The symme y o op ical ield in pho onic c ys al ib e wi h igonal symme y
365
THE SYMMETRY OF OPTICAL FIELD IN PHOTONIC CRYSTAL FIBRE WITH
TRIGONAL SYMMETRY
I. Tu ek, D. Káik, I. Ma inek
Depa men o Physics, Facul y o Elec ical Enginee ing, Uni e si y o Zilina
e-mail: u ek@ el.u c.sk
Summa y
Some pho og aphs o in ensi y o op ical ield o a pho onic c ys al ib e a e p esen ed in he con ibu ion.
P esen ed pho og aphs documen ha he symme y o pho onic c ys al c ea ing he cladding o ib e is mani es ed in he
symme y o dis ibu ion o he op ical ield in ensi y. In case when mo e modes a e exci ed in he ib e he symme y o he
gene a ed ield can be di e en as he symme y o he e en ual modes. How he symme y may be changed is illus a ed by a
model example.
1. INTRODUCTION
A pho onic c ys al is a anspa en medium wi h
a pe iodic a angemen o he e ac i e index
inhomogeni ies. These inhomogeni ies can be
dis ibu ed in such s uc u e which (due o B agg
sca e ing) does no allow o some op ical wa es ( he
wa es wi h some wa e ec o s) o p opaga e
h ough he medium. This means, ha some egions
o ene gies (wa eleng hs) and wa e ec o s a e
pe mi ed o he s o bidden. I is simila o beha iou
o solids whe e exis egions o allowed ene gies o
elec ons and egions wi h o bidden ene gies. These
ene ge ic egions a e, in solid s a e physics, named
as alence and conduc ing band o allowed alues
o ene gies and as he bandgap o o bidden
ene gies, espec i elly. In analogy, also he egion o
wa eleng hs o ligh which can no p opaga e
h oug h he op ical s uc u e is named as bandgap –
he pho onic bandgap [1].
I is possible o p epa e a 2D pho onic c ys al,
which con ains a long ”de ec ” (inhomogeni y o he
pho onic c ys al) wi h small c oss sec ion (Fig. 1). I
he c oss-sec ion is small, bu g ea enough o allow
ha an op ical wa e wi h he wa e ec o om
o bidden egion can p opaga e along i , he de ec
beha es as he co e o a op ical ib e, because ligh
o hese wa eleng hs canno lea e he a ea o he
de ec . I means ha he s uc u e se es as a op ical
ib e – he Pho onic C ys al Fib e (PCF).
The s uc u e o he pho onic c ys al ib e
de e mines he ib e p ope ies [2], including phase
eloci y, dispe sion and also he dis ibu ion o he
ligh in ensi y in he “co e” o he ib e. The
dispe sion and he di e ence o he phase eloci ies
o he i s and second modes we ha e desc ibed in
ou p e ious pape s [3,4]. In his pape we p esen
some esul s o in es iga ion o he op ical ield
pa e n o he pho onic c ys al ib e made by
CENTAURUS Technologies, Sydney. The
in es iga ed ib e is pe o med by a se o ai holes in
silica glass wi h igonal a angemen and his co e is
pe o med by a missing hole in he cen e o he
ib e [3]. The a angemen o he ai holes can be
seen in he pho og aph gi en in Fig. 3.a. The
diame e o he hole is 2,6µm and he pi ch is 7,1µm.
The in es iga ed ib e belongs o he g oup o PCFs
called index-guiding PCFs. This name is ela ed o
he ac ha he e ac i e index o ib e co e is
highe han a e age index o he cladding egion,
c ea ed by pho onic c ys al wi h pe iodically
dis ibu ed ai holes. The e o e, in he i s
app oxima ion, o he cha ac e isa ion o hese
ib es can be used he heo y o s ep-index ib es
wi h he e ac i e index o he “cladding” equal o
a e age alue in he su ounding s uc u e [5].
Beside he index-guiding ib es he e a e also so
called band-gap guiding PCFs [6]. The e ac i e
index o he co e in hese ib es is smalle han
a e age index o cladding. The s uc u e o he
pho onic c ys al su ounding he co e o such ib es
can cause su icien e lec ion o he wa es
impinging om he co e also when he cladding is
ealised as an ai hole s uc u e.
Fig. 1 A scheme o pho onic c ys al wi h a long de ec .
The de ec –
„missing“
inhomogeni y
Ad ances in Elec ical and Elec onic Enginee ing
366
2. DESCRIPTION OF EXPERIMENT AND
RESULTS
The in es iga ion o p o ile o he op ical ield
in he ib e we ealised by pho og aphy o he end
ace o he ib e a e i s magni ica ion by an
app op ia e mic oscope.. The used se up is shown in
Fig. 2.
The illumina ion o he on ace o he ib e
was changed du ing he in es iga ion by changing i s
posi ion in he ocal plane o he illumina ing
sys em. I led o di e en ampli udes o exci ed
modes and ga e qui e b oad a ie y o obse ed
op ical ield shape in he ib e. Be ween ob ained
pic u es he e we e pic u es o ield, which has
e iden six old symme y. Such symme y o he
ield a he end o ib e exci ed by halogen lamp and
by He-Ne lase , espec i ely, a e shown in Fig. 3. In
he used egion o wa eleng hs ( isible ligh ) wo
modes can p opaga e in he ib e. In o de o show
he ield o ien a ion wi h espec o he ib e
s uc u e, he end ace o he pho og aphed ib e was
illumina ed by whi e ligh (Fig. 3.a). Tha is why he
ai holes a e displayed as small black ci cles. No e,
he pho og aph gi es he pa e n o op ical ield
in ensi y i.e. he squa e o he elec ical ield o he
wa e. Due o he symme y o he h ee old
symme y o he op ical ield he pic u e is six old
one. Bu his does no mean ha six old symme y o
he in ensi y mus be in ol ed by h ee old
symme y o he ield.
