# On the transmittance of metallic superlattices in the optical regime and the
true refraction angle.

Pedro Pereyra  Ciencias Básicas, UAM-Azcapotzalco, México D.F., México

###### Abstract

Transmission of electromagnetic fields through $(dielectric/metallic)^{n}$
superlattices, for frequencies below the plasma frequency $\omega_{p}$, is a
subtle and important topic that is reviewed and further developed here.
Recently, an approach for metallic superlattices based on the finite periodic
systems theory was introduced Pereyra2020 . Unlike most, if not all, of the
published approaches that are valid in the $n\rightarrow\infty$ limit, the
finite periodic approach is valid for any natural number $n$, and allows one
to determine analytical expressions for scattering amplitudes and dispersion
relations. It was shown, for frequencies below $\omega_{p}$ and large
metallic-layer thickness, that under the common assumption that fields inside
conductors move along the so-called ”true” angle that defines the orientation
of the constant-phase planes, anomalous results appear with an apparent parity
effect. This issue is addressed here and it is shown that those results are
due to the lack of unitarity and the underlying phenomena of absorption and
loss of energy. Two compatible approaches are presented here to solve the lack
of unitarity and to account for the absorption phenomenon. We show that by
keeping the complex angles, the principle of flux conservation is fully
satisfied, above and below $\omega_{p}$. The results above $\omega_{p}$ remain
the same as in Ref. [Pereyra2020, ]. This approach, free of assumptions, where
all the information of the scattering process is preserved, gives us light to
improve the formalism when the real angle assumption is made. In fact, we show
that by taking into account the induced currents and the requirement of flux
conservation, we end up with an improved approach, with new Fresnel and
transmission coefficients, fully compatible with those of the complex-angle
approach. The improved approach allows one also to evaluate the magnitude of
the induced currents and the absorbed energy, as functions of the frequency
and the superlattice parameters. We determine the plasmonic resonant
frequencies, and present preliminary results of the metallic superlattices
response to electromagnetic pulses and wave packets, particularly, in the
optical domain. We calculate the reflection and transmission coefficients as
well as the phase time $\tau(\omega)$. We show that the predicted space-time
positions agree extremely well with the actual positions of the wave-packet
centroids.

Keywords: Transmittance of Electromagnetic Fields; Metallic Superlattices;
Plasmon Resonances in Optical Regime; Induced Currents and Absorption; Phase
Time in Metallic Superlattices.

##  I Introduction

The interest in the response of metallic structures to electromagnetic fields
(EMFs), has grown as the possibilities of application of their properties
increase. The research activity evolved along different trails, determined, by
the dimension, shape, size and order of the metallic structure. The scattering
of light by small metallic particles, or cylindrical and rectangular rods,
require different approaches than the scattering by layered metallic
structures. The geometrical differences of the scatterer systems imply,
naturally, the use of different mathematical tools; For example, scattering
matrices $S$, in lower dimensional cases, and the transfer matrices $M$, for
layered systems.

The ancient and primordial questions on the colors of the rainbow and why the
sky is blue challenged the minds of all ages; Natural philosophers dating back
to the Greeks (like Aristotle and Ptolemy), Arab physicists (like Al Kindī and
Ibn Al-Haytham), and recently mighty minds like Da Vinci and Newton, explained
these enigmatic phenomena, one way or another, in terms of the interaction of
light and tiny particles in the atmosphere.Aristotle ; Ptolemy ; AlKindi ;
AlHazen ; DaVinci ; Newton  The development of electromagnetic theory led, as
early as the 19th century, to the formulation of strikingly elegant, and
rigorous theories on the scattering of light by small particles. In this
endeavor, the seminal and influential articles by Lorenz, Rayleigh, Mie, and
Debye stand out.Lorenz1880 ; Rayleigh ; Maxwell-Garnett ; Mie ; Debye  The
interest in the scattering of light by small particles grew rapidly after the
World War II, when applied science and engineering began to produce small
particles with various shapes for different purposes, usually through chemical
methods. To characterize and to understand the optical properties of these
systems, with more realistic shaped particles, extensive numerical methods
were applied, and the Lorenz-Mie theory, for ideal spherical particles, became
not only an insightful reference, but also a starting first-order
approximation, in rather involved calculations; Most of the theoretical
descriptions are based on numerical calculations; Countless articles have been
published dealing with non-spherical particles.vanHulst1957 ; Kerker ;
vanHulst1980 ; Bohren ; special ; Haes ; Kelly ; Kempa ; Haragushi ; Kjeldsen
; Mishchenko ; Gregorchuk ; Ershov ; Trautmann ; Zhang ; Trendafilov ; KJHuang
As in other fields of science, the experimental and applied research on
plasmonic phenomena in metallic structures is ahead of, and move faster than,
the theoretical understanding and accurate calculations, particularly, when
systems contain many non-spherical particles, multiple scattering processes,
shape and size dispersion and, perhaps, also the presence of random variables.

In the last thirty years, as the ability to produce low-dimensional structures
grew, interest in periodic arrangements of spherical particles, cylindrical
and rectangular rods, and even layered metal structures, led to the profuse
field of photonic crystals. Not only does periodicity entail the possibility
for simpler systems to analytically solve the light scattering problem for
systems with a large number of scatterers, but introduces one of the most
important known properties of periodic quantum systems, the phase coherence
that is behind the band and gap structures. An important amount of properties
and physics of photonic crystals, containing metallic inclusions with
spherical and cylindrical symmetries,Pendry1994 ; BottenI2000 ; BottenII2000 ;
Jacak ; Bordo ; Raza ; Mayergoyz ; Davis  have been reasonably explained,
although accurate calculations are difficult to perform because of the
complexity of the actual systems and of their intricate response to
electromagnetic fields. Nevertheless, more accurate and appropriate
theoretical descriptions are possible for layered metallic structures, as was
shown in Ref. [Pereyra2020, ] and is further developed here.

An important class of systems with properties similar to those of the widely
studied systems in the photonic crystals field, but more feasible to produce,
are the flat layered $dielectric/metal$ structures, in particular periodic
arrays $(dielectric/metal)^{n}$ where $n$ is finite and the layers thicknesses
are chosen at will. In an attempt to study the physical properties of these
structures, here called metallic superlattices, many theoretical works were
published, practically all of them assuming infinite or semi-infinite
superlattices.Camley1984 ; Vigneron1985 ; Xue1985 ; Wallis1987 ; Wendler1987 ;
Mochan1988 ; Trutschel1989 ; Sheng1992 ; Nazarov1994 ; Pendry1994 ; Quinn1995
; BottenI2000 ; BottenII2000 ; Lyndin ; Bria2004 ; Inan1999  In Ref.
[Pereyra2020, ], a comprehensive theory was published where the finiteness is
rigorously respected, and the well-known properties of electromagnetic fields
inside conducting layers are taken into account.Stratton1941  Among those
properties, the fact that the wave vector ${\bf k}_{c}$ and the refraction
angle $\theta_{c}$ are complex quantities, is particularly significant; the
constant-amplitude ($pz=const$) and constant-phase ($k_{cx}x+qz=const$) planes
are distinct, and, as a consequence the assumption that the EMFs propagate
along the so-called real or “true” angle $\psi=\tan^{-1}(-k_{cx}/q)$ is made,
with $q$ and $p$ wave numbers as defined in equation (9). In the theoretical
approach of Ref. [Pereyra2020, ], based on the theory of finite periodic
systems, it is assumed that the electromagnetic fields, inside metals, move
along the true angle $\psi$, and the transmittance as well as the plasmons’
resonant frequencies are determined for almost any set of superlattice
parameters, any number of unit cells, and for frequencies above and below the
plasma frequency $\omega_{p}$.PlasmaFreq  But, anomalous results were noticed
for frequencies below $\omega_{p}$, and further research was offered. In this
paper, we exhibit and solve the problem, and show that it is, essentially,
related to the loss of flux due to neglected currents at the surfaces of the
metallic layers.

