# Spin-momentum locking breakdown on plasmonic metasurfaces

Fernando Lorén<EMAIL_ADDRESS>Instituto de Nanociencia y Materiales de
Aragón (INMA), CSIC-Universidad de Zaragoza, 50009 Zaragoza, Spain
Departamento de Física de la Materia Condensada, Universidad de Zaragoza,
50009 Zaragoza, Spain  Cyriaque Genet  University of Strasbourg and CNRS, CESQ
& ISIS (UMR 7006), 8, allée G. Monge, 67000 Strasbourg, France  Luis Martín-
Moreno<EMAIL_ADDRESS>Instituto de Nanociencia y Materiales de Aragón (INMA),
CSIC-Universidad de Zaragoza, 50009 Zaragoza, Spain  Departamento de Física de
la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, Spain

###### Abstract

We present a scattering formalism to analyze the spin-momentum locking in
structured holey plasmonic metasurfaces. It is valid for any unit cell for
arbitrary position and orientation of the holes. The spin-momentum locking
emergence is found to originate from the unit cell configuration.
Additionally, we find that there are several breakdown terms spoiling the
perfect spin-momentum locking polarization. We prove that this breakdown also
appears in systems with global symmetries of translation and rotation of the
whole lattice, like the Kagome lattice. Finally, we present the excitation of
surface plasmon polaritons as the paramount example of the spin-momentum
locking breakdown.

##  I Introduction

Metasurfaces based on plasmonic arrays have been demonstrated to have a
plethora of applications [1, 2] such as sensing [3], imaging [4, 5], or
telecommunications [6]. In particular, geometric phase metasurfaces (GPMs)
have gained significant attention in the last years due to their ability to
manipulate the polarization of light waves in a controllable manner [7, 8, 9,
10, 11]. One important property of these metasurfaces is that they can exhibit
spin-momentum locking (SML), which refers to the coupling between the
polarization and the momentum of the involved light waves [12].

Despite the evidenced applicability of these plasmonic GPMs and numerous
numerical studies, no first principles rigorous theoretical analysis had been
developed. There have been studies for continuously space-variant structures
[13] and for structures with translation and rotation symmetries of the whole
lattice under stringent conditions for the direction of the electric field
[14]. Recently, we have applied a scattering formalism to study holey
plasmonic GPMs that present a chiral arrangement in the unit cell [15].

This article presents a general analysis of the SML on GPMs, extending our
previous study to lattices that present full translation and rotation
symmetry. In particular, we apply it to the Kagome lattice, which has been
considered as a platform for GPMs [14, 16] and also studied due to its
relevance in antiferromagnets [17, 18]. Our results provide a comprehensive
understanding of the SML mechanism on GPMs and have important implications for
designing and optimizing these metasurfaces. Based on this general formalism,
we demonstrate that the appearance of the SML breakdown is ubiquitous for any
system, revealing the interplay between the SML and the linear character of
the surface plasmon polaritons (SPPs). The SML breakdown appears in systems
with and without global rotation symmetries, both of which will be considered
below.

##  II Theoretical formalism

The general derivation of the scattering formalism used in this paper is
provided in the Supplemental Material of [15]. In this section, we present the
essential elements required to comprehend the relevant terms of the formalism,
along with the article’s results.

We consider a general plasmonic metasurface, this is, a metal slab
characterized by a periodically repeated unit cell with an arbitrary number of
elements ($N$) distributed in. A huge variety of shapes can be considered [19,
20, 21, 22] yet we will focus on one of the simplest ones, rectangular
dimples, which corresponds to the study of our metasurfaces by reflection.
Analyzing them by transmission, if we had considered holes, would lead to the
same main results. Each dimple has a short side $a$, a long side $b$, and
depth $d$. Furthermore, each dimple is defined by its position
($\vec{r}_{\alpha}=(x_{\alpha},y_{\alpha})^{T}$) and the angle with respect to
the $\vec{u}_{x}$ direction ($\theta_{\alpha}$), where $\alpha$ is the index
associated with each dimple.

Figure 1: Scheme of the excitation of the metasurfaces. $\mathbf{i}_{0}$ is
the amplitude of the incident plane wave and $\mathbf{r}_{m}$ are the
reflection coefficients of the Bragg modes.

An electromagnetic (EM) plane wave is impinging our metasurface with an in-
plane wavevector
$\vec{k}^{in}=k^{in}_{x}\,\vec{u}_{x}+k^{in}_{y}\,\vec{u}_{y}$ and an incident
polarization $\sigma_{in}$, and our goal is to compute the reflection
coefficients into the different Bragg orders (see Figure 1). For this purpose,
we employ the coupled-mode method (CMM), which has been extensively used in
the study of EM properties in metallic dimple arrays [23, 22, 15]. The CMM
expands the EM fields in plane waves in the free space regions and waveguide
modes inside the dimples, and finds the electric field amplitudes by properly
matching the EM fields at the interfaces.

The reciprocal lattice vectors that define our unit cell in the Fourier space
are $\vec{G}_{1}$ and $\vec{G}_{2}$. The Bragg modes are characterized by an
in-plane wavevector
$\vec{k}_{m}=\vec{k}^{in}+m_{1}\,\vec{G}_{1}+m_{2}\,\vec{G}_{2}$ and a
polarization $\sigma$. We will combine the integers $m_{1}$ and $m_{2}$ into a
single index: $m=(m_{1},m_{2})$, for notational simplicity.

