PHYSICAL REVIEW A 101, 053828 (2020)
Electrodynamics of conductive oxides: Intensity-dependent anisotropy, reconstruction
of the effective dielectric constant, and harmonic generation
Michael Scalora,
1
Jose Trull ,
2
Domenico de Ceglia,
3
Maria Antonietta Vincenti,
4
Neset Akozbek,
5
Zachary Coppens,
5
Laura Rodríguez-Suné,
2
and Crina Cojocaru
2
1
Charles M. Bowden Research Center, CCDC AVMC, Redstone Arsenal, Alabama 35898-5000, USA
2
Department of Physics, Universitat Politècnica de Catalunya, 08222 Terrassa, Spain
3
Department of Information Engineering, University of Padova, Via Gradenigo 6/a, 35131 Padova, Italy
4
Department of Information Engineering, University of Brescia, Via Branze 38, 25123 Brescia, Italy
5
AEgis Technologies Inc., 401 Jan Davis Dr., Huntsville, Alabama 35806, USA
(Received 13 January 2020; accepted 31 March 2020; published 13 May 2020)
We study electromagnetic pulse propagation in an indium tin oxide nanolayer in the linear and nonlinear
regimes. We use the constitutive relations to reconstruct the effective dielectric constant of the medium, and show
that nonlocal effects induce additional absorption resonances and anisotropic dielectric response: longitudinal
and transverse effective dielectric functions are modulated differently along the propagation direction, and
display different epsilon-near-zero crossing points with a discrepancy that increases with increasing intensity.
We predict that hot carriers induce a dynamic redshift of the plasma frequency and a corresponding translation
of the effective nonlinear dispersion curves that can be used to predict and quantify nonlinear refractive index
changes as a function of incident laser peak power density. Our results suggest that large, nonlinear refractive
index changes can occur without the need for epsilon-near-zero modes to couple with plasmonic resonators.
At sufficiently large laser-pulse intensities, we predict the onset of optical bistability, while the presence of
additional pump absorption resonances that arise from longitudinal oscillations of the free electron gas give way
to corresponding resonances in the second and third harmonic spectra. A realistic propagation model is key to
unraveling the basic physical mechanisms that play a fundamental role in the dynamics.
DOI: 10.1103/PhysRevA.101.053828
I. INTRODUCTION
Typical plasmonic resonators consist of metallic nanopar-
ticles or nanostructures where free electrons oscillate in res-
onance with light. These resonances can produce strong field
amplification and enhanced scattering (absorption) cross sec-
tions, which are key properties for applications in sensing,
detection, energy harvesting, and generic light manipulation
at the nanoscale. However, metals can be either too absorptive
or inadequate in a given wavelength range, and alternative
replacements must be sought. In this work we explore linear
and nonlinear propagation effects that manifest themselves but
are not limited to free-electron systems that may display an
epsilon-near-zero (or ENZ) crossing of the real part of the
dielectric constant. In particular, we study the basic properties
of simple layers composed of degenerate semiconductors like
indium tin oxide (ITO) only a few tens of nanometers in
thickness in order to ascertain basic physical characteristics
that may transfer to more complicated nanostructured geome-
tries. Generally, free-electron systems are centrosymmetric
and are described by a simplistic Drude model. However,
experiments show that in reality these materials possess a
combined Lorentz-Drude-like dielectric response [1] that can
be tuned by controlling doping levels and annealing tem-
peratures. This dual material aspect simultaneously compli-
cates and enriches the dynamics, whose understanding and
description thus require theoretical models that are more com-
prehensive than what may be required in ordinary photonic
structures.
