boundary element analysis of dielectric waveguides

T. Lu and D. O. Yevick Vol. 19, No. 6 /June 2002/J. Opt. Soc. Am. A 1197

Boundary element analysis of dielectricwaveguides

Tao Lu

Department of Electrical and Computer Engineering, Queen’s University, Kingston, Ontario K7L 3N6, Canada

David O. Yevick

Department of Physics, University of Waterloo, Waterloo, Ontario N2L 3G7, Canada

Received October 16, 2001; revised manuscript received December 4, 2001; accepted December 7, 2001

We apply the boundary element method to the analysis of optical waveguides. After summarizing constantand linear element algorithms for both two- and three-dimensional simulations, we introduce a new recursiveseries procedure for constructing the diagonal matrix elements. We then demonstrate that our method can beemployed to minimize the reflectivity of optical waveguide antireflection coatings with both straight andangled facets. © 2002 Optical Society of America

OCIS codes: 130.0130, 220.0220, 230.0230, 060.0060.

1. INTRODUCTIONIn this paper, we apply the boundary element method(BEM) to the reflectivity of antireflection-coated (AR-coated) optical waveguide facets. While initially appliedin mechanics,1 the algorithm has been frequently used tosolve microwave problems in metallic and subsequentlyoptical waveguides2 as well as dielectric waveguide scat-tering problems.3,4 In the last case, the procedure hasbeen found to be highly efficient for integrated opticalstructures with piecewise-homogeneous refractive-indexprofiles.

The BEM employs the Green’s theorem to relate thefield distribution inside a homogeneous waveguide regionto the field on the region boundaries and is valid for one-way, bidirectional, and wide-angle propagation problems.In the BEM, only the boundaries and not the entire re-gion is discretized. Further, exact boundary conditionscan in many cases be imposed for open-boundary prob-lems. Below, we present a discussion of both two-dimensional (2-D) and three-dimensional (3-D) BEM for-malism that includes several new procedures forcomputing the required field quantities at boundarypoints.

2. TWO-DIMENSIONAL BOUNDARYELEMENT METHOD FORMALISMIn this section, we consider waveguides with piecewise-homogeneous refractive-index distributions that are in-variant along the transverse ( y) direction and furtherspecify the z axis as the propagation axis. Consider Fig.1, which displays a piecewise-homogeneous dielectricstructure in which G is the region boundary, lj is thelength of element j, r is an observation point, and r8 is afield point lying on the boundary G. In the case in whichobservation point r also lies on the boundary, an artificialhalf-circle Ge with infinitesimal radius centered at the ob-

0740-3232/2002/061197-10$15.00 ©

servation point should be added to avoid the singularity ofthe boundary integral.1 We adopt the convention thatgiven the normal vector

n 5]f~x, z !

]xi 1

]f~x, z !

]zk (1)

of the boundary f(x, z) 5 0, the region that the normalvector points toward will be designated the ‘‘right’’ (R)boundary region while the ‘‘left’’ (L) region is the opposingregion. We also define the direction of the boundary asthe direction associated with a 90° counterclockwise rota-tion of the normal vector. A given electric or magneticfield component U(r) 5 U(x, z) at any observation pointr then satisfies the integral equation

gU~r! 1 EG2Ge

U~r8!]G~r8,r!

]n8dl8

5 EG2Ge

G~r8, r!]U~r8!

]n8dl8 1 S~r!. (2)

The coefficient g 5 u/(2p) is dependent on the position ofthe observation point r such that g 5 1/2 if the observa-tion point lies on a smooth boundary while g 5 1 if thepoint is inside a homogeneous region. Additionally,

G~r8, r! 5 G~r! 51

4jH0

~2 !~kr!,

]G~r8, r!

]n5

]G~r!

]n5

jk

4H1

~2 !~kr!cos /~ r, n !. (3)

Here r 5 ur8 2 ru, r 5 (r8 2 r)/ur8 2 ru, Hm(2) is the

mth-order Hankel function of the second kind, and k isthe wave vector for the specified region. To obtain thesource function S(r) from the known input field, we firstdenote the input field incident on source boundary Gs by

2002 Optical Society of America

1198 J. Opt. Soc. Am. A/Vol. 19, No. 6 /June 2002 T. Lu and D. O. Yevick

Fig. 1. 2-D regions and integration over region boundaries. The input field is incident on boundary Gs .

