Pulsed electromagnetic field radiation from a narrow slot antenna with a dielectric layer

Authors


Abstract

[1] Analytic time domain expressions are derived for the pulsed electromagnetic field radiated by a narrow slot antenna with a dielectric layer in a two-dimensional model configuration. In any finite time window of observation, exact pulse shapes for the propagated, reflected, and refracted wave constituents are constructed with the aid of the modified Cagniard method (Cagniard-DeHoop method). Numerical results are presented for vanishing slot width and field pulse shapes at the dielectric/free space interface. Applications are found in any system whose operation is based on pulsed electromagnetic field transfer and where digital signals are detected and interpreted in dependence on their pulse shapes.

1. Introduction

[2] With the rapid development of communication systems whose operation is based upon the transfer of pulsed electromagnetic fields and the detection and subsequent interpretation of the pertaining digital signals, there is a need for mathematical analysis of model configurations where the influence of (a number of) the system parameters on the performance shows analytic expressions in analytic time domain that characterize the physical behavior. The present paper aims at providing such a tool with regard to the pulsed radiation behavior of a narrow slot antenna covered with a dielectric layer in a two-dimensional setting. The source exciting the structure is modeled as a prescribed distribution of the transverse electric field across a narrow slot of uniform width in a perfectly electrically conducting planar screen. The pulse shape of the exciting field is arbitrary. In front of this slotted plane, there is a homogeneous, isotropic dielectric slab of uniform thickness. The structure further radiates into free space. Using the combination of a unilateral Laplace transformation with respect to time and the spatial slowness representation of the field components that is known as the modified Cagniard method (Cagniard-DeHoop method), analytic time domain expressions are obtained for the electric and the magnetic field as a function of position and time. The representation appears as the superposition of a number of propagating, reflecting, and refracting wave constituents in the slab and is, within any finite time window of observation, exact. It is immediately clear that the pulse shapes of these constituents (that successively reach a receiving observer) are distorted versions of the activating source signature. Parameters in this respect are the pulse shape of the excitation (characterized by the pulse risetime and the pulse time width of a unipolar pulse), the thickness and the dielectric properties of the slab, as well as the position of observation relative to the exciting slot.

[3] The analytic expressions are readily evaluated numerically. Results are presented for vanishing slot width and field pulse shapes at the dielectric/free space interface for a variety of parameters, all chosen such that the pulse time width is smaller than the travel time needed to traverse the slab. In this way, the study can focus on the changes in pulse shape that occur in the individual successive wave constituents. In line with the International Electrotechnical Vocabulary (IEV) of the International Electrotechnical Committee (IEC 60050–IEV) (http://www.electropedia.org), the signature of the excitation is taken to be a unipolar pulse characterized by its pulse amplitude, pulse risetime, and pulse time width. The power exponential pulse provides a convenient mathematical model to accommodate these parameters.

[4] Apart from this, the obtained expressions can serve the purpose of benchmarking the performance of purely computational techniques that have to be called upon in the more complicated configurations met in practice, in particular the ones in patch antenna design, where the field calculated in the present paper represents the field “incident” on the geometry of the patches.

2. Description of the Configuration and Formulation of the Field Problem

[5] The configuration examined is shown in Figure 1. In Figure 1, the position is specified by the right-handed orthogonal Cartesian coordinates {x1, x2, x3}. The time coordinate is t. Partial differentiation with respect to xm is denoted by ∂m; ∂t is a reserved symbol denoting partial differentiation with respect to t. The configuration consists of an unbounded electrically perfectly conducting screen equation image = {(−∞ < x1 < −w/2) equation image (w/2 < x1 < ∞), −∞ < x2 < ∞, x3 = 0} with a feeding aperture equation image = {−w/2 < x1 < w/2, −∞ < x2 < ∞, x3 = 0} of width w ↓ 0. The covering dielectric slab occupies the domain equation image1 = {−∞ < x1 < ∞, −∞ < x2 < ∞, 0 < x3 < d}. The structure radiates into the vacuum half-space &#55349;&#56479;0 = {−∞ < x1 < ∞, −∞ < x2 < ∞, d < x3 < ∞}. The spatial distribution of electric permittivity and magnetic permeability is

