# Space-Time Metasurfaces for Perfect Power Combining of Waves

X. Wang1 V. S. Asadchy2 S. Fan2 S. A. Tretyakov1 1Department of Electronics
and Nanoengineering, Aalto University, P.O. Box 15500, FI-00076 Aalto, Finland
2Ginzton Laboratory and Department of Electrical Engineering, Stanford
University, Stanford, California 94305, USA

###### Abstract

In passive linear systems, complete combining of powers carried by waves from
several input channels into a single output channel is forbidden by the energy
conservation law. Here, we demonstrate that complete combination of both
coherent and incoherent plane waves can be achieved using metasurfaces with
properties varying in space and time. The proposed structure reflects waves of
the same frequency but incident at different angles towards a single
direction. The frequencies of the output waves are shifted by the metasurface,
ensuring perfect incoherent power combining. The proposed concept of power
combining is general and can be applied for electromagnetic waves from the
microwave to optical domains, as well as for waves of other physical nature.

##  I Introduction

Combining electromagnetic waves from multiple input ports to a single output
port is highly desired in optical and microwave engineering. It would overcome
power limits of conventional sources, enabling high-power laser and long-
distance free-space communications. The most straightforward approach for wave
combining is based on beam splitters or power dividers Pozar (2011). However,
these passive components allow complete combination of coherent beams only
when the beams have specific phase and amplitude relations among them. This
constraint originates from the passivity of the system He and Liu (2017);
Hanna  _et al._ (2016). In general, input ports of a passive combiner are not
matched and, when excited by incident waves, parasitic reflections appear.
Only when the combiner is coherently illuminated from all the input ports
simultaneously, with specific phases of each input wave, the reflections will
interfere destructively, and all the input power can be directed to the output
port. Such phase- and amplitude-sensitive operation is analogous to that of
recently proposed coherent absorbers and lasers Baranov  _et al._ (2017).
However, in most practical scenarios, incident coherent waves have arbitrary
phases and amplitudes and, therefore, the condition of destructive
interference of parasitic reflections is not satisfied, and the combining
efficiency drops significantly.

For incoherent incident waves, the passivity constraint does not hold since
the waves do not interfere, and the combined output intensity is merely a sum
of the input intensities. This opportunity has been widely used in high-power
lasers and electromagnetic waves. It is, for example, certainly possible to
combine the power of two waves with orthogonal polarizations, but in this
case, the number of incoming waves is limited to two Uberna  _et al._ (2010).
One can also combine waves with different frequencies, which is also known as
wavelength beam combining. Using dispersive elements (prisms or gratings),
beams from different directions propagating at different frequencies can be
collimated at the same output direction Minott and Abshire (1987); Smits
(1993); Madasamy  _et al._ (2009). For example, using one Belousov  _et al._
(1991); Chann  _et al._ (2005) or a pair of grating structures Madasamy  _et
al._ (2009), the beams from an incoherent laser array are overlapped in one
direction through materials with normal dispersion. Nevertheless, the emission
wavelengths from each source must be accurately controlled. In addition, this
method cannot be used to combine low-terahertz and microwave beams with small
frequency differences $\Delta f$ because the required size of gratings would
be enormous due to scaling as $1/\Delta f$ Smits (1993); Thumm and Kasparek
(2002). Therefore, it is highly desirable to have a compact system to
efficiently combine powers from different ports without restrictions on
amplitudes, phases, and frequency differences for the input beams.

In this paper, we propose a method for perfect power combining for both
coherent and incoherent plane waves using reflective space-time modulated
metasurfaces. Spatiotemporal metasurfaces recently demonstrated great
potential for a plethora of applications, such as Doppler-type frequency shift
Shaltout  _et al._ (2015), isolation Wang  _et al._ (2020); Cardin  _et al._
(2020); Hadad  _et al._ (2015), circulation Shi  _et al._ (2017),
nonreciprocal phase shifting Wang  _et al._ (2020) and beam splitting Taravati
and Kishk (2019). We show that such metasurfaces can provide complete power
combining in reflection for waves incident from several predefined directions
having identical frequencies but arbitrary phases and amplitudes. The spatial
modulation in the metasurface ensures that all the incident waves are
reflected into a single output channel. Due to the temporal modulation,
incident same-frequency plane waves are reflected as waves with slightly
different frequencies, allowing perfect incoherent power combining in the
output channel. We provide several metasurface designs for complete power
combining with weak frequency conversion (using slow temporal modulations). In
all the cases, the combined power at the output channel can be captured by a
single receiver with a finite bandwidth.

##  II results

In what follows, we describe periodic metasurfaces as multi-port systems
characterized by scattering matrix $\bar{\bar{S}}$ Asadchy  _et al._ (2017).
Propagating Floquet harmonics for a given illumination direction represent
orthogonal channels and are equivalent to physical ports of a multi-port
system.

