## 2 Hollow-Core Woodpile Coupler Design  
The woodpile structure is based on the 3D photonic crystal lattice, which consists of a combination of high-index dielectric bricks (silicon with \(e_{r}=11\) in this work) immersed in a vacuum background, stacked layer-by-layer, each layer rotated \(90^{\circ}\) with respect to the layer below and with an offset of half of the lattice period from the layer \(2h\) below, as shown in Figure 2a. The fundamental dimensions \(w\), \(h\) and \(d\), which represent the brick width, height and spacing between adjacent brick centers, are highlighted. It is worth pointing out that the woodpile numerical design is dimensionless and can be easily scaled to any working frequency by choosing the structure period \(d\); the other rod dimensions are determined in simulations with the objective of maximizing the band gap. The presented woodpile coupler operates at the wavelength \(\lambda_{0}=5\) um (\(f_{0}\simeq 60\) THz), and the design procedure can be summarized as follows. As a first step, the band diagram (normalized frequency vs. normalized wavevectors) of the woodpile structure is calculated through the use of the MIT Photonic Bands (MPB) software [13]. In order to obtain the maximum band gap, the normalized brick dimensions \(w=0.2\sqrt{2}/d\), \(h=0.25\sqrt{2}/d\) are used. The band gap can be centered at the desired working frequency by choosing the lattice constant \(d\) through the expression \(f_{\lceil\text{THz}\rfloor}=f_{norm}c/d\). By imposing \(d=2.082\,\upmu\)m, we obtain a band gap centered at around 60 THz, with fundamental woodpile dimensions \(w=0.588\,\upmu\)m, and \(h=0.736\,\upmu\)m. As a second step, we obtain a hollow core accelerating waveguide by removing a specific amount of dielectric material from the structure creating the so-called \"defect\" region, where the beam will propagate [14; 15]. The accelerating waveguide, whose projected band diagram is visible in Figure 3, is tuned in order to support a confined TM\({}_{01}\)-like mode and has dimensions \(w_{d}=2.429\,\upmu\)m and \(h_{d}=2.209\,\upmu\)m (see Figure 2a). The last step is the mode converter creation: it consists of two side-coupled launching TE\({}_{10}\)-like hollow-core waveguides, whose waves are combined through the use of two  
Figure 2: (**a**) Accelerating waveguide front view with fundamental geometrical parameters \(w\), \(h\), \(d\) and hollow-core defect dimensions \(w_{d}\), \(h_{d}\). (**b**) Structure slice in the \(xz\) plane. It is composed of two mode converters (from TE\({}_{10}\)-like to TM\({}_{01}\)-like mode and vice versa) and an accelerating waveguide. One of the two bricks composing the short wall of the exit mode converter is also highlighted by a green circle.  
properly tuned dielectric bricks acting as \"shorts\" in order to perform the conversion into the accelerating TM\({}_{01}\)-like mode. Figure 2b shows the complete structure in a back-to-back configuration (\(xz\) cut plane): the mode conversion and the accelerating sections are divided by a black dash-dotted line, while the accelerating waveguide axis is highlighted with blue dash-dotted lines. The traveling wave is picked up at the accelerator output by a second mode converter in order to be absorbed without reflection on a load or amplified and re-injected at the input port [16; 17].