Biomedical Engineering Reference
In-Depth Information
color, as exhibited by periodic multilayered
structures--which are sometimes called
Bragg
filters.
.
Diffraction gratings provide another simple
and widely exploited manifestation of the Bragg
phenomenon. Most commonly, a
diffraction grat-
ing
is a periodically corrugated sheet of a metal
that is much thicker than the skin depth. When
monochromatic light is incident on the grating
at an angle
θ
with respect to the mean plane of
the grating, then reflected light can be seen
propagating in possibly more than one different
directions. These directions are oriented at
angles
exhibited
[54]
. For visible effects, the lattice
constant must be in the range of 100 nm-1
μ
m,
which can be accessed with conventional nano-
fabrication and self-assembly techniques
[53]
. A
natural photonic crystal is the precious opal,
which is formed by a spontaneous organization
of colloidal silica spheres on a crystalline lattice.
Periodic arrays of micron-scale cavities in the
wings of many beetles, such as the Japanese
jewel beetle
Chrysochroa fulgidissima
, function
optically as inverse opals
[55]
. Both opals and
inverse opals are made artificially as well. A
splendid two-dimensional example of artificial
opal is presented in
Figure 11.8
, which shows a
θ
M
=
COS
−
1
(COSθ
+
M
λ/
D
),
(
M
=
0,
±
1,
±
2, ...),
(11.3)
where
d
is the period of the grating. The direc-
tion for
m
=
0 is the specular direction. Only a
few of the angles
θ
m
predicted by Eq.
(11.3)
being
real (i.e., |cos
θ
m
| ≤ 1), there are only a finite
number of nonspecular directions.
Periodicity need not be one-dimensional. It
can exist in two dimensions as well as in three
dimensions. For optical purposes, we can say
that the relative permittivity tensor of a periodic
material has certain periodic symmetries. Now-
adays these materials are called
photonic crystals
.
These materials display a
photonic bandgap
, i.e.,
a
λ
-range in which propagation of light through
the photonic crystal is not allowed in a certain
direction
[54]
. An infinitely thick one-dimen-
sional photonic crystal does not display a com-
plete bandgap, which means that there is no
λ
-range, howsoever narrow, in which propaga-
tion is inhibited in all directions. Many incom-
plete bandgaps are exhibited by two- and
three-dimensional photonic crystals. A complete
bandgap is possible in some three-dimensional
photonic crystals.
Most photonic crystals are fabricated artifi-
cially. The wavelength of light must be compa-
rable to the periodicity of the photonic crystal
for the Bragg phenomenon and bandgaps to be
FIGURE 11.8
Photograph of a monolayer of close-
packed carboxylate-modified polystyrene spheres of 510-nm
diameter deposited on a silicon wafer. Courtesy of A. Shoji
Hall (Pennsylvania State University, USA).