an+efficient+numerical+approach+for+determiningthe+dispersion+characteristics+of+dual...

8/12/2019 An+Efficient+Numerical+Approach+for+Determiningthe+Dispersion+Characteristics+of+Dual ModeEllip+Tical Co

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1926 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 13, NO. 9, SEPTEMBER 1995

An Efficient Numerical Approach for Determiningthe Dispersion Characteristics of Dual-ModeEllip ical-Core Optical FibersTzong-Lin Wu, Student Member IEEE, and Hung-chun Chang, Member IEEE

Abstract-Based on vectorial formulations which combinethe surface integral equation method and the finite-element method, a novel numerical approach is proposedfor calculating the dispersion coefficients of dual-modeelliptical-core fibers with arbitrary refractive index profiles.By differentiating the original formulations involving thepropagation constant p and the guided mode fields H , andH , once and twice with respect to the normalized frequencyV , the new formulations for { d p / d V , d H , / d V, d H , / d V }and for { d 2 p / d V Z , 2H , / d V Z , 2 H y / d V 2 } are obtained,respectively. Once {p,H , , H , } is solved through theeigenvalue procedure which dominates the computing time,only a few matrix manipulations are required to obtain{ d p l d V , d H , /d V , d H y / d V ) and { d 2 p / d V 2 , d 2H , / d V 2 , d 2H ,/ d V 2 } . Some numerical examples are examined to see theinfluence of different refractive index distributions with dipson the dispersions of the four nondegenerate LPll modes forelliptical-core fibers.

I. INTRODUCTIONUAL-MODE elliptical-core (e-core) optical fibers haveD een used extensively in many sensors and optical de-

vices such as intermodal couplers, modal filters, and acousto-optic frequency shifter [11-[3]. Recently, a chromatic dis-persion compensation technique that employs the dual-modefiber with small ellipticity was proposed by Poole et al.[4]. This technique is based on the principle of convertingthe LPol mode from the communication fiber operated near1.55 pm to the LPll modes in an e-core compensatingfiber and using the large negative dispersions of the LPl lmodes close to their cutoff wavelengths to compensate forthe dispersion in the communication fiber. It has been shownthat the dispersion of the LPll mode for the circular-corecompensator is sensitive to the variation of the refractiveindex profile using a scalar waveguide theory [5], but, to ourknowledge, the dispersion of the higher-order modes for the e-core compensator has not been studied theoretically in detail.Because the e-core compensator is operated near the cutoff

Manuscript received November 17, 1994; revised April 7, 1995. This workwas supported in part by the National Science Council of the Republicof China Grant NSC83-0417-E002-009 and in part by TelecommunicationsLaboratories, Ministry of Transportation and Communications, Republic ofChina, Grant TL-83-5101.T. L. Wu is with the Department of Electrical Engineering, National TaiwanUniversity, Taipei, Taiwan 106-17, Republic of China.H.-C Chang is with the Department of Electrical Engineering and theGraduate Instituteof Electro-optical Engineering,National Taiwan University,Taipei, Taiwan 106-17, Republic of China.IEEE Log No. 9413045

wavelength, the difference in the dispersion among the fournondegenerate LP1 modes, which include the first-higher-order modes of 2 and y polarizations (LP;';e, , LPyTt)and thesecond-higher-order modes of 2 and y polarizations (LPi'f;,LPyf i) , is significant. Rigorous vectorial waveguide theory isthus needed in order to investigate the dispersions of the fournondegenerate LPll modes for the e-core fibers. Some relatedanalysis results have recently been reported by Poole et al.[6]. In this paper, a novel numerical approach based on thevectorial waveguide theory is proposed to efficiently calculatethe dispersions of the e-core fibers.

The computation of dispersion coefficient requires the firstand second derivatives of the propagation constant with respectto wavelength. A simple way to evaluate the dispersion isthe direct numerical calculation, based on finite differences,of the first and second derivatives from the propagaion con-stant versus wavelength data. However, such simple methodrequires large computational efforts and may result in greaterrors [7]. Some improved numerical approaches based onthe scalar theory for circular optical fibers, such as thatusing the Rayleigh quotient to obtain the first derivative ofthe propagation constant [81, that solving three differentialequations which include the first and second derivatives ofthe propagation constant [9], and that based on the matrixperturbation method [lo], have been proposed. Besides, basedon the variational finite-element formulation of the scalartheory, an algorithm [111, in which formulas for the first andsecond derivatives of the propagation constant with respectto V are explicitly derived by intrinsically differentiating thescalar wave equation with respect to V and by using a reactionformula, has been presented for calculating the dispersionsof optical fibers. This algorithm which eliminates the needof numerical differentiation and is efficient in computationaleffort can only treat the dispersion of circular-core fiber withcircularly symmetric refractive index profile (1-D problem).

