scattering. When the observation point moves away from the incident angle, the incoherent scattering increases and then fluc- tuates due to the random phase situation. The fluctuations are characteristic of random scattering, since the bistatic scattering cross section per unit volume will fluctuate from sample to sample. When we increase the number of particles in the simulations, the multiple scattering simulations show increased scattering effects at the incident angle, which is caused by higher-order scattering. In Figure 7, the phase function of independent scattering, sticky particles, and nonsticky cylinders is compared. The sticky cylin- ders show an angular distribution pattern similar to that of the nonsticky cylinders, but a stronger amplitude than both indepen- dent scattering and nonsticky cylinders. The stronger scattering effect of the sticky cylinders is expected because the cylinders adhere to each other to form larger aggregates. ACKNOWLEDGMENT This work is supported by NASA contract NAG5-9835 from the NASA Goddard Space Flight Center and by the City University of Hong Kong Research Grant 9380034. REFERENCES 1. L. Tsang, J.A. Kong, and R. Shin, Theory of microwave remote sensing, Wiley–Interscience, New York, 1985. 2. L. Tsang, J.A. Kong, and K.-H. Ding, Scattering of electromagnetic waves: Theories and applications, Wiley–Interscience, New York, 2000. 3. L. Tsang, J.A. Kong, K.-H. Ding, and C.O. Ao, Scattering of electro- magnetic waves: Numerical simulations, Wiley–Interscience, New York, 2001. 4. L. Tsang and J.A. Kong, Scattering of electromagnetic waves: Ad- vanced topics, Wiley–Interscience, New York, 2001. 5. L. Zurk, Electromagnetic wave propagation and scattering in dense, discrete random media with application to remote sensing of snow, Ph.D. dissertation, University of Washington, 1995. 6. L. Tsang, C.E. Mandt, and K.-H. Ding, Monte Carlo simulations of the extinction rate of dense media with randomly distributed dielectric spheres based on solution of Maxwell’s equations, Optics Lett 17 (1992), 314 –316. 7. J. Guo, L. Tsang, K.-H. Ding, A.T.C. Chang, and C.-T. Chen, Fre- quency dependence of scattering by dense media of small particles based on Monte Carlo simulation of Maxwell’s equations, IEEE Transactions Geoscience Remote Sensing 40 (2002), 153–161. 8. C.H. Chan and L. Tsang, A sparse-matrix canonical grid method for scattering by many scatterers, Microwave Optical Technol Lett 8 (1995), 114 –118. 9. C.C. Huang, L. Tsang, and C.H. Chan, Multiple scattering among vias in planar waveguides using the SMCG method, Proc IMS, Seattle, Washington, 2002. 10. C.C. Lu and W.C. Chew, A multilevel algorithm for solving a bound- ary integral equation of wave scattering, Microwave Optical Technol Lett 7 (1994), 466 – 470. 11. W.C. Chew, Waves and fields in inhomogeneous media, IEEE Press, New York, 1994. © 2003 Wiley Periodicals, Inc. FULL-WAVE FDTD DESIGN AND ANALYSIS OF WIDEBAND MICROSTRIP-TO-WAVEGUIDE TRANSITIONS Ca ` ndid Reig, 1 Enrique A. Navarro, 2 and Vicente Such 2 1 Departament d’Enginyeria Electro ` nica Universitat de Vale ` ncia Dr. Moliner, 50 46100-Burjassot, Spain 2 Departament de Fı´sica Aplicada Universitat de Vale ` ncia Dr. Moliner, 50 46100-Burjassot, Spain Received 22 January 2003 ABSTRACT: Wideband transitions are designed and analysed by using two different approaches of the finite-difference time-domain (FDTD) method, in combination with the theory of nonuniform transmission lines. These transitions consist of a ridged waveguide-based taper be- tween a shielded microstrip and a standard X-band rectangular waveguide. In the first step, a full-wave 2D-FDTD scheme is used to calculate the dispersion characteristics, as well as the geometry depen- dence of the impedance in the double ridged waveguide. Once these design curves have been obtained, the stepped transmission line trans- former theory is used to design the tapers. In a former step, the nonuni- form 3D-FDTD technique is applied, the transitions are simulated and the method is validated. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 38: 317–320, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11048 Key words: microstrip-to-waveguide transition; 2D-FDTD design; inho- mogeneous FDTD; ridged waveguide; impedance definition INTRODUCTION Transitions from microstrip lines to waveguides are essential in millimetre-wave hybrid and monolithic integrated applications. Such structures generally involve complicated geometries and thus their design and analysis are complex. Among the most popular geometries of transition we can find coupling through slots [1], antennae [2], and stepped transitions [3]. Most of the previous work about design and analysis techniques has been mostly em- pirical and based on the circuit impedance concept. A typical example is found in [3], where characterisation of both microstrip and ridge waveguides was described and used to model a micro- strip-to-ridge waveguide discontinuity and ridge waveguide step junctions. On the other hand, the finite-difference time-domain (FDTD) method has been demonstrated to be a feasible and powerful tool for the analysis of microwave problems where Figure 7 Comparison of the phase function of sticky and nonsticky cylinders. The sticky case has stickiness parameter of 0.1. For both cases, 300 cylinders occupy a fractional volume of 10% and ka = 0.1 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 38, No. 4, August 20 2003 317