The Role of a Discontinuous Free-Electron Density in
Harmonic Generation from Metal Surfaces
Michael Scalora
1
, Maria Vincenti
2
, Domenico de Ceglia
3
, Neset Akozbek
4
, Mark Bloemer
1
,
Jose Trull
5
and Crina Cojocaru
5
1
Charles M. Bowden Research Center, AMRDEC, RDECOM, Redstone Arsenal, AL 35898-5000, U.S.A.
2
Department of Information Engineering, University of Brescia, Via Branze 38, 25123 Brescia, Italy
3
National Research Council, AMRDEC, Charles M. Bowden Research Center, Redstone Arsenal, AL 35898-5000, U.S.A.
4
AEgis Technologies Inc., 401 Jan Davis Dr. 35806, Huntsville, AL, U.S.A.
5
Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya,
Rambla Sant Nebridi, 08022 Terrassa, Spain
Keywords: Harmonic Generation, Nonlinear Frequency Conversion, Plasmonics, Electron Cloud, Metal Optics.
Abstract: We discuss a dynamical model of harmonic generation that arises from surfaces that demarcate regions of
discontinuous free electron densities. These circumstances can arise from a simple metal mirror, to more
complex structures such as layered structures composed of different metals or a metal and a conducting
oxide. Using a modified hydrodynamic model we examine the simple case of a metal mirror, assuming the
surface is characterized by an electron cloud that spills out into vacuum and shields the internal portions of
the remaining medium. We also assess the relative importance of additional nonlinear sources that arise
when a free electron discontinuity is present, and show that under the right circumstances both second and
third harmonic generation can be very sensitive to the nature, density and thickness of the free electron
cloud. Our findings suggest the possibility to control surface harmonic generation through surface charge
engineering.
1 INTRODUCTION
The study of second harmonic generation (SHG)
from surfaces has simultaneously captivated and
frustrated researchers since the early days of
nonlinear optics: captivated because it is an ideal
tool to study surfaces; frustrated because while
different theoretical models may yield similar
angular dependence of the generated SH signal,
there is disagreement on the magnitude of the
predicted SH signal, with results sometimes
differing by several orders of magnitude. It may be
said that this apparent, model-dependent
inconsistency is perhaps symptomatic of a
combination of incomplete knowledge of surface
composition and the relative importance of a number
of unfolding physical phenomena. When the
material in question is centrosymmetric, as for noble
metals, or if it generally lacks a bulk second order
nonlinear coefficient, SHG is particularly sensitive
to surface properties such as roughness, sample
thickness, deposition methods, dielectric constant,
and other prominent surface features that may be
accidental or otherwise. For example, the resulting
electron effective mass appears to be somewhat
sensitive to the particular deposition method
employed (Dryzek and Czapla, 1985). Through the
detection of surface plasmon modes, it has also been
shown that a simple metal layer may be denser on
the side of the substrate compared to the air side,
leading to a position dependent dielectric function
(Maaroof et al., 2009) and large discrepancies
between actual and tabulated values. At the same
time, the dielectric “constant” itself may become a
function of both frequency and wave vector via the
excitation of nonlocal effects, e.g. electron gas
pressure (Sipe et al., 1980; Sipe and Stegeman,
1982).
In addition to questions regarding deposition
processes and surface preparation, substantial issues
have been raised about the methods employed in
predicting the electrodynamics that unfolds in
nanoscale systems. Over the years, technological
progress has led to a steady miniaturization
98
Scalora, M., Vincenti, M., Ceglia, D., Akozbek, N., Bloemer, M., Trull, J. and Cojocaru, C.
The Role of a Discontinuous Free-Electron Density in Harmonic Generation from Metal Surfaces.
DOI: 10.5220/0006645800980104
In Proceedings of the 6th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2018), pages 98-104
ISBN: 978-989-758-286-8
Copyright © 2018 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
progression that has produced devices having near-
atomic size, in turn generating questions about the
very applicability of classical electrodynamics at
those length scales. Classical electrodynamics is
based on a process that turns the rapidly fluctuating
microscopic fields found near individual atoms into
macroscopic fields averaged over a volume of space
that may contain myriad of such atoms, or dipoles,
so much so that the medium loses its granularity and
becomes a continuum, necessitating only the mere
application of boundary conditions (Jackson, 1999).
If unaltered, this simplified picture obviously fails if
the macroscopic theory is in turn applied to systems
with features that are only a few atomic diameters in
size. This is already the case for typical nanowire
and/or nanoparticle systems that are now easily
fabricated with features so small and so closely
spaced that the electronic wave functions spilling
outside their respective surfaces may begin to
overlap. The diameter of a typical noble metal atom
is approximately (Fig. 1), while the electronic
cloud forming and shielding a flat, noble metal
surface may extend several Ås into free space.
The study of light interactions with optically
thick metal layers (mirrors) below their plasma
frequencies (long wavelengths), where the dielectric
constant is negative, is limited only to the study of
reflection due to the large negative dielectric
constant and the absence of propagation modes. At
the nanometer scale, transmission through thick
structures that may contain a large total metal
thickness (100nm or more) has been shown to be
possible by exploiting simple cavity/interference
phenomena and surface plasmon excitation (Ebbesen
et al., 1998). The typical, linear optical response of
metals is modelled almost exclusively using the
simple Drude model, which assumes the metal is
essentially a cloud of free electrons that responds
and is driven by an incident electric field. The
resulting dielectric response then depends only on
frequency.
Free electron systems like Indium Tin Oxide
(ITO) or Cadmium Oxide (CdO) are characterized
by absorption that is smaller compared to that of
noble metals, especially in the range where the real
part of the dielectric constant crosses the axis and
takes on near-zero values. The consequences of
these peculiar dispersive properties are important
because they can trigger novel, low-intensity
nonlinear optical phenomena that usually require
high local fields may be of some relevance to
surface harmonic generation and thus shed new light
on the process. The mechanism is triggered by the
requirement that the longitudinal component of the
displacement vector of a TM-polarized field be
continuous, which for homogeneous, flat structures
is exemplified by the relationship:
zz
in in out out
EE

