A detailed numerical analysis is performed to deduce the structure parameters. Preliminary, for a periodical woodpile PhC with a constant layer-to-layer distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=\frac{\sqrt{2}}{4}a$$\end{document}d=24a (i.e. the rods from the consecutive layers just touch one another) the local band structure is numerically calculated with the plane-wave expansion method38 and the result is shown in Fig. 2(a). The calculated band structure reveals a complete PBG with a gap/mid-gap ratio (the ratio of gap width to the central frequency of the gap) of 11%, while the gap/mid-gap ratio of the partial PBG in the propagation direction Γ-X′ (stacking direction) is equal to 39%. Moreover, the frequencies at the bottom and top of this partial PBG are defined as the lower and upper cut-off frequencies and their variation with respect to the layer-to-layer distance is plotted in Fig. 2(b). Despite their similarity in terms of their layer-to-layer distance relations, the two cut-off regions differ in an important respect: To obtain a local PBG at the lower/upper bands, the layer-to-layer distance should be increased/decreased; respectively (see upper left inset of Fig. 1(b)). Consequently, the direction of the gradual layer-to-layer distance variation (whether it decreases or increases along the propagation direction) depends on the operational frequency region: The upper/lower cut-off regions require a decreasing/increasing layer-to-layer distance, respectively (indicated by red and green arrows in Fig. 1(b)). Apart from this apparent distinction, another difference between the two cut-off regions becomes evident if the partial PBGs in other directions are examined. The PBGs in the Γ-K direction, which corresponds to the direction along the rod axes are superimposed on Fig. 2(b). Unlike the Γ- X′ direction, the wave propagation in the Γ-K direction is nondegenerate for modes with polarization vectors along the rod axis and along the stacking direction, which we will refer to as the transverse electric-like (TE-like) and transverse magnetic-like (TM-like) modes, respectively. As can be seen from Fig. 2(b), the TM-like PBG comprises the whole upper cut-off region in the examined layer-to-layer distance range, whereas the TE-like PBG encloses only a portion of the examined range. On the other hand, for all utilized layer-to-layer distances, the lower cut-off region lies outside the coverage of both PBGs, which indicates that in the case of vertically scattered light, the corresponding structure based on the lower cut-off operation will be vulnerable to radiation losses. Furthermore, the PBG variations in the same range for propagation directions corresponding to the X, U, U′, K′, W, W’, W″, and L symmetry points35, whose positions in the Brillouin zone are shown on the lower right inset of Fig. 2(a), are calculated and given in Fig. 2(c). For directions with nondegenerate modes, the complete PBGs comprising simultaneously both TE-like and TM-like PBGs have been taken into account. Consequently, the local dispersion analyses in Fig. 2(b) and (c) imply that a complete confinement is not possible, since modifying the layer-to-layer distances will further break the low rotational symmetry of the lattice structure39, which in turn will increase the structural anisotropy. Nevertheless, a close inspection of the PBG variations reveals that for the layer-to-layer distance range \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1.15,1.35\}\ast \frac{\sqrt{2}}{4}a$$\end{document}{1.15,1.35}∗24a, the localized mode near the cut-off region is still confined in most directions.Figure 2Bandstructure analysis of the woodpile PhC structure. (a) Numerically calculated band structure of the woodpile PhC with Al2O3 rods. The lower right inset depicts the first Brillouin zone of the woodpile structure. (b) The dependency of the layer-to-layer distance onto the longitudinal and transverse cut-off frequencies. The TM-like and TE-like PBGs in the transverse directions are highlighted with light green and red colored areas. (c) The PBG variations for other directions. The dashed white line indicates the upper cut-off frequencies in the Γ- X′ direction.