First, we show that linear time-invariant systems (both passive and active)
cannot provide complete power combining for coherent plane waves with
arbitrary phases and amplitudes. To keep the discussion general, we consider a
multi-port system shown in Fig. 1, which can describe power combining in
various scenarios: metasurfaces, electrical networks, waveguides, single
scatterers (described by a T-matrix), etc. The multi-port system is excited at
input ports $i$ ($1\leq i\leq N$) with some field amplitudes $a_{i}=A_{i}{\rm
e}^{j\phi_{i}}$, where $\phi_{i}$ are real and $A_{i}$ are positive real
quantities. The output port $N+1$ is not excited, that is, $a_{N+1}=0$. From
the scattering matrix of the system $S$, we can find the output field at this
port as $b_{N+1}=\sum_{i=1}^{N}S_{N+1,i}\,a_{i}$. The efficiency of power
combining expressed as the ratio of the output power in port $N+1$ to the
total input power at ports from 1 to $N$ reads

$\eta(a_{i})=\frac{\left|\sum_{i=1}^{N}S_{N+1,i}\,A_{i}\,{\rm
e}^{j\phi_{i}}\right|^{2}}{\sum_{i=1}^{N}A_{i}^{2}}.$ (1)

Figure 1:  A general time-invariant power combiner with $N$ input ports and
one output $(N+1)$-st port.

As seen from Eq. (1), the combining efficiency strongly depends on the
amplitudes and phases of the input fields, for both passive and active
systems. For any $S$-matrix there in fact exist input fields for which the
power combining efficiency is exactly zero. For arbitrary input fields
$a_{i}$, the efficiency is always limited by (see Sec. 1 of the Supplement 1)

$\eta(a_{i})\leq\sum_{i=1}^{N}|S_{N+1,i}|^{2}.$ (2)

In active systems, the combining efficiency can be larger than unity for some
input fields $a_{i}$. On the other hand, in passive systems, the efficiency
obeys $\eta(a_{i})\leq 1$. The inequality becomes equality only if the
following three conditions hold: (i) port $N+1$ is matched, that is,
$S_{N+1,N+1}=0$; (ii) there is no energy dissipation in the system; (iii) all
the input fields satisfy the same condition $a_{i}=CS_{N+1,i}^{*}$, where $C$
is a complex constant and “∗” denotes complex conjugate.

In the special case when a passive multi-port network is excited only at one
port $i$ ($1\leq i\leq N$), one can calculate the average efficiency (averaged
over all $N$ ports) of power coupling into the output port $N+1$ as (see Sec.
2 of the Supplement 1)

$\eta_{\rm coupl}=\frac{1}{N}\sum_{i=1}^{N}|S_{N+1,i}|^{2}\leq\frac{1}{N}.$
(3)

The average efficiency of power combining can reach its maximum value
$\eta_{\rm coupl}={1}/{N}$ only in the absence of power dissipation and if the
output port is matched. We see that with increasing the number of input ports
$N$, the average coupling efficiency linearly decreases. Furthermore, complete
power combining from multiple input ports becomes exceedingly difficult with
increasing $N$ due to the requirement of mutual coherence of the incident
waves. Obviously, the phase-dependent operation is not desired in practical
applications of power combining. In linear time-invariant systems, the problem
cannot be avoided because the monochromatic outgoing waves at the output port
always interfere.

Figure 2: The principle of complete power combining using space-time
metasurfaces. The metasurface totally reflects incident plane waves coming
from three different directions (three scenarios) into the same output
direction at an angle $\theta_{\rm r}$. The metasurface imparts both
tangential spatial and temporal momenta onto incident waves, modifying the
isofrequency curves.

In order to overcome these fundamental constraints on power combining, we
propose to exploit a metasurface possessing both spatial and temporal
modulations of its electromagnetic properties. Such a time-varying multi-port
system introduces a frequency shift between the output waves making them
incoherent. Due to this incoherence, the device can fully combine waves of the
same frequency from multiple input ports to a single output port, and operate
efficiently for both single- and multi-port input waves with arbitrary
amplitudes and phases. The operational principle of the metasurface is
illustrated in Fig. 2. Plane waves of the same frequency $\omega_{0}$ incident
at different angles on the metasurface (illustrated by three separate
scenarios) are completely reflected to the same direction, at an angle
$\theta_{\rm r}$. The temporal modulation of the metasurface adds different
frequencies (temporal momenta) to the three reflected waves: $0$,
$\Delta\omega_{2}$, and $\Delta\omega_{3}$, respectively. Therefore, the
output waves in all the scenarios become incoherent (for coherent inputs) and
their powers can be perfectly combined. The identical reflection angles for
the output waves are ensured by proper spatial modulation of the metasurface:
the reflected waves gain different spatial momenta, $0$,
$\Delta\mathbf{k}_{2}$, $\Delta\mathbf{k}_{3}$, respectively, such that they
are deflected towards the same direction.

We model the metasurface as an impedance sheet positioned on a grounded
dielectric substrate, as shown in Fig. 3. Such geometry was previously
exploited both for time-invariant Wang  _et al._ (2018) and time-varying Wu
_et al._ (2020); Li  _et al._ (2020); Cardin  _et al._ (2020) metasurfaces.
Here, we model the impedance sheet as a spatiotemporally modulated sheet
capacitance $C(z,t)$. Such an impedance sheet can be realized at microwave
frequencies by a metallic pattern with embedded varactors (voltage-controlled
capacitors). For optical frequencies, temporal modulations of similar type can
be realized by voltage-biased $pn$-junction electrical diodes Lira  _et al._
(2012), and optically pumping epsilon-near zero materials such as AZO
(aluminum-doped zinc oxide) for ultrafast modulation Kinsey  _et al._ (2015);
Clerici  _et al._ (2017).

