Submicrometer diameter micropillar cavities with high quality factor and ultrasmall mode volume

We theoretically demonstrate high Quality factor ( Q~ 3×10 6 ) micropillar cavities with record low mode volumes [ V ~0.1( λ / n ) 3 ] based on the TiO 2 /SiO 2 material system. The proposed cavities have Q / V three orders of magnitude larger than previously reported ones. We show that our cavity embedded with diamond nanocrystal provides a feasible platform for cavity quantum electrodynamics experiments in the strong coupling limit.

recently for large diameter (d=4µm) pillars, resulting in a relatively large mode volume [V>50(λ/n) 3 ] [6]. However, for applications in cQED, coupling between an emitter and a photon localized in the cavity requires a large Rabi frequency g that is proportional to 1/ V . In order to decrease the mode volume, it is necessary to decrease the pillar diameter and thereby improve the radial confinement of light. However, this can lead to a significant reduction in the Q factor in the traditional micropillar designs [7], and the best Q factors reported in the case of submicron diameter micropillars have been theoretically limited to approximately 2,000 [8].
In this letter, we for the first time theoretically demonstrate sub-micron diameter micropillar cavities with an ultra-high Q/V that is three orders of magnitude larger than previously reported. This is achieved by simultaneously increasing the Q (Q~3×10 6 ) and reducing the mode volume [V~0.1(λ/n) 3 ], using a bandgap tapering method developed recently in 1D photonic crystal structures [9,10]. While the proposed approach is general and can be applied to a range of material systems and applications, our cavities are designed to operate at 637nm wavelength and therefore are suitable for coupling to nitrogen-vacancy (NV) color centers embedded within diamond nanocrystals positioned at the middle of the cavity. NV color centers have recently attracted significant attention as promising quantum emitters [11]. NVs emission is broad band (630nm-750nm) with stable zero-phonon line at 637nm visible even at room temperature.
Our micropillar is based on two distributed Bragg mirrors (DBR) that consists of TiO 2 (n TiO2 =2.4) and SiO 2 (n SiO2 =1.5) alternating layers, and a TiO 2 spacer of thickness s sandwiched between them [ Fig. 1(a)]. The micropillar cavity Q is inversely proportional to the cavity losses that in turn can be separated into two components: the transmission losses due to the finite length of the DBRs, and the scattering losses at the spacer/mirror interfaces. The former can be minimized (for a given number of TiO 2 /SiO 2 pairs) using a "quarter stack" DBR that consists of TiO 2 /SiO 2 layers with thickness λ/4n eff , where λ=637nm. This also maximizes light confinement along the pillar axis, resulting in a minimized mode volume. For example, in the case of a micropillar with diameter d=340nm, the thicknesses of TiO 2 and SiO 2 layers in DBR are 82.6nm and 147.4nm, respectively, resulting in total DBR periodicity of a=230nm. Next, the thickness of the spacer is chosen (s=220nm) in order to position the cavity resonance at 637nm, and the quality factor (Q) of such cavity is obtained using finite-difference time-domain method (FDTD). We find Q~100 which is consistent with previous reports [7]. The low Q is attributed to scattering losses arising from the mode profile mismatch between the localized cavity mode and evanescent Bloch mode inside the DBRs [12]. Fig. 1(a) illustrates the profile mismatch for E r component. To suppress this mode mismatch and the resulting scattering losses, we use the mode matching technique previously developed for 1D photonic crystal cavities [9]. We substitute the uniform center segment with a single TiO 2 /SiO 2 pair with the same aspect ratio as in the DBR but a smaller thickness w. When w=0.67a, the cavity resonates at λ=637nm with Q=6,000, a 60fold improvement over the conventional design.
In order to increase the Q factor further, we incorporate more TiO 2 /SiO 2 segments with varying the thicknesses w i (i is the segment number). This can also be seen as a "tapered DBR" approach, where each taper section further reduces the mode mismatch in order to suppress the scattering losses. In Fig. 2(a), we use 4 tapered segments and 20 DBR pairs at each side. In order to set the resonating wavelength at 637nm, the thickness of each taper segment is precisely tuned to w 1 =215.3nm, w 2 =202.4nm, w 3 =191.0nm, and w 4 =180.8nm. The resulting mode has a Q factor of 250,000 and a mode volume of 0.07(λ/n) 3 , which represents at least three orders of magnitude enhancement of Q/V compared to any previous micropillar designs. As shown in Fig. 2(b), this high-Q mode has an anti-node at the central TiO 2 segment and therefore is ideally suited for coupling to quantum emitters, such as an NV center in diamond or a semiconductor nanocrystal, embedded within this layer. We also find a second-order cavity mode [ Fig. 2(c)] at wavelength of 685nm with a respectable Q=110,000. The Q factor of the fundamental mode can be enhanced by increasing the tapering process. For instance, we obtain Q=3,000,000 and V=0.10(λ/n) 3 with 10 taper segments. However, increasing the cavity length pulls higher-order modes from the dielectric band into the bandgap, as shown in Fig. 2(d) and (e). These higher order modes can potentially couple to the emitter placed at the center of the cavity and the resulting multi-mode cavity is not desirable. Therefore from here on, we only consider 4-taper-segment cavities, which limits our Q to 250,000.
