Modal analysis of coupling problems in optical fibers

162 IEEK TRANSACTIONS ON MICROWAVESTHEORY AND TECHNIQUES, VOL. Ml’T-%], NO. 1, JANUARY 1975 Modal AnalYsis of Coupling Problems inopticd Fibers ANGEL CARDAMA, MEMBER, IEEE, AND EDWARD T. KORNHAUSER, SENIOR MEMBER) IEEE Absfracf—A modal analysis of the problems of excitation of the dominant mode in ah optical fiber by incident plane waves and Gaussian beams has been carried out, and the results applied to the effect on transmission of misalignment in fiber junctions due to offsets, tilts, and gaps. The results in casesof matched media confirm the accuracy of previous theoretical treatments using the Born approximation, which in turn show good agreement with experimental results. In addition, the modal analysis gives more precise solutions when there is amismatch of media and makes possible the treatment of some problems to which the Born approximation is not applicable. I. INTRODUCTION O VER THE 1’AST five years the advances inade in optical signal processing techniques and in the development of low-loss glasses for optical fibers have made ‘the utilization of the enormous bandwith available in optical communications an attractive possibility. However, if such fibers are actually to be employed as transmission channels, in the manner of transmission lines and n~icrowave waveguides, one must be able to couple, splice, bifurcate, etc., in away analogous to that used with those older channels. This is not atrivial technical problem, because the core of the fiber, where the fields are concentrated, is typically only afew microns in diameter so that the difficulty, for example, of aligning two fibers when splicing is critical. The excitation of propagating modes on a fiber by various types of source illumination has been studied by Snyder [1]–[3] and Marcuse [4J both using the Born approximation. Furthermore, splicing techniques have been developed [5]–[7] and some measurements made [7]-[9] of the effects of imperfect alignment at the interface between two uniform fibers. Most recently Cook et al. [9] have also given atheoretical analysis of the effects of misalignment in splicing on the transmission coefficient, again making use of the Bonn approximation. The importance of such atheoretical calculation lies in the fact that it would enable one to set meaningful standards of precision which must be adhered to in the practical means used for splicing. Ma,nllscriPt received April 4, 1974; revised July 21, 1974. This work was supported by the National Science Foundation under Grant GK 10971 to Brown University. A. Cardama was with the Division of Engineering, Brown Universit y, Providence, R. 1. He is now with the E.T.S. Ing. Telecomunicaci6n, Polytechnic University of Barcelona, Barcelona, Spain. E. T. Kornhauser is with the Division of Engineering, Brown University, Providence, R. I. 02912. However, all of the theoretical calculations mentioned employ the Born approximation, i.e., the assumption that the fields at the illuminated cross section of the fiber consist entirely of those in the incident wave. That there must be a discrepancy [10] resulting from the use oi this approximation is indicated by the fact that Marcuse [4] calculated two values of transmission coefficient corresponding to the two possible boundary conditions to be used (continuity of tangential Eor of tangential H) and arbitrarily took their geometric mean. Although this mean result was always quite reasonable, in some cases one of the two boundary conditions led to atransmission coefficient larger than unit y, which is clearly impassible. Consequently, this paper will reexamine the basic excitation problems, making use of amore rigorous modal expansion of the fields, which in turn is based on the general techniques for hybrid modes developed by Yaghjian [11], [12]. The solutions will show that, rather surprisingly, the result of taking the geometric mean of the two Born approximation coefficients was remarkably accurate. h’inall y, the results and techniques will be applied to the problems of the effect on efficiency of