Us(r8). The total field distribution on Gs can be writtenas the sum of an incident and a reflected field:Utotal(r8) 5 Us(r8) 1 U(r8), where U(r8) at the sourceboundary is the reflected field distribution. The result-ing source function may then be expressed as

S~r! 5 2EGs

Us~r8!]G~r8, r!

]n8dl8

1 EGs

G~r8, r!]Us~r8!

]n8dl8. (4)

A. Scalar and Vectorial FormulationsAs the boundary node i shown in Fig. 1 is shared by bothsides of the boundary, the field component on node i sat-isfies Eq. (2) on each side of the boundary. In the scalarformulation as introduced in Ref. 5, the field componentand its directional derivative are both assumed to be con-tinuous across each boundary. This assumption, whichleads to a scalar formalism, is highly accurate for weaklyguided waveguides where the field components and theirdirectional derivatives are nearly continuous acrossboundaries between media with different refractive indi-ces.

For waveguides with a high index contrast, we nowpresent a more accurate vectorial formulation that incor-porates the electric field discontinuities at the bound-aries. After introducing a boundary mesh and applyingcontinuity relations at the boundary points, we can re-write Eq. (2) in the following form for both scalar and vec-torial formulations:

g rf~ri! 5 2(jÞi0

EG j

]Gr~rj 2 ri!

]njr f~rj!dl

2 (jE

G j

Gr~rj 2 ri!aFr c ~rj!dl 1 SR~ri!,

g lf~ri! 5 2(jÞi

EG j

]Gl~rj 2 ri!

]njl f~rj!dl

1 (jE

G j

Gl~rj 2 ri!aFl c ~rj!dl 1 SL~ri!.

(5)

In Eq. (5), the superscripts l and r denote the field com-ponent and its directional derivative on the left (l) andright (r) sides of the boundary node designated by i, andkl and kr below are similarly wave vectors in the left andright regions. Also, SR(ri) and SL(ri) are the excitationfunctions5 in the left and right boundary regions. In oursubsequent discussion, the subscript F in aF

l,r is TM forthe TM mode, TE for the TE mode, and SC for the scalarformulation, with

aTMl,r 5 kl,r

2 , aTEl,r 5 aSC

l,r 5 1. (6)

Finally, for TE fields,

f~r! 5 Eyl ~r! 5 Ey

r~r!,

c ~r! 5]Ey

l ~r!

]n5 2

]Eyr~r!

]nr , (7)

while for TM fields,

f~r! 5 Hyl ~r! 5 Hy

r~r!,

c ~r! 51

kl2

]Hyl ~r!

]nl 5 21

kr2

]Hyr~r!

]nr . (8)

Thus f and c are independent variables continuousacross the boundary. While, in the scalar formalism, fand c can be any field component and its directional de-rivative, the other field components in the vectorial for-malism must be computed from the Hy and Ey field dis-tributions.

B. Constant Element AlgorithmThe constant element algorithm assumes a uniform fielddistribution within each boundary element.1 The fieldwithin each meshed element is then represented by thefield at a central node point i as in Fig. 1. Equations (5)can in this case be expressed as

1

2f i 5 2(

jÞif jE

G j

]Gr~rj 2 ri!

]njdl

2 (j

c jaFr E

G j

Gr~rj 2 ri!dl 1 SRi ,


1

2f i 5 2(

jÞif jE

G j

]Gl~rj 2 ri!

]njdl

1 (j

c jaFl E

G j

Gl~rj 2 ri!dl 1 SLi . (9)

For N boundary elements, this yields 2N linear equationswith 2N unknowns. Defining

V 5 S f1

f2

]

fN

c1

c2

]

cN

D , S 5 S SR1

SR2

]

SRN

SL1

SL2

]

SLN

D , (10)

we can rewrite Eq. (9) in matrix form as

HV 5 S. (11)

The matrix H is composed of submatrices Gl,r and Hl,r:

H 5 FHr Gr

Hl 2GlG . (12)