In this paper, stemming from the same principle as in[ l l ] and based on vectorial formulations which combinethe surface integral equation method SIEM) and the finite-element method (FEM) [12], a novel numerical approach isproposed for calculating the waveguide dispersions of e-corefibers with arbitrary refractive index profiles (2-D problem).First, the propagation constant p and the guided mode fieldsH , and HY are solved by the combined formulations ofSIEM and FEM and the continuity requirement of E, and

0733-8724/95$04.00 1995 IEEE

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WU AND CHANG DISPERSION CHARACTERISTICSOF DUAL-MODE ELLIPTICAL-CORE OPTICAL FIBERS 1927

H , at the core-cladding boundary. Second, new formula-tions about {p,h y , where the dot denotes differentiationwith respect to the normalized frequency V , with the solved{p,H , , H y } being substituted are derived by differentiatingthe original formulations with respect to V. Matching thecontinuity of fi and fi, at the boundary, {b,hx.?hy}sobtained. Similarly, the formulations about { p ,fi,, H y } withthe solved {p ,H ,, H y }and (8, i x ,Hy}eing substituted canbe derived by differentiating the original formulations twicewith respect to V and { p ,B:,By}s obtained by matching thecontinuity requirement of E, and fi, at the boundary. Whenp,p, and p are obtained, the dispersion can be calculatedby the definition. Most of the computational efforts are spentin solving { p ,H,, Hy}.Only some matrix manipulations arerequired in obtaining { ~ , h , , h y }nd { f i ,&,ay}nce{ p ,H ,, H y } is solved.

The detailed mathematical formulations about {p ,H, , Hy ,{b,Az,fiy}, nd {p,BZ,By}re presented in Section II.The solution procedure described by matrix representation isdiscussed in Section 111. In Section IV, as a numerical example,the influence of the refractive index distributions, includingdifferent cy-power profiles and the index dip effect, on thedispersions of the four nondegenerate LPll modes for the e-core compensators is investigated. The vectorial mode fieldpatterns of these four modes are also presented. Section Vgives the conclusions.

11. FORMULATIONSAs described in [12], based on a full-wave formulation,a combined method employing the surface integral equa-tion method and the finite-element technique can treat the

propagation characteristics of inhomogeneous optical waveg-uides with arbitrary cross-sections. Although similar numericaltechniques have been proposed for optical waveguides witharbitrary cross-sections [13], [14], they can only treat thepropagation characteristics of homogeneous waveguides.

Fig. 1 gives the sketch of an inhomogeneous dielectricwaveguide (core region) embedded in a homogeneous sur-rounding medium (cladding region) with an elliptical cross-section. In the homogeneous cladding with the core-claddingboundary I? the transverse magnetic field F = H , or H y )and its normal derivative d F / d n ( = d H , /d n or d H y / d n ) onI? can be related through a surface integral equation

where kd is related to the wave number in free space koand the propagation constant of the guided mode by kd =p2- k i d ) 1 / 2with ~d being the relative permittivity of thehomogeneous cladding, r and 2 are the position vectors inthe 2-D vector space with 2 on r as shown in Fig. 1, G =(112~)KO kdlF - 21) with KO eing the modified Besselfunction of the second kind and order zero denotes the 2-DGreen function in the homogeneous cladding, and P s denotes

Y-. t

Fig. 1. Elliptical-core fiber with a and b being the major and minor axes,respectively.

the Cauchy principal value integral with the singularity atthe point of r' = r' being removed. How the P s integralis approximated will be described in next section. Throughthe integral equation. ( l ) , the transverse magnetic field at anarbitrary position r' in the cladding region can be describedby the field on r and its normal derivative along A. WhenF ( 3 is chosen as the field on the boundary I?, a completerelation between F and d F / d n on r for the cladding regionis established in (1).