When the transfer matrix that propagates EMFs across a metallic layer, of
thickness $d_{c}$, factorizes into an attenuation factor $e^{-pd_{c}}$ and a
matrix that accounts for the gained phases $\pm iqd_{c}$, the matrix becomes
subunitary and the flux conservation principle is broken. The factorization,
compelled by the requirement of finiteness, needs to include the surface
current in order to restore the unimodularity of the transfer matrix. In this
paper we face this problem and present two fully compatible approaches to deal
with the transmission of EMFs through metallic layers, valid also in the
optical regime. In the first approach, we deal with the complex angle and
complex wave vector assuming that the phases gained in the metallic layers are
given by $\pm i{\mathbb{k}}_{c}\cdot{\mathbb{d}}_{c}$, independent of whether
${\mathbb{k}}_{c}$ is real or complex. In this approach, the unimodular nature
of the transfer matrices, thus the principle of flux conservation, is
rigorously preserved. In the second approach, we consider again the real-angle
approach but include now the induced currents at the metallic layers and
impose a flux conservation requirement. This allows us to determine the
magnitude of the induced currents and to define an absorption factor $a$. The
calculation of the transmittance shows that the predictions of both approaches
agree completely. An advantage of the second approach is that it allows to
obtain an insight on the induced currents and the absorption factor.

In Section 2, we review the boundary conditions and show the origin of the
lack of unitarity in the transfer matrices. We show that, for frequencies
below the plasma frequency $\omega_{p}$, the assumption of EMFs moving along
the true angle and neglecting, at the same time, the induced currents (in the
metallic layers), lead to a lack of unitarity. In section 3 we present the
complex angle approach (CAA) and show that, applying the TFPS without being
disturbed by the complex nature of the refracted angle, i. e. without any
reference to the constant phase and constant plane projections, we obtain an
approach that works well below and above $\omega_{p}$. In section 4 we turn
into the true angle approach but now taking into account in the boundary
conditions the currents induced by the electric polarization of the right and
left moving fields and, on top of this, we impose the flux conservation
requirement in a way to include the absorbed energy. This approach led us not
only to determine the magnitude and phase of the effective induced currents,
but also to present a unified and more insightful approach that works equally
well for frequencies below and above the plasma frequency $\omega_{p}$. The
explicit transmittance calculations show that, in the low frequencies
($\omega<\omega_{p}$) domain, this approach agrees with the complex angle
approach predictions. In the high frequencies ($\omega>\omega_{p}$) domain,
both approaches coincide with the predictions of the approach in Ref.
[Pereyra2020, ].

We will see that in the domain of frequencies $\omega<\omega_{p}$, where the
reflection is large, a photonic band structure, with almost complete
transmission, emerges, when the dielectric width $d_{a}$ increases. We will
see also that the low frequency resonances describe highly localized polarons
with large mean-life time. As in Ref. [Pereyra2020, ], qualitative and
quantitative differences are observed in the transmittance features above and
below $\omega_{p}$, with strong dependence on the incidence angle and the
superlattice parameters. We will see that the resonant dispersion relation,
derived in the TFPS, predicts the band widths and the frequencies of the
plasmonic resonances.

In this paper we present also results of reflected, transmitted and tunneling
times of Gaussian electromagnetic wave packets by metallic superlattices
(MSLs). A detailed analysis of the space-time evolution of Gaussian wave
packets will be publish elsewhere.

##  II Unitarity deficit in the constant-phase direction

In this section we will show that the assumption of EMFs moving along the true
angle $\psi$ and the requirement of finite EMFs, lead to break the principle
of flux conservation, for frequencies below the plasma frequency $\omega_{p}$.
To make clear this effect, let us assume an electromagnetic field with, say,
parallel polarization and incidence angle $\theta_{j}$, moving across a
superlattice $(D_{1}/M_{2}/D_{3})^{n}$, where $n$ is the number of unit cells.
The unit cell $D_{1}/M_{2}/D_{3}$ comprises two dielectric ($D_{j}$) layers
characterized by electric permittivities $\epsilon_{j}$ and magnetic
permeabilities $\mu_{j}$, and a metal ($M_{2}$) layer, with dielectric
function written asYang

$\epsilon_{2}(\omega)=\epsilon_{\infty}-\frac{E_{p}^{2}}{\hbar^{2}\omega^{2}}+i\frac{\sigma}{\omega},$
(1)

For specific calculations we will consider silver parameters:
$\epsilon_{\infty}=5.7$ and $E_{p}=9eV$, with plasma frequencyPlasmaFreq ;
Pendry2000  $\omega_{p}=E_{p}/(\hbar\sqrt{\epsilon_{\infty}})\simeq
5.729\times 10^{15}s^{-1}$. We will assume also that
$\mu_{1}=\mu_{3}=\mu_{2}=1$. Regardless of whether the electric parameters are
real or complex, the electric and magnetic fields in layer $j$, see figure 1,
can be written as

$\displaystyle{\mathbb{E}}_{j}(r,t)$ $\displaystyle=$
$\displaystyle{\mathbb{E}}_{rj}+{\mathbb{E}}_{lj}=\mathcal{E}_{rj}{\mathbb{n}}_{rj}e^{i({\mathbb{k}}_{rj}\cdot{\mathbb{r}}-\omega
t)}+\mathcal{E}_{lj}{\mathbb{n}}_{lj}e^{-i({\mathbb{k}}_{lj}\cdot{\mathbb{r}}+\omega
t)}\hskip 21.68121pt{\rm for}\hskip 21.68121ptj=1,2,3$ (2)
$\displaystyle{\mathbb{H}}_{j}(r,t)$ $\displaystyle=$
$\displaystyle{\mathbb{H}}_{rj}+{\mathbb{H}}_{lj}=\frac{{\mathbb{k}}_{rj}\times{\mathbb{n}}_{rj}}{\omega\mu_{j}}\mathcal{E}_{rj}e^{i({\mathbb{k}}_{rj}\cdot{\mathbb{r}}-\omega
t)}+\frac{{\mathbb{k}}_{lj}\times{\mathbb{n}}_{lj}}{\omega\mu_{j}}\mathcal{E}_{lj}{\mathbb{n}}_{l}e^{-i({\mathbb{k}}_{lj}\cdot{\mathbb{r}}+\omega
t)}$ (3)

where ${\mathbb{n}}_{rj}$ and ${\mathbb{n}}_{lj}$ are the polarization vectors

${\mathbb{n}}_{rj}=(\cos\theta_{j},0,\sin\theta_{j})\hskip 21.68121pt{\rm
and}\hskip 21.68121pt{\mathbb{n}}_{lj}=(\cos\theta_{j},0,-\sin\theta_{j}),$
(4)

of the right and left moving fields, and the corresponding wave vectors are

${\mathbb{k}}_{rj}=k_{j}{\mathbb{u}}_{rj}=\omega\sqrt{\mu_{j}\epsilon_{j}}(-\sin\theta_{j},0,\cos\theta_{j})\hskip
21.68121pt{\rm and}\hskip
21.68121pt{\mathbb{k}}_{lj}=k_{j}{\mathbb{u}}_{lj}=\omega\sqrt{\mu_{j}\epsilon_{j}}(-\sin\theta_{j},0,-\cos\theta_{j}).$
(5)

For a simplified notation, we define the field phases

$\displaystyle\phi_{rj}$ $\displaystyle=$ $\displaystyle
k_{rjx}x+k_{rjz}z-\omega t=-k_{j}x\sin\theta_{j}+k_{j}z\cos\theta_{j}-\omega
t$ (6) $\displaystyle\phi_{lj}$ $\displaystyle=$ $\displaystyle
k_{ljx}x-k_{ljz}z-\omega t=-k_{j}x\sin\theta_{j}-k_{j}z\cos\theta_{j}-\omega
t.$ (7)

Figure 1: The right and left moving electromagnetic waves in the superlattice
layers.

