Numerical computation of complex image Green‘s functions for multilayer dielectric media: near-field zone and the interface region

N. Hojjat S. Safavi-Naeini Y. L. Chow

Indexing terms: Complex images, Green S functions, Multilayer dielectric media

Abstract: The paper is primarily concerned with the problems arising in connection with the complex image Green’s function computation in the near-field zone and over the interface in multilayer dielectric media. The numerical difficulties are investigated related to the cases where the source and observation points are located along an axis perpendicular to lamination and the observation point is located on the interface between two dielectric media. Methods for resolving these difficulties when using complex image representations are presented. For near-field computation a range is determined within which the surface waves must not be extracted from the spectral representation. Over the interface plane, to obtain a more accurate result, the complex images should be placed in the medium identical to the unbounded region or the medium with a lower dielectric constant.

1 Introduction

The complex image representation of the spatial Green’s function have been applied to circuit analysis and planar antennas in multilayer media by several authors. Although the computational advantages of this method over conventional spectral or space domain approaches have generally been established, there are a few issues related to the numerical imple- mentation of the method which still remain to be clari- fied [l-61.

In this paper, we are primarily concerned with the problems arising in connection with the complex image field computation in the near zone and over the inter- face.

0 IEE, 1998 IEE Proceedings online no. 19982255 Paper first received 23rd December 1997 and in revised form 8th September 1998 N. Hojjat is with the Department of Electrical and Computer Engineer- ing, University of Tehran, North Kargar Avenue, P.O. Box 14395/515, Tehran, Iran S. Safavi-Naeini is with the Department of Electrical and Computer Enginering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 Y.L. Chow is with the Department of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

2

The general form of a typical component of the dyadic Green’s function for multilayer media [2] has the fol- lowing spectral-domain representation:

Near zone field and surface wave extraction

Q , z ) = /+” F ( @ ) H p ( k p P ) w k p (1) -m

where z is the co-ordinate variable in the direction per- pendicular to the layers’ interfaces. The geometry of the problem is shown in Fig. 1.

field P point

Fig. 1 Microstrip structure imder consideration

One common approach to derive the complex image representation of eqn. 1 is first to extract the surface wave and quasi-dynamic contribution, and then express the remaining part in terms of a set of spherical waves originating from a small number of complex point sources or images located at complex positions [2-51. The result is

(2) where the first summation on the right-hand side includes the complex and quasi-dynamic images’ con- tributions and the second summation contains surface waves,

It should be clarified here that, although in some work 11, 61 the surface wave terms were not extracted, the effect of this extraction on the accuracy of the numerical results, especially for the near field, was not investigated.

The main contribution of this paper in this area is the rigorous analysis o f these effects. It is demonstrated that the surface wave poles would lead to erroneous results in near-field regions.

449 IEE Proc-Microw. Antennas Propag.. Vol. 145. No. 6. December 1998

The difficulty with eqn. 2 is that for zf z 0, when p - 0, the surface wave terms blow up owing to the singu- lar behaviour of the Hankel functions at the origin.

To study the root of this problem and determine the region of validity of eqn. 2 we can asymptotically expand the Sommerfeld integral of eqn. 1 using SDP (steepest descent method). It can be argued that each pole of the integrand, k,, has a significant contribution to the field only in the angular region B > Op [7], where Op = sin-'(kO/kpp) and 0 = tan-'(p/zf), as shown in Fig. 1. This condition may be rewritten as

p > ~f tan [sin-'(kO/kpp)] (3) Eqn. 3 shows that when the source and the field points are not on the same plane (z, * 01, the effects of the poles are important only when p is greater than the Idis- tance given in eqn. 3. For example, for the microstrip layer of Fig. 1 with zf = lmm, E, = 12.6, h = lmm, f = lOGHz when log(kop) < -0.05, the first TM pole, which is the only pole of the structure, does not have any important contribution to the solution.

Based on the above argument, it is now clear that to compute the field in the near-zone region the extract ion of the surface wave poles, as in eqn. 2, is not only nec- essary but also leads to erroneous results. In other words, the complex image terms in eqn. 2 cannot com- pensate for the singular behaviour of the surface wave terms and, as a result, the computed field data are highly inaccurate. In fact, the complex spherical waves emanating from the complex-image source cannot rep- resent the logarithmic singularity of the zeroth order Hankel function properly, although, in principle, it is possible to provide a more accurate representation for the near field, by using a longer sampling interval in a Prony approximation process or a higher-order approximating scheme.

As mentioned earlier, the surface wave term would result in a nonphysical singular behaviour of Green's function for small p. To resolve the above difficulty, the simplest way is to use complex image representa- tion for the whole spectral function without extraclion of surface waves [l].

