1.7 Photonic Crystals
19
Fig. 1.14 Light line of air
and a bulk dielectric
The cause of bandgap
It has been observed that a low-frequency mode tends to concentrate more of
its field in the high dielectric region, while a high-frequency mode tends to
concentrate more of its field in the low dielectric region, in order to maintain
orthogonality between themselves (read Joannopoulos 2008 [37]). If one finds
this assertion difficult to accept, he should scrutinize the light line of air and a
bulk dielectric, shown in the adjacent figure. A mode of the same wave vector
is of a lesser frequency in the dielectric medium than in the air (Fig. 1.14). The
fact remains true for the inhomogeneous structures of the dielectric as well.
In Fig. 1.13, the modes of the first band concentrate their energy mainly in
the dielectric region, and thus have lower energy and lower frequency, whereas
the second band accommodates most of its energy in the air region, and hence
has high energy and high frequency. As an obvious consequence, there is
a substantial difference between the frequencies of the first band (a.k.a. the
dielectric band) and the second band (a.k.a the air band), which manifests
as the bandgap. The size of the bandgap depends directly on the dielectric
contrast (i.e., difference between the high and low dielectric constants) of the
photonic crystal. The greater the contrast, the broader the gap.
1.7.3 Photonic Crystal Waveguides
As mentioned above, any frequency lying in the bandgap region is forbidden from
propagation. This property can be exploited to achieve confined propagation of light
in a narrow channel of a homogeneous medium sandwiched between two bandgap
structures. Such kind of a device is called a photonic crystal waveguide. Figure 1.15
illustrates the working of photonic crystal waveguides, where (a) and (b) illustrate a
1D PhC waveguide, while (c) and (d) present a 2D PhC waveguide. In the geometries
shown in Fig. 1.15a and c, one can observe a narrow air region sandwiched between
19
Fig. 1.14 Light line of air
and a bulk dielectric
The cause of bandgap
It has been observed that a low-frequency mode tends to concentrate more of
its field in the high dielectric region, while a high-frequency mode tends to
concentrate more of its field in the low dielectric region, in order to maintain
orthogonality between themselves (read Joannopoulos 2008 [37]). If one finds
this assertion difficult to accept, he should scrutinize the light line of air and a
bulk dielectric, shown in the adjacent figure. A mode of the same wave vector
is of a lesser frequency in the dielectric medium than in the air (Fig. 1.14). The
fact remains true for the inhomogeneous structures of the dielectric as well.
In Fig. 1.13, the modes of the first band concentrate their energy mainly in
the dielectric region, and thus have lower energy and lower frequency, whereas
the second band accommodates most of its energy in the air region, and hence
has high energy and high frequency. As an obvious consequence, there is
a substantial difference between the frequencies of the first band (a.k.a. the
dielectric band) and the second band (a.k.a the air band), which manifests
as the bandgap. The size of the bandgap depends directly on the dielectric
contrast (i.e., difference between the high and low dielectric constants) of the
photonic crystal. The greater the contrast, the broader the gap.
1.7.3 Photonic Crystal Waveguides
As mentioned above, any frequency lying in the bandgap region is forbidden from
propagation. This property can be exploited to achieve confined propagation of light
in a narrow channel of a homogeneous medium sandwiched between two bandgap
structures. Such kind of a device is called a photonic crystal waveguide. Figure 1.15
illustrates the working of photonic crystal waveguides, where (a) and (b) illustrate a
1D PhC waveguide, while (c) and (d) present a 2D PhC waveguide. In the geometries
shown in Fig. 1.15a and c, one can observe a narrow air region sandwiched between