We describe the behavior of the metal using the Perfect Electric Conductor
(PEC) approximation, which assumes that the metal’s dielectric constant tends
toward negative infinity. This simplification allows for a clearer description
of the system’s physics, as demonstrated in [23, 15]. The effects of field
penetration in the metal and associated losses are discussed in Appendices A,
C, and E, showing that the results obtained within the PEC approximation are
qualitatively accurate.

It is convenient to express the polarization of each Bragg mode on the
circular polarization (CP) basis to study the SML provided by our metasurface.
We represent the reflection coefficients as spinors to contain both spin
components: $\mathbf{r}_{m}=(r_{m}^{+},r_{m}^{-})^{T}$, where $\pm$ denote the
right- and left-handed polarization (or spin), each of them defined within the
plane perpendicular to the wavevector associated to the Bragg mode $m$. This
representation is chosen because the spin of a plane wave is conserved upon
reflection by a mirror [24, 25, 26, 27] (while the helicity changes sign).

The reflection coefficients in the CP basis with respect to the propagation
directions satisfy the following equations

$\mathbf{r}_{m}=-\delta_{m0}\,\mathbf{i}_{0}+C_{m0}\,Y_{0}\,\mathbf{i}_{0}-\sum_{m^{\prime}}C_{mm^{\prime}}\,Y_{m^{\prime}}\,\mathbf{r}_{m^{\prime}}.$
(1)

The first term is the specular reflection, being $\mathbf{i}_{0}$ the
amplitude of the incident plane wave and $\delta_{m0}$ the Kronecker delta.
$C_{mm^{\prime}}$ are the geometric couplings [15], which are $2\times 2$
matrices operating in polarization space. They couple different Bragg modes
($m^{\prime}$ with $m$) via scattering with the plasmonic metasurface and
encode the geometry of the unit cell through the overlaps between the Bragg
and the waveguide modes.

$Y_{m^{\prime}}$ are also $2\times 2$ matrices representing the modal
admittances. They relate the in-plane magnetic field to the electric one and,
in the CP basis, can be written as
$Y_{m^{\prime}}=\bar{Y}_{m^{\prime}}\,\mathbb{1}+\Delta_{m^{\prime}}\,\sigma_{x}$,
where $\mathbb{1}$ and $\sigma_{x}$ are the $2\times 2$ unit matrix and the
Pauli matrix that swaps spin states, respectively. In terms of the linear p
(transverse magnetic) - s (transverse electric) polarized basis,
$\bar{Y}_{m^{\prime}}\equiv(Y_{m^{\prime}p}+Y_{m^{\prime}s})/2$ and
$\Delta_{m^{\prime}}\equiv(Y_{m^{\prime}p}-Y_{m^{\prime}s})/2$. For a plane
wave with frequency $\omega$ and in-plane wavevector
$k_{m^{\prime}}=|\vec{k}_{m^{\prime}}|$ propagating in a uniform medium with
dielectric constant $\epsilon$, the modal admittances are
$Y_{m^{\prime}p}=\epsilon/q_{m^{\prime}z}$ and
$Y_{m^{\prime}s}=q_{m^{\prime}z}$, where $q_{m^{\prime}z}=\sqrt{\epsilon-
q_{m^{\prime}}^{2}}$ ($q_{m^{\prime}}=c\,k_{m^{\prime}}/\omega$ and $c$ is the
speed of light). Notice that $\Delta_{0}=0$ at normal incidence, while both
$\bar{Y}_{m^{\prime}}$ and $\Delta_{m^{\prime}}$ diverge at the Rayleigh
points (i.e., whenever a diffractive order becomes tangent to the metal-
dielectric interface).

The geometric couplings allow us to explore the SML emergence because they
provide the coupling between two different Bragg modes and their corresponding
CP components. They can be written as

$C_{mm^{\prime}}=R^{k(m)\leftarrow z}C_{mm^{\prime}}^{z}R^{z\leftarrow
k(m^{\prime})}.$ (2)

The interaction of the Bragg modes is ruled through the dimples, so the in-
plane EM fields are the ones playing a role in the couplings. Therefore, the
origin of the SML resides in the properties of the geometric couplings in the
CP basis but with respect to the $\vec{u}_{z}$ direction,
$C_{mm^{\prime}}^{z}$. However, each Bragg mode is transversal so its
polarization is defined with respect to the propagation direction. Then, we
need the $R$’s to encapsulate the change of basis with respect to the
$\vec{u}_{z}$ and the propagation direction.

The matrix that changes basis from the $\vec{u}_{z}$ direction to the
propagation direction of the $m$-th Bragg mode is $R^{k(m)\leftarrow
z}=\frac{1}{2}\left[(\sqrt{q_{mz}^{2}+q_{m}^{2}}/q_{mz}+1)\,\mathbb{1}+(\sqrt{q_{mz}^{2}+q_{m}^{2}}/q_{mz}-1)\,\sigma_{x}\right]$.
The presence of $\sigma_{x}$ in $R^{k(m)\leftarrow z}$ implies its occurrence
in $C_{mm^{\prime}}$, leading to the swapping of spin states.

On the other hand, the expression for $C_{mm^{\prime}}^{z}$ is:

$C_{mm^{\prime}}^{z}=C^{\prime}\sum_{\alpha=0}^{N-1}S_{m\alpha}S_{m^{\prime}\alpha}^{*},$
(3)

where $C^{\prime}$ is the dimple cross-section, which depends on the dimple
area and depth, and the impedance of the waveguide mode; and $S_{m\alpha}$ is
a geometrical factor that measures how well a given EM plane wave overlaps
with the fundamental mode in the dimple (details in Appendix A and [15]).

