GENERALIZED SCATTERING MATRIX FOR OPTICAL STRUCTURES Suit Mehrotra,Reea umbhare ad Girish P. Saraph Dept. of Electrical Egieerig Idia Istitute of Techology Bombay Mumbai 476 suit,shaku,girishs@ee.iitb.ac.i Abstract-The Geeralized Scatterig Matrix (GSM) method is commoly used for aalyzig differet RF structures ad waveguide discotiuities. The method is exteded here to model optical structures. The basic GSM formulatio is bechmarked usig the High Frequecy Structure Simulator (HFSS) simulatios i the RF domai. A similar aalysis is employed for optical devices ad ca be used to determie the electromagetic fields ad the Poytig vector at every poit of structure. The GSM is used to aalyze Off-axis excitatio i multimode fibers. This ca be used to reduce iter-modal dispersio i multimode fiber ad lower the modal oise. The scatterig matrix treatmet ca also be used to aalyze dielectric discotiuities i the form of a periodic structure, kow as fiber Bragg Gratig (FBG). The aalysis ca be exteded for the costructio of Bad-Stop filter by varyig the periodicity of the FBG structure.. INTRODUCTION The Scatterig Matrix formulatio is a geeral method for aalyzig electromagetic propagatio i terms of orthogoal modes from low frequecies to RF/microwave frequecies ad all the way to frequecies i the optical regio. It is a very powerful ad accurate techique for aalyzig various types of waveguide discotiuities ad the cascaded structures []. It is computatioally efficiet as compared to other methods ad suitable for ay waveguide dimesios or operatig wavelegths. The techique has bee used for studyig power couplig, reflectio ad resoaces i microwave devices, i the past. I this paper scope of this techique has bee exteded to optical devices by defiig eigemodes for optical fibers ad dielectric slab waveguides. Due to small refractive idices differece betwee core ad claddig of the commoly used optical fibers, they satisfy the weakly guided coditios. Uder which, the vector EM wave equatio ca be expressed i the scalar form ad the eige modes ca be expressed as LP modes. The aalysis of may WDM devices such as Bad-stop filters, directioal couplers, Add-drop MUXs ad itegrated optical chips is a potetial applicatio of GSM techique.. SCATTERING MATRIX FORMULATION The problem of electromagetic scatterig at waveguide discotiuities has bee treated by may exact ad approximate methods. The variatio method has provided quite accurate mode solutios to a wide variety of waveguide discotiuity problems. Sice the advet of fast digital computers iterest has shifted to umerically orieted techiques. Oe of the most geeral ad rigorous method is the scatterig matrix treatmet based o modal expasio []. The procedure used i GSM is based o the geeralized etwork formulatio for multi-port problem, where each eige mode i a structure acts like a separate port. To illustrate the procedure, its applicatio to aalyze trasverse discotiuity betwee two ifiitely log uiform waveguides has bee cosidered. The scatterig matrix treatmet of the juctio is described i the followig sectio... Scatterig Matrix Let the juctio be excited from waveguide by ay arbitrary field, which is resolved ito umber of orthogoal eige modes. It geerates forward travelig modes whose amplitudes ad phases are ot iflueced by reflected wave from the juctio. Assumig expasio of trasverse electric ad magetic fields E ad H, i terms of the eige-modes e r ad i h espectively for i waveguide as: Waveguide Waveguide A A B B Fig.. Defiitio of Mode Amplitudes r = M r E ( A + B ) e i i i i = ()
r M ( A B ) r () H = i i h i i= z i Where A i ad B i are the complex mode amplitudes of the forward ad backward propagatig waves of the i th eige mode i waveguide. ei ad hi are the trasverse basis vectors or eige-modes. z i is the characteristic impedace. M ad N are the total umber of modes cosidered(propagatig ad o-propagatig) i the waveguides ad respectively. For the solutio to coverge [3]: M/N a /a (3) Applyig cotiuity coditios for trasverse electric ad magetic fields across the commo aperture area, yields followig relatios betwee the forward ad backward mode amplitudes i two waveguides: P[ A + B ] = I [ A + B ] (4) Z PY [ A B ] = I [ A B ] Where I is idetity matrix: A ad B are vectors cotaiig the ukow complex mode amplitude coefficiets (A l, A l,..a kl ), (B l,b l,..b kl ) foegios l=, ad z is a M M diagoal matrix of the impedace i the regio ad y is a N N diagoal matrix of - the modal admittace i regio (y = z ). The elemets of the mode-couplig matrix P are give by []: P e r.r e ds (5) ji = j i CA Where itegratio is performed over the commo aperture area betwee the waveguides, ad the ormalizatio is chose such that e e ds = δ e e ds = δ (6) s i. ' ' ; ' i ii j. j s jj Equatios (4) ca be solved for B s as B = S A (7) Where B B = ; A A = ; S S S = B A S S Here S ad S are matrices represetig the amplitudes of reflected ad trasmitted