Modal analysis of coupling problems in optical fibers

162 IEEK TRANSACTIONS ON MICROWAVESTHEORY AND TECHNIQUES, VOL. Ml’T-%], NO. 1, JANUARY 1975 Modal AnalYsis o Coupling P oblems inop icd Fibe s ANGEL CARDAMA, MEMBER, IEEE, AND EDWARD T. KORNHAUSER, SENIOR MEMBER) IEEE Abs ac —A modal analysis o he p oblems o exci a ion o he dominan mode in ah op ical ibe by inciden plane wa es and Gaussian beams has been ca ied ou , and he esul s applied o he e ec on ansmission o misalignmen in ibe junc ions due o o se s, il s, and gaps. The esul s in caseso ma ched media con- i m he accu acy o p e ious heo e ical ea men s using he Bo n app oxima ion, which in u n show good ag eemen wi h expe i- men al esul s. In addi ion, he modal analysis gi es mo e p ecise solu ions when he e is amisma ch o media and makes possible he ea men o some p oblems o which he Bo n app oxima ion is no applicable. I. INTRODUCTION O VER THE 1’AST i e yea s he ad ances inade in op ical signal p ocessing echniques and in he de- elopmen o low-loss glasses o op ical ibe s ha e made ‘ he u iliza ion o he eno mous bandwi h a ailable in op ical communica ions an a ac i e possibili y. Howe e , i such ibe s a e ac ually o be employed as ansmission channels, in he manne o ansmission lines and n~ic o- wa e wa eguides, one mus be able o couple, splice, bi u ca e, e c., in away analogous o ha used wi h hose olde channels. This is no a i ial echnical p oblem, because he co e o he ibe , whe e he ields a e concen- a ed, is ypically only a ew mic ons in diame e so ha he di icul y, o example, o aligning wo ibe s when splicing is c i ical. The exci a ion o p opaga ing modes on a ibe by a i- ous ypes o sou ce illumina ion has been s udied by Snyde [1]–[3] and Ma cuse [4J bo h using he Bo n app oxima ion. Fu he mo e, splicing echniques ha e been de eloped [5]–[7] and some measu emen s made [7]-[9] o he e ec s o impe ec alignmen a he in e - ace be ween wo uni o m ibe s. Mos ecen ly Cook e al. [9] ha e also gi en a heo e ical analysis o he e ec s o misalignmen in splicing on he ansmission coe icien , again making use o he Bonn app oxima ion. The im- po ance o such a heo e ical calcula ion lies in he ac ha i would enable one o se meaning ul s anda ds o p ecision which mus be adhe ed o in he p ac ical means used o splicing. Ma,nllsc iP ecei ed Ap il 4, 1974; e ised July 21, 1974. This wo k was suppo ed by he Na ional Science Founda ion unde G an GK 10971 o B own Uni e si y. A. Ca dama was wi h he Di ision o Enginee ing, B own Uni- e si y, P o idence, R. 1. He is now wi h he E.T.S. Ing. Tele- comunicaci6n, Poly echnic Uni e si y o Ba celona, Ba celona, Spain. E. T. Ko nhause is wi h he Di ision o Enginee ing, B own Uni e si y, P o idence, R. I. 02912. Howe e , all o he heo e ical calcula ions men ioned employ he Bo n app oxima ion, i.e., he assump ion ha he ields a he illumina ed c oss sec ion o he ibe consis en i ely o hose in he inciden wa e. Tha he e mus be a disc epancy [10] esul ing om he use oi his app oxima ion is indica ed by he ac ha Ma cuse [4] calcula ed wo alues o ansmission coe icien co e- sponding o he wo possible bounda y condi ions o be used (con inui y o angen