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600 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGA TION, VOL. A P - 3 3 , NO. 6 , JUNE 1985

Dipole and Slot Elements and Arrays on Semi-Infinite Substrates

MASANOBU KOMINAMI, DAVID M. OZAR, EMBER, IEEE, AND DANIEL H. CHAUBERT, SENIOR MEMBER, IEEE

Abstract-The printed dipole or slot ntenn a on a semi-infinite substrate

and infinite phased arrays of these elements are nvestigated. The solutionis based on the moment method in the Fourier transform domain. The

generalized impedance or adm ittance matrix can be expressed in rapidly

converging infinite-integral or infinite-summation forms, allowing the

accnrate determination of the current distributions. Using the present

formulation, the input impedance, resonan t length, and radiation pattern

fo r the isolated antennas, and the reflection coeffic ient for nfinite phased

arrays, are calculated.

TI. INTRODUCTION

HIS PAPER presents rigorous solutions for the driving point

impedance and radiation patterns of dipole and slot antenna

elements printed on a semi-infinite dielectric substrate, and for

infinite phased arrays of these elements on semi-infinite sub-

strates. The semi-infinite substrate is of practical interest because

a number of millimeter wave imaging or phased arrays are being

proposed [l], [2] where the elements (e.g., dipoles, microstrip

patches, slots) are printed on the planar surface of a dielectric

lens. The lens then forms a substrate for the antenna elements

as well as active microw ave devices (e.g., field-effect transistor

(FET) amplifiers., mixers, phase sh ifters, etc.). Sinc e the lens is

electrically large, the antenna elements act as if they are at the

interface of an air-dielectric half-space. The fields at the curved

side of the lens locally form a plane wave, and matching layers

can be used to minimize reflections back to the antenna. Some of

the radiation goes into the ai r side of the interface bu t, as will

be shown , more of t he pow er is delivered to the dielectric side.Printed antennas seem to be a likely choice for such mono-

lithic, or “active aperture” antennas, and printed dipoles [3],

[ 4 ] , [5] and microstrip patches [ 6 ] , [7] on grounded dielectric

substrates have eceived significant attention. Printed slots [8 ]

have received somewhat less atten tion , but are of nterest for

monolithic applications because the ground plane facilitates

the integration of active devices.

The problem of an antenna at the interface of two differentdielectric half-spaces is of course closely related to the classic

Sommerfeld problem ofan antenna on th e surface of a lossy

earth. ,More recently, the patterns of resonant dipoles and loops

over a lossless dielectric half-space havealso been studied [ 9 ] ,

Presented here is amomentmethod solution forprinted

dipoles or slots on semi-infmite substrates. This amroach uses

[101*

of printed dipoles [3], [4] and m icrostrip patches l11 on grounded

dielectric substates. A solution for an infinite array of printeddipoles or slots on sem i-in ffite substrates is also presented here.

The momentmethod solution is used to calculate the active

impedance versus canangle, and isan extension of previous

solutions for an infinite array of printed dipoles [5] and micro-

strip patches [71 .Section I1 describes the basic theory and equations for he

Fourier transform domain momentmethod and modifications

for the case of an i n f i t e phased m a y of printed antennas on

semi-infinite substrates. Section 111 presents numerical results

for inpu t impedance, resonant length and resistance, and radia-

tionpattern s for isolated elemen ts. This section also presents

results forhe reflection coefficient magn itude variation of

an infiite phased array with angle for E-plane, H-plane, and

diagonal plane scan. Two dielectric constants were selected:

E, = 2.55 forpolytetrafluorethylene (PTFE) andDuroid-type

materials and e, = 12.8 for A lumina and GaAs-type materials.

11. THEORY

A. Isolated Elements

The geometries of theprinted dipole and slot elements are

shown in Figs. l(a) and l(b), respectively. Two s e m i - in f~ t e

regions (1 and 2), have the common boundary z = 0. The upperregion l), z > 0, is air and characterized by the parameters

e l = eo and p 1 = p o : the lower region (2), z < 0, is a homo-

geneous and isotropic medium characterized by the parameters

e2 = E ~ E eOerO1 - tan 6) and p2 = p o , where tan 6 is the

loss tangent. B oth elements are of length L and width W nd are

center-fed by an ideal delta-gap generator as shown in Figs. l(a)

and l(b). The delta gap generator is known to yield good numeri-

cal results for radiation and impedance of th n wire antennas

fed by balanced lines or coaxial lines with appropriate baluns.