The components of G and H are given in terms of a dimen-sionless coordinate j, which varies from 21 to 11 on ele-ment j, by

gijl,r 5 aF

l,rE21

11 1

4jH0

~2 !~kr!djlj

2,

hijl,r 5 E

21

11 1

4jkH1

~2 !~kr!@cos /~ r, n !#djlj

2. (13)

In this expression, rj 5 rj0 1 jj(lj/2), rj0 is the mid-point of element j, j is the vector tangent to the element,and lj is the element length. The variable r is related toj according to r 5 urj 2 ri0u. For i Þ j, we have toevaluate the above integrals by Gaussian quadrature. Ifthe nodes i and j are identical, hii

l,r 5 1/2. giil,r is an im-

proper integral (Ref. 6, p. 374) whose integrand becomesinfinite at j 5 0. The series form of the Cauchy principalvalue gii

l,r is then constructed by expanding the Hankelfunction3:

giil,r 5

1

2jk (k51

`~21 !k

~2k 1 1 !22k~k! !2 S kli

2 D 2k11

3 F1 2 i2

pS ln

kli

41 gE 2

1

2k 1 12 (

m51

k 1

m D G1 F1 2 i

2

pS ln

kli

42 1 1 gED G kli

2, (14)

in which gE 5 0.5772157 is the Euler constant.

C. Linear Element AlgorithmIn the linear element algorithm, f(j) and c (j) are linearlyinterpolated within each element from the field strengthat the two edge nodes. If we denote the two nodes at the

element vertices by 1 and 2, so that the values of f(j) andc (j) at the ends of the element are f1 , f2 , c1 , and c2 ,we have1

f~j! 51 2 j

2f1 1

1 1 j

2f2 ,

c ~j! 51 2 j

2c1 1

1 1 j

2c2 . (15)

The Green’s function can then be expressed as

grf i 1 (e51

N

~h1e,r h2

e,r!S f1e

f2e D

5 2(e51

N

~ g1e,r g2

e,r!S c1e

c2e D 1 SRi ,

g lf i 1 (e51

N

~h1e,l h2

e,l!S f1e

f2e D

5 (e51

N

~ g1e,l g2

e,l!S c1e

c2e D 1 SLi . (16)

The matrix elements in this equation are

h1e

h2e J 5 E

21

1 1

2~1 6 j!

jk

4H1

~2 !~kr!@cos /~ r, n !#djle

2,

g1e

g2e J 5 E

21

1 1

2~1 6 j!

1

4jH0

~2 !~kr!djle

2. (17)

If node i does not belong to element e, the matrix ele-ments are obtained by Gaussian quadrature. If node ibelongs to element e, g1

e and g2e are the Cauchy principal

values of the improper integrals (17), since the integrandis infinite at node i. As an element has two nodes at bothends, two cases exist. In the first of these, node i is ver-tex 1 of element e, so that r 5 (le/2)(j 1 1) and h1

e

5 h2e 5 0. The series expressions for g1

e and g2e are

then3

g1e 5 (

k51

`~21 !k

~2k 1 1 !22k~k! !2 ~kle!2k11

3 F1 2 i2

pS ln

kle

21 g 2

1

2k 1 1 D 12i

p (m51

k 1

mG1 F1 2 i

2

pS ln

kle

22 1 1 g D G

3 ~kle! 2kle

4kjH1

~2 !~kle! 11

2kp,

g2e 5

1

4kjH1

~2 !~kle! 21

2k2lep. (18)

If node i is instead vertex 2 of e, we solve for g18 and g28simply by interchanging the values of g1

e and g2e above.


3. THREE-DIMENSIONAL BOUNDARYELEMENT METHOD FORMALISMIn the 3-D case, analytic expressions for the Green’s func-tion and its directional derivative are

G~r! 51

4prexp~2jkr!,

]G~r!

]n5 2

1 1 jkr

4pr2 exp~2jkr!cos /~ r, n !. (19)

As a result, U(x, y, z) can be expressed throughout theproblem domain as the following integral of the field dis-tribution over the region boundary (Se denotes an infini-tesimal hemisphere around observation points on the re-gion boundary):

gU~r! 1 ES2Se

U~r8!]G~r8, r!

]n8ds8

5 ES2Se

G~r8, r!]U~r8!