Thus, the field components in each layer $j$-th of the superlattice are

$\displaystyle E_{jx}$ $\displaystyle=$
$\displaystyle(\mathcal{E}_{rj}e^{i\phi_{rj}}+\mathcal{E}_{lj}e^{i\phi_{lj}})\cos\theta_{j}$
(8) $\displaystyle E_{jz}$ $\displaystyle=$
$\displaystyle(\mathcal{E}_{rj}e^{i\phi_{rj}}-\mathcal{E}_{lj}e^{i\phi_{lj}})\sin\theta_{j}$
(9) $\displaystyle H_{jy}$ $\displaystyle=$
$\displaystyle\frac{k_{j}}{\omega\mu_{j}}(\mathcal{E}_{rj}e^{i\phi_{rj}}-\mathcal{E}_{lj}e^{i\phi_{lj}}).$
(10)

Figure 2: Constant-phase and constant-amplitude planes for the electromagnetic
fields propagating in the conductor media.

Applying the boundary condition for the electric fields at the interface
$1|2$, we obtain the the well-known Snell’s law

$k_{2}\sin\theta_{2}=k_{1}\sin\theta_{1},$ (11)

with $k_{2}$ and $\sin\theta_{2}$ complex. Because of the complex nature of
these quantities, it has been common to represent the electromagnetic field
phases in the conductor asStratton1941

$\displaystyle\phi_{r2}$ $\displaystyle=$
$\displaystyle(k_{2R}+ik_{2I})(-x\sin\theta_{2}+z\cos\theta_{2})-\omega
t=a_{2}x+qz+i(b_{2}x+pz)-\omega t$ (12) $\displaystyle\phi_{l2}$
$\displaystyle=$
$\displaystyle(k_{2R}+ik_{2I})(-x\sin\theta_{2}-z\cos\theta_{2})-\omega
t=a_{2}x-qz+i(b_{2}x-pz)-\omega t.$ (13)

Here $k_{2x}=a_{2}+ib_{2}=-k_{1}\sin\theta_{1}$, and $k_{2z}=q+ip$, with

$\displaystyle
q=\rho\left(k_{2R}\cos\frac{\chi}{2}-k_{2I}\sin\frac{\chi}{2}\right)\hskip
21.68121pt{\rm and}\hskip
21.68121ptp=\rho\left(k_{2I}\cos\frac{\chi}{2}+k_{2R}\sin\frac{\chi}{2}\right),$
(14)

and

$\displaystyle\rho=\left(1+\frac{2k_{1}^{2}(k^{2}_{2I}-k^{2}_{2R})\sin^{2}\theta_{1}+k^{4}_{1}\sin^{4}\theta_{1}}{k_{2}^{2}}\right)\hskip
21.68121pt{\rm and}\hskip
21.68121pt\chi=\tan^{-1}\left(\frac{2k_{1}^{2}k_{2I}k_{2R}\sin^{2}\theta_{1}}{k_{2}^{2}+2k_{1}^{2}(k^{2}_{2I}-k^{2}_{2R})\sin^{2}\theta_{1}}\right).$
(15)

To recognize that, in the propagation of the electromagnetic waves inside
conductors, one can distinguish directions of constant-phase and directions of
constant-amplitude, as sketched in figure 2. The constant-phase and constant-
amplitude planes, defined by

$a_{2}x\pm qz=constant\hskip 21.68121pt{\rm and}\hskip 21.68121ptb_{2}x\pm
pz=constant,$ (16)

respectively, propagate along the normals making angles $\psi$ and
$\psi^{\prime}$ with the $z$-axis, defined by

$\tan\psi=-\frac{a_{2}}{q}\hskip 21.68121pt{\rm and}\hskip
21.68121pt\tan\psi^{\prime}=-\frac{b_{2}}{p},$ (17)

respectively. An important and well-known consequence of the complex phases is
the attenuation of the electromagnetic fields, with the aftermath of loss of
energy through the longitudinal and transverse currents induced by the
electric fields. The rather common assumption made in Ref. [Pereyra2020, ]
that the electromagnetic fields move along the direction determined by the
true or real angle $\psi$,Inan1999  neglecting the longitudinal and transverse
currents, supposedly concentrated only at the surfaces, led one to write the
transfer matrix that connects field vectors at the left and right of the
conductor as

$M^{\prime}_{c}=\frac{1}{2\kappa\mu_{1}\cos\psi+2\xi}\left(\begin{array}[]{cc}\alpha_{l}&\beta_{l}\cr\beta_{l}^{*}&\alpha_{l}^{*}\end{array}\right)\left(\begin{array}[]{cc}e^{i(q+ip)d_{c}}&0\cr
0&e^{-i(q-ip)d_{c}}\end{array}\right)\frac{1}{2k_{1}\mu_{2}\cos\theta_{1}}\left(\begin{array}[]{cc}\alpha_{l}^{*}&-\beta_{l}\cr-\beta_{l}^{*}&\alpha_{l}\end{array}\right),$
(18)

$p$ and $q$ are as defined before,
$\kappa=\bigl{(}q^{2}+k_{1}^{2}\sin^{2}\theta_{i}\bigr{)}^{1/2}$,
$\xi=k_{1}\mu_{1}\sec\theta_{1}\tan\psi$ and

$\displaystyle\alpha_{l}$ $\displaystyle=$ $\displaystyle
k_{1}\mu_{2}\sec\theta_{1}+\kappa\mu_{1}\cos\psi+\xi+i\,p\mu_{1}$ (19)
$\displaystyle\beta_{l}$ $\displaystyle=$ $\displaystyle
k_{1}\mu_{2}\sec\theta_{1}-\kappa\mu_{1}\cos\psi-\xi+i\,p\mu_{1}.$ (20)

Note that, to keep the physical quantities finite, one of the important
matrices in (18), the transfer matrix that connects the fields at the
beginning and the end of the metallic layer, was written as

$M_{m}=\left(\begin{array}[]{cc}e^{i(q+ip)d_{c}}&0\\\
0&e^{-i(q-ip)d_{c}}\end{array}\right),$ (21)

with a phase $-i(q-ip)d_{c}$ instead of $-i(q+ip)d_{c}$ for the left moving
EMF.

Figure 3: Determinant of the transfer matrix $M^{\prime}_{c}$ in (18), and the
real angle as functions of frequency, conductor-layer width and incidence
angle. (a) The determinant of the unit-cell transfer matrix, $\det
M^{\prime}_{c}$, decreases as the conductor layer width $d_{c}$ increases. (b)
The cosine of the real angle $\psi$. We show here that even though that for
$\omega>\omega_{p}$ it is close to 1, as is well known, for
$\omega<\omega_{p}$ is close to zero, except for $\theta_{i}=0$.