transmission at the junction of two similai optical fibers of three types of defect in their alignment: tilt of one axis with respect to the other, offset of their axes, and swiall gaps between the fiber cores. Numerical results will be presented for all of these problems and compared with the corresponding Ilml approximation results. II. lJORIIULATIOX The surface modes of the infinite circular dielectric rod have been extensively studied [2], [13], [14], and their orthogonalit ywell established [1.5]. The fact that the modes are hybrid and the existence of acontinuous spectrum malce the exact solution of diffraction and scattering problems for afiber \vith infinite outer diameter (od) a formidable task. In fact, there is no kM)~YD exact solution even for the axially symmetric caseJ which reduces to a scalar problem [16], [17]. An exact solution would require solving an integral equation, and one possible means of overcoming the difficulty of its solution would be to approximate the integrals by infinite summations and solve the resulting linear system. Anatural way of achieving this is to enclose the dielectric rod in aconcentric perfectly conducting cylinder with radius large compared to the core radius and the wavelength. As the radius of the metallic pipe increases, the surface modes are unaffected and the nonsurface type modes become acontinuum CARDAMA AND KORNHAUSI?R: MODAL ANALYSIS 163 (Appendix). This method will allow us to solve the excitation and fiber coupling problems by anormal-mode analysis, as in astandard waveguide discontinuity yproblem, and to take into account the reflected energy. To find the scattered and transmitted fields for awave incident at z=O, Fig, 1, we will have to expand the fields on both sides of the interface as an infinite summation of the modes in each structure. Continuity of the transverse components of the fields at z=Ogives ~a~e~ +E, = ~ a.e. nm –~aJh2+Hi=~a.k (1) nm where E,, Hi are the transverse incident fields, em, h~ the transverse fields of the surface and nonsurface modes of the fiber, and efi’, h.’ are the forward traveling modes of the homogeneously filled circular waveguide [18] in the launching case or the hybrid modes of the fiber in the fiber junction case. The set of modal coefficients am, which will give the efficiencies of excitation, can be obtained from (1) by cross multiplication by hn’ and en’, integration over the cross-sectional area Aat the leit of the interface, and making use of the orthogonalit yof the modes, giving afiWnml +A. = ~ aJII~’1 m —a.’AT..’ +B. =~kN%”L (2a) m these two equations can be added to obtain alinear system for the modal coefficients of the transmitted modes ~~l%lIWm”+N.m[=A.+B., )/ =1,2,. . . (2b) where Nnnt =Jen’ xh: ..2da are the normalization factors of 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 calculated from the expressions of the incident fields, and the cross-normalization factors, N~”II and M~n, have been calculated by Yaghjian [12] by reducing the surface integrals to line integrals and are given in the Appendix. Fig. 1. The summation over nincludes for the launching case all TE and TM modes of the homogeneous circular metallic waveguide arranged in order of increasing eigenvalues, and in the fiber junction case all the propagating surface modes and all the nonsurface modes (Appendix). To obtain anumerical solution of (2b), we shall have to truncate the system and solve the resulting finite system, adding more equations until the modal coefficients obtained become stable within the accuracy of the computations, and the addition of new equations does not produce further variations. Practical fibers are made with the radius of the cladding large compared to the core radius and usually are externally coated. This in fact reduces the continuous spectrum to adiscrete one but does not affect significantly the surface modes. The preceding model can be viewed not only as asolution for the transversally infinite fiber but as an exact study of the propagation and excitation of modes in afiber with ametallic external coating. III. EXCITATION