BIREFRINGENCE IN SINGLE-MODE OPTICAL FIBRES DUE TO CORE ELLIPTICITY

Transcription

1 Philips J. Res. 33, , 1978 R992 BIREFRINGENCE IN SINGLE-MODE OPTICAL FIBRES DUE TO CORE ELLIPTICITY by D. L. A. TJADEN Abstract The splitting-up of the degenerate HEll fundamental mode of a stepindex optical fibre, due to a slight elliptical deformation of the core cross-section, is considered. A first-order perturbation analysis is applied with respect to both the ellipticity parameter and the index contrast parameter. The results, which differ from those earlier given in the literature, show that the group delay difference between both modes equals zero at a value of the reduced frequency parameter v ~ Introduetion One possible cause of group dispersion in single-mode optical fibres is splitting-up of the degenerate HEll ground mode due to deviations of the circular symmetry of the configuration. Particularly relevant in this respect are moreor-less elliptical deformations of the core cross-section. Rigorous analysis of the modes of a dielectric step-index waveguide with elliptical core cross-section has been the subject of various papers 1-3), all using expansions in terms of Mathieu functions. Of these only Yeh 3) gives actually numerical results, obtained by extremely involved computational efforts. These results, however, do not cover the parameter range of present interest (small ellipticity, small index difference between core and cladding). In later papers by Schlosser 4.5), Yeh 6), and Dyott and Stern 7) various approximative approaches to the problem were attempted. The obvious inconsistency of these results, which were sometimes obtained by quite involved calculations, induced the present author to carry out a new analysis, which is the subject of this paper. Throughout the paper we use dimensionless Cartesian coordinates (x, y, z) reduced by the average core radius a. The z axis coincides with the fibre axis. In addition, we introduce polar coordinates in the (x, y) plane according to x = r cos cp, y = r sin ip, The elliptical core boundary, with semi-axes a (1 + q) and a (1 - q) respectively, is represented as r = rc(cp) = 1 + q cos 2cp + O(q2), q-o, (1) and we restrict our analysis to first-order terms in power series expansions with respect to q. 254 Phlllps Journalof Research Vol. 33 Nos. 5/6 1978'

2 Birefringence in single-mode optical fibres We will use a harmonic time factor exp (- iwt) and consider waves propagating in the positive z direction with propagation constant {J, thus giving rise to a z dependence of the electromagnetic field according to a factor exp (ia{jz). The magnetic permeability is /-to and the dielectric permittivities of the core and the cladding are denoted by e 2 and el respectively. We put and, in accordance with the usual notation, introduce reduced parameters u, v and w by v = aco (/-to el c5)"!-, u = a (w 2 /-to e 2 - {J2)t, and (3) (2) satisfying We introduce unit vectors i, j and k along the x, y and z axis, respectively. Furthermore we put and () () V,=i-+j- ()X ()y Et = i Ex + j El' = el(x, y) exp (ia{jz - Then el should satisfy the vector wave equation V / el + u 2 et = 0, V I 2 el - w 2 el = 0, (r < rc(ep)) (r> rc(ep)) and vanish for r -- 00, with the requirement of continuity at r = rc(ep) of n x et> n (eel), VI X et> and (V t + VI ele). el' Here n is the unit normal to the core-cladding interface, whereas V I2 represents the two-dimensional vector Laplacian. For all practical fibres c5 «1 and it is well known 8) that the solutions and corresponding eigenvalues are then well represented by those of the so-called scalar (or "weak-guidance") approximation according to which all Cartesian transverse field components are proportional to a scalar function 1p satisfying with V 1 21p d- U 0 2 1p = 0, V 1 2 1p - W 0 2 1p = 0, (r < rc(ep)) (r > rc(ep)) iwt). (4) (5) (6) (7) Phillps Journal of Research Vol. 33 Nos. 5/

3 D. L. A. Tjaden, At the core boundary r = rc(ep) both 'IjJ and "ilt'ijj must be continuous. In the limit of this scalar approximation the field solutions are linearly polarized with an arbitrary polarization direction and as such this approximation is too crude for our purpose. As we have shown in a previous paper 9) its solution, however, easily enables us to find the first-order term in a power series expansion U = Uo + (jul +... (8) of the normalized propagation constant u. Our analysis will thus consist of two steps. First (in sec. 2) we will solve the scalar problem, correct to the first order in q. Next (in sec. 3), we will derive an expression for UI in (8), finally leading to a term of the order qö in a double power series expansion of U with respect to q and ö. In the appendix we will merely state the result of a rather involved calculation starting directly from (5) with the exact form of the boundary conditions. This result, valid for general values of Ö, enables us to check our outcomes with those of Yeh 3). 2. Scalar approximation We attempt solutions of (6) in the form 'IjJ= 00 L qk k=o (/.k(u O r, ep), 00 L qk Yk(WO r, ep), k=o furthermore assuming U o and Wo to depend on q according to By virtue of (8) we have Uo = U + quo 1 +, Wo = W + qwoi +. (9) (10) (11) The functions (/.k(e, ep) are solutions of ()2(/.k 1 (l(/.k 1 (l2(/.k (/.k = O. ()e 2 e (le e 2 ()ep2 (12) As they are regular at (! = 0 they are linear combinations ofterms Jm(e) cos mep and Jm(e) sin mep (m = 0, 1, 2,... ). Similarly the functions Yk(e, ep) are linear combinations of terms Km(e) cos mep and Km(e) sin mep. 256 PhIllpgJournalof Research Vol.33 Nos. 5/6 1978

