SEM model for physics-oriented analysis of time-domain radiation properties of antenna-based sensors

Measurement 46 (2013) 985–992

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A hybrid FDTD/SEM model for physics-oriented analysis of time-domain radiation properties of antenna-based sensors D. Caratelli a, A. Lay-Ekuakille b,⇑, P. Vergallo b, A. Massaro c, A. Yarovoy a a

Microwave Sensing, Signals and Systems, Delft University of Technology, The Netherlands Department of Innovation Engineering, University of Salento, Italy c Center of Biomolecular Nanotechnology, Italian Institute of Technology, Italy b

a r t i c l e

i n f o

Article history: Received 6 July 2012 Received in revised form 23 September 2012 Accepted 15 October 2012 Available online 23 November 2012 Keywords: Antenna-based sensors Locally conformal finite-difference timedomain technique Singularity expansion method Spherical harmonic expansion Incomplete spherical bessel functions Time-domain gain Effective height

a b s t r a c t A semi-analytical methodology is presented for the accurate analysis of time-domain radiation characteristics of antenna sensors. A locally conformal finite-difference time-domain technique is adopted to derive a minimal pole/residue spherical harmonic expansion of the equivalent currents excited on a suitable Huygens surface enclosing the sensing device. In this way, by using the singularity expansion method, the time-domain gain and effective height of the structure can be evaluated analytically, in terms of the newly introduced incomplete spherical Bessel functions, as the superposition of non-uniform spherical wave contributions attenuating along with the time and space according to the complex poles accounting for the natural resonant processes occurring in the device. The accuracy of the developed technique is assessed by application to an ultrawideband bow-tie antenna-based sensor for millimeter-wave radar measurements. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The monitoring of natural processes is of essential importance in many fields of science and engineering. As a matter of fact, the development of efficient antennabased sensing systems is a key issue in different emerging applications, such as radar imaging, environmental monitoring, non-destructive testing, and nanotechnology [1–5]. Advances in sensor technology generally depend upon enhancements in the performance of electronic devices, as well as material characteristics. On the other hand, an accurate sensor characterization is instrumental in order to improve the design reliability. Therefore, rigorous modeling procedures useful to analyze the radiation properties of complex sensing devices are highly desirable. Unfortunately, except for a few classes of structures, such as aperture antennas [6], analytical models for the repre⇑ Corresponding author. Tel.: +39 0832 297822; fax: +39 0832 297827. E-mail address: [email protected] (A. Lay-Ekuakille). 0263-2241/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.measurement.2012.10.022

sentation of the transient electromagnetic field distribution excited in Fraunhofer region are not available in the scientific literature. Therefore, in most cases the evaluation of useful far-field parameters, such as time-domain gain and effective height [7,8], is carried out by using intensive numerical techniques relying on brute-force computation. However, such an approach does not provide an integral physical insight into the mechanisms which are responsible for the electromagnetic behavior of antenna sensors, and requires large computational times and storage of a large amount of data. To overcome this limitation, a suitable semi-analytical formulation based on a locally conformal finite-difference time-domain (FDTD) scheme and the singularity expansion method (SEM) is described in this paper. Using the proposed procedure, the time-domain radiated field is described analytically in terms of incomplete spherical Bessel functions as the superposition of non-uniform spherical wave contributions related to the natural resonant processes occurring in the structure under analysis. In this way, one can gain useful information

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which can be exploited to optimize the performance of antenna-based sensors for a wide variety of applications. 2. Time-domain radiation properties of the antenna sensor Let us consider an antenna sensor operating in freespace and driven through a transmission line by a matched voltage generator vg (t) having internal resistance Rg (see Fig. 1). Denoting by Sh a spherical surface of radius Rh enclosing the structure, the relevant time-domain electromagnetic field radiated in Fraunhofer region is readily obtained as vector slant-stack transform (SST) of the equivalent electric and magnetic currents IS (#, u, t) = [JS (#, u, t) MS (#, u, t)] excited along Sh [7]. At any time, IS (#, u, t) can be conveniently approximated as a finite superposition of spherical harmonics:

IS ð#; u; tÞ ’

N X n X

wn;m ðtÞY m n ð#; uÞ:

ð1Þ

n¼0 m¼n

Therefore, provided that the considered antenna sensor is excited by a finite-duration pulse, the following modified SEM-based representation of the vectorial current expansion coefficients in (1) can be adopted:

wn;m ðtÞ ’

K X cn;m;k esn;m;k t u ðtÞ;

ð2Þ

k¼1

with u (t) being the usual Heaviside unit-step distribution, and where sn, m, k = rn,m,k + jxn,m,k, cn,m,k = [an,m,k bn,m,k] denote, respectively, the complex poles and vector residues relevant to the natural resonant processes occurring in the device. According to [7], the transient electric field radiated by the device can be expressed as the convolution integral of the propagating input current wave at the sensor terminals i+ (t) = vg (t)/(2Rg) and the effective height in transmit T mode h ð^r; sÞ, namely:

g0 þ T Eðr; sÞ ¼  i ðsÞ  h ð^r; sÞ: 4prc0

ð3Þ

Hence, under the assumption of impulse source T (t) = 2Rgd (t), one easily obtains that h ð^r; sÞ ¼ d d 4prc0 E ðr; sÞ=g0 , where the superscript is used to denote the field quantities due to the input current excitation i+ (t) = d (t). In this way, it is straightforward to show that:

vg

T h ð^r; sÞ ¼

Z Z Sh



1

g0

0 J_dS ð#0 ; u0 ; s þ th cos cÞdS

^r 

Z Z Sh

_ d ð#0 ; u0 ; s þ t h cos cÞdS0 ; M S

ð4Þ

with the dot denoting the time derivative, and where th = Rh/c0, cosc = sin#sin#0 cos (u  u0 ) + cos#cos#0 . In (4), s = t  r/c0 is the spherical-wave delayed time, and c0, g0 are the speed of light and the wave impedance in free space, respectively. A trivial reasoning leads to the conclusion that the impulse equivalent currents IdS ð#; u; tÞ ¼ d  JS ð#; u; tÞ MdS ð#; u; tÞ satisfy the integral equation d v g ðtÞ  IS ð#; u; tÞ=ð2Rg Þ ¼ IS ð#; u; tÞ. As discussed in [7,8], the time-domain gain is defined by simply extending the expression of the corresponding frequency-domain parameter as follows: 4p r 2

GT ð^rÞ ¼

g0

kEðr; sÞ  1e k22

Rg ki ðsÞk22 þ

;

ð5Þ

1e denoting the field polarization direction and k  k2 the usual energy (L2) norm. So, taking advantage of (3) one obtains:

GT ð^rÞ ¼

½Riþ ðÞ  g T ð^r; Þ ð0Þ ; Riþ ð0Þ

ð6Þ

where the continuous auto-correlation operator R þ1 RX ðsÞ ¼ 1 X ðfÞX ðf þ sÞdf ¼ X ðsÞ  X ðsÞ has been introduced, and:

g T ð^r; sÞ ¼

g0 4pc20 Rg

RhT

1e

ð^r; sÞ

ð7Þ

is the so-called gain correlator [7], which is an intrinsic energy-related characteristic of the antenna sensor which describes the relevant radiation properties in Fraunhofer region. 3. Near-field modeling of the antenna sensor

Fig. 1. Antenna sensor enclosed by a spherical Huygens surface Sh. The radiating structure is assumed to be connected to a uniform transmission line excited by a real voltage generator vg(t) with internal resistance Rg.

In the presented modeling approach, the rigorous evaluation of the equivalent electric and magnetic currents IS (#, u, t) excited along Sh requires an accurate time-domain full-wave procedure. In this respect, a suitable technique is the FDTD method [9,10]. However, in the conventional formulation of the algorithm [11], each cell in the computational grid is implicitly assumed to be filled in by a homogeneous material. For this reason, the adoption of Cartesian meshes can result in reduced numerical accuracy where structures having complex geometry are to be modeled. In this context, the hereafter detailed locally conformal FDTD scheme provides a clear advantage over the use of the stair-casing approach or unstructured and stretched space lattices, potentially suffering from significant numerical dispersion and/or instability [11]. Such scheme is based on the definition of suitable effective

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material tensors accounting for the local electrical and geometrical properties of the sensor under analysis. Let us consider a three-dimensional domain D filled by a linear, isotropic, non-dispersive material, having permittivity e (r), magnetic permeability l (r), and electrical conductivity r (r). In such a domain, a dual-space, non-uniform lattice formed by a primary and secondary Cartesian mesh is introduced. The primary mesh MD is composed of space-filling hexahedrons having nodes ri,j,k. Thereby, upon denoting the triplet ði; j; kÞ 2 N30 as ! for shortness, the edge lengths between adjacent vertices in MD turn to be Dn! ¼ ^n  ðr!þKn  r! Þ, where Kn and ^n are the fundamental triplet and the unit vector respectively, relevant to the n coordinate axis (n = x, y, z). On the other hand, the secondf D (see Fig. 2) consists of closed hexaheary or dual mesh M drons whose edges penetrate the shared faces of the primary cells and connect the relevant centroids e r ! . As a consequence, the dual edge lengths are found to be ‘n! ¼ ^ n  ðe r!  e r !Kn Þ ¼ ðDn!Kn þ Dn! Þ=2. As usual, the electric field components are defined along each edge of a primary lattice cell, whereas the magnetic field components are assumed to be sampled at midpoints of the edges of the secondary lattice cells. The governing relationship between the E- and H-field components is given by Maxwell’s equations in integral form, namely the Faraday–Neumann’s and Ampere’s laws. In particular, the enforcement of the Ampere’s law on the general dual-mesh e n (see Fig. 2), recell face e S n! , having boundary @ e S n! ¼ C ! sults in the integral equation:

I eC n

H ðr; tÞ  d‘ ¼

Z Z

!

þ

r ðrÞEn ðr; tÞdS

eS n Z! Z

@ @t

eS n

e ðrÞEn ðr; tÞdS:

ð8Þ

!

Under the assumption that the spatial increments Dn! (n = x, y, z) of the computational grid are small compared to the minimum operating wavelength, the contour integral appearing on the left-hand side of (8) can be evaluated by the mean value theorem, neglecting infinitesimal terms of higher order. Besides, the partial derivative operator in

(8) can be approximated by using a central-difference approximation, which is second order-accurate where the E- and H-field components are staggered in the time domain. Therefore, upon introducing the normalized field quantities:



nþ12

Hn j!

¼ ‘ n!

2   1 Hn er !  ^nDn! ; t nþ1 ; 2 2

ð9Þ ð10Þ

with tm = mDt, Dt being the time step selected according to the Courant–Friedrichs–Lewy (CFL) stability condition [11], after some algebraic manipulation the following explicit time-stepping equation is obtained:

E n jnþ1 ¼ !

en j!  12 r  n j ! Dt n Dt nþ1 E n j! þ ðr  HÞn j! 2 ; 1 1 en j! þ 2 r en j! þ 2 r  n j ! Dt  n j ! Dt ð11Þ

nþ1=2 where ðr  HÞn jnþ1=2 ¼ Hf j!nþ1=2  Hf j!K  Hg jnþ1=2 þ ! ! g nþ1=2 Hg j!Kf denotes the finite-difference expression of the n-component of the normalized magnetic field curl. By applying the duality principle in the discrete space f D Þ, the FDTD-update equations of the H-field can ðMD ; M be easily derived as:

nþ12

Hn j!

n1

¼ Hn j! 2 

Dt

l n j!

ðr  EÞn jn! ;

ð12Þ

with Sn! being the general primary-mesh cell face orthogonal to the n coordinate axis. In (11) and (12), the information regarding the local physical properties, as well as geometrical non-conformability to the adopted Cartesian mesh of the sensor under analysis, is transferred to the following position- dependent effective permittivity, electrical conductivity, and permeability tensors:



  Z Z   e 1  ðrÞdS;  ¼ eS n r n  Dn! r e

!

l n j! ¼

Fig. 2. Cross-sectional view of the FDTD computational grid in presence of a non-conformal inhomogeneous structure.



1 en jn! ¼ Dn! En r! þ ^nDn! ; tn ;

1 ‘ n!

ð13Þ

!

Z Z

l ðrÞdS:

ð14Þ

Sn!

It is worth noting that the computation of the effective material parameters (13) and (14) can be conveniently carried out before the FDTD-method time marching starts. As a consequence, unlike in conformal techniques based on stretched computational grids, no additional correction is required in the core of the numerical algorithm. In order to assess the effectiveness of the developed scheme, and show the enhancement in terms of numerical accuracy over the conventional stair-case modeling approach, several test cases have been considered. For the sake of brevity, only the characterization of a metallic cavity loaded with a cylindrical dielectric resonator is presented and discussed in the Appendix. The obtained results clearly demonstrate the suitability of the proposed method to model complex electromagnetic devices featuring non-conformal geometries and heterogeneous electrical properties.

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In the numerical simulations carried out within this study, a ten-cell uniaxial–perfectly–matched–layer (UPML) absorbing boundary condition [12] has been adopted to simulate the extension of the space lattice to infinity. In particular, a quartic polynomial grading of the UPML conductivity profile has been selected in such a way as to obtain a nominal value RPML ’ 104 of the spurious reflection level at the truncation of the computational mesh and, hence, achieve an excellent numerical accuracy in the near-field modeling of the electromagnetic structure under analysis. 4. Evaluation of the time-domain gain of the antenna sensor The closed-form evaluation of the antenna effective height is carried out by introducing the spherical harmonic representation of IdS ð#; u; tÞ as evaluated by means of the detailed locally conformal FDTD procedure. In this way, by

using (2) and the orthonormality property of Y m n ð#; uÞ , it follows that v g ðtÞ  wdn;m ðtÞ=ð2Rg Þ ¼ wn;m ðtÞ, which is Laplace-transformed to V g ðpÞWdn;m ðpÞ=ð2Rg Þ ¼ Wn;m ðpÞ. Under the assumption that V g ðpÞ ¼ Lfv g ðtÞg – 0 at any point in the complex p-plane and jWn,m (p)j = O (jVg (p)j) as p ? 1, the following pole/residue model can be adopted: K X cdn;m;k 2Rg ; Wn;m ðpÞ ¼ Wdn;m ðpÞ ’ cdn;m;0 þ V g ðpÞ p  sn;m;k k¼1

ð15Þ

h i d the poles sn,m,k and vector residues cdn;m;k ¼ adn;m;k bn;m;k being computed by means of a suitable time-domain vector-fitting procedure [13–15]. By combining the time-domain equivalent representation of (15) with Eq. (4), and exploiting the analytical properties of the classical Bessel functions and spherical harmonics, the expression of the effective height relevant to the antenna sensor is found to be, after some mathematical manipulations, as:

    N X n  @ X ð1Þn s s u 1 hn;m;0 Pn @ s n¼0 m¼n 2t h th th   X     # K s s s u 1þ þ u 1þ hn;m;k in sn;m;k t h ; min 1; th th th k¼1

T h ð^r; sÞ ’ Ah

_ sÞY m ð#; uÞ;  dð n

It is to be pointed out that the polynomial terms in the SEM expansion (16) describe the instantaneous effect of the driving voltage on the radiated electric field, whereas the damped wave contributions are related to complex resonant processes occurring in the structure. In particular, the incomplete modified spherical Bessel functions account for the very early transient when only the portion of the surface Sh corresponding to the solid angle Xh = {(#0 ,u0 ): cosc > s/th} intersected by the half space s þ ^r  r0 =c0 > 0 gives contribution to the electromagnetic field in the observation point r = {r > Rh,#,u}. In the limit s ? th each of the terms in (sn,m,kth, min{1, s/th}) in (16) approaches to a time-independent quantity, meaning that the observation point starts to collect wave contributions from the whole Huygens sphere enclosing the antenna sensor under analysis. By using (16) and (17), the correlation integral RhT 1e ð^r; sÞ can be evaluated analytically and, in this way, the gain correlator g T ð^r; sÞ can be determined without any limitation involving the time s or the observation spatial direction ^r. 5. Example of FDTD/SEM modeling of a millimeter-wave antenna sensor In the presented study, the proposed formulation has been applied to the modeling of a resistively loaded bowtie antenna sensor for millimeter-wave applications. A sketch of the structure geometry is shown in Fig. 3. As it appears, the device features two circularly ended flairs, having length lf = 4 mm and angular aperture Hf = 75°, whose electrical conductivity rf is function of the radial distance q from the central delta gap, where a voltage source with internal resistance Rg = 150 X is used to excite the radiating element. To properly enlarge the sensor bandwidth (thus reducing the late-time ringing due to spurious multiple reflections between the flair truncations and the feed point), the resistive loading has been assumed to have a Wu-King-like distribution [18–20]. The near-field full-wave analysis of the considered sensing device has been performed by means of the detailed FDTD-based sub-cell method. In doing so, the individual structure has been meshed on a graded space

ð16Þ d bn;m;k

R2h

where hn;m;k ¼ þ  ^r=g0 , and Ah ¼ 4p is the surface area of the Huygens sphere. In (16), Pn () denotes the Legendre polynomial of order n [16], whereas R1 in ðn; wÞ ¼ 12 w enz P n ðzÞdz is the incomplete modified spherical Bessel function of order n generalizing the canonical complete function in (n) = in (n,1) [14]. In particular, using the sum formula for Pn (z) [16], in (n, w) can be evaluated in closed form as: adn;m;k

in ðn; wÞ ¼ in ðnÞ  en 

n  1 q1



nþ1  X q¼1

q1

C ðq; nð1 þ wÞ; 2nÞ ð2nÞq R z2



n

a1 f

ð17Þ

;

with C ða; z1 ; z2 Þ ¼ z1 f e df being the generalized incomplete Gamma function [17].

Fig. 3. Three-dimensional view of an ultrawideband loaded bow-tie antenna sensor. Device characteristics: lf = 4 mm, Hf = 75°, tf = 0.1 mm. The local coordinate system adopted to express the field quantities is sketched.

D. Caratelli et al. / Measurement 46 (2013) 985–992

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lattice with maximum spatial increment Dnmax = kmin/ 15 ’ 0.1 mm, where kmin denotes the free-space wavelength at the maximum operating frequency fc = 200 GHz in the spectrum of the excitation signal, which is a Gaussian pulse defined by:

vg

"   2# t  t0 u ðtÞ; ðtÞ ¼ V g exp  tg

ð18Þ

where

t0 ¼ Tg ¼ 10

pffiffiffiffiffiffiffiffiffiffiffi ln 10 ’ 2:415 ps: pfc

ð19Þ

The selection of the time parameter Tg according to (19) results in the significant spectral content of the source pulse, measured at 10 dB level, within the frequency band up to fc. The energy delivered by the voltage generator, assumed to feature internal resistance Rg, is accounted for by adding a local current density term in the finite-difference equations used to update the transient distribution of the electric field at the feed section of the sensor probe [21]:

E n jnþ1 !g ¼

en j!g  en j!g þ þ

Dt 2Rg Dt 2Rg

E n jn!g

h i 1 Dt nþ1 ðr  HÞn j!g 2 þ I g jnþ2 ; D t en j!g þ 2R g

ð20Þ

!g being the index triplet relevant to the driving point, and where I g jnþ1=2 ¼ v g ðt nþ1=2 Þ=Rg denotes the discrete-valued excitation current. In this way, the voltage and current accepted at the input terminals of the feeding line can be computed as follows:

v in

ðt n Þ ¼ 

 iin t nþ1 ¼ 2

Z I

CV

CI

E ðr; t n Þ  d‘ ’ E n jn!g ;

