180 J. Opt. Soc. Am. A/Vol. 6, No. 2/February 1989 Markku Ilmari Oksanen
Perturbational analysis of curved anisotropic optical fibers
Markku Ilinari Oksanen
Electromagnetics Laboratory, Helsinki University of Technology, Otakaari 5 A, SF-02150 Espoo, Finland
Received February 17, 1988; accepted September 16, 1988Birefringence due to a uniform bending of an anisotropic optical fiber is studied. The analysis is perturbational andvalid only for a large radius of curvature. First, the theory is applied to a general inhomogeneous weakly guidinganisotropic waveguide leading to second-order differential vector equations for the transversal fields and a formulafor the change of the propagation constant. This is seen to be proportional to (a/Ro)2 , where a is the radius of the fi-ber and Ro is the radius of the curvature. Next, the concept of linearly polarized modes in the weakly guiding fiber isadopted, and the analysis is applied to isotropic step-index and parabolic-index fibers together with the correspond-ing anisotropic waveguides. Field solutions and expressions for the birefringence are discussed for each of the cases.
1. INTRODUCTION
One of the sources of birefringence in single-mode opticalwaveguides is bending.1' 2 This phenomenon is a stress ef-fect3 that depends on the outer radius d of the waveguideand on the radius of curvature according to (d/RO)2. Whenthe waveguide is bent, the outer portion of the waveguidecross section is in tension, and it presses laterally on theinner portion, which is in compression.' As a result of this,waveguide material becomes anisotropic. The linear bire-fringence due to that effect has been estimated in manypapers (see, e.g., Refs. 1-3).
In this paper, birefringence in curved isotropic and aniso-tropic fibers is studied. The analysis concentrates on geo-metrical effects on the birefringence, thus neglecting modifi-cations in the waveguide cross section or in the dielectricproperties. The cross section and the (an)isotropy are as-sumed to remain circular and constant, respectively. Theanisotropy generated by the bend4 is taken to be muchsmaller than the primary anisotropy of the fiber. The iso-tropic curved step-index fiber was analyzed in Refs. 5 and 6.It has been observed that the geometry effect has a minorrole when compared with the stress effect that is due to thebend. However, the analytic equation for the bending bire-fringence in Ref. 5 is incorrect. The analysis in Ref. 5 madeuse of scalar field theory, which predicts zero birefringence,in contrast to the finite birefringence claimed in Ref. 5. Theerror in Ref. 5 was mentioned in Ref. 6 also, in which it wasshown that the geometrical birefringence is due to the polar-ization corrections to the scalar fields.
This study is a direct extension of our previous asymptoticanalysis of the straight anisotropic fiber.7 For this reasonwe have adopted the same notation as in Ref. 7 for the fieldsand for normalized parameters used in the optical wave-guide theory. A bent fiber radiates, which causes the fieldsand the propagation constant to become complex. The ef-fect of radiation, however, is not included in this analysis.
The structure of the present paper is as follows. In Sec-tion 2 a curved anisotropic waveguide is introduced with thebasic differential equation for the transverse field. In Sec-tion 3 the field and the propagation constant are expressedas asymptotic series with respect to the normalized differ-ence between the dielectric constants in the fiber and with
respect to the normalized radius of curvature. The analysismakes use of a weak-guidance assumption, under which thedielectric constant in the inner part of the waveguide differsslightly from that in the outer part of the guide. What thenfollows is a systematic representation of field equations andformulas for propagation factors in the curved anisotropicwaveguide. Section 3 is concluded with a short discussion ofthe concept of birefringence. In Section 4 isotropic step-index and parabolic-index fibers are considered. Two dif-ferent kinds of anisotropy in curved waveguides that can betransformed from the corresponding isotropic guides areanalyzed in Section 5.
2. CURVED ANISOTROPIC DIELECTRICWAVEGUIDE
We consider an open dielectric waveguide that is bent by acurvature of radius Ro in the cylindrical coordinate system0-Rq5Z, (Fig. 1). The guiding direction is along the axis ofthe waveguide, which is the s axis of the local toroidal coordi-nate system 0'-psO. The two coordinate systems are relatedby the following equations:
R =Ro + p coso, X = p coso,
Z = p sinO = y,
0 = -s/Ro.