In contrast to noble metals, conducting oxides display
lower losses and may thus substitute or even supplant metals
in certain applications and spectral wavelength ranges. To
date, many aspects related to pulse propagation phenomena in
free-electron systems like noble metals or conducting oxides
remain incomplete. In what follows we describe a model that
simultaneously accounts for: (i) the intrinsic nonlinearities of
background bound charges; (ii) nonlocal effects (pressure and
viscosity of the electron gas); (iii) pump depletion; (iv) the
dynamics that ensue from including an intrinsic, temperature-
dependent effective mass (in the case of conducting oxides)
or free-charge density (in the case of noble metals or semi-
conductors) and related nonlinearities that ultimately manifest
themselves in the form of effective χ
(3)
, χ
(5)
, and higher-
order nonlinear contributions; and (v) surface and magnetic
nonlinearities that are almost always neglected in favor of bulk
nonlinearities. As an example of this theoretical deficiency in
conducting oxides, and to some extent in metals and semicon-
ductors, the nature and magnitude of nonlinear index of re-
fraction changes as a function of incident pump intensity have
not yet been clarified [2,3]. Differing explanations have been
provided regarding the source of third-order phenomena [3,4],
and practically no good insight into second-order, surface, and
magnetic phenomena outside of the context found in Refs. [1]
and [5]. In Ref. [3] third-order phenomena responsible for
nonlinear index changes were attributed to the free-electron
cloud. In Ref. [4] third-harmonic generation (THG) was at-
tributed exclusively to the background crystal. In Ref. [6],
2469-9926/2020/101(5)/053828(11) 053828-1 ©2020 American Physical Society
MICHAEL SCALORA et al. PHYSICAL REVIEW A 101, 053828 (2020)
THG from an ITO nanolayer was studied theoretically and
experimentally using a generic, dispersionless χ
(3)
having
no specified origin. Finally, in Ref. [7], the simultaneous
generation of negatively refracted and phase conjugate beams
was experimentally recorded from a structure consisting of
gold nanoantennas patterned on top of a 40-nm-thick ITO
layer displaying an ENZ crossing point. However, the the-
oretical effort tackled only generic aspects of a third-order
nonlinearity present only in the ITO layer, and no detailed
field dynamics. In summary, the picture that emerges from
the detailed microscopic model discussed below is somewhat
more complicated than it would appear in Refs. [3,4,7].
In a recent paper [1] experimental and theoretical results
on second and third-harmonic generation (SHG and THG)
were reported near the ENZ condition of an ITO nanolayer,
which manifested itself near 1240 nm. The pulse propagation
model that was used comprised a hydrodynamic descrip-
tion of the material equations that takes into account free
and bound charges, nonlocal effects, a time-dependent free-
electron plasma frequency, surface, magnetic, and convective
second and third-order nonlinearities, as well as the inclu-
sion of linear and nonlinear contributions of the background
medium to the dielectric constant. A direct comparison of the
SHG spectra and the angular dependence of SH conversion ef-
ficiencies showed good qualitative and quantitative agreement
with experimental results. Good qualitative and quantitative
agreement was also found for the angular dependence of third-
harmonic generation for incident laser pulse power densities
in the 1 GW/cm
2
range. In our present effort we provide
further details about the model by: (i) expanding the range of
investigation well into the IR range; (ii) examining the linear
regime in order to ascertain the multifaceted contributions of
nonlocal effects; and (iii) extending our predictions into the
high-intensity regime in an attempt to distinguish between
bound and free (hot) electron contributions.
The local dielectric constant of any material may be ex-
pressed as a superposition of Lorentz and Drude oscillators,
which in the simplest case of two polarization species (one
free and one bound electron contribution, as in the case of
ITO [1]) may be written as follows:
ε
ITO
(ω) = 1 −
ω
2
p
ω
2
+ iγ
f
ω
−
ω
2
p,b
ω
2
− ω
2
0,b
+ iγ
b
ω
, (1)
where ω
2
p
=
4πn
0, f
e
2
m
∗
f
is the free-electron plasma frequency;
n
0, f
the free-electron density; e the electronic charge; γ
f
the free-electron damping coefficient; m
∗
f
the effective free-
electron mass; ω
p,b
is the bound electron plasma frequency,
defined similarly to the free-electron counterpart to which
there corresponds a bound electron density n
b
and mass m
∗
b
;
ω
0,b
is the resonance frequency; and γ
b
the bound electron
damping coefficient.