4 Birefringence in single-mode optical fibres and According to the boundary conditions at r = rc(cp) we have L qk [ak (uo rc(cp), cp) - Yk (WO rc(cp), cp)] = k=o (13) ro :L qk [uo a/ (uo rc(cp), cp) - Wo Yk' (WOrc(cp), cp)] = 0, (14) k=o in which the primes denote differentiations with respect to e. We now insert (1) and (10), expand the functions ak and Yk in Taylor series about respectively e = U and e = W, and collect terms with equal powers of q. From the zeroorder terms we have ao(u, cp) - Yo(W, cp) = 0 and (15) Uao'(U, cp) - WYo'(W, cp) = 0, thus leading to the familiar solutions for the circular fibre. As we are interested in the behaviour of the fundamental mode we take the solutions and (16) in which the constant Bo is to be determined later by a normalization condition and in which U and Ware subject to (11) and to (17) Using (16), (17), and some familiar Bessel function identities we find from the first-order terms in the expansions of (13) and (14) Ua/CU, cp) - Wy/(W, cp) = Bo v 2 cos 2cp, (18) in which we abbreviated Jm(U) = I"" KIII(W) = Km. From these equations together with (11) and (17) it is not difficult to derive that UOl = WOl = 0, (19) Phlllps Journalof Research Vol. 33 Nos. 5/

5 D. L. A. Tjaden and where BI is a constant. Substitution of (10), (16) and (19) to (21) in (9) gives Jo(Ur) UWK2 J2(Ur) cos 2cp Jo 2J l s, 'lfj(r,cp) = (Bo + BI q) -- + Bo q + O(q2), (r < rc(cp» (22) Ko(Wr) UWJ2 KiWr) cos 2cp 'lfj(r,cp) = (Bo + BI q) + Bo q + O(q2),. Ko 2J l K, whereas (r > rc(cp» (23) We will normalize 'lfj(r,cp), thus fixing the values of the constants Bo and BI, by requiring (24) 21< ex> J dcp J 'lfj2(r,cp)r dr = 1 + O(q2). (25) o 0 We find readily that BI = 0, together with 27t B02 {!J0 2 (Ur)/J02 r dr +l K02(Wr)/K02 r dr} = 1. (26) These integrals are easily evaluated and, requiring Bo > 0, we find that 1 UK o Bo=--- V7t vkl (27) 3. First-order term in IS Adopting the notation introduced by Yeh 3) we denote the modes in which the fundamental (HEll) mode of a circular fiber is split due to an elliptical deformation as the eheu-mode and the ohell-mode. The subscripts e and 0 refer to the even and odd symmetries of the electric field with respect to the long axis. 258 PhIllps Journal of Research Vol. 33 Nos. 5/6 1978

6 Birefringence in single-mode optical fibres If q > 0 the ellipse's long axis is along the x axis. Then, the coefficient UI in (8) is given by 9) for the.hell-mode, 1 ' ê)21p2 UI = - 4u o f f ê)x2 drr, (28) where the integration extends over the core region r < rc(fij) in the (x, y) plane. With ê)21p2fê)x 2 cp) we have We write, 1 21< I UI = - 4U f cp)r dr (I, cp)q'cos 2fIJ] + O(q2). (29) o for r < rc(fij), in which, according to (22) and (27) 1p2 = P(r) + qq(r) cos 2cp + O(q2), (30) and 1 W 2 P(r) = - -- Jo2(Ur) 7t v2ji2 (31) = (cos cp~- sin cp ~)2 [P(r) + qq(r)cos2f1j] + O(q2). ' (32) ör r ê)fij Substitution of (32) in (29) gives after some calculations 7t UI = - - 8U {2P'(I) + q [P"(I) - P'(I) + Q'(1) + 2Q(1) - 2Q(0)l} + O(q2). (33) Finally, substitution of (31) in (33) gives, with the help of some Bessel function relations, where, in accordance with (17), s; Jo F=--=- (35) WK I UJ I Phillps Journalof Research Vol. 33 Nos. 5/

7 D. L. A. Tjaden A similar treatment of the oreii-mode would, at first sight, require evaluation of (28) with ê)21jj2jbx 2 replaced by ê)21jj2jby2. It is easily seen, however, that the correct result is simply found by replacing q by - q in (34). 4. Phase and group retardation From (3) we have for the propagation constant fj fj = wn (1 <5 ~)t. c 1 + <5 v 2 where n = (e 2 Je O )t is the core refraction index and c = (Po eo)-t. Expansion in powers of ö gives with (8) (36) (37) Denoting the propagation constants for the erel1- and ohel1-modes by fje and fjo respectively, we find from (34) for the leading term in an expansion of fje- fjo (38) where The group delay difference per unit length is found as where dfj.fj n - = - <5 2 q G(v), dw c d G(v) = - [vgiv)]. dv From (11) and (35) it may be derived that du U - = - (1 - W 2 F 2 ), dv v dw W _ = - (1 + U 2 F 2 ) dv v and (42) df 1 _ = - _(1 - W 2 F 2 ) (1 + U 2 F 2 ). dv v (40) (41) 260 PhIllps Journal of Research Vol. 33 Nos. 5/6 1978