(1)
(2)
(3)
The waveguide is modeled by a symmetric dielectric dyadic,
f(P) = E2{ + a2[f(p)1Ž Oa + KW(p)],
where I is the unit dyadic, u, is the unit vector along the axisof the waveguide, and the subscript 2 refers to the outerregion of the guide. The parameter a is a small quantitythat tends to zero in the weakly guiding optical fiber. Thefunction f(p) defines the dielectric profile of the waveguide.K is a two-dimensional dyadic in the transverse plane, fromwhich we have K * CI = UO - K = 0. An isotropic waveguide is aspecial case of the more general structure [Eq. (4)] and canbe constructed by defining K = f(p) (f - UOI).
The waveguide problem can be formulated by writing theguided fields as E(R, Z, 0) = [e(R, Z) + e,(R, Z)tzO]exp(-jv0)
Vol. 6, No. 2/February 1989/J. Opt. Soc. Am. A 181
Z
y
p
R,x
Fig. 1. Uniformly bent optical waveguide and the related coordi-nate systems.
and H(R, Z, 0) = [h(R, Z) + hp(R, Z)a0jexp(-jv0) and byinserting them into Maxwell's equations. After somestraightforward juggling with the algebra, we have
VXe+ jwpihoai 0, (5)
V X h-j e, u 0, (6)
V(eIR) X a,, + jve X t,, + jcoAIRh = 0, (7)
V(hoR) X uz, + jvh X ao -jIRe - e =0. (8)
Here, e = a,/ e * and up e = * h = 0.From Eqs. (5) and (8) we eliminate the transverse field
vector h:
h =oRu~to XE f e/v - V(Ru./ *-V X elwA)lv. (9)
By inserting this into Eqs. (6) and (7) and eliminating theazimuthal component ep, we obtain the equation for thetransverse field vector e:
V(Rto - V X (Rtz, X e * e)/e,) + Rtto X V(Ri2 - V X e)
+ co2 IR2e * e-v 2e = 0. (10)
By rearranging terms we get the fundamental equation forthe transverse field in a curved anisotropic waveguide
V2e - V(V - e) + V[V (e - e)/E.,+ 2aRV * (E * e)/(RE.) + k 2 -* e = O, (11)
where the dyadic k,2 is defined as
k 2 = W2,e- (v/R)2( - aA). (12)
The propagation constant / of the waveguide is related tothe azimuth index v by the relation = v/IRO.
3. ASYMPTOTIC SOLUTION IN THE WEAK-GUIDING LIMIT
A. Asymptotic SeriesAsymptotic solutions of Eq. (11) can be sought by writing allterms as series of powers of expansion parameters and byinserting them into Eq. (11). When coefficients of differentpowers of the parameter are equated, equations of differentasymptotic orders for the unknown coefficients are ob-tained. Here natural expansion parameters are a and 1/Ro.The former refers to the weakly guiding waveguide resultingfrom solutions in the limit a - 0, and the latter refers touniformly bent waveguides in the limit of large radius of
curvature 1/Ro - 0. Because the medium is an even func-tion of a [Eq. (4)], it can be concluded that in a straightwaveguide transverse fields are even functions and the prop-agation constant is an odd function of a. 7 This is obviouslyvalid also for curved waveguides. On the other hand, sym-metry in the waveguide structure implies that the propaga-tion constant is an even function of 1/Ro.8 To combinethese, we write the following asymptotic series for the propa-gation constant in a straight waveguide7:
= K/a + ad + a'32 +..., (13)
and in a curved waveguide
/3 = As + (K/a + af3 + a 304)(6/RO)2 + * (14)
Parameters K and 6 are defined so that K = acz,(Ae2)1/2 (Ref.
7) and 6 = K2a3 /a 2, or 6 = W21Ue2a3, which shows that 6 isindependent of a. The parameter a is a transverse dimen-sion of the waveguide, the radius of the core of the circularfiber. The parameter K is related to the normalized fre-quency V by the relation K = V/a. K is useful also fornoncircular and anisotropic waveguides and is used here inthe asymptotic equations. a 2
= A = (n,2 - n22)/n 2
2, where 1
and 2 denote the media of the inner and outer regions of theguide, respectively. Finally, V = k 2aA1 2.