Figure 3: Scattering from a space-time modulated impedance sheet positioned on
a grounded substrate.

The modulation of surface capacitance has a traveling-wave form with spatial
period $D$ (along the $z$-direction) and temporal period $T$, and therefore
the modulation velocity is $v_{\rm M}=D/T$. Due to the periodicity, the
modulation function can be expanded into the Fourier series

$C(z,t)=\sum_{m=-\infty}^{+\infty}c_{m}e^{-jm(\beta_{\rm M}z-\omega_{\rm
M}t)},$ (4)

where $\beta_{\rm M}=2\pi/D$ and $\omega_{\rm M}=2\pi/T$ are the spatial and
temporal modulation frequencies, respectively, and $c_{m}=c_{-m}^{*}$ to
ensure $C(z,t)$ is real function.

We consider a TE (transverse electric) plane wave incident from
$\theta=\theta_{\rm i}$ at a frequency $\omega_{0}$. The incident wavenumber
in free space is denoted as $k_{0}$. Due to the spatial modulation, the
incident wave is scattered into infinitely many spatial harmonics with
tangential wavenumbers $k_{zn}=k_{0}\sin\theta_{\rm i}+n\beta_{\rm M}$, and
due to the time modulation, the propagation frequency of wave in the $n$-th
spatial channel is $\omega_{n}=\omega_{0}+n\omega_{\rm M}$ Wang  _et al._
(2020). The wavevector of the $n$-th harmonic must respect the dispersion
relation in free space, $k_{n}=|\omega_{n}|\sqrt{\epsilon_{0}\mu_{0}}$. If
$|k_{zn}|<|k_{n}|$, the harmonic is propagating wave and scattered into free
space. If $|k_{zn}|>|k_{n}|$, the harmonic is evanescent. Once the spatial and
temporal modulation periods, the frequency of the incident waves and the
incidence directions are known, the directions of scattering channels (free-
space propagating modes) can be determined according to the Floquet theorem.
When the capacitance function $C(z,t)$ and the substrate properties
(permittivity $\varepsilon_{\rm r}$ and thickness $d$) are chosen, we
calculate the amplitudes and phases of all the scattering harmonics using the
mode-matching method (see Sec. 3 of the Supplement 1).

The schematics of a general multi-port metasurface combiner is depicted in
Fig. 4, where ports from $1$ to $N$ are the input ports, and port $N+1$ is the
output port. The directions of the input ports are denoted by the incident
angles $\theta_{n}$, where $n\in[1,N]$ represents the input port index. The
output port is along the specular direction of port 1. Thus, when illuminated
from port 1, the reflection to port $N+1$ corresponds to the zero-order
harmonic of the metasurface. Consecutively, a wave incident from port $n$
($n\in[1,N]$) is reflected to port $N+1$ as the ($n-1$)-th Floquet harmonic.
According to the Floquet-Bloch theorem, the tangential wavevectors of
incidence and its $n-1$ reflection mode are related by expression

$k_{0}\sin\theta_{n}+(n-1)\beta_{\rm M}=\alpha_{n}k_{0}\sin\theta_{1},$ (5)

where $\alpha_{n}=|\frac{\omega_{0}+(n-1)\omega_{\rm M}}{\omega_{0}}|$. Next,
we specify the incidence angles of port 1 and port $N$ (denoted as
$\theta_{1}$ and $\theta_{N}$), and thus the spatial period can be determined
from Eq. (5):

$D=\frac{(N-1)\lambda_{0}}{\alpha_{N}\sin\theta_{1}-\sin\theta_{N}}.$ (6)

By substituting Eq. (6) into Eq. (5), one can uniquely determine the angles of
port $n$ for $n\in[2,N-1]$:

$\sin\theta_{n}=\alpha_{n}\sin\theta_{1}-\frac{(n-1)(\alpha_{N}\sin\theta_{1}-\sin\theta_{N})}{N-1}.$
(7)

As soon as the spatial period, modulation frequency, and the directions of all
ports are fixed, one can optimize the space-time modulation profile and
achieve full reflection from all input ports to the output port $N+1$.

Figure 4: Schematics of a general $N+1$ metasurface power combiner. The output
port is along the specular direction of incidence from port $1$, i.e.,
$\theta_{\rm 1}=\theta_{N+1}$. All the angles are counted from the metasurface
normal in the anticlockwise direction.

According to the Manley–Rowe relations (Collin, 2007, Chap. 11.2), while the
energy in time-varying systems is not conserved, the total number of photons
entering a lossless system must be equal to that leaving the system. Since our
metasurface does not include energy dissipating components, it will conserve
the photon flux. Due to the periodicity and infinite extent of the
metasurface, the normal components of the total incident photon flux per unit
area $P_{{\rm i},x}/(\hbar\omega_{\rm i})$ and that of the total reflected
photon flux $P_{{\rm r},x}/(\hbar\omega_{\rm r})$ must be equal for any
illumination. Here, $P_{x}$ denotes the normal component of the Poynting
vector, and it is proportional to $E_{\rm i}^{2}\cos\theta_{\rm i}$ for
incident and to $E_{\rm r}^{2}\cos\theta_{\rm r}$ for reflected plane waves.
To achieve complete power combining using the metasurface, it is sufficient to
ensure that, when illuminated from a single port $n$ ($n\in[1,N]$), the
metasurface will reflect all the photons into port $N+1$. Therefore, we
optimize the metasurface so that for single-port excitation the following
relation approximately holds

