# Extremely thin perfect absorber by generalized multipole bianisotropic
effect

Hao Ma  Research Institute of Superconductor Electronics (RISE) & Key
Laboratory of Optoelectronic Devices and Systems with Extreme Performances of
MOE, School of Electronic Science and Engineering, Nanjing University, Nanjing
210023, China  Purple Mountain Laboratories, Nanjing 211111, China  Andrey B.
Evlyukhin  Institute of Quantum Optics, Leibniz Universität Hannover,
Welfengarten 1, 30167 Hannover, Germany  Andrey E. Miroshnichenko  School of
Engineering and Information Technology, University of New South Wales at
Canberra, Northcott Drive, Canberra ACT 2610, Australia  Fengjie Zhu  Research
Institute of Superconductor Electronics (RISE) & Key Laboratory of
Optoelectronic Devices and Systems with Extreme Performances of MOE, School of
Electronic Science and Engineering, Nanjing University, Nanjing 210023, China
Siyu Duan  Research Institute of Superconductor Electronics (RISE) & Key
Laboratory of Optoelectronic Devices and Systems with Extreme Performances of
MOE, School of Electronic Science and Engineering, Nanjing University, Nanjing
210023, China  Jingbo Wu  Research Institute of Superconductor Electronics
(RISE) & Key Laboratory of Optoelectronic Devices and Systems with Extreme
Performances of MOE, School of Electronic Science and Engineering, Nanjing
University, Nanjing 210023, China  Purple Mountain Laboratories, Nanjing
211111, China  Caihong Zhang  Research Institute of Superconductor Electronics
(RISE) & Key Laboratory of Optoelectronic Devices and Systems with Extreme
Performances of MOE, School of Electronic Science and Engineering, Nanjing
University, Nanjing 210023, China  Purple Mountain Laboratories, Nanjing
211111, China  Jian Chen  Research Institute of Superconductor Electronics
(RISE) & Key Laboratory of Optoelectronic Devices and Systems with Extreme
Performances of MOE, School of Electronic Science and Engineering, Nanjing
University, Nanjing 210023, China  Purple Mountain Laboratories, Nanjing
211111, China  Biao-Bing Jin<EMAIL_ADDRESS>Research Institute of
Superconductor Electronics (RISE) & Key Laboratory of Optoelectronic Devices
and Systems with Extreme Performances of MOE, School of Electronic Science and
Engineering, Nanjing University, Nanjing 210023, China  Purple Mountain
Laboratories, Nanjing 211111, China  Willie J. Padilla
<EMAIL_ADDRESS>Duke University, Department of Electrical and Computer
Engineering, Durham, NC 2770 USA  Kebin Fan<EMAIL_ADDRESS>Research
Institute of Superconductor Electronics (RISE) & Key Laboratory of
Optoelectronic Devices and Systems with Extreme Performances of MOE, School of
Electronic Science and Engineering, Nanjing University, Nanjing 210023, China
Purple Mountain Laboratories, Nanjing 211111, China

###### Abstract

Symmetry breaking plays a crucial role in understanding the fundamental
physics underlying numerous physical phenomena, including the electromagnetic
response in resonators, giving rise to intriguing effects such as directional
light scattering, supercavity lasing, and topologically protected states. In
this work, we demonstrate that adding a small fraction of lossy metal (as low
as $1\times 10^{-6}$ in volume), to a lossless dielectric resonator breaks
inversion symmetry thereby lifting its degeneracy, leading to a strong
bianisotropic response. In the case of the metasurface composed of such
resonators, this effect leads to unidirectional perfect absorption while
maintaining nearly perfect reflection from the opposite direction. We have
developed more general Onsager-Casimir relations for the polarizabilities of
particle arrays, taking into account the contributions of quadrupoles, which
shows that bianisotropy is not solely due to dipoles, but also involves high-
order multipoles. Our experimental validation demonstrates an extremely thin
terahertz-perfect absorber with a wavelength-to-thickness ratio of up to
25,000, where the material thickness is only 2% of the theoretical minimum
thickness dictated by the fundamental limit. Our findings have significant
implications for a variety of applications, including energy harvesting,
thermal management, single-photon detection, and low-power directional
emission.

##  I Introduction

Metamaterials have brought about a paradigm shift in the approach to the
manipulation of electromagnetic waves by allowing for interaction not only
with the electric component of light but also with the magnetic component,
which was previously challenging at high frequencies. The subwavelength nature
of metamaterial unit-cells, i.e., meta-atoms, allows their scattering to be
well described by the response of electric dipoles (EDs) and magnetic dipoles
(MDs). The geometry of the meta-atoms enables the independent control of the
ED ($\bm{p}$) and MD ($\bm{m}$) moments, showcasing its versatility in shaping
the electromagnetic responses, including transparency [1, 2, 3, 4], perfect
absorption[5, 6, 7], directional scattering, [8, 9], anapole effects [10] and
exceptional points [11].

