Skorobogatiy et al. Vol. 19, No. 12/ December 2002/ J. Opt. Soc. Am. B 2867

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1 Skorobogatiy et al. Vol. 19, No. 1/ Deember / J. Opt. So. Am. B 867 Analysis of general geometri saling perturbations in a transmitting waveguide: fundamental onnetion between polarization-mode dispersion and group-veloity dispersion Maksim Skorobogatiy, Mihai Ibanesu, Steven G. Johnson, Ori Weisberg, Torkel D. Engeness, Marin Soljačić, Steven A. Jaobs, and Yoel Fink OmniGuide Communiations, One Kendall Square, Building 1, Cambridge, Massahusetts 139 Reeived November 8, 1; revised manusript reeived May 17, We develop a novel perturbation theory formulation to evaluate polarization-mode dispersion (PMD) for a general lass of saling perturbations of a waveguide profile based on generalized Hermitian Hamiltonian formulation of Maxwell s equations. Suh perturbations inlude eliptiity and uniform saling of a fiber ross setion, as well as hanges in the horizontal or vertial sizes of a planar waveguide. Our theory is valid even for disontinuous high-index ontrast variations of the refrative index aross a waveguide ross setion. We establish that, if at some frequenies a partiular mode behaves like pure TE or TM polarized mode (polarization is judged by the relative amounts of the eletri and magneti longitudinal energies in the waveguide ross setion), then at suh frequenies for fibers under elliptial deformation its PMD as defined by an intermode dispersion parameter beomes proportional to group-veloity dispersion D suh that D, where is a measure of the fiber eliptiity and is a wavelength of operation. As an example, we investigate a relation between PMD and group-veloity dispersion of a multiple-ore step-index fiber as a funtion of the ore lad index ontrast. We establish that in this ase the positions of the maximum PMD and maximum absolute value of group-veloity dispersion are strongly orrelated, with the ratio of PMD to group-veloity dispersion being proportional to the ore lad dieletri ontrast. Optial Soiety of Ameria OCIS odes: 6.31, 6.4, 6.8, INTRODUCTION Reently, muh effort has been plaed on understanding the detrimental effets of polarization-mode dispersion (PMD) beause of the fiber imperfetions in high-bit-rate transmission systems (see Ref. 1 and referenes therein). The task of quantifying PMD involves two problems. The first problem is to understand signal propagation along a fiber with a stohastially varying birefringene. The seond problem is to quantify the loal birefringene by assuming a partiular type of perturbation and employing some method to find the new perturbed modes. In this paper we analyze the PMD indued by geometri imperfetions that fall into a general lass of saling perturbations, inluding elliptiity or a uniform saling of a fiber ross setion, as well as hanges in the horizontal or vertial size of a planar waveguide. In general, PMD arises when an imperfetion splits an originally degenerate mode (suh as the doubly degenerate linearly polarized fundamental mode of a silia fiber or a square planar buried waveguide) into two losely spaed modes, eah arrying substantial signal power and traveling at slightly different group veloities. A standard way to quantify loal PMD is to define an intermode dispersion parameter, whih equals the differene between the inverse of the group veloities of the newly split modes. There has been a signifiant amount of researh on the estimation of suh quantities as the loal birefringene indued by perturbations in the fiber profile. Most of that study is urrently a part of several standard textbooks in the field. 