7 Ultrafast and Nonlinear Plasmon Dynamics
239
in the frequency domain, where χ(ω) is the dielectric susceptibility, with the
frequency-dependence in general arising from material resonances. χ(ω) is related
to the relative dielectric permittivity by ε(ω) = 1 + χ(ω), and the complex index
of refraction ˜
n(ω) = n(ω) + iκ(ω) =
∪
ε(ω). Both χ(ω) and ε(ω) are tensor
properties, but we will initially consider the medium to be isotropic.
Alternatively, the optical response can be described by an induced electrical
current j (ω) as
j (ω) = σ (ω)E(ω)
(7.2)
with electrical conductivity σ (ω). The relationship between the typically complex
σ (ω) = σ 1 (ω) + iσ 2 (ω) and ε(ω) = ε 1 (ω) + iε 2 (ω) is given by
σ (ω) = −iε 0 ω[ε(ω) − 1].
(7.3)
7.1.3 Time Domain Description
The standard frequency domain description of linear optics and the induced
polarization from above has an equivalent time domain formalism. In this case the
optical polarization at time t and location r is the result of the interaction of the
optical electic field E(r, t) with the medium at earlier times and possibly different
locations (non-local response):
P(r, t) = ε 0
≈
−≈
R(r, r
∼
, t
∼
)E(r
∼
, t − t
∼
)d
3 r
∼ dt
∼
(7.4)
where R(r, r ∼ , t ∼ ) is the response function of the system. The response function
encodes the memory of the system, with causality dictating that for t < 0,
R(r, r ∼ , t) = 0. Additionally, time invariance means that the dynamical response
of the system is unchanged by a time offset. For most situations discussed in this
chapter, we can neglect the spatial dependence of the response function. 1 The linear susceptibility in the frequency domain can then be derived from the response
function as
χ(ω) =
≈
−≈
R(t
∼
)e
iωt ∼ dt
∼
.
(7.5)
1 In general the dielectric function is wave vector dependent, ε(k, ω). However, for the regime
discussed here, we can apply the local approximation ε(k = 0, ω) = ε(ω). Non-local effects and
the associated spatial dispersion become significant for ω = v F k, where v F is the Fermi velocity.
This corresponds to k > 1 nm −1 , i.e. structure sizes of a few nanometers at optical frequencies
[4]. Note, however, that this effect is different from the finite-size effect, which can also alter the
dielectric function when structure sizes become less than the characteristic scattering length or the
onset of quantum confinement.
239
in the frequency domain, where χ(ω) is the dielectric susceptibility, with the
frequency-dependence in general arising from material resonances. χ(ω) is related
to the relative dielectric permittivity by ε(ω) = 1 + χ(ω), and the complex index
of refraction ˜
n(ω) = n(ω) + iκ(ω) =
∪
ε(ω). Both χ(ω) and ε(ω) are tensor
properties, but we will initially consider the medium to be isotropic.
Alternatively, the optical response can be described by an induced electrical
current j (ω) as
j (ω) = σ (ω)E(ω)
(7.2)
with electrical conductivity σ (ω). The relationship between the typically complex
σ (ω) = σ 1 (ω) + iσ 2 (ω) and ε(ω) = ε 1 (ω) + iε 2 (ω) is given by
σ (ω) = −iε 0 ω[ε(ω) − 1].
(7.3)
7.1.3 Time Domain Description
The standard frequency domain description of linear optics and the induced
polarization from above has an equivalent time domain formalism. In this case the
optical polarization at time t and location r is the result of the interaction of the
optical electic field E(r, t) with the medium at earlier times and possibly different
locations (non-local response):
P(r, t) = ε 0
≈
−≈
R(r, r
∼
, t
∼
)E(r
∼
, t − t
∼
)d
3 r
∼ dt
∼
(7.4)
where R(r, r ∼ , t ∼ ) is the response function of the system. The response function
encodes the memory of the system, with causality dictating that for t < 0,
R(r, r ∼ , t) = 0. Additionally, time invariance means that the dynamical response
of the system is unchanged by a time offset. For most situations discussed in this
chapter, we can neglect the spatial dependence of the response function. 1 The linear susceptibility in the frequency domain can then be derived from the response
function as
χ(ω) =
≈
−≈
R(t
∼
)e
iωt ∼ dt
∼
.
(7.5)
1 In general the dielectric function is wave vector dependent, ε(k, ω). However, for the regime
discussed here, we can apply the local approximation ε(k = 0, ω) = ε(ω). Non-local effects and
the associated spatial dispersion become significant for ω = v F k, where v F is the Fermi velocity.
This corresponds to k > 1 nm −1 , i.e. structure sizes of a few nanometers at optical frequencies
[4]. Note, however, that this effect is different from the finite-size effect, which can also alter the
dielectric function when structure sizes become less than the characteristic scattering length or the
onset of quantum confinement.