In the inhomogeneous core region with the relative permit-tivity distribution E, x, ) , the magnetic fields of the guidedmodes satisfy the following source-free equation

-+

By making the dot product of the left-hand side of (2) withan arbitrary vector function l f c which is independent of3 nd integrating the scalar product over the entire space,the differential equation. (2) can be transformed into thevariational-equation formulations. After some manipulationsas shown in [12], the variational equations with the arbitraryfields I f c being chosen as H:? and H;& respectively, are

a H , a H ; a H , a H :(IC P ~ ) H , H ;- a x a x a y a yand

a H , a H ; a H y a H i/ k g c c - p 2 ) H H c - x a x a y a y

By making a variation of (3a) and (3b) with respect to H ;and H i , one obtains the differential equation of H , and H yin (2). For guided modes, the boundary condition that E, andH , are continuous at the boundary r should be satisfied. Fromthe Maxwell equations j w & = V x I and V 3 = 0, E,and H , can be expressed as



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1928 JOURNAL OF LIGHTWAVE TECHNOLOGY, OL. 13, NO. 9, SEPTEMBER 1995

and

where E = E O E , or E = EoEd with EO being the permittivityof the vacuum, H , and He are the magnetic fields normalto and tangential to the boundary, respectively, and d / d nand denote the partial derivatives along the normaland the tangential directions, respectively. Explicitly, H,, =H , cos (B)+Hysin(8) and Hl = H, sin(B)-Hy cos(I3),whereI is the angle between A and the x direction. By discretizing(1) and (3) with the boundary element method and the finiteelement method, respectively, the normal derivative fields( a H , / a n and a H y / a n ) on r can be expressed explicitelyin terms of the boundary fields ( H , and H y ) in their cor-responding regions. The tangential derivative fields ( d H , /&and aHy / a f ) in (4) are approximated by a three-point finitedifference. Through matching the continuity of E, and H,,respectively, in (4a) and (4b) which are now in terms of H ,and H y only, the propagation characteristics of the guidedmodes can be obtained.

The integral equations. (1) and (3) and the continuityrequirement 4) on r give a complete description of the elec-tromagnetic behaviour of the guided mode p,H , , H y ) .Whenwe perturb (l), (3), and (4) with respect to the normalizedfrequency V , similar descriptions about the electromagneticbehaviour of the perturbed field (b,fix,fiy)an be con-structed. Described in the following is the details for theperturbation process.

To calculate the first derivative of ,B with respect to thenormalized frequency V ,we differentiate l) , 3), and (4) withrespect to V . n doing so, we obtain that #(= fix or f iy) ndb on I? should satisfy the integral equation

( 5 )that Ikxl y and in the core region can be related as

8HxaHg 8HxaHg\ k:c, - p2)HXH,C x a x a y a y

aHydHy" aHyally"/ ( k : ~ ~p2)HH" - x ax a y a y+ ( -xHy --H,"dxdyE c 1a x E ,

+ l k ~ k ~ ~ ,2/3p)HyHGdxdy= 0 (6b)and that a / a V ( j w E , ) and a / a V ( j p H , ) on r can be ex-pressed as

aand

(7b)where the dot denotes the derivative with respect to V andthe waveguide material is assumed to be nondispersive (i.e.

/aV = 0). Note that in (6) the arbitrary fields ( H i and H:)are not changed as V varies. By differentiatingG and dG/dnwith respect to V , the closed forms of G and d G / d n can beexpressed as

a aHn aHew ( j D H * )= n + l

(Sa)i dG = - tK1 ( k d t )27rand

(8b)where t = I?- 21, A is the unit vector along the normaldirection of the cladding boundary at position 2 and Ki is themodified Bessel functions of the second kind and order i.

Similarly, to calculate the second derivative of p withrespect to V , we differentiate (perturb) 3,6), and (7) withrespect to V . In doing so, we obtain that F and p at thecladding boundary should satisfy the integral equation

- --G - k d k d A (?- '?)Ko(kdt )dn 2T

+ dG(kd ,?,2) d;A F ~ P S,F(r1) dn- d F 9 ) dr3+ 2~ S,F(r >- 2i ( k d ,F, r ' ) ~

G ( k d ,F,2 ) rdG(kd ,?,2)d2

d n- d ( 2 ) d r ;- d G ( k d , T , J ) d2

d n- d F ( 2 ) d;+ P J F ( r ' )r (9)i ( k d ,F, r')that I?,, fiYand in the core region can be related as

aHxaHg aHx8 H gk:ec - p2)I?,H: - x a x a y ay+ g )% L H , " d x d y+ J 4 k o k o Ec - j ) H , H ; d x d y = 0

a y E+ 2 ( k & - b2+ k o i O ~ , p)H,H; dxdy

(loa)