]n8ds8 1 S~r!. (20)

Here S(r) is a source function whose derivation is similarto that in the 2-D cases, except that line integrals are re-placed by surface integrals. The constant g 5 1 when Uis located inside the homogeneous region. As in the 2-Dcase, we define n 5 ¹f(x, y, z) to be the normal vector ofboundary surface f(x, y, z) 5 0. We further define theregion toward which the normal vector points as the‘‘right’’ region of the boundary and the opposing region asthe ‘‘left’’ region. Further, if U is located on the boundaryof the problem domain S, we have g 5 V/(4p), while ifthe node is on a smooth section of the boundary, g5 1/2.

As in the 2-D case, the scalar formulation can be em-ployed when the field component U and its directional de-rivative can be approximated by functions that are con-tinuous across the region boundaries, as is usually thecase for weakly guided waveguides. To improve the nu-merical accuracy, we here present an alternative semivec-torial formulation that more accurately describes the con-tinuity relations across refractive-index discontinuities.To simplify the formulation, we assume that all theboundaries are parallel to either the xy, xz, or yz plane,which is generally the case for the AR-coated waveguideend facets to be considered in the results section (Section4). For other refractive-index distributions, such as cor-ner reflectors, an exact full-vectorial formulation wouldgenerally be more appropriate. Consider accordingly aquasi-TE input field Ey , and denote the field componentat the boundary where the normal vector of the boundaryn(r)'y by Eyi(r) and the perpendicular field componentby Ey' . From the continuity relation at the boundary,we have

f i 5 Eyil 5 Eyi

r ,

f' 5 kl2Ey'

l 5 kr2Ey'

r ,

c 5]Ey

l

]nl 5 2]Ey

r

]nr . (21)

A similar analysis applies to the quasi-TM input mode ifwe identify Ex as the field component U, change the sub-script y to x, and interchange i and ' in Eqs. (21). As inthe 2-D case, we find

g raFr ~r!f~r! 5 2(

jÞi0

ESj

]Gr~r!

]njr aF

r ~rj!f~rj!ds

2 (jE

Sj

Gr~r!c ~rj!ds 1 SR,

g laFl ~r!f~r! 5 2(

jÞi0

ESj

]Gl~r!

]njl aF

l ~rj!f~rj!ds

1 (jE

Sj

Gl~r!c ~rj!ds 1 SL, (22)

where aF again represents aSC for scalar formulation, aTEfor TE semivectorial formulation, and aTM for TM semi-vectorial formulation. For these three choices,

aSCl,r 5 1,

aTEl,r 5 H 1, n~r!'y

1

kl,r2 , n~r!iy

,

aTMl,r 5 H 1, n~r!'x

1

kl,r2 , n~r!ix

. (23)

To evaluate the integral over each element, we approxi-mate the field distribution over the entire boundary ofeach element by interpolating its values on the gridpoints with a particular shape function. The choice ofthe shape functions largely determines the accuracy ofthe BEM formulation. As in the 2-D case, we examinetwo interpolation strategies associated with constant andlinear elements.

A. Constant Element AlgorithmThe constant element algorithm again assumes that thefield distribution is invariant within each element. De-fining f i and c i as the values of f and c at the centralnode i of element e, which is placed at the center of ele-ment e such that f(re) ' f i and c (re) ' c i , then yields

1

2aF

r ~ri!f i 5 2(jÞi

f jESj

aFr ~rj!

]Gr~r!

]njr ds

2 (j

c iESj

Gr~r!ds 1 SRi ,

1

2aF

l ~ri!f i 5 2(jÞi

f jESj

aFl ~rj!

]Gl~r!

]njl ds

1 (j

c jESj

Gl~r!ds 1 SLi . (24)

We set g 5 1/2, since node i is taken to be the observationpoint inside the planar element, which spans a solid angleV 5 2p. With i P $1 ,..., N%, there are a total of 2N in-


dependent linear equations of forms (24) with 2N un-knowns. We can rewrite Eqs. (24) in matrix form as inthe 2-D case. The matrix elements, obtained from Eqs.(24), are given by

hijl,r 5 2aF

l,rESj

1 1 jkr

4pr2 exp~2jkr!@cos/~ r, n !#ds

gijl,r 5 E

Sj

1

4prexp~2jkr!ds. (25)

