# Towards Analytical Electromagnetic Models for Reconfigurable Intelligent
Surfaces

Tiebin Mi, Jianan Zhang, Rujing Xiong, Zhengyu Wang, Robert Caiming Qiu T. Mi,
J. Zhang, R. Xiong, Z. Wang, and R. Qiu are with the School of Electronic
Information and Communications, Huazhong University of Science and Technology,
Wuhan 430074, China (e-mail:
{mitiebin,zhangjn,rujing,wangzhengyu,caiming}@hust.edu.cn).

###### Abstract

Physically accurate and mathematically tractable models are presented to
characterize scattering and reflection properties of reconfigurable
intelligent surfaces (RISs). We take continuous and discrete strategies to
model a single patch and patch array and their interactions with multiple
incident electromagnetic (EM) waves. The proposed models consider the effect
of the incident and scattered angles, polarization features, and the topology
and geometry of RISs. Particularly, a simple system of linear equations can
describe the multiple-input multiple-output (MIMO) behaviors of RISs under
reasonable assumptions. It can serve as a fundamental model for analyzing and
optimizing the performance of RIS-aided systems in the far-field regime. The
proposed models are employed to identify the advantages and limitations of
three typical configurations. One important finding is that complicated beam
reshaping functionality can not be endowed by popular phase compensation
configurations. A possible solution is the simultaneous configurations of
collecting area and phase shifting. Numerical simulations verify the
effectiveness of the proposed configuration schemes.

###### Index Terms:

Reconfigurable intelligent surfaces, system model, scattering, far-field,
phase compensation, arbitrary beam reshaping.

##  I Introduction

Reconfigurable intelligent surfaces (RISs) have received considerable
attention for their potential to enhance the communication and sensing
performance of wireless networks [1, 2, 3, 4, 5]. RISs consist of an array of
tunable electromagnetic (EM) unit cells [6]. Such a tuning varies widely in
the literature, among which switch diode and continuous tuning varactor based
controlling are popular [7, 8, 9, 10]. The scattering and reflection
properties of EM waves impinging upon RISs can be controlled by software [3,
11, 12, 13], depending on configuration, topology, and geometry.

For communication purposes, RISs are employed to offer coverage enhancement by
signal-to-noise ratio (SNR) improvement and interference suppression. Channel
estimations for RIS-aided communication were discussed in [14, 15, 16, 17].
The authors of [2, 18, 19] showed that cooperative active and passive
beamforming could significantly improve the coverage and energy consumption.
The impact of discrete phase shifting was studied in [20, 21, 22]. On the
other hand, for environmental awareness purposes, RISs are expected to offer
high-resolution sensing and localization capabilities. RISs can be equipped
with embedded sensors or active unit cells [23, 24, 25, 26]. A sensing scheme
was studied in [27], which minimized the Cramér-Rao lower bound for high
accuracy direction estimation.

A fundamental component of analyzing and optimizing RIS-aided systems is the
development of simple but accurate models to characterize the scattering and
reflection properties. There are two possible methodologies to model the
interactions of RISs to EM waves impinging on them.

The first class, taking a continuous perspective, treats RISs as continuous
surfaces subject to appropriate electromagnetic boundary conditions [28, 29,
30, 31]. The field supported by RISs is strictly governed by Maxwell’s
equations and the associated boundary conditions. This class is often used to
identify the theoretical limits of performance independent of the specific
implementation of RISs. However, the strategy is not particularly useful for
the analysis and design because it is difficult to establish the relationship
between boundary conditions and unit cell pattern, configuration, topology,
and geometry [32].

The second class, taking a discrete perspective, calculates the response of
RISs as a summation of the scattering from all unit cells [2, 8, 33]. A widely
used but greatly oversimplified model is to assume that unit cells act as
diffusive scatterers able to change the amplitude and phase of the incident EM
waves, i.e.,

$E^{s}=\gamma e^{j\Omega}E^{i},$ (I.1)

where $E^{i}$ and $E^{s}$ are the incident and reflected scalar electric
fields, respectively, $\gamma\in(0,1]$ is the reflection coefficient, and
$\Omega\in[0,2\pi)$ is the phase shifting to be optimized [34, 35].

Moreover, for the transmission model of RIS-aided systems, consider
communication from a single-antenna source to a single-antenna destination as
an example. Suppose the RIS consists of $N$ unit cells. The deterministic
channel from the transmitter to RIS is denoted by
$\mathbf{h}^{i}=[h^{i}_{1},\cdots,h^{i}_{N}]^{\top}$. The channel between RIS
and receiver is denoted by
$\mathbf{h}^{s}=[h^{s}_{1},\cdots,h^{s}_{N}]^{\top}$. The configuration is
represented by a diagonal matrix
$\mathbf{W}=\text{diag}\bigl{(}\gamma_{1}e^{j\Omega_{1}},\cdots,\gamma_{N}e^{j\Omega_{N}}\bigr{)}$.
The received signal at the destination is given by the summation of the
scattered fields, i.e.,

$y=\sum_{n=1}^{N}\gamma_{n}e^{j\Omega_{n}}h^{i}_{n}h^{s}_{n}x={\mathbf{h}^{s}}^{\top}\mathbf{W}\mathbf{h}^{i}x,$
(I.2)

where $x$ is the transmitted signal at the source.

Most widely used models, e.g., (I.1) and (I.2), are typically rudimentary to
facilitate theoretical analysis and numerical simulation. A clear disadvantage
of (I.1) is that it is not derived from the physics and electromagnetic points
of view. The interaction with incident EM waves (e.g., angles, frequencies,
polarizations) is entirely ignored. The main weakness of (I.2) is that it does
not contain any information about the topology and geometry of RISs. The model
assumes that channel states are perfectly known, which becomes quite
challenging without dedicated signal processing capability at the passive unit
cells.

A significant limitation of current research is the lack of physically
accurate and mathematically tractable models to describe the input/output
behaviors of RISs. A complete description of the fields supported by RISs at
any point in space requires Maxwell’s equations and the associated boundary
conditions. The electromagnetic nature requires us to rethink models from
_first principles_ to ensure performance prediction and algorithmic design
build on a solid foundation.

Inspired by the two classes of methodologies mentioned above, this paper uses
continuous and discrete strategies to model a single patch and patch array.
Specifically, by leveraging physical optics, which approximate boundary
conditions that concern fields on a closed surface [36], we introduce a
physically accurate formula for the electric field scattered by a rectangular
metallic patch with near-zero thickness. This strategy is appropriate because
unit cells below 10 GHz are mainly based on simple planar shapes [37], e.g.,
rectangular patches [9, 7, 38, 39].

On the other hand, for patch arrays, the phase discrepancies introduced by
configuration and interelement path length difference are essential for
beamforming functionality. The path length differences are closely related to
the topology and geometry of RISs. We derive mathematically tractable models
to characterize the overall input/output behaviors of RISs by applying the
superposition principle on the fields scattered by all unit cells. The
proposed models are both physically rigorous and easy to calculate. They can
serve as fundamental models for analyzing and optimizing the performance of
RIS-aided systems.

###  I-A Contributions

The principal contribution of this work is to provide mathematically concise
models to characterize the scattering and reflection properties of RISs
consistent with electromagnetic theory. We focus on analytical methods to
model a single patch and patch array and their interactions with multiple
incident EM waves. Perhaps the most important finding is that a simple system
of linear equations can describe the multiple-input multiple-output (MIMO)
behaviors of RISs under reasonable assumptions. To the best of our knowledge,
no studies have been performed to investigate the MIMO behaviors of RISs. It
can serve as a fundamental model for analyzing and optimizing the performance
of RIS-aided systems in the far-field regime.

The development of physically motivated and mathematically tractable models
facilitates theoretical analysis. The second contribution is that the proposed
models identify the advantages and limitations of three typical
configurations. We clarify that the anomalous mirror effect is a significant
limitation of continuous phase compensation configuration, especially when
multiple incident EM waves exist. Another important finding is that a possible
solution to complicated beam reshaping functionality is the simultaneous
configurations of collecting area and phase shifting. Such configurations have
the power to redistribute the incident waves to (almost) arbitrary radiation
patterns.

###  I-B Outline

The remainder of this paper is organized as follows. We start by considering
how a unit cell responds to incident EM waves in Section II. Physical optics
techniques are employed to calculate the bistatic scattered field of a single
patch. Section III is devoted to the interaction of a patch array with
external illuminations. Section IV focuses on analytical models of linear
RISs. One interesting finding is that, under reasonable assumptions, a system
of linear equations can describe the MIMO behavior of RISs. Section V studies
the performance of linear RISs with three typical configurations. We identify
that the anomalous mirror effect is the inherent limitation of continuous
phase compensation configuration.

###  I-C Reproducible Research

The simulation results can be reproduced using code available at:
https://github.com/mitiebin/RIS_Models.git

###  I-D Notation

We describe the electric and magnetic fields in a spherical coordinate system.
The strength of the incident electric field is denoted by
$E^{i}(\theta^{i},\phi^{i})$. The boldface letters stand for vector fields.
For example, we denote by $\mathbf{E}^{s}(r^{s},\theta^{s},\phi^{s})$ the
scattered electric field at $(r^{s},\theta^{s},\phi^{s})$. We will write it
simply $\mathbf{E}^{s}$ when no confusion can arise. The corresponding
$(r,\theta,\phi)$ components are denoted by $E^{s}_{r}$, $E^{s}_{\theta}$ and
$E^{s}_{\phi}$, respectively. We denote by $\lambda$ the wavelength throughout
the paper. A Cartesian coordinate system is used to describe the topology and
geometry of RISs. We use $\mathbf{e}_{x}$, $\mathbf{e}_{y}$, and
$\mathbf{e}_{z}$ to denote the unit vectors in $x$, $y$, and $z$ directions.

##  II Scattering from a single planar unit cell

In this section, we consider the fundamental principles that govern the
interaction of each unit cell with incident EM fields. To characterize the
scattered fields observed from multiple angles, models taking into account the
incident and reflective directions simultaneously are necessary. Borrowing the
concept of bistatic scattering cross section [36, 40], we begin by calculating
the bistatic scattered field of a single patch. We assume throughout the paper
that the incident field consists of multiple uniform plane waves (narrow
banded) along with various directions. In the far-field regime, the electric
and magnetic fields at any point are perpendicular to each other and the
propagation direction.

###  II-A The bistatic scattered field from a rectangular patch

For the convenience of modeling, this paper assumes that unit cells are
approximated well by rectangular perfect electric conducting plates [9, 7, 38,
39]. We first consider a patch illuminated by one incident EM wave originating
from $(\theta^{i},\phi^{i})$. The strength/magnitude of the incident electric
field is denoted by $E^{i}(\theta^{i},\phi^{i})$. We will write it simply
$E^{i}$ when no confusion can arise. The incident wave is linearly polarized
with perpendicular polarization, as illustrated in Fig. 1. The scattered field
can be evaluated using physical optics techniques by neglecting the edge
effects (suppose the induced current density on the patch is the same as that
on an infinite conducting flat plate).

