13
5 Introduction to Fibre Bragg Gratings This chapter is a short introduction to fibre Bragg gratings aimed at providing a fundamental understanding of the spectral properties of the filters designed using this technology. The concepts presented in this chapter are important for the analysis and optimisation of add-drop multiplexers based on gratings inscribed in the waist of fibre-couplers (chapter 8).

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mathematical deign and simulation of fibre bragg grating

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Page 1: fibr bragg grating design

5���

Introduction�to�Fibre�

Bragg�Gratings��

This� chapter� is� a� short� introduction� to� fibre� Bragg� gratings� aimed� at� providing� a�

fundamental� understanding� of� the� spectral� properties� of� the� filters� designed� using�

this�technology.�The�concepts�presented�in�this�chapter�are�important�for�the�analysis�

and�optimisation�of�add-drop�multiplexers�based�on�gratings�inscribed�in�the�waist�of�

fibre-couplers�(chapter�8).�

Page 2: fibr bragg grating design

5�-�Introduction�to�Fibre�Bragg�Gratings� � 51�

5.1� Phase�Matching�Conditions�

Fibre�gratings�allow�the�transfer�of�power�between�modes�of�an�optical�fibre.�This�is�

achieved�by�perturbing�the�phase�of�one�mode�such�that�it�matches�the�phase�of�the�

other,�“phase�matching�condition”.�Fibre�gratings�are�usually�written�in�bare�fibres�

where� the�acrylate�coating� is� removed.�This�means� the�optical� fibre�behaves� like�a�

three-layer� structure�with�different� effective� refractive� indexes� in� the� core,� n1,� and�

the�cladding,�n2,�with�a�final�outer�cladding�being�air,�n3=1.�For�a�single�mode�fibre�

with�these�parameters�the�core-guided�mode�has�a�propagation�constant�βco�given�by,�

12222

nnn coco λπ

λπβ

λπ <=< � � � � � � � � � � � � (5.1)�

and� the� cladding� modes� that� are� guided� by� the� cladding-air� structure� have�

propagation�constants�that�fall�in�the�range:�

2322

nn cl λπβ

λπ << �� � � � � � � � � � � � � �������(5.2)�

and� finally� there� are� radiation� modes� that� can� have� propagation� constants� in� the�

limit:�

32

0 nrad λπβ << �� � � � � � � � � � � �� � � ������(5.3)�

With� the� introduction�of�a�periodic�variation�of� the�effective� index�along� the� fibre�

length,� the� first� order� phase� matching� between� the� fundamental� and� backward�

propagating�fibre�modes�(fundamental�or�cladding�modes)�occurs�when�[76]:�

Λ=− πββ 2

21 ������ � � � � � � � � � � � � � � �(5.4)�

Page 3: fibr bragg grating design

5�-�Introduction�to�Fibre�Bragg�Gratings� � 52�

For� the�case�of�coupling�into�the�backward�propagating�fundamental�mode,�β2=-β1�

and�the�resonance�condition�yields:�

Λ= πβ1 �� � � � � � � � �������� � � � � � � � (5.5)�

In�expressions�(5.4)�and�(5.5)�Λ�is�the�period�of�the�effective�index�modulation�and�

β1,�β2�are�respectively,� the�propagation�constants�of� the�fundamental�mode�and�the�

mode� the� reflected� light� is� coupled� into.� Gratings� that� couple� to� backward�

propagating� modes� are� known� as� reflection� or� Bragg� gratings.� Typically� these�

devices� are� based� on� coupling� between� the� forward� and� backward� fundamental�

modes.��

�Figure�5.1�–�Schematic�representation�of�the�modes�existing�in�uncoated�single�mode�fibres�

and�the�matching�condition�for�the�core�mode�reflection.��

For� long� period� gratings� (both� β1� and� β2� are� positive)� the� phase� condition� for�

forward� coupling� from� the� fundamental� mode� into� forward� propagating� cladding�

modes�is�given�by:�

Λ=− πββ 2

21 � � � � � � � � � � � � � � �������� (5.6)�

Page 4: fibr bragg grating design

5�-�Introduction�to�Fibre�Bragg�Gratings� � 53�

5.2� Mathematical�Description�of�Bragg�Gratings�

This� section� describes� a� simple� approach� for� obtaining� the� spectral� properties� of�

fibre�Bragg�gratings.�For�an�extensive�review�of� the� theory�and�properties�of� fibre�

Bragg�gratings�the�references�[77-79]�are�suggested.�

5.2.1� Coupled�mode�equations�

Coupled�mode�theory�has�been�successfully�used�to�describe�the�spectral�properties�

of�Bragg�gratings�[78].�Refractive�index�variations�with�a�period�Λ�along�the�length�

of�a�fibre�are�generally�expressed�as:��

( ))(2cos)()( 0 zznnzn θπ +Λ∆+= � � � � � � � � ��������������� (5.7)�

the� functions�∆n(z)�and�θ(z)�are� slowly�varying� functions�compared� to� the�grating�

period� Λ,� n0� is� the� refractive� index� of� the� core,� and� ∆n(z)� the� envelope� of� the�

refractive� index�modulation.�The�parameter,�θ(z),� defines� locally,� the�phase�of� the�

effective�index�modulation,�which�is�used�to�describe�phase�shifts�or�grating�chirp.�

For� simplicity� this� introduction� will� consider� unchirped� gratings� only,� therefore�

θ(z)=0.� Along� the� grating� the� forward� propagating� wave,� v1,� and� backward�

propagating�wave,�v2,�are�related�by�the�coupled�mode�equations�[80]:�

1*

22

211

)(

)(

vziqvidzdv

vziqvidzdv

−+=

+−=

δ

δ� � � � � � � � � � � � �������� (5.8)�

where� the� amplitudes� of� the� waves� v1� and� v2� are� related� to� the� amplitudes� of� the�

forward�and�backward�propagating�electric�field,�A(z)�and�B(z)�respectively:��

Page 5: fibr bragg grating design

5�-�Introduction�to�Fibre�Bragg�Gratings� � 54�

zi

zi

evzB

evzAδ

δ

+

=

=

2

1

)(

)(� � ,� � � � � � � � � � � � �������� (5.9)�

q(z)�is�the�coupling�coefficient�given�by:�

)(2

)(0

znn

zq ∆Λ

= � � � � � � � � � � � � � ������(5.10)�

and� δ� represents� the� detuning� from� the� Bragg� grating� resonance� wavelength,�

λBragg=2n0Λ,�defined�as:�

Λ−= π

λπδ 0

2n � � � � � � � � � � � � � ������ � � (5.11)�

In�the�case�of�Bragg�gratings�where�∆n�varies�along�the�grating�length�the�spectral�

characteristics� can�be�obtained�by� solving� the�differential� coupled�mode� equations�

(5.8).�The�particular�case�of�a�uniform�grating�has�been�solved�analytically�[81],�the�

reflection�coefficient�ρ=v1(δ)/v2(δ)�and�reflectivity�R=|ρ|2�at� the�start�of� the�grating�

(z=0)�are:�

)cosh()sinh()sinh(

)(LiL

Lqγγγδ

γδρ+

−= � � � � � � � � � � ������ � (5.12)�

222

2

)(cosh)(sinh

)(qL

LR

δγγδ−

= � � � � � � � � � � � ������ � (5.13)�

where�γ2=q2−δ2�.�

Page 6: fibr bragg grating design

5�-�Introduction�to�Fibre�Bragg�Gratings� � 55�

Some� important� features� can� be� inferred� from� these� results.� Firstly� it� can� be�

demonstrated� that� the� maximum� reflectivity� Rmax� occurs� when� the� resonance�

condition�is�observed,�i.e.,�δ=0�and�is�given�by�

)(tanh2max qLR = � � � � � � � � � � � � � ������ � (5.14)�

and� secondly� the� spectral� bandwidth,� ∆λzeros,� defined� as� the� two� first� zeros� in�

reflectivity�calculated�using�(5.13)�yielding�[78]:�

2

0

1

���

∆+∆=∆

nLnn Braggzeros

λλ

λ� � � � � � � � � � � ������ � (5.15)�

For� strong� gratings� where� ∆nL>>λBragg� the� normalised� bandwidth� is� approximated�

by:�

0nnzeros ∆≈∆

λλ

� � � � � � � � � � � � � � ������ � (5.16)�

and�for�weak�gratings�where�∆nL<<λBragg�the�normalised�bandwidth�is�approximated�

by:�

LnBraggzeros

0

λλ

λ =∆�� � � � � � � � � � � � ������ � � (5.17)�

When�writing�gratings� in� fibres,� equation�(5.15)�provides�useful� information�about�

the� induced�effective� index�change�simply�by�measuring� the�spectral�bandwidth�of�

the�grating.�Similarly�for�uniform�gratings,�the�induced�refractive�index�change�can�

also�be�calculated�using�(5.14),�by�measuring�the�maximum�reflectivity�at�the�Bragg�

wavelength.�

Page 7: fibr bragg grating design

5�-�Introduction�to�Fibre�Bragg�Gratings� � 56�

To� fully� understand� the� dispersive� properties� of� fibre� Bragg� gratings� the�

concept�of�group�or�time�delay�must�be�introduced.�For�a�uniform�grating�the�time�

delay� can� be� determined� from� the� phase� of� the� reflection� coefficient� ρ� defined� in�

(5.12).� If�θρ=phase(ρ),� then�the�time�delay,�τρ,� for� light�reflected�from�a�grating�is�

defined�as�[78]:��

λθ

πλ

ωθ

τ ρρρ d

d

cd

d

2

2

−== � � � � � � � � � � � � ������ � (5.18)�

and�the�effective�length,� leff,� that� light�at�a�particular�wavelength�travels�within�the�

grating�before�it�returns�to�the�origin�can�be�calculated�from�leff=cτρ/n0.� In�uniform�

gratings,�the�minimum�time�delay�occurs�at�the�Bragg�wavelength.�For�wavelengths�

near� the� edges� of� the� grating� bandwidth� and� the� sidelobes� of� the� reflectivity,� the�

dispersion� is� greatest� with� the� time� delay� varying� rapidly� with� wavelength.� Thus,�

large�time�delays�are�characteristic�of�this�regime�and�are�due�to�these�wavelengths�

suffering�multiple�reflections�before�exiting�the�grating�(as�in�a�Fabry-Perot�cavity).�

Figure�5.2�shows�the�reflectivity�spectrum�and�the�time�delay�for�a�uniform�grating�

with�a�strength,�qL=4,�and�a�grating�length�of�L=20mm.�The�maximum�reflectivity,�

which�can�be�calculated� from�(5.14),� corresponds� to� the�minimum� time�delay.�For�

wavelengths� near� the� first� reflectivity� zeros,� the� time� delay� is� maximum�

corresponding�to�several�round-trips�before�the�light�exists�the�grating.�

Page 8: fibr bragg grating design

5�-�Introduction�to�Fibre�Bragg�Gratings� � 57�

0

100

200

300

400

500

1549.75 1549.85 1549.95 1550.05 1550.15 1550.25Wavelength�(nm)

Gro

up�d

elay

�(ps

)

0

0.2

0.4

0.6

0.8

1

Ref

lect

ivity

�Figure� 5.2� -� Calculated� reflection� spectra� (dotted� line)� and� group� delay� (solid� line)� for� a�

uniform�grating�with�qL=4.�

5.3� Apodisation�

In� order� to� increase� side-lobe� suppression� to� achieve� the� required� discrimination�

between� adjacent� wavelength� channels� (at� least� 30dB)� in� WDM� systems,� fibre�

gratings� are� generally� apodised.� This� is� achieved� by� tapering� the� refractive� index�

modulation,� ∆n(z),� at� both� ends� of� the� grating� structure.� The� reflectivity� of� an�

apodised� grating� can� be� calculated� by� defining� an� effective� length,� Leff,� for� the�

grating� calculated� using� the� following� expression� [79],� which� describes� the�

normalised�coupling�strength.�

Page 9: fibr bragg grating design

5�-�Introduction�to�Fibre�Bragg�Gratings� � 58�

�=L

eff dzzqLq0

max )( � � � � � � � � � � � � ������ � � (5.19)�

The�reflectivity�at�the�grating�resonance�wavelength�is�calculated�by�substituting�Leff�

in� (5.14)� and� using� q=qmax.� When� comparing� gratings� with� different� apodisations,�

the� quantity� defined� by� (5.19)� must� be� equal� for� each.� Thus� to� achieve� the� same�

normalised� coupling� strength� for� the� same� maximum� grating� refractive� index�

modulation,�∆nmax,�or�coupling�strength,�qmax,�the�length�of�the�gratings�is�multiplied�

by�L/Leff.��

Inter-channel� cross-talk� of� grating� based� add-drop� multiplexers� depend� upon�

side-lobe� suppression� and� the� grating� spectrum.� Ideally� a� square� filter� with� high�

reflectivity� and� –50dB� side-lobes� is� required.� Recently� these� filters� have� been�

determined� using� a� numerical� inverse� scattering� method� [50]� and� demonstrated�

experimentally�[82].�In�OADMs�based�on�gratings�inscribed�in�the�coupler�waist,�the�

fabrication� limitations� for� the� grating� length� play� a� vital� role� in� the� choice� of�

apodisation�and�the�consequent�add-drop�performance�discussed�further�in�chapter�8.�

Figures� 5.3� and� 5.4� compare� the� reflectivity� spectrum� and� penetration� depth�

respectively,� for�gratings�with� the� same�normalised�coupling�strength�qLeff=4.�The�

black�line�corresponds�to�a�uniform�grating,�the�blue�a�Blackman�apodised�grating,�

and�the�red�line�to�a�sine2�apodised�grating.�The�Blackman�apodised�grating�offers�

the�best�side-lobe�suppression�although�it�has�the�highest�penetration�depth�into�the�

grating.�The�actual� lengths�of� the�gratings� to�obtain� the� same�normalised�coupling�

strength,� for� each� of� the� apodisations� were;� Blackman:� 47.6mm;� sine2:� 40mm;�

Uniform:�20mm.�

Page 10: fibr bragg grating design

5�-�Introduction�to�Fibre�Bragg�Gratings� � 59�

-80

-60

-40

-20

0

1549.8 1549.9 1550 1550.1 1550.2Wavelength�(nm)

Ref

lect

ivity

�(dB

)

Blackman

sin2

Uniform

�Figure� 5.3� –� Reflectivity� spectrum� of� gratings� with� different� apodisations.� Black� line:�

Uniform�apodised�grating;�Blue�line:�Blackman�apodised�grating;�Red�line:�sine2�apodised�

grating.�

0

5

10

15

20

25

30

35

1549.8 1549.9 1550 1550.1 1550.2Wavelength�(nm)

Pen

etra

tion�

leng

th�(m

m)

Blackman

sin2

Uniform

�Figure�5.4�–�Penetration�depth�spectrum�of�the�same�gratings�as�in�Figure�5.3.�

Page 11: fibr bragg grating design

5�-�Introduction�to�Fibre�Bragg�Gratings� � 60�

5.4� Transfer�Matrix�

For�modelling�the�spectral�properties�of�gratings�with�arbitrary�apodisation�and�chirp�

profiles,� a� simple� method� exists,� whereby� the� grating� is� described� using� N� sub�

matrices� representing� N� uniform� sections� of� the� grating;� these� matrices� are� then�

multiplied� to� obtain� the� total� grating� response� [78,� 83].� The� solution� of� the�

propagation�equation�(5.8)�for�a�uniform�medium�of�length�∆z�and�constant�coupling�

coefficient�q�can�be�expressed�in�terms�of�the�well-known�transfer�matrix�[78],�MT:�

��

���

�=�

���

∆+∆+

),(),(

),(),(

1

1

2

1

δνδν

δνδν

z

zM

zz

zzT �

���

���

∆+∆∆

∆∆−∆=

)sinh()cosh()sinh(

)sinh()sinh()cosh(

zss

izszs

zssq

zss

izsMT δ

δ

Where� s=|q|2−δ2.� The� output� amplitudes� of� the� entire� grating� can� be� found� by�

multiplying�the�transfer�matrices�correspondent�to�each�of�the�N�individual�sections:�

��

���

�=�

���

)0()0(

)()(

1

1

2

1

νν

νν

TML

L;���������� 11 ... T

NT

NTT MMMM ⋅⋅⋅= − �

Throughout� this� thesis� the� above� method,� in� conjunction� with� an� appropriate�

discretisation� algorithm� [50]� was� employed� to� efficiently� model� the� spectral�

characteristics�of�the�gratings�investigated.�To�increase�the�numerical�efficiency�by�

reducing�the�computation�time,�the�matrix�MT�was�expressed�as�a�product�of�simpler�

matrices�[50].�The�scattering�process�is�desbribed�as�a�localised�event�in�the�centre�

of� each� individual� grating� section.� Taking� MT� in� the� limit� |q|→∞� while�keeping� a�

finit�product�q∆z�we�can�calculate�a�simplified�matrix� that�describes� the�scattering�

process�MS(∆z),�in�the�section�of�length�∆z:����

Page 12: fibr bragg grating design

5�-�Introduction�to�Fibre�Bragg�Gratings� � 61�

����

����

∆∆

∆∆=∆

)cosh()sinh(

)sinh()cosh(

)( *

zqzqqq

zqqq

zq

zM S �

The�propagation�along�the�grating�has�also�to�be�taken�into�account.�The�propagation�

matrix�MP(∆z,δ),�is�calculated�taking�MT�in�the�limit�|q|→0�giving:�

��

���

�=∆ ∆+

∆−

zi

zi

P e

ezM δ

δ

δ0

0),( �

The�transfer�matrix�MT�can�be�approximated�with�an�error�O(∆3)� in�terms�of� these�

two�matrices�as:�

( ) .,2

,2

��

� ∆∆

��

� ∆≈ δδ zMzM

zMM PSPT �

5.5� Photosensitivity�

To� write� strong� gratings� in� short� fibre� lengths,� the� photosensitivity� of� the�

germanium-doped�fibre�core�should�be� increased� to�achieve� larger� refractive� index�

changes.�This� issue�is�especially�important�when�writing�gratings�in�fibre� tapers�or�

couplers�where�the�photosensitive�area�and�therefore�the�overlap�with�the�core�mode�

is�reduced.�Increased�photosensitivity�is�commonly�achieved�by;�loading�the�optical�

fibres� with� hydrogen� or� deuterium� under� high� pressures� [84],� brushing� the� optical�

fibres/waveguides� with� an� hydrogen� flame� [85]� and,� increasing� germanium�

concentration� and� adding� codopants� such� as� fluorine� or� boron� to� reduce� NA.� The�

physical�origin�of� the�photosensitivity�in�optical�fibre�is�still�a�matter�for�extensive�

Page 13: fibr bragg grating design

5�-�Introduction�to�Fibre�Bragg�Gratings� � 62�

discussions�and�is�out�of�the�context�of�this�thesis.�For�good�reviews�concerning�the�

photosensitization�process,�readers�are�referred�to�[86-88].���

5.6� Summary�

A�brief�introduction�to�fibre�Bragg�gratings�was�presented�in�this�section.�The�phase�

matching�between�forward�and�backward�propagating�fundamental�fibre�modes�can�

be� achieved� with� a� periodic� variation� of� the� effective� index.� The� interaction� is�

quantified�using� the�well� known�coupled�mode� equations.�Analytical� solutions� for�

these�equations�exist�for� the�simplest�case�of�a�uniform�grating.�For�useful�devices�

with�low�side�lobes,�in�order�to�accurately�discriminate�between�adjacent�channels,�

different�fibre�apodisations�are�used.�The�spectral�response�of�gratings�with�arbitrary�

apodisations� is� obtained�by� solving� the� coupled� mode�equations�using� an� efficient�

scattering�matrix�model.�The�concept�of�time�delay�and�penetration�depth�of�light�in�

the�grating�were�introduced�as�well�in�order�to�optimise�the�performance�of�add-drop�

multiplexers�based�on�gratings�inscribed�in�the�waist�of�fibre�couplers,�discussed�in�

chapter�8.