Fast Calculations of Dyadic Green’s Functions

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    Journal of Computational Physics 165, 121 (2000)doi:10.1006/jcph.2000.6583, available online at http://www.idealibrary.com on

    Fast Calculations of Dyadic Greens Functionsfor Electromagnetic Scattering in a

    Multilayered MediumWei Cai and Tiejun Yu

    Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223E-mail: [email protected]

    Received November 15, 1999; revised April 28, 2000; published online November 3, 2000

    In thispaper, weintroduce a novel acceleration method for the calculation ofdyadicGreens functions for the mixed potential integral equation formulation of electro-

    magnetic scattering of scatterers embedded in a multilayered medium. Numericalresults are provided to demonstrate the efciency and accuracy of the proposedmethod. c 2000 Academic Press

    Key Words: electromagnetic scattering; dyadic Greens functions; mixed potentialintegral equations; Hankel transforms; fast algorithms; window functions.

    1. INTRODUCTION

    Integral equation formulation of electromagnetic scattering of conductive surfaces is apopular approach for many applications, including the parametric extraction for IC inter-connects and computer packaging simulations [1] and the performance of multilayeredantenna calculations [2]. The main advantage of the integral formulation [2, 3] is reduc-tion of unknowns and its exibility in handling very complex geometries of the scatteringsurface and the automatic enforcement of Sommerfeld exterior decaying conditions by the

    construction of proper Greens functions in the multilayered medium usually encounteredin those applications [4].

    However, the calculation of the dyadic Greens functions in a multilayered medium hasbeen one of the bottlenecks in the effort to increase the speed of integral equation methods;the other major bottleneck has been the solution of the impedance matrix resulting from theboundary element methods or method of moments [2]. Extensive research has been done onaccelerating the calculation speed of the dyadic Greens functions in a multilayered medium[3, 57]. The key difculty is the calculation of the Sommerfeld integral appearing in theHankel transformation, which denes the time domain dyadic Greens functions in termsof their Fourier spectral forms. Such difculty comes from several factors: (a) the existence

    1

    0021-9991/00 $35.00Copyright c 2000 by Academic Press

    All rights of reproduction in any form reserved.

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    2 CAI AND YU

    of surface wave poles in the spectral form of the Greens functions [3], which affects theselection of the contour path in the Sommerfeld integral; (b) the slow decay of the spectralGreens functions, especially when the eld location and source location are close, as in

    many of the VLSI and MMIC applications; and (c) the oscillatory behavior of the Hankeltransformation kernel (i.e., J 0( z)) and the highly oscillatory prole of the spectral Greensfunctions in a complex integration contour.

    Several methods have been proposed to address the difculties mentioned above. Forexample, the Prony type method originally proposed in [5], and later modied in [6], triesto extract the surface wave poles from the spectral form of the Greens function; thenexponential functions are used to approximate the remaining part of the Greens functionin the spectral domain. The main problem with this approach is the requirement of poleextraction. Techniques to extract the pole have been found to be very unstable and poleextraction is close to impossible when many layers are considered. Also, approximation of spectral Greens functions by exponentials via a Prony technique is not efcient for high-frequency problems or when the source and observation points are not on the same layer ina multilayered structure. Another approach [7] is to try to nd a steep descent path for theSommerfeld integration, which again is not easy for many layered media.

    In this paper, wepresent a novel method which utilizes a window function as a convolution

    kernel to the time domain Greens function. The effect of this window function is to modulatethe decay of the integrand in the Sommerfeld integration. The idea of convolution with awindow function in the time domain is equivalent to a low-pass lter in the frequencydomain used in signal processing and Gabor transformation in wavelet theory [8]. A similarapproach has been used to recover high-order approximations of discontinuous functionsfrom their Fourier coefcients in spectral methods [9]. The fast decay rate of the windowfunction in the spectral domain effectively creates a steep descent path for the integrationwithout the existence or the information of the location of possible steep descent paths forthe spectral Greens functions. Extensive numerical results have conrmed the effectivenessof this method, especially when the observation and source locations are close, whereas theSommerfeld integration will converge extremely slow. A comprehensive code WDS (WaveDesign Simulator) [10], which uses this window function technique, has been used to carryout 3-D full wave analysis of RF components and scattering of general objects in arbitrarymultilayered media.