Figure 1: A rectangular metallic patch of size $a\times b$ is illuminated by
EM waves, which are linearly polarized with perpendicular polarizations. The
scattered electric field is observed at $(r^{s},\theta^{s},\phi^{s})$. (a) One
incident wave. (b) Muliple incident waves originating from
${(\theta^{i}_{1},\phi^{i}_{1}),\ldots,(\theta^{i}_{M},\phi^{i}_{M})}$.

######  Theorem II.1.

Suppose a patch of size $a\times b$ is illuminated by one uniform plane wave
$E^{i}(\theta^{i},\phi^{i})$. Let $\Gamma$ be the reflection coefficient. In
the far-field regime, the $(r,\theta,\phi)$ components of the scattered
electric field $\mathbf{E}^{s}$ at $(r^{s},\theta^{s},\phi^{s})$ are given as

$\begin{dcases*}\begin{aligned}
E^{s}_{r}(r^{s},\theta^{s},\phi^{s})\simeq&0,\\\
E^{s}_{\theta}(r^{s},\theta^{s},\phi^{s})\simeq&C\frac{A}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}\cos\theta^{i}\cos\theta^{s}\bigl{(}\cos\phi^{i}\sin\phi^{s}-\sin\phi^{i}\cos\phi^{s}\bigr{)}\text{Sa}(a,b;\theta^{s},\phi^{s};\theta^{i},\phi^{i}),\\\
E^{s}_{\phi}(r^{s},\theta^{s},\phi^{s})\simeq&C\frac{A}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}\cos\theta^{i}\bigl{(}\sin\phi^{i}\sin\phi^{s}+\cos\phi^{i}\cos\phi^{s}\bigr{)}\text{Sa}(a,b;\theta^{s},\phi^{s};\theta^{i},\phi^{i}),\end{aligned}\end{dcases*}$
(II.1)

where $A=ab$, $C=-j(1-\Gamma)/2$, and

$\displaystyle\text{Sa}(a,b;\theta^{s},\phi^{s};\theta^{i},\phi^{i})=\frac{\sin\bigl{(}\frac{\pi
a}{\lambda}\bigl{(}\sin\theta^{s}\cos\phi^{s}+\sin\theta^{i}\cos\phi^{i}\bigr{)}\bigr{)}}{\frac{\pi
a}{\lambda}\bigl{(}\sin\theta^{s}\cos\phi^{s}+\sin\theta^{i}\cos\phi^{i}\bigr{)}}\frac{\sin\bigl{(}\frac{\pi
b}{\lambda}(\sin\theta^{s}\sin\phi^{s}+\sin\theta^{i}\sin\phi^{i})\bigr{)}}{\frac{\pi
b}{\lambda}(\sin\theta^{s}\sin\phi^{s}+\sin\theta^{i}\sin\phi^{i})}.$

###### Proof.

See Appendix A. ∎

######  Remark II.2.

For the role of the term $e^{-j2\pi r^{s}/\lambda}/r^{s}$ in (II.1), we find
that $e^{-j2\pi r^{s}/\lambda}$ actually represents the propagation delay due
to the distance $r^{s}$. It is well known that, for narrow banded signals, the
propagation delays generally correspond to phase shifts. Another property of
EM waves is that the electric field is expected to decrease as $1/r^{s}$ in
free space as the distance $r^{s}$ increases. The term $1/r^{s}$ thus serves
as an attenuation factor.

######  Remark II.3.

As expected, the strength of the scattered electric field is proportional to
plate area $A$. At the receiving stage, the patch converts the incident power
density into received power by the collecting area. We emphasize that the unit
cell commonly consists of sub-wavelength-scaled patterns. The actual
collecting/effective area is not equal but closely related to its physical
size.

######  Remark II.4.

The term $\text{Sa}(a,b;\theta^{s},\phi^{s};\theta^{i},\phi^{i})$ essentially
characterizes the intrinsic directivity of the patch. When $a\ll\lambda$ and
$b\ll\lambda$, $\text{Sa}(a,b;\theta^{s},\phi^{s};\theta^{i},\phi^{i})\simeq
1$. That implies that a unit cell behaves like an isotropic point source
without a strong intrinsic scattering directivity when sub-wavelength-sized
(e.g., $\lambda/5\times\lambda/5$ or even small). On the other hand, when
$a\gg\lambda$ and $b\gg\lambda$, the patch performs specular reflection
(Snell’s law), although it has multiple reflection lobes. The reason is that,
if we choose $\theta^{s}=\theta^{i}$ and $\phi^{s}=\phi^{i}\pm\pi$,
$\text{Sa}(a,b;\theta^{s},\phi^{s};\theta^{i},\phi^{i})=1$.

With Theorem (II.1), the strength of the scattered electric field at
$(r^{s},\theta^{s},\phi^{s})$ is

$\displaystyle|\mathbf{E}^{s}(r^{s},\theta^{s},\phi^{s})|=$
$\displaystyle\sqrt{(E^{s}_{r}(r^{s},\theta^{s},\phi^{s}))^{2}+(E^{s}_{\theta}(r^{s},\theta^{s},\phi^{s}))^{2}+(E^{s}_{\phi}(r^{s},\theta^{s},\phi^{s}))^{2}}$
(II.2) $\displaystyle\simeq$ $\displaystyle|C|\frac{A}{\lambda
r^{s}}E^{i}(\theta^{i},\phi^{i})\cos\theta^{i}\bigl{|}\text{Sa}(a,b;\theta^{s},\phi^{s};\theta^{i},\phi^{i})\bigr{|}$
$\displaystyle\sqrt{\cos^{2}\theta^{s}\bigl{(}\cos\phi^{i}\sin\phi^{s}-\sin\phi^{i}\cos\phi^{s}\bigr{)}^{2}+\bigl{(}\sin\phi^{i}\sin\phi^{s}+\cos\phi^{i}\cos\phi^{s}\bigr{)}^{2}}.$

The bistatic RCS is then given as

$\displaystyle\sigma(\theta^{s},\phi^{s};\theta^{i},\phi^{i})=$
$\displaystyle\lim_{r^{s}\to\infty}4\pi(r^{s})^{2}\frac{|\mathbf{E}^{s}(r^{s},\theta^{s},\phi^{s})|^{2}}{E^{i}(\theta^{i},\phi^{i})^{2}}$
(II.3) $\displaystyle=$ $\displaystyle
4\pi|C|^{2}\bigl{(}\frac{A}{\lambda}\bigr{)}^{2}\cos^{2}\theta^{i}\bigl{[}\cos^{2}\theta^{s}\bigl{(}\cos\phi^{i}\sin\phi^{s}-\sin\phi^{i}\cos\phi^{s}\bigr{)}^{2}$
$\displaystyle+\bigl{(}\sin\phi^{i}\sin\phi^{s}+\cos\phi^{i}\cos\phi^{s}\bigr{)}^{2}\bigr{]}\text{Sa}^{2}(a,b;\theta^{s},\phi^{s};\theta^{i},\phi^{i}).$

Multiple comparisons are made using CST Studio Suite to evaluate the
performance of the proposed models (II.1) and (II.2). CST is a high-
performance package for analyzing EM components and systems. The plot data
from CST are exported and reproduced in Matlab for better illustration. We
demonstrate in Fig. 2 the principal plane bistatic RCS of a square patch of
size $5\lambda\times 5\lambda$, where $f=1/\lambda=3.3\text{GHz}$,
$\theta^{i}=0^{\circ}$ and $\phi^{i}=0^{\circ}$. The blue curves are the RCS
raised by the proposed formula (II.3). The dash-dotted red curves are the RCS
solved by CST. The 3D RCS plots are also illustrated in Fig. 3. The
comparisons show a good agreement, especially for the side lobes close to the
main lobe, which validate the effectiveness of the proposed models.

Figure 2: Principal plane bistatic RCS of a square patch of size
$5\lambda\times 5\lambda$. The blue curves are the RCS raised by the proposed
model (II.3). The dash-dotted red curves are the RCS solved by CST. The
comparisons show a good agreement, especially for the side lobes close to the
main lobe. (a) $xoz$-plane ($\phi^{s}=0^{\circ},180^{\circ}$). (b) $yoz$-plane
($\phi^{s}=90^{\circ},270^{\circ}$).

Figure 3: The 3D bistatic RCS of a square patch. The comparisons show a good
agreement, especially for the side lobes close to the main lobe. (a) The RCS
raised by the proposed model (II.3). (b) The RCS solved by CST.

###  II-B Scattering introduced by multiple incident waves

Several EM waves may illuminate the patch simultaneously, especially when the
spatially-continuous field consists of several principal incident waves, as
demonstrated in Fig. 1. The total electric current density induced on the
plate can be obtained by the superposition of the individual fields related to
each incident wave.

######  Proposition II.5.

Suppose a patch of size $a\times b$ is illuminated by $M$ uniform plane waves.
The strengths of the incident EM waves are
$E^{i}(\theta^{i}_{m},\phi^{i}_{m}),m=1,\ldots,M$. The scattered electric
field observed at $(r^{s},\theta^{s},\phi^{s})$ is given as

$\begin{dcases*}\begin{aligned}
E^{s}_{r}(r^{s},\theta^{s},\phi^{s})\simeq&0,\\\
E^{s}_{\theta}(r^{s},\theta^{s},\phi^{s})\simeq&C\frac{A}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}\sum_{m=1}^{M}E^{i}(\theta^{i}_{m},\phi^{i}_{m})\cos\theta^{i}_{m}\cos\theta^{s}\bigl{(}\cos\phi^{i}_{m}\sin\phi^{s}-\sin\phi^{i}_{m}\cos\phi^{s}\bigr{)}\\\
&\text{Sa}(a,b;\theta^{i}_{m},\phi^{i}_{m};\theta^{s},\phi^{s}),\\\
E^{s}_{\phi}(r^{s},\theta^{s},\phi^{s})\simeq&C\frac{A}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}\sum_{m=1}^{M}E^{i}(\theta^{i}_{m},\phi^{i}_{m})\cos\theta^{i}_{m}\bigl{(}\sin\phi^{i}_{m}\sin\phi^{s}+\cos\phi^{i}_{m}\cos\phi^{s}\bigr{)}\\\
&\text{Sa}(a,b;\theta^{i}_{m},\phi^{i}_{m};\theta^{s},\phi^{s}).\end{aligned}\end{dcases*}$

We illustrate in Fig. 4 the scattered electric fields of a square patch of
size $5\lambda\times 5\lambda$ illuminated by two incident waves originating
from $(\theta^{i}_{1}=15^{\circ},\phi^{i}_{1}=-45^{\circ})$ and
$(\theta^{i}_{2}=45^{\circ},\phi^{i}_{2}=135^{\circ})$, respectively. Let
$E^{i}(\theta^{i}_{1},\phi^{i}_{1})=1$ and
$E^{i}(\theta^{i}_{2},\phi^{i}_{2})=0.5$. The normalized scattered electric
field is plotted in Fig. 4. The 3D scattered field is plotted in Fig. 4. For
better illustration, we also plot the incident uniform plane waves as two
spikes. These results indicate that a patch of large size performs specular
reflections.