$E_{{\rm r},N+1}=\sqrt{\frac{\omega_{\rm r}}{\omega_{\rm
i}}\frac{\cos\theta_{n}}{\cos\theta_{N+1}}}E_{{\rm i},n}.$ (8)

Here, $\omega_{\rm i}=\omega_{0}$ is the frequency at the input port $n$ and
$\omega_{\rm r}=\alpha_{n}\omega_{0}$ is the frequency at the output port
$N+1$. For illumination from multiple ports ($n\in[1,N]$), all the photons
will be automatically reflected into port $N+1$ if (8) holds, due to the
incoherence of the output waves.

To satisfy (8) for ports $n\in[1,N]$, we employ numerical tools to optimize
the metasurface parameters. There are in total $N$ optimization objectives for
a metasurface with a given substrate thickness and substrate permittivity. We
use ‘fmincon’ function in MATLAB for optimizing the Fourier coefficients of
the modulation function $c_{m}$ of the metasurface. The reflected fields are
calculated using the mode-matching method (see Sec. 3 of the Supplement 1).

Next, to demonstrate the universality of the proposed concept based on an
example, we design a 3-port ($N=2$) metasurface power combiner. Without loss
of generality, we specify the angles of ports $1$ and $2$ as
$\theta_{1}=+45^{\circ}$ and $\theta_{2}=0^{\circ}$, respectively. The output
port $3$ is at the angle $\theta_{3}=-\theta_{1}=-45^{\circ}$ (all the angles
are counted from the metasurface normal in the anticlockwise direction, as
shown in Fig. 4). The incident frequency is $\omega_{0}$ and the modulation
frequency is assumed much smaller than the incident frequency (arbitrarily
chosen as $\omega_{\rm M}=0.044\omega_{0}$), such that only ports $1$, $2$,
and $3$ support propagating waves in free space, and other higher-order
harmonics excited at the metasurface are evanescent. The advantage of the slow
modulation scenario is that such a metasurface can be relatively simply
implemented Lira  _et al._ (2012); Cardin  _et al._ (2020); Wu  _et al._
(2020); Taravati  _et al._ (2017); Hadad  _et al._ (2016); Correas-Serrano
_et al._ (2018); Guo  _et al._ (2019). Moreover, the converted frequencies at
the output port only slightly differ from the incident frequency, so that all
power can be efficiently received even by a receiver with a small bandwidth.
Based on these parameters, the spatial modulation period is determined from
Eq. (6) as $D=1.355\lambda_{0}$. The substrate thickness is $d=8.13\times
10^{-4}\lambda_{0}$, and $\epsilon_{\rm r}=3.55$. To find the proper
modulation function of the capacitive impedance sheet that can realize full
photon delivery from ports $1$ and $2$ to port $3$, two optimization
objectives are set in numerical optimization according to Eq. (8). The first
one ensures full specular reflection from port $1$ to port $3$, i.e., $E_{{\rm
r},3}=E_{\rm i,1}$. The second objective is $E_{{\rm r},3}\approx 1.215E_{\rm
i,2}$, which ensures full anomalous reflection from port $2$ to port $3$.
Since the number of objectives is small, we introduce only three Fourier terms
in the modulation function $c_{0}$, $c_{\pm 1}$, and $c_{\pm 2}$. One of the
optimized solution is

$\begin{split}C(z,t)&=0.061[1-0.268\cos(\beta_{\rm M}z-\omega_{\rm M}t)\\\
&-0.01\cos(2\beta_{\rm M}z-2\omega_{\rm
M}t)]\cdot\omega_{0}^{-1}~{}\mathrm{[F]}.\end{split}$ (9)

Note that this modulation function is independent from the relative phases of
waves input to different ports. As soon as the modulation function is
obtained, the amplitudes of all the scattering harmonics for each incidence
can be calculated following the mode-matching method.

Figure 5 shows the scattering harmonics for two illumination scenarios: from
port $1$ and from port $2$. One can see that the two objectives defined in the
optimization are reached with negligible errors, and undesired waves scattered
into input ports have negligible amplitudes. In addition to the realized power
combining of propagating modes, strong evanescent modes (shown by dashed
lines) are excited at the metasurface, but they do not carry power in the
normal direction. Nevertheless, the excitation of evanescent modes is
essential to satisfy the boundary conditions that simultaneously ensure
desired scattering for both illumination scenarios.

Figure 5: Normalized amplitudes of scattering harmonics for illuminations from
port $1$ (top) and port $2$ (bottom). Here, index $n$ inside the brackets
denotes the scattered harmonic orders that correspond to different ports in
the top and bottom subfigures, as depicted in the insets. The scattering
harmonics of higher orders are negligibly small.