COEFFICIENTS .4. Truncated [lniform Plane Wave For this case, the incident fields at z=Oare taken as those of auniform plane wave in amedium of dielectric constant e~ illuminating acircle of radius cconcentric with the fiber and propagating at an angle 6with respect to the axis of the fiber z. The plane of the fiber axis and the direction of incidence is t,alwm to be the J, zplane, so that Ei =EOexp (–jk3 zsin 0)~ (7) with 1cS2=0J3.w, ii =ksin e+2cm e, and an @time dependence is assumed. We will consider only ~-polarized plane waves, which excite only modes with gl (0), VZ(0) given by (A6) with n=1. Expressions for the other polarization could be similarly derived. Also we will assume the fields incident at z=O, but if the source w-ere at z=Z.(z. <O) with e%or adifferent medium in the region Z. <z<0, the problem could be treated in asimilar way by writing the continuity equations at the z=Z. and z=Ointerfaces and eliminating coefficients until we are left with asystem relating incident and transmitted fields, as will be done in studying the effects of gaps in fiber joints. 164 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, JANUARY 1975 I?oranx-polarized uniform plane wave, we obtain EZ =–EOexp (–jA?”cos@) sin O(9a) E, =EOexp (–jAYcos+) COSOCOS4 (9b) E+ =–EO exp (–~Ar cos 4) cos Osin 4(9C) with A=%sine (10) the Hi field components are obtained from (8) and (9). We shall lhave to solve the system (2). Substitution into (3), (4) of the fields of the homogeneous] yfilled circular waveguide [18] gives for An and B. the following expressions. For nodd, TEln modes 13m2E,Cos~02“ An= –j— // exp (–jAr cos 4) wo’Yn 00 [ tll(~nr) 1 .-—COS24 +J/ (7J’) Sill’ 4YCh d4 (11) ~nr with -Y. =Pl~’/b (n =2m —1, m=1,2, ●..), where PI~’ is the mth order zero of J{, bis the radius of the guide, and I% is the propagation constant of the nth mode. Use of the recurrence relations for the Bessel fuctions gives ch A. =–j ~2~ cos eHexp (—jAr cos 4) [Jo (?’TJ”) #u/’loy* 00 +J2(w) C(JS~4]T dr c14. (12) Using the associated series exp (—~Ar cos 4) =Jo (AT) +2~(—j) ‘Jk (Ar) cos ~4 k=l (13) and integrating over @gives A, . . –j% (t,/po) ‘% Coseln’ (14) with ! I.1 =‘[Jo(~.r)Jo(Ar) –J2(7.r)J,(Ar) ]r d? (15) o which can be integrated analytically to give [ –[C/ (yn’ –A’)]{ fi[Jo(w) +J2(w) ~l(AC) 1=1 =–~.[Jo(Ac) +J2(Ac) VI(W) ), A#Y.. (2/@Jl’(At), A=7$$ (16) B. is obtained in the same way as Bz =–j *(eJ/po) %rl.1 (17) for normal 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 etticiency in afiber with nl =1.50, n~ =1.49, b=25 ~, a=1?2 p, xo =0.9 p, excited by atruncated 1miform plane wave at obhque incidenee versus radius of illuminated area. Similar] yfor neven, TM1. modes, we have .4n =–j :(e3/#o) %cm 191n’ (19) ~.EO B~= –j— (dM) 112rIn2 (20) ‘Y71 with ~. =P1~/b (n =%n, n~ =1,!2,. ..), where Pl~ is the mth order zero of J1. I.?, =/‘[JO(~.rkJo(Ay) +Jz(7.7)J2(Ar) ]r dr (21) o and 1 –[c/ (’y.’ –A’) ]{ AIJo(-rnc) –~2(7mc) uI(Ac) —Tn[~o (AG) —R72( AG)]~I(TnG) },A#~. In’ = 1 (c2/2) {J02(Ac) at normal incidence +J,’(Ac) +J22(Ac) –~3(flc)~1(fk) ], A=yn (~~) ~,=CJl(-ync) nA=O. (23) ‘Y. ‘ Substitution in (2) gives asystem with the modal coefficients in the fiber as unknowns. Fig. 2represents numerical CARDAMA AND KORNHAUSER: MODAL ANALYSIS a) a:2, n,:l Lg v=2.4143 c) a:2, n~.1 V=2.4143 “10 100 I 165 for the a=2P fiber which is just below cutoff of the TMOI mode [13], and at 4° deviation from normal incidence it ’60 KM \,,oodrops to about half that value for the matched case. As 60 the fiber radius decreases, so does the maximum of the curves. The effect of tilts increases with the refractive 40 2index of the medium of incidence. Fibers of small radks 20 need a much greater illuminating area, as expected, i.e., most of the power is in the cladlng. The outer pipe does #r o123 4p not have anoticeable effect in the fiber well above cutoff. b) a:l, n3:lL9 V=1.207 I!