11. zyxwvutsrqpo P. Myslinski, D. Nguyen, and J. Chrostowski, “Effects of Concen- tration and Clusters in Erbium-Doped Fiber Amplifiers,” Paper No. WP3, Conference on Optical Fiber Communication, OFC‘95, San Diego, CA, Feb.-March, 1995. 12. A. M. Vengsarkar, D. J. DiGiovanni, W. zyxwvuts A. Reed, K. W. Quoi, and K. L. Walker, “Measurements of Erbium Confinement in Optical Fibers: A Differential Mode-LaunchingTechnique,” zyxwvuts Opt. Lett., Vol. 17, No. 18, Sept. 1992, pp. 1277-1279. Received 9-21-95 Microwave and Optical Technology Letters, 11/2, 61-64 zyxwvut 0 1996 John Wiley zyxwvutsrqp & Sons, Inc. CCC 0895-2477/96 COMPUTING THE QUALITY FACTOR OF RESONATORS USING THE ALGORITHM FINITE-DIFFERENCE - TIME-DOMAIN Paul H. Harms, Yuval Shimony, and Raj Mittra Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana. Illinois 61801 KEY TERMS Unloaded qualityfactor, resonators, finite-difference-time-domain method ABSTRACT The unloaded Q zyxwvutsrq of complex cavity structures is computed by using the finite-difference-time-domain technique (FDTD). The FDTD technique computes the cavity zyxwvutsrqpon fields that are integrated in time and space to yield the stored energy and power loss frvm which the quality factor is obtained. A comparison of the Q computed from this method with analytical and measured results is given. zyxwvutsrqp 0 1996 John wiley & zyxwvut Sons, Inc. 1. INTRODUCTION In the design of cavity resonators for microwave oscillators, for example, for mobile phone transmitters, the unloaded quality factor is an important parameter for determining its resonant characteristics. For complex resonant structures the Q must be computed with a numerical approach, and al- though much work [l-51 has been done in this area, it still remains a challenging problem. The finite-difference-time-domain (FDTD) method is well suited for modeling complex geometries and is adapted in this work to compute the unloaded Q for microwave cavities. The analysis presented in [5] also employs the FDTD tech- nique, but it computes the quality factor in a significantly different manner, namely, by using the discrete Fourier trans- form (DFT) of the time-domain fields. In contrast, the ap- proach presented herein employs time-averaged fields ex- tracted from the FDTD analysis for computing the Q. In Section II, the theory of the methodology is described. In Section I11 numerical results are presented for several cavi- ties and are comparcd with analytical and measured results where available. II. THEORY Assuming the dominant mode is not degenerate, the cavity is excited at the dominant resonant frequency with a sinusoidal signal in time. Once the initial transients are gone, the unloaded Q is calculated from the time-domain fields by using the equation where the numerator corresponds to the total energy, the denominator corresponds to the power loss, and R, is the surface resistance [6]. The field magnitudes squared are time averaged over a complete cycle of the sinusoidal fields. The integrations are performed at each time step by summing over the fields in the mesh weighted by the appropriate unit cell dimension(s1. Three memory locations are required to store the total energy and power loss per time step and the Q per period. Additional memory is also required to store the pointers to the metal surfaces for computing the power loss. The outline of the procedure for a specific case is described in Section 111. The magnetic fields are employed for computing both the stored energy and power loss, thereby avoiding errors caused by the half time-step difference between the electric and magnetic fields. If dielectric losses are considered, a similar approach can be taken using the displacement current term, mE, to compute the dielectric power loss and the electric energy term to compute the stored energy. 111. NUMERICAL RESULTS A. The Quality Factor of a Rectangular Cavity with Different Loads. The method will be demonstrated with a rectangular copper cavity of dimensions 2.286 cm X 2.286 cm X 1.016 cm under four different load conditions. Initially, the cavity is empty; next, a small block of copper (0.254 cm per side) is placed inside against the lower wall, as shown in Figure 1. For the third case, the small block of copper is lengthened to Metallic or Dielectric Block (4 Metallic Rod I a (b) Figure 1 Rectangular resonator with (a) metallic or dielectric ( zyxwv C, = 36.7) block (0.254-cm cube) at the center of the lower wall and (b) a metallic rod (0.254-cm square) at the center of the cavity extending from the lower to the upper wall (a = 2.286 cm, b = 1.016 cm) 64 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 11, No. 2, February 5 1996