5 However, most of these treatments were geared toward understanding low-index ontrast, weakly guiding systems suh as silia optial fiber, and are not diretly appliable to high ontrast systems suh as Bragg fibers, photoni rystal fibers, and integrated-optis waveguides that are emerging as an integral part of stateof-the-art optial ommuniation systems. Although several approahes have been suggested for the perturbative treatment of arbitrary fiber profiles, 3,6 9 they were either developed for speifi index profile geometries or their validity deteriorated substantially with an inrease in the index ontrast. Here we derive a generalized Hermitian Hamiltonian formulation of Maxwell s equations in waveguides, as well as develop a perturbation theory for the general lass of saling perturbations that for fibers inlude elliptiity and uniform saling of an arbitrary index profile, whereas for planar waveguides it treats the hanges in their horizontal or vertial sizes. Beause of the Hermitian nature of the formulation most of the results from the welldeveloped perturbation theory of quantum mehanial systems an be applied diretly to light propagation in the waveguides. Suh a formulation provides an intuitive way to understand PMD and birefringene in elliptially perturbed fibers. In partiular, we show that the regions in whih the mode is predominantly TE or TM polarized, as judged by the magnitudes of the longitudinal eletri and magneti energies in the fiber ross setion, orre //1867-9$15. Optial Soiety of Ameria

2 868 J. Opt. So. Am. B/ Vol. 19, No. 1/ Deember Skorobogatiy et al. spond to the regions of high PMD. Moreover, within the same limit, we show that the PMD of suh a polarized mode is proportional to the group-veloity dispersion, whih might beome of onern in the ase of dispersionompensating fibers. Finally, we demonstrate these findings for the example system of a dual-ore, step-index fiber. We find that the point of maximum absolute value of PMD orrelates strongly with the point of maximum absolute value of dispersion in the 1 % range of ore lad dieletri ontrasts. Moreover, for small ore lad dieletri ontrasts the ratio of PMD to mode dispersion inreases proportional to suh a ontrast. Finally, the results of our perturbative formulation are ompared to numerial alulations by use of a general plane-wave expansion method 1 and exellent orrespondene between theoretial and numerial results is found. Preditions of a standard perturbation theory are shown to math our results in a region of small ore lad dieletri ontrast whereas for large ore lad dieletri ontrast substantial deviations from the orret results are found.. GENERALIZED HERMITIAN HAMILTONIAN FORMULATION OF THE PROBLEM OF FINDING EIGENMODES OF A WAVEGUIDE A. Hamiltonian Formulation in First we derive a Hamiltonian formulation, also present in Ref. 11, for the guided eigenfields of a generi waveguide that exhibits translational symmetry in the longitudinal ẑ diretion. We start with a well-known expression of Maxwell s equations in terms of the transverse and longitudinal fields. 4 Assuming that the form of the fields is Ex, y, z, Ex, Hx, y, z, t expiz it (1) Hx, y and introduing transverse and longitudinal omponents of the fields, E E z, E z ẑe z, ẑ E ẑ, () Maxwell s equations, E i H, H i E, H, E, (3) take the form z i ẑ H t t E z, H t z i ẑ t H z, (4) ẑ t i H z, ẑ t H t i E z, t E z z, t H t H z. z Eliminating the E z and H z omponents from Eqs. (4) by using Eqs. (5) and following the derivation in Refs. 11 and 1 we arrive at (5) (6) z i Et t H ẑ t 1 ẑ t ẑ t 1 ẑ t Et H t. (7) Substituting Eq. (1) into Eq. (7), and after some rearrangement employing the identity ẑ ( t ) t (ẑ ), we arrive at the following eigenproblem: ẑ t t ẑ 1 ẑ t Et H t ẑ 1 (8) ẑ t H t. ẑ Et In this form the operators at left and at right are Hermitian, thus defining a generalized Hermitian eigenproblem 13 and allowing for all the onvenient properties that pertain to suh a form, inluding real eigenvalues for guided modes as well as orthogonality of the modes that orrespond to the different s (for more disussion see Refs. 11 and 1). Defining operators  and Bˆ as ẑ Bˆ, ẑ  t ẑ 1 ẑ t t ẑ 1 (9) ẑ t,