& 80 OZo” 60 Lo 3 20 : 0123 ~P d) a:l, n~:t V=1.207 u Fig. 3. HEII mode launching efficiency in afiber with nl =1.50, nz =1.49, b. =25 P, a=1,2 P, XO =0.9 P, excited by aGaussian beam at obhque incidence versus Gaussian width u. TABLE I Born ApproxiModal Solution Modal Solution mation O(K) for nj =1for n, =1.49 (Geometric mean) 0.5 33.83 percent 35.24 percent 34.16 percent 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 Note: HE,I mode launching efficiency in afiber with nl =1.50, ~z =1.49,. b=Mp, a=2p, AO =0.9 pexcited by aGaussian beam of width uat normal incidence from amedium with refraction index ns. solution of that system and gives the eficiency of excitation of the HE1l mode versus radius of the illuminated area for various angles of incidence. Amore extensive set of curves for different fibers, outer radius and medium of incidence together with the corresponding Born approximations can be found in [19]. The transmitted HEII mode was found to be in phase with the incident fields; values of Bessel functions were calculated to six digits accuracy and the ratio of imaginary to real part of the modal coefficient is of the order of 10–6. Maximum launching cfliciencies Oi SO percent with matched media and 77 percent for vacumn were obtained B. Gaussian Beam We shall now consider an incident Gaussian beam propagating at an angle 0with respect to the axis of the fiber z. Again, the plane of the axis and the direction of incidence is taken to be the x, zplane, and the center of the beam is displaced a, distance d on the positive x-axis. For small angles of incidence we have (24) Hi is given by (8) with D=d/u2 (26) and Agiven by (10). We shall consider only x-polarized beams and treat oblique incidence and offsets separately. Ii both were considered, it would be necessary to perform numerical integrations with Bessel functions of the complex argument {r. If the center of the beam were not on the x-axis but at apoint (d,cr), then the argument of &CO@ in (24) would be replaced by Dr cos (t#J—a) and both polarizations of the HE1l mode would be excited. The formalism to treat those cases is the same as that for the cases A=O or D=O, but the computations become much more cumbersome. For oblique incidence (D =0) we obtain the expressions for An and B. given by (14), (17), (19), (20) but with –J,(7.T)J2(A7)]Y dr (27) +J2(7N)J2 (Ar) IT Cb. (28) Both of these expressions require numerical integration, and if both tilts and offsets were to be considered sinmltaneously, amultiplicative factor exp (–dz/2u2) would appear and {would replace A. Fig. 3shows two of the curves obtained [19]. Maximum values of efficiency are 99.7 percent for a=2~ and 9S.7 percent for a=1.5P. As the radius of the core adecreases, the maXhIml decreases and the effect of tilts increases. 166 “1. 100 80 60 40 D -fo 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 v=1.207 u Fig.4. HE,, mode launching efficiency inafiberwithn, =1.50, nz =1.49, b=25A+ a=1,2A xo =13.9P, excitedh aGaussian beam with the axis displaced dversus Gaussian width ~. Again, the efficiency for the vacuum was about 4percent less than for m=m. The results are insensitive to changes in the pipe radius except for fibers with very small core radius. ABorn approximation was carried out for the same cases, and the geometric mean of the two values obtained for the modal coefficient was in very good agreement for matched media, but for incidence from the vacuum was about 4percent higher than the corresponding modal solution (Table I). Since the theoretical calculations of Cook et al. [9] are also based on the Born approximation, the same degree of precision presumably also applies to them. For beam offsets (A =O), using the associated series exp (D cos 0) =10(D) + we again obtain A. and B. given but with –J2(-ynr)I’(Dr) ]r d?