IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 11, NOVEMBER 2003 3183 IV. DISCUSSION AND CONCLUSION The results show that resistive loading applied over short distances can be effective in controlling edge effects in the scattering from high conductivity surfaces such as sea water. The Taylor-taper loading de- veloped by Haupt and Liepta [6] proved superior to the power-law taper overall. As expected, the stronger off-specular backscattering with the power-law taper demonstrated in Fig. 3 interferes with the true surface scattering, giving poorer accuracy. However, the sidelobe level is not the only limiting factor in the use of the tapers, as evidenced by the small errors that appear in the VV scattering in Fig. 7 as the grazing angle decreases below 10 . This loss of accuracy is consistent with the behavior of the surface current in Fig. 4. Due to the downward slope of the leading edge of the surface, the local illumination grazing angle on the leading edge is less than 30 in this region, which is also the grazing angle at which the current on the flat plate becomes significantly per- turbed in Fig. 4. The oscillations in the error in Fig. 7 appear to be due to the oscillations in the current moving across the major scattering feature (the crest jet) as the grazing angle changes. These oscillations were consistently observed for the later waves in the LONGTANK se- ries using both the complete wave profiles (as shown here), or using the shorter “isolated-crest” profiles that were also considered in the mul- tipath study in [4]. The largest error observed in any case when the local grazing angle was more than 20 was 1.5 dB, and occurred with an isolated-crest surface where the loading was quite close to the crest scattering feature. Moreover, errors of several dB appeared in test cases where the local grazing angle on the loaded section was permitted to move below 20 (not shown). This is clearly a limiting factor in the use of resistive loading for control of edge effects. The current oscillations may be reduced by increasing the surface length over which the loading is added. However, simply increasing the length of the unloaded sec- tion has no effect on the maximum current perturbation. The limitations in the edge loading identified in this study do not re- sult because the surfaces that are of finite conductivity are considered. In fact, the VV current oscillations in Fig. 4 actually increased some- what when a perfectly conducting surface was considered, behavior that was reflected by greater errors in the wave-profile cross section (greater than 2 dB in some cases where the local grazing angle on the loading was more than 20 ). No explanation is offered for this behavior. (The HH accuracy was unchanged.) Finally, the resistive-loading ap- proach is significantly more efficient than even the MM/GTD tech- nique for the 2-D calculations, requiring less than 30% of the CPU time for the 31 grazing angles presented in Fig. 7. The advantage fur- ther increases as more angles are added. This makes resistive loading an attractive approach for scattering from 3-D surfaces, provided that the surface geometry allows the local grazing angle on the loaded sur- face sections to remain above 20 . REFERENCES [1] E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Amer., vol. 83, no. 1, pp. 78–82, Jan. 1988. [2] J. De Santo, G. Erdmann, W. Hereman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimen- sional perfectly reflecting surfaces,” Waves Random Media, vol. 8, no. 4, pp. 385–414, Oct. 1998. [3] H. Kim and J. T. Johnson, “Radar images of rough surface scattering: comparison of numerical and analytical models,” IEEE Trans. Antennas Propagat., vol. 60, pp. 94–100, Feb. 2002. [4] J. C. West and Z. Zhao, “Electromagnetic modeling of multipath scattering from breaking water waves with rough faces,” IEEE Trans. Geosci. Remote Sens., vol. 40, no. 3, pp. 583–592, Mar. 2002. [5] Y. Oh and K. Sarabandi, “Improved numerical simulation of electro- magnetic wave scattering from perfectly conducting random surfaces,” Proc. Inst. Elect. Eng.—Microwave Antennas Propagation, vol. 144, pp. 256–260, Aug. 1997. [6] R. L. Haupt and V. V. Liepa, “Synthesis of tapered resistive strips,” IEEE Trans. Antennas Propagat., vol. AP-35, p. 1217, Nov. 1987. [7] J. D. Kraus and K. R. Carver, Electromagnetics, 2 ed. New York: Mc- Graw-Hill, 1973, pp. 449–451. [8] J. C. West, “Integral equation formulation for iterative calculation of scattering from lossy rough surfaces,” IEEE Trans. Geosci. Remote Sens., vol. 38, pp. 1609–1615, July 2000. [9] P. Wang, Y. Yao, and M. P. Tulin, “An efficient numerical tank for non- linear water waves, based on the multi-subdomain approach with BEM,” Int. J. Num. Meth. Fluids, vol. 20, no. 12, pp. 1315–1336, June 1995. Extension of the ADI-FDTD Method to Debye Media S. González García, R. Godoy Rubio, A. Rubio Bretones, and R. Gómez Martín Abstract—This communication describes an extension of the alternating direction implicit finite-difference time-domain (ADI-FDTD) method to an- alyze problems involving Debye media. Index Terms—Alternating direction implicit finite-difference time- domain (ADI-FDTD), dispersive media, finite-difference time-domain (FDTD). I. INTRODUCTION The alternating direction implicit finite-difference time-domain (ADI-FDTD) method to solve Maxwell curl equations [1], [2] represents, due to its unconditional stability, a new alternative to the traditional FDTD method. It is specially well suited for problems involving geometries needing different details of discretization (narrow slots, high-permittivity materials, etc.) treated with nonuni- form meshing techniques [3], [4], since the use of locally small spatial increments does not imply, for stability reasons, the use of unnecessarily small time increments, achieving in many practical problems significant reductions in CPU time compared to the FDTD method [1]–[6]. In this communication, we propose a technique to incorporate Debye materials into the ADI formulation. The communication is organized as follows. In Section II, we formu- late Maxwell time-domain equations for Debye media in a way conve- nient for the ADI algorithm presented in this communication, which is described in Section III. In Section IV, a simple normal incidence problem is simulated to validate the method. II. DISPERSIVE MATERIAL EQUATIONS There are several approaches to the treatment of dispersive media in FDTD: convolution methods [7], the auxiliary differential equation form [8], etc. In this paper we use the latter as the starting point to build an extension of the ADI-FDTD to Debye dispersive media. Manuscript received October 17, 2002; revised January 15, 2003. This work was supported in part by the Spanish National Research Project TIC-2001- 3236-C02-01 and TIC-2001-2364-C03-03. The authors are with the Department of Electromagnetism and Matter Physics, University of Granada, 18071 Granada, Spain (e-mail: salva@ugr.es). Digital Object Identifier 10.1109/TAP.2003.818770 0018-926X/03$17.00 © 2003 IEEE