The symmetry of optical field in photonic crystal fibre with trigonal symmetry

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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Le ., ol.22, 961-963, 1997 [6] Langsgaa d, J., Hansen, K.P., Nielsen, M.D., Hansen, T.P., Riishede, J., Hougaa d, K., So ensen, T., La sen, T.T., Mo ensen, N.A., 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)