# Optical absorption in tilted geometries as an indirect measure of
longitudinal plasma waves in layered cuprates

Niccolò Sellati<EMAIL_ADDRESS>Department of Physics and ISC-
CNR, “Sapienza” University of Rome, P.le A. Moro 5, 00185 Rome, Italy  Jacopo
Fiore  Department of Physics and ISC-CNR, “Sapienza” University of Rome, P.le
A. Moro 5, 00185 Rome, Italy  Claudio Castellani  Department of Physics and
ISC-CNR, “Sapienza” University of Rome, P.le A. Moro 5, 00185 Rome, Italy
Lara Benfatto<EMAIL_ADDRESS>Department of Physics and ISC-CNR,
“Sapienza” University of Rome, P.le A. Moro 5, 00185 Rome, Italy

(August 28, 2024)

###### Abstract

Electromagnetic waves propagating in a layered superconductor with arbitrary
momentum with respect to the main crystallographic directions display an
unavoidable mixing between longitudinal and transverse degrees of freedom.
Here we show that this basic physical mechanism explains the emergence of a
well-defined absorption peak in the in-plane optical conductivity for light
propagating at small tilting angles with respect to the stacking direction in
layered cuprates. More specifically, we show that this peak, often interpreted
as a spurious leakage of the $c$-axis Josephson plasmon, is instead a
signature of the true longitudinal plasma mode occurring at larger momenta. By
combining a classical approach based on Maxwell’s equations with a full
quantum derivation of the plasma modes based on the modelling of the
superconducting phase degrees of freedom, we provide an analytical expression
for the absorption peak as a function of the tilting angle and light
polarization. We suggest that an all-optical measurement in tilted geometry
can be used as an alternative way to access plasma-wave dispersion, usually
measured by means of large-momenta scattering techniques like RIXS or EELS.

##  I Introduction

In superconductors the breaking of the continuous gauge symmetry below the
superconducting (SC) critical temperature is accompanied by the emergence of
two collective modes, associated with the amplitude (Higgs) or phase
(Goldstone) fluctuations of the complex SC order parameter, whose absolute
value at equilibrium defines the spectral gap for single-particle excitations
[1]. While the former is a massive excitation, the latter is massless at long
wavelength, reflecting the infinity of possible ground states connected by a
global change of the order-parameter phase. Nonetheless, the coupling of the
SC phase to the density is directly affected by long-range Coulomb
interactions between charged electrons. This effect moves the phase mode to
the plasma energy scale [2], that is usually much larger than the spectral
gap. As a consequence, optical signatures at the plasma energy scale, i.e. at
the zero of the dielectric function, are usually unaffected by the SC
transition. A rather different phenomenology is instead observed in
anisotropic layered superconductors, i.e. systems where the pairing mainly
occurs within planes stacked along the $c$ direction, and SC order is
established below $T_{c}$ thanks to a weak Josephson-like inter-plane
interaction. The hallmark of this category is represented by high-temperature
cuprates [3], where the marked anisotropy has been experimentally proven by
different optical probes, starting from linear optics, which measures two
well-separated energy scales for the plasma modes at long-wavelength for
electric fields propagating in the CuO2 planes or perpendicular to them. In
these systems the incoherent quasiparticle hopping along the stacking
direction makes the $c$-axis response badly metallic: in contrast, below
$T_{c}$ the opening of a sizeable spectral gap along with the weak inter-layer
pair hopping leave a rather sharp SC plasma edge at a frequency $\omega_{c}$
of few THz in the optical reflectivity, that clearly testifies the emergence
of a well-defined SC Josephson plasmon. Even though this feature has been
experimentally observed already in the late ’90s [4, 5, 6, 7, 8, 9], a renewed
interest in the physics of Josephson plasmons emerged more recently thanks to
non-linear THz spectroscopy from one side [10, 11, 12, 13, 14, 15], and to
nano-plasmonic from the other side [16, 17]. In both cases it becomes
theoretically relevant understanding the momentum dependence of the plasmon
dispersion at generic momentum, i.e. not along the main crystallographic axes.
In this configuration one immediately realizes that the anisotropy leads to a
non-trivial response of the system, due to the fact that the current induced
by the external electric field is no more parallel to the field itself. As it
has been extensively discussed in details in Ref.s [18, 19, 20], this
mechanism leads to a mixing of the longitudinal and transverse response inside
the material, making the distinction between plasmons and polaritons blurred
at momenta smaller than a scale
$\bar{k}\sim\sqrt{\omega_{ab}^{2}-\omega_{c}^{2}}/c$ set by the anisotropy
between in-plane $\omega_{ab}$ and out-of-plane $\omega_{c}$ plasma
frequencies. Since usually $\omega_{ab}\gg\omega_{c}$, the effect is relevant
for non-linear Josephson plasmonics in the THz regime [21, 22, 23, 24], but
does not affect e.g. the measurements of plasmons in RIXS [25, 26, 27, 28, 29]
or EELS [30, 31, 32], that usually measure momenta in a fraction of the
Brillouin-zone. In the present manuscript we investigate an additional
consequence of the above-mentioned mixing, showing how even linear optics can
be used to disentangle the longitudinal-transverse mixing in a reflection or
transmission geometry which highlights the emergence inside the material of a
longitudinal response induced by an external transverse electromagnetic wave.
The effect manifests as an absorption peak at a scale nearby $\omega_{c}$ for
an electromagnetic wave travelling at small angle with respect to the $c$
direction. This feature has been measured in the past in different samples of
electron-doped cuprates [33, 34, 35, 36] below $T_{c}$, and it has been often
interpreted as a leakage of the $c$-axis plasmon into the in-plane response
[37]. Even more interestingly, the peak position has been shown to change by
varying the wave polarization in the plane of incidence, challenging
considerably the interpretation of the results. Here we provide a full
theoretical description of the microscopic mechanism behind the anomalous
absorption peak, and we show that it is a direct consequence of the plasmon-
polariton mixing in an anisotropic layered superconductor. We argue that this
effect can be used to indirectly probe the plasmon dispersion that usually
appears in RIXS and EELS experiments at much larger momenta and, by changing
the light polarization, to extract the in-plane and out-of-plane plasma
frequencies. Our findings are benchmarked against existing experimental data
for cuprates. On a more general ground, our results offer a novel perspective
on the possibility to access collective polariton modes in complex materials
by properly engineering optical measurements.

##  II Anisotropic linear response of layered system

As we discussed in the Introduction, several experiments in electron-doped
cuprates [33, 34, 35, 36] have shown the emergence of a peak in the in-plane
conductivity below $T_{c}$ at a frequency close to the one of the out-of-plane
plasma edge, whose position moves by changing the light polarization. This
peak is often interpreted as a spurious effect due to the leakage of the
$c$-axis plasmon into the in-plane response [37], and light polarization is
used to remove the effect [36]. However, in Ref. [35] the problem has been
investigated in details by growing on purpose a sample with a stacking
direction tilted with respect to the light wave-vector, and a preliminary
interpretation has been provided linked to such a tilted geometry. Here we
will follow the same reasoning, and we will study the response for propagating
wave-vector at tilted angle with respect to the stacking direction. To fix the
notation, in the following we will use the convention by which the SC sheets
are parallel to the $ab$-plane and stacked along the $c$-axis. We then assume,
without loss of generality, that the momentum k of the propagating wave is
along the $ac$-plane ($k_{b}=0$). The angle between k and the $c$-axis is
denoted $\eta$ and the angle between the transverse current and the $b$-axis
is denoted $\phi_{\text{J}}$ (see Fig. 1 for the notation followed in this
manuscript). Even though in such tilted geometries the discussion of the
Fresnel conditions at the sample/air boundary is not straightforward, we will
postpone this analysis to the last Section, and we will focus here on the
behavior inside the sample. We are then interested in determining the measured
conductivity, defined as the ratio between the current J induced in the field
direction and the modulus of the electric field E itself.
Because of anisotropy, the charge mobility within the planes is much higher
than in between stacked layers and the current J in the material is in general
not parallel to E unless propagation occurs along the the principal axes of
the crystal $(a,b,c)$. Indeed, in general one can write the conductivity
tensor as

$\displaystyle\begin{pmatrix}\text{J}_{a}\\\ \text{J}_{b}\\\
\text{J}_{c}\end{pmatrix}=\begin{pmatrix}\sigma_{ab}&0&0\\\ 0&\sigma_{ab}&0\\\
0&0&\sigma_{c}\end{pmatrix}\begin{pmatrix}\text{E}_{a}\\\ \text{E}_{b}\\\
\text{E}_{c}\end{pmatrix},$ (1)

where $\sigma_{ab}$ and $\sigma_{c}$ are the in-plane and out-of-plane
conductivities respectively. In the following we simplify the tensorial
notation by writing the reference frame in which a quantity is considered as
its subscript, e.g. Eq. (1) reads
$\textbf{J}_{abc}=\hat{\boldsymbol{\sigma}}_{abc}\textbf{E}_{abc}$. If the
wave is propagating perpendicularly to the planes ($\eta=0$), the electric
field oscillates within the SC sheets and one directly extracts $\sigma_{ab}$
from the measured transmissivity/reflectivity; analogously, with a wave
propagating within the planes ($\eta=\pi/2$) one can measure $\sigma_{c}$.
However, for a generic value of the propagation angle the measured
conductivity will be a combination of the two quantities. In other words, for
k at generic angle $\eta$ the current J will develop both longitudinal and
transverse components with respect to the momentum.