0 5

5 m

8 U

0.5

0

- - 0 8 -0.5 -

-1 .o

0 - m

8 U - - 0 8 -0.5- -

'Y -1.5l -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0

W k O P )

Fig.2 Scalar potential of an HED by three methods: comnplex imtiges: sum of complex images (CIS) (with surface-wave poles extractedJ and sur- face wave (SW); and exact numerical integration +++ surface wave part 0000 complex images part . . . . . . . . . . only complex images

~ CIS (surface wave extracted) + SW HsW numerical integration

450

2.2

0.4 1 0.2 I I -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0

log (ked Fig.3 Scalar potential of a VED by three methody: complex images: sum of complex images (CIS) (with surface-wave poles extracted) and sur- face wave (SW); and exact numerical integration +++ surface wave part 0000 complex images part . . . . . . . . . . only complex images __ CIS (surface wave extracted) + SW H+H numerical integration

To verify this conclusion, the results for the scalar (charge) potentials for both an HED (horizontal elec- tric dipole) and a VED (vertical electric dipole) for the geometry of Fig. 1 obtained by two approaches (with and without surface wave extraction) are plotted in Figs. 2 and 3 (zl = lmm, E, = 12.6, h = lmm, f = 10GHz). The results of this method (no surface wave extraction for the near field) are in excellent agreement with those of the exact numerical integration of eqn. 1 up to log(kop) < 1.5. Beyond this range, we can use the conventional eqn. 2 or just use the surface wave terms. The effect of the length of the sampling path, zo, on the approximation for the VED scalar potential is shown in Fig. 4. In Fig. 5, the surface wave contribution is compared with the rigorous numerical calculation of the HED scalar potential integral, and it is observed that the surface waves dominate when log(kop) > 1.5. For a more accurate approximation, in the transition region 1 < log(k0p) < 1.5, the extraction of surface wave poles is recommended.

W % P )

Fig.4 0000 numerical integration - . . . , . . . .

Effect of to on accuracy of eqn. 2 for scalar potential of a VED

CIS (surface wave extracted) + SW, so = I CIS (surface wave extracted) + SW, zo = 3

IEE Proc.-Microw. Anfennus Propug., Vol. 145, No. 6, December I998

Fig. 5 tion for an HED 0000 numerical integration - _ - - surface wave behaviour

Compurison of surjuce waves contribution and numerical integru-

3 medium on the accuracy of the field evaluation over the interfaces

Effects of the complex image embedding

In this Section, we consider the case where the field point is on the planar interface between two dielectric layers. Obviously for those components of the potential which are continuous across the interface, the choice of the observation point in either of the surrounding lay- ers, when it is sufficiently close to the interface plane, should not change the result. In the complex image method, however, if the complex images are assumed to be embedded in the medium of the field (observa- tion) point, the computed results do not remain contin- uous across the interface plane.

This purely numerical discontinuity is a result of the inexact nature of the method. The precision of the complex image method is, to a large extent, determined by the accuracy of the spectral function approximation scheme and, in fact, the irregularity of the spectral function. A reasonable measure of this irregularity is the configuration (number and location) of the branch cuts and the poles.

h2=0 3 mm I z,*=125

1 tr'=2'1 1 11 hl=0,7mm

Fig. 6 Typical two-layer structure

To simplify our examination of the root of this error and its minimisation, let us consider the example illus- trated in Fig. 6. The components of the potential Green's function for the geometry shown in Fig. 6 have the general form given in eqn. 1, where the spec- tral function, E(k,, z), has one branch cut correspond- ing to the unbounded medium (free space) [8]. To apply the complex image method we rewrite eqn. 1 in the following form:

where k,; = d(Srjko2 - kP2) is the wave constant along the z-direction in the medium E,.;. The complex image part of F, i.e. F,, can be represented as

IEE Proc.-Microw. Antennas Propag., Vol. 145, No. 6, December 1998

IC, == &#to,

(5) It is reasona.ble to choose ki to be the same as that of the medium containing the observation point [3]. With this choice it is obvious that, as the observation point crosses the interface, the complex images' embedding medium changes, which causes the approximated func- tions and the sampling path in the k,-plane to change as well (Fig. 7). The approximation path in the kZi plane is chosen as k,; = { tk j + jkmaxzO(t - l)}.

