Classification of Defect Modes in Hexagonal Lat- Lat-ticeLat-tice

3.2.1 Symmetry Group of Photonic Crystals

The eigenvalue equations of Maxwell’s equations in photonic crystals are given by[20]:

L_HH(r) ≡ ∇ × 1

ε_r(r)∇ × H(r)

= ω²

c²H(r), (3.9)

L_EE(r) ≡ 1

ε_r(r)∇ ×

∇ × E(r)

= ω²

c²E(r), (3.10)

Analogue to the case in quantum mechanics, the group of these equations is formed by the space group of the photonic crystal which transforms ε_r(r) into itself when the group of Schr¨odinger’s equation is determined by the potential function V (r).

One can prove that any symmetry operator belonging to the point group of 3D pho-tonic crystals commutes with L_E and L_H. This exhibits the ability to classify the modes of equations (3.10) E_kn(r) and H_kn(r) according to the irreducible representations of the group M_k in momentum space, just as the treatment to Schr¨odinger’s equation in quantum mechanics.

However, there is something important to be noted. The symmetry of the magnetic field and that of the electric field are generally different from each other, as the former is an axial vector field whereas the latter is a polar vector field. The character for magnetic field differs from that of electric field by a multiple of the determinant of the operation.

The determinant is +1 when the operation is proper transformation, wheras a determinant of −1 is derived for improper transformations. That means the characters of improper transformations for magnetic fields need to change sign from the origin.

Electromagnetic fields can be classified into pure TE or TM modes when the structure has mirror symmetry and continuous translational symmetry in z-direction. In other words, the dielectric constant of the structure remains constant and extends to infinity in z direction. Now the eigenvalue problems for TE and TM modes in 2D photonic crystals are given by[20]:

L⁽²⁾_H H_z(rk) ≡ − ∂ respectively, where rk denotes the in-plane position vector (x, y).

Like the 3D case of photonic crystals, we should note that L⁽²⁾_H and L⁽²⁾_E commute with the 2D symmetry operations belonging to the point group of the 2D photonic crystal.

Therefore the electromagnetic fields can be classified according the irreducible represen-tations of the k-group M_k.

Since no 2D symmetry operation changes Ez or Hz, we can observe the symmetry properties of the modes just from the scalar field E_z for the TM polarization and the scalar field Hz for the TE polarization for 2D photonic crystals for convenience.

3.2.2 Symmetry of 2D Photonic Crystals of Finite Thickness

In actual case the 2D photonic crystal slabs are surrounded by air in the z direction.

The symmetry group of this structure is D6v, which is a direct product of C6c and C1h

point groups:

D6h = C6v× C1h

where C_1h consists of the identity operation and the mirror reflection by the middle plane of the slab, ˆσ_z. The character table of C_1h is shown in Table 3.2. Thus the middle plane of the slab can be taken as a symmetry boundary for σ_z = +1 to save the computation time. On the contrary, it can be taken as an asymmetry boundary for σ_z = −1. The modes for σ_z = +1 are the so-called TE-like modes. And those for σ_z = −1 are TM-like modes. The dominant components of electromagnetic fields of TE-like modes are E_x, E_y, and H_z while other components are small and close to zero for a slab thickness around λ/2. Here we take the λ to be around the mid-gap. The situation of TM-like modes is similar, the dominant components are H_x, H_y, and E_z.

From now on we will focus on the discussion of symmetry to TE-like modes, which is pure TE mode and has only magnetic component H_z in the middle plane of the slab.

3.2.3 Symmetry of Defect Modes

A small variation of dielectric constant in a small region within the photonic crystals can be seen as a perturbation of modes of photonic crystals without defects. Enlarging a hole creates acceptor modes in photonic crystal band gap, which is the analogue to the acceptors in electronic states in a crystal. Similarly, reducing the radius of hole creates donor modes in analogy to donors in electronic states in a crystal[21]. The easiest way to create donor modes is to remove single air hole from array of holes of hexagonal lattice arrangement. For the coordinate system in which origin is at the defect center, the space group and the point group are still C_6v as before. The defect modes are attributed to the irreducible representations of the C_6v point group. There are four one-dimensional representations and two two-dimensional representations for the C_6v point group.

The modes at photonic band edges are used as a symmetry basis to generate approxi-mated forms of defect modes[4]. The minimum in air band occurs at M -points. Therefore donor modes can be approximated by the unperturbed Bloch modes in air band edges.

On the contrary, acceptor mode can be approximated by the unperturbed Bloch modes in dielectric band edges.

Let us find the symmetry basis of unperturbed Bloch modes at the band edges first.

A symmetry basis for the modes of the photonic crystal at the M -point can be found by projecting the seed Bloch modes B_?kM to the representations of the group of the wave vector M_kM(=C_2v). The B_A^M₂ corresponds to the dielectric band mode and B_B^M₁ to the air band mode. A set of basis functions for modes in air band edges can be found:

CB_B^M₁ = ˆz

where the superscript stands for the wave vectors and the subscript stands for the repre-sentations of the corresponding group of those wave vectors. Similarly, the basis functions for modes in the dielectric band edges should be:

V B_A^K0

The representation of the CB_B1^M basis under the operations of C_6v can be written as

This representation equals to E1⊕ B₁⁰⁰. Using the projection operators on CB_B^M₁, a set of basis functions for E₁ and B₁⁰⁰ appears as follows:

B^d1_B00

1 mode transforms like a hexapole, whereas the doubly degenerate modes B_E^d1 transform as an (x, y)-dipole pair.