262
M. B. Raschke et al.
also found for other susceptibility components, with their relative magnitude varying
with fundamental frequency. In this early work, the nonlinear response of Au was
measured to be χ
(2)
s∞∞∞ ∝ 2 × 10 −8 m/V, approximately four times larger than that of
Ag, close to 1 eV. Later measurements on thin films found that Ag has the strongest
SHG intensity of the metals, with Au slightly smaller and Cu approximately 50 % of
the Ag response [32]. With careful angle and polarization-dependent measurement
enabling separation of the bulk and surface responses in the experiment, the χ
(2)
s∞∞∞
component was determined to be approximately 200 times larger than χ
(2)
B for Au,
and ∝100 χ
(2)
B in Ag, at a fundamental photon energy of 1.55 eV. Interestingly,
Al has an inherently high bulk nonlinear response, with χ
(2)
B values an order of
magnitude higher than Au and Ag. However, it also suffers from a short skin depth
and high losses in the visible and NIR, in addition to a tendency to oxidize, so is
generally not considered as suitable for plasmonic applications.These experiments
also demonstrated a wide variation in surface susceptibility values depending on
surface roughness and growth conditions of the thin films, an effect that is the subject
of Sect. 7.3.4. Hence, the development of both accurate quantitative experiments and
an accurate quantitative theory has remained difficult.
While the discussion to this point has been limited to the nonlinear response
of metal involving single particle excitations, the next section is concerned with
nonlinear interactions involving SPPs, where the surface-sensitivity of the secondorder response becomes particularly important for the enhancement of the nonlinear
signal.
7.3.3 Nonlinear Wavemixing with Surface Plasmons
The momentum mismatch between the incident and emitted light and in plane SPP
wavevectors as shown in Fig. 7.4 means that linear excitation of SPPs on planar
surfaces typically requires an effective momentum change of the incident field in
the form of a coupling element or increase in index of refraction, for example a
grating or Kretschmann prism, respectively. However, the phase-matching conditions
associated with the different wavevectors participating in a nonlinear wavemixing
process provide additional flexibility, allowing free-space launching of SPPs via the
generated nonlinear polarization, or SPPs as the source for the nonlinear output field,
or both. While even-order processes are intrinsically surface-confined, interactions
involving SPPs are also in practice limited to the near-surface region given by the skindepth, even for odd-order processes. Any of the participating wavemixing fields can
be an SPP, with a wavevector that is defined by the surface dispersion relation of the
specific SPP frequency. Figure 7.11 shows a set of different possible configurations.
The surface-parallel components of the free-space k-vectors k i = ω i n(ω i ) sin θ i /c,
where θ i is the angle with respect to the surface normal and n(ω i ) is the index
of refraction of the dielectric medium, can then be summed appropriately with the
M. B. Raschke et al.
also found for other susceptibility components, with their relative magnitude varying
with fundamental frequency. In this early work, the nonlinear response of Au was
measured to be χ
(2)
s∞∞∞ ∝ 2 × 10 −8 m/V, approximately four times larger than that of
Ag, close to 1 eV. Later measurements on thin films found that Ag has the strongest
SHG intensity of the metals, with Au slightly smaller and Cu approximately 50 % of
the Ag response [32]. With careful angle and polarization-dependent measurement
enabling separation of the bulk and surface responses in the experiment, the χ
(2)
s∞∞∞
component was determined to be approximately 200 times larger than χ
(2)
B for Au,
and ∝100 χ
(2)
B in Ag, at a fundamental photon energy of 1.55 eV. Interestingly,
Al has an inherently high bulk nonlinear response, with χ
(2)
B values an order of
magnitude higher than Au and Ag. However, it also suffers from a short skin depth
and high losses in the visible and NIR, in addition to a tendency to oxidize, so is
generally not considered as suitable for plasmonic applications.These experiments
also demonstrated a wide variation in surface susceptibility values depending on
surface roughness and growth conditions of the thin films, an effect that is the subject
of Sect. 7.3.4. Hence, the development of both accurate quantitative experiments and
an accurate quantitative theory has remained difficult.
While the discussion to this point has been limited to the nonlinear response
of metal involving single particle excitations, the next section is concerned with
nonlinear interactions involving SPPs, where the surface-sensitivity of the secondorder response becomes particularly important for the enhancement of the nonlinear
signal.
7.3.3 Nonlinear Wavemixing with Surface Plasmons
The momentum mismatch between the incident and emitted light and in plane SPP
wavevectors as shown in Fig. 7.4 means that linear excitation of SPPs on planar
surfaces typically requires an effective momentum change of the incident field in
the form of a coupling element or increase in index of refraction, for example a
grating or Kretschmann prism, respectively. However, the phase-matching conditions
associated with the different wavevectors participating in a nonlinear wavemixing
process provide additional flexibility, allowing free-space launching of SPPs via the
generated nonlinear polarization, or SPPs as the source for the nonlinear output field,
or both. While even-order processes are intrinsically surface-confined, interactions
involving SPPs are also in practice limited to the near-surface region given by the skindepth, even for odd-order processes. Any of the participating wavemixing fields can
be an SPP, with a wavevector that is defined by the surface dispersion relation of the
specific SPP frequency. Figure 7.11 shows a set of different possible configurations.
The surface-parallel components of the free-space k-vectors k i = ω i n(ω i ) sin θ i /c,
where θ i is the angle with respect to the surface normal and n(ω i ) is the index
of refraction of the dielectric medium, can then be summed appropriately with the