3 Skorobogatiy et al. Vol. 19, No. 1/ Deember / J. Opt. So. Am. B 869 and introduing Dira notation H t, we introdue more ompat notation for the generalized eigenproblem of Eq. (8):  Bˆ, (1) with an orthogonality ondition between guided modes and having the form Bˆ,. (11) In the following we analyze perturbations that are uniform along the ẑ axis. If a perturbation suh as a small uniform resaling (a uniform inrease in the sizes of all transverse dimensions of a struture by a small multipliative parameter) is introdued into the system, it will modify operator Â. Denoting the orretion to an original operator by Â, the new eigenvalues (up to the first order) an be expressed with the standard perturbation theory 14 :  Bˆ (1) In general, at a partiular frequeny, knowledge of alone is not enough to haraterize an eigenmode of a fiber uniquely. Additional labels are needed, suh as angular index m, whih indiates the type of angular dependene. In the ase of an elliptial perturbation (equivalent to the resaling of the axes suh that one of them inreases while the other dereases by some small multipliative fator), a split ours in the doubly degenerate eigenmodes haraterized by the same but having opposite angular indies m and m. For the ase of a doubly degenerate mode, Eq. (1) is not diretly appliable. Instead, the split is obtained from degenerate perturbation theory. 14 Then, new linearly polarized nondegenerate eigenmodes, whih we denote and, up to the phase are found to be to the first order 1,m,m ), (13) and the perturbed eigenvalues are,mâ,m,m Bˆ,m,mÂ,m.,m Bˆ,m (14) The intermode dispersion parameter is defined to be the mismath of the inverse group veloities of the perturbed modes that an be expressed in terms of the frequeny derivative 1 v 1 g v g e, (15) where e ( ). The PMD of a doubly degenerate mode that exhibits splitting aused by a perturbation is defined to be proportional to. 15 The desirable ondition of zero PMD at a partiular frequeny then implies a zero value of the frequeny derivative of the degeneray split e, or equivalently e must be stationary at suh a frequeny. Finally, Eq. (7) permits perturbations that are, in general, nonuniform along the ẑ axis. In this ase one has to resort to an analog of the time-dependent perturbation theories of quantum mehanis. 14 B. Hamiltonian Formulation in An alternative formulation of a generalized Hermitian Hamiltonian eigenproblem in terms of rather than is also possible. Following the same derivation as in Subsetion.A, we an also eliminate omponents E z and H z from Eqs. (4) by using Eqs. (6). Following an expliit substitution of Eq. (1) into Eqs. (6) and after some manipulations, we arrive at ẑ D t ẑ B t 1 1 t t 1 (16) 1 Dt t t B t, where D t, B t H t, and operators at left and at right are Hermitian. Unfortunately, the formulation in for the transverse fields does not offer perturbative treatment of ẑ dependent perturbations as there is no analog of Eq. (7) for Eq. (16). However, ẑ independent perturbations an be suessfully treated by the same approah as desribed in Subsetion.A. 3. GENERIC PERTURBATION OF THE DIELECTRIC PROFILE (WEAK PERTURBATION THEORY) We now derive an expression for the matrix elements of the perturbation orresponding to a generi hange in the dieletri profile of a fiber. In all that follows we assume nonmagneti materials with 1. Starting with the definitions in Eqs. (9) for operators  and Bˆ and assuming perturbations of the form, we arrive at a modified generalized Hamiltonian eigenproblem: where  o  Bˆ, (17)  t ẑ ẑ t. (18) Expanding Dira notation into an integral form we proeed with an evaluation of the matrix elements:

4 87 J. Opt. So. Am. B/ Vol. 19, No. 1/ Deember Skorobogatiy et al. Bˆ ds S H t ẑ ẑ H t Sdsẑ, * H t,, H t, *, (19) where the integration is performed over the waveguide ross setion. Using the vetor identity a b b a b a, () the divergene theorem, the square integrability of the fields, and Eqs. (4) and (5), after some manipulation involving integration by parts we derived similar to the vetor perturbation theories developed in Refs., 5, 7, and 16. The problem of validity of suh a form of perturbation theory in the ase of high-index ontrast systems stems from its inadequay in treating the field disontinuities at the perturbed interfaes of substantially dissimilar dieletris. By literally moving the high-index ontrast interfaes the hange in the dieletri profile beomes large in absolute value near the highindex ontrast interfaes. The eigenmodes of an unperturbed fiber do not onstitute