The zeroth-order term is K/a = W(A)1/2 = k 2, whereas 03and 02 represent lowest-order eigenvalues of the straight-waveguide problem. /33 and 34 are the corrections to thepropagation factor in the curved guide.
The field e is represented by the following series:
e = eo + (6/R 0 )el + (6/R 0 )2e2
+ a 2[eao + (6/Ro)ea, + (6/RO)2ea2+ * *] + * * -= Eo + a 2Ea + .. e (15)
where the first three terms are field solutions in the limit a0 and the terms in the parentheses correspond to the
lowest-order correction due to nonzero a. If Roa2/a is de-fined as a normalized radius, then the term (6/1o) = (k2 a)2(a/Ro) is independent of a 2
.
B. Basic EquationsTo obtain lowest-order equations from Eq. (11), we start byexpanding the term
V[V - (e - e)/e,] = V{V * EO + a2[V * E- fV * EO
+ V * (K - E0 )1 + .. .} (16)
and inserting it into Eq. (11). Next we write Eq. (11) in thelocal coordinate system and disperse V2e and V * e: V2e =Vt2e + (1/R)Oe/IR - (1/R2)e URUR and V -e = Vt -e + e * UR/
R. Vt2 = o2/op2 + (1/p)O/Op + (1/p2)02/a02 and Vt * e = Oex/OR
+ Oey/0Z. The partial derivates are defined as 0/0R = cos a0/Op - (1/p)sin O0/0O and a/OZ = (1/p)cos 00/00 + sin 0al/p.
The electric field components in the global coordinatesystem are related to those in the local system by the rela-tions (Fig. 1)
eR = ex, ez = ey, e. =-e, (17)
e, = ep cos 0 - eo sin 0, ey = ep sin 0 + eo cos 0. (18)
The lowest-order equation with respect to a, that is, a - 0,can now be written in the local coordinate system:
Markku Ilmari Oksanen
0o
182 J. Opt. Soc. Am. A/Vol. 6, No. 2/February 1989 Markku Ilmari Oksanen
Vt2EO + k,00 ' - E0 = 0, (19)
where the dyadic is
k, 002 = limj[W 2
Y E2 - (Ro/IR)2/21(1 - rt'ai) + a2W2 W2K).a-O
By inserting the expansion of /3 and taking the limit, weobtain
k = kco2 + k,,', (20)
with
kco2 = K2K - 2K/, (1-uq~u,/,) (21)
and
k~l2 = {2 p cos 0/a3(6/RO) -2[(K/a)2
+ K(/3, + / 3)] (6/RO)21(-u _ el ) . (22)
The first-order equation is missing, and the second-orderequation of the order of a 2 is the differential equation for thefield Ea,
Vt2 Ea + kc002 * Ea = Vt[fVt * E -Vt - (K -E0)]
- (0Eo/OR + 2 URVt * EO)(6/RO)/(K2a 3 )
+ (/32 + 2K/2 - 4 p cos O3, (6/RO)/(Ka3)
+ {3(p cos 0)2 /(K2 a6 ) + 2[K(/ 2 + 034)
+ 103,3 ] (6/Ro) 2 )E0, (23)
where the terms up to the order of (6/Ro)2 have been re-tained.
C. Straight Anisotropic WaveguideIn this subsection we consider asymptotic equations for thestraight waveguide. Because this was studied in Ref. 7 wegive only the main results here.
The lowest-order equation (a - 0) from Eq. (19) is
Vt2eo + K2 K(p) eo = 2K/3leo. (24)
This is the eigenvalue equation for the straight waveguidewith the eigenvalue fl, and the eigenvector eo. The operatorin Eq. (24) is self-adjoint, provided that the dyadic K(p) issymmetric, which was our assumption in Eq. (4).
