24
1 Electromagnetics for Zero-Index Metamaterials
peak starts to appear in the said region and intensifies as the array densifies. It should
be noted that with a change in periodicity (a) there is no significant shift in the
position of Mie peak, attributed to the fact that it only depends on the rods’ radius
and permittivity, which are invariant in this case. Hence, in denser arrays where
the lattice constant is substantially smaller compared to λ Mie , the effective medium
theory becomes applicable and the structure acquires metamaterial character.
Furthermore, Fig. 1.18c shows the effect of the second important parameter, the
permittivity. Here the permittivity has been varied from 12 to 50, keeping r and a
constants at 170 nm and 850 nm, respectively. It can be seen that λ Mie undergoes
redshift as the permittivity increases. For = 12, λ Mie is the same as above, which
is not very large compared to a = 850 nm. But as increases, so does λ Mie , making the structure more suitable for application of the effective medium theory and
consideration as a metamaterial. In this way, it has been proved that to make the
photonic crystal act as a metamaterial, either the periodicity should be reduced or the
permittivity should be increased. We believe that the above discussion is sufficiently
rigorous to explain the difference between a photonic crystal and a metamaterial and
the transition from one to the other.
1.8.2 Photonic Crystals as Metamaterials
One of the most interesting phenomena achieved by metamaterials is the negative refractive index. Conventional design of negative-index metamaterials included
metallic split-ring resonators or metallic fishnet-type structures. Contrarily, the new
generation variants are purely dielectric and exhibit no power dissipation [79–84].
The simplest all-dielectric route to the negative refractive index is the abovementioned rods-in-air-type photonic crystal. Figure 1.19a shows the photonic band diagram of a square lattice of silicon rods of periodicity a = 1 µm and radius r = 0.25a
and dielectric constant = 12.25. By using bands 2 and 4, the refractive index has
been calculated around the center of the Brillouin zone and shown in Fig. 1.19c. One
can notice the spectral regions of negative- and positive-index values, as well as an
empty region between them where no real value of refractive index exists. This forbidden region corresponds to the photonic bandgap between the two bands. It indicates
that if a frequency belonging to the negative-index regime is allowed to travel through
the photonic crystal, it will undergo negative refraction. This claim gets confirmed
in Fig. 1.19e (a far-field plot), where a wave of frequency ω = 760 × 10
12 rad/s traveling from a prism of the photonic crystal to air emerges on the same side of the
normal as the incident beam, a clear manifestation of negative refraction.
As negative index is achieved, one may wonder if it is possible to bring the
negative- and positive-index curves of Fig. 1.19c close together to get a single continuous curve passing through the zero of y-axis at a particular frequency. The answer
is: Yes, if one chooses a suitable radius of the rods, such that the photonic bands 2 and
4 intersect at a common point, obliterating the gap. This condition was fulfilled for
radius r = 0.2a. Figure 1.19b shows the band diagram for r = 0.2a case, in which
1 Electromagnetics for Zero-Index Metamaterials
peak starts to appear in the said region and intensifies as the array densifies. It should
be noted that with a change in periodicity (a) there is no significant shift in the
position of Mie peak, attributed to the fact that it only depends on the rods’ radius
and permittivity, which are invariant in this case. Hence, in denser arrays where
the lattice constant is substantially smaller compared to λ Mie , the effective medium
theory becomes applicable and the structure acquires metamaterial character.
Furthermore, Fig. 1.18c shows the effect of the second important parameter, the
permittivity. Here the permittivity has been varied from 12 to 50, keeping r and a
constants at 170 nm and 850 nm, respectively. It can be seen that λ Mie undergoes
redshift as the permittivity increases. For = 12, λ Mie is the same as above, which
is not very large compared to a = 850 nm. But as increases, so does λ Mie , making the structure more suitable for application of the effective medium theory and
consideration as a metamaterial. In this way, it has been proved that to make the
photonic crystal act as a metamaterial, either the periodicity should be reduced or the
permittivity should be increased. We believe that the above discussion is sufficiently
rigorous to explain the difference between a photonic crystal and a metamaterial and
the transition from one to the other.
1.8.2 Photonic Crystals as Metamaterials
One of the most interesting phenomena achieved by metamaterials is the negative refractive index. Conventional design of negative-index metamaterials included
metallic split-ring resonators or metallic fishnet-type structures. Contrarily, the new
generation variants are purely dielectric and exhibit no power dissipation [79–84].
The simplest all-dielectric route to the negative refractive index is the abovementioned rods-in-air-type photonic crystal. Figure 1.19a shows the photonic band diagram of a square lattice of silicon rods of periodicity a = 1 µm and radius r = 0.25a
and dielectric constant = 12.25. By using bands 2 and 4, the refractive index has
been calculated around the center of the Brillouin zone and shown in Fig. 1.19c. One
can notice the spectral regions of negative- and positive-index values, as well as an
empty region between them where no real value of refractive index exists. This forbidden region corresponds to the photonic bandgap between the two bands. It indicates
that if a frequency belonging to the negative-index regime is allowed to travel through
the photonic crystal, it will undergo negative refraction. This claim gets confirmed
in Fig. 1.19e (a far-field plot), where a wave of frequency ω = 760 × 10
12 rad/s traveling from a prism of the photonic crystal to air emerges on the same side of the
normal as the incident beam, a clear manifestation of negative refraction.
As negative index is achieved, one may wonder if it is possible to bring the
negative- and positive-index curves of Fig. 1.19c close together to get a single continuous curve passing through the zero of y-axis at a particular frequency. The answer
is: Yes, if one chooses a suitable radius of the rods, such that the photonic bands 2 and
4 intersect at a common point, obliterating the gap. This condition was fulfilled for
radius r = 0.2a. Figure 1.19b shows the band diagram for r = 0.2a case, in which