8 Birefringence in single-mode optical fibres With the help of these relations it is found that G(v) U 2 W = --{l- 2 2(U 2 - W2)F+ (5U 2-5W 2-3U 2 W 2 )F2 2v 4 Both functions Gp(v) and G(v) are shown in fig. 1. For v < 0.5 they are practically zero. In the region of interest, about v = 2, Gp(v) ~ 0.22 whereas G(v) decreases and crosses zero at v ~ 2.478, which is just outside the region of single-mode operation (v < 2.4). Q6 (43) f o 0 05 Fig. 1. Normalized phase (Gp) and group (G) delay vs normalized frequency parameter v. 5. Final remarks Our result (38) has the same form as that obtained by Schlosser 5), but for Gp(v) he finds (in our notation) U4 W2 (1 + W2 F) G (v) =. (44) p 2v6 J 1 2 His earlier calculation 4), upon which this result is based, follows rather different lines from ours and involves certain approximations the ultimate effect of which can not easily be estimated. In a very recent paper by Snyder and Young 10) a result is given which in our notation would imply that U2 W2 G (v) = -_ (1 + 2W2 F2 - U2 W2 F3). (45) p 2v4 PhIllps Journalof Research Vol. 33 Nos. 5/

9 D. L. A. Tjaden Here the approach seems to be rather similar to ours but unfortunately no details of the calculation are given. Assuming n = 1.5, () = 0.006, and q = 0.1 (i.e. an axial ratio of the core cross-section of 1.22) we find for v = 2 a group delay difference of 2.9 ps/km. At v = 2.3 we find a value of 0.8 ps/km only, whereas both refs 5 and 10 would predict a value of 4.7 ps/km at v = 2.3. Our expressions hold for all RE 1n -modes, provided that the proper roots of (17) are substituted. Similar expressions could be obtained for the ER 1n -modes. A rather different type of analysis is required for the TE., TM n and RE 2n mode group of elliptically deformed fibres 10.U). Philips Research Laboratories Eindhoven, October 1978 Appendix A perturbation analysis with respect to q, similar to that of sec. 2, can be applied directly to the vector wave equation (5). A characteristic equation is thus obtained for the HE 1n -modes and the ER 1n -modes, correct to the first order in q. If we put we find that Jo(u) 1 Ko(w) 1 r/1 = and 'YJ2 = ' (46) uj 1 (u) u 2 wk 1 (w) W ± tq() [ 'YJ1 + 'YJ2 - ( :~ - ::) (1 - u 2 w 2 'YJ1 'YJ2) ] = 0, (47) where the upper and lower signs refer to the even and odd modes respectively. As it should, eq. (34) follows again from this result for ()-+ O. Equation (47) enables us to make a comparison with the results obtained by Yeh 3) which are for () = l.s. For this purpose we choose his figures 2 and 6 which show, for the ereu-mode and the oheu-mode respectively, the relationship between (1 + q) u and (1 + q) w, for various values of the ellipticity. In Yeh's paper the latter is expressed by a parameter ~o = -t In q. The smallest nonzero value of q considered is q ;;;;0.1353, corresponding to ~o = 1. In figures 2 and 3 a few of Yeh's curves are reproduced, together with the corresponding curves obtained by numerical solution of (47). For q = the agreement is still reasonable. 262 Philips Journalof Research Vol. 33 Nos. 5/6 1978

10 Birefringence in single-mode optical fibres 3 q=o r ît+qlw i 2.HE,,-mode 5 = trom ret eq.(47) ~~----~~~~~----~3~----~ (1+q)u Fig. 2. Comparison with Yeh's results 3) for ehell-mode. i t+q) w 3.HE" -mode 5 = trom ref eq.(47) ~~--_'~~--2~---3~--~4 -(1+q)u Fig. 3. Comparison with Yeh's results ê) for.hell-mode. REFERENCES 1) L. A. Lyubimov, G. 1. Veselov and N. A. Bei, Radio Engng and Electr. Phys. 6, , ) G. Piefke, A.E.D. 18, 4-8, ) C. Yeh, J. Appl, Phys. 33, , ) W. Schlosser, A.E.D. 19, 1-8, ) W. O. Schlosser, Bell Syst. Tech. J. 51, , ) C. Yeh, Opt. and Quantum Electronics 8, 43-47, ) R. B. Dyott and J. R. Stern, Electronics Letters 7, 82-84, ) A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-17, , ) D. L. A. Tj aden, Philips J. Res. 33, , ) A. W. Snyder and W. R. Young, J. Opt. Soc. Am. 68, , ) D. L. A. Tjaden, Philips J. Res., to be published. Phlllps Journalof Research VoI.33 Nos. 5/