If the free-electron effective mass changes approxi-
mately linearly with temperature, as shown below using the
two-temperature model, the equations of motion are repro-
duced from Ref. [1], and may be written as follows:
¨
P
f
+ ˜γ
f
˙
P
f
=
n
0, f
e
2
λ
2
0
m
∗
0
c
2
E −
eλ
0
m
∗
0
c
2
(∇ · P
f
)E +
⎛
⎝
l=1,2,3
(−
˜
)
l
(E · E)
l
⎞
⎠
E +
eλ
0
m
∗
0
c
2
˙
P
f
× H
+
3E
F
5m
∗
0
c
2
∇(∇ ·P
f
) +
1
2
∇
2
P
f
−
1
n
0, f
eλ
0
[(∇ ·P
f
)
˙
P
f
+ (
˙
P
f
· ∇ )
˙
P
f
], (2)
¨
P
b
+ ˜γ
b
˙
P
b
+ ˜ω
2
0,b
P
b
+ P
b,NL
=
n
0,b
e
2
λ
2
0
m
∗
b
c
2
E +
eλ
0
m
∗
b
c
2
(P
b
· ∇ )E +
eλ
0
m
∗
b
c
2
˙
P
b
× H. (3)
Time and space have been scaled such that temporal and
spatial derivatives are carried out with respect to the fol-
lowing coordinates: ς = y/λ
0
, ξ = z/λ
0
, and τ = ct /λ
0
,
where in our case λ
0
= 1 μm is a convenient reference
wavelength. It follows that the coefficients are also scaled:
˜γ
f ,b
= γ
f ,b
λ
0
/c,˜ω
2
0,b
= ω
2
0,b
λ
2
0
/c
2
. Equation (2) describes
the free-electron polarization, P
f
;
n
0, f
e
2
λ
2
0
m
∗
0
c
2
E −
eλ
0
m
∗
0
c
2
E(∇ · P
f
)
are Coulomb terms generated by the continuity equation;
(
l=1,2,3
(−
˜
)
l
(E · E)
l
)E, where
˜
is a constant of pro-
portionality, follows from the expansion of the effective
mass as a function of temperature and absorption [1,8]
(as the summation index indicates, looking ahead to our
results our parameter choices demand we retain hot elec-
tron nonlinearities up to seventh order);
eλ
0
m
∗
0
c
2
˙
P
f
× H arises
from the magnetic Lorentz force;
3E
F
5m
∗
0
c
2
(∇(∇ · P
f
) +
1
2
∇
2
P
f
)
represent pressure and viscosity, respectively, where E
F
=
¯h
2
2m
∗
0
(3π
2
n
0, f
)
2/3
is the Fermi energy and we have ne-
glected damping terms that tend to broaden absorption res-
onances [9], ending with the first-order convective contribu-
tion
1
n
0, f
eλ
0
[(∇ ·
˙
P
f
)
˙
P
f
+ (
˙
P
f
· ∇ )
˙
P
f
]. Equation (3) in turn de-
scribes the dynamics of bound electrons; P
b
is the bound elec-
trons’ polarization; P
b,NL
= ˜αP
b
P
b
−
˜
β(P
b
· P
b
)P
b
+··· is
the bound electron’s nonlinear polarization, depicted here up
to third order;
n
0,b
e
2
λ
2
0
m
∗
b
c
2
E +
eλ
0
m
∗
b
c
2
(P
b
· ∇ )E are Coulomb terms,
followed by the magnetic Lorentz term,
eλ
0
m
∗
b
c
2
˙
P
b
× H.The
coefficients ˜α and
˜
β are tensors that reflect crystal symmetry.
In what follows we assume ITO is centrosymmetric ( ˜α = 0)
and isotropic, so
˜
β is a constant. Equations (2) and (3)are
integrated together with the vector Maxwell equations, where
the total polarization is expressed as the vector sum of all
polarization components, in this case P
Total
= P
f
+ P
b
.Itis
important to point out that in the sections that follow we never
specify either the dielectric constant or the index of refraction.
