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ADP013470TITLE: Solution of the General Helmholtz Equation Starting fromLaplace' s Equation

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187 ACES JOURNAL, VOL. 17, NO. 3, NOVEMBER 2002

SOLUTION OF THE GENERAL HELMHOLTZ EQUATION STARTINGFROM LAPLACE'S EQUATION

Tapan K. Sarkar and Young-seek ChungDepartment of Electrical Engineering and Computer Science

121 Link HallSyracuse University

Syracuse, New York 13244-1240.Phone: 315-443-3775, Fax: 315-443-4441.

Email: [email protected], Homepage: http://web.svr.edv/~-ksarkar

Magdalena Salazar PalmaGrupo de Microondas y Radar, Dpto. Senales, Sistemas y Radiocomunicaciones

ETSI Telecomunicacion, Universidad Politecnica de MadridCiudad Universitaria, 28040 Madrid, Spain.

E-mail: salazar@ gmr.ssr.upm.es

ABSTRACT: In this paper we illustrate how to solve conventional methods, it has been found that fine gridsthe general Helmholtz equation starting from and a large number of elements must be employed toLaplace's equation. The interesting point is that the get satisfactory accuracy [3]. This requires largeHelmholtz equation has a frequency term whereas computer core storage, and more computational timeLaplace's equation is the static solution of the same especially for the iteration scheme of the nonlinearboundary value problem. In this new formulation the Poisson's problem where the value at each grid pointfrequency dependence is manifested in the form of an needs to be updated at each step of the iteration.excitation. A new boundary integral method for Further, the BIM formulations are in most casessolving the general Helmholtz equation is developed, limited to homogeneous Helmholtz equation and tiedThis new formulation is developed for the two- closely to the particular problem at hand [6]. In thisdimensional Helmholtz equation with the method of paper, a simple approach to solve the homogeneousmoments Laplacian solution. The main feature of this and nonhomogeneous Helmholtz equations isnew formulation is that the boundary conditions are proposed. The technique is based on the computationsatisfied independent of the region node of Laplacian potential by the method of momentsdiscretizations. The numerical solution of the present (MoM) [9], without resorting to different formulationsmethod is compared with finite difference and finite using Hankel functions as it is commonly done in BIMelement solutions of the same problem. Application of [10]. Besides its generality to solve Laplace's,this method is also presented for the computation of Poisson's, and Helmholtz's equations in one singlecut-off frequencies for some canonical waveguide code implementation, the present method willstructures. considerably reduce the number of domain grids

compared to the finite difference methods and doesI. INTRODUCTION not require any interpolation. The accuracy of the

MOM solution from this new formulation will beThe two dimensional Helmholtz's equation is an compared with the solutions of fine difference methodimportant equation to be solved in many numerical (FDM) and finite element method (FEM), using theelectromagnetics problems such as waveguide related ELLPACK implementation [4].problems and appears in a variety of physicalphenomena and engineering applications, such as, II. MATHEMATICAL FORMULATIONacoustic radiation [1], heat conduction [2], and waterwave propagation [3]. In semiconductor device Consider the following two-dimensional ellipticmodeling, Helmholtz's equation arises frequently as equation for a smooth function T defined in a 2-Dan intermediate step in the solution of the nonlinear region defined by 91 which is bounded by a contour CPoisson's problem. To solve these problems diverse so thatnumerical methods have been reported which includefinite difference [4], finite element [5], and boundary 2integral methods (BIM) [6-8]. Using these V'P(x,y)+X(x,y)I(x,y)=F(x,y) (1)

1054-4887 © 2002 ACES

SARKAR, CHUNG, PALMA: SOLUTION TO HELMHOLTZ EQUATION FROM LAPLACE'S EQUATION 188

where X and F are known functions in the domain 91. 0p (x, y) =

The general form of (1) includes, as specializations,the following cases: 1 ) Kdy,

1) Laplace's equation, with k = 0 and F = 0 - O(x, y')n ,'K

2) Poisson's equation, with ?, = 0 and F • 0. 2r (xx'W +(y-y)

3) Helmholtz's equation, with X • 0 and F # 0. (8)So only one general formulation addresses the solutionto all of the three cases above. In addition we illustrate where the spatial sets (x, y) and (x', y') denote thehow to use the frequency independent solution of the spatial coordinates of the field and source coordinates,Laplace's equation to solve the general Helmholtz respectively, and K is an arbitrary constant. Here,equation. On the contour C the boundary condition can f n ( K ) represents the value of the scalar potential 0be of Dirichlet, Neumann, or mixed type, as given by at infinity for the two dimensional case. For the 3Dthe general form case, this term does not exist, as the potential at

infinity is zero. Hence the Green's function is simply

aI + DT =y (2) 1/R instead of the natural log function. The value ofan ( the parameter K is chosen to be 100 for the 2-D

problem as the reference potential at infinity is not

where cx, j3, y, are known spatial functions and aT/Dn zero but finite.

