PRODUTIVIDAD Y COMPETITIVIDAD

sults of a realistic numerical bandstructure calculation using the frequency-domain plane-wave method [117]. We focus on the number and locations of singularities in the absence of optical activity as these are the crucial quantities to determining the Chern number once optical activity is introduced. We specifically consider a biaxial dielectric in which cylindrical air holes are drilled to form a square lattice.

FIGURE3.7: Comparison of the locations of the C-points given by the lattice Hamiltonian and a frequency-domain plane-wave simulation askis varied. In both cases C-points occur along the linesky=0 (left)

andky= π_a (right). Solid lines representkxlocations of C-points from

the lattice Hamiltonian while the shading represents bounds on kx

for the C-points from the plane-wave simulations. The chosen shad- ing bounds are such that the magnitude of the splitting of the two most paraxial bands is less than a∆kz = 2π×e−6. The plane-wave

simulation has cylindrical air holes in the dielectric background. The radius of each of the air holes considered isr=0.15a.

In figure 3.7 we examine how the locations of the C-points are affected by variation of the lattice spacing to wavelength ratio. The solid black contour lines give the locations predicted byB(k), while the shaded regions provide bounds on the C-point locations from the numerical simulation. We see that the B

3.3. Degeneracy Structure of Square Photonic Crystals 61 a reasonable qualitative account of the numerical results. Each picture considers the two previously mentioned high symmetry lines of the first Brillouin zone upon which the C-points appear. The left panel of figure 3.7 is along the high symmetry lineky = 0 which features two C-points in both approaches at all values of lattice spacing to wavelength ratio considered. Both approaches have the zone centre C-point and agree that the second one is adjacent to the −X boundary of the Brillouin zone. The right panel of figure 3.7 shows the C-point positions along the lineky= π_a. There are some deviations between the two approaches along this high symmetry line, in particular, the numerics indicate two C-points at all lattice spacing to wavelength ratios considered while the extra two only emerge above a critical

a

λ for the lattice Hamiltonian B(k). We attribute this difference to the different

treatment of scattering in the two methods, and perhaps also to higher-order terms neglected in the paraxial Hamiltonian of the homogeneous material. Figure 3.7 suggests that the theory represented byB

(k)is a reasonable qualitative descriptor

of the desired 2Dphotonic crystal structures, but to further confirm this we turn to comparing iso-frequency surfaces generated from each approach.

Figure 3.8 shows two iso-frequency diagrams plotted following a path along high symmetry lines in the first Brillouin zone. The top figure is that produced from the HamiltonianB(k), and the bottom one the result of the numerical simulation. These surfaces are compared at ka ' 4π such that we are in a regime towards

the top of the scale in figure 3.7, where each approach predicts four C-points. We note the striking similarity between the surfaces generated by each approach, in terms of their overall morphology as well as the number and locations of the conical intersections of the surfaces. As expected from figure 3.7, we see that the extra C-point along the ky = 0 line in each approach are slightly displaced from each other. In the lattice Hamiltonian model the additional C-point along the ky = 0 line is just inside the −X boundary, whereas in the numerical simulation this C-point is further inward along the−XΓ line. We further note that the lattice Hamiltonian generally underestimates the overall dispersion of the surfaces as well as the polarisation splitting of the bands. A quantitative measure of the differences of the two methods could have been calculated. However, from a topological point of view, the important features of the iso-frequency surface are the number and locations of the degeneracies. As such, a quantitative measure was not calculated.

Given the overall reasonable correspondence between the two approaches we conclude that the Hamiltonian B

(k) is a qualitatively accurate descriptor of 2D

photonic crystals composed primarily of biaxial material. This conclusion is based on the two approaches showing

1. Agreement on the number of C-points of the iso-frequency surfaces over most lattice spacing to wavelength ratios.

-X _Γ X M L M' _-X 1.96 1.98 2.00 2.02 2.04

a k

2_π

FIGURE3.8: A comparison between the iso-frequency surfaces gen- erated from the lattice Hamiltonian (top) and those generated from a frequency domain plane-wave simulation (bottom). The iso- frequency surfaces are plotted as a path along high-symmetry lines of the first Brillouin zone is followed. The high symmetry points are indicated in figure 3.3. These surfaces are compared atka'4π.

2. Agreement on the pinning of those C-points to high symmetry lines of the first Brillouin zone.

3. Approximate agreement on the locations of the C-points along those high sym- metry lines.

Owing to this we argue that the topological phase diagrams of the two models will be similar to each other. In the regime prior to the emergence of the extra C-points inB

(k)the phase diagram of the numerical simulation will be richer than

that of the lattice model. Having established thatB(k)is an adequate qualitative descriptor of these structures we now turn to the introduction of each form of optical activity separately. These additions will allow us to examine the topological phase diagrams of 2Dphotonic crystals composed of anisotropic optically active materials.