Knowing the amplitudes of all the scattering harmonics, the photon flux
reflectance (the ratio of the normal components of the reflected and incident
photon fluxes per unit area) at each port can be calculated as $R=E_{\rm
r}^{2}\cos\theta_{\rm r}\omega_{\rm i}/(E_{\rm i}^{2}\cos\theta_{\rm
i}\omega_{\rm r})$. Table 1 shows the calculated values. As is seen, almost
all the incident photons are reflected to port $3$ for illuminations from
ports $1$ and $2$. For completeness, the last column of the table presents the
results for the case when the metasurface is illuminated at port $3$. The
total number of photons is conserved for each illumination scenario (the sum
of reflectances in each table column equals unity), as expected for a lossless
time-modulated metasurface. Generally speaking, the designed metasurface can
be described by a normalized scattering matrix whose entries relate the input
and output tangential electric fields normalized by the square roots of the
corresponding frequencies Wang  _et al._ (2021). Since the metasurface is
lossless, this scattering matrix is unitary. Since the metasurface exhibits
both spatial and temporal modulations, the scattering matrix has an infinite
number of elements. The truncated matrix for the designed metasurface can be
found in Sec. 4 of the Supplement 1.

Table 1: Photon flux reflectances from the metasurface modulated at $\omega_{\rm M}=0.044\omega_{0}$. $R_{pq}$ is the reflectance for incidence from port $q$ to port $p$. Illumination from port 1 |  Illumination from port 2 |  Illumination from port 3
---|---|---
$R_{11}=0.087\%$ |  $R_{12}=0.000\%$ |  $R_{13}=8.365\%$
$R_{21}=0.087\%$ |  $R_{22}=0.197\%$ |  $R_{23}=73.508\%$
$R_{31}=99.826\%$ |  $R_{32}=99.803\%$ |  $R_{33}=18.127\%$

An important characteristic of a power combining device is its bandwidth.
Figure 6 shows the reflection efficiencies from ports $1$ and $2$ to port $3$,
when the incident frequency is swept around $\omega_{0}$. It should be noted
that, when the incident frequency changes, the direction of port $2$ also
changes according to Eq. (6) because the modulation parameters ($D$ and
$\omega_{\rm M}$) and $\theta_{1}$ are assumed to be constant. From Fig. 6 one
can estimate the relative bandwidth of the metasurface to be in the range of
2%. Increasing the modulation speed can widen the bandwidth of the metasurface
Williamson  _et al._ (2020).

Modulation of the metasurface capacitance can be replaced by modulation of the
permittivity of a material film. At optical frequencies, permittivity
modulation can be achieved based on the nonlinear Kerr effect using a strong
light pump Guo  _et al._ (2019). Alternatively, one can discretize the
required permittivity in space and apply local voltage control at each sub-
cell Lira  _et al._ (2012), which is also used in the microwave range Hadad
_et al._ (2016); Taravati  _et al._ (2017). In the Supplement 1 (Sec. 6), we
show that by discretizing one spatial period into at least ten sub-cells, the
above calculated combining efficiency can be achieved.

Figure 6: Left axes: photon flux reflectance from ports 1 and 2 to port 3.
Right Axes: variation of the angle of the port 2 direction as the incident
frequency changes.

The above results show power combining from two input ports. The proposed
method can be generalized to design power concentrators with an arbitrary
number of input ports. As the numbers of ports increases, the numbers of
objectives in optimization also increases, and one needs to introduce more
Fourier terms in the modulation function. Alternatively, one can introduce
additional reactive components in the surface impedance to increase the design
freedom. Section 5 of the Supplement 1, includes a design of a 4-port power
combining metasurface.

##  III conclusion

In summary, we have introduced the concept of perfect power combining using
space-time metasurfaces. Using proper traveling modulation in the metasurface,
arbitrary numbers of incident plane waves can be redirected towards one output
port with nearly 100% efficiency. The presented method is general and can be
applied from microwave to optical frequencies. Microwave power combining
metasurfaces can be realized based on varactors (capacitors with varying
capacitance) controlled by an external time-varying voltage. Such
implementations of different devices were reported in Lira  _et al._ (2012);
Cardin  _et al._ (2020); Wu  _et al._ (2020); Taravati  _et al._ (2017); Hadad
_et al._ (2016). In optical frequencies, the traveling-wave modulation can be
achieved based on temporally modulating two-dimensional materials such as
graphene Wang  _et al._ (2020); Correas-Serrano  _et al._ (2018), epsilon-
near-zero materials such as AZO (aluminum-doped zinc oxide), Kinsey  _et al._
(2015); Clerici  _et al._ (2017), as well as nonlinear materials exhibiting
Kerr effect Guo  _et al._ (2019).

Since the designed metasurface power combiner has a finite bandwidth, it is
possible to extend the concept to complete and phase-independent combination
of incident light pulses, provided that their spectrum is within the device
bandwidth. Perfect pulse combination is essential for obtaining high-intensity
laser sources and has important applications for laser-plasma accelerators
Leemans and Esarey (2009); Esarey  _et al._ (2009), particle sources (e.g.,
ion) Haseroth and Hora (1996), secondary radiation generation Daido  _et al._
(2012), and so on. As is seen from Fig. 6, the incident angles $\theta_{2}$
for different frequency components of the pulse at port $2$ must be different
to provide high combination efficiency. Such spatial dispersion of the
incident pulse can be generated using conventional techniques with additional
diffraction elements Zhou  _et al._ (2017).

###### Acknowledgements.

This work was supported in part by the Academy of Finland (project 330260) and
the U.S. Air Force Office of Scientific Research MURI project (Grant No.
FA9550-18-1-0379).