Figure 4: The scattered electric fields of a square patch of size
$5\lambda\times 5\lambda$ illuminated by two incident waves originating from
$(\theta^{i}_{1}=15^{\circ},\phi^{i}_{1}=-45^{\circ})$ and
$(\theta^{i}_{2}=45^{\circ},\phi^{i}_{2}=135^{\circ})$, respectively. (a) The
normalized scattered electric field. For better illustration, the two incident
uniform plane waves are plotted as two spikes. (b) The 3D scattered electric
field.

##  III Phase discrepancy due to multiple unit cells

This section focuses on the interaction of patch arrays with external
illuminations. For ease of description, we limit the discussion to RISs
consisting of multiple rectangular metallic patches with near-zero thickness.
We emphasize that the phase discrepancies corresponding to unit cells are
essential for various beamforming functionalities. There are two equivalent
ways to introduce phase discrepancies. One is the phase shifting due to the
configuration. The other is the interelement path length difference, closely
related to the topology and geometry of RISs.

A Cartesian coordinate system is used to describe the topology and geometry.
Assume that a planar or even conformal RIS, consisting of multiple rectangular
metallic patches located at $\mathbf{p}_{n}=[x_{n},y_{n},z_{n}]^{\top}$,
$n=1,\ldots,N$, is illuminated by one uniform plane wave originating from
$(\theta^{i},\phi^{i})$. We first address the phase discrepancy related to the
$n$-th unit cell, whose collecting area is $A_{n}$.

Suppose there is a virtual unit cell located at the origin. We denote by
$\mathbf{E}_{\mathbf{o}}^{s}(r^{s},\theta^{s},\phi^{s};t)$ the scattered time-
varying electric field through this virtual patch. Let
$d_{\mathbf{p}_{n}}=d_{\mathbf{p}_{n}}^{i}+d_{\mathbf{p}_{n}}^{s}$ stand for
the path length difference between the origin and $\mathbf{p}_{n}$, as
illustrated in Fig. 5. With the narrowband assumption, the scattered electric
field of the unit cell at $\mathbf{p}_{n}$ can be written as

$\displaystyle\mathbf{E}_{\mathbf{p}_{n}}^{s}(r^{s},\theta^{s},\phi^{s};t)=\mathbf{E}_{\mathbf{o}}^{s}(r^{s},\theta^{s},\phi^{s};t-d_{\mathbf{p}_{n}}/c)\simeq\mathbf{E}_{\mathbf{o}}^{s}(r^{s},\theta^{s},\phi^{s};t)e^{-j2\pi
d_{\mathbf{p}_{n}}/\lambda}.$ (III.1)

As discussed in Section II-A, the complex exponential term $e^{-j2\pi
d_{\mathbf{p}_{n}}/\lambda}$ is the propagation delay created by the incident
and scattered path length difference.

To calculate $d^{i}_{\mathbf{p}_{n}}$ and $d^{s}_{\mathbf{p}_{n}}$, let
$\mathbf{u}(\theta,\phi)=[\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta]^{\top}$.
We can figure out

$\begin{dcases*}\begin{aligned}
d_{\mathbf{p}_{n}}^{i}=&-\bigl{(}\mathbf{e}_{x}x_{n}+\mathbf{e}_{y}y_{n}+\mathbf{e}_{z}z_{n}\bigr{)}\cdot\bigl{(}\mathbf{e}_{x}\sin\theta^{i}\cos\phi^{i}+\mathbf{e}_{y}\sin\theta^{i}\sin\phi^{i}+\mathbf{e}_{z}\cos\theta^{i}\bigr{)}=-\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{i},\phi^{i}),\\\
d_{\mathbf{p}_{n}}^{s}=&-\bigl{(}\mathbf{e}_{x}x_{n}+\mathbf{e}_{y}y_{n}+\mathbf{e}_{z}z_{n}\bigr{)}\cdot\bigl{(}\mathbf{e}_{x}\sin\theta^{s}\cos\phi^{s}+\mathbf{e}_{y}\sin\theta^{s}\sin\phi^{s}+\mathbf{e}_{z}\cos\theta^{s}\bigr{)}=-\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{s},\phi^{s}).\end{aligned}\end{dcases*}$

Combining (II.1) and (III.1) yields the $(r,\theta,\phi)$ components of the
scattered electric field of the unit cell at $\mathbf{p}_{n}$

$\begin{dcases*}\begin{aligned}
E^{s}_{\mathbf{p}_{n},r}(r^{s},\theta^{s},\phi^{s})\simeq&0,\\\
E^{s}_{\mathbf{p}_{n},\theta}(r^{s},\theta^{s},\phi^{s})\simeq&C\frac{A_{n}}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}(\theta^{i},\phi^{i})\cos\theta^{i}\cos\theta^{s}\bigl{(}\cos\phi^{i}\sin\phi^{s}-\sin\phi^{i}\cos\phi^{s}\bigr{)}\\\
&\text{Sa}(a_{n},b_{n};\theta^{s},\phi^{s};\theta^{i},\phi^{i})e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{i},\phi^{i})/\lambda}e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{s},\phi^{s})/\lambda},\\\
E^{s}_{\mathbf{p}_{n},\phi}(r^{s},\theta^{s},\phi^{s})\simeq&C\frac{A_{n}}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}(\theta^{i},\phi^{i})\cos\theta^{i}\bigl{(}\sin\phi^{i}\sin\phi^{s}+\cos\phi^{i}\cos\phi^{s}\bigr{)}\\\
&\text{Sa}(a_{n},b_{n};\theta^{s},\phi^{s};\theta^{i},\phi^{i})e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{i},\phi^{i})/\lambda}e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{s},\phi^{s})/\lambda}.\end{aligned}\end{dcases*}$

Figure 5: Phase discrepancy due to the interelement path length difference. A
RIS is illuminated by one uniform plane wave (in blue) originating from
$(\theta^{i},\phi^{i})$. The scattered electric field is observed at
$(r^{s},\theta^{s},\phi^{s})$ (in red). The incident and scattered path length
differences between the origin and $\mathbf{p}_{n}$ are
$d_{\mathbf{p}_{n}}^{i}=-\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{i},\phi^{i})$
and
$d_{\mathbf{p}_{n}}^{s}=-\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{s},\phi^{s})$,
respectively.

On the other hand, the sub-wavelength-scaled pattern of each unit cell is used
to change the value of the surface current density. We denote by
$e^{j\Omega_{n}},n=1,\ldots,N$, the phase shifting operations. The total
electric field can be calculated by the superposition of the individual
fields.

######  Proposition III.1.

Suppose a RIS, consisting of multiple rectangular patches located at
$\mathbf{p}_{n}=[x_{n},y_{n},z_{n}]^{\top}$, $n=1,\ldots,N$, is illuminated by
one uniform plane wave $E^{i}(\theta^{i},\phi^{i})$. The scattered electric
field observed at $(r^{s},\theta^{s},\phi^{s})$ is given as

$\begin{dcases*}\begin{aligned}
E^{s}_{r}(r^{s},\theta^{s},\phi^{s})\simeq&0,\\\
E^{s}_{\theta}(r^{s},\theta^{s},\phi^{s})\simeq&C\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}(\theta^{i},\phi^{i})\cos\theta^{i}\cos\theta^{s}\bigl{(}\cos\phi^{i}\sin\phi^{s}-\sin\phi^{i}\cos\phi^{s}\bigr{)}\\\
&\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j\Omega_{n}}\text{Sa}(a_{n},b_{n};\theta^{s},\phi^{s};\theta^{i},\phi^{i})e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{i},\phi^{i})/\lambda}e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{s},\phi^{s})/\lambda},\\\
E^{s}_{\phi}(r^{s},\theta^{s},\phi^{s})\simeq&C\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}(\theta^{i},\phi^{i})\cos\theta^{i}\bigl{(}\sin\phi^{i}\sin\phi^{s}+\cos\phi^{i}\cos\phi^{s}\bigr{)}\\\
&\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j\Omega_{n}}\text{Sa}(a_{n},b_{n};\theta^{s},\phi^{s};\theta^{i},\phi^{i})e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{i},\phi^{i})/\lambda}e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{s},\phi^{s})/\lambda}.\end{aligned}\end{dcases*}$
(III.2)

The strength of the scattered electric field at $(r^{s},\theta^{s},\phi^{s})$
is

$\displaystyle|\mathbf{E}^{s}(r^{s},\theta^{s},\phi^{s})|=$
$\displaystyle\sqrt{(E^{s}_{r}(r^{s},\theta^{s},\phi^{s}))^{2}+(E^{s}_{\theta}(r^{s},\theta^{s},\phi^{s}))^{2}+(E^{s}_{\phi}(r^{s},\theta^{s},\phi^{s}))^{2}}$
(III.3) $\displaystyle\simeq$
$\displaystyle|C|\frac{E^{i}(\theta^{i},\phi^{i})}{r^{s}}\cos\theta^{i}\sqrt{\cos^{2}\theta^{s}\bigl{(}\cos\phi^{i}\sin\phi^{s}-\sin\phi^{i}\cos\phi^{s}\bigr{)}^{2}+\bigl{(}\sin\phi^{i}\sin\phi^{s}+\cos\phi^{i}\cos\phi^{s}\bigr{)}^{2}}$
$\displaystyle\Bigl{|}\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j\Omega_{n}}\text{Sa}(a_{n},b_{n};\theta^{s},\phi^{s};\theta^{i},\phi^{i})e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{i},\phi^{i})/\lambda}e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{s},\phi^{s})/\lambda}\Bigr{|}.$

The bistatic RCS is given as

$\displaystyle\sigma(\theta^{s},\phi^{s};\theta^{i},\phi^{i})=$
$\displaystyle\lim_{r^{s}\to\infty}4\pi(r^{s})^{2}\frac{|\mathbf{E}^{s}(r^{s},\theta^{s},\phi^{s})|^{2}}{E^{i}(\theta^{i},\phi^{i})^{2}}$
(III.4) $\displaystyle\simeq$ $\displaystyle
4\pi|C|^{2}\cos^{2}\theta^{i}\bigl{[}\cos^{2}\theta^{s}\bigl{(}\cos\phi^{i}\sin\phi^{s}-\sin\phi^{i}\cos\phi^{s}\bigr{)}^{2}+\bigl{(}\sin\phi^{i}\sin\phi^{s}+\cos\phi^{i}\cos\phi^{s}\bigr{)}^{2}\bigr{]}$
$\displaystyle\Bigl{|}\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j\Omega_{n}}\text{Sa}(a_{n},b_{n};\theta^{s},\phi^{s};\theta^{i},\phi^{i})e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{i},\phi^{i})/\lambda}e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{s},\phi^{s})/\lambda}\Bigr{|}^{2}.$

######  Remark III.2.

The proposed equation (III.3) exhibits the inter-unit cooperative beamforming
gain, the most crucial feature of RIS. The beamforming gain can compensate for
the power loss over the distance. A general rule of thumb from (III.3) is that
if the scattered strength at the distance $r^{s}$ is of the same order as the
incident strength, we must have $\lambda r^{s}/\sum A_{n}=\mathcal{O}(1)$.

When there exist multiple incident waves illuminate the RIS, combining the
individual scattered fields yields the overall scattered electric field.