-. Fig. 4shows the effect of offset misalignments and (29) (~o) (30) (31) indicates that they are more critical than tilts. Alignments of cores of Ior 2microns in fiber joints is not an easy task, especial] yin the field; also in the launching system there is the possibility of some misalignment. In fibers close to cutoff of the TMOI mode an offset equal to the core radius reduces the efficiency by more than half. As the radius of the core is decreased the effect decreases, as expected. Also as uincreases the effect becomes less important but then the efficiency drops and the effect of tilts increases. Born approximations for these cases can be found in [19]. MIcROWAVE THEORY ANI) TECHNIQUES, JANUARY 1975 zy-b---~”’ reg I1-0 r@g II ;2,20,2. reg 111 ,1 0-, m Fig. 5. IV. THE FIBER BUTT JOINT WITH AGAP The problem depicted in Fig. 5is that of two identical metallic coated fibers aligned on the same axis, but whose ends are separated by the gap region O<z<ZOwhich has arefractive index ns. ln region I, z<0, there is a set of forward-moving incident modes with amplitudes a~O,k=l,2, ”””, k, and backward-moving reflected modes with coefficients a~fl). Region II, O<z<zO, is characterized by aset of TE and TM circular waveguide modes with forward-directed modal coefficients ati(2) and backward-directed ones b~f2J;and in region III, z>ZII,aset of transmitted hybrid modes ai(3J is excited. If one takes the cross product ot the modal fields L and e%with the equations for continuity at z=Oof tangential Eand H, respectively, and then integrates over the cross section, the result 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) where the cross-normalization factors N~~ and ill~n are defined as in (5), (6), and where .~mm=((em X&).iclf-Z. (34) ‘A An exactly similar procedure applied at z=ZOyields ‘ae(2)Nmmlexp (—,jb~’zo) —bfi~2)Nfi.’ exp (j/3~’zo) = ~ a,(3JML” exp (–[email protected]) (35) 1=1 ~n(2)~nn{ exp (‘,~~.’~o) +bn‘z).~~~’ exp (.&’20) = ~ a,@)Nn2 exp (–.j13tzo) (36) 1=1 where ,8. is the propagation constant of the nth mode and the primes refer to region II. The reflected mode coefficients and those in region II may be eliminated from the four sets of equations obtained, leaving { mUm,nUZ,n exp [j(O.’ –/3z)zo] –Vn,.VZ,n exp [–j(@n’ +Bz)201 Sat(3) ~ ~Nn%t 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 efficiency for a gap of length z, in afiber with nl =1..50, nz =1.49, b=25 Aa=1.5 p, ~o =0.9 u. 0: Experimental results [8] for afiber of 3.7-M diameter core, nl =1:6171, nz =1.6038, at x=0.6328 presented on the same normahzed separation-core radii scale. where U.,. =iV~m +M~n and V~,n =llf~n —N%m. When there is only asingle incident mode, the HEll, all terms except the first one on the right hand side are zero, and that one is 2a1°N11. Nume~ical solutions for this case have been found by truncating the system of equations to m~40 for four different sizes of fiber with ng =nj =1.49, corresponding to a gap filled with nlatching oil, and m=1corresponding to an air gap. The results are shown in Fig. 6, which also includes acomparison with the experimental results of Bisbee [81. Even though his ratio a/A is somewhat different, the agreement is good, as it was for his data on offsets as well; however, it should be emphasized that, unlike the other problems previously considered, no Born approximation solution can be obtained for thk one, and the theoretical values are the only ones available. It might also be noted that the reason the curves are relatively insensitive to the value of a/X is that the absolute transverse distribution of the fields remains nearly the same even though more of it is contained in the cladding at smaller a/L The same technique could be applied to the case of lateral displacement as well, but the lack of rotational symmetry would make the calculations of the crossnormalization factors much more complicated, requiring numerical integration. V. SUMMARY AND CONCLUSIONS The effect of including aconducting boundary at ~“=b was found to be insignificant as long as x=b/a >4; there was less than 0.3 percent difference between the calculated launching efficiwzcies for pipes of S-P and Q$LL radius. The type of coating used on the fiber seems to be irrelevant. The maximum transmission coefficient for excitation by anormally incident uniform plane wave was 81 percent, as compared with avalue of 80 percent obtained by Snyder [3] with aBorn approxilnation. The corresponding values for excitation by aGaussian beam were 99.6 percent, as compared with avalue of 99.7 percent reported byMarcuse [4] when the beam is incident from amedium matched