Modeling OAM Transmission in Waveguides with COMSOL Multiphysics

Transcription

1 Modelg OAM Trasmsso Wavegudes wth COMSOL Multphyscs A. Caglero 1*, A. De Vta, R. Gaffoglo 1 ad B. Sacco 1 Departmet of Physcs, Uversty of Toro, Toro, Italy Cetre for Research ad Techologcal Iovato, RAI Radotelevsoe Italaa, Toro, Italy * Correspodg author: adrea.caglero@uto.t Abstract: The recet years have wtessed a growg terest the possblty of ehacg the formato trasfer per ut badwdth by explotg the Orbtal Agular Mometum (OAM) of lght both free-space ad guded scearos. I the proposed paper, the propagato of sutable OAM superpostos of wavegude egemodes gudg structures wth a crcular symmetry s aalyzed both theoretcally ad wth the ad of the software COMSOL Multphyscs, leadg to the estmato of the power atteuato costats due to the fte coductvty of the metallc gude walls. Keywords: Crcular wavegude; coaxal cable; COMSOL Multphyscs; Orbtal Agular Mometum (OAM). 1. Itroducto As emphaszed lterature [1], the freespace propagato of OAM beams s affected by a over-quadratc power decay wth dstace. However, ths egatve aspect does ot cocer guded propagato of waves carryg OAM, whch s stead free from ths lmtato. Ths fact has favored the terest towards the possblty of explotg the OAM wavegude scearos, where vortex beams ca be thought as sutable superpostos of the stadard Trasverse lectrc (T) or Trasverse Magetc (TM) modes supported by the cosdered gudg structures []. I Optcs, the possblty to excte OAM modes fbers has bee expermetally verfed frst partcular kd of fbers [3,4], wth rgshaped cores, ad the covetoal optcal fbers [5], pavg the way to the realzato of OAM-based multplexg schemes [5,6]. The guded propagato of OAM modes has bee also cosdered at the rado frequeces ad the mcrowave bad, sce wavegudes are ofte part of radatg systems [7,8]; as a example, ths s the case of the hor ateas, whch cosst of a flared metal wavegude shaped lke a hor [9]. Accordg to the cosderatos reported [], hollow metallc wavegudes wth a crcular symmetry, trval combatos of the covetoal egemodes are able to carry agular mometum, cludg both a OAM ad a SAM (Sp Agular Mometum) cotrbuto. I ths work, the propagato of these modes crcular wavegudes ad coaxal cables wll be vestgated both theoretcally ad also by meas of umercal smulatos performed wth the RF module of the software COMSOL Multphyscs. The preseted aalyss wll explctly prove how such OAM superpostos ca be supported by the cosdered gudg structures at frequeces greater tha the correspodg cutoff values. The, the power atteuato law relatve to the propagato of dfferet OAM modes sde a crcular wavegude wth copper walls wll be preseted, as a result of a realstc electromagetc smulato.. Crcular wavegude Ay perturbato sde a crcular wavegude wth metallc walls ca be descrbed terms of T ad TM modes, properly obtaed by solvg the Maxwell's macroscopc equatos wth the boudary codtos dctated by the wavegude geometry. Therefore, these electromagetc modes, whose mathematcal expresso s derved [1,11], costtute a complete set of felds sde the crcular wavegude. I ths secto the T modes for a crcular wavegude wth uform cross secto ad perfectly coductg walls are explctly reported. The, t wll be show how to obta propagatg modes carryg OAM by supermposg parwse such covetoal egemodes. By defto, a trasverse electrc mode has o electrc feld the drecto of propagato,.e., z. Accordg to [11], the trasverse COMSOL Multphyscs ad COMSOL are ether regstered trademarks or trademarks of COMSOL AB. Mcrosoft s a regstered trademark of Mcrosoft Corporato the Uted States ad/or other coutres. xcerpt from the Proceedgs of the 16 COMSOL Coferece Much