    The rest of the paper is divided into the following sections: Section 2 gives a brief

    introduction to the dyadic Greens functions in a multilayered medium; in Section 3, Greensfunctions for the vector and scalar potentials used in the mixed potential integral equationsare described; Section 4 gives the window-function-based acceleration technique and errorestimations; Section5 provides several numerical examples to demonstrate the effectivenessand accuracy of the proposed methods; and a conclusion is given in Section 6. The appendixcontains technique derivations and proofs.

    2. DYADIC GREENS FUNCTION IN MULTILAYERED MEDIA

    In this section we present the setup for the dyadic Greens functions G E (r | r ), G H (r | r )in a multilayered medium. As weonly consider time-harmonic elds, a time-harmonic factore j t is assumed in all eld quantities. The medium considered is shown in Fig. 1. It is a strat-ied structure consisting of N +1 dielectric layers separated by N planar interfaces parallelto the x y plane of a Cartesian coordinate system and located at z = d i , i =0, 1, . . . , N .

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    CALCULATION OF DYADIC GREENS FUNCTION 3

    FIG. 1. A three-dimensional scatterer embedded in an N +1 layered medium.

    The medium of the mth layer is characterized by permeability m and permittivity m .Components of dyadic Greens functions G E (r | r ), G H (r | r ) represent the electromag-netic elds at location r excited by current Hertz dipole at r ; namely, G st E is the s -component

    of the electric eld generated by a t -oriented current Hertz dipole, s , t = x, y, z.

    2.1. Two-Dimensional Fourier and Hankel Transformations

    As the multilayered medium is radially symmetric in the x y plane, we can apply thetwo-dimensional Fourier transform to the Maxwell equations [11] to obtain the componentsof the dyadic Greens function in the Fourier transform (spectral) domain. The followingidentities will be used in nding the Greens function in time domain once the spectral formof the Greens function is obtained:

    F { f ( x, y)} = f (k x , k y) =1

    2

    f ( x, y)e j[k x x+k y y] dx dy (2.1)

    F 1{ f (k x , k y)} =f ( x, y) =1

    2

    f (k x , k y)e j[k x x+k y y] dk x dk y. (2.2)

    The Fourier integrals (2.1)(2.2) can be conveniently expressed in terms of the Hankel trans-form if f ( x, y) = f () is a radially symmetric function of . Introducing polar coordinatesin both the transform and space domains,

    x = cos , y = sin ,k x = k cos , k y =k sin ,

    where

    = x2 + y2, = arctan y x

    ,

    k = k 2 x +k 2 y, = arctank yk x

    ,

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    we can show that

    f (k ) = F { f () } =S0[ f () ](k ) (2.3)and

    f () = F 1{ f (k )} =S0[ f (k )](), (2.4)

    where the n th-order Hankel transform Sn [ f (k )] for integer n 0, is dened as

    Sn [ f (k )]()

    =

    0

    f (k ) J n (k ) k n+1 dk (2.5)

    and the roles of and k can be switched and where J n ( z) is the n th-order Bessel function.

    2.2. Dyadic Greens Functions G E , G H

    From the Maxwell equations [11], the electric eld is shown to satisfy the vector equation

    E k 2

    E = j J (2.6)where J is the current source. Ina source-free region, wehavethe followingvectorHelmholtzequation:

    2E +k 2E = j J . (2.7)

    The dyadic electric Greens function G ., t E is the solution to (2.7) when the source isa t -directed electric dipole, namely J (r ) = 1 j (r r ) t . The magnetic dyadic Greensfunction G H can be obtained by

    G H (r | r ) = 1

    j G E (r | r ). (2.8)

    3. DYADIC GREENS FUNCTIONS G A(r | r ) , GV (r | r )

    In this section, we describe the dyadic Greens functions for both vector potentials andscalar potentials. These are used for the mixed potential integral equation formulation(MPIE) of scattering problems [3, 4].

