2.5 Accidental-Degeneracy-Induced Dirac Cones in Photonic Crystals
39
What about the velocity of light inside a zero-index medium?
One may wonder that zero refractive index means infinite velocity of light
inside the metamaterial, which sounds unphysical. This is a wrong perception!
One must understand that there are two types of velocities associated with
electromagnetic waves: the phase velocity and the group velocity. The phase
velocity is given as v p = ω/k, whereas the group velocity is given as v g =
dω/dk. The wave vector k, being zero, affects only the phase velocity and the
wavelength (λ = 2π/k), making both of them infinite. The phase velocity is
a mathematical entity, has no physical meaning, hence can be infinite. It is the
group velocity that is the actual rate of transfer of energy from one point to the
other inside the medium. The group velocity is calculated from the slope of the
dispersion curve. For the metamaterial shown in Fig. 2.8, the group velocity
at the Dirac point, calculated from the dispersion curves of Fig. 2.9a, is 0.33c,
where c is the velocity of light. Hence, no physical law gets violated here!
The abovementioned square lattice of rods-in-air type was the all-dielectric zeroindex metamaterial [27, 60, 92]. As the research in this area progressed, several
other designs for a variety of applications were developed, a few of which are shown
in Fig. 2.13. Besides the square lattice of rods, there can be hexagonal/triangular or
honeycomb lattices too (Fig. 2.13a, b). The rod-based structures are operational for
TM polarization, while, for TE polarization, complementary structures, like holes
in a dielectric slab (Fig. 2.13 c–e), have been introduced [64, 114]. The lattice of
air columns can be of square, hexagonal, or honeycomb type, similar to the case of
rods. In addition to these, a few exotic designs such as the star/flower-shaped holes
providing high Q-factor [115] and complex designs such as honeycomb lattice of
rods embedded inside a hexagonal lattice of holes have also been used for special
applications like second-harmonic generation [112, 113]. Designs can be as many
as a creative and innovative mind can think of, depending on the utility and ease of
fabrication.
2.6 Reflection and Refraction by Zero-Index Metamaterials
When light travels from one medium to another, depending on the impedance of the
two media, some part of it gets reflected, some part gets absorbed, and the rest gets
transmitted [10]. The role of impedance matching in electromagnetic power transmission is well acknowledged [7, 116–119]. The greater the impedance mismatch,
the greater the reflection, and the lesser the transmission. The refractive indices of the
two media govern the direction of the transmitted ray with respect to incident rays,
according to Snell’s law of refraction. Snell’s laws of refraction and reflection have
been in use for a long time for systems involving natural and positive-index media,
and have recently been found suitable for negative-index media as well. Zero-index
media are the latest addition to this list, and in this chapter we shall see how light
gets reflected or refracted when zero refractive index is involved.
39
What about the velocity of light inside a zero-index medium?
One may wonder that zero refractive index means infinite velocity of light
inside the metamaterial, which sounds unphysical. This is a wrong perception!
One must understand that there are two types of velocities associated with
electromagnetic waves: the phase velocity and the group velocity. The phase
velocity is given as v p = ω/k, whereas the group velocity is given as v g =
dω/dk. The wave vector k, being zero, affects only the phase velocity and the
wavelength (λ = 2π/k), making both of them infinite. The phase velocity is
a mathematical entity, has no physical meaning, hence can be infinite. It is the
group velocity that is the actual rate of transfer of energy from one point to the
other inside the medium. The group velocity is calculated from the slope of the
dispersion curve. For the metamaterial shown in Fig. 2.8, the group velocity
at the Dirac point, calculated from the dispersion curves of Fig. 2.9a, is 0.33c,
where c is the velocity of light. Hence, no physical law gets violated here!
The abovementioned square lattice of rods-in-air type was the all-dielectric zeroindex metamaterial [27, 60, 92]. As the research in this area progressed, several
other designs for a variety of applications were developed, a few of which are shown
in Fig. 2.13. Besides the square lattice of rods, there can be hexagonal/triangular or
honeycomb lattices too (Fig. 2.13a, b). The rod-based structures are operational for
TM polarization, while, for TE polarization, complementary structures, like holes
in a dielectric slab (Fig. 2.13 c–e), have been introduced [64, 114]. The lattice of
air columns can be of square, hexagonal, or honeycomb type, similar to the case of
rods. In addition to these, a few exotic designs such as the star/flower-shaped holes
providing high Q-factor [115] and complex designs such as honeycomb lattice of
rods embedded inside a hexagonal lattice of holes have also been used for special
applications like second-harmonic generation [112, 113]. Designs can be as many
as a creative and innovative mind can think of, depending on the utility and ease of
fabrication.
2.6 Reflection and Refraction by Zero-Index Metamaterials
When light travels from one medium to another, depending on the impedance of the
two media, some part of it gets reflected, some part gets absorbed, and the rest gets
transmitted [10]. The role of impedance matching in electromagnetic power transmission is well acknowledged [7, 116–119]. The greater the impedance mismatch,
the greater the reflection, and the lesser the transmission. The refractive indices of the
two media govern the direction of the transmitted ray with respect to incident rays,
according to Snell’s law of refraction. Snell’s laws of refraction and reflection have
been in use for a long time for systems involving natural and positive-index media,
and have recently been found suitable for negative-index media as well. Zero-index
media are the latest addition to this list, and in this chapter we shall see how light
gets reflected or refracted when zero refractive index is involved.