The eigenvalue can then be solved by applying, e.g., avariational technique.7 The following stationary functionalcan be derived:
K2 J eO * K(p) * e0 dS - J (Ve0 ):(Ve0 )dS
/1 =2K J eo * e0 dS
The integration extends over the whole transverse plane. Inthe special case of an isotropic fiber, K(p) is a multiple of atwo-dimensional unit dyadic, and Eq. (24) reduces to anequation of the scalar Schr6dinger type. The problem isnow degenerate, because there exist two linearly polarizedsolutions eo of Eq. (24) for each eigenvalue /3. The eigen-function is eo = a4(p), where a is a constant vector. Toobtain polarization, we must make use of Eq, (23), Thissimplifies, in the limit 6/Ro - 0, to the form of the order ofa2,
Vt2eao + kcoo2 -e (a 2 + 2 K 2)eo + VtfVt * eo - Vt I (K * eO)].
(26)
The field eao now gives polarization correction to thestraight-guide field. The eigenvalue /2 is a solution of theintegral 7
(27)After Eq. (27) is evaluated, the component eao can be solvedfor from Eq. (26). The result is not unique, because anarbitrary multiple of eo can be added to any solution of Eq.(26). The total field must then be normalized to obtain theunknown coefficient. For the isotropic step-index fiber andpower-law-index fibers, solutions for eao fields are avail-able.9 These depend on the special choice of the polariza-tion in eo, thus breaking the degeneracy of the fundamentalmode in a noncircular fiber or between the HE and EHmodes in a circular fiber.9
D. Curved Anisotropic WaveguideHaving solved the first eigenvalues of the straight wave-guide, we consider changes in the propagation factor in acurved guide and in the limit of weak guidance a -> 0.Returning to Eq. (19), we can determine the lowest-orderasymptotic equation with respect to 6/Ro. Obviously, it isthe same equation as Eq. (24) for the straight guide. Thefirst-order equation (a -> 0) of the order of 6/Ro reads as
Vt 2el + k, 02 -el = - 2 p cos Oeo/a3, (28)
where the unknown term is the field el. This equationshows that for those modes for which eo = a46o, such as HELmmodes in a circular fiber, polarization does not change be-cause of the bending. The same result was observed also inRefs.10 and 11. Equation (28) can also be derived by apply-ing the effective straight-waveguide approximation method."
The next equation of the order of (6/R?0 )2 at the limit a -> 0is of the following form:
After performing the integration f [e2 * Eq. (24) - eo * Eq.(29)]dS over the whole transverse plane xy and noting thatf (e2 * Vt2eo - eo - Vt2e2)dS = 0, owing to the exponentialdecay of the fields of the bound modes, we obtain the follow-ing correction to the propagation constant:
J 2p cos Oeo * eldS
K/33= ° -k22 -K_ ,.J eO * e0 dS
(30)
The case in which a $4 0 in the curved waveguide is discussedin Appendix A. The square of the propagation constant cannow be approximated up to the orders of a 2 and (6/Ro)2:
/3 -3)1 + 21k22 + K(/1 + 03)
+ a2[10103 + K(/ 2 + /4)] (6/Ro)2, (31)
Vol. 6, No. 2/February 1989/J. Opt. Soc. Am. A 183
where f/32 = k22 + 2K/, + a 2(/3 2 + 2K/ 2). Different terms
can be calculated from Eqs. (25), (27), (30), and (A2); Eq.(A2) is given in Appendix A. The field in the curved wave-guide is correspondingly
e = eo + a 2eao + (6/RO)(el + a 2eal) + (6/?R0 ) 2e2 , (32)
with the components being the solutions of Eqs. (24), (26),(28), (29), and (Al) (see Appendix A).
E. BirefringenceBy definition, birefringence is a measure of the differencebetween the propagation factors of the two orthogonallypolarized waves of the same mode, denoted Oa and /b:
B = (Oa - /b)/k 2 - (/a2 - /b 2 )/2k 22 = a 2 (Oa2
- /b 2)/2K2 ,
(33)
where the relation defined by - is approximatively valid,provided that the propagation factors do not differ muchfrom each other and from k2. Compared with relation (31),
the above square-term formula is more convenient.In waveguides with nondegenerate propagation constants,
such as anisotropic guides with or without a nonsymmetricalcross section, the birefringence is proportional to a 2 and (a6/Ro)
For degenerate modes, such as those in an isotropic noncir-
cular waveguide, /3la equals /3lb,9 and we have
B a4(/32a - 2b)/K + a 4(/2a - /2b + /4a - / 4b)(6/Ro)2/K.
