Network Oriented Dyadic Time Domain Green's Function for a Sequentially Excited Infinite Planar Array of Dipoles in Free Space

Abstract

During the past few years we have performed an effective wideband analysis for characterizing the electrodynamic behavior of phased array antennas, infinite periodic structures, frequency selective surfaces and related applications, with emphasis on gaining physical insight into the phenomenology of short-pulse radiation. The present contribution shows the current status of our network-oriented dyadic time domain Green's function (TD-GF) for a planar array of sequentially excited dipoles that constitutes a prototype study of sequentially short-pulsed radiation by infinite periodic arrays. The dispersive effects of the TD-Floquet waves (FW) and the consequences of a TE and TM field decomposition are discussed in details. In this network formulation, the equivalent TD transmission line (TL) voltages and currents excited by equivalent TD current generators are determined. An expression for the TM voltage TL-GF is provided for the first time in terms of incomplete Lipschitz-Hankel integrals and Bessel functions together with some remarks for its evaluation. As shown in previous publications for the scalar potentials, real-TD physical observables are determined by pairing TD Floquet waves with positive and negative indices. It is shown that the vectorial TD radiated field is reconstructed at any time with the superposition of a few TD-FWs.

Full text

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2228IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 8, AUGUST 2007
Network Oriented Dyadic Time Domain Green’s
Function for a Sequentially Excited Infinite Planar
Array of Dipoles in Free Space
Filippo Capolino, Senior Member, IEEE, Leopold B. Felsen, Fellow, IEEE, and Alessandro Della Villa
This paper is dedicated to Prof. Leopold B. Felsen, mentor and
friend. The paper contains several of his ideas and he checked
and wrote parts of it. Unfortunately he did not see the final
version.
Abstract—Duringthepastfewyearswehaveperformedaneffec-
tive wideband analysis for characterizing the electrodynamic be-
havior of phased array antennas, infinite periodic structures, fre-
quency selective surfaces and related applications, with emphasis
on gaining physical insight into the phenomenology of short-pulse
radiation.Thepresentcontributionshowsthecurrentstatusofour
network-oriented dyadic time domain Green’s function (TD-GF)
for a planar array of sequentially excited dipoles that constitutes a
prototype study of sequentially short-pulsed radiation by infinite
periodic arrays. The dispersive effects of the TD-Floquet waves
(FW) and the consequences of a TE and TM field decomposition
are discussed in details. In this network formulation, the equiva-
lent TD transmission line (TL) voltages and currents excited by
equivalent TD current generators are determined. An expression
for the TM voltage TL-GF is provided for the first time in terms
of incomplete Lipschitz-Hankel integrals and Bessel functions to-
gether with some remarks for its evaluation. As shown in previous
publicationsfor thescalarpotentials, real-TDphysicalobservables
aredeterminedbypairingTDFloquetwaveswithpositiveandneg-
ative indices. It is shown that the vectorial TD radiated field is re-
constructed at any time with the superposition of a few TD-FWs.
IndexTerms—Arrays,Greenfunction(GF),periodicstructures,
short pulse radiation, time domain (TD) analysis.
I. INTRODUCTION
E
relevant in the contest of multilayer metamaterials), suggests
that an analytic framework parameterized directly in the time
domain(TD)mightleadtobetterproblem-matchedphysics,and
NGINEERING interest in ultrawideband/short pulse phe-
nomena pertaining to phased periodic planar arrays (also
Manuscript received June 9, 2006; revised December 31, 2006. This work
was supported by the EU-funded project METAMORPHOSE (FP6/NMP3-CT-
2004–500252).
F. Capolino and A. Della Villa are with the Department of Information
Engineering, University of Siena, Via Roma 56, 53100 Siena, Italy (e-mail:
capolino@dii.unisi.it).
L.B.Felsen,(deceased),waswiththeDepartmentofAerospaceandMechan-
ical Engineering and the Department of Electrical and Computer Engineering,
Boston University, Boston, MA 02215 USA, and also with Polytechnic Univer-
sity, Brooklyn, NY 11201 USA.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAP.2007.901841
thereby to better numerical convergence. Moreover, a network
(transmissionline) approach has conventionally been utilized in
the analysis and design of such structures. These circumstances
have motivated the present investigation of a TD network-ori-
ented representation of the vector electromagnetic field radi-
ated by an infinite periodic array of sequentially-pulsed elec-
tric dipoles, that constitutes a prototype study of sequentially
short-pulsed radiation by infinite periodic arrays.
Wehavepreviouslyinvestigated canonicalTD dipole-excited
Green’s functions (GF) for infinite [1] and truncated [2] peri-
odic line arrays, and for infinite [3] and semi-infinite [4] peri-
odic planar arrays. The radiated field there has been expressed
and parameterized in terms of a few TD Floquet waves (FWs).
A generalized frequency domain (FD)-TD transform for linear
arrays of dipoles is in [5]. The resulting scalar TD-GF has al-
ready been used advantageously to construct a fast TD method
ofmomentsalgorithm forwideband analysisofinfinite periodic
structures [6].
A principal feature of the network-oriented approach is that
E (TM) and H (TE)-type TD-FW modes can be separated and
treated individually. The field is expressed in terms of TD trans-
mission line (TL) Green’s functions that obey standard network
theory. Therefore, possible infinite planar vertically inhomoge-
neous media may readily be incorporated into the formalism.
Interesting causality issues accompany such E and H mode de-
compositions: it is found that individually, each E and H mode
is noncausal. Causality on the total TD-FW vector mode field
is recovered by summing the E and H mode contributions. In
order to parameterize the TD-FW behavior, we begin with the
solution in the FD, with subsequent inversion to the TD. The
formal aspects of the analysis follow the traditional lines in
[7], to which we refer frequently. Thus, the transverse
vector FW mode fields are expressed in terms of transverse
FW-modescalareigenfunctions.Longitudinalfieldpropagation
is described by voltage and current TL-GFs,
TD, these TL-GFs can be evaluated in terms of Bessel func-
tions and incomplete Lipschitz-Hankel integrals (see [8] for a
summary of the main features for the scalar case combined with
a preliminary analysis for the vectorial case). Numerical ex-
amples of radiation from infinite planar arrays of dipoles with
short-pulse band-limited excitationdemonstrate the accuracy of
the TD-FW algorithm, and illustrate the rapid convergence of
the (TD-FW)-based field representation since only a few FW
terms are required for describing the off-surface field radiated
by the planar array. Results for the case of an array simultane-
ouslyexcited(nonphased)havealreadybeenusedinacombined
th
and; in the
0018-926X/$25.00 © 2007 IEEE