.
()in out
is the dielectric constant inside (outside) the
medium, and
()
z
in out
E
is the corresponding
longitudinal component of the electric field
amplitude inside (outside) the material. It is easy to
see that if
0
in
, it follows that
z
in
E 
.
Additionally, nonlocal effects in these materials can
become more pronounced compared to noble metals,
as both field penetration inside the medium and field
derivatives are correspondingly more prominent,
leading to significant deviations from the predictions
of local electromagnetism. For subnanometer
spacing between metal objects quantum tunnelling
phenomena have to be taken into account.
As a representative example, we start our
discussion with the seemingly simple problem of a
gold film interface as seen from the atomic scale. In
Fig. 1a we depict a typical noble metal atom, which
is characterized by a nearly-free, s-shell electron that
orbits at an approximate distance r
s
~ 1.5Å from the
nucleus, and d-shell electrons whose orbits extend
out approximately r
d
~ 0.5Å from the nucleus. The
simple picture that emerges even from a cursory
look at Fig.1b, which schematically represents atoms
distributed at and just below the surface of a
hypothetical metallic medium, is one of a negatively
charged electron cloud that spills outside the ionic
surface (the dashed line in Fig. 1b) and screens the
inside portions of the metal. Upon further reflection
it becomes obvious that the interior sections of the
medium contain a combination of free and bound
charges, which are schematically shown in Fig. 1c
and are represented as Lorentz oscillators, which
present their own surface to the incoming
electromagnetic wave. Therefore, it becomes
plausible to assume that the reasons for the
discrepancies between experimental results and most
theoretical models, and between theoretical models
themselves, may to some extent reside in the failure
to accurately describe the spatial distribution of the
electron cloud that spills outside the medium’s ionic
surface, to account for all surfaces (free and bound
electrons alike) in and around the transition region
indicated by the dashed line. In addition, nonlinear
optical phenomena due to anharmonic spring
behavior is necessarily confined to the volume
below the dashed line in Fig. 1, i.e. the free electron
gas shields nonlinear third order effects arising from
bound electrons.
The Role of a Discontinuous Free-Electron Density in Harmonic Generation from Metal Surfaces
99
Figure 1: (a) A typical noble metal atom. The radii represent the maximum amplitude of orbital wave functions calculated
using a many-body approach. (b) The dashed line represents a perfectly aligned last row of atoms in a medium that extend
to the left. (c) Schematic representation of bound, d-shell electrons as nonlinear Lorentz oscillators.
Figure 2: Left Panel: depiction of two types of exponential decay of the electron cloud that covers the metal surface. Both
decays yield a mean density approximately 10% of the value at the hard, ionic surface, i.e. the dashed line in Fig. 1. The
main difference between the two density profiles is the spatial extension into vacuum, i.e. -5Å. Right Panel: Once the
average density and spatial extension into vacuum have been chosen, the electromagnetic problem is solved by introducing
two surfaces and an external layer that contains only free charges, and an internal medium that is now screened and contains
both free and bound charges. The bulk, third order nonlinear coefficient is assumed to originate only in bound charges,
which are described as collections of nonlinear Lorentz oscillators.
2 THE MODEL
It can be shown that for real metals the charge
density decays exponentially with distance from the
ionic surface (Lang and Kohn, 1970; Lang and Kohn
1971; Kenner et al., 1972). Therefore, we adopt a
similar model, at first assuming some type of
exponential decay of the free charge density from
the surface (Fig. 2, left panel and relative inset), and
subsequently assigning average value and thickness
to an external charge density that in our modified
vision forms a single, uniform layer composed only
of free charges (see Fig. 2, right panel). Fig. 2 shows
the modified configuration of Fig. 1.
PHOTOPTICS 2018 - 6th International Conference on Photonics, Optics and Laser Technology
100
A gold mirror may thus be thought of as a two-
layer system: a free electron layer between 0.2Å-
0.5Å thick that covers the remaining thickness of
material where one finds a mix of both free and
bound electrons. Of course, a similar free-electron
layer should be considered on the right side of the
mirror, but its effects are negligible for thick layers.
For the moment we assume the bulk, third order
nonlinear coefficient is intrinsic to bound, d-shell
electrons described as nonlinear Lorentz oscillators,
depicted in Fig. 1c, which as indicated above
shielding occurs. However, free electrons are also
capable of generating a third harmonic signal via a
cascaded process. Therefore, the classical
description that we weave together attacks the
problem as a classical boundary value problem
where the composition of individual layers and their
thicknesses are chosen according to the quantization
of atomic orbitals. The local dielectric constant of
the free electron layer is Drude-like and given by:
2
,
( ) 1
pf spillout
spillout
f
i