ð21Þ

 nþ1 H r; tnþ1  d‘ ’ ðr  HÞn j!g 2 ;

ð22Þ

2

where CV is an open contour extending along the driving point, and CI a closed contour path wrapping around the generator. Hereupon, the input impedance is readily evaluated as:

Z in ðf Þ ¼ ejpf Dt

DFT fv in ðtn Þgðf Þ n o ; DFT iin ðt nþ1 Þ ðf Þ

Fig. 4. Frequency-domain behavior of the input reflection coefficient featured by the millimeter-wave antenna sensor shown in Fig. 3. The sensing device, characterized by a nearly flat input impedance around Rg = 150 X, is well matched to the feeding line in the frequency band between fmin ’ 11.2 GHz and fmax ’ 63.2 GHz.

By using the developed SEM-based approach, the timevariant spherical harmonic expansion coefficients of the surface equivalent currents excited along the Huygens sphere with radius Rh = 4.5 mm have been computed onthe-fly in step with the numerical FDTD simulation, and then fitted to the modified pole/residue expansion (2) with orders N = 10 and K = 20. The resulting distribution of the sensor poles in the complex frequency domain is shown in Fig. 5. The pair of conjugate poles with minimum damping coefficient (r ’ 5.2 ns1) can be readily noticed. The relevant spherical harmonic mode of orders n = 1 and m = 0 is characterized by the resonant frequency f = x/ (2p) ’ 17.2 GHz where the response of the device peaks (see Fig. 4), and the electrical length of each flair is about p/2. As it can be easily inferred, the pair of dominant poles primarily determines the late-time characteristics of the structure, whereas the poles with the larger damping coefficients account for the relevant early transient behavior. Subsequently, the proposed non-uniform spherical wave representation in terms of incomplete spherical Bessel functions has been applied to the evaluation of the T effective height h ð^r; sÞ and, hence, the time-domain gain

ð23Þ

2

with DFT fgðf Þ being the usual discrete Fourier transform operator, and where the correction term ejpfDt accounts for the half-time-step staggering inherent in the FDTD-method marching algorithm. Therefore, the reflection coefficient at the input terminals of the device is given by:

Cin ðf Þ ¼

Z in ðf Þ  Rg : Z in ðf Þ þ Rg

ð24Þ

As it can be noticed in Fig. 4, the considered bow-tie antenna sensor features, by virtue of the selected resistive loading, a good impedance matching (at 10 dB return-loss level) in a very wide frequency band between fmin ’ 11.2 GHz and fmax ’ 63.2 GHz. This noticeable characteristic is surely useful to meet demanding requirements of modern pulse radar applications [18].

Fig. 5. Distribution in the complex frequency plane of the main poles of the resistively loaded bow-tie antenna sensor shown in Fig. 3.

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GT ð^rÞ of the sensor under the mentioned assumption of Gaussian pulse excitation, which implies:

Riþ ðsÞ ¼

t g V 2g

rffiffiffiffi

8R2g

2

p

e

2

s2 2t

g

erfc

! jsj  2t 0 pffiffiffi ; tg 2

ð25Þ

Rz 2 with erfc ðzÞ ¼ 1  p2ffiffipffi 0 ef df denoting the usual complementary error function [17]. As expected from the theory [22], and shown in Fig. 6, the radiation pattern of the device features a stretched donut-like shape with almost uniform level along the H-plane and deep nulls along the bow-tie axis. Furthermore, it is worth noting that the peak value of the gain is GTmax ’ 3 dB since half the power accepted by the device is dissipated along the flairs. In order

(a)

(b) Fig. 7. Block diagram (a) of a time-domain microwave imaging system based on an array and (b) of bow-tie antenna sensors.

(a)

to assess the reliability of the obtained results, and gain a physical insight into the mechanisms affecting the radiation properties of the considered sensor, the relevant time-domain gain has been evaluated by using the definition in (5) and adopting a brute-force numerical integration approach under the assumption that the electric current density excited along the antenna flairs is fairly described by the following Wu-King-like distribution:

    . þ . ^ 1 i ; Jf ð.; tÞ ¼ . t lf c0

ð26Þ

and, hence, the transient electric field distribution in Fraunhofer region can be determined as:

Eðr; sÞ ¼

(b) Fig. 6. Angular behavior of the time-domain gain GT ð^rÞ along the E-plane (a) and the H-plane (b) of the millimeter-wave antenna sensor shown in Fig. 3.

g0 ^ ^ rr 4prc0

Z Z Sf

J_f



.0 ; s þ

 1 ^r  .0 dS0 ; c0

ð27Þ

Sf = {(.,u):0 6 . 6 lf,ju ± p/2j 6 Hf/2} being the surface of the radiating flairs (see Fig. 3). Simple visual inspection of Fig. 6 shows that the analytically based results obtained with the developed hybrid FDTD/SEM technique are in good agreement with the heuristic model, a marginal discrepancy being observed only in the angular region around the end-fire direction # = u = p/2 as an effect of a reduced spurious current reflection occurring at the antenna open ends but not accounted for in (26). This comparison demonstrates clearly the reliability and accuracy and physical soundness of the proposed modeling approach. By virtue of the compact size and good performance in terms of large fractional bandwidth and stable radiation

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identify the presence and location of significant scatterers. As a result, a qualitative image of the object under investigation is finally obtained. As outlined in [23,24], the stable response featured by the considered sensor over a broad frequency band (see Fig. 4) is instrumental in achieving a good tradeoff between spatial resolution and penetration depth in the radar target, which in turn is essential to obtain a good reconstruction quality in the imaging procedure. 6. Conclusions

(a)

(b) Fig. 8. Geometry of a dielectric loaded rectangular cavity (a), and behavior of the relevant fundamental resonant frequency fr as function of the FDTD mesh size Dh (b). Shown is the confidence region where the relative deviation dr = jfr  mrj/mr with respect to the reference resonant frequency mr = 1.625 GHz [15] is smaller than 0.1%. Structure characteristics: a = b = 50 mm, c = 30 mm, D2 = 36 mm, h2 = 16 mm, D1 = 24 mm, h1 = 7 mm, Ds = 11 mm, hs = 0 mm. The relative permittivities of the dielectric puck and the cylindrical foam support are er2 ¼ 37 and er1 ¼ 1, respectively.

properties, the considered bow-tie antenna sensor can be conveniently adopted, in array configuration, to design time-domain millimeter-wave imaging systems (see Fig. 7a). In such systems, an ultrawideband (UWB) radio signal, synthesized by a suitable impulse generator, is radiated in order to illuminate a target, and the reflected/scattered signal is measured at different locations. This procedure is performed for all the possible combinations of transmit-receive sensor pairs in the array structure (see Fig. 7b) by using a switching matrix. The acquired signals are measured by means of the data acquisition module, and the whole measurement is automated by using a processing unit. The information about amplitude and time of arrival of the reflected signals is afterwards utilized to