The only reason we even contemplate writing Eq. (1)isto
illustrate the point that the ellipsometric data extracted from
any sample may be expressed in that form. However, the
reason we use Eq. (1) to fit the linear dielectric constant data
053828-2
ELECTRODYNAMICS OF CONDUCTIVE OXIDES: … PHYSICAL REVIEW A 101, 053828 (2020)
FIG. 1. (a) Calculated local and nonlocal pump absorption spectra at 60° angle of incidence. The vertical arrows denote the locations of
additional resonances that are triggered by longitudinal oscillations of the electron gas. The horizontal arrow represents a blueshift of the
resonance triggered by a changing, effective dielectric constant. (b) Absorption calculated in two ways: “full nonlocal,” corresponding to the
curve in (a), and by using a hybrid approach consisting of using the fields calculated from the full model, and the local dielectric constant of
Eq. (1). This comparison illustrates that total absorption must be calculated by including the additional piece of dielectric constant triggered
by nonlocal effects, a quantity that depends on the spatial derivatives of the polarization, i.e., local charge density. Left inset: geometry of the
interaction.
in the local approximation is for the sole purpose of extracting
the necessary effective electron masses, densities, resonance
frequencies, and damping coefficients that are required in
order to integrate Eqs. (2) and (3) in the time domain in the
nonlocal and nonlinear regime. Once this set of coefficients is
established for a given sample, the effective dielectric constant
may then be faithfully reproduced yielding curves that match
Eq. (1) in the linear and local regime, while predictions can be
made in the nonlocal and nonlinear regime, as we will see in
Sec. II B.
We conclude this section with a word about the equations
of motion and the method of integration. Equation (2)was
first derived in the context of harmonic generation from metal
surfaces [10], without hot electron or viscosity contributions
[see Eq. (10) below] while Eq. (3) has been discussed in the
context of harmonic generation from semiconductor surfaces
and nanowire arrays (Ref. [11], and references therein). Both
Eqs. (2) and (3) generally separate into three equations for
complex envelope functions, each representing a harmonic
field. The equations are then solved together with Maxwell’s
equations in the time domain in a two-dimensional spatial
grid using a split-step, fast Fourier transform (FFT) pulse
propagation method described in detail in the references,
using
FORTRAN programming code. The calculation of spa-
tial derivatives is accurate to all orders using spectral meth-
ods, while using finite differences computational accuracy
may extend only up to second order. Unlike finite-difference
methods, the FFT-based, split-step method is unconditionally
stable, a crucial point that avoids possible phase errors that
may occur if, for example, the free-space causality condition
δz = cδt is not preserved. For planar structures and incident
plane waves, the computational grid (8×100 000) can be
reduced drastically across the transverse coordinate (8 points),
while for accuracy at large angles the entire pulse should
be contained within the longitudinal grid (100 000 points).
Ultimately, a single pulse propagation event is carried out
using identical scaled time and spatial steps (to avoid phase er-
rors) so that δτ =δξ =10
−3
. Since the FFT approach implies
periodic boundary conditions, and allows for inhomogeneities
between the longitudinal and transverse steps sizes, the trans-
verse spatial step is chosen to be δς = 0.1 to maximize the
distance between transverse edge points on the grid. These
scaled quantities correspond to δt ≈ 3.3×10
−18
s
, δz ≈ 1nm,
and δy ≈ 100 nm. Execution times can last several hours on
an ordinary desktop computer, depending on the angle of
incidence.
II. NONLOCAL EFFECTS
A. Linear absorption
At low incident power densities, in a two-dimensional
geometry (invariant in the x direction; see Fig. 1 for an eluci-
dation of the geometry) only linear, nonlocal effects survive.
The free-electron component Eq. (2) may then be rewritten as
follows:
¨
P
f
+ ˜γ
f
˙
P
f
=
n
0, f
e
2
λ
2
0
m
∗
0
c
2
E +
3E
F
5m
∗
0
c
2
∂
∂y
ˆ
j +
∂
∂z
ˆ
k
∂P
y
∂y
+
∂P
z
∂z
+
1
2
∂
2
∂y
2
+
∂
2
∂z
2
(P
y
ˆ
j + P
z
ˆ
k)
, (4)
where
ˆ
j and
ˆ
k are unit vectors along y and z, respectively.
