1 Abstract— Two-dimensional (2-D) boundary integral equation analysis of a notched circular microdisk resonator is presented. Results obtained provide accurate description of optical modes, free from the staircasing and discretization errors of other numerical techniques. Splitting of the double degenerate Whispering-Gallery (WG) modes and directional light output is demonstrated. The effect of the notch depth and width on the resonance wavelengths, Q-factors, and emission patterns is studied. Further improvement of the directionality is demonstrated in an elliptical notched microdisk. Applications of the notched resonators to the design of microdisk lasers, oscillators, and biosensors are discussed. Index Terms— optical resonators, semiconductor microdisk lasers, integral equations, whispering gallery modes. I. INTRODUCTION IELECTRIC or semiconductor resonators shaped as circular cylinders and thin disks are, together with spherical particles, among the structures able to support the high-Q WG modes. Semiconductor microdisk lasers are very attractive light sources offering small mode volumes and ultralow threshold currents [1]. Perfectly circular microcavities provide very high optical confinement, which results in record Q-factors of the WG modes [2,3], however, they have two important drawbacks that limit their applications. These are, first, non-directive emission patterns with many identical beams, because a mode field in the disk plane depends on the azimuthal angle ϕ as either ϕ m cos or ϕ m sin (m=0,1,2,…). Second, each mode with 0 > m is double degenerate that leads to appearance of closely located doublets in the spectra of realistic resonators due to fabrication errors (sidewall roughness and shape imperfections, etc.) [3-5]. This work has been supported by the UK Engineering and Physical Sciences Research Council (EPSRC) under grants GR/R90550/01 and GR/S60693/01P, and the Royal Society under grant IJP-2004/R1-FS. The authors are with the George Green Institute for Electromagnetics Research, University of Nottingham, Nottingham NG7 2RD, UK (e-mail: eezsb@gwmail.nottingham.ac.uk or SBoriskina@gmail.com ). A. I. Nosich is also with the Institute of Radio Physics and Electronics NASU, Kharkov 61085, Ukraine. To ensure a single-mode operation of the microdisk laser, it is desirable to stabilize the lasing mode against the fabrication imperfections [6] and either suppress all the parasitic modes (i.e., spoil their Q-factors) or detune their resonant frequencies away from that of the lasing mode [7]. As the lasing mode, we consider a fundamental transverse electric (TE) first-radial- order WG mode (one of the modes of a doublet) with the frequency at or near the spontaneous emission peak of the cavity material [2]. Several types of parasitic modes can be supported in a microdisk resonator, such as modes of the orthogonal (TM) polarization, higher-radial-order WG modes, and the other first-radial-order WG mode of a doublet. TM-polarized emission is not usually observed in thin microdisks of several microns in diameter [3]. In high-index- contrast microdisks, the first radial-order WG-mode field is concentrated inside the microdisk very close to its rim. All of the higher-radial-order WG modes penetrate deeper inside the cavity. They can be suppressed by either decreasing the cavity radius and thus increasing their diffraction losses [8], or by removing material from the interior of the disk, which disturbs only the high-radial-order WG modes [7]. However, the former approach leads to increasing the diffraction losses of the lasing mode as well, and neither of them efficiently suppresses or shifts in frequency the second nearly degenerate first-radial-order WG mode of a doublet. Recently, a suppression of such a parasitic mode using a circular microcavity with a rotationally periodic modification to its rim - a microgear laser cavity - has been reported [9]. Enhancement of the lasing WG mode Q factor in such a cavity enabled microgear lasers with low threshold currents to be fabricated [10]. However, for the microlaser applications, another important design parameter is the directionality of the light output [11]. The emission from a thin circular microdisk mostly occurs in the disk plane. Unfortunately, due to rotational symmetry of the circular microdisk or microgear resonators, lateral light directionality cannot be achieved. One of the ways to extract the light from the resonant cavity is to use output evanescent-field couplers of various geometries [12]. Alternatively, microcavity shape deformations that destroy the rotational symmetry can be introduced [13-16], which include elongation, projections, notches, and openings. In this paper, we perform, for the first time to our knowledge, a detailed and accurate 2-D numerical study of the Q-factor and emission pattern control of the WG modes in notched microdisk resonators Svetlana V. Boriskina, Member, IEEE, Trevor M. Benson, Senior Member, IEEE, Phillip Sewell, Senior Member, IEEE, and Alexander I. Nosich, Fellow, IEEE Journal-ref: IEEE J. Select. Topics Quantum Electron., Jan./Feb. 2006 © 2006 IEEE D