1940 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 18, NO. 18, SEPTEMBER 15, 2006 Fast Calculation of Cavity-Mode Characteristics of Photonic Crystal Cavities Zexuan Qiang and Weidong Zhou, Member, IEEE Abstract—A fast approach based on effective index perturba- tion method is proposed to evaluate the intrinsic characteristics of photonic-crystal-slab-based microcavity with two-dimensional finite-difference time-domain (2-D FDTD) technique. For two widely used single defect structures, less than 2% computational error was obtained in calculating the defect mode frequencies. Accurate prediction of cavity modal properties and resonant peak frequencies is feasible based on 2-D FDTD simulation by adjusting the effective index to match the dielectric band edge for donor-like defect mode. The correlation between the modified effective index and the cavity (lasing) mode with the highest quality factor offers an efficient tool in the design of defect mode based photonic crystal microcavities. Index Terms—Cavity quality factor, defect-mode cavity, effective index method (EIM), effective index perturbation (EIP), photonic crystal slabs (PCSs). I. INTRODUCTION T WO-DIMENSIONAL (2-D) photonic-crystal-slab (PCS) microcavities have been a good candidate to perform single photon sources [1]–[3], biosensor [4], and high-sensi- tivity filter [5], due to their ultrasmall mode volume , high quality factor , and enhanced spontaneous emission [6]. Sig- nificant progress has been made in the computational techniques for the design of PCS-based microcavity, and in the under- standing of these cavity characteristics. Works reported to date are mostly based on three-dimensional (3-D) finite-difference time-domain (FDTD) technique [7]–[9], which can accurately simulate the characteristics of these devices. However, the fully vectorial 3-D FDTD approach is extremely time and computer memory consuming. The effective index method (EIM) [7], [10] has proved to be very effective and efficient in predicting the cavity properties with reduced dimensionality (from 3-D to 2-D), where only the effective index of fundamental guided mode of the unperturbed slab is considered. EIM is most effec- tive for the low-index-contrast PCS. However, it becomes less accurate when it is applied to high index contrast PCSs, where high index contrast is favorable for reduced vertical cavity loss and better mode confinement. Efforts have been reported to ad- just the effective index by matching the photonic band diagram Manuscript received May 1, 2006; revised July 6, 2006. This work was supported by the Air Force Office of Scientific Research (AFOSR) under the SPRING and NANO Programs. Z. Qiang is with the Nanofab Center, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX 75019-0072 USA, and also with the Institute of Optical Communication Engineering, Nanjing University, Nanjing 210093, China (e-mail: zxqiang@uta.edu). W. Zhou is with the Nanofab Center, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX 75019-0072 USA (e-mail: wzhou@uta.edu). Digital Object Identifier 10.1109/LPT.2006.882289 (dispersion plot) with frequency offset [11] and effective index modification [12]. However, to the best of our knowledge, no work has been reported in EIM-based approaches, on the correlation of shifted photonic band diagram with the cavity defect mode and the corresponding quality factor . In this letter, we introduce an effective index perturbation (EIP) technique in determining the suitable effective index for accurate prediction of cavity modal property and defect mode locations based on 2-D and 3-D plane-wave-expansion (PWE) techniques and the 2-D FDTD method. In donor-like defect mode cavities formed in air-column-based PCSs, very good agreement in defect mode locations was obtained with the EIP technique by matching the dielectric band edges simulated from 2-D and 3-D PWE techniques. The highest mode can also be correctly predicted, which is also very important in photonic crystal microcavity design, where the mode with the highest cavity could be most important in determining the cavity characteristics, including lasing and sensing. II. EFFECTIVE INDEX PERTURBATION It is well-known that the air-band mode is more affected by removing an air hole and finally a donor state is excited and pulled into the bandgap from the air band [13]. In standard EIM, a relatively large error in predicting the resonant mode locations is often seen as a result of large offset in simulated photonic bandgap. This leads us to believe accurate prediction of reso- nant peak locations from 2-D FDTD is feasible by properly ad- justing the effective index to matching the simulation photonic bandgaps. In this study, we found the suitable perturbation in ef- fective index ( ) can lead to very small computation errors in resonant peak locations for these donor-like modes by matching the dielectric band edge. The procedure to choose the suitable is illustrated in Fig. 1. First, the dielectric band edge of 2-D PCS without defect is calculated by 3-D PWE. The stan- dard effective refractive index of unperturbed slab waveguide is also done at this step by conventional methods [14]. Second, the 2-D PCS is transformed into an ideal 2-D PC structure with the high refractive index replaced by the slab effective re- fractive index . The dielectric band edge of 2-D PC without defect is calculated by 2-D PWE. The calculated frequency error is computed as . If the value of is greater than 2%, an EIP is done to adjust the effective index based on (with sign): . Step 2 is repeated with perturbated until is less than 2%. Fi- nally, we use the satisfied to calculate the quality factor and resonant wavelength by using the 2-D FDTD method in 2-D PCs with defect. Note that the criterion of 2% is set so that only one or two iterations are needed to get the accurate effec- tive index. Higher accuracy with smaller tolerance could trans- 1041-1135/$20.00 © 2006 IEEE