(35)
On the other hand, if we consider the basic mode in a circu-larly symmetric and straight fiber, the propagation con-stants of the orthogonally polarized waves are identical, re-
gardless of the difference between the dielectric constants ofthe fiber. This can be seen, up to the order of a 2, from Eqs.(25) and (27); these solutions for isotropic guides are inde-pendent of polarization. Thus the birefringence of the basicmode is zero in an isotropic fiber. In a curved fiber, 04
depends on the polarization (see Appendix A), and the bire-fringence of the basic mode is of the order of a 4 (6/Ro) 2
A2(a/Ro)2, which was found in Ref. 6, too.By assuming that the vector corrections eao and ea, are
negligibly small, we can simplify the analysis considerably;then, by combining Eqs. (19) and (23) and using both scalarand vector wave equations, we can derive an approximativeanalytical formula for the geometrical bending birefringencein an isotopic curved fiber.6 It was shown that this birefrin-
gence is due to the polarization corrections to the scalar wavetheory. The analysis in Ref. 5 is incorrect, and the birefrin-gence is calculated incorrectly because, if only the scalarfields are included, the birefringence is zero to order (a/IRo)2 .
4. ISOTROPIC FIBER
Here we apply the theory for a bent round isotropic fiber.The following analysis is limited in two aspects: (1) the
cross section of the fiber is assumed to be circular and (2)
dielectric parameters are not affected by the bend. To see
how the bend modifies the refractive index of a fiber, werefer to Refs. 1-4. The normalized propagation constant bis defined in the optical waveguide theory asl3
=2 - k22 /2 - k2
2
k,2_ k2 2 K 2 (36)
After inserting the expansion of /2 [relation (31)], we obtain
b = 2/3,K + F(6/RO)2/K2 + a2G/K 2 + a2E(6/RO)2/K 2 - bo + 6b,
(37)
where bo and 6b represent the first two terms, i.e., a = 0.The subscript 0 refers to the straight fiber, and the change 6bcomes from the bending. The coefficients are defined as F= 2[(K/a) 2 + K(/3 + /3)], G = /12 + 2K/2 , and E = 2[/1/3 +
K(/ 2 + /34)]. In the isotropic fiber in the basic HE,, mode,where eo = a4o and el = afl and a denotes the polarizationvector, F reduces to
f 2p cos 0f oldS
F - , (38)a
3 J 4i02dS
where the integration extends over the whole transverseplane. The factor /3 in the isotropic waveguide can beobtained from Eq. (25),
K2J K(p)io 2 dS - J (V4o)2dS
/1 = (39)
2K J i' 2 dS
and the correction term /32 can be obtained from Eq. (27).Solutions for this component are derived in Ref. 9 for step-index and infinitely and finitely clad parabolic profiles at thebasic mode. Higher-order modes also were considered forstep-index fiber in Ref. 12.
A. Step-Index FiberFor the round step-index fiber we have f(p) = 1 inside thecore (p < a) and f(p) = 0 in the cladding (p > a). The zeroth-order transverse fields can be solved from Eq. (24), and thesolutions are the well-known scalar fields expressible interms of Bessel and modified Bessel functions.'3 The eigen-value reads as uJ,(u)/Jo(u) = wKi(w)/Ko(w) in the limit A =(n, 2
- n22)/n 2
2-0.13 The parameters u and w are functions
of bo and V: u = V(1- bo)1/2 and w = Vbol/2 and thenormalized frequency V = k2aA"/2 .
The first-order fields involve solutions of the form7(AnpnJo + BnpnJl)cos O and F(CnpnKo + DnpnK,)cos 0.
Inserting these into Eq. (28) and finding the constants An,Bn, Cn, and Dn leads to the following functions for the HE,,mode:
The boundary conditions require that Bo = K2(w)/2uKo(w),Do = J2 (u)/2wJo(u). These solutions are same as thosederived in Ref. 11 except for a multiple of 1/2 that is due to adifferent definition of A. In Ref. 11 A equals (n12 - n22)/
2n, 2, which causes the difference. The total field eo + (6/
Markku Emari Oksanen
184 J. Opt. Soc. Am. A/Vol. 6, No. 2/February 1989 Markku Ilmari Oksanen
Ro)ej is the same as in Ref. 11. The change in the normal-ized propagation constant 6b can now be calculated by usingEq. (38). The normalized result is shown in Fig. 2. Thecurve is exactly the same as that shown in Ref. 14, which iscalculated by an entirely different method, namely, a varia-tional technique.