Figure 1:  Sketch of the notation used in the manuscript to define the
reference frames. The crystalline orientation defines the frame $(a,b,c)$, the
direction of the momentum defines $(t,b,l)$. The angles $\eta$ and
$\phi_{\text{J}}$ are also represented. The $tb$-plane is highlighted in blue.

To see this explicitly, we perform a rotation of angle $\eta$ around the
$b$-axis to move in the reference frame $(t,b,l)$ in which $l$ labels the
longitudinal components and $t$ labels the transverse component with respect
to the momentum in the $ac$-plane, while preserving the second transverse
component $b$. In this frame Eq. (1) transforms into
$\textbf{J}_{tbl}=\hat{\boldsymbol{\sigma}}_{tbl}\textbf{E}_{tbl}$, where the
conductivity tensor now reads

$\displaystyle\hat{\boldsymbol{\sigma}}_{tbl}=\begin{pmatrix}\sigma_{ab}\cos^{2}\eta+\sigma_{c}\sin^{2}\eta&0&(\sigma_{c}-\sigma_{ab})\sin\eta\cos\eta\\\
0&\sigma_{ab}&0\\\
(\sigma_{c}-\sigma_{ab})\sin\eta\cos\eta&0&\sigma_{ab}\sin^{2}\eta+\sigma_{c}\cos^{2}\eta\end{pmatrix}.$
(2)

Notice that the components $\text{J}_{t}$ and $\text{J}_{l}$ are coupled to
both $\text{E}_{t}$ and $\text{E}_{l}$, as one expects in an anisotropic
crystal, whereas the transverse $\text{J}_{b}$ component only couples to
$\text{E}_{b}$. As we will detail below, what one determines experimentally is
an effective conductivity defined as the ratio between the transverse current
and the transverse electric field. According to Ampere’s law
$\frac{4\pi}{c}\textbf{J}+\frac{\varepsilon_{\infty}}{c}\frac{\partial\textbf{E}}{\partial
t}=0$, with $c$ the light velocity and $\varepsilon_{\infty}$ the background
dielectric constant, so that the current in the longitudinal direction is
compensated by the displacement current. We then obtain the relation
$4\pi\text{J}_{l}-i\omega\varepsilon_{\infty}\text{E}_{l}=0$ that can be used
to eliminate the longitudinal component and write a system that only takes the
transverse components $t$ and $b$ into consideration,
$\textbf{J}_{tb}=\hat{\boldsymbol{\sigma}}_{tb}\textbf{E}_{tb}$. The
transverse conductivity tensor reads

$\displaystyle\hat{\boldsymbol{\sigma}}_{tb}=\begin{pmatrix}\sigma_{t}&0\\\
0&\sigma_{ab}\end{pmatrix},$ (3)

where

$\displaystyle\sigma_{t}(\omega,\eta)=\frac{-i\omega\varepsilon_{\infty}/4\pi\big{(}\sigma_{ab}\cos^{2}{\eta}+\sigma_{c}\sin^{2}\eta\big{)}+\sigma_{ab}\sigma_{c}}{-i\omega\varepsilon_{\infty}/4\pi+\sigma_{ab}\sin^{2}\eta+\sigma_{c}\cos^{2}\eta}.$
(4)

For an electric field polarized along $t$ ($\text{E}_{b}=0$), Eq. (4)
immediately gives the conductivity we were looking for,
$\sigma_{t}=\text{J}_{t}/\text{E}_{t}$. This expression was first derived in
Ref. [34], and its real part displays a peak with central frequency that moves
with $\eta$. To show it explicitly, we replace
$\sigma_{ab}=-\frac{\varepsilon_{\infty}\omega_{ab}^{2}}{4\pi i\omega}$ and
$\sigma_{c}=-\frac{\varepsilon_{\infty}\omega_{c}^{2}}{4\pi i\omega}$, where
$\omega_{ab}$ and $\omega_{c}$ are the in-plane and out-of-plane plasma
frequencies respectively: one then immediately sees that the real part of
$\sigma_{t}(\omega,\eta)$ peaks at frequency $\omega_{l}(\eta)$, that reads

$\displaystyle\omega_{l}^{2}(\eta)=\omega_{ab}^{2}\sin^{2}\eta+\omega_{c}^{2}\cos^{2}\eta.$
(5)

As we will discuss below, $\omega_{l}$ does not define a plasma mode of the
system: this can be immediately understood already within a classical
approach, by writing explicitly the dielectric function corresponding to the
conductivity (4). By using
$\sigma_{ab}=-\frac{i\omega}{4\pi}(\varepsilon_{ab}-\varepsilon_{\infty})$ and
$\sigma_{c}=-\frac{i\omega}{4\pi}(\varepsilon_{c}-\varepsilon_{\infty})$ we
can write the in-plane $\varepsilon_{ab}$ and out-of-plane $\varepsilon_{c}$
dielectric functions of the SC system as:

$\displaystyle\varepsilon_{ab}(\omega)=\varepsilon_{\infty}\left(1-\frac{\omega_{ab}^{2}}{\omega^{2}}\right),$
(6)

and

$\displaystyle\varepsilon_{c}(\omega)=\varepsilon_{\infty}\left(1-\frac{\omega_{c}^{2}}{\omega^{2}}\right).$
(7)

Thus Eq. (4) can be recast as
$\sigma_{t}=-\frac{i\omega}{4\pi}(\varepsilon_{t}-\varepsilon_{\infty})$,
where

$\displaystyle\varepsilon_{t}(\omega,\eta)=\varepsilon_{\infty}\frac{\big{(}\omega^{2}-\omega_{ab}^{2}\big{)}\big{(}\omega^{2}-\omega_{c}^{2}\big{)}}{\omega^{2}\big{(}\omega^{2}-\omega^{2}_{l}(\eta)\big{)}}.$
(8)

In Eq. (8) the frequency $\omega_{l}(\eta)$ in Eq. (5) appears as a divergence
of the dielectric function, while the plasma frequencies in the long-
wavelength limit appear as usual as zeros of the dielectric function. This
already proves that the scale $\omega_{l}$ does not identify a true plasma
mode. However, as we will demonstrate below, it turns out that $\omega_{l}$
provides a good approximation for the finite-momentum longitudinal plasmon of
the layered system at large momenta, i.e. in the momentum region where
retardation effects are no more relevant. As a consequence, the present
results show that the optical absorptive peak in the tilted geometry, that
appears as linear response in the long-wavelength limit, can be used to
indirectly access the plasma-wave dispersion at large momenta. Notice that in
principle Eq. (4) is valid in general for any collective mode in an
anisotropic uniaxial system, provided that the corresponding expressions of
$\sigma_{ab}(\omega)$ and $\sigma_{c}(\omega)$ are used.
Even though these considerations solve the problem of defining a transverse
conductivity at tilted angles for electric field polarized along $t$, two main
issues remain open. The first one regards the connection between the frequency
of the peak (5) and the real plasma modes of the anisotropic superconductor.
The second point is to link these results to the measured quantity in an
experiment with a generic polarization of the electric field. The first matter
will be discussed in the next Section, by using a quantum formalism based on
the description of the electromagnetic modes via the SC phase degree of
freedom. The second issue will be the subject of the last Section, where we
will explicitly study the Fresnel problem for transmission/reflection through
a tilted-grown sample and we further discuss the dependence of the measurement
on the polarization $\phi$ of the external incident electric field.

##  III Linear response of generalized plasma modes

###  III.1 Effective action description of plasma modes

To gain a better physical insight into the results of the previous section, we
will take advantage of the description of the plasma modes in the SC state
obtained via the phase degrees of freedom. Indeed, as it has been recently
discussed in Ref.s [18, 19], this is a rather powerful and elegant approach to
describe the interplay between longitudinal and transverse plasma waves in a
layered superconductor, that leads to generalized plasma modes with mixed
character at low momenta. We summarize here the main ingredients of the
derivation, referring the reader to Ref.s [18, 19] and references therein for
a detailed derivation of the layered phase-only model.
Below the critical temperature $T_{c}$ the neighboring SC planes interact with
a Josephson-like coupling [10, 23, 3, 38, 39, 40, 41] that is much weaker than
the in-plane phase stiffness. Following the notation set above, we denote the
in-plane superfluid stiffness by $D_{ab}$ and the out-of-plane one by $D_{c}$
and we write the Gaussian action for the phase fluctuations $\theta$ as [42,
43, 44]