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WU AND CHANG: DISPERSION CHARACTERISTICS OF DUAL-MODE ELLIPTICAL-CORE OPTICAL FIBERS 1929

Given V+ Match boundary continuity of (E,, H,) with (4)then with V and {H, , Hy,p }Match boundary continuity of g,,i,) with (7)then with V , { H , , H y , P } , and {b,&,fiy}+ Match boundary continuity of (I?,, g , ) with (11) { ,fi,, ky}

+ solve (1) and (3){ p ,H,, Hy}+ solve (5) and (6)+ (8, ?,, Hy}+ solve (10) and (1 1)

and

+ 2 ( k - 8' + kol&c - p p ) H y H ; d x d y+ 4 k o k o ~ ,- O,b)HyHGdxdy = 0 (lob)

and that a 2 / a V 2 ( j ~ E Z )nd a 2 / a V 2 ( j p H , ) on I? can beexpressed as

and

The closed forms of G and d G / d n in (9) can be derived as

and

+ k;(KO(kdt ) - k d t K l ( k d t ) ) ] (12b)respectively, by differentiating (8) again with respect to V .

The remaining work is to solve the propagation charac-teristics { p ,H,, Hy}, (8, ?,, fiy},and (8,B,,BY}.heintegral equations (l), ( 5 ) , and (9) on the cladding boundaryr and the variational equations (3), (6), and (10) in thecore region are discretized by the boundary element methodand the finite-element method, respectively. By enforcing thecontinuity requirement of (E,, H , ), ?,, I?,), and (8, & )expressed as (4), (7), and (1 1) on the core-cladding boundaryr, the propagation characteristics 0, ,, Hy} 8 ,&, hY},and {b, ,,Hy} can be solved, respectively. Once i isobtained, the waveguide dispersion D , can be calculated by

- --p 2- 27rcwhere c is the velocity of light in free space.

111. NUMERICALROCEDURESFor clarity, we summarize the numerical procedure for

solving p as shown at the top of the page. In the following,the steps of the numerical procedures are described in detailwith the matrix-form representations.

Steps 1 and 2 describe the numerical procedures forsolving the propagation characteristics of the guided modesp,H,, Hy)for a given normalized frequency V .Steps 3 and4 describe the procedures for solving the first-order perturbedpropagation characteristics (b,H, , hy)f the guided modesfor the given V and the solved p,H,, Hy) in steps 1 and 2.Steps 5 and 6 describe the procedures for solving the second-order perturbed propagation characteristics p, ,, I?, of theguided modes for the given V and the solved p,H,, Hy) insteps 1 and 2 and @,a,, y)n steps 3 and 4.Step 1: Express the normal derivatives of the fields on theboundary in terms of the boundary fields for the cladding andcore regions by discretizing (1) and (3), respectively. Basedon the boundary-element method with the pulse bases for(l), the relation between the boundary fields and their normalderivatives on the cladding boundary for the cladding regioncan be expressed as

M M(14)

1- i = j-jc;ej - fj Gije3.7=1 i= l

3 %

for i = 1,2, . . ,M , where M is the total number of dividedsegments on r, and e j is the arc length of the j-th segment.f j and fj , which are both unknowns, denote the transversemagnetic field and its normal derivative, respectively, on theboundary r at node j. Similarly, the matrix elements Gij andGn. denote the values of the Green function and its normaldenvative, respectively, on the boundary r between node iand node j . Note that the first summation in (14) excludes thepoint i = due to the fact that in the Cauchy principal integralthe singular value GG for i = j is removed. By some linearalgebra operations on the M equations for i = 1,2, . . .,M in(14), a matrix form representation of (14) can be written as

15)where f = [ f i z . .. ~ ] ~ ,n = [ f r ;, . . . I T , and thematrix elements of [Md]are related to the Green function andits normal derivative as in (14). Because f represents either thex component (H, ) or the y component (Hy)of the transversemagnetic field on the cladding boundary I? we denote f as h:for the H , component on r and as hz for the Hy component.Similarly, f, is denoted as hz for the normal derivative of the

.

f n ] ~1 = [Mdl [ f l M X

H , component on r and as hd,, for that of the Hycomponent.Authorized licensed use limited to: National Taiwan University. Downloaded on February 23, 2009 at 04:10 from IEEE Xplore. Restrictions apply.