equation image

The corresponding electromagnetic wave speeds are c0 = (equation image0μ0)1/2 and c1 = (equation image1μ1)1/2. The antenna aperture is fed by the uniformly distributed, x2-independent, electric field

equation image

where V0(t) is the feeding “voltage.” Since the excitation, as well as the configuration, are independent of x2, the nonzero components of the electric field strength {E1, E3}(x1, x3, t) and the magnetic field strength H2(x1, x3, t) satisfy in &#55349;&#56479;0 and &#55349;&#56479;1 the source-free field equations:

equation image
equation image
equation image

The interface boundary conditions require that

equation image
equation image

while the excitation condition is

equation image

with H(x) denoting the Heaviside unit step function. It is assumed that V0(t) starts to act at t = 0 and that prior to this instant, the field vanishes throughout the configuration.

Figure 1.

Configuration with indication of the critical angle θc = arcsin(c1/c0). Positions of observation points {B, C, D} along the line {x3 = d} are not true-to-scale with those chosen in section 6.

3. Field Representations

[6] Analytic time domain expressions for the field components will be constructed with the senior (second) author's modification of the Cagniard method (Cagniard-DeHoop method) [Cagniard, 1939; Cagniard, 1962; De Hoop, 1960; De Hoop and Frankena, 1960; Langenberg, 1974; De Hoop, 1979; De Hoop and Oristaglio, 1988; De Hoop, 2000]. The method employs a unilateral Laplace transformation with respect to time of the type

equation image

in which s is taken to be real-valued and positive. In fact, Lerch's theorem [Widder, 1946] states that uniqueness of the inverse transformation is ensured under the weaker condition that equation image0(s) is specified at the sequence of real s values {sequation imageequation image; sn = s0 + nh, s0 > 0, h > 0, n = 0,1,2,…}. The next step is to use the slowness representation of the field quantities

equation image

that involves imaginary values of the complex slowness parameter p. Using (9) and (10), the field equations (3)–(5) transform into

equation image
equation image
equation image

the interface boundary conditions (6) and (7) into

equation image
equation image

and the excitation condition (8) into

equation image

In what follows we shall consider the limiting case of vanishing slot width. Then, (16) reduces to

equation image

The slowness-domain field quantities follow from (11)–(17) by expressing them in the form

equation image
equation image

in which

equation image

with Re[γ0,1(p)] > 0 for all pequation imageequation image. Using these expressions in (14), (15), and (17) it is found that

equation image
equation image
equation image

in which

equation image
equation image
equation image

Via the convergent expansion

equation image

the slowness-domain field quantities can be written as the superposition of constituents, each of which admits a analytic time domain representation attainable with Cagniard-De Hoop method.

[7] For convenience of the reader, the procedure as based on the time Laplace transformation as used by Cagniard [1939] and De Hoop [1960] is briefly reviewed in Appendix A. The corresponding analysis as based on the time Fourier transform, which can be found in the work of Chew [1990, section 4.2].

4. The Time Domain Field at the Vacuum/Dielectric Interface

[8] In this section we focus on those field components at the interface x3 = d that are continuous across this interface, i.e. E1 and H2. The radiated field in equation image0 is subsequently easily expressed in terms of these field values. Using the results of section 3, we express them as

equation image

with

equation image

The corresponding time domain expressions of each of these constituents follow upon the application of the Cagniard-DeHoop method (see Appendix A).

5. Radiated Field in the Absence of a Dielectric Slab

[9] To illustrate the pulse distortion that results from the presence of the dielectric slab we compare the relevant results with the ones applying to the radiation in the absence of the slab. The latter are well known:

equation image

for x3 > 0 and with E1(x1, 0, t) = V0(t)δ(x1), and

equation image
equation image

where equation image, denotes time convolution and r = (x12 + x32)1/2 > 0.