If node i differs from node j, the above integrals can benumerically evaluated by, e.g., Gaussian quadrature.For diagonal matrix elements, we have hii

l,r 5 aFl,r/2, since

n' r for a planar element. However, we now introduce amore accurate and efficient method for evaluating gii

l,r

based on a recursive procedure for expanding the expo-nential term into a series that is integrated term by term.That is, consider first the triangular boundary element eshown in Fig. 2(a), and divide the element e into threesubtriangles that share a common vertex at node i. Thematrix element gii

l,r can then be expressed as the sum ofthree integrals over the subtriangles:

giil,r 5 (

k51

3

gk 5 (k51

3 EDk

1

4prexp~2jkr!ds. (26)

Integrating over r and expanding the exponential terminto a Taylor series yield

g1 5 EDr1

1

4prexp~2jkr!ds

5 E0

u01

duE0

r0~u! 1

4prexp~2jkr!rdr

5 21

4pjk (n51

`~2jkr1 sin u1

1!n

n!pn~u1

1, u01!,

g2 5 21

4pjk (n51

`~2jkr2 sin u2

2!n

n!pn~u2

2, u02!,

g3 5 21

4pjk (n51

`~2jkr3 sin u3

3!n

n!pn~u3

3, u03!, (27)

where we have defined a new function

pn~u1 , u2! 5 E0

u2 du

sinn~u 1 u1!, (28)

which is evaluated recursively as

p0~u1 , u2! 5 u2 ,

p1~u1 , u2! 5 ln tanu1 1 u2

22 ln tan

u1

2,

pn~u1 , u2! 5cos u1

~n 2 1 !sinn21u1

2cos~u1 1 u2!

~n 2 1 !sinn21~u1 1 u2!

1n 2 2

n 2 1pn22~u1 , u2!. (29)

It can further be proved that for arbitrary u i , u j , and uk ,with u i 1 u j 1 uk 5 p, as in a triangular element,pn(u i , uk) 5 pn(u j , uk). As a result, we can simplify theseries solution for the linear algorithm discussed in Sub-section 3.B. A similar technique applies to quadrilateralelements as shown in Fig. 2(b). Arbitrary polygonal ele-ments can similarly be divided into simple triangles,where each triangle has a vertex on node i. Thus, in allsuch cases, gii

l,r can be evaluated as a sum of integralsover triangular elements.

B. Linear Element AlgorithmWe now discuss the linear interpolation of triangular ele-ments. We first introduce the dimensionless coordinates(h1 , h2 , h3), with 0 , h i , 1, through the formula r5 (k51

3 hkrk, where (r1, r2, r3) are the coordinates of thethree vertices of the triangular element. Clearly, onlytwo of these variables are independent, since h3(h1 , h2)5 1 2 h1 2 h2 . Additionally, r 2 r3 5 h1r13e131 h2r23e23 can be interpreted geometrically as additionof two vectors, shown in Fig. 3. Assuming a linear varia-tion for f and c on element e, we can then rewrite Eqs.(22) as

Fig. 2. Subdivision of different elements into simple triangles centered at A0 . (a) Subdivision of the triangular element into T1 , T2 ,and T3 and (b) subdivision of the quadrilateral element into T1 , T2 , T3 , and T4 .


graFr f i 5 2(

eÞi

N

~h1e,r h2

e,r h3e,r!S f1

e

f2e

f3eD

2 (e51

N

~ g1e,r g2

e,r g3e,r!S c1

e

c2e

c3eD 1 SRi ,

g laFl f i 5 2(

eÞi

N

~h1e,l h2

e,l h3e,l!S f1

e

f2e

f3eD

1 (e51

N

~1e,l g2

e,l g3e,l!S c1

e

c2e

c3eD 1 SLi , (30)

where the coefficients h1e , h2

e , h3e , g1

e , g2e , and g3

e are cal-culated from

S h1e

h2e

h3eD 5

aF

4pE

De

21 2 jkr

r2 exp~2jkr!@cos /~ r, n !#

3 S h1

h2

h3

D ds,

S g1e

g2e

geeD 5

1

4pE

De

1

rexp~2jkr!S h1

h2

h3

D ds. (31)