A simple way to have a measure of the lack of flux is the unimodularity of the
transfer matrix. In Figure 3 (a), we plot the determinant of the unit-cell
transfer matrix $M^{\prime}$, shown in Eq. (18) and used in Ref. [Pereyra2020,
], as a function of frequency $\omega$, and for different values of the
conducting layer width $d_{c}$. It is clear in this figure that, for
frequencies below $\omega_{p}$, the determinant of $M^{\prime}$ is less than 1
and decreases as the layer width $d_{c}$ increases. Another quantity whose
behavior needs to be clarified is the real or true angle $\psi$ that defines
the direction of constant-phase planes. It is generally assumed that its
values, measured from the normal to the interface, are small. That is true for
frequencies above $\omega_{p}$. But, that is not true for $\omega<\omega_{p}$,
as shown in Figure 3 (b). Except for $\theta_{i}=0$, where $\tan\psi$ is
rigorously zero, $\psi$ is close to $\pi/2$ for $\theta_{i}\neq 0$ and
$\omega<\omega_{p}$, as shown in the inset of the figure. This means that when
the electric field has a large component parallel to the interface, the planes
of constant-phase tend to propagate parallel to the interface, for small
frequencies and finite conductivity.

Our purpose in the next sections is to solve the lack of unitarity in the
optical regime. We will show that both the finiteness and the lack of
unitarity can be overcome. We will present two approaches, with compatible
results. We shall first consider a complex-angle approach, and in section IV,
we will present an improved real-angle approach, where, besides the true angle
assumption, we will take into account, explicitly, the induced currents. In
the complex-angle approach, we keep the complex angle and the complex wave
number at the conducting layers. In this case, the unit-cell transfer matrix
is automatically unimodular. In the improved real-angle approach, the induced
currents account for the absorption factor and restore the flux-conservation
principle.

##  III Complex-angle approach

Even though the recognition of constant-phase and constant-amplitude planes
for EMFs inside conductors helps one to understand important properties of
these fields in this kind of media, there is no need, in principle, in the
transfer matrix approach to assume that the EMWs move along the real or true
angle. In fact, we can equally well work with the complex angle $\theta_{2}$,
be compatible with the principle of flux conservation, and we can determine
the transmission and reflection coefficients of metallic superlattices for
frequencies below and above the plasma frequency $\omega_{p}$. In this
section, we will present this complex-angle approach, and obtain some results.

Te boundary conditions at the interface $1|2$, at $z=z_{l}$, are expressed by
the following relation

$\left(\begin{array}[]{c}E_{r2x}\cr
E_{l2x}\end{array}\right)_{z_{l}^{+}}=\left(\begin{array}[]{c}\mathcal{E}_{r2}e^{i\phi_{r2}}\cos\theta_{2}\cr\mathcal{E}_{l2}e^{i\phi_{l2}}\cos\theta_{2}\end{array}\right)_{z_{l}^{+}}=\frac{1}{2}\left(\begin{array}[]{cc}1+B_{l}&1-B_{l}\cr
1-B_{l}&1+B_{l}\end{array}\right)\left(\begin{array}[]{c}\mathcal{E}_{r1}e^{i\phi_{r1}}\cos\theta_{1}\cr\mathcal{E}_{l1}e^{i\phi_{r1}}\cos\theta_{1}\end{array}\right)_{z_{l}^{-}}=M_{l}(z_{l}^{+},z_{l}^{-})\left(\begin{array}[]{c}E_{r1x}\cr
E_{l1x}\end{array}\right)_{z_{l}^{-}}$ (22)

where the subindices $z_{l}^{-}$ and $z_{l}^{+}$ mean evaluation at
$z_{l}-\epsilon$ and $z_{l}+\epsilon$, in the $\lim\epsilon\rightarrow 0$. The
parameters $B_{l}$ and $cos\theta_{2}$ are

$B_{l}=\frac{k_{1}\mu_{2}\cos\theta_{2}}{k_{2}\mu_{1}\cos\theta_{1}}\hskip
21.68121pt{\rm and}\hskip 21.68121pt\cos\theta_{2}=\rho e^{i\,\chi/2}$ (23)

Similarly, the boundary conditions at the interface $2|3$, where $z=z_{r}$,
lead to

$\left(\begin{array}[]{c}E_{r3x}\cr
E_{l3x}\end{array}\right)_{z_{r}^{+}}=\frac{1}{2}\left(\begin{array}[]{cc}1+B_{r}&1-B_{r}\cr
1-B_{r}&1+B_{r}\end{array}\right)\left(\begin{array}[]{c}E_{r2x}\cr
E_{l2x}\end{array}\right)_{z_{r}^{-}}=M_{r}(z_{r}^{+},z_{r}^{-})\left(\begin{array}[]{c}E_{r2x}\cr
E_{l2x}\end{array}\right)_{z_{r}^{-}}.$ (24)

Here

$B_{r}=\frac{k_{2}\mu_{1}\cos\theta_{1}}{k_{1}\mu_{2}\cos\theta_{2}}.$ (25)

The matrices $M_{l}$ and $M_{r}$ are formally similar to the well-known
matrices of Fresnel amplitudes. In Ref. [Pereyra2020, ], we had $\psi$ instead
of $\theta_{2}$. In the following, we will omit the subindices $z_{x}^{\pm}$
for the matrices. The relation between the electromagnetic fields at the left
and right hand side of the conducting layer is

$\left(\begin{array}[]{c}E_{r3x}\cr
E_{l3x}\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{cc}1+B_{r}&1-B_{r}\cr
1-B_{r}&1+B_{r}\end{array}\right)\left(\begin{array}[]{cc}e^{i\,\varphi_{c}}&0\cr
0&e^{i\,\varphi_{c}}\end{array}\right)\frac{1}{2}\left(\begin{array}[]{cc}1+B_{l}&1-B_{l}\cr
1-B_{l}&1+B_{l}\end{array}\right)\left(\begin{array}[]{c}E_{r1x}\cr
E_{l1x}\end{array}\right)=M_{c}\left(\begin{array}[]{c}E_{r1x}\cr
E_{l1x}\end{array}\right),$ (26)

where $\varphi_{c}=k_{2}d_{c}\cos\theta_{2}$, when the conducting layer width
is $d_{c}$. Multiplying the matrices we obtain the transfer matrix for a
conductor layer

$M_{c}=\left(\begin{array}[]{cc}\alpha_{c}&\beta_{c}\\\
\gamma_{c}&\delta_{c}\end{array}\right),$ (27)

with

$\displaystyle\alpha_{c}$ $\displaystyle=$
$\displaystyle\cos\varphi_{c}+i\displaystyle{\frac{k_{1}^{2}\cos^{2}\theta_{2}+k_{2}^{2}\cos^{2}\theta_{1}}{2k_{1}k_{2}\cos\theta_{1}\cos\theta_{2}}}\sin\varphi_{c}\hskip
43.36243pt\beta_{c}=i\displaystyle{\frac{k_{2}^{2}\cos^{2}\theta_{1}-k_{1}^{2}\cos^{2}\theta_{2}}{2k_{1}k_{2}\cos\theta_{1}\cos\theta_{2}}}\sin\varphi_{c}$
(28) $\displaystyle\delta_{c}$ $\displaystyle=$
$\displaystyle\cos\varphi_{c}-i\displaystyle{\frac{k_{1}^{2}\cos^{2}\theta_{2}+k_{2}^{2}\cos^{2}\theta_{1}}{2k_{1}k_{2}\cos\theta_{1}\cos\theta_{2}}}\sin\varphi_{c}\hskip
43.36243pt\gamma_{c}=-i\displaystyle{\frac{k_{2}^{2}\cos^{2}\theta_{1}-k_{1}^{2}\cos^{2}\theta_{2}}{2k_{1}k_{2}\cos\theta_{1}\cos\theta_{2}}}\sin\varphi_{c}.$
(30)

Therefore, the transfer matrix of the unit cell $D_{1}/M_{2}/D_{3}$, whose
layer widths are $d_{a}/2$, $d_{c}$ and $d_{a}//2$, is

