Biomedical Engineering Reference
In-Depth Information
in order to maintain a constant force between the tip and the sample. Due to its
working mechanism, AFM has a limited applicability in natural photonic structures
as the structural information beneath the sample surface cannot be probed.
8.4.5
Theoretical Treatments
For natural photonic structures, analytical treatments are only possible for some
particular cases, e.g., thin films [
23
]. To give a quantitative description of the optical
properties for nontrivial natural photonic structures, numerical simulations are
needed. Analytical and numerical results can provide reflection, scattering, trans-
mission, and absorption spectra which can be directly compared with experimental
results. Other optical quantities, such as photonic band structures and photon density
of states, can be also obtained. With these results, we may uncover interesting
mechanisms of structural coloration, light steering strategies, and even new optical
phenomena.
With the development of computational algorithms and computer itself, various
computational methods have been proposed and implemented to solve Maxwell's
equations numerically. Commonly used methods include transfer matrix method
(TMM) [
31
-
33
], finite-difference time-domain (FDTD) method [
34
,
35
], and plane-
wave expansion (PWE) method [
29
].
TMM was originally developed for multilayers. In TMM, the fields of the
adjacent layers are related by a transfer matrix that is composed of two matrices:
one is to propagate the fields in a distance in the uniform medium and the other is
to make the fields from one side of an interface to the other. From the total transfer
matrix, the fields at one side of a multilayer can be related to those at the other
side. TMM can be extended to deal with photonic structures of higher dimensions
[
32
,
33
].
FDTD is a popular tool which is widely used for electromagnetic computations
[
35
]. The central idea of FDTD is to solve Maxwell's equations based on dis-
cretizing time and space into finite grids. The electric and magnetic fields at all
points within the computational domain can be calculated. FDTD is a versatile
and powerful technique which can be applied to complicated photonic structures.
But the spatial grid discretizations must be sufficiently fine, which will increase the
computational time and computer memory.
For periodic photonic structures, i.e., photonic crystals, their photonic band
structures can offer important information in understanding the optical response.
PWE is widely used for calculating photonic band structures [
29
]. Within the
framework of PWE, the electric and magnetic fields are expanded by plane waves,
and Maxwell's equations are then cast into eigen-problems. By imposing the Bloch
theorem, photonic band structures can be obtained by solving eigen-equations.
Photonic band structures can also be calculated by other methods such as TMM
and FDTD provided that the Bloch theorem is imposed.