represents the normal derivative. It is implied that a An expression similar to (8) can be derived to

consistent set of boundary conditions has been chosen approximate the Laplacian potential 0h,. The potential

for the problem. oh can be assumed to be produced by some equivalentThe proposed scheme to solve the given boundary sources consisting of electrical charges, a located on

value problem starts by assuming the term XkP to be a the contour C [12]. Then the potential 0/, at any point

known function, and including it with the given (x, y) can be obtained from o(x, y) using the followingexcitation function F, and thereby reducing (1) to the integral [9]familiar Poisson's equation

0, (X, y) =V2'P(x, y) = -E(x, y) (3)

with .I f (x',y')nK dl'2)r C •(-'2+yy)

E(x, y) = 2(x, y)'f(x, y) - F(x, y). (4) (x- +(yy(9)

The solution to Poisson's equation in (3) can be where 1' is the arc length on the contour C.expressed as The boundary condition of the homogeneous

potential 0/, is obtained from (2) and (5) asI1"1= Oh"b~p(5)

where Oh is the solution to the homogeneous Poisson's ah+/8- --- kXoAP + f/J -). (10)equation (Laplace's equation) a

V2Oh = 0 (6) It can be seen that (6) along with the boundarycondition of (10), constitute a similar boundary-value

and Op is the particular integral, i.e., problem. A similar problem was addressed in [9]almost thirty years ago.

V 20P =-e(x, y) (7) This completes the formulation of the problem.

The characteristic features of this formulation are:a. The frequency term appears in an explicit

Here we use the particular solution of the Poison's form.equation given by [12] b. The Green's function and the unknowns are

independent of frequency. The Green'sfunction is in fact the static Green'sfunction.


Next we illustrate how to solve these coupled ci (x, y)=equations both for the homogeneous part and theparticular integral. tr K (15)f Vgn _X) , I'_ )

III. NUMERICAL SOLUTION USING AcJ (XX')2+(2- )2THE METHOD OF MOMENTS (MOM)

Using (4), (11) and (14), the Helmholtz potential T atby ord10wer toysolve the aboove equaion vent. an arbitrary field point (x, y) can now be expressed asby (8)-(10) we employ the Method of Moments. q~,y

To evaluate the above integral in (8) we P(x, y) =

divide the domain 9t into N sub-regions. The M N

midpoint coordinates of each of the sub-regions A91i OiCi (x, y) + Y( - F;) ai (x, y)are denoted by (x2i, Y2i). Then the potential op at the i = 1 i =

field point (x, y) is approximated by (16)

N where for simplicity we have used the followingp (x, y) -- 8(X 2 i, Y2 i )ai (x, y) (11) abbreviations in (4):

i =1

where ?i = X (x 2i, Y2i)

ai (x, y) =Fi = F (x2i, Y2i)Ti = T (x2i Y2i)

1 [fffn .K_ dx'dy, The unknown function P(x, y) in (16) of Helmholtz2r A (x - ) T +(y-y,)2 equation can now be evaluated once the unknown

(12) terms, ai and TPi, are determined. Next a system oftwo matrix equations is derived and solved for the

Hence it is assumed that O(x, y) is constant within unknowns a, and Ti. The first matrix equation of this

each sub-region A91i and is equal to the value 9 system is readily obtained by satisfying (16) at the

(X2i,Y2i). midpoints (x2/, Y2j) of each of the N sub-region A91i.

Next we show how to numerically evaluate (9). Using matrix notation, we obtainHere, we take recourse to MOM. We expand thesurface unknown a using a pulse expansion. For [ij=Lpjj][' +Lqjj 11&T-F;] (17)purposes of illustration and simplicity let us choosepoint matching. Through the use of a pulse-expansionand point-matching techniques we will solve the where P1 i -- c•(x 21, Y2j), and qjj = ai(x2.i, Y2j).

and oin-mathin tehniqes e wll slvetheThe second matrix equation is now obtained bypresent problem given by (9). Furthermore, if the enforcing the boundary condition of (10) as follows.contour C is segmented by M straight lines of length enforcine boundary c o 1 0 as follows.AC, between points i and i + 1, then a can be We define j- (xj,y1 ), j= 1, 2 ... , M, tobetherepresented by the step approximation midpoints of ACj. The boundary conditions areM