ial Eo o angen ial H) and a bi a ily ook hei geome ic mean. Al hough his mean esul was always qui e easonable, in some cases one o he wo bounda y condi ions led o a ansmission coe icien la ge han uni y, which is clea ly impassible. Consequen ly, his pape will eexamine he basic exci a ion p oblems, making use o amo e igo ous modal expansion o he ields, which in u n is based on he gene al echniques o hyb id modes de eloped by Yagh- jian [11], [12]. The solu ions will show ha , a he su p isingly, he esul o aking he geome ic mean o he wo Bo n app oxima ion coe icien s was ema kably accu a e. h’inall y, he esul s and echniques will be applied o he p oblems o he e ec on e iciency o ansmission a he junc ion o wo similai op ical ibe s o h ee ypes o de ec in hei alignmen : il o one axis wi h espec o he o he , o se o hei axes, and swiall gaps be ween he ibe co es. Nume ical esul s will be p esen ed o all o hese p oblems and compa ed wi h he co esponding Ilml app oxima ion esul s. II. lJORIIULATIOX The su ace modes o he in ini e ci cula dielec ic od ha e been ex ensi ely s udied [2], [13], [14], and hei o hogonali ywell es ablished [1.5]. The ac ha he modes a e hyb id and he exis ence o acon inuous spec- um malce he exac solu ion o di ac ion and sca e ing p oblems o a ibe i h in ini e ou e diame e (od) a o midable ask. In ac , he e is no kM)~YD exac solu ion e en o he axially symme ic caseJ which educes o a scala p oblem [16], [17]. An exac solu ion would e- qui e sol ing an in eg al equa ion, and one possible means o o e coming he di icul y o i s solu ion would be o app oxima e he in eg als by in ini e summa ions and sol e he esul ing linea sys em. Ana u al way o achie - ing his is o enclose he dielec ic od in aconcen ic pe ec ly conduc ing cylinde wi h adius la ge compa ed o he co e adius and he wa eleng h. As he adius o he me allic pipe inc eases, he su ace modes a e una ec ed and he nonsu ace ype modes become acon inuum CARDAMA AND KORNHAUSI?R: MODAL ANALYSIS 163 (Appendix). This me hod will allow us o sol e he exci a- ion and ibe coupling p oblems by ano mal-mode anal- ysis, as in as anda d wa eguide discon inui y yp oblem, and o ake in o accoun he e lec ed ene gy. To ind he sca e ed and ansmi ed ields o awa e inciden a z=O, Fig, 1, we will ha e o expand he ields on bo h sides o he in e ace as an in ini e summa ion o he modes in each s uc u e. Con inui y o he ans- e se componen s o he ields a z=Ogi es ~a~e~ +E, = ~ a.e. nm –~aJh2+Hi=~a.k (1) nm whe e E,, Hi a e he ans e se inciden ields, em, h~ he ans e se ields o he su ace and nonsu ace modes o he ibe , and e i’, h.’ a e he o wa d a eling modes o he homogeneously illed ci cula wa eguide [18] in he launching case o he hyb id modes o he ibe in he ibe junc ion case. The se o modal coe icien s am, which will gi e he e iciencies o exci a ion, can be ob ained om (1) by c oss mul iplica ion by hn’ and en’, in eg a ion o e he c oss-sec ional a ea Aa he lei o he in e ace, and making use o he o hogonali yo he modes, gi ing a iWnml +A. = ~ aJII~’1 m —a.’AT..’ +B. =~kN%”L (2a) m hese wo equa ions can be added o ob ain alinea sys em o he modal coe icien s o he ansmi ed modes ~~l%lIWm”+N.m[=A.