eigemodes i waveguides ad respectively, uder excitatio from waveguide. S ad S represet the amplitude of trasmitted ad reflected modes whe excitatio is from the waveguide. Rearragig Eq (4) ito the the scatterig matrix form yields: S =[I+z P T y P] - [I-z P T y P] S =[I+z P T y P] - z P T y (8) S =[I+ Pz P T y ] - P S =[I+ Pz P T y ] - [I- Pz P T y ].. Cascaded Structures Oe of the potetial applicatios of Scatterig Matrix method is aalysis of complicated structures/etworks, by decomposig them ito several simpler juctios ad the cascadig the juctios. Sice the higher-order cut-off modes are icluded, the iteractio betwee the juctios ca be correctly described eve if the distace betwee them is small. I cascaded waveguides the middle guides give a trasitio matrix that modifies the trasmitted amplitudes of propagatig modes from first iterface by multiplyig each term with a phase lag. T i = exp(-jβ i l) Where l is the legth of middle waveguide. Whereas, the amplitudes of the cutoff modes decay expoetially with the guide legth. The the overall scatterig matrix of the cascaded juctio is give by S = S A +(S A S B S A T )/(-S A S B T ) S = (S A S B T )/(-S A S B T ) S = (S B S A T)/(-S A S B T ) (9) S = (S B S B S A T )/(-S A S B T )+S B Where S A ad S B are the scatterig matrix at juctio A ad B. Aother approach for fidig the GSM of two cascaded juctio is give i Ref.[6] which has bee verified to give the same result. S = as b S + a (SaS b) S I SaS b Sa SaS b Sb Sb Where S a = T S a T ; I T = ; T ii = exp(-jβ i l) T [ ] [ ].3. Bechmarkig usig Rectagular Metallic Waveguide The verificatio of power couplig for rectagular waveguides i microwave regio usig GSM method is carried out. A quatitative compariso of S ad S values obtaied from the GSM ad HFSS simulatios is preseted. For this the height ad width of secod waveguide is chaged i steps of mm.iitial dimesios of waveguide are 3.6x.) cm ad workig frequecy at 5.GHz.
s S.5..5..5.99.98.97.96 Variatio i value of S Vs Legth of ier WG 3 4 5 Legth of ier WG i cm Fig.. Quarter Wave Trasformer HFSS GSM.95 4 6 8 Variatio i mm Fig. 3. Effect of variatios i height of waveguide o reflectio ad trasmissio parameter. The above fig shows that, the eergy balace coditio (S *S +S *S =) for ay set of dimesios of two waveguides is satisfied. This result is preseted i followig table. TABLE I Eergy Balace for Differet Dimesio of Waveguides Var.i dim.(mm) S S Power.4.999. 4.85.996.999 8.84.98. 3. OPTICAL ANALOG I this chapter, the modal aalysis of optical fiber i terms of LP modes is treated ad the applicatio of GSM is explaied. The exact solutio of Maxwell equatios for a cylidrical dielectric waveguide leads to TE, TM ad hybrid modes, (EH ad HE). The aalysis ca be simplified uder the weakly guidig approximatio[4] which leads to form two sets of liearly polarized (LP) modes with x ad y polarizatio. The field equatios for the LP mode ca be writte i the followig form [5]. cos φ E x = E( r) exp( ωt βz) () si φ Where, E x is the domiat trasverse electric field compoet. It has periodic depedece o azimuthal agleφ.the radial depedece of electric field E(r) ca be expressed by Bessel fuctio (ur/a) iside the core ad modified Hakel fuctio (wr/a) outside the core as follows: ( ur / a) E( r) = E fo<a (core) () ( u) ( wr / a) E( r) = E fo>a (claddig) () ( w) Where, E is amplitude of the electric field, a is radius of the fiber core, u ad w are radial propagatio costat ad claddig decay parameter. u +w =V =ka( - ) (3) The field matchig coditios at the boudary (r = a) requires that: ± ( u) ± ( w) u = ± w (4) ( u) ( w) Where m th eige solutio of the equatio correspods to the LP m mode. Thus the propagatio characteristics (u, w ad β ) of various modes ad their depedece o the optical wavelegth ad the fiber parameters are determied. The orthogoality coditio is satisfied which gives the solutio to scalar wave equatio. π E xm ( r, φ )E x' m' ( r, φ ) rdrdφ = δ ' δ mm' (5) Thus, the couplig matrix for the propagatig modes of two coected optical fibers is determied from Eq (5) ad the procedure explaied i sectio is followed to get the scatterig matrix of the juctio. 4. MODELLING OF OPTICAL STRUCTURES The GSM techique described above ca be used for aalyzig optical structures or etworks cosistig of series of optical juctios coupled together. 4.. Couplig of Sigle-Mode to Multi-Mode fiber I the multi-mode fibers the power is carried by all propagatig modes travelig with differet group velocities. This causes pulse broadeig due to iter-modal dispersio ad limits the badwidth-distace product for a give optical lik. It also leads to modal oise for a coheret source. If oly higher order modes of the multimode fiber are selectively excited the the iter-modal dispersio ca be reduced by three fold. The selective excitatio of higher-modes i a multi-mode fiber by a sigle-mode fiber is studied here usig off-axis excitatio.