"= oiP (l) (13) enforced at each Pj . Substitution of (11) and (14) into

i= 1 (10) gives the following set of equations

where Pi() is a pulse function equal to 1 on ACi and M N

zero elsewhere and a, is its unknown amplitude. = (1 - F;)bjiSubstituting (13) into (9), we obtain an approximation (18)

for 0h 1, 2,...,MM

•h(X, y) --- OiCi (X, y) (14)(14)1where 1j = (j ) and

where Wji Ci Ci1 (19)an ((Xy)1


(" ~ai"• illustrated through numerical examples in thei= I (20) next section.

ba l 5) Once Tj. is determined at the grid points,

Equation (18) can now be written in matrix notations using Gaussian elimination for instance the

as potential at any other point is obtained usingordinary matrix multiplications as shown by(17). No interpolations are needed.

=wji ][i ] rj -bji L[,bTij [P-F] (21) Next, we illustrate these points through somenumerical examples.

Observe that (17) and (21) form a system of two III. STATIC APPLICATIONSequations in two unknowns aFi and Ti, from whichthey can be solved. We use (21) to obtain an Example 1-Water Wave Propagation: As a firstexpression for a and then substitute it into (17), and example, we consider the problem of water waveafter simple matrix manipulations we obtain the propagation in a rectangular basin 100m x 100m [3].following equation for Ti Denoting by TP the water evaluation, then the wave

propagation is governed by (1) with F = 0, and X = k2[A][Ti]=[B] (22) where k is the wave number. The boundary conditions

where used are displayed in Figure 1. The solution by MOM/l 1 is compared to the solutions obtained by finite

[A]=[I]+ ([P]i [wji [bji -[qji[A difference method (FDM) and finite element method(FEM) for various mesh sizes inside the domain. The

(23) minimum number of nodes required for each methodto converge to the exact solution at any arbitrary pointis optimized. Figure 2 compares the solutions

[B] = (Pji1Wji1 j+ Pjil computed along the line x = 90 by the Method of(24) Moments utilizing (lxl), (2x2) and (4x4) grids.

_[wj,|-,ibj, lqjil,]) ] .Figure 3 shows the solutions obtained by the finite-[w1.]

1 [. b g.,[qI. ])[.] element method utilizing (5x5), (1OxlO), and (20x20)grids. Figure 4 shows the results obtained by the finite

and [I] denotes the N x N identity matrix, difference method utilizing (5x5), (15x15), andFrom the last three equations some general (35x35) grids. In all these figures, the solid line

comments can be made: represents the exact solution. Referring to these1) Analytic expressions can easily be derived figures, one can easily observe that a considerably

for the evaluation of all the terms of the smaller number of nodes is employed with the presentmatrices [A] and [B] that include the integrals method than with the conventional methods. The(12) and (15). This is true at least for the 2-D minimum number of the domain nodes required tocase. obtain a satisfactory accuracy using MOM for the

2) The matrix elements [wji], which are a part of present method is 16 (4x4), compared to 400 (20x20)the system matrix of the moment method, are for FEM or up to 1000 nodes for FDM.similar to the matrix elements obtained in [9].For different Helmholtz problems with Example 2-MOST Modeling: The nonlineardifferent boundary conditions, [wji] remains Poisson's equation plays a key role in numericalunchanged. This observation is important in modeling of semi-conductor devices. Many importantan iteration scheme where the boundary characteristics of VLSI devices can be extracted fromconditions are kept the same. the solution of Poisson's equation. The most common

3) Problems with multiple right-hand sides are approach to the numerical solution of the nonlinearsolvable with minimum additional Poisson's equation is based on the application ofcomputation time since the function F in (17) Newton's method to simultaneous discretizedappears only as a term of matrix [B] in (24). equations [151. This approach often requires large

4) The domain and the contour discretization etons [1r T is apoh ofte rqie larschemes are totally independent. This feature storage especially for fine meshes, as is the case forcan offer the flexibility to handle boundary two-dimensional modeling of the MOST [14].discontinuities without the need for excessivedomain grid generations. This will be


4¢ =f,(x)C ...... ', ...-.. -... -..-.......... . .. • ... 7.. ..

