****footnotetext: K.K. and E.S. equally contributed to this work as co-first
authors.

]Last modified:

# Gravitational lensing aided luminosity distance estimation for compact
binary coalescences

Kyungmin Kim∗∗<EMAIL_ADDRESS>Department of Physics, Ewha Womans
University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Republic of Korea
Eungwang Seo∗∗<EMAIL_ADDRESS>SUPA, School of Physics and
Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom  Department
of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong
Kong  Chunglee Kim<EMAIL_ADDRESS>Department of Physics, Ewha Womans
University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Republic of Korea

([)

###### Abstract

The luminosity distance is a key observable of gravitational-wave
observations. We demonstrate how one can correctly retrieve the luminosity
distance of compact binary coalescences if the gravitational-wave signal is
strongly lensed. We perform a proof-of-concept parameter estimation for the
luminosity distance supposing (i) strong lensing produces two lensed
gravitational-wave signals, (ii) the advanced LIGO-Virgo network detects both
lensed signals as independent events, and (iii) the two events are identified
as strongly lensed signals originated from a single compact binary
coalescence. Focusing on the maximum magnification allowed in the given
lensing scenario, we find that the strong lensing can improve the precision of
the distance estimation by up to a factor of two compared to that can be
expected for the signal experiencing no lensing. Our results imply that strong
lensing of gravitational waves can be helpful for better constraining the
distance to the source, and furthermore, the Hubble constant.

_Introduction._ —The luminosity distance $D_{L}$ to a source has significant
implications in astronomy as well as in cosmology. It is one of the direct
observables available when gravitational-wave (GW) signals are detected. The
precision of parameter estimation for an observed GW signal is in general
subject to the signal-to-noise ratio (SNR) of the data obtained by a detector
with given sensitivity Christensen and Meyer (2022). As $D_{L}$ is directly
proportional to the strain amplitude of the GW signal, any increase in SNR of
the data would be helpful to better constrain the distance to the source.

This work is motivated by the fact that GWs can be strongly lensed Ohanian
(1974); Bliokh and Minakov (1975); Bontz and Haugan (1981); Thorne (1983);
Deguchi and Watson (1986); Schneider  _et al._ (1992); Nakamura and Deguchi
(1999); Takahashi and Nakamura (2003). When some conditions are satisfied,
lensing of GWs can magnify the GW signal from a source. This implies the
increase of the SNR of an observed GW signal. Based on forecast studies (Ng
_et al._ , 2018; Li  _et al._ , 2018; Oguri, 2018; Mukherjee  _et al._ , 2020;
Wierda  _et al._ , 2021), it is expected to observe $\sim\\!\mathcal{O}(1)$
strongly lensed GW events per year with design sensitivities of the Advanced
LIGO Aasi  _et al._ (2015) and the Advanced Virgo Acernese  _et al._ (2015).
Although there has been no confirmed strongly lensed GW event from the
previous observing runs yet (Liu  _et al._ , 2021; Hannuksela  _et al._ ,
2019; Abbott  _et al._ , 2021a, 2023), searching for strongly lensed GW
signals has been ongoing.

The precision of distance measure of a GW source is important information for
follow-up observations and for understanding astrophysics of the source.
Furthermore, the distance measure can govern the quality of a Hubble constant
$H_{0}$ estimation Christensen and Meyer (2022). The Hubble–Lemaître law is
$H_{0}\\!=\\!v/d$ Lemaître (1927); Hubble (1929), where $v$ is the recessional
velocity of an astronomical source and $d$ is the distance to the source. The
law can be re-written by $d=D_{L}$ when $D_{L}$ is measured by GW
observations. Many studies have discussed methods and implications of $H_{0}$
measurement enabled by observing GWs Schutz (1986); Holz and Hughes (2005);
Dalal  _et al._ (2006); Nissanke  _et al._ (2010, 2013); Vitale and Chen
(2018); Borhanian  _et al._ (2020); Borhanian and Sathyaprakash (2022); Ghosh
_et al._ (2022). These studies consider GW signals from compact binary
coalescences (CBCs) consisting of black holes or neutron stars. Among the
known CBCs, GW170817 Abbott  _et al._ (2017a) is the most successful example
for being used to estimate $H_{0}$ based on a GW observation Abbott  _et al._
(2017b). This provides an independent constraint on $H_{0}$ in addition to
those from the electromagnetic (EM) observations, e.g., the cosmic microwave
background observation Ade  _et al._ (2016), the Type Ia supernova survey
Riess  _et al._ (2016), or measuring time delays between strongly lensed
multiple images of quasars Bonvin  _et al._ (2017).

In this Letter, we examine the best precision in distance measurement
achievable with the advanced HLV detector network sensitivity for strongly
lensed GW signals. We assume a binary blak hole (BBH) as a GW source and
conduct a proof-of-concept parameter estimation (PE) for the $D_{L}$. We
compare posterior probability density functions (PDFs) of $D_{L}$, $p(D_{L})$
for lensed and unlensed GW signals with different detector sensitivities.

_Strong lensing of GWs._ —We adopt the lens configuration described in
Takahashi and Nakamura (2003). We consider a lens is located between a BBH and
an observer (i.e. the GW detector network on Earth). We assume two strongly
lensed GW signals are generated and propagated toward the observer as an
originally unlensed GW signal radiated from a BBH passes through the lens. In
this work, we assume a galaxy-like lens and apply the thin-lens approximation.
Then we can obtain the lensed GW signal $h_{l}(f)$ from $h_{u}(f)$ by a simple
relation: $h_{l}(f)\\!=\\!F(f)h_{u}(f)$, where $F(f)$ is an amplification
factor that determines the lensing characteristics. As we consider a galaxy-
like lens, $F(f)$ can be obtained in the geometrical optics limit.

In this work, we consider two lens models, the point-mass (PM) and the
singular isothermal sphere (SIS). An amplification factor for both lens models
is given as $F(f)\\!=\\!\sqrt{|\mu_{+}|}\\!-\\!i\sqrt{|\mu_{-}|}e^{2\pi
if\Delta t}$. Here, $\mu_{+}$ and $\mu_{-}$ are individual magnification
factors corresponding to each lensed GW signals $h^{\textrm{I}}_{l}(f)$ and
$h^{\textrm{II}}_{l}(f)$, respectively. Also, $\Delta t$ is the time delay
between the arrival times of $h^{\textrm{I}}_{l}(f)$ and
$h^{\textrm{II}}_{l}(f)$ to an observer. For each lens model, $\mu_{\pm}$ and
$\Delta t$ can be written as follows:

PM: $\displaystyle\mu_{\pm}~{}$
$\displaystyle=\frac{1}{2}\pm\frac{y^{2}+2}{2y\sqrt{y^{2}+4}}~{},$ SIS:
$\displaystyle\mu_{\pm}~{}$ $\displaystyle=\pm 1+\frac{1}{y}~{},$ (1)

and

PM: $\displaystyle\Delta t~{}$
$\displaystyle=\frac{4GM_{lz}}{c^{3}}\left[\frac{y\sqrt{y^{2}\\!+\\!4}}{2}+\ln\left\\{\frac{\sqrt{y^{2}\\!+\\!4}+y}{\sqrt{y^{2}\\!+\\!4}-y}\right\\}\right]~{},$
SIS: $\displaystyle\Delta t~{}$ $\displaystyle=\frac{8GM_{lz}y}{c^{3}}~{}.$
(2)

In Eqs. (1) and (2), $y$ denotes the parameterized source position following
the lens configuration used in Takahashi and Nakamura (2003). The range of $y$
can be constrained to be $[0.1,~{}1.0)$ in this work. The expected occurrence
rate of strongly lensed GWs Abbott  _et al._ (2021b); Lai  _et al._ (2018)
sets the lower limit of $y\\!\geq\\!0.1$. The upper limit is given by the SIS
model, i.e., the $y$-dependent validity of $F(f)$ requires $y\\!<\\!1$ in
order to produce two lensed signals with SIS Takahashi and Nakamura (2003).
The expression of $F(f)$ implies that PM always produces two lensed signals
with any $y$. As shown in Eq. (2), the time delay is proportional to a
redshifted mass $M_{lz}\\!=\\!M_{l}(1+z_{l})$ of a lens at redshift $z_{l}$.
As a representative value, we set $M_{lz}\\!=\\!10^{11.5}M_{\odot}$, which
results in a time delay from weeks to months between $h^{\textrm{I}}_{l}(f)$
and $h^{\textrm{II}}_{l}(f)$ in the range of $y$ considered in this work.

Table 1: Parameters used to generate an unlensed GW signal $h_{u}(f)$ from a BBH along with prior assumptions.  Parameter | Unit | Value | Prior
---|---|---|---
distribution
Component masses, $m_{1}$ & $m_{2}$ | $M_{\odot}$ | 30.0 | Uniform
Chirp mass, $\mathcal{M}$ | $M_{\odot}$ | 26.1 | Uniform
Luminosity distance, $D_{L}$ | Gpc | 3 | Power-law
Right Ascension | rad | 1.3750 | Uniform
Declination | rad | -1.2108 | Isotropic

_Parameter estimation for unlensed signals._ —Let us label an unlensed signal
$h_{u}(f)$ is the true (unchanged) GW signal from a CBC when there is no lens
between the source and an observer. We use the Bilby pipeline Speagle (2020);
Ashton  _et al._ (2019); Romero-Shaw  _et al._ (2020) and perform PE for an
artificially generated $h_{u}(f)$. Bilby utilizes the Dynesty nested sampler
Skilling (2006). We consider design power spectral densities (PSDs) Abbott
_et al._ (2020) of the detector network consisting of advanced LIGO-Hanford
(H), advanced LIGO-Livingston (L), and advanced Virgo (V); the HLV detector
network hereafter. As we aim to find the best possible precision for the
distance estimation via GW observation, a “zero-noise” model is assumed for a
baseline PE with $h_{u}(f)$ Rodriguez  _et al._ (2014); Favata  _et al._
(2022).

In order to generate templates and injection GW signals used in PE, we use
IMRPhenomD waveform model Husa  _et al._ (2016); Khan  _et al._ (2016). As for
a CBC source, we assume an equal-mass BBH with
$m_{1}\\!=\\!m_{2}\\!=\\!30M_{\odot}$. The chirp mass of the binary is
$\mathcal{M}\\!=\\!(m_{1}m_{2})^{3/5}(m_{1}\\!+\\!m_{2})^{-1/5}\\!\simeq\\!26M_{\odot}$.
We set the $D_{L}$ to the BBH to be $3\\!~{}\textrm{Gpc}$******We label
$D_{L}$ as a true distance for an unlensed signal and $D_{L}+,D_{L}-$ for
apparent distances for two lensed signals originated from the same source with
different magnification. All are observables and subject to biases.. For the
sky location and the polarization angle, we adopt the observed median values
of GW150914 Abbott  _et al._ (2016). We assume an arbitrary event time that
can make the inclination angle to the L detector become zero. Other
parameters, such as spins, eccentricity, or higher-order modes, are ignored
for simplicity. By choosing these parameters, we have the network SNR of 8 for
$h_{u}(f)$ given the HLV design PSDs with zero-noise.

We assume priors of parameters as follows. A power-law distribution in a
comoving volume with a power-law index $\alpha\\!=\\!2$ Veitch  _et al._
(2015) is used for the distance prior with a range of $[0.1,5]\textrm{Gpc}$.
For other source parameters, including chirp mass with a range of
$\mathcal{M}\\!=\\![10,100]M_{\odot}$, we adopt prior distributions described
in Romero-Shaw  _et al._ (2020). Details of selected parameters are summarized
in Table 1.

Figure 1: Posterior PDFs for chirp mass $\mathcal{M}$ (left) and luminosity
distance $D_{L}$ (right) recovered from an unlensed GW signal. Black solid
lines represent injected parameters. Vertical orange dotted lines are lower
and upper bounds of the 99% C.I., respectively.

Fig. 1 presents $p(\mathcal{M})$ and $p(D_{L})$ for $h_{u}(f)$ with the HLV
detector network at design PSD and zero-noise. For lensed signals,
$p(\mathcal{M})$ is almost the same but $p(D_{L})$ becomes different by strong
lensing (see Fig. 3).

_Parameter estimation for lensed signals._ —When performing PE analyses, we
assume both the waveform model for a GW signal and a lens model generating two
lensed GW signals are known. In this case, the precision of distance
estimation is governed by the SNR. As for the most optimistic scenario, we
then find the maximum amplification condition for SIS and PM lens models,
respectively.

Two lensed signals $h^{\textrm{I,II}}_{l}(f)$ are generated by incorporating
$y\\!=\\!0.1$ in Eq. (1) for each lens model. This is when $\mu_{\pm}$ is
maximum. Let us assume the physical association of $h^{\textrm{I,II}}_{l}(f)$
is identified. By calculating a joint likelihood using both signals, one can
retrieve the posterior for the true luminosity distance $p(D_{L})$ and other
source parameters.

When detected, the two lensed signals $h^{\textrm{I,II}}_{l}(f)$ would likely
to be identified as two independent events separated by time in GW
observation. If the SNRs of both signals are large enough, the most likely
values of $\mathcal{M}$ obtained from two lensed signals would be almost
identical within the uncertainty attributed to the sensitivity of the detector
network and analysis pipelines. However, the inferred distances for the two
lensed signals are different depending on individual amplification factors.
Hence, two _apparent_ luminosity distances
$D_{L\pm}\\!=\\!D_{L}/\sqrt{|\mu_{\pm}|}$ are obtained from PE analysis for a
given GW signal when each signal is analyzed individually. In order for
reflecting this realistic observation scenario, we inject apparent distances
$D_{L\pm}$ to simulate two lensed signals $h^{\textrm{I,II}}_{l}(f)$ but use
the same value for the chirp mass, i.e.,
$\mathcal{M}_{+}\\!=\\!\mathcal{M}_{-}\\!=\\!\mathcal{M}\\!=\\!26.1M_{\odot}$.
We ignore time dilation and phase shift, as they do not affect luminosity
distance estimation in our scenario. We also assume the apparent sky locations
of two lensed signals are the same because the subtle differences of the
lensed signals in sky cannot be distinguished by the sensitivities of the
current advanced detector network.

Figure 2: Relation between the relative magnification factor $\mu_{\rm rel}$
and the source position $y$. The green and blue lines are results from the
point-mass (PM) and singular isothermal sphere (SIS) models, respectively.

As for the PE analysis, we use the Golum Janquart  _et al._ (2021) pipeline
that is constructed based on Bilby. The pipeline enables us to infer the
apparent source parameters of strongly lensed signals, for instance,
$D_{L\pm}$ and $\mathcal{M}_{\pm}$ for $h^{\textrm{I,II}}_{l}(f)$,
respectively. In addition, it allows us to infer lensing parameters such as
the relative magnification factor $\mu_{\textrm{rel}}$.

Based on what is discussed earlier, we use the same injection parameters with
the unlensed signal except $D_{L_{\rm\pm}}$. PE for lensed signals involves
assumptions on lensing parameters and $D_{L_{\rm\pm}}$ in addition to typical
assumptions used for an unlensed signal. We assume uniform prior distributions
for lensing parameters, e.g., $\mu_{\mathrm{rel}}$, time delay, and the Morse
phase shift parameter Janquart  _et al._ (2021). As for other parameters
including the distance and chirp mass, we apply the same distributions used in
the PE for $h_{u}(f)$.

In the geometrical optics limit, $\mu_{\textrm{rel}}$ can be rewritten by the
apparent distances for both PM and SIS models, that is,
$\mu_{\textrm{rel}}\\!\equiv\\!\left|\mu_{-}/\mu_{+}\right|\\!=\\!(D_{L+}/D_{L-})^{2}$.
The likelihood of $\mu_{\textrm{rel}}$ ($\mathcal{L}(\mu_{\mathrm{rel}})$) can
then be obtained from the ratio
$[\mathcal{L}(D_{L+})/\mathcal{L}(D_{L-})]^{2}$. It is straightforward to
compute $p$($\mu_{\mathrm{rel}}$) given the uniform prior for
$\mu_{\mathrm{rel}}=[0.01,0.9]$. The minimum value of $\mu_{\mathrm{rel}}$ is
chosen in order to avoid zero. Also this range is well within the allowed
constraints but would not create any railing in posteriors.

We obtain the maximum likelihood value $\mu_{\mathrm{rel,~{}max}}$ from the
$p$($\mu_{\mathrm{rel}}$) and find corresponding
$y$($\mu_{\mathrm{rel,~{}max}}$) from Fig. 2. Also, it is straightforward to
calculate $\mu_{\pm}$ using Eq. (1). When $\mu_{\pm}$ and $D_{L\pm}$ are at
hands, we can obtain $D_{L}$ by $D_{L}\\!=\\!\sqrt{|\mu_{+}|}D_{L+}$ or
$D_{L}\\!=\\!\sqrt{|\mu_{-}|}D_{L-}$.

It is shown in Janquart  _et al._ (2021) that we can constrain the lensing and
source parameters better by _reweighting_ the posterior samples of individual
lensed signals. For the same CBC, $p(D_{L})$ can be obtained from a reweighted
posterior for the apparent distance $p(D_{L+})$ by
$p(D_{L})\\!=\\!\sqrt{\left|\mu_{+}\right|}p(D_{L+})$. The individual
magnification factor $\mu_{+}(y)$ is determined by
$y\\!=\\!y(\mu_{rel,~{}max})$.

One can choose either $h^{\textrm{I}}_{l}$ or $h^{\textrm{II}}_{l}$ when
combining the two likelihoods in order to determine the true distance
posterior of the source. In this work, we choose $h^{\textrm{I}}_{l}$ assuming
(i) $h^{\mathrm{I}}_{l}$ arrives earlier than $h^{\mathrm{II}}_{l}$ and (ii)
$h^{\mathrm{I}}_{l}$ is experienced stronger magnification than
$h^{\mathrm{II}}_{l}$. We perform the same reweighting procedure—described in
Equation (15) and Appendix A of Janquart  _et al._ (2021)—and calculate
posteriors of lensing and apparent source parameters of $h^{\textrm{I}}_{l}$.

Different detector sensitivities are considered in this work: (i) the HLV
design PSDs with zero-noise (Case A) and (ii) the HLV O3a PSDs (based on the
first three months of O3) ******The corresponding PSD data for L, H, and V can
be found from https://dcc.ligo.org/LIGO-T2000012/public. with a Gaussian noise
(Case B).

Figure 3: Posteriors of $\mu_{\mathrm{rel}}$, $\mathcal{M}_{+}$, and $D_{L+}$
obtained for $h^{\mathrm{I}}_{l}$ based on PM (green) and SIS (blue) models.
Black solid lines indicate $\mu_{\mathrm{rel}}(y\\!=\\!0.1)$ and injected
values of $\mathcal{M}_{+}$, and $D_{L+}$. Vertical dashed lines are the lower
and upper bounds of the 99% C.I.’s centered around a median value of each
parameter.

(a) HLV design PSDs with zero noise

(b) HLV O3a PSDs with a Gaussian noise realization

Figure 4: One dimensional posterior PDFs of $D_{L}$ recovered from
$h^{\mathrm{I}}_{l,\textrm{PM}}(f)$ (green) and
$h^{\mathrm{I}}_{l,\textrm{SIS}}(f)$ (blue). The distance posterior from an
unlensed signal $p(D^{\textrm{UL}}_{L})$ is shown in orange solid lines for
comparison. We compare results from the HLV design PSDs with zero-noise (left)
and HLV O3a PSDs with a Gaussian noise (right). Black solid lines are injected
value of $3~{}\textrm{Gpc}$, and dashed lines indicate lower and upper bounds
of the 99% C.I.’s.

_Results._ —The maximum magnification ($y=0.1$) of two lens models implies
$\mu^{\textrm{SIS}}_{+}\\!=\\!11$ and $\mu^{\textrm{PM}}_{+}\\!\simeq\\!5.5$
or $\textrm{SNR}_{\textrm{SIS}}\\!=\\!24.6$ and
$\textrm{SNR}_{\textrm{PM}}\\!=\\!16.3$, respectively. This implies a distance
posterior based on the SIS model can be better constrained than that of PM.
For example, Fig. 3 shows PE results for $h^{\mathrm{I}}_{l}$. Posteriors
$p(\mu_{\textrm{rel}})$ and $p(\mathcal{M}_{+})$ show prominent peaks at
around the injected values, i.e., $\mu_{\textrm{rel}}\\!\simeq\\!0.81$ and
$\mathcal{M}_{+}\simeq 26.1M_{\odot}$. Posteriors based on the SIS model are
better constrained than those from the PM model. This implies
$D^{\textrm{SIS}}_{L+,\rm peak}<D^{\textrm{PM}}_{L+,\rm peak}$.

Fig. 4 shows effects of strong lensing in distance estimation. Fig. 4(a) (Case
A) shows $p(D_{L})^{\prime}s$ obtained by the HLV design PSDs with zero-noise.
Results show that the 99% C.I. width of $p(D_{L})$ can be better constrained
by almost a factor of two when both $h^{\textrm{I}}_{l}(f)$ and
$h^{\textrm{II}}_{l}(f)$ are detected. Fig. 4(b) (Case B) shows similar
results for the O3a PSDs with a Gaussian noise. As seen from Fig. 3,
posteriors from lensed signals are better constrained than that of the
unlensed signal.

In Table 2, we compare widths of PDFs at the 99% C.I. (labeled as
$\mathcal{W}_{99}$) in Fig. 4. We define a ratio
$\mathcal{R}_{99}\\!\equiv\\!\mathcal{W}^{\textrm{UL}}_{99}/\mathcal{W}^{\textrm{X}}_{99}$,
where X is either UL, PM, or SIS. If detected, strong lensing is definitely
helpful to constrain the luminosity distance better for a given detector. For
example, considering the O3a PSD with Gaussian noise, the $\mathcal{W}_{99}$
in $p(D_{L})$ is reduced by 37% (PM) and 42% (SIS) with respect to the results
based on the unlensed signal. If comparing the widths at the 67% C.I., we
expect 19% (PM) and 34% (SIS) improvements by strong lensing.

Table 2: Quantitative comparisons of the distance estimation for unlensed, PM,
and SIS models based on Fig. 4. The 2nd column presents $\mathcal{W}_{99}$
that is the width of $p(D_{L})$ at the 99% C.I. The 3rd column shows
$\mathcal{R}_{99}$, a ratio between $\mathcal{W}_{99}$ values (see text for
definition). The 4th and 5th columns present widths and ratios at the 67% C.I.
of $p(D_{L})$. Case A: HLV design PSDs with zero-noise
---
Signal | $\mathcal{W}_{99}~{}\textrm{[Mpc]}$ | $\mathcal{R}_{99}$ | $\mathcal{W}_{67}~{}\textrm{[Mpc]}$ | $\mathcal{R}_{67}$
Unlensed | 2,697 | 1.00 | 1,085 | 1.00
Lensed (PM) | 1,429 | 1.89 | 653 | 1.66
Lensed (SIS) | 1,281 | 2.11 | 602 | 1.80
|  |  |  |
Case B: HLV O3a PSDs with a Gaussian noise
Signal | $\mathcal{W}_{99}~{}\textrm{[Mpc]}$ | $\mathcal{R}_{99}$ | $\mathcal{W}_{67}~{}\textrm{[Mpc]}$ | $\mathcal{R}_{67}$
Unlensed | 2,090 | 1.00 | 871 | 1.00
Lensed (PM) | 1,527 | 1.37 | 732 | 1.19
Lensed (SIS) | 1,470 | 1.42 | 651 | 1.34

_Discussions._ —Identification of two lensed signals is assumed, but in
practice, this is a challenge in data analysis for O4 and future observations.
Also, any mismatch due to the GW waveform or lens model would introduce
systematics biases that can further degrade the quality of PE.

We consider two simplest lens models and assume two lensed signals are
produced by a lens in the geometrical optics limit. Definitions and/or the
relation between $y$ and $\mu_{\textrm{rel}}$ can be nontrivial when there are
more lens parameters needed to describe a amplification factor $F(f)$. More
than two lensed signals are predicted by models such as singular-isothermal-
ellipsoid Kormann  _et al._ (1994) or the Navarro-Frenk-White model Navarro
_et al._ (1997) or a more complex macrolens containing multiple microlenses
Diego  _et al._ (2019); Diego (2020); Mishra  _et al._ (2021). Future studies
on more realistic and/or complicated GW lensing are useful to understand the
effects of strong lensing in the context of GW parameter estimation in more
details.

More precise and accurate distance estimation is valuable not only for
understanding the formation and evolution of various CBC populations but also
for constraining a Hubble constant. When strongly lensed GW signals are
detected, the PE procedure presented in this Letter can be used to estimate
$D_{L}$ with better precision that can be directly used for the Hubble’s law.
Strongly lensed GW signals are also likely to be detected with the next
generation detectors Piórkowska  _et al._ (2013); Cao  _et al._ (2014) such as
the Einstein Telescope Punturo  _et al._ (2010) and the Cosmic Explorer Reitze
_et al._ (2019). More precise distance estimation by strong lensing can shed
lights on the Hubble tension in the next decades.

###### Acknowledgements.

_Acknowledgements_ This work is supported by the National Research Foundation
of Korea (NRF) grants funded by the Ministry of Science and ICT of Korea
Government (NRF-2020R1C1C1005863, NRF-2021R1F1A1062969, and
NRF-2021M3F7A1082056). E.S. is partially supported by grants from the Research
Grants Council of the Hong Kong (Project No. CUHK 24304317), The Croucher
Foundation of Hong Kong, the Research Committee of the Chinese University of
Hong Kong, and the Science and Technology Facilities Council (Grant No.
ST/L000946/1). We are grateful for computational resources provided by the
LIGO Laboratory and supported by the National Science Foundation Grants
PHY-0757058 and PHY-0823459. K.K. also appreciates the warm hospitality of
Korea Astronomy and Space Science Institute, where a part of the work has been
done. K.K. and E.S. equally contributed to this work as co-first authors.

## References

  * Christensen and Meyer (2022) N. Christensen and R. Meyer, “Parameter estimation with gravitational waves,” Rev. Mod. Phys. 94, 025001 (2022), arXiv:2204.04449 [gr-qc] .
  * Ohanian (1974) H. C. Ohanian, “On the focusing of gravitational radiation,” Int. J. Theor. Phys. 9, 425–437 (1974).
  * Bliokh and Minakov (1975) P. V. Bliokh and A. A. Minakov, “Diffraction of light and lens effect of the stellar gravitation field,” Ap&SS 34, L7–L9 (1975).
  * Bontz and Haugan (1981) R. J. Bontz and M. P. Haugan, “A diffraction limit on the gravitational lens effect,” Ap&SS 78, 199–210 (1981).
  * Thorne (1983) K. S. Thorne, “The theory of gravitational radiation: An introductory review,” in  _Gravitational Radiation_ , edited by N. Deruelle and T. Piran (Amsterdam: North Holland, 1983).
  * Deguchi and Watson (1986) S. Deguchi and W. D. Watson, “Diffraction in gravitational lensing for compact objects of low mass,” Astrophys. J. 307, 30–37 (1986).
  * Schneider  _et al._ (1992) P. Schneider, J. Ehlers, and E. E. Falco,  _Gravitational Lenses_ (Springer, New York, 1992) p. 112.
  * Nakamura and Deguchi (1999) T. T. Nakamura and S. Deguchi, “Wave Optics in Gravitational Lensing,” Prog. Theor. Phys. Suppl. 133, 137–153 (1999).
  * Takahashi and Nakamura (2003) R. Takahashi and T. Nakamura, “Wave effects in the gravitational lensing of gravitational waves from chirping binaries,” Astrophys. J. 595, 1039 (2003).
  * Ng  _et al._ (2018) K. K. Y. Ng, K. W. K. Wong, T. Broadhurst, and T. G. F. Li, “Precise ligo lensing rate predictions for binary black holes,” Physical Review D 97, 023012 (2018).
  * Li  _et al._ (2018) S.-S. Li, S. Mao, Y. Zhao, and Y. Lu, “Gravitational lensing of gravitational waves: A statistical perspective,” Mon. Not. Roy. Astron. Soc. 476, 2220–2229 (2018).
  * Oguri (2018) M. Oguri, “Effect of gravitational lensing on the distribution of gravitational waves from distant binary black hole mergers,” Mon. Not. Roy. Astron. Soc. 480, 3842–3855 (2018).
  * Mukherjee  _et al._ (2020) S. Mukherjee, B. D. Wandelt, and J. Silk, “Probing the theory of gravity with gravitational lensing of gravitational waves and galaxy surveys,” Mon. Not. Roy. Astron. Soc. 494, 1956–1970 (2020).
  * Wierda  _et al._ (2021) A. R. A. C. Wierda, E. Wempe, O. A. Hannuksela, L. V. E. Koopmans, and C. Van Den Broeck, “Beyond the detector horizon: Forecasting gravitational-wave strong lensing,” Astrophys. J. 921, 154 (2021).
  * Aasi  _et al._ (2015) J. Aasi _et al._ (LIGO Scientific), “Advanced LIGO,” Class. Quant. Grav. 32, 074001 (2015), arXiv:1411.4547 [gr-qc] .
  * Acernese  _et al._ (2015) F. Acernese _et al._ (VIRGO), “Advanced Virgo: a second-generation interferometric gravitational wave detector,” Class. Quant. Grav. 32, 024001 (2015), arXiv:1408.3978 [gr-qc] .
  * Liu  _et al._ (2021) X. Liu, I. M. Hernandez, and J. Creighton, “Identifying strong gravitational-wave lensing during the second observing run of advanced ligo and advanced virgo,” Astrophys. J. 908, 97 (2021).
  * Hannuksela  _et al._ (2019) O. A. Hannuksela, K. Haris, K. K. Y. Ng, S. Kumar, A. K. Mehta, D. Keitel, T. G. F. Li, and P. Ajith, “Search for gravitational lensing signatures in LIGO-Virgo binary black hole events,” Astrophys. J. Lett. 874, L2 (2019), arXiv:1901.02674 [gr-qc] .
  * Abbott  _et al._ (2021a) R. Abbott, T. D. Abbott, S. Abraham, Fausto Acernese, K Ackley, A Adams, C Adams, RX Adhikari, VB Adya, C Affeldt, _et al._ , “Search for lensing signatures in the gravitational-wave observations from the first half of ligo–virgo’s third observing run,” Astrophys. J. 923, 14 (2021a).
  * Abbott  _et al._ (2023) R. Abbott _et al._ (LIGO Scientific, VIRGO, KAGRA), “Search for gravitational-lensing signatures in the full third observing run of the LIGO-Virgo network,” (2023), arXiv:2304.08393 [gr-qc] .
  * Lemaître (1927) G. Lemaître, “Un Univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques,” Annales de la Société Scientifique de Bruxelles 47, 49–59 (1927).
  * Hubble (1929) E. Hubble, “A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae,” Proceedings of the National Academy of Science 15, 168–173 (1929).
  * Schutz (1986) B. F. Schutz, “Determining the Hubble Constant from Gravitational Wave Observations,” Nature 323, 310–311 (1986).
  * Holz and Hughes (2005) D. E. Holz and S. A. Hughes, “Using gravitational-wave standard sirens,” Astrophys. J. 629, 15–22 (2005), arXiv:astro-ph/0504616 .
  * Dalal  _et al._ (2006) N. Dalal, D. E. Holz, Scott A. Hughes, and Bhuvnesh Jain, “Short grb and binary black hole standard sirens as a probe of dark energy,” Phys. Rev. D 74, 063006 (2006), arXiv:astro-ph/0601275 .
  * Nissanke  _et al._ (2010) S. Nissanke, D. E. Holz, S. A. Hughes, N. Dalal, and J. L. Sievers, “Exploring short gamma-ray bursts as gravitational-wave standard sirens,” Astrophys. J. 725, 496–514 (2010), arXiv:0904.1017 [astro-ph.CO] .
  * Nissanke  _et al._ (2013) S. Nissanke, D. E. Holz, N. Dalal, S. A. Hughes, J. L. Sievers, and C. M. Hirata, “Determining the Hubble constant from gravitational wave observations of merging compact binaries,” (2013), arXiv:1307.2638 [astro-ph.CO] .
  * Vitale and Chen (2018) S. Vitale and H.-Y. Chen, “Measuring the Hubble constant with neutron star black hole mergers,” Phys. Rev. Lett. 121, 021303 (2018), arXiv:1804.07337 [astro-ph.CO] .
  * Borhanian  _et al._ (2020) S. Borhanian, A. Dhani, A. Gupta, K. G. Arun, and B. S. Sathyaprakash, “Dark Sirens to Resolve the Hubble–Lemaître Tension,” Astrophys. J. Lett. 905, L28 (2020), arXiv:2007.02883 [astro-ph.CO] .
  * Borhanian and Sathyaprakash (2022) S. Borhanian and B. S. Sathyaprakash, “Listening to the Universe with Next Generation Ground-Based Gravitational-Wave Detectors,” (2022), arXiv:2202.11048 [gr-qc] .
  * Ghosh  _et al._ (2022) T. Ghosh, B. Biswas, and S. Bose, “Simultaneous Inference of Neutron Star Equation of State and Hubble Constant with a Population of Merging Neutron Stars,” (2022), arXiv:2203.11756 [astro-ph.CO] .
  * Abbott  _et al._ (2017a) B. P. Abbott _et al._ (LIGO Scientific, Virgo), “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” Phys. Rev. Lett. 119, 161101 (2017a), arXiv:1710.05832 [gr-qc] .
  * Abbott  _et al._ (2017b) B. P. Abbott _et al._ (LIGO Scientific, Virgo, 1M2H, Dark Energy Camera GW-E, DES, DLT40, Las Cumbres Observatory, VINROUGE, MASTER), “A gravitational-wave standard siren measurement of the Hubble constant,” Nature 551, 85–88 (2017b), arXiv:1710.05835 [astro-ph.CO] .
  * Ade  _et al._ (2016) P. A. R. Ade _et al._ (Planck), “Planck 2015 results. XIII. Cosmological parameters,” Astron. Astrophys. 594, A13 (2016), arXiv:1502.01589 [astro-ph.CO] .
  * Riess  _et al._ (2016) A. G. Riess _et al._ , “A 2.4% Determination of the Local Value of the Hubble Constant,” Astrophys. J. 826, 56 (2016), arXiv:1604.01424 [astro-ph.CO] .
  * Bonvin  _et al._ (2017) V. Bonvin _et al._ , “H0LiCOW – V. New COSMOGRAIL time delays of HE 0435$-$1223: $H_{0}$ to 3.8 per cent precision from strong lensing in a flat $\Lambda$CDM model,” Mon. Not. Roy. Astron. Soc. 465, 4914–4930 (2017), arXiv:1607.01790 [astro-ph.CO] .
  * Abbott  _et al._ (2021b) R. Abbott _et al._ (LIGO Scientific, VIRGO), “Search for Lensing Signatures in the Gravitational-Wave Observations from the First Half of LIGO–Virgo’s Third Observing Run,” Astrophys. J. 923, 14 (2021b), arXiv:2105.06384 [gr-qc] .
  * Lai  _et al._ (2018) K.-H. Lai, O. A. Hannuksela, A. Herrera-Martín, J. M. Diego, Tom Broadhurst, and T. G. F. Li, “Discovering intermediate-mass black hole lenses through gravitational wave lensing,” Phys. Rev. D 98, 083005 (2018).
  * Speagle (2020) J. S. Speagle, “DYNESTY: a dynamic nested sampling package for estimating Bayesian posteriors and evidences,” Mon. Not. Roy. Astron. Soc. 493, 3132–3158 (2020), arXiv:1904.02180 [astro-ph.IM] .
  * Ashton  _et al._ (2019) G. Ashton _et al._ , “BILBY: A user-friendly Bayesian inference library for gravitational-wave astronomy,” Astrophys. J. Suppl. 241, 27 (2019), arXiv:1811.02042 [astro-ph.IM] .
  * Romero-Shaw  _et al._ (2020) I. M. Romero-Shaw, C. Talbot, S. Biscoveanu, V. D’emilio, G. Ashton, C. P. L Berry, S. Coughlin, S. Galaudage, C. Hoy, M. Hübner, _et al._ , “Bayesian inference for compact binary coalescences with bilby: Validation and application to the first ligo–virgo gravitational-wave transient catalogue,” Mon. Not. Roy. Astron. Soc. 499, 3295–3319 (2020).
  * Skilling (2006) J. Skilling, “Nested sampling for general Bayesian computation,” Bayesian Analysis 1, 833 – 859 (2006).
  * Abbott  _et al._ (2020) B. P. Abbott, R Abbott, T. D. Abbott, S. Abraham, F. Acernese, K. Ackley, C. Adams, V. B. Adya, C. Affeldt, M. Agathos, _et al._ , “Prospects for observing and localizing gravitational-wave transients with advanced ligo, advanced virgo and kagra,” Living reviews in relativity 23, 1–69 (2020).
  * Rodriguez  _et al._ (2014) C. L. Rodriguez, B. Farr, V. Raymond, W. M. Farr, T. B. Littenberg, D. Fazi, and V. Kalogera, “Basic Parameter Estimation of Binary Neutron Star Systems by the Advanced LIGO/Virgo Network,” Astrophys. J. 784, 119 (2014), arXiv:1309.3273 [astro-ph.HE] .
  * Favata  _et al._ (2022) M. Favata, C. Kim, K. G. Arun, J.C. Kim, and H. W. Lee, “Constraining the orbital eccentricity of inspiralling compact binary systems with Advanced LIGO,” Phys. Rev. D 105, 023003 (2022), arXiv:2108.05861 [gr-qc] .
  * Husa  _et al._ (2016) S. Husa, S. Khan, M. Hannam, M. Pürrer, F. Ohme, X. J. Forteza, and A. Bohé, “Frequency-domain gravitational waves from nonprecessing black-hole binaries. i. new numerical waveforms and anatomy of the signal,” Phys. Rev. D 93, 044006 (2016).
  * Khan  _et al._ (2016) S. Khan, S. Husa, M. Hannam, F. Ohme, M. Pürrer, X. J. Forteza, and A. Bohé, “Frequency-domain gravitational waves from nonprecessing black-hole binaries. ii. a phenomenological model for the advanced detector era,” Phys. Rev. D 93, 044007 (2016).
  * Abbott  _et al._ (2016) B. P. Abbott _et al._ (LIGO Scientific, Virgo), “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc] .
  * Veitch  _et al._ (2015) J. Veitch, V. Raymond, B. Farr, W. Farr, B. Aylott, K. Blackburn, _et al._ , “Parameter estimation for compact binaries with ground-based gravitational-wave observations using the LALInference software library,” Phys. Rev. D 91, 042003 (2015), arXiv:1409.7215 [gr-qc] .
  * Janquart  _et al._ (2021) J. Janquart, O. A. Hannuksela, K. Haris, and C. Van Den Broeck, “A fast and precise methodology to search for and analyse strongly lensed gravitational-wave events,” Mon. Not. Roy. Astron. Soc. 506, 5430–5438 (2021).
  * Kormann  _et al._ (1994) R. Kormann, P. Schneider, and M. Bartelmann, “Isothermal elliptical gravitational lens models.” Astron. Astrophys. 284, 285–299 (1994).
  * Navarro  _et al._ (1997) J. F. Navarro, C. S. Frenk, and S. D. M. White, “A Universal density profile from hierarchical clustering,” Astrophys. J. 490, 493–508 (1997), arXiv:astro-ph/9611107 .
  * Diego  _et al._ (2019) J. M. Diego, Otto A Hannuksela, P. L. Kelly, G Pagano, T Broadhurst, K. Kim, T. G. F. Li, and G. F. Smoot, “Observational signatures of microlensing in gravitational waves at ligo/virgo frequencies,” Astron. Astrophys. 627, A130 (2019).
  * Diego (2020) J. M. Diego, “Constraining the abundance of primordial black holes with gravitational lensing of gravitational waves at ligo frequencies,” Phys. Rev. D 101, 123512 (2020).
  * Mishra  _et al._ (2021) A. Mishra, A. K. Meena, A. More, S. Bose, and J. S. Bagla, “Gravitational lensing of gravitational waves: effect of microlens population in lensing galaxies,” Mon. Not. Roy. Astron. Soc. 508, 4869–4886 (2021).
  * Piórkowska  _et al._ (2013) A. Piórkowska, M. Biesiada, and Z.-H. Zhu, “Strong gravitational lensing of gravitational waves in Einstein Telescope,” JCAP 10, 022 (2013), arXiv:1309.5731 [astro-ph.CO] .
  * Cao  _et al._ (2014) Z. Cao, L.-F. Li, and Y. Wang, “Gravitational lensing effects on parameter estimation in gravitational wave detection with advanced detectors,” Phys. Rev. D 90, 062003 (2014).
  * Punturo  _et al._ (2010) M. Punturo,  _et al._ , “The Einstein Telescope: A third-generation gravitational wave observatory,” Class. Quant. Grav. 27, 194002 (2010).
  * Reitze  _et al._ (2019) D. Reitze,  _et al._ , “Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO,” Bull. Am. Astron. Soc. 51, 035 (2019), arXiv:1907.04833 [astro-ph.IM] .


Implementation of conformal digital metasurfaces for THz polarimetric sensing

Javad Shabanpour, Sina Beyraghi, Fardin Ghorbani, and Homayoon Oraizi

Iran University of Science and Technology, Tehran, Iran

<EMAIL_ADDRESS>
###### Abstract

Monitoring and controlling the state of polarization of electromagnetic waves
is of significant interest for various basic and practical applications such
as linear position sensing and medical imaging. Here, we propose the first
conformal digital metamaterial absorber to detect the polarization state of
THz incident waves. The proposed polarimeter is capable of characterizing four
independent polarization states of (TE, TM, $\pm{45^{\circ}}$ linear, and
RCP/LCP) by observing the reflectivity of the structure with respect to the x-
and y-direction. Besides, the proposed structure displays a strong
absorptivity above 90% up to the incidence angle of $50^{\circ}$ for oblique
incident waves with different polarizations. By mere changing the bias voltage
of two orthogonal VO2 microwires via two independent computer-programmed
multichannel DC network, distinct conditions for reflected waves occurs under
excitations of different polarizations, whereby the polarization state of the
incident wave may readily be estimated. We believe that the proposed
metasurface-based polarimeter can pave the way for polarization detection
applications on curved surfaces.

In recent years, Terahertz (THz) waves have reached maturity and have
witnessed remarkable attention due to their distinct features that make them
beneficial in various sensing and imaging technologies[1]. Artificial
metasurfaces, as periodic arrangements of engineered sub-wavelength
structures, can be structured for complex and functional manipulation of
electromagnetic (EM) waves[9]. They can be exploited to modify the
permittivity and permeability of materials by purposefully controlling the
local properties of phase, amplitude, and polarization of EM waves. Beyond the
scope of analog metamaterials, digital metamaterials have experienced rapid
evolution compared to traditional wave manipulations and outstandingly provide
a broader range of wave-matter functionalities which make them fascinating in
various practical applications such as smart surfaces, power intensity
control, and programmable structures[2, 3, 17].

Fig. 1:  a) The schematic diagram of conformal metamaterial absorber for
polarization detection application. (b), schematic view of the unit-cell. (c)
schematic view of the back of the unit-cell where Vx and Vy are two patches
that serve as positive electrodes for applying DC bias voltage.

Given that each polarization can represent an independent information in
communication systems, the polarization can be considered as one of the
fundamental attributes characterizing the features of EM waves, and the
measurement of polarization divulges rich information about the nature of
scattering, radiation, and absorption phenomena. A polarization-dependent
metamaterial absorber is an ideal building block for polarimetric sensing
systems which can equip an inspiring platform for solving some crucial THz
challenges such as linear position sensing, security scanning, and medical
imaging[4, 5, 16].

After introducing the first perfect metamaterial absorber (MMA) by Landy et
al. in 2008[6], as a counterpart to conventional absorbers, MMAs have become
research hotspots and have also been of great attention for solving EM
interference challenges, such as the stealth technologies, radar absorbing and
thermal emitter[7]. Generally, the maximum absorption is obtained for the
normal incidence. However, when a metamaterial absorber is built on a curved
surface, its absorption efficiency drops drastically. Accordingly, the
flexibility and incident angle insensitivity are two main factors for
designing MMAs for curved surfaces that have been followed by some works in
recent years[8].

The necessity of polarimetric sensors for curved surfaces has led us to design
a first conformal MMA for detection of the polarization states of THz
wavefronts. Our proposed phase/polarization encoded anisotropic digital
metasurface can exhibit the perfect absorptivity in wide incident angles based
on tuning the proper electrical resistivity of vanadium dioxide (VO2).

VO2 is a well-known tunable material that undergoes an ultrafast reversible
phase transition from insulator monoclinic to metallic tetragonal phase above
the critical temperature around ${{\rm{T}}_{\rm{c}}}{\rm{=340K}}$. This metal-
insulator transition (MIT) can be provoked by thermal, optical, or electrical
activation which can occur within an order of several nanoseconds or even in
picoseconds range for optical stimuli[10]. Due to the dramatic changes of
electrical and optical properties of VO2 across the two phases, it has been
identified as a notable substance in tunable metamaterial devices over a broad
spectral range and has myriad applications in the THz regions, such as THz
waves modulator, angle-insensitive absorber, and programmable ultrafast meta-
devices[11].

In this paper, a THz polarimetry device is proposed as depicted in Fig. 1,
which comprises two perpendicular VO2 microwires whose reflection
characteristics can be dynamically tuned in a real-time manner independently
for two orthogonal x- and y-polarized excitations. The proposed VO2-
integrated anisotropic metasurface (VIAM) can retain the perfect absorptivity
in wide oblique incidences for THz waves that make them suitable for a curved
surface. By digitally changing the spatial voltage distribution provided by
two independent multichannel DC bias voltages, and observing the reflection
properties of scattered beams via two receiving antennas, the state of
incident polarization can be determined. To demonstrate the main mechanism of
VIAM, the relationship between spatial coding sequences and the polarization
of the incident beam is investigated. Besides, by analyzing the induced
electric field and the power loss density of the proposed VIAM, we show that
increasing the electrical conductivity of VO2 plays an effective role in
dissipating the incoming EM energy especially when the oblique incident angle
increases. To the best knowledge of the authors, our proposed VIAM is the
first conformal polarimetric device. We believe that the integrated structural
design, ultrafast switching time, and wide coverage of incident-angle
characteristics pave the way for many scientific and engineering applications,
such as linear position sensing and future practical THz devices.

Fig. 1b shows the sketch representation of the proposed meta-particle composed
of four layers of VO2 microwires, polyamide, gold plate, and polyimide from
top to bottom respectively. Two pieces of separated patches are located
beneath the bottom substrate as positive electrodes which are independently
connected to two DC bias voltage networks as shown in Fig. 1c. The gold plate
acts as a negative electrode which is connected to two lateral metalized
electrodes for each VO2 microwires through two metallic via holes. Through
this mechanism, all the biasing wires and networks can be located beneath the
structure without interfering with the EM properties of the metasurface. The
geometrical dimensions are P=50$\mu m$,H=17$\mu m$ and T=40$\mu m$
respectively. The thickness and the width of the VO2 microwires are 1$\mu m$
and 4$\mu m$.

Typical VO2 thin films show electrical conductivity in the range of $10\sim
100\,\,{\rm{S/m}}$ with the relative permittivity of about 9 and after
occurring the structural transformation, its electrical conductivity reaches
as high as an order of ${10^{5}}$ in the metallic state[12]. We have performed
full-wave simulations by the commercial CST software. For the evaluation of
reflection properties of the infinite array of VIAM, the periodic boundary
conditions are utilized along the x and y directions and the open boundary
condition is employed along the z-axis. Meanwhile, linearly and circularly
polarized plane waves at different incidence angles illuminate the VIAM
structure.

Fig. 2:  Simulated results of reflectivity of the designed VIAM structure
under the excitation of a) TM polarized THz incident waves. b) TE polarized
THz incident waves. c,d) Circular polarized THz incident waves for receiving
antenna on X and Y directions respectively. e,f) $+45^{\circ}$ linear
polarized THz incident waves for receiving antenna on X and Y directions
respectively.

Since the thickness of a ground plane (gold layer) is much larger than the
penetration depth of the incident THz wavefronts ($T(\omega)=0$), the
absorptivity of the VIAM can be calculated by Eq. 1, where $A(\omega)$ and
$\Gamma(\omega)$ are the absorption and reflectance, respectively[13].

$A(\omega)=1-\Gamma(\omega)$ (1)

When VO2 is in the dielectric steady-state phase and having the lowest
electrical conductivity, the reflection characteristics of the VIAM is in its
highest value. On the other hand, at the intermediate temperatures, the
transmitted waves are dissipated due to the significant VO2 ohmic losses, and
the maximum absorption efficiency is achieved. Therefore, we set
${\sigma_{off}}=10S/m$ ( ”0” digital state) and
${\sigma_{on}}=5\times{10^{4}}S/m$ ( ”1” digital state). The conductivity of
VO2 microwires along the x and y-direction are marked by ${\sigma_{x}}$ and
${\sigma_{y}}$, and by independently changing the DC bias voltage, four
digital states of 0/0, 0/1, 1/0, and 1/1 correspond to
${\sigma_{off}}/{\sigma_{off}}$, ${\sigma_{off}}/{\sigma_{on}}$,
${\sigma_{on}}/{\sigma_{off}}$, ${\sigma_{on}}/{\sigma_{on}}$ will be attained
where the binary codes before and after the slash symbol ( / ) indicates the
digital codes for the VO2 microwires along the x and y-directions,
respectively.

Table 1: Reflectivity for four digital states at the frequency of 2.4 THz and
the corresponding distinct polarization states.

Under the normal incidence, numerical simulations have been conducted for
different polarized incident waves, and the reflection spectra of the proposed
VIAM for four different digital states are illustrated in Fig. 2. The
absorptivity of VIAM is novelty employed to characterize the four independent
polarization states (TE, TM, $\pm{45^{\circ}}$ linear, and RCP/LCP).
Consequently, the proposed structure demonstrates a wideband polarization-
dependence of absorption for distinct independent polarization, thus, the high
contrast can be realized for sensing applications. Consider the TM polarized
waves as an example (depicted in the first row of Table. 1), since the VIAM
has anisotropic structural nature, just those VO2 films which are parallel to
x-direction are excited and the reflection amplitude response for cross-
polarization is zero. Therefore, the receiving antenna towards the y-direction
receives nothing for all the four spatial code distributions. Observe in
Table. 1 that the polarization cross-talk between two orthogonal directions is
negligible, and changing the digital state of VO2 microwire in one orthogonal
direction does not affect the electromagnetic response in the other direction
for the LP incident THz wave.

For another example, consider RCP THz incident waves illuminating the
structure. For a digital state of 0/0, all the EM energy reflected by RCP, and
according to the relationship between CP and LP polarization, we observe that
half of the EM radiation reflected towards x-direction, and the remaining EM
energy reflected towards y-direction. For the digital state of 1/1, all the
incident THz wave energy is effectively absorbed by the proposed VIAM due to
the ohmic losses of VO2 films. For the digital states of 0/1, a quarter of the
incident wavefronts is reflected by RCP, and the other quarter of the wave
energy is reflected by LCP, and these two wavefronts have the same phase
difference. Therefore, half of the incident THz wavefront is reflected towards
x-direction and the remaining wave energy is absorbed by the structure. In the
same way, for the 1/0 digital state, half of the EM energy absorb by the
structure and the remaining THz wavefront is reflected toward y-direction. As
a final comment, all the conditions presented in Table. 1 are distinct, and by
observing the reflected energies through x- and y-polarization, one can
readily determine the state of the incident THz wavefronts polarization. It
should be noted that due to the nature of the structure, our designed VIAM can
not differentiate between LCP and RCP as well as $+45^{\circ}$ LP and
$-45^{\circ}$ LP.

Fig 3: The simulated result of absorptivity for different cases of a) (TE, TM,
$\pm{45^{\circ}}$ linear)-polarized oblique incident THz wavefronts. b) RCP
and LCP oblique incident THz wavefronts, when the electrical conductivity of
VO2 microwires are 50000 S/m.

When an MMA is fabricated on a curved surface, only those meta-atoms at the
center of the absorber are illuminated at normal incidence, while, the other
meta-particles on the curved surface are illuminated at oblique incidence.
However, by increasing the incident angle, the degradation of absorptivity is
inevitable[14]. For instance, under TE and TM oblique incidence, the
reflection coefficients can be obtained by[15]:

${\Gamma_{{\rm{TE}}}}(\omega)=\frac{{Z(\omega)\cos{\theta_{i}}-{Z_{0}}\cos{\theta_{t}}}}{{Z(\omega)\cos{\theta_{i}}+{Z_{0}}\cos{\theta_{t}}}}$
(2)
${\Gamma_{{\rm{TM}}}}(\omega)=\frac{{Z(\omega)\cos{\theta_{t}}-{Z_{0}}\cos{\theta_{i}}}}{{Z(\omega)\cos{\theta_{t}}+{Z_{0}}\cos{\theta_{i}}}}$
(3)

where ${Z(\omega)}$ and ${{Z_{0}}}$ indicate the impedances of the VIAM and
free space, and ${{\theta_{i}}}$ and ${{\theta_{t}}}$ are the incident and
transmission angles, respectively. We will show that the near-unity absorption
up to $50^{\circ}$ incident angles which is crucial for polarization detection
applications can be achieved with our elaborately designed VIAM structure. The
absorption spectra of the VIAM under different incident angles up to 50
degrees for (TE, TM, $\pm{45^{\circ}}$ linear)-polarized THz wavefronts is
depicted in Fig. 3a. All the numerical simulations are conducted for the
electrical conductivity of VO2 microwires equal to
${\sigma_{on}}=5\times{10^{4}}$. To better understand the main mechanism of
absorption, notice that by increasing the incident angle, according to Eq. 4,
the power loss density decreases for the electrical conductivity of VO2 in the
off-state.

${P_{loss}}=\int{\sigma{{\left|E\right|}^{2}}dv}$ (4)

To compensate for this decline, the only way is to increase the electrical
conductivity of VO2 microwires to reach the value of on-state by changing the
DC bias voltage. Therefore, the power loss is mostly due to the ohmic losses
of VO2 microwires, and changing the electrical conductivity of VO2 plays a
beneficial role in dissipating the incoming EM energy. By increasing the
incident angle of THz incident wavefronts, the absorbing bandwidth becomes
narrow, and our designed VIAM can retain its absorptivity up to $50^{\circ}$
in the band of 1.6-3 THz. Accordingly, our proposed structure shows a wideband
absorptivity above 90% in 1.6–3 THz for oblique incidences up to $50^{\circ}$
for different cases of incident polarization (LP, $\pm{45^{\circ}}$ linear,
and CP (See Fig. 3b)).

In summary, we have introduced the first conformal metamaterial absorber for
polarimetric detection that can characterize the four independent polarization
states of (TE, TM, $\pm{45^{\circ}}$ linear, and RCP/LCP). Our proposed VIAM
structure displays a strong absorptivity above 90% up to the incidence angle
of $50^{\circ}$. By encoding the VIAM in four distinct digital states through
changing the independent DC bias voltages for two orthogonal VO2 microwires,
our designed structure is successfully characterized the polarization of THz
incident wavefronts with an ultrafast switching time between distinct digital
states. We believe that the presented conformal metasurface-based polarimeters
may find great potentials for solving some crucial THz challenges, such as
linear position sensing, security scanning, and medical imaging.

## References

  * [1] Yardimci, Nezih Tolga, and Mona Jarrahi. ”High sensitivity terahertz detection through large-area plasmonic nano-antenna arrays.” Scientific reports 7 (2017): 42667.
  * [2] Shabanpour, Javad. ”Programmable anisotropic digital metasurface for independent manipulation of dual-polarized THz waves based on a voltage-controlled phase transition of VO 2 microwires.” Journal of Materials Chemistry C 8.21 (2020): 7189-7199.
  * [3] Shabanpour, Javad. ”Full Manipulation of the Power Intensity Pattern in a Large Space‐Time Digital Metasurface: From Arbitrary Multibeam Generation to Harmonic Beam Steering Scheme.” Annalen der Physik 532.10 (2020): 2000321.
  * [4] Wang, Bingnan, et al. ”Metamaterial absorber for THz polarimetric sensing.” Terahertz, RF, Millimeter, and Submillimeter-Wave Technology and Applications XI. Vol. 10531. International Society for Optics and Photonics, 2018.
  * [5] Sadamoto, Kota, et al. ”Terahertz polarimetric sensing for linear encoder.” 2018 43rd International Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz). IEEE, 2018.
  * [6] Landy, N. I., Sajuyigbe, S., Mock, J. J., Smith, D. R., & Padilla, W. J. (2008). Perfect metamaterial absorber. Physical review letters, 100(20), 207402.
  * [7] Al-badri, Khalid Saeed Lateef. ”Multi Band Metamaterials Absorber for Stealth Applications.” Law, State and Telecommunications Review 11.1 (2019): 133-144.
  * [8] Shabanpour, Javad, Sina Beyraghi, and Homayoon Oraizi. ”Reconfigurable honeycomb metamaterial absorber having incident angular stability.” Scientific Reports 10.1 (2020): 1-8.
  * [9] Rajabalipanah, Hamid, et al. ”Asymmetric spatial power dividers using phase–amplitude metasurfaces driven by huygens principle.” ACS omega 4.10 (2019): 14340-14352.
  * [10] Shabanpour, Javad, Sina Beyraghi, and Ahmad Cheldavi. ”Ultrafast reprogrammable multifunctional vanadium-dioxide-assisted metasurface for dynamic THz wavefront engineering.” Scientific Reports 10.1 (2020): 1-14.
  * [11] Hashemi, Mohammed Reza M., et al. ”Electronically-controlled beam-steering through vanadium dioxide metasurfaces.” Scientific reports 6 (2016): 35439.
  * [12] Wang, Dacheng, et al. ”Switchable ultrathin quarter-wave plate in terahertz using active phase-change metasurface.” Scientific reports 5.1 (2015): 1-9.
  * [13] Lee, Dongju, et al. ”Incident angle-and polarization-insensitive metamaterial absorber using circular sectors.” Scientific reports 6 (2016): 27155.
  * [14] Jang, Youngsoo, Minyeong Yoo, and Sungjoon Lim. ”Conformal metamaterial absorber for curved surface.” Optics express 21.20 (2013): 24163-24170.
  * [15] Wu, Chihhui, et al. ”Metamaterial-based integrated plasmonic absorber/emitter for solar thermo-photovoltaic systems.” Journal of Optics 14.2 (2012): 024005.
  * [16] Gholamian, Meysam, Javad Shabanpour, and Ahmad Cheldavi. ”Highly sensitive quarter-mode spoof localized plasmonic resonator for dual-detection RF microfluidic chemical sensor.” Journal of Physics D: Applied Physics 53.14 (2020): 145401.
  * [17] Shabanpour, Javad, and Homayoon Oraizi. ”Some useful approximations for calculation of directivities of multibeam power patterns of large planar arrays.” arXiv preprint arXiv:2006.10423 (2020).