2 Fgure 1. Crcular wavegude oreted alog the z- axs ad smulated wth the software COMSOL Multphyscs. electrc feld compoets of a T mode propagatg sde a crcular wavegude of radus a ca be wrtte as: kc B J ( ) j where the harmoc tme depedece has bee omtted, B s a costat wth magetc feld dmesos, A ad B are arbtrary admesoal costats, whle j ζ j / a, beg ζ 1, ζ,..., ζ j the o-ull postve zeros of the fucto J' (ζ), arraged by creasg magtude. The dces ad j dcate a trasverse electrc mode propagatg sde the wavegude wth a waveumber α j gve by: beg ε r ad μ r the relatve permttvty ad permeablty of the teror medum. It should be oted that such mode ca exsts sde the wavegude f ad oly f the operatg frequecy ω s greater tha the correspodg cutoff frequecy / j ϕ j j z [ Acos( ϕ) + Bs( ϕ) ] e (1) j k cb J ' ( ) j jz [ As( ϕ) + B cos( ϕ) ] e () α j ε rµ rk j, εµ, where ε ad μ are the absolute permttvty ad permeablty. Sce the lowest zero of the frst dervatve of the Bessel fucto J s ζ 11, t follows that the T mode whch ca propagate sde the wavegude at the lowest frequecy s the T 11. (3) The two set of modes T cj, wth A ad B, ad T sj, wth A ad B, completely spa the orthogoal set of trasverse electrc modes sde the wavegude. By redefg these modes such a way that the parameters A ad B fulfll A B or A -B, the followg ew complete set of orthogoal modes ca be obtaed []: T T. (4) cj Usg the expressos for the T modes reported (1) ad (), the electrc feld compoets relatve to the superpostos defed (4) ca be explctly wrtte as: ϕ j 1 m j kc B K k cb K J' where K s a ew admesoal costat. I [] t has bee demostrated that the modes defed (5),(6) carry a et agular mometum whch s, o average, ћ per photo. Ths quatty cludes both a OAM ad a SAM (Sp Agular Mometum) cotrbuto. It s deed mpossble for a electromagetc mode to propagate sde the wavegude carryg all ts agular mometum ether the form of sp or OAM. The combatos defed (4) represet othg but a partcular bass choce the space of the wavegude egemodes ad the mode desty of the two sets s the same []. I order to vsualze these modes ad to prove ther propagato a crcular wavegude, the aalytc expresso of the electrc feld (5),(6) s provded put to a crcular wavegude modeled wth the RF module of the software COMSOL Multphyscs (see Fg. 1). The radus of the crcular wavegude s assumed to be a. m, the teror rego s flled wth ar ad a Perfect lectrc Coductor (PC) codto s mposed o the walls. Fg. ad 3 show the testy profles of two superpostos lke (4), wth j 1 ad, 1, at a operatg frequecy f 1 GHz, suffcet to allow the propagato of these modes sde the cosdered wavegude. It s worth otg that the above preseted approach could be exteded to the TM modes, for whch superpostos carryg agular mometum are defed complete aalogy to (4). J sj mϕ jz ( ) e e (5) j mϕ jz ( ) e e (6) j xcerpt from the Proceedgs of the 16 COMSOL Coferece Much

3 Fgure. Normalzed electrc feld testy for the superposto T c1 - T s1 evaluated wth the software COMSOL Multphyscs a crcular wavegude of radus a. m. The real part of the vector feld over the mddle trasverse cross secto s also reported. Fgure 3. Normalzed electrc feld testy for the superposto T c11 - T s11 evaluated wth the software COMSOL Multphyscs a crcular wavegude of radus a. m. 3. Coaxal cable A coaxal cable s made by a er coductg wre sequetally surrouded by a tubular sulatg layer ad a exteral metallc sheld. Ths kd of structure ca be vewed as a type of wavegude able to support the propagato of the Trasverse lectrc ad Magetc (TM) mode (ether electrc or magetc feld the drecto of propagato), by vrtue of the presece of two coductors. Whe t exsts, lke a coaxal cable, the TM mode s the fudametal oe, havg o cutoff frequecy. Despte ths, order to obta modes able to carry agular mometum, t s ecessary to resort to the hgher- order T (or TM) modes, whose formulato for the coaxal cable wll be troduced the followg. Accordg to the approach preseted [11], the electrc feld compoets of a T mode propagatg sde a coaxal cable wth outer radus a ad er radus b ca be wrtte as: Y j kc B [ Y ' ( a) J ( ) J ' ( a) ( )] Acos( ϕ) + Bs( ϕ ) j j j j j z [ ] e (7) xcerpt from the Proceedgs of the 16 COMSOL Coferece Much

