The construction of the first Brillouin zone for a two-dimensional triangular lattice is shown below.
Figure 2.5: Triangular Lattice a) Lattice generated by the basis vectors b) The cor- responding reciprocal lattice and reciprocal vectors c) First Brillouin zone (Wigner- Seitz primitive cell) and irreducible region denoted as hexagon and triangle, re- spectively.
The lattice vectors and corresponding reciprocal lattice vectors for triangular lattice are centred at the origin (Γ). The points at the corner and face are known as K and M respectively. The diagram on the right (Figure 2.5 c)) shows the Brillouin zone of the triangular lattice
~a1 = a 1 2, √ 3 2 ~b1 = 2π a √ 3 2 , 1 2 (2.38) ~a2 = a( 1 2, − √ 3 2 ) ~b2 = 2π a √ 3 2 , − 1 2 (2.39)
The coordinates of the symmetry point of the first Brillouin zone are:
Γ = (0, 0), M = 0,√2π 3a , K = 2π 3a, 2π √ 3a
The filling fraction, f = √2π 3 rr a 2 (2.40)
A 2D photonic crystal is periodic along two of its axes and homogeneous along the third axis. The multilayer film only reflects light at normal incidence but two- dimensional photonic crystals can reflect incident light from any direction in the plane. The modes propagating in photonic crystals can be classified as transverse electric, TE or transverse magnetic, TM according to their polarization directions. If we consider a photonic crystal that has discrete transitional symmetry in the xy- plane and continuous transitional symmetry in the z-direction, the TE or TM polar- ization can be distinguished from whether the electric field or magnetic field vectors are in line with the z-direction. The definition of TE and TM polarization is how- ever different from the conventional definition. TM polarization defines the wave orientation where the electric field is parallel to the rod axis. The magnetic field os- cillates transverse to the z-direction Ez, Hx, Hy are non-zero, in TM polarization.
TE polarization defines the wave orientation where the electric field is perpendicu- lar to the rod axis where Hz, Ex, Ey are non-zero.
There are two types of photonic crystals: high-index materials are grown in a low- index medium and low-index materials drilled in a high-index medium. We may distinguish these structures as pillar type and hole type crystals, respectively, since in most conditions the low-index material is the air. The most studied photonic crys- tals consist of arrays of cylindrical pillars (rods) or circular holes in air or dielectric in square and triangular (hexagonal) lattice configurations.
Figure 2.6: Directions and electromagnetic field vectors for TE and TM polariza- tions considered for 2D photonic crystals.
a square lattice array and air holes in dielectric medium in a triangular lattice array. In 2D photonic crystals, high-ε dielectric materials in a low-ε medium lead to large TM band-gaps because most of the energy is concentrated in high-ε. Therefore, pillar type crystals exhibit large TM, small TE band gaps with small filling factors. On the other hand low-ε material in a high-ε material lead to TE band-gaps in a connected lattice, therefore hole-type crystals exhibit both TE and TM bands when the size of radius is large enough.
The irreducible Brillouin zone is represented with special directions in a crystal structure called high symmetry points. These are used in order to characterize the dispersion of electromagnetic radiation inside the structure. In the band-gap dia- grams for 2D and 3D photonic crystals, k-vectors are often expressed in Greek letters corresponding to the high symmetry points in the crystal. Usually computa-
tion starts from the centre of the Brillouin zone denoted by Γ and by scanning the all possible angles through the symmetry points, the contour is completed by returning to the centre where the k-vector is zero. For instance, for 2D photonic crystals in square array these symmetry points are Γ-X-M where in the interval Γ-X, kx in-
creases while ky remains zero, ω = c kx, in the X-M interval, kxremains constant
while ky increases, ω = c
q
kx2+ ky2 and M-Γ both kx and ky are decreasing.
ω = ck, ~k = kxx + kˆ yy.ˆ
Silicon is one of the most commonly used materials in semiconductors and photonic crystals due to its high refractive index and low losses. The dielectric constant of silicon is 11.7 (24). Figure 2.7 shows the band-gap diagram of Silicon (Si) pillars of square lattice array in air. The radius of the rods is 0.2a.