######  Proposition III.3.

Suppose a RIS, consisting of multiple rectangular patches located at
$\mathbf{p}_{n}=[x_{n},y_{n},z_{n}]^{\top}$, $n=1,\ldots,N$, is illuminated by
multiple uniform plane waves. The strengths of the incident EM waves are
$E^{i}(\theta^{i}_{m},\phi^{i}_{m}),m=1,\ldots,M$. The scattered electric
field observed at $(r^{s},\theta^{s},\phi^{s})$ is given as

$\begin{dcases*}\begin{aligned}
E^{s}_{r}(r^{s},\theta^{s},\phi^{s})\simeq&0,\\\
E^{s}_{\theta}(r^{s},\theta^{s},\phi^{s})\simeq&C\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}\sum_{m=1}^{M}E^{i}(\theta^{i}_{m},\phi^{i}_{m})\cos\theta^{i}_{m}\cos\theta^{s}\bigl{(}\cos\phi^{i}_{m}\sin\phi^{s}-\sin\phi^{i}_{m}\cos\phi^{s}\bigr{)}\\\
&\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j\Omega_{n}}\text{Sa}(a_{n},b_{n};\theta^{i}_{m},\phi^{i}_{m};\theta^{s},\phi^{s})e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{i}_{m},\phi^{i}_{m})/\lambda}e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{s},\phi^{s})/\lambda},\\\
E^{s}_{\phi}(r^{s},\theta^{s},\phi^{s})\simeq&C\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}\sum_{m=1}^{M}E^{i}(\theta^{i}_{m},\phi^{i}_{m})\cos\theta^{i}_{m}\bigl{(}\sin\phi^{i}_{m}\sin\phi^{s}+\cos\phi^{i}_{m}\cos\phi^{s}\bigr{)}\\\
&\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j\Omega_{n}}\text{Sa}(a_{n},b_{n};\theta^{i}_{m},\phi^{i}_{m};\theta^{s},\phi^{s})e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{i}_{m},\phi^{i}_{m})/\lambda}e^{j2\pi\mathbf{p}_{n}^{\top}\mathbf{u}(\theta^{s},\phi^{s})/\lambda}.\end{aligned}\end{dcases*}$
(III.5)

######  Remark III.4.

Proposition (III.3) presents a pretty generic model for RISs consistent with
electromagnetic theory in the far-field regime. The only assumption is that
unit cells can be approximated well by rectangular metallic patches. Unlike
other popular models, the proposed model takes into account the effect of the
incident and scattered angles, polarization features, and the topology and
geometry of RISs.

##  IV Uniform linear RISs

We note that the physics-based models (II.1) and (III.5) are too vague to
offer much insight. To shed new light on the input/output behaviors, this
section studies uniform linear RISs consisting of patches at
$\mathbf{p}_{n}=[0,(n-1)d,0]^{\top}$, $n=1,\ldots,N$. Here $d$ is the unit
cell spacing.

For a quick demonstration, suppose a linear RIS is illuminated by uniform
plane waves lying in the $yoz$ plane, as illustrated in Fig. 6. The electric
field can be simplified considerably if the scattered waves are observed in
the same plane. To reduce the ambiguity of the evaluation angles,
$\theta^{i},\theta^{s}\in[-\pi/2,\pi/2]$ are both measured against the
$z$-axis in what follows.

Let $\text{Sa}(b;\theta^{s};\theta^{i})$ stand for the sampling function
$\frac{\sin\bigl{(}\frac{\pi
b}{\lambda}(\sin\theta^{s}+\sin\theta^{i})\bigr{)}}{\frac{\pi
b}{\lambda}(\sin\theta^{s}+\sin\theta^{i})}$. We obtain from (II.1)
immediately the scattered field $E^{s}_{\mathbf{o}}(r^{s},\theta^{s})$ of the
patch at the origin

$\begin{dcases*}E^{s}_{r}(r^{s},\theta^{s})\simeq 0,\\\
E^{s}_{\theta}(r^{s},\theta^{s})\simeq 0,\\\
E^{s}_{\phi}(r^{s},\theta^{s})\simeq C\frac{A}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}(\theta^{i})\cos\theta^{i}\text{Sa}(b;\theta^{s};\theta^{i}).\end{dcases*}$
(IV.1)

The scattered vector electric field is reduced to a scalar field. The strength
is written as

$\displaystyle\bigl{|}E^{s}_{\mathbf{o}}(r^{s},\theta^{s})\bigr{|}=\sqrt{(E^{s}_{r})^{2}+(E^{s}_{\theta})^{2}+(E^{s}_{\phi})^{2}}\simeq\frac{|C|}{r^{s}}\frac{A}{\lambda}E^{i}(\theta^{i})\cos\theta^{i}\bigl{|}\text{Sa}(b;\theta^{s};\theta^{i})\bigr{|}.$

We now turn to phase difference created by the incident and scattered path
length difference. We find from Fig. 6 that, the path length difference
between $\mathbf{p}_{n}$ and the origin is

$d_{\mathbf{p}_{n}}=d_{\mathbf{p}_{n}}^{s}+d_{\mathbf{p}_{n}}^{i}$

and

$\begin{dcases*}d_{\mathbf{p}_{n}}^{s}=-(n-1)d\sin\theta^{s},\\\
d_{\mathbf{p}_{n}}^{i}=-(n-1)d\sin\theta^{i}.\end{dcases*}$

With the narrowband assumption, the scattered time-varying electric field is
then given by

$E^{s}_{\mathbf{p}_{n}}(r^{s},\theta^{s})\simeq
C\frac{A_{n}}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}(\theta^{i})\cos\theta^{i}\text{Sa}(b_{n};\theta^{s};\theta^{i})e^{j2\pi(n-1)d\sin\theta^{i}/\lambda}e^{j2\pi(n-1)d\sin\theta^{s}/\lambda}.$

The combination of the scattered fields of all patches yields the overall
field observed at $(r^{s},\theta^{s})$.

######  Proposition IV.1.

Suppose a linear RIS with spacing $d$ is illuminated by one uniform plane wave
whose strength is $E^{i}(\theta^{i})$. The scattered electric field observed
at $(r^{s},\theta^{s})$ is given as

$E^{s}(r^{s},\theta^{s})\simeq C\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}(\theta^{i})\cos\theta^{i}\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j\Omega_{n}}\text{Sa}(b_{n};\theta^{s};\theta^{i})e^{j2\pi(n-1)d\sin\theta^{i}/\lambda}e^{j2\pi(n-1)d\sin\theta^{s}/\lambda}.$
(IV.2)

######  Remark IV.2.

Proposition (IV.1) provides a natural and intrinsic characterization of the
single-input single-output (SISO) behaviors of linear RISs. It determines, if
there is only one incident wave, what is observed at $(r^{s},\theta^{s})$. How
much will the incident wave be attenuated (or amplified) at
$(r^{s},\theta^{s})$?

To gain further understanding of the SISO behavior of RISs, we define the
steering function111The steering function is used to describe some inherent
properties of RISs. Since the term $\cos\theta^{i}$ in (IV.2) is related to
the particular polarization features of the EM waves impinging on RISs, it is
not included in the steering function. of linear RISs as

$\displaystyle T(\theta^{s};\theta^{i})=$
$\displaystyle\lim_{r^{s}\to\infty}\frac{r^{s}E^{s}(r^{s},\theta^{s})}{e^{-j2\pi
r^{s}/\lambda}E^{i}(\theta^{i})\cos\theta^{i}}$ (IV.3) $\displaystyle\simeq$
$\displaystyle
C\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j\Omega_{n}}\text{Sa}(b_{n};\theta^{s};\theta^{i})e^{j2\pi(n-1)d\sin\theta^{i}/\lambda}e^{j2\pi(n-1)d\sin\theta^{s}/\lambda}.$

The bistatic RCS is then given by

$\sigma(\theta^{s};\theta^{i})=4\pi\cos^{2}\theta^{i}\bigl{|}T(\theta^{s};\theta^{i})\bigr{|}^{2}.$
(IV.4)

When there exist multiple incident waves, the superposition of individual
fields at $(r^{s},\theta^{s})$ leads to the overall scattered field. We thus
have a proposition to characterize the multiple-input single-output (MISO)
behaviors of RISs.

######  Proposition IV.3.

Suppose a linear RIS is illuminated by multiple uniform plane waves. The
strengths of the incident EM waves are $E^{i}(\theta^{i}_{m}),m=1,\ldots,M$.
The scattered electric field observed at $(r^{s},\theta^{s})$ is given as

$E^{s}(r^{s},\theta^{s})\simeq C\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}\sum_{m=1}^{M}E^{i}(\theta_{m}^{i})\cos\theta_{m}^{i}\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j\Omega_{n}}\text{Sa}(b_{n};\theta^{s};\theta^{i}_{m})e^{j2\pi(n-1)d\sin\theta_{m}^{i}/\lambda}e^{j2\pi(n-1)d\sin\theta^{s}/\lambda}.$
(IV.5)

Figure 6: Phase discrepancy due to the interelement path length difference for
linear RISs. The RIS is illuminated by uniform plane waves lying in the $yoz$
plane. The incident EM wave (in blue) is originating from $\theta^{i}$. The
scattered electric field is observed at $(r^{s},\theta^{s})$ (in red). The
incident and scattered path length differences between the origin and the unit
cell at $(n-1)d$ are $d_{\mathbf{p}_{n}}^{i}=-(n-1)d\sin\theta^{i}$ and
$d_{\mathbf{p}_{n}}^{s}=-(n-1)d\sin\theta^{s}$, respectively.

###  IV-A A canonical representation for the MIMO behaviors of RISs

Several equations, e.g., (III.4), (IV.3), (IV.4), and (IV.5), are proposed to
characterize RISs’ output observed at a single point. We now address how to
describe the MIMO behaviors of RISs.

