Purcell factors and β−factors in photonic crystal waveguides

We introduce a theoretical formalism to calculate the spontaneous emission rate enhancements (Purcell effect) and propagation mode β-factors from a quantum dot in a planar-photoniccrystal waveguide. Impressive Purcell factors and broadband enhanced β−factors are predicted. c © 2007 Optical Society of America OCIS codes: (230.7400) Waveguides, slab; (270.5580) Quantum electrodynamics Confining single quantum dots (QD’s) in high-index-contrast photonic crystals (PC’s) has recently became an active area of research, which is largely due to the continued success in the fabrication of such structures. The ability to couple single QD’s to nanophotonic materials and photonic dots is not only of fundamental importance but can also lead to novel applications including single-photon sources [1], low-threshold lasers and optical amplifiers. With regard to enhancing the emission rate of a single photon emitter, it is well established that a modified electromagnetic vacuum of the surrounding environment can have a profound influence on the emission rates of a localized emitter. Therefore, an increase in the local density of states (LDOS) created by a cavity can lead to enhanced spontaneous emission of a single photon through the Purcell effect [2]. Since the pioneering works of Yablonovitch [3] and John [4], it is also known that periodic dielectric media such as PC’s can create photonic bandgaps that may substantially influence the LDOS that a photon feels. From a practical viewpoint, the planar semiconductor PC’s currently show enormous promise since they exhibit large quality factors (Q ∼ 10) and very small effective mode volumes (Veff ∼ 0.05μm) [5]. Indeed PC nanocavities are now facilitating new regimes of cavity-QED [6, 7]. However, to better exploit the single photon emission potential of QD’s in planar PC’s, improvements in the microcavity excitation, enhanced cavity coupling and efficient extraction of the emitted photons are all required. While enhanced emission and ‘strong coupling’ in closed cavity defect modes are to be expected, less obvious it what can be achieved with a waveguide mode or an open cavity where excitation and extraction can be also extremely efficient, and everything can be performed in the plane and fully integrated. Having complete knowledge of the photon emission factor into an optical mode from a single QD is a highly desired property, yet presents a considerable challenge for an open cavity system as one must fully understand photon coupling to all available modes that contribute to a particular frequency. In addition, one needs to determine the LDOS as a function of position. For regular planar waveguiding structures, it has proven useful to separate out the various mode contributions in order to highlight the underlying mechanisms of spontaneous emission [8]. Recently, signatures of spontaneous emission enhancement of single QD emission in a PC waveguide have been demonstrated experimentally [9], though the Purcell factors were rather modest since the QD was coupled to a leaky mode with a large effective mode volume. Nevertheless, there is a lot of interesting physics behind this effect, and it also relates to earlier predictions about wire waveguides having a divergent-like LDOS [10]. In this work, we present a general theoretical formalism with rigorous calculations to study Purcell factors and propagation mode β−factors for planar PC waveguides, where the β is defined as the fraction of the emitted light 0 0.1 0.2 0.3 0.4 0.5 180 190 200 210 220 230 k (2π/a) f [ T H z] (a) Y−Y 0 X − X 0 Y−Y 0 (nm) X − X 0 (n m ) −200 0 200 −100 0 100 5 10 15 20 25 30 (c) (b)