4 Fgure 4. Coaxal cable oreted alog the z-axs ad smulated wth the software COMSOL Multphyscs. ϕ j Y ' where B s a costat wth the dmesos of a magetc feld, A, B ad K are admesoal costats, whle 1,,..., j are the solutos, sorted by creasg magtude, of the followg trascedetal equato: ( a) Y ' ( b) J ' ( b) Y ' ( a). (9) J ' k cb [ Y ' ( a) J ' ( ) J ' ( a) ( )] As( ϕ) + Bcos( ϕ ) j j j z [ ] e (8) A frequecy ω greater tha the cutoff value j / εµ allows the propagato of the T j mode sde the coaxal cable, wth a waveumber: αj ε rµ rk j. (1) j j Followg the same approach preseted the prevous secto for the crcular wavegude, the T compoets (7), (8), relatve to the coaxal cable, ca be supermposed parwse accordg to the combato defed (4), leadg to: ϕ J' J' j 1 m kc B [ Y' ( a) J ( ) mϕ j z ( a) Y ( )] e e (11) j j k cb K j [ Y' The propagato of these modes ca be verfed wth the ad of the software COMSOL Multphyscs by properly feedg the put port of a coaxal cable (Fg. 4) wth the cosdered superpostos of T modes. The er ad outer coductors of the smulated coaxal cable are modeled wth a Perfect lectrc Coductor codto mposed o ther surfaces, whle the er rego s flled wth a Teflo delectrc layer (ε r.1, μ r 1). Moreover, the rad of the two coductors are assumed to be: a.16 m ad b.5 m. Fg. 5 ad 6 show the electrc feld testy for the combatos T cj - T sj, wth j 1 ad, 1, propagatg sde the cosdered coaxal cable at a operatg frequecy f 1 GHz. j j ( a) J ' ( ) mϕ j z ( a) Y ' ( )] e e (1) j K j j j + + Fgure 5. Normalzed electrc feld testy for the T superposto evaluated wth the software COMSOL Multphyscs a coaxal cable flled wth Teflo. The real part of the vector feld over the mddle trasverse cross secto s also reported. xcerpt from the Proceedgs of the 16 COMSOL Coferece Much

5 Fgure 6. Normalzed electrc feld testy for the 1 T superposto evaluated wth the software COMSOL Multphyscs a coaxal cable flled wth Teflo. 4. Power atteuato law The wavegudes cosdered so far have bee deally supposed to be made of perfect coductors. Isde ths kd of wavegudes the electromagetc modes ca propagate above the cutoff wthout sufferg ay atteuato. However, practce, perfect coductors do ot exst, but rather there are good coductg materals, such as metals, wth fte coductvtes o the order of S/m. These great, but lmted, values of the metal coductvty causes the feld to peetrate a lttle the wavegude walls; as a cosequece, the coducto electros of the metal layer are set to moto, dsspatg a part of the eergy va the Joule effect. Hece, real wavegudes wth metallc walls, the supported electromagetc modes udergo some atteuato due to the coducto (ohmc) losses. Moreover, f the cosdered wavegude s flled wth a lossy delectrc materal, the so-called delectrc losses troduce a further cotrbuto to the wavegude atteuato. Ayway, restrctg the preset aalyss to the case of wavegudes flled wth ar, such kd of losses ca be eglected. If P represets the power assocated to a mode wth dces (, j) at some referece pot alog the gudg drecto (.e., z ), the the power P at some other pot z s related to P by: P β z ( z) P e j, (13) where β j s the ohmc losses atteuato coeffcet. I order to estmate the atteuato coeffcet relatve to the propagato of a superposto T cj T sj sde a real gudg structure, a crcular wavegude wth copper walls s smulated wth the software COMSOL Multphyscs. The modeled wavegude s flled wth ar, the radus s assumed to be a. m ad the operatg frequecy f s fxed at 1 GHz. By meas of a parametrc sweep, COMSOL provdes the estmato of the S 1 P(z)/P parameter for dfferet values of the wavegude legth z, betwee λ ad 1λ, beg λ the wavelegth. Fg. 7 shows the S 1 parameters as a fucto of z logarthmc scale, for three dfferet T mode superpostos wth dces j 1 ad, 1,. For each of the cosdered cases a lear ft s performed eablg the estmato of the correspodg power atteuato costats. The same procedure s the repeated for the propagato of sgle T j modes, wth j 1 ad, 1,. As ca be see from Table 1, the atteuato coeffcets estmated for the T modes cocde wth those obtaed for the OAM superpostos. As expected, ths result shows that the wavegude mode combatos able to carry agular mometum exhbt the same behavour terms of power budget as the covetoal modes. xcerpt from the Proceedgs of the 16 COMSOL Coferece Much