.8 LP S.6.4 LP LP LP. 3 4 5 Offset from the ceter (e-6) Fig 4. Offset couplig variatio The above figure clearly shows that couplig to higher order modes icreases as the sigle mode fiber is moved off-axis i the core of multi-mode fiber. Thus off-axis excitatio of multimode fiber ca be used to selectively couple power ito higher order modes. 4.. Fiber Bragg Gratig A umber of preset day optical etworks use WDM devices based o FBG, which ca be aalyzed usig GSM. The FBG uses periodic variatio i refractive idex of core to get selective reflectio at a give wavelegth. Ref[7] Fig.7. Uiform Fiber Bragg Gratig Aother mm log FBG from Ref. [8] is aalyzed where value is chaged from 3. -4 to.5-3. The results are i good agreemet with those give i [8] ad are preseted i followig figure: Fig 5. Fiber Bragg Gratig A umber of periodic trasitios from to i the core ca be cascaded together to study the FBG structure. The results show a liear icrease i reflectio coefficiet S with chage i the refractive idex ad it has bee verified with the theoretical results, ad the agreemet is exact. Ref[8] Fig. 8. Reflectio Characteristics of a mm log FBG Fig 6. Reflectio parameter variatio A uiform FBG usig sigle mode step idex fiber is cosidered. The result shows good agreemet i term of stop-bad to the theoretical ad experimetal results preseted i Ref.[7]. The GSM results show higher levels of side-lobes due to step discotiuous rather tha siusoidal variatio i the refractive idex profile. 4.3. Effect of Chage i Number of Periods o Reflectio Coefficiet I order to study the effect of icrease i umber of periods o reflectio properties of FBG the fiber of dimesios as metioed i previous sectio are selected. The period legth is kept costat ad effect of chage i umber of periods o reflectio coefficiet S is studied. As the umbers of periods were chaged from to 6 the reflected power icreased as show i the figure.
S.8.6.4. 6 periods 3 periods periods periods badwidth.the formulatio follows the scalig orms ad bad stop filter for three differet frequecy bad ca be desiged by appropriate variatio i the period legths. This gratig could be used i a WDM Add-drop Multiplexer to select or drop multiple wavelegths simultaeously. 544 546 548 55 55 554 556 Wavelegth (m) Fig. 9. Effects of chage i umber of periods o S It is observed that iitially the variatio i the value of S is icreased liearly ad the it saturated expoetially to a maximum value of.9999 ever crossig. These results are preseted i followig plots: Fig.. Bad Stop Filter 5. CONCLUSION Fig.. Variatio i value of S with umber of periods As it ca be see that the icrease i umber of juctios beyod 5 will have egligible effect o value of S. The above plot shows that GSM techique ca be effectively used eve i o-lieaegio. The expoetial saturatio of S meas that the trasmissio coefficiet moves towards zero expoetially as the eergy is always coserved. The expoetial decay implies that the log curve should have a costat slope. 4.4 Bad-Stop Filter The FBG is used to desig a stop-bad filter over a bad of wavelegths istead of sigle wavelegth. I above desigig the period legth i all three sectios is desiged for three differet frequecies. Thus a bad of No. of Periods 8 7 6 5 4 3 λ λ λ3 Wavelegth Fig.. No. of period selected at desiged wavelegth frequecy is reflected from the complete structure ad we get a adjustable The mode couplig aalysis of optical devices usig GSM techique has bee show. The techique is computatioally more efficiet ad gives very accurate results. The aalysis of a cascaded structure ca be carried out by decomposig it i terms of simple juctios. Thus the method is much superior to existig oes ad has wide applicatio i microwave ad optical etworks. 6. REFERENCES [].Esteba ad.m.rebollar, Geeralized scatterig matrix of geeralized two-port discotiuities, IEEE Tras. Microwave Theory Tech.,vol. MTT-39, p.75,99. [].M. Neilso, P. E. Latham, M.Capla, ad W. Lawso, Determiatio of the resoat frequecies i a complex cavity usig the scatterig matrix formulatio, IEEE Tras Microwave Theory Tech., vol.37, p. 65, 989. [3] S.W.Lee, W.R.oes ad..campbell, Covergece of umerical solutios of iris type discotiuity problem, IEEE Tras Microwave Theory Tech., vol. MTT-9, p58,97. [4] D.Gloge, Weakly guidig fibers, Appl.Opt.,,pp.5-58,97. [5]. M. Seior, Optical fiber commuicatio, Pritice Hall Idia. [6] T.Itoh, Numerical Techiques for Microwave ad Millimeter-wave passive structures, ho Willey ad Sos, 989. [7] C.R.Giles, Lightwave applicatios of Fiber Bragg Gratig, oural of Lightwave Techology, vol. 5, pp. 39-43,Aug997. [8] T.Erdoga, Fiber Bragg Gratig Spectra, oural of Lightwave Techology, vol. 5, pp. 77-94,Aug997.