Next, we optimize the diameter of our micropillar cavity to minimize its mode volume. It can be seen in Fig. 3(a) that the smallest mode volume is obtained at d=340nm. For d<340nm, V increases due to the reduced confinement in the axial direction: the effective mode index contrast between TiO 2 and SiO 2 is reduced for small diameters, and therefore the width of the bandgap decreases, resulting in deeper penetration of the cavity mode into the DBRs, thus increasing V.
For d>340nm, however, the mode is almost completely confined within the pillar and the n eff of each segment approach the refractive index of the material. Therefore, the width of the bandgap remains approximately constant as the pillar diameter increases and the axial confinement remains the same. However, V increases due to the larger mode cross-section (radial confinement increases). The trade-off between radial and axial confinement results in an optimized diameter of d=340nm. For the 4-taper-segment cavity, we also find the cavity modes with TE 01 and TM 01 profiles at wavelengths of 578nm and 492nm, respectively. Both of these modes peak at the central TiO 2 segment. However, the lateral electric field density profiles shown in Fig. 3(b) indicate that these modes have a node at the center of the micropillar and therefore will not couple efficiently to the nano-emitter placed at the center. Moreover, these modes are detuned from the emission spectrum of an NV center and therefore are not of interest. It is also important to note that if the pillar diameter increases further, additional higher-order modes, with the same azimuthal order as HE 11 (e.g. EH 11 ), are allowed. These modes can couple to the fundamental HE 11 cavity mode and thus introduce additional loss mechanism [7], and reduce the Q factor of the fundamental cavity mode.
Finally, we investigate the potential of our microcavity for applications in cQED.
Diamond nanocrystals (size<50nm) with single NV color centers could be embedded within the central TiO 2 layer in the cavity. It is interesting to note that refractive index of TiO 2 is very close to that of diamond, and therefore the amount of light scattering at the interface between diamond and TiO 2 is negligible. The light-matter interaction process can be characterized by the coupling strength (Rabi frequency) g, the emitter spontaneous emission rate γ, and the photon loss rate κ.
The common emission life time of a diamond NV center is τ ≈ 20ns, leading to respectively. In a diamond NV center, the zero-phonon line only contributes about 5% of the total emission, the rest being emitted into the phonon sideband. Therefore, only ≈ 5% of the total emission is coupled to the cavity mode, so g must be scaled by a factor of 1 / 20 , resulting in 0 / 20 2 7.1 g g GHz     . By comparing the characteristic frequencies, we conclude that in our system, g > κ, γ, and that the system is well into the strong coupling regime. Furthermore, it is interesting to note that the system is in the strong coupling regime even when Q>30,000, which can be obtained with only 10 pairs of DBR mirrors on each side of the tapered section.
In conclusion, we have demonstrated that high Quality factor micropillar cavities can be realized with sub-micron diameter pillars. We have engineered the cavities with a record low mode volume of V=0.07(λ/n) 3 and a Quality factor of 250,000. We expect, however, that realistic fabricated structures will have reduced quality factors due to fabrication-related imperfections, including surface roughness, slanted walls and material absorption. One possible approach to overcome these problems is based on oxide-aperture design [13]. We predicted that by embedding a diamond nanocrystal with an NV color center at the middle of the cavity, the strong coupling limit of light-matter interaction can be achieved. Our method can be easily adapted to different material systems and enable realization of an ultra-high Q/V cavity in AlAs/GaAs platform suitable for realization of low-threshold VCSELs, for example.
This work was supported in part by NSF grant ECCS -0708905 "NIRT: Photon and Plasmon Engineering in Active Optical Devices based on Synthesized Nanostructures" and Harvard NSEC center. The authors acknowledge the assistance provided by Murray W.
McCutcheon. Yinan Zhang would also like to dedicate this work to Honglie Yin and Erfang Zhang.