The value of r again satisfies r 5 ure(h1 , h2 , h3) 2 riu.If node i is not a vertex of element e, the integrals in the

expressions for hvk

e and gvk

e (vk 5 1, 2, 3) can be computednumerically by Gaussian quadrature. If node i is insteadat the vertex vi of element e, then hvk

e 5 gaF for vi 5 vk

and hvk

e 5 0 for vi Þ vk . gvk

e can be evaluated through aseries expansion of the integrand. Without loss of gener-ality, we assume that node i is vertex 3 of the element e,so that in the (h1 , h2 , h3) coordinate system, ri

5 (0, 0, 1)T. Introducing polar coordinates according toFig. 3, we have

r

sin u5

h1r13

sin~Q3 2 u!5

h2r23

sin u. (32)

The quantity g3e can then be expressed as a function of g1

e

and g2e according to

Fig. 3. Linear element algorithm for the triangular element;any triangular element can be mapped in (h1 , h2) coordinates.

g3e 5

1

4pE

De

exp~2jkr!

r~1 2 h1 2 h2!ds

5 g0e 2 g1

e 2 g2e . (33)

After expanding the exponential term into a Taylor seriesand integrating term by term, we obtain

g0e 5

1

4pE

De

exp~2jkr!

rds

51

4p~2jk !(n51

`~2jkr13 sin Q1!n

n!pn~Q1 , Q3!

51

4p~2jk !(n51

`

an~13! . (34)

The coefficients an(lm) , l, m P 1, 2, 3 and l Þ m, are de-

fined as

an~lm ! 5

~2jkrlm sin Q l!n

n!pn~Q l , Qm!. (35)

The quantities

pn~Q1 , Q3! 5 E0

Q3 du

sinn~0 1 Q1!, (36)

can then be obtained recursively as described in Subsec-tion 3.A. From the expression for (h1 , h2) in polar coor-dinates, the integral of g1

e becomes

g1e 5

1

4pE

De

exp~2jkr!

rh1ds

51

4pE

0

Q3

duE0

~r13 sin Q1!/sin~Q11u!

3exp~2jkr!

r

r sin~Q3 2 0 !

r13 sin Q3rdr. (37)

This yields a series solution for g1e that we can write as,

noting that g2e is identical to g1

e after interchanging(Q1 , Q2) and (r13 , r23),

g1e 5

r23 cos Q2

4pr12~2jk !(n51

` n

n 1 1an

~13!

1sin Q1 exp~2jkr13! 2 sin Q2 exp~2jkr23!

4pr12~2jk !2

1sin Q2 2 sin Q1

4pr12~2jk !2 ,

g2e 5

r13 cos Q1

4pr12~2jk !(n51

` n

n 1 1an

~13!

1sin Q2 exp~2jkr23! 2 sin Q1 exp~2jkr13!

4pr12~2jk !2

1sin Q1 2 sin Q2

4pr12~2jk !2 ,


g3e 5 g0

e 2 g1e 2 g2

e

51

4p~2jk !(n51

`

an~13!F1 2

n

n 1 1cos~Q1 2 Q2!G .

(38)

If node i is vertex 2 of element e, a series solution for gke is

then obtained by selecting (h1 , h3) as the independentvariables and interchanging indices as 3 → 1, 1 → 2, and2 → 3. If node i is instead located at the third vertex ofelement e, (h2 , h3) are taken as the independent vari-ables and the indices are interchanged according to 2→ 1, 3 → 2, and 1 → 3.

4. NUMERICAL RESULTSWe now apply the above formalism to several representa-tive waveguide reflection problems. In two dimensions,we consider reflection from a slab waveguide facet withsingle and double AR-coated layers. In the 3-D case, weinstead apply the semivectorial formalism to rectangularwaveguide facets. Here we examine the modal reflectiv-ity with different single-layer AR-coated waveguides andobserve the expected difference between the TM and TEmodal reflectivities. We subsequently investigate themode conversion that results when the end facet of thewaveguide is tilted.

