Page 152 - Principles and Applications of NanoMEMS Physics

140 Chapter 3

§ 1 · G G § ω · 2 G G
∇ × ¨ ¨ G ∇ ¸ × ¸ H () r = ¨ ¸ H () , (189)
r
© ε () r ¹ © c ¹

if the dielectric profile defining a PBC is scaled as follows,
G
G
G
ε () r → () r ' ε = ε () sr , where s is the scaling factor, then it can be shown
that the scaled PBC will obey the equation,
§ 1 · G G § ω · 2 G G
∇ × ¨ ¨ G ∇ ' ¸ × ¸ H ( /'r ) s = ¨ ¸ H ( /'r ) s , (190)
© () 'r ' ε ¹ © cs¹

from where one derives that the properties corresponding to the scaled PBC
G G G G
are derived from those of the unscaled one as follows: () 'r'H = H ( s'r ) and
ω ' ω s / . Thus, once the PBC solutions are known at one length scale, they
=
are automatically known at all others. As a practical application, microwave-
length-scale PBCs may be exploited as vehicles to study to optical-scale
PBC concepts.
Similarly, there is no fundamental value of dielectric constant, therefore,
it may be shown that whenever the dielectric constant is uniformly scaled
throughout a PBC as follows: () r →ε G () r ' ε G = ε () sr G 2 , where s is the scaling
factor, then the scaled PBC will obey the equation,

G
G
s ·
G
∇ × § ¨ 1 G ∇ × · ¸ H () r = § ω ¸ 2 H () r G . (191)
¨
¨ ' ε ¸
© () r ¹ © c ¹
This means that, upon scaling the dielectric constant, the mode geometry
remains unchanged, but the frequency scales as: ω → ω' = s ω . Thus,
multiplying the dielectric constant by a factor of 1/9 will result in
multiplying the frequency of their modes by three.
Lastly, the properties of PBCs depend on parameters such as filling
fraction, the contrast between host and lattice dielectric constants, and the
number of layers employed. Fig. 3-28 shows the computed transmission
coefficient for an eleven-layer PBC as the index of refraction n = ε is
2
increased from 1.2 to 2.98.