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CAPOLINO et al.: NETWORK ORIENTED DYADIC TD-GF FOR INFINITE PLANAR ARRAY2229
Fig. 1. Geometry of an infinite planar periodic array of dipoles. ? , ? : in-
terelement spacing along ? and ?, respectively. ???? ? ??: phase gradient of
the excitation (i.e., the wavefront) along the direction ? (see [3]). ?
phase speed along ? .
? ???:
(TD-FW)-FDTDalgorithm, shownin[9],for theanalysisof pe-
riodic arrays of complex scatterers.
The results of this paper lay the foundation to include layered
media in the TD-FW analysis to widen the TD-FW applications
already shown in [6] and [9].
The nonphased case (all array dipoles simultaneously ex-
cited) of the TD analysis shown in this paper is strictly related
the TD analysis of wave propagation in closed waveguides (see,
for instance, [10]).
II. STATEMENT OF THE PROBLEM
The canonical problem for the network formulation under
study consists of the generic infinite periodic array of electric
dipoles, shown in Fig. 1, oriented along the constant vector
withperiodicities
andalongthe
tively. The observation point is denoted by
transverse coordinate
intheperiodicarrayislocatedat
+and
symbols definevectorquantitiesand
tors along , , , respectively. The FW-based modal FD and
TD fields due to the array are related by the Fourier transform
pair
,
and directions,respec-
, with
. Theth dipole source
,where
. Here, bold face
,denoteunitvec-
,
,
(1)
in which
denotes time-dependent quantities.
The phased FD and sequentially pulsed TD dipole array cur-
rents
and, respectively, are given by
can be either a scalar or a vector quantity; a caret
(2)
where
and
,
. Moreover,denotes the
ambient wavenumber,
the ambient permittivity and permeability) denotes the ambient
wave speed, and
and
(withandrepresenting
, with
(3)
are the interelement phase gradients along
[3].IntheFD,thecompositelinearphasingonthearrayisalong
thedirection
,perpendicularto
rotated through the angle
with respect the
this translates into interelement excitation delayed by
andalong and , respectively; i.e., the sequentially
pulsed dipole elements located at
time
. The important nondimensional single
parameter
and , respectively
(seeFig.1),and
axis. In the TD,
are turned on at
(4)
combines both phasings
impressed phase speed along . Here, we treat the case
which implies excitation phase speeds
spondingprojected phasespeeds
ambient wavespeed [1], [3]. The nonphased case
sponds to simultaneous excitation of all the dipoles.
and[3]. In (4),is the
(and corre-
) larger thantheand
corre-
III. FD GREEN’S FUNCTION: DECOMPOSITION INTO TE AND
TM MODES AND PREPARATION FOR FOURIER INVERSION
The FD electric and magnetic vector fields
at location , excited by the array of time-harmonic
transverse dipole sources given in (2), oriented along
represented a sum of FW modes
and
, are
(5)
(6)
Each
th indexed FW is explicitly separated into its E and
H parts
(7)
(8)
The steps detailed in Appendix A lead to the following con-
venient form for the transverse components
the modes
and
and of
(9)
(10)
and
(11)
(12)

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2230IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 8, AUGUST 2007
Fig. 2. Frequency and time domain FW-based transmission line (TL) voltage
and current Green’s functions excited by a unit current generator at ? .
where the factor is given by
(13)
The present analysis is limited to transverse-to- dipoles, the
simplercasewithdipolesorientedalongthe axiscanbetreated
analogously. Here,
H voltages at
excited by a delta current generator at
Fig.2),while
isthecurrentat excitedbythesame
generator [7, Sec. 2.3c]. The vector wavenumber
and are the E and
(see
(14)
(15)
combines the two Floquet-type dispersion relations
(16)
(17)
with
the vector component transverse to , and
-independentpartofthevectordispersionrelationin(14).Flo-
quet waves with transverse propagation constants
or, with
or evanescent, respectively, along the -direction (we have as-
sumed a lossless environment).
In (9)–(12) [as well as in (64)–(68)] we have introduced the
FD admittance and voltage TL Green’s functions
. The subscript “ ” ondenotes
represents the
, are propagating
(18)
shown in Fig. 2, where the longitudinal Floquet wavenumber is
(19)
Fig. 3. Topology of the ? plane for the evaluation of the inverse Fourier trans-
form.
with definitions
(20)
(21)
The quantity
duced by a voltage generator at
derivation in Appendix A. The top Riemann
is defined by
or, respectively [7, p. 207], [1], [3]. The charac-
teristic impedances for E and H modes are
represents the current at
, and it is here used in the
pro-
-sheet in Fig. 3
or, andfor
(22)
The -components of the fields can be determined from the
transverse components.
IV. TD GREEN’S FUNCTION: FOURIER INVERSION OF THE FD
MODAL EXPANSION
The total TD field is obtained as a superposition of the
TD-FWs
(23)
(24)
obtained by Fourier-inverting the FD modes in (5) and (6). Ac-
cordingly, each TD mode is decomposed into its E and H com-
ponents
(25)
(26)
Here, each
of (7) and (8) and is carried out by first analyzing the critical
points in the complex
-plane and then applying standard ana-
lytic Fourier transforms.
th TD-FW is determined by Fourier inversion

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CAPOLINO et al.: NETWORK ORIENTED DYADIC TD-GF FOR INFINITE PLANAR ARRAY2231
A. Analysis of Singularities in the Complex-Plane
1) Physical Branch Points,
modaladmittanceandvoltageTLGreen’sfunctionsin(18)con-
tainthelongitudinalwavenumber
fore they contain
-dependent branch points at
shown in Fig. 3 that cause
are physical because they are related to the usual radiation con-
dition at
[3]. Branch cuts are defined by imposing
, and they separate the two Riemann sheets
where the spectral wave components exponentially decay (the
top one, where
) or grow (the bottom one) at
.
2) Nonphysical Poles,
: Introducing the frequency shift
,thevector
into
: The expressions for the
givenin(19).There-
vanish. They
in(14)istransformed
(27)
and
(28)
The
th E and H mode inversion integrals arising from (9)
and (10) have poles at frequencies
, i.e., for
, where
(29)
with the sign
axis (see Fig. 3). Note that the poles move to
phased case
. These poles are nonphysical since their
contributionviolatescausalityfortheindividualEandHmodes.
However, when the
th E and H mode contributions are super-
posedtoobtainthetotal
thTD-FWfield,thepolesingularities
cancelout,andthesumin(7)iscausal,havingonlybranchpoint
singularities above the
-integration path in Fig. 3. The cancel-
lation is demonstrated by first observing that (see Appendix B)
denoting the pole above/below the real
for the non-
(30)
Next, the two characteristic impedances in (22), evaluated at
, become
(31)
where we have used the fact that, by definition, the
rected wavenumber in (19) for
. This implies that for
the nonphysical poles),
di-
becomes
(near
(32)
Therefore, (9) and (10) for
, are such that
, which proves the pole
cancellation, and restores causality for the sum of the E and H
modes in (7).
B. TD Transmission Line Green’s Functions
A TD version of the TL Green’s functions, schematized in
Fig.2,isobtainedbyFourier-invertingtheFD-TLGreen’sfunc-
tions in (18)
(33)
in which
branch points at
of the integration path is chosen in accord with the radiation
condition at
(causality) for any
path from
to is shifted below the branch cuts (see
Fig. 3), where
in accord with the radiation condition specified in the text after
(21) (see also [7, p. 35] where, to ensure the existence of the
Fourier transform pair in (1), the
contour of integration in (33) is shifted slightly below the real
-axis into). The dashed area in Fig. 3 represents
the side of the branch cut where
Note that, because of the
-dependency in (22), the evaluation
of
is analogous to that of
1) The Nondispersive Case
sive case
,
voltage and current are evaluated exactly as
is given in (19). The integrand in (33) has
, as shown in Fig. 3, and the indentation
; therefore the integration
orforor
variable and therefore the
(see also [5]).
.
: For the nondisper-
, and the TD
(34)
where
(35)
defines the turn-on time at which the impulsive wavefront
reaches the observer at
space impedance.
2) The Dispersive Case,
higher order
-mode voltages and currents are evaluated
as in Appendix C (see also [9] for the nonphased case, and
[11]–[13] for similar dispersive problems), yielding the closed
form expressions
, andis the free
or: Dispersive,
(36)
(37)
(38)

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2232IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 8, AUGUST 2007
with
(39)
Here,
tionsofthefirstkind,and
roth-orderincompleteLipschitz-Hankelintegralofthefirstkind
(see [14] and references therein for more details). In (36)–(39)
and are the zero-th and first order Bessel func-
istheze-
(40)
(41)
The evaluation of
carried out in this paper for the first time. The main steps of
its derivation are shown in Appendix C. Note that all the
TD-TL GF have the same turn-on time
step function
.
is rather complex and it is
th
, defined by the unit
C. Fourier Inversion of the E-H-Decomposed
The TD electric and magnetic fields in (25) and (26), decom-
posed into their E and H components, are now rewritten as
th FD-FWs
(42)
(43)
in which
denotes time convolution, and
(44)
The retarded time
(45)
in the TL GFs in (42) and (43) [cfr. (36)–(38)] appears because
of the
TD dyads in (42) and (43) are obtained by Fourier-inverting
a corresponding group of terms in the FD-FW expressions in
(9)–(12). We evaluate here explicitly only
factor inin (9)–(12). The
(46)
From (27)
(47)
where we have used the shifted frequency
(27). After decomposition of (46) into three integrals with dis-
tinct
dependency, we obtain the expression
defined just before
(48)
with TD coefficients
(49)
(50)
(51)
where
commonfactor
mentionedshiftoffrequency.Notethat
tion since it has been evaluated by applying Cauchy’s theorem
with residue contributions from the nonphysical poles at
in (29), both above and below the
located slightlybelowthereal axis(Fig. 3).The remainingdyad
for the electric field in (42) is then evaluated by using the prop-
erty
that leads to
is a function ofas shown in (28), and the
arisesfrom the
isanoncausalfunc-
integration path which is
(52)
Similarly, those for the magnetic field in (43) are evaluated
using
andthat yield
(53)
(54)
Note that in (42) and (43) the TD-TL Green’s functions de-
fined in (36)–(38) are causal since the signal arrives at
turn-on time
() (see [3] for a physical interpre-
tation of
andreference systems in terms of moving coordi-
nates.) The E,H-decomposed TD field expressions in (42) and
(43) are noncausal since they are determined by a time convo-
lution between casual TD-TL Green’s functions and noncausal
dyads that are time-spread around
strated in Section IV-A, causality is recovered by summing the
E and H constituents in (25) and (26). This can easily be seen
directly in the TD for the magnetic field in free space where the
TL Green’s functions
for the E and H cases coincide: the
sum of the E and H components is equivalent to the sum of the
TD dyads
is not spread around
.
1) The Case of the Array Elements Simultaneously Excited
(Nonphased): When the array dipoles are simultaneously ex-
cited (the nonphased case)
dyads reduce to
at the
. However, as demon-
, which
and the TD coefficients in the
(55)