Coaxial and m icrostrip line feeds fo r slots can also be approxi-

mated by a delta gap generator. The agreement between meas-

ured and calculated results in [5] substantiates he use of t h i s

source model forprinted elements. The electromagnetic fields

in each homogeneous region 1 = 1, 2) are described by two

scalar potential functions \ k l and @ I that satisfy the Helmholtzequation and boundary conditions at he interface. The field

components are related to \k l and aZ y

Galerkin’s method applied in the Fourier transform domain. E, = V X V X (;al) wp , V X iq,)

The solution is similar in principle t o the previously treated case

.(1)

HI = B X X i9,) ue lV X ( Dl) (2)

Manuscriptreceived October26, 1984;revised aw1 1984. This work where ; s a U n i t vector in the Z-direction. For an eXp jut)was s~pp0rte-dy heNational Sci enc e FoundationunderGrantECS-8206420, time dependence, the scalar potentials can be expressed in theand heNationalAeronauticsandSpaceAdministration,Langley ResearchCenter, under Grant NA G-1 -163 .M. Kominami was with theDepartment of ElectricalndComputer

Engineering,University of Massachusetts,Amherst, MA 01033, on eave from * d X , y > =the University of Osaka Prefecture, Osaka, Japan.

Computer Engineering, University of Massachusetts, Amherst, MA 01033. dk, cik,

forms

A,@,, ky)e-j(k,,+kyY+711zl)47i

D. M.Pozar and D. H. Schaubert are witli the Departmentof Electrical and

0018-926X/85/0600-0600S01.00 1985 IEEE

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KOMINAMI er a [ . : SEMI-INFINITEUBSTRATES 601

Z fields E, and HI inhe boundary conditions (5a) and (5b) forhedipole and (6a) and (6b) fo r the slot.

The functions are

Y1

72

A1 =--A2

Y2kxEfor dipole

E, weo(Y2 +W l ) J q

+ k;) '--kyfix ( 7 )

(a) b)

Fig. 1 . Geometry of antennas on a semi-infinite substrate. a) pri nteddipole.(b) Slot antenna.

for slotr1w: + k;) '

kY E, for dipole

where quantitieswith tilde are Fourier transforms or cor-

is defined as

where I and B I are functions to be and 71 k the responding quantitieswithout tilde. The Fourier transform

propagation constant in the z direction given by

7 = k; k t - k; (Im ~ l ) 0 ,k l= 0fi1).

At the air-dielectric interface (z = 0), theontinuity condi- &kx , k,) =/ / $(x7y)e-i(kxX+kry) x y. ( 9 )

tions for the tangential electric and magnetic field componentshave to be satisfied as follows. (Because of the similarity of the The remaining boundary conditions (5C) and (6C) are enforced

dipole and slot solutions, bot h are treatedhere in parallel.) to yield the following set of equations:

Dipole case:

i X ( H l - H 2 ) =J, (on dipole)

0, (elsewhere)

(El + E,) X =(on dipole)

Eo X 2, (elsewhere)

E l x i =M, (onlot)

0, (on conductor)

(E, E2) X i = 0 (6b)

; X ( H , H2) =J (an slot)

Jo, (on conductor)

where

J electric current density on he printed dipole

E, source electric field for the dipole

Eo electric field on the interface exce pt dipoleM magnetic current density in the slot

J, source electric current fo r the slot

JO electric current on the conductor.

It is assumed that he dipole or slot width W is very small

compared to the wavelength in free space and in the dielectric,

and therefore we consider only the axial compo nent of the cur-

rent and field distributions. The unknown functions appearing

in he scalar potentials can be determined by substituting the

Hence, 10) and_ ll) contain four unkn owns? ,,fix, goox,nd

J o x . However, Eo , and J o x wi be eliminated later in the solu-

tion process based on the momentmethod procedure. These

spectral domain algebraic equations correspond to Pocklington's

integral equations in the space domain.

The mo ment metho d solution used here is a Galerkin method

applied in theFourier transform domain [121. The unknown

current densities J,(x, y on the printed dipoles and M,(x, y )

in the slot are expanded in a set of N basis functions:

- -

N

M A X , V I = q A ( X , r) 13)j = 1

where f i ( x , y is the f t h basis function and I? and Vi are its

unknow n am plitude for he dipole and slot, respectively. Two

types of expansion modes are considered: entire domain basis

(EDB) and piecewise sinusoidal (PWS). The expansion function

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IEEERANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. A P - 3 3 , NO. 6, JUNE 1985

(24)

The integral in 19) is obtained hroughnumerical integra-

tion. To improv e he com putational efficiency of the solution,

we use the method of rewriting the Green's function as a sum of

aclosed-formexpressionanda fast converging integral [131.