6. Illustrative Numerical Results

[10] This section contains some illustrative numerical results for the case of excitation with the power exponential pulse with parameters c0tw/d = 0.9236, tw/tr = 1.8473, ν = 2, shown in Figure 2a (with Figure 2b showing its spectral diagram) and vanishing slot width. In any finite time window of observation, only a finite number of terms in the summation in (28) yield a nonzero contribution, while in the range of critical refraction, only a subset of these contributions have a head wave part. The objective of our analysis is to compare the pulse shapes of the different constituents with the ones that the narrow slot antenna would radiate into a half-space with the properties of equation image0. This comparison is carried out on the vacuum/dielectric interface.

Figure 2.

The power exponential excitation signature: (a) pulse shape and (b) spectral diagram. The represented quantities are V0(t)/Vmax (normalized V0(t)), c0t/d (normalized time), equation image0ω)/equation image0(0), (normalized equation image0ω)), ω/ωcorner (normalized angular frequency).

[11] The properties of the slab are taken as {ε1, μ1} = {4ε0, μ0}. All time convolution integrals contain inverse square-root singularities at one of the end points. These are numerically handled via a stretching of the variable of integration according to τ = TBW cosh(u), with 0 < u < cosh−1(t/TBW) for body wave constituents and τ = TBW sin(v), with sin−1(THW/TBW) < v < π/2 for head wave contributions. Four positions of observation at x3 = d have been selected: (1) x1/d = 0, (2) x1/d = 0.5, (3) x1/d = 2.8, (4) x1/d = 5.0. In accordance with Figure 1, only the last two observation points are in the range of critical refraction. The time window of observation is taken as 0 < c0t/d < 20. The arrival times of the different contributions are shown in Table 1.

Table 1. Arrival Times of Time Domain Constituents
Order nc0THW[n]/dc0TBW[n]/d
ABCDABCD
04.53216.72312.02.23615.946410.1980
17.996210.19626.06.08288.207311.6619
213.660310.010.049911.461214.1421
317.124414.014.035715.078517.2047
4>2018.018.027818.8510>20

[12] On account of the field equations, one might expect that in the first instance the pulse shapes of the field components would contain replicas and the time derivative of the exciting pulse shape. Outside the range of the occurrence of the head waves (in the frequency domain analysis of electromagnetic fields also denoted as lateral waves), this is more or less the case. In ranges where head wave contributions do occur, the situation is entirely different as shown in Figures 3–6. For pulse time widths of excitation that exceed the travel time to traverse the slab, the intermix of wave constituents overlapping in time leads to a signal in which the exciting pulse is no longer recognizable.

Figure 3.

Time domain responses at x1/d = 0. The represented quantities are E1d/Vmax normalized E1), (μ00)1/2H2d/Vmax (normalized H2), and c0t/d (normalized time).

Figure 4.

Time domain responses at x1/d = 0.5. The represented quantities are E1d/Vmax (normalized E1), (μ00)1/2H2d/Vmax (normalized H2), and c0t/d (normalized time).

Figure 5.

Time domain responses at x1/d = 2.8. The represented quantities are E1d/Vmax (normalized E1), (μ00)1/2H2d/Vmax (normalized H2), and c0t/d (normalized time).

Figure 6.

Time domain responses at x1/d = 5.0. The represented quantities are E1d/Vmax (normalized E1), (μ00)1/2H2d/Vmax (normalized H2), and c0t/d (normalized time).

7. Conclusion

[13] Analytic time domain analytic expressions have been constructed for the head and body wave constituents of the time domain electromagnetic field radiated by a pulse-excited narrow slot in a perfectly conducting ground plane with a dielectric slab. Illustrative numerical results clearly show the changes in pulse shape arising from the multiple reflections, in particular in the range where head wave contributions occur. The results can be used to quantify these changes with a view of their acceptability in any transmission system where pulsed electromagnetic fields are the carrier of the information. In addition, the analytic time domain expressions can serve the purpose of providing benchmark results in the testing of purely computational software for the evaluation of time domain electromagnetic fields.

Appendix A: The Cagniard-DeHoop Method

[14] The generic form of the wave constituents in the interior of the dielectric slab and on its boundary is

equation image

where equation image(p) has the branch cuts in accordance with Re[γ0,1(p)] > 0, i.e. {1/c0,1 < ∣Re(p)∣ < ∞, Im(p) = 0}, and X and Z are the propagation paths in the x1- and x3-directions. We assume that c0c1. Under the application of Cauchy's theorem and Jordan's lemma of complex function theory, the path of integration in (A1) is replaced with one along the modified Cagniard path (the Cagniard-DeHoop path) defined through

equation image

where τ is real-valued.

A1. Body Wave Path

[15] The body wave path follows from (A2) as the hyperbolic arc {p = pBW (X, Z, τ)} equation image {p = pBW* (X, Z, τ)}, where

equation image

for TBW < τ < ∞ with

equation image

as the arrival time of the body wave constituent. Along this path

equation image

for TBW < τ < ∞ and the Jacobian of the mapping from p to τ is

equation image

for TBW < τ < ∞. The body wave path replaces the path of integration in (A1) as long as the intersection of this path with the real p axes does not lie on the branch cut associated with γ0(p), i.e., for points of observation in the range ∣X∣/(X2 + Z2)1/2 < c1/c0. For points of observation outside this range, the body wave path has to be supplemented with a connecting loop integral along the branch cut associated with γ0(p). This loop integral yields the head wave (or lateral-wave) contribution to the wave constituent.

A2. Head Wave Path

[16] The parametrized head wave path follows from (A2) as the loop {p = pHW (X, Z, τ)} equation image {p = pHW* (X, Z, τ)}, where

equation image

for THW < τ < TBW with

equation image

[17] as the arrival time of the head wave constituent. Along this path

equation image

for THW < τ < TBW and the Jacobian of the mapping from p to τ is

equation image

for THW < τ < TBW. The corresponding wave constituents follow from combining the two complex conjugate parts of the paths and applying Schwarz' theorem of complex function theory.

A3. Time Domain Body Wave Constituent

[18] The s-domain body wave constituent follows from (A1) as

equation image

Again, on account of Lerch's uniqueness theorem of the unilateral Laplace transformation [Widder, 1946] the time domain constituent then follows as

equation image

where (A6) has been used and equation image, denotes convolution with respect to time.

A4. Time Domain Head Wave Constituent

[19] The s-domain head wave constituent follows from (A1) as

equation image

On account of Lerch's uniqueness theorem of the unilateral Laplace transformation [Widder, 1946] the time domain constituent then follows as

equation image

where (A10) has been used. These results are used in the main text.

Appendix B: The Power Exponential Pulse

[20] A convenient pulse type to model a unipolar pulse excitation is the power exponential pulse [Quak, 2001]:

equation image

for ν = 0, 1, 2, … where Vmax is the pulse amplitude, ν is the rising exponent of the pulse and tr is the pulse risetime. Note that V0(tr) = Vmax. The pulse time width tw follows from

equation image

as

equation image

The time Laplace transform of (B1) is

equation image

The spectral amplitude of V0(t) follows from (B4) as

equation image

From

equation image

it follows that both

equation image

and

equation image

In the spectral diagram (where ∣equation image0(iω)∣ is plotted against ∣ω∣, both on logarithmic scales), the right-hand sides of (B7) and (B8) are straight lines that are denoted as the spectral bounds of ∣equation image0(iω)/equation image0(0)∣. The two spectral bounds intersect at their corner point

equation image

Acknowledgments

[21] The work described in this paper and the research leading to these results have received funding from the European Community's Seventh Framework Program FP7/2007-2013, under grant agreement 205294, HIRF SE project. The short term scientific mission of M. Š. at International Research Centre for Telecommunications and Radar (IRCTR) has been financed by COST IC0603 ASSIST. The authors would like to thank the anonymous reviewers for their careful reading of the manuscript and for making valuable suggestions for improving the paper.

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