We continue to assume appropriately scaled Cartesian coordi-
nates, but for clarity we have retained the usual notation. With
the spatial derivatives such that
∂
∂y
→ i
˜
k
y
and
∂
∂z
→ i
˜
k
z
, after
direct Fourier transformation of Eq. (4) and upon separation
of the polarization’s vector components, we may write:
˜
P
y
=
n
0, f
e
2
λ
2
0
/(m
∗
0
c
2
)
˜
E
y
− η
˜
k
y
˜
k
z
˜
P
z
−˜ω
2
− i ˜γ
f
˜ω +
3
2
η
˜
k
2
y
+
η
2
˜
k
2
z
˜
P
z
=
n
0, f
e
2
λ
2
0
/(m
∗
0
c
2
)
˜
E
z
− η
˜
k
y
˜
k
z
˜
P
y
−˜ω
2
− i ˜γ
f
˜ω +
3
2
η
˜
k
2
z
+
η
2
˜
k
2
y
(5)
η =
3E
F
5m
∗
0
c
2
˜ω =
ω
ω
0
˜
k
y,z
= λ
0
k
y,z
.
053828-3
MICHAEL SCALORA et al. PHYSICAL REVIEW A 101, 053828 (2020)
For typical noble metals and conductive oxides, η ≈ 10
−5
.
For planar structures, each plane wave represented in Eqs. (5)
refracts at an angle dictated by the magnitudes of
˜
k
y
and
˜
k
z
.
Equations (5) may be solved and put into the usual form:
˜
P
y
˜
P
z
=
χ
yy
χ
yz
χ
zy
χ
zz
˜
E
y
˜
E
z
. While the off-diagonal elements are gen-
erally nonzero, for uniform layers they tend to perturb the
system. Therefore, for the purposes of our discussion, off-
diagonal elements will be neglected, in view of the relatively
small magnitude of η. The form of Eqs. (5) thus demon-
strates that even if the medium is assumed to be isotropic via
Eq. (1), nonlocal effects intervene by introducing an intrinsic
anisotropy [12] that affects propagation and eventually non-
linear interactions at all angles of incidence, and as we will
see below, by triggering dramatic modulation of the transverse
dielectric constant. Below we examine both consequences in
some detail.
Modifications of the dielectric constant due to nonlocal
effects are usually understood and described almost exclu-
sively in terms of a blueshift of the main plasmonic resonance,
in this case centered near the ENZ wavelength, and by the
generation of additional absorption resonances that can be
correlated directly to longitudinal, resonant oscillations of
the free-electron gas prompted by radiation pressure [9]. In
Fig. 1(a) we depict linear pump absorption spectra for 100-fs,
p-polarized pulses incident at a 60° angle on a 20-nm-
thick ITO film suspended in vacuum, for local and nonlocal
regimes. For the pump field, nonlocal effects manifest them-
selves primarily with the aforementioned blueshifted main
peak (horizontal arrow) and additional absorption resonances,
highlighted by the perpendicular arrows near 700 and 900 nm.
In general, absorption cannot be calculated analytically due
to the presence of dynamic pressure and viscosity terms.
In Fig. 1(a) we calculate absorption as the total scattered
(transmitted and reflected) pump energy subtracted from the
total energy contained in the incident pulse. This approach is
exact, since it is based on energy conservation.
An aspect that is often overlooked, but is nevertheless
associated with modifications of the dielectric constant, is
depicted in Fig. 1(b), where we compare the total, nonlocal
absorption shown in Fig. 1(a) with the absorption calculated
using the standard Poynting theorem, but by using the local
dielectric constant. The discrepancy between the curves is
obvious in both amplitude and the near absence of additional
absorption peaks, and is due to the fact that the imaginary
parts of the effective susceptibilities derived from Eqs. (5)
are modified in nontrivial ways. Figure 1(b) thus strongly
suggests that care should be exercised when either linear
or nonlinear absorption are being considered and evaluated
anytime nonlocal effects are relevant. Finally, we note that
peak locations and amplitudes in Fig. 1 depend on incident
angle (see Fig. 11 below).
