Biomedical Engineering Reference
In-Depth Information
7
Biodetection Using Silicon Photonic Crystal
Microcavities
P.M. Fauchet, B.L. Miller, L.A. DeLouise, M.R. Lee, and H. Ouyang
7.1 Photonic Crystals: A Short Introduction
7.1.1 Electromagnetic Theory
Lord Rayleigh was the first to study electromagnetic wave propagation in one-
dimensional (1D) periodic media and to identify the angle-dependent, narrow
band in which light propagation is prohibited. However, it was not until a full
century later, when Yablonovitch [1] and John [2] in 1987 combined Maxwell's
equations with solid-state physics theorems to introduce the concept of pho-
tonic bandgaps in two and three dimensions. Many subsequent developments
in fabrication, theory, and applications (e.g., fiber optics, integrated optics,
and negative refraction materials) have since followed.
As shown in Fig. 7.1, photonic crystals (PhCs) are periodically structured
media [3] in one, two, or three dimensions. PhCs can be designed to produce
photonic bandgaps. Light with photon energies or frequencies that fall inside
this bandgap cannot propagate through the PhC. The periodicity in length
scale is proportional to the wavelength of light inside the bandgap. PhCs are
the electromagnetic analog of crystalline atomic lattices, in which interference
in the electron wavefunction produces the forbidden bands. Hence, the study
of PhCs is also governed by the Bloch-Floquet theorem.
To study the propagation of light, Maxwell's equations are solved as an
eigenproblem. We assume that light propagates in a mixed dielectric medium
with no free charges or currents. Maxwell's equations are then given by:
r
,t
)+
1
c
∂
H
r
,t
)
∂t
(
∇
H
(
r
,t
)=0
∇×
E
(
=0
(7.1)
ε
(
r
)
∂
E
r
,t
)
∂t
(
∇ε
(
r
)
E
(
r
,t
)=0
∇×
H
(
r
,t
)
−
=0
,
c
where
ε
(
r
)=
εr
+
a
i
is the dielectric function,
a
i
is the primitive lattice vector
in all three dimensions, and
c
is the speed of light in vacuum. By combining