6 Fgure 7. S 1 parameter as a fucto of the gude legth z for a crcular wavegude wth copper walls ad radus a. m. I the leged the results of the correspodg lear ft: S 1 -β j + c, wth c, are also reported. Fally, order to provde a further theoretcal cofrmato of the obtaed results, a formula for the estmato of the atteuato coeffcet a realstc crcular wavegude s cosdered. Accordg to the approach reported [1], for a crcular wavegude of radus a, the power atteuato coeffcet relatve to the propagato of a T j mode ca be evaluated theoretcally va the followg formula: β T j ωµ σ f aη 1 f c f f, (14) where σ s the metal coductvty, η µ / ε s the mpedace of the er medum, ζ j s the T mode egevalue, ω π f s the operatg agular frequecy, whle f j s the cutoff frequecy assocated to the cosdered mode ad defed by: ζ j f j. (15) π a εµ It should be oted that, for the cosdered crcular wavegude flled wth ar ad surrouded by copper walls, σ S/m, ε ε, μ μ c + ζ j ad η correspods to the vacuum mpedace η 377 Ω. The results reported Table 1 clearly show that a very good agreemet betwee the theoretcal ad umercal results has bee acheved. 5. Coclusos I ths work, the propagato of T superpostos carryg agular mometum two possble guded scearos has bee vestgated wth the help of COMSOL Multphyscs. It s mportat to emphasze that the possblty to excte such modes RF wavegudes s of paramout mportace ot oly for guded trasmssos, but also Table 1: Ohmc losses atteuato coeffcet for a crcular wavegude wth radus a. m ad copper walls. β 1 (db/m) 1 COMSOL (OAM) COMSOL (T) Th. formula (14) xcerpt from the Proceedgs of the 16 COMSOL Coferece Much

7 for more geeral applcatos, sce most of the ateas ad the RF systems are fed va wavegudes ad coaxal cables. Although t s kow that the above defed OAM modes do ot crease the wavegude mode desty ad preset aalogous propagato propertes as the stadard T modes (see Table 1), they actually represet a atural bass set for crcular symmetrc geometres. Therefore, t s reasoable to expect that they mght brg techologcal smplfcatos all cotexts were the crcular symmetry s preserved. [1] S. J. Orfads, lectromagetc Waves ad Ateas, (14) [Ole]. Avalable: [13] C. A. Balas, Advaced geerg lectromagetcs, Wley, New York (1989). 6. Refereces [1] M. Tamagoe et al., Commet o Reply to Commet o codg may chaels o the same frequecy through rado vortcty: frst expermetal test, New Joural of Physcs 15 (13). []. Bergld ad G. Björk, Humblet's Decomposto of the lectromagetc Agular Momet Metallc Wavegudes, I Trasactos o Mcrowave Theory ad Techques 6, (14). [3] N. Bozovc et al., Cotrol of orbtal agular mometum of lght wth optcal fbers, Optcs Letters 37, (1). [4] S. Ramachadra ad P. Krstese, Optcal vortces fber, Naophotocs, (13). [5] H. Huag et al., Mode dvso multplexg usg a orbtal agular mometum mode sorter ad MIMO-DSP over a graded-dex few-mode optcal fbre, Scetfc Reports 5 (15). [6] N. Bozovc et al., Terabt-Scale Orbtal Agular Mometum Mode Dvso Multplexg Fbers, Scece 34, (13). [7] D. Zhag et al., Geeratg -plae optcal orbtal agular mometum beams wth slco wavegudes, I Photocs Joural 5 (13). [8] W. L. We et al., Two moopole ateas for geeratg rado OAM waves crcular wavegude, 1th uropea Coferece o Ateas ad Propagato (ucap 16). [9] COMSOL Multphyscs 5.1, RF Module, (15). [1] J. D. Jackso, Classcal lectrodyamcs, thrd edto, Wley (1999). [11]. Botta ad T. Bressa, lemet d elettromagetsmo avazato, Arace (9). xcerpt from the Proceedgs of the 16 COMSOL Coferece Much