B. Induced anisotropy and reconstruction of linear
and nonlinear effective dielectric constants
We now wish to discuss a method that allows extrac-
tion of the approximate, effective linear and/or nonlinear
responses of the medium under consideration, and to evaluate
the intrinsic anisotropy suggested by Eqs. (5) in the general
FIG. 2. Longitudinal and transverse local dielectric constants
retrieved using Eqs. (6)and(7), as indicated by the labels. The
dashed curves represent our measured data purposely fitted using a
local Drude-Lorentz model. The equations reproduce the data quite
well over the entire range, including both Drude and Lorentz regions.
In this local case the dielectric constant is isotropic.
case of oblique incidence. Although the dielectric constants
expressed in Eqs. (1)or(5) are never explicitly specified or
introduced, they may be recovered by integrating the system
comprising Eqs. (2) and (3) and Maxwell’s equation in the
time domain, and by exploiting the macroscopic constitutive
relations. For instance, following the development that leads
to Eqs. (5), assuming that the off-diagonal elements continue
to be negligible, for a nearly monochromatic incident field we
may write approximate expressions for the total polarizations:
P
y
≈ χ
yy
E
y
, and P
z
≈ χ
zz
E
z
.Itfollowsthat
ε
yy
≈ 1 +4πP
y
/E
y
, (6)
and
ε
zz
≈ 1 +4πP
z
/E
z
. (7)
It is understood that fields and polarizations are functions
of position, so that both ε
yy
and ε
zz
in Eqs. (6) and (7)are
spatially modulated by the ratio of the fields. These relations
hold in both linear and nonlinear regimes, conditional on
near monochromaticity of the incident pulse. For practical
purposes, a field may be said to be nearly monochromatic if
its spatial extension is much larger than the structure under
study, and if there are no sharp spectral features that may
either span the bandwidth of the incident pulse, or that more
generally may intrude in the spectral region of interest. Both
conditions are satisfied for ordinary dispersive systems like a
20-nm-thick ITO layer being illuminated by pulses that have
a spatial extension in excess of 30 µm(∼100 fs in duration).
In Fig. 2 we plot the complex dielectric function re-
trieved experimentally via spectroscopic ellipsometry (Wool-
lam, VASE 250–1700 nm) at multiple angles of incidence
(60°–70°) for the 20-nm ITO layer grown on both fused silica
and silicon substrates investigated in Ref. [1], purposely fitted
053828-4
ELECTRODYNAMICS OF CONDUCTIVE OXIDES: … PHYSICAL REVIEW A 101, 053828 (2020)
0
1
2
3
0
0.01 0.02
Im[ε
zz
(z)]
Re[ε
zz
(z)]
Position (μm)
Re[ε
zz
(z)], Im[ε
zz
(z)]
-2.5
0
2.5
5.0
0.005 0.010 0.015
0.020
Im[ε
yy
(z)]
Re[ε
yy
(z)]
Position (μm)
Re[ε
yy
(z)], Im[ε
yy
(z)]
vacuum
vacuum
(b)
0.005 0.0
10 0.015 0.020
Position (μm)
vacuum
vacuum
(a)
FIG. 3. (a) Complex longitudinal and (b) transverse dielectric constants calculated using Eqs. (6)and(7). The angle of incidence is 60°.
In addition to conspicuous edge effects clearly visible in (a), the imaginary component of the transverse dielectric constant takes on negative
values, an indication of transient, local gain, quickly offset by local losses. The alternating sign of the effective imaginary dielectric constant
is more likely an indication that local currents alternate sign within a distance of a Fermi wavelength.
with the local, isotropic permittivity model of Eq. (1), in
order to test our propagation model as outlined above. Using
a Lorentz-Drude model ensures that the retrieved dielectric
constant is consistent with the Kramers-Kronig relations. Also
shown in Fig. 2 are the effective, complex dielectric constants
retrieved by integrating Eqs. (2) and (3) in the time domain,
followed by calculating the dielectric constants defined in
Eqs. (6) and (7), at low power densities (1 MW/cm
2
) and in
the local approximation (no pressure and viscosity terms.) The
dielectric functions are evaluated when the peak of the pulse
reaches the ITO layer, as denoted by the labels.
