We consider a 3D woodpile PhC composed of cylindrical alumina (Al2O3) rods with a radii and height equal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=\frac{\sqrt{2}}{8}a$$\end{document}r=28a and h = 8.463a, respectively, where a is the in-plane period. The Al2O3 rods have a dielectric constant equal to ε rod = 9.8 at microwave frequencies. The general schematics and the main operational principle of the proposed configuration is shown in Fig. 1(a). As mentioned above, the period in the stacking direction is continuously varied. We consider a slow (adiabatic) variation of the PhC period, which allows to consider local dispersion relation of it at every layer24. In result, the frequency of the PBG becomes smoothly varying along the propagation direction, as shown in Fig. 1(b). Therefore, a wave entering the structure will gradually slow down inside the PhC as its frequency will progressively approach the edge of the local PBG. Once the PBG is reached, the wave will “stop”, i.e. become temporarily localized, and afterwards will propagate back. Since the local PBG is distinct for every spatial position, each frequency component will be localized at different positions along the PhC, as illustrated in the left inset of Fig. 1(a).Figure 1Proposed 3D PhC configuration for rainbow trapping. (a) Schematic illustration of the chirped woodpile PhC structure. For better visualization the number of periods of the illustrated PhC structure is reduced. The lower inset depicts the cross sectional view of the structure in the yz- plane. (b) The variation of the PBG along the propagation direction is representatively depicted. Due to the gradual variation of the layer-to-layer distances, the incident wave will be spatially separated into its frequency components along the propagation direction, thus forming a “trapped rainbow”, as it is representatively shown as a cross sectional view on the lower left of (a). The upper right inset in (b) representatively shows the change of the dispersion curves in the propagation direction, where the layer-to-layer distance of the solid black dispersion curve is smaller than that of the dashed blue dispersion curve. The black arrows reveal the propagation direction of the incoming wave.