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# Data Analytics for Smart cities: Challenges and Promises

Farid Ghareh Mohammadi1,Farzan Shenavarmasouleh1, M. Hadi Amini2, and Hamid R.
Arabnia1
1:Department of Computer Science, Franklin College of arts and sciences,
University of Georgia, Athens, Georgia, 30601
2: School of Computing and Information Sciences, College of Engineering and
Computing,
Florida International University, Miami, FL 33199
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The explosion of advancements in artificial intelligence, sensor technologies,
and wireless communication activates ubiquitous sensing through distributed
sensors. These sensors are various domains of networks that lead us to smart
systems in healthcare, transportation, environment, and other relevant
branches/networks. Having collaborative interaction among the smart systems
connects end-user devices to each other which enables achieving a new
integrated entity called Smart Cities. The goal of this study is to provide a
comprehensive survey of data analytics in smart cities. In this paper, we aim
to focus on one of the smart cities important branches, namely Smart Mobility,
and its positive ample impact on the smart cities decision-making process.
Intelligent decision-making systems in smart mobility offer many advantages
such as saving energy, relaying city traffic, and more importantly, reducing
air pollution by offering real-time useful information and imperative
knowledge. Making a decision in smart cities in time is challenging due to
various and high dimensional factors and parameters, which are not frequently
collected. In this paper, we first address current challenges in smart cities
and provide an overview of potential solutions to these challenges. Then, we
offer a framework of these solutions, called universal smart cities decision
making, with three main sections of data capturing, data analysis, and
decision making to optimize the smart mobility within smart cities. With this
framework, we elaborate on fundamental concepts of big data, machine learning,
and deep leaning algorithms that have been applied to smart cities and discuss
the role of these algorithms in decision making for smart mobility in smart
cities.

keywords: Smart cities, Smart Mobility, Making Decision, Artificial
Intelligence, Data Science, Machine Learning, Deep Learning.

##  1 Introduction

$\noindent\diamond$Motivation The explosion of advancement in artificial
intelligence, sensor technologies, like Internet of Things (IOT), and wireless
communication activates ubiquitous sensing through distributed sensors in
various domains of networks that lead us to smart systems in healthcare,
transportation, environment and other relevant branches/networks. Having
collaborative interaction among the smart systems connects end-user devices to
each other enabling a new integrated entity called Smart Cities.

In the last decade, the Internet of Things (IoT) devices have been connected
among different independent agents and heterogeneous networks as with
communication technologies [1] [2]. The Connected high-performance sensors,
and end-user devices, Internet of Things, is the trigger leveraging the
networks in transitioning from urban cities towards smart sustainable cities.
The goal of smart cities is to address upcoming challenges of conventional
cities by offering integrated management systems with a combination of
intelligent infrastructures [3].

Making decisions in smart cities is challenging due to high direct/indirect
dimensional factors and parameters. In this paper, we aim to focus on one of
the smart cities important branches, namely Smart Mobility, and its positive
ample impact on smart cities decision making processes. Intelligent decision
making systems in smart mobility offer many advantages, such as saving energy
[4], relaying city traffic [5], and more importantly, reducing air pollution
[6] by offering real-time useful information and imperative generated
knowledge. Making an optimal decision in time in smart mobility with a wide
variety of smart devices and systems is challenging. You cannot make a
promising decision when your data is not frequently collected. Consequently, a
training process of decision support systems still is challenging due to the
lack of data [7]. In this paper, we first address current challenges in smart
cities and provide an overview of potential solutions to these challenges.
Then, we offer a framework of these solutions, called universal smart cities
decision making, with three main sections of data capturing, data analysis,
and decision making to optimize the smart mobility within smart cities.

Interestingly, large cities are losing their own urban style and turning into
smart ones. Smart cities are growing due to advanced technology, especially
Artificial Intelligence(AI). The more people live in large cities, the more
need to have an integrated system to cost-efficiently handle the ample growth
in urbanization. The proliferation of population offers smart development
challenges in such cities and enables enormous pressure on society to create
innovative, smart, and sustainable environments. Therefore, today’s developed
cities (or smart cities) are in need of integrated smart policies and novel
innovative solutions to enhance the monitoring functionality in order to
facilitate urban living conditions [8].

Smart cities are created to enable advanced capabilities, such as sustainable
energy systems and electrified transportation networks, and interact with
information and communication technologies (ICTs) to enhance the efficiency of
the cities being generated [9, 10]. An example of responding to new changes
for smart cities may be seen in the progress being made in smart healthcare
systems for emergency care cases. To enable Hospitals to monitor and control
their patients and let the specialists offer better solutions to their
patients, they need to use a smart healthcare system [11] [12].

To develop the smart healthcare system, we need to use healthcare networks
that are the inter and intra connection among the healthcare components like
IOT devices and sensors to enhance the process of monitoring and consequent
services for patients [13, 4]. The performance of such a system heavily
depends on the quality of this network communication or online synchronization
with other associated networks (other smart systems in smart cities) that
actively contribute to the service operation in a positive way. For example, a
healthcare network may take advantage of networks’ resources, like energy, in
case the system fails to run an operation due to lack of enough power supply.
In this situation, the best solution is connecting this network with other
networks for the benefit of both patients and their health and service
management reliability [4].

