Analysis of simulated reflection characteristics of uniform and apodized fiber bragg gratings

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. REFERENCES [1] ERDOGAN, T.: Fibe g a ing spec a. Jou nal o Ligh wa e Technology, ol.15, no.8, 1997, pp. 1277-1294 [2] OUELLETTE, F.: Fibe B agg g a ings. Spie’s Oemagazine – The SPIE Magazine o Pho onics Technologies and Applica ions, ol.1, no.1, 2001, pp. 38-41 [3] ROSTAMI, A., YAZDANPANAH- GOHARRIZI, A.: A new me hod o classi ica ion and iden i ica ion o comples ibe B agg ga ing using gene ic algo i hm. P og ess in Elec omagne ic Resea ch, PIER 75, 2007, 329 - 356 [4] KANG, Y., YE, CH., LUO, Z., SI, M.: In es iga ion on e lec ion spec a o chi ped and phase-shi ed ibe B agg g a ings. 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