Directional aspects of vegetation linear and circular polarization
biosignatures

C.H. Lucas Patty1*, Antoine Pommerol1, Jonas G. Kühn2,3, Brice-Olivier
Demory2, Nicolas Thomas1

1 Physikalisches Institut, Universität Bern, CH-3012 Bern, Switzerland
2 Center for Space and Habitability, Universität Bern, CH-3012 Bern,
Switzerland
3 Département d’Astronomie, Université de Genève, CH-1290 Versoix, Switzerland

<EMAIL_ADDRESS><EMAIL_ADDRESS>
## Abstract

Homochirality is a generic and unique property of all biochemical life and is
considered a universal and agnostic biosignature. Upon interaction with
unpolarized light, homochirality induces fractional circular polarization in
the light that is scattered from it, which can be sensed remotely. As such, it
can be a prime candidate biosignature in the context of future life-detection
missions and observatories. The linear polarizance of vegetation is also
sometimes envisaged as a biosignature, although it does not share the
molecular origin as circular polarization. It is known that the linear
polarization of surfaces is strongly dependent on the phase angle. The
relation between the phase angle and circular polarization stemming from
macromolecular assemblies, such as in vegetation, however, remained unclear.
We demonstrate in this study using the average of 27 different species that
the circular polarization phase angle dependency of vegetation induces
relatively small changes in spectral shape and mostly affects the signal
magnitude. With these results we underline the use of circular
spectropolarimetry as a promising agnostic biosignature complementary to the
use of linear spectropolarimetry and scalar reflectance.
Spectropolarimetry, Biosignatures, Remote Sensing, Homochirality,
Photosynthesis, Life Detection

##  1 Introduction

While in recent decades the use of polarimetry for aerosol particle and cloud
characterization has rapidly been established, optical remote sensing of
surfaces using polarimetry is not broadly established. Instrumentation for the
acquisition of sensitive linear polarimetry has matured towards the point that
they can be readily applied for such purposes. Interpretation of the linear
polarizance, however, is difficult especially due to the strong phase angle
dependency.

In recent years, circular spectropolarimetric instruments have also been
developed allowing the sensitive acquisition of circular polarization of
vegetation in the field [1] and from a swift moving helicopter [2]. The
circular polarizance by biopigments provides a unique spectropolarimetric
biosignature and is generally absent in the light scattered of abiotic
surfaces [1, 2, 3]. While abiotic materials can also create circular
polarization, such as through multiple scattering by clouds or aerosols, these
interactions result in a weaker and spectrally flat circular polarization
signal [4]. Rather than being the result of surface scattering, the circular
polarizance of photosynthetic organisms, including algae and bacteria,
primarily results from interactions with chiral pigment-protein complexes
inside the cells [5, 6, 7].

Terrestrial biochemistry is based on chiral molecules which determine both the
functioning and structure of biological systems. Unlike abiotic chemistry,
where these molecules occur in equal, racemic, concentrations of both
versions, living nature almost [8] exclusively utilizes these molecules in
only one of the mirror image configurations (homochirality). Biological
macromolecules and biomolecular architectures are often also homochiral. The
$\alpha$-helix, for example, the most prevalent secondary structure of
proteins, is almost [9] exclusively right-hand-coiled. Homochirality is a
prerequisite for self-replication and thus for life as we know it [10, 11,
12]. It is likely that homochirality is a universal feature of life [13, 14,
5, 3] and may serve as an agnostic trait of life itself. Nonetheless, a lack
of homochirality does not necessarily indicate an absence of life and false-
negative scenarios can be possible [15].

The chirality of molecules and molecular architectures endows them a specific
response to electromagnetic radiation [16, 5]. When unpolarized incident
light, such as emitted from the Sun [17], is reflected off a homochiral
material it induces a wavelength dependent fractional circular polarization
signal, which can thus be detected remotely. It has been demonstrated that the
circular polarizance of vegetation carries the same information as circular
dichroism [18, 19], which is the differential absorbance of left- and right-
handed circularly polarized incident light, and has has proven to be an
indispensable tool in biomolecular research.

In vegetation, the chlorophylls are organized in chiral protein-pigment
aggregates. It is the chirality of the supramolecular structure, that
dominates the circular polarization spectrum rather than the chirality of the
constituent molecules themselves. While the signals resulting from smaller
scale chirality are superimposed, the total polarization spectrum is not the
sum of these spectra alone [20, 21]. These supramolecular chiral aggregates
can create very intense polarization signals (polymer and salt induced (psi-
type) polarization) that can be up to two orders of magnitude larger than the
polarization of the constituents. Additionally, psi-type polarization is
characterized by non-conservative anomalously shaped bands that range beyond
the wavelengths of molecular absorbance. Due to the direct relation between
the circular polarizance and the macromolecular structure it is an highly
sensitive indicator of the membrane macro-organization of the photosynthetic
apparatus and its functioning [22, 23].

As such, vegetation circular polarizance could provide sensitive monitoring of
vegetation physiology [24]. It has been demonstrated that circular
polarization can be very indicative of stress factors such as drought and can
in principle also be used to detect different physiological stress responses
[18, 24]. While vegetation circular polarizance is typically less than 1%,
making its use challenging for instrumentation, polarization remote sensing
could prove to be a very powerful complementary tool in assessing climate
change, desertification, deforestation and the general monitoring of
vegetation physiological state in vulnerable regions [2].

Additionally, also linear dichroism spectroscopy has contributed significantly
to the understanding of molecular organization. In principle, linear
polarizance would yield the same results as linear dichroism. However, the
retrieval of structural information through linear spectropolarimetry is
dependent on the molecular alignment of the sample. In a randomly oriented
sample this information is therefore averaged out, even if the sample
possesses intrinsically anisotropic molecular architectures [22, 23]. The
linearly polarized proportion of the light reflected of vegetation generally
stems from leaf surface polarization and the light scattered by leaf internal
structures is only polarized to a small extent [25, 26]. As such, linear
polarizance will not carry any information on molecular arrangements. It has
however been demonstrated that remote sensing using linear polarization can be
informative of plant physiological status [27, 28, 29], and can aid in
estimating leaf chlorophyll content [30]. The linear polarizance of vegetation
can be a sensitive biosignature [31, 32] that can be complementary to the
scalar reflectance and circular polarizance.

Beside leaf optical properties such as the optical index of refraction from
the cuticle and/or epidermis and the surface roughness, the linear polarizance
of leaves is highly dependent on the angle of the incident light [33, 25].
This is largely due to the increase in specular reflection by the leaf surface
as a function of the angle of incidence [34]. This effect becomes stronger in
the pigment absorption bands, where the main contribution to the total
reflection comes from the specular reflectance on the surface and the
scattered light by the leaf internal structures will be absorbed.
Reciprocally, the linear polarizance is lowest in the near-infrared, where no
significant absorption occurs and the contribution of internally scattered
unpolarized light is the greatest.

The relation between the circular polarizance of vegetation and the angle of
incidence and leaf optical properties has not been elucidated. In the past,
various studies have investigated the circular polarization scattering on a
molecular scale, indeed the circular polarizance is not only influenced by
differential absorption but also by differential scattering. Circular
Intensity Differential Scattering (CIDS) confirmed the existence of chirally
organized macrostructures which show extraordinary large polarizance if the
helical pitch and diameter commensurate with the wavelength [35, 36, 37, 38].
Also in leaves, this effect was thought to play an important role on small
scale circular polarizance, i.e. leading to different polarization signals for
normal leaf tissue and veins [19]. As of yet, however, no systematic
experimental study on the effect of incident angles on vegetation circular
polarizance has been carried out. While some studies demonstrate the
possibilities to detect the circular polarizance in the laboratory or in the
field, these studies have either used an integrating sphere or did not record
the angles of incidence and reflection (e.g. [3, 1, 2]).

In the current study we present, to the best of our knowledge, for the first
time the systematic evaluation of the relation between the angle of incidence
and the full-Stokes polarizance of leaves. A better understanding of this
phenomenon will be crucial in the further development of possible future
remote sensing applications using circular polarizance and will ultimately aid
in interpreting spectropolarimetric signals in future life-detection missions.

##  2 Materials and methods

Figure 1: Schematic of the measurement setup. The detector always detects at
nadir direction. $\theta$ represents the phase angle, $\varphi$ represents the
azimuthal angle.

The polarimeter (Hinds Instruments II/FS42-47, USA) is based on dual
photoelastic modulator (PEM) modulation and is able to record the full
normalized Stokes vector with minimal risk of systematic crosstalk. It
consists of 2 PEMs oriented at 45∘ modulated at 42 and 47 kHz resonance
frequencies, a prism analyzer that is 22.5∘ to the PEMs and a photomultiplier
tube (Hamamatsu R955, Japan).

Polarization in general is described in terms of the Stokes vector
$\textbf{S}=(I,Q,U,V)$, where $I$ is the absolute intensity, $Q$ and $U$ are
the unnormalized linear polarization parameters and $V$ is the unnormalized
circular polarization parameter. We describe the complete polarization states
in terms of the normalized Stokes parameters $Q/I$, $U/I$ and $V/I$, where
$Q/I$ denotes the difference between linear polarization normal and parallel
to the plane of scattering, $U/I$ denotes the difference but with a 45∘ offset
and $V/I$ denotes the difference between right-handed and left-handed
circularly polarized light. A value of 1 or -1 indicates 100% polarized light.

Light from a 100 watt halogen light source was fed to a monochromator (Oriel
instruments MS257, USA), with a FWHM of 10 nm, which is consequently fiber
launched and collimated and depolarized in a head mounted on a rotating arm.
The polarimeter was always detecting at nadir direction. The angle of the
incident light, which is similar to the phase angle $\theta$, varied between
10-75∘ with an increment of 2.5-7.5∘ varying per sample. The measured area was
approximately 2 cm in diameter.

The leaves measured in this study were all picked from vegetation growing in
and around the city of Bern, Switzerland, or are taken from a private indoor
collection (see also Table 1 for a list of the species). The vegetation
species used in this study do not represent any biome. The leaves were rinsed
and padded dry prior to each measurement. The leaves where flattened and
attached to a rotation stage which can change the azimuth angle $\varphi$ by
$0-360^{\circ}$ and was generally sampled in steps of 45∘. All leaves were
measured with the apex at $\varphi=0^{\circ}$ and where applicable the leaf
surface left of the midrib was measured. The petioles were placed in a small
tube containing water in order to prevent dessication during the measurements,
which had a duration of 24-72 hours per leaf depending on the measurement
parameters.

Table 1: The measured vegetation species. | Vegetation species |
---|---|---
Acer platanoides | Ficus maclellandii | Prunus lusitanica
Allium ursinum | Hedera helix | Quercus robur
Buddleja davidii |  Hosta sp. | Reynoutria japonica
Calathea makoyana | Hosta crispula |  Rosa sp.
Capsicum chinense | Hosta plantaginea | Rumex obtusifolius
Castanea sativa | Lonicera maackii | Schefflera actinophylla
Cornus alba | Mandevilla sanderi | Scindapsus pictus
Dypsis lutescens | Prunus avium | Syringa vulgaris
Epimedium alpinum | Prunus laurocerasus | Viola odorata

As Stokes $I$ was not recorded quantitatively in this study, we present a
typical vegetation reflectance spectrum in Figure 2, to allow for comparison
between the location of the pigment absorption bands and the presented
polarizance data.

Figure 2: Typical vegetation reflectance spectrum for comparison.

##  3 Results

###  3.1 Species

Multiple different species were measured (see also Table 1) in this study. We
show some representative full-Stokes polarization spectra of some of these
species in Figure 3 and present the average of 27 species as the vegetation
polarization biosignature in the rest of this study. The averaged full-Stokes
polarization spectra of all the leaves is shown in Figure 4, while the
polarization spectra of all individual leaves is shown in the appendix.
Differences between the species can be observed, most notable in terms of
polarization magnitude. For instance, Rosa sp. has both a very low $Q/I$ and
$V/I$ even at high phase angles, with a maximum $Q/I$ of $0.30$ at a phase
angle of $75^{\circ}$ and $V/I$ of $9.1*10^{-4}$ at a phase angle of
$60^{\circ}$. On the other hand, Viola odorata has a much higher linear and
circular polarizance, with a $Q/I$ of $0.53$ at a phase angle of $75^{\circ}$
and a $V/I$ of $5.9*10^{-3}$ at a phase angle of $45^{\circ}$.

For some species such as Acer platanoides and Viola odorata the positive psi-
type circular polarization band remained virtually unchanged with changes in
measurement geometry, whereas this effect was much larger for instance for
Rosa sp. and Calathea makoyana. Contrarily, the phase angle dependency of the
negative psi-type band was very small for Calathea makoyana but much larger
for Acer platanoides. Important here is the difference between these two
bands, which we describe as the polarization edge below.

Figure 3: The polarization spectra $Q/I$, $U/I$ and $V/I$ of Acer platanoides,
Rosa sp., Calathea makoyana and Viola odorata averaged over azimuth. The
colors represent the different phase angles and the shaded areas denote the
standard deviation.

###  3.2 Effects of the phase angle

We show in Figure 4 (A,B and C) the average linear and circular polarization
spectra ($Q/I$, $U/I$ and $V/I$ respectively) of 27 different leaves per phase
angle. Figure 4 (C,D and E) represents the same data visualized in a 2-D
display. Selected wavelengths versus the phase angle are shown in Figure 5 for
$Q/I$ (A), $U/I$ (B) and $V/I$ (C). The wavelengths that were selected are the
maximum and minimum magnitude of the linear polarizance (670 nm and 750 nm)and
the maximum magnitudes of the positive psi-type band in the blue (530 nm) and
the positive and negative psi-type bands in the red (670 nm and 700 nm
respectively).

Figure 4: The average polarization spectra $Q/I$ (A,& D) $U/I$ (B & E) and
$V/I$ (C & F) of 27 leaves of different species averaged over azimuth. In the
upper panels, the colors represent the different phase angles and the shaded
areas denote the standard error.

For all wavelengths, Q/I gradually increases with increasing increment angles
(see also Figure 4 (A and D) and Figure 5 (A)). The spectral features of $Q/I$
closely resembles the inverse scalar reflectance spectrum, with low
polarizance in the green (550 nm) and in the near-infrared ($>700$ nm).
Additionally, the linear polarizance was especially large around the
chlorophyll absorption band. This general inverse relation between the linear
polarization and the albedo is consistent with the Umov effect [39] and
results from the attenuation of multiple scattered light in the leafs interior
(see also Figure 2 for the general vegetation reflectance). For 670 nm, the
wavelength where the linear polarizance has the largest amplitude, the
polarizance increased from on average $6*10^{-3}$ (or 0.6%) at a 10∘
inclination angle to a polarizance of 0.39 (or 39%) at a phase angle of 75∘.
For 750 nm, this increased from $9*10^{-4}$ (0.09%) to 0.03 (3%).

The normalized Stokes parameter U/I shows a different spectral dependency on
the phase angle (Figure 4 (B and E) and Figure 5 (B))). Essentially, the $U/I$
appears similar to $Q/I$ at larger phase angles. At 670 nm, where $U/I$ has
the largest magnitude, the polarizance is $3*10^{-4}$. Unlike $Q/I$, however,
$U/I$ stays relatively small ($7*10^{-4}$) and free of any spectral features
until a phase angle of $30^{\circ}$. Between $30^{\circ}$ and $67.5^{\circ}$
the polarizance gradually increases, reaching a maximum value of $5.9*10^{-3}$
after which it flattens out.

Figure 5: The polarizance, $Q/I$(A), $U/I$(B) and $V/I$ (C), versus phase
angle for selected wavelengths. Shaded areas denote the standard error.

The circular polarizance V/I is in comparison with the linear polarizance
($Q/I$ and $U/I$) much less sensitive to changes in the phase angle, as can be
seen in Figure 4 (C and F) and Figure 5 (C). A small positive band can be
observed around $\sim$510-540 nm (Figure 4 (C and F)), which is the psi-type
band in blue stemming from the macro-organization of the pigment system. Two
more psi-type bands can be observed: a negative band with the largest
magnitude around 670 nm and a positive band with a peak magnitude around 700
nm. Interestingly, while the positive band has its highest magnitude around
$52.5^{\circ}$, with an amplitude of $1.9*10^{-3}$ a similar response is not
observed for the negative band. The amplitude of the negative band flattens
out after $60^{\circ}$ with a maximum amplitude of $-1.1*10^{-3}$. Typically
the blue psi-type circular polarizance band is much smaller than the psi-type
bands at longer wavelengths, as can also be seen in our data. While the band
keeps a similar spectral shape, the whole band increasingly becomes more
negative with increasing phase angles.

###  3.3 Effects of the azimuthal rotations

All leaves were measured over a full $360^{\circ}$ rotation in steps of
$45^{\circ}$. In Figure 6 we show the non-azimuthal averaged measurements per
wavelength and phase angle for Stokes $Q/I$. Figures 7 and 8 show these
results for $U/I$ and $V/I$ respectively. The 26 polar plots present the
different wavelengths that were measured. For each plot, the radius $r$
represents the phase angle, while the azimuthal rotation is provided by
$\varphi$.

Figure 6: Polar plots per wavelength of the normalized Stokes parameter $Q/I$
for the average of 27 leaves. The phase angle is given by the radial
coordinate and the azimuth by the angular coordinate.

For $Q/I$ the variation in polarizance per azimuthal rotation is minimal
although slightly increased with increasing phase angles. The difference in
polarizance was the highest around the chlorophyll absorbance band (670 nm).
For a phase angle of $75^{\circ}$, the polarizance at an azimuth of
$0^{\circ}$ and $180^{\circ}$ was $0.43$, while this was $0.37$ for an azimuth
of $45^{\circ}$ and $225^{\circ}$. At smaller phase angles no significant
differences were observed.

Figure 7: Polar plots of the normalized Stokes parameter $U/I$ per wavelength
for the average of 27 leaves. The phase angle is given by the radial
coordinate and the azimuth by the angular coordinate.

For $U/I$ the differences in polarizance per azimuthal rotation were larger
compared to $Q/I$ as is shown in Figure 7. On average, the polarizance was the
highest at an azimuth of $135^{\circ}$ and $315^{\circ}$, while smallest at an
azimuth of $45^{\circ}$ and $225^{\circ}$. For example, at a phase angle of
$75^{\circ},U/I$ was -0.012 at $315^{\circ}$ and only $-1.9*10^{-3}$ at
$45^{\circ}$. At an azimuth of $45^{\circ}$ and $225^{\circ}$, Stokes $U/I$
was positive at lower phase angles ($10-37.5^{\circ}$). For low phase angles
($10-30^{\circ}$) this results in an azimuthal average with no significant
polarization signals (see also Figure 4). For many individual species, also at
higher phase angles both positive and negative $U/I$ was observed depending on
the azimuthal rotation. This results in a per species $U/I$ signal, averaged
over azimuth, that also at high phase angles is not always significant (see
also the appendix for the rotational averages of the individual species).

Figure 8: Polar plots of the normalized Stokes parameter $V/I$ per wavelength
for the average of 27 leaves. The phase angle is given by the radial
coordinate and the azimuth by the angular coordinate.

Stokes $V/I$ (see also Figure 8) shows small variations in polarizance per
azimuthal rotation. While at lower phase angles there are no significant
variations, at phase angles above $45^{\circ}$ the polarizance was more
slightly more negative at an azimuth of $0^{\circ}$ and $180^{\circ}$ and at
an azimuth of $135^{\circ}$ and $315^{\circ}$. On the other hand, the
polarizance was more slightly more positive at an azimuth of $45^{\circ}$ and
$225^{\circ}$ and at an azimuth of $90^{\circ}$ and $270^{\circ}$. For $V/I$
these variations seem primarily related to shifts in the zero line. When the
difference between the positive and negative psi-type band per azimuth is
taken no significant variations between the azimuthal rotations are observed.

In Figure 9 we show the values for Stokes $Q/I$ (A and D), $U/I$ (B and E) and
$V/I$ (C and F) per azimuth versus the average over all rotations. In the
upper panels (A,B and C) the color of the data points represent the azimuth
angles, while in the lower panels (D, E and F) the same data is shown with the
colors representing the phase angles.

Figure 9: The normalized Stokes parameters $Q/I$ (A and D), $U/I$ (B and E)
and $V/I$ (C and F) per leaf azimuthal measurement vs the leaf average over
all azimuthal angles. In panels A, B and C the colors indicate the azimuth
angle. Panels D, E and F show the same data but the colors indicate the
different phase angles.

It is clearly visible that some azimuthal directions have a higher or lower
polarizance than the rotational average. An extreme for a direction in one of
the Stokes parameters does not lead to a similarly high or low value in the
other parameters. The differences between the azimuthal directions and the
average increase with increasing phase angles (Figure 9 D, E and F) as is also
shown in Figure 4.

###  3.4 Polarization Edge

Figure 10: A 2D histogram showing the polarization edge of the linear
polarization (DoLP) (A) and the circular polarization (DoCP) (B) versus the
phase angle for all measurements. The red line represents the average of 27
different leaves and the red shaded area represents the standard error.

Similar to the vegetation red-edge, which is the steep increase in scalar
reflectance between the chlorophyll absorption band in the red (around
approximately 670 nm, see Figure 2) and the region of maximum reflectance in
the near-infrared (reached after approximately 750 nm, see Figure 2), the
strength of the polarization signals can be described by their ’polarization
edge’. We have grouped $Q/I$ and $U/I$ together in the degrees of linear
polarization ($DoLP$) (i.e. $DoLP=\sqrt{Q^{2}+U^{2}}/I$) which makes more
sense from a biosignature point of view. For the $DoLP$ we have measured the
polarization edge by taking the value at 670 nm, where the polarizance is
maximal for both $Q/I$ and $U/I$, and subtracting it with the background
polarization at 750 nm. For the circular polarization, we compute the
polarization edge as the difference between the negative (670 nm) and positive
(700 nm) psi-type circular polarizance band, where the absolute value is
expressed as the degrees of circular polarization ($DoCP$). Note that we did
not correct for the background polarization at 750 nm as due to the split
signal this will only lead to a total offset in $V/I$ but not a change in
magnitude. Additionally, the magnitude at 750 nm was less than $5*10^{-5}$
(see also Figure 5 (C). We show the polarization edges in Figure 10 of the
linear polarization A and of the circular polarization B shown in degrees of
circular polarization. The red line represents the average of 27 leaves while
the edges per measurement are grouped into bins, with the amount per bin
represented by the colorbar.

For the $DoLP$ the largest difference between the minimum and maximum
polarizance was observed at $75^{\circ}$, where the edge had a magnitude of
$0.35$. As the maximum polarizance was still increasing with increasing phase
angles it is likely that the peak polarizance and edge magnitude occurs at
even higher phase angles. For the $DoCP$, the maximum edge was measured at
$60^{\circ}$, with an average value of $2.7*10^{-3}$. We show the maximum and
the minimum extend of the polarization edges per species in Figure 11.

Figure 11: The maximum and minimum polarization edge over phase angle of the
linear polarization (DoLP) (A) and the circular polarization (DoCP) (B) for
all measured species averaged over azimuth. Data are sorted on the average
edge value.

###  3.5 Non-vegetation polarizance

For comparison with the vegetation polarization data, we show in Figure 12 the
linear (A and B) and the circular polarization spectra (C) versus the phase
angle of 3 sand samples. Again, Figure 12 (C,D and E) represents the same data
visualized in a 2-D display. Selected wavelengths, similar to those in Figure
5, versus the phase angle are shown in G, H and I for $Q/I$, $U/I$ and $V/I$
respectively.

Figure 12: The average polarization spectra $Q/I$ (A,& D) $U/I$ (B & E) and
$V/I$ (C & F) of 3 samples of sand averaged over azimuth. In the upper panels,
the colors represent the different phase angles and the shaded areas denote
the standard error. In the bottom panels, the colors represent different
wavelengths and the shaded areas denote the standard error.

Similar to vegetation, $Q/I$ strongly increases with increasing phase angles.
Also for sand, the linear polarizance correlates to its reflectance spectrum
which in the visible gradually increases with increasing wavelength. At small
phase angles (see also Figure 12 (G)), $Q/I$ was negative, thus the
polarization was oriented parallel to the plane of scattering. $Q/I$ changed
sign at $\sim 17^{\circ}$ for all measured wavelengths. We observed a similar
negative $Q/I$ at small phase angles (typically below $22.5^{\circ}$) for some
leaves, but on average $Q/I$ was positive for all measured wavelengths (see
Figure 4 (A)).

Stokes $U/I$ is approximately 0 for all wavelengths below a phase angle of
$37.5^{\circ}$ but then increases in overall with increasing phase angles and
is spectrally featureless. The circular polarizance, too, is not showing any
spectral features. Especially towards the blue $V/I$ increases slightly with
increasing phase angles, with a maximum of $-2.2*10^{-4}$.

##  4 Discussion and conclusions

We present in this study the systematic investigation of full-Stokes
spectropolarimetry on leaves and its dependency on the phase angle.
Importantly, while some studies and models exist for the linear polarizance of
vegetation, studies showing the relationship between vegetation circular
polarizance and phase angle are lacking. It has been demonstrated that
circular polarimetry is able to provide an unambiguous means to detect
homochiral matter, considered an important universal agnostic biosignature [3,
5]. The results in this study demonstrate that the characteristic
spectropolarimetric properties of vegetation remain relatively unchanged with
changes in measurement geometry, underlining the potential of circular
spectropolarimetry as a means of detecting life beyond Earth.

Accounting only for the molecular phenomena, the vegetation circular
polarizance should in principle be insensitive to phase angle variations.
While the phase angle is important at the molecular scale and in oriented
isolated constituents, which can result in very different spectral shapes [40,
41], it can be expected that these variations even out at larger scales where
the chloroplasts are more randomly distributed. Leaves, however, are
geometrically complex structures with many different cellular interfaces and
covered by a extracellular lipid structure with varying chemical compositions
that can all potentially affect its spectropolarimetric response [33, 26].

We demonstrate in the present study that the circular polarizance of
vegetation is relatively insensitive to alterations in the phase angle. These
results confirm previous circular spectropolarimetric field observations of
vegetation canopy under overcast and cloudless sky conditions, showing a
circular polarization spectrum that was virtually identical [1]. Still, the
magnitude of the signal varied with changes in phase angle, where the largest
differences between the negative and positive band were observed at a phase
angle of $60^{\circ}$ and the smallest at a phase angle of $10^{\circ}$
(Figure 10). The spectral shape of the typical split psi-type polarization,
however, remained preserved. As such, integrating over a planetary disc should
not yield spectral changes due to possible phase angle dependency.
Additionally, it has been suggested that the vegetation circular polarizance
could be a powerful tool in assessing vegetation physiology remotely [18, 2].
Remote sensing using circular spectropolarimetry will be able to directly
probe the macromolecular structure of the photosynthetic apparatus and thus
its functioning [18, 24] and as such could provide a powerful complementary
tool in assessing effects of climate change and the monitoring of vegetation
physiological state in vulnerable regions [2].

It should be noted that in some cases significant spectropolarimetric
differences between species can be observed. For instance, in a Hosta cultivar
(see Appendix) we observed a negative psi-type polarization band with a
magnitude of well over $-5*10^{-3}$ but with a postive band of only
$1*10^{-3}$. On the other hand we measured for a Viola cultivar (see Figure 3)
a positive band with a magnitude of $6*10^{-3}$ but with a corresponding
negative band less than $-1*10^{-4}$. Importantly, the two psi-type bands have
a different physical origin, where the negative band is preferentially
associated with the stacking of the thylakoid membranes inside the
chloroplasts, and where the positive band is mainly associated with the
lateral organization of the chiral macro-domains [24, 23]. Similar
differences, where some species have much larger negative or positive bands,
have been reported before, see e.g. [7, 2]. While the circular polarizance
does not always provide a specific species signature and it will thus be
difficult to determine the species based on the signal alone, it could
potentially aid in enhancing the contrast in existing distinguishing
techniques.

The linear polarizance of vegetation, too, can serve as a biosignature [31,
32]. Importantly, the linear polarization created by vegetation could be used
to enhance the contrast between the reflected light of an exoplanet and the
very bright emission by its host star [32] As has been reported before (see
e.g. [34, 33, 25]), vegetation linear polarization is strongly dependent on
the phase angle. At a phase angle of $10^{\circ}$, we measured an average
linear polarization of $6.4*10^{-3}$. At $75^{\circ}$ the linear polarizance
was much higher with a value of $0.35$ which on average continued to increase.
Likely the maximum average polarizance lies between $75^{\circ}$ and two times
the Brewster angle (which for leaves is $55.4^{\circ}$ on average [42]).

In contrast to vegetation circular polarizance, the linear polarizance is more
dependent on the pigments present in the leaf, rather than their molecular
organization. The structural geometry of leaves can vary greatly between
species and is not homogeneous. Some leaves have a thick shiny cuticle while
others are abundant in trichomes (fine epidermal outgrowths or hairs).
Importantly, a leaf surface is generally far from optically smooth [26].
Larger structures are shown along the veins which can undulate from the
surface but many plants have surfaces with angular aspects in multiple
directions. Such phenomena can affect the obtained leaf linear polarizance
[26, 25]. Additionally, the interior anatomy of leaves can vary greatly
between species, and can vary between leaves of the same species. While the
effects of these variations on the polarization response has not been studied
in depth, it is clear that they can effect the optical properties greatly and
can have a large effect on scalar reflectance and transmittance [42]. It has
additionally been demonstrated that linear polarization can be informative of
plant physiological status [27, 28, 30].

Similar to the scalar reflectance, edges mimicking biotic matter can readily
emerge in the spectropolarimetry of abiotic matter, whereas circular
polarizance is less prone to such confounding factors. Consequently, while
circular polarization signals of comparatively small magnitude (but can be up
to a couple percent for certain eukaryotic algae [7]), it has the potential to
deliver a more robust and specific biosignature. The scalar reflectance,
linear polarizance and circular polarizance are, however, very complementary
tools. In the end, a confident detection of extraterrestrial life candidates
will presumably have to consist of a vast range of spectroscopic techniques.

Overall, we have successfully demonstrated that the circular polarization
biosignatures of vegetation are relatively insensitive to changes in phase
angle, providing a promising outlook on future life detection or remote
sensing applications. It will be important to compare the results in this
study with controlled measurements of vegetation taken in the field.
Additionally, it will be also key to incorporate the circular polarizance-
phase angle relationship in (exo)planetary models with realistic surface and
atmospheric components [43]. While we have only measured vegetation in this
study, more primitive phototrophs also show remarkable polarization features
[6]. Although it is not expected that measurements on these organisms yield
different results for the circular polarization phase angle dependency, such
study would be highly interesting and important in understanding the nature of
especially circular polarization as a biosignature.

## Acknowledgments

This work has been carried out within the framework of the National Centre of
Competence in Research (NCCR), PlanetS, supported by the Swiss National
Science Foundation (SNSF).

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##  5 Supplementary Material

See pages - of appendixsmall.pdf


$\displaystyle+\left(\sum_{S^{\prime}\in\\{N,V,O\\}}\left(\sum_{\theta^{t^{*}-1}\in\Theta_{NS^{\prime}}^{t^{*}-1}}f_{\theta^{t^{*}-1}}-\sum_{\theta^{t^{*}}\in\Theta_{NS^{\prime}}^{t^{*}}}f_{\theta^{t^{*}}}\right)p_{S^{\prime}C}\right)\gamma$
$\displaystyle\leq\sum_{S^{\prime}\in\\{N,V,O\\}}k_{S^{\prime}}\left(\sum_{\theta^{t^{*}-1}\in\Theta_{NS^{\prime}}^{t^{*}-1}}f_{\theta^{t^{*}-1}}-\sum_{\theta^{t^{*}}\in\Theta_{NS^{\prime}}^{t^{*}}}f_{\theta^{t^{*}}}\right)$

Note that as $p_{S^{\prime}D}$ and $p_{S^{\prime}C}$ increase,
$k_{S^{\prime}}$ decreases. This indicates that, the more prone to
experiencing unexpected disruptive events, the more sensitive our optimal
inspection time is to a change in the penalties from unexpected disruptive
events occurring.

We next consider what values $d$ and $c$ can take such that a pre-specified
time period $t^{*}$ is the optimal inspection time. By Corollary 2, we know
that $V^{1}(b^{1},\sigma^{1}_{t^{*}-1})\leq V^{1}(b^{1},\sigma^{1}_{t^{*}})$
and $V^{1}(b^{1},\sigma^{1}_{t^{*}})\geq V^{1}(b^{1},\sigma^{1}_{t^{*}+1})$ is
sufficient to imply that $V^{1}(b^{1},\sigma^{1}_{t^{*}})\geq
V^{1}(b^{1},\sigma^{1}_{t})$ for all $t$. Thus, we only need to compare the
value functions for $t^{*}-1$, $t^{*}$ and $t^{*}+1$ to determine bounds on
$d$ and $c$.

First, to ensure that we should wait to inspect no later than $t^{*}$, we
devise conditions on $d$ and $c$ based on $V^{1}(b^{1},\sigma^{1}_{t^{*}})\geq
V^{1}(b^{1},\sigma^{1}_{t^{*}+1})$.

$\displaystyle
V^{1}(b^{1},\sigma^{1}_{t^{*}})-V^{1}(b^{1},\sigma^{1}_{t^{*}+1})\geq 0$
$\displaystyle\sum_{S^{\prime}\in\\{N,V,O\\}}(k_{NS^{\prime}}^{t^{*}}-k_{NS^{\prime}}^{t^{*}+1})\geq
0$
$\displaystyle\sum_{S^{\prime}\in\\{N,V,O\\}}k_{S^{\prime}}\left(\sum_{\theta^{t^{*}}\in\Theta_{NS^{\prime}}^{t^{*}}}f_{\theta^{t^{*}}}-\sum_{\theta^{t^{*}+1}\in\Theta_{NS^{\prime}}^{t^{*}+1}}f_{\theta^{t^{*}+1}}\right)\geq
0$
$\displaystyle\sum_{S^{\prime}\in\\{N,V,O\\}}(1-(d+1)p_{S^{\prime}D}-(c+1)p_{S^{\prime}C})\left(\sum_{\theta^{t^{*}}\in\Theta_{NS^{\prime}}^{t^{*}}}f_{\theta^{t^{*}}}-\sum_{\theta^{t^{*}+1}\in\Theta_{NS^{\prime}}^{t^{*}+1}}f_{\theta^{t^{*}+1}}\right)\geq
0$
$\displaystyle\left(\sum_{S^{\prime}\in\\{N,V,O\\}}\left(\sum_{\theta^{t^{*}}\in\Theta_{NS^{\prime}}^{t^{*}}}f_{\theta^{t^{*}}}-\sum_{\theta^{t^{*}+1}\in\Theta_{NS^{\prime}}^{t^{*}+1}}f_{\theta^{t^{*}+1}}\right)p_{S^{\prime}D}\right)d$
$\displaystyle+\left(\sum_{S^{\prime}\in\\{N,V,O\\}}\left(\sum_{\theta^{t^{*}}\in\Theta_{NS^{\prime}}^{t^{*}}}f_{\theta^{t^{*}}}-\sum_{\theta^{t^{*}+1}\in\Theta_{NS^{\prime}}^{t^{*}+1}}f_{\theta^{t^{*}+1}}\right)p_{S^{\prime}C}\right)c$
$\displaystyle\leq\sum_{S^{\prime}\in\\{N,V,O\\}}(1-p_{S^{\prime}D}-p_{S^{\prime}C})\left(\sum_{\theta^{t^{*}}\in\Theta_{NS^{\prime}}^{t^{*}}}f_{\theta^{t^{*}}}-\sum_{\theta^{t^{*}+1}\in\Theta_{NS^{\prime}}^{t^{*}+1}}f_{\theta^{t^{*}+1}}\right).$

If $d$ and $c$ satisfy the above inequality, then it is better to inspect in
time period $t^{*}$ than it would be delay inspection.

Likewise, we can derive conditions on $d$ and $c$ such that we should inspect
no earlier than $t^{*}$ based on $V^{1}(b^{1},\sigma^{1}_{t^{*}})\geq
V^{1}(b^{1},\sigma^{1}_{t^{*}-1})$. Using the same derivation as above, we
arrive at:

$\displaystyle\left(\sum_{S^{\prime}\in\\{N,V,O\\}}\left(\sum_{\theta^{t^{*}}\in\Theta_{NS^{\prime}}^{t^{*}}}f_{\theta^{t^{*}}}-\sum_{\theta^{t^{*}-1}\in\Theta_{NS^{\prime}}^{t^{*}-1}}f_{\theta^{t^{*}-1}}\right)p_{S^{\prime}D}\right)d$
$\displaystyle+\left(\sum_{S^{\prime}\in\\{N,V,O\\}}\left(\sum_{\theta^{t^{*}}\in\Theta_{NS^{\prime}}^{t^{*}}}f_{\theta^{t^{*}}}-\sum_{\theta^{t^{*}+-}\in\Theta_{NS^{\prime}}^{t^{*}-1}}f_{\theta^{t^{*}-1}}\right)p_{S^{\prime}C}\right)c$
$\displaystyle\leq\sum_{S^{\prime}\in\\{N,V,O\\}}(1-p_{S^{\prime}D}-p_{S^{\prime}C})\left(\sum_{\theta^{t^{*}}\in\Theta_{NS^{\prime}}^{t^{*}}}f_{\theta^{t^{*}}}-\sum_{\theta^{t^{*}-1}\in\Theta_{NS^{\prime}}^{t^{*}-1}}f_{\theta^{t^{*}-1}}\right).$

If $d$ and $c$ satisfy the above inequality, then it is better to inspect in
time period $t^{*}$ than it would be inspect earlier. Thus, if $d$ and $c$ lie
in the polytope defined by the two inequalities, then it is optimal to inspect
in time period $t^{*}$.

##  Appendix D Additional Computational Study

(a) $d=18$, $s=0.01$

(b) $d=18$, $s=0.02$

Figure 4: Histograms of ETD inspection rule recommendations for randomly
generated matrices

Rule | $t_{E}$ | $t_{V}$ | $t_{VC}$ | 24 | 60 | 120 | NoIns
---|---|---|---|---|---|---|---
% Caught, $d=10$ | 46.530% | +5.601% | +5.601% | -37.880% | -46.513% | -46.530% | -
Value (no IC), $d=10$ | 4.6743 | +0.128% | - | -12.062% | -16.856% | -16.871% | -16.871%
Value (IC), $d=10$ | 4.4856 | - | -0.145% | -9.352% | -13.258% | -13.374% | -
% Caught, $d=14$ | 46.070% | +30.244% | +30.244% | -37.488% | -46.047% | -46.070% | -
Value (no IC), $d=14$ | 3.0582 | +16.470% | - | -51.592% | -66.555% | -66.578% | -66.578%
Value (IC), $d=14$ | 2.8713 | - | +16.386% | -50.009% | -64.382% | -64.403% | -
% Caught, $d=18$ | 46.601% | +36.149% | +36.149% | -37.787% | -46.579% | -46.601% | -
Value (no IC), $d=18$ | 1.6318 | +84.894% | - | –160.526% | -205.050% | -205.142% | -205.142%
Value (IC), $d=18$ | 1.4439 | - | +94.376% | -171.535% | -218.727% | -218.824% | -

Table 15: Percent of Unexpected Disruptive Events Caught and Average Value
Accumulated with Each Inspection Rule for $\tilde{P}^{2}(ni)$ with $s=0.01$

Rule | $t_{E}$ | $t_{V}$ | $t_{VC}$ | 24 | 60 | 120 | NoIns
---|---|---|---|---|---|---|---
% Caught, $d=10$ | 45.903% | +5.542% | +5.542% | -36.848% | -45.872% | -45.903% | -
Value (no IC), $d=10$ | 4.5517 | +0.204% | - | -10.058% | -14.838% | -14.856% | -14.856%
Value (IC), $d=10$ | 4.3683 | - | -0.094% | -7.330% | -11.265% | -11.281% | -
% Caught, $d=14$ | 46.091% | +29.706% | +29.706% | -36.881% | -46.063% | -46.091% | -
Value (no IC), $d=14$ | 3.0497 | +13.605% | - | -46.191% | -61.091% | -61.157% | -61.157%
Value (IC), $d=14$ | 2.8672 | - | +13.382% | -44.357% | -58.622% | -58.684% | -
% Caught, $d=18$ | 46.103% | +35.745% | +35.745% | -37.068% | -46.065% | -46.103% | -
Value (no IC), $d=18$ | 1.5778 | +80.980% | - | -156.832% | -200.615% | -200.780% | -200.780%
Value (IC), $d=18$ | 1.3936 | - | +90.342% | -167.580% | -213.928% | -214.100% | -

Table 16: Percent of Unexpected Disruptive Events Caught and Average Value
Accumulated with Each Inspection Rule for $\tilde{P}^{2}(ni)$ with $s=0.02$

Rule | $t_{E}$ | $t_{V}$ | $t_{VC}$ | 24 | 60 | 120 | NoIns
---|---|---|---|---|---|---|---
% Caught, $d=18$ | 43.217% | -41.990% | -43.217% | +2.406% | -38.030% | -43.059% | -
Value (no IC), $d=18$ | 10.9907 | +7.139% | - | -0.812% | +6.541% | +7.414% | +7.466%
Value (IC), $d=18$ | 10.8724 | - | +8.636% | -0.871% | +7.561% | +8.579% | -
% Caught, $d=22$ | 43.432% | +2.478% | +2.478% | +2.478% | -38.283% | -43.286% | -
Value (no IC), $d=22$ | 9.4174 | -0.096% | - | -0.096% | -4.358% | -5.149% | -5.156%
Value (IC), $d=22$ | 9.2988 | - | -0.152% | -0.251% | -3.309% | -3.942% | -
% Caught, $d=26$ | 43.114% | +22.641% | +22.641% | +2.487% | -37.955% | -42.974% | -
Value (no IC), $d=26$ | 7.7032 | +4.618% | - | +0.946% | -18.996% | -21.822% | -21.912%
Value (IC), $d=26$ | 7.5860 | - | +4.163% | +0.895% | -17.944% | -20.620% | -
% Caught, $d=30$ | 43.186% | +32.031% | +32.031% | +2.416% | -37.940% | -43.037% | -
Value (no IC), $d=30$ | 6.0871 | +17.488% | - | +2.249% | -41.802% | -47.436% | -47.532%
Value (IC), $d=30$ | 5.9699 | - | +17.136% | +2.206% | -40.927% | -46.413% | -
% Caught, $d=34$ | 43.371% | +38.178% | +38.178% | +2.466% | -38.086% | -43.219% | -
Value (no IC), $d=34$ | 4.6578 | +41.288% | - | +4.386% | -78.301% | -89.282% | -89.542%
Value (IC), $d=34$ | 4.5378 | - | +41.558% | +4.392% | -78.071% | -89.006% | -

Table 17: Percent of Unexpected Disruptive Events Caught and Average Value
Accumulated with Each Inspection Rule for $\tilde{P}^{3}(ni)$ with $s=0.01$

Rule | $t_{E}$ | $t_{V}$ | $t_{VC}$ | 24 | 60 | 120 | NoIns
---|---|---|---|---|---|---|---
% Caught, $d=18$ | 45.272% | -42.632% | -45.272% | +2.332% | -37.909% | -44.490% | -
Value (no IC), $d=18$ | 11.7223 | +18.104% | - | -1.059% | +14.699% | +20.384% | +23.580%
Value (IC), $d=18$ | 11.6058 | - | +24.820% | -1.107% | +15.702% | +21.582% | -
% Caught, $d=22$ | 45.140% | +2.359% | +2.359% | +2.359% | -37.824% | -44.270% | -
Value (no IC), $d=22$ | 10.2258 | -0.628% | - | -0.628% | +6.477% | +11.255% | +15.580%
Value (IC), $d=22$ | 10.1099 | - | -0.684% | -0.684% | +7.324% | +12.519% | -
% Caught, $d=26$ | 44.755% | +21.713% | +21.713% | +2.394% | -37.573% | -43.939% | -
Value (no IC), $d=26$ | 8.6925 | -1.137% | - | +0.2222% | -5.215% | -0.978% | +2.625%
Value (IC), $d=26$ | 8.5782 | - | -1.587% | +0.162% | -3.566% | +0.329% | -
% Caught, $d=30$ | 45.051% | +30.695% | +30.695% | +2.415% | -37.748% | -44.285% | -
Value (no IC), $d=30$ | 7.2452 | +4.766% | - | +1.159% | -19.512% | -18.117% | -14.008%
Value (IC), $d=30$ | 7.1269 | - | +4.296% | +1.107% | -18.429% | -16.774% | -
% Caught, $d=34$ | 45.031% | +36.747% | +36.747% | +2.396% | -37.813% | -44.270% | -
Value (no IC), $d=34$ | 5.8057 | +19.708% | - | +2.689% | -43.952% | -45.051% | -40.493%
Value (IC), $d=34$ | 5.6868 | - | +19.526% | +2.641% | -42.701% | -43.919% | -

Table 18: Percent of Unexpected Disruptive Events Caught and Average Value
Accumulated with Each Inspection Rule for $\tilde{P}^{3}(ni)$ with $s=0.02$