Biomedical Engineering Reference
In-Depth Information
where
e
is the electron charge,
v
=
v
z
z
is its velocity (along the
z
-
direction),
E
ind
(
r
e
(
t
),
t
) is the induced electric field along the path
of the electron,
r
e
(
t
), as a function of time
t
. The last equality is
the scattering probability per unit frequency,
(
ω
), where
ω
is the
photon/plasmon energy.
Using the properties of Fourier transforms, asserting that the
field is real, gives:
∞
−∞
{
exp(
i
ω
t
)
E
ind
(
r
e
(
t
),
t
)
}
dt
e
π
ω
(
ω
)
=
(12.7)
where
is the real part.
Theelectronmaybeconsideredasamovingpointcharge,sothat
the current density can beexpressed:
j
(
r
,
ω
)
=−
ev
∞
−∞
δ
(
r
−
r
e
(
t
))exp(
−
i
ω
t
)
dt
(12.8)
where
δ
(
r
−
r
e
(
t
)) is the Dirac delta function. Using this expression
for the current density as a source, it is possible to calculate the
induced field through the dyadic Green's function, and thereby find:
{
exp
i
ω
(
t
−
t
)
G
zz
r
e
(
t
),
r
e
(
t
);
ω
}
dt dt
(12.9)
where both integrals are over all time,
is the imaginary part, and
the trajectory of the electron is chosen to be along the
z
direction.
The
z
component of the dyadic Green's function is the particular
solution to the equation:
e
2
v
z
μ
0
π
(
ω
)
=
∇×∇×
G
z
(
r
,
r
)
G
z
(
r
,
r
)
−
r
)
z
k
2
−
=
δ
(
r
(12.10)
where
k
2
r
is the relative permittivity (assuming
a non-magnetic material). When integrating over the trajectory of
the electron, the integral in Eq. 12.9 can be taken over the
z
,
z
co-
ordinates by notingthat
t
=
z
/
v
z
and
t
=
=
r
ω
2
/
c
2
,and
z
/
v
z
.
12.3.2
EELS for Nanoplasmonics
As is clear from the theory of the previous subsection, EELS allows
for mapping out the induced electric response of a nanostructure