The op ical ield o he undamen al mode in a
PCF ha e he same phase in all c oss-sec ion like
LP
01
mode o a s ep-index ib e which ield is
ci cula ly symme ic. The six old symme y o his
mode can be simple a esul o he exis ence o he
ib e inhomogeni ies (holes), posi ions o which a e
shown in Fig. 3.a. Howe e , besides ields wi h
six old symme y we obse ed also ield wi h
signi ican h ee old symme y (Fig. 4).
The exis ence o such ield can be explained as
ollows: A change o he phase o he ield in he
ib e is gi en by i s phase eloci y ( he phase
cons an o he pa icula mode). As i can be seen
om Fig. 5 he phase eloci y is 1/cos (α) mul iple
o speed o a plana wa e in ha medium, whe e α
is he angle be ween he di ec ion o pa icula
wa es o ming he op ical mode and axis o he
Fig. 2 Se up o he expe imen .
Fig. 4 The pho og aphy o ield wi h h ee old
symme y.
a)
b)
Fig. 3 Pho og aphs o he op ical ield a he end ace o PCF when ib e was exci ed by halogen lamp (a) and by He-Ne
lase (b), espec i ely.
The symme y o op ical ield in pho onic c ys al ib e wi h igonal symme y
367
ib e. Because hese angles ha e di e en alues o
pa icula modes, phase cons an s o hese modes a e
di e ed, oo.
In he case o op ical ield o med by pa icula
symme ically dis ibu ed plane wa es he phase o
he ield depends on coo dina e by he same way. I
means ha he dependence o he phase cons an on
angle α illus a ed in Fig. 5 is alid ou side he ib e,
as well as inside i . Only he angle α is g ea e in
compa ison wi h angle inside he ib e because o
he di e ence o e ac i e indexes inside and
ou side he ib e. So, he ela i e phase o he ields
o med by he pa icula modes nea he end ace o
he ib e depends on he dis ance om he ib e end.
Tha is why de ocusing he mic oscope, i.e.
changing he dis ance be ween mic oscope objec i e
and he ib e end, causes he changes o he shape o
he imagined op ical ield o he app op ia e
posi ion change o he mic oscope. The change can
be so signi ican ha he six old ield symme y can
be ans o med o h ee old one.
Fo illus a ion o his e ec le us assume ha wo
modes a e gene a ed in ib e: undamen al mode
wi h ci cula symme y and h ee old symme y
mode. Field dis ibu ion o hese modes in 3D g aph,
as well hei in ensi y dis ibu ion a e shown in
Fig. 6.
I ollows om an elemen a calcula ion ha in he
places whe e phases o he ields o he i s and he
second mode a e shi ed by (n+1/2)π he in ensi y o
hei sum is he sum o he pa icula in ensi ies
(Fig. 7).
In he place whe e modes ha e equal o
opposi e phases he ampli udes o he in e e ing
modes should be summed up algeb aically. Fo an
app op ia e a io o hei ampli udes i can be
adically mani es ed in dis ibu ion o he in ensi y
o he inal ield. I illus a es he in ensi y
dis ibu ion shown in Fig. 8.
Fig. 5 Illus a ion o phase cons an dependence on di ec ion
o wa e p opaga ion.
1
0.002
M
1
3.723 10 6
MM MM
0.303
0.303
N
0.092
4.341 10 11
K K
Fig. 6 Pa e n o undamen al mode (M) and his in esi y in 3D (MM) and 2D plo (MM) and pa e n o mode wi h
igonal symme y (N) and his squa e (K).
Ad ances in Elec ical and Elec onic Enginee ing
368
The pho og aphs gi en in Fig. 3 and Fig. 4 show ha
o app op ia e posi ion o he scope (app op ia e
ocusing o he mic oscope) bo h si ua ions shown in
Fig. 7 and Fig. 8 can be obse ed.
3. CONCLUSION
The pic u es o modal ield dis ibu ion p esen ed in
his con ibu ion show ha he op ical ield in he
in es iga ed ib e has no cylind ical symme y. The
ield dis ibu ion is signi ican ly in luenced by
igonal a angemen o ai hole o he PCF s uc u e.
This esul documen s ha he ib e a ibu es PCF
a e de e mined by p ope ies o pho onic c ys al
e lec ing he ligh wa es p opaga ing h ough he
ib e co e also when he ib e is index-guiding one.
I documen s ha he name “index guiding ib es”
does no e lec he ib e p ope ies and should be
accep ed eally only as a name o kind o pho onic
c ys al ib es.
Acknowledgemen
This wo k has been pa ly suppo by Slo ak
Na ional G an Agency No. 1/2048/05 and by
Science and Technology Assis ance Agency con ac
No. APVT 20-013504.
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Issa, N.A., Lyy ikäinen, K.: Op ics Exp ess 12,
3465, (2004)
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[5] Bi ks, T.A., Knigh , J.C., Russell, P.S.J.: Op .
Le ., ol.22, 961-963, 1997
[6] Langsgaa d, J., Hansen, K.P., Nielsen, M.D.,
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B oeng, J., Jensen, J.B., Bja kle , A.:
P oceedings SBMO/IEEE MTT-S IMOC 259-
264, 2003
1
9.54 10 6
Fig.7 In ensi y dis ibu ion when phase di e ence o undamen al and he igonal mode is
π
/2 shown in
3D (a) and 2D g aph (b).
1.227
8.393 10 13
Fig.8 In ensi y dis ibu ion when bo h modes ha e he same phase in 3D and 2D g aph.
a)
b)
a) b)