Figure 6.7: Schematic illustration of a photonic crystal waveguide (W1) formed by introducing a line defect along the Γ - K direction of a triangular lattice of metallic rods in air. The defect breaks the periodicity in the x-axis while it is still periodic in the y-axis. The supercell used in dispersion calculations is highlighted. The width of W1 waveguide, w = a√3 − 2r, a is the lattice constant and r is the rod radius.
We consider a line defect consisting of a missing row of rods along the Γ - K dir- ection of the triangular lattice. The line defect breaks the symmetry in the Γ - M direction, but the structure is still symmetric along the Γ - K direction, where the lattice period is a. The dispersion diagram is obtained using FEM with a com- bination of periodic boundary conditions and absorbing boundary conditions over a supercell. In the supercell, the line defect consists of certain number of layers separated by a missing rod in the middle. The height of the supercell is a lattice period long. Similar to the square lattice waveguide, the Bloch wave vector along the waveguide axis represents the guided mode wave-vectors. In a triangular lattice structure the line defect is introduced along the Γ - K direction rather than the Γ - M direction, since the waveguide walls are smoother. The width of waveguide in triangular lattice array is w = a√3(N + 1)/2 − 2r for the so-called WN waveguide where N is the number of removed rows.
The dispersion diagram is defined by the frequency of eigenmodes as a function of wavevector along the central axis of the waveguide. The dispersion diagram is represented by band-diagrams of mode frequencies as a function of the Bloch vec- tor. Bloch vectors for a triangular lattice waveguide were given in earlier chapters. To obtain the dispersion characteristics, eigenvalue calculations are carried out by means of FEM for E-polarization of the photonic crystal waveguide consisting of a triangular lattice array of copper cylinders embedded in air. The eigenvalue problem for the waveguide is solved along the (1,1) direction, namely the Γ - K direction, of the triangular lattice. The radius of the rods is r = 0.2a, where a is set to 50 µm. W1 and W3 waveguides are considered. In W3 waveguide 3 rows of rods are removed in order to ensure the symmetry. Even though there are studies for disper- sion calculations and the guided modes of metallic photonic crystal waveguides in square lattice pattern, there very are few studies conducted on dispersion relations of metallic photonic crystal waveguides with triangular lattice pattern (10).
In the previous chapter, the photonic band structure of a triangular lattice was cal- culated. The complex eigenvalue problem was solved for wave vector ~k, for a given frequency ω in the unit cell of triangular lattice by setting periodic boundary condi- tions. The calculated band-gap diagram showed that there are two band-gaps. The first band-gap is wider extends from 0 to 3.764 THz and the second band is very narrow between 7.019 and 7.157 THz corresponding to 0 - 0.673 (ωa/2πc) and 1.1698 - 1.928 (ωa/2πc), respectively, in terms of normalized frequencies.
Figure 6.8: Photonic band structure of a W1 waveguide where a row of rods is removed from the triangular lattice composed of metal cylinders in air. Dispersion curves for waveguide are obtained in the in the Γ - K (1,1) direction of the triangular lattice. Blue solid lines show the band-gap structure of metallic triangular lattice in the same direction. Inset shows the supercell used in the dispersion calculations for W1 waveguide.
for W1 and W3 linear waveguides, respectively, while the inset at the right-bottom side of the figures indicates the supercell used in the calculation. The guided modes and leaky modes are depicted with black dots. The band structure of the triangular lattice in the Γ - K direction is shown in the figures with blue lines as guideline.
Figure 6.9: Photonic band structure of a W3 waveguide where a row of rods is removed from the triangular lattice composed of metal cylinders in air. Dispersion curves for waveguide are obtained in the Γ - K (1,1) direction of the triangular lattice. Blue solid lines show the band-gap structure of metallic triangular lattice in the same direction. Inset shows the supercell used in the dispersion calculations for W3 waveguide.
As is expected from the dispersion diagram of W1 waveguide, there is only one mode in the wider band-gap of the crystal. The parity of this mode is even, which can be seen from Figure 6.10 depicting the field distribution of waveguide cross- section within the supercell. This guided mode exists from cut-off frequency (1.926 THz) to nearly the band-gap at the edge of the Brilllouin zone at 3.412 THz.
From the dispersion diagram of the W1 waveguide in Figure 6.8, the fundamental mode is folded at around 3.5 THz and creates a mini stop-band between 3.412 THz and 3.666 THz. Below the first band indicated by a blue line, the even mode which extends from 3.804 to 3.666 THz crosses with the odd mode, which extents from 3.674 to 4.27 THz at 4.136 THz for k = 0.38. Another mode crossing occurs between an even and an odd mode at 4.747 THz for k = 0.3. The odd mode lies between 4.426 and 5.048 THz, while the even mode extends from 5.16 to 4.748 THz.
Figure 6.10: Visualisation of E-field distribution of guided modes in a W1 line defect waveguide for the lowest order mode which is an even mode at 3.412 THz for k = 0.5.
The dispersion diagram of the W3 waveguide, with the same lattice parameters as that of the W1 case, clearly shows the multi-mode behaviour of the waveguide. There are several even and odd modes within the band-gap of the triangular lat- tice. The number of allowed guided modes increases as the width of the waveguide increases. As can be seen from the dispersion diagram of the W3 waveguide in Figure 6.9, there are three guided modes in the photonic band-gap of the structure. The first mode, an even mode, lies between the cut-off frequency (0.914 THz) and 3.115 THz. This mode is a fundamental mode and is folded back at the edge of Brilllouin zone at 3.115 THz and constitutes the third mode, which extends from 2.722 to 3.145 THz. The second mode is an odd mode between 1.824 and 3.436 THz. The waveguide operates in single mode within the frequency range of 0.914 - 1.824 THz.
Figure 6.11: Electric field distributions of the guided modes within the band-gap of the W3 waveguide. a) for k = 0.25, for frequencies corresponding to 1.755, 2.357, 3.097 THz and b) for k = 0.5 at frequencies 3.115, 3.436 and 3.145 THz respectively for the first three lowest order guided modes, starting from the one which has the lowest frequency.
The electric field distribution of modes appears in the W3 waveguide within the band-gap. Two even and one odd mode are shown in Figure 6.11 for a) k = 0.25 and b) k = 0.5, respectively. The symmetry of the modes can be clearly distin- guished from the figure. In the band-gap of the lattice structure, the guided mode has its energy confined inside the defect and interacts with the first row of rods that forms the wall of the waveguide.
In the dispersion diagram of the triangular W3 waveguide in Figure 6.9, the lowest order mode is also the fundamental mode and is folded around 3.1 THz and cre- ates a very narrow band, between 3.115 and 3.145 THz. Modes having the same symmetry avoid crossing and form mini stop-bands.