Principles of Fluorescence Spectroscopy

864 RADIATIVE-DECAY ENGINEERING: SURFACE PLASMON-COUPLED EMISSION
Figure 26.5. Calculated wavelength-dependent reflectivity for a 47nm-thick
gold film. From [22].
is the propagation constant, which is sometimes called the
wavevector. This value is given by
(26.2)
where λ = λ 0 /n is the wavelength, λ 0 is the wavelength in a
vacuum, n is the refractive index of the medium, and k 0 is
the propagation constant of the wave in a vacuum. It is
understood that the physical values are given by the real
part of eq. 26.1. Hence the electric field is described by
(26.3)
For SPR we need to consider the electric field along the xaxis
at the metal–water interface. This component is given
by
(26.4)
To satisfy Maxwell's equations the electric fields have to be
continuous across the interface, which requires k x to be
equal in both media.
The phenomenon of SPR can be understood by considering
the propagation constant of the electromagnetic wave
in the metal along the x-axis. In the metal film the field is
described by eqs. 26.3 and 26.4 with k x = k r + ik im being the
complex wavevector along the x-axis. For a metal the propagation
constant for the surface plasmon is given by
k SP ω
c (
k 2π
λ
ε mε s
ε m ε s
nω
c nk 0
E(r,t) E 0 exp(cos ωt k r)
E(x,t) E 0x exp(cos ωt k xx)
) 1/2
εmεs k0 ( εm εs ) 1/2
(26.5)
where ε m and ε s are the dielectric constant of the metal (m)
and sample (s), respectively. ε s refers to the effective dielectric
constant in the region of the evanescent field (Figure
26.2). Because the real part of ε m is larger than the imaginary
part the propagation constant can be approximated by
εrεs kSP k0 ( εr εs (26.6)
The incident light can excite a surface plasmon when
its x-axis component equals the propagation constant for the
surface plasmon (Figure 26.6). The propagation constant
for the incident light in the prism (p) is given by
k p k 0n p
and the component along the x-axis is equal to
k x k 0n p sinθ p
) 1/2
(26.7)
(26.8)
where θ p is the incidence angle in the prism. Hence the conditions
for SPR absorption is satisfied when
k SP k x k 0n p sinθ sp
(26.9)
where θ sp is the angle of incidence in the metal for surfaceplasmon
resonance to occur. These considerations show
Figure 26.6. Schematic showing propagation constants in a prism and
a thin film.