$\displaystyle
S_{G}[\theta]=\frac{1}{8}\sum_{q}\big{[}\kappa_{0}\Omega_{m}^{2}+D_{ab}k_{ab}^{2}+D_{c}k_{c}^{2}\big{]}|\theta(q)|^{2},$
(9)

where $q=(i\Omega_{m},\textbf{k})$ is the imaginary-time 4-momentum, with
$\Omega_{m}=2\pi mT$ the bosonic Matsubara frequencies,
$k_{ab}=\sqrt{k_{a}^{2}+k_{b}^{2}}$ and $k_{c}$ are the in-plane and out-of-
plane momentum respectively and $\kappa_{0}$ is the compressibility. In the
following we denote by $\lvert\mathbf{k}\rvert^{2}=k_{ab}^{2}+k_{c}^{2}$. We
introduce the electromagnetic field A by performing in Eq. (9) the minimal
coupling substitution $i\textbf{k}\theta\to i\textbf{k}\theta+2e\textbf{A}/c$,
where $-e$ is the charge of the electron, and we also add the action of the
free electromagnetic field [1],

$\displaystyle S_{\text{e.m.}}[\textbf{A}]=\frac{1}{8\pi
c^{2}}\sum_{q}\big{[}\varepsilon_{\infty}\Omega_{m}^{2}|\textbf{A}(q)|^{2}+c^{2}|\textbf{k}\times\textbf{A}(q)|^{2}\big{]},$
(10)

Both the minimal coupling substitution and Eq. (10) are written in the Weyl
gauge in which the scalar potential is zero. We then recast the coupling
between the phase fluctuations and the electromagnetic field by performing the
substitution

$\displaystyle\boldsymbol{\psi}(q)=i\textbf{k}\theta(q)+\frac{2e}{c}\textbf{A}(q).$
(11)

These gauge-invariant fields provide a full description of the plasma modes
once the phase fluctuations are integrated out [18, 19]. In the basis
$\boldsymbol{\psi}_{abc}=\begin{pmatrix}\psi_{a}&\psi_{b}&\psi_{c}\end{pmatrix}^{T}$,
and by taking the limit for infinite compressibility that is appropriate for
cuprates,

$\displaystyle S[\boldsymbol{\psi}_{abc}]=\frac{1}{32\pi
e^{2}}\sum_{q}\boldsymbol{\psi}_{abc}^{T}(-q)\begin{pmatrix}\Omega_{m}^{2}\varepsilon_{ab}+c^{2}k_{c}^{2}&0&-c^{2}k_{a}k_{c}\\\
0&\Omega_{m}^{2}\varepsilon_{ab}+c^{2}|\textbf{k}|^{2}&0\\\
-c^{2}k_{a}k_{c}&0&\Omega_{m}^{2}\varepsilon_{c}+c^{2}k_{a}^{2}\end{pmatrix}\boldsymbol{\psi}_{abc}(q),$
(12)

where we have set the in-plane momentum along the $a$-direction ($k_{b}=0$)
without loss of generality, such that $\psi_{b}$ is decoupled, in full analogy
with the case of Eq. (2) above. In the action we have defined, in the
Matsubara formalism, the in-plane dielectric function

$\displaystyle\varepsilon_{ab}(i\Omega_{m})=\varepsilon_{\infty}\left(1+\frac{\omega_{ab}^{2}}{\Omega_{m}^{2}}\right),$
(13)

and the out-of-plane dielectric function

$\displaystyle\varepsilon_{c}(i\Omega_{m})=\varepsilon_{\infty}\left(1+\frac{\omega_{c}^{2}}{\Omega_{m}^{2}}\right),$
(14)

where the plasma frequencies are linked to the in-plane and out-of-plane
superfluid stiffness, $\omega_{ab}^{2}=4\pi e^{2}D_{ab}/\varepsilon_{\infty}$
and $\omega_{c}^{2}=4\pi e^{2}D_{c}/\varepsilon_{\infty}$ respectively.
Indeed, these go back to Eq.s (6) and (7) once the analytic continuation
$i\Omega_{m}\to\omega+i0^{+}$ is performed. Notice that the dielectric tensor
is diagonal in the basis $\boldsymbol{\psi}_{abc}$, as $(a,b,c)$ is the
reference frame of the principal axes of the crystal. By their definition in
Eq. (11), the gauge invariant fields are formally proportional to currents,
and we can then apply within the effective-action framework the same procedure
used above for the classical approach, i.e. a change of the reference frame to
describe a transverse dielectric tensor. We thus perform a rotation around the
$b$-axis that combines the $\psi_{a}$ and $\psi_{c}$ components into
transverse $\psi_{t}$ and longitudinal $\psi_{l}$ components with respect to
the momentum k. The matrix that performs the change of basis
$\boldsymbol{\psi}_{abc}\to\boldsymbol{\psi}_{tbl}=\begin{pmatrix}\psi_{t}&\psi_{b}&\psi_{l}\end{pmatrix}^{T}$
reads

$\displaystyle U=\begin{pmatrix}k_{c}/|\textbf{k}|&0&k_{a}/|\textbf{k}|\\\
0&1&0\\\ -k_{a}/|\textbf{k}|&0&k_{c}/|\textbf{k}|\end{pmatrix},$ (15)

and Eq. (12) transforms in this basis as

$\displaystyle S[\boldsymbol{\psi}_{tbl}]=\frac{1}{32\pi
e^{2}}\sum_{q}\boldsymbol{\psi}_{tbl}^{T}(-q)\mathcal{D}_{tbl}^{-1}\boldsymbol{\psi}_{tbl}(q),$
(16)

where the matrix of the coefficients reads

$\displaystyle\mathcal{D}_{tbl}^{-1}=\begin{pmatrix}\Omega_{m}^{2}(\varepsilon_{ab}k_{c}^{2}+\varepsilon_{c}k_{a}^{2})/|\textbf{k}|^{2}+c^{2}|\textbf{k}|^{2}&0&\Omega_{m}^{2}(\varepsilon_{c}-\varepsilon_{ab})k_{a}k_{c}/|\textbf{k}|^{2}\\\
0&\Omega_{m}^{2}\varepsilon_{ab}+c^{2}|\textbf{k}|^{2}&0\\\
\Omega_{m}^{2}(\varepsilon_{c}-\varepsilon_{ab})k_{a}k_{c}/|\textbf{k}|^{2}&0&\Omega_{m}^{2}(\varepsilon_{ab}k_{a}^{2}+\varepsilon_{c}k_{c}^{2})/|\textbf{k}|^{2}\end{pmatrix}.$
(17)

Before moving forward and studying the linear response, we here provide a
brief review of the generalized plasma modes that Eq. (16) describes, useful
in the following to provide a physical interpretation of the finite-frequency
peak in the real part of the conductivity. The action identifies two
longitudinal-transverse mixed modes and one decoupled purely transverse mode
along the $b$-direction. The former ones cannot be studied separately, as the
anisotropy of layered superconductors is such that the $\psi_{t}$ and
$\psi_{l}$ components are coupled for generic direction of the momentum, that
is, the off-diagonal elements of Eq. (17) are nonvanishing. On physical
grounds, this is a manifestation of retardation effects: as already seen in
the previous section, at generic wavevector the current induced in the system
is not parallel to ${\bf E}$. This induces a longitudinal electric field in
the system in response to a transverse perturbation, making longitudinal and
transverse response unavoidably mixed. Since the displacement current scales
as $\partial{\bf E}/\partial(ct)$, the corrections coming from retardation
effects are also named relativistic, as they vanish for $c\to\infty$. The
dispersion relations of the two modes obtained from Eq. (17) read

$\displaystyle\omega_{\pm}^{2}(\textbf{k})=\frac{1}{2}\left(\omega_{ab}^{2}+\omega_{c}^{2}+\frac{c^{2}}{\varepsilon_{\infty}}|\textbf{k}|^{2}\pm\sqrt{(\omega_{ab}^{2}-\omega_{c}^{2})^{2}+\frac{c^{4}}{\varepsilon_{\infty}^{2}}|\textbf{k}|^{4}-2\frac{c^{2}}{\varepsilon_{\infty}}(k_{a}^{2}-k_{c}^{2})(\omega_{ab}^{2}-\omega_{c}^{2})}\right),$
(18)

A detailed discussion of the properties of the generalized plasma modes of
single-layer anisotropic superconductors can be found in Ref. [18].
Nonetheless, it is important here to stress the main physical outcomes of the
present derivation. The generalized dispersions (18) describe two regular
functions of the momenta that give $\omega_{+}(\textbf{k}\to 0)\to\omega_{ab}$
and $\omega_{-}(\textbf{k}\to 0)\to\omega_{c}$. For generic propagation
direction $\eta$ and for momenta
$|\textbf{k}|\lesssim\bar{k}=\sqrt{\varepsilon_{\infty}(\omega_{ab}^{2}-\omega_{c}^{2})}/c$
these modes have mixed longitudinal/transverse character, with a degree of
mixing that is maximum at $\eta=\pi/4$ and vanishes as one moves along the
main crystallographic directions ($k_{a}=0$ or $k_{c}=0$), as one immediately
realizes by the structure of the off-diagonal matrix elements of Eq. (17)
scaling as $k_{a}k_{c}$. Explicitly neglecting this coupling, i.e. setting the
off-diagonal elements to zero, would result in having the two modes uncoupled,
one of which purely transverse and the other purely longitudinal. In this case
the dispersion relation of the latter, that is by definition the plasma mode
of the system, can be found by setting to zero the bottom-right element of
$\mathcal{D}_{tbl}^{-1}$:

$\displaystyle\varepsilon_{ab}(\omega)\frac{k_{a}^{2}}{|{\bf
k}|^{2}}+\varepsilon_{c}(\omega)\frac{k_{c}^{2}}{|{\bf
k}|^{2}}=\varepsilon_{ab}(\omega)\sin^{2}\eta+\varepsilon_{c}(\omega)\cos^{2}\eta=0,$
(19)

where we have performed the analytic continuation
$i\Omega_{m}\to\omega+i0^{+}$ and used $k_{c}=|\textbf{k}|\cos\eta$ and
$k_{a}=|\textbf{k}|\sin\eta$. Using the definitions of the dielectric
functions in Eq.s (13) and (14), the solution of Eq. (19) is exactly the
frequency

$\omega_{l}^{2}({\bf k})=\omega^{2}_{ab}\frac{k_{a}^{2}}{|{\bf
k}|^{2}}+\omega^{2}_{c}\frac{k_{c}^{2}}{|{\bf
k}|^{2}}\equiv\omega_{ab}^{2}\sin^{2}\eta+\omega_{c}^{2}\cos^{2}\eta,$ (20)

defined in Eq. (5). In addition, one can easily see from Eq. (18) that in the
limit $c\to\infty$, i.e. in the regime where $\bar{k}/|\textbf{k}|\to 0$,
retardation (or relativistic) effects can be neglected and one obtains

$\omega_{-}({\bf k})\to\omega_{l}({\bf
k}),\quad|\textbf{k}|\gg\bar{k}=\sqrt{\varepsilon_{\infty}(\omega_{ab}^{2}-\omega_{c}^{2})}/c.$
(21)

In other words, the expression $\omega_{l}(\eta)$ defines the longitudinal-
plasmon dispersion in a layered superconductor that one obtains by neglecting
retardation effects, as one usually does in the standard RPA approach where
only Coulomb interactions are included [42, 43, 45, 44, 46, 47, 48]. We also
note in passing that the limit of $\omega_{l}({\bf k})$ for $\textbf{k}\to 0$
is non-regular as it depends on the direction $\eta$ of the momentum. As shown
above, this is not the case for the real electromagnetic mode $\omega_{-}$,
that is regular at $|\textbf{k}|=0$. In Fig. 2(a) we show
$\omega_{-}(\textbf{k})$ and $\omega_{l}(\eta)$ for small values of the
propagation angle: as one can see, as $|\textbf{k}|$ overcomes the $\bar{k}$
scale $\omega_{-}$ rapidly approaches the $\omega_{l}$ limit and the mode
becomes longitudinal. By using realistic values of plasma frequencies in
cuprates one sees that $\bar{k}\sim\mu\text{m}^{-1}$. As such, this scale is
two orders of magnitude smaller than the momenta usually accessible in RIXS
[25, 26, 27, 28, 29] or EELS [30, 31, 32] experiments, that are not sensitive
to the relativistic regime and probe the plasmon dispersion given by Eq. (20).

###  III.2 Interpretation of the conductivity peak of plasmons

From the action in Eq. (16) we can perform the integration of $\psi_{l}$ and
work with an action of the transverse components
$\boldsymbol{\psi}_{tb}=\begin{pmatrix}\psi_{t}&\psi_{b}\end{pmatrix}^{T}$
only. This procedure is equivalent to using Ampere’s law as a condition to
eliminate the longitudinal components, see Eq. (2) and the discussion below.
One is left with an action that reads

$\displaystyle S[\boldsymbol{\psi}_{tb}]=\frac{1}{32\pi
e^{2}}\sum_{q}\boldsymbol{\psi}_{tb}^{T}(-q)\begin{pmatrix}\Omega_{m}^{2}\varepsilon_{t}+c^{2}|\textbf{k}|^{2}&0\\\
0&\Omega_{m}^{2}\varepsilon_{ab}+c^{2}|\textbf{k}|^{2}\end{pmatrix}\boldsymbol{\psi}_{tb}(q),$
(22)

where

$\displaystyle\varepsilon_{t}(i\Omega_{m},\eta)=\frac{\varepsilon_{ab}\varepsilon_{c}}{\varepsilon_{ab}\sin^{2}\eta+\varepsilon_{c}\cos^{2}\eta},$
(23)

is a dielectric function that describes the transverse linear response of the
superconductor along the $t$-axis. Indeed, by making use of the relation
$\varepsilon_{\alpha}=\varepsilon_{\infty}+4\pi i\sigma_{\alpha}/\omega$
between the optical conductivity and the dielectric function along the
direction $\alpha$ [49] one recovers $\sigma_{t}$ as in Eq. (4). Remarkably,
the denominator of $\varepsilon_{t}$ can be brought back to the left-hand side
of the characteristic equation (19) for the uncoupled longitudinal mode.
Indeed, by using the explicit expressions in Eq.s (13) and (14) for the in-
plane and out-of-plane dielectric functions of plasma modes and performing the
analytic continuation $i\Omega_{m}\to\omega+i0^{+}$, one can rewrite Eq. (23)
as
$\varepsilon_{t}(\omega,\eta)=\varepsilon_{\infty}(\omega^{2}-\omega_{ab}^{2})(\omega^{2}-\omega_{c}^{2})/\omega^{2}(\omega^{2}-\omega_{l}^{2}(\eta))$,
exactly as in Eq. (8) above.

Figure 2:  (a) Dispersion of the Josephson plasma mode
$\omega_{-}(\textbf{k})$ (solid lines) and $\omega_{l}(\eta)$ (dashed lines)
for different small propagation angles, having chosen
$\omega_{ab}/\omega_{c}=100$. (b) Real part of the conductivity $\sigma_{t}$
in the case of superconducting plasma modes for corresponding values of $\eta$
of panel (a). The conductivity spectra are normalized to the maximum value of
the peak at $\eta=0.5^{\circ}$. Phenomenological damping parameter is taken as
$\gamma=0.1\omega_{c}$.

While this result has been formally obtained already within the Maxwell’s
classical formalism of the previous Section, we can now identify the energy of
the peak in the transverse conductivity at vanishing momentum as the value of
the longitudinal plasma mode in the high-momentum regime, i.e. the same
usually probed by EELS and RIXS, since $\omega_{l}(\eta)$ is a good
approximation of the dispersion of the lower mode $\omega_{-}(\textbf{k})$ for
$|\textbf{k}|\gg\bar{k}$, see Eq. (21).
The real part of the conductivity $\sigma_{t}$ is shown in Fig. 2(b), where we
also introduced a finite damping parameter $\gamma$ when performing the
analytic continuation $i\Omega_{m}\to\omega+i\gamma$. We emphasize once more
that such a peak is not a direct manifestation of the Josephson plasmon of the
superconductor [34, 33, 35, 36] which, as discussed above, for vanishing
momentum is at frequency $\omega_{c}$ for every direction $\eta$. Indeed,
plasma modes appear as zeroes of the dielectric function and do not lead to
finite-frequency peaks in the conductivity. Instead, the absorptive peak at
$\omega_{l}(\eta)$ is a manifestation of the mixing mechanism between in-plane
and out-of-plane plasma modes described in the previous section, as the
dielectric function in Eq. (8) comes directly from the action for the coupled
modes Eq. (16). On a more general ground, our derivation clarifies that a
signature of longitudinal nature appears in the transverse response whenever
the longitudinal mode is coupled to the transverse one without directly
participating in the detection, that is, the degree of freedom is integrated
out.
It is worth mentioning that our derivation is not restricted to electron-doped
cuprates, in which the peak has already been experimentally reported [34, 33,
35, 36], but it is in principle valid for any single-layer superconductor,
like the hole-doped LSCO. We also point out that the results could be extended
to bilayer superconductors like YBCO, that display two Josephson plasmons at
frequencies $\omega_{c1}$ and $\omega_{c2}$. Indeed, by using the out-of-plane
bilayer dielectric function [5]
$\varepsilon_{c}=\varepsilon_{\infty}(\omega^{2}-\omega_{c1}^{2})(\omega^{2}-\omega_{c2}^{2})/\omega^{2}(\omega^{2}-\omega_{T}^{2})$,
with $\omega_{T}^{2}=\omega_{c1}^{2}d_{2}+\omega_{c2}^{2}d_{1}$ and $d_{1,2}$
the intra- and inter-bilayer spacings, one predicts two absorptive peaks in
the conductivity centered at the high-momenta values of the dispersions of the
Josephson modes [19]. The high-energy one follows the same trend of the peak
in single-layer superconductors, moving with $\eta$ from $\omega_{c1}$ to
$\omega_{ab}$. The low-energy one quickly moves from $\omega_{c2}$ to
$\omega_{T}$ even for small values of $\eta$, and does not disappear for
$\eta=\pi/2$ [4, 5, 6, 7, 8, 9, 11, 13].