    In a MPIE formulation, the electromagnetic elds can be expressed in terms of a vectorpotential A and a scalar potential V e, i.e.,

    E = j A V e (3.1)H = 1 A , (3.2)

    where

    2A +k 2A

    1 A = J e (3.3)

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    CALCULATION OF DYADIC GREENS FUNCTION 5

    and where J e is the electric current source and k 2 =2 . Equation (3.3) simplies to thefollowing equation when is a constant:

    2

    A +k 2

    A = J e . (3.4)If the Lorentz gauge condition is used to relate V e to A , i.e.,

    A = j V e , (3.5)then we have

    2

    V e +k 2

    V e = 1

    e, (3.6)

    where e is the charge density related to the electric current J e by the continuity equation

    J e + j e =0. (3.7)The vector potentials used to represent the magnetic eld H in (3.2) are not unique

    and there are many ways of dening the potentials. Two of the most used approaches are

    Sommerfeld potentials [12] and transverse potentials [13, 14]. In formulating these poten-tials, we take into account that only two components of the magnetic eld are independent,thus only two components of these vector potentials are sufcient. In the Sommerfeld po-tential formulation, it is stipulated that the electromagnetic elds from a horizontal electricdipole (HED) can be represented by a horizontal component of A in the same direction of the HED and the z-component of A ; elds of a VED (vertical electric dipole) will be repre-sented by only the z-component of A . In contrast, for the transverse potential formulation,HED generated electromagnetic elds will be represented by two transverse componentsof A while a VED generated eld is represented by only the z-component of A . Otherpotentials include HertzDebye potentials [15].

    Sommerfeld potential. The dyadic Greens function G A for the Sommerfeld vectorpotential A [12] has the form

    G A = xG x x A + zG zx A x + yG yy A + zG

    zy A y + zG zz A z. (3.8)

    Transverse potentials. The dyadic Greens function G A for the transverse vector poten-tial A [13, 14] has the form

    G A = xG xx A + yG yx A x + xG

    xy A + yG

    yy A y + zG zz A z. (3.9)

    The scalar potential G V (r | r ) and the components in the Sommerfeld and transversepotentials can be obtained in the spectral domain; explicit formulas can be found in [3, 10].

    4. FAST CALCULATION OF DYADIC GREENS FUNCTIONS G A( , z; z ) , G V ( , z; z )

    The spectral components of the vector and scalar potential Greens function G A(, z; z )and G V (k , z; z ) in Section 3 can be shown [3, 10] to consist of terms such as

    jk x G (k , z; z ), jk yG (k , z; z ) (4.1)

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    where G (k , z; z ) is a function of k . Thus, the inverse Fourier transform of (4.1) can beevaluated as

    F

    1

    { j k x

    G (k , z; z )} =

    xF

    1

    {

    G (k , z; z )} =

    x S0(

    G (k , z; z )) (4.2)F 1{ jk yG (k , z; z )} =

    y

    F 1{G (k , z; z )} =

    yS0(G (k , z; z )). (4.3)

    The partial derivatives with respect to x or y can be approximated by a nite differenceformula with appropriate accuracy. Therefore, we only have to discuss an accelerationalgorithm for calculating the Hankel transformation

    G (, z; z ) = S0(G (k , z; z )) = C G (k , z; z ) J 0( k )k dk , (4.4)where the contour C should be in the rst quadrant of the complex wavenumber k spacefrom 0 to .

    The main difculty involving the calculation of function G ( , z; z ) via the Hankel trans-form (4.4) is the fact that the integrand decays very slowly when z = z , or z z (i.e., whenthe source point and the observation point are on or nearly on the same horizontal plane).

    In order to accelerate the calculation of G (, z; z ) when z z or z = z , we introduce thefollowing mth-order window function a ( x, y) = a () with a support size a :

    a () =1

    a

    2 m, if a

    0, otherwise .(4.5)

    We have the following lemma regarding the window function a ().

    LEMMA 1. For any cylindrical symmetrical function f (), we have the identity

    f ( x, y)

    a ( x, y) = S0[ f (k ) a (k )](), (4.6)where

    f (k ) = S0[ f () ](k )and

    a (k ) = S0[ a () ](k ).To recover the value of f ( x, y) from its Hankel transform, we also need the following

    result.