+B., )/ =1,2,. . . (2b) whe e Nnn =Jen’ xh: ..2da a e he no maliza ion ac- o s o Lhe TE and TM modes and An =/(E, XhJ) .2 da (3) A B. = ~ (en’ XH~).%da (4) A / N.m =(e.’ Xh~) ..2da (5) A M%9L=/(em Xh;)”2 da. (6) A A. and B. can be calcula ed om he exp essions o he inciden ields, and he c oss-no maliza ion ac o s, N~”II and M~n, ha e been calcula ed by Yaghjian [12] by educing he su ace in eg als o line in eg als and a e gi en in he Appendix. Fig. 1. The summa ion o e nincludes o he launching case all TE and TM modes o he homogeneous ci cula me al- lic wa eguide a anged in o de o inc easing eigen alues, and in he ibe junc ion case all he p opaga ing su ace modes and all he nonsu ace modes (Appendix). To ob ain anume ical solu ion o (2b), we shall ha e o unca e he sys em and sol e he esul ing ini e sys em, adding mo e equa ions un il he modal coe icien s ob ained become s able wi hin he accu acy o he com- pu a ions, and he addi ion o new equa ions does no p oduce u he a ia ions. P ac ical ibe s a e made wi h he adius o he cladding la ge compa ed o he co e adius and usually a e ex- e nally coa ed. This in ac educes he con inuous spec um o adisc e e one bu does no a ec signi ican ly he su ace modes. The p eceding model can be iewed no only as asolu ion o he ans e sally in ini e ibe bu as an exac s udy o he p opaga ion and exci a ion o modes in a ibe wi h ame allic ex e nal coa ing. III. EXCITATION COEFFICIENTS .4. T unca ed [lni o m Plane Wa e Fo his case, he inciden ields a z=Oa e aken as hose o auni o m plane wa e in amedium o dielec ic cons an e~ illumina ing aci cle o adius cconcen ic wi h he ibe and p opaga ing a an angle 6wi h espec o he axis o he ibe z. The plane o he ibe axis and he di ec ion o incidence is ,alwm o be he J, zplane, so ha Ei =EOexp (–jk3 zsin 0)~ (7) wi h 1cS2=0J3.w, ii =ksin e+2cm e, and an @ ime dependence is assumed. We will conside only ~-pola ized plane wa es, which exci e only modes wi h gl (0), VZ(0) gi en by (A6) wi h n=1. Exp essions o he o he pola iza ion could be simila ly de i ed. Also we will assume he ields inciden a z=O, bu i he sou ce w-e e a z=Z.(z. <O) wi h e%o adi e en medium in he egion Z. <z<0, he p oblem could be ea ed in asimila way by w i ing he con inui y equa ions a he z=Z. and z=Oin e aces and elimina ing coe icien s un il we a e le wi h asys em ela ing inciden and ansmi ed ields, as will be done in s udying he e ec s o gaps in ibe join s. 164 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, JANUARY 1975 I?o anx-pola ized uni o m plane wa e, we ob ain EZ =–EOexp (–jA?”cos@) sin O(9a) E, =EOexp (–jAYcos+) COSOCOS4 (9b) E+ =–EO exp (–~A cos 4) cos Osin 4(9C) wi h A=%sine (10) he Hi ield componen s a e ob ained om (8) and (9). We shall lha e o sol e he sys em (2). Subs i u ion in o (3), (4) o he ields o he homogeneous] y illed ci cula wa eguide [18] gi es o An and B. he ollowing exp essions. Fo nodd, TEln modes 13m2E,Cos~02“ An= –j— // exp (–jA cos 4) wo’Yn 00 [ ll(~n ) 1 .