Page 150 - Principles and Applications of NanoMEMS Physics

138 Chapter 3

studied structure. From these applications it is known that PBCs can act as
perfect mirrors for light whose frequency lies within a well-defined range,
namely, when kd (where d is the lattice constant) is an odd multiple of π / 2 ,
and that they may localize modes when endowed with defects [162]. The
application of PBCs in the context of routing and controlling the propagation
of light waves, for example, requires their realization in, at least, 2-D. Next,
we deal with multi-dimensional PBCs.

3.2.2.2 Multi-dimensional PBC Physics

The properties of 2- and 3-D PBCs may be formulated in terms of the
coherent scattering properties of 2- and 3-D lattices [64]. Fig. 3-27(a)
typifies a 2-D triangular-lattice PBC consisting of cylinders of dielectric
constant ε embedded in a host of dielectric constant ε .
2 1
Top View
Top View

a a

Host t
Hos ε ε 1 1
Lattic
Lattice e ε ε 2 2
(a)

& &
k k
a a cosθcos θ & &
a a & &
θ θ
' θ' θ k′ k′

a a cosθ'cosθ '
(b)
G G
Figure 3-27. (a) Sketch of 2-D PBC with lattice constants a and a consisting of cylinders
1 2
of dielectric constant ε embedded in a host of dielectric constant ε . (b) Detail of an
2 G 1 G
incoming wave with wave vector k impinging on two objects separated a distance a , and
G
scattered along wave vector 'k .