(1)
is the plasma frequency and
f
is the
related damping coefficient. The local dielectric
constant of the interior bulk section also contains a
Drude portion that describes free electrons, and at
least two Lorentz oscillator contributions (this
allows one to model a more accurate medium
response down to approximately 200nm or so) that
take into account the contribution to the dielectric
constant by d-shell electrons, as follows:
2
,
2
22
12
2 2 2 2
01 01 02 02
( ) 1

pf bulk
bulk
f
pp
i
ii


(2)
1,2p
are the bound electrons’ plasma frequencies
and
01,2
the related damping coefficients. As we
will see below, in addition to bound electrons the
total linear dielectric function is
augmented/modified dynamically by an additional
term, a second order spatial derivative of the free
electron polarization that describes electron gas
pressure in the two relevant regions of space
depicted in Fig. 2. Most of what occurs at the
surface and the evolution of the harmonically
generated signals may, under the right
circumstances, be determined entirely by the density
of the thin, external layer of free charges. The full
dynamical equation of motion that describes
harmonic generation from the free electron gas only,
modified to account for a discontinuous charge
density (i.e. a spatial derivative,) nonlocal effects,
magnetic contributions and convection may be
written as follows:
2
2
0 0 0 0
* * 2 * 2
00
0
*2
0 0 0
1
11













f f f
ff
f f f f
F
f f f
n e e e
c
m m c m c
ne
E
n
n e n
mc
PP
E P E P B
P P P P
P P P
(3)
where
f
P
is the free electron polarization;
E
and
B
are the propagating electric and magnetic fields,
respectively;
*
m
is the free electron’s effective mass;
0
n
is the background charge density with no applied
field;
F
E
is the Fermi energy; c is the speed of light
in vacuum. The equation is scaled with respect to
dimensionless time, longitudinal and transverse
coordinates (2-D),
0
/,ct

0
/,z

0
/,yy
respectively, where
0
1 m

is chosen as a
convenient reference wavelength. The effect of a
discontinuous free charge density between the
external and internal free electron distributions is
described by a new term that appears inside the
bracketed expression on the right hand side, i.e.
00
1/
ff
nn•PP
. Eq.(3) represents a simple
Drude model when
2
2
00
ne
mc



E
is the only driving
term, augmented by a number of linear and
nonlinear source terms as follows: the magnetic
Lorentz force,
*2
0
/
f
e m c
PH
; a Coulomb
term,
*2
0
/
f
e m c
EP
that describes
redistribution of free charges at and near each
boundary, according to the strength of the
derivatives; convective terms,
0
0
1
f f f f f f
n
n