A hybrid FDTD/SEM model for the accurate analysis of time-domain radiation properties of antenna sensors has been developed. By using the presented methodology, the analytical expressions of the time-domain effective height and gain of complex sensing devices can be derived. The proposed methodology is based on the vectorial pole/residue spherical harmonic expansion of the equivalent currents excited along a Huygens surface enclosing the device, and derived by means of a full-wave locally conformal FDTD technique. In this way, the sensor response can be written, in terms of incomplete spherical Bessel functions, as the sum of input-voltage-driven wave contributions and a number of dumped oscillations related to the complex resonances occurring in the structure. Therefore, the developed modeling approach can provide a meaningful insight into the time-domain radiation properties of the structure under analysis, which in turn is essential for a physics-oriented performance optimization. The presented analysis method has been assessed by application to a UWB antenna-based sensor for time-domain millimeter-wave imaging applications, and the obtained results have been found to be in excellent agreement with the theory. Acknowledgment This research has been partly carried out in the framework of the Sensor Technology Applied in Reconfigurable systems for sustainable Security (STARS) project funded by the Dutch government. For further information: http://www.starsproject.nl/. Appendix A. Validation of the locally conformal FDTD scheme In order to validate the accuracy of the proposed locally conformal FDTD scheme, a number of test cases have been considered. Here the results relevant to the evaluation of the fundamental resonant frequency of a dielectric resonator enclosed in a metallic cavity are presented. The structure under consideration (see Fig. 8a) has been already analyzed in [25]. It consists of a perfectly conducting cavity of dimensions a = b = 50 mm and c = 30 mm, loaded with a cylindrical dielectric (ceramic) puck with diameter D2 = 36 mm, height h2 = 16 mm, and relative dielectric constant er2 ¼ 37. The puck is placed above a cylindrical support made of foam ðer1 ¼ 1Þ, and having diameter D1 = 24 mm, height h1 = 7 mm. Furthermore, a metallic screw of diameter Ds = 11 mm is inserted perpendicularly

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at the center of the top wall for the fine tuning of the cavity by proper selection of the stub length hs. Since the dielectric permittivity of the resonator is rather high, the effect of the orthogonal Cartesian FDTD mesh being not conformal to the resonator shape is expected to be noticeable. In particular, the structure has been analyzed by means of the conventional FDTD algorithm featuring the traditional staircase approximation of the resonators contour, and by means of the weighted averaging approach proposed in [26], and the locally conformal FDTD technique described in Section 3. To this end, a cubic computational mesh having fixed spatial increment Dh has been adopted. The numerical results obtained from the mentioned FDTD schemes have been compared against the ones reported in [25] resulting from the use of a commercial transmission line matrix (TLM) method-based solver. As it appears in Fig. 8b, this example demonstrates clearly the suitability of the developed technique to efficiently model metal-dielectric structures with curved boundaries. As a matter of fact, the proposed locally FDTD scheme introduces a significant improvement in accuracy over the staircasing approximation, converging very quickly to the reference value. Such feature is thus of crucial importance to optimize the analysis and design of complex sensing systems. References [1] F.C. Chen, W.C. Chew, Time domain ultra-wideband microwave imaging radar system, J. Electromagn. Waves Appl. 17 (2) (2003) 313–331. [2] F.I. Rial, H. Lorenzo, M. Pereira, J. Armesto, Analysis of the emitted wavelet of high-resolution bowtie GPR antennas, Sensors 9 (2009) 4230–4246. [3] S.C. Mukhopadhyay, Novel planar electromagnetic sensors: modeling and performance evaluation, Sensors 5 (2005) 546–579. [4] C.K.M. Fung, N. Xi, B. Shanker, K.W.C. Lai, H. Chen, Dipole and bowtie nano-antenna for carbon nanotube (CNT) based infrared sensors, in: Proceedings IEEE Nanotechnology Materials and Devices Conference, Traverse City, USA, 2009, pp. 87–90. [5] H. Chen, N. Xi, K.W.C. Lai, L. Chen, C.K.M. Fung, J. Lou, Plasmonicresonant bowtie antenna for carbon nanotube photodetectors, Int. J. Opt. (2012), http://dx.doi.org/10.1155/ 2012/318104. [6] G. Marrocco, M. Ciattaglia, Approximate calculation of time-domain effective height for aperture antennas, IEEE Trans. Antennas Propag. 53 (3) (2005) 1054–1061. [7] A. Shlivinski, E. Heyman, R. Kastner, Antenna characterization in the time domain, IEEE Trans. Antennas Propag. 45 (7) (1997) 1140–1149. [8] E.G. Farr, C.E. Baum, Extending the definitions of antenna gain and radiation pattern into the time domain, Sens. Simul. Note 350 (1992).

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