# Cm-Level Photonic-Crystal-Like Subwavelength Waveguide Platform with High Integration Density
## 1 Introduction
Because the 3D FDTD simulation is very time consuming (hundreds of hours) due to very large data volume/size and calculation memory, we had to transform the 3D structure simulation into the 2D simulation by using the effective index method [14], as some authors do for integrated optics and waveguides. In the \(z\) (thickness) direction, it is a sandwich structure of air/silicon/silica. The equivalent refractive index of the \(z\)-direction sandwich structure is firstly calculated, and then is assigned to the refractive index of two-dimensional planar silicon. By means of the effective index method, it realizes the equivalent two-dimensional model for the three-dimensional structure. The confinement/divergence in thickness direction are approximately and equivalently considered in the simulation. In the processing of the effective index method, the effective refractive indices of the air/silicon/silica structure were obtained as, respectively, 3.38 and 3.42, for the thicknesses of core layer of 1 micron and 1.5 micron. The relative error is very small, compared with the refractive index of bulk Si of 3.47. Therefore, in the following 2D FDTD simulation, we used the effective refractive index of 3.47 approximately. In addition, due to high index contrast between silicon and air or silica, most of the light field is tightly confined in the silicon layer in thickness-direction. Thus, the divergence in the direction is so small that it has only little influence on simulation result. Therefore, in the following study, we applied the two-dimensional finite-difference time domain (2D FDTD) method with the boundary conditions for perfect matching absorption in the lateral direction, and the fundamental slab TE mode was considered in the 2D simulations. The grid size of simulation was 5 nm for both \(x\) and \(y\) directions, and the perfectly matched layer PML [15] absorption boundary with a reflection coefficient below \(-\)70 dB was applied to outside of the waveguide, where there were, respectively 1 \(\upmu\)m and 1.5 \(\upmu\)m spacing in \(x\) and \(y\) directions away from the boundaries of the waveguide. To make sure that the PhC-like waveguide can perform the distinguished properties of cm-level propagation length, subwavelength confinement and low crosstalk at telecommunication wavelengths, the geometric parameters of the WG were calculated by the following processes: (1) The parameters of the structure were repeatedly scanned many times, and were finally optimized to be as follows: \(Period=0.43\,\upmu\)m, \(R=0.13\,\upmu\)m, and \(d_{1}=0.745\,\upmu\)m, with one column of air holes and silver sidewall at each side; (2) The thickness (\(d_{2}=0.1\,\upmu\)m) of silver sidewalls was chosen, because it is much larger than the skin depth (21-22 nm) of silver and is thick enough to screen or isolate electromagnetic crosstalk between adjacent waveguides. In the simulation, \(\epsilon_{a}=1\) and \(\epsilon_{s}=12.04\) were, respectively, the relative permittivities of air and silicon, whose material dispersions on wavelength are very weak for them and can be ignored due to no absorption within the wavelength range of the simulation (1-2 \(\,\upmu\)m). However, the metal silver is of strong material dispersion. Thus, its relative permittivity \(\epsilon_{m}\) can be described by the well-known E. Palik model Palik (1960) to satisfy the dependence of refractive index (permittivity) on wavelength in simulation. We further discuss performances of different photonic lattice structures with different column numbers of air holes on both sides of the waveguide. For hybrid waveguides with multiple columns of air holes, there are typically two types of lattices of air holes in the lateral direction. One is a triangular lattice of circular air holes (the number of air hole columns to be \(0.5,1.5,2.5,3.5\dots\), respectively, on both sides of the waveguide), while the other one is a square lattice (\(0.5,1.0,1.5,2.0,2.5,3.0\dots\), respectively). When the air-hole column number is 0, the waveguide is a pure MDM waveguide.