11institutetext:  Department of Physics and Astronomy, University of Turku,
FI-20014 Turku, Finland 22institutetext: Aalto University Metsähovi Radio
Observatory, Metsähovintie 114, 02540 Kylmälä, Finland 33institutetext: Aalto
University Department of Electronics and Nanoengineering, P.O. BOX 15500,
FI-00076 AALTO, Finland 44institutetext: European Southern Observatory, Karl-
Schwarzschild-Str. 2, 85748 Garching b. München, Germany 55institutetext:
Gemini Observatory/NSF’s NOIRLab, Casilla 603, La Serena, Chile
66institutetext: School of Sciences, European University Cyprus, Diogenes
street, Engomi, 1516 Nicosia, Cyprus 77institutetext: Finnish Centre for
Astronomy with ESO (FINCA), FI-20014 University of Turku, Finland
88institutetext: Department of Physics and Earth Science, University of
Ferrara, via Saragat 1, I-44122 Ferrara, Italy 99institutetext: INFN, Sezione
di Ferrara, via Saragat 1, I-44122 Ferrara, Italy 1010institutetext: INAF,
Osservatorio Astronomico d’Abruzzo, via Mentore Maggini snc, 64100 Teramo,
Italy 1111institutetext: Departamento de Ciencias Fisicas, Universidad Andres
Bello, Avda. Republica 252, Santiago, Chile

# Spectropolarimetry of Type II supernovae

(II) Intrinsic supernova polarization and its relations with the
photometric/spectroscopic properties

T. Nagao  , F. Patat<EMAIL_ADDRESS>A. Cikota 44 D. Baade 55 S.
Mattila 44 R. Kotak 1166 H. Kuncarayakti 11 M. Bulla 1177 88991010 B. Ayala
1111

(Received April ??, 2023; accepted ?? ??, 2023)

The explosion processes of supernovae (SNe) are imprinted in their explosion
geometries. The recent discovery of several highly aspherical core-collapse
SNe is significant, and regarded as the key to understanding the explosion
mechanism of core-collapse SNe. Here, we study the intrinsic polarization of
15 hydrogen-rich core-collapse SNe and explore the relation with the
photometric and spectroscopic properties. Our sample shows diverse properties
of the continuum polarization. The polarization of most SNe has a low degree
at early phases but shows a sudden rise to $\sim 1$ % degree at certain points
during the photospheric phase as well as a slow decline during the tail phase,
with a constant polarization angle. The variation in the timing of peak
polarisation values implies diversity in the explosion geometry: some SNe have
aspherical structures only in their helium cores, while in other SNe these
reach out to a significant part of the outer hydrogen envelope with a common
axis from the helium core to the hydrogen envelope. Other SNe show high
polarization from early phases and a change of the polarization angle around
the middle of the photospheric phase. This implies that the ejecta are
significantly aspherical to the outermost layer and have multi-directional
aspherical structures. Exceptionally, the Type IIL SN 2017ahn shows low
polarization at both the photospheric and tail phases. Our results show that
the timing of the polarization rise in Type IIP SNe is likely correlated with
their brightness, velocity and the amount of radioactive Ni produced: brighter
SNe with faster ejecta velocity and a larger 56Ni mass have more extended-
aspherical explosion geometries. In particular, there is a clear correlation
between the timing of the polarization rise and the explosion energy, that is,
the explosion asphericity is proportional to the explosion energy. This
implies that the development of a global aspherical structure, e.g., a jet,
might be the key to realising an energetic SN in the mechanism of SN
explosions.

###### Key Words.:

supernovae: general – Polarization

##  1 Introduction

Core-collapse supernovae (CCSNe) are catastrophic explosions of massive stars
at their deaths. This phenomenon is closely related to the evolution of
galaxies through the processes such as the chemical enrichment of galaxies and
the induction of star formation. Despite great efforts, the explosion
mechanism has not fully been understood yet. The most promising picture is the
neutrino-driven explosion, in which a star blows up due to neutrino heating
from a proto-neutron star (e.g., Janka et al. 2007; Janka 2012). Even though
some recent multi-dimensional simulations have been successful in launching
the explosion, they are unable to reproduce some of the basic observed
properties, e.g., the $10^{51}$ erg of explosion energy (e.g., Buras et al.
2006; Marek & Janka 2009; Takiwaki et al. 2012; Hanke et al. 2013; Melson et
al. 2015). The results of these simulations indicate that some multi-
dimensional hydrodynamic instabilities, such as convective motion (e.g.,
Herant et al. 1994) and the standing-accretion-shock-instability (SASI; e.g.,
Foglizzo 2002; Blondin et al. 2003), might play a crucial role in producing
energetic explosions, because the random movement of the ejecta gas due to the
instabilities can increase the neutrino heating efficiency in the gain region.
Additional effects to enhance some instabilities have also been proposed to
obtain realistic SN explosions. For instance, effects from general relativity,
rotation, magnetic fields, and inhomogeneities in the progenitor core. The
asymmetries resulting from the instabilities in the inner core might introduce
Rayleigh-Taylor (RT) instability at the C+O/He and He/H composition interfaces
and create global asymmetries in the ejecta (e.g., Wongwathanarat et al.
2015).

Investigating explosion geometries of SNe can provide insights on the
explosion mechanism. In particular, it is critically important to clarify
relations between the explosion asphericity and the SN properties (e.g.,
explosion energy, ejecta mass and progenitor radius). The proposed explosion
scenarios can be tested by comparing these relations with those from
theoretical simulations. The most reliable and direct way for investigating
the geometry of a spatially unresolved SN is polarimetry. For an SN with a
spherically symmetric photosphere, no polarization is expected due to
cancellation of the orthogonal Stokes vectors. Thus, the detection of a non-
null continuum polarization provides direct and independent evidence for an
overall asphericity.

Hydrogen-rich CCSNe (hereafter Type II SNe) are classified into two subtypes,
Type IIP and IIL, based on the light-curve shapes. Type IIP SNe are the most
common class of CCSNe ($\sim 50$ % of all CC events; e.g., Li et al. 2011).
They show constant optical brightness until $\sim 100$ days after explosion
(the photospheric phase), which is followed by an exponential flux decline
(the tail phase) after a sudden drop (e.g., Anderson et al. 2014; Faran et al.
2014a; Sanders et al. 2015; Valenti et al. 2016). In general, a low level of
polarization ($\sim$0.1 %) has been measured during the photospheric phase,
while a rapid increase of continuum polarization ($\sim 1.0$ %) is seen at the
beginning of the tail phase in some Type IIP SNe (e.g., Leonard et al. 2001,
2006; Chornock et al. 2010; Kumar et al. 2016). This polarimetric feature can
be explained by an asymmetric helium core being revealed when the outer
spherical hydrogen envelope becomes optically thin as indicated by the light
curve falling off the plateau (e.g., Leonard et al. 2006). On the other hand,
Dessart & Hillier (2011) demonstrated that the small continuum polarization
during the photospheric phase can be, in some cases, reproduced by optical-
depth effects irrespective of the magnitude of the asphericity of the hydrogen
envelope, based on the polarimetric calculations from axially symmetric SN
ejecta artificially constructed using 1-dimensional inputs from non-local-
thermodynamic-equilibrium radiative-transfer simulations. Although only a few
Type IIP SNe have been observed in polarimetry with a sufficient temporal
coverage, this high polarization is regarded as supporting evidence for the
asymmetric explosions suggested by some numerical simulations (e.g., Janka
2012). Recently, Nagao et al. (2019) found an unprecedented, highly-extended
aspherical explosion in SN 2017gmr indicated by an early rise of polarization,
hence clearly showing that asymmetries are present not only in the helium core
but also in a substantial portion of the hydrogen envelope. This demonstrates
the existence of an intrinsic diversity in the aspherical structure of Type
IIP SN explosions.

Type IIL SNe are a relatively rare subtype of Type II SNe, which show a linear
decline in their brightness during the photospheric phase (e.g., Faran et al.
2014b). There has not been many Type IIL SNe observed with polarimetry. Nagao
et al. (2021) reported that a low polarization has been observed during the
entire evolution from the photospheric to the tail phases in SN 2017ahn, while
SN 2013ej showed a high polarization from the beginning of the photospheric
phase. Based on the polarization analysis of SNe 2017ahn and 2013ej, Nagao et
al. (2021) claim that there are at least two subtypes of Type IIL SNe in terms
of their explosion schemes: some SNe may come from different explosions than
Type IIP SNe (a spherical explosion; e.g., SN 2017ahn) and others from similar
explosions (an aspherical explosion; e.g., SN 2013ej).

In this study, we investigate the polarimetric properties of 15 Type II SNe
using unpublished and published spectropolarimetric data obtained with the
Very Large Telescope (VLT) and other telescopes. We have divided the analysis
into two papers. In Nagao et al. (2023; hereafter Paper I), we present the
full description of our SN sample, the observations, data reduction
techniques, and our method to estimate the interstellar polarization (ISP). We
also investigate the properties of the ISP components of 11 well-observed SNe
from our sample in Paper I. Here, we discuss the properties of the intrinsic
SN polarization of 15 SNe from our SN sample, their photometric and
spectroscopic properties, and the relations between these properties, with the
aim of exploring possible relations between the explosion asphericity and the
explosion physics. Throughout this work, the classification between Type IIP
and IIL is based on their LC shapes, i.e., quantified by the value of the
decline rate during the photospheric phase ($s_{p}$; see Section 3.1). In the
following sections, we show polarimetric (Section 2), photometric and
spectroscopic (Section 3) properties of our sample. In Section 4, we discuss
relations between the polarimetric properties and the other observational
properties of the SNe. Our conclusions are finally presented in Section 5.

##  2 The properties of the intrinsic SN polarization

In this section, we discuss the continuum polarization in the intrinsic SN
polarization, which is derived by subtracting the ISP component from the
observed polarization spectrum. The full description of our SN sample, the
observations, the data reduction techniques, and the method to estimate the
interstellar polarization are presented in Paper I. As for the VLT sample, we
calculate the continuum polarization in the same way with Chornock et al.
(2010), Nagao et al. (2019) and Nagao et al. (2021). We independently take the
sums of the ordinary fluxes and the extraordinary fluxes within the wavelength
ranges 6800-7200 Å as well as 7820-8140 Å, using the reduced 50Å-binned
polarization spectra (see Section 3.1 in Paper I). From these averaged fluxes
of the ordinary and extra-ordinary beams, we calculate the Stokes $Q$ and $U$
values and thereby the polarization degree and angle for the continuum
polarization. The derived values are discussed in Section 2.1 and shown in
Figures 1-11. It is noted that, for SNe 1999em, 2004dj, 2006ov and 2007aa, we
have instead adopted the derived values for the ISP and the intrinsic SN
polarization from the literature (Leonard et al. 2001, 2006; Chornock et al.
2010). As the continuum polarization of SN 1999em, we adopted the $V$-band
polarization values that were measured by calculating the debiased, flux-
weighted averages of $q$ and $u$ over the intervals 6050-6950 Å for the first
epoch and 5050-5950 Å for the other epochs (Leonard et al. 2001). For SN
2004dj, we adopted the continuum polarization values that were derived as the
median values of the Stokes parameters over the interval 6800-8200 Å (Leonard
et al. 2006). The continuum polarization of SNe 2006ov and 2007aa were derived
in the same way with the one in this paper.

In addition to the ISP, there are two possible origins for the continuum
polarization observed in Type II SNe: electron scattering in an aspherical
photosphere (the aspherical-photosphere origin) and dust scattering in
aspherically distributed circumstellar material (the dust scattering origin;
see, e.g., Nagao et al. 2019, 2021, and references therein). In the case of
the aspherical-photosphere origin the polarization is wavelength independent,
while in the dust scattering origin some wavelength dependence is expected
(see Nagao et al. 2018, for details). Since all polarization spectra in our
sample do not show any clear wavelength dependence (see Figures 20-30 in
Appendix B) at any phase, we conclude that the continuum polarization of all
the SNe at all the epochs in our sample originate from aspherical structures
of their photosphere. Such asphericities can be created by an aspherical
explosion (e.g., SN 2017gmr; Nagao et al. 2019), an ejecta interaction with
aspherical CSM (e.g., SN 2013ej; Nagao et al. 2021) or other unknown
processes.

###  2.1 Individual SNe in the VLT sample

The VLT sample consists of the following 11 SNe: SNe 2017gmr, 2017ahn, 2013ej,
2012ec, 2012dh, 2012aw, 2010hv, 2010co, 2008bk, 2001du, and 2001dh. The
individual objects are discussed in the following sub-sections.

####  2.1.1 SN 2017gmr

The data of SN 2017gmr were already discussed in Nagao et al. (2019). As shown
in that work, the intrinsic continuum polarization of SN 2017gmr shows high
degrees ($\sim 0.8$ %) with a constant angle ($\sim 95\degr$) from the middle
of the photospheric phase to the beginning of the tail phase (see Figure 1).
This indicates that the asymmetry is not confined to the helium core but
extends to a significant part of the hydrogen envelope, and that the SN has a
global aspherical structure from the helium core to the hydrogen envelope,
e.g., a jet-like structure. Since the SN did not show any features due to a
strong CSM interaction (see Andrews et al. 2019), we interpret the origin of
this polarization as an aspherical structure of the ejecta originated from an
aspherical explosion (see also discussions in Nagao et al. 2019). In addition,
since the polarization evolution in the tail phase is similar to other Type
IIP SNe that are believed to be aspherical explosions ($t^{-2}$ evolution,
which can be interpreted as being due to geometrical dilution that reduces the
optical depth of the ejecta; e.g., SN 2004dj Leonard et al. 2006), it is
natural to interpret the origin as a common feature, i.e., aspherical
explosions. This conclusion is supported by the asymmetric lines seen in the
nebular spectra of SN 2017gmr (e.g., Andrews et al. 2019; Utrobin et al.
2021).

Figure 1: Polarization degree and angle of the continuum polarization in SN
2017gmr.

####  2.1.2 SN 2017ahn

The data of SN 2017ahn were already presented in Nagao et al. (2021). As
discussed in that paper, the continuum polarization of this object remains low
from the photospheric to tail phases (Figure 2), which implies a purely
spherical explosion or a polar viewing angle of an aspherical explosion. We
note that, at the first two epochs (and possibly also at the other epochs as
well), the polarization might show a component in the bluer wavelengths
($\lambda<6000$ Å) with $\sim 0.5$ % degree of polarization (see Figure 21).
This might originate from some line polarization, some dust scattering
component or an incomplete ISP subtraction.

Figure 2: Polarization degree and angle of the continuum polarization in SN
2017ahn.

####  2.1.3 SN 2013ej

The data of SN 2013ej were already discussed in Nagao et al. (2021). The
continuum polarization of this SN has two components with different angles,
possibly consisting of an interaction component (originating from the
aspherical CSM interaction) and an explosion component (arising from an
aspherical explosion). Following the discussion in Nagao et al. (2021), we
assume that the signal at the first epoch comes purely from the CSM
interaction ($P=0.40$ % and $\theta=78.4\degr$). The explosion component,
which is derived by subtracting the interaction component from the observed
polarization, is shown in Figure 3 and Table 6. Hereafter, we consider only
the explosion component as the continuum polarization of SN 2013ej.

Figure 3: Polarization degree and angle of the continuum polarization in SN
2013ej. The explosion component is shown with a red color.

####  2.1.4 SN 2012ec

This SN shows low polarization degrees at least until the latest observation
(Phase -54.49 days; see Figure 4). This implies that the shape of the
photosphere is likely close to spherical, at least until this epoch.

Figure 4: Polarization degree and angle of the continuum polarization in SN
2012ec.

####  2.1.5 SN 2012dh

SN 2012dh shows a relatively low polarization level ($P<0.3$ %) until the
epoch of the last observation (phase -54.37 $\pm$ 16.40 days), having erratic
polarization angles (see Figure 5). This implies that the shape of the
photosphere is quite spherical, at least, until this phase. Here, the large
errors bars in the phases originate from the large uncertainty of the end of
the photospheric phase.

Figure 5: Polarization degree and angle of the continuum polarization in SN
2012dh.

####  2.1.6 SN 2012aw

The data of SN 2012aw were already discussed in Dessart et al. (2021), even
though they assumed slightly different values for the ISP (see Paper I). As SN
2013ej, SN 2012aw has at least two components showing the polarization-angle
change with time: a component with $\theta\sim 35\degr$ and a component with
$\theta\sim 125\degr$ (see Figure 6). The first two epochs appear to be
dominated by the first component with a relatively constant polarization angle
$P\sim 0.35$ %, while the later epochs seem to be affected by the second time-
evolving component with another constant polarization angle, $\theta\sim
125\degr$. This behavior of the continuum polarization is similar to that of
SN 2013ej (Nagao et al. 2021). As in that case, the first and second
components might be due to a combination of an aspherical CSM interaction and
an aspherical explosion, respectively. However, as opposed to SN 2013ej, SN
2012aw did not show strong interaction features in its photometric and
spectroscopic properties at early phases. In addition, the two components are
close to perpendicular. This might suggest the presence of two-component
ejecta driven by a ”jet” explosion (a collimated high-energy component and a
canonical, bulk SN component).

Assuming the signals in the first epoch represents purely one component
($P\sim 0.35$ % and $\theta=34.8\degr$), we derive the remaining component by
subtracting this first component from the original polarization signals (see
Figure 6). In fact, the remaining component shows a relatively constant
polarization angle ($\theta\sim 120\degr$) as well as a polarization rise
similar to the behaviours of SNe 2017gmr and 2013ej. Thus, we assume that the
remaining component is related to an aspherical explosion, as in other SNe,
e.g., SNe 2017gmr and 2013ej. We note that the angle of the second component
at the second epoch differs from those at the other epochs. This may be due to
the large uncertainty originating from the subtraction of similar vectors in
the Q-U plane, which we do not take into account for the error estimation of
the polarization angles.

The difference of the polarization angles of the first component and the
explosion component is $\sim 90\degr$ for SN 2012aw, while $\sim 20\degr$ for
SN 2013ej (see Tables 6 and 9). If the origin of the first component for SN
2012aw is the same as for SN 2013ej, which is supposed to come from an
aspherical CSM interaction (Nagao et al. 2021), this might indicate that the
orientations of the aspherical structures of the explosion and the CSM
distribution are independent. In turn, this suggests that the two
asphericities (the first and second polarization components) are produced by
different mechanisms. Unlike SN 2013ej, SN 2012aw does not show any clear
evidence for a strong CSM interaction in the LC and spectra during the
photospheric phase. Therefore, the origin of the first component may be
different in the two objects. The number of SNe showing such a component is
still too small to allow any firm conclusion on its nature. Hereafter, we
consider only the second component (red points in Figure 6) as the continuum
polarization of SN 2012aw.

Figure 6: Polarization degree and angle of the continuum polarization in SN
2012aw. The aspherical explosion component is shown with a red color.

####  2.1.7 SN 2010hv

SN 2010hv shows low polarization degrees during most parts of the photospheric
phase (see Figure 7), even though there is a large uncertainty in the phases.
This implies that either most of the hydrogen envelope is quite spherical or
that the viewing angle is close to polar for an aspherical structure.

Figure 7: Polarization degree and angle of the continuum polarization in SN
2010hv.

####  2.1.8 SN 2010co

SN 2010co shows a polarization rise during the photospheric phase as SNe
2017gmr, 2013ej and 2012aw. This implies that this SN also has an aspherical
structure extending to the hydrogen envelope.

Figure 8: Polarization degree and angle of the continuum polarization in SN
2010co.

####  2.1.9 SN 2008bk

SN 2008bk shows relatively low polarization degrees during the photospheric
phase, while it develops high polarization in the tail phase. It is not clear
whether the timing of the polarization rise corresponds to the luminosity drop
as in the other SNe, because that part of the polarimetric evolution
(occurring between our fith and sixth epoch) is not covered by the available
data. However, the general behavior is similar to that expected for the
classical scenario (e.g., SN 2004dj; Leonard et al. 2006): an aspherical He
core surrounded by a spherical hydrogen envelope. Since the polarization
degree around 60 days after the luminosity drop is high ($P\sim 1.17$ %)
compared with the other SNe in our sample (see Figure 12), the actual
polarization peak might be also high ($\sim 2$ %) at the timing just after the
luminosity drop, if this SN also follows the $t^{-2}$ evolution. This might
imply a very aspherical He core or just be due to the viewing angle effect of
a similar spherical core in our sample.

Figure 9: Polarization degree and angle of the continuum polarization in SN
2008bk.

####  2.1.10 SN 2001du

Although the data of SN 2001du are rather scanty, this object shows low
polarization degrees at least at the start of the photospheric phase unlike
SNe 2013ej and 2012aw. This behavior is seen in most Type II SNe.

Figure 10: Polarization degree and angle of the continuum polarization in SN
2001du.

####  2.1.11 SN 2001dh

The continuum polarization of SN 2001dh shows a very different behavior with
respect to those of all the other SNe that have been observed so far, in that
it displays a component with a single polarization angle ($\theta\sim
160\degr$), which declines from the $\sim 1$% level during the photospheric
phase. The 1% of the continuum polarization at the start of the photospheric
phase is the highest level among all observed Type II SNe ever. In addition,
the marked decline has not been observed in any other SNe. Since it is
difficult for an aspherical explosion to create some aspherical structure only
in the outermost layers, this polarization decline might be due to
cancellation of polarization components by multiple aspherical structures as
in SNe 2013ej and 2012aw.

Figure 11: Polarization degree and angle of the continuum polarization in SN
2001dh.

###  2.2 Diversity in the continuum polarization

Figure 12: Time evolution of the continuum polarization of our SN sample.
Here, for SNe 2013ej and 2012aw, the explosion components after the
subtraction of the interaction components are plotted.

Figure 12 shows the time evolution of the continuum polarization of the SNe in
our sample. All the SNe, with the only exception of SN 2017ahn, show
relatively high polarization degrees at some epoch ($\sim 1$ %), although
there is large diversity in the temporal evolution. This supports that an
aspherical structure is a common feature in Type II SNe (e.g., Wang & Wheeler
2008).

A polarization level as large as $\sim 1$ % corresponds to an asphericity with
a minimum axis ratio of 1.2:1.0 in the electron-scattering ellipsoidal
atmosphere model by Höflich (1991), even though the peak polarization degree
depends on the viewing angle. Given its behavior, SN 2017ahn might be an
exceptionally spherical explosion, or a common aspherical structure viewed
along the polar axis. In general, the timing of the polarization rise does not
depend on the viewing angle unless we observe it somehow away from an axis of
an aspherical structure. Thus, the different timings of the polarization rise
mean that the extent of an aspherical structure in the hydrogen envelope is
different for different SNe. As discussed above, SN 2001dh shows a behavior
that is very different with respect to the others. This might imply a multi-
directional explosion or an external factor, like an aspherical CSM
interaction.

The observed diversity in the continuum polarization points to diverse
explosion geometries in Type II SNe, which can be categorized as follows:

  * •

Group 1 (SNe 2008bk, 2007aa, 2006ov and 2004dj): this group includes SNe which
show low polarization degrees ($<0.5$ %) at the photospheric phase and a
polarization rise ($\sim 1$ %) at the photospheric to tail transitional phase.
This is a classical picture of the continuum polarization in Type II SNe
(e.g., Leonard et al. 2006). This is explained in terms of an aspherical
helium core enshrouded within a spherical hydrogen envelope. We note that SN
2006ov does not have observations during the photospheric phase, so it is not
clear whether the polarization is low during the photospheric phase. However,
it shows a clear polarization rise at the transitional phase, and thus may
belong to this group.

  * •

Group 2 (SNe 2017gmr, 2013ej, 2012aw and 2010co): objects in this group show
an increase of the polarization ($>0.5$ %) already during the photospheric
phase. This early rise implies that SNe have a very extended aspherical
structure: asymmetries are not confined to the helium core but extend to a
significant part of the hydrogen envelope.

  * •

Group 3 (SN 2017ahn): SNe in this group show low polarization at all times,
from photospheric to tail phases. Since the properties of SN 2017ahn deviate
from those of the other Type II SNe (Tartaglia et al. 2021), this might be
generated by an explosion mechanism which is different from the majority of
the explosions. Alternatively, this SN might be a similar explosion but, by
chance, it was viewed along the axis of the aspherical structure. Since there
is only such a case, it is difficult to draw any firm conclusion. Future
larger samples may allow one to disentangle between possible options.

The polarization behaviour of SN 2001dh does not conform to any of the above
groups. Since the observational data are very limited, we cannot conclude
whether the high polarization originates just from an aspherical CSM
interaction like SN 2013ej (belonging to one of above groups), or its
explosion geometry is really aspherical to the outer ejecta edge (therefore
belonging to a new group). As for the remaining SNe 2012ec, 2012dh, 2010hv,
2001du, 2001dh and 1999em, the data are not sufficient to decide to which of
the above groups they belong to.

In the tail phase, the continuum polarization seems to have a behavior that is
similar for all objects, i.e., a decline that approximately follows the so-
called $t^{-2}$ evolution (see Leonard et al. 2006; Nagao et al. 2019), which
corresponds to the optical depth decreasing in an optically-thin, homologously
expanding material. Although the number of SNe that have observations in the
tail phase is limited, this indicates that the inner helium cores of Type II
SNe are generally aspherical. This implication is consistent with the
polarimetric observations of stripped-envelope SNe (e.g., Patat et al. 2001;
Kawabata et al. 2002; Tanaka et al. 2009; Maund et al. 2009; Reilly et al.
2016).

SNe 2013ej and 2017gmr, which have data both in the photospheric and in the
tail phases, show a constant polarization angle throughout the observations.
This implies a global aspherical structure from the inner to outer parts of
the ejecta, which could be produced, for instance, by a collimated explosion.

###  2.3 Characteristic value for the continuum polarization

Figure 13: Same as Figure 12 but with the time after explosion normalized by
the length of the photospheric phase, for SNe in Groups 1 and 2 (left panel)
and for the other SNe (right panel). The horizontal dashed lines express zero
degree of polarization. The dotted lines are the extrapolation of the first
two points that start to show an increase of the polarization degree, whose
intersection with the dashed line ($P=0$) is defined as the timing of the
polarization rise ($t_{\rm{pol}}$).

In general, the polarization of Type II SNe shows low degrees at early
photospheric phases, and then it increases to a $\sim 1.0\%$ level at a
certain time during the photospheric phase: some SNe show an early rise, while
in others this occurs just at the end of the photospheric phase (see Figure
12). The polarization degree depends on the shape and asymmetry of the
photosphere and also on the viewing angle. On the other hand, the time of the
polarization rise corresponds to the time when the photosphere recedes back to
the outer edge of the aspherical structure, and it is therefore linked to the
extent of this structure. Since the mass of the hydrogen envelope of Type II
SNe progenitors SNe varies from object to object, to allow for a meaningful
comparison, we normalise the time of the polarization rise to the length of
the photospheric phase, and we adopt the time of the polarization rise as a
characteristic property of the continuum polarization ($t_{\rm{pol}}$). In
practice, we estimate this time by extrapolating using the first two points
that start to show an increase of the polarization degree for the SNe in Group
1 and 2 (see the left panel of Figure 13). For the other SNe, whose data do
not show a clear polarization rise, we use the last data point in the
photospheric phase as an upper limit. The derived values of $t_{\rm{pol}}$ are
shown in Table 1, together with other parameters (see below).

Table 1: The type, polarization group and characteristic polarimetric,
photometric and spectroscopic parameters for the 15 SNe in our sample.

Name SN Type Polarization $t_{\rm{pol}}$ $M_{V}(t_{\rm{ph}}/2)$,
$L(t_{\rm{\rm{ph}}}/2)$ $v(t_{\rm{ph}}/2)$ $M_{\rm{Ni}}$ group [mag],
$10^{42}$[erg/s] $10^{3}$[km/s] [M⊙] SN 2017gmr IIP 2 0.41 -18.1$\pm$0.2,
5.6$\pm$1.1 5.6$\pm$0.1 0.269$\pm$0.015 SN 2017ahn IIL 3 (0.83-1.00)
-17.3$\pm$0.4, 2.7$\pm$1.2 (4.7$\pm$0.1) 0.038$\pm$0.002 SN 2013ej IIL 2 0.08
-16.7$\pm$0.2, 1.6$\pm$0.3 4.5$\pm$0.1 0.019$\pm$0.001 SN 2012ec IIP ?
(0.49-1.00) -16.5$\pm$0.1, 1.3$\pm$0.1 3.8$\pm$0.1 0.034$\pm$0.002 SN 2012dh
IIL ? (0.39-1.00) -16.3$\pm$0.5, 1.1$\pm$0.7 (4.6$\pm$0.1) 0.024$\pm$0.002 SN
2012aw IIP 2 0.12 -16.7$\pm$0.1, 1.6$\pm$0.1 3.6$\pm$0.1 0.061$\pm$0.004 SN
2010hv II 1 or 3 (0.75-1.00) - (4.2$\pm$0.1) - SN 2010co II 2 0.28 $\pm$ 0.14
- (4.6$\pm$0.1) - SN 2008bk IIP 1 (0.98-1.00) -14.7$\pm$0.1, 0.25$\pm$0.02
(1.9$\pm$0.1) 0.007$\pm$0.001 SN 2007aa IIP 1 0.76 -16.7$\pm$0.2, 1.6$\pm$0.3
3.1$\pm$0.1 0.042$\pm$0.012 SN 2006ov IIP 1 0.79 -14.8$\pm$0.4, 0.27$\pm$0.12
(1.7$\pm$0.1) 0.002$\pm$0.001 SN 2004dj IIP 1 0.82 -16.0$\pm$0.2,
0.81$\pm$0.26 2.9$\pm$0.1 0.016$\pm$0.001 SN 2001du II ? (0.05-1.00) -
(5.8$\pm$0.1) - SN 2001dh II 2? - - (3.2$\pm$0.1) - SN 1999em IIP 1 or 2
(0.36-1.00) -16.7$\pm$0.2, 1.6$\pm$0.3 3.0$\pm$0.1 0.058$\pm$0.002

##  3 Photometric and spectroscopic properties

###  3.1 Photometric properties

Figure 14: V-band light curves of our sample (left panel) and the same light
curves with the time after explosion nomalized by the length of the
photospheric phase (right panel). The vertical line and dashed lines indicate
the timing of the middle and end of the photospheric phase, respectively. The
blue dotted line is the extrapolation of the first point. Figure 15:
Schematics of the function used for the light curve fitting (see Eq. (4))

Figure 14 shows the light curves (LCs) of our SN sample, which are corrected
for reddening (see Table 1 in Paper I). The SNe have diverse properties in
terms of brightness and duration of the photospheric phase. For characterizing
the LC shapes, we fit the time evolution of the absolute magnitude ($M(t)$)
with the following artificial function composed by two power law functions and
a sigmoid function (see Figure 15):

$\displaystyle M(t)=\frac{-a_{0}}{1+e^{\frac{t-t_{\rm{ph}}}{w_{0}}}}+M_{t}$
$\displaystyle+\left\\{\begin{array}[]{ll}s_{p}\left[t-\left(t_{\rm{ph}}-w_{0}\right)\right]&(t\leq
t_{\rm{ph}}-w_{0})\\\ 0&(t_{\rm{ph}}-w_{0}\leq t\leq t_{\rm{ph}}+w_{0})\\\
s_{t}\left[t-\left(t_{\rm{ph}}-w_{0}\right)\right]&(t\geq
t_{\rm{ph}}+w_{0})\end{array}\right.$ (4)

where $M_{t}$, $a_{0}$, $t_{\rm{\rm{ph}}}$, $w_{0}$, $s_{\rm{p}}$ and
$s_{\rm{t}}$ are free parameters. The best-fit values for these parameters are
summarized in Table 2 for each SN.

As a characteristic value for the SN brightness we adopt the V-band absolute
magnitude in the middle of the photospheric phase
($M_{V}(t_{\rm{\rm{ph}}}/2)$). For deriving the typical value of the SN
bolometric luminosity from the V-band magnitude we simply assume a 6000 K
black-body spectrum. The values of $L(t_{\rm{\rm{ph}}}/2)$ converted from
$M(t_{\rm{\rm{ph}}}/2)$ are presented in Table 1. For deriving the Ni mass for
each SN, also summarized in Table 1, we adopt the empirical method proposed by
Hamuy (2003), using the following equations:

$M_{\rm{Ni}}=(7.866\times
10^{-44})L_{t}\exp\left[\frac{(t_{\rm{ph}}+w_{0})/(1+z)-6.1}{111.26}\right]\rm{M}_{\odot},$
(5)

where $L_{t}$ is the bolometric luminosity at $t=t_{\rm{ph}}+w_{0}$ which, in
turn, is estimated using the following empirical equation by Hamuy (2001):

$\log_{10}L_{t}=\frac{-\left(M_{t}-A_{V}+BC\right)+5\log\left(10\rm{pc}\right)-8.14}{2.5}.$
(6)

Here, for the the bolometric correction ($BC$) we have adopted the value
$BC=0.26\pm 0.06$ during the tail phase (Hamuy 2001).

In this work, we have classified the SNe between Type IIP and IIL based on the
values of $s_{p}$. The SNe with $s_{p}<1$ are classified as Type IIP, while
the others are as Type IIL. Here, the derived value of $s_{p}$ for SN 2006ov
indicates it as a Type IIL SN in our criteria ($s_{p}>1$; see Section 3.1).
However, this is just due to the limited photometric data during the
photospheric phase (see Section 3.1). The photometric and spectroscopic
properties of SN 2006ov in Chornock et al. (2010) suggest a class of Type IIP.
Thus, we classify it as a Type IIP SN in this work. As for the SNe that do not
have photometric observations (SNe 2010hv, 2010co, 2001du and 2001dh), we
label them just as Type II in this paper. The photometric classification of
the SNe is shown in Table 1.

Table 2: Best-fit values from the LC fitting

Name $t_{\rm{ph}}$ $a_{0}$ $w_{0}$ $M_{t}$ $s_{p}$ $s_{t}$ SN 2017gmr 92.92
$\pm$ 0.13 1.50 $\pm$ 0.01 5.58 $\pm$ 0.09 -16.39 $\pm$ 0.01 0.56 $\pm$ 0.01
1.29 $\pm$ 0.02 SN 2017ahn 61.62 $\pm$ 0.41 1.65 $\pm$ 0.05 4.04 $\pm$ 0.22
-14.59 $\pm$ 0.04 3.96 $\pm$ 0.06 1.75 $\pm$ 0.10 SN 2013ej 100.47 $\pm$ 0.08
2.38 $\pm$ 0.01 3.73 $\pm$ 0.06 -13.47 $\pm$ 0.01 1.90 $\pm$ 0.02 1.17 $\pm$
0.01 SN 2012ec 106.70 $\pm$ 0.67 2.22 $\pm$ 0.07 7.13 $\pm$ 0.56 -14.01 $\pm$
0.06 0.45 $\pm$ 0.04 1.11 $\pm$ 0.02 SN 2012dh (115.4 $\pm$ 16.4)
1.53${}^{+0.18}_{-0.17}$ (5.8) -13.53${}^{+0.19}_{-0.22}$
2.18${}^{+0.02}_{-0.06}$ 1.16 $\pm$ 0.04 SN 2012aw 133.83 $\pm$ 0.33 2.01
$\pm$ 0.02 9.19 $\pm$ 0.32 -14.35 $\pm$ 0.01 0.51 $\pm$ 0.01 1.01 $\pm$ 0.02
SN 2008bk 130.10 $\pm$ 0.65 2.82 $\pm$ 0.02 10.75 $\pm$ 0.26 -12.05 $\pm$ 0.01
0.02 $\pm$ 0.01 1.06 $\pm$ 0.01 SN 2007aa 103.20 $\pm$ 1.22 2.10 $\pm$ 0.31
3.31 $\pm$ 0.76 -14.34 $\pm$ 0.26 0.61 $\pm$ 0.24 0.15 $\pm$ 0.85 SN 2006ov
121.37 $\pm$ 0.33 3.85 $\pm$ 0.07 2.08 $\pm$ 0.16 -10.58 $\pm$ 0.06 1.78 $\pm$
0.18 0.23 $\pm$ 0.09 SN 2004dj 101.77 $\pm$ 0.52 2.54 $\pm$ 0.04 8.01 $\pm$
0.31 -13.22 $\pm$ 0.02 0.47 $\pm$ 0.06 0.80 $\pm$ 0.02 SN 1999em 122.34 $\pm$
0.12 2.17 $\pm$ 0.01 4.13 $\pm$ 0.09 -14.15 $\pm$ 0.01 0.49 $\pm$ 0.01 0.65
$\pm$ 0.03 Notes. The fitting function according to Equation 4 and Figure 15.

###  3.2 Spectroscopic properties

Figure 16: Fe II velocities of our SN sample (left panel) and the velocities
with the time after explosion nomalized by the length of the photospheric
phase (right panel). The vertical lines are the same as in Figure 14. The
dotted lines are the extrapolation of the first point.

In this paper we do not discuss the spectroscopic properties of our sample.
However, for the purposes of our analysis, we have extracted the expansion
velocities. In particular, we have derived the ejecta velocity at the middle
of the photospheric phase derived using the Fe II $\lambda 5169$ line
($v(t_{\rm{\rm{ph}}}/2)$), and used it as a characteristic value. The
velocities were estimated from the absorption minima, using the spectra
presented in Paper I. For SNe 2004dj and 1999em, we adopted the Fe velocity
derived by Takáts & Vinkó (2012). The uncertainties on the absorption minima
position are evaluated from the spectral the dispersion (e.g., $\sim 3.3$ Å
pixel-1 for G300V/FORS2/VLT). The derived velocities are shown in Figure 16
and the values of $v(t_{\rm{\rm{ph}}}/2)$ are provided in Table 1.

###  3.3 SN explosion properties

We estimate the basic SN physical properties (explosion energy:
$\widetilde{E}_{\rm{sn}}$, ejecta mass: $\widetilde{M}_{\rm{ej}}$, and pre-
explosion progenitor radius: $\widetilde{R}_{\rm{p}}$) from the SN photometric
and spectroscopic properties (typical luminosity: $\widetilde{L}_{\rm{sn}}$,
typical duration: $\widetilde{t}_{\rm{sn}}$, typical ejecta velocity:
$\widetilde{v}_{\rm{sn}}$, and the 56Ni mass: $M_{\rm{Ni}}$). Here, we are not
interested in deriving precise values for these physical quantities but rather
in obtaining some relative scales for the quantities. Thus, we use a simple
one-zone model of homologously expanding gas with radiative diffusion, as
proposed by Kasen & Woosley (2009). This is characterised by the following
equations:

$\displaystyle\widetilde{M}_{\rm{ej}}$ $\displaystyle=$
$\displaystyle\frac{4\pi
c\sigma_{\rm{SB}}T_{I}^{4}}{\kappa_{\rm{es}}}\widetilde{t}_{\rm{sn}}^{4}\widetilde{L}_{\rm{sn}}^{-1}\widetilde{v}_{\rm{sn}}^{3},$
(7) $\displaystyle\widetilde{E}_{\rm{sn}}$ $\displaystyle=$
$\displaystyle\frac{2\pi
c\sigma_{\rm{SB}}T_{I}^{4}}{\kappa_{\rm{es}}}\widetilde{t}_{\rm{sn}}^{4}\widetilde{L}_{\rm{sn}}^{-1}\widetilde{v}_{\rm{sn}}^{5},$
(8) $\displaystyle\widetilde{R}_{\rm{p}}$ $\displaystyle=$
$\displaystyle\frac{\kappa_{\rm{es}}}{2\pi
c\sigma_{\rm{SB}}T_{I}^{4}}\widetilde{t}_{\rm{sn}}^{-2}\widetilde{L}_{\rm{sn}}^{2}\widetilde{v}_{\rm{sn}}^{-4}\left[1-\widetilde{L}_{\rm{sn}}^{-1}\widetilde{t}_{\rm{sn}}^{-2}(E_{\rm{Ni}}t_{\rm{Ni}}+E_{\rm{Co}}t_{\rm{Co}})\right],$
(9)

where $E_{\rm{Ni}}\simeq 0.6\times 10^{50}(M_{\rm{Ni}}/M_{\odot})$ erg and
$E_{\rm{Co}}\simeq 1.2\times 10^{50}(M_{\rm{Ni}}/M_{\odot})$ erg are the total
energy by 56Ni and 56Co decay, and $t_{\rm{Ni}}\simeq 8.8$ days and
$t_{\rm{Co}}\simeq 113$ days are the decay time. Here, we simply assume that
$\widetilde{t}_{\rm{sn}}=t_{\rm{\rm{ph}}}$,
$\widetilde{L}_{\rm{sn}}=L(t_{\rm{\rm{ph}}}/2)$ and
$\widetilde{v}_{\rm{sn}}=v(t_{\rm{\rm{ph}}}/2)$.

We note that, since we assume a constant brightness by the radiative
diffusion, we cannot apply this analysis to Type IIL SNe in general. However,
SN 2013ej is also proposed as a Type IIP SN based on its photometric and
spectroscopic properties (e.g., Dhungana et al. 2016). In fact, its LC is very
similar to those of Type IIP SNe except the early bump during the first $\sim
40$ days, which is understood as a result of a CSM interaction. Thus, we
conducted this analysis for the Type IIP SNe that have the photometric and
spectroscopic information (SNe 2017gmr, 2012ec, 2012aw, 2008bk, 2007aa,
2006ov, 2004dj and 1999em) and the Type IIL SN 2013ej.

The derived values for $\widetilde{E}_{\rm{sn}}$, $\widetilde{M}_{\rm{ej}}$
and $\widetilde{R}_{\rm{p}}$ are summarized in Table 3.

Table 3: Characteristic SN explosion properties from the SN photometric and
spectroscopic properties.

Name $\widetilde{M}_{\rm{ej}}$ $\widetilde{E}_{\rm{sn}}$
$\widetilde{R}_{\rm{p}}$ [M⊙] $10^{51}$ [erg/s] [100$R_{\odot}$] SN 2017gmr
5.32${}_{-1.13}^{+1.70}$ 1.66${}_{-0.40}^{+0.61}$ 1.63${}_{-1.63}^{+4.87}$ SN
2013ej 13.20${}_{-2.84}^{+4.21}$ 2.66${}_{-0.67}^{+1.00}$
2.37${}_{-1.02}^{+1.43}$ SN 2012ec 12.45${}_{-2.05}^{+2.50}$
1.79${}_{-0.37}^{+0.47}$ 2.11${}_{-0.58}^{+0.75}$ SN 2012aw
21.28${}_{-3.05}^{+3.61}$ 2.74${}_{-0.52}^{+0.65}$ 2.63${}_{-0.63}^{+0.81}$ SN
2008bk 17.88${}_{-4.08}^{+5.25}$ 0.64${}_{-0.20}^{+0.28}$
0.98${}_{-0.33}^{+0.49}$ SN 2007aa 4.81${}_{-1.31}^{+2.01}$
0.46${}_{-0.15}^{+0.23}$ 7.38${}_{-3.76}^{+5.89}$ SN 2006ov
8.98${}_{-3.85}^{+10.43}$ 0.26${}_{-0.13}^{+0.37}$ 2.58${}_{-1.99}^{+4.50}$ SN
2004dj 7.35${}_{-2.44}^{+4.88}$ 0.61${}_{-0.23}^{+0.48}$
2.93${}_{-2.01}^{+3.63}$ SN 1999em 8.60${}_{-2.08}^{+3.13}$
0.77${}_{-0.22}^{+0.35}$ 6.06${}_{-3.08}^{+4.74}$ Notes. These values were
estimated using Equations 7-9.

##  4 Relations between polarimetric and spectro-photometric properties

In this section, we discuss relations between the polarimetric and
photometric/spectroscopic parameters of eleven SNe that have both sets of
observations (eight Type IIP and three Type IIL SNe).

Figure 17 shows the relations between the polarimetric ($t_{\rm{pol}}$) and
photometric parameters ($t_{\rm{\rm{ph}}}$, $a_{0}$, $w_{0}$, $M_{t}$,
$s_{\rm{p}}$ and $s_{\rm{t}}$). There seems to be no significant correlation
between polarization degree and the other photometric parameters. However, the
brightness at the tail phase $M_{t}$ (and thus the 56Ni mass, see below) and
the brightness gap between the photospheric and tail phases ($a_{0}$) might be
correlated with $t_{\rm{pol}}$ in Type IIP SNe, although the limited sample
does not allow for a firm conclusion.

Figure 18 shows the relations between the polarimetric parameter
($t_{\rm{pol}}$) and the characteristic values from the photometric and
spectroscopic observations ($M(t_{\rm{\rm{ph}}}/2)$, $v(t_{\rm{\rm{ph}}}/2)$
and $M_{\rm{Ni}}$). In the Type IIP sample, there might be some correlations:
SNe that are brighter and faster and have larger amount of 56Ni tend to have
more extended aspherical structures. Since the Type IIL sample is limited, we
cannot conclude anything about their behavior. However, they do not seem to
follow these correlations. We emphasize that the ejecta velocity estimated
from the Fe II absorption minimum is related to the ejecta velocity along the
line of sight, and is thus affected by the viewing angle. Most SNe show $\sim
1$ % of polarization degrees at their polarization peaks, implying aspherical
structures with a minimum ellipsoidal axial ratio of 1.2:1.0. If the parts of
the ejecta forming the Fe II line have such an asphericity, the estimated
velocities can have an uncertainty of $\pm\gtrsim 10$% just because of the
uncertainties of the viewing angle. This effect is more important for SNe that
have more aspherical structures in the line forming regions, i.e. the outer
parts of the ejecta above the photosphere. Moreover, the Fe distribution does
not necessarily have to be consistent with the overall aspherical structure.
In some extreme cases, e.g., SN 2017gmr, there is a global aspherical
structure from the inner core to the outer envelope, which might be created by
a jet-like explosion and might produce a more aspherical Fe distribution. A
larger sample is required to statistically confirm this correlation between
the ejecta velocity and the polarimetric parameter taking into account this
effect.

Figure 19 shows the relations between the polarimetric property
($t_{\rm{pol}}$) and the SN explosion parameters ($\widetilde{M}_{\rm{ej}}$,
$\widetilde{E}_{\rm{ej}}$, $\widetilde{R}_{\rm{p}}$). The polarimetric
property ($t_{\rm{pol}}$) is correlated with the explosion energy
($\widetilde{E}_{\rm{ej}}$) rather than the ejecta mass
($\widetilde{M}_{\rm{ej}}$) and the progenitor radius
($\widetilde{R}_{\rm{p}}$): More energetic SNe tend to have more extended
aspherical structure in their explosions. We finally note that the absolute
values of the SN explosion parameters are not accurate because of limitation
of the one-zone model (see Section 3.3). Therefore, only a relative comparison
between the various SNe is possible. In addition, the largest uncertainties
for the SN explosion parameters come from the uncertainty of the viewing
angles which translates into a significant uncertainty on the expansion
velocities.

Figure 17: Relations between the polarimetric property ($t_{\rm{pol}}$) and
photometric properties ($t_{\rm{\rm{ph}}}$, $a_{0}$, $w_{0}$, $M_{t}$,
$s_{\rm{p}}$ and $s_{\rm{t}}$). The black points represent Type IIP SNe, while
gray points do Type IIL SNe.

Figure 18: Relations between the polarimetric property ($t_{\rm{pol}}$) and
the characteristic values from the photometric and spectroscopic observations
($M(t_{\rm{\rm{ph}}}/2)$, $v(t_{\rm{\rm{ph}}}/2)$ and $M_{\rm{Ni}}$). The
black points represent Type IIP SNe, while gray points do Type IIL SNe.

Figure 19: Relations between the polarimetric property ($t_{\rm{pol}}$) and
the SN explosion parameters ($\widetilde{M}_{\rm{ej}}$,
$\widetilde{E}_{\rm{ej}}$, $\widetilde{R}_{\rm{p}}$). The black points
represent Type IIP SNe, while gray point does the Type IIL SN 2013ej.

##  5 Conclusions

In this article, we have investigated the continuum polarization of 15 Type II
SNe. Our analysis shows that the objects in the sample have diverse
properties. Most SNe show low polarization at early phases with a sudden rise
to $\sim 1\%$ degree at a certain point during the photospheric phase,
followed by a slow decline during the tail phase, with a constant polarization
angle. The timing of the polarization rise varies for different SNe. We
interpret this as the signature of different explosion geometries: some SNe
have aspherical structures only in their helium cores, while the asphericity
of other SNe reaches out to a significant part of the outer hydrogen envelope,
with a common axis through the helium core and the hydrogen envelope. Some SNe
show high polarization already at early phases and a change of the
polarization angle. This indicates multi-directional aspherical structures in
their explosions, which might originate from a combination of a common
aspherical SN explosion and an aspherical CSM interaction or from a multi-
directional SN explosion. An exception in our sample is the Type IIL SN
2017ahn, which has low polarization during both the photospheric and tail
phases. Since this object is unique in our sample, we cannot conclude whether
this SN is produced by a completely spherical explosion or just the result of
a different viewing angle for a similarly aspherical explosion.

The timing of the polarization rise in Type IIP SNe appears to be correlated
with their brightness, ejecta velocity, and Ni mass: SNe with higher
brightness, higher ejecta velocity and larger Ni production show more extended
asphericities. In particular, there is a clear correlation between the timing
of the polarization rise and the explosion energy: the asphericity in the
explosions is proportional to the explosion energy (see Section 4 and Figure
19). This implies that the emergence of a global aspherical structure, e.g., a
jet-like structure, might be the key ingredient in the explosion mechanism to
produce an energetic SN.

###### Acknowledgements.

We thank Keiichi Maeda, Masaomi Tanaka and Giuliano Pignata for useful
discussions. T.N. was supported by a Japan Society for the Promotion of
Science (JSPS) Overseas Research Fellowship and was funded by the Academy of
Finland project 328898. H.K. was funded by the Academy of Finland projects
324504 and 328898. S.M. was funded by the Academy of Finland project 350458.
This research has made use of NASA’s Astrophysics Data System Bibliographic
Services.

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##  Appendix A Polarization degrees and angles of the continuum polarization
for the VLT sample

Table 4: SN 2017gmr

$\displaystyle\begin{array}[]{lccccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\rm{Date}&\rm{MJD}&\rm{Phase}^{a}&\rm{Days\;from\;explosion}&\rm{Pol.\;degree}&\rm{Pol.\;angle}\\\
(\rm{UT})&(\rm{days})&(\rm{days})&(\rm{days})&(\rm{per\;cent})&(\rm{degrees})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr 2017-10-19.60&58045.60&-46.41&+46.51&0.18\pm 0.01&87.5\pm
2.0\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr 2017-11-10.24&58067.24&-24.77&+68.15&0.71\pm 0.02&93.4\pm
0.7\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr 2017-12-12.86&58099.86&+7.85&+100.77&0.81\pm 0.01&99.7\pm
0.7\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr 2017-12-21.76&58108.76&+16.75&+109.67&0.59\pm 0.01&97.4\pm
0.7\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr 2018-01-16.93&58134.93&+42.92&+135.84&0.39\pm 0.02&93.9\pm
2.1\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$

Notes.

Table 5: SN 2017ahn

$\displaystyle\begin{array}[]{lccccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\rm{Date}&\rm{MJD}&\rm{Phase}^{a}&\rm{Days\;from\;explosion}&\rm{Pol.\;degree}&\rm{Pol.\;angle}\\\
(\rm{UT})&(\rm{days})&(\rm{days})&(\rm{days})&(\rm{per\;cent})&(\rm{degrees})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr 2017-03-24.75&57836.75&-16.63&+44.99&0.13\pm 0.02&11.0\pm
3.4\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr 2017-03-31.38&57843.38&-10.00&+51.62&0.09\pm 0.03&28.2\pm
4.3\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr 2017-04-24.06&57867.06&+13.68&+75.30&0.16\pm 0.07&29.4\pm
6.0\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr 2017-04-28.73&57871.73&+18.35&+79.97&0.15\pm 0.08&39.2\pm
6.8\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$

Notes.

Table 6: SN 2013ej

$\displaystyle\begin{array}[]{lcccccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\rm{Date}&\rm{MJD}&\rm{Phase}^{a}&\rm{Days\;from\;explosion}&\rm{Pol.\;degree}&\rm{Pol.\;angle}\\\
(\rm{UT})&(\rm{days})&(\rm{days})&(\rm{days})&(\rm{per\;cent})&(\rm{degrees})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr 2013-08-01.38&56505.38&-91.99&+8.48&0.40(0.00)\pm
0.01&78.4(-)\pm 0.6\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr
2013-08-27.28&56531.28&-66.09&+34.38&0.75(0.37)\pm 0.01&84.0(90.0)\pm 0.3\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr 2013-09-17.21&56552.21&-45.16&+55.31&1.03(0.67)\pm 0.01&88.9(94.9)\pm
0.2\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr 2013-09-29.24&56564.24&-33.13&+67.34&1.24(0.90)\pm
0.01&92.1(97.9)\pm 0.2\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
2013-10-29.23&56594.23&-3.14&+97.33&1.19\pm 0.01&97.2\pm 0.3\\\ \vskip 3.0pt
plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
2013-12-04.56&56630.56&+33.19&+133.66&0.54\pm 0.02&106.1\pm 1.5\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
2014-01-09.38&56666.38&+69.01&+169.48&0.64\pm 0.05&123.5\pm 3.6\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$

Notes.

Table 7: SN 2012ec

$\displaystyle\begin{array}[]{lcccccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\rm{Date}&\rm{MJD}&\rm{Phase}^{a}&\rm{Days\;from\;explosion}&\rm{Pol.\;degree}&\rm{Pol.\;angle}\\\
(\rm{UT})&(\rm{days})&(\rm{days})&(\rm{days})&(\rm{per\;cent})&(\rm{degrees})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr 2012-09-08.28&56178.28&-71.42&+35.28&0.05\pm
0.02&131.5\pm 5.9\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2012-09-25.21&56195.21&-54.49&+52.21&0.12\pm
0.02&134.5\pm 5.1\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\end{array}$

Notes.

Table 8: SN 2012dh

$\displaystyle\begin{array}[]{lcccccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\rm{Date}&\rm{MJD}&\rm{Phase}^{a}&\rm{Days\;from\;explosion}&\rm{Pol.\;degree}&\rm{Pol.\;angle}\\\
(\rm{UT})&(\rm{days})&(\rm{days})&(\rm{days})&(\rm{per\;cent})&(\rm{degrees})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr 2012-07-18.03&56126.03&-94.35\pm 16.40&+21.05&0.05\pm
0.01&76.3\pm 3.9\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2012-08-11.98&56150.98&-69.40\pm
16.40&+46.00&0.25\pm 0.02&21.5\pm 2.2\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
2012-08-24.99&56163.99&-56.39\pm 16.40&+59.01&0.12\pm 0.03&173.0\pm 6.7\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr 2012-08-27.01&56166.01&-54.37\pm 16.40&+61.03&0.19\pm 0.05&173.7\pm
5.2\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$

Notes.

Table 9: SN 2012aw

$\displaystyle\begin{array}[]{lcccccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\rm{Date}&\rm{MJD}&\rm{Phase}^{a}&\rm{Days\;from\;explosion}&\rm{Pol.\;degree}&\rm{Pol.\;angle}\\\
(\rm{UT})&(\rm{days})&(\rm{days})&(\rm{days})&(\rm{per\;cent})&(\rm{degrees})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr 2012-04-01.19&56018.19&-117.74&+16.09&0.35\pm
0.01&34.8\pm 1.5\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2012-05-01.04&56048.04&-87.89&+45.94&0.31\pm
0.02&22.8\pm 2.6\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2012-05-18.07&56065.07&-70.86&+62.97&0.15\pm
0.01&86.3\pm 1.6\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2012-05-27.01&56074.01&-61.92&+71.91&0.16\pm
0.01&102.6\pm 3.3\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2012-06-16.03&56094.03&-41.90&+91.93&0.45\pm
0.01&117.8\pm 0.5\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2012-07-02.01&56110.01&-25.92&+107.91&0.75\pm
0.01&124.1\pm 0.4\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2012-07-15.98&56123.98&-11.95&+121.88&1.00\pm
0.03&126.3\pm 1.4\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\end{array}$

Notes.

Table 10: SN 2010hv

$\displaystyle\begin{array}[]{lcccccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\rm{Date}&\rm{MJD}&\rm{Phase}^{a}&\rm{Days\;from\;explosion}&\rm{Pol.\;degree}&\rm{Pol.\;angle}\\\
(\rm{UT})&(\rm{days})&(\rm{days})&(\rm{days})&(\rm{per\;cent})&(\rm{degrees})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr 2010-09-21.15&55460.15&-73.45\pm 24.4&+41.95\pm
8.00&0.06\pm 0.05&89.5\pm 4.6\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
2010-09-24.14&55463.14&-70.46\pm 24.4&+44.94\pm 8.00&0.11\pm 0.05&120.6\pm
4.1\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr 2010-10-15.07&55484.07&-49.53\pm 24.4&+65.87\pm 8.00&0.06\pm
0.04&40.8\pm 4.4\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2010-11-29.56&55529.56&-4.04\pm
24.4&+111.36\pm 8.00&0.11\pm 0.02&18.1\pm 3.7\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\end{array}$

Notes.

Table 11: SN 2010co

$\displaystyle\begin{array}[]{lcccccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\rm{Date}&\rm{MJD}&\rm{Phase}^{a}&\rm{Days\;from\;explosion}&\rm{Pol.\;degree}&\rm{Pol.\;angle}\\\
(\rm{UT})&(\rm{days})&(\rm{days})&(\rm{days})&(\rm{per\;cent})&(\rm{degrees})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr 2010-06-05.73&55352.73&-73.27\pm 27.90&+42.13\pm
11.50&0.21\pm 0.06&115.1\pm 4.2\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
2010-07-08.21&55385.21&-40.79\pm 27.90&+74.61\pm 11.50&0.83\pm 0.10&175.2\pm
5.2\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$

Notes.

Table 12: SN 2008bk

$\displaystyle\begin{array}[]{lcccccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\rm{Date}&\rm{MJD}&\rm{Phase}^{a}&\rm{Days\;from\;explosion}&\rm{Pol.\;degree}&\rm{Pol.\;angle}\\\
(\rm{UT})&(\rm{days})&(\rm{days})&(\rm{days})&(\rm{per\;cent})&(\rm{degrees})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr 2008-06-02.36&54619.36&-60.24&+69.86&0.19\pm
0.01&148.3\pm 2.5\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2008-07-01.31&54648.31&-31.29&+98.81&0.44\pm
0.01&142.2\pm 0.9\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2008-07-24.29&54671.29&-8.31&+121.79&0.23\pm
0.01&133.6\pm 0.8\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2008-07-30.28&54677.28&-2.32&+127.78&0.31\pm
0.01&117.6\pm 2.0\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2008-08-06.25&54684.25&+4.65&+134.75&0.43\pm
0.01&102.3\pm 1.0\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2008-09-28.10&54737.10&+57.50&+187.60&1.17\pm
0.01&126.3\pm 0.4\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2008-11-19.03&54789.03&+109.43&+239.53&0.35\pm
0.02&112.7\pm 1.9\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2008-12-20.04&54820.04&+140.44&+270.54&0.32\pm
0.02&108.2\pm 2.5\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2009-01-01.04&54832.04&+152.44&+282.54&0.32\pm
0.03&110.9\pm 2.7\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\end{array}$

Notes.

Table 13: SN 2001du

$\displaystyle\begin{array}[]{lcccccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\rm{Date}&\rm{MJD}&\rm{Phase}^{a}&\rm{Days\;from\;explosion}&\rm{Pol.\;degree}&\rm{Pol.\;angle}\\\
(\rm{UT})&(\rm{days})&(\rm{days})&(\rm{days})&(\rm{per\;cent})&(\rm{degrees})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr 2001-08-30.32&52151.32&-109.28\pm 16.90&+6.12\pm
0.50&0.05\pm 0.04&143.6\pm 10.2\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
2001-09-13.36&52165.36&-95.24\pm 16.90&+20.16\pm 0.50&0.13\pm 0.04&3.7\pm
8.5\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$

Notes.

Table 14: SN 2001dh

$\displaystyle\begin{array}[]{lcccccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\rm{Date}&\rm{MJD}&\rm{Phase}^{a}&\rm{Days\;from\;explosion}&\rm{Pol.\;degree}&\rm{Pol.\;angle}\\\
(\rm{UT})&(\rm{days})&(\rm{days})&(\rm{days})&(\rm{per\;cent})&(\rm{degrees})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr 2001-08-10.18&52131.18&-83.72\pm 29.80&+31.68\pm
13.40&0.97\pm 0.05&154.6\pm 1.9\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
2001-08-19.16&52140.16&-74.74\pm 29.80&+40.66\pm 13.40&0.89\pm 0.05&151.8\pm
2.3\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr 2001-08-29.05&52150.05&-64.85\pm 29.80&+50.55\pm 13.40&0.74\pm
0.07&154.7\pm 4.0\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr 2001-09-11.09&52163.09&-51.81\pm
29.80&+63.59\pm 13.40&0.70\pm 0.14&162.2\pm 8.0\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\hline\cr\end{array}$

Notes.

##  Appendix B Spectropolarimetric data of the FORS/VLT sample

Figure 20: Spectropolarimetry of SN 2017gmr after ISP subtraction. The red
hatching shows the adopted wavelength range for the continuum polarization
estimate. Figure 21: Same as Figure 20, but for SN 2017ahn. Figure 22: Same as
Figure 20, but for SN 2013ej. Figure 23: Same as Figure 20, but for SN 2012ec.
Figure 24: Same as Figure 20, but for SN 2012dh. Figure 25: Same as Figure 20,
but for SN 2012aw. Figure 26: Same as Figure 20, but for SN 2010hv. Figure 27:
Same as Figure 20, but for SN 2010co. Figure 28: Same as Figure 20, but for SN
2008bk. Figure 29: Same as Figure 20, but for SN 2001du. Figure 30: Same as
Figure 20, but for SN 2001dh.


# Microscopic analysis of the spin-momentum locking on a Geometric Phase
Metasurface

Fernando Lorén  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  Gian L.
Paravicini-Bagliani  Université de Strasbourg, CNRS, Institut de Science et
d’Ingénierie Supramoléculaires, UMR 7006, F-67000 Strasbourg, France  Sudipta
Saha  Université de Strasbourg, CNRS, Institut de Science et d’Ingénierie
Supramoléculaires, UMR 7006, F-67000 Strasbourg, France  Jérôme Gautier
Université de Strasbourg, CNRS, Institut de Science et d’Ingénierie
Supramoléculaires, UMR 7006, F-67000 Strasbourg, France  Minghao Li
Université de Strasbourg, CNRS, Institut de Science et d’Ingénierie
Supramoléculaires, UMR 7006, F-67000 Strasbourg, France  Cyriaque Genet
<EMAIL_ADDRESS>Université de Strasbourg, CNRS, Institut de Science et
d’Ingénierie Supramoléculaires, UMR 7006, F-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 revisit spin-orbit coupling in a plasmonic Berry metasurface comprised of
rotated nanoapertures, which is known to imprint a robust far-field
polarization response. We prove that spin-momentum locking is a property that
emerges from the geometry of the unit cell and does not require global
rotation symmetries. We theoretically find, and confirm with Mueller
polarimetry measurements, that spin-momentum locking is an approximate
symmetry. The breakdown is ascribed to the elliptical projection of circularly
polarized light into the planar surface and is maximal when surface waves are
excited.

Introduction.– Chiral light-matter interactions Bliokh et al. (2015); Lodahl
et al. (2017), form the core of recent discussions in quantum optics and
material science. They appear in non-reciprocal systems, where isolators and
circulators have opened new possibilities for the design of on-chip signal
processing systems and quantum networks Mahmoodian et al. (2016); Yuan et al.
(2015); Seif and Hafezi (2018). Recently, such interactions have been
investigated in the field of nano-optics, with the appropriate designs of two-
dimensional nanoantennas or plasmonic arrays Shitrit et al. (2011); O’Connor
et al. (2014); Shitrit et al. (2013a, b); Dahan et al. (2010); Frischwasser et
al. (2011); Lin et al. (2013). Chiral metasurfaces, sometimes known as “Berry”
or “Geometric Phase Metasurfaces” (GPM) Bomzon et al. (2002), have found many
applications for the selective manipulation of quantum emitters, in particular
in the field of valleytronics Sun et al. (2019); Guddala et al. (2019); Jha et
al. (2018). There, valley excitons can be selectively excited and detected by
the spin angular momentum of the emitted light due to the metasurface’s spin-
momentum locking (SML) mechanism. This allows routing valley degrees of
freedom into optical cavity modes, opening ways for new valley-photon
interfaces Gong et al. (2018); Chervy et al. (2018); Rong et al. (2020).

Despite their applicative potential and the fascinating connections they draw
with many fundamental issues in optics, plasmonic GPMs have been elusive to a
rigorous and exhaustive theoretical description. Previous theoretical attempts
have either considered (i) adiabatic processes in systems with optical
elements presenting a continuous spatial modulation Bomzon et al. (2002), or
(ii) a group theory analysis in the Kagome lattice, restricted to waves with
an electric field perpendicular to the surface, that ascribed SML to the
simultaneous presence of translation and rotation symmetries of the whole
lattice Shitrit et al. (2013b). None of these approaches cover the typical
case of GPMs with discrete elements that present chirality within the unit
cell but without global rotation symmetries Ding et al. (2014); Guddala et al.
(2019); Yu and Capasso (2014); Genevet et al. (2017); Yu et al. (2011); Karimi
et al. (2014); Lin et al. (2014). An issue that has not been addressed yet is
how SML can be reconciled with the surface plasmon polariton (SPP) linear
polarization, with a transverse spin density with respect to the propagation
direction.

This letter explores the interplay between polarization and optical momenta on
a plasmonic holey GPM. We propose a scattering formulation that clarifies how
SML appears. Our formulation has three strong assets. First, it shows that the
SML appears from the geometry of the unit cell. Second, it demonstrates that
the SML rules are approximate for the spin due to the elliptical projection
onto the GPM of the circularly polarized (CP) diffraction orders. Third, it
enables a direct comparison with experimental data. The validity of our
findings is confirmed by the excellent agreement between theory and the
experimental determination of the polarization states associated with each of
the diffracted Bragg modes with a Mueller polarimetry performed in the optical
Fourier space.

Figure 1: (a) Scanning electron microscope (SEM) image of the periodic
metasurface, composed of rectangular grooves in a gold film. (b) Schematic
bird’s-eye view of the unit cell. Nominal parameters in panel (a): $a=80$ nm,
$b=220$ nm, $d=60$ nm, $L=460$ nm.

Theoretical formalism.– We consider the simplest GPM, characterized by a
periodic array of rectangular grooves in a metal slab. Fig. 1(a) presents an
SEM image of the periodic gold metasurface that we will experimentally study
later, and Fig. 1(b) is a representation of the unit cell, which is
periodically repeated in both $\vec{u}_{x}$ and $\vec{u}_{y}$ directions. Each
groove has short side $a$, long side $b$ and depth $d$. The distance between
the centers of the two nearest slits is $L$. The unit cell comprises N such
grooves, with centers aligned along the $\vec{u}_{x}$ axis. The $\alpha$-th
groove is rotated at an angle $\theta_{\alpha}$ with respect to the
$\vec{u}_{x}$ axis. We consider the case where $\theta_{\alpha}$ 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 (in the particular case $n_{w}=0$, the array becomes a non-
chiral square lattice of rectangular grooves). The system depicted in Fig.
1(b) has $n_{w}=1/2$, as each rectangular groove presents a mirror symmetry
with respect to the middle of the long axis. Although the grooves perform a
step-wise rotation within the unit cell, the whole lattice does not support a
global rotation symmetry.

We consider an electromagnetic plane wave impinging onto the structure with an
in-plane wavevector along the $x$-direction,
$\vec{k}^{in}=k_{x}^{in}\vec{u}_{x}$, and compute the reflection coefficients
into the different Bragg modes that can be diffracted off the periodic
metasurface. For that, we use the coupled-mode method (CMM), which has been
widely used in the study of electromagnetic (EM) properties in holey metallic
films Garcia-Vidal et al. (2010). The CMM expands the EM fields in plane waves
in the free space regions and waveguide modes inside the grooves and finds the
field amplitudes by adequately matching the fields at the interfaces.

The equations derived from the CMM are usually written in terms of the
amplitudes of the waveguide modes Martín-Moreno et al. (2001); Martín-Moreno
and García-Vidal (2008); de León-Pérez et al. (2008). However, we find it more
convenient to derive the equations directly for the reflection coefficients
for studying the SML mechanism. Although less efficient computationally, this
provides a more transparent description, as it avoids the near-field to far-
field decoding needed to extract scattering coefficients from waveguide mode
amplitudes.

Bragg modes are characterized by an in-plane momentum
$\vec{k}=\vec{k}^{in}+\vec{G}$, where $\vec{G}$ is a reciprocal lattice
vector, in this case $\vec{G}=2\pi m/(NL)\,\vec{u}_{x}+2\pi
n_{y}/L\,\vec{u}_{y}$. To develop a minimal model for studying the SML, and
simplify the presentation, in the main text we consider only Bragg modes with
$n_{y}=0$. As shown in the Supp. Mat. sup , this does not change the main
results because along the $y$-axis there is no breaking of inversion symmetry.
We collect the two circular polarizations for the reflected $m$-th Bragg mode
in the spinor $\mathbf{r}_{m}\equiv(r_{m}^{+},r_{m}^{-})$, where $\pm$ denote
the right- and left-handed polarizations. We choose the spin representation
because the spin of a plane wave is conserved upon reflection by a mirror
Bliokh et al. (2013); Cameron et al. (2012, 2014, 2017) (while the helicity
changes sign).

The CMM can take into account the dielectric constant in the metal via the
implementation of the surface impedance boundary conditions (SIBC). The
general expressions can be found in the Supp. Mat. sup , and are the ones used
below in comparison with the experimental data. Here, we present the
expressions for the more straightforward case when the metal is treated as a
perfect electric conductor (PEC), as this is sufficient to discuss the
physics:

$\displaystyle\mathbf{r}_{m}=$
$\displaystyle-\delta_{m0}\,\mathbf{i}_{0}+\bar{Y}_{0}\,C_{m0}\,\mathbf{i}_{0}-\sum_{m^{\prime}}\bar{Y}_{m^{\prime}}\,C_{mm^{\prime}}\,\mathbf{r}_{m^{\prime}}$
(1)
$\displaystyle+\Delta_{0}\,C_{m0}\,\sigma_{x}\,\mathbf{i}_{0}-\sum_{m^{\prime}}\Delta_{m^{\prime}}\,C_{mm^{\prime}}\,\sigma_{x}\,\mathbf{r}_{m^{\prime}}.$

The first term takes into account the specular reflection ($\mathbf{i}_{0}$ is
the amplitude of the incident wave). The coefficients $C_{mm^{\prime}}$, which
we call “geometric couplings”, are $2\times 2$ matrices operating in
polarization space. They couple different Bragg modes via scattering with the
GPM and encode the geometry of the unit cell through the overlaps between
Bragg and waveguide modes (see the Supp. Mat. sup ). The coefficient
$\bar{Y}_{m}=(Y_{mp}+Y_{ms})/2$ is the average of the modal admittances in the
linear $s$ (transverse electric) - $p$ (transverse magnetic) polarized basis,
while $\Delta_{m}=(Y_{mp}-Y_{ms})/2$. The modal admittances relate the in-
plane magnetic field to the electric one. For a plane wave with frequency
$\omega$ and in-plane wavevector $k_{m}$ propagating in a uniform medium with
dielectric constant $\epsilon$, $Y_{mp}=\epsilon/q_{mz}$ and $Y_{ms}=q_{mz}$,
where $q_{mz}=\sqrt{\epsilon-(ck_{m}/\omega)^{2}}$ (c being the speed of
light).

In the second line of Eq. 1 where $\sigma_{x}$ is the Pauli matrix that swaps
spin states.

Notice that $\Delta_{0}=0$ at normal incidence, while both $\bar{Y}_{m}$ and
$\Delta_{m}$ diverge at the Rayleigh points (i.e., whenever a diffractive
order becomes tangent to the metal-dielectric interface, as then $q_{mz}=0$).

Spin-Momentum Locking.– The geometric couplings have an analytical expression.
When the groove dimensions are much smaller than the wavelength, we find

$C_{mm^{\prime}}=C\left(\delta_{m,m^{\prime}+n_{0}N}\,\mathbb{1}+\sum_{s=\pm}\delta_{m,m^{\prime}+n_{0}N-2n_{w}s}\,\sigma_{s}\right),$
(2)

where $n_{0}$ is an integer, $C$ is a constant that only depends on the
properties of a single groove, $\mathbb{1}$ is the $2\times 2$ identity matrix
and $\sigma_{\pm}$ are Pauli matrices that increase and decrease spin,
respectively.

If all $\Delta_{m}$ were zero, the previous expressions would give rise to the
following Bragg laws. The term proportional to $\mathbb{1}$ preserves spin and
the associated Bragg law $k_{x}^{out}=k_{x}^{in}+n_{0}G^{0}$, with
$G^{0}=2\pi/L$, would be the one that would appear if all grooves were
parallel (and it is denoted as “standard” Bragg law Shitrit et al. (2013b)).
The terms inside the summatory change spin by $\pm 1$ and shift the standard
Bragg law by a term that depends both on the spin change and the winding
number: $k_{x}^{out}=k_{x}^{in}+n_{0}G^{0}\mp k_{g}$, where $k_{g}\equiv
2\pi\,2n_{w}/(NL)$ is the geometric momentum. This condition is denoted spin-
orbit Bragg law Shitrit et al. (2013b) and corresponds to the exact spin-
momentum locking mechanism.

Notice that the standard and spin-orbit Bragg laws have arisen in a system
without a global lattice symmetry that combines translation and rotation.
Thus, SML is a feature of the basis of the unit cell and not of the symmetry
of the whole lattice. This constitutes the first significant result of this
Letter.

However, for Bragg modes away from normal incidence $\Delta_{m}\neq 0$, and
the terms proportional to $\sigma_{x}$ in Eq. 1 must be considered. These
terms flip circular polarization before the geometric couplings are applied.
Thus the momentum conservation laws that originate from Eq. 2 still hold, but
the exact link between changes in momentum and spin, previously described when
assuming $\Delta_{m}=0$, now breaks down.

The physical origin of the breakdown terms resides in that the polarization of
the transversal EM field is defined on the plane perpendicular to the
wavevector. At the same time, it is the in-plane component of the EM field
that is relevant in the interaction with the holey metasurface. Thus, in
general, an EM circularly polarized wave (with respect to its wavevector) has
an elliptical projection onto the planar surface. It is thus described as a
combination of the two circular polarizations with respect to the surface
normal.

Mathematically, for a PEC, the breakdown term is maximum for waves with
$q_{mz}=0$, while for a real metal, this occurs when the Bragg mode coincides
with the SPP of the flat surface. Considering SML breakdown terms is thus
essential when surface resonances are excited (spoof SPPs in the case of a
PEC, SPPs of the corrugated surface in the case of a real metal Pendry et al.
(2004)).

Figure 2: Reflection coefficients for the two polarizations (spin $+$ in red
and spin $-$ in blue) and two diffraction orders. The incident EM wave has
spin $+$ and in-plane momentum $k_{x}^{in}$. (a, b) Approximate calculation
neglecting SML breakdown terms ($\Delta_{m}=0$). (c, d) Full calculation.
Chosen values: $\omega=2.1\;eV$, $a=80$ nm, $b=220$ nm, $d=60$ nm, $L=460$ nm,
$n_{w}=1/2$ and $N=6$, based on the experimental sample shown in Fig. 1. The
metal is considered as a PEC. The chosen resonance is such that
$k_{x}^{SPP}\approx k_{x}^{in}+G^{0}+k_{g}$.

To illustrate the relevance of the breakdown terms, we consider an incident
left-handed (spin $+$) polarized wave, impinging onto the GPM described in
Fig. 1. We compute the reflection coefficients at a fixed frequency as a
function of $k_{x}^{in}$, in the vicinity of an SPP resonance. Fig. 2 renders
the results for two different Bragg orders ($r_{k_{x}^{in}+G^{0}}$ and
$r_{k_{x}^{in}+G^{0}+k_{g}}$) and the two circular polarizations (spin $+$ in
red and spin $-$ in blue). The left panels (a and b) are computed artificially
forcing $\Delta_{m}=0$ (taking $Y_{ms}=Y_{mp}$), while the right panels (c and
d) are the full calculations. As expected, when $\Delta_{m}=0$ the coefficient
$r_{k_{x}^{in}+G^{0}}$ is non-zero only for spin $+$, while the Bragg mode
that has gained an extra geometric momentum $k_{g}$ has an associated spin
reduction, and is thus non-zero only for spin $-$. When the breakdown terms
are considered, as they should, both polarizations become finite for any Bragg
order. Away from the SPP resonance, the effect of the breakdown terms is
small. However, the breakdown is strong at resonance when the excitation
process involves the transfer of geometric momentum. This constitutes the
second significant result of this Letter.

Mueller polarimetry.– A more detailed analysis of the metasurface’s full
polarization response can be obtained with Mueller polarimetry that measures,
artifact-free, the polarization states of the light beams incident on and
scattered off the metasurface. The 4x4 Mueller matrix $M$ is defined by
$\mathbf{S}_{out}=M\,\mathbf{S}_{in}$, where $\mathbf{S}_{in/out}$ are the
input and output polarization states described by the Stokes vectors Menzel et
al. (2010); Pérez and Ossikovski (2017); Fujiwara (2007). Using the
experimental scheme presented in the Supp. Mat. sup , $M$ can be measured as a
function of photon energy $\omega$ and $k_{x}^{in}$. Among the sixteen
components of the Mueller matrix, we concentrate on the component $M_{30}$,
which provides the difference in reflected intensities between the $+$ and $-$
polarizations when the system is illuminated with unpolarized light.

For a flat interface, in-plane momentum is conserved, and $M$ can be measured
for all $\omega$ and $k_{x}^{in}$ at once using an objective whose back focal
plane is fully illuminated with a collimated white light beam. However, the
Mueller matrix can be defined and measured for each diffracted order $n$ in a
periodically corrugated interface. Here we will analyze
$M_{30}^{n}(\omega,k_{x}^{in})$, obtained from the polarization properties of
the reflected light at $k_{x}^{out}=k_{x}^{in}+nk_{g}$, and restricting
ourselves to $n=0,1,2$, for reasons explained below.

Figure 3: Mueller polarimetry for the system described in Fig. 1(a). The
grooves are filled with $SiO_{2}$, which also forms a 4 nm layer above the
gold metasurface. (a) Scheme of the chosen Bragg modes. (b-d) Experimental
$M_{30}^{n}(\omega,k_{x}^{in})$, for a reflected wave with
$k_{x}^{out}=k_{x}^{in}+nk_{g}$. (e-g) Theoretical
$M_{30}^{n}(\omega,k_{x}^{in})$. The calculations have been performed within
the SIBC approximation, and the groove dimensions have been phenomenologically
enlarged by 1.25 times the skin depth, in order to consider the field
penetration in the metal Martín-Moreno et al. (2001).

The experimental results for the GPM shown in Fig. 1 are presented in Figs.
3(b-d). Panels Figs. 3(e-g) show the numerical simulations performed with the
CMM framework within the SIBC approximation. The calculations reproduce the
main features found in the experiments. All panels show inverted parabolic
features related to the resonant excitation of the SPP of the metasurface. Let
us concentrate on the plasmonic resonances, as they are the main reason for
analyzing metallic GPM. The resonant reflection processes can be understood in
a two-step scattering process. First, the incident light scatters with the
surface picking up momentum $n_{0}G^{0}+n_{1}k_{g}$ and adding spin $-n_{1}$
(where $n_{1}=0,\pm 1$). If this scattered wave is an SPP, it only has a
$p$-polarization component. A subsequent scattering of the SPP with the
surface brings it into one of the considered diffraction orders (for that, the
picked up momentum must be $-n_{0}G^{0}+n_{2}k_{g}$, which adds spin
$-n_{2}$). Taking into account that the SPP excitation can mathematically be
represented as a projector onto the $p$-linear polarization, this two-step
resonant scattering process results in the following rules:

$\displaystyle k_{x}^{SPP}=k_{x}^{in}+n_{0}G^{0}+n_{1}\,k_{g},$ (3)
$\displaystyle k_{x}^{out}=k_{x}^{in}+(n_{1}+n_{2})\,k_{g},$ (4)
$\displaystyle\mathbf{\sigma}_{out}=\sigma_{-n_{2}}\cdot\mathbf{p}\,(\mathbf{p}^{T}\cdot\sigma_{-n_{1}}\cdot\mathbf{i}_{0})$
(5)

where $\mathbf{p}=2^{-1/2}\,(1,1)^{T}$ and $\mathbf{i}_{0}$ are the
$p$-polarization and the incident polarization, respectively (in the circular
polarization basis), and we have written $\sigma_{0}\equiv\mathbb{1}$.

Let us start by considering how this reasoning apply to the case
$k_{x}^{out}=k_{x}^{in}+k_{g}$, rendered in Figs. 3(c) and (f). The central
parabola corresponds to the case $n=1,\,n_{1}=0,n_{2}=1$. As $n_{1}=0$, the
unpolarized incident light maintains its spin after the interaction with the
surface and thus can excite the linearly polarized SPP. The second scattering
with the surface removes the $nG^{0}$ momentum. Still, it adds $+k_{g}$, thus
decreasing the spin of the $p$-polarized wave, ending with spin $-$, in
agreement with the experimental and computed results. The parabola displaced
to a smaller $k_{x}^{in}$ arises from $n=1,\,n_{1}=1$. Then the incident
photon has spin $-$ after the first scattering, still being able to excite the
SPP. As the second scattering with the grating has $n_{2}=0$, it conserves
spin, and the reflected photon is linearly polarized (so, it has $M_{30}=0$,
in accordance with the experimental data). Both parabolas thus involve
resonant excitation of an SPP and two interactions with the holey array,
described by a succession of one standard and one spin-orbit Bragg law, the
difference being in the order of the “standard” and “geometric” processes.

Figs. 3(d,g) represent the case when the outcoming momentum is $+2k_{g}$
larger than the incident one. The existence of this diffraction order is a
direct confirmation of the presence of the SML breakdown, as an exact SML
would imply that the addition of $+2k_{g}$ momentum must be accompanied by a
reduction of spin in -2 units, which is not possible (as $\sigma_{-}^{2}=0$
for spin 1/2 spinors). In this case, both experiment and calculation show only
one plasmonic resonance, resulting in a spin $-$ in the reflected wave. The
reason is that the two interactions with the array now involve spin-orbit
Bragg laws. The first interaction of the incident wave with the holey surface
and the resonant excitation of SPP is like in Figs. 3(c,f), but the second
interaction also adds geometrical momentum and thus decreases spin, producing
a reflected wave with spin $-$.

Finally, Figs. 3(b,e) represent the case of specular reflection:
$k_{x}^{out}=k_{x}^{in}$. In this case, on top of the processes described by
Eqs. (3-5) the specular reflection must also be considered, and it dominates
the signal. For instance, the left parabola corresponds to processes where
spin $+$ light is only partially reflected in the specular direction. In
contrast, spin $-$ light gets fully reflected. Thus, overall,
$M_{30}^{0}(\omega,k_{x}^{in})$ is negative in that spectral region.

Conclusion.– Experiment and theory together demonstrate that spin-momentum
locking in Geometric Phase Metasurfaces appears when elements in the unit cell
have a non-zero winding number. Global rotational symmetries are thus not
required for such locking mechanisms to emerge. Nonetheless, with or without
global symmetries, Spin-Momentum Locking is an approximate symmetry. SML
breakdown terms yield a reflection into Bragg orders that would otherwise not
exist, as reflected waves that pick up two units of geometrical momentum. The
origin of the SML breakdown is that circularly polarized light gets an
elliptical projection onto the surface away from normal incidence. This
breakdown is relevant when linearly polarized surface resonances (as surface
plasmon polaritons) are excited. Experimental results on excitation of
plasmonic resonances, obtained with Mueller polarimetry, can be explained as a
two-step interaction with the metasurface, with the plasmon acting as a
polarization filter. Considering the crucial role played by spin-momentum
locking in integrated quantum optical systems Lodahl et al. (2017), our
elucidation of the mechanism and its relation to the near-field will
undoubtedly renew the vision of the corresponding scattering process. This is
important in the current applicative perspectives discussed currently in the
context of optovalleytronic systems Li et al. (2021), non-linear hybrid
metasurfaces Hu et al. (2019), and topology-based high-resolution sensors Ding
et al. (2017).

Acknowledgements.– We acknowledge Project PID2020-115221GB-C41, financed by
MCIN/AEI/10.13039/501100011033, and the Aragon Government through Project
Q-MAD. We also acknowledge the Interdisciplinary Thematic Institute QMat, as
part of the ITI 2021 2028 program of the University of Strasbourg, CNRS and
Inserm. This work was supported in part by IdEx Unistra (ANR 10 IDEX 0002), by
SFRI STRAT’US project (ANR 20 SFRI 0012), by the Labex NIE (ANR-11-LABX-0058
NIE) and CSC (ANR-10-LABX-0026 CSC) Projects, and by the University of
Strasbourg Institute for Advanced Study (USIAS; ANR-10-IDEX-0002-02) under the
framework of the French Investments for the Future Program. G.L.P.B.
acknowledges support from the Marie Sklodowska-Curie actions of the European
Commission (project 882173, MaP).

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