The Green's function in a homogeneous medium of relative per-mittivity E can be writtenas

the edge singul rity is enforced in 14).

UsingGalerkin's method in the spectraldomain, (10) and

11) reduce t o the following matrix equations to be solved for

unknowns1; and Vis:

[ P [ I D ] = [ P I

[Y ] V ] = [I

(22)

f i ( x , y , E,,(x, y andJ,,(x, y are defined on the antenna

E o x ( x , y and J o y @ , y are defined outside he

ent, by using Parseval's relation, (21) and (22) can be writtens follows:

where k , = w d b x . For largevalues o f f l ( = d m j ,Q(kx, ,) i n (20) and Qh kx, ,) in (25) behave asymptoticGy

as

Q(kx k y1

Therefore, if the relative permittivity of the hom ogeneo us mediu m

is selected as = 1 + ~ , ) / 2 , hen Q(k,, k,,) wiu have the same

asymptotic form as Qh(kx, ,). The integralsZf and Y: of 19)

are then split into tw o parts as follows:

+ \ [Q(k.xyb)-Q ~, y)l

?j*(kX, k,)&> k,) dk dk , (28)

where the econd integral converges rapidly since Q(k,, k,)

approaches Qh(kx , ,) for largevalueof 0. The fust integral

is equal to the mpedancematrixelement ofan antenna in a

homogeneousmediumand iswell know n foi both entire do-

main an d piecewise sinusoidal basis functions. (See, for examp le,

The expression for the far field can be obtained by evalu ateg

the Fourier transform asymptotically byhe methodof stationary

phase :

1141.I

,-jkR

E&) = {zx(k,, k,) cos 9 + E, , k y ) sin 91 (29)

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KOMINAMI er al.: SEMI-INFINITESUBSTRATES 603

with

k , = k sin cos Q

k , = k sin O sin Q

y = k cos 8

and

k , (air: I e I 5 K/Z)

G r k o dielectric: I O - n 6 nJ2 .

Once the unknown complex coefficients Zy and V;' are

determined by solving (17) and 18), the radiation field can be

obtained from(29) and 30).

The directive properties of an antenna are described in terms

of pow er gain, given as follows:

where-

and Pin is the tota l npu t power to the elements. The power

division in each medium is of particular interest. In order to

compare the power radiating int o he dielectric and into he

ai r consider the power density ratio for 0 = 0 and 0 = K . Then,

from (29), 30),and (32), we get

(33)

This indicates that the broadside power density division in eachmedium varies :/ for both printed dipole and slot antennas

on semi-infinite substrates. In addition, or the case of a slot

antenna, t he tota l power radiated i nto the two half-spaces also

vanes as € : I 2 , since the slot antenna pattern is the same in air

as in the dielectric.

B. Infinite Phased Arrays

The structure to be analyzed, shown in Fig. 2, consists of a

rectangular array of printed dipole or slot antennas on a semi-infinite substrate. The array is planar and infinite in extent

and the elements are fed in an equi-amplitude, progressive-phase

fashion. The active impedance can be determined using the mo-

ment metho d in conjunction with theperiodic-structure approach

Because the structure is periodic theantenna element cur-

rents can be expanded in a Fourier serieswhose components

represent space harmonics, and the set of equations IO) and 1 1)

can be applied to the infinite array problem. Since the array

elements are assumed flat, the current distribution onhe elements

is confined t o a single plane. I t is furth er assumed that these cur-

rents are unidirectional, flowing only t h 2 length of th e elements,

as was the case for a single dipole or slot.A Fourier series expansion of the current density can be writ-

~ 5 1 .

Fig. 2. Geometry of the iniinite array of antennason a semi-infinite substrate.

ten as

m m

c

(34)

(3 5)

2 r n

bk , , =- + kov

u = sin 0 cos Q = sin sin 4.

In order to evaluate the coefficients, both sides of (34) and (35)

are multiplied by exp (jkxmzx+ jk , ,a) and then integrated over

t he cell area show n in Fig. 2. Use s made of the ortho gona lity

property of the exponentials, and the result is

(37)

where FX(k,k , , ) and i x ( k x k , ) are Fourier transforms of the

electric current for a single dipole and magnetic current ora

single slot, respectively.