    LEMMA 2. Let f ( x, y) be a C 2 function. Then it can be shown that

    f ( x, y)

    a ( x, y) = M 0 f ( x, y) + M 2( f x x( , 0) + f yy( , 0)), (4.7)where 0 = x

    2 + y2 and

    M 0 = x2+ y2 a a ( x, y) dx dy = a 2

    m +1 M 2 = x2+ y2 a a ( x, y) x2 dx dy .

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    CALCULATION OF DYADIC GREENS FUNCTION 7

    Proof. The estimate (4.7) can be obtained by using a second-order Taylor expansionfor the function f ( x , y ) at point ( x, y) and using the fact that terms such as ( x x) and( y y) will vanish in the convolution with the radial symmetrical function a ( x, y) .

    As a result of (4.6) and (4.7), we can approximate f ( x, y) as

    f ( x, y) =1

    M 0S0[ f (k ) a (k )]() +O (a

    2) as a 0. (4.8)

    Applying (4.8) to G ( , z ; z ) , we obtain the following algorithm.ALGORITHM 1 (Fast algorithm for G ( , z; z )). For > a ,

    G (, z; z ) =1

    M 0W 0() +O (a

    2) as a 0, (4.9)

    where

    W 0() = S0[G (k , z; z ) a (k )](). (4.10) Remark 1. Algorithm 1 requires condition > a as otherwise the Greens function will

    be unsmooth and the estimate in (4.7) will not be valid.

    Therefore, to apply the approximation (4.8) to function G (, z ; z ) for a , we willrewrite G (, z ; z ) as

    G (, z; z ) =G 2( , z; z )/ r 2 , (4.11)

    where r

    = x2

    + y2

    +( z

    z )2 . From the singularity property of the vector and scalar

    potential Greens function [16], we can assume that G 2(, z; z ) = r 2G ( , z; z ) is a smoothfunction, and the approximation (4.8) thus can be used. Meanwhile, we have the followingidentity.

    LEMMA 3. Let G 2( , z; z ) = r 2G (, z; z ), with r = 2 +( z z )2. ThenG 2(, z; z ) a ( x, y) = r

    2W 0() 2 W 1() +W 2(), (4.12)

    where

    W 1() = S0[G (k , z; z )a (k )]() (4.13)W 2() = S0[G (k , z; z )a (k )]() (4.14)

    and

    a (k ) = S0[ a () ](k ) = a

    0 a () J 0(k ) d (4.15)

    a (k ) = S1[ a () ](k ) = a

    0 a () J 1(k ) 2 d (4.16)

    a (k ) = a

    0 a () J 0 k 3 d . (4.17)

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    Proof. The proof of (4.12) can be found in the Appendix.

    Combining (4.7) and (4.12), we have the following approximation scheme to the Greensfunction.

    ALGORITHM 2 (Fast Algorithm for G (, z; z )). If > 0, a 0, then

    G (, z; z ) =1

    M 0r 2[r 2W 0() 2 W 1() +W 2() ] +O (a

    2). (4.18)

    Next, we will address the issue of how to calculate W 0(), W 1() , and W 2() . First weneed the following spectral properties of a () and a (k ) and a (k ) .

    LEMMA 4. The Hankel transform a (k ) for a () and a (k ) and a (k ) have the following decaying properties , i.e. , as |k | :

    | a (k )| =o(|ak |m ) (4.19)|a (k )| =o(|ak |

    m+1) (4.20)

    |a (k )| =o(|ak |m ). (4.21)Proof. We will outline the proof of (4.19), the other two decay conditions can be shown

    similarly. Using the equivalence between the 2-D Fourier transform and the Hankel trans-form (2.3), we can rewrite a (k ) as

    a (k ) = F ( a )( k , 0) =1

    2

    a ( x, y)e jk x dx dy

    which, after integration by parts with respect to x variable m times, becomes

    a (k ) =(1)m

    2( j k )m

    m a m x

    ( x, y)e jk x dx dy =o1

    k m.