A. Two-Dimensional Step-Index Antireflection-CoatedWaveguideWe first employ the BEM to analyze 2-D symmetric slabwaveguides with AR-coated end facets. We will examinethe modal reflectivity as a function of the coating thick-ness and the angle described by the end facet with respectto the propagation axis. We will compare our simulationresults with the results of a finite-difference time-domain(FD-TD) method simulation with the perfectly matchedlayer (PML) boundary condition.7

We consider first the single-mode waveguide of Fig. 4,in which the refractive indices of the guiding region andthe cladding region are set to ng 5 3.6 and nc 5 3.564,respectively. Further, the incident wavelength is l05 1.55 mm, and the waveguide width is D 5 1.46 mm.The width of the calculation window is 8 mm. The bound-ary is discretized with grid points that are spaced a dis-tance of 0.072 mm and 0.02 mm along the x and the z di-rection, respectively. In Fig. 4(a), a single AR-coatedlayer is deposited on the end facet. The refractive indexof the coating is nAC 5 Aneff 5 1.8938, which ensures animpedance match to the air region, where neff is the con-ventional effective index of the TE0 waveguide mode.The coating thickness L that yields minimum reflection isl0 /(4nAC).7 We launch the TE0 mode at the front end ofthe waveguide (Z 5 0 mm) and measure the modal reflec-tivity at the same end. The input mode profile is com-puted by the conventional finite-difference method.4 InFig. 5, we display the power reflectivity as a function ofcoating thickness. Clearly, the lowest power reflectivityof 243.34 dB is obtained at a thickness L 5 0.206 mm, inclose agreement with published results.7 An identicalcalculation performed by using a standard 2-D FD-TDsoftware package required 7115 s on a 400-MHz AMD-k6system as opposed to 105 s with our boundary elementprogram. This calculation has been performed for boththe constant and linear element algorithms. In the caseof the linear element algorithm, we apply a nonuniformgrid to minimize the error associated with the fact thatthe directional derivative of U is undefined at the wave-guide corners. The power reflectivity as a function ofgrid point spacing as calculated with the BEM and FD-TDmethods is shown in Fig. 6. As shown in the figure, thelinear element algorithm with uniform grid spacing con-verges faster than the constant element algorithm, whilethe nonuniform grid scheme for the linear element algo-rithm converges most rapidly. As expected, the linear el-

Fig. 4. 2-D slab waveguides: (a) single-layer AR coating and (b) double-layer AR coating.


ement algorithm with nonuniform grid spacing yields thebest results of the three algorithms.

Introducing an angled end facet of the waveguide di-rects the reflected field away from the propagation axis,

Fig. 5. Modal reflection versus coating thickness for a single-layer AR-coated slab waveguide as calculated with both the BEMand the FD-TD method incorporating the PML boundary condi-tion.

Fig. 6. Discretization error: power reflectivity versus gridpoint spacing dx for the constant BEM and for the linear BEMwith uniform and nonuniform grids.

which decreases the modal reflectivity of the fundamentalmode. Figure 7 shows the reflected modal power as afunction of the tilt angle. The reflectivity minimum is ob-tained at a tilt angle of 7°, as predicted by Ref. 8. At thisangle, the reflected modal power is 10 dB less than thatfor the corresponding untilted facet.

A double-layer AR coating can also be employed toachieve a lower reflectivity over a large wavelength regioncompared with that for a single-layer AR coating, as dem-onstrated in Fig. 8. The reflective indices of the two coat-ing layers in this example are chosen such that nAC15 nAC2Aneff ' 2.76494, nAC2 5 1.46, L1 5 l0 /(4nAC1)

Fig. 7. Power reflectivity versus tilt angle for a single-layer AR-coated waveguide.

Fig. 8. Modal reflectivity versus wavelength for single-layer anddouble-layer AR-coated slab waveguides as calculated with boththe BEM and the FD-TD method with the PML boundary condi-tion.


Fig. 9. 3-D view of a single-layer AR-coated rectangular waveguide.

5 0.138 mm, and L2 5 l0 /(4nAC2) 5 0.266 mm in orderto ensure an impedance match between the waveguideand the air interface.

B. Three-Dimensional Rectangular WaveguidesWe now study the modal reflectivity of AR-coated 3-Drectangular waveguides. We will also illustrate the modeconversion that results at a tilted output facet.

