# Cavity Design in Woodpile Based 3D Photonic Crystals
## 1 Introduction  
There has been significant interest in three-dimensional photonic crystal (3D PhC) structures in recent years aiming to exploit full photonic band gaps (PBG) [1; 2; 3] and the unprecedented confinement of light at defects in these structures. A variety of structures have shown complete photonic bandgaps including dielectric spheres in a face-centered-cubic (FCC) lattice [4; 5; 6], rod-connected diamond (RCD) structures [7; 8], 'Yablonovite' PhC [9] and woodpile based PhC structure [10]. Various defects can be introduced in such perfect crystals to create localized photonic modes such as optical waveguides [11; 12; 13; 14; 15] and nanocavities [16; 17; 18; 19; 20; 21; 22]. Light trapped in such defects interacts strongly with light emitters making them suitable for ultra-low-threshold lasers [23; 24; 25; 26], enhanced optoelectronic sensors [27; 28; 29; 30], ultrafast and low-power all-optical switches [31; 32; 33; 34] and quantum information processing devices with non-linearity at the single photon level [35; 36; 37; 38]. Although much effort has been made in applications of 1D or 2D PhCs [39; 40; 41; 42], they still lose the confinement of light in the third dimension but rely on a total internal reflection. This leads to out-of-plane losses and manufactural restrictions such as suspended membranes or low index substrates. In a 3D PhC structure, the cavity mode is strongly associated with the surrounding environment which could result in a rich variety of mode patterns. Recently, fabrication of 3D photonic structures by direct laser writing (DLW) using two-photon polymerization (TPP) [43] has been reported creating woodpile [44; 45; 46; 47] and RCD structures [48]. Notably, the woodpile structure is a promising structure which could be possible to mass produce with lower cost methods via layer by layer 2D lithographic approaches [13; 49]. Although current applications are rather limited by the challenges of fabrications, some pioneering work has been proposed and implemented for various applications such as photon guiding [13; 49] and spontaneous-emission control [50].  
In this study, we choose a woodpile structure formed by rectangular rods as the template. Design of the defect in such woodpile based 3D PhC structures is investigated through comparing the woodpile lattice with the RCD lattice [8]. Although a woodpile formed by rectangular rods may not hold the widest photonic bandgap record as an inverse woodpile [10] or a RCD structure does [6; 7; 8], such discussion about the position of a defect based on the lattice still could apply more generally toother kinds of structures [21]. We found that, by carefully choosing the position of the defect, we could create a cavity with high symmetry which supports well confined cavity modes and also demonstrate that such a cavity in a woodpile based 3D PhC structure works in a similar way to a [PERSON] laser resonator [51]. Moreover, choosing the specific defect position can help to simplify the analysis of the mode pattern as a result of the combination of the cavity with and without the 3D PhC surroundings.  
This paper is organized as follows. In Section 2, we introduce the simulation methods applied throughout this paper. We then detail the designed woodpile structure in Section 3 with optimized parameters used in our simulations and the position we choose for the cavity. The principle of choosing this position is also discussed. In Section 4, we show the numerical results to demonstrate the tunable cavity mode and mode volume.