P P P P P P
,
where, as stated earlier, the last term is newly
derived; and a linear electron gas pressure term
proportional to
f
P
that leads to a k-
dependent dielectric constant. Similarly to Eq.(3),
The Role of a Discontinuous Free-Electron Density in Harmonic Generation from Metal Surfaces
101
each species of bound electrons is described by a
nonlinear oscillator equation of the following type:
2
1 01 1 01 1 1 1 1 1
22
01 0 0
1
* 2 * 2
11
bb
b
n e e
m c m c


P P P P P P
E P H
(4)
Bound electrons are characterized by their own
effective mass,
*
1b
m
; resonance frequency,
01
;
density,
01
n
; damping,
01
; and third order nonlinear
spring constant
1
b
, which as the form suggests, is
generally proportional to
(3)
and leads directly to
self-phase modulation and third harmonic
generation. Ultimately, the material equations of
motion yield a polarization that is the vectorial sum
of each contribution, namely
12
...
Total f
P P P P
, which is in turn inserted
into Maxwell’s equations to solve for the dynamics.
3 RESULTS: GOLD MIRROR
In Fig. 3 we show the typical behavior of SH
conversion efficiency vs. incident angle when the
internal and external electron densities are identical
(no density discontinuities), for incident pulses tuned
to three different carrier wavelengths. The
calculations are carried out by simultaneous
integration of Eqs.(3-4) together with Maxwell’s
equations. Each vector equation contains one spatial
component for each coordinate, and one for each
harmonic. We note that curve shape does not change
much for the three cases, except for slight shifts of
the maximum as field penetration changes with
incident wavelength. These shapes are similar to the
shapes predicted by other models (Sipe et al., 1980;
Sipe and Stegeman, 1982).
On the other hand, the predicted maximum
conversion efficiencies can differ dramatically
compared to other models and to experimental
results (Akhmediev et al., 1994; O’Donnell and
Torre, 2005), unless significant adjustments are
made to free and bound effective electrons masses
and densities. Alternatively, since effective masses
and densities are often known precisely and are not
usually taken as free parameters, one may look at the
subtleties of surface composition, i.e. density and
thickness of the outer patina, as discussed above, in
an attempt to reconcile predictions and observations.
In Fig. 4a we show the results of a calculation of
SH conversion efficiencies as a function of the free
electron layer density in the spillout region, for
electron cloud layer thickness of 0.5nm and angle of
incidence fixed at 68°. This angle corresponds to the
approximate location of maximum SH efficiency in
Fig. 3. We also contrast the results with and without
the new, extra convective term that includes the
spatial derivative of the free electron density
highlighted above.
Figure 3: Second Harmonic conversion efficiency vs
incident angle for 100fs pulses incident on a gold mirror,
tuned as indicated in the color-coded caption, as a function
of incident angle. Peak pump intensity is approximately
2GW/cm
2
.
Two distinct peaks appear, at
0.13
spillout bulk
nn
,
and
0.03
spillout bulk
nn
: these values correspond to
( ) 0
in

and
(2 ) 0
in

, respectively, resulting in
an increase in conversion efficiency between four
and six orders of magnitude compared to the case
where there is no spillout region (no density
discontinuity, which is indicated by the solid blue
line,) due to the enhancement of the local fields at
the respective
0
conditions. The impact of the
extra convective nonlinear term in the equations of
motion is also clear, with conversion efficiencies
that can change by as much as two orders of
magnitude. The results summarized in Figs. 4 thus
strongly suggest that variations in surface layer
charge density can have significant consequences on
the predicted outcome. In fact, it is reasonable to
assume that the disposition of surface charges will
be different for every sample fabricated, due to
different conditions that may evolve inside the
fabrication chamber, possibly due to local
temperature and pressure fluctuations that lead to
accidental roughness and thus, different effective
local parameters. The issue of THG from the gold
mirror is also as interesting and perhaps as subtle as
SHG. As mentioned earlier, the outer free electron
patina is not assumed to possess an intrinsic third
PHOTOPTICS 2018 - 6th International Conference on Photonics, Optics and Laser Technology
102
Figure 4: (a) Reflected second harmonic conversion efficiency for 50fs pulses incident at 68° on the gold mirror shown in
Fig. 2, which contains an external free-electron-only layer of variable density. (b) Comparison of the results in (a) with a
different angle of incidence and thinner outer free electron layer. The outer electron density drives the interaction, not the
thickness.
order coefficient, b (proportional to
(3)
), but it can
nevertheless trigger THG via a frequency mixing
that evolves from explicit nonlinear sources present
on the right side of Eq.(3), namely convection,
Coulomb, and magnetic Lorentz components.
Normally, however, the presence of a bulk b
anywhere inside the structure overwhelms these
contributions, so much so that surface generated (i.e.
free electron) THG can always be neglected. Here,
we report that surface generated THG can in fact
overwhelm the TH signal produced by the bulk,
provided the right conditions are met: for example, if
the density of the outer layer is such that the
condition
( ) 0
in