an adequate basis to approximate large hanges of the fields in the perturbed region and a standard perturbation theory breaks down. With the development of integrated optis, as well as mirostrutured and photoni rystal fibers, there is also  o ds S H t t ẑẑ t t ẑ 1 ẑ t Et H t S ds Ez H z 1 H z 1Ez, (1) H t H t where  o Bˆ, ()  ds S S H t ds Ez H z H t Et t ẑ ẑ t H t Ez. (3) H z H t Given the eigenmodes of a struture, one an use Eqs. (19), (1), and (3) to evaluate the new eigenvalues of the eigenmodes given by Eqs. (1) and (14). As we demonstrate later in this paper, when applied to the problem of elliptial perturbations of a fiber, the theory derived above gives a orret result only in the limit of small elliptiity as well as small index hanges over the ross setion of a waveguide. We, therefore, refer to this form of the perturbation theory as a weak perturbation theory. Its form and region of appliability is a need for a perturbation theory that is valid for arbitrary index ontrast. Some perturbation theory formulations for arbitrary ontrast are known, but most of them are partiular to some analyti geometries with sharp boundaries between layers. 6,8,9 Although urrent development of a vetorial perturbation theory 3,16 might also be appliable to an arbitrary index ontrast, there is no detailed disussion of that issue by the authors. In the following, we desribe a vetorial perturbation theory based on the generalized Hermitian Hamiltonian formulation of Eqs.

5 Skorobogatiy et al. Vol. 19, No. 1/ Deember / J. Opt. So. Am. B 871 (9) and (18) for a general saling of a struture. Our theory is valid for an arbitrary index profile as long as the perturbation is small. 4. GENERIC SCALING PERTURBATION OF THE DIELECTRIC PROFILE (STRONG PERTURBATION THEORY) Here we develop a vetor perturbation theory for the general saling of a waveguide refrative-index profile. We lassify a perturbation as general saling, as depited in Fig. 1 if the oordinates (x, y) of the unperturbed waveguide an be related to the oordinates of the perturbed waveguide (x saled, y saled )by x saled x1 x, y saled y1 y. (4) We start with a perturbed struture defined in a regular Eulidian oordinate system. To formulate a onverging perturbation theory we perform a oordinate transformation so that in the new (generally urvilinear) oordinate system the initially perturbed geometri struture beomes unperturbed. Then we use the modes of an unperturbed struture to form an adequate basis set for further expansions. Proeeding in suh a way we effetively substitute the problem of evaluating new fields in a struture with shifted dieletri interfaes by a problem with unhanged geometry but a slightly hanged form of underlying eletromagneti equations. We start with a perturbed waveguide and a generalized Hermitian Hamiltonian formulation [Eqs. (9), (19), and (1)], where the derivatives in Eq. (1) should be understood as the derivatives over the oordinates x saled and y saled. We then transform into the oordinate system in whih a saled waveguide beomes unperturbed using the transformations in Eqs. (4). Thus, starting with Eq. (1) written in saled oordinates (x saled, y saled ), we reexpress the same operator in terms of the unperturbed oordinates (x, y). First, we note that if F is a vetor field then 17 xˆ ŷ ẑ t,saled F det y saled x saled F x,saled F y,saled F z xˆ ŷ ẑ det1 x x 1 y y F x F y F z t F Ô, (5) where the operator Ô is defined in terms of x x /(1 x ) and y y /(1 y )by Fig. 1. General saling perturbation defined by a saling of oordinates x saled x(1 x ) and y saled y(1 y ): (a) the partiular ase of x y orresponds to uniform elliptial perturbation of a fiber, and (b) saling perturbations an also be used to analyze size variations of planar waveguides. xˆ ŷ ẑ Ô det x y (6) x y F x F y F z. Substituting Eqs. (5) and (6) into Eq. (1), we find  saled  o Â, (7) where  o has the same form as in Eq. (1) and is written for a waveguide with an unperturbed index profile, whereas the perturbation operator A is  t ẑẑ Ô Ôẑẑ t t ẑ 1 ẑ Ô Ô ẑ 1 ẑ t. (8)