For planar structures and arbitrary angle of incidence,
the fields are uniform along the transverse coordinate, and
so it suffices to perform an average along the longitudinal
coordinate: ε
yy,zz
(λ)=
1
L
L
0
ε
yy,zz
(λ, z)dz, where L is
layer thickness. This procedure yields effective parameters
and is equivalent to implementing a kind of numerical
dielectric constant retrieval method on the sample. The results
depicted in Fig. 2 show excellent agreement between the
experimentally retrieved data and our theoretical predictions,
a fact that engenders confidence in our theoretical framework.
Based on the results shown in Fig. 2, one may also
conclude that medium response is local and isotropic,
i.e.,ε
yy
(λ)=ε
zz
(λ), as expected, notwithstanding the
presence of the ENZ crossing point.
The introduction of nonlocal effects causes ε
yy
and ε
zz
to
display unusual spatial inhomogeneities (Fig. 3), while the
effective dispersions ε
yy
(λ) and ε
zz
(λ) exhibit discordant
ENZ crossing points (Fig. 4). Mindful of our assumptions
above, in Fig. 3 we plot the complex dielectric constants
as functions of position via Eqs. (6) and (7), inside and
just outside the medium, for the propagation snapshot that
corresponds to the peak of the pulse reaching the ITO layer.
The carrier wavelength of the pulse is ∼1230 nm, and the
incident angle is 60°. Besides edge effects, in Fig. 3(a)
Re[ε
zz
(z)] displays the expected drop to near-zero values
inside the medium. Perhaps surprisingly at first, however, in
Fig. 3(b) the complex ε
yy
(z) exhibits previously unreported,
quite dramatic oscillatory behavior with periodicity of only a
few nanometers.
Thicker layers exhibit similar oscillations near entry and
exit surfaces, accurately reflecting the facts that pressure
and viscosity are felt mostly near interfaces, and that their
neglect comes at the risk of inaccurate depictions of both
edge effects and boundary conditions. Even in a cursory
examination of Fig. 3(b) one cannot avoid ascertaining that
locally it is possible for Im[ε
yy
(z)] to be negative, which
ordinarily might suggest rapid, local gain, offset by equally
rapid, local loss. However, another, perhaps more physically
meaningful way to view these rapid oscillations is to note
that since we are dealing with mostly free electrons, non-
local effects induce currents that alternate direction inside
the layer on the scale of the Fermi wavelength, as predicted
and reported for a cadmium oxide layer [9]. In general, the
connection between conductivity and dielectric constant is
easily established, and may be quantified as follows: σ
yy
=
−iω
0
ε
yy
−1
4π
=
ω
0
4π
{Im[ε
yy
] − i(Re[ε
yy
] − 1)}, and similarly for
σ
zz
. The sign of the imaginary part thus governs the direction
of local current flow. Whether or not these oscillations can
ultimately be measured, possibly by probing the layer with
a soft x-ray beam, is a fact presently not easily determined.
However, from an effective medium standpoint, i.e., ellipsom-
etry, their overall significance may be dismissed just as one
might dismiss the significance of a phase velocity that exceeds
the speed of light, an ordinary occurrence in metals. The fact
is that from an effective medium standpoint, the averages
Im[ε
yy,zz
(λ)]=
1
L
L
0
Im[ε
yy,zz
(λ, z)] dz are greater than zero
in all cases we have investigated.
For illustration purposes, in Fig. 4(a) we plot only the
magnitudes of the total, local, and nonlocal effective dielec-
053828-5
![FIG. 11. (a) Pump absorption and (b) transmitted THG spectra for three different incident angles, as indicated by the labels. Reflected THG displays similar behavior. The sensitivity to incident angle is particularly obvious in the THG spectra, where both resonant (60°) and antiresonant (30°) behavior can be excited. The minimum predicted at 30° has been experimentally observed and reported in Ref. [1].](/figures/fig-11-a-pump-absorption-and-b-transmitted-thg-spectra-for-d1uz7vsr.webp)