##  IV Fresnel equations at normal incidence on a tilted-grown sample

To link the results obtained in the previous sections to experiments we must
consider the measured quantity, that is the electric field transmitted or
reflected through the sample with respect to the incident wave, and link it to
the conductivity $\sigma_{t}$. Moreover, one might argue that due to the fact
that the system is anisotropic, both angles $\eta$ and $\phi_{\text{J}}$ that
define the current propagation within the material differ respectively from
$\eta_{in}$, the angle between the incident momentum of the external wave and
the normal to the planes, and $\phi$, angle between the $b$-axis and the
electric field that describes its polarization. To this aim, we must write the
Fresnel conditions at the boundaries of the sample. In this section we analyze
the configuration in which a THz pulse is at normal incidence on a thin-film
layered superconductor, grown with tilted planes at a small angle $\eta$ [34,
33, 35, 36], and we show that in this case the theoretical results can be
easily related to the experiments, see Appendix A.

Figure 3:  (a) Sketch of the experimental configuration with a THz wave at
normal incidence on a tilted-grown sample of thickness $d$. The reference
frame $(t,b,l)$ for this configuration is also shown, to highlight the
direction of the external electric field $\textbf{E}^{in}$ that defines the
polarization angle $\phi$. (b) Real part of the measured conductivity as a
function of frequency and polarization angle as in Eq. (30). Solid black line
corresponds to $\omega_{r}(\eta,\phi)$ as in Eq. (32). In this plot
$\eta=0.25^{\circ}$, $d=0.150\text{ }\mu\text{m}$,
$\omega_{ab}/\omega_{c}=100$ and $\gamma=0.1\omega_{c}$. (c) Fit of
experimental data from Ref. [36] with $\sigma(\omega,\eta,\phi)$ for different
polarization angles. Fitting parameters are extracted at once from the three
measurements: $\omega_{ab}/2\pi=60\text{ THz}$, $\omega_{c}/2\pi=0.6\text{
THz}$, $\eta=0.26^{\circ}$, $\gamma=0.075\text{ THz}$.

Following the notation set above, we define the reference frame $(t,b,l)$ in
such a way that the $tb$-plane corresponds to the interface and the $l$-axis
is perpendicular to it, see Fig. 3(a). At normal incidence $\eta_{in}=\eta$
immediately, as the momentum of the wave does not change its direction while
crossing the interface. Within the material, the $b$-polarized and
$t$-polarized electric fields are decoupled and travel with different values
of the wave-vector, see Eq. (16) and the discussion below. In particular, from
Eq. (17) the equations of motion read
$|\textbf{k}|^{2}=\omega^{2}\varepsilon_{ab}/c^{2}$ for the former and
$|\textbf{k}|^{2}=\omega^{2}\varepsilon_{t}/c^{2}$ for the latter [22], with
$\varepsilon_{t}$ defined in Eq. (23). We then impose the continuity of the
tangential components of the electric field, $\text{E}_{t}$ and
$\text{E}_{b}$, and of the magnetic field, $\text{B}_{t}$ and $\text{B}_{b}$,
at the interface $l=0$ and at $l=d$, with $d$ the sample thickness. By solving
the system set by these conditions one finds the transmission and reflection
coefficients for the $t$ and the $b$ components of the field in the thin-film
configuration, that read

$\displaystyle\text{T}_{t}=\frac{\mathcal{T}_{t}\mathcal{T}_{t}^{\prime}e^{in_{t}\omega
d/c}}{1-\mathcal{R}_{t}^{2}e^{2in_{t}\omega d/c}},$ (24)
$\displaystyle\text{R}_{t}=\frac{\mathcal{R}_{t}(1-e^{2in_{t}\omega
d/c})}{1-\mathcal{R}_{t}^{2}e^{2in_{t}\omega d/c}},$ (25)
$\displaystyle\text{T}_{b}=\frac{\mathcal{T}_{b}\mathcal{T}_{b}^{\prime}e^{in_{b}\omega
d/c}}{1-\mathcal{R}_{b}^{2}e^{2in_{b}\omega d/c}},$ (26)
$\displaystyle\text{R}_{b}=\frac{\mathcal{R}_{b}(1-e^{2in_{b}\omega
d/c})}{1-\mathcal{R}_{b}^{2}e^{2in_{b}\omega d/c}},$ (27)

where $n_{\alpha}=\sqrt{\varepsilon_{\alpha}}$ is the refractive index along
the direction $\alpha$, $\mathcal{T}_{\alpha}=2/(1+n_{\alpha})$ is the
transmission coefficient going from vacuum to the material,
$\mathcal{T}^{\prime}_{\alpha}=2n_{\alpha}/(1+n_{\alpha})$ is analogously the
transmission coefficient from the sample to the vacuum, and
$\mathcal{R}_{\alpha}^{2}=1-\mathcal{T}_{\alpha}\mathcal{T}_{\alpha}^{\prime}$
accounts for the Fabry-Perot interference within the thin film. The ratios
$\text{T}_{t}/\text{T}_{b}$ and $\text{R}_{t}/\text{R}_{b}$ carry the
information on the rotation of the polarization of the transmitted or
reflected wave. By the definition of the dielectric function $\varepsilon_{t}$
in Eq. (23) one has that $\varepsilon_{t}\simeq\varepsilon_{ab}$ under the
assumption of small tilt angle of the planes. Then $n_{t}\simeq n_{b}$ and the
ratios are approximately 1: one can thus conclude that the polarization of the
transmitted or reflected wave does not differ significantly from the one of
the incident wave in the experiment. With the same reasoning
$\sigma_{t}\simeq\sigma_{ab}$, so that the transverse current is approximately
parallel to the field, see Eq. (3), and we can thus conclude that
$\phi\simeq\phi_{\text{J}}$. In an experiment the measured quantity (see
Appendix A) is either the transmissivity

$\displaystyle\text{T}=\text{T}_{b}\cos^{2}\phi+\text{T}_{t}\sin^{2}\phi,$
(28)

or, analogously, the reflectivity

$\displaystyle\text{R}=\text{R}_{b}\cos^{2}\phi+\text{R}_{t}\sin^{2}\phi.$
(29)

From these quantities one can define the measured transverse conductivity.
Indeed, under the assumption of film-thickness $d$ much smaller than the
wavelength of the radiation inside the material and its penetration depth, one
finds [36]

$\displaystyle\sigma(\omega,\eta,\phi)=\frac{2}{Z_{0}d}\left(\frac{1}{\text{T}}-1\right),$
(30)

where $Z_{0}=4\pi/c$ is the impedance of free space. This proportionality
establishes the link between the measured quantity and the theoretical
conductivity we were looking for. Moreover, in the case of cuprates one can
numerically estimate $\text{T}\ll 1$ in Eq. (28), and then approximate
$\sigma\propto 1/\text{T}$. Since also $\sigma_{ab}\propto 1/\text{T}_{b}$ and
$\sigma_{t}\propto 1/\text{T}_{t}$, from (28) one can express the measured
transverse conductivity as

$\displaystyle\sigma(\omega,\eta,\phi)\simeq\frac{\sigma_{ab}\sigma_{t}}{\sigma_{ab}\sin^{2}\phi+\sigma_{t}\cos^{2}\phi}.$
(31)

With $\sigma_{t}$ from Eq. (4) and using the expressions of $\sigma_{ab}$ and
$\sigma_{c}$ for the superconductor, one finds that the real part of the
conductivity has a peak at a resonance frequency $\omega_{r}(\eta,\phi)$ that
depends on both $\eta$ and $\phi$:

$\displaystyle\omega_{r}^{2}(\eta,\phi)=\frac{\omega_{ab}^{2}\sin^{2}\eta\sin^{2}\phi+\omega_{c}^{2}(1-\sin^{2}\eta\sin^{2}\phi)}{1-\sin^{2}\eta\cos^{2}\phi+\left(\frac{\omega_{c}}{\omega_{ab}}\right)^{2}\sin^{2}\eta\cos^{2}\phi}.$
(32)

In Fig. 3(b) we show the real part of Eq. (30) as a function of the external
polarization angle and we compare the peak emerging in the measured
conductivity with Eq. (32). Indeed, the approximated expression (31) provides
an excellent description of the experimental data in Ref.s [34, 33, 35, 36],
and the frequency Eq. (32) establishes a link between the peak of the
experimental conductivity and the plasma frequencies $\omega_{ab}$ and
$\omega_{c}$, which can then be extracted as fitting parameters given the
angles $\eta$ and $\phi$. In Fig. 3(c) we fit experimental data from Ref. [36]
to provide an estimate of the in-plane and out-of-plane plasma frequencies of
the overdoped $\text{La}_{1.87}\text{Ce}_{0.13}\text{CuO}_{4}$ ($T_{c}=21$ K)
at $5$ K. So far, it was only empirically observed in Ref. [33] that the data
could be well fitted by using an effective conductivity
$\sigma_{t}(\omega,\eta_{\text{eff}})$ having the same functional form of Eq.
(4), but with an effective tilt angle $\eta_{\text{eff}}=\eta\sin\phi$. This
result actually follows from Eq. (31) in the case of small angle $\eta$
between the momentum and the $c$-axis of the crystal, which is indeed the
configuration of Ref. [33]. In this case, the frequency of the peak in Eq.
(32) can be approximated as