Substituting the Fourier transforms of (34) and (35) together

with 36) and (37) into (10) and (1 I), we obtain the following

set of equations:

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604 IEEERANSACTIONS O N ANTENNAS A N D PROPAGATION, VOL. AP-33, NO. 6, JUNE 1 9 8 5

The moment method can now be applied to these equations.

Using t he same procedure as for the isolated elements, the matrix

elements Z j nd Y:. for the nfinite arrays can be expressed as

?(kxrn, k y n ) j j ( k x m , k y n (40)

where Q ( k x , k,,) is defined in (20). The voltage element ‘9and the current element 1; are given by (23) and (24), respec-

tively.

III. NUMERICAL RESULTSBased on the previously developed analysis, numerical compu ta-

tions have been performed for printedantenna elements and

arrays on semi-infmite substrates. Twoubstrate materials

-PTFE ( E , = 2.55, tan 6 = 0.001- .003;X -band) and Gallium

Arsenide (e, = 12.8, tan 6 = 0.0 02 ; X-band)-were chosen for

comparison in this paper.

The input admittance (Yi, = G + B ) of printed dipoles has

been plotted against element length in Fig. 3 for two EDB, three

PWS, and five PWS expansion m odes. From these results it seems

that there is not much difference between these three expansion

sets for isolated elements. Therefore, 2 EDB expansion modes

were used for subsequent isolated element calculations.Fig. 4 shows the nput impedances of (a) aprinted dipole

and (b) a slot antenna versus length L , for wo different sub-

strates. The impedances calculated from Booker’s relation are

also shown in these figures. It is given as

where ee is the effective permitivity discussed n the previous

section. Note that Booker’s relation should not be expected

to be satisfied exactly since the printed dipole is no t the strict

dual of the p rinted slot but, as can be seen, Booker’s relation is

approximately satisfied with the mean permittivity. Similarresults have been noted in [161.

Fig. 5 shows the required length for the first resonance of aprinted dipole and a slot element versus substrate dielectric con-

stant e,, for W/L = 0.02. Also shown is the resonant length

givenby 0.48 A [17] , where the correction factor A is modi-

fied fo r a flat element of width W (whose equivalent radius is

W/4) on a dielectric interface, as

As can be seen, agreement i s very good for bothelements, and the

correction factor (42) should be useful for designing flat elements

on semi-mfiite substrates. Fig. 6 shows the resonant resistance

of printed dipoles and slots versus dielectric constant, e for

W/L = 0.02. The limiting values of resonant len gth and resistance

Pr in ted d ipo le

EDB m o d e s W/L = 002( =12.8) - _-- 3 PWSodesan 6 0.002

1

0. 0.2 0.3 0 4 0.5

LENGTH ( L/X,)

Fig. 3. Input admittance of the printed dipole on a semi-infinite substrate for

different basis functions.

~~

c: Pr in tedipo le -‘dipole

----- Booker ’ s e lo l i onW/L = 0.02

t a n 6 = 0.002

0. 2 0.3 0.4 0.5

LENGTH ( L/X,)

a)

- I Ic: 4

I I

01 0 2 0 3 0 4 0 5

LENGTH L/X,

c b )

Fig. 4. Input impedance of antennas on a semi-infinite substrate. (a) printe

dipole. @ Slot antenna.

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K O M I N ~ It a[ . :SEMI-INFINITE SUBSTRATES 605

E - plane no H - plane

Resonantength I

0 2 -Printed dipole

-.- .48A

1 3 5 7 9 13

PERMITTIVITY, € r

Fig. 5 . Resonant length of antennas on a semi-infinite substrate.

100 500

'\Resonantesistance

4300

PERMITTIVITY, € r

Fig. 6 . Resonant resistance of antennas on a semi-infinite substrate.

of very thi n dipoles on a hick grounde d substrate asgiven in

[181 agree quite well with the results in Figs. 5 and 6.

In Figs. 7(a) and 7(b), power gain pa tte rns for resonant dipoles

and slots with W /L = 0.02 are shown for er = 1.0, 2.55 and 12.8.