It is worth noting that the non-reducible term $\text{Sa}(\cdot)$, a product
of two sampling functions, leads to fundamental challenges (non-linearity) to
simplifying the scattering model (IV.5) further. Nevertheless, when
$b_{n}\ll\lambda$, the sampling function is constant approximately, i.e.,
$\text{Sa}(b_{n};\theta^{s};\theta^{i})\simeq 1$. The MISO representation
(IV.5) could be expressed in matrix form

$\displaystyle E^{s}(r^{s},\theta^{s})\simeq$
$\displaystyle\frac{C}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}\sum_{n=1}^{N}A_{n}e^{j\Omega_{n}}e^{j2\pi(n-1)d\sin\theta^{s}/\lambda}\sum_{m=1}^{M}e^{j2\pi(n-1)d\sin\theta_{m}^{i}/\lambda}\cos\theta_{m}^{i}E^{i}(\theta_{m}^{i})$
(IV.6) $\displaystyle=$ $\displaystyle\frac{C}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}\begin{bmatrix}1&\cdots&e^{j2\pi(N-1)d\sin\theta^{s}/\lambda}\end{bmatrix}\begin{bmatrix}A_{1}e^{j\Omega_{1}}&&0\\\
&\ddots&\\\ 0&&A_{N}e^{j\Omega_{N}}\end{bmatrix}$
$\displaystyle\begin{bmatrix}1&\cdots&1\\\ \vdots&\ddots&\vdots\\\
e^{j2\pi(N-1)d\sin\theta_{1}^{i}/\lambda}&\cdots&e^{j2\pi(N-1)d\sin\theta_{M}^{i}/\lambda}\end{bmatrix}\begin{bmatrix}\cos\theta_{1}^{i}E^{i}(\theta_{1}^{i})\\\
\vdots\\\ \cos\theta_{M}^{i}E^{i}(\theta_{M}^{i})\end{bmatrix}.$

We are now in a position to show the linear representation to characterize the
MIMO behaviors of RISs. Given observation points
$\\{(r_{1}^{s},\theta_{1}^{s}),\cdots,(r_{T}^{s},\theta_{T}^{s})\\}$, as
illustrated in Fig. 7, we stack all of the observed electric fields
$\\{E^{s}(r_{1}^{s},\theta_{1}^{s}),\ldots,E^{s}(r_{T}^{s},\theta_{T}^{s})\\}$
to form a column vector denoted by
$\mathbf{E}^{s}(\mathbf{r}^{s},\mathbf{\Theta}^{s})$. With a slight abuse of
notations, the incident fields
$\\{E^{i}(\theta_{1}^{i}),\ldots,E^{i}(\theta_{M}^{i})\\}$ are stacked into a
column vector $\mathbf{E}^{i}(\mathbf{\Theta}^{i})$. We then have a canonical
representation to describe the MIMO behaviors

$\displaystyle\begin{bmatrix}E^{s}(r_{1}^{s},\theta_{1}^{s})\\\ \vdots\\\
E^{s}(r_{T}^{s},\theta_{T}^{s})\end{bmatrix}=$
$\displaystyle\frac{C}{\lambda}\begin{bmatrix}\frac{e^{-j2\pi
r_{1}^{s}/\lambda}}{r_{1}^{s}}&&0\\\ &\ddots&\\\ 0&&\frac{e^{-j2\pi
r_{T}^{s}/\lambda}}{r_{T}^{s}}\end{bmatrix}\begin{bmatrix}1&\cdots&e^{j2\pi(N-1)d\sin\theta_{1}^{s}/\lambda}\\\
\vdots&\ddots&\vdots\\\ 1&\cdots&e^{j2\pi(N-1)d\sin\theta_{T}^{s}/\lambda}\\\
\end{bmatrix}\begin{bmatrix}A_{1}e^{j\Omega_{1}}&&0\\\ &\ddots&\\\
0&&A_{N}e^{j\Omega_{N}}\end{bmatrix}$ (IV.7)
$\displaystyle\begin{bmatrix}1&\cdots&1\\\ \vdots&\ddots&\vdots\\\
e^{j2\pi(N-1)d\sin\theta_{1}^{i}/\lambda}&\cdots&e^{j2\pi(N-1)d\sin\theta_{M}^{i}/\lambda}\end{bmatrix}\begin{bmatrix}\cos\theta_{1}^{i}&&0\\\
&\ddots&\\\
0&&\cos\theta_{M}^{i}\end{bmatrix}\begin{bmatrix}E^{i}(\theta_{1}^{i})\\\
\vdots\\\ E^{i}(\theta_{M}^{i})\end{bmatrix}.$

Figure 7: Multiple EM waves originating from
$\theta_{1}^{i},\ldots,\theta_{M}^{i}$ illuminate the RIS simultaneously. The
scattered field are observed at mutiple points
$(r_{1}^{s},\theta_{1}^{s}),\ldots,(r_{T}^{s},\theta_{T}^{s})$. A simple
system of linear equations (IV.7) can describe the MIMO behaviors under
reasonable assumptions.

To shorten notations, we let

$\displaystyle\mathbf{L}(\mathbf{r}^{s})$
$\displaystyle=\text{diag}\Bigl{(}\frac{e^{-j2\pi
r_{1}^{s}/\lambda}}{r_{1}^{s}},\frac{e^{-j2\pi
r_{2}^{s}/\lambda}}{r_{2}^{s}},\cdots,\frac{e^{-j2\pi
r_{T}^{s}/\lambda}}{r_{T}^{s}}\Bigr{)},$
$\displaystyle\textbf{cos}(\mathbf{\Theta}^{i})$
$\displaystyle=\text{diag}\bigl{(}\cos\theta_{1}^{i},\cos\theta_{2}^{i},\cdots,\cos\theta_{M}^{i}\bigr{)},$
$\displaystyle\mathbf{W}$
$\displaystyle=\text{diag}\bigl{(}A_{1}e^{j\Omega_{1}},A_{2}e^{j\Omega_{2}},\cdots,A_{N}e^{j\Omega_{N}}\bigr{)}.$

Given knots $x_{1},\ldots,x_{m}$, denote by ${\text{Vander}}_{m\times
n}\bigl{(}x_{1},\ldots,x_{m}\bigr{)}$ the Vandermonde matrix

$\begin{bmatrix}1&x_{1}&\cdots&x_{1}^{n-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
1&x_{m}&\cdots&x_{m}^{n-1}\end{bmatrix}.$

Let

$\displaystyle\mathbf{V}(\mathbf{\Theta}^{s})$
$\displaystyle=\text{Vander}_{T\times N}\bigl{(}e^{j2\pi
d\sin\theta_{1}^{s}/\lambda},\cdots,e^{j2\pi
d\sin\theta_{T}^{s}/\lambda}\bigr{)},$
$\displaystyle\mathbf{V}(\mathbf{\Theta}^{i})$
$\displaystyle=\text{Vander}_{M\times N}\bigl{(}e^{j2\pi
d\sin\theta_{1}^{i}/\lambda},\cdots,e^{j2\pi
d\sin\theta_{M}^{i}/\lambda}\bigr{)}.$

######  Proposition IV.4.

Suppose a linear RIS is illuminated by multiple uniform plane waves, whose
strengths are $E^{i}(\theta^{i}_{m}),m=1,\ldots,M$. When $b_{n}\ll\lambda$, a
canonical representation for the MIMO behaviors is given as

$\mathbf{E}^{s}(\mathbf{r}^{s},\mathbf{\Theta}^{s})=\frac{C}{\lambda}\mathbf{L}(\mathbf{r}^{s})\mathbf{V}(\mathbf{\Theta}^{s})\mathbf{W}{\mathbf{V}(\mathbf{\Theta}^{i})}^{\top}\textbf{cos}(\mathbf{\Theta}^{i})\mathbf{E}^{i}(\mathbf{\Theta}^{i}).$
(IV.8)

######  Remark IV.5.

The proposed MIMO representation (IV.8) is physically accurate and
mathematically tractable compared with other popular models. The advantage is
that a simple system of linear equations is employed to describe the MIMO
behaviors under reasonable assumptions. It can serve as a fundamental model
for analyzing and optimizing the performance of RIS-aided systems in the far-
field regime.

######  Remark IV.6.

The two Vandermonde matrices $\mathbf{V}(\mathbf{\Theta}^{s})$ and
$\mathbf{V}(\mathbf{\Theta}^{i})$ are the steering matrices associated with
the incident and scattered directions, respectively. They are well-conditioned
if the knots are more or less equally spaced on unit circle [41]. The matrix
$\mathbf{L}(\mathbf{r}^{s})$ serves as the large-scale distance attenuation
and phase delay factors. The diagonal matrix
$\textbf{cos}(\mathbf{\Theta}^{i})$ is related to the particular polarization
features of the EM waves impinging on RISs.

######  Remark IV.7.

The weight matrix $\mathbf{W}$ consists of two individual components
(collecting area and phase shifting). It is worth noting that the collecting
area provides us with a new degree of design freedom. The simultaneous
configuration of collecting area and phase shifting is essential for the
capability of complicated beamforming. It also converts many constrained
optimization problems in the union of tori (highly non-convex) to linear space
(a regular convex set) as a by-product.

##  V Advantages and limitations under three typical configurations

This section studies the advantages and limitations of RISs under three
typical configurations. We first consider the random phase configuration,
which avoids the overhead caused by direction-of-angle (DOA) and strength
estimations. Unlike the other two configuration schemes, random phase shifting
is a well-accepted benchmark to quantify the performance enhancement [42, 43,
44].

The continuous phase compensation configuration, which is by far the most
popular approach to steering the impinging waves, gives rise to the
generalized Snell’s law. The main disadvantage of this configuration is the
anomalous mirror effect, especially when multiple incident EM waves exist.

In some applications, e.g., high-resolution sensing, RISs are required to
manipulate incident EM waves to generate specific radiation patterns. We
finally show the capability of complicated beamforming functions, e.g., beam
focusing, endowed by simultaneous configurations of collecting area and phase
shifting. Such configuration has the power to redistribute the incident waves
to (almost) arbitrary radiation patterns under reasonable constraints.

It is worth pointing out that an essential difference exists between the
proposed simultaneous configuration of collecting area and phase shifting and
other dynamic amplitude and phase controlling schemes [23, 25, 26, 45, 24]. To
achieve the capability of dynamic amplitude controlling with a wide range,
active components, e.g., active-load impedances, are popular choices. We
attempt to change amplitude by designing the geometry of unit cells [46].

Before proceeding, we summarize in Table I the capability and overhead of the
proposed three typical configurations.

TABLE I: Performance and overhead of the propsed three typical configurations Configuration |  Functionality & Capability |  Estimation overhead |  Beafroming complexity
---|---|---|---
Random discrete phase shifts |  Statistical isotropic scattering patterns |  – |  Low
Continuous phase compensation |  Anomalous reflection & Generalized Snell’s law |  Directions of arrival and departure |  Middle
Simultaneous configuration of collecting area and phase shifting |  Arbitrary beam reshaping functionality, e.g., beam splitting, beam focusing. |  Full information of incident waves, e.g., directions and amplitudes, |  High

###  V-A Scattering via random discrete phase shifting

The collecting area is fixed for the first configuration, but the phase
shifting is discrete and randomly chosen. The advantage lies in that RISs with
random phase shifting will exhibit isotropic gain statistically. To this end,
we assume $\\{\Omega_{n}\\}$ are i.i.d. and
$\text{Pr}(\Omega_{n}=\pi)=\text{Pr}(\Omega_{n}=-\pi)=1/2$. We first analyze
the _statistical_ SISO behaviors.