B. Parabolic-Index FiberAs an example of an inhomogeneous guide we consider around parabolic-index fiber with a homogeneous cladding.
- aq/ol/V2)cos 6, p - a
p > a,(42)
where Io' denotes the first derivative of '0 with respect to p/a:
The dielectric function is now f(p) = 1 - (p/a)2 for p < a andf(p) = 0 for p 3 a. The solution of the straight-waveguide eq.(24) has the following form for the HE,, mode in the limit ofweak guidance' 5 :
expl[1 - (p/a)2] V/21M[A, 1, V(p/a) 2]
To = M(A, 1, V) , (41)KO(wp/a)/KO(w) p > a
wth A = 1/2 - V(1 - bo)/4. M denotes the Kummer hyper-geometric function. The eigenvalue equation now reads asV[2M(A + 1, 2, V)/M(A, 1, V) - 1] =-wnKj(w)/n 2Ko(w).The field component 0l can be solved from Eq. (28):
I -Ns -
cN -
(44)
The 6b curve of this fiber is shown in Fig. 3. The regionwhere the curve grows rapidly occurs here in a region ofhigher V than in the step-index case. This comes from thestronger 4j component at lower V values in the parabolic-index fiber. Also, dispersion curves of the straight fiberbehave differently. For a step-index fiber at V = 1.5 wehave bo = 0.2292, whereas this parameter equals 0.045 in aparabolic-index fiber. Thus the parabolic-index fiber is al-most at its virtual cutoff at V = 1.5. For a step-index fiber asimilar characteristic appears at V values closer to 1.
5. ANISOTROPIC CURVED FIBER
A. Anisotropic Fiber with Isotropic CladdingThe first example of an anisotropic fiber is a guiding struc-ture whose dielectric dyadic is of the same form as thatgenerated by the bend
Kdp) = (K'b + KVWW)f(p) with v * wv = 0.
001-toZ
_n
0In
u0To
o.To
0 I _. I ' _ r1 2 3 4 5 6
VFig. 2. Change of the normalized propagation constant 6b for acurved isotropic step-index fiber [Eq. (37)] multiplied by N =(4Roa/a) 2 in a weak-guiding limit A - 0.
(45)
More-general anisotropic fibers can be constructed from Eq.(4). The fiber consists of an anisotropic core and an outerisotropic cladding. The optical axes are denoted by the unitvectors b and Cv, which lie in the plane of the bend andperpendicular to it, respectively. As was shown in Ref. 7,the vector Schr6dinger Eq. (24) can be separated into twoscalar equations for the two modes eo, = beo, and eow = (veo,.If the solutions of the isotropic fiber are denoted by to(K, p)and Al3(K), the solutions of the anisotropic fiber can be ex-pressed as7
which is a useful approximation for perturbational anisotro-py, provided that analytical formulas for bo and 6b are avail-able. A prime denotes a derivative with respect to the argu-ment.
The normalized birefringence is calculated at differentanisotropy ratios in Figs. 8-11. Figures 8 and 9 show thestraight-fiber curves, whereas Figs. 10 and 11 show changesdue to the bending. When Figs. 8 and 9 are compared withthe straight-fiber curves of Ref. 7, which were evaluated byapplying variational methods and trial functions, the differ-ence between the curves is of the order of 1%, except for theparabolic-index curve at e = K,/Kw = 1.01. In Ref. 7 the formof this curve differs from the others given in that paper and
0 1 2
Fig. 7. Samefiber.
KJ = 1.50
jr; = 1.01
K1.00
1. *. IC=0.80 --
.................................
3 4 5 6
Ka
ao0
Co0
as Fig. 5 for a curved anisotropic parabolic-index
0:1m .}; p=1.01 --------------
0- i=0.80
0 1
KaVTw
Fig. 8. Normalized birefringence Bno [Eq. (53)] for a straight aniso-tropic step-index fiber at various values of the anisotropy ratio A =Kv/Kw.
