Analysis o simula ed e lec ion cha ac e is ics… 21
ANALYSIS OF SIMULATED REFLECTION CHARACTERISTICS OF UNIFORM
AND APODIZED FIBER BRAGG GRATINGS
E. Gemzický, J. Mülle o á
Depa men o Enginee ing Fundamen als, Facul y o Elec ical Enginee ing, Uni e si y o Žilina,
ul.kp . J. Nálepku 1390, 031 01 Lip o ský Mikuláš, Slo akia, mail: [email protected]
Summa y In his pape , he simula ions o he e lec ance o he uni o m and apodized ibe B agg g a ings (FBG) a e
p esen ed. The simula ions a e based on he coupled–mode heo y. The simula ed e lec ances o FBGs wi h di e en leng hs
and he modula ions o he e ac i e index we e desc ibed. The in luence on he bandwid h o he e lec ed spec a a he
B agg wa eleng h and he maxima o he e lec ed ene gy we e in es iga ed. FBGs wi h se e al ypes o apodized a ia ions
o he e ac i e indices we e modeled o show how he sidelobes can be sup essed.
1. INTRODUCTION
The ibe B agg g a ing (FBG) is an op ical
de ice wi h a pe iodic a ia ion o he e ac i e
index along he p opaga ion di ec ion in he co e o
he ibe . As he co e o he op ical ibe is due o he
doping by ge manium highly pho osensi i e, FBGs
can be ab ica ed by he exposing he ibe co e o
UV adia ion. This induces he changes o e ac i e
index along he co e o he ibe . The esul ing
e ac i e index changes depend on he UV ligh
exposu e and exposu e pa e n. Se e al echniques
a e commonly used o ab ica e FBGs: he poin -by-
poin echnique, he in e e ome ic echnique and
he phase mask echnique [1-3].
FBGs ake he ad an ages o a simple s uc u e,
low inse ion loss, high wa eleng h selec i i y,
pola iza ion insensi i i y and ull compa ibili y wi h
gene al single mode communica ion op ical ibe s.
P ope ly manu ac u ed FBGs o e high e lec ances
(g ea e han 75% [4]) and na ow bandwid hs a he
B agg wa eleng h. All his makes hem sui able o
applica ions in ibe op ical communica ions, e.g. as
WDM demul iplexe s, ibe lase echnique and ibe
senso sys em [2], e.g. o s ain a empe a u e
measu emen s.
This pape is de o ed o he simula ions o
e lec ance pe o mances o he e lec ion FBGs
wi h he a e age ibe co e e ac i e index o 1.447.
The e ac i e index modula ion and he FBG leng h
we e changed o show he in luence on he spec al
e lec ance o he B agg wa eleng h o 1550 nm.
2. THE COUPLED-MODE THEORY
An FBG can be conside ed as a weakly coupled
wa eguide s uc u e. The coupled-mode heo y is
mos gene ally used o analyze he ligh p opaga ion
in a weakly coupled wa eguide medium. The
coupled-mode equa ions ha desc ibe he ligh
p opaga ion in he g a ing can be acqui ed by using
he coupled-mode heo y.
The e ac i e index modula ion along he ibe
axis z (Fig. 1) can be ep esen ed by he exp ession
(
)
)(zVnzn
na
= (1)
whe e n
a
is he a e age e ac i e index o he ibe
co e, V
n
(z) is he modula ion o he e ac i e index
de ined as
( )










⋅+= z
d
zgzV
n
π
2
cos1)( (2)
whe e g(z) is he apodiza ion unc ion, d is he B agg
g a ing pe iod and is he inge isibili y.
Acco ding o Eq. 2, he modula ion o he e ac i e
index V
n
depends on he apodiza ion unc ion g(z).
A small amoun o he inciden ligh ene gy is
e lec ed a each pe iodic e ac i e index change
(Fig. 1). Then we ha e ei he he e lec ion FBG o
he ansmission FBG. The e lec ion FBGs wo k
p ope ly wi h sho e B agg g a ing pe iods whils
he ansmission FBGs a e long-pe iod g a ings.
Fig. 1. The change o he e ac i e index o he FBG
All e lec ed ligh wa es a e combined in o he one
e lec ed a a pa icula wa eleng h ha is ela ed o
he B agg condi ion de ined by
dn
a B agg
⋅⋅= 2
λ
(3)
The wa eleng h
B agg
λ
a which his e lec ion
occu s is called he B agg wa eleng h and depends
on he e ac i e index and he B agg g a ing pe iod
d. Longe pe iods can be used o achie e he b oade
bandwid h and FBGs o his ype a e called long-
pe iod FBGs. They ypically ha e B agg g a ing
pe iods o he o de om ~100 mic ome e s o a
millime e .
22 Ad ances in Elec ical and Elec onic Enginee ing
Only he wa eleng hs ha sa is y Eq. 3 a e
e lec ed. The e lec ance o he inpu ligh achie es
a peak a he B agg wa eleng h.
The cha ac e is ics o he FBG e lec ed
spec um can be modelled by se e al me hods. The
basic idea o he coupled-mode heo y is ha
elec ical ield o he op ical ibe wi h he a ia ion
o he e ac i e index can be ep esen ed by he
linea combina ion o he modes o he ield
dis ibu ion wi hou a ia ions [5, 6].
The modal ields o he op ical ibe can be
ep esen ed by
(
)
(
)
,...2,1exp),(,, =±⋅=
±±
jziyxezyxE
jj j
β
(4)
whe e e
±j
is he ampli ude o ans e se elec ic ield
o he j
h
p opaga ion mode and ± symbolizes he
di ec ion o he p opaga ion. 
j
is called he
p opaga ion cons an . Nex we assume a ime
dependence exp(-i ). The ans e se componen o
he elec ic ield a posi ion z in he a iable ibe
can be desc ibed by he linea supe posi ion o he
ideal guided modes o he in a iable ibe . Tha can
be w i en as
( )
(
)
( )
)exp(),(
exp)(
exp)(
,,,
iyxe
zizA
zizA
zyxE
j
jjj
jj
ω
β
β
−⋅
⋅