6 87 J. Opt. So. Am. B/ Vol. 19, No. 1/ Deember Skorobogatiy et al. We assume that the eigenmodes of the unperturbed waveguide are known, and we will use them to alulate the matrix elements in Eqs. (1) and (14) by using Eq. (8) as the perturbation. In the following we evaluate the matrix element  for a generi saling perturbation haraterized by x and y. Using the vetor identity in Eq. () as well as Maxwell s equations in Eq. (8) after some manipulation involving integration by parts, we found that the matrix elements of Eq. (8) take the form,m Â,m S ds Ex E y H x H y y x y x y x x y y y x x Ex E y H x H y, (9) whih is valid for arbitrary geometries, inluding ylindrial fibers and planar retangular waveguides. For a ylindrially symmetri fiber the eigenmodes of the fiber are haraterized by their wave vetor and a orresponding angular index m so that, in ylindrial oordinates (,, z), the fields take the form Er, Hr, t E expiz it im. (3) H,m,m After a hange of oor- Beause of this symmetry of fibers it is advantageous to operate in suh ylindrial oordinates. dinates and expliit integration over angle, Eq. (9) transforms into E,m Â,m dr Er H,m x y m,m x y Er E m,m i i i i i i i i. (31) H,m The form of the matrix elements suggests that generi saling an be deomposed into uniform saling with a saling parameter s ( x y )/ and uniform elliptiity with elliptiity parameter e ( x y )/. We now address these ases of uniform saling and uniform elliptiity in greater detail.

7 Skorobogatiy et al. Vol. 19, No. 1/ Deember / J. Opt. So. Am. B 873 A. Uniform Saling Perturbation For uniform saling, x y, the matrix element orresponds to the first term in Eq. (31), whih implies that uniform saling diretly ouples only modes with the same angular index, m m. In partiular, the firstorder orretion to the propagation onstant of a mode (,m) that is due to uniform saling is, by means of Eq. (1), s,m Â,m E dr Er H,m Er E H,m dre z H z, (3) s D, (34) where D() ( /)( / ) is the group-veloity dispersion of the mode. B. Uniform Elliptiity Perturbation For uniform elliptiity, x y, the matrix element orresponds to the seond term in Eq. (31). It shows that uniform elliptiity diretly ouples only modes with angular momentum states different by, i.e., m m. In partiular, for the ase of an m 1 doubly degenerate mode, PMD arises beause of the elliptiity-indued split between the originally degenerate m 1 and m 1 modes. The first-order orretion of Eq. (14) to the split in the propagation onstants of modes (, 1) and (,1) beause of uniform elliptiity beomes e,1 Â,1 Er E dr H,1 i Er i E 1 i i 1 dre z H z H,1 where the final expression was derived with the help of the mode orthogonality ondition in Eq. (11), representations in Eqs. (19), (1), and () of operators A, B and eigenvalue. From the expression in Eq. (3) we see that the shift in the propagation onstant of the mode that is due to uniform saling is proportional to the time-average longitudinal (magneti and eletri) energy in the ross setion of a struture. Another interesting and independent result about the hange in the propagation onstant of a mode under uniform saling is that its frequeny derivative is proportional to the group-veloity dispersion of the mode. To derive this result, we onsider a dispersion relation for some mode of the waveguide, f(). From the form of Eq. (8), it is obvious that, if we uniformly resale all the transverse dimensions in a system by a fator of (1 ), the new s for the same will satisfy f(1 )/(1 ). Expanding this expression in a Taylor series and olleting terms of the same order in, we derive expressions for s valid up to seond order in : s. (33) Taking the derivative of Eq. (33) with respet to, we obtain up to seond order in : ImE r *E *H, (35) where the E and the H are those of the (, 1) mode. In derivation we used the fat that the fields of the degenerate modes (, m) and (, m) an be related by E z m E z m, E r m E r m, E m E m, H z m H z m, m m, H m H m, and the total eletri and magneti energies are equal dre z drh z H t. (36) We find that, for high-index ontrast profiles, e is generally dominated by the diagonal term dr(e z H z ), whereas for low-index ontrast profiles the ross terms of Eq. (35) also beome important. For any index ontrast, an important onlusion about the PMD of a struture an be drawn when either the eletri or the magneti longitudinal energy dominates substantially over the other. If a mode behaves like a pure TE (dre z drh z ) or TM (drh z dre z ) mode, the split that is due to uniform saling [see Eq. (3)] beomes almost idential to the split in the degeneray of the modes that is due to uniform elliptiity perturbation [see Eq. (35)] and thus s

