Biomedical Engineering Reference
In-Depth Information
at the origin while the Hankel functions satisfy the outgoing wave
requirement outside the cylinder. In the azimuthal direction, the
exponential functions
e
±
i
m
φ
(
m
=
0, 1, 2, . . .) fulfill the periodic
condition. Equivalently, the linear combination of them, that is,
the trigonometric functions,
cos
(
m
φ
)and
sin
(
m
φ
), can be used as
eigen functions as well. These trigonometricfunctions present clear
pictures of specific collective oscillations that propagate along the
nanowire surface. Therefore, it is convenient to adapt the trigono-
metric functions when discussing plasmonic waveguides. The wave
vectors of the SPPs characterize several important features of the
modes, including the propagation constant and the penetration
depth away from the metal-dielectric interface. The propagation
constant
k
m
,
is complex, with its real and imaginary part defining
thephaseconstantandattenuationconstant,respectively.Bysolving
the transcendental equation involving Bessel functions and their
derivatives, the complex propagation constants of four low-order
SPPs on a Ag nanowire can be obtained, as shown in Fig. 6.1. The
permittivity of Ag at the vacuum wavelength
λ
0
=
632.8 nm,
ε
=
−
18.36
+
0.4786i, is interpolated from the experimental data by
Johnson and Christy [22]. The surrounding matrix is assumed to be
oil with
ε
M
D
=
2.25. As having been discussed, for example, by D. E.
Chang et al. [23], the
m
=
0 mode exists in a nanowire of arbitrary
size. On the contrary, the high-order modes (
m
≥
2) have distinct
cutoffsizesbelowwhichthemodescannotsustain.The
m
=
1mode,
however, is not strictly cutoff but suffers from a rapid expansion of
its mode area at the thin wire limit. As shown in Fig. 6.1(a), when
the radius of the nanowire is smaller than 50 nm, the real part
of its propagation constant approaches to that of the light in the
surrounding medium. This means that on a very thin wire, the
m
=
1 mode becomes similar to a plane wave propagating parallel to the
wire axis. Therefore, this mode becomes di
cult to excite under
normal (to the wire axis) incidence. This property is important in
the formation of chiral SPPs as will be discussed below. As the
radius of the nanowire increases, the real part of the propagation
constant of the
m
=
0 mode decreases, whereas those for the other
modes increase. Notice that for very thick wire, the cylinder surface
is similar to a flat interface so that all the modes will eventually
approach the SPP on asingle metal-oil interface.
