# Cavity Design in Woodpile Based 3D Photonic Crystals
## 5 Discussion and Conclusions  
In all cases in this paper, the dipole is oscillating along the \(z\)-axis. In a more general situation, a dipole could be in any orientation and other \(Ex\) or \(Ey\) modes (in which case the \(Ex\) or \(Ey\) component could be much greater than \(Ez\) component) could be excited. The reason only \(z\)-axis orientated case is discussed here is, as we decrease the defect size in \(xy\)-plane, we found that the \(Ex\) and \(Ey\) modes are dramatically suppressed and only \(Ez\) modes survive in such a cavity (see Figure S1 in the supplementary materials). It means the confined cavity mode is polarized along the \(z\)-axis when the defect size \(L\) is small enough. In general, the resulting resonant mode in a cavity surrounded by PhC structure with complete photonic bandgap could be considered a combination between cavity mode itself without PhC structure and the confinement of complete photonic bandgap. However, the mode distribution in a 3D PhC structure is strongly affected by the surrounding environment. It leads to a great complexity of the investigation due to kinds of potential defect positions. In this paper we demonstrate that, by carefully choosing the position of the defect at the lattice center, a cavity with high symmetry can support well confined Gaussian-like cavity modes and the analysis of such cavities can be greatly simplified by using symmetrical lattices. We find that high Q-factor cavities can be obtained when the resonant frequency close to the center of the photonic bandgap and small mode volumes of the resonant modes can be achieved by reducing the cavity size. Furthermore, we can tune the size of the cavity and manually choose the cavity mode with desired frequency. Additionally, the patterns of the resonant modes investigated in this paper can then be used for the examination of mode orders. In the future, we could introduce light emitters in such cavities to control the spontaneous-emission and design waveguides to couple cavity modes out. As an example, we provide our preliminary work on the fabrication of polymer woodpile structures with incorporated defects using DLW method in the supplementary materials (Figure S2). In the long term, the investigation of manipulating the light propagation and confinement in woodpile could lead to the ultimate template for photonic integrated circuits and optical computers.  
The following are available online at [[http://www.mdpi.com/2076-3417/8/7/1087/s1](http://www.mdpi.com/2076-3417/8/7/1087/s1)]([http://www.mdpi.com/2076-3417/8/7/1087/s1](http://www.mdpi.com/2076-3417/8/7/1087/s1)), Figure S1: Amplitude of the electric field (\(Ex\), \(Ey\) and \(Ez\) component) of \(Ex\) mode (a-c) and \(Ez\) mode (d-f) as a function of time and their corresponding fast Fourier transform when defect size \(L\) = 0.25c, Figure S2: Scanning electron microscopy images of the fabricated polymer woodpile structures using the DLW method, (a) with various defects introduced (b) with dye doped defects embedded.  
Conceptualization, X.Z., and Y.-L.D.H.; Data curation, X.Z. and M.P.C.T.; Formal analysis, X.Z.; Funding acquisition, Y.-L.D.H. and J.G.R.; Investigation, X.Z. and M.P.C.T.; Project administration, Y.-L.D.H. and J.G.R.; Software, X.Z. and M.P.C.T.; Supervision, Y.-L.D.H. and J.G.R.; Visualization, X.Z.; Writing (original draft), X. Z; Writing (review & editing), X.Z., Y.-L.D.H. and J.G.R.  
This research was funded by the Engineering and Physical Sciences Research Council (EPSRC) grant number EP/M009033/1 and EP/M024458/1.  
This work was carried out using the computational facilities of the Advanced Computing Research Centre, University of Bristol, Bristol, UK and funded by the Engineering and Physical Sciences Research Council (EPSRC) grant EP/M009033/1 and EP/M024458/1.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.