1 j k",

L k ,

", 0

Fig.7 Sampling paths for complex images schemes in kz and k, planes

For the specific example illustrated in Fig. 6, this means that if the field point is chosen in medium 1, then Ej is (equal to 2jk,,E and the complex image expansion is in terms of terms-of kZl . If the field point is chosen in medium 2, then F j is equal to 2jkz2F and the complex image expansion is in terms of terms of

It is important to note that in both cases the spectral function 2jkzjP has two branch cuts. One is correspond- ing to free space, resulted from kzO, and the second one is due to th.e multiplying factor k,, or kz2. As men- tioned elsewlhere [3], a part of the approximation error is produced by this second nonphysical branch cut. It should be clarified that the function to be expanded is again sampled along a path where it is single-valued and the Sommerfeld identity can be employed. The dif- ficulty with this extra branch cut is that it increases the complexity of the spectral function to be approximated. The function E itself is smoother than 2jkZiF. The term k,; causes a ,singularity in the derivative of 2jkzjP. This makes approximation by the Prony method less accu- rate, although it may not cause much problem for more rigorous approximation methods [5, 61. If kZi is chosen as k;:o, the complexity of the function will be reduced and the approximation will become more accu- rate.

For the two-layer medium of Fig. 6, the A,,y compo- nent of the Green's dyadic has been computed over the interface plane between the media 1 and 2 by three dif- ferent appro.ximation schemes: (a) complex images are placed in free space, ki = ko (b) complex images are placed in the medium 1, k j =

k-2.

d(&r I

45 1

(c) complex images are placed in the medium 2, X:, =

The value of zo has been chosen as 3 for the three cases.

All the results are compared with the rigorous ones obtained from the direct numerical calculation of the Sommerfeld integral (eqn. 1) and are plotted in Figs. 8, 9 and 10. It is observed that when the complex images are placed in either the unbounded medium (free space, k, = k,) or in the medium with lower dielectric constant (k, = d(crl)ko), the results are in excellent agreement with those of direct numerical integration. However, the complex image results are not as good in the case where they are placed in the medium with higher die- lectric constant (k, = d(cr2)kO) beyond log(kop) > 0. The increase in the accuracy for large p with decreasing the sampling interval (compare the z, = 1 case with that of to = 3 in Fig. 8) is essentially due to the increase in density of sampling points.

d(&r2)k0. much difference between the lengths of the sampling intervals in these two cases. In fact, for zo = 1 the length of the sampling interval in the k,,-plane is less than that in the kzl-plane for, z, = 3, but still the approximation based on sampling in the krl-plane is more accurate (Figs. 8 and 9).

- 1 I

-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 W k o d

fi .10 in Fee spuce ~

0000 numerical integration

A,,y potential evaluated by complex images when they are placed

complex image in kzg plane

Therefore, the under-sampling could not be the only reason for less accuracy in approximation in the kz2- plane compared to that in the kzl-plane. In fact, the more complex form of the spectral function in the kz2- plane has stronger effect in decreasing the approxima- tion accuracy.

I -2 -1.5 -1.0 -0.5 0 0.5 1.CI

log(kgp) Fi 8 A potential evaluated by complex images when they are placed in Eyer u i i ra higher dielectric constant ~

. . . . . . . . . 0000 numerical integration

complex image in kn2 plane, T,, = 1 complex image in kr2 plane, q = 3

-0.8

-0.9

-1 .o

-1.1 ' I 0 500 1000 1500

Real part of spectrum of A,, estimated by Prony method in kz,- kP

Fig. 11 plane

~

. . . . . . . . . 0000

spectral function in e, = 12.5 Prony approximation in the kz2 plane, T,, = 3 Prony approximation in the kz2 plane, T , ~ = 1

-2.0 -1.5 -1.0 -0.5 0 0.5 1.0

log(koP)

Fig.!? AX" potential evaluated by complex images when they are pi'aced in ayer wit a lower dielectric constant 0000 numerical integration ~ complex image in terms of kZ1

Considering our path for zo = 3, it could be observed that when the approximation is based on the sampling in the kzl-plane, the path in the k -plane begins from k , = 0 and ends at k, = 10.7ko, ancfwhen the sampling is performed in the kz2-plane, the path begins from k, = 0 and ends at k, = 11.18ko. Thus, there is not actually

452

The above behaviour can be better understood by looking at the spectral representation of the potential function and its various approximations, as plotted in Figs. 11, 12 and 13. The most accurate approximation is obtained when k, = ko (Fig. 13) because, in this case, the spectral function has only one branch cut (corre- sponding to the unbounded medium) which is also the same as the only physical branch cut from E. The spec- tral function approximation based on the complex images located in the medium with the dielectric con- stant closest to that of the unbounded medium, i.e. medium 1, or k, = d(&,Jk0 (Fig. 12) is also very accu-

IEE Proc.-Microw. Antennas Propug., Vol. 145, No. 6, December 1998

-1.1 ‘ I 0 500 1000 1500

kP

Fig. 12 k,,-plune __ . . . . . . . . . .

