In this chapter we have examined the propagation of light through homogeneous anisotropic optically active materials within several frameworks and at varying degrees of generality. In particular, we considered dielectrics which are optically biaxial and additionally exhibit either the Faraday effect or chirality.
This examination began by developing the wave equation as a 2×2 matrix eigenvalue problem for the transverse displacement field. Within this framework we explored the refractive index surfaces of biaxial optically active materials, which characterise the retardation experienced by light of a particular polarisation travelling in a given direction in these materials. These surfaces have an associated polarisation texture, describing the natural vibrational state of the displacement field in the plane transverse to the propagation direction. In the presence of either form of optical activity, this natural vibrational state is generally that of elliptic polarisation. We have seen that there are, however, two noteworthy possible departures from this general state. These departures are directions in which the polarisation state is either that of circular or linear polarisation. The former occur in isolated directions in reciprocal space and are known as C-points. The C-points originate from the polarisation degeneracies of pure biaxial materials. The latter occur along lines in reciprocal space which are known as L-lines. L-lines are contours along which the optical activity effects vanish.
In tandem, these polarisation features allow us to understand the assignation of a value of a topological invariant to each of the refractive index surfaces. The invariant in question is the Chern number. The values of the Chern number of the refractive index surfaces can be read off from examination of the polarisation texture of each of the surfaces. The realisation of a non-zero value of this invariant relies on the topological index of each of the C-points combining rather than cancelling. To achieve this there has to be an asymmetry between the number of left and right circularly polarised C-points on each surface. Whether or not this occurs depends on both the variant of optical activity considered and the precise details within each variant. We saw that, for one of these optical activity variants, a non-zero Chern number is not possible due to the Faraday effect contribution being an odd function of the propagation direction. However, in the case of chiral biaxial materials, it is possible to have topologically non-trivial index surfaces. As well as being novel in and of themselves, these conclusions will be of interest for later comparative analysis.
We are additionally interested in the local behaviour of the refractive index around one of these C-points, rather than the global refractive index surfaces
2.7. Conclusion 49 themselves. To this end we pursued a Hamiltonian which describes the evolu- tion of the field in directions close to an optic axis. This derivation was achieved within the framework of the paraxial approximation, an intrinsically local approach. This Hamiltonian is a central result of this chapter and being to second order in the transverse wavevector it is therefore an extension of that derived by Jeffrey [98]. The Hamiltonian derived by Jeffrey [98] has already proved a powerful tool, being used to analyse the minutiae of the diffractive evolution of an incident field through homogeneous anisotropic optically active media [105, 108, 109]. Our motivation in extending the Hamiltonian of Jeffrey [98] shall become more transparent in the following chapters.
In these chapters we adapt the derived Hamiltonian to describe the propagation of light through periodic optical media, known as photonic crystals, which are com- posed of biaxial optically active materials. In these arrangements the vector field is defined on a torus rather than a sphere. In such a setting the Poincaré-Hopf theorem dictates that the total topological index of the vector field on the torus is null as com- pared to a total topological index of two on the sphere. This requirement therefore dictates equal numbers of degeneracies of positive and negative topological index on the torus of the Brillouin zone. The alternative consequence of the Poincaré-Hopf theorem, dictated by geometry, will have interesting implications when we look at the topological invariants of the iso-frequency surfaces of these materials. It is within this interplay between the Hamiltonian derived in this chapter and the geometric en- forcement of the periodicity that the motivation for the extended Hamiltonian will become manifest, as we shall now explore.
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Chapter 3
Square Patterned Photonic Crystals
3.1
Introduction
In this chapter we study periodic optical materials known as photonic crystals. These regular and repeating structures, which are periodic on the scale of the wavelength, can be patterned in one, two or three dimensions [15]. In figure 3.1 we show an illustration of forms of photonic crystals with varying dimensions of patterning. In these figures the different coloured blocks represent materials of distinct refractive indices. In the case of 1D photonic crystals, the patterning is restricted to repeating blocks along the patterning direction. For higher dimen- sional photonic crystals more complex patterning geometries are possible. The available patterning arrangements are linked to the possible Bravais lattices for that dimension. In figure 3.1 each of the photonic crystals features a simple Bravais lattice (linear, square and cubic) along with a two point basis specifying the place- ment of the two different dielectrics within each fundamental block of the structures.
FIGURE3.1: A cartoon showing the periodic dielectric function of one (left), two (middle) and three (right) dimensional photonic crystals (PhCs).
In any of the cases the dielectric function of the structure obeys the periodicity requirement
whereRis a translation by an integer number of fundamental blocks of the structure [110]. These possible translationsRcan be expressed in terms of sums of the minimal translationsaas R= D
∑
i=1 niai, (3.2)whereni ∈ Z, Dis the number of dimensions of patterning andai determines the Bravais lattice geometry. In this chapter we will be concerned exclusively with 2D photonic crystals which are patterned in the simplest square geometry. In particular, we will consider photonic crystals composed primarily of anisotropic optically active materials. Figure 3.2 below shows the entrance face of a square-patterned 2D photonic crystal structure.
FIGURE 3.2: An illustration of a section of the entrance face of a square-patterned 2D photonic crystal.
Just as for periodic electronic systems these photonic crystals possess electro- magnetic dispersion relations ω(k) describing the totality of possible frequencies
and wavevectors at which light can travel through the material. Our focus in this chapter, and in chapter 4, will be to determine the topological invariants associated with the band-structures of 2D photonic crystals. In particular, we examine the iso-frequency surfaces of these photonic crystals, which are constant frequency slices of the full ω(k) dispersion relation. The iso-frequency surface is akin to
a Fermi surface of a solid, and is the fundamental construct for analysing the refractive, reflective and diffractive properties of monochromatic light incident on a photonic crystal [16]. In this context a non-zero Chern number of a particular photonic crystal iso-frequency surface could indicate the presence of edge states for which incident light follows the boundary of the 2Dpatterned plane.
Two-dimensional photonic crystals with topologically non-trivial band struc- tures have been theorised [9, 10, 31, 111] and realised [32, 111]. These realisations have commonly followed a conventional developmental route, set in scenarios where the propagation direction and polarisation degrees of freedom straightfor- wardly decouple, and are reliant on a specific patterning geometry so as to produce
3.2. Square Photonic Crystal Geometry 53