Since each unit cell re-scatters the incident EM waves independently, we see

$\displaystyle\mathbb{E}\bigl{[}E^{s}(r^{s},\theta^{s})\bigr{]}=$
$\displaystyle C\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}(\theta^{i})\cos\theta^{i}\sum_{n=1}^{N}\frac{A_{n}}{\lambda}\mathbb{E}\bigl{[}e^{j\Omega_{n}}\bigr{]}\text{Sa}(b_{n};\theta^{s};\theta^{i})e^{j2\pi(n-1)d\sin\theta^{i}/\lambda}e^{j2\pi(n-1)d\sin\theta^{s}/\lambda}$
$\displaystyle=$ $\displaystyle 0.$

By a similar argument, we find

$\mathbb{E}\Bigl{[}\bigl{|}E^{s}(r^{s},\theta^{s})\bigr{|}^{2}\Bigr{]}=|C|^{2}\bigl{(}\frac{1}{r^{s}}\bigr{)}^{2}\bigl{(}E^{i}(\theta^{i})\cos\theta^{i}\bigr{)}^{2}\sum_{n=1}^{N}\Bigl{(}\frac{A_{n}}{\lambda}\Bigr{)}^{2}\text{Sa}(b_{n};\theta^{s};\theta^{i})^{2}.$
(V.1)

When $A_{n}=A$,

$\mathbb{E}\Bigl{[}\bigl{|}E^{s}(r^{s},\theta^{s})\bigr{|}^{2}\Bigr{]}=N|C|^{2}\bigl{(}\frac{1}{r^{s}}\bigr{)}^{2}\bigl{(}E^{i}(\theta^{i})\cos\theta^{i}\bigr{)}^{2}\Bigl{(}\frac{A}{\lambda}\Bigr{)}^{2}\text{Sa}(b;\theta^{s};\theta^{i})^{2}.$
(V.2)

Moreover, when $b_{n}\ll\lambda$,
$\text{Sa}(b_{n};\theta^{s};\theta^{i})\simeq 1$. Then

$\mathbb{E}\Bigl{[}\bigl{|}E^{s}(r^{s},\theta^{s})\bigr{|}^{2}\Bigr{]}=N|C|^{2}\bigl{(}\frac{1}{r^{s}}\bigr{)}^{2}\bigl{(}E^{i}(\theta^{i})\cos\theta^{i}\bigr{)}^{2}\Bigl{(}\frac{A}{\lambda}\Bigr{)}^{2}.$
(V.3)

For the steering and RCS functions, we see

$\mathbb{E}\Bigl{[}\bigl{|}T(\theta^{s};\theta^{i})\bigr{|}^{2}\Bigr{]}=|C|^{2}\sum_{n=1}^{N}\Bigl{(}\frac{A_{n}}{\lambda}\Bigr{)}^{2}\text{Sa}(b_{n};\theta^{s};\theta^{i})^{2}$

and

$\displaystyle\mathbb{E}\bigl{[}\sigma(\theta^{s};\theta^{i})\bigr{]}=4\pi\cos^{2}\theta^{i}\mathbb{E}\Bigl{[}\bigl{|}T(\theta^{s};\theta^{i})\bigr{|}^{2}\Bigr{]}=4\pi|C|^{2}\cos^{2}\theta^{i}\sum_{n=1}^{N}\Bigl{(}\frac{A_{n}}{\lambda}\Bigr{)}^{2}\text{Sa}(b_{n};\theta^{s};\theta^{i})^{2}.$
(V.4)

When $A_{n}=A$ and $b_{n}\ll\lambda$,

$\mathbb{E}\bigl{[}\sigma(\theta^{s};\theta^{i})\bigr{]}=4\pi
N|C|^{2}\cos^{2}\theta^{i}\Bigl{(}\frac{A}{\lambda}\Bigr{)}^{2}.$ (V.5)

Figure 8: Statistical bistatic RCS of RISs under random phase configuration.
The RIS consists of $100$ unit cells with size $0.1\lambda\times 0.1\lambda$
and spacing $d=0.5\lambda$. For any given incident angle $\theta^{i}$, RISs
scatter the impinging power field equally in all directions, resulting in a
(statistically) constant scattering/radiation pattern.

We now turn to the statistical MISO behaviors. For simplicity, let
$\hat{\mathbf{E}}^{i}$ stand for
${\mathbf{V}(\mathbf{\Theta})}^{\top}\textbf{cos}(\mathbf{\Theta})\mathbf{E}^{i}(\mathbf{\Theta})$.
Then (IV.6) becomes

$\displaystyle E^{s}(r^{s},\theta^{s})=$
$\displaystyle\frac{C}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}[1,\cdots,e^{j2\pi(N-1)d\sin\theta^{s}/\lambda}]\mathbf{W}{\mathbf{V}(\mathbf{\Theta})}^{\top}\textbf{cos}(\mathbf{\Theta})\mathbf{E}^{i}(\mathbf{\Theta})$
$\displaystyle=$ $\displaystyle\frac{C}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}[1,\cdots,e^{j2\pi(N-1)d\sin\theta^{s}/\lambda}]\mathbf{W}\hat{\mathbf{E}}^{i}.$

Since $\mathbb{E}\bigl{[}\mathbf{W}\bigr{]}=\mathbf{0}$,
$\mathbb{E}\bigl{[}E^{s}(r^{s},\theta^{s})\bigr{]}=0$. Similarly, we see

$\displaystyle\mathbb{E}\Bigl{[}\bigl{|}E^{s}(r^{s},\theta^{s})\bigr{|}^{2}\Bigr{]}=\Bigl{(}\frac{|C|}{\lambda
r^{s}}\Bigr{)}^{2}\mathbb{E}\Bigl{[}\bigl{|}[1,\cdots,e^{j2\pi(N-1)d\sin\theta^{s}/\lambda}]\mathbf{W}\hat{\mathbf{E}}^{i}\bigr{|}^{2}\Bigr{]}=|C|^{2}\Bigl{(}\frac{1}{r^{s}}\Bigr{)}^{2}\sum_{n=1}^{N}\Bigl{(}\frac{A_{n}}{\lambda}\Bigr{)}^{2}\Bigl{|}\hat{E}^{i}_{n}\Bigr{|}^{2}.$
(V.6)

When $A_{n}=A$,

$\mathbb{E}\Bigl{[}\bigl{|}E^{s}(r^{s},\theta^{s})\bigr{|}^{2}\Bigr{]}=|C|^{2}\Bigl{(}\frac{1}{r^{s}}\Bigr{)}^{2}\Bigl{(}\frac{A}{\lambda}\Bigr{)}^{2}\bigl{\|}\hat{\mathbf{E}}^{i}\bigr{\|}^{2}.$
(V.7)

We conclude through (V.1)-(V.5) that, under the random phase configuration,
RISs scatter the incident power field equally in all directions giving a
(statistically) constant scattering/radiation pattern, as illustrated in Fig.
8. We conclude through (V.6) and (V.7) that the RIS has a statistical
isotropic scattering pattern even when multiple incident EM waves exist.

We thus refer to RISs with random phase configuration as statistical isotropic
RISs. Although the random phase configuration cannot effectively use the
available degrees of freedom, it is a well-accepted benchmark for quantifying
the other two configurations.

###  V-B Anomalous reflection via phase compensations

Phase compensation configuration is the most popular approach approach to
steer an incident EM wave to a preferred reflective direction. We now address
the advantages and limitations. We say linear RISs work in phase compensation
configuration mode if the phase shifting of each unit cell can be written as
$\Omega_{n}=-2\pi(n-1)d\Delta/\lambda+2k\pi$, $n=1,\ldots,N$.

Consider SISO behaviors first. Suppose a linear RIS is illuminated by one
uniform plane wave originating from $\theta^{i}$. Then, with (IV.6), the
scattered field at $(r^{s},\theta^{s})$ is given as

$\displaystyle E^{s}(r^{s},\theta^{s})=$ $\displaystyle C\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}(\theta^{i})\cos\theta^{i}\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j\Omega_{n}}e^{j2\pi(n-1)d(\sin\theta^{i}+\sin\theta^{s})/\lambda}$
$\displaystyle=$ $\displaystyle C\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}(\theta^{i})\cos\theta^{i}\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j2\pi(n-1)d(\sin\theta^{i}+\sin\theta^{s}-\Delta)/\lambda}.$

It is easily seen that

$\bigl{|}E^{s}(r^{s},\theta^{s})\bigr{|}\leq\frac{|C|}{r^{s}}\bigl{(}E^{i}(\theta^{i})\cos\theta^{i}\bigr{)}\sum_{n=1}^{N}\frac{A_{n}}{\lambda}.$
(V.8)

For any given $\theta^{i}$, if there exists a $\theta^{s}$ such that
$\Delta=\sin\theta^{i}+\sin\theta^{s}$, equality holds in (V.8). The scattered
field achieves its global maximum at $\theta^{s}$. That means the RIS is
configured to reflect the incident EM wave originating from $\theta^{i}$ to
$\theta^{s}$. If all unit cells have the same collecting area $A$, the
received power at $(r^{s},\theta^{s})$ is given as

$\bigl{|}E^{s}(r^{s},\theta^{s})\bigr{|}^{2}=N^{2}|C|^{2}\bigl{(}\frac{1}{r^{s}}\bigr{)}^{2}\bigl{(}E^{i}(\theta^{i})\cos\theta^{i}\bigr{)}^{2}\Bigl{(}\frac{A}{\lambda}\Bigr{)}^{2}.$
(V.9)

######  Remark V.1.

The proposed equation (V.9) demonstrates several desired properties with phase
compensation configurations. The most important one is the inter-unit
cooperative beamforming gain. This highlights the fact that the maximum power
reflected by RISs could grow quadratically with the total collecting area of
all unit cells (not the physical aperture of RIS). Therefore, large RISs could
be leveraged to compensate for the significant power loss over the two paths
via beamforming. Comparison of (V.9) and (V.2) shows that the power gain
brought by phase compensations becomes $N$ times greater than random phase
shifting.

####  V-B1 Grating lobes

It is well known that, when $d/\lambda>1/2$, grating lobes will occur in the
visible region for some incident angles. In fact, a necessary and sufficient
condition for a grating lobe at $\widetilde{\theta^{s}}\neq\theta^{s}$ is that
there exists two integers $k\neq 0$ and $k^{\prime}$ such that

$d(\sin\theta^{i}+\sin\theta^{s}-\Delta)/\lambda=d(\sin\theta^{i}+\sin\widetilde{\theta^{s}}-\Delta)/\lambda+k=k^{\prime}.$
(V.10)

A necessary condition for (V.10) is
$d(\sin\theta^{s}-\sin\widetilde{\theta^{s}})/\lambda=k$. To ensure the
existence of such a grating lobe at $\widetilde{\theta^{s}}$, we must have
$d/\lambda>1/2$.

We demonstrate the scattered field for various values of $d/\lambda$ in Fig.
9. It illustrates the concept of grating lobes. The RIS, consisting of $100$
unit cells with size $0.1\lambda\times 0.1\lambda$, is configured to steer the
incident wave originating from $30^{\circ}$ to $-50^{\circ}$ by setting
$\Delta=\sin(30^{\circ})+\sin(-50^{\circ})$. The incident wave is plotted in
blue. The scattered electric fields at $r^{s}=100\lambda$ are plotted in red,
from which we can see that the main lobes are centered around $-50^{\circ}$.
For the case of $d/\lambda=0.7$ (the solid red curve), there does exist a
grating lobe centered around $41.49^{\circ}$.

Figure 9: Grating lobe. The RIS is configured to steer the incident wave from
$30^{\circ}$ to $-50^{\circ}$. The spike plotted in blue is the EM wave
impinging upon RIS. The scattered fields at $r^{s}=100\lambda$ are plotted in
red, from which we can see that the main lobes are both centered around
$-50^{\circ}$. For the case of $d/\lambda=0.7$ (the solid red curve), there
does exist a grating lobe centered around $41.49^{\circ}$.

####  V-B2 Anomalous reflection

A significant limitation of beam steering through continuous phase
compensation configurations is the anomalous mirror effect.