$\displaystyle M=\left(\begin{array}[]{cc}e^{i\,\theta_{a}/2}&0\\\
0&e^{-i\,\theta_{a}/2}\end{array}\right)\left(\begin{array}[]{cc}\alpha_{c}&\beta_{c}\\\
\gamma_{c}&\delta_{c}\end{array}\right)\left(\begin{array}[]{cc}e^{i\,\theta_{a}/2}&0\\\
0&e^{-i\,\theta_{a}/2}\end{array}\right)=\left(\begin{array}[]{cc}\alpha&\beta\\\
\beta^{*}&\alpha^{*}\end{array}\right)$ (39)

Here

$\theta_{a}=d_{a}k_{1}\cos\theta_{1}.$ (40)

It is worth noticing that in this representation, both flux and time reversal
invariance are preserved. This is so because keeping the complex angle, all
the components of the electromagnetic field, i.e. the transmitted, reflected
and absorbed components, are fully taken into account. As will be seen below,
the transmission coefficients in the high frequency domain, for
$\omega>\omega_{p}$ are exactly the same as in the real-angle approach.
However, in the optical regime, where $\omega<\omega_{p}$, some properties
remain, but others like the apparent parity effect and transparency,
disappear.

Figure 4: Transmission coefficient as a function of $\omega$ and $d_{a}$, for
a metallic superlattice with $n=12$, $d_{c}=15$nm and $\theta_{i}=\pi/3$. In
(a) the bands move to lower frequencies as $d_{a}$ grows. In (b) we see more
clearly the resonant transmission for a fix value of the conductor layer width
$d_{c}$. In the upper part of (b), bands and resonances predicted by the
resonant dispersion relation (44).

As was pointed out in Ref. [Pereyra2020, ], to determine the scattering
properties of the electromagnetic fields through metallic superlattices, using
the theory of finite periodic systemsPereyra1998 ; Pereyra2002 ; Pereyra2005 ;
Simanjuntak2007 ; Pereyra2008 ; Pereyra2012  outlined there, it is essential
to know the unit-cell transfer matrix, and applying the general formulas of
the TFPS one can, straightforwardly, obtain the transmittance and reflectance
through metallic superlattices, as well as to determine the resonant band
structure of the surface plasmon polaritons. The transmission and reflection
coefficients of a superlattice with $n$ unit cells, are obtained from

$T_{n}=\frac{1}{|\alpha_{n}|^{2}}\hskip 21.68121pt{\rm and}\hskip
21.68121ptR_{n}=\frac{|\beta_{n}|^{2}}{|\alpha_{n}|^{2}},$ (41)

where

$\alpha_{n}=U_{n}-\alpha^{*}U_{n-1}\hskip 21.68121pt{\rm and}\hskip
21.68121pt\beta_{n}=\beta^{-1}U_{n-1},$ (42)

and $U_{n}$ the Chebyshev polynomial of the second kind and order $n$,
evaluated at the real part of $\alpha$. The resonant frequencies and band
widths are determined by the resonant dispersion relationPereyra2002 ;
Pereyra2005 ; Pereyra2017 ; Pereyra2020

$\cos\frac{\nu+(\mu-1)n}{n}\pi=(\alpha_{R})_{\mu,\nu}\hskip 21.68121pt{\rm
with}\hskip 21.68121pt\mu=1,2,3,...\;\;\nu=1,2,...,n-1.$ (43)

$\mu$ and $\nu$ are the quantum numbers of the resonant frequencies
$\omega_{\mu,\nu}$, of the $\nu$-th resonance of the band $\mu$. Generally
$\mu=$1, 2, 3, … and $\nu=$1, 2, …, $n$-1.

In terms of the physical quantities defined in this approach, the resonant
dispersion relation is

$\displaystyle\cos\frac{\nu\\!+\\!(\mu\\!-\\!1)n}{n}\pi\\!=Re\left[(\cos
2\theta_{a}+i\sin
2\theta_{a})\left(\cos\varphi_{c}+i\frac{k_{1}^{2}\cos^{2}\theta_{2}+k_{2}^{2}\cos^{2}\theta_{1}}{2k_{1}k_{2}\cos\theta_{1}\cos\theta_{2}}\sin\varphi_{c}\right)\right]_{\omega_{\mu,\nu}}$
(44)

In Figures 4 and 5 we show the trends of the transmission coefficient as
function of the frequency and of the layer widths. The frequencies vary from 0
to $1.5\times 10^{16}$Hz, i.e. frequencies below and above $\omega_{p}$. In
Figure 4, the transmission is plotted as a function of the dielectric-layer
width $d_{a}$, while in Figure 5 as a function of the conducting layer width
$d_{c}$. For these examples and the others in this report, we consider air in
the dielectric layers and silver in the metallic ones.

For frequencies above $\omega_{p}$, which for silver is of the order of
$5.72\times 10^{15}$Hz, the transmission coefficients are, in all cases,
exactly the same as those in the real-angle approach of Ref. [Pereyra2020, ],
however, for frequencies below $\omega_{p}$, the results are different. Below
the plasma frequency, we have now narrower bands and thinner resonant states,
implying larger mean-life times for the resonant states and larger tunneling
times for electromagnetic waves whose frequencies are resonant. In electronic
and electromagnetic field transport, the transmission resonances result from a
complex and coherent superposition of electromagnetic fields facilitated by
the superlattice periodicity. These coherent superposition of fields imply the
participation of collective photon-driven electron oscillations, the so-called
plasmon polaritons. The long standing resonances correspond to localized
plasmon polaritons. From figures 4 and 5 it is clear that the increasing of
$d_{a}$ and the increasing of $d_{c}$ have opposite effects on the extension
of the complete-reflection domain, at low frequencies. While increasing
$d_{a}$ the reflection domain diminishes, increasing $d_{c}$ the complete-
reflection domain grows.

Figure 5: Transmission coefficient as a function of $\omega$ and $d_{c}$, for
a metallic superlattice with $n=11$, $d_{a}=150$nm and $\theta_{i}=\pi/12$. In
(a), the band width diminishes as the conductor layer width $d_{c}$ increases.
This effect is shown in (b) for three values of $d_{c}$, indicated with white
arrow in (a).

In Figures 4 (a) and (b), we have the transmission coefficient as a function
of $\omega$ and $d_{a}$, for a metallic superlattice with $n=12$, $d_{c}=15$nm
and $\theta_{i}=\pi/3$. In Figure 4 (a), we see that increasing $d_{a}$ the
bands move to lower frequencies, as in quantum systems when quantum-well
widths increase, and the band widths become narrower. To visualize the
resonant-band features better, we plot in Figure 4 (b) the transmission
coefficient evaluated at $d_{a}=300$nm, indicated with the white arrow in (a).
We also plot, in the upper part of this graph, the resonant levels predicted
by Eq. (44). These resonances, that result from complex coherent
superpositions of multiple reflected and transmitted fields plus collective
electron oscillations are, in generally, extended electromagnetic states, with
large mean-life time, nevertheless the collective electron oscillations occur
mainly at the surfaces of the metallic layers. These surface excitations
correspond to the so-called localized surface plasmon polaritons.

Figure 6: Transmission coefficient as a function of $\omega$ and $\theta_{i}$,
for a metallic superlattice with $n=12$, $d_{a}=150$nm and $d_{c}=5$nm. In
(b), the transmission coefficient for $\theta_{i}=0$, $\pi/4$ and $pi/2$.
These graphs correspond to those in (a) indicated with white arrows.