49~ ~ ~ ~ ~ ~ ~ ~~~6-4 = rxmdr .TeFM ouinfo xml .FA '!-

0 0.3 0 u (,A0. 0.1 0.t 0.7 00 . 9

='X Figure 4. The FDM solution for Example 1.

cos(ax) if 20<x<80

where f (x) -2 if X<80 In this section the MOM is applied to solve thehe{ inonlinear Poisson's equation that arises in the MOST

1 if x > 80 modeling. We consider the MOST structure of Figure

f(x) = sin(ay), f0 (x) = cos(ax), and a=21r 5 made on a p-type substrate with doping NA. Underthe low current approximation the potential 1 is

Figure 1. Physical structure and the associated boundary conditions governed by the Poisson's equation [16]

associated with Example 1.

.. .... . .......... D2T DT

a. n. . + a y 2 = e - e - N ( 2 5 )

where T=(I/VT is the normalized potential, VT is the., thermal potential, ni is the intrinsic carrier

S... '"..00".-,-0concentration, and LD is the Debye length.

-•.•,--- ,• o:_ o• :, ;_ ,:................=.V -V •0 01 012 0. 0. i . 0.7 0.8 09 4D= V, -Vlb

Figure 2. Solution obtained by the present method. D =VS - V, tox 9ox = V, NT.

O. oxo D 90

.L YFigure 5.Tepyia tutr n h soitdbudr

"t 10. 0 - •...... 0: 0.4...... 4........ .i ....... 0".0 ......... . Fi.......lgure 5. The physical structure and the associated boundary

,,. 10•)o.0 condition for MOST modeling.

Figure 3. The FEM solution for Example 1. The boundary conditions adopted along the edgesof the device are the same as those used by [16], [121,and are displayed in Figure 5. On the oxide-semiconductor interface the following boundarycondition is assumed

VG-D-V 3 =Es (26)toxY


where So,, and S, are, respectively, the oxide and The accuracy of the solution obtained by thesilicon permittivities, VFB is the flat band voltage, and present technique is demonstrated by comparing itt0, is the oxide thickness. with the FDM (5-point) solution. For both methods,

To solve the above elliptic problem we proceed MOM and FDM, we let the values of the electric

first by dividing the boundary into M segments. Finer potential to be updated at each mesh point by means of

segments are used on the top edge of the device to an explicit formula, that is, without the solution ofhandle its boundary discontinuities. Using the simultaneous algebraic equations [15]. We assign T =boundary conditions, and the technique described in 0, as an initial guess for all the inner nodes. Once the[9], matrix [wji] is computed, then inverted and stored. convergence is attained for the domain nodes, then theIndependently, the base region is also divided into N electric potential at any other point in the device isfinite elements or cells. We seek to determine the determined by a matrix multiplication as shown inelectric potential at each midpoint of the cells using (17), and without the need of any interpolation. For(25) which is nonlinear. To solve it we set up an numerical computations, the following data were used:iterative procedure, based on Newton's linearization The thickness of the oxide layer was to, = 0.5 gim, the[15]. At the k-th iteration we replace the right-hand flat band voltage is VFB = -1 V and the doping profileside of (25) by its Fourier expansion about 1 k. Then is assumed to be uniform with NA = 1018 cm-', ni =the following Helmholtz's equation is obtained 1.5x1010 cm-3, thermal potential, VT = 0.0258 V, and

R = 0.1. The results of the computations are shown ina

2 T(k+) a2 T(k+l) (k) (kFigure 6, where the distribution of the electric

+ -G )G( )T) + potential at the thermal equilibrium is plotted alongx) •ydifferent lines parallel to the x-axis. The solid lines

FT t(k)• represent the solution obtained using the FDM and (o)symbol is reserved for the solutions obtained by the

(27) present method. Close agreements can be observedwhere F and G are given by between the two methods. However, for FDM 721

nonuniform mesh points are employed to reduce the

F( (k)• =total number of nodes {for uniform meshes over 4900(70x70) nodes ought to be used}. Finer meshes were

_ ( ) ((k) T NA chosen in the depletion region and near the junctionsL(T) +1e- + +(T -)e to reproduce accurately the fast variation of the

D ni electric potential [16] and the surface discontinuitiesof the potential; whereas for MOM, the number of the

(28) uniform meshes was 529 (23x23) or 400 for

nonuniform cells. The number of iterations required

G -J•)=n(e-T(') + ) (29) for the convergence was 51 with the present technique2 as compared to 108 iterations with FDM to reach thepoint at which the absolute maximum between two

The iteration scheme starts by taking some initial subsequent iterations was less than 0.05 (tolerance).guess value for Ti(0) so that (27) can be solved for the Next we apply this method to the solution of the

first approximation T("). Then V1(I) is used to find the cutoff-frequencies of various waveguides.

second approximation Tp Z. The procedure is repeated 5" - r

until the norm Tk+l) - Ttk) is less than a

desired tolerance. However, this iteration scheme "4

often diverges [13], [15], and some damping factor oz • _

was found necessary to improve the convergence ofthe iterative solution. At the beginning of the k-thiteration step, the following formula was used for all -... ._.__ --inner nodes - ', o ., -6's

T (+1) = (1 - R)T(k - 1) + R 1 j(k) (30) Figure 6: The distribution of the electric potentials along the lines

parallel to the x-axis.

where R ( < 1) is the relaxation factor.


