In the previous chapter, we calculated the photonic band structure of lattice and compared them with the transmission characteristics of lattice arrays. After study-
ing dispersion diagrams of 2D metallic photonic crystal waveguides in the previous section, propagation and transmission characteristics will be discussed in this sec- tion. We will also compare the calculated the dispersion diagrams of waveguide structures with their transmission characteristics.
For a waveguide, our interest is to investigate the amount of reflected energy, and wave distribution. The losses of metals are included by implementing a complex expression for material properties. Here, we investigate the losses for different metals. As it mentioned earlier, we will compare four highly conductive metals for use in photonic crystal structures for high transmission performance in the THz range. The THz frequency range will be scanned for transmission of wave-guiding structures.
The spectral behaviour of the waveguides depends on the dispersion characteristics and can be tuned by the rod radius and the lattice constant. As discussed in the pre- vious chapter, increasing the size of the lattice constant resulted in a decrease in the size of the band-gap of the photonic crystal lattice structure. This simultaneously results in a decrease in the transmission bandwidth and total transmission of the photonic crystal waveguides. Therefore, in the following calculations and design, the lattice constant is set to 50 µm.
In the transmission analysis of the lattice array, PML boundaries were used in the calculations to prevent reflection in any directions since multiple scattering occurs when a wave is incident on the metallic structure. The PML regions need to be included in the computational area. Enlargement of the computation domain results in an increase of mesh elements. Therefore, for a large frequency sweep, the imple- mentation of the calculations is slower as it requires more CPU time and memory. This issue becomes a significant problem for 3D simulations. To analyse transmis- sion characteristics of metallic photonic crystal waveguides, the calculations are
carried out by replacing PML boundaries with absorbing boundary conditions. For input and output boundaries, a type of absorbing boundary condition is called matched boundary condition is used that also make S-parameter calculations pos- sible.The matched boundary condition is mainly used at boundaries that do not represent a physical boundary. If the electric field is an eigenmode of the boundary the boundary is exactly non-reflecting (11).
ˆ
ez· ~n × (∇ × Ezeˆz) − iβEz = −2iβE0z (T M waves) (6.1)
ˆ
ez· ~n × (∇ × Hzˆez) − iβHz = −2iβH0z (T E waves) (6.2)
where β is the propagation constant of the electromagnetic wave.
Calculations from the previous chapter have shown that the first row of photonic crystals at each side of the waveguiding channel is the most important for minim- izing in-plane scattering losses. Therefore, the absorbing boundary condition at the boundaries at each side of the waveguide is enough to prevent reflections.
The purpose of this section is to validate the modelling of waveguiding structure for designing the desired functionality. FEM is well suited for resolving photonic crystal geometry and is used for calculating the transmission spectra, displaying wave propagation and analysing the reasons for propagation losses. It is used for wave propagation in linear defects and translating the general concepts of disper- sion relation and of mini stop-bands as seen in the following section transmission spectra.
In the following sections, dispersion diagrams and the transmission spectra of square lattice waveguides (W1, W2) and triangular lattice waveguides (W1, W3) are com- pared. The transmission of waveguides is interrupted by the transmission dips. The
reasons for these dips are discussed by analysing the dispersion diagram in parallel to the transmission spectra of waveguides.
Since our interest is on the waveguiding devices in the THz range, in order to design low loss waveguiding devices, we calculated the transmission, reflection and ab- sorption of each waveguides. Waveguide attenuations are calculated for square and triangular lattice W1 waveguides for different materials.
6.5.1
Validation of the Two-Dimensional Approximation
In the following sections, transmission characteristics of photonic crystals made of metallic rods in air are considered. Before proceeding with the transmission calcu- lation of metallic photonic crystal waveguides, in order to validate our 2D numerical simulations for waveguiding structures a comparison has been conducted between 2D and 3D simulations. A W1 square lattice waveguide is modelled in 3D and transmission and reflections results are compared with the 2D simulation results. In 3D simulations the metallic rods are sandwiched between two parallel metallic plates of a perfect conductivity, separated by the height of the rods of 50 µm. The 2D simulation is based on the same photonic crystal pattern except the structure is projected in the plane defined by the direction of propagation and direction perpen- dicular to the direction of the electric field. Transmission and reflection spectra in range of 1 - 6 THz are calculated for a simple waveguide. A W1 waveguide in a 10x10 rod square lattice photonic crystal is tested and results are presented in Fig- ure 6.12. Black lines represent the transmission spectra, while red lines represent the reflection results. 3D transmission and reflection spectra are represented with black and red solid lines, respectively. Similarly, dashed lines represent the results for 2D simulations.
Figure 6.12: Transmission/reflection spectrum simulated in 3D (black/red solid line) and in 2D (black/red dash line) for a linear waveguide formed by removing one row of rods from a metallic PhC. The PhC is characterized by a 50 µm lattice period in square lattice pattern of 50 µm height rods, with a radius of 0.2a.
As can be seen in Figure 6.12 the transmission spectrum of 2D geometry coin- cides with that of the 3D geometry and is comparable to band-gap diagram results and reported works (4; 12; 13). The transmission characteristic of a waveguide matches the band-gap and dispersion characteristics of W1 and W2 waveguides. The losses in linear waveguides are caused by the band-gap characteristics of the metallic photonic crystals, as the frequencies where losses are occur appear in dis- persion characteristics of metallic photonic crystals (14). As long as the height is kept as small as half of the wavelength, which is the case for the remaining results of this study, 2D and 3D simulations give very similar results. It is worth noting that the positions and width of the photonic band-gap are well reproduced by 2D