# Cm-Level Photonic-Crystal-Like Subwavelength Waveguide Platform with High Integration Density
## 1 Introduction
### A Triangular Lattice of Circular Air-Holes
#### 2.1.1 Performance of PHc-Like Waveguides with Different Number of Columns  
Figure 1 shows the corresponding waveguide structures, which consist of \(0.5,1.5\) columns of air holes at both sides, respectively, with two narrow metal walls as the waveguide claddings and the basic parameters of photonic crystal-like waveguides are also briefly described. The light diffusing or leaking from the waveguide core firstly passes partly through and is partly scatted by the air-hole structure in planar direction. After being near-completely reflected by the metal, the transmission part passes partly through and is partly scatted by the original air-hole structure again, which scatters (or reflects) and transmits the light twice. The effect is similar to doubling the structure of the air holes, which is similar to the symmetrical imaging of a mirror P. L.  
efficient transmission can be maintained regardless of the number of columns of air holes on both sides. For this reason, the wavelength of \(1.55\,\upmu\)m is chosen in the following numerical investigation, and the results are shown in Figure 3.  
Figure 1: The schematic of different waveguide structures and their equivalent mirror structures with numbers of air-hole columns of \(0.5,1.5\), respectively, at each side of the waveguide cores. Basic parameters of the waveguides are: \(d_{1}=0.745\,\upmu\)m, \(d_{2}=0.1\,\upmu\)m, \(d_{3}=\frac{\sqrt{3}}{2}\times\textit{Period}\), \(\textit{Period}=0.43\,\upmu\)m, \(R=0.13\,\upmu\)m, \(\epsilon_{a}=1\) and \(\epsilon_{s}=12.04\). The metal is silver.  
Figure 2: The transmission spectra of the triangular lattice waveguides with different column numbers of air holes.  
One can see that the waveguide mode is close to the two metal walls for the case of 0.5 column of air holes on each side, as shown in Figure 3a. The waveguide mode is mainly confined by only 0.5 column of air holes and metallic sidewalls on both sides, resulting in very large propagation loss (up to \(-122.9\) dB/cm), as shown in Figure 4 below, due to the ohmic absorption of the metal. As the number of air-hole columns increases, the field confinement of the waveguide is strengthened, which means the field is more focused or distributed in waveguide core and fewer amount of light contacts with the lossy metal. Therefore, the propagation loss decreases with the increase of columns of air holes and approaches toward zero, because the light-metal interaction becomes weaker, as shown in Figures 3 and 4.  
Figure 4: Propagation loss and integration density versus the column number of triangular lattice air holes waveguides at the wavelength of \(1.55\,\upmu\).  
Figure 3: The \(E_{z}\) fields of waveguides with different column numbers of air holes ((**a–d**) 0.5, 1.5, 2.5, 3.5 columns, respectively) at the wavelength of \(1.55\,\upmu\)m. The dotted lines represent the boundaries between metal and silicon.  
Figure 5 shows simulation results of light propagation in the waveguides of air holes without metallic sidewall, in order to demonstrate the effect of metallic sidewalls. It can be seen that part of light wave leaks from the core and spreads outside. The propagation losses of the waveguides with 1.0, 2.0 and 3.0 columns of air holes without metallic sidewall are calculated to be, respectively \(-132.6\) dB/cm, \(-77.5\) dB/cm and \(-20.5\) dB/cm from the simulation at \(1.55\,\upmu\)m wavelength, which are much larger than the values of propagation losses shown in Figure 4. Basically, there are two kinds of photonic crystals. One is the structure of periodic dielectric cylinders with complete photonic bandgap [19]. The other is the structure of a solid dielectric filled peripherally with periodic air holes, most of which only has partial bandgap. Our waveguide is the combination of the latter with an MDM structure. Ohmic absorption of light on metallic sidewalls and then the waveguide loss can be reduced by introducing only a few periodic air holes between the core and metallic sidewall to effectively weaken the light-metal interaction.