IV. SOLUTION OF THE GENERAL Comparing (31) and (32) with (1), and alsoHELMHOLTZ EQUATION FOR the boundary conditions of the TM, and TE, cases with

HOMOGENEOUSLY FILLED those of the general equations, we can draw theWAVEGUIDES following analogies as outlined in Table I.

By examining (32) thatIn waveguides, solution of the Helmholtz [B] = [pji][wji]-'[yj]

equation determines the electromagnetic fieldconfiguration within the guides. It is convenient to +([Pj11][wji]-[bji]-[qji])[F] _divide the possible field configurations within thewaveguides into two sets, namely TM waves and TE it can be inferred that [B] = 0, since F = 0 and y=0 forwaves, each of which is governed by similar TM, and TE, cases.Helmholtz equations. In the case of TM, (22) reduces to the form

If we consider a waveguide in which thedirection of propagation of the wave is along the z- [A] - [Ezi] = 0 (33)direction, then the Helmholtz equations are as follows.

TMz Case (H, - 0): In the case of TE, (22) reduces to the form

V 2 Ez (x, y) + (C)2 pE - K2z )Ez (x, y) = 0 (31) [A]. [Hzi] 0 (34)

with the appropriate boundary conditions E, = 0, on Here again, Ezi and Hzi refer to the values of E, and H,the conductor walls and for the at the midpoints of the subregions of the discretized

TE, Case (Ez - 0): waveguide cross section.

For (33) and (34), nontrivial solutions existV2 Hz (x, y) + - K )Hz (x, y) =0 (32) for [Ei] and [Hj,] only if the matrix [A] is singular.

The condition for nontrivial (i.e., nonzero) solutions to

with appropriate boundary conditions aHl/ln = O,on exist for [Ezi] and [Hzi] it is essential that

the conductor walls where DHl/Jn represents thenormal derivative. Here, det[A] = 0 (35)

Ez z-component of the electric field; where det[A] stands for determinant of [A]. We knowH, z-component of the magnetic field; from (23) that(o angular frequency = 21cf,f frequency of interest; [A] = ([Pji]I [wji ]'[bi]_[qi +[]p. permeability of the homogeneous

medium;E permittivity of the homogeneous and we also know that for the cases of the TMz and

medium; TEz wavesK, propagation constant in the z-

direction. 2 = cae/- K'. (36)

Table ISpecial cases of the General Helmholtz Equation

General Equation (3) TMz Equation (1) TEz Equation (2)V )g 2 2 ).H2=V2+kTJ=F 2Ez+( F- -t Kz ). Ez= 2Hz+((co gtE - Kz HZ=

T EZ Hz22 2k F V,(0) 2gj.--Kz VO)Lt - KZ

F 0 0•T +P'a /an=7 Ez =0 aHlz In=0

7 0 0


Hence, given the frequency at which the Helmholtz Therefore, the propagation constants of differentequation is to be solved, det [A] would be a function of modes in the waveguide are given by the following:Ký, the roots of which will provide the values of Kz for F )2which det [A] = 0. Once these K, values are known, For (K >0,the eigenvector of [A] corresponding to the minimum 2 1eigenvalue gives the nontrivial solutions for [Eki] and KEpropagating modes[Hi,] in case of TMz and TEz cases, respectively.