$\displaystyle\omega_{r}^{2}(\eta,\phi)\simeq\omega_{ab}^{2}\sin^{2}\eta_{\text{eff}}+\omega_{c}^{2}\cos^{2}\eta_{\text{eff}}=\omega_{l}^{2}(\eta_{\text{eff}}),$
(33)

where again $\eta_{\text{eff}}=\eta\sin\phi$.
At first, one might as well consider the configuration in which the THz pulse
is incident with a small angle on a $c$-axis grown sample. However, computing
the Fresnel conditions in this case results in featureless transmissivity and
reflectivity, and no peak appears in the real part of the conductivity (see
Appendix A for details)

##  V Conclusions

In this manuscript we studied the optical absorption in layered
superconductors in a tilted geometry, where the light propagates inside the
sample by forming a small angle with the stacking direction. We showed that
such a geometry makes it possible to observe with optics, that is essentially
a zero-momentum probe, a direct signature of the plasmon dispersion at momenta
of the order of a fraction of the Brillouin zone, that is usually probed by
RIXS or EELS. The basic physical mechanism behind this observation is the
intrinsic mixing between transverse and longitudinal electromagnetic modes in
a layered material, due to the anisotropy between the in-plane and out-of-
plane response. Such mixing, that is absent when light propagates along the
main crystallographic axes, leads to the emergence of an absorption peak in
the transverse optical conductivity in tilted geometry. Interestingly, we can
show analytically that the peak frequency moves as a function of the tilting
angle according to the functional law that the physical longitudinal plasmon
displays at momenta larger than the scale where transverse/longitudinal mixing
is relevant. In cuprates, where the SC c-axis plasmon is weakly affected by
Landau damping due to the opening of a large spectral gap below $T_{c}$, the
peak is well defined at small tilting angle, and it has been indeed observed
in several electron-doped cuprates [34, 33, 35, 36]. Here we argue that the
same effect can be seen in any layered sample, provided that the appropriate
Fresnel geometry is implemented. In addition, we provide an analytical
expression for the peak frequency as a function of both tilting angle and
light polarization, that can be used to obtain from a single-set of
measurements the relevant scales for plasma excitations in these systems. It
is worth stressing that in the last few years, after charged plasmons have
been detected for the first time with high-resolution RIXS [25, 26, 27, 28,
29] and EELS [30, 31, 32] experiments, an intense discussion emerged on the
nature of charge fluctuations in these correlated materials [31, 32]. The all-
optical measurement proposed here is in principle a bulk probe, it is not
affected by the lack of sensitivity at small momenta connected to plasmon
measurements via charge-detecting probes, and it allows for a precise control
on the momentum value, that can be problematic e.g. to EELS [32]. As a
consequence, the experimental verification of this idea could provide an
additional knob to explore charge fluctuations in cuprates, and their possible
interplay with other collective modes of the systems.

## Acknowledgements

We acknowledge useful discussion with N.P. Armitage. We acknowledge financial
support by EU under project MORE-TEM ERC-Syn grant agreement No. 951215 and by
Sapienza University of Rome under Project Ateneo 2022 RP1221816662A977 and
Project Ateneo 2023 RM123188E357C540.

## Appendix A

Figure 4:  Sketch of the two possible experimental configurations discussed in
this Appendix. In both cases the reference frame $(x,y,z)$ is defined such
that $xy$ is the vacuum-sample interface and $xz$ is the plane of incidence.
For graphical purposes, only TM polarized waves are depicted. (a) Geometry of
the configuration in which the incident THz pulse is at normal incidence on a
thin-film crystal grown with layers tilted at angle $\eta$ with respect to the
vacuum-sample interface, as in Section 3.3. In this case the $(x,y,z)$
reference frame corresponds to the $(t,b,l)$ frame introduced in the main
text. (b) Geometry of the configuration in which the incident field is at
oblique incidence with angle $\eta_{in}$ on a thin-film sample with planes
parallel to the interface. In this case the $(x,y,z)$ reference frame
corresponds to the crystallographic $(a,b,c)$ frame.

In this Appendix we derive the transmitted electric field in two experimental
configurations in which the electromagnetic wave travels with a finite angle
with respect to the stacking direction of the planes, by means of standard
Fresnel-like boundary conditions applied on a uniaxial film. Let us consider a
transmission experiment on a superconductor placed in the region $0<z<d$, as
in Fig. 4. The electric field satisfies Maxwell’s equations

$\displaystyle\begin{cases}\nabla^{2}\mathbf{E}-\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{E}}{\partial
t^{2}}=0&z<0,\text{ }z>d\\\
\nabla^{2}\mathbf{E}-\nabla(\nabla\cdot\mathbf{E})-\frac{1}{c^{2}}\frac{\partial^{2}(\hat{\boldsymbol{\varepsilon}}\mathbf{E})}{\partial
t^{2}}=0&0<z<d\\\ \end{cases},$ (34)

where $\hat{\boldsymbol{\varepsilon}}$ is the dielectric tensor of the
uniaxial material. By expanding the electric field on a basis of plane waves,
Eq. (34) becomes a linear system for the Cartesian Fourier components of the
electric field

$\displaystyle\begin{cases}(\lvert\mathbf{k}\rvert^{2}-\omega^{2}/c^{2})\delta_{\alpha\beta}\text{E}_{\beta}=0&z<0,\text{
}z>d\\\
(\lvert\mathbf{k}\rvert^{2}\delta_{\alpha\beta}-k_{\alpha}k_{\beta}-\omega^{2}\varepsilon_{\alpha\beta}/c^{2})\text{E}_{\beta}=0&0<z<d\\\
\end{cases}.$ (35)

A propagating solution is allowed whenever the determinant of this system is
zero. For the experimental configuration of Fig. 4(a) in which the frame
$(x,y,z)$ corresponds to $(t,b,l)$, as in Section 3.3, the incoming momentum
is along the $z$ (or $l$) direction. In this geometry the dielectric tensor is
defined as (see Eq.s (2) and (17))

$\displaystyle\hat{\boldsymbol{\varepsilon}}=\begin{pmatrix}\varepsilon_{ab}\cos^{2}{\eta}+\varepsilon_{c}\sin^{2}{\eta}&0&(\varepsilon_{ab}-\varepsilon_{c})\cos{\eta}\sin{\eta}\\\
0&\varepsilon_{ab}&0\\\
(\varepsilon_{ab}-\varepsilon_{c})\cos{\eta}\sin{\eta}&0&\varepsilon_{ab}\sin^{2}{\eta}+\varepsilon_{c}\cos^{2}{\eta}\\\
\end{pmatrix}.$ (36)

From Eq. (35) one can notice that the subspace associated with $\text{E}_{y}$
is decoupled from the one associated with $\text{E}_{x}$ and $\text{E}_{z}$.
As a consequence, one can show immediately that $y$-polarized electric fields
in the material propagate with wave-vector
$\lvert\mathbf{k}\rvert^{2}=\omega^{2}\varepsilon_{ab}/c^{2}$, while $x$\- and
$z$-polarized electric fields propagate with wave-vector
$\lvert\mathbf{k}\rvert^{2}=\omega^{2}\varepsilon_{t}/c^{2}$, where

$\displaystyle\varepsilon_{t}=\frac{\varepsilon_{ab}\varepsilon_{c}}{\varepsilon_{ab}\sin^{2}{\eta}+\varepsilon_{c}\cos^{2}{\eta}},$
(37)

as in Eq. (23) in the main text. Conversely, for the experimental
configuration of Fig. 4(b) in which the frame $(x,y,z)$ corresponds to the
crystallographic frame $(a,b,c)$, the dielectric tensor is diagonal
$\hat{\varepsilon}_{\alpha\beta}=\varepsilon_{\alpha}\delta_{\alpha\beta}$
(see Eq.s (1) and (12)). As the wave-vector belongs to the $xz$-plane, again
in this case the equation for $\text{E}_{y}$ is decoupled from the other two
components. One can then in both cases solve the system separately for the
transmission of the $y-$ and for the mixed $xz-$ polarized components of
$\mathbf{E}$. In the following we will refer to the former component as the
Transverse Electric (TE) field and to the latter as the Transverse Magnetic
(TM) field, as one would commonly do in the oblique incidence configuration in
which the plane of incidence is $xz$. Notice that the transmitted wave
$\textbf{E}^{\text{T}}$ is generically polarized along a direction
$\phi^{\prime}$ that differ from the polarization $\phi$ of the incident wave
$\textbf{E}^{in}$, although the experiment is still set so to measure the
outgoing $\phi$-polarized wave $\textbf{E}^{\text{T}^{\prime}}$. Assuming that
the TE and TM transmission coefficients are known, such that
$\text{E}^{\text{T}}_{\text{TE,TM}}=\text{T}_{\text{TE,TM}}\text{E}^{in}_{\text{TE,TM}}$
(see Fig. 5 for the notation), one finds the measured transmitted field
polarized along $\phi$ as

$\displaystyle\text{E}^{\text{T}^{\prime}}$
$\displaystyle=\text{E}^{\text{T}}\cos{(\phi^{\prime}-\phi)}$
$\displaystyle=\text{E}^{\text{T}}\cos{\phi^{\prime}}\cos{\phi}+\text{E}^{\text{T}}\sin{\phi^{\prime}}\sin{\phi}$
$\displaystyle=\text{T}_{\text{TE}}\text{E}^{in}_{\text{TE}}\cos{\phi}+\text{T}_{\text{TM}}\text{E}^{in}_{\text{TM}}\sin{\phi}$
$\displaystyle=(\text{T}_{\text{TE}}\cos^{2}{\phi}+\text{T}_{\text{TM}}\sin^{2}{\phi})\text{E}^{in}\equiv\text{T}\text{E}^{in}.$
(38)

Analogously, one can express the reflected field in a similar way. For the
configuration discussed in the Section 3.3, see Fig. 4(a), the TE and TM
components stand for the $b$ and $t$ components respectively, and one recovers
Eq. (28) of the main text. We now compute explicitly the transmission
coefficients $\text{T}_{\text{TE}}$ and $\text{T}_{\text{TM}}$ for the two
configurations separately.