−⋅+
⋅
=
→
−
+
→

(5)
Whe e )(zA
j
+
and )(zA
j
−
a e slowly a ying
ampli udes o he j
h
backwa d and o wa d
a elling wa es espec i ely and ),( yxe
j
→
is he
ans e se mode ield. The coupled-mode equa ions
can be simpli ied in he wo modes ha hey a e
desc ibed as [6]
)()()()(
ˆ
)( zSzikzRzi
dz
zdR +=
σ
(6)
)()()()(
ˆ
)( zRzikzRzi
dz
zdS
∗
−−=
σ
(7)
[
]
)2/(exp)()(
φδ
−=
+
zizAzR is he o wa d mode
and
[
]
)2/(exp)()(
φδ
+−=
−
zizAzS is he e e se
mode.
σ
ˆ is a gene al “DC” sel -coupling coe icien
called he local de uning and
( )
zgzVzk
n
)()(
λ
π
= is
he “AC” coupling coe icien – local g a ing
s eng h. The coupled-mode Eq. 6, 7 a e used in he
simula ion o he spec al esponse o he FBG.
Local g a ing s eng h and local de uning a e
undamen al pa ame e s in he calcula ion o he
spec al esponse o he FBGs. The gene al “DC”
sel -coupling coe icien
σ
ˆ can by desc ibed by
dz
d
φ
σδσ
2
1
ˆ−+= (8)
whe e dz
d
φ
2
1 desc ibes possible chi p o g a ing
pe iod and
φ
is he g a ing phase. The de uning
δ
can by desc ibed by








