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1 Electromagnetics for Zero-Index Metamaterials
Fig. 1.13 Comparison of the band structure of three types of photonic crystal. Inset: Unitcell
of a variety of structures ranging from bulk material to three-dimensional diamondlattice photonic crystal, computed using MPB. Similar to an electronic bandgap,
a photonic band structure has a photonic bandgap region in which no propagating
mode exists. Firstly, Fig. 1.13a shows the band structure for a homogeneous silicon
medium, which is nothing but the light line for silicon folded back into the Brillouin
zone, due to the periodic nature of the wave vector with a period of 2π/a. This is
a compact way of representing the dispersion relation based on the fact that all the
possible modes are already present within the Brillouin zone. It can be seen that
the bands are continuous without any discontinuity. Figure 1.13b shows the band
diagram for a one-dimensional photonic crystal, which is made up of a quarter-wave
stack of silicon and air regions. A quarter-wave stack means that the thickness of
each layer is equal to one-fourth of the wavelength in that medium. One-dimensional
photonic crystals of such design show maximum bandgap. A broad bandgap region
can be observed in Fig. 1.13b. Similarly, Fig. 1.13c and d shows the band diagrams for
square lattice of silicon rods in air (a 2D photonic crystal) and a diamond (i.e., facecentered cubic) lattice of silicon spheres (a 3D photonic crystal). A well-pronounced
bandgap can be seen in each one of them.
1 Electromagnetics for Zero-Index Metamaterials
Fig. 1.13 Comparison of the band structure of three types of photonic crystal. Inset: Unitcell
of a variety of structures ranging from bulk material to three-dimensional diamondlattice photonic crystal, computed using MPB. Similar to an electronic bandgap,
a photonic band structure has a photonic bandgap region in which no propagating
mode exists. Firstly, Fig. 1.13a shows the band structure for a homogeneous silicon
medium, which is nothing but the light line for silicon folded back into the Brillouin
zone, due to the periodic nature of the wave vector with a period of 2π/a. This is
a compact way of representing the dispersion relation based on the fact that all the
possible modes are already present within the Brillouin zone. It can be seen that
the bands are continuous without any discontinuity. Figure 1.13b shows the band
diagram for a one-dimensional photonic crystal, which is made up of a quarter-wave
stack of silicon and air regions. A quarter-wave stack means that the thickness of
each layer is equal to one-fourth of the wavelength in that medium. One-dimensional
photonic crystals of such design show maximum bandgap. A broad bandgap region
can be observed in Fig. 1.13b. Similarly, Fig. 1.13c and d shows the band diagrams for
square lattice of silicon rods in air (a 2D photonic crystal) and a diamond (i.e., facecentered cubic) lattice of silicon spheres (a 3D photonic crystal). A well-pronounced
bandgap can be seen in each one of them.