The polarization plays an important role on wave propagation in 2D photonic crys- tals. As can be seen from the band-gap diagrams in Figure 2.7, wave propagation is affected by the polarization of light. There are no TE gaps for square lattice in the frequency range displayed with the given ratio. However, there are two gaps clearly seen in the band structure for TM mode, as high-ε regions lead to TM gaps. In the reverse dielectric configuration where air holes are drilled in silicon medium, in triangular lattice array, the band diagrams are shown in Figure 2.8. A triangular lattice of air columns of radius r = 0.48a exhibits a band-gap for both polarizations for a short range of frequencies around 0.5(ωa/2πc) whereas the TE and TM gaps overlap and encompass a complete band-gap.
A complete photonic band gap is a range of ω in which there are no propagating (real ~k) solutions of Maxwell’s equations for any ~k, surrounded by propagating states above and below the gap. There are also incomplete gaps, which only exist over a subset of all possible wave vectors, polarizations, and/or symmetries. In or- der for a complete band gap to arise in 2D or 3D, an additional requirement must be
Figure 2.7: The photonic band structure for a square array of dielectric columns embedded in air, ε1 = 11.7 and ε2 = 1, with r = 0.2a. The photonic band gaps are
shown by shaded areas a) TM polarization b) TE polarization.
met. In each symmetry direction of the crystal (and each k point) there may be band gaps. However, these band gaps do not necessarily overlap in frequency (or even lie
Figure 2.8: The photonic band structure for a triangular of dielectric columns em- bedded in air, ε1 = 1 and ε2 = 11.7, with r = 0.48a. The photonic band gaps are
shown by shaded areas a) TM polarization b) TE polarization.
between the same bands) as seen in the diagram above. An overlap is more likely if a band-gap is sufficiently large, which implies a minimum ε contrast (typically
at least 4:1 in 3D structures). Since mid-gap frequency for 1D photonic crystals is ≈ cπ/(a√ε) and varies inversely with the period a, large band-gaps are obtained for the crystals whose periodicity is nearly constant in all directions. Therefore, the largest gaps typically arise for triangular lattices in 2D structures and face-centred cubic (fcc) lattices in 3D structures which have the most nearly circular/spherical Brillouin zones (4).
In order to have a clearer idea on the band-gap positions not only for a particu- lar size of crystal but also for all possible sizes, we can utilize diagrams called band-gap maps. In these, diagram shows the size and positions of band-gaps of a 2D photonic crystal for a given lattice type and filling factor and dielectric contrast. Gap maps are generated from band gap figures, by determining the gaps for each r/a value. As holes/pillars overlap for r/a ≥ 0.5 data for larger r/a are not shown in band-gap map figures. The x-axis of the diagram is the radius of rods or the holes in the inverse configuration, y-axis is the frequency; both are normalized to lattice constant. From the diagram, frequency interval of largest band-gap and correspond- ing r/a value can be determined. As the diagram is normalized by lattice constant, it must be scaled to the desired levels.
Here, we present band-gap maps for two photonic crystal structures: silicon pil- lars in air in square lattice array and air holes in silicon in triangular lattice array in Figure 2.9 and Figure 2.10, respectively. The locations of the band gaps are shown as a function of r/a for both TE and TM polarizations. In this study, both band-gap diagrams and band-gap maps are obtained by means of the PWE method, which implemented on a MATLAB platform.
TE gaps are very sparse, because the isolated patches of high-ε regions lead to the TM gaps and the connectivity of high dielectric constant regions. In square lat- tice structures it is difficult to produce the complete photonic band gaps because of
Figure 2.9: Band-gap map for a square array of silicon in air, red and black regions show band-gap islands of TE and TM modes, respectively.
high spatial symmetry. On the other hand, triangular lattice structures exhibit com- plete band-gaps. TE and TM bands overlap when the radii are large enough. Such overlap can be very desirable for some applications. Otherwise conditions lead to polarization selectivity, which is also important for many applications.