    To derive the estimate (4.20) for a (k ) , we rst use the identity J 0( x) = J 1( x) andintegrate by parts with respect to to rewrite a (k ) as

    a (k ) = a

    0() J 0(k ) d

    where () = a () +2 a () . Then, the same proof can be used to get (4.20).The fast decay condition of (4.19)(4.21) ensures that a short integration contour can be

    selected without sacricing the accuracy of approximation of Algorithms 1 and 2.

    4.1. Selection of Contour C and Window Order m and Support a

    It is well known that the spectral form G (k , z; z ) has surface wave poles which arelocated in the fourth quadrant of the complex k plane. Therefore, we should deform theintegration contour in the denition of W i (), i =0, 1, 2, to a complex contour which staysaway from the surface poles. A simple contour C , suggested in Fig. 2, consists of four

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    CALCULATION OF DYADIC GREENS FUNCTION 9

    FIG. 2. Contour C for Algorithms 1 and 2.

    straight segments L i , 1 i 4. The last segment L4 will be nite and is determined by thedecay properties of (4.19)(4.21).

    To maintain the second-order accuracy O (a 2) of Algorithms 1 and 2, we will choose the L4 portion of the contour C in Fig. 2 to satisfy the following minimum condition, whileusing the decay condition (4.19)(4.21):

    L4 C

    a 2(m s 3)1

    ms3. (4.22)

    Here we assume that for k [k max , ) , we have for some integer s and constant C

    |G (k , z; z )|C (k ) s .

    The window support a is the primary parameter to consider. It is determined by thesecond-order accuracy estimate (4.18) for the algorithm for the dyadic Greens function.Once a is selected, the length of the contour L4 should be given by (4.22).

    The order of the window function m in principle should be large to have faster decaysof a (k ) , a (k ) , and a (k ) according to (4.19)(4.21). However, for smaller valueof k , windows with lower order may have smaller magnitude in the spectral domain (seeFigs. 35). In our numerical tests, a window a () of order 5 (m =5) was a good overallchoice.

    4.2. Calculations of a (k ), a (k ), and a (k )

    In practice, these functions can be precalculated for a range of k as determined by theestimate (4.22). For small argument, they can be calculated directly by Gauss quadratures,while for larger k we can use the identities

    a (k ) =m

    i=0C mi (1) i

    1a 2i k 2i+2 I

    02i+1(ak )

    a (k ) =m

    i=0C mi (1)

    i 1a 2i k 2i+3

    I 12i+2(ak )

    a (k ) =m

    i=0C mi (1)

    i 1a 2i k 2i+4

    I 02i+3(ak ),

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    FIG. 3. The spectral decay of window functions a (k ), a =1 of order m =3, 5, 7. Inset is the shape of window function a (), a =1 of order m =3, 5, 7.

    FIG. 4. Spectral decay of window functions a (ak ), a =1 of order m =1, 3, 5, 7.

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    CALCULATION OF DYADIC GREENS FUNCTION 11

    FIG. 5. Spectral decay of window functions a (ak ), a =1 of order m =1, 3, 5, 7.

    where for positive or zero integers ,

    I ( z) = z

    0 J () d .

    For small arguments of z = ak , I ( z) and (4.15)(4.17) can be calculated directly withGaussian quadrature, while for large values of z, we can use the formulas

    I ( z) = 2

    + +12 +12 +

    z[( + 1) J ( z) S 1, 1( z) J 1( z) S, ( z)],

    where S, ( z) are the Lommel functions (see Appendix A.2).