However, the constitutive equations indicate that the electric polarization
and magnetization may not be inherently independent of each other. A possible
cross-coupling between polarizations is termed bianisotropic polarizability
(BAP) and enables coupling of electric to magnetic polarization
($\alpha_{em}$) and magnetic to electric polarization ($\alpha_{me}$), thereby
providing an additional degree of freedom for manipulating electromagnetic
waves. The bianisotropic effect can be induced by breaking the symmetry of
metasurfaces (MSs), therefore leading to fascinating phenomena, such as
asymmetric absorption, asymmetric reflection, enhanced harmonics generation,
asymmetric parametric generation, and trapped modes with high-Q resonance [12,
13, 14, 15, 16, 17, 18, 19, 20, 21]. Further, the reciprocity theorem requires
$\overline{\overline{\alpha}}_{me}=-\overline{\overline{\alpha}}_{em}^{T}$,
which is called the Onsager-Casimir condition, where $T$ denotes the transpose
operation. However, a significant geometric change is often required to
achieve a large asymmetric response [12, 21, 22, 23, 24, 25, 26]. For
instance, to obtain an asymmetric absorption, in one study the volume
difference between the top and bottom patterns had to exceed 40% [12, 24]. An
added complexity is that for structures which lack inversion symmetry (IS),
fabrication can be challenging.

Although the behavior of MSs can often be well described by dipoles, meta-
atoms may not always be deeply subwavelength in the short wavelength range.
Therefore, a full description of the resonator scattering needs to go beyond
the simple dipole approximation, i.e., high-order multipoles should be
included. Controlling the interaction of dipoles and high-order multipoles has
enabled a new route to shape electromagnetic waves, such as the generalized
Kerker effect, hybrid anapoles for transparency, and bound state in the
continuum [7, 27, 28, 29, 30, 31, 32]. Further, the coupling among high-order
multipoles could provide more physical effects such as trapped multipole modes
[33], exceptional points [34], and the lattice anapole effect [35].

In this letter, we present an extremely thin asymmetric perfect absorber which
exhibits an extreme bianisotropic response far exceeding simple dipole
coupling. By breaking the IS with an extremely subwavelength lossy disk on top
of a lossless cylinder as illustrated in Figure 1a, the MS also exhibits
nearly perfect reflection under backward illumination. The developed theory
reveals that the bianisotropy is not only caused by dipole cross-coupling, but
also enhanced by quadrupoles. We experimentally demonstrate an asymmetric
terahertz perfect absorber by patterning lossy Niobium nitride (NbN) disks on
dielectric cylinders as shown in the inset of Figure 1b. The patterned disk
absorbs 99.999% of the power, and achieves a remarkable wavelength-to-
thickness ratio of 25,000, thereby surpassing the theoretically established
limits of absorber thickness. Further, the volume filling fraction of the
demonstrated absorber is two orders of magnitude smaller than other absorbing
thin films in literature, as shown in Figure 1b. This breakthrough not only
illuminates new pathways for exploring light-matter interactions but also
promises a new generation of high-performance metasurface devices.

##  II Results & Discussion

To obtain asymmetric perfect absorption from an initially symmetric MS,
different reflection under forward and backward incidence has to be achieved.
Here, we break IS through the introduction of a lossy conductive disk on top
of a cylindrical resonator, as depicted in Figure 1a. In Figure1c and e, we
compare the simulated scattering response of a metasurface with and without
IS. For MSs consisting of lossless silicon cylinders with IS about its center
point, the reflections with illumination from the top ($-k$) direction (dashed
green curve) and bottom ($+k$) direction (solid red curve) are identical, as
shown in Figure1(b). The transmission remains the same due to reciprocity. In
this case, two fundamental modes, namely the EH111 and HE111 modes [5], are
excited with their electric field distributions shown in Figure 1d. The
symmetric and anti-symmetric profiles of the modes indicate that there is no
cross-coupling between the scattering responses with even and odd parity.
However, for the broken symmetry case where a 4.4-nm thick lossy disk with a
radius of 28.5 $\mu$m is added to one side (Figure 1e), the forward reflection
$r_{+k}$ is significantly suppressed, while the backward reflection $r_{-k}$
remains close to 1, as shown in Figure1e. Notably, despite the conductive
disk’s thickness being a mere 8.8$\times 10^{-6}$ fraction of the operational
wavelength of 500 $\mu$m, its influence on the bianisotropic response is
profound. Here the disk is modeled as gold with a conductivity of
$\sigma=2\times 10^{6}$ S/m, which is over one order of magnitude lower than
its bulk value, given its few-nanometer thickness [36]. Because of the
conductive layer, the electric field at the top surface of the cylinder is
partially screened, leading to a significantly damped resonance at 0.61 THz,
as shown in Figure 1f. In this case, the field of the first mode near 0.6 THz
changes from an ellipse-like shape to a cone-like shape, and the zero electric
field plane of the second mode near 0.61 THz shifts lower, as shown in Figure
1f. In comparison to the symmetric case, the asymmetric field distribution
indicates that the scattering response consists of mixed even and odd parity
states.

