdf

d

jT

TAtzA

zdf

djT

zdf

dt

p

p

p

p

t

),( tzA

t

X(f,0)Multiplication by

)( fHfibre

( )tfeE

tx

T

t

0

2

0

2cos

)0,(

20

2

p

t

),0( tA

X(f,z)

( )F

( )1-F

Chromatic Dispersion: rms pulse width for z>0


88/93





0

1

T

f DOptical source with low linewidth ( ):

2

2

2

2

0

2

0)

)2(

11()( z

df

d

TTzT

p

Optical source with linewidth not to be neglected ( ):

2

2

2

2

2

2

2

0

2

0)

2

1()

)2(

11()( z

df

dfz

df

d

TTzT

p

pD

0

1

Tf D

22

2

2

2

0

2

0)

2

1()

)2(

11()( z

df

d

d

dz

df

d

TTzT

lpl

pD

p p

Influence of dispersion on the Bit Rate


89/93





p

Rule usually adopted :

5

)( BITt

zT

)(

2.0

zT

Br

Optical source with low linewidth ( ):

2

0

2

0)

1()( zDT

TzT wD

Optical source with linewidth not to be neglected ( ):

0

1

Tf D

0

1

Tf D

zDzdf

d

d

dzT ll

lpl

DDD

2

1~)(

zD

Brll DD

2.0

Minimum value for zDT wD0zD

Br

wD2

2.0

Chromatic Dispersion: Counter measures


90/93





Use of Dispersion CompensatingFibers (DCF)

Use of Gratings(Ex. Fiber Bragg Gratings FBG)

Use of Soliton Propagation(exploiting the nonlinearities of the fiber)

(SMF +D) (DCF -D)

(SMF +D) (SMF +D)

(FBGD)

t t

Dispersion compensating Fibers


91/93





Through a relatively high doping ofGeO2 it is possible to abtain anegative value ofD [ps/(nm km)]

L1(es G652)

L2


92/93





Each channel needs a slightly different compensation

DG652

l(mm)

1.3 1.55

l(mm)

Spettro

WDM

In the band B it is:

DG652 B

~ D(l0) + (dD/dl) (ll

0)

It must then be:

DDCF B = -(L1/L2) DG652 B

=-(L1/L2) D(l0) -(L1/L2) (dD/dl) (ll0)

B

l0 l1DDCF

l(mm)

Pulse Enlargement per unit bandwidth


93/93

It is represented the quantity

(ps/nm)

z

z

L1 L2 L1 L2 L1

Channel 1 WDM

Channel i WDM

Channel N WDM

Ideal Case

Real Case

z

dD0

)(

z

dD0

)(

z

dD0

)(

1st Tranche Slides

Documents

Transcript of 1st Tranche Slides