In Figures 5 (a) and (b), we have the transmission coefficient as a function
of $\omega$ and $d_{c}$, for a metallic superlattice with $n=11$,
$d_{a}=150$nm and $\theta_{i}=\pi/12$. In this example, we have also a
resonant transmission band for $\omega<\omega_{p}$. In Figures 5 (a), we see
that the band width diminishes as the conductor layer increases. To visualize
the resonant behavior better, we plot in Figure 5 (b) the transmission
coefficients for $d_{c}=15$nm, $d_{c}=30$nm and $d_{c}=60$nm. They correspond
to those layers’ widths indicated with white arrows in Figure 5 (a). Here also
the resonant transmission bands are within the visible light domain.

The resonances positions and band widths depend not only on the layers’
widths, but also depend strongly on the incidence angle. In figures 6 (a) and
(b), we plot the transmission coefficient as a function of $\omega$ and
$\theta_{i}$, for a metallic superlattice with $n=12$, $d_{a}=150$nm and
$d_{c}=5$nm. In Figure 6 (b), we see the transmission coefficient for
$\theta_{i}=0$, $\pi/4$ and $pi/2$, which correspond to those values of
$\theta_{i}$ indicated with white arrows in (a). It is clear from these graphs
the enormous qualitative and quantitative differences in the transmission
coefficient and the resonant features as functions of the incidence angle. As
was shown in Ref. [Pereyra2020, ] the transmission resonances become delta
type when the incidence angle is equal or even close to $\pi/2$.

Figure 7: The reflection coefficient $R_{n}$ as a functions of the wavelength
$\lambda$. In (a) for a single layer of silver. In (b) for a metallic
superlattice with $n=10$. At the plasma wavelength $\lambda_{p}$, we see a
characteristic Fano-like resonance, due to the interference between
propagating and evanescent modes.

Since $R_{n}=1-T_{n}$, we generally omit the calculation of the reflection
coefficient. Nevertheless, it may be helpful to visualize its behavior, in
particular, the effect that the number of layers may have in the optical
regime. In Figure 7, we have the reflection coefficient of a single silver
layer (a) and for a superlattice with $n=10$. It is clear that the
superlattice not only imply resonances, but also well defined bands of
complete reflection and, for some configurations, of complete transmission.

Playing with the superlattice parameters, the incidence angle and the
electromagnetic field polarization, we can obtain an endless variety of
optical responses for the electromagnetic fields that fall upon the surface of
a metallic superlattice. In the next section we will return to the real angle
approach and compare with the results of this approach.

##  IV Improved real-angle approach

As explained before, the lack of flux in the real-angle approach, indicates
the need to include the induced currents at the metallic layers. We will
include the induced currents, assume that inside the conductor the
electromagnetic waves move along the real angle (also called true angle) of
the constant-phase planes, and finally impose the conservation of flux. It is
well known that the inclusion of currents, induced in the metallic layers and
responsible of the plasmonic resonances, is compatible with the Maxwell
equation

${\mathbb{\nabla}}\times{\mathbb{H}}={\mathbb{J}}+\epsilon\frac{\partial\mathbb{E}}{\partial
t}$ (45)

which contemplates the possibility of induced currents, the plasmon
oscillations, which in perfect conductors are localized at the surface, but in
lossy conductors, respond to the local field. We will assume that in each
layer $j$ we have currents, induced be the right and left moving fields, whose
magnitudes are proportional to the magnitude of the local electric fields

$J_{rj}(x,z)=s_{R}{E}_{rj}(x,z)\hskip 14.45377pt{\rm and}\hskip
14.45377ptJ_{lj}(x,z)=s_{L}{E}_{lj}(x,z),$ (46)

Here the amplitudes $s_{L}$ and $s_{R}$ are proportional to the conductivity
$\sigma$ and to an attenuation factor determined below, that accounts for the
loss of energy. Taking into account the surface currents

$\displaystyle
J_{rxl}=s_{R}{E}_{r2}(x,z_{l})\cos\psi=s_{R}{E}_{r2x}\big{|}_{z_{l}},\hskip
21.68121pt{\rm and}\hskip
21.68121ptJ_{lxl}=s_{L}{E}_{l2}(x,z_{l})\cos\psi=s_{L}{E}_{l2x}\big{|}_{z_{l}},$
(47)

at $z=z_{l}$, and

$\displaystyle
J_{rxr}=s_{R}{E}_{r2}(x,z_{r})\cos\psi=s_{R}{E}_{r2x}\big{|}_{z_{r}},\hskip
21.68121pt{\rm and}\hskip
21.68121ptJ_{lxr}=s_{L}{E}_{l2}(x,z_{r})\cos\psi=s_{L}{E}_{l2x}\big{|}_{z_{r}},$
(48)

at $z=z_{r}$, for the boundary conditions. The transfer matrix of the
conducting layer that connects the field vectors at its left and right hand
sides becomes

$M_{\jmath
c}=\frac{1}{2\kappa\mu_{1}\cos\psi+2\xi}\left(\begin{array}[]{cc}\alpha_{l}-s_{R}&\beta_{l}-s_{L}\cr\beta_{l}^{*}+s_{R}&\alpha_{l}^{*}+s_{L}\end{array}\right)M_{m}(z_{r}^{-},z_{l}^{+})\frac{1}{2k_{1}\mu_{2}\cos\theta_{1}}\left(\begin{array}[]{cc}\alpha_{l}^{*}-s_{L}&-\beta_{l}-s_{L}\cr-\beta_{l}^{*}+s_{R}&\alpha_{l}+s_{R}\end{array}\right),$
(49)

with $p=\rho(\epsilon_{R}\sin\gamma+\epsilon_{I}\cos\gamma)$,
$\kappa=\bigl{(}q^{2}+k_{1}^{2}\sin^{2}\theta_{i}\bigr{)}^{1/2}$,
$\xi=k_{1}\mu_{1}\sec\theta_{1}\tan\psi$. The matrix-elements

$\displaystyle\alpha_{l}$ $\displaystyle=$ $\displaystyle
k_{1}\mu_{2}\sec\theta_{1}+\kappa\mu_{1}\cos\psi+\xi+i\,p\mu_{1}$ (50)
$\displaystyle\beta_{l}$ $\displaystyle=$ $\displaystyle
k_{1}\mu_{2}\sec\theta_{1}-\kappa\mu_{1}\cos\psi-\xi+i\,p\mu_{1}$ (51)

and the transfer matrix $M_{m}$, that connects the electromagnetic fields at
the far left ($z_{l}^{+}$) and far right ($z_{r}^{-}$) inside the metallic
layer is written as mentioned before as

$M_{m}=e^{-pd_{c}}\left(\begin{array}[]{cc}e^{iqd_{c}}&0\cr
0&e^{-iqd_{c}}\end{array}\right).$ (52)

with an attenuation factor $e^{-pd_{c}}$, which depends on the attenuation
constant $p$ and the conducting layer $d_{c}$, and a transfer matrix that
accounts for the phases, $qd_{c}$ and $-qd_{c}$, gained by the electromagnetic
fields that propagate along the real angle $\psi$, with wave number $q$.

