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mathematical deign and simulation of fibre bragg grating
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�
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).�
�
�
�
�
�
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)�
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)�
�
�
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:��
�
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�.�
�
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.�
�
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.�
�
�
�
�
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.�
�
�
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.�
�
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.�
�
�
�
�
�
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:����
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�
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.