In the processes of making and turning cities into smart cities, we can
identify several representative features. According to a research study [14],
the scientists investigated a period of 10 years of work from 2008 to 2018
discovering that smart cities might have some features in common. The most
interesting features 1 [14, 15, 9] are smart economy, smart people, smart
governance, smart mobility, smart environment, smart energy, and smart living
that are shown in figure 1. As shown in the figure 1, although each feature
represents the importance of itself to smart cities, they all are inter-
related to each other. Each feature has direct or indirect impact on others.
Furthermore, proposed decision-making solutions used in smart cities are
Multi-criteria decision-making (MCDM), Mathematical Programming (MP),
Artificial Intelligence (AI), and Integrated Method (IM). In this study, we
aim to discover AI solutions, particularly Machine learning and deep learning
approaches rather than others, for smart mobility.

Figure 1: Smart cities main features

##  2 Role of Machine Learning in Smart Cities

Current advanced technologies in sensors and Internet of Things (IoT) devices
make it essential in leveraging Artificial Intelligence, particularly Machine
Learning, to model the data for further application. [16, 17]. The IoT devices
are considered as the most important and unavoidable parts of smart cities.
These devices provide a huge amount of data depending on which applications
are going to be used, such as healthcare and transportation applications [18].

Internet of Things technologies have proliferated in many fields, such as
smart healthcare [13, 13, 19] and smart home systems like Alexa [20, 21],
specifically in urban cities, turning them into smart cities. Thus, the huge
usage of Internet of Things technologies plays a pivot role in generating big
data which requires solutions to analyze and keep track of smart cities
activities. This big data analysis [22] provides invaluable knowledge in order
to integrate all smart cities’ sources like IOT and networks. As smart cities
and their data grow, this analysis process may become challenging for future
decision-making. In the next section, we address this problems and discuss
possible solutions have been proposed thus far. Researchers in [23] proposed a
new solution to leverage citizen-centered big data analysis to apply to smart
cities. Their solution is to determine a path for implementation of citizen-
centered big data analysis for the sake of decision-making. This solution’s
main goal is to provide imperative perspectives: data-analysis algorithms and
urban governance issues [23].

Furthermore, researchers in [24] provided several deep learning applications
in smart cities, such as smart governance, smart urban modeling, smart
education, smart transportation,intelligent infrastructures, and smart health
solutions. Additionally, the challenges of using deep learning towards smart
cities are also addressed. However, still the problem of decision making in
smart cities remains challenging. In the next section, we address the problems
and highlight the possible solutions.

##  3 Smart Cities Data Analytics Framework

There are plenty of research studies accomplished in smart cities like [25],
which is an intelligent decision computing solution for crowd monitoring. In
this section, we dig into smart cities applications, and the foundation of
techniques are developed upon by establishing a universal smart cities data
analytics framework, which is elaborately depicted in figure 2. This framework
has three main sections: Data Capturing, Data Analysis, and Decision Making.
We elaborate on each in separate sections.

###  3.1 Data Capturing

Smart cities have been engulfed with too much data which requires the
management department to control and monitor the cities, but this department
is cost and time inefficient. Hopefully, due to domains (features of smart
cities), these data ,which are grouped automatically into domains, create
particular big data for each domain separately. IOT devices used to gather
data for healthcare are completely different from the ones that are developed
specially for traffic control (i.e that is why we need to categorize the data
into groups of domains to create certain technical big data for each domain).
According to figure 2, we provide 6 different samples of sensors and IOT
devices, such as smart phones, smart cameras, smart thermometers, smart users
with sensors, smart cars, and smart houses. The variations of sensors enable a
data center to receive different types, ranges, and values of data from
objects. Here, we highlight the current upcoming challenges with associate
solutions, respectively.

$\noindent\diamond$ Challenges: The process of data gathering in smart cities
has been enhanced yet remains challenging due to lack of enough equipment in
every place we aim to monitor and make decisions for that region, like
relaying traffic information. No matter our focus, whether in smart healthcare
mobility or other areas of interest, IOT devices and other sensors may still
fail to capture all data correctly or may miss data entirely due to their
limited storage and inefficient time-scales.

In addition to that, data have proliferated significantly and are produced
from heterogeneous sources. Therefore, the types of data vary, from video and
images to digits or strings, and need particular procedures to convert all of
the data into single unit measurements. These measurements enable us to run
machine learning algorithms and other deep learning algorithms on the data
readily to make optimum decisions.

$\noindent\diamond$ Solutions: Pre-processing plays a pivotal role in managing
missing information and values within the generated dataset. There are some
basic and advanced approaches [26] to handle the missing values, and also,
there are tools and techniques to select important and relevant features like
IFAB [27] using Artificial Bee colony to get rid of irrelevant features and
ensure the genuineness and reproducibility of the results [28].

To handle the heterogeneous problem, Data Engineering [22] is responsible for
managing and analyzing the input data and adding labels to data left
unlabeled. This requires experts and time, thus it is not cost and time
efficient. Therefore, leveraging Big Data algorithms helps to tune and analyze
the data properly.