The resonant lengthsareL,=0.357ho(~,=2.55)and0.177ho(e,=

12.8) for the printed dipole and are L , = 0.361 h0(e, = 2.55)

and 0.185 (E, = 12.8) for the slot. For the dipole theH -plan e

pattern in the d ielectric has a maximum at he critical angleB C = 71 - h - l ( f i r - ' ) and the E-plane pattern has aminimum

there. Both patterns have a null at the nterface except he H -

plane patte rn for E, = 1.0, as discussed by Rutledge e t al. [2 ] .For the slot, the H-plane pattern has a null at the interface but

the E -plane pattern has no null there, as note d by Brewitt-Taylor

et al. [ 9 ] . A maximum and/or minimum at the critical angle

does not o ccur for slot antenn as because the condu cting plane

effectively isolates the two media.In the mom ent method formulation surface wave fields and

space wave fields are easily separated from the Somm erfeld-type

integral expression for the total fields of a current source on a

grounded dielectric slab [ l I] -the surface waves coming from the

residues of the surface wave poles. But this sepa ratio n doe s not

apply for the printed antenna elementsn semi-infinte substrates,

since the Somm erfeld-type integral given by 19) has only virtual

poles on the mprop er Riemann sheet [19] . Thus, the semi-

M i t e substrate is advantageous w hen comp ared to the dielectric

slab since no power will be lost to surface waves.

.

900

RESONANT PRINTED DIPOLE(a)

H - plane E - plane

180'

RESONANT SLOT

(b)

Fig. 7. Power pattern for resonant element. (a) Printed dipole. (b) slotantenna.

Next, some numerical results for infinite arrays will be shown.

The current on the printed antenna elements is no longer sym-metric, so the symmetric entire domain modes of (15) are n o t

sufficient: andseven PWS expansion modes (16)wereused in

this calculation. The arrayspacingwasselected so that grating

lobeswill not occur within he critical angle. To avoidgrating

lobes within OC(=n- in-' f i r - ' ) the spacing should be

(43)

For E = 2.55, a = b = 0.3851 A and for e, = 12.8, a = b =

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06 IEEE TRANSACTIONS O N ANTENNAS AND PROPAGATION, VOL. AP-33,NO. 6, U N E 985

I I

141 228

Printed Dlpole J

E r = 2.55

1 0 -

L = 0365Xo

o = b = 0 8 5 1 X.

0 8 - w = 0 . O 1 X a

I R I o 6 - N S = 7N = M = ? 5 0

0 4 -

150-

I R I

air ielectr ic

SCAN ANGLE, e(a )

I 1 ~~

125.96' 151.39' 163.77 1

L = 0 .1 8 2 Xa

0 8 1 O O I X o

Printed Dipole

E. = 128

0 b = 0 2184Aa

N M = f50

0 4 -

0 2 -

30' . 60° 90 120' 150'

alr ielectrlc

SCAN ANGLE, e(C)

I R I

141.228°

0 = b = 0 851X0

0 6 -

Ns = 7

N = M =?50

0 4 -

o l rlelectr ic

I D-plane broting l obe bbundoryA( - l , - l ) -I,O) O,-l)125.95' 151.39O 163.79'

o i r i e lec t r i c

SCAN ANGLE, 9

d)

Fig. 8. Reflection coefficient magnitude of an infinite array. (a) Resonant printed dipole for E = 2.55 . @) Resonant slot antenna for e,= 2.55. c) Resonant printed dipole for E, = 12.8. (d) Resonant slot antenna for E = 12.8.

X The resonant length L , and resonant resistance

t broadside R,, for printedantenna elements are calculated

(a) printed dipole: E, = 2.55, L , = 0.3650h0, R, = 60.2 R

(b) slot: E, = 2.55, L , = 0.3636X0, R, = 352.7 R

(c) printed dipole: E, = 12.8, L , = 0.1820X0, R, = 25.9 R

(d) slo t: e, = 12.8, L , = 0 . 1 7 7 7 b , R, = 270.8 R.

elements, th e reflection coefficient

versus scan angle for E-plane, H-plane , and a diagonal

can plane are shown in Figs. S(a), 8(b), 8(c) and 8(d), respec-

ively. The reflection coefficient is calculated as

44)

Z = Zi, at broadside, and Zi, is the input impedance

U computations are made using

s upper limits for the series in (34) and (35). For scanning

in the ai r region, the reflection coefficient magnitude increased

montonically with scan angle up to 8 = 90 . Fo r scanning in th e

dielectric region, the magnitude variations with scan angleare

more complicated. As discussed for isolated elements, the excita-

tion of surface waves on the semi-infinite substrate is negligibleor nonexistent. Thus, in these figures, there is no surface wave-induced blind spot, which is important to he problem of an

infiite array of printed dipoles and patches on a grounded di-

electric substrate [SI,7] . The unity reflection coefficient

magnitude angles 6 = 141.228' for E, = 2.55 and 6 = 163.77'

for e, = 12. 8 correspond to the critical angle for th e propagation

from region 2 to 1: and 6 = 125.95' and 151.39' in the D-

plane of e, = 12.8 correspond to th e grating lobe bound ary of

Floquet modes (-1, -1) and (-1,O) or (0, I) , respectively.