8 874 J. Opt. So. Am. B/ Vol. 19, No. 1/ Deember Skorobogatiy et al. Defining the PMD parameter D e /(), we know from the disussion in previous setions and Eq. (37), that D e is loosely bound from the above by the value of the group-veloity dispersion D. Thus, for modes in whih eletri and magneti longitudinal energy densities are omparable, D e D, whereas for pure TE and TM modes D e D. In Fig. 4 we plot the dispersion urves D() and D e () for different values of the index of refration of the ore n 1.5,., 1.6, keeping the index of refration of the ladding fixed and equal to n l 1.5. When the ore lad index ontrast is small, the PMD parameter Fig.. Dual-ore dieletri waveguide. The inner ore is a dieletri ylinder of radius R 1 and index n 1, whereas the outer ore is a ring of index n with inner and outer radii R and R 3, respetively. Two ores are separated by ladding with an index n l. e. As PMD is proportional to e / and taking into aount Eq. (34) for the frequeny derivatives of s, we arrive at the onlusion that, for suh modes, PMD is proportional to the dispersion of a mode: e s D. (37) 5. PMD AND DISPERSION OF A DUAL- CORE STEP-INDEX FIBER In the following we onsider a relationship between PMD and dispersion in a dual-ore step-index fiber (Fig. ). In partiular, we are interested in how PMD behaves, ompared with group-veloity dispersion D, as the ore lad index ontrast inreases. An inner ore of a dual-ore dieletri waveguide is a dieletri ylinder of radius R 1 a and a variable index n 1, whereas an outer ore is a ring of index n (n 1 n l )/, with inner and outer radii R 5a and R 3 6a, respetively. Two ores are separated by ladding with an index n l 1.5. For a given ore index of refration n 1, we hose the radius of an inner ore a in suh a way that the absolute value of group-veloity dispersion of a fundamental m 1 mode reahes its maximum at 1.55 m. In Fig. 3 we plot a split e m1 m1 in a propagation onstant of a doubly degenerate m 1 fundamental mode of a dual-ore fiber with n 1.5 that is due to a uniform elliptial perturbation of %. e is measured in units of /a, where a.46 m. The solid urve orresponds to e as alulated by the first-order perturbation theory in Eq. (35). For omparison, we omputed the dispersion relations of an elliptially perturbed fiber by using iterative solutions of a general plane-wave expansion implemented in a frequenydomain ode, i.e., MIT photoni bands (MPB). 1 The rosses orrespond to e as alulated with the MPB. Exellent agreement was observed between our firstorder perturbation theory and the MPB ode aross a wide frequeny range. Fig. 3. Value of a split e m1 m1 in a propagation onstant of a doubly degenerate m 1 fundamental mode of a dual ore (n 1.5) fiber (see Fig. ) that is due to a uniform elliptial perturbation of %. e is measured in units of /a, where a.46 m. The solid urve orresponds to e as alulated by the first-order perturbation theory in Eq. (35). The rosses orrespond to e as alulated by the frequeny-domain plane-wave expansion ode MPB. Exellent agreement was observed between the first-order perturbation theory and the frequeny-domain ode aross a wide frequeny range. Fig. 4. PMD parameter D e (dotted urves) and group-veloity dispersion D (solid urves) plotted for the m 1 fundamental mode as a funtion of frequeny for different ore lad index ontrasts. The value of saling fator a was hosen to maximize the absolute value of the negative dispersion of the waveguide for the m 1 fundamental mode at 1.55 m. For a fixed n lad 1.5, varying the dieletri onstant of the ore leads to the D e urves that approah the D urves for the high values of index ontrast.

9 Skorobogatiy et al. Vol. 19, No. 1/ Deember / J. Opt. So. Am. B CONCLUSION We have presented a novel perturbation theory approah to treat general saling perturbations in a waveguide with an arbitrary dieletri profile. This formulation, unlike muh previous researh, holds equally well for disontinuous high-index ontrast profiles. We have demonstrated that the PMD of a fiber that is due to an elliptial perturbation an be onveniently haraterized by a PMD parameter D e that is proportional to intermode dispersion parameter. In the ase of low-index ontrast or a mixed polarization mode (judged by the eletri and magneti energies in the longitudinal diretion), this PMD parameter is muh smaller than dispersion (D e D), whereas for a high-index ontrast or a purely polarized mode D e D in general. The address for M. Skorobogatiy is Fig. 5. Ratio of the PMD parameter and group-veloity dispersion D e /D plotted as a funtion of the ore lad dieletri ontrast relative to the dieletri onstant of the ore. The solid urve orresponds to the results of a strong perturbation theory (irles). The rosses orrespond to a numerial simulation that was performed with the frequeny-domain plane-wave expansion ode MPB, and they losely math a urve that is due to a strong perturbation theory. Weak perturbation theory (squares) predits orret results only for the relative dieletri ontrasts that are less than 1%. D e is muh smaller than the group-veloity dispersion D over the whole single/mode frequeny range, with the negative minimum of D approximately oiniding with the negative minimum of the PMD parameter. When the index ontrast is inreased, the PMD parameter starts to approah the group-veloity dispersion, and for large index ontrasts the urves of the PMD parameter and the group-veloity dispersion almost overlap. In Fig. 5, the ratio of the PMD parameter to groupveloity dispersion at 1.55 m, D e /D, is plotted versus the ratio of the ore lad dieletri ontrast to the ore dieletri onstant ( ore lad )/ ore. The same urve omputed by the frequeny-domain MPB ode is plotted in Fig. 5 as rosses. The results of our (strong) perturbation theory are in exellent agreement with the MPB alulations for small as well as large dieletri ontrasts. We also note that for small index ontrast, the ratio of the PMD parameter to group-veloity dispersion is diretly proportional to the relative dieletri ontrast. For high-index ontrasts, the differene between the PMD parameter and dispersion approahes zero linearly with the ratio of lad-to-ore dieletri onstants. To demonstrate inadequaies of the standard (weak) perturbation theory derived in Setion 3, we performed alulations of the same D e /D urve using Eq. (3). As shown in Fig. 3 as squares, this urve mathes the orret result only for the relative dieletri ontrast smaller than 1% whereas it gives erroneous results for higher dieletri ontrasts. REFERENCES AND NOTES 1. S. J. Savori, Pulse propagation in fibers with polarizationmode dispersion, J. Lightwave Tehnol. 19, (1).. A. W. Synder and J. D. Love, Optial Waveguide Theory (Chapman & Hall, London, 1983). 3. C. Vassallo, Optial Waveguide Conepts (Elsevier, Amsterdam, 1991). 4. J. D. Jakson, Classial Eletrodynamis (Wiley, New York, 1998), p D. Maruse, Theory of Dieletri Optial Waveguides (Aademi, New York, 1974). 6. M. Lohmeyer, N. Bahlmann, and P. Hertel, Geometry tolerane estimation for retangular dieletri waveguide devies by means of perturbation theory, Opt. Commun. 163, (1999). 7. D. Chowdhury, Comparison between optial fiber birefringene indued by stress anisotropy and geometri deformation, IEEE J. Sel. Top. Quantum Eletron. 6, 7 3 (). 8. V. P. Kalosha and A. P. 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