Specifically, for a linear RIS, we can choose
$\Delta=\sin\theta^{i}+\sin\theta^{s}$ such that the incident wave from
$\theta^{i}$ is steered to $\theta^{s}$, i.e.,
$d(\sin\theta^{i}+\sin\theta^{s}-\Delta)/\lambda=0$. It is easy to check that,
for any $|\Delta|<2$, there are infinitely many pairs
$(\widetilde{\theta^{i}};\widetilde{\theta^{s}})\neq(\theta^{i};\theta^{s})$
satisfying
$d(\sin\widetilde{\theta^{i}}+\sin\widetilde{\theta^{s}}-\Delta)/\lambda+k=0$,
where $k$ is an integer. That means the RIS also steers the incident wave from
$\widetilde{\theta^{i}}$ to $\widetilde{\theta^{s}}$. We theoretically
demonstrate the generalized Snell’s law of reflections (anomalous mirror
effect).

We illustrate the anomalous mirror effect in Fig. 10. There exist two EM waves
impinging upon the RIS. It is configured to steer the wave from $30^{\circ}$
to $-50^{\circ}$. From the scattered field observed at $r^{s}=100\lambda$, we
find two main lobes exist in the angular range. The higher one centered around
$50^{\circ}$ is introduced by the incident wave from $30^{\circ}$. The lower
one centered around $\theta=52.59^{\circ}$ is related to the incident wave
from $70^{\circ}$. RIS operates as an anomalous mirror.

Figure 10: (a) Anomalous reflections. The RIS is configured to steer the one
originating from $30^{\circ}$ to $-50^{\circ}$. There are two main lobes in
the angular range (in red). The higher one centered around $50^{\circ}$ is
introduced by the incident wave from $30^{\circ}$. The lower one around
$\theta=52.59^{\circ}$ is related to the incident wave from $70^{\circ}$. (b)
Arbitrary beam reshaping. The RIS is configured to perform energy focusing.
The main lobe centered around $\theta=52.59^{\circ}$ in (a) is removed.

To have a better understanding of the difference between specular and
anomalous reflections, we turn to the bistatic RCS functions. It is easy to
check that the steering function is

$\displaystyle
T(\theta^{i},\theta^{s})=C\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j\Omega_{n}}e^{j2\pi(n-1)d\sin\theta^{i}/\lambda}e^{j2\pi(n-1)d\sin\theta^{s}/\lambda}=C\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j2\pi(n-1)d(\sin\theta^{i}+\sin\theta^{s}-\Delta)/\lambda}.$
(V.11)

The bistatic RCS function is then given by

$\displaystyle\sigma(\theta^{i},\theta^{s})=4\pi\cos^{2}\theta^{i}\bigl{|}T(\theta^{s};\theta^{i})\bigr{|}^{2}=4\pi|C|^{2}\cos^{2}\theta^{i}\biggl{|}\sum_{n=1}^{N}\frac{A_{n}}{\lambda}e^{j2\pi(n-1)d(\sin\theta^{i}+\sin\theta^{s}-\Delta)/\lambda}\biggr{|}^{2}.$
(V.12)

We plot the steering and bistatic RCS functions in Figs. 11 and 12 to compare
the difference between specular and anomalous reflections. We first set
$\Delta=0$ to make the RIS operate as a specular mirror. The steering and RCS
functions are plotted in Fig. 11. It is then configured to steer the incident
wave from $30^{\circ}$ to $-50^{\circ}$ by setting
$\Delta=\sin(30^{\circ})+\sin(-50^{\circ})$ as plotted in Fig. 12. A
comparison of Figs. 11 and 12 shows that the steering and RCS functions of
anomalous reflection can be considered as non-linear translations of the
corresponding two functions of specular reflection.

Figure 11: The steering and bistatic RCS functions of specular reflection. We
set $\Delta=0$ to make the RIS operate as a specular mirror. (a) Steering
function. (b) Bistatic RCS function.

Figure 12: The steering and bistatic RCS functions of anomalous reflection.
The RIS is configured to steer the incident wave from $30^{\circ}$ to
$-50^{\circ}$. The steering and RCS functions are non-linear translations of
the corresponding two functions of specular reflection. (a) Steering function.
(b) Bistatic RCS function.

###  V-C Arbitrary beam reshaping

We now discuss how to redistribute the incident power to produce an arbitrary
(desired) power radiation pattern. The problem could be formulated as, given
an incident field $\mathbf{E}^{i}(\mathbf{\Theta}^{i})$, how to choose
$\mathbf{W}$ so that the actual scattered field
$\mathbf{E}^{s}(r^{s},\mathbf{\Theta}^{s})$ approximates the desired field
$\mathbf{E}^{s}_{d}(r^{s},\mathbf{\Theta}^{s})$, i.e.,

$\underbrace{\mathbf{E}^{s}_{d}(r^{s},\mathbf{\Theta}^{s})}_{\text{Desired}}\approx\mathbf{E}^{s}(r^{s},\mathbf{\Theta}^{s})=\frac{C}{\lambda}\frac{e^{-j2\pi
r_{s}/\lambda}}{r_{s}}\mathbf{V}(\mathbf{\Theta}^{s})\overbrace{\mathbf{W}}^{\text{Configuration}}\underbrace{{\mathbf{V}(\mathbf{\Theta}^{i})}^{\top}\textbf{cos}(\mathbf{\Theta}^{i})\mathbf{E}^{i}(\mathbf{\Theta}^{i})}_{=\hat{\mathbf{E}}^{i},\
\text{Known}}.$

A most straightforward approach is to minimize the $\ell_{2}$-norm of the
difference between desired and actual output, i.e.,

$\mathbf{W}={\arg\min}_{\mathbf{W}}\bigl{\|}\mathbf{E}^{s}(r^{s},\mathbf{\Theta}^{s})-\mathbf{E}^{s}_{d}(r^{s},\mathbf{\Theta}^{s})\bigr{\|}.$
(V.13)

For simplicity of notations, we let
$\hat{\mathbf{E}}^{i}={\mathbf{V}(\mathbf{\Theta}^{i})}^{\top}\textbf{cos}(\mathbf{\Theta}^{i})\mathbf{E}^{i}(\mathbf{\Theta}^{i})$
and denote by $\beta$ the constant $\frac{NC}{\lambda}\frac{e^{-j2\pi
r_{s}/\lambda}}{r_{s}}$. It is easily seen that (V.13) is equivalent to

$\min_{\mathbf{W}}\bigl{\|}\frac{\beta}{N}\mathbf{V}(\mathbf{\Theta}^{s})\mathbf{W}\hat{\mathbf{E}}^{i}-\mathbf{E}^{s}_{d}(r^{s},\mathbf{\Theta}^{s})\bigr{\|}.$
(V.14)

To illustrate the potential of the simultaneous configuration of the
collecting area and phase shifting, we consider a particular situation where
the spacing $d=\lambda/2$ and the scattered field are sampled at regular grids
$\theta^{s}_{n}=\arcsin(-1+2(n-1)/N)$, $n=1,\ldots,N$. The advantage lies in
the fact that the Vandermonde matrix $\mathbf{V}(\mathbf{\Theta}^{s})$ is
perfectly conditioned because the knots are equally spaced on the unit circle
[41]. In fact, $\mathbf{V}(\mathbf{\Theta}^{s})/\sqrt{N}$ is an angular
discrete Fourier transform (DFT) matrix, i.e.,
$\mathbf{V}(\mathbf{\Theta}^{s})\mathbf{V}(\mathbf{\Theta}^{s})^{H}=\mathbf{V}(\mathbf{\Theta}^{s})^{H}\mathbf{V}(\mathbf{\Theta}^{s})=N\mathbf{I}$.

If we further write
$\hat{\mathbf{E}}^{s}_{d}=\mathbf{V}(\mathbf{\Theta}^{s})^{H}\mathbf{E}^{s}_{d}(r^{s},\mathbf{\Theta}^{s})$,
(V.14) is equivalent to

$\min_{\mathbf{W}}\bigl{\|}\beta\mathbf{W}\hat{\mathbf{E}}^{i}-\hat{\mathbf{E}}^{s}_{d}\bigr{\|}.$

A solution is given as

$W_{n}=A_{n}e^{j\Omega_{n}}=\left\\{\begin{array}[]{ll}\beta^{-1}\hat{E}^{s}_{d,n}/\hat{E}^{i}_{n},&\hbox{$\hat{E}^{i}_{n}\neq
0$;}\\\ 0,&\hbox{$\hat{E}^{i}_{n}=0$.}\end{array}\right.$

For generic scattered steering matrix $\mathbf{V}(\mathbf{\Theta}^{s})$,
provided $\mathbf{V}(\mathbf{\Theta}^{s})^{H}\mathbf{V}(\mathbf{\Theta}^{s})$
is invertible (i.e., $\mathbf{V}(\mathbf{\Theta}^{s})$ is a full column rank
matrix), we let
$\hat{\mathbf{E}}^{s}_{d}=\bigl{(}\mathbf{V}(\mathbf{\Theta}^{s})^{H}\mathbf{V}(\mathbf{\Theta}^{s})\bigr{)}^{-1}\mathbf{V}(\mathbf{\Theta}^{s})^{H}\mathbf{E}^{s}_{d}(r^{s},\mathbf{\Theta}^{s})$.
A feasible solution to (V.14) is given as

$W_{n}=A_{n}e^{j\Omega_{n}}=\left\\{\begin{array}[]{ll}\beta^{-1}\hat{E}^{s}_{d,n}/\hat{E}^{i}_{n},&\hbox{$\hat{E}^{i}_{n}\neq
0$;}\\\ 0,&\hbox{$\hat{E}^{i}_{n}=0$.}\end{array}\right.$ (V.15)

######  Remark V.2.

If $\mathbf{V}(\mathbf{\Theta}^{s})$ is ill-conditioned or some
$\hat{E}^{i}_{n}$ is close to zero, the scattered field
$\mathbf{E}^{s}(r^{s},\mathbf{\Theta}^{s})$ can approximate the desired
$\mathbf{E}^{s}_{d}(r^{s},\mathbf{\Theta}^{s})$, but
$\hat{\mathbf{E}}^{s}_{d}$ or $\mathbf{W}$ will have a large norm. This means
that the RIS with this configuration is more sensitive to perturbations. Low
rank approximates of $\mathbf{V}(\mathbf{\Theta}^{s})$ or the truncated
version of $\hat{\mathbf{E}}^{i}$ should be used. This ensures that the norms
of $\hat{\mathbf{E}}^{s}_{d}$ and $\mathbf{W}$ will not be unnecessarily
large.

We now employ an example to demonstrate the arbitrary beam reshaping
capability endowed by the simultaneous configuration of the collecting area
and phase shifting. To make a fair comparison with phase compensation as
illustrated in Fig. 10, the RIS is expected to focus two incident waves from
$30^{\circ}$ and $70^{\circ}$ to produce a single main lobe centered around
$-50^{\circ}$. The configuration is given through (V.15). The final scattered
electric field at $r^{s}=100\lambda$ is plotted in Fig. 10, from which we find
the main lobe centered around $\theta=52.59^{\circ}$ in Fig. 10 is removed. We
thus demonstrate the functionality of energy focusing through the simultaneous
configuration of collecting area and phase shifting.