Figure 5:  (a) Incoming electric field $\mathbf{E}^{in}$ at a generic
polarization angle $\phi$ with respect to the TE axis (corresponding to the
$y$ axis and to the crystallographic $b$ axis in both configurations of Fig.
4). (b) Transmitted electric field $\mathbf{E}^{\text{T}}$ found by means of
the appropriate set of boundary conditions. The transmitted field still lies
in the TM-TE plane due to conservation of the parallel component of the
incident momentum, but $\mathbf{E}^{\text{T}}$ is generically polarized with
angle $\phi^{\prime}\neq\phi$. (c) Comparison between the directions of the
transmitted field $\textbf{E}^{\text{T}}$ and of the projected field
$\textbf{E}^{\text{T}^{\prime}}$ along the incoming polarization, that is the
measured quantity in a transmission experiment.

For the first configuration (Fig. 4(a))
$\text{T}_{\text{TE}}\equiv\text{T}_{b}$ and
$\text{T}_{\text{TM}}\equiv\text{T}_{t}$, thus the TE and TM components
propagate with different refractive indices, $n_{b}=\sqrt{\varepsilon_{ab}}$
and $n_{t}=\sqrt{\varepsilon_{t}}$ respectively. Imposing the continuity of
the tangential components of the electric E and magnetic B fields, one
recovers the usual expression for transmission at normal incidence on a slab,
namely

$\text{T}_{\alpha}=\frac{\mathcal{T}_{\alpha}\mathcal{T}^{\prime}_{\alpha}e^{in_{\alpha}\omega
d/c}}{1-\mathcal{R}_{\alpha}^{2}e^{2in_{\alpha}\omega d/c}},$ (39)

where $\mathcal{T}_{\alpha}=2/(n_{\alpha}+1)$ is the transmission coefficient
from vacuum $n=1$ to a medium with refractive index $n_{\alpha}$,
$\mathcal{T}^{\prime}_{\alpha}=2n_{\alpha}/(n_{\alpha}+1)$ is analogously the
transmission coefficient from medium to vacuum and the denominator accounts
for the Fabry-Perot interference inside the slab of thickness $d$, with
$\mathcal{R}^{2}_{\alpha}=1-\mathcal{T}_{\alpha}\mathcal{T}_{\alpha}^{\prime}$.
Eq. (39) can also be expressed as

$\text{T}_{\alpha}=\frac{2n_{\alpha}}{2n_{\alpha}\cos{\zeta_{\alpha}}-i(n_{\alpha}^{2}+1)\sin{\zeta_{\alpha}}},$
(40)

where $\zeta_{\alpha}=n_{\alpha}\omega d/c$. In the approximation $d\ll\lvert
n_{\alpha}\rvert\omega/c$, i.e. the thickness of the film is much smaller than
the wavelength of the radiation inside the material
$\lambda=\text{Re}(n_{\alpha})\omega/c$ and the skin depth
$\delta=\text{Im}(n_{\alpha})\omega/c$, one can take at first order in
$\zeta_{\alpha}$

$\frac{1}{\text{T}_{\alpha}}\approx
1-i\frac{n_{\alpha}^{2}+1}{2n_{\alpha}}\zeta_{\alpha}=1-i\frac{(n_{\alpha}^{2}+1)\omega
d}{2c}.$ (41)

Using the relation between refractive index and conductivity
$\sigma_{\alpha}=\frac{\omega}{4\pi i}(n_{\alpha}^{2}-1)$ one can rewrite Eq.
(41) as

$\frac{1}{\text{T}_{\alpha}}=1+\frac{4\pi d\sigma_{\alpha}}{2c}-i\frac{2\omega
d}{c}\approx 1+\frac{4\pi d\sigma_{\alpha}}{2c},$ (42)

where again we considered $\omega d/c\ll 1$. Consequently, one finds

$\sigma_{\alpha}=\frac{2}{Z_{0}d}\left(\frac{1}{\text{T}_{\alpha}}-1\right),$
(43)

where $Z_{0}=4\pi/c$. This relation between the conductivity and the
transmissivity is valid along both the $b$ and $t$ directions. On the other
hand, one can imagine to extract an experimental conductivity from the
experimental transmissivity T as in Eq. (Appendix A) by applying the same
relation, see e.g. Ref. [36] where Eq. (30) is used.
In the second configuration (Fig. 4(b)) the interface is parallel to the
$ab$-plane of the crystal and one needs to solve a wider set of continuity
conditions. Indeed, one must impose the continuity of the tangential
components of the electric E and magnetic B fields as for the previous
configuration, but also the continuity of the normal component of the
displacement field D [50, 51]. To understand how the transmission occurs in
this case, let us first recall the results expected for an isotropic film,
where the propagation of the electromagnetic wave inside the sample is defined
by a unique refractive index $n$. In this case one easily finds that
$\text{T}^{\text{(iso)}}\equiv\text{E}^{\text{T}^{\prime}}/\text{E}^{in}$
reads:

$\text{T}^{\text{(iso)}}(\eta_{in},n)=\frac{\mathcal{T}\mathcal{T}^{\prime}e^{in\omega
d\cos{\eta}/c}}{1-\mathcal{R}^{2}e^{2in\omega d\cos{\eta}/c}},$ (44)

where $\eta$ is the propagation angle inside the material, see Fig. 4(b),
while $\eta_{in}$ is the external angle of incidence. Here we defined as
before $\mathcal{T}=2\cos{\eta_{in}}/(n\cos{\eta_{in}}+\cos\eta)$ as the
transmission coefficient from vacuum to the medium and analogously
$\mathcal{T}^{\prime}=2n\cos{\eta}/(n\cos{\eta_{in}}+\cos{\eta})$ as the
transmission coefficient from medium to vacuum, while again the denominator of
Eq. (44) accounts for the Fabry-Perot interference inside the slab of
thickness $d$, with $\mathcal{R}^{2}=1-\mathcal{T}\mathcal{T}^{\prime}$. In
Eq. (44) we made explicit the dependence of the transmissivity T on the
incident angle and on the refractive index $n$ only. Indeed, the propagation
angle inside the sample is automatically defined by these two quantities
thanks to the Snell’s law, which states that $\sin\eta=\sin{\eta_{in}}/n$
(notice that in the isotropic case one has to define $\eta$ with respect to
the normal to the interface as there are no planes, but we maintain the
notation to press the analogy between the two cases). However, in the uniaxial
case of Fig. 4(b) Snell’s relation is not valid, since due to anisotropy of
the refractive indices the components $k_{ab}$ and $k_{c}$ of the momentum are
rescaled differently. This makes the transmission coefficient a function of
$\eta_{in},n_{b}$ and $n_{c}$. More specifically, for the TM wave one finds
that $\text{T}_{\text{TM}}$ has an expression analogous to Eq. (44), provided
that one replaces $n\to n_{b}=\sqrt{\varepsilon_{ab}}$ and
$\sin\eta\to\sin\eta_{in}/n_{c}=\sin\eta_{in}/\sqrt{\varepsilon_{c}}$:

$\text{T}_{\text{TM}}(\eta_{in},n_{b},n_{c})=\frac{\mathcal{T}\mathcal{T}^{\prime}e^{in_{b}\omega
d\cos{\eta}/c}}{1-\mathcal{R}^{2}e^{2in_{b}\omega d\cos{\eta}/c}},$ (45)

with $\mathcal{T}$ and $\mathcal{T}^{\prime}$ retaining the same functional
dependence on $\eta,\eta_{in}$ as before. In this situation, the argument of
the complex exponential in Eq. (45) reads:

$\zeta=n_{b}\frac{\omega d}{c}\cos{\eta}=n_{b}\frac{\omega
d}{c}\sqrt{1-\frac{\sin^{2}\eta_{in}}{n_{c}^{2}}}.$ (46)

One can check that for THz frequencies around the Josephson plasma frequency
$\omega\sim\omega_{c}$ the divergence in the square root possible for
$\eta_{in}\neq 0$ is weakened by the residual quasiparticle damping $\gamma$,
so that we obtain $\zeta\ll 1$. Then, evaluating $1/\text{T}_{\text{TM}}$ from
(45) at small $\zeta$ one gets