Once [Ez,] and [/-j] are determined at the grid (42)points, using Gaussian elimination for instance, the 2values of E, and Hz at any other point can be obtained For (KZ) <0,using ordinary matrix multiplications, as explained in

K ' vIZ- nonpropagating modes

IVa. Calculation of Propagation Constants for (43)Different Waveguide Modes

Results of propagation constants of various modes in aIt is evident that for the existence of nontrivial rectangular waveguide computed by this method are

(nonzero) solutions for [Eki] and [Hj], it is necessary shown next.that (35) be satisfied. Let us define a matrix [Z] suchthat IV b. Calculation of Cutoff Frequencies for

Different Waveguide Modes[Z] = ([j]wi] [j][j]. (37)

The cutoff frequencies for the various propagatingmodes in the waveguide are given byHence, (35) becomes

det([Z] [•j] + [I]) = 0 (38) fci V (o' 2p - (K)-2 (44)

27twhich can be rewritten as where fi =- cutoff frequency of the ith mode. Here,

1) velocity of light in the homogeneous medium

ZlI =0. (39) -1/V-9 and it can be deduced from (41) that

c _ -1

Equation (39) is similar to the characteristic equation EVi[zI (45)

of matrix [Z], with its eigenvalues given by -1/fi. i = 1, 2,..., NKnowing that Xi above =-0o3 2 - Kz2 for TM- and

TE, cases, it can be concluded that Using (45) in (44), we find the cutoff frequencies forthe first N propagating modes as

-1 1 ~ VZ(K)2 _oj2,ue = EVi[zl' j ) -1('(40) f;= ' i ,2...N

f=, 2... 1,~ry 2 -ix2 211 1 1

(46)

where KZ is the propagation constant of the ith mode The cutoff wave number kC of the ith mode can beand EVr[Z] - ith eigenvalue of [Z]. calculated from the cutoff frequency using the relation

kc' _ 2n. I f i = i1, 2,..... N

Equation (40) can be rearranged as V

(K')2 =a. •.+ EVZ (41) This method thereby provides a straightforwardSEVilzj "approach to find the cutoff frequencies (and, hence,cutoff wavenumbers) of any waveguide structure


without resorting to scanning over a wide range of was divided into 96 subregions and the boundary wasfrequencies, as is done in the Ritz-Galerkin and discretized into 112 subcountours. The maximumsurface integral-equation methods. matrix size involved in computations was 96 x 96.

The computational time involved in finding the cutoffV a. Results: Rectangular Waveguide wavenumbers of the first 96 modes in each case on a

Sun SPARC 10 workstations was 18 s.Consider a rectangular waveguide. For the

waveguide in Figure 7, the region was divided into100 subregions and the boundary was discretized into 1.0 cm96 subcontours. The maximum matrix size involvedin the computations was 100 x 100. Results have beendisplayed in Table II for the cutoff wave numbers ofthe first eight TMz/TEZ modes. The computational 0.5 cmtime involved in finding the cutoff wavenumbers ofthe first 100 modes on a Sun SPARC 10 workstationwas 16 s.

Figure 8. A single-ridge waveguide.

Table 1llE Cutoff wavenumbers for air-filled ingle-ridge wav guide

Mode Mode k, k, Diff.No. published computed %

(rad/cm) (rad/cm)1. T M 12.1640W 12.2338 0.57

m- 2. TM, 2.293817 12,4106 0.954cm 3. TM 13.9964'7 14.2152 1.56

Figure 7. A rectangular waveguide. 4. TM• 15.587117 15.8221 1.50

5. TEz 2.2566' 2.2688 0.54Table I 6. TE 4.9436 17 5.0149 1.44

Cutoff wavenumbers for air-filled rectangular wave uide 7. T E 6.518917 6.6289 1.68Mode Mode ke actual ký Diff. 8. TE 1 7.564217 7.7097 1.92No. (rad/cm) computed %

(rad/cm)1-0 TEZ 0.7857 0.7921 0.81 VI. CONCLUSION0-1 TEý 1.0476 1.0536 0.571-1 TE, 1.3095 1.3239 1.00 An efficient technique based on MoM formulation

TM, _ for solving a general Helmholtz equation starting from2-0 TE, 1.5714 1.5827 0.72 Laplace's equation is presented. The main feature of2-1 TE, 1.8886 1.9108 1.10 this new formulation is that the boundary conditions

TM, are satisfied independent of the discretizations of the0-2 TEZ 2.0952 2.1095 0.68 regions and the nodes. This feature was found1-2 TMZ, 2.2377 2.2610 1.00 especially useful when the boundary conditions have

TM _ discontinuities. Considerable reduction in the domain3-0 TE 2.3571 2.3896 1.30 grids is realized with the present method as compared

to the conventional methods such as finite differencemethod or the finite element methods. In addition, one

Vb. Single-Ridge Waveguide need not use a frequency dependent Green's functionwhich can reduce the computational cost significantly.

A single ridge waveguide is a popular means ofgetting higher bandwidth. The first four TMz and TEz VII. REFERENCESmode cutoff wavenumbers were computed for thesingle-ridge hollow waveguide shown in Figure 8. [1] L. G. Copley, "Fundamental results concerningResults have been displayed in Table III and compared integral representation in acoustic radiation," J.with published data. For the waveguide, the region Acoust. Soc. Am., Vol. 44, pp. 28-32, 1963.