##  VI Conclusion

In this paper, we have employed continuous and discrete strategies to model a
single patch and patch array and their interactions with multiple incident EM
waves. We introduced a physically accurate formula to calculate the electric
field scattered by a rectangular metallic patch. Several mathematically
tractable models were proposed to characterize the input/output behaviors of
patch array. Particularly, a simple system of linear equations was used to
describe the MIMO behavior of linear RISs. Based on the proposed models, we
have evaluated the performance of RISs under three typical configurations. We
proved that a good solution to complicated beam reshaping functionality is
through the simultaneous configuration of collecting area and phase shifting.

##  Appendix A Proof of Theorem II.1

###### Proof.

We follow similar steps as in [36, Ch. 6.8]. Denote by $\mathbf{E}^{i}$ the
incident electric field on the plate. The electric and magnetic fields could
be written in Cartesian coordinates as

$\mathbf{E}^{i}=E^{i}\bigl{(}-\mathbf{e}_{x}\sin\phi^{i}+\mathbf{e}_{y}\cos\phi^{i}\bigr{)}e^{-j2\pi(-x\sin\theta^{i}\cos\phi^{i}-y\sin\theta^{i}\sin\phi^{i}-z\cos\theta^{i})/\lambda}$

and

$\mathbf{H}^{i}=\frac{E^{i}}{\eta}\bigl{(}\mathbf{e}_{x}\cos\theta^{i}\cos\phi^{i}+\mathbf{e}_{y}\cos\theta^{i}\sin\phi^{i}-\mathbf{e}_{z}\sin\theta^{i}\bigr{)}e^{-j2\pi(-x\sin\theta^{i}\cos\phi^{i}-y\sin\theta^{i}\sin\phi^{i}-z\cos\theta^{i})/\lambda}.$

Here $\eta$ is the intrinsic impedance of the medium.

The starting point is the electric current density (measured in $A/m^{2}$)
induced on the plate, which is given as

$\mathbf{J}^{s}=\mathbf{n}\times\mathbf{H}^{\text{total}}|_{z=0}=\mathbf{e}_{z}\times\bigl{(}\mathbf{H}^{i}+\mathbf{H}^{r}\bigr{)}|_{z=0}.$

The reflected magnetic field $\mathbf{H}^{r}$ could be represented by
$\mathbf{H}^{r}=-\Gamma\mathbf{H}^{i}$, where $\Gamma$ is the reflection
coefficient of the surface. We then write $\mathbf{J}^{s}$ in Cartesian
coordinates as

$\displaystyle\mathbf{J}^{s}=$
$\displaystyle\mathbf{e}_{z}\times\bigl{(}\mathbf{H}^{i}+\mathbf{H}^{r}\bigr{)}|_{z=0}=\mathbf{e}_{z}\times(1-\Gamma)\mathbf{H}^{i}|_{z=0}$
$\displaystyle=$
$\displaystyle(1-\Gamma)\frac{E^{i}}{\eta}\bigl{(}-\mathbf{e}_{x}\cos\theta^{i}\sin\phi^{i}+\mathbf{e}_{y}\cos\theta^{i}\cos\phi^{i}\bigr{)}e^{-j2\pi(-x\sin\theta^{i}\cos\phi^{i}-y\sin\theta^{i}\sin\phi^{i})/\lambda}.$

Or equivalently,

$J^{s}_{x}=\begin{dcases*}-(1-\Gamma)\frac{E^{i}}{\eta}\cos\theta^{i}\sin\phi^{i}e^{-j2\pi(-x\sin\theta^{i}\cos\phi^{i}-y\sin\theta^{i}\sin\phi^{i})/\lambda},&for
$(x,y)\in S$;\\\ 0,&for elsewhere,\end{dcases*}$
$J^{s}_{y}=\begin{dcases*}(1-\Gamma)\frac{E^{i}}{\eta}\cos\theta^{i}\cos\phi^{i}e^{-j2\pi(-x\sin\theta^{i}\cos\phi^{i}-y\sin\theta^{i}\sin\phi^{i})/\lambda},&for
$(x,y)\in S$;\\\ 0,&for elsewhere,\end{dcases*}$

and

$J^{s}_{z}=0,\quad\hbox{for everywhere}.$

The structure of unit cells is designed to change surface current density
$\mathbf{J}^{s}$ properly. We emphasize that the reflection coefficient
$\Gamma$ is determined by the surface impedance. For a perfect electric
conductor, the magnetic current density (measured in $V/m^{2}$) induced on the
plate is $\mathbf{M}^{s}=\mathbf{0}$, i.e., $M^{s}_{x}=M^{s}_{y}=M^{s}_{z}=0$,
for everywhere.

Next we will evaluate [36, eq. (6-123)]

$\displaystyle\mathbf{N}^{s}=$ $\displaystyle\iint_{S}\mathbf{J}^{s}e^{j2\pi
r^{\prime}cos\psi/\lambda}d\Omega=\iint_{S}\bigl{(}\mathbf{e}_{x}J^{s}_{x}+\mathbf{e}_{y}J^{s}_{y}+\mathbf{e}_{z}J^{s}_{z}\bigr{)}e^{j2\pi
r^{\prime}cos\psi/\lambda}d\Omega,$ $\displaystyle\mathbf{L}^{s}=$
$\displaystyle\iint_{S}\mathbf{M}^{s}e^{j2\pi
r^{\prime}cos\psi/\lambda}d\Omega=\iint_{S}\bigl{(}\mathbf{e}_{x}M^{s}_{x}+\mathbf{e}_{y}M^{s}_{y}+\mathbf{e}_{z}M^{s}_{z}\bigr{)}e^{j2\pi
r^{\prime}cos\psi/\lambda}d\Omega.$

The integrations are performed over the entire region occupied by the plate,
as illustrated in Fig. 1. Since $\mathbf{M}^{s}=\mathbf{0}$, it is obvious
that $\mathbf{L}^{s}=\mathbf{0}$, i.e.,
$L^{s}_{\theta}=L^{s}_{\phi}=L^{s}_{r}=0$.

We now turn to the evaluation of $\mathbf{N}^{s}$. With the rectangular-to-
spherical component transformation

$\begin{bmatrix}\mathbf{e}_{r}\\\ \mathbf{e}_{\theta}\\\
\mathbf{e}_{\phi}\end{bmatrix}=\begin{bmatrix}\sin\theta\cos\phi&\sin\theta\sin\phi&\cos\theta\\\
\cos\theta\cos\phi&\cos\theta\sin\phi&-\sin\theta\\\
-\sin\phi&\cos\phi&0\end{bmatrix}\begin{bmatrix}\mathbf{e}_{x}\\\
\mathbf{e}_{y}\\\ \mathbf{e}_{z}\end{bmatrix},$

we see that

$\begin{dcases*}\begin{aligned}
N^{s}_{\theta}=&\iint_{S}\bigl{[}J^{s}_{x}\cos\theta^{s}\cos\phi^{s}+J^{s}_{y}\cos\theta^{s}\sin\phi^{s}-J^{s}_{z}\sin\theta^{s}\bigl{]}e^{j2\pi
r^{\prime}\cos\psi/\lambda}d\Omega,\\\
N^{s}_{\phi}=&\iint_{S}\bigl{[}-J^{s}_{x}\sin\phi^{s}+J^{s}_{y}\cos\phi^{s}\bigl{]}e^{j2\pi
r^{\prime}\cos\psi/\lambda}d\Omega.\end{aligned}\end{dcases*}$

Using the fact
$r^{\prime}\cos\psi=\mathbf{r}^{\prime}\cdot\mathbf{e}_{r}=x\sin\theta^{s}\cos\phi^{s}+y\sin\theta^{s}\sin\phi^{s}$,
we have

$\displaystyle N^{s}_{\theta}=$
$\displaystyle(1-\Gamma)\frac{E^{i}}{\eta}\cos\theta^{i}\cos\theta^{s}\bigl{(}\cos\phi^{i}\sin\phi^{s}-\sin\phi^{i}\cos\phi^{s}\bigr{)}$
$\displaystyle\int_{-\frac{a}{2}}^{\frac{a}{2}}e^{j2\pi
x(\sin\theta^{s}\cos\phi^{s}+\sin\theta^{i}\cos\phi^{i})/\lambda}dx\int_{-\frac{b}{2}}^{\frac{b}{2}}e^{j2\pi
y(\sin\theta^{s}\sin\phi^{s}+\sin\theta^{i}\sin\phi^{i})/\lambda}dy$

Using the integral $\int_{-\frac{c}{2}}^{\frac{c}{2}}e^{j\alpha
x}dx=c\biggl{[}\frac{\sin\bigl{(}\frac{\alpha}{2}c\bigr{)}}{\frac{\alpha}{2}c}\biggr{]}$,
the above expression could be written as

$\displaystyle N^{s}_{\theta}=$
$\displaystyle(1-\Gamma)ab\frac{E^{i}}{\eta}\cos\theta^{i}\cos\theta^{s}\bigl{(}\cos\phi^{i}\sin\phi^{s}-\sin\phi^{i}\cos\phi^{s}\bigr{)}\text{Sa}(a,b;\theta^{s},\phi^{s};\theta^{i},\phi^{i}).$

In the same vein, we have

$\displaystyle N^{s}_{\phi}=$
$\displaystyle\iint_{S}\bigl{[}-J^{s}_{x}\sin\phi^{s}+J^{s}_{y}\cos\phi^{s}\bigr{]}e^{j2\pi
r^{\prime}\cos\psi/\lambda}d\Omega$ $\displaystyle=$
$\displaystyle(1-\Gamma)ab\frac{E^{i}}{\eta}\cos\theta^{i}\bigl{(}\sin\phi^{i}\sin\phi^{s}+\cos\phi^{i}\cos\phi^{s}\bigr{)}\text{Sa}(a,b;\theta^{s},\phi^{s};\theta^{i},\phi^{i}).$

Then, in the far-field regime, the $(r,\theta,\phi)$ components of the
scattered electric field $\mathbf{E}^{s}$ are [36, eq. (6-122)]

$\begin{dcases*}\begin{aligned}
E^{s}_{r}(r^{s},\theta^{s},\phi^{s})\simeq&0,\\\
E^{s}_{\theta}(r^{s},\theta^{s},\phi^{s})\simeq&C\frac{A}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}\cos\theta^{i}\cos\theta^{s}\bigl{(}\cos\phi^{i}\sin\phi^{s}-\sin\phi^{i}\cos\phi^{s}\bigr{)}\text{Sa}(a,b;\theta^{s},\phi^{s};\theta^{i},\phi^{i}),\\\
E^{s}_{\phi}(r^{s},\theta^{s},\phi^{s})\simeq&C\frac{A}{\lambda}\frac{e^{-j2\pi
r^{s}/\lambda}}{r^{s}}E^{i}\cos\theta^{i}\bigl{(}\sin\phi^{i}\sin\phi^{s}+\cos\phi^{i}\cos\phi^{s}\bigr{)}\text{Sa}(a,b;\theta^{s},\phi^{s};\theta^{i},\phi^{i}).\end{aligned}\end{dcases*}$

∎

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