The geometry shown in Figure 1e permits the asymmetric metasurface to achieve
perfect absorption ($A_{+k}$) under $+k$ illumination and nearly zero
absorption under $-k$ incidence as shown in Figure 1g; a result attributable
to the symmetry breaking introduced by the patterned conductive disk. The
absorption is strongly influenced by both the structural geometry and the
disk’s conductivity – details can be found in Section 4 of the Supporting
Information. In Figure 2a, the reflection coefficients ($r_{+k}$, $r_{-k}$)
are illustrated, along with their dependence on disk radius at 0.6 THz. To
assess the variation between the reflection from both sides, we establish an
asymmetric factor, denoted as $\beta=(|r_{-k}|-|r_{+k}|)/(|r_{+k}|+|r_{-k}|)$.
With a smaller disk radius, both $r_{+k}$ and $r_{-k}$ approach unity, thereby
resulting in a nearly null asymmetric factor. However, for $r_{d}$ exceeding
10 $\mu$m, the forward reflection $r_{+k}$ drops precipitously to zero while
the backward reflection $r_{-k}$ remains over 0.94. Here, the asymmetric
factor ascends to its maximum of 1 at $r_{d}$= 28.5 $\mu$m.

It is crucial to underscore that achieving a nearly perfect asymmetric factor
with such a minimal volume filling fraction (VFF) – here a value of 1.8$\times
10^{-7}$ – is unprecedented. In this context, VFF is expressed as
$\rm{VFF}=t_{d}S_{d}/\lambda^{3}$, where $t_{d}$ denotes the thickness of the
absorbing film, and $S_{d}$ indicates the film’s surface area within one
wavelength area ($\lambda^{2}$). Given the negligible change in transmission
at 0.6 THz, the structure exhibits perfect absorption under $+k$ directional
illumination and nearly null absorption ($A_{-k}$) under the $-k$ illumination
at maximum $\beta$, as evidenced in Figure 1g. This asymmetric perfect
absorption ties directly into the asymmetric field distribution within the
structure. Under the $+k$ illumination, the excited electric and magnetic
field at 0.6 THz predominantly localizes near and within the disk (see Figure
2b). As the disk is the sole lossy component in the structure, the incident
power will largely be dissipated within it as
$P_{abs}=\frac{1}{2}\int\sigma|E|^{2}dV$, leading to significant energy
confinement within the thin film.

Given that intrinsic silicon has an electron concentration of 1011 /cm3, we
confirm that the conductive disk absorbs an astounding 99.999% of the total
incident power, as highlighted in Figure2d. Remarkably, the disk represents
only 0.44% of the theoretical thickness limit for absorbers (refer to section
5 in Supporting Information). When subjected to $-k$ illumination, the
electric field—represented in Figure 2d—demonstrates a slight increase in
intensity but remains mostly concentrated within the cylinder’s low-loss area.
As a result, the backward absorption $A_{-k}$ is approximately $\sim 2$%.
Despite the relatively constant magnetic field distribution within the
cylinder, its intensity decreases by a factor of ten relative to the $+k$
incidence. Consequently, the exceptionally thin, asymmetric bianisotropic
metasurface achieves perfect absorption. Combined with an extraordinary
wavelength-to-thickness ratio of 1.13$\times 10^{5}$, this underscores the
impressive effectiveness of the proposed structure in realizing perfect
absorption with an exceptionally thin profile.

Clearly, asymmetric reflection and absorption responses must arise from
distinct total scattering cross-section (SCS) and total absorption cross-
section (ACS) spectra under opposite illuminations. As shown in the top panel
of Figure 2e, under $+k$ incidence the cylinder exhibits strong and equal SCS
(red curve) and ACS (green curve) values. At the frequency of 0.602 THz, the
two cross-section curves intersect at a value of 0.123 mm2 – approximately the
unit cell area. This intersection signifies a critical coupling point,
enabling optimal absorption [37]. The far-field scattering pattern, as shown
in the inset of the top panel of Figure 2e, exhibits a significant forward
scattering at the frequency indicated by the dashed line, but with zero
backward scattering due to the null reflection (Figure 2f). However, when the
metasurface is illuminated in the $-k$ direction at 0.602 THz, the absorption
cross-section is nearly zero (dashed green curve) while the SCS (dashed red
curve) is large and near a maximum value, as shown in the lower panel of
Figure 2e. The far-field radiation pattern under the $-k$ illumination shows
similar forward and backward scattering values, as shown in Figure 2g, which
is drastically different from that under forward incidence. Such a radiation
pattern is very close to that without the conductive disk. Also noteworthy is
the nearly identical forward scattering in the normal direction for both
illumination directions, attributable to the structure’s reciprocity.

In Figure 1e, we have shown that the asymmetric reflection arises from the
mixing of scattering response with even and odd parity. To further elucidate
the bianisotropy-induced asymmetric perfect absorption [12, 38, 39], we
developed a bianisotropic model, which includes the multipole contributions
using multipole decomposition and $\alpha$-Tensor methods [40, 41, 42, 43].
The multipole moments are calculated using an expansion in spherical
coordinates with plane waves incident in the $+k$ and $-k$ directions,
respectively. It is widely recognized that the parity of multipoles,
determined by spherical harmonics, can be classified as even or odd.
Importantly, the parity is closely linked to the Onsager-Casimir relations
[44, 45, 46, 47]. To show the cross-coupling between the even and odd
multipoles, with an assumption of an $x$-polarized incident field Ex, we
define two effective multipole moments based on the formulas for the normally
scattered field of an arbitrarily shaped particle as [48, 35]

$\displaystyle M_{o}^{\pm}$
$\displaystyle=p_{x}^{\pm}+(k_{0}/2ic)Q_{yz}^{m,\pm}$ (1) $\displaystyle
M_{e}^{\pm}$ $\displaystyle=m_{y}^{\pm}+(k_{0}c/6i)Q_{xz}^{e,\pm}$ (2)

where $M_{o}^{\pm}(M_{e}^{\pm})$ represents the effective moments contributed
from multipoles with odd (even) parity, the superscripts + (-) indicate the
incident wave propagating along $+k$ ($-k$) direction, respectively; $p_{x}$,
$Q_{zx}^{e}$ are the induced ED and magnetic quadrupole with odd parity,
respectively; $m_{y}$, and $Q_{zy}^{m}$ are the induced MD and electric
quadrupole (MQ) with even parity, respectively [49]. Then the direction-
dependent effective polarizabilities can be defined as
$\alpha_{o}^{\pm}=M_{o}^{\pm}/\varepsilon_{0}E_{x}$ and
$\alpha_{e}^{\pm}=\pm\eta_{0}M_{e}^{\pm}/E_{x}$, where $E_{x}$ is the incident
electric field at the center of the mass of the cylinder, $\varepsilon_{0}$ is
the vacuum permittivity. In the absence of a conductive disk, the cylinder
exhibits two identical opposite reflections at 0.6 THz, with an amplitude of
unity and a phase of -$\pi/2$ with respect to the incidence. The mid-plane of
z=0 is set as the phase reference plane. By using the transmission and
reflection equations based on multipole moments (Section 1 in Supporting
Information), we can derive the same effective directional polarizabilities,
as shown in the second row of Table 1. However, for the cylinder without the
IS, the two effective moments are mixed together [50]. Then, under the
condition of forward perfect absorption, the effective directional
polarizabilities are the same with
$\alpha_{o}+=\alpha_{e}^{+}=\frac{1}{k_{0}\rho}$, where $k_{0}$ is the
wavenumber of the resonant frequency, and $\rho=1/S_{\rm L}$ is the inverse
area of the unit cell. For the condition of near-unity backward reflection, we
can obtain
$\alpha^{-}_{o}=-(\alpha^{-}_{e})^{*}=\frac{-1}{k_{0}\rho}+\frac{i}{k_{0}\rho}$
as shown in the third row of the Table.1. These conditions achieve excellent
agreement with simulations, as shown in Figure 3a and b, indicated by the
intersection of the dashed grey lines. We note that these conditions are also
the same as those required to achieve perfect absorption and perfect
reflection in the dipole limit [7].

The scattering behavior of subwavelength elements is primarily governed by
dipoles. However, in our structure, we find that dipole contributions alone
are insufficient to account for the observed phenomenon of perfect absorption
and nearly perfect reflection at resonance. Rather, when the effect of
multipoles is incorporated we achieve excellent agreement between retrieved
values and simulation data, as displayed in Figure 3c (See Section 3 in
Supporting Information). Further, bianisotropic perfect absorption calculated
using only dipoles violates the Onsager-Casimir conditions (see Section 3 in
Supporting Information). To fully describe the bianisotropic effect, the
contribution of high-order multipoles cannot be easily ignored [26, 50].
Incorporating $\bm{\alpha}$-tensor method [51, 52, 43], we can rewrite the
constitutive equations using the effective multipole moments as

$\begin{split}\frac{1}{\varepsilon_{0}}M_{o}=\alpha^{oo}E_{x}+\alpha^{oe}(\eta_{0}H_{y})\\\
\eta_{0}M_{e}=\alpha^{eo}E_{x}+\alpha^{ee}(\eta_{0}H_{y})\end{split}$ (3)

where $E_{x}$ and $H_{y}$ are the incident field at the position of the
multipoles, and the equivalent polarizabilities under the x-polarized
incidence are given as in Section 1 in Supporting Information. Based on
Lorentz reciprocity [53] we can prove that $\alpha^{eo}=-\alpha^{oe}$, which
obeys the Onsager-Casimir conditions.

The bianisotropic response of the asymmetric metasurface (MS) evidently does
not solely arise due to dipole cross-coupling, and indeed the dipole to
quadrupole coupling also significantly contributes to this response. If the
dipoles are dominant, Equation 3 can be relaxed to the standard constitutive
equations with typical Onsager-Casimir conditions. Using Eqs. (13)-(16) of
Supporting Information and the Onsager-Casimir condition
($\alpha^{eo}=-\alpha^{oe}$), we obtain generalized expressions:

$\displaystyle t^{\pm}$
$\displaystyle=1+\frac{ik_{0}\rho}{2}\Big{(}\alpha^{oo}+\alpha^{ee}\Big{)}$
(4) $\displaystyle r^{\pm}$
$\displaystyle=\frac{ik_{0}\rho}{2}\Big{(}\alpha^{oo}\pm
2\alpha^{oe}-\alpha^{ee}\Big{)}\>.$ (5)

These two expressions are very similar to those described using only dipoles
[24], however, they include the scalar polarizabilities of the effective
multipole moments (3) with odd and even parity, providing contributions to the
transmission coefficient $t$ and reflection $r$ coefficient (Eqs. (13) and
(14) of Supporting Information). Figure 3d shows the calculated equivalent
polarizabilities as a function of the disk conductivity at 0.6 THz. For small
values of the conductivity we find that the IS of the structure is still
approximately held. The nearly perfect reflection and zero transmission enable
$\rm{Re}\\{\alpha^{oo}\\}=-\rm{Re}\\{\alpha^{ee}\\}\approx-1/k_{0}\rho$ and
$\rm{Im}\\{\alpha^{oo}\\}\approx\rm{Im}\\{\alpha^{ee}\\}\approx 1/k_{0}\rho$,
which are consistent with those listed in the second row of Table.1. The
equivalent BAPs $\alpha^{oe}$ and $\alpha^{eo}$ are approximately zeros, which
are expected. However, as the disk conductivity is larger than $10^{5}$ S/m,
the IS is significantly broken. As a result, the bianisotropic terms become
significant. Additionally, we find that the conditions
$\rm{Re}\\{\alpha^{oo}\\}=-\rm{Re}\\{\alpha^{ee}\\}$ and
$\rm{Im}\\{\alpha^{oo}\\}+\rm{Im}\\{\alpha^{ee}\\}=\frac{2}{k_{0}\rho}$, which
arise from the reciprocity and near-zero transmission at the resonance, are
always satisfied. As the conductivity reaches 2$\times 10^{6}$ S/m, i.e.,
forward perfect absorption state,
$\rm{Re}\\{\alpha^{oo}\\}=-\rm{Re}\\{\alpha^{ee}\\}\approx\frac{-1}{2k_{0}\rho}$,
$\rm{Im}\\{\alpha^{oo}\\}=\rm{Im}\\{\alpha^{ee}\\}\approx\frac{1}{k_{0}\rho}$,
and $\alpha^{oe}=-\alpha^{eo}\approx\frac{1}{2k_{0}\rho}$, which are also
consistent with the theoretical prediction as shown in Table.1. We note that
the imaginary part of the equivalent BAPs should be zero if the backward
incidence is perfectly reflected, as listed in Table 1. The small deviation of
the retrieved imaginary parts of the equivalent BAPs from zeros (Figure 3d),
is caused by the imperfect backward reflection. Nonetheless, the equivalent
BAPs always satisfy the Onsager-Casimir relations.

Thus, from the general Eqs. (4), (5), and the above discussion, we arrive at
the following scenario. The bianisotropic portions of the polarizability
($\alpha^{eo}=-\alpha^{oe}$) do not contribute to the transmission
coefficient, so transmission suppression does not depend on the direction of
incidence. In this case, the suppression of transmission can only be
associated with the excitation of resonant modes in particles and an increase
of electromagnetic fields in them. On the contrary, the reflection coefficient
explicitly depends on the bianisotropic polarizabilities. For the negative
direction of incidence, this contribution leads to constructive interference
in the reflection and its enhancement, and for the positive direction of
incidence, to destructive interference in the reflection and its significant
suppression. In the latter case, the reflection is suppressed along with the
transmission, which leads to the absorption of the accumulated electromagnetic
energy in the particles.

Disk radius $r_{d}$ |  $\alpha^{\pm}_{o},\alpha^{\pm}_{e}$ |  $\alpha^{oo},\alpha^{ee}$ |  $\alpha^{oe},\alpha^{eo}$
---|---|---|---
$r_{d}$=0, $r_{+k}=r_{-k}=-i\ \ \&\ \ t=0$ |  $-\textrm{Re}\\{\alpha^{\pm}_{o}\\}=\textrm{Re}\\{\alpha^{\pm}_{e}\\}=\frac{1}{k_{0}\rho}$ $\textrm{Im}\\{\alpha^{\pm}_{o}\\}=\textrm{Im}\\{\alpha^{\pm}_{e}\\}=\frac{1}{k_{0}\rho}$ |  $\textrm{Re}\\{\alpha^{oo}\\}=-\textrm{Re}\\{\alpha^{ee}\\}=\frac{-1}{k_{0}\rho}$ $\textrm{Im}\\{\alpha^{oo}\\}=\textrm{Im}\\{\alpha^{ee}\\}=\frac{1}{k_{0}\rho}$ |  $\alpha^{oe}=\alpha^{eo}=0$
$r_{d}$=$r_{opt}$, ($r_{+k}=t=0\ \ \&\ \ r_{-k}\approx-i$) |  $\alpha^{+}_{o}=\alpha^{+}_{e}=\frac{i}{k_{0}\rho}$
$\alpha^{-}_{o}=-(\alpha^{-}_{e})^{*}=\frac{-1}{k_{0}\rho}+\frac{i}{k_{0}\rho}$ |  $\alpha^{oo}=-(\alpha^{ee})^{*}\approx\frac{-1}{2k_{0}\rho}+\frac{i}{k_{0}\rho}$ |  $\alpha^{oe}=-\alpha^{eo}\approx\frac{1}{2k_{0}\rho}$
Table 1: Conditions of the effective polarizabilities at the resonant
frequency of 0.6 THz to achieve symmetric perfect reflection (second row) and
asymmetric perfect absorption (third row).

To experimentally validate the extremely thin perfect absorber, we fabricated
an array of free-standing cylinders (103 $\mu$m radius, 80 $\mu$m height, 310
$\mu$m periodicity). The high-resistive silicon cylinders are connected with
10 $\mu$m beams. Instead of gold, we utilized a 20-nm thick niobium nitride
(NbN) with a room temperature conductivity of $5.5\times 10^{5}$ S/m as the
disk. (Section 6 in Supporting Information) The thickness of the NbN thin film
was characterized by atomic force microscopy. Figure 4a shows the 3D view of
an NbN disk after reactive ion etching. The average surface roughness of the
disk is about 0.26 nm. As shown in Figure 4b, the average height of the disk
is about 23.1 nm, which is close to the designed 20 nm. Terahertz time-domain
spectroscopic measurements were performed for samples with various radii
(Section 7 in Supporting Information), and Figure 4c-f shows the measured
reflection and transmission spectra for the samples with a disk radius of 10
and 28 $\mu$m, respectively. For the $r_{d}$ = 10 $\mu$m sample, all
scattering parameters exhibit two resonances at 0.605 THz and 0.69 THz (Figure
4c), which are primarily due to the weakly perturbed EH111 and HE111 modes,
respectively. Due to the weak symmetry breaking, both reflections $r_{+k}$ and
$r_{-k}$ are very similar with measured peak amplitude larger than 0.8. The
calculated absorption (Figure 4d) for both incidences is very small since most
of the power is reflected. With increasing the disk radius, the transmission
$t$ at 0.605 THz does not change significantly. For $r_{d}$ = 28.5 $\mu$m
(Figure 4e), the measurements exhibit a single resonance in the backward
reflection $r_{-k}$ (solid blue curve) with an amplitude above 0.9 and near-
zero transmission (solid black curve) at the resonance. However, the forward
reflection $r_{+k}$ (solid red curve) drops to zero due to the IS breaking,
leading to an asymmetric perfect absorption A+k at 0.605 THz (Figure 4f). The
measured results are in excellent agreement with the simulations (dashed
curves). Notably, our ultrathin perfect absorber achieves an extraordinary
wavelength-to-thickness ratio exceeding 25,000, representing only about 2% of
the theoretical limit dictated by causality (Section 5 in Supporting
Information).

Our proposed absorber demonstrates adaptability and is extendable beyond thin
films and circular geometries. In Figure 5 we show that it is possible to
achieve perfect absorption at 0.6 THz, so long as a specific relationship
between the disk conductivity $\sigma$ its thickness $t_{d}$ (red circles) is
followed. Therefore, the surface impedance $Z_{s}=1/\sigma t_{d}$ plays an
important role in attaining perfect absorption. The constancy of $Z_{s}$ for
perfect absorption indicates a constant surface carrier concentration,
assuming the carrier mobility does not change. Notably, our design with
conductivity on the order of $10^{7}$ S/m enables the realization of atomic-
level thickness for perfect absorption. Compared to other thin-film absorbers
as shown in Figure 1(b) [54, 55, 56, 57, 58, 59, 60, 61, 62], our NbN disk
exhibits the smallest VFF with a value of 1.06$\times 10^{-6}$, surpassing
graphene-based absorber by two orders of magnitude.

##  III Conclusion

We have experimentally demonstrated an asymmetric absorber with the thickness
of only 4 $\times 10^{-5}$ of the operational wavelength. Through breaking of
the cylinder’s inversion symmetry, nearly 100% of the absorbed power was
confined to the conducting disk. Our analysis reveals that such a significant
asymmetric response primarily arises from the bianisotropic effect among the
dipoles and quadrupoles. Additionally, we extended the Onsager-Casimir
relations to encompass quadrupoles, providing a more comprehensive
understanding of the system. These findings not only advance our knowledge of
subwavelength particle scattering but also open up exciting possibilities for
various novel applications that demand extremely confined energy, such as
energy harvesting, sensitive detection, and controlled scattering
manipulation.

##  IV Methods

Numerical Simulations: The reflection and transmission spectra are calculated
using commercial electromagnetic simulation software based on the finite-
element method. In the modeling, the silicon is set as a Drude-type material
with a carrier density of 1$\times 10^{10}$ cm-3. The gold disk is modeled as
a Drude metal with a DC-conductivity of 2$\times 10^{6}$ S/m and a collision
frequency of 2$\pi\times 2\times 10^{13}$ rad/s. Periodic boundaries are used
for the metasurface structure. To ensure the accuracy of the simulations, the
mesh layers are set to at least 4 in thickness. The multipole moments are
calculated from the fields with the integration origin set as the mass center
of the cylinder. Similarly, in the modeling for the NbN-based structures, the
NbN thin film was modeled with a DC-conductivity of $5,5\times 10^{5}$ S/m.
The far-field radiation patterns are calculated based on the electric field
distribution excited by a plane wave using periodic conditions.
Fabrication: First, an ultrathin with a thickness of 20 nm NbN film was
sputtered on a cleaned SOI (silicon on insulator) substrate. Then the NbN disk
was defined using CF4 gas in the reactive ion etching process. After removing
the etching mask, the sample was coated with AZ9260 as the etching mask for
following the deep reactive ion etching (DRIE) process. Next, to achieve a
free-standing structure, the backside of the sample with a thickness of around
500 µm was again etched in the DRIE system. And the 1-µm buried oxide layer
was removed in buffered oxide etch (BOE) solution for about 10 mins. Finally,
the free-standing silicon disk array was cleaned using acetone and isopropyl
alcohol.
Characterization: Three samples with different disk radii $r_{d}$ were
characterized by a fiber-based THz time-domain spectroscopy. In the
transmission measurements, the transmission coefficient $t(\omega)$ from the
samples is measured with the air as the reference. In the reflection
measurement, a 3-mm high-resistive silicon is used as a beam splitter to split
the incident the reflected beams. Using this setup, the terahertz beam is
normally incident on the samples, which are placed on a sample mount with an
open window in the center to ensure no beam is reflected from the mount to the
sample. To obtain the reflection coefficient r($\omega$), a gold mirror is
used as the reference. The frequency dependent absorptivity was calculated as
A($\omega$) = 1-$|t(\omega)|^{2}$-$|r(\omega)|^{2}$. The disk profile was
measured using the Bruker Dimension ICON.

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##  V Acknowledgements

This work is supported by the National Natural Science Foundation of China
(62275118, 62288101), the Fundamental Research Funds for the Central
Universities, and the Research fund for Jiangsu Key Laboratory of Advanced
Techniques for Manipulating Electromagnetic Waves; A.B.E. acknowledges support
from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
under Germany’s Excellence Strategy within the Cluster of Excellence PhoenixD
(EXC 2122, Project ID No. 390833453); The work of A.E.M. was supported by the
Australian Research Council (DP200101353); W.J.P. acknowledges support from
the US Department of Energy (DOE) (DESC0014372)

Figure 1: Concept illustration and comparison of demonstrated absorber
performance with other literature. (a) Illustration of the extremely thin
absorber consisting of an array of lossless dielectric cylinders with metallic
disks patterned on the top. The structure shows perfect absorption under
forward illumination (red arrow) and nearly perfect reflection under backward
illumination. (b) Comparison of the volume filling fraction of our absorbing
disk and other thin-film absorbers, including graphene [59], MoS2 [58], Ge
[54], Au2Pd2 [60], WeSe2 [56], PtSe2 [61], AgInSbTe [62], ITO [55], Bi [57].
The inset shows a scanning electron microscopic image of the fabricated free-
standing asymmetric metasurface, where the scale bar is 100 $\mu$m. c)-f)
Comparison between metasurfaces with (c,d) and without (e,f) IS. (c) & (e)
Simulated transmission and reflections under the forward ($+k$) and backward
($+k$) illuminations. The dimensions of the cylinders are $r_{c}=103$ $\mu$m,
$h=80$ $\mu$m and $p=350$ $\mu$m. The asymmetric MS in (e) is achieved by
patterning 4.4-nm thick metallic disks with $r_{d}=28.5$ $\mu$m on top of the
cylinders. The insets in (c) and (e) illustrate the symmetric and asymmetric
metasurfaces. (d) & (f) shows the electric field ($E_{x}$) distributions at
the two eigenmodes at the frequencies indicated by the dashed orange and gray
dashed vertical lines, respectively, of (c) and (e). (g) Simulated absorption
spectra under $+k$ (red) and $-k$ (black) illuminations with the
$r_{d}=28.5\mu$m. Figure 2: Asymmetric response of the symmetry broken
metasurface. (a) The asymmetric reflections and the asymmetric ratio varying
with the disk radius.(b) & (c) The distribution of the electric field Ex in
the ZX plane, the magnetic field Hy in the ZY plane, and the power loss
density in the disk under the $+k$ (b) and $-k$ (c) illuminations at 0.6 THz.
(d) Calculated absorbed power in the disk (red) and the cylinder (gray),
respectively. (e) simulated total scattering cross sections (SCS) and total
absorption cross sections (ACS) under the $+k$ (top panel) and $-k$ (bottom
panel) illuminations for the structure with rd = 28.5 $\mu$m. The insets show
the sliced radiation patterns along the YZ plane, for two opposite incidences,
respectively.(f) & (g) show the corresponding radiation patterns at the YZ
plane (H plane) for the two cases, respectively.  Figure 3: Multipole
expansion analysis of the asymmetric MS. (a) and (b) are the calculated
effective directional polarizabilities for $+k$ and $-k$ incidences,
respectively. (c) The comparison of the retrieved forward and backward
reflections among the simulation, the dipole model (d), and the multipole
model (m). (d) Calculated equivalent multipole BAPs varying with the
conductivity of the disk. The inset shows the equivalent polarizabilities as a
function of the disk conductivity.  Figure 4: Experimental Results.(a) 3D view
of the measured NbN disk profile using atomic force microscopy. (b) The
measured height profile of the disk along the white line as indicated in the
inset. (c) & (e) The measured (solid) and simulated (dashed) forward
reflection (red), backward reflection (blue), and transmission (gray), for the
samples with a disk radius of 10 and 28 $\mu$m, respectively. (d) & (f) are
the calculated forward $A_{+k}$ (red) and backward $A_{-k}$ (black) absorption
spectra for the two samples. The insets are the microscopic images of the
fabricated samples. Figure 5: The relationship between the disk thickness and
the metal conductivity to achieve perfect absorption at the resonance of 0.6
THz.