[2] R. A. Altenkirch, M Rezayat, R. Eichhorn, and [15] W. F. Ames, Nonlinear Partial DifferentialF. J. Rizzo, "A study of heat conduction forward Equations in Engineering. New York:of flame in solids, by the boundary integral Academic, 1965.equation method," Trans. ASME Ser. C. J. Heat [16] J. D. Kendall and A. R. Boothroyd, "A two-Transfer, Vol. 104, pp. 734-740, 1982. dimensional analytical threshold voltage model

[3] M. Kawahara and K. Kashiyama, "Boundary for MOSFET's with arbitrarily dopedtype finite element method for surface wave substrates," IEEE Electron Device Lett., Vol.motion based on the trigonometric function EDL-7, pp. 401-403, 1986.interpolation," Int. J. Numer. Meth. Eng., Vol.21, pp. 1833-1852, 1985.

[4] J. R. Rice and R. F. Boisvert, Solving Elliptic Tapan Kumar SarkarProblems Using ELLPACK. New York: (http:llweb.syr.edu/-tksarkar)Springer-Verlag, 1985. received the B. Tech. degree from

[5] P. L. Arlett, A. K. Bahrani, and 0. C. the Indian Institute of Technology,Zienkiewicz, "Application of finite elements to Kharagpur, India, in 1969, thethe solution of Helmholtz's equation," Proc. M.Sc.E. degree from the UniversitylEE, Vol. 115, No. 12, pp. 1762-1766, 1968. of New Brunswick, Fredericton,

[6] M. Rezayat, F. J. Rizzo, and D. J. Shippy, "A Canada, in 1971, and the M.S. andunified boundary integral equation method for Ph.D. degrees from Syracuse University; Syracuse, Newclass of second order elliptic boundary value York in 1975.problems," J. Austral. Math. Soc. Ser., Vol. B- From 1975 to 1976 he was with the TACO25, pp. 501-517, 1984. Division of the General Instruments Corporation. He

[7] K. Nagaya, and T. Yamaguchi, "Method for was with the Rochester Institute of Technology,solving eigenvalue problems of the Helmholtz Rochester, NY, from 1976 to 1985. He was a Researchequation with an arbitrary shaped outer Fellow at the Gordon McKay Laboratory, Harvardboundary and a number of eccentric inner University, Cambridge, MA, from 1977 to 1978. He isboundaries of arbitrary shape," J. Acoust. Soc. now a Professor in the Department of Electrical andAm., Vol. 90, pp. 2146-2153, 1991. Computer Engineering, Syracuse University; Syracuse,

[8] P. J. Harris, A boundary element method for NY. His current research interests deal with numericalHelmholtz equation using finite part solutions of operator equations arising inintegration," Comp. Meth. Appl. Mech. Eng., electromagnetics and signal processing with applicationVol. 95, pp. 331-342, 1992. to system design. He obtained one of the "best solution"

[9] R. F. Harrington, K. Pontoppidan, P. awards in May 1977 at the Rome Air DevelopmentAbrahamsen, and N. C. Abertsen, "Computation Center (RADC) Spectral Estimation Workshop. He hasof Laplacian potentials by an equivalent source authored or coauthored ten books including the latestmethod," Proc. lEE, Vol. 116, No. 10, pp. ones "Iterative and Self Adaptive Finite-Elements in1715-1920, 1969. Electromagnetic Modeling" and "Wavelt Applications in

[10] C. A. Brebbia, The Boundaor Element Method for Engineering Electromagnetics" which were published inEngineers. New York: Wiley, 1978. 1998 and 2002, respectively by Artech House, more than

[11] R. F. Harrington, Field Computation by Moment 210journal articles and numerous conference papers andMethods. New York: Macmillian, 1968. has written chapters in 28 books. The book on "Smart

[12] E. Arvas, R. 1. Turkman, and P. S. Antennas" will be published by John Wiley in MarchNeelakantaswamy, "MOSFET analysis through 2003.numerical solution of Poisson's equation by the Dr. Sarkar is a registered professional engineermethod of moments," Solid-State Elect., Vol. in the State of New York. He received the Best Paper30, No. 12, pp. 1355-1361, 1987. Award of the IEEE Transactions on Electromagnetic

[13] G. De Mey, "The boundary element method for Compatibility in 1979 and in the 1997 National Radarmodeling semiconductor components under low Conference. He received the College of Engineeringcurrent approximation," in Proc. Int. Conf Research Award in 1996 and the chancellor's citationSimulation of Semiconductor Devices and for excellence in research in 1998 at SyracuseProcesses, Pineridge Press, 1984, pp. 261-266. University. He was elected Fellow of the IEEE in 1991.

[14] W. Fichtner, D. J. Rose, and R. E. Bank, He was an Associate Editor for feature articles of the"Semiconductor device simulation," IEEE Tran. IEEE Antennas and Propagation Society Newsletter, andElectron. Devices, Vol. ED-30, No. 9, pp. 1018- he was the Technical Program Chairman for the 19881030, 1983. IEEE Antennas and Propagation Society International

Symposium and URSI Radio Science Meeting.


Currently he is a distinguished lecturer for the IEEE integrated circuits; and network and filter theory andAntennas and Propagation society. He is on the editorial design. For a number of times she has been a visitingboard of Journal of Electromagnetic Waves and professor at the Department of Electrical EngineeringApplications and Microwave and Optical Technology and Computer Science, Syracuse UniversityLetters. He has been appointed U.S. Research Council (Syracuse, New York, USA).Representative to many URSI General Assemblies. He She has authored three scientific books and awas the Chairman of the Intercommission Working total of 15 contributions for chapters and articles inGroup of International URSI on Time Domain books published by international editorial companies,Metrology (1990-1996). Dr. Sarkar is a member of 25 papers in international journals, and 125 papers inSigma Xi and International Union of Radio Science international conferences, symposiums, andCommissions A and B. He received the title Docteur workshops, plus a number of national publications andHonoris Causa from Universite Blaise Pascal, Clermont reports. She has delivered a number of invitedFerrand, France in 1998 and the Friends of Clennont presentations, lectures, and seminars. She has lecturedFerrand award in 2000. in several short courses, some of them in the frame of

European Community Programs. She has participatedYoung-Seek Chung received the in 45 projects and contracts, financed by international,B.S., M.S. and Ph.D. degrees in European, and national institutions and companies.Electrical Engineering from She has been a member of the Technical ProgramSeoul National University, Seoul, Committee of several international symposiums and

p i t, Korea, in 1989, 1991, and 2000, has acted as reviewer for different internationalrespectively, scientific journals, symposiums, and editorialFrom 1991 to 1996, he was with companies. She has assisted the Comisi6nthe Living System Laboratory, Interministerial de Ciencia y Tecnologia (National

LG Electronics, From 1998 to 2000, he was a Board of Research) in the evaluation of projects. SheTeaching Assistant in Electrical Engineering at the has also served in several evaluation panels of theSeoul National University. Since 2001, he has been Commission of the European Communities. In the pastworking at the Syracuse University with Dr. Tapan K. she has acted several times as topical editor for theSarkar. His current interests are the numerical analysis disk of references of the triennial Review of Radioand the inverse scattering using the time domain Science. She is a member of the editorial board oftechniques. three scientific journals. She is a registered engineer in

Spain. She is a senior member of the Institute ofMagdalena Salazar-Palma was Electrical and Electronics Engineers (IEEE). She hasborn in Granada, Spain. She served as vicechairman and chairman of IEEE Spainreceived the degree of Ingeniero Section MTT-S/AP-S (Microwave Theory andde Telecomunicacitn and the Techniques Society / Antennas and PropagationPh.D. degree from the Society) Joint Chapter and chairman of IEEE SpainUniversidad Politicnica de Section. She is a member of IEEE Region 8Madrid (Madrid, Spain), where Nominations and Appointments Committee, IEEEshe is a Profesor Titular de Ethics and Member Conduct Committee, and IEEE

Universidad of the Departamnento de Sefiales, Sistemnas Women in Engineering Committee (WIEC). She isy Radiocomnunicaciones (Signals, Systems and acting as liaison between IEEE Regional ActivitiesRadiocommunications Department) at the Escuela Board and IEEE WIEC. She has received twoTicnica Superior de Ingenieros de Telecomunicaci6n individual research awards from national institutions.of the same university. She has taught courses onelectromagnetic field theory, microwave and antennatheory, circuit networks and filter theory, analog anddigital communication systems theory, numericalmethods for electromagnetic field problems, as well asrelated laboratories.

She has developed her research within theGrupo de Microondas y Radar (Microwave and RadarGroup) in the areas of electromagnetic field theory,computational and numerical methods for microwavestructures, passive components, and antenna analysis;design, simulation, optimization, implementation, andmeasurements of hybrid and monolithic microwave

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