K.= 1.50
K.=;1.01so
0
0 Co-o 6.
Co0*
6'
0cIt01
0.
Fig. 6.fiber.
6
InC4
0
In
0
00-oZ
0q
Markku Ilmari Oksanen
y. = 1 -,0
Vol. 6, No. 2/February 1989/J. Opt. Soc. Am. A 187
0'--I1
/2 = 1.50
A = 1.01
Al= 0.80
0 1 2 3
Kavrw
Fig. 9. Same as Fig. 8fiber with ji = Ku/Kw.
a6-
I . . . I
4 5 6
for a straight anisotropic parabolic-index
outer radius of the fiber was 62.5 Am. The radius of thecurvature (Ro/a) varied between 1000 and 3000. It wasfound in Ref. 6 that the bending birefringence due to thestress is approximately 3 decades larger than the geometricalbirefringence. More exactly, at Ro/a = 2000 the geometricalbirefringence is approximately 0.2 m-1 and the stress bire-fringence is approximately -110 m-1. These values arenonnormalized (B = pa - 13b) values. From Fig. 8, the non-normalized values of the straight anisotropic fiber can becalculated to be approximately 9900 m- 1 (u = 0.80), -510m-1 (pu = 1.01), and -26700 m-1 (,u = 1.50). These are muchlarger than the stress-induced values of the curved isotropicfiber. This means that the anisotropy that is due to thebending stress is much smaller than the original anisotropyof the straight fiber. When the anisotropic fiber is bent, thebirefringence changes according to Fig. 10. The nonnor-malized values are now -10 m'1 (,u = 0.80), 2 m-1 (Ai = 1.01),and 50 m- 1 (Ai = 1.50). Thus the geometrical effects on thebirefringence in a curved anisotropic fiber are extremelysmall compared with the primary birefringence of thestraight anisotropic fiber. The effects of bending on thedielectric parameters must be included in the analysis only ifthe initial anisotropy is much smaller than , = 1.01.
B. Anisotropic Curved Fiber with Anisotropic Core andAnisotropic CladdingThe second example is that of a structure in which theanisotropy extends over the whole transverse plane of thefiber. In this case, the dyadic K assumes the form
clT0
0O -t
"-, I
mz'0Co
0
to6I
0T
= 1.50
A = 1.01
A = 0.80
IT ; ' I 2
0 1 2 3 4
KaVfW
5 6
Fig. 10. Change of the normalized birefringence in a curved aniso-tropic step-index fiber with u = K,/K-.
from that shown in Fig. 9. Normalized 6B curves in bothfibers are quite similar. In fact, when the ordinates areshifted properly the curves almost meet.
It is interesting to compare these results with the stress-induced geometry-induced birefringences of the isotropicstep-index fiber. In Ref. 6 the fiber parameters were chosenas X = 1.3 ,um, a = 2.4 ,m, A = 0.018, and V = 2.26, and the
R
o-
I
O It
"~k 1
m -z e0
o-
o-
o,
t = 1.50
U = 1.01 -------- -----
/2I= 0.80 --
,/'//.I
.I: ~ ~ ~ ~ __ ,__I
I-_ I . I , I , I I0 1 2 3 4 5 6
Kav'Kw
Fig. 11. Same as Fig. 10 for a curved anisotropic parabolic-indexfiber with i = Kv/Kw.
Co6-
6-0C:m
0-
q0
Markku Ilmari Oksanen
188 J. Opt. Soc. Am. A/Vol. 6, No. 2/February 1989
in the core and
K = K w VV + KW WW (57)
in the cladding, we obtain the propagation factor in thestraight anisotropic fiber,7
31i(K) = KKi'/2 + (Ki - K') 1/2Ij[(K- K- )1K],
i = v, w, (58)
and the perturbation term in the curved fiber,
I33i(K) = -KKi'/2 + (Ki - Ki ) 1/2 3 [(Ki- Ki-)1/2K] (59)
or
K
Fig. 12. Sketch of the dispersion curve of an anisotropic fiber forthe basic mode.