Both $\sigma_{x}$ appearances (in $Y_{m^{\prime}}$ and $C_{mm^{\prime}}$)
contribute to the mixing of the spin components of the Bragg modes, reducing
the SML contrast and producing what we coined as spin-momentum locking
breakdown in [15]. We have shown that the SML breakdown terms are ubiquitous
to any configuration independent of whether they host or not global rotation
symmetries. The paramount example of the relevance of the SML breakdown is the
excitation of SPPs because both $\Delta_{m^{\prime}}$, and the factor
$(\sqrt{q_{mz}^{2}+q_{m}^{2}}/q_{mz}-1)$) appearing in $R$, rise and become as
large as $\bar{Y}_{m^{\prime}}$ and $(\sqrt{q_{mz}^{2}+q_{m}^{2}}/q_{mz}+1)$).

In the succeeding sections, we will describe two different, although related,
structures: without and with global rotation symmetries. For both, we will
present the SML mechanism derived from their geometric couplings and the SML
breakdown effects.

##  III Spatially rotated dimples along $\vec{u}_{x}$ direction

Figure 2: (a) Unit cell with $N=3$ spatially rotated dimples and $n_{w}=1$,
and scheme of the considered full array, showing $3\times 3$ unit lattices.
(b) Geometric couplings with respect to the $\vec{u}_{z}$ direction,
considering $m^{\prime}_{1}=m^{\prime}_{2}=m_{2}=0$ and spinor $+$. The spin
$+/-$ component is represented in red/blue. (c, d) $|\Delta\mathbf{r}_{m}|$
with respect to $m=m_{1}$ and $m_{2}=0$, computed by taking a normal incident
plane wave with spin $+$. Chosen geometrical parameters: $L=460\,nm$,
$a=80\,nm$, $b=220\,nm$, and $d=60\,nm$. (c) is computed away from an SPP
resonance, with incoming energy $\omega=3\,eV$. (d) is for an SPP resonance
associated to the Bragg modes $m_{1}=\pm 3$, with incoming energy
$\omega=2.692\,eV$. We break the y-axis for a better observation of every
Bragg mode.

We consider a rectangular unit cell of $N=3$ dimples evenly spaced along the
$\vec{u}_{x}$ of the unit cell, with $L$ being the distance between the
centers of the two nearest dimples, in both $x$\- and $y$\- directions. We
consider that $\theta_{\alpha}$ varies linearly with $\alpha$:
$\theta_{\alpha}=2\pi n_{w}\alpha/N$, where the winding number $n_{w}$ defines
the number of complete $2\pi$ rotations along the unit cell. We have selected
the following set of geometrical parameters for the remainder of the paper:
$L=460\,nm$, $a=80\,nm$, $b=220\,nm$, and $d=60\,nm$. These particular
parameters align with those employed in the experiments detailed in [9, 15].
However, we note that the specific dimensions and inter-distances of the
dimples, although influencing the dimple cross-section, do not impact the
system’s topological properties.

The system is depicted in Figure 2a, where the winding number is $n_{w}=1$.
The case presented in [15] is similar and the appearance of SML breakdown was
already demonstrated. The choice of $N=3$ and $n_{w}=1$ is based on the system
considered in the next section, the Kagome lattice, whose unit cell can be
seen as three clusters of three dimples each, with winding numbers of
$n_{w}=1$ as well. Another reason for considering $N=3$ is because the
rotation steps of $2\pi/3$ are very far from the adiabatic and continuous
condition required to apply the Berry phase formalism, which was conceived to
analyze adiabatic and continuous deformations of a closed spatial path [28].

Notice that although the dimples perform a step-wise rotation along the unit
cell, the whole lattice does not support global rotation symmetry.

For this case, the reciprocal lattice vectors are:
$\vec{G}_{1}=2\pi/(NL)\,\vec{u}_{x}$ and $\vec{G}_{2}=2\pi/L\,\vec{u}_{y}$.
Considering $m_{2}=m_{2}^{\prime}=0$ is enough to explore the underlying
physics because there is no inversion symmetry breaking along the
$\vec{u}_{y}$ direction [14, 15]. Thus, we consider $m=m_{1}$,
$m^{\prime}=m_{1}^{\prime}$ and $k_{y}^{m}=k_{y}^{m^{\prime}}=0$. Besides, the
small-dimple approximation simplifies the overlapping integrals by considering
the dimples much smaller than the wavelength. Then, $C_{mm^{\prime}}^{z}$
reads

$\displaystyle C_{mm^{\prime}}^{z}$
$\displaystyle=C\sum_{\alpha=0}^{2}e^{i2\pi\alpha(m^{\prime}-m)/N}\begin{pmatrix}1&e^{-i2\pi
2n_{w}\alpha/N}\\\ e^{i2\pi 2n_{w}\alpha/N}&1\\\ \end{pmatrix}$ (4)
$\displaystyle=CN\left(\delta_{m,m^{\prime}+n_{0}N}\,\mathbb{1}+\sum_{s=\pm}\delta_{m,m^{\prime}+n_{0}N-2n_{w}s}\,\sigma_{s}\right),$

where $n_{0}$ is any integer, $\sigma_{\pm}$ are Pauli matrices that increase
and decrease spin, respectively, and $C=4abC^{\prime}/(\pi^{2}A_{uc})$, being
$A_{uc}$ the area of the unit cell.