−=
B agg
a
n
λλ
πδ
11
2 (9)
Fo uni o m FBGs, he chi p dz
d
φ
equal ze o and
he local de uning equals he de uning
δ
. The
e lec ed ampli ude spec um can be ob ained and
desc ibed by
( )
)(cosh)(sinh
ˆ
)(sinh
2222
22
LL
Lk
BBB
B
γγγσ
γ
λ
+
= (10)
whe e
(
)
λ
is he ampli ude e lec ance and
22
ˆ
σγ
−= k
B
i
22
ˆ
σ
>k o
22
ˆki
B
−=
σγ
i
22
ˆ
σ
<k.
3. THE AMPLITUDE REFLECTANCE OF
FIBER BRAGG GRATINGS
The e lec ance spec a o he e lec ion FBGs
we e MATLAB simula ed acco ding o he Eq. 10.
The e ec i e e ac i e index n
a
and he B agg
g a ing pe iod d a e cons an o he uni o m B agg
g a ing (g(z) = cons ) and he B agg wa eleng h
B agg
λ
=
1550 nm. The bandwid h
B agg
λ
∆
a he
B agg wa eleng h a -50 dB and he ampli ude
e lec ance
B agg
a he B agg wa eleng h we e
in es iga ed. Fig. 2 shows he ampli ude e lec ance
o a uni o m FBG o h ee di e en leng hs o he
FBG wi h he ollowing pa ame e s: d=535.59 nm,
n
a
=1.447, V
n
=3.10
-4
. The bandwid h
B agg
λ
∆=
0.5 nm o L=5 mm,
B agg
λ
∆=
0.32 nm
o L=10 mm and
B agg
λ
∆=
0.24 nm o L=20 mm.
The simula ions show ha he inc ease o L esul s
in he educ ion o he bandwid h by 1550 nm and in
he inc ease o
B agg
. The sidelobes a e p esen on
bo h sides o he peak a he B agg wa eleng h ha
a e undesi able o communica ion and senso
applica ions.
Analysis o simula ed e lec ion cha ac e is ics… 23
Fig. 2. Calcula ed e lec ances (a [%], b [dB]) o he
uni o m FBG (g(z)=cons ). and a iable L, d=535.59 nm,
na =1.447, Vn=3.10-4
Then we in es iga ed he in luence o he
e ac i e index modula ion V
n
on he ampli ude
e lec ance spec a o he FBG wi h he uni o m
B agg g a ing o h ee di e en pa ame e V
n
o he
FBG wi h he ollowing pa ame e s: d=535.59 nm,
n
a
=1.447, L=10 mm (Fig. 3).
Fig. 3. Calcula ed e lec ances (a [%], b [dB]) o he
uni o m FBG (g(z)=cons .) o a iable Vn , d=535.59 nm,
na =1.447, L=10 mm
Th ee di e en alues o he modula ion o he
e ac i e index V
n
we e used in he simula ions. The
simula ion shows ha he modula ion V
n
a ec s no
only he bandwid h, bu he maximum e lec ance in
B agg
λ
oo. The inc ease in V
n
causes he inc ease in
B agg
, which is highly desi able o e lec ion
FBGs. On he o he hand, a he same ime he
bandwid h
B agg
λ
∆
ises which could be ha m ul
o WDM demul iplexe s. We no ice ha a
V
n
=3.10
-4
, he e lec ance
B agg
=83% and he
bandwid h
B agg
λ
∆=
0.34 nm, whe eas a V
n
=1.10
-3
B agg
=100%,
B agg
λ
∆=
0.74 nm. The e o e, in
his case he op imiza ion is ecommended o a
speci ic applica ion.
FBGs wi h he uni o m index p o ile as he
WDM demul iplexe s a e epo ed o su e om
high c oss alk le el [7, 8] especially due o he
sidelobe exis ence. Two ypes o e ac i e index
p o ile – he Gaussian p o ile and he sinc p o ile -
we e inse ed in o he simula ion o show ha he
le el o he undesi able sidelobes can be dec eased.
The apodized e ac i e index ep esen s he
si ua ion when he unc ion anishes smoo hly a he
edges o he g a ing. Then he sidelobes decay and
he c oss alk be ween wo adjacen channels can be
educed.
The Gaussian-apodized p o ile is desc ibed by
( )












−
⋅−=
2
2
exp L
Lz
azg (11)
whe e a is he Gauss wid h pa ame e . The simula ed
e lec ance spec um is in Fig. 4. We no ice ha he
e lec ance
B agg
= 100% and
B agg
λ
∆=
0.44 nm.
Fig. 4. The Gaussian- apodized index p o ile ( a), he
calcula ed e lec ance [%] (b), [dB] (c)
The apodized e ac i e index p o ile desc ibed by
he sinc unc ion is cha ac e ized by mo e signi ican
comp ession o he e ac i e index a he edges o
he FBG han he Gaussian p o ile (Fig. 4, 5). The
sinc-apodized p o ile is de ined by he unc ion
(
)
))2(sinc( Lzzg −⋅=
π
(12)
Fo his index p o ile (Fig. 5) a conside able
educ ion o sidelobes was achie ed and he
bandwid h
B agg
λ
∆=
0.26 nm. The e o e, FBGs o
24 Ad ances in Elec ical and Elec onic Enginee ing
his ype a e sui able candida es o WDM
demul iplexe s wi h low c oss alk.
Fig. 5. The sinc-apodized index p o ile (a), he calcula ed
e lec ance [%](b), [dB] (c)
Fig. 6. The compa ison o he calcula ed e lec ance (a
[%], b [dB]) o he FBG o he uni o m index p o ile, he
Gaussian and he sinc apodiza ion.
The compa ison o he ampli ude e lec ance o
uni o m (non-apodized), Gaussian-apodized and
sinc-apodized FBGs is shown in Fig.6. No e ha he
mos na ow e lec ed spec al band wi h supp essed
sidelobes is achie ed by he sinc-apodized FBG,
al hough he e lec ance in
B agg
λ
is educed in
compa ison wi h o FBGs o uni o m o Gaussian
p o ile.
4. CONCLUSION
In his pape e lec ances o FBGs wi h di e en
g a ing leng hs and di e en modula ions o he
e ac i e index we e simula ed. The esul s show
ha he inc ease o he g a ing leng h causes he
bandwid h dec ease and he inc ease o he
e lec ance a he B agg wa eleng h
.
When he
e ac i e index modula ion inc eases, hen he
e lec ance and he bandwid h inc ease oo.
Al hough he e lec ance a he B agg wa eleng h o
he sinc-apodized g a ing is educed in compa ison
wi h he non-apodized FBG, a conside able
educ ion o he sidelobes and o he bandwid h a
he B agg can be achie ed. This could be o in e es
o na ow-band wa eleng h selec i e il e ing.
Acknowledgemen
This wo k was pa ly suppo ed by he Slo ak
Resea ch and De elopmen Agency unde he
p ojec APVV COST-0041-06.
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