is met for the pump field. To
demonstrate this we perform calculations in both the
presence and in the absence of a bulk
(3)
, which in
our specific case means a non-zero b coefficient in
Eq. 4, as a function of outer layer density, and report
the results in Fig. 5. When the bulk nonlinearity is
present (
0b
), the reflected TH signal is predicted
to displays a single peak, which occurs near the
( ) 0
in

condition. Removing the bulk nonlinearity
(
0b
everywhere) reveals two additional, surface
generated TH peaks of much lower amplitude,
whose locations correspond to the two remaining
conditions that tend to maximize TH conversion
efficiency:
(2 ) 0
in

and
(3 ) 0
in

, which as
mentioned earlier, produce amplified local fields.
The fact that the maximum produced by the
( ) 0
in

condition persists even after b is set equal
to zero signifies that this particular TH peak is
independent of
b
, and is solely due to surface
phenomena, while the remaining two peaks are
easily overwhelmed. While satisfying the
( ) 0
in

at the pump field appears to yield the most efficient
conversion efficiency, designing a surface charge
having those characteristics is probably an arduous
process, given the much reduced surface charge
density required.
Figure 5: Reflected third harmonic conversion efficiency
for 50fs pulses incident at 45° on the gold mirror shown in
Fig. 2 as a function of free-electron-only layer of variable
density, with and without a bulk nonlinearity. When the
condition
( ) 0
in

is satisfied for the outer free electron
gas layer, THG displays a maximum in either case. This is
an indication that surface THG dominates even in the
presence of a non-zero bulk
b
coefficient.
The Role of a Discontinuous Free-Electron Density in Harmonic Generation from Metal Surfaces
103
4 CONCLUSIONS
We have shown that surface harmonic generation
from a simple gold mirror can be a very sensitive
function of the parameters that characterize the
external free electron cloud that spills outside the
ionic surface and screens the internal portions of the
medium. In addition our calculations suggest that a
discontinuous free electron gas density can introduce
nonlinear, convective sources that can significantly
increase/modify nonlinear conversion efficiencies.
These results demonstrate that it may well be
possible to engineer a surface charge and actively
control it through charge injection to maximize
harmonic generation for a number of technological
applications.
REFERENCES
Akhmediev, N. N., Mel’nikov, I. V. and Robur, L. J.
(1994). Second-Harmonic Generation by a Reflecting
Metal Surface", Laser Physics 4(6), p.1194; O’Donnell
K. A. and Torre, R. (2005). Characterization of the
second-harmonic response of a silverair interface",
New Journal of Physics 7(1), p.154.
Dryzek, J. and Czapla, A. (1985). Free-electron
parameters of sputtered noble metal films. Journal of
Materials Science Letter, 4 p.154.
Ebbesen, T. W., Lezec, H. J., Ghaemi, H. F., T. Thio, T.
and Wolff P. A., (1998). Extraordinary
optical transmission through sub-wavelength hole
arrays. Nature, 391(6668), p.667.
Jackson, J. D. (1999). The Classical Electromagnetic
Field (Wiley, New York).
Lang, N. D. and Kohn, W. (1970). Theory of Metal
Surfaces: Charge Density and Surface Energy,
Physical Review B, 1(12), p. 4555; Lang, N. D. and
Kohn, W. (1971). Theory of Metal Surfaces: Work
Function. Physical review B, 3(4), p.1215; Kenner, V.
E., Allen, R. E. and Saslow, W. M. (1972). Screening
of external fields and distribution of excess charge
near a metal surface. Physics Letters A, 38(4) p.255.
Maaroof, A.I., Gentle, A., Smith, G. B. and Cortie, M. B.
(2009). Bulk and surface plasmons in highly
nanoporous gold films. Journal of physics D: Applied
Physics, 40 (18), p.5675.
Sipe, J. E., So, V. C. Y., Fukui, M. and Stegeman, G. I.
(1980). Analysis of second-harmonic generation at
metal surfaces, Physical Review B, 21(10) p 4389;
Sipe, J. E. and Stegeman, G. I. (1982). In Surface
Polaritons: Electromagnetic Waves at Surfaces and
Interfaces, edited by V. M. Agranovich and D. Mills
North-Holland, Amsterdam.
PHOTOPTICS 2018 - 6th International Conference on Photonics, Optics and Laser Technology
104