Figure 2: Smart cities data analytics framework

###  3.2 Data Analysis

The smart cities promises lead us to an ample proliferation and generation in
data from all aspects of the domains and branches. Therefore, such huge
amounts of data are at the core of the services generated by the IoT
technologies [29]. This section of the framework, Data Analysis, is imperative
because its results lead us to make proper decisions. If this process is not
accomplished, the decision made will not be efficient. Thus, a large number of
research studies have enhanced the process and yielded better results. In the
early era of smart cities, there were only limited data generated everyday due
to the lack of sensors.‘Therefore, typical machine learning algorithms were
sufficient for data analysis to make a model that can handle the situation and
provide enough information to make a decision. However, thanks to
technologies, the number of sensors and IOT objects have proliferated and thus
we have huge amounts of data that require big data algorithms like Spark and
Hadoop to handle the data. [30]

Additionally, due to the huge quantity of data, researchers used deep learning
algorithms, especially Transfer learning and Meta-learning [7] and some other
famous machine learning techniques to learn within Reinforcement Learning like
Q-learning [31, 32, 33] for generating smart systems [34, 35].

####  3.2.1 Big Data Algorithms and challenges

Due to the Big Data revolution, the enormous volume of high performance
computations are unavoidable in such smart cities. Thus, big data algorithms
are getting one of the important functioning pieces of smart cities.

$\noindent\diamond$ Transportation System: Transportation systems are
generated using advanced technologies where applications of big data vary and
are important [36]. Scenarios are established in the research studies to offer
the following options to smart people: suggesting the best travel time for any
given trip, providing real time traffic information, and predicting movement
patterns according to personality (daily routing path) or spatio-temporal
routines, enhancing crash analysis, planning bus routes, improving taxi
dispatch, and optimising traffic time during big occasions [36]. For example,
Zhu _et al_ in [37], developed an approach to use two algorithms: Bayesian
inference and Random forest to execute in real-time to predict the probability
of crash occurrence to decrease crash risks in smart cities. Moreover,
researchers [38] have established a combination of supervised and regression
algorithms, such as multivariate adaptive regression splines, regression
trees, and logistic regression, to study motor vehicle accident injury
behaviour.

$\noindent\diamond$ Urban Governance Learning from human mobility patterns
discloses the movement and trend of a large population, the most popular
applications of which are in domains like crime prediction [36], disaster
evacuation, big events management, and safety estimation. To that end,
knowledge about big events derived from bus trajectories, including event
start time and end time, can be beneficial for the event planning and
management department [39].

####  3.2.2 Machine learning process and challenges

The learning process based on given input where data are not sufficient does
not produce reliable and robust results [7]. Machine learning algorithms get
stuck in local minimum or maximum easily due to a large amount of data which
leads to a problem called over-fitting [7]. Furthermore, machine learning
algorithms fail to learn positions and states (classes) when the number
occurrence in the data is far less than others. For instance, within the smart
healthcare domain, when capturing data, the probability of having all desired
classes looks low and the number of rare diseases would be insufficient for
learning.

$\noindent\diamond$ Solutions: In paper [40], researchers established an
approach called Meta-Sense, i.e. learn to sense rather than sense to learn.
This approach took advantage of one of the important data analytic promises
called zero-shot learning (ZSL) [41, 7] . There are many applications of ZSL
in computer vision [42] and motion detection in healthcare application [43]
Furthermore, researchers in [44] proposed a solution based on ZSL called Zero-
Virus providing a deep understanding for an intelligent transportation system
to generate the best rout for drivers. Zero-virus does not need any vehicle-
tracklets annotation, thus it is the most volatile real-world traffic scheme.

$\noindent\diamond$ Machine learning statistic records: Iskandaryan _et al_ in
[6] investigated a survey to study sensor data and analyze the impact of air
quality in Smart Cities using supervised machine learning algorithms. They
established four important and popular categories of algorithms associated
with the number of research studies: first, the most popular ML algorithm,
neural network (NN), second, Logistic regression, third, ensemble algorithm,
followed by others. Furthermore, the results in [6] demonstrated that the
number of publications associated with machine learning algorithms with the
application of smart cities, particularly smart environment by predicting and
preventing the air pollution risks, have increased progressively.
Additionally, researchers in [6] evaluated the number of metrics that have
been used in publications to examine the performance of each algorithm. The
two most popular metrics are Root Mean Square Error (RMSE) and Mean Absolute
Error (MAE).

####  3.2.3 Deep learning process and challenges

Feasible smart cities are established by technology driven foundations and
their initiatives are on different branches and domains, in which each of them
may requires systems with high-performance computing resources and
technologies. Such systems have pros and cons such as saving energy and
reducing the air pollution and reducing diagnose time, but (writing something
bad). One of the most popular technologies used to tackle such huge amounts of
data is Deep Learning (DL). DL is a special algorithm within machine learning
and is efficiently used to obtain required knowledge from the input data,
extracting the patterns which govern the whole data and also classify them.
Several research studies are successfully applying DL on smart cities [24],
such as urban modeling for smart cities, intelligent infrastructure for smart
cities, and smart urban governance. Here, we aim to focus on a particular
application of deep learning which is smart mobility and transportation.

AI definitely pushed all the science a step forward by making the systems and
processes of scientific inquiry as smart as possible, for example autonomous
transportation systems [24] in smart mobility. Making a decision about whether
the object seen is a human being or not is challenging [45]. Object detection
is one of the challenging issues in smart mobility that surely boosts and
facilitates automation in transportation systems. Consequently, this
positively enhances and improves smart mobility in Smart Cities. Researchers
in [45] explored analysis of decent object detection solutions like deