Closed-Form Time-Domain Expressions for the 2D Pulsed EM Field Radiated by an Array of Slot Antennas of Finite Width Martin Stumpf #1 , Adrianus T. De Hoop ∗2 , Ioan E. Lager †3 #
Department of Radio Electronics, Brno University of Technology Purkynova 118, 612 00 Brno, Czech Republic 1
∗
[email protected]
Laboratory of Electromagnetic Research, Delft University of Technology 4 Mekelweg, 2628 CD Delft, the Netherlands 2
†
[email protected]
International Research Centre for Telecommunications and Radar, Delft University of Technology 4 Mekelweg, 2628 CD Delft, the Netherlands 3
[email protected]
Abstract—Closed-form time-domain expressions are derived for the pulsed EM field radiated by a planar array of 2D slot antennas of finite width through the use of the CagniardDeHoop technique. Each radiating slot is excited by a uniform electric field crossing the aperture and having a prescribed pulse shape. Mutual coupling between the apertures is neglected. With the results, the time-domain beam-steering and beam-shaping properties of the array can be studied. Parameters in this respect are: the collection of pulse shapes and amplitudes of the excitation and the time delays between the excitation of the successive slots. Numerical illustrations are shown.
various amplitudes, pulse rise times, pulse time widths and mutual time delays, beam steering and beam shaping can be influenced. Combined with the mutual geometrical positioning of the slots, there is a substantial amount of parameters whose influence on the antenna’s time-domain performance can be studied with the expressions obtained. Some illustrative numerical results show the time evolution in space of the twocomponent Poynting vector associated with the field in the form of vector (color) density plots.
I. I NTRODUCTION A field of exploration that is of growing importance is the development of direct time-domain pulsed-field radar return identification methods [1], [2]. First results of these methods typically deal with responses from certain classes of objects to an incident, linearly polarized, plane wave with a ramp function electric field pulse shape, observed in the far-field region of the scatterer, under the application of some physicaloptics field approximation on the boundary surface of the object. One of the first generalizations of the method involves the taking into account of the properties of the transmitting antenna and allowing the scattering object not necessarily to be in the far-field region of the antenna (where a local plane-wave approximation applies). The interaction of the pulsed transmitted field with the scatterer can then be analyzed through the time-domain field/source reciprocity theorem [3]. For this, the time-domain expression of the field transmitted by the radar antenna is needed. With this kind of application in mind, the present paper calculates the pulsed EM field radiated by a planar array of slot antennas in a 2D setting, neglecting the mutual coupling between the slots. Under this assumption, the Cagniard-DeHoop technique [4], [5], provides closed-form analytic expressions for all field components, expressed in terms of the transverse electric field distibution in the apertures. By exciting the slots with pulse shapes of
II. D ESCRIPTION OF THE CONFIGURATION AND FORMULATION OF THE FIELD PROBLEM
The antenna array configuration is shown in Fig. 1. In it, position is specified by the orthogonal Cartesian coordinates {x1 , x2 , x3 } with respect to the origin O and three base vectors {i1 , i2 , i3 } that form a right-handed system. The time coordinate is denoted by t. Partial differentiation with respect to xm is denoted by ∂m ; ∂t is a reserved symbol for differentiation with respect to time. x3 vacuum half-space
D0 {0 , μ0 }
A1 a1
O PEC ground plane Fig. 1.
S
A2 b1
V1
a2
b2
x1
V2
The antenna configuration.
