2.5 Accidental-Degeneracy-Induced Dirac Cones in Photonic Crystals
37
(a) r/a=0.18
(b) r/a=0.20
(c) r/a=0.22
Fig. 2.10 Role of geometrical parameters in obtaining a well-defined Dirac cone
zoomed-in view of the second, the third, and the fourth bands. The resemblance of
these double cones to those of the graphene, shown in Fig. 2.5, is noticeable, the only
difference being an extra horizontal band between them. The normalized frequency
(ωa/2πc) corresponding to the Dirac point is 0.541. The degeneracy here is rightly
called “accidental” because it is possible only for a particular value of radius-toperiodicity ratio (r/a). The only way to determine the suitable value of r/a ratio is
to scan through different values and choose the best one, for which a well-defined
Dirac cone is being formed. The crucial role of r/a ratio is illustrated in Fig. 2.10.
It can be observed that the Dirac cone obtained for r/a = 0.2 gets ruined, even for
a slight variation of r/a to 0.18 or 0.22.
A Dirac cone is an important sign of zero-index character. However, it should
be noted that the existence of a Dirac cone is necessary but not sufficient condition
for the exhibition of the effectively zero refractive index. An additional important
factor is the electric field distribution of the three modes corresponding to the tripledegenerate Dirac point. If the field distribution is a combination of monopole and
dipole modes, the conical dispersion can be translated to zero refractive index with
both the permittivity and permeability tending to zero at the Dirac point [27, 60,
92, 111]. Figure 2.11 shows the three types of distributions of the z-component of
electric field E z at the Dirac point. Figure 2.11a shows the electric monopole mode,
Fig. 2.11b shows the transverse magnetic dipole mode, and Fig. 2.11c shows the
longitudinal magnetic dipole mode [70]. In light of the above observations, i.e.,
37
(a) r/a=0.18
(b) r/a=0.20
(c) r/a=0.22
Fig. 2.10 Role of geometrical parameters in obtaining a well-defined Dirac cone
zoomed-in view of the second, the third, and the fourth bands. The resemblance of
these double cones to those of the graphene, shown in Fig. 2.5, is noticeable, the only
difference being an extra horizontal band between them. The normalized frequency
(ωa/2πc) corresponding to the Dirac point is 0.541. The degeneracy here is rightly
called “accidental” because it is possible only for a particular value of radius-toperiodicity ratio (r/a). The only way to determine the suitable value of r/a ratio is
to scan through different values and choose the best one, for which a well-defined
Dirac cone is being formed. The crucial role of r/a ratio is illustrated in Fig. 2.10.
It can be observed that the Dirac cone obtained for r/a = 0.2 gets ruined, even for
a slight variation of r/a to 0.18 or 0.22.
A Dirac cone is an important sign of zero-index character. However, it should
be noted that the existence of a Dirac cone is necessary but not sufficient condition
for the exhibition of the effectively zero refractive index. An additional important
factor is the electric field distribution of the three modes corresponding to the tripledegenerate Dirac point. If the field distribution is a combination of monopole and
dipole modes, the conical dispersion can be translated to zero refractive index with
both the permittivity and permeability tending to zero at the Dirac point [27, 60,
92, 111]. Figure 2.11 shows the three types of distributions of the z-component of
electric field E z at the Dirac point. Figure 2.11a shows the electric monopole mode,
Fig. 2.11b shows the transverse magnetic dipole mode, and Fig. 2.11c shows the
longitudinal magnetic dipole mode [70]. In light of the above observations, i.e.,