It is easy to verify that up to a phase $\varphi$, with negligible effects on
the results as shown below, the requirement of flux conservation implies that

$\displaystyle s_{R}=\frac{\kappa\tanh{(pd_{c})}}{\cos\psi}\hskip
21.68121pt{\rm and}\hskip
21.68121pts_{L}=\frac{\kappa\tanh{(pd_{c})}}{\cos\psi}e^{i(\pi+\varphi)}.$
(53)

Since $s_{R}\propto\sigma$, we can define an absorption or flux-loosing factor
$a$, such that $s_{R}=\sigma a$, thus the attenuation factor

$\displaystyle
a=\frac{\kappa\tanh{(pd_{c})}}{\sigma\cos\psi}\simeq\frac{\kappa\tanh{(d_{c}/\delta_{c})}}{\sigma\cos\psi}$
(54)

where $\delta_{c}$ is the skin depth, provides the magnitude of the induced
currents and the amount of flux lost exciting the plasmonic resonances. Given
the transfer matrix $M_{\jmath c}$ it is easy to obtain the unit-cell transfer
matrix

$\displaystyle M_{\jmath}=\left(\begin{array}[]{cc}e^{i\,\theta_{a}/2}&0\\\
0&e^{-i\,\theta_{a}/2}\end{array}\right)\left(\begin{array}[]{cc}\alpha_{\jmath
c}&\beta_{\jmath c}\\\ \gamma_{\jmath c}&\delta_{\jmath
c}\end{array}\right)\left(\begin{array}[]{cc}e^{i\,\theta_{a}/2}&0\\\
0&e^{-i\,\theta_{a}/2}\end{array}\right)=\left(\begin{array}[]{cc}\alpha_{\jmath}&\beta_{\jmath}\\\
\beta_{\jmath}^{*}&\alpha_{\jmath}^{*}\end{array}\right)$ (63)

where $\theta_{a}=d_{a}k_{1}\cos\theta_{1}$ and $\alpha_{\jmath c}$,
$\beta_{\jmath c}$,… are the matrix elements of $M_{\jmath c}$. Having the
unit-cell transfer matrix, we can apply the theory of finite periodic
systems.Pereyra1998 ; Pereyra2002 ; Pereyra2005 ; Pereyra2017  The
transmission and reflection coefficients of a superlattice with $n$ unit
cells, are obtained from

$T_{\jmath n}=\frac{1}{|\alpha_{\jmath n}|^{2}}\hskip 21.68121pt{\rm
and}\hskip 21.68121ptR_{\jmath n}=\frac{|\beta_{\jmath
n}|^{2}}{|\alpha_{\jmath n}|^{2}},$ (64)

where

$\alpha_{\jmath n}=U_{n}-\alpha_{\jmath}^{*}U_{n-1}\hskip 21.68121pt{\rm
and}\hskip 21.68121pt\beta_{\jmath n}=\beta_{\jmath}^{-1}U_{n-1},$ (65)

with $U_{n}$ the Chebyshev polynomial of the second kind and order $n$,
evaluated at the real part of $\alpha_{\jmath}$.

Figure 8: Transmission coefficients as functions of $d_{a}$ and $\theta_{i}$
plotted here to compare with those in figures 4 (a) and 6 (b). The agreement
is perfect.

In figures 8 (a) and (b) we plot the transmission coefficients with the same
parameters as in those of figures 4 (a) and 6 (b). The agreement below and
above the plasma frequency is excellent. However, at $w_{p}$ and close to this
frequency the behavior is different. For the graphs in Figure 8 we considered
the phase $\varphi=0$. This phase accounts for the effective phase gain due to
the multiple internal reflections. As shown in Figure 9, the effect of
$\varphi$ is practically negligible and null for frequencies above
$\omega_{p}$. Figure 9 (b) is just a zoom of Figure 9 (a).

Figure 9: The effect of phase $\varphi$ in the transmission coefficients. The
graph in (b) is a zoom of the graph in (a). The effect of the phase $\varphi$
is negligible, and null for frequencies above $\omega_{p}$.

For the purpose of calculating only transmission and reflection coefficients
there is no advantage using in this real-angle approach, compared with the
more accurate complex-angle approach. However, the improved real-angle
approach does inform on the role that the induced currents play in the energy-
absorption phenomenon, for frequencies below $\omega_{p}$, which implies the
optical domain.

In Figure 10 (a) we plot the reflection coefficient of a single layer of
silver for different thicknesses, and in Figure 10 (b), the absorption factor
$a$, which represents the strength of the absorbed flux exciting the plasmon
oscillations. These currents attenuate the transmission and enhance the
reflection coefficient. As shown in Figure 10 (b) the absorption factor $a$
grows when the conducting layer increases. In Figure 11 we show the effect of
the incidence angle and conducting layer width on the absorption factor $a$
and on the reflection coefficient $R_{n}$ for a superlattice with $n=12$ and
dielectric width $d_{a}=400$nm. As shown by these results the absorption
factor is very sensitive to the incidence angle and, of course, to the
conducting layer width. We see that increasing the incidence angle from
$\theta_{i}=\pi/6$ to $\theta_{i}=\pi/4$ the factor $a$, hence the induced
currents, grows by a factor of 2.

Figure 10: Reflection and absorption factor for a single layer when the
incidence angle is $\theta_{i}=\pi/4$. In (a), the reflection coefficient as
function of the frequency, for different thicknesses of the silver layer. In
(b), the absorption factor $a$ grows as the layers’ thicknesses $d_{c}$
increase. The absorption factor vanishes at $\omega=0$ and
$\omega=\omega_{p}$, and its maximum shifts to higher frequencies as $d_{c}$
grows.  Figure 11: Reflection and absorption factor for a superlattice with
$n=$12, and two values of the incidence angle. The incidence angle at the left
is $\pi/6$, and $\pi/4$ at the right. The bandwidths and the absorption factor
grow with the incidence angle $\theta_{i}$ and the conducting layer
thicknesses $d_{c}$.

Before we conclude this section, let us see how is the resonant dispersion
relation, and particularly how are the dispersion relation predictions below
the plasma frequency $w_{p}$. As mentioned before the resonant dispersion
relation derived in the theory of finite periodic systems isPereyra2005 ;
Pereyra2002 ; Pereyra2017 ; Pereyra2020

$\cos\frac{\nu+(\mu-1)n}{n}\pi=(\alpha_{\jmath R})_{\mu,\nu}\hskip
21.68121pt{\rm with}\hskip 21.68121pt\mu=1,2,3,...\;\;\nu=1,2,...,n-1.$ (66)

In terms of the physical quantities defined in this approach, this relation
(for $\varphi=0$) becomes

$\displaystyle\cos\frac{\nu\\!+\\!(\mu\\!-\\!1)n}{n}\pi\\!=\left(\cosh{p{d_{c}}}\cos
qd_{c}\cos\theta_{a}-f_{1}\cosh{p{d_{c}}}\sin
qd_{c}\sin\theta_{a}+f_{2}\sinh{p{d_{c}}}\cos
qd_{c}\sin\theta_{a}\right)_{\mu,\nu}$ (67)

with

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle\frac{\mu_{2}\sin
2\psi}{2\mu_{1}\sin
2\theta_{1}}\left(1+\frac{p^{2}\mu_{1}^{2}}{k_{1}^{2}\mu_{2}^{2}}\cos^{2}\theta_{1}+\frac{\mu_{1}^{2}\sin^{2}2\theta_{1}}{\mu_{2}^{2}\sin^{2}2\psi}\right)$
(68) $\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle\frac{\mu_{1}\sin
2\theta_{1}}{2\mu_{2}\sin 2\psi}\left(\tan qd_{c}\tanh
pd_{c}+\frac{p}{k_{1}}\frac{\sin 2\psi}{\sin\theta_{1}}\right)$ (70)

A quantity that is also useful is the imaginary part of $\alpha_{\jmath}$,
that can be written as

$\displaystyle\alpha_{\jmath I}=\cosh{p{d_{c}}}\cos
qd_{c}\sin\theta_{a}+f_{1}\cosh{p{d_{c}}}\sin
qd_{c}\cos\theta_{a}-f_{2}\sinh{p{d_{c}}}\cos qd_{c}\cos\theta_{a}.$ (71)

Figure 12: Transmission and reflection coefficients for a superlattice with
$n=$8, $d_{a}=50$nm, $d_{c}=10$nm, and incidence angle $\theta_{i}=\pi/4$. In
the left, upper part, we have also the predicted resonant frequencies from the
dispersion relation. In the right the absorption factor.