$\frac{1}{\text{T}_{\text{TM}}}\approx 1-i\frac{\omega
d}{2c}\frac{\varepsilon_{ab}\cos^{2}{\eta_{in}}+1-\sin^{2}{\eta_{in}}/\varepsilon_{c}}{\cos{\eta_{in}}}.$
(47)

Similar reasonings can be made for the TE component, that is expressed as in
Eq. (45), provided that
$\sin\eta\to\sin\eta_{in}/n_{b}=\sin\eta_{in}/\sqrt{\varepsilon_{ab}}$. Also
in this case one can approximate the transmission coefficient along this
direction as:

$\frac{1}{\text{T}_{\text{TE}}}\approx 1-i\frac{\omega
d}{2c}\frac{\varepsilon_{ab}\cos^{2}{\eta_{in}}+1-\sin^{2}{\eta_{in}}/\varepsilon_{ab}}{\cos{\eta_{in}}}.$
(48)

Even though Eq.s (47) and (48) still depend on a combination of
$\varepsilon_{ab}$ and $\varepsilon_{c}$, these structures do not lead to the
pole observed in the transverse dielectric function $\varepsilon_{t}$, as
opposed to Eq. (41) obtained in the first configuration. In the end, by
explicit numerical computation with realistic parameter values for cuprates of
the transmissivity $\text{T}_{\text{TM}}$ and $\text{T}_{\text{TE}}$ in Eq.
(45), with the corresponding definitions of $\eta$, we verified that the
corresponding conductivities, expressed as in Eq. (43), are featureless, and
no finite-frequency peaks are observed.

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Appendix Tasker’s Goals in Dynamic qa Generation When tasking human authors
with adversarial writing of questions, wallace2019trick emphasizes the
importance of “who” the authors should be: talented and eager question writers
with specific goals; they should aim to generate questions that stump
computers but seem normal enough for humans to answer. To make this work, they
recruit members of the Quizbowl community who have deep trivia knowledge and
craft question for Quizbowl tournaments jennings2007brainiac. However, their
challenge was to convey what is ”normal” to authors and stimulate examples
that can elucidate the weaknesses of qa models.

Merging Trivia Question Generation and Dynamic Adversarial Generation Process
merge Many QA datasets are now too easy for modern models as models have
become more powerful rogers2023qa. However, even these easy QA datasets have
serious data flaws min2020ambigqa, yu-etal-2023-crepe, which suggests that
creating question-answer pairs is a very challenging task. This is also a norm
for questions written for human players, where more than 100,000 questions are
produced annually. To create effective and challenging enough questions, the
professional experts (e.g., writing staff) take a rigorous editing pass on the
questions to decide whether they are adequate enough to guarantee players a
fair game lelkes2021quiz, pollard2006student. They follow strict guidelines to
be selected to be used in the quiz matches. We propose to merge the above
pipelines to help improve data creation for robust QA models by adding an
editing step to ensure that grammatical errors and nonfactual questions
(following the norms of Trivia questions) do not exist in the pool.

Variational Inference for IRT models To discover the IRT parameters that best
explain the whole data, difficulty $\theta_{j}$ and
discriminability$\gamma_{j}$, we turn to variational inference for the full
generative process, an approximation method for intractable posterior
distribution in Bayesian inference natesan2016bayesian, lalor2019learning. The
parameters $\theta$ and $\beta$ follow Gaussian prior distributions and make
inferences through joint posterior distribution $\pi(\theta,\beta|Y)$
natesan2016bayesian.

Topic Categories of Questions app:topics We ask the question writers to tag
their questions with the categories below. With reference to specific
categories and examples, we encourage them to be as creative and diverse as
possible when authoring the questions. In the interface, they can monitor how
many questions they wrote per category. They are required to submit packets
with a specific amount of questions in each category.

1pt Question |  Answer
---|---
1pt Art |  Questions about works: Mona Lisa, Raft of the Medussa, B) Questions about forms: color, contour, texture, C) Questions about artists: Picasso, Monet, Leonardo da Vinci, D) Questions about context: Renaissance, post-modernism, expressionism, surrealism
Literature Movement |  A) Questions about works: novels (1984), plays (The Lion and the Jewel), poems (Rubaiyat), criticism (Poetics), B) Questions about major characters or events in literature: The Death of Anna Karenina, Noboru Wataya, the Marriage of Hippolyta and Theseus
Literary Movement |  A) Cross-cutting questions (appearances of Overcoats in novels), B) Common link questions (the literary output of a country/region)
Geography |  A) Questions about location: names of capital, state, river, B) Questions about the place: temperature, wind flow, humidity
History |  A) When: When did the First World war start?, B) Who: Who is called Napoleon of Iran?, C) Where: Where was the first Summer Olympics held?, D) Which: Which is the oldest civilization in the world?
Science |  Questions about terminology: The concept of gravity was discovered by which famous physicist?, Questions about the experiment, Questions about theory: The social action theory believes that individuals are influenced by this theory.
TV and Film |  Quotes: What are the dying words of Charles Foster Kane in Citizen Kane?, Title: What 1927 musical was the first ”talkie”?, Plot: In The Matrix, does Neo take the blue pill or the red pill?
Music |  Singer: What singer has had a Billboard No. 1 hit in each of the last four decades?, Band: Before Bleachers and fun., Jack Antonoff fronted what band?, Title: What was Madonna’s first top 10 hit?
Lifestyle |  Clothes: What clothing company, founded by a tennis player, has an alligator logo?, Decoration: What was the first perfume sold by Coco Chanel?
Sports |  Known facts: What sport is best known as the ‘king of sports’? Nationality: What’s the national sport of Canada? Sport player: The classic 1980 movie called Raging Bull is about which real-life boxer? Country: What country has competed the most times in the Summer Olympics yet hasn’t won any kind of medal?
1pt |
Table : Categories of questions along with the subcategories and corresponding
examples.

##  Question Type Annotation

In Table , we list the problematic question types that we ask the annotators
to annotate. These are illustrated with descriptions and examples to help them
better understand each question.

1pt Question Type |  Description |  Examples
---|---|---
1pt Lacks Factuality |  Requires information is factual |  “Trump, the first woman president of the United States, is charged against federal laws” is non factual as the gender of Trump is male
Lacks Specificity
(False Presupposition) |  Requires more information to be answered with clarity |  ’What is the color of Flamingo’s feathers?’ is ambiguous as Pink and White could be two possible answers depending on when they are born
Subjectivity |  Contains clues that are highly subjective |  “What’s the name of Christopher Columbus’s most famous ship?” Possible answers could be either Santa Maria, La Nina, Santa Clara. Also, as “Most famous” can mean many different things, the revised question could be “Which of Columbus’s ships was stripped of its timbers to build a fort called La Navidad in northern Haiti?”
Ambiguity &
Multiple acceptable answers |  Can be answered with multiple answers |  Nikolas Alexandrovitch Romanov, Nikolas II, Nikolai II Alexandrovich Romanov: all of these are acceptable as answers.
1pt |  |
Table : We list the problematic question types that we ask to annotate. The
four types are illustrated with descriptions and examples to help them better
understand each question, and help determine whether each question has good
quality.

##  Adversarial Type Annotation

In Table , we list adversarial types (techniques) to determine how each
question is using them to stump the models. The annotators are given the
description and examples to better understand the reasons why the models may
have been stumped. They are expected to tag the examples with the model
prediction and question.

1pt Question Type |  Adversarial Type
---|---
1pt Composing seen clues |  Contains clues that need to be integrated for the question to be answered
Logic and Calculation |  Requires mathematical or logical operators
Multi-Step Reasoning |  Requires multiple reasoning steps between entities. For eg: A building dedicated to this man was the site of the “I Have A Dream” speech. A reasoning step is required to infer : “I have a dream” speech -¿ Lincoln Memorial -¿ Abraham Lincoln
Negation |  Contains “not” or “non-” and “no” or any negation entities that may confuse the model to answer
Temporal Misalignment |  Contains a specific year, month, or timely event that the model got confused about or does not know.
Location Misalignment |  Contains a location that the model got confused about or does now know.
Commonsense Knowledge |  Requires information that cannot be answered without commonsense
Domain Expert Knowledge |  Requires information that cannot be answered without domain expert knowledge
Novel Clues |  Contains information that exists in the question but is not required to answer. These confuse the models.
Crosslingual |  Contains multilingual aspects that confuse the model.
1pt |

Table : We list adversarial types (techniques) to determine how each question
is using them to stump the models. The annotators are given the description
and examples to better understand the reasons why the models may have been
stumped. They are expected to tag the examples with the model prediction and
question.

##  Question Examples Annotated with Question and Adversarial Types

Table LABEL:Tab:adv_good shows question examples that are annotated with
question and adversarial types. The highlights in the question correspond to
either adversarial types or question types that are highlighted with the same
color.

[][Machine Discriminability and
Difficulty][width=0.98]figures/2pl_machine_diff_disc.pdf

Figure : 2pl results