The SML mechanism is derived exactly from Eq. 4. The first term corresponds to
the spin-preserving processes and the associated Bragg law is
$k_{x}^{out}=k_{x}^{in}+n_{0}\,G^{0}$, with $G^{0}=2\pi/L$. Two Bragg modes
with a difference in indices proportional to N can be coupled if the spin is
preserved. The second term describes the spin-flipping processes and the
associated Bragg law is $k_{x}^{out}=k_{x}^{in}+n_{0}G^{0}\mp k_{g}$, where
$k_{g}=2\pi 2n_{w}/(NL)$ is the geometric momentum. Two Bragg modes with a
difference in indices proportional to $N\pm 2\,n_{w}$ can be coupled if the
spin is changed to $\mp 1$, which is exactly the spin-to-momentum conversion
of the SML.

To illustrate this, we come with spin $+\equiv(1,0)^{T}$ and represent both
spin components of the normalized amplitudes of the geometric couplings in the
CP basis. This is,
$\mathbf{c}_{m_{1},0}\equiv(c_{m_{1},0}^{+},c_{m_{1},0}^{-})^{T}=C^{z}_{m0}\cdot(1,0)^{T}/(CN)$.

In Figure 2b, we represent $|c_{m_{1},0}^{\pm}|$. The SML is evident. Spin is
preserved for $m_{1}=0,\pm 3$, which are multiples of $N$; and spin is flipped
for $m_{1}=2,-1$, which are $2\,n_{w}$ and $2\,n_{w}-N$ respectively. Hence,
the exact SML mechanism arises from the geometric couplings with respect to
the $\vec{u}_{z}$ direction, $C_{mm^{\prime}}^{z}$.

When computing the full EM system (reflection coefficients), breakdown terms
appear in both geometric couplings and modal admittances. Additionally, there
is the contribution from the specular reflection. As we want to study the
interaction of the light with the dimple lattice, we define
$\Delta\mathbf{r}_{m}=\mathbf{r}_{m}+\delta_{m0}\,\mathbf{i}_{0}$, which
removes the specular reflection from the zero order for a better observation
of the SML breakdown.

In Figure 2c we represent $|\Delta r^{\pm}_{m_{1},0}|$ for an incoming plane
wave impinging normally to the metasurface with spin $+$ and energy
$\omega=3\,eV$. The consequences of the SML breakdown terms are already
noticeable: all the Bragg modes are a combination of both CP states, and the
perfect SML does not hold anymore but is recognizable. Since at that frequency
SPP resonances are not excited, the general behavior is still similar to the
perfect SML.

Note that we are studying a plasmonic metasurface and the breakdown is maximum
when a plasmonic resonance is excited [15]. Thus, we show the reflection
coefficients when we are at a plasmonic resonance in Figure 2d. We represent
$|\Delta r^{\pm}_{m_{1},0}|$ at a SPP resonance associated to the Bragg modes
$m_{1}=\pm 3$. We use an incoming plane wave impinging normally to the
metasurface with spin $+$ and energy $\omega=2.692\,eV$. The consequences of
the SML breakdown terms are now predominant: $|\Delta r^{\pm}_{\pm 3,0}|$ are
very large and both spin components are similar, which is characteristic of
the linearly $p$ polarized character of the SPP. Moreover, the perfect SML
behavior cannot be recognized because of SML breakdown, being spoiled and
mixed both spin components of all the Bragg modes.

##  IV Kagome lattice

Figure 3: (a) Unit cell of the $\sqrt{3}\times\sqrt{3}$ Kagome lattice,
defined by the dashed line with the three clusters highlighted in red. (b)
Geometric couplings with respect to the $\vec{u}_{z}$ direction, considering
$m^{\prime}_{1}=m^{\prime}_{2}=m_{2}=0$ and spinor $+$. The spin $+/-$
component is represented in red/blue. (c, d) $|\Delta\mathbf{r}_{m}|$ with
respect to $m=m_{1}$ and $m_{2}=0$. Computed by taking a normal incident plane
wave with spin $+$. Chosen values: $L=460\,nm$, $a=80\,nm$, $b=220\,nm$, and
$d=60\,nm$. (c) Computed away from an SPP resonance, with incoming energy
$\omega=3\,eV$. (d) Computed at an SPP resonance associated with the Bragg
modes $m_{1}=\pm 3$, with incoming energy $\omega=2.694\,eV$. We break the
y-axis for a better observation of every Bragg mode.

In this section, we present the main result of the article: the appearance of
the SML breakdown in a system with combined translation and rotation symmetry
of the whole lattice. This is the staggered (or $\sqrt{3}\times\sqrt{3}$)
Kagome lattice (KL) [29, 18, 14]. The reciprocal lattice vectors of the KL
are: $\vec{G}_{1}=2\pi/(3L)\,\vec{u}_{x}$ and
$\vec{G}_{2}=\pi/(3L)(-\vec{u}_{x}+\sqrt{3}\vec{u}_{y})$. We will analyze its
geometric couplings, as well as the reflection coefficients.

This symmetry is important because it has been used in other works [14] to
study the appearance of SML via group theory arguments, although restricted to
waves with an electric field perpendicular to the surface and at normal
incidence.

Figure 3a shows a schematic representation of the considered KL. The unit cell
is defined by the dashed lines and is composed of $N=9$ dimples, defined by
the positions of their centers and their angles with respect to the
$\vec{u}_{x}$ direction (see Table 1 in Appendix B). These nine dimples can be
subdivided in three similar clusters
$\\{\alpha\\}=\\{\\{0,1,2\\},\\{3,4,5\\},\\{6,7,8\\}\\}$. The dimples in each
cluster are distributed forming an equilateral triangle, with angles that are
step-wisely rotated with a winding number of $n_{w}=1$.

Each triangular cluster has the same number of dimples and the same winding
number as the rectangular unit cell of the previous section. However, they
have different spatial distributions. Consequently, the involved Bragg modes
in the KL host similar, but different, coupling processes.

The geometric couplings in the CP basis with respect to the $\vec{u}_{z}$ in
the PEC and small-dimple approximations are,

$\displaystyle C_{mm^{\prime}}^{z}=$
$\displaystyle\,C\sum_{\alpha=0}^{8}e^{i(\vec{k}_{m^{\prime}}-\vec{k}_{m})\vec{r}_{\alpha}}\begin{pmatrix}c_{++}&c_{+-}\,e^{-i2\theta_{\alpha}}\\\
c_{-+}\,e^{i2\theta_{\alpha}}&c_{--}\\\ \end{pmatrix}$ (5) $\displaystyle=$
$\displaystyle\,C\,A^{mm^{\prime}}\begin{pmatrix}c_{++}\,\delta_{m_{1}+m_{2},m_{1}^{\prime}+m_{2}^{\prime}+3\,n_{0}}&-c_{+-}\,\delta_{m_{1}+m_{2},m_{1}^{\prime}+m_{2}^{\prime}+3\,n_{0}-2\,n_{w}}\\\
-c_{-+}\,\delta_{m_{1}+m_{2},m_{1}^{\prime}+m_{2}^{\prime}+3\,n_{0}+2\,n_{w}}&c_{--}\,\delta_{m_{1}+m_{2},m_{1}^{\prime}+m_{2}^{\prime}+3\,n_{0}}\\\
\end{pmatrix},$

where we have defined
$c_{\sigma\sigma^{\prime}}=(\vec{k}_{m}\cdot\vec{\sigma})(\vec{\sigma}^{\prime}\cdot\vec{k}_{m^{\prime}})/(k_{m}k_{m^{\prime}})$,
being $\vec{\sigma}=\vec{u}_{x}+i\sigma\vec{u}_{y}$, with $\sigma=\pm$. These
$c_{\sigma\sigma^{\prime}}$ are the projections of the Bragg modes $m$ and
$m^{\prime}$ with the circular polarizations $\sigma$ and $\sigma^{\prime}$,
respectively. The Kronecker deltas provide the selection rules between these
Bragg modes, being $n_{0}$ an integer. Besides, depending on the Bragg modes
to be coupled, the coupling amplitude is different: $|A^{mm^{\prime}}|=N$ if
both $\Delta_{1}$ and $\Delta_{2}$ are even, and $|A^{mm^{\prime}}|=N/3$ in
the rest of cases; being $\Delta_{1/2}=m_{1/2}^{\prime}-m_{1/2}$. This is
inferred from the sum over the dimples in the unit cell, in the first line of
Equation 5.

Equation 5 rules two different processes. One process (given by the diagonal
elements of $C^{z}_{mm^{\prime}}$) conserves spin. The corresponding Bragg
law, called standard Bragg law [14] is
$\vec{k}^{out}=\vec{k}^{in}+m_{1}\,\vec{G}_{1}+m_{2}\,\vec{G}_{2}$ such that
$m_{1}+m_{2}=3\,n_{0}$ (notice that the incident plane wave corresponds to
$m_{1}^{\prime}=m_{2}^{\prime}=0$) . The other process flips spin (off-
diagonal elements of $C^{z}_{mm^{\prime}}$). The corresponding Bragg law,
called spin-orbit Bragg law [14], satisfies another condition:
$m_{1}+m_{2}=3\,n_{0}\mp 2\,n_{w}$, which is exactly the SML mechanism.

Figure 3b shows the SML mechanism derived from the geometric couplings. We
represent $\mathbf{c}_{m_{1},m_{2}}=C^{z}_{m0}\cdot(1,0)^{T}/(CN)$, where
$(1,0)^{T}$ is the spinor for the spin $+$. Although we have considered both
$m_{1}$ and $m_{2}$ in the calculation, we take $m_{2}=0$ for a simpler
representation. We observe the feature of the coupling amplitudes
$A^{mm^{\prime}}$ of the different processes. It is easy to observe that the
SML mechanism that we described above is satisfied.

Once we have shown how the SML arises from the geometric couplings for the KL,
we look at $\Delta\mathbf{r}_{m}$. In Figure 3c we represent $|\Delta
r^{\pm}_{m_{1},0}|$ for an incoming plane wave with spin $+$, energy
$\omega=3\,eV$ and normal to the metasurface. Since SPPs are not excited at
that frequency, the general behavior is similar to the perfect SML, although
we already see some signatures of the breakdown. The amplitude relation
between the different modes is no longer exactly satisfied, and we also
observe small amplitudes of modes that should be zero if SML were exact.

Finally, in Figure 3d, we show the reflection coefficients when a plasmonic
resonance is excited. We represent $|\Delta r^{\pm}_{m_{1},0}|$ at a SPP
resonance associated to the Bragg modes $m_{1}=\pm 3$. We use an incoming
plane wave with spin $+$, energy $\omega=2.694\,eV$ and impinging normally to
the metasurface. The SML breakdown terms have acquired a governing relevance.
$|\Delta r^{\pm}_{\pm 3,0}|$ are very large and both spin components are
similar, which is characteristic of the linearly $p$ polarized character of
the SPP. From these resonantly excited modes, successive couplings with other
modes can occur. In consequence, we cannot recognize anymore the expected SML
because both spin components of all the Bragg modes are spoiled and mixed.

The physical interpretation is as follows: the EM fields carry CP light
perpendicular to the propagation direction of the plane waves. However, the
system has a particular symmetry perpendicular to the planar metasurface
($\vec{u}_{z}$ direction). This mismatching results in that when the CP light
gets projected onto the planar surface, it becomes elliptical (which is a
combination of the two CP states) and then, the SML is spoiled.

##  V Conclusion

We have shown that even a system with combined translation and rotation
symmetry of the whole lattice suffers spin-momentum locking breakdown. The
physical interpretation lies in the elliptical projection onto the planar
metasurface of the circularly polarized light. Therefore, together with the
results obtained in [15], this shows that any system, with or without global
lattice symmetries, presents breakdown of the SML. Nonetheless, we stress that
the breakdown terms are often small, so the SML is a useful concept. However,
in some cases such as the plasmonic resonances, breakdown terms become very
relevant. Plasmon resonances are, thus, the paramount example of SML
breakdown.

Despite the occurrence of this breakdown, it presents an opportunity to
optimize the system in order to minimize it. Additionally, other applicative
perspectives could be renewed by the consideration of the results presented in
this work, such as optovalleytronic systems [30], non-linear hybrid
metasurfaces [31], and topology-based high-resolution sensors [32].

## Acknowledgements

We acknowledge Project PID2020-115221GB-C41 was financed by
MCIN/AEI/10.13039/501100011033 and the Aragon Government through Project
Q-MAD.

This work is part of the Interdisciplinary Thematic Institute QMat of the
University of Strasbourg, CNRS, and Inserm. It was supported by the following
programs: IdEx Unistra (ANR-10-IDEX-0002), SFRI STRATUS project
(ANR-20-SFRI-0012), and USIAS (ANR-10-IDEX-0002-02), under the framework of
the French Investments for the Future Program.

##  Appendix A Details of the theoretical formalism

Here, we extend the calculations presented in the main text and introduce the
required quantities such as $C^{\prime}$ and the overlapping integrals.

We present the formalism within the surface impedance boundary conditions
(SIBC) approximation. The SIBC approximation provides a more accurate
derivation because it considers the real dielectric constant of the metal
$\epsilon_{M}(\omega)$, via the Lorentz-Drude model [33], and also the
penetration of the EM fields into the metal through the surface impedance
$z_{s}=1/\sqrt{\epsilon_{M}}$. Yet, we consider
$z_{s}=1/\sqrt{\epsilon_{M}+1}$, which is a phenomenological correction that
leads to the exact dispersion relation of surface plasmon polaritons (SPPs) in
a metal-vacuum interface. The reflection coefficients are now

$f^{+}_{m}\,\mathbf{r}_{m}=-f_{0}^{-}\,\delta_{m0}\,\mathbf{i}_{0}+C_{m0}\,Y_{0}\,\mathbf{i}_{0}-\sum_{m^{\prime}}C_{mm^{\prime}}\,Y_{m^{\prime}}\,\mathbf{r}_{m^{\prime}},$
(6)

where the SIBC signatures are encapsulated in the geometric couplings and in
the quantities $f_{m}^{\pm}$, which are $2\times 2$ matrices in the CP basis
with respect to the propagation of the $m$-th Bragg mode, that depend on the
surface impedance $z_{s}$ such that:

$f_{m}^{\pm}=\frac{1}{2}\begin{pmatrix}f_{mp}^{\pm}+f_{ms}^{\pm}&f_{mp}^{\pm}-f_{ms}^{\pm}\\\
f_{mp}^{\pm}-f_{ms}^{\pm}&f_{mp}^{\pm}+f_{ms}^{\pm}\\\ \end{pmatrix},$ (7)

with $f_{m\sigma}^{\pm}=1\pm z_{s}Y_{m\sigma}$ and $\sigma=\\{ p,s\\}$.

The dependence of the metal approximation in the geometric couplings is
encapsulated in the constant $C^{\prime}$:

$C^{\prime}_{SIBC}=\frac{1}{Y}\frac{f^{+}\,f^{-}\,(1+\Phi)}{f^{+}-f^{-}\,\Phi},$
(8)

whereas

$C^{\prime}_{PEC}=\frac{1}{Y}\frac{1+\Phi}{1-\Phi},$ (9)

being $Y$ the modal admittance of the fundamental waveguide mode,
$f^{\pm}=1\pm z_{s}\,Y$, $\Phi=-e^{i2k_{z}^{w}d}$ and $k_{z}^{w}$ is the
propagation constant along the z-direction of the fundamental waveguide mode.
For a rectangular dimple with long side $b$, filled with a material with
dielectric constant $\epsilon_{d}$,
$k_{z}^{w}=\sqrt{\epsilon_{d}(\omega/c)^{2}-k_{w}^{2}}$, with $k_{w}=\pi/b$.

We posed in the main text that the geometric couplings depend on the
overlapping integrals $S_{m\sigma\alpha}$ between the Bragg modes
(characterized by $m$ and $\sigma$) and waveguide modes (characterized by the
dimple index $\alpha$). A general expression for the overlapping integrals is
intricate because of the dependence on the in-plane momenta and the size of
the dimples (it can be found in [15]). However, if we consider the small-
dimple approximation for which the dimple size is smaller than the wavelength,
they read

$S_{m\sigma\alpha}=\sqrt{\frac{ab}{2A_{uc}}}\frac{4}{\pi}\,v_{m\sigma\alpha}\,e^{-i\vec{k}_{m}\vec{r}_{\alpha}},$
(10)

where $A_{uc}$ is the area of the unit cell, $\sigma$ is the polarization of
the considered Bragg mode, and
$v_{mp\alpha}=(k_{x}^{m}\cos{\theta_{\alpha}}+k_{y}^{m}\sin{\theta_{\alpha}})/k_{m}$
and
$v_{ms\alpha}=(-k_{y}^{m}\cos{\theta_{\alpha}}+k_{x}^{m}\sin{\theta_{\alpha}})/k_{m}$,
being $k_{x}^{m}$ and $k_{y}^{m}$ the $x$ and $y$ components of the in-plane
momentum $\vec{k}_{m}$, respectively.

With these expressions, one can easily achieve the geometric couplings for
both systems presented in the main text (see Equations 4 and 5).

##  Appendix B Kagome lattice elements

In Tab. 1, we present the defining quantities for all the dimples comprising
the analyzed Kagome lattice. We label each dimple with an index $\alpha$ and
show its center position and its angle.

$\alpha$ | $x_{\alpha}$ | $y_{\alpha}$ | $\vec{r}_{\alpha}$ | $\theta_{\alpha}$
---|---|---|---|---
$0$ | $2L$ | $\sqrt{3}L$ | $2\vec{R}_{1}/3+\vec{R}_{2}/6$ | $\pi/2$
$1$ | $3L$ | $\sqrt{3}L$ | $\vec{R}_{1}$ | $7\pi/6$
$2$ | $5L/2$ | $3\sqrt{3}L/2$ | $5\vec{R}_{1}/6+\vec{R}_{2}/3$ | $11\pi/6$
$3$ | $0$ | $\sqrt{3}L$ | $\vec{R}_{1}/2$ | $7\pi/6$
$4$ | $L$ | $\sqrt{3}L$ | $\vec{R}_{1}/3+\vec{R}_{2}/3$ | $11\pi/6$
$5$ | $L/2$ | $3\sqrt{3}L/2$ | $\vec{R}_{1}/6+2\vec{R}_{2}/3$ | $\pi/2$
$6$ | $L$ | $2\sqrt{3}L$ | $\vec{R}_{1}/3+5\vec{R}_{2}/6$ | $11\pi/6$
$7$ | $2L$ | $2\sqrt{3}L$ | $2\vec{R}_{1}/3+2\vec{R}_{2}/3$ | $\pi/2$
$8$ | $3L/2$ | $5\sqrt{3}L/2$ | $\vec{R}_{1}/2+\vec{R}_{2}$ | $7\pi/6$
Table 1: Centers positions (also in terms of the direct lattice vectors
$\vec{R}_{1}=3L\vec{u}_{x}+\sqrt{3}L\vec{u}_{y}$ and
$\vec{R}_{2}=2\sqrt{3}L\vec{u}_{y}$) and angles for the $N=9$ dimples
constituting the unit cell of the $\sqrt{3}\times\sqrt{3}$ KL represented in
Figure 3a.

##  Appendix C SIBC approximation in the KL

Figure 4: $\Delta\mathbf{r}_{m}$ with respect to $m=(m_{1},m_{2})$. The spin
$+/-$ component is represented in red/blue, and we have considered $m_{2}=0$.
It has been computed taking a normal incident plane wave with spin $+$. Chosen
geometrical parameters: $L=460\,nm$, $a=80\,nm$ $b=220\,nm$, and $d=60\,nm$.
We have considered the SIBC approximation, phenomenologically enlarging the
dimple dimensions by 1.25 times the skin depth to consider the EM field
penetration in the metal [19]. a) Away from any plasmonic resonance, with
$\omega=3\,eV$ energy for the incident plane wave. b) At the plasmonic
resonance associated with the Bragg modes $m_{1}=\pm 2$, with
$\omega=1.73\,eV$ energy for the incident plane wave. The shadowed region
indicates modes that are outside the light cone.

In this section, we expand on the SML breakdown cases that we studied in the
main text for the Kagome lattice. We compute the effect of considering the
SIBC approximation and finite-size dimples. This is shown in Figure 4, where
we consider a representative case of non-resonant excitation and another case
of a resonant plasmonic excitation. In both cases, we represent $\Delta r_{m}$
and sweep $m_{1}$. We observe the effect of the SIBC approximation at first
glance. The zero order is larger than the rest (except when we excite an SPP
and the resonant modes govern). Furthermore, the SML breakdown is evident in
both figures, although the underlying SML can be noticed in the orders
$m_{1}=-1,2$ of Figure 4a where the spin $-$ component is larger than the spin
$+$ one, for instance. Besides, in Figure 4b we observe the same behavior of
very large $|\Delta r_{\pm 2}^{\pm}|$ as we presented in the main text for
$|\Delta r_{\pm 3}^{\pm}|$. Therefore, in the SIBC approximation, the SML
becomes less evident because of the metal absorption.

In this case and below, we have kept $m_{2}=0$ not only for the representation
but also for the simulation. This does not affect the physical behavior
because the $\vec{G}_{1}$ direction presents a breaking of the inversion
symmetry [14, 15].

##  Appendix D Analysis of the incident momentum in the KL

Figure 5: Absolute value for both spin components (red/blue is spin $+/-$) for
two reflection coefficients $r_{3,0}$ and $r_{2,0}$. We take $m_{2}=0$,
$k_{y}^{in}=0$, $\omega=1.79\,eV$ and incoming spin $+$. (a, b) are computed
neglecting SML breakdown terms. (c, d) are computed considering the SML
breakdown terms (full calculation). We take the PEC and the small-dimple
approximations. Chosen geometrical parameters: $L=460\,nm$, $a=80\,nm$
$b=220\,nm$, and $d=60\,nm$. The shadowed region indicates that reflection
coefficients are outside the light cone for that incident momentum
$k_{x}^{in}$.

We have focused on the KL by analyzing its SML, the breakdown terms, and its
dependence on being or not at a plasmonic resonance. For the latter analysis,
we have varied the energy and kept the normal incidence. However, we can also
excite different SPP resonances by varying the incident momentum. This section
will show how the reflection coefficients behave when the incident momentum is
varied away from the normal.

Figure 5 represents the absolute value of both spin components for two
reflection coefficients: $r_{3,0}$ and $r_{2,0}$, with respect to the incident
momentum in the $x$ direction: $k_{x}^{in}$. We have chosen the representative
values of $k_{y}^{in}=0$, $\omega=1.79\,eV$ and $\sigma_{in}=+$. Figures 5a
and 5b show that there is only one spin component for each mode, which is in
excellent agreement with the SML features derived from the geometric couplings
$C^{z}_{mm^{\prime}}$. The three small peaks for each subfigure correspond to
plasmonic resonances which, given that breakdown terms have been neglected,
preserve the SML. However, when we perform the full calculation, considering
all SML breakdown terms, both spin components are non-negligible and the SML
is spoiled (see Figures 5c and 5d). Besides, when the corresponding plasmonic
resonance is associated with the Bragg mode that we are representing via the
reflection coefficient, there is an enhancement of the latter. This was also
seen in the $|\Delta\mathbf{r}_{m}|$ plots of the main text.

Logically, the SML breaks down when a plasmonic resonance is excited because
the SPPs are linearly $p$ polarized. However, this breakdown persists even
when $k_{x}^{in}$ is increased away from resonance. The reason is that for
larger $k_{x}^{in}$, the Bragg modes associated with these reflection
coefficients ($r_{3,0}$ and $r_{2,0}$) are evanescent. Given this and
considering that both breakdown sources (modal admittances and the change of
basis matrices) depend on the momentum in the $\vec{u}_{z}$ direction of the
corresponding Bragg mode $q_{mz}$, it is easy to infer that the evanescent
modes introduce a strong breakdown as well.

##  Appendix E Analysis of the approximations in the KL

The results presented in the main text are computed in the PEC and small-
dimple approximations. On the other hand, in Figure 4 we showed what happens
if we calculate the same quantities but in the SIBC approximation and with
finite-size dimples. A global comparison is still lacking. For this reason, in
Figure 6, we display the five possibilities: neglecting the SML breakdown
terms (blue), PEC and small-dimple (red), PEC and finite-size (yellow), SIBC
and small-dimple (purple), and SIBC and finite-size (green).

Figure 6: Both spin components of $\Delta\mathbf{r}_{m}$. a)
$\Delta\mathbf{r}_{m}^{+}$, spin $+$ component. b) $\Delta\mathbf{r}_{m}^{-}$,
spin $-$ component. They have been computed taking a normal incident plane
wave with spin $+$ and energy $\omega=3\,eV$, and we have considered
$m_{2}=0$. Chosen geometrical parameters: $L=460\,nm$, $a=80\,nm$ $b=220\,nm$,
and $d=60\,nm$.

Along the main text and the rest of the appendices, we have dealt with two of
the five approximations detailed in Figure 6. In Figures 2c, 2d, 3c, 3d, 5c
and 5d, we considered the PEC and small-dimple approximations, or what we call
“full calculation”. Besides, in Figure 4, we used the SIBC and finite-size
approximations. Therefore, we present Figure 6 to compare them and add the
rest of the possible combinations: neglecting SML breakdown terms, PEC with
finite-size, and SIBC with small-dimple approximations.

The effects of the different approximations are observed in Figure 6,
representing both spin $+/-$ components of the reflection coefficients. Blue
dots represent the case of neglecting SML breakdown terms; because of that,
some modes are zero (not seen). This approximation is equivalent to the
behavior of the geometric couplings $C^{z}_{mm^{\prime}}$. The rest of the
approximations represent different levels of SML breakdown. The smallest SML
breakdown is obtained when the metal is considered as a PEC and the dimples
are very small, whereas the maximal breakdown appears when the metal is real
and the dimples are finite-sized. Moreover, a general pattern appears: the
effect of the dimple size is less relevant than the effect of the PEC
approximation. That is to say, choosing small dimples or finite dimples only
provides a small deviation over the reflection coefficients. However, a
greater difference appears between the PEC and the SIBC approximations.

Note that we have stayed away from any plasmonic resonance for this comparison
because the plasmonic resonance locations depend on the considered metal
approximations.

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