    5. NUMERICAL RESULTS

    In this section, we validate Algorithm 2 for the fast calculation of dyadic Greens functionsin homogeneous and multilayered media. In all cases, the frequency in the Greens function f =1 GHz.5.1. Window Functions in Spectral Domain

    The window function a ( x, y) of order m has a compact support in the physical domainwhile in the spectral domain it decays in an algebraic order 1 / k m ; so do both a (k )and a (k ) used in Algorithm 2. Figure 3 shows both window functions a ( x, y) of order m =3, 5, 7 with support size a =1 (lower left insert) and their frequency decays,

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    12 CAI AND YU

    respectively. Figures 4 and 5 show the frequency decays of a (k ) and a (k ) of orderm =1, 3, 5, 7 with support a =1, respectively.5.2. Second-Order Accuracy and Efciency of Algorithm 2

    Algorithm 2 is second-order accurate, O (a 2) . The contour integrations in the denitionof W i (), i =0, 1, 2, of Algorithm 2 are only evaluated over a truncated nite path as inFig. 2. By choosing the L4 portion of contour C according to (4.22), we can maintainan overall error of O (a 2) . To verify the second-order accuracy, we apply Algorithm 2 tocalculate the scalar potential for the free-space Greens function

    G v (, z; z ) =1

    4e jk R

    R , (5.1)

    where R = |r r |.Figure 6 shows the real and imaginary parts of calculated results for G v ( , z; z ), z =

    0.001m , z =0.002 m. The lines are the exact solutions given by (5.1) and the symbols arethe calculation by Algorithm 2 with L4 =20/ a in the contour of Fig. 2. A window a ()of order m =5 and support size a =0.005 m is used. Figure 7 shows the errors in log-scaleof three applications of Algorithm 2 with three different window sizes a

    =w, 2w , and 4w ,

    where w =0.005 m. It is clear to see that the convergence rate is O (a 2) .Finally, Fig. 8 shows the savings of Algorithm 2 over the direct Sommerfeld integration

    of (4.4). In Fig. 8, G f denotes the calculation results of Algorithm 2 (squares, diamonds,

    FIG. 6. Free-space scalar Greens function G V (, z; z ), z =0.001 m, z =0.002 m. Lines indicate analyticresults; symbols are for calculations with Algorithm 2 with window a () of order m =5 and window supporta =0.005 m.

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    CALCULATION OF DYADIC GREENS FUNCTION 13

    FIG. 7. Free-space scalar Greens function G V (, z; z ) , z =0.001 m, z =0.002 m. Relative errors withAlgorithm 2 with window a () of order m =5 and window support a =h , 2h , and 4h , where h =0.005.

    FIG. 8. Greens function G V (, z; z ), z =0.001 m, z =0.002 m. Errors of Algorithm 2 with windowfunction a () of order m =5 and L 4 = L, 2 L( ), and 4 L( ) ; L =5/ a ; window support size a =0.005. Errorsof direct numerical integration of (4.4) using L 4 =240 L( ), 400 L(6 ).

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    14 CAI AND YU

    FIG. 9. Half free space over a PEC at z

    =0.

    down triangles) with the last portion of contour (2) L4 = L , 2 L, and 4 L, where L =5/ aand a =0.005 m is the support size of the window function of order m =5. G s denotesthe direct integration of Sommerfeld integration (4.4), and upper triangles and stars arethe results with a truncated contour L4 =240 L , 400 L, respectively. From Fig. 8, we cansee that results of Algorithm 2 with L4 =2 L are already better than those of the directSommerfeld integration with L4 =400 L; thus the contours used in Algorithm 2 are 200times shorter than those used in direct integrations (4.4).

    5.3. Greens Functions for a Half Space over a PEC Ground Plane

    In this test case, we calculate the vector potential Greens function G A(, z; z ) for a half free space over a perfect conductor (PEC) ground plane at z =0 in Fig. 9, where analyticforms of the Greens functions are available by using mirror images [11].

    Figure 10 contains the calculated real and imaginary part of the scalar potential Greensfunction G V (, z; z ), z =0.001 m, z =0.002 m. Again, window a () of order m =5with support size a =0.005 m is used and L4 =20/ a in the contour C of Fig. 2.

    5.4. G A(, z; z ), G V (, z; z ) for a Five-Layered MediumIn this test case, we calculate the dyadic Greens functions G A( , z; z ), G V ( , z; z ) forthe ve-layered medium depicted in Fig. 11. Four dielectric layers are used between the

    open air and a PEC ground plane. The relative dielectric constants for the four dielectriclayers are, from top to bottom, 1 =9.6, 2 =12.5, 1 =2.4, and 1 =3.6, respectively.Their corresponding thickness are h 1 =0.001 m, h 2 =0.003 m, h 1 =0.002 m, and h1 =0.0015 m, respectively.