We first consider step-index rectangular waveguideswith the profile of Fig. 9. The guiding region of all thewaveguides below has a core refractive index ng 5 3.52,while the cladding region index is nc 5 3.20. The widthand the thickness of the waveguides are 0.4 mm3 0.4 mm, 0.4 mm 3 1.5 mm, and 0.15 mm 3 3.0 mm,respectively.

The modal reflectivity of both TE and TM input modeswith AR coating will now be computed by using the semi-vectorial formulation. The discretization parameters areDx 5 0.1 mm, Dy 5 0.1 mm, and Dz 5 0.025 mm. Eachcalculation required approximately 21 min on our com-puter system. In this case, we could not perform a CPUtime comparison, as we were unable to obtain appropriate

Fig. 10. Power reflectivity versus coating thickness in the rect-angular and square waveguides.

3-D FD-TD software. Figure 10 shows the modal reflec-tivity versus the coating thickness for the 0.4-mm3 0.4-mm and 0.4-mm 3 1.5-mm waveguide geometries.The refractive index of the AR coating has been set tonAC 5 1.92. The input mode profile is calculated by aboundary integral method similar to that of Ref. 9. Weobserve that the computed TE00 and TM00 modal reflec-tivities are identical in this square waveguide case, as re-quired by the profile symmetry. The results for the re-flectivity in the case of a 0.4-mm 3 1.5-mm waveguidegeometry are, however, clearly different for TE and TMmodes. Our values for the two polarizations are in closeagreement with those of Ref. 10. Figure 11 shows the re-flected field distribution in the 0.4-mm 3 0.4-mm squarewaveguide excited by its TE00 mode. This calculationyields a power reflectivity of 6.6 3 1024, which is identi-cal to the modal reflectivity of the TM00 mode because ofthe symmetric property of square waveguides. As the re-flected field is computed only 0.1 mm from the coating in-terface, there is a significant contribution from the radia-tion modes in the reflected field profile. However, inlonger waveguides, these backward radiation modes de-cay rapidly, leaving only the guided mode.

As noted in Subsection 4.A, tilting the facet anglecauses the reflected power to shift from the center of thewaveguide, thus decreasing the overlap between the in-put and reflected fields.10 Figure 12 shows the resultingdecrease in reflectivity with increasing tilt angle for the0.15-mm 3 3.0-mm rectangular waveguide introduced inthe preceding paragraph. The magnitude of the reduc-tion generally increases with increasing guide width.This decrease is of special interest in this waveguide,since reflectivity minima exist at tilt angles of approxi-mately 7°, as predicted in Ref. 8. Figure 13 displays themode conversion to the higher-order waveguide modes re-sulting from the tilted output facet.

5. CONCLUSIONSIn this paper, we have simulated the reflection of an elec-tric field from coated and uncoated waveguide facets with


Fig. 11. Reflected field in a square waveguide (0.4 mm3 0.4 mm). The refractive index of the AR coating has been setto nAC 5 1.92, and the semivectorial formalism is used.

Fig. 12. Power reflectivity versus tilt angle for rectangularwaveguides with different geometries.

Fig. 13. Modal reflectivity spectrum of a 0.15-mm 3 3.0-mmrectangular waveguide (TE mode).

the aid of the boundary element method. Our calcula-tions are based on solving a banded sparse-matrix equa-tion for the reflection coefficient in the framework of theBEM. The off-diagonal matrix elements in this equationwere obtained through Gaussian integration, while a new,efficient series solution was found for the diagonal matrixelements in both two and three dimensions. This methodwas applied to both constant and linear elements for sca-lar and semivectorial electric fields.

While our results show that the BEM is an accurate,efficient, and versatile method for simulating wavepropagation problems in dielectric waveguides consistingof a few homogeneous layers, the method is generally im-practical for continuously varying refractive-index distri-butions. The extension of our BEM formulation tosuch index distributions through integration with othermode-solving procedures, as well as to quadratic or evencubic elements, would be of obvious interest. A full vec-torial procedure would also be worthy of futureinvestigation.

ACKNOWLEDGMENTSSupport for this work was provided by the CanadianInstitute for Photonic Innovations, Nortel Networks,and the National Sciences and Research Council ofCanada.

Corresponding author Tao Lu may be contacted bye-mail, [email protected].

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boundary element analysis of dielectric waveguides

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