than the threshold) coupling is inhibited and more than 70% of the injected power remains in the excited channel. The numerical parameters used to obtain these results are as follows. Transversal mesh has been set for both methods (RK- FDBPM and CN-FDBPM) to x =y = 0.05 m. This is the coarsest transversal mesh that ensures good agreement with the results reported in [3, 4]. Once the transversal mesh has been set, adequate selection of the remaining simulation parameters is needed. In the case of the RK-FDBPM, it is only necessary to ensure the method’s stability by choosing a sufficiently small propagation step (in this case, z = 0.01 m). Using this prop- agation step, the simulation was carried out in 12.1 h. When using the CN-FDBPM, three numerical parameters, which greatly influ- ence the simulation results and the computational effort, must be set: the number of Gauss–Seidel iterations ( GS it ), the number of nonlinear iterations ( NL it ), and the propagation step size ( z ). Adequate selection of these parameters is heavily dependent on the nonlinearity of the device and usually becomes a difficult and lengthy task, which is done manually and requires an expert user. In this case, after some previous trials, it was observed that convergence to the correct results (shown in Fig. 6) was obtained with the following parameter setting: 11 Gauss–Seidel iterations GS it , two nonlinear iterations NL it , and propagation step size z = 0.01 m. With these settings, the measured CPU time was 24.8 h, that is, more than twice the time expended by the RK-FDBPM algorithm, although the overall time employed in properly setting the CN-FDBPM parameters and verifying convergence has not been accounted for. The nonlinear directional coupler of Figure 5 has also been studied with the help of commercially available FDBPM software (Optiwave) based on an alternate direction im- plicit (ADI) technique. The obtained results closely agree with those shown in Figure 6, but the computational effort, 31 h 53 min, was higher than that of the proposed RK-FDBPM method. 4. CONCLUSION A fast Runge–Kutta-based FDBPM technique for the analysis of nonlinear optical dielectric waveguides has been presented. This technique is the semivectorial extension of the previously pub- lished scalar RK-FDBPM method [8], and essentially exhibits its same advantages: ease of use and lower computational effort. A detailed comparison between the proposed Runge–Kutta-based technique and the Crank–Nicholson-based technique, frequently cited in the bibliography, has been performed. From the obtained results it can be seen that the new technique offers similar accu- racy, but with much lower computational effort in all the tested situations (2D and 3D waveguides), with the reduction of the CPU time ranging from 2–10 times the original, depending on the nonlinearity level. In addition to higher computational efficiency, the new method is easier to use due to the reduced number of simulation parameters to be set. In all the analyzed devices, the obtained results have been compared with those obtained using other numerical techniques (finite-element BPM for the 2D device and commercially available software in the 3D coupler), showing very good agreement between the results in all cases, and always with lower computational effort in the case of the proposed RK- FDBPM. ACKNOWLEDGMENTS This work was supported by the Spanish C.I.C.Y.T. under project TIC2000-1245. The authors wish to thank Dr. J. G. Wangu ¨emert- Pe ´rez for valuable discussions. REFERENCES 1. R. Scarmorzzino, A. Gopinath, R. Pregla, and S. Helfert, Numerical techniques for modeling guided-wave photonic devices, IEEE J Sel Topics Quantum Elect 6 (2000). 2. S.S.A. Obbaya, B.M.A. Rahman, and H.A. El-Mikati, New full-vec- torial numerically efficient propagation algorithm based on the finite- element method, IEEE J Lightwave Technol 18 (2000). 3. A. Cucinotta, S. Selleri, and L. Vicenti, Nonlinear finite-element semivectorial propagation method for three-dimensional optical waveguides, IEEE Photon Technol Lett 11 (1999). 4. S.S.A. Obbaya, B.M.A. Rahman, and H.A. El-Mikati, Full-vectorial finite-element beam propagation method for nonlinear directional cou- pler devices, IEEE J Quantum Elec 36 (2000). 5. M.S. 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The achievable band- width with bow-tie feed is around 115% (more than 2:1) for VSWR  2 and a similar configuration could achieve a bandwidth of 45% for VSWR  2 with radial stub feed. The effect of feeding mechanisms on the dimensions of the slots is discussed. A comparison of measured re- sults for various feed configurations is presented. © 2004 Wiley Period- icals, Inc. Microwave Opt Technol Lett 40: 77–79, 2004; Published on- line in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ mop.11289 Key words: slot antenna; bow-tie feed; radial stub feed; wideband 1. INTRODUCTION Microstrip slot antennas have the advantage of being able to produce radiation patterns over wide bandwidth. Various feeding MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 40, No. 1, January 5 2004 77