to the cladding. In fact, if one takes the geometrj cmean of the two possible Born approximation solutions, the results are surprisingly close (within 0.2 percent )to the modal solution in all cases where the media were matched; but if the medium of incidence is vacuum,, the Born approximation solution is about 4percent higher (Table I). The practical effect of tilts or offsets of the incident beam is quite pronounced; an offset of’ one core radius or atilt of 4° reduces the coupling efficiency to less than 50 percent for the cases treated. However, a gap of 25 radii is required to produce acomparable effect when matching oil is used or about 15 radii for an air gap, so that in practice the gap is apt to be a much less serious problem. The effect of a gap of given length is insensitive to the core diameter when it is of the order of awavelength because the lateral extent of the fields does not change very much. However, as Cook et al. [9] have pointed out, asmaller core diameter makes the effect of offsets, as measured in core diameters, less critical and of angular displacements more critical when one finally approaches the situation where the fields begin to spread laterally. Although ~he gap problem was solved by rigorous application of the boundary conditions to the modtil expansions, in the tilt and offsd problems in fiber junctions COW@Ut21tkJOd symplicity prompted an approximai e treatment, replacing the incident HEII mode in the fiber by aGaussian beam. This should, however, give excellent results for the case of incidence from an actual fiber for that value of ucorresponding to maximum transmission at normal incidence and no offsd,, since at this point the two types of incident fields are w’ry nearly identical. That choice of akappropriate because the two fibers were assumed idwltical, and coupling is maximum when the incident wave is very nearly matched to the transmitted mode. This si.mp]ification is also borne out by the experimental results of Cook et al. [9], who got good agreement with their Bc,rn approximation calculations. Arelated problem, that of the radiation into auniform medium fr(m] the termination of afiber, has been treated, yielding atransmission coefficient of 96 percent for radiation into avacuum and 100.0 percent for amatched medium. In conclusion, the more rigorous modal analysis has shown that the Born approximatioli does give accurate results in all cases treated using matched media, and that, even though it requirm somewhat greater computational effort, the modal approach would be advantageous in problems ~rhere serious mismatches of refractive index occur or ~vhere the Born approximation is inapplicable, as in the case of the gap. APPENDIX The set of functions em, h,,,, are the fields of the modes of adielectric rod of permittivity El and radius ain a medium of’ lo~ver permittivity 6Z surrounded by aperfectly conducting pipe of radius b and cam be obtained from the usual longitudinal formulation [1 S]. l’or the range of the propagation constant k~2<~2 < kiz(kl,j’ =W2pOel,z), we obtain the surface-type modes, whose existanlce is independent of the surrounding pipe and which have fields 1ocalized to the vicinity of the core with cutoff condition D=k~. Their longitudinal com- 168 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TIK’HNIQUES, JANUARY 1975 ponents are given by The nofisurface type modes are obtained in the range I Jfi(qr), &~ ?&andthelongitudinalcw mponentsare givenby r<a / Jn(mr), Kn(xw)l.(sr) –In~xw)Kn(:r) r<a E,= g,(@) J.(zJ) K.(xto)l.(to) –1.(XW)K.(W) ‘Ez=gl(0) J.(u) J~(a,r).V~(x31) –.~”~(a,r)J~(xv) a<r<b (Ala) J~(v).Vn(xv) ‘.}T.(~)J. (xV) ‘ . Hz= (q/p”) ’/’(kl)Plg2(dJ)J) where \ 6=1 – e2/el =1 – nz2/n12 g,(4,={;192(@)={_:;} with n a normegative integer. The eigenvalue equation is with V2 ~ Pl=— ‘U2W2 m+‘712&2 (A3) u=ala w=ata ~2—v2=v2 (A4) and the eigenvalue equation (A5) F’ =(&’//{/) P, (A6) ‘ (A7) F2=g(–71+ (1 –6)7?371) ~3= .Yn’(v) Uivn(u) J.’(v] /&”.’ (V) –J.(xv) /1~.(xV) (A8) 7’ =Jn(v)/Nn(v) –J.(xV)/f~. (XV) (A9) (A12b) (A13) (.414) (A15) (A16) (A17) (AIS) (A19a) (A19b) Jn’ (U) K.’(w) ‘1 =d.(u) ‘2 =toK. (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 f(Ldkk’ – P.” –%’ cefiefig’e~) .fi cll As bincreases, these modes become the surface modes The following surface integrals have been reduced by (A1O) Yaghjian [11] to simple line integrals The index ~~?is used for the modes of the fiber, and n refers to the modes of the homogeneous cylindrical waveguide. (Alla) &I~’I =!(en Xh.’) .2 da A (Allb) of the open rod. I.n the limiting case x~cc:.%-+ 1, &-+ 1, and (Al ), (A7) become the expressions for the open fiber +3 f(fl.’~l.,’e~ +~~e~.h~). ?dl. [~]. Use of the ~symptotic expressions for the Bessel %’2 –pm2 . functions yields for x>> 1(~~()) –.–.l --@x –l+ex The cross-normalization coefhcient $’- l–e, .$2‘---’ l+e~’ for w>> 1 N.. =!(en’ Xh~) .2 da gl -~ 1+2/x2= &=1—2/x2~, for ~w <<1 A and n z1. is obtained from (A20) with ~Land minterchanged: CARDAMA AND KORNHAUSER: MODAL ANALYSIS 169 (A21) where the contour cincludes both sides faces, and p.” =colqlen —bn’z pmz =wzpo% —f?mz. For afiber with qin the core and q in have of all the inter- (AJ2) (A~:~) the cladding, we Mm” =jra (11 — 2—czlmt -yn )(wpohmzhnt.’ ‘Y. 2—ff2m2 (161’/62) —j7ratie3 — 2—~1m2 ) 2—~2m2 enz’em, (a–) I’. Y. (A25 ) where 7.2 =w7.ll)E3 —/%’2 O!lmz =Wzpoq —Bmz a2m2 =w2poez —/3m2. (A26) (A~7) (A~~) All field components have the angular dependence removed and are evaluated ,at r=a, exeept e~~(a–), which denotes the radial electric field of the core at 7“=a. REFERENCES [1] i!. W. Snyder, ~;Surf ace ~aveg[lide modes along asemi-infinite dielectric fiber excited bv ariane wave,” J. Opt. SOC. Artier., [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [19] [1{)] vol. 56, pp. 601–606, Ma; 1!)66. .— “Asymptotic expressions for eigenkmctious and eigenvalu~s of adielectric or optical waveguide, ”IEEE l“rans. .Vicrowave Theo,{] Tech. (1969 SymPost um Issue ), vol. hlTT-17 , pp. 113 G1138, IIec. 1960. I( ,,.citati~~l a,ld fiCatterillg of modes 011 a dielectric or ——, E.. optical fiber, ”IEEE Trans. !l[icrowavc Theorjj Tech. (196.9 Symposz um Issue),, vol. MTT-17, pp. 1138-1144, Jlec. 1969. 1>. Nlarciwe,, “.~xcltation of the domiklant mode of arotmd fiber by a(;ausslau beam, ”f;(’1[ ~,/JSt. Tech. ~., VO1. 49, pp. 1695– 1703, CM 1970. D. L. 13isbee, “Optical fiber joining technique,” Bell Syst. Tech. 7., vol. .5CI,pp. 3153-31S8, I)ec. 1971. It. B. Dvott. J. R. Stern. aud J. H. Steward. “Fusion ionctions fol glass%bre waveg~lides,” E/cct/on. htt., vol. 8, pp. “290-292, June 1972. C’. G. Someda, i‘Simple, 10 W-IOSS joints between single-mode optical fibers,” Ile// S,yst. 7’ech. J., vol. 52, pp. .583-596, Apr. 1!)73. 1). L. Bisbee, “Measuremel~ts of loss dLle to offsets and end separations of optical fibers, ”Bell SW+. Twh. J., vol. .50, pp. 31.59–3168, l>ec. 1071. J. S. Cook, W. L. Mammel, and R. J. (irow, “Effect of misalignments on coupling efhciel lcy Of single-mode optical fiber b(ltt joints, ”Bell Sysf. Tech. .J., vol. 3?, pp. 1439–1448, Oct. ,mm.. mon, 1964-. A. D. Yaghjian, ‘Wyhri(i modes and the (’dielectric rod antenna, ”Ph. 1). dissertation, Brewn Univ., Providence, Ii. 1., June 1970. A. D. Yaghjian and 1<;.T. Komhauser; “A modal analysis df the dielectric rod ar)telllla excited by the IIEII mode, ”IEEE Trans. Antennas Propagut., vol. AP-20, pp. 12!2-128, Mar. 1972. E. Snitze r, “Cylindrical dielectric waveg~lide modes,” J. Opt. Sot. Amer., vol. 51, pp. 4{)1-498, ~a~ 1961. 1). Clloge, ‘( Weakl~ giliding fibers, ”Appl. Opt., vol. 10, pp. Z252-Z2.58; Oct. 1971. lt. lc. Chllin, F’ic[d Theory of Gui~led Wcwcs. New York: \lc~rraw-’[~ill, 1960, ch. 11. C. M. Angulo and W. S. C. Chang, (‘A variational expression for the terminal admittance of asemi-infinite dielectric rod,” IRE Trans. Antcnnus Pro)moal., vol. iiF’-7, TIP. 307–~M J(I1v 1959. . . A. L. Jones, “Coupling of optical fibers a]d scattering in fibers,” J. Opt. Sot. Amw., vol. .55, pp. 261–271, l[ar. 196.5. C. C. Johnson, Ft eld and 1}-sue Ekctrod,ynam tcs. New York: McC~rawHill, 1965. A. Cardama, “Modal a]mlysis of problems ill optical fibers, ” Ph.D. dissertation, Brow-l) Univ., Providence, Ii. I., June 1973.