In Figure 12 we plot the transmission and reflection coefficients, together
with the absorption factor and the resonances and bands predicted by the
dispersion relation. These graphs show not only the ability of this approach
to calculate the essential quantities like the transmission and reflection
coefficients, and the accurate prediction of the resonance frequencies
spectrum, it shows also the appropriateness of this approach to determine,
through the absorption factor $a$, the strength of the absorbed energy
consumed to excite the plasmonic polaritons as response to the incident
electromagnetic field. This information provides additional insight in the
physics of the metallic superlattices.

In the applications based on the response of metallic-structures to
electromagnetic fields, the response to electromagnetic pulses might be of
interest. Since the details of the electromagnetic fields inside the metallic
structure is a bit complex, we will just present here a couple of results.

##  V Reflection and transmission of Gaussian pulses by metallic superlattices

As illustrative examples of use of the above mentioned results, we will
present the transmission and reflection of an electromagnetic pulse by a
metallic superlattice. If the electromagnetic pulses are Gaussian wave packets
defined by

$\displaystyle\Psi_{E}(x,z,t,\theta_{i})=\int_{-\infty}^{\infty}dke^{-\gamma(k-k_{o})^{2}}e^{i(\bf{k\cdot
r}_{o}-\omega t)}{\bf{\cal
E}}(k,z,\theta_{i})=\int_{-\infty}^{\infty}dke^{-\gamma(k-k_{o})^{2}}{\bf
E}(k,z,\theta_{i},t)$ (72)

Figure 13: Space time evolution of a Gaussian wave packet through metallic
superlattices with $n=$ 1, 2, and 10, in the left, middle and right,
respectively. In the upper panels we show both the transmission coefficients
and the Gaussian wave packets at the frequencies at which they were defined at
$z_{o}$ and $t=$0\. In the lower panel, we have the reflected and transmitted
wave packets at $t=2|z_{o}|/c+\tau$, where $\tau$ the tunneling time of the
wave packet components of frequency $\omega_{o}$. The green and red Gaussian
curves, define the positions at which the reflected and transmitted packets
should be if $\tau$ is truly the tunneling and reflecting time. Notice that
the reshaping of the transmitted and reflected packets are strongly determined
by the transmission coefficients in the frequencies domain where the packets
are defined, at a resonance in the middle and at a gap in the right.

where $k_{o}$ defines the peak of the Gaussian pulse and $\bf{r}_{o}$ its
position at $t=0$. For $z<0$, the $z$ component of the electromagnetic field
is written as

$\displaystyle E_{z}(k,z,t,\theta_{i})={\cal
E}_{o}\left(e^{ik\cos\theta_{i}(z+z_{o}-v_{g}t)}-\frac{\beta_{n}^{*}}{\alpha_{n}^{*}}e^{-ik\cos\theta_{i}(z-z_{o}+v_{g}t)}\right)\hskip
21.68121ptz<0$ (73)

where $\alpha_{n}$ and $\beta_{n}$ are the matrix elements of the $n$-unit
cells transfer matrix $M_{n}$. For $z>L=nl_{c}$, with $l_{c}$ the SL unit cell
length, the electromagnetic field is written as

$\displaystyle E_{z}(k,z,t,\theta_{i})={\cal
E}_{o}\left(\alpha_{n}-\beta_{n}\frac{\beta_{n}^{*}}{\alpha_{n}^{*}}\right)e^{ik\cos\theta_{i}(z-L+z_{o}-v_{g}t)}\hskip
21.68121ptz>L$ (74)

The possibilities of total, partial or zero reflection of an electromagnetic
pulse by metallic superlattices are important for applications and strongly
determined by the transmission or reflection coefficients of each component of
the Gaussian packet, in particular by the domain of frequencies where the wave
packet is defined. Hence, the transmission coefficients for the Gaussian pulse
components are different. This difference manifests also in the shape of the
reflected and transmitted pulses, as shown in the lower panels. In the figure
we plot the $z$-components as functions of $z$ only. We assume also that the
fields are in parallel polarization and the incidence angle is
$\theta_{i}=\pi/4$.

To visualize the effect of the metallic SL on a Gaussian pulse we consider
three cases where the Gaussian packets are similar, but the characteristics of
the domain of frequencies where they are defined are different, and the number
of unit cells in the metallic SLs is different. The number of unit cells in
(a), (b) and (c), is 1, 2 and 10, respectively.

In figure 13 the transmitted and reflected wave packets are shown at
$t=2|z_{o}|/c+\tau$, assuming that the fields outside the SL move with
velocity $c$, and the centroid with wavenumber $k_{o}$ spends a time equal to
phase time $\tau$ inside the SL.Spielmann1994 ; Pereyra2000  In the lower
panels we have also the envelopes of the wave packets at $t=0$, green, (in
position $-z_{o}$) and at $t=2|z_{o}|/c+\tau$, red. The green and red Gaussian
curves define the positions at which the reflected and transmitted packets
should be found if $\tau$ is truly the tunneling or reflecting time. In each
case, the tunneling time $\tau$ of the packet peaks is indicated. The
prediction is correct, as was shown in other papers.Simanjuntak2007 ;
Pereyra2011  In case b), and slightly in case a), the asymmetry of the
tunneling times implies that the components of one tail (those with smaller
frequency) move faster than those in the other tail. For this reason, in order
to plot the whole transmitted and reflected packets in case b), we increase
the distance $|z_{o}|$.

Since, the scattering process is different for each wave packet component
$E(k,z,\theta_{i})$, the wave packets are, generally, distorted, unless the
whole wave packet is defined in a frequency domain such that almost all
components have the same transmission coefficients. This is the case in figure
13 (c), where the wave packet is defined in a gap, with $T_{n}\simeq$0, being
thus almost completely reflected. In figures 13 (a) and (b), because of the
frequency domains, the wave packet in a) is partially transmitted and
partially reflected. The wave packet in b), defined at a resonance, is
strongly distorted. The components close to the Gaussian peak are transmitted
while those in the tails are reflected.

##  VI Conclusions

We have shown that the anomalous results and apparent parity effects reported
in Ref. [Pereyra2020, ], are consequence of the common assumption that
electromagnetic fields move along the direction of propagation of the
constant-phase planes, the finiteness requirement, and the neglection of the
induced currents. We have shown that this assumptions imply a lack of
unitarity related to the underlying phenomena of absorption and loss of
energy. To cure this problem, we introduced two approaches. On the one hand,
we have shown that by keeping the complex angles, the principle of flux
conservation is fully satisfied, above and below $\omega_{p}$. The anomalous
results for frequencies below $\omega_{p}$ disappear, and the predictions
above $\omega_{p}$ obtained in Ref. [Pereyra2020, ], remain. The complex-angle
approach presented here, preserves all the information of the scattering
process in the metallic superlattice, and gave us light to improve the
formalism when the real angle assumption is made. In fact, we have shown here
that taking into account the induced currents and the requirement of flux
conservation, we end up with an improved approach, with new Fresnel and
transmission coefficients, fully compatible with those of the complex-angle
approach. The improved approach allows also to evaluate the magnitude of the
induced currents and the absorbed energy, as functions of frequency and the
superlattice parameters. We determine the plasmonic resonant frequencies, and
present preliminary results of the response of metallic superlattices to
electromagnetic pulses and wave packets, particularly, in the optical domain.
We calculate the reflection and transmission coefficients as well as the phase
time $\tau(\omega)$. We show that the predicted space-time positions agree
extremely well with the actual positions of the wave-packet centroids.

##  VII Acknowledgements

The author acknowledges comments of H. P. Simanjuntak.

## References

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