If the cladding is isotropic, then K' = 0, and the equationsreduce to Eqs. (46), (49), and (50). For the guided mode topropagate, Ki must be greater than Ki'; otherwise the modebecomes leaky. If this is realized to only one of the polariza-tions, the fiber behaves as a single-mode guide, as was no-ticed in Refs. 7, 16, 17. Figures 12 and 13 show sketches ofhow the transformation works with the dispersion curve ofthe straight fiber and with that of the perturbed componentthat is due to the curvature.
The birefringence can now be written from Eq. (52) bychanging KU to KV - Ku' and changing Kw to KW - KW' and byadding the term (K,' - KW')a
2/2. The normalized birefrin-
gence can then be calculated by using the definition [Eq.(53)]
° = -2Ba K,- IC .)
db _
____j- KI I
Fig. 13. Construction of the geometry-induced change in the dis-persion curve for a curved anisotropic fiber by a transformationfrom the curve for an isotropic fiber.
K = K,(P)VD + K(P)WWV, (55)
and, again, optical axes lie in the plane of the bend andperpendicular to it.
Next we restrict the analysis to the structures for whichK,(P) and Kw(p) are the same function except for a multipleconstant, so that the fiber can be analyzed by a transforma-tion from the isotropic fiber. For nonseparable guides wemust work with general formulas (25) and (30).
By writing
K = KVV + K WW
Ki > Ki, i = V, W. (61)
Its straight-fiber properties were discussed in Ref. 7.In the bent fiber we have an additional term that can be
obtained from Eq. (59) by making the above parameterchanges in the terms within the parentheses and by takinginto account the guidance condition in Eq. (61). If we defineK, = MUKw and Kv' = /IKW' we get the curves shown in Figs. 10 and11 where Kw is replaced Kw - Kw' and 6BKW2 by 6BKW(K -Kw )-
6. CONCLUSIONS
The curved anisotropic round fiber was analyzed by apply-ing asymptotic power-series expansions. The method wasdemonstrated with examples that include isotropic step-index and parabolic-index fibers as well as their anisotropiccounterparts, whose solutions can be obtained throughtransformation relations from the isotropic guides. Lowest-order field solutions, propagation constants, and birefrin-gence formulas were given. It was shown that lowest-orderfields of the HEm modes in a curved isotropic guide do notdepend on the polarization, and the birefringence is, to afirst approximation, zero in that guide. Only by includingpolarization correction can the birefringence be proved to benonzero. If the fiber is initially anisotropic, the effects ofgeometrical and bending stresses on the birefringence aresmall. However, when the anisotropy is perturbational, thebending stress must be taken into account.
x.
Markku Elmari Oksanen
Vol. 6, No. 2/February 1989/J. Opt. Soc. Am. A 189
APPENDIX A: HIGHER-ORDER EQUATIONS
Differential equations in the curved waveguide of the orderof a2 can be derived directly from Eq. (23). The equation ofthe order of (6/Ro), and thus for the field component eal, isgiven by
Vt2eai + keo2 * e.
= (01 + 2K32)el + Vt(fVt * el) - V[Vt * (K * el)]
- (aeo/aR + 2uRVt - eo)/(K 2a3)
- 2p cos Oeao/a3- 4p cos 0l 1eo/(Ka 3 ). (Al)
Solving this gives us the polarization correction to the per-turbed field el [Eq. (28)].
The next equation of the orders of (6/RO)2 and a 2 is morecomplicated:
+ 13(p cos 0)2 /(K 2 a6 ) + 2[K(032 + 034) + 103P] leo
- 4p cos O1le 1/(Ka3). (A2)
With the help of Eq. (24) and by using the method describedin Subsection 3.D, we can obtain the unknown parameter 034without solving for the field e2a. If the right-hand side ofEq. (A2) is denoted by A, then we obtain S eo -AdS = 0.This can be solved for 434. The solution depends on thepolarization, also for the fundamental mode, in a circularfiber. However, before this can be calculated, field compo-nents eo, eao, el, eal, and e2 must be determined.
ACKNOWLEDGMENTS
This research was financially supported by the Academy ofFinland and by Emil Aaltonen Foundation and the Jennyand Antti Wihuri Foundation.
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