Fig. 9 shows the ratio of the power radiating into the dielec-

tric and into the ar for an i n f ~ t e rray of printed dipoles on

a semi-infiite substrate. The ratio is constant in H-plane anddecreases monotonically with scan angle in the E- and D-planes.

The result is almost the same for the infinite slot array. As shown

in the figure, the broadside power division in each medium

varies as e;/* for the i n f a t e phased array. The ratio is different

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607

I I I I

4

8 12.8

SCAN ANGLE IN AIR, eFig. 9 . Ratio of the powers radiated into the dielectric and into the air as a

function of scan angle.

fromhe value forhesolated lements, because the rray

field is given by the sum of radiated fields of each of the ele-

ments, and propagateaway from the array as a plane wave.

IV. CONCLUSION

The mpedanc e and radiation characteristics of dipolesand

slots printed on a se mi-i nfiite substrate have been investigated.

A solution for an infinite phased array of these printed elementsis also presented. The current distributions have been obtained

by employingamomentmethod to solve thespectraldomain

algebraic equationsorresponding to Pocklington’s integral

equations in the space domain.

Booker’s relation, the resonant length and resistance, and the

radiation power pattern for the isolated elements are comp uted.

Further,he eflection oefficien ts and the roadside power

division for infM tephased arraysare calculated.

The present method can be easily extended to other printed

geometries on a semi-infinite substrate (bow-ties, ring, twin-slot,

etc.) and to mutual coupling problems by obtaining the Fourier

transforms of the appropriate current distributions.

ACKNOWLEDGMENT

The authors would like t o than k Professor Yngvesson of the

University of Massachusetts for his interest in the mplications

of ths wo rk, and Professo r Jelen ski of the U niversity of Massa-

chusetts (visiting) for helpfuldiscussions.

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D. M. Porn, “Input impedance and mutual coupling of rectangularmicrostrip antennas,” IEEE Trans. Antennas Propa gat., vol. AP-30,

T. Itoh and W. Menzel, “A full-wave analysis method for openmicrostrip struc tures,” IEEE Trans.Antennas Propagat., vol. AP-29,

D. M. Pozar, “Improved computational efficiency for the momentmethod solution of printed dipoles and patches,” J . Electromagn. SOC.,vol. 3 no. 3-4, pp. 299-309, July-Dec. 1983.W. .Stutzman and G . A. Thiele, Antenna Theory and Design. NewYork: Wiley, 1981, pp. 329-332.A. A. Oliner and R.G . Malech, “Periodic-structure approach: Largeslots and dipoles,” in Microwa ve Scanning Antennas, Vol. ZI, Array

Academic, 1966, pp. 247-268.

Theorynd Practice, R. C. Hansen, Ed. New York, London:

D. B. Rutledge, D. P. Neikirk and D. P. Kasilingam, “Integrated circuitantennas,” in Infrared and Millimeter W aves, vol. 10, K. I.Button,Ed. New York: Academic, 1983.J. D.Kraus, Antennas. New York: McGraw-Hill, 1950.P. B. Ka tehi and N. G . Alexopoulos, “On the effect of substratethickness and permittivity on printed circuit dipole properties,” IEEETrans. Antennas Pro paga t., vol. AP-31, pp. 34-39, Jan. 1983.A. Baiios, Dipole Radiation in the Presence of a Conducting Hau-Space. New York: Pergamon, 1966, p p . 53-62.

,e

AP-29, pp. 99-105, Jan 1981.

pp. 232-246, Mar. 1984.

pp. 1191-1196, NOV.1982.

p ~ .3-69, Jan. 1981.

Masanobu Kominami, fora photograph and biography please see page 792 of

the September 1981 issue of this TRANSACTIONS.

David M. ozar (S’74-M’80), for a photograph and biography please see page

4 of the January 1985 issue of this TRANSACTIONS.

Daniel H. Schanbert (S’68-M’74-SM’79), for a photograph and biography

please see page 85 of the January 1985 issue of this TRANSACITONS.

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