IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 48, NO. 3, MARCH 2000 839 Letters__________________________________________________________________________________________ A Novel Finite-Difference Time-Domain Wave Propagator Funda Akleman and Levent Sevgi Abstract—In this letter, a novel time-domain wave propagator is intro- duced. A two-dimensional (2-D) finite-difference time-domain (FDTD) al- gorithm is used to analyze ground wave propagation characteristics. As- suming an azimuthal symmetry, surface, and/or elevated ducts are repre- sented via transverse and/or longitudinal refractivity and boundary pertur- bations in 2-D space. The 2-D FDTD space extends from (bottom) to (top), vertically and from (left) to (right), horizontally. Perfectly matched layer (PML) blocks on the left, right, and top terminate the FDTD computation space to simulate semi-open propa- gation region. The ground at the bottom is simulated either as a perfectly electrical conductor (PEC) or as a lossy second medium. A desired, initial vertical field profile, which has a pulse character in time, is injected into the FDTD computation space. The PML blocks absorb field components that propagate towards left and top. The ground wave components (i.e., the di- rect, ground-reflected and surface waves) are traced longitudinally toward the right. The longitudinal propagation region is covered by a finite-sized FDTD computation space as if the space slides from left to right until the pulse propagates to a desired range. Transverse or longitudinal field pro- files are obtained by accumulating the time-domain response at each alti- tude or range and by applying discrete Fourier transformation (DFT) at various frequencies. Index Terms—FDTD methods, propagators. I. INTRODUCTION Ground wave propagation has been and will continue to be one of the important options for communication over long distances near the earth's surface. The model environment is a spherical earth with var- ious ground characteristics, above which exists a radially inhomoge- neous atmosphere. Having an excitation and observer anywhere on or above the ground, this model has served as a canonical problem. It is very complex and a full wave numerically computable analytical so- lution has not appeared yet. Operating frequency, medium parameters (permittivity, permeability, and conductivity) and boundary conditions (geometry) may totally change physical characteristics. For example, while the propagation is limited by the line-of-sight at microwaves, be- yond the horizon propagation is possible at high (HF 3–30 MHz) and lower frequencies. The problem, available analytical approximate (ray and mode) solutions and numerical techniques such as split-step para- bolic equation (SSPE) have been outlined in [1] together with the hy- bridization of ray and mode methods. The effects of transverse as well as longitudinal refractivity profiles have also been investigated either analytically [2], [3], or numerically [4]–[6]. Here, a very effective technique is introduced to model various ground wave propagation characteristics. It is a general technique that can be applied to broad range of propagation problems. The technique is based on a two-dimensional (2-D) finite-difference time-domain [7] (FDTD) algorithm, where broad-band pulse propagation is simulated and traced over long distances by a virtual rectangular window that Manuscript received August 26, 1999; revised December 2, 1999. F. Akleman is with the Communication Engineering Department, ITU Elec- tronics, 80626 Maslak/Istanbul, Turkey. L. Sevgi is with TUBITAK-MRC Information Technologies Research Insti- tute (ITRI), 41470 Gebze/Kocaeli, Turkey. Publisher Item Identifier S 0018-926X(00)02458-3. circulates the FDTD space longitudinally back and forth until the range of interest is reached. II. A TIME-DOMAIN WAVE PROPAGATOR The 2-D FDTD wave propagation region and the FDTD computation space are pictured in Fig. 1. The structure is assumed to have azimuthal symmetry. The transverse and longitudinal field components are and , respectively, which models the classical 2-D TMz ground wave propagation over earth's surface. Time-domain ground wave propagation over earth's surface is sim- ulated as follows. • The propagation region (see Fig. 1(a)) is much larger than the FDTD computation space. Therefore, the FDTD computation space covers this region like a moving computation subregion. • An by (number of transverse and longitudinal cells, re- spectively) FDTD computation space is terminated by perfectly matched layer (PML) blocks [8], [9] from left , right and top. A perfectly reflecting conductor (PEC) or lossy ground termination is used at the bottom. • Taking and (total of 250 000 FDTD cells) corresponds to a ( : minimum wavelength) space with a typical spatial discretization. • Various refractivity (e.g., exponential, linear, bilinear or trilinear) profiles over earth's surface is introduced via relative permittivity . • Initial altitude profile is injected via the necessary field compo- nents. The profile may be a line source or an antenna pattern at a given altitude. Whatever the spatial field distribution, the initial field has a pulse character in time, providing broad-band analysis via a single FDTD run. • One-way propagation is traced via a 2-D rectangular window as shown in Fig. 1(b). The content of this propagation window is the pulse, which carries information related to the three wave com- ponents; direct, ground-reflected, and surface waves. • This virtual propagation window moves from left to right in FDTD computation space and circulates back to the left when reaches the right-most end, which is the initial profile of the next FDTD computation space. The process and FDTD simulations repeat until the wave longitudinally propagates to a desired range. • Keeping in mind the number of FDTD cells traced during the circulation of the propagation window, the transverse and/or lon- gitudinal propagation characteristics are obtained. The direct, ground-reflected and surface waves are traced in time domain, where wave fronts and their interferences (wave maxima and minima) appear as 2-D images. A typical example is given in Fig. 2 for a bilinear refractivity profile. Here, a Gaussian altitude profile is fed as the initial field distribution inside a 500 500 FDTD compu- tation space corresponding to 50 m 50 m physical space (with 0.1 m spatial discretization). The initial field distribution has a pulse char- acter in time (i.e., first derivative of a Gaussian function with 200 MHz bandwidth at 200 MHz center frequency). A 500 250 virtual window circulates 20 times as if the longitudinal number of cells in FDTD com- putation space is 5000. Instant snapshots are taken at different simula- tion times and are plotted as the field profiles at different ranges. As 0018–926X/00$10.00 © 2000 IEEE