Transient radiation by periodic structures: accuracy of the (time domain-floquet wave)-FDTD algorithm

Author(s): Marrocco, G; Capolino, F | Abstract: Transient radiation by periodic structures using a time domain (TD) transmission line network formalism based on the recently developed time-domain Floquet-wave (TD-FWs) is analyzed. The efficiency and the accuracy of the given method depend on the dimension of the finite-difference time-domain (FDTD) computational domain and, on how many FW modes may be used in the signal reconstruction within the frequency range of interest. This paper emphasizes the accuracy problem by numerical analysis of a simple periodic structure whose analytical solution can be easily computed, and a frequency selective surface (FSS).


Introduction
Transient radiation by periodic structures is analyzed using a time domain (TD) transmission line network formalism based on the recently developed time-domain Floquet-waves (TD-FWs) [1], [2], [3]. The FDTD algorithm is applied to the array element cell to compute pulsed modal current generators that form the initial condition for TD-FW propagation. The transient radiated field is expressed as a sum of modal voltages and currents evaluated using previously developed propagation of TD-FWs, which form an efficient field representation for this class of problems. The efficiency and the accuracy of the method depend on the dimension of the FDTD computational domain and, on how many FW modes may be used in the signal reconstruction within the frequency range of interest. In preliminary works [1], [2], [3], [4], the authors described the basic principles and some preliminary applications to both narrow-band and wide-band arrays. Here, the accuracy problem is emphasized by numerical analysis of a simple periodic structure whose analytical solution can be easily computed, and a frequency selective surface (FSS) [5]. The analysis will help to define guidelines for the FDTD domain size and the maximum number of TD-FW modes needed.
Referring to Fig. 1, we consider an infinite array with periods d, and d,, along z and y, respectively. We assume array elements with arbitrary geometry, simultaneously excited by short-pulse generators. The FDTD method together with appropriate boundary conditions is used to evaluate the magnetic field at a reference plane 10cated at t = z', which defines the transverse equivalent electric current distribution (1) PIP P A in terms of time dependent E and H modal current generators iE(t) and iE(t), together with the TEM generator &o(t). The TD generators are computed by projecting the total current distribution, evaluated by the FDTD method, onto the vector basis set which are numerically implemented as finite summations, with the vector modal basis sampled at the FDTD grid points. Transient radiation at z > z' is evaluated using modal TD-FW voltage Vw(z, 2, t ) and current i p g ( z , 2, t ) propagators, excited by the modal current generators iw(t) at z', as shown in Fig.la. The magnetic field (analogous considerations hold for the electric field) is evaluated as in which we have also included the TEM propagator HOO(T, t ) . In particular, the current fm(z, z', t ) is evaluated by convolution between the current generator Zw(t) and the impulse current response f~z ' " ' p ( z , terms of more involved incomplete Lipschitz-Hankel integrals. Physical insight can be gained from frequency asymptotics applied to Fourier inversion of a FD-FW [2], which leads to the definition of the dominant local instantaneous frequencies ww,i(t) = (-l)%&/-, i = 1,2. They are real &er turn-on (t > t o ) , and determine the time-dependent dominant frequencies which govern TD-FWs. At turn-on, the pqth TD-FW has ww,i(t + t o )co, while ww,i(t + m) + Gw, implying that at turn-on the pqth TD-FWs cannot be excited by its band-limited generator ipg(t), and that after to the frequency of the pqth TD-FW localizes around its cutoff frequency Gpp. Therefore, the pqth TD-FW, contributes to the total radiated field when its cutoff frequency Gw is lower than the maximum frequency of the spectrum of its generator im(t). This furnishes a simple criterium to determine the total number of TD-FWs needed in (3).
Note that the algorithm does not require storage of the FDTD time-samples for each FDTD spatial location in the periodic cell. One needs to store only their projections tw(t) in

I I i;yy Lz ) 11
where 11g(Lz)112 = dc:Ll Ig(L.,tn)12, has been computed versus the distance L, for lower order modes. As shown in Fig.lb, the error ePq for the TEM, and TElo modes is low for every L,, while increasingly larger errors are observed for higher order modes, for larger L,, especially for the TEzl, TE30 and TE31 modes. Therefore, modal currents should be best computed at an equivalence plane (see Fig.la) which is not too distant (1-5 cells) from the array plane, with the important benefit to keep small the FDTD computational domain. The error increase with the mode order is directly related to the sampling rate of the periodic modes, which are sinusoidal functions of the variable p, for the discrete approximation of the integrals in (2). This error could be reduced by a finer FDTD grid or by using a quadrature method more accurate than the finite summations. Following the discussion in [4], it has been experimentally found that the highest ( p , y)-mode, whose equivalent current can still be computed with the typical FDTD accuracy is limited t o p 5 d,/(20A) and y 5 dy/(20A). In this case p 5 2, y 5 0.95 in agreement with the results in Fig.1. The reconstruction of the magnetic field H, using the TD-FW-FDTD algorithm in Sec.11, is analyzed in Fig.2. Equivalent current generator iw(t) are computed by the FDTD method at an equiv- alence plane one cell away (Lz = 2mm) from the array plane. Fig.2 shows currents jw(q, L,, t ) at a distance z1 = 5011 = 100mm from the array plane ( q -L, from the equivalence plane). Currents i,(tl, L,, t ) are computed using convolution between the generators iw, and both the exact ("exact convolution") and asymptotic ("asymp totic convolution") impulsive propagators f z ( t 1 , L,, t). These are compared with an extended FDTD estimation comprising z1 (numerically expensive), and the "exact" reference solution i g p ( z l r 0, t ) in (4). The "exact convolution" provides an excellent agreement with the "exact" reference solution, whereas, as expected, the "asymptotic convolution" provides less accurate results in the early transient. Finally, in Fig.2b the magnetic field H, has been regenerated at point (50mm, 24mm, 21) by summing TD-FWs. The early transient and the dominant oscillation of the signal's tail are correctly regenerated by the superposition of the sole TEM, and TElo modes, while higher order modes (those whose cut-off frequency G,/27r 515 GHz, i.e., (pl 5 2, IpI 5 2), have been required to more accurately reproduce the oscillating late transient. It has been experienced that a superposition of higher order modes, with cut-off outside the simulation bandwidth does not improve further the accuracy of the reconstructed signal.
Application to FSS. The method has been applied to the analysis of a doubly periodic frequency selective surface (FSS). The elementary cell with size d x d consists of two concentric square loops [5] with (d/16) x (d/16) crossection and mutual spacing d/32. The structure has been illuminated by a TEM, wave with normal incidence and gaussian time waveform with fmax=4c/(3d). The modal current generators :,(t) for the reflected field have been computed at a an equivalent plane d/16 (two FDTD cells) from the FSS plane. Preliminary results in Fig.3 show lower order modal current generators (TEM, TElo, and TEol) in the time domain and in the frequency domain (in the FD, we have normalized by the spectrum of the incident field). As expected, mainly the TEM, current is excited. As in the previous case, the signal reconstructed (not shown here) at t > L, by the TD-FW-FDTD algorithm is in accord with an extended FDTD computation (numerically expensive).