Real part of spectrum of A , estimated by Prony method in

spectral function in e, = 2.1 Prony approximation in the k;l plane

3.0

- 2.5 = x

2.0 1

1.5

1.0

0.5

- - - -

- -

-0.81 /

0

-0.1

-0.2

-0.3

- -0.4

3 -0.5- P

1%

500 1000 1500 kP

0

Fig. 13 plane ___ spectral function . . . . . . . . . .

Real part of spectrum of A , estimated by Prony method in kzo-

Prony approximation in the kz, plane, N = 5, = 1

-

/J - -

/’ -

70 I

0 20 40 60 80 100 l-t(XlO0)

Fig. 14 ble in kz2-plane . . . . . . . . . . .

Potential spectrum as a function of Prony path sampling varia-

Prony approximation in the k,2 plane, N = 5, “0 = 1

rate mainly because of the small distance between the two branch cuts. The main problem with the third approximation scheme (ki = q(cr2)kO), where the com- plex images are embedded in the medium with the highest contrast with respect to the unbounded region,

IEE Proc.-Microw. Antennas Propug., Vol. 145> No. 6, December 1998

is that the two branch cuts are relatively far apart, and the spectral approximation is inaccurate between these two points, as shown in Fig. 11. As a further verifica- tion of this conclusion, the spectral representations of the scalar potential for the same structure obtained by the above three approximating schemes, along with the rigorous numerical integration results and the corre- sponding spatial domain data, are presented in Figs. 14, 15 and 16.

nl I - 0 20 40 60 80 100

14 X(lO0)

Potential spectrum as a function of Prony path sampling varia- Fig. 15 ble in kzl-plane . . . . . . . . . . -

Prony approximation in the kz, plane, N = 5, r, = 1 spectral function in e, = 2.1

I 0 0.5 1.0

log(kgP)

Fig. 16 Spatid Green S function of scalar potential of an NED on inter- face of two dielectric media of Fig. 6 at f = 3OGHz by dflerent complex- images schemes 0000 numerical integration ~

. . . . . . . . . ++++

complex image in k;, plane complex image in k,, plane complex image in kzz plane

4 Conclusions

The numerical difficulties related to the near zone and interface field computation using complex image repre- sentations have been investigated and the methods to resolve them have been presented.

For near-field computation, a range was determined within which the surface waves must not be extracted from the spectral representation. Over the interface planes, to obtain an accurate and unique result, the complex images should be placed in the medium with a lower dielectric constant or the unbounded region.

453

5 Acknowledgments 3

This work was supported by the Research Council of the University of Tehran (Iran), the National Science and Engineering Research Council (NSERC) of Can- ada and the Communication and Information Technol- ogy Ontario (CITO).

References 6

FANG, D.G., YANG, J.J, and DELISLE, G.Y.: ‘Discrete image theory for horizontal electric dipoles in a multilayered medium’, IEE Proc. Microw. Antennas Propug., 1988, 135, pp. 297-303 CHOW, Y.L., YANG, J.J, FANG, D.J., and HOWARD, <;.E.: ‘A closed-form spatial Green’s function for the thick microstrip substrate’, IEEE Trans., Microw. Theory Tech., 1991, 39, pp. ,588- 592

7

8

AKSUN, M., and MITTRA, R.: ‘Derivation of closed-form Green’s functions for a general microstrip geometry’, IEEE Trans., Microw. Theory Tech., 1992, 40, pp. 2055-2061 KIPP, R.A., and CHAN, H.C.: ‘Complex image method for source in bounded regions of multilayer structures’, IEEE Trans., Microw. Theory Tech., 1994, 42, pp. 860-865 DURAL, G., and AKSUN, M.: ‘Closed-form Green’s functions for general sources and stratified media’, IEEE Trans., Micron’. Theory Tech.. 1995, 43, pp. 860-865 AKSUN, M.1.: ‘A robust approach for the derivation of closed- form Green’s functions’, IEEE Truns., Microiv. Theory Tecl7.. 1996, 44, pp. 651-658 MOSIG, J.R., and GARDIAL, F.E.: ‘A dynamical radiation model for microstrip structures’, Adv. Electron. Electron Phys., 1982, 59, pp. 139-237 CHEW, W.C.: ‘Waves and fields in inhomogenous media’ (Van Nostrand Reinhold, New York, 1990)

454 IEE Proc.-Microw. Anfennus Propug., Vol. 145, No. 6 , December 1998