# Reciprocal Metasurfaces for On-axis Reflective Optical Computing

Ali Momeni2, Hamid Rajabalipanah3, Mahdi Rahmanzadeh4, Ali Abdolali31, Karim
Achouri5, Viktar Asadchy6 and Romain Fleury21
2 Laboratory of Wave Engineering, School of Electrical Engineering, Swiss
Federal Institute of Technology in Lausanne (EPFL), Lausanne, Switzerland.
3 Applied Electromagnetic Laboratory, School of Electrical Engineering, Iran
University of Science and Technology, Tehran 1684613114, Iran.
4Electrical Engineering Department, Sharif University of Technology, Tehran
11155-4363, Iran.
5Nanophotonics and Metrology Laboratory, Swiss Federal Institute of Technology
Lausanne (EPFL), 1015 Lausanne, Switzerland.
6Ginzton Laboratory and Department of Electrical Engineering, Stanford
University, Stanford, California 94305, USA.
1 Corresponding author’s email<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

Analog computing has emerged as a promising candidate for real-time and
parallel continuous data processing. This paper presents a reciprocal way for
realizing asymmetric optical transfer functions (OTFs) in the reflection side
of the on-axis processing channels. It is rigorously demonstrated that the
presence of Cross-polarization Exciting Normal Polarizabilities (CPENP) of a
reciprocal metasurface circumvents the famous challenge of Green’s function
approach in implementation of on-axis reflective optical signal processing
while providing dual computing channels under orthogonal polarizations.
Following a comprehensive theoretical discussion and as a proof of concept, an
all-dielectric optical metasurface is elaborately designed to exhibit the
desired surface polarizabilities, thereby reflecting the first derivative and
extracting the edges of images impinging from normal direction. The proposed
study offers a flexible design method for on-axis metasurface-based optical
signal processing and also, dramatically facilitates the experimental setup
required for ultrafast analog computation and image processing.

###### Index Terms:

Optical Signal Processing, all-dielectric Metasurfaces, Surface
Polarizabilities.

##  I Introduction

Over the past few years, the growing demands for ultrafast and large-scale
signal, image, and information system processing as well as the saturation of
digital computational capacities have brought about the emergence of new
proposals for optical analog computing schemes [1, 2, 3]. In traditional
approaches, mathematical processing is mainly performed in the digital domain
by using traditional integrated electronic-based technologies for implementing
logics [4, 5, 6, 7], differentiators and integrators [8, 9]. Unfortunately,
these computing hardwares have some major disadvantages related to operational
speed and power consumption [10, 11]. Recently, Silva et. al [11] have
proposed the construction of analog computational metamaterials, based on two
distinct approaches: the spatial Fourier transfer and Green’s function (GF)
methods [12].

Figure 1:  (a) Schematic sketch of the proposed spatial metasurface processor
for performing real-time on-axis reflective analog signal processing (b)
Demonstration of the symmetries/asymmetries of the angular scattering around
the boresight direction for nonreciprocal/reciprocal metasurfaces.

The former solution, executed in the spatial Fourier domain, was initially
accompanied with additional bulky optical components (4f correlators),
hindering miniaturization [11]. This shortcoming was later avoided by directly
realizing the spatial impulse response of interest by means of the nonlocal
transmission or reflection response of artificial structures [13, 14, 15, 16].
A large number of studies presented resonant and non-resonant structures to
expand this idea for advanced all-optical signal processing architectures [17,
18, 19]. The authors in [14] have demonstrated that the nonlocality of
artificial structures can be engineered to enable signal manipulation in the
momentum domain over an ultrathin platform for performing basic mathematical
operations in transmission mode. Another progress in this area, discussed in
[13], has shown that the interference effects associated with surface plasmon
excitations at a single metal-dielectric interface can perform spatial
differentiation and edge detection. More recently, a theoretical work has
demonstrated that the Laplacian operator required to do spatial
differentiation in transmission mode can be obtained using the guided
resonances of a photonic crystal slab [20]. Besides, different innovative
approaches have been also reported, for instance manipulating the complex-
valued electromagnetic wave propagating through specially designed recursive
paths [21], or the use of topological insulators as a way to increase the
robustness to geometrical tolerances [22].

In the GF-based optical signal processing methods, the metasurfaces often aim
at realizing transfer functions (TF) with either odd-/or even-symmetric
properties (see Fig. 1a). For odd/even TFs, the reflection or transmission
coefficient of the metasurface processor should be an asymmetric/symmetric
function of the incident angle. Due to fundamental limitations arising from
the nonlocal behavior of reflection/transmission responses in the wavevector
domain, the existing GF-based analog computing proposals have exploited
complex oblique illumination setups (challenging to align) to implement odd-
symmetric operations [23, 17, 13, 18, 24]. In most practical situations,
realizing on-axis signal processing operators with asymmetric optical transfer
functions (OTFs) is a key requirement for increasing the compatibility with
standard image processing/recognition schemes, such as image sharpening and
edge detection [25, 26, 16, 27]. In addition, reflection-type processing
systems are superior to transmission-type ones in terms of compactness.
Besides, although using both reflection and transmission channels of the
metasurfaces would enhance the degrees of freedom for performing parallel
processing, the on-axis reflective channels are still unavailable. The reason
is attributed to the fact that realizing an odd-symmetric angular dispersion
for co-polarized components of these channels violates the well-known
reciprocity theorem [28]. Nevertheless, up to now prior proposals for
accomplishing on-axis optical signal processing have only considered
transmissive configurations [14, 28, 25, 29, 27, 30].

Figure 2:  (a) The synthesized (dashed line) and exact (solid line) transfer
functions obtained by a nonreciprocal anisotropic metasurface differentiator
characterized with the following polarizabilities:
$\alpha_{\text{ee}}^{xx}=1\times 10^{-20}$,
$\alpha_{\text{ee}}^{xz}=2.73\times 10^{-17}$,
$\alpha_{\text{ee}}^{zx}=2.19\times 10^{-17}$,
$\alpha_{\text{ee}}^{zz}=1.21\times 10^{-18}$, and
$\alpha_{\text{mm}}^{yy}=1\times 10^{-16}$. All quantities are in m3. (b) The
Gaussian-shape incident field as well as (c) the derivative reflected field
along the boresight direction.

Motivated by such theoretical developments, here, we propose a reciprocal way
for exploiting the on-axis reflective processing channels in bianisotropic
metasurfaces. We demonstrate that taking into account the cross-polarization
reflections caused by the normal polarizabilities of bianisotropic
metasurfaces brings in new degrees of freedom in controlling the meta-atom
scattering. We reveal a simple appealing opportunity to circumvent the
coercive reflection symmetry in the wavevector domain without resorting to
intricate non-reciprocal options or bulky Fourier lenses. We show that using
the cross-polarized channels enabled by a certain group of normal
polarizabilities, the odd-symmetric operations can be implemented in on-axis
reflective scenarios. Compared with recent works [25, 28, 30, 31], the main
contributions of this paper are:
1) By considering the general form of the surface polarizabilities of a
reciprocal bianisotropic metasurface, we extract the general rules for
performing standard on-axis optical signal processing tasks in the reflection
side. Some remarks about the transmission configuration are also presented.
2) Realizing an all-dielectric optical metasurface that exposes the desired
collective polarzabilites and creates a cross-polarized reflective processing
channel.

##  II Theoretical Investigation

###  II-A Scattering coefficients of a metasurface

Fig. 1a illustrates the employed computational metasurface comprising a
uniform array of polarizable meta-atoms in the x-y plane (z=0). Let us
consider that p-polarized (or s-polarized) light with a transverse field-
profile of s${}_{\text{in}}$(x) impinges normally on the metasurface processor
(along the $-$z direction). This incident light is reflected by the
metasurface with a beam-profile s${}_{\text{out}}$(x) in the $+$z direction.
Here, p- and s-polarized refer to the polarized lights whose electric field is
oriented along x and y directions, respectively, at normal incidences. In the
spatial Fourier domain, each beam profile can be spectrally represented by a
superposition of plane waves, as spatial Fourier components (SPF), i.e.,
s${}_{\text{in}}$(x)=$\int\limits_{2W}{\tilde{s}_{\text{inc}}^{\text{}}\left({{k}_{x}}\right)}~{}\text{exp}\,(j{{k}_{z}}{{z}+jk_{x}x_{\text{}}})\,d{{k}_{x\text{}}},$
and
s${}_{\text{out}}$(x)=$\int\limits_{2W}{\tilde{H}(k_{x})\tilde{s}_{\text{inc}}^{\text{}}\left({{k}_{x}}\right)}~{}\text{exp}\,(-j{{k}_{z}}{{z}+jk_{x}x_{\text{}}})\,d{{k}_{x\text{}}}$.
Here, $W$ denotes the spatial bandwidth of the input signal and
kx=k0sin$\theta$ specifies the x component of the wavenumber of the plane wave
harmonic illuminating the metasurface with the incidence angle $\theta$.
Moreover, $\tilde{H}(k_{x})$ refers to the transfer function describing an
operator of choice, that we aim at synthesizing. Due to the sub-wavelength
periodicity of the metasurface along the x direction, the tangential wave
vector kx must be continuous at the interface. Therefore, the incident plane
wave with kx only generates a reflected propagating plane wave with the same
kx. Furthermore, the beam-profile transformation between the incident and
reflected lights can be spectrally described by using a spatially-dispersive
transfer function representing the mathematical operator of interest,
${\tilde{H}}\left({{k}_{x}}\right)={{{s}_{out}}\left({{k}_{x}}\right)}/{{{s}_{in}}\left({{k}_{x}}\right)}$.
Mathematically speaking, the $k_{x}$-dependency of the transfer function
should be imparted by the angular response of the scattering parameter
(reflection or transmission coefficient) of the employed metasurface [32]. It
should be noted that the challenges accompanied with reflective optical signal
processing of on-axis illuminations are more severe than those of transmissive
configurations. For instance, designing asymmetric reflective OTFs by using
co-polarization channels inherently requires non-reciprocity [28]. Although
both reflection and transmission problems will be investigated, the focus of
the paper is the reflection mode. In this case, the output field of the
metasurface processor is
s${}_{\text{out}}$(x)=$F^{-1}${s${}_{\text{inc}}$(kx)$\tilde{R}$(kx)}. The
metasurface can be theoretically treated as a bianisotropic homogeneous sheet
in its general form with four distinct sets of collective polarizabilities
(${{\overline{\overline{{\hat{\alpha}}}}}_{\text{ee}}}$,
${{\overline{\overline{{\hat{\alpha}}}}}_{\text{em}}}$,
${{\overline{\overline{{\hat{\alpha}}}}}_{\text{me}}}$,
${{\overline{\overline{{\hat{\alpha}}}}}_{\text{mm}}}$) [33]. Keeping the
tensorial format of polarizability components in mind, the reflection
coefficient (the transfer function) exerted by the metasurface boundary would
be different for s- and p-polarized signals, and we write

$\overline{\overline{{\tilde{H}}}}({{k}_{x}})\equiv\left[\begin{matrix}{{{\tilde{R}}}^{s\to
s}}\left({{k}_{x}}\right)\,&{{{\tilde{R}}}^{s\to
p}}\left({{k}_{x}}\right)\,\,\\\ {{{\tilde{R}}}^{p\to
s}}\left({{k}_{x}}\right)\,\,&{{{\tilde{R}}}^{p\to
p}}\left({{k}_{x}}\right)\,\\\ \end{matrix}\right]\,\,\,$ (1)

where $\tilde{R}$ alludes to the nonlocal reflection coefficient of the
bianisotropic metasurface in the wavevector domain. The first and second
superscripts refer to the polarization state of the input and output fields,
respectively. Indeed, for an arbitrary input signal having a specific
polarization state, the bianisotropic metasurface may suggest two possible
transfer functions resulting in two different output signals with s or p
polarization in the reflection mode. Each tensorial component of the transfer
function in Eq. 1 is affiliated to a certain combination of surface
polarizabilities, in an explicit or implicit manner. To unveil these
dependencies, we need constitutive parameters which do not change with
different conditions of the external excitation and only depend on the
physical parameters of the metasurface i.e., the shapes and sizes of the meta-
atoms.

With the advent of metasurfaces, a new generation of 2D computational
interfaces has emerged to spatially shape the optical fields over deeply sub-
wavelength volumes [34, 35, 36]. Several characterization frameworks based on
local tensorial scattering coefficients [37, 38, 39, 40, 41], surface
impedance formulation [42, 43], generalized sheet transition conditions
(GSTCs) by susceptibility tensors [44, 45] as well as individual/effective
polarizabilities [46, 47, 33, 48], have been fostered to represent
metasurfaces in their general bianisotropic forms. These methods do consider
normal components; however, in most studies the normal components are omitted.
However, normal components are pivotal when complex spatially-dispersive
behavior are demanded in reflection or transmission mode [32]. We thus take
these components fully into account and follow the so-called Tretyakov-
Simovski formalism relating the fields and polarizations in the general form
[47, 33]

$\displaystyle\textbf{p}={{\overline{\overline{{\hat{\alpha}}}}}_{\text{ee}}}{{\textbf{E}}_{\text{inc}}}+{{\overline{\overline{{\hat{\alpha}}}}}_{\text{em}}}{{\textbf{H}}_{\text{inc}}}$
(2)
$\displaystyle\textbf{m}={{\overline{\overline{{\hat{\alpha}}}}}_{\text{me}}}{{\textbf{E}}_{\text{inc}}}+{{\overline{\overline{{\hat{\alpha}}}}}_{\text{mm}}}{{\textbf{H}}_{\text{inc}}}$
(3)

in which, ${{\overline{\overline{{\hat{\alpha}}}}}_{\text{ee}}}$,
${{\overline{\overline{{\hat{\alpha}}}}}_{\text{em}}}$,
${{\overline{\overline{{\hat{\alpha}}}}}_{\text{me}}}$, and
${{\overline{\overline{{\hat{\alpha}}}}}_{\text{mm}}}$ are called collective
surface polarizabilities: electric, magnetoelectric, electromagnetic, and
magnetic ones, respectively. P and M also represent the electric and magnetic
polarization densities induced on the metasurface. The transversal components
of the reflected and transmitted electric fields can be expressed by [47]:

$\displaystyle{{\textbf{E}}_{\text{ref,t}}}=-{{\left({{\overline{\overline{I}}}_{t}}+{{\overline{\overline{Z}}}_{{}_{bo}^{to}}}{{\overline{\overline{Y}}}_{{}_{to}^{bo}}}\right)}^{-1}}\left({{\overline{\overline{I}}}_{t}}-{{\overline{\overline{Z}}}_{{}_{bo}^{to}}}{{\overline{\overline{Y}}}_{{}_{to}^{bo}}}\right)\cdot\,{{\textbf{E}}_{\text{inc,t}}}-{{\left({{\overline{\overline{I}}}_{t}}+{{\overline{\overline{Z}}}_{{}_{bo}^{to}}}{{\overline{\overline{Y}}}_{{}_{to}^{bo}}}\right)}^{-1}}\cdot\left[j\omega\left({{\overline{\overline{Z}}}_{{}_{bo}^{to}}}\cdot{{\textbf{P}}_{t}}\pm\textbf{n}\times{{\textbf{M}}_{t}}\right)\pm
j\left({{\textbf{k}}_{t}}\frac{{{P}_{n}}}{\varepsilon}\mp{{\overline{\overline{Z}}}_{{}_{bo}^{to}}}\cdot\left({\textbf{k}_{t}}\times\textbf{n}\right)\frac{{{M}_{n}}}{\mu}\right)\right]$
(4)
$\displaystyle{{\textbf{E}}_{\text{tran,t}}}={{\left({{\overline{\overline{I}}}_{t}}+{{\overline{\overline{Z}}}_{{}_{to}^{bo}}}{{\overline{\overline{Y}}}_{{}_{bo}^{to}}}\right)}^{-1}}\left({{\overline{\overline{I}}}_{t}}+{{\overline{\overline{Z}}}_{{}_{to}^{bo}}}{{\overline{\overline{Y}}}_{{}_{bo}^{to}}}\right)\cdot\,{{\textbf{E}}_{\text{inc,t}}}-{{\left({{\overline{\overline{I}}}_{t}}+{{\overline{\overline{Z}}}_{{}_{to}^{bo}}}{{\overline{\overline{Y}}}_{{}_{bo}^{to}}}\right)}^{-1}}\cdot\left[j\omega\left({{\overline{\overline{Z}}}_{{}_{to}^{bo}}}\cdot{{\textbf{P}}_{t}}\mp\textbf{n}\times{{\textbf{M}}_{t}}\right)\mp
j\left({\textbf{k}_{t}}\frac{{{P}_{n}}}{\varepsilon}\pm{{\overline{\overline{Z}}}_{{}_{to}^{bo}}}\cdot\left({\textbf{k}_{t}}\times\textbf{n}\right)\frac{{{M}_{n}}}{\mu}\right)\right]$
(5)

Here, P=p/$S$ and M=m/$S$ are polarization surface densities wherein $S$
denotes the unit cell area. Also, the top/bottom sign corresponds to the wave
propagating along $\mp$z direction, and $to$ and $bo$ in the subscripts refer
to top/bottom medium. For the sake of simplicity, we assume the metasurface is
surrounded by free-space. Thus, ${\overline{\overline{Z}}}$ and
${\overline{\overline{Y}}}$=${\overline{\overline{Z}}}^{-1}$ denoting
respectively the dyadic impedance and admittance of the corresponding medium,
can be written as [49]

$\overline{\overline{Z}}=\eta\left[\begin{matrix}\cos\theta{{\cos}^{2}}\phi+{{{\sin}^{2}}\phi}/{\cos\theta}\,&\left(\cos\theta-1/\cos\theta\right)\sin\phi\cos\phi\,\,\\\
\left(\cos\theta-1/\cos\theta\right)\sin\phi\cos\phi\,\,&\cos\theta{{\sin}^{2}}\phi+{{{\cos}^{2}}\phi}/{\cos\theta}\,\\\
\end{matrix}\right]\,\,\,$ (6)

in which, $\eta$ is the free-space impedance and $\theta$ and $\phi$ are the
elevation and azimuth angles of the incident wave. The formulations above may
be applied for any special case of external illumination; for example, s- or
p- polarized incident waves or a superposition of these two. To exploit the
full potential of the metasurface boundary, each polarizability tensor of Eqs.
(2), (3) includes both normal and tangential components, i.e., 36 scalar
polarizabilities which can be written as

$\overline{\overline{{{\alpha}}}}_{\text{ee}}=\left[\begin{matrix}{\alpha_{\text{ee}}^{xx}}&\alpha_{\text{ee}}^{xy}&{\alpha_{\text{ee}}^{xz}}\\\
\alpha_{\text{ee}}^{yx}&{\alpha_{\text{ee}}^{yy}}&\alpha_{\text{ee}}^{yz}\\\
{\alpha_{\text{ee}}^{zx}}&\alpha_{\text{ee}}^{zy}&{\alpha_{\text{ee}}^{zz}}\\\
\end{matrix}\right],\overline{\overline{{{\alpha}}}}_{\text{em}}=\left[\begin{matrix}\alpha_{\text{em}}^{xx}&{\alpha_{\text{em}}^{xy}}&\alpha_{\text{em}}^{xz}\\\
{\alpha_{\text{em}}^{yx}}&\alpha_{\text{em}}^{yy}&{\alpha_{\text{em}}^{yz}}\\\
\alpha_{\text{em}}^{zx}&{\alpha_{\text{em}}^{zy}}&\alpha_{\text{em}}^{zz}\\\
\end{matrix}\right]\\\ \\\
\overline{\overline{{{\alpha}}}}_{\text{me}}=\left[\begin{matrix}\alpha_{\text{me}}^{xx}&{\alpha_{\text{me}}^{xy}}&\alpha_{\text{me}}^{xz}\\\
{\alpha_{\text{me}}^{yx}}&\alpha_{\text{me}}^{yy}&{\alpha_{\text{me}}^{yz}}\\\
\alpha_{\text{me}}^{zx}&{\alpha_{\text{me}}^{zy}}&\alpha_{\text{me}}^{zz}\\\
\end{matrix}\right],\overline{\overline{{{\alpha}}}}_{\text{mm}}=\left[\begin{matrix}{\alpha_{\text{mm}}^{xx}}&\alpha_{\text{mm}}^{xy}&{\alpha_{\text{mm}}^{xz}}\\\
\alpha_{\text{mm}}^{yx}&{\alpha_{\text{mm}}^{yy}}&\alpha_{\text{mm}}^{yz}\\\
{\alpha_{\text{mm}}^{zx}}&\alpha_{\text{mm}}^{zy}&{\alpha_{\text{mm}}^{zz}}\\\
\end{matrix}\right]$

With substituting Eqs. 2, 3, 7, 8 into Eqs. 4, 5 and after some algebraic
manipulations, the most general form for the co-polarized reflection and
transmission coefficients of the metasurface can be obtained as:

$\displaystyle R_{\pm}^{s\to
s}=\frac{-j\omega}{2}\left[\frac{\eta}{\cos\theta}\alpha_{\text{ee}}^{yy}\pm\alpha_{\text{em}}^{yx}\mp\tan\theta\alpha_{\text{em}}^{yz}\mp\alpha_{\text{me}}^{xy}-\frac{\cos\theta}{\eta}\alpha_{\text{mm}}^{xx}+\frac{\sin\theta}{\eta}\alpha_{\text{mm}}^{xz}\mp\tan\theta\alpha_{\text{me}}^{zy}-\frac{\sin\theta}{\eta}\alpha_{\text{mm}}^{zx}+\frac{\sin\theta\tan\theta}{\eta}\alpha_{\text{mm}}^{zz}\right]$
(7) $\displaystyle T_{\pm}^{s\to
s}=1-\frac{j\omega}{2}\left[\frac{\eta}{\cos\theta}\alpha_{\text{ee}}^{yy}\pm\alpha_{\text{em}}^{yx}\mp\tan\theta\alpha_{\text{em}}^{yz}\pm\alpha_{\text{me}}^{xy}+\frac{\cos\theta}{\eta}\alpha_{\text{mm}}^{xx}-\frac{\sin\theta}{\eta}\alpha_{\text{mm}}^{xz}\mp\tan\theta\alpha_{\text{me}}^{zy}-\frac{\sin\theta}{\eta}\alpha_{\text{mm}}^{zx}+\frac{\sin\theta\tan\theta}{\eta}\alpha_{\text{mm}}^{zz}\right]$
(8) $\displaystyle R_{\pm}^{p\to
p}=\frac{-j\omega}{2}\left[\eta\cos\theta\alpha_{\text{ee}}^{xx}\mp\alpha_{\text{em}}^{xy}-\eta\sin\theta\alpha_{\text{ee}}^{xz}\pm\alpha_{\text{me}}^{yx}-\frac{1}{\eta\cos\theta}\alpha_{\text{mm}}^{yy}\mp\tan\theta\alpha_{\text{me}}^{yz}+\eta\sin\theta\alpha_{\text{ee}}^{zx}\mp\tan\theta\alpha_{\text{em}}^{zy}-\eta\sin\theta\tan\theta\alpha_{\text{ee}}^{zz}\right]$
(9) $\displaystyle T_{\pm}^{p\to
p}=1-\frac{j\omega}{2}\left[\eta\cos\theta\alpha_{\text{ee}}^{xx}\mp\alpha_{\text{em}}^{xy}-\eta\sin\theta\alpha_{\text{ee}}^{xz}\mp\alpha_{\text{me}}^{yx}+\frac{1}{\eta\cos\theta}\alpha_{\text{mm}}^{yy}\pm\tan\theta\alpha_{\text{me}}^{yz}-\eta\sin\theta\alpha_{\text{ee}}^{zx}\pm\tan\theta\alpha_{\text{em}}^{zy}+\eta\sin\theta\tan\theta\alpha_{\text{ee}}^{zz}\right]$
(10)

These equations reveal the angular dispersion behavior of the reflection and
transmission coefficients for a non-reciprocal bianisotropic metasurface in
its general form with 36 tonsorial components and will be instrumental for the
discussions presented in the following sections.

###  II-B Nonreciprocal on-axis differentiator

Figure 3:  (a) The array of polarizable meta-atoms for performing optical
signal processing. (b) The proposed reciprocal bianisotropic meta-atom
exposing normal polarizability components. The geometry was chosen in such a
way that both symmetries in xy- and yz- planes are broken. The geometrical
parameters are L1=410 nm, L2=205 nm, L3=205 nm, L4=350 nm, w1=695 nm, w2=315
nm, w3=410 nm, H1=760 nm, H2=455 nm, H3=250 nm, and H4=730 nm. (c) The
schematic setup for testing the optical signal processing performance of the
reflective metasurface in which P and BS stand for the polarizer and beam
splitter, respectively.

The main goal of this paper is to search for the best group of polarizability
components whereby the reflection or transmission coefficient in Eqs.
(9)-(12), $\tilde{\textbf{R}}$(kx) or $\tilde{\textbf{T}}$(kx), obeys the
angular dispersion of the desired mathematical operator. The first-order
differentiation operation is a good representative for the family of
asymmetric OTFs. Referring to $\tilde{H}$(kx)=jkx for the first-order
differentiation operator, our purpose is to realize an on-axis reflective
channel whose transfer function, $\tilde{R}$(kx), has indispensably a non-
local odd-symmetric treatment around $\theta=0$ without polarization rotation,
i.e., R($\theta$)=R($-\theta$). Let us consider the conceptual illustration of
Fig. 1b in which the possibility of R($\theta$)$\neq$R($-\theta$) is
graphically investigated. A first-order, polarization-preserving,
differentiation operation requires that the co-polarized reflection
coefficient imparted on the plane wave coming from $R_{2\to 1}$ channel
($\theta$ illumination) essentially differs from that seen from $R_{1\to 2}$
channel ($-\theta$ illumination), which is prohibited by reciprocity. So,
resorting to a non-reciprocal metasurface is inevitable to implement a
polarization-preserved first-order differentiator for on-axis illuminations
[28]. This observation can be verified through Eqs. (9), (11) in which the
necessary condition to keep terms with odd functionality from $\theta$ is
$\alpha_{\text{em}}^{yz}\neq-\alpha_{\text{me}}^{zy}$ or
$\alpha_{\text{mm}}^{xz}\neq\alpha_{\text{mm}}^{zx}$ for s-polarized
incidences and $\alpha_{\text{ee}}^{xz}\neq\alpha_{\text{ee}}^{zx}$ or
$\alpha_{\text{me}}^{yz}\neq-\alpha_{\text{em}}^{zy}$ for p-polarized
incidences. These constraints for polarization-preserved reflective optical
processing obviously break the Casimir-Onsager reciprocity relations [50].
However, as can be directly inferred from Eqs. (10), (12), polarization-
preserving on-axis asymmetric OTFs in transmission regime can be simply
implemented by using a reciprocal metasurface having non-zero
$\alpha_{\text{mm}}^{xz}$ and $\alpha_{\text{ee}}^{xz}$ for s- and p-polarized
incidences, respectively.

As a representative example, let us consider the following synthesis problem:
find the surface polarizabilities of a nonreciprocal metasurface for which a
p-polarized input field with an illumination angle of
$\theta_{\text{inc}}=0^{\circ}$ reflects into a co-polarized beam whose
profile is the first-order derivative of the one of the input. A Gaussian-
shape beam profile with a spatial bandwidth of $W=0.2k_{0}$ is considered as
the input field. In order to obtain the required surface polarizabilties, the
metasurface synthesis will be treated as a non-linear optimization problem
which is solved numerically. Firstly, according to the scattering role of each
polarizability component, the neutral components are assumed to be zero. The
involved polarizabilities are $\alpha_{\text{{ee}}}^{xx}$,
$\alpha_{\text{{ee}}}^{xz}$, and $\alpha_{\text{{ee}}}^{zx}$,
$\alpha_{\text{{ee}}}^{zz}$, and $\alpha_{\text{{mm}}}^{yy}$. Then, we seek
for those surface polarizabilities for which the difference between the
desired GF, $\tilde{G}_{\text{des}}(k_{x})$=jkx, and the angle-dependent
reflection coefficient of the metasurface, $\tilde{R}(k_{x})$, is minimized
within a pre-determined angular range. The cost function is defined as the sum
of squares of the differences at a finite number of N samples for both real
and imaginary parts
$f={{w}_{\text{re}}}{{\sum\limits_{i=1}^{N}{\left(\operatorname{Re}\left[{{{\tilde{G}}}_{\text{des}}}\left({{k}_{x,i}}\right)-{\tilde{R}}\left({{k}_{x,i}}\right)\right]\right)}}^{2}}+{{w}_{\text{im}}}{{\sum\limits_{i=1}^{N}{\left(\operatorname{Im}\left[{{{\tilde{G}}}_{\text{des}}}\left({{k}_{x,i}}\right)-{\tilde{R}}\left({{k}_{x,i}}\right)\right]\right)}}^{2}}$.
Here, $w_{\text{re}}$ and $w_{\text{im}}$ represent the weight coefficients
established to selectively adjust the contribution of real and imaginary
parts, respectively. At this stage, the metasurface processor is synthesized
with the optimized polarizabilities shown in the caption of Fig. 2. It should
be noted that, since $\alpha^{\text{xz}}_{\text{ee}}$ is not equal to
$\alpha^{\text{zx}}_{\text{ee}}$, the synthesized metasurface is non-
reciprocal. The resultant transfer functions along with the input and output
fields are displayed in Figs. 2a-c. These figures show good agreement between
the desired and synthesized transfer functions and output beam profiles.
Indeed, the reflected field propagating out along the boresight direction is
the 1${}^{\text{st}}$-order derivative of the Gaussian-shape input field. The
accuracy of differentiation is 99.5%, described by the Pearson correlation
coefficient between the simulated and exact reflected field amplitudes. The
results confirm that at the expense of a complex fabrication, a suitably
designed nonreciprocal anisotropic metasurface can be thought of as a
reflective optical 1${}^{\text{st}}$-order differentiator working for input
fields coming from normal direction.

###  II-C Reciprocal on-axis differentiator

As an alternative solution, incorporating cross-polarization channels to the
problem can elaborately circumvent the nettlesome restrictions arising from
reciprocity. In this case, the polarizability components are engineered so
that the polarization state of the incident light coming from Port 1 be
rotated when it is captured at Port 2 (see Fig. 1b). Hereafter, the cross-
polarized reflection of $R^{s\to p}$ or $R^{p\to s}$ serves as the transfer
function of our processing channel meaning that the output signal has
orthogonal polarization with respect to that of the input wave. It should be
noted that such a metasurface, if designed, will exhibit $R_{1\to 2}^{s\to
p}$$\neq$$R_{2\to 1}^{s\to p}$ which no longer quashes the reciprocity
condition. Similar deduction can be made for $R_{1\to 2}^{p\to
s}$$\neq$$R_{2\to 1}^{p\to s}$. Due to the insensitivity of conventional
optical sensors, including the human eyes, to the polarization of light,
polarization rotation does not degrade the overall performance of optical
differentiator, instead, it dramatically facilitates the realization
procedure. Based on Eqs. (2)-(8), the cross-polarized reflection and
transmission coefficients of the metasurface read as:

Figure 4:  (a), (b), (c) The tangential collective polarizabilties of the
designed optical metasurface given in Fig. 3a. Evaluation of (d) zero cross-
polarized reflection and (e) asymmetric angular dispersion conditions on the
surface polarizabilities. The working frequency is $f$=177 THz.

$\displaystyle R^{s\to
p}=\frac{-j\omega}{2}\left[\eta\alpha_{\text{ee}}^{xy}\pm\cos\theta\alpha_{\text{em}}^{xx}\mp\sin\theta\alpha_{\text{em}}^{xz}\pm\frac{\alpha_{\text{me}}^{yy}}{\cos\theta}+\frac{1}{\eta}\alpha_{\text{mm}}^{yx}-\frac{\tan\theta}{\eta}\alpha_{\text{mm}}^{yz}+\eta\tan\theta\alpha_{\text{ee}}^{zy}\pm\sin\theta\alpha_{\text{em}}^{zx}\mp\sin\theta\tan\theta\alpha_{\text{em}}^{zz}\right]$
(11) $\displaystyle T^{s\to
p}=-\frac{j\omega}{2}\left[\eta\alpha_{\text{ee}}^{xy}\pm\cos\theta\alpha_{\text{em}}^{xx}\mp\sin\theta\alpha_{\text{em}}^{xz}\mp\frac{\alpha_{\text{me}}^{yy}}{\cos\theta}-\frac{1}{\eta}\alpha_{\text{mm}}^{yx}+\frac{\tan\theta}{\eta}\alpha_{\text{mm}}^{yz}-\eta\tan\theta\alpha_{\text{ee}}^{zy}\mp\sin\theta\alpha_{\text{em}}^{zx}\pm\sin\theta\tan\theta\alpha_{\text{em}}^{zz}\right]$
(12) $\displaystyle R^{p\to
s}=\frac{-j\omega}{2}\left[\eta\alpha_{\text{ee}}^{yx}\mp\frac{1}{\cos\theta}\alpha_{\text{em}}^{yy}-\eta\tan\theta\alpha_{\text{ee}}^{yz}\mp\cos\theta\alpha_{\text{me}}^{xx}+\frac{1}{\eta}\alpha_{\text{mm}}^{xy}\pm\sin\theta\alpha_{\text{me}}^{xz}\mp\sin\theta\alpha_{\text{me}}^{zx}+\frac{\tan\theta}{\eta}\alpha_{\text{mm}}^{zy}\pm\sin\theta\tan\theta\alpha_{\text{me}}^{zz}\right]$
(13) $\displaystyle T^{p\to
s}=-\frac{j\omega}{2}\left[\eta\alpha_{\text{ee}}^{yx}\mp\frac{1}{\cos\theta}\alpha_{\text{em}}^{yy}-\eta\tan\theta\alpha_{\text{ee}}^{yz}\pm\cos\theta\alpha_{\text{me}}^{xx}-\frac{1}{\eta}\alpha_{\text{mm}}^{xy}\mp\sin\theta\alpha_{\text{me}}^{xz}\mp\sin\theta\alpha_{\text{me}}^{zx}+\frac{\tan\theta}{\eta}\alpha_{\text{mm}}^{zy}\pm\sin\theta\tan\theta\alpha_{\text{me}}^{zz}\right]$
(14)

Figure 5:  The phase and amplitude of (a) co-polarized and (b), s$\to$p and
(c) p$\to$s reflection coefficients. The phase and amplitude of the exact
transfer function for implementing the first-order differentiation operation
are also presented. The solid- and dashed-lines stand for the exact and the
synthesized transfer functions, respectively, and the working frequency is
$f$=177 THz.

It should be remembered that we are seeking for those reciprocal collective
polarizabity tensors which can be potentially effective to satisfy two
important conditions at the same time: exciting the cross-polarization
reflected fields and making the cross-polarized reflection to be an asymmetric
function of kx (or $\theta$) variable. As clearly seen from Eqs. (13), (15),
although the presence of 9 surface polarizabilities leads to generation of
cross-polarized reflection fields for each of s- and p-polarized
illuminations, only 4 of them deal with odd-order terms (with respect to
$\theta$). Indeed, the necessary condition for reciprocal realization of on-
axis asymmetric OTFs in the cross-polarized reflective channel is the presence
of

rule I:

$\alpha_{\text{em}}^{xz}$ or $\alpha_{\text{mm}}^{yz}$ or
$\alpha_{\text{ee}}^{zy}$ or $\alpha_{\text{em}}^{zx}$ (s-polarized
illumination)

$\alpha_{\text{ee}}^{yz}$ or $\alpha_{\text{me}}^{xz}$ or
$\alpha_{\text{me}}^{zx}$ or $\alpha_{\text{mm}}^{zy}$ (p-polarized
illumination)

This means that the role of normal polarizabilities and bianisotropy is
undeniable for a reciprocal metasurface to build asymmetric angular dispersion
behavior for the reflection around $k_{x}$=0. We call these surface
polarizability components Cross-polarization Exciting Normal Polarizabilities
(CPENP) throughout the paper. The cross-polarized transmission processing
channel can also be enabled if the surface polarizabilities
$\alpha_{\text{mm}}^{yz}$ or $\alpha_{\text{ee}}^{zy}$ for s-polarized
illumination and $\alpha_{\text{ee}}^{yz}$ or $\alpha_{\text{mm}}^{zy}$ for
p-polarized input signals contribute to the overall scattering of the
metasurface. For the specular case of first-order differentiation operator,
$R^{\text{cr}}$$($$k_{x}$$=0$$)$=0 meaning that no cross-polarized field
should exist upon illuminating by on-axis plane waves. Mathematically
speaking, Eqs. (13), (15) implies that

rule II:

$\eta\alpha_{\text{ee}}^{xy}\pm\alpha_{\text{em}}^{xx}\pm\alpha_{\text{me}}^{yy}+\frac{1}{\eta}\alpha_{\text{mm}}^{yx}=0$
(s-polarized illumination)

$\eta\alpha_{\text{ee}}^{yx}\mp\alpha_{\text{em}}^{yy}\mp\alpha_{\text{me}}^{xx}+\frac{1}{\eta}\alpha_{\text{mm}}^{xy}=0$
(p-polarized illumination)

must be satisfied, because otherwise, the metasurface may expose non-zero
cross-polarized reflection at $\theta$$=$$0$. In summary, among all possible
types of bianisotropy and longitudinal polarizabilities, specific solutions
are desired which have been comprehensively discussed above. As an important
deduction from Fig. 1b, the reciprocity enforces $R^{s\to p}(\theta)$=$R^{p\to
s}(-\theta)$ meaning that the mirror version of each mathematical operator
realized by $R^{s\to p}$ channel, $\tilde{H}$(kx), will also be constructed by
$R^{p\to s}$ channel, $\tilde{H}$($-$kx). Indeed, the proposed approach is
inherently dual-polarized and can be applied to both $s$\- and $p$-polarized
input signals for many types of transfer functions like spatial
differentiation and integration for which $\tilde{H}$($-$kx) still yields a
usable output wave. This interesting feature which has been rarely reported in
the literature may find great potential applications in dual-polarized optical
computations [30, 16].

Figure 6:  (a) Field profile of the input signal, formed by a combination of
trapezoidal, Gaussian, and sinusoidal functions (b), (c) the s$\to$p and
p$\to$s reflected field profiles, respectively. The exact derivative signals
have been also presented for the sake of comparison. The metasurface is
illuminated from normal direction and the working frequency is $f$=177 THz.

##  III Dual-polarized Optical Signal and Image processing

As known, the macroscopic scattering features of any metasurface is mainly
determined by properties of its constituent meta-atoms in the microscopic
scale [45, 47]. In this section, we demonstrate that the proposed all-
dielectric optical metasurface shown in Fig. 3a exposes the same collective
polarizabilities required for performing first-order spatial differentiation
at the macroscopic view. The meta-atom of Fig. 3a comprises two non-identical
L-shaped dielectric particles with a relative 90-degree rotation. The meta-
atoms are made of silicon (Si) material whose relative permittivity is
extracted from Palik’s book [51]. A comprehensive parametric study has been
accomplished to search for the best group of parameters making the angular
dispersion of the cross-polarized reflection coefficient of the dielectric
particles as close to the desired transfer function as possible. The
dielectric inclusions are periodically repeated with the periodicity of 1.1
$\mu$m and 0.9 $\mu$m along the both x and y directions, respectively. The
symmetry/asymmetry properties of the angular scattering at microscopic scale
has a straightforward relation with the meta-atom structural symmetries. As a
consequence, the geometrical asymmetries of the meta-atoms in both transverse
and normal directions yield a reciprocal bianisotropic metasurface whose CPENP
are effectively excited at oblique incidences [32, 46]. In order to implement
the first-order differentiator, we design an all-dielectric meta-atom deprived
of any geometrical symmetry with respect to $xyz$ coordinates.

Based on the retrieval procedure presented in [52, 46], the in-plane
collective polarizabilities of the designed metasurface have been extracted.
In fact, after the dipole approximation of meta-atoms in a polarizability
retrieval setup, the metasurface is illuminated by four normally incident
plane waves, and the tangential polarizations can be described in terms of
induced dipole moments. By applying an equivalent surface model for the
bianisotropic sheet, the tangential collective polarizabilities can be
calculated as a function of the reflection/transmission coefficients from the
array. The retrieved results are displayed in Figs. 4a-c. Moreover, the zero
cross-polarized reflection condition of the surface polarizabilities at the
normal incidence has been assessed in Fig. 4d. As seen, the rule II is
satisfied in the vicinity of 177 THz at which the metasurface creates no on-
axis cross-polarized fields at the reflection side. Hereafter, we intend to
distinguish which one of the CPENP is provided by the designed metasurface.
After some mathematical manipulations on Eqs. (13), (15) and with applying the
reciprocity constraints, we have:

$\displaystyle\alpha_{\text{em}}^{xz}=-\alpha_{\text{me}}^{zx}=\frac{1}{j2\omega\sin\theta}\left[\left(R_{\theta,+}^{s\to
p}-R_{-\theta,+}^{s\to p}\right)+\left(T_{\theta,+}^{s\to
p}-T_{-\theta,+}^{s\to p}\right)\right]$ (15)
$\displaystyle\alpha_{\text{em}}^{zx}=-\alpha_{\text{me}}^{xz}=\frac{1}{-j2\omega\sin\theta}\left[\left(R_{\theta,+}^{s\to
p}-R_{-\theta,+}^{s\to p}\right)+\left(T_{\theta,-}^{s\to
p}-T_{-\theta,-}^{s\to p}\right)\right]$ (16)
$\displaystyle\eta\alpha_{\text{ee}}^{zy}-\frac{1}{\eta}\alpha_{\text{mm}}^{yz}=\frac{1}{-j2\omega\tan\theta}\left[\left(R_{\theta,+}^{s\to
p}-R_{\theta,-}^{s\to p}\right)-\left(T_{\theta,+}^{s\to p}-T_{\theta,-}^{s\to
p}\right)\right]$ (17)

Four s-polarized oblique plane waves illuminating the metasurface along
forward and backward directions serve to extract some information regarding
the normal components of the collective polarizabilities. By using an
arbitrary small incident wave angle $\theta$$=$$4^{\circ}$, all required
scattering parameters in Eqs. (17)-(19) have been simulated and the left-hand
side parameters are plotted in Fig. 4e. As can be seen, the results clearly
demonstrate that around 177 THz, among eight possible CPENP, the impact of
$\eta\alpha_{\text{ee}}^{zy}-\frac{1}{\eta}\alpha_{\text{mm}}^{yz}$ is more
tangible while the role of $\alpha_{\text{em}}^{zx}$ can be neglected. Indeed,
the engineered geometry of the designed reciprocal meta-atom makes it as a
suitable choice for realizing on-axis asymmetric OTFs in the reflection side.

For instance, the cross-polarized reflection coefficient of the array can be
tailored to mimic the angular trend of R${}^{s\to
p}(\theta)$=$j\zeta$sin$(\theta)$. The parameter $\zeta$ stands for the gain
of differentiator which depends on the value of surface polarizabilites.
Through engineering the meta-atom geometry, one can achieve a cross-polarized
response that approximates the nonlocal behavior of the first-derivative
operation with a phase jump at kx=0. The phase and amplitude of the co- and
cross-polarized reflection coefficients at f=177 THz are plotted in Figs.
5a-c, respectively. The results are numerically recorded by using CST full-
wave commercial software, where periodic boundary conditions are applied to
the x- and y-directed walls and Floquet ports are considered along the z
direction. The first deduction is that the co-polarized reflection coefficient
possesses a symmetric response with respect to $k_{x}$ due to the reciprocity
theorem (Fig. 5a). The second inference is that, as expected from the
presented analysis, the cross-polarized reflection for both s- and p-polarized
incidences, possesses an odd-symmetric spatial trend around kx=0. Owing to the
reciprocity constraint and $\tilde{R}^{s\to p}(k_{x})$=$-\tilde{R}^{s\to
p}(-k_{x})$, the results of Figs. 5b, c differ only in the phase sign. The
exact amplitude and phase of the transfer function representing the first-
order differentiation operation are fairly compared with those synthesized by
the designed metasurface. An excellent agreement between the cross-
polarization reflection of the designed metasurface and the exact transfer
function has been achieved as long as the normalized spatial bandwidth of the
input signals lies within $|k_{x}/k_{0}|$$<$0.25. This range essentially
provides the maximum spatial resolution of input signals that can be correctly
processed by the designed optical metasurface.

To inspect the differentiation performance of the employed metasurafce
processor, a complex signal which is a combination of trapezoidal, Gaussian,
and sinusoidal functions (see Fig. 6a), has been utilized as the beam profile
of the on-axis incidence. The details for the schematic setup are shown in
Fig. 3c. The polarizers are utilized to filter out the unwanted polarization
in each case and the beam splitter is aimed at directing the reflected fields
to the camera. The cross-polarized reflections egressing the metasurface from
s$\to$p and p$\to$s normal processing channels are captured as output signals
and the corresponding results are illustrated Figs. 6b, c, respectively. The
output signals were in this manner: Each plane wave impinging on the
metasurface will be reflected with a certain reflection coefficient
corresponding to its incident wave angle, $\tilde{R}(k_{x})$. In this case,
the angular spectrum of the output fields is given by

$\displaystyle\tilde{g}_{\text{ch1}}(k_{x})=\tilde{R}^{s\to
p}(k_{x})\tilde{f}_{\text{ch1}}(k_{x})$ (18)
$\displaystyle\tilde{g}_{\text{ch2}}(k_{x})=\tilde{R}^{p\to
s}(k_{x})\tilde{f}_{\text{ch2}}(k_{x})$ (19)

in reflection mode. Here, the subscripts denote the number of processing
channel. Thus, the synthesized output fields can be numerically calculated as:

$\displaystyle E_{\text{ref,
ch1}}=\int\limits_{k_{0x}-W}^{k_{0x}+W}{\tilde{R}^{s\to
p}(k_{x})\tilde{f}_{\text{ch1}}^{\text{}}\left({{k}_{x}}\right)}~{}\text{exp}\,(-j{{k}_{x}}{{x}_{\text{}}}-j{{k}_{z}}{{z}_{\text{}}})\,d{{k}_{x\text{}}}$
(20) $\displaystyle E_{\text{ref,
ch2}}=\int\limits_{k_{0x}-W}^{k_{0x}+W}{\tilde{R}^{p\to
s}(k_{x})\tilde{f}_{\text{ch2}}^{\text{}}\left({{k}_{x}}\right)}~{}\text{exp}\,(-j{{k}_{x}}{{x}_{\text{}}}-j{{k}_{z}}{{z}_{\text{}}})\,d{{k}_{x\text{}}}$
(21)

As can be seen, the all-dielectric optical metasurface successfully implements
the first-order differentiation of the input signal at both output
terminations. Indeed, when the input signal is reflected by the metasurface,
the flat regimes are filtered out, the linear segments are converted to flat
ones, the sine is converted to the cosine one, and finally, the descending and
ascending parts of the Gaussian signal are distinguished. To evaluate the
performance of our metasurface differentiator, the numerically-obtained
results are compared with the exact first-derivative of the input signals in
the same figures. The results are exactly the responses expected from a first-
derivative operation with only 2$\%$ error according to
${{e}_{f}}={\left(\left\|s_{\text{out}}^{\text{exact}}(x)-s_{\text{out}}^{\text{simul}}(x)\right\|\right)}/{\left\|s_{\text{out},x}^{\text{ideal}}(x)\right\|}$.
Therefore, the effective role of CPENP in constructing odd-symmetric nonlocal
reflection suggests a new reciprocal way to realize first-order spatial
differentiation for normal illuminations. It should be mentioned that the
designed metasurface is suitable for optical applications [33].

Figure 7:  (a), (c) The s-polarized sinc-shape incident field profile together
with (b), (d) the p-polarized reflected derivative field profile. The exact
derivative signals have been also presented for the sake of comparison. The
reciprocal metasurface is illuminated from normal direction and the working
frequency is $f$=177 THz.  Figure 8:  (a), (c) The p-polarized sinc-shape
incident field profile together with (b), (d) the s-polarized reflected
derivative field profile. The exact derivative signals have been also
presented for the sake of comparison. The reciprocal metasurface is
illuminated from normal direction and the working frequency is $f$=177 THz.

The metasurface processor is dual-polarized and the processing channel can
also carry a couple of input/output signals, separately. Two distinct 1D
signals (sinc and sinusoidal functions) representing the beam-profiles of s-
and p-polarized on-axis incidences are considered as the input signals (see
Figs. 7a, c and Figs. 8a, c). As can be noticed from Figs. 7b, d and Figs. 8b,
d, both orthogonal channels can separately serve to construct the first-order
derivative of the corresponding input fields at the output termination.

One of the other advanced applications of the analog optical computing is
image processing. Among all possible processing functions, edge detection
plays a crucial role in the image segmentation and the other image pre-
processing steps [27, 25, 16]. To recognize objects inside a specified image,
the spatial differentiation enables us to extract the boundaries between two
regions of different intensities [27]. Medical and biological image processing
has been gaining significant attention for different medical imaging
modalities including X-ray, CT, and MRI [53, 54, 27]. Recently, the image
processing of biological or ordinary holograms such as various contrast
filters and edge detection schemes can be found in literature [27, 14, 14, 28,
25]. In this way, we project two differently-polarized images including
ordinary and medical ones from normal direction to the designed all-dielectric
metasurface differentiator each of which is carried by one of the processing
channels (see Figs. 9a, b). The reflected images are numerically simulated and
the corresponding transverse field profiles are displayed in right side of
(see Figs. 9). As expected, the 1D edge-detector metasurface successfully
exposes all outlines of the normally incident image along the vertical
orientations, exhibiting its higher sensibility to fine details as a first-
order derivative operator. Besides, it reveals a dual-polarized processing
ability. This is while all the prevalent GF-based edge detection schemes fail
to operate in on-axis scenarios [13, 17] or need bulky Fourier lenses to do
their mission upon illuminating by normally input signals [17].

Figure 9:  Edge Detection of images by exploiting the proposed all-dielectric
metasurface processor. (a), (b) Photograph of the s- and p-polarized images
whose edges are aimed to be detected. Resulting edge-detected images when 1D
all-dielectric metasurface differentiates the input image along x direction,
revealing its vertical edges.

##  IV Conclusion

To conclude, for the first time, we exploited the full macroscopic potential
of a reciprocal bianisotropic metasurface to realize dual-polarized asymmetric
OTFs at on-axis reflective processing channels. We analytically demonstrated
how the cross-polarization channel governed by the CPENP can simply break the
angular reflection of the metasurface around $\theta$=0 and mimic the kx-
asymmetric spatially-dispersive transfer functions. Motivating by the existing
relation between the angular scattering and geometrical mirror symmetries in
the microscopic scale, the realization possibility of the proposed design was
assessed through proposing an all-dielectric meta-atom and retrieving its
important surface polarizabilities. The numerical simulations illustrated that
the proposed metasurface reflects the first-derivative of the input signals,
with either $s$ or $p$ polarization, coming from the normal direction, without
using any bulky Fourier lens. The 1D edge detection performance of the
metasurface differentiator was also evaluated where it successfully exposed
all boundaries along vertical direction. The analytical design method
presented in this paper can be extended to 2D scenarios by involving ky
wavenumber into the relations and obtaining explicit or implicit expression
for $\tilde{R}$(kx,ky). Therefore, the designed all-dielectric metasurface
offers a simple and ultra-thin normally-oriented channel to perform optical
signal/image processes without complicated settings arising from oblique
illuminating setups or resorting to nonreciprocal media. Meanwhile, the
proposed reflective processing system is more compatible with the current
demands on the integrated devices than the transmissive ones while offering a
promising vision for realizing parallel computations using both reflection and
transmission channels.

Acknowledgements - A.M. and R.F. acknowledge support from the Swiss National
Science Foundation (SNSF) under the Eccellenza award number 181232.

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