The antenna array consists of an electrically perfectly conducting screen S with the collection of N non-overlapping feeding slots ∪N n=1 An , with An = {an < x1 < bn , −∞ < x2 < ∞, x3 = 0}, subject to the condition a1 < b1 < . . . < aN < bN . The antenna radiates into the vacuum half-space
∂1 H2 − 0 ∂t E3 = 0 ∂3 H2 + 0 ∂t E1 = 0
(1) (2)
∂1 E3 − ∂3 E1 − μ0 ∂t H2 = 0
(3)
in D0 and t > t0 , where t0 is the instant at which the excitation is started. The excitation condition is N Vn (t) lim E1 (x1 , x3 , t) = [H(x1 − an ) − H(x1 − bn )] x3 ↓0 b − an n n=1 (4) Here, H(x) denotes the Heaviside unit step function. III. T HE T IME - DOMAIN FIELD EXPRESSIONS Expressions for the field components are obtained with the aid of the Cagniard-DeHoop technique. This technique employs the unilateral Laplace transformation with respect to time ˆ1 , E ˆ3 , H ˆ 2 }(x1 , x3 , s) = {E ∞ exp(−st){E1 , E3 , H2 }(x1 , x3 , t)dt (5) t=t0
where s is taken to be real and positive, and the wave slowness representation along x1 ˆ1 , E ˆ3 , H ˆ 2 }(x1 , x3 , s) = {E i∞ s ˜1 , E ˜3 , H ˜ 2 }(p, x3 , s)dp exp(−spx1 ){E (6) 2πi p=−i∞ Using (5) and (6) in (1)–(4), we arrive at ˜3 , H ˜ 2 }(p, x3 , s) = ˜1 , E {E
N Vˆn (s) 1 0 p , 1, − b − an sp γ(p) γ(p) n=1 n
{exp[spbn − sγ(p)x3 ] − exp[span − sγ(p)x3 ]}
(7)
where γ(p) = (1/c20 −p2 )1/2 with Re[γ(p)] > 0, which entails branch-cuts along {1/c0 < |Re(p)| < ∞, Im(p) = 0}. Using the generic form of a wave constituent as discussed in the Appendix, the time domain field is obtained as N Vn (t) (t) ∗ b − an n=1 n H(t − Tb;n ) 1 (x1 − bn )t 1 x3 (x1 − bn ) , 1, 2 )1/2 π c20 t2 − x23 μ0 c20 t2 − x23 (t2 − Tb;n H(t − Ta;n ) 1 x3 (x1 − an ) 1 (x1 − an )t − , 1, 2 2 2 2 2 2 2 )1/2 π c0 t − x3 μ0 c0 t − x3 (t2 − Ta;n 1/2
0 x3 + 1, 0, [H(x1 − bn ) − H(x1 − an )]δ(t − ) μ0 c0 (8)
{E1 , E3 , H2 }(x1 , x3 , t) =
IV. I LLUSTRATIVE NUMERICAL E XAMPLES In this section we present some illustrative numerical results for the case where each of the slots is excitated with a power exponential pulse [6] (Fig. 2) V (t) = Vmax (t/tr )ν exp[−ν(t/tr − 1)]H(t − T )
(9)
where the pulse amplitude Vmax , the pulse rise time tr , the rising exponent ν > 0 and pulse starting time T can arbitrarily be chosen. For (9), the pulse rise time, the pulse time width tw and the rising exponent are interrelated via tw /tr = Γ(ν) exp(ν)/ν ν . 1 ν =2
0.5
V (t)/Vmax
D0 = {−∞ < x1 < ∞, −∞ < x2 < ∞, x3 > 0}. We neglect mutual coupling. Under this condition, the radiated field is the superposition of the fields radiated by the different slots. Since the excitation, as well as the configuration, are independent of x2 , the non-zero components of the electric field strength {E1 , E3 }(x1 , x3 , t) and the magnetic field strength H2 (x1 , x3 , t) satisfy the source-free field equations
0
−0.5
−1 0
Fig. 2.
2
4 t/tr
6
8
The power exponential excitation signature.
All cylindrical wave field constituents contain timeconvolution integrals with inverse square-root singularities at the arrival time of the wave. These are numerically handled via a stretching of the variable of integration according to τ = Ta cosh(u), with 0 < u < cosh−1 (τ /Ta ). In the (color) vector density plots Figs. 3–6 we show the time evolution of the two-component Poynting vector S1 = −E3 H2 , S3 = E1 H2 , normalized to the magnitude |S|ref of the maximum value of the Poynting vector as it would be carried by a T EM -mode in a parallel-plate waveguide that would be feeding the radiating apertures, i.e., |S|ref = (0 /μ0 )1/2 maxn ([Vmax;n /(bn − an )]2 ; n = 1, . . . , N ) for an array with N = 5, bn − an = w (n = 1, . . . , 5) and an+1 = bn + w/2 (n = 1, . . . , 4). The excitation pulses are taken to have all the same amplitudes and shapes with ν = 2 and c0 tw /w = 1.1084. Four different delays between the pulse starting values have been considered: (A) c0 (Tn+1 − Tn )/w = 0, (B) c0 (Tn+1 − Tn )/w = 0.5, (C) c0 (Tn+1 − Tn )/w = 1.0, (D) c0 (Tn+1 − Tn )/w = 1.5, all for n = 1 . . . , 4 and T1 = 0, and two observation times (a) c0 t/w = 4, (b) c0 t/w = 8. The plots clearly show the beam steering associated with the time delays in excitation. More complicated is the (time-varying) spatial distribution of the field. V. C ONCLUSION Closed-form analytic expressions have been constructed for the 2D pulsed EM field radiated by an array of slot antennas of finite width. It is shown that each of slots generates cylindrical waves emanating from its edges in addition to a
9
1
7
7
4
0.13
5
0.06 0.03
4 3
2
2
1
1 −6
−4
−2
Aa
0 x1 /w
2
4
6
0.01
0
0
−6
−4
−2
Ba
9
1
0 x1 /w
2
4
0
6
9
8
8
7
7
1
6
3
3 2
1
1 −6
Ab
−4
−2
0 x1 /w
2
4
6
0
0
−6
Bb
Fig. 3. Normalized Poynting vector of the field at (Aa) c0 t/w = 4, (Ab) c0 t/w = 8 for an array of N = 5 equally spaced slots of width w and spacing (an+1 − bn )/w = 0.5, excited with zero time delays.
plane wave contribution. The expressions are used to illustrate the phenomena of beam steering and beam shaping associated with time delayed pulsed excitations of the slots. The computational code is straightforward, which allows its use in other applications. A PPENDIX The application of the Cagniard-DeHoop technique to the configuration of Section II leads to the generic integral (s = time Laplace-transform parameter, p = slowness parameter along x1 ) ˆ b (x1 , x3 , s) − W ˆ a (x1 , x3 , s)] Fˆ (x1 , x3 , s) = Vˆ (s)[W
−4
−2
0 x1 /w
2
4
6
Fig. 4. Normalized Poynting vector of the field at (Ba) c0 t/w = 4, (Bb) c0 t/w = 8 for an array of N = 5 equally spaced slots of width w and spacing (an+1 − bn )/w = 0.5, excited with time delays c0 (Tn+1 − Tn )/w = 0.5 (n = 1, . . . , 4).
small radius. This leads for the integration to the same result. ˆ a is, under the application of Subsequently, the integral in W Jordan’s lemma and Cauchy theorem replaced with one along the path Ca ∪ Ca∗ , where [5] (x1 − a)τ + ix3 (τ 2 − Ta2 )1/2 Ca = pa (τ ) = , Ta < τ < ∞ (x1 − a)2 + x23 (12) in which Ta = [(x1 − a)2 + x23 ]1/2 /c0
(10)
(13)
The corresponding time-domain expression follows as
in which ˆ ˆ a (x1 , x3 , s) = V (s) W 2πi i∞ G[γ(p)] exp{−s[p(x1 − a) + γ(p)x3 ]}dp, p p=−i∞
0.05 0.03 0.01
4
2
0
0.1
5
|S|/(NSn )
4
|S|/(NSn )
0.09 0.05 0.02 0.01
5
x3 /w
6
x3 /w
6
3
0
1
|S|/(NSn )
0.05 0.03 0.01
|S|/(NSn )
0.1
5
x3 /w
8
6
x3 /w
9
8
Wa (x1 , x3 , t) = Wacyl (x1 , x3 , t) + Waplane (x1 , x3 , t)
(14)
where (11)
where γ(p) = (1/c20 − p2 )1/2 with Re[γ(p)] > 0, which entails branch-cuts along {1/c0 < |Re(p)| < ∞, Im(p) = 0}. ˆ b (x1 , x3 , s) is a similar expression with a replaced with W b. The right-hand side of (10) has no singularity at p = 0. However, the desired Cagniard-DeHoop technique [5] can only be applied to the separate terms, each of which has a simple pole at p = 0. To accomodate this situation, the path of integration is replaced with one that is indented to the right with a semi-circular arc with center at p = 0 and vanishingly
Wacyl (x1 , x3 , t) = γ[pa (t)]G{γ[pa (t)]} H(t − Ta ) 1 Re π pa (t) (t2 − Ta2 )1/2
(15)
represents a cylindrical wave emanating from the edge {x1 = a, x3 = 0} of the slot and the pole contribution Waplane (x1 , x3 , t) = G(1/c)H(x1 − a)δ(t − x3 /c0 )
(16)
represents a plane wave emanating from the aperture of the slot. With this, the time-domain wave constituent arising from
9
1
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8 7
0.05
4
0.02
x3 /w
5
0.12
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0.06
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0.02
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3
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1
0 −6
−4
−2
0 x1 /w
Ca
2
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0.23
6
0
0
−6
−4
−2
Da
9
1
0 x1 /w
2
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9
8
|S|/(NSn )
0.11
|S|/(NSn )
x3 /w
7
0.22
6
0
1
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5
0.07 0.04
4
0.01
5
0.09 0.04
4
0.02
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1
1
0 −6
−4
−2
0 x1 /w
Cb
2
4
6
0.18
6
x3 /w
0.15
0
Fig. 5. Normalized Poynting vector of the field at (Ca) c0 t/w = 4, (Cb) c0 t/w = 8 for an array of N = 5 equally spaced slots of width w and spacing (an+1 − bn )/w = 0.5, excited with time delays c0 (Tn+1 − Tn )/w = 1.0 (n = 1, . . . , 4).
(10) is
|S|/(NSn )
6
|S|/(NSn )
x3 /w
1
8
0 −6
Db
−4
−2
0 x1 /w
2
4
6
Fig. 6. Normalized Poynting vector of the field at (Da) c0 t/w = 4, (Db) c0 t/w = 8 for an array of N = 5 equally spaced slots of width w and spacing (an+1 − bn )/w = 0.5, excited with time delays c0 (Tn+1 − Tn )/w = 1.5 (n = 1, . . . , 4).
R EFERENCES (t)
F (x1 , x3 , t) = V (t) ∗ Wbcyl (x1 , x3 , t) + Wbplane (x1 , x3 , t)
− Wacyl (x1 , x3 , t) − Waplane (x1 , x3 , t) (17) These results are used to construct the numerical results in the main text. ACKNOWLEDGMENT The results reported in this paper were carried out during the short term scientific mission of M.S. at International Research Centre for Telecommunications and Radar (IRCTR) financed by COST IC0603 ASSIST.
[1] S. Nag and L. Peters, “Radar images of penetrable targets generated from ramp profile functions,” IEEE Trans. Antennas Propag., vol. 49, pp. 32– 40, Jan. 2001. [2] C. C. Chen and L. Peters, “Radar scattering and target imaging obtained using ramp-response techniques,” IEEE Antennas Propag. Mag., vol. 49, pp. 13–27, Jun. 2007. [3] A. T. de Hoop, I. E. Lager, and V. Tomassetti, “The pulsed-field multiport antenna system reciprocity relation and its applications - a time-domain approach,” IEEE Trans. Antennas Propag., vol. 57, pp. 594–605, Mar. 2009. [4] A. T. de Hoop, “A modification of Cagniard’s method for solving seismic pulse problems,” Applied Scientific Research, vol. B8, pp. 349–356, 1960. [5] ——, “Pulsed electromagnetic radiation from a line source in a two-media configuration,” Radio Science, vol. 14, pp. 253–268, 1979. [6] D. Quak, “Analysis of transient radiation of a (traveling) current pulse on a straight wire segment,” in Proc. 2001 IEEE EMC International Symposium, Montreal, Que., Canada, Jul. 2001, pp. 849–854.