    Window a () of order m =5 with support size a =0.001 m is used in Algorithm 2 andthe L4

    =20/ a is the last portion of the contour C in Fig. 2. Figure 12 shows the magnitude

    and imaginary and real parts (top to bottom, lines are from integration of Hankel transform(4.4); symbols are results of Algorithm 2) of the scalar Greens function G V (, z; z ) with z = 0.0035 m, z = 0.0062 m. Figure 13 shows the component G xx A ( , z; z ) for thevector potential A along with the magnitude and imaginary and real parts of this component(top to bottom, lines are from integration of Hankel transform (4.4); symbols are resultsfrom Algorithm 2). Figure 14 shows similar results for the component G zz A (, z; z ) .

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    CALCULATION OF DYADIC GREENS FUNCTION 15

    FIG. 10. G V (, z; z ), z = 0.001 m, z =0.002 m for a half free space over a PEC ground plane at z =0.Lines are the analytic results; symbols are the results of Algorithm 2 with window a () of order m =5 andsupport size a =0.005 m, with L4 =20/ a in the contour C of Fig. 2.

    5.5. Comparison with Method of Complex ImagesIn the last numerical test, we compare our method with the method of complex images

    of [5, 6]. The method of complex images has been extensively used in the engineeringcommunities for the calculation of dyadic Greens functions. The idea is based on theSommerfeld identity

    e jkr r =

    j2

    C

    dk H (2)0 (k )

    e jk z| z|k z

    , (5.2)

    where k z = k 2 k 2 and C is an integration contour from to in the rst and thirdquadrant of the complex k -plane.

    FIG. 11. A ve-layer medium: four dielectrics layered between the air ( z =0) and a PEC ground plane.

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    FIG. 12. ScalarGreens functions G V (, z; z ), z = 0.0035 m, z = 0.0062 m for the ve-layered mediumin Fig. 9. Lines are results of direct integration of (4.4) for comparison; symbols are results of Algorithm 2 withwindow a () of order m =5 and window size a =0.001 m, with L 4 =20/ a in the contour C of Fig. 2.

    FIG. 13. Vector potential Greens functions G xx A (, z; z ), z = 0.0035 m , z = 0.0062 m for the ve-layered medium in Fig. 9. Lines are results of direct integration of (4.4) for comparison; symbols are results of Algorithm 2 with window a () of order m =5 and window size a =0.001 m, with L 4 =20/ a in the contourC with Fig. 2.

    16

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    CALCULATION OF DYADIC GREENS FUNCTION 17

    FIG. 14. Vector potential Greens functions G zz A (, z; z ), z = 0.0035 m, z = 0.0062 m for the ve-layered medium in Fig. 9. Lines are results of direct integration of (4.4) for comparison; symbols are results of Algorithm 2 with window a () of order m =5 and window size a =0.001 m, with L4 =20/ a in the contourC of Fig. 2.

    FIG. 15. Error comparison of the method of complex images (top line with 10 complex images) and ourwindow-based fast algorithm (bottom three lines with windoworder m =5 andsupport a =4h , 2h , h respectively,where h =0.001 and L4 =10/ a ).

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    18 CAI AND YU

    FIG. 16. Half space (airsoil with 5% water content).

    The Greens function in the spectral domain is approximated by a sum of exponentialfunctions, namely

    G (k , z; z )

    n

    i=1a i

    e jkr ir i , (5.3)

    where a i is the complex magnitude and r i = 2 +( z + z jb i )2 is the complex distanceof the complex images. The coefcients a i and bi are obtained using a Prony method to tsampled values of the G (k ) along a nite portion of the integration path C .

    We have found out that such an approach will give good results when the source andobservation points are on the same layer of a multilayered structure. However, the method

    of complex images does not perform well when source and observation points are not on thesame level, and the Prony procedure for obtaining the coefcients a i and bi also becomesunstable and inefcient when high-frequency problems are considered.