-—COS24 +J/ (7J’) Sill’ 4YCh d4 (11) ~n wi h -Y. =Pl~’/b (n =2m —1, m=1,2, ●..), whe e PI~’ is he m h o de ze o o J{, bis he adius o he guide, and I% is he p opaga ion cons an o he n h mode. Use o he ecu ence ela ions o he Bessel uc ions gi es ch A. =–j ~2~ cos eHexp (—jA cos 4) [Jo (?’TJ”) #u/’loy* 00 +J2(w) C(JS~4]T d c14. (12) Using he associa ed se ies exp (—~A cos 4) =Jo (AT) +2~(—j) ‘Jk (A ) cos ~4 k=l (13) and in eg a ing o e @gi es A, . . –j% ( ,/po) ‘% Coseln’ (14) wi h ! I.1 =‘[Jo(~. )Jo(A ) –J2(7. )J,(A ) ] d? (15) o which can be in eg a ed analy ically o gi e [ –[C/ (yn’ –A’)]{ i[Jo(w) +J2(w) ~l(AC) 1=1 =–~.[Jo(Ac) +J2(Ac) VI(W) ), A#Y.. (2/@Jl’(A ), A=7$$ (16) B. is ob ained in he same way as Bz =–j *(eJ/po) % l.1 (17) o no mal incidence (0 =O), A=O, and CJ1(’ync) Inl –A=O. (18) Y. ‘ “1. 100 80 60 m g,o. Lo 2 20 4 6 0123 ~Pc a) a=2, n$=1.L9 V=2.4143 c) a=2, n,=l V=2.4143 ●I. 100 80 60 ❑e.@ 2 40 20 L a6 0123LPC b) a=l ,ns:l,49 V=1.207 80 60 k e,00 2 Lo L 20 6 8 0 123 ~P d] a=!, n,=l V=1.207 c Fig. 2. HE,, mode launching e iciency in a ibe wi h nl =1.50, n~ =1.49, b=25 ~, a=1?2 p, xo =0.9 p, exci ed by a unca ed 1mi o m plane wa e a obhque incidenee e sus adius o illumi- na ed a ea. Simila ] y o ne en, TM1. modes, we ha e .4n =–j :(e3/#o) %cm 191n’ (19) ~.EO B~= –j— (dM) 112 In2 (20) ‘Y71 wi h ~. =P1~/b (n =%n, n~ =1,!2,. ..), whe e Pl~ is he m h o de ze o o J1. I.?, =/‘[JO(~. kJo(Ay) +Jz(7.7)J2(A ) ] d (21) o and 1 –[c/ (’y.’ –A’) ]{ AIJo(- nc) –~2(7mc) uI(Ac) —Tn[~o (AG) —R72( AG)]~I(TnG) },A#~. In’ = 1 (c2/2) {J02(Ac) a no mal incidence +J,’(Ac) +J22(Ac) –~3( lc)~1( k) ], A=yn (~~) ~,=CJl(-ync) nA=O. (23) ‘Y. ‘ Subs i u ion in (2) gi es asys em wi h he modal coe i- cien s in he ibe as unknowns. Fig. 2 ep esen s nume ical CARDAMA AND KORNHAUSER: MODAL ANALYSIS a) a:2, n,:l Lg =2.4143 c) a:2, n~.1 V=2.4143 “10 100 I 165 o he a=2P ibe which is jus below cu o o he TMOI mode [13], and a 4° de ia ion om no mal incidence i ’60 KM ,,ood ops o abou hal ha alue o he ma ched case. As 60 he ibe adius dec eases, so does he maximum o he cu es. The e ec o il s inc eases wi h he e ac i e 40 2index o he medium o incidence. Fibe s o small adks 20 need a much g ea e illumina ing a ea, as expec ed, i.e., mos o he powe is in he cladlng. The ou e pipe does # o123 4p no ha e ano iceable e ec in he ibe well abo e cu o . b) a:l, n3:lL9 V=1.207 I!& 80 OZo” 60 Lo 3 20 : 0123 ~P d) a:l, n~: V=1.207 u Fig. 3. HEII mode launching e iciency in a ibe wi h nl =1.50, nz =1.49, b. =25 P, a=1,2 P, XO =0.9 P, exci ed by aGaussian beam a obhque incidence e sus Gaussian wid h u. TABLE I Bo n App oxi- Modal Solu ion Modal Solu ion ma ion O(K) o nj =1 o n, =1.49 (Geome ic mean) 0.5 33.83 pe cen 35.24 pe cen 34.16 pe cen 1. 8072 84.08 1..5 83.81 95.78 99.75 2. 99.60 90.12 93.84 2.5 93.69 78.10 81.32 3. 81.19 65.81 68..52 3.5 68.40 35.09 57.36 4’. 57.26 46.25 48.15 48.07 No e: HE,I mode launching e iciency in a ibe wi h nl =1.50, ~z =1.49,. b=Mp, a=2p, AO =0.9 pexci ed by aGaussian beam o wid h ua no mal incidence om amedium wi h e ac ion index ns. solu ion o ha sys em and gi es he e iciency o exci a- ion o he HE1l mode e sus adius o he illumina ed a ea o a ious angles o incidence. Amo e ex ensi e se o cu es o di e en ibe s, ou e adius and medium o incidence oge he wi h he co esponding Bo n ap- p oxima ions can be ound in [19]. The ansmi ed HEII mode was ound o be in phase wi h he inciden ields; alues o Bessel unc ions we e calcula ed o six digi s accu acy and he a io o imagina y o eal pa o he modal coe icien is o he o de o 10–6. Maximum launching c liciencies Oi SO pe cen wi h ma ched media and 77 pe cen o acumn we e ob ained B. Gaussian Beam We shall now conside an inciden Gaussian beam p op- aga ing a an angle 0wi h espec o he axis o he ibe z. Again, he plane o he axis and he di ec ion o incidence is aken o be he x, zplane, and he cen e o he beam is displaced a, dis ance d on he posi i e x-axis. Fo small angles o incidence we ha e (24) Hi is gi en by (8) wi h D=d/u2 (26) and Agi en by (10). We shall conside only x-pola ized beams and ea oblique incidence and o se s sepa a ely. Ii bo h we e conside ed, i would be necessa y o pe o m nume ical in eg a ions wi h Bessel unc ions o he complex a gu- men { . I he cen e o he beam we e no on he x-axis bu a apoin (d,c ), hen he a gumen o &CO@ in (24) would be eplaced by D cos ( #J—a) and bo h pola iza- ions o he HE1l mode would be exci ed. The o malism o ea hose cases is he same as ha o he cases A=O o D=O, bu he compu a ions become much mo e cumbe some. Fo oblique incidence (D =0) we ob ain he exp essions o An and B. gi en by (14), (17), (19), (20) bu wi h –J,(7.T)J2(A7)]Y d (27) +J2(7N)J2 (A ) IT Cb. (28) Bo h o hese exp essions equi e nume ical in eg a ion, and i bo h il s and o se s we e o be conside ed sinml- aneously, amul iplica i e ac o exp (–dz/2u2) would appea and {would eplace A. Fig. 3shows wo o he cu es ob ained [19]. Maximum alues o e iciency a e 99.7 pe cen o a=2~ and 9S.7 pe cen o a=1.5P. As he adius o he co e adec eases, he maXhIml dec eases and he e ec o il s inc eases. 166 “1. 100 80 60 40 D - o 20 /---; 0123 kp ,al a:2, nazl L9 V=2.4143 J IEEE TRANSACTIONS ON “1. 100 E 80 q:o 60 2 3 Lo L 20 01234A b] a=l, n,:l L9 =1.207 u Fig.4. HE,, mode launching e iciency ina ibe wi hn, =1.50, nz =1.49, b=25A+ a=1,2A xo =13.9P, exci edh aGaussian beam wi h he axis displaced d e sus Gaussian wid h ~. Again, he e iciency o he acuum was abou 4pe cen less han o m=m. The esul s a e insensi i e o changes in he pipe adius excep o ibe s wi h e y small co e adius. ABo n app oxima ion was ca ied ou o he same cases, and he geome ic mean o he wo alues ob ained o he modal coe icien was in e y good ag ee- men o ma ched media, bu o incidence om he acuum was abou 4pe cen highe han he co esponding modal solu ion (Table I). Since he heo e ical calcula ions o Cook e al. [9] a e also based on he Bo n app oxima- ion, he same deg ee o p ecision p esumably also applies o hem. Fo beam o se s (A =O), using he associa ed se ies exp (D cos 0) =10(D) + we again ob ain A. and B. gi en bu wi h –J2(-yn )I’(D ) ] d?-. Fig. 4shows he e ec o o se misalignmen s and (29) (~o) (30) (31) indi- ca es ha hey a e mo e c i ical han il s. Alignmen s o co es o Io 2mic ons in ibe join s is no an easy ask, especial] yin he ield; also in he launching sys em he e is he possibili y o some misalignmen . In ibe s close o cu o o he TMOI mode an o se equal o he co e adius educes he e iciency by mo e han hal . As he adius o he co e is dec eased he e ec dec eases, as expec ed. Also as uinc eases he e ec becomes less impo an bu hen he e iciency d ops and he e ec o il s inc eases. Bo n app oxima ions o hese cases can be ound in [19]. MIcROWAVE THEORY ANI) TECHNIQUES, JANUARY 1975 zy-b---~”’ eg I1-0 @g II ;2,20,2. eg 111 ,1 0-, m Fig. 5. IV. THE FIBER BUTT JOINT WITH AGAP The p oblem depic ed in Fig. 5is ha o wo iden ical me allic coa ed ibe s aligned on he same axis, bu whose ends a e sepa a ed by he gap egion O<z<ZOwhich has a e ac i e index ns. ln egion I, z<0, he e is a se o o wa d-mo ing inciden modes wi h ampli udes a~O,k=l,2, ”””, k, and backwa d-mo ing e lec ed modes wi h coe icien s a~ l). Region II, O<z<zO, is cha ac- e ized by ase o TE and TM ci cula wa eguide modes wi h o wa d-di ec ed modal coe icien s a i(2) and back- wa d-di ec ed ones b~ 2J;and in egion III, z>ZII,ase o ansmi ed hyb id modes ai(3J is exci ed. I one akes he c oss p oduc o he modal ields L and e%wi h he equa ions o con inui y a z=Oo angen ial Eand H, espec i ely, and hen in eg a es o e he c oss sec ion, he esul is ~&@~ONh~ -N~~a. (1) = ~ a%@JN~m—~&@JN~”l k=l .n=l n=l (32) ~&a~ON~~ +.V~~an”) =~an(2ji14~n +cb~’2)M~n, k=l a=1~=1 ~= 172, . . . (33) whe e he c oss-no maliza ion ac o s N~~ and ill~n a e de ined as in (5), (6), and whe e .~mm=((em X&).icl -Z. (34) ‘A An exac ly simila p ocedu e applied a z=ZOyields ‘ae(2)Nmmlexp (—,jb~’zo) —b i~2)N i.’ exp (j/3~’zo) = ~ a,(3JML” exp (–[email p o ec ed]) (35) 1=1 ~n(2)~nn{ exp (‘,~~.’~o) +bn‘z).~~~’ exp (.&’20) = ~ a,@)Nn2 exp (–.j13 zo) (36) 1=1 whe e ,8. is he p opaga ion cons an o he n h mode and he p imes e e o egion II. The e lec ed mode coe icien s and hose in egion II may be elimina ed om he ou se s o equa ions ob- ained, lea ing { mUm,nUZ,n exp [j(O.’ –/3z)zo] –Vn,.VZ,n exp [–j(@n’ +Bz)201 Sa (3) ~ ~Nn% 1 = ~ &na%hONw, m=1,2,3,. .0 (37) 1=1 ~=1 h=l CARDAMA AND KORNHAUSER: MODAL ANALYSIS 167 a) a:2, V=2.4143 b) a:15, V= 1.810 Fig. 6. —: Coupling e iciency o a gap o leng h z, in a ibe wi h nl =1..50, nz =1.49, b=25 Aa=1.5 p, ~o =0.9 u. 0: Expe imen al esul s [8] o a ibe o 3.7-M diame e co e, nl =1:6171, nz =1.6038, a x=0.6328 p esen ed on he same no mahzed sepa a ion-co e adii scale. whe e U.,. =iV~m +M~n and V~,n =ll ~n —N%m. When he e is only asingle inciden mode, he HEll, all e ms excep he i s one on he igh hand side a e ze o, and ha one is 2a1°N11. Nume~ical solu ions o his case ha e been ound by unca ing he sys em o equa ions o m~40 o ou di e en sizes o ibe wi h ng =nj =1.49, co esponding o a gap illed wi h nla ch- ing oil, and m=1co esponding o an ai gap. The esul s a e shown in Fig. 6, which also includes acompa i- son wi h he expe imen al esul s o Bisbee [81. E en hough his a io a/A is somewha di e en , he ag eemen is good, as i was o his da a on o se s as well; howe e , i should be emphasized ha , unlike he o he p oblems p e iously conside ed, no Bo n app oxima ion solu ion can be ob ained o hk one, and he heo e ical alues a e he only ones a ailable. I migh also be no ed ha he eason he cu es a e ela i ely insensi i e o he alue o a/X is ha he absolu e ans e se dis ibu ion o he ields emains nea ly he same e en hough mo e o i is con ained in he cladding a smalle a/L The same echnique could be applied o he case o la e al displacemen as well, bu he lack o o a ional symme y would make he calcula ions o he c oss- no maliza ion ac o s much mo e complica ed, equi ing nume ical in eg a ion. V. SUMMARY AND CONCLUSIONS The e ec o including aconduc ing bounda y a ~“=b was ound o be insigni ican as long as x=b/a >4; he e was less han 0.3 pe cen di e ence be ween he calcula ed launching e iciwzcies o pipes o S-P and Q$LL adius. The ype o coa ing used on he ibe seems o be i ele an . The maximum ansmission coe icien o exci a ion by ano mally inciden uni o m plane wa e was 81 pe cen , as compa ed wi h a alue o 80 pe cen ob ained by Snyde [3] wi h aBo n app oxilna ion. The co esponding alues o exci a ion by aGaussian beam we e 99.6 pe cen , as compa ed wi h a alue o 99.7 pe cen epo ed byMa cuse [4] when he beam is inciden om amedium ma ched o he cladding. In ac , i one akes he geome j cmean o he wo possible Bo n app oxima ion solu ions, he esul s a e su p isingly close (wi hin 0.2 pe cen ) o he modal solu ion in all cases whe e he media we e ma ched; bu i he medium o incidence is acuum,, he Bo n ap- p oxima ion solu ion is abou 4pe cen highe (Table I). The p ac ical e ec o il s o o se s o he inciden beam is qui e p onounced; an o se o ’ one co e adius o a il o 4° educes he coupling e iciency o less han 50 pe cen o he cases ea ed. Howe e , a gap o 25 adii is equi ed o p oduce acompa able e ec when ma ching oil is used o abou 15 adii o an ai gap, so ha in p ac ice he gap is ap o be a much less se ious p oblem. The e ec o a gap o gi en leng h is insensi i e o he co e diame e when i is o he o de o awa eleng h because he la e al ex en o he ields does no change e y much. Howe e , as Cook e al. [9] ha e poin ed ou , asmalle co e diame e makes he e ec o o se s, as measu ed in co e diame e s, less c i ical and o angula displacemen s mo e c i ical when one inally app oaches he si ua ion whe e he ields begin o sp ead la e ally. Al hough ~he gap p oblem was sol ed by igo ous applica ion o he bounda y condi ions o he mod il expansions, in he il and o sd p oblems in ibe junc- ions COW@U 21 kJOd symplici y p omp ed an app oximai e ea men , eplacing he inciden HEII mode in he ibe by aGaussian beam. This should, howe e , gi e excellen esul s o he case o incidence om an ac ual ibe o ha alue o uco esponding o maximum ansmission a no mal incidence and no o sd,, since a his poin he wo ypes o inciden ields a e w’ y nea ly iden ical. Tha choice o akapp op ia e because he wo ibe s we e assumed idwl ical, and coupling is maximum when he inciden wa e is e y nea ly ma ched o he ansmi ed mode. This si.mp]i ica ion is also bo ne ou by he expe i- men al esul s o Cook e al. [9], who go good ag eemen wi h hei Bc, n app oxima ion calcula ions. A ela ed p oblem, ha o he adia ion in o auni o m medium (m] he e mina ion o a ibe , has been ea ed, yielding a ansmission coe icien o 96 pe cen o adia- ion in o a acuum and 100.0 pe cen o ama ched medium. In conclusion, he mo e igo ous modal analysis has shown ha he Bo n app oxima ioli does gi e accu a e esul s in all cases ea ed using ma ched media, and ha , e en hough i equi m somewha g ea e compu a- ional e o , he modal app oach would be ad an ageous in p oblems ~ he e se ious misma ches o e ac i e index occu o ~ he e he Bo n app oxima ion is inapplicable, as in he case o he gap. APPENDIX The se o unc ions em, h,,,, a e he ields o he modes o adielec ic od o pe mi i i y El and adius ain a medium o ’ lo~ e pe mi i i y 6Z su ounded by ape - ec ly conduc ing pipe o adius b and cam be ob ained om he usual longi udinal o mula ion [1 S]. l’o he ange o he p opaga ion cons an k~2<~2 < kiz(kl,j’ =W2pOel,z), we ob ain he su ace- ype modes, whose exis anlce is independen o he su ounding pipe and which ha e ields 1ocalized o he icini y o he co e wi h cu o condi ion D=k~. Thei longi udinal com- 168 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TIK’HNIQUES, JANUARY 1975 ponen s a e gi en by The no isu ace ype modes a e ob ained in he ange I J i(q ), &~ ?&and helongi udinalcw mponen sa e gi enby <a / Jn(m ), Kn(xw)l.(s ) –In~xw)Kn(: ) <a E,= g,(@) J.(zJ) K.(x o)l.( o) –1.(XW)K.(W) ‘Ez=gl(0) J.(u) J~(a, ).V~(x31) –.~”~(a, )J~(x ) a< <b (Ala) J~( ).Vn(x ) ‘.}T.(~)J. (xV) ‘ . Hz= (q/p”) ’/’(kl)Plg2(dJ)J) whe e 6=1 – e2/el =1 – nz2/n12 g,(4,={;192(@)={_:;} wi h n a no mega i e in ege . The eigen alue equa ion is wi h V2 ~ Pl=— ‘U2W2 m+‘712&2 (A3) u=ala w=a a ~2— 2= 2 (A4) and he eigen alue equa ion (A5) F’ =(&’//{/) P, (A6) ‘ (A7) F2=g(–71+ (1 –6)7?371) ~3= .Yn’( ) Ui n(u) J.’( ] /&”.’ (V) –J.(x ) /1~.(xV) (A8) 7’ =Jn( )/Nn( ) –J.(xV)/ ~. (XV) (A9) (A12b) (A13) (.414) (A15) (A16) (A17) (AIS) (A19a) (A19b) Jn’ (U) K.’(w) ‘1 =d.(u) ‘2 = oK. (w) ~, =i.’(w) /Kn’(w) –In(xw) /K.(~w) In(w) /K.(w) –ln(xw)/&(x1o) ~, =].’(w) /Kn’(w) –~.’ (xw)/Kn’ (xW) in(w)/Kn(w) –in’(xW)/Kn’(Xw) ‘ ‘ ~.7’cJ (Ldkk’ – P.” –%’ ce ie ig’e~) . i cll As binc eases, hese modes become he su ace modes The ollowing su ace in eg als ha e been educed by (A1O) Yaghjian [11] o simple line in eg als The index ~~?is used o he modes o he ibe , and n e e s o he modes o he homogeneous cylind ical wa eguide. (Alla) &I~’I =!(en Xh.’) .2 da A (Allb) o he open od. I.n he limi ing case x~cc:.%-+ 1, &-+ 1, and (Al ), (A7) become he exp essions o he open ibe +3 ( l.’~l.,’e~ +~~e~.h~). ?dl. [~]. Use o he ~symp o ic exp essions o he Bessel %’2 –pm2 . unc ions yields o x>> 1(~~()) –.–.l --@x –l+ex The c oss-no maliza ion coe hcien $’- l–e, .$2‘---’ l+e~’ o w>> 1 N.. =!(en’ Xh~) .2 da gl -~ 1+2/x2= &=1—2/x2~, o ~w <<1 A and n z1. is ob ained om (A20) wi h ~Land min e changed: CARDAMA AND KORNHAUSER: MODAL ANALYSIS 169 (A21) whe e he con ou cincludes bo h sides aces, and p.” =colqlen —bn’z pmz =wzpo% — ?mz. Fo a ibe wi h qin he co e and q in ha e o all he in e - (AJ2) (A~:~) he cladding, we Mm” =j a (11 — 2—czlm -yn )(wpohmzhn .’ ‘Y. 2— 2m2 (161’/62) —j7 a ie3 — 2—~1m2 ) 2—~2m2 enz’em, (a–) I’. Y. (A25 ) whe e 7.2 =w7.ll)E3 —/%’2 O!lmz =Wzpoq —Bmz a2m2 =w2poez —/3m2. (A26) (A~7) (A~~) All ield componen s ha e he angula dependence emo ed and a e e alua ed ,a =a, exeep e~~(a–), which deno es he adial elec ic ield o he co e a 7“=a. REFERENCES [1] i!. W. 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