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By employing the FEM with the triangular elements andlinear bases for both the unknow fields ( H , and H y ) andthe arbitrary fields ( H i and H,") and taking the variation onthe arbitrary fields (see [12] for details), (3a) and (3b) arediscretized as hz ,n 2M x1 hd 2Mx1

respectively, and by the finite difference with (21) as in step1 [h;,,,h;,,lT is expressed as[ = [ M , . ] P ]

where N is the entire number of nodes in the core regionincluding the nodes on I? hi = I z ; , ~ , . .. h M]Tdenotes the column vector of nodal unknowns inside thecore region (excluding the nodes on I?) with j polarizationj= x,y), and [Q] is a matrix determined from (3) as [Md]

in (15) is derived from (1).The relation between the associated fields and their normal

derivatives on the core boundary I? can be obtained by thefinite difference with (16) 1121 as.I = [ M . ] P ]d 2M x1 (17)

2Mx1

where the column vecotors [Td] and [ I are determined by thesolved/Ihe boundary nodal fields [f,, f] in steps 1-2, and thefirst derivatives of the Green function and its normal derivativeon the cladding boundary I' G nd d G / d n ) with respect toV, and the column vectors [ ycj and [ I are determined by pand the nodal fields h in the core region.Step 4: Solve ,b and h. By substituting (20) and (22) into(7) and enforcing the continuity requirement of & and k, tthe boundary I? one obtains

where hi,, is the column vector with its elements beingthe normal derivatives of the j-polarized fields on the coreboundary r. For the sake of convinience, in the followingdiscussion we define h = [h;, h i ,h;, h:IT.

Step 2: Solve for p and h. By substituting (15) and (17)at the core-cladding boundary I?, the relation between h i andh: can be expressed as

In (23), there are 2M + 1 unknowns (hi,h: and 8).butonly 2M equations. By differentiating (19) with respect to V,another constraint

M(24)in (4) and enforcing the continuity requirement of E, and H , C(h h A ) = 0i = l

is obtained. Only one solution of 8, i ,h:} can be solvedFor guided modes, the eigenvalue p is determined fromdet(R(/?)) = 0 and the eigenvector [hi,h:IT is nontrivial.The bisection method serves to locate the desired p. Becausestep 1 is repeated at each trial in the bisection root search, itis the most time-comsuming step. Once p and [hi,h;IT withthe constraint

M + h:,i2) = 1i = l

are obtained, [hi, ,, h;,,lT and [hf,hZIT can be found from(15) and (16), respectively. The reason for the constraint on[hi,hY,ITwith normalized norm will become clear in the nextfew steps.

Step 3: Discretize 5 ) and (6) and obtain the relation be-tween [f,] and [f]and the relation between [hf,,, hZ,JTand[hi,h;IT. By the same method with the same basis functionsand adopting the same nodes as in step 1, (5) and (6) can betransformed into

from (23) and (24).Step 5: Discretize (9) and (10) and obtain the relationbetween [fn] nd [f]and the relation between [hf,,,hz,n]T nd[h:, hZ]l'.By the same method with the same basis functionsand adopting the same nodes as in steps 1 and 3, (9) and (10)can be transformed into

and= [Q]p]

Z(N-M)xl hd 2(N-M )x l

respectively, and by the finite difference with (26) as in steps1 and 3 [hz,n,hZ,n]Ts expressed as

where [TA] and [SA] are determined by the first and secondderivatives of the Green function and its normal derivative onthe cladding boundary I? (G, d G / d n , G, d G / d n ) with respectto V, and the solved p, p, fn, f]: and [fn,f] in steps 1 4 ?;I,and [S ] are determined by p, p, h, and h which have been

f n ] ~ ~ ~[ M d ] [ f ] ~ x lb[rd]+ [ad] (20)and

= [ l [ I(N-M)x 1 hs 2 ( N - M ) x l+B[Sll + [a21 (21) solved in steps 1-4.Authorized licensed use limited to: National Taiwan University. Downloaded on February 23, 2009 at 04:10 from IEEE Xplore. Restrictions apply.


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WU AND CHANG: DISPERSION CHARACTERISTICS OF DUAL-MODE ELLIPTICAL-CORE OPTICAL FIBERS 1931

_ - - _ _ < -

_ _ * . I

I ' ', 4

1

. , * * . :, , I ; I

' I' I '

I 14 } , 'I # , # # *I : . * - -

TEm

+

. -. - - --

LPIY

* -

- -TMm

+

. .

Fig. 2.as the fiber is deformed from a circle a / b = 1 to an ellipse (a/b = 1.05).(a), (b), (c) and (d) show the patterns changing from T E o l to LyT:;, T M o l to Lyy eg , HEgl to L y t i and HE:, to L y ? d z , respectively,

Step 6: Solve i and h. By substituting (25) and (27) into refractive index profile is defined as(11) and enforcing the continuity requirement of Ez and kzat the boundary l7, one obtains - (nf - nz)R* for R 5 1elsewhere2(R)

[R(P)12MX2M [ IY + 'I + (?I = I (28) where R = (. /a2 + y2/b2)1/2.Note that E,(R)= n2(R)and Ed = ni. When a = 1, 2, and 00, the index profile is a2 ~ x 1By differentiating (24) again with respect to V, the constraintbecomes

i=lOnly one solution of (8, : hz} can be solved from (28) and(29).Based on the full-wave vectorial formulations of the SIEMand FEM, the approach described above can efficiently cal-culate the first and second derivatives of the propagationconstants with respect to V of the guided modes for opticalwaveguides with inhomogeneous core and uniform cladding.Most of the computational time is spent in the first two stepswhen searching for and h. In steps 3-6, only some matrixgenerations, matrix multiplications, and matrix inversions arerequired.

Iv. RESULTS AND DISCUSSIONSConsider a dual-mode e-core fiber with the major and minor

axis radii being a and b, respectively, as shown in Fig. 1. The

triangular, parabolic, and step distribution, respectively. Thechange of the vectorial magnetic field patterns of the fourhigher-order LPll modes as the fiber deforms from a circle( a / b = 1) to an ellipse ( a / b = 1.05) is shown in Fig. 2with the parameters being A = (nf - nE)/2nf = 2%, V( 2 7 r b / X ) d m = 4 and a = 2. Fig. 2(a), (b), (c), and(d) shows the patterns changing from TEol to LP??, , T M o lto Lc;;:, HE& to LP$: and HE;, to LPyf,,d,, espectively.Each arrow represents the orientation and the relative strengthof the magnetic field at the point specified by the arrow root.It can be seen that the small deformation of core geometryresults in the linearly polarized field patterns with two lobeorientations.To check the correctness of this novel numerical approach,we compare the dispersions, which are calculated by ourapproach, of the LP:\;e modes for the e-core fiber with theresults by Poole et al. [6] which are also based on a vectorialwaveguide theory. The e-core parameters used in [6] are e2 a - ) / ( a+b) = 10%and a = 00 (step index distribution).The cutoff wavelength of the LET; mode is designed to beat 1600 nm for the e-core fiber. Fig. 3shows the dispersionsof the LyT:, and LET:: modes versus wavelength for threedifferent index steps A = O S % , 2%, and 4%, respectively.



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-100-200EE

- - index distribution on the waveguide dispersions of the TEol,- mol,nd HE21 modes for the circular fiber with A =2%. The radius b = 2.2 pm is chosen such that the cutoffwavelength is 1.623 pm for the HE21 mode. It can be seen

A 0.5 evenP11 y

LP11,xe ~ n Thiswork) -

I \-...I for a = 2 and a = co cases. Assume that in the presence ofthe dip the index profileFig. 3. Dispersion versus wavelen gth for the step-index e-core fiber withthree different index deoths (A = 0.5 .2% and 4%).ComDarision between be written as- \the results in [6] and ours is also shown.

- a =---.---;= - - - -_____ --- - - - - - -50 --250 -

ScalarTheory in Ref. [ l l ]-400153 1.535 1.54 1.545 155 1.555 156 1.565

hclrmFig. 4. Dispersion versus wavelen gth for the circular fiber with three differ-ent a- po we r distributions. The results calculated by the scalar theory in [ l l ]are also presented.

Our results are shown as the solid and dashed curves, whilethose in [6] are plotted as dots and crosses. It can be seen thatthe dispersions increase with the increase of the index stepand that for A = 0.5% the dependence of the dispersion onthe polarization is indistinguishable. Our results are generallyconsistent with [6], except that for fibers with larger indexsteps A = 2% and 4%) the difference between the dispersionsfor the two polarizations in our results is smaller than thatpredicted by [6]. This discrepancy becomes more significantfor larger wavelength which is close to cutoff. It should benoted that the cutoff wavelengths for the Ley:; and LET::modes splits as the index step increases. We design the minoraxis b of the e-core fiber such that the cutoff wavelength of theLP:T, mode is 1600 nm. However, how the cutoff wavelength1600 nm was related to the modes was not clearly stated in [6],making the comparision of the exact values of the dispersionmore difficult.

where ni(R) ,which can be any a-power distribution, is therefractive index profile in the absence of the dip and p andd define the fractional dip depth and the fractional dip width,respectively. A case of the profile with ni(R)being the stepdistribution a = co) s shown in Fig. 5. The dispersions ofthe four LPl l modes versus wavelength from 1.53 pm to 1.56pm for three different dip parameters d = p = 0, d = p =0.2, and d = p = 0.3) with a = co, / b = 1.05, A = 2% andb = 2.137 pm are shown in Fig. 6. For each pair of curvesthe upper curve is for x polarization and the lower curve isfor y polarization for each case with different dip parameter.It can be seen that the index dip causes the increase of thedispersions of the LPll modes. The reason is that due to thedip the fractional power guiding in the core decreases andthe compensator is operated much near the cutoff wavelength.The influence of the dip on the dispersions of the compensatorwith a = 2, a / b = 1.05, A = 2%, and b = 3.06 pm is shownin Fig. 7. The dip effect of increasing the dispersion valuesis the same as in the previous case a = co).Note that thedispersion discussed in Figs. 3, 5, and 6 is the summation ofthe waveguide dispersion calculated by the approach describedin Section I11 and the material dispersion obtained from theempirical Sellmeier equation with 13% GeO2 doping in silicafor the core and pure silica for the cladding [151. Comparedwith the waveguide dispersion, the contribution of the materialdispersion is relatively small for the e-core Compensator whichis operated near the cut-off wavelength.

V . CONCLUSIONBased on the full-wave vectorial formulations of combining

the surface integral equation method SIEM) and the finite-element method FEM), novel approach for determiningthe dispersion characteristics of the dual-mode elliptical-corefibers with arbitrary index profile in the core is proposedwithout resorting to numerical differentiation. Most computing



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1933WU AND CHANG DISPERSION CHARACTERISTICS OF DUAL-MODE ELLIPTICAL-CORE OPTICAL FIBERS

2n 1

2n 2

IIIIIIII

-1 0 1R

Fig. 5.index dip.The index profile of the step a co)distribution with the Gaussian

320 -340 p = d 10.3p .2 a / b 1.05

-d .0 b = 2.1368360 I1.53 1.535 1.54 1.545 1.55 1.555 1.56

P mFig. 6. Dispersion versus wavelength for three different dip parameters forthe case of step index profile. For each pair of curves the upper curve is forz polarization and the lower curve is for y polarization for each case withdifferent dip parameter.

efforts are spent in solving the propagation constant ,b andthe guided mode fields H , and Hy.Once they are obtained,only some matrix manipulations are required to obtain thefirst and second derivatives of the propagation constant withrespect to V. By this proposed approach, the influence of therefractive index distribution parameters, such as the differenta-powers and the index dip effect, on the dispersions of thefour nondegenerate LPll modes for the e-core compensatorsare investigated as numerical examples. The vectorial fieldpatterns of these four modes for the e-core fibers have alsobeen presented.

REFERENCES[ l ] J. N. Blake, B. Y. Kim, and H. J. Shaw, Fiber-optic modal couplerusing periodic microbending, Opt. Lett. , vol. 11, pp. 177-179, 1986.[2] W. V. Sorin, B. Y. Kim, and H. J. Shaw, Highly selective evane-sent modal filter for two-mode optical fibers, Opt. Lett., vol. 11,pp. 581-583, 1986.[3] B. Y Kim, J. N. Blake, H. E. Engan, and H. J. Shaw, All-fiberacousto-optic frequency shifter, Opt. Lett., vol. 11, pp. 389-391, 1986.

I

p = d .3D . d I a 2p - d - 0 b 3.0Clum

1.53 1.535 1.54 1.545 1.55 1.555 1.56-300

a (w)Fig. 7. The same as in Fig. 5 but for the case of parabolic index profile.

[4] C. D. Poole, J. M. Wiesenfeld, A. R. McCormick, and K. T. Nelson,Broadband dispersion compensation by using the higher-order spatialmode in a two-mode fiber, Opt. Lett., vol. 17, pp. 985-987, 1992.[5] A. M. Vengsarkar, W. A. Reed, and C. D. Poole, Effect of refractive-index profiles on two-mode optical fiber dispersion compensators, Opt.Lef t . , vol. 17, pp. 1503-1505, 1992.[6] C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar,Optical fiber-based dispersion compensation using higher order modesnear cutoff, J. Lightwave Technol., vol. 12, pp. 1746-1758, 1994.[7] R. A. Sammut, Analysis of approximations for the mode dispersion inmonomode fiber, Electron. Lett., vol. 15, pp. 590-591, 1979.[8] W. L. Mammel and L. G. Cohen, Numerical prediction of fibertransmission characteristics from arbitrary refractive-index profiles,Appl . Opt . , vol. 21, pp. 699-703, 1982.[9] E. K. Sharma, A. Sharma, and I. C. Goyal, Propagation characteristicsof single mode optical fibers with arbitrary index profiles: A simplenumerical approach, IEEE J. Quantum Electron., vol. QE 18, pp.1484-1489, 1982.[lo] A. Sharma and S. Banerjee, Chromatic dispersion in single mode fiberwith arbitrary index profiles: A simple method for exact numericalevaluation, J. Lightwave Technol.,vol. 7, pp. 1919-1923, 1989.[ll] H. Y. Lin, R.-B. Wu, and H.-C. Chang, n efficient algorithm fordetermining the dispersion characteristics of single-mode fibers, J.Lightwave Technol., vol. 10, pp. 705-711, 1992.[12] C.-C. Su, A combined method for dielectric waveguides using thefinite-element technique and the surface integral equations method,ZEEE Trans. Microwave Theory Tech., vol. MlT-34 , pp. 1140-1 146,1986.[13] L. Eyges, P. Gianino, and P. Wintersteiner, Modes of dielectricwaveguides of arbitrary cross sectional shape, J. Opt. Soc. Am., vol.69, pp. 1226-1235, Sept. 1979.[I41 N. Morita, A method extending the boundary condition for analyzingguided modes of dielectric waveguides of arbitrary cross-sectionalshape, IEEE Trans. Microwave Theory Tech., vol. MTT-30, pp. 6-12,1982.[I51 J. W. Fleming, Material dispersion in lightguide glasses, Electron.Lett., vol. 14, pp. 326329, 1978.

Tzong-Lin Wu (S95) was born in Hsinchu, Tai-wan, Republic of China, on June 23, 1969. He re-ceived the B.S.E.E. and Ph.D. degrees from NationalTaiwan University, Taipei, Taiwan, in 1991 and1995, respectively.His research interests include thetheory of optical fibers and modeling of fused fibercouplers.



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Hung -chun C ha ng (S78-M83) was born inTaipei, Taiwan, Republic of China, on February8, 1954. He received the B.S. degree from NationalTaiwan University, Taipei, R.O.C., in 1976, and theM.S. and Ph.D. degrees from Stanford University,Stanford, CA, in 1980 and 1983, respectively, allin electrical engineeringFrom 1978 to 1984, he was with the Space,Telecommunications, and Radioscience Laboratoryof Stanford University. In August 1984, he joinedthe faculty of the Electrical Engineering Department

of National Taiwan University, where he is currently a Professor. He servedas Vice-chairman of the EE Department from 1989 to 1991. Since August1992, he has been Chairman of the newly established Graduate Instituteof Electro-Optical Engineering at the same university. His current researchinterests include the heory, design, and application of guided-wave structuresand devices for fiber optics, integrated optics, optoelectronics, and microwaveand millimeter-wave circuits.Dr. Chang is a member of Sigma Xi, the Phi Tau Phi Scholastic HonorSociety, the Chinese Institute of Engineers, the Optical Society of America,ChindSRS (Taipei) National Committee (a Standing Committee member since1988) and Commission H of the US. ational Committeeof the InternationalUnion of Radio Science (URSI). In 1987, he was among the recipients of theYoung Scientists Award at the URSI XXIInd General Assembly. In 1993, hewas one of the recipients of the Distinguished Teaching Award sponsored bythe Ministry of Education of the Republic of China.

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