7 Ultrafast and Nonlinear Plasmon Dynamics
245
the SPP is determined by the local dielectric properties of the bulk (for structures
with dimensions above the onset of finite size effects).
This surface wave phenomenon manifests itself in two dis tinct ways: either in
the form of propagating SPP modes, or as localized SPP oscillations.The former
surface-bound wave allows for energy propagation and transport over distances at the
dielectric-metal interface. In the latter, the additional restoring force introduced to the
electron motion by geometric constraints of, for example, noble metal nanoparticles,
results in spatially localized resonant charge density oscillations.These can lead to
large optical polarizabilities and local optical field enhancement. Signatures of both
propagating and localized SPP modes can be observed in random, percolated, or
clustered media.
7.1.8 Propagating Surface Plasmon Polaritons
From the wave equation with appropriate boundary conditions at a metal/dielectric
interface, the dispersion relationship for propagating SPPs is given by
k
2
∗ (ω) =
ω 2
c 2
ε m (ω)ε s
ε m (ω) + ε s
(7.10)
with dielectric permittivity of the metal ε m (ω) and its surrounding ε s (assumed
to be frequency independent in the spectral range of interest). 4 Specifically for
the metal/vacuum interface the resonant condition ε(ω) = −1 results in ω sp =
ω p /
∪
1 + ε ≈ for the surface plasmon resonance for a Drude metal. 5
Figure 7.4 shows the ω versus k dispersion relationship for the ideal Drude surface
plasmon polariton with and without loss. The SPP is characterized by surface parallel
wave vectors that are large compared to light at optical frequencies. Only in the
region near k ∗ → ω/c does the surface plasmon couple to free-space electromagnetic
radiation. For higher frequencies, the excitation requires additional momentum via,
for example, direct evanescent excitation, grating coupling, or increased index of
refraction of the adjacent medium.
Momentum conservation relates the propagating in-plane and evanescent outof-plane wavevectors to the incident free space wavevector via k 2
∞,i + k 2
∗,i = ε i k 2
0
for both the metal (i = 1) and adjacent dielectric (i = 2). Im (k ∗ ) describes the
finite propagation length along the interface. Unlike localized SPP resonances discussed below, due to the large electric field component of propagating SPPs that
penetrates into the dielectric medium, lifetimes generally exceed the Drude damping
time. k ∞,i governs the spatial extent of the evanescent field in the surface normal
4 A wide range of interesting phenomena result for the case of frequency dependent or resonant
surrounding media, but are beyond the scope of this chapter.
5 In treatments of this subject the contribution of core electrons is frequently neglected, using
ε ≈ = 1. This results in ε sp = ω p /
∪
2 and a plasmonic bandgap in the range of ω sp < ω < ω p .
245
the SPP is determined by the local dielectric properties of the bulk (for structures
with dimensions above the onset of finite size effects).
This surface wave phenomenon manifests itself in two dis tinct ways: either in
the form of propagating SPP modes, or as localized SPP oscillations.The former
surface-bound wave allows for energy propagation and transport over distances at the
dielectric-metal interface. In the latter, the additional restoring force introduced to the
electron motion by geometric constraints of, for example, noble metal nanoparticles,
results in spatially localized resonant charge density oscillations.These can lead to
large optical polarizabilities and local optical field enhancement. Signatures of both
propagating and localized SPP modes can be observed in random, percolated, or
clustered media.
7.1.8 Propagating Surface Plasmon Polaritons
From the wave equation with appropriate boundary conditions at a metal/dielectric
interface, the dispersion relationship for propagating SPPs is given by
k
2
∗ (ω) =
ω 2
c 2
ε m (ω)ε s
ε m (ω) + ε s
(7.10)
with dielectric permittivity of the metal ε m (ω) and its surrounding ε s (assumed
to be frequency independent in the spectral range of interest). 4 Specifically for
the metal/vacuum interface the resonant condition ε(ω) = −1 results in ω sp =
ω p /
∪
1 + ε ≈ for the surface plasmon resonance for a Drude metal. 5
Figure 7.4 shows the ω versus k dispersion relationship for the ideal Drude surface
plasmon polariton with and without loss. The SPP is characterized by surface parallel
wave vectors that are large compared to light at optical frequencies. Only in the
region near k ∗ → ω/c does the surface plasmon couple to free-space electromagnetic
radiation. For higher frequencies, the excitation requires additional momentum via,
for example, direct evanescent excitation, grating coupling, or increased index of
refraction of the adjacent medium.
Momentum conservation relates the propagating in-plane and evanescent outof-plane wavevectors to the incident free space wavevector via k 2
∞,i + k 2
∗,i = ε i k 2
0
for both the metal (i = 1) and adjacent dielectric (i = 2). Im (k ∗ ) describes the
finite propagation length along the interface. Unlike localized SPP resonances discussed below, due to the large electric field component of propagating SPPs that
penetrates into the dielectric medium, lifetimes generally exceed the Drude damping
time. k ∞,i governs the spatial extent of the evanescent field in the surface normal
4 A wide range of interesting phenomena result for the case of frequency dependent or resonant
surrounding media, but are beyond the scope of this chapter.
5 In treatments of this subject the contribution of core electrons is frequently neglected, using
ε ≈ = 1. This results in ε sp = ω p /
∪
2 and a plasmonic bandgap in the range of ω sp < ω < ω p .