In Yee’s lattice, it is difficult to form the unit cell of hexagonal lattice which is a rhombus in orthogonal grids. Thus we use a rectangular domain containing two unit cells shown in Figure 4.4:
We assume the in-plane periodicity extends to infinity for the convenience to apply Bloch’s boundary conditions with a specific wave vector at all four sides of the domain.
The procedure of Bloch’s boundary conditions is discussed in Section 2.6. Open boundary conditions such as PML ABCs are applied to the upper and lower boundaries in the directions outward and inward to this paper.
source monitor dual
phase shift
Figure 4.4: Double unit cell and the locations of sources and monitors.
A short impulse is used as the excitation source. One should note that the source should have a dual source at the other unit cell in the domain. And there is a phase shift of e−ik·a between them, where a is the lattice vector of unit cell. To avoid neglecting all possible modes, time monitors are randomly chosen at three positions to record the time signals. Then we collect the three time signals and add them as a new signal.
Normalize this signal with maximal amplitude. Apodize them using Gaussian function to exclude the undesired frequencies. Chirp z-transform the signal with a range of frequency of interest to observe the frequency response. To obtain the band structure we have to repeat the procedure of Bloch’s boundary conditions with wave vectors along the edges of irreducible Brillouin zone in k-space. As an example, the band structure of a hexagonal lattice is shown in Figure 4.5. The TE-like photonic band gap is ranged from a/λ = 0.288 to a/λ = 0.364. Note that the missing points may result from the abandoning of small signals of 1/1000 smaller than the maximum peak.
0 19 38
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Normalized frequency (a/λ)
M K
Γ Γ
light cone
light cone Photonic Band Gap
0.364
0.288
Figure 4.5: The band structure by the FDTD method. (r/a = 0.3, d/a = 0.4, n = 3.4)
4.2 Defect Systems
In this section, the refractive index of the material is fixed to be 3.4. This implies that here we are not concerned with the dispersive media and lossy materials so that n is taken to be a frequency-independent real constant. The ratio between the slab thickness and the lattice constant d/a is chosen to be 0.4. The defect is formed by removing a single air hole from array of holes of hexagonal lattice arrangement on a dielectric slab. The number of layers surrounding the defect is five.
We have also designed some new structures by gradually increasing or decreasing the hole radii as the surrounding layers go outward the defect. Figure 4.6 shows these designs. The ranges of photonic band gaps of photonic crystal slabs of hexagonal lattice without defects vs different r/a are listed in table 4.1 for references of our structures. The calculation procedure has been described in previous sections.
Table 4.1: Comparison of photonic band gaps using different methods. The frequencies are taken to be normalized (a/λ).
method r/a 0.26 0.28 0.30 0.32 0.34
2D PWE air band edge 0.303 0.318 0.336 0.356 0.378
(n
ef f= 2.65)
dielectric band edge 0.262 0.265 0.269 0.275 0.282 3D PWE air band edge 0.330 0.335 0.337 0.340 0.343 dielectric band edge 0.283 0.286 0.291 0.297 0.304 3D FDTD air band edge 0.330 0.346 0.364 0.385 0.408 dielectric band edge 0.279 0.282 0.288 0.293 0.3054.2.1 Resonant Frequencies of Defect Modes
The PWE method is still useful to determine the resonant frequencies of the defect cavity by the use of so-called supercell scheme, which defines a larger size of unit cell containing the defect region. The spirit of supercell scheme lies in the periodic arrange-ment of supercell and the enough space apart from each other for small coupling between defect modes. Figure 4.7 shows an example of the band structure of the defect system by 2D and 3D PWE methods. A supercell containing 7 by 7 unit cells and a single defect is used. In 3D PWE method a margin of air of twice broader than the slab thickness is
(a) Type A (b) Type B
(c) Type C
(d) Type D (e) Type E
Figure 4.6: Various patterns by changing hole radii. The radius of holes of most inner layer is 0.3a. (a) r/a gradually decrease by 0.02 per layer outward. (b) r/a gradually decrease by 0.01 per layer outward. (c) typical pattern without changing the hole radii.
(d) r/a gradually increase by 0.01 per layer outward. (e) r/a gradually increase by 0.02 per layer outward.
Frequency (ωa/2πc=a/λ)
0.0 0.1 0.2 0.3 0.4
Γ K M Γ
(a) 2D PWE (n
ef f= 2.65)
Frequency (ωa/2πc=a/λ)
0.0 0.1 0.2 0.3 0.4
Γ K M Γ
(b) 3D PWE
Figure 4.7: The band structures of defect system Type C(ar is fixed to be 0.3) by the 2D and 3D PWE methods.
used in the direction perpendicular to the slab. In 2D case, Two degenerate defect modes of normalized frequency around 0.31 in the photonic band gap are observed. In 3D case this frequency is around 0.323.
Now adding a defect provides the existence of the defect modes inside the gap though it breaks the translational symmetry. So Bloch’s boundary conditions can no more be applied to obtain the band structure by the FDTD method. Although the band diagrams of defect structures cannot be simulated by the FDTD method with Bloch’s boundary
100 200 300 400 500 600 700 800 900 1000
Figure 4.8: Excitation Signal. (α = 1024 and f0 = 0.015)
conditions. There are still some tricks to locate the frequencies of the defect modes inside the photonic band gap.
We use an excitation of a magnetic point source located at a low symmetric point in the cavity in order to excite all defect modes in the cavity. The polarization of the source is set to be TE polarization in which we are interested. The temporal distribution is a short Gaussian pulse covering the spectrum of the whole band gap and with a central frequency located at the mid-gap frequency. The source signal can be expressed as Hz(t) = sin(2πf0t) · e−α(t−t0), where f0 is taken around the mid-gap frequency, t and t0 is in unit of ∆t, and α is the coefficient for apodizing the original sine function exponentially.
Figure 4.8(a) shows the signal as a function of time steps and Figure 4.8(b) shows the corresponding spectrum. Parameters used in the calculation are listed in table 4.2.
Some data processes are performed to analyze the results. First of all, we record the time evolution of the Hz field at three chosen low symmetry positions inside the cavity for 33792 time steps, to detect all possible defect modes. The positions of the source and
source monitors monitor 1
monitor 3
monitor 2
Figure 4.9: The locations of source and monitors.