    In Figure 15, we present the results of both the method of complex images and ourmethod (with m =5 and L4 =10/ a and three different values of a =4h , 2h , and h , withh =0.001) for the two-layer structure of Fig. 16 (air over soil with 5% water content). Thetop line is the relative error in logrithmic scale with complex images (10 complex images);the other three lines depict the relative error in logrithmic scale with decreaing size of a .

    6. CONCLUSION

    We haveproposed a novel accelerationmethodforcalculation of theSommerfeld integralsin the denition of dyadic Greens functions for electromagnetic scattering of conductorsembedded in multilayered media. The introduction of a window function effectively reducesthe cost of the calculation of the Sommerfeld integration by forcing a faster decay of theintegrands. The efciency and accuracy of the calculation of the multilayered Greens func-tions are critical to the speed of the method of moments for the electromagnetic scatteringof conductors in multilayered media. Numerical results presented in this paper and also in[10] show the effectiveness and accuracy of the proposed method. Because Algorithm 2involves three integrals, compared with just one in Algorithm 1, a combination of the twoalgorithms can give better efciency. Moreover, the fast Hankel transformation approach

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    CALCULATION OF DYADIC GREENS FUNCTION 19

    proposed in [17] can be combined with Algorithms 1 and 2 for further speeding up of thecalculation.

    APPENDIX

    A.1. Proof of Lemma 3 (4.12)

    From the denition of the convolution, we have with z = z z

    G 2(, z; z ) a ( x, y)

    =1

    2

    ( x

    2

    + y2

    + z2)G ( x , y , z; z ) a ( x x , y y ) dx dy

    = r 2[G (, z; z ) a ( x, y)] 2 x[G (, z; z ) x a ( x, y)]

    2 y[G ( , z; z ) y a ( x, y)] +G (, z; z )(( x2+ y

    2) a ( x, y))

    = r 2W 0() 2 xV 1 2 yV 2 +W 2(). (A.1)

    In arriving at the second equation, we have used identities x 2

    = x2

    +2 x( x

    x)

    +( x x)2 and y 2 = y2 +2 y( y y) +( y y)2. Using the integral denition of the Besselfunction,

    J n ( z) = jn

    0e j z cos cos n d ,

    we can show that

    V 1 =G ( , z; z ) x a ( x, y) = cos 0 G (k , z; z ) J 1( k )a (k ) k dk (A.2)and

    V 2 =G (, z; z ) y a ( x, y) = sin 0 G (k , z; z ) J 1( k )a (k ) k dk , (A.3)where a (k ) is dened in (4.16). Therefore,

    xV 1 + yV 2 = 0 G (k , z; z ) J 1( k )a (k ) k dk = S0[G (k , z; z )a (k )]() = W 1(). (A.4)

    Similarly, we can show that

    W 2() =G (, z; z )(( x2+ y

    2) a ( x, y)) = S0[G (k , z; z )a (k )](), (A.5)

    where a (k ) is given in (4.17). Substituting (A.4) and (A.5) into (A.1), we have the proof of (4.12).

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    20 CAI AND YU

    A.2. Lommel Function

    If + or is a positive integer, then S, ( z) has a nite polynomial expansion interms of 1 / z, i.e.,

    S, ( z) = z 1

    n

    k =0

    ( ia +1) ( ib +1)( ia k +1) ( ib k +1)

    (1)k 2 z

    2k

    with n =min(ia , ib ), ia =1 +2 , ib =

    1 2 .For other values of , , we have the asymptotic approximations [19]

    S, ( z) z

    1

    3F 0 1, ia , ib , 4

    z2

    = z 1 1

    ( 1)2 2 z2 +

    [( 1)2 2][( 3)2 2] z4

    [( 1)2 2][( 3)2 2][( 5)2 2]

    z6 + ,

    where 3F 0(1, ia , ib ,

    4

    z2 ) is the hypergeometric function [19].

    ACKNOWLEDGMENTS

    The authors acknowledge the support of DARPA with a grantAFOSR F49620-96-1-0341 and NSF grant CCR9972251 for the work reported in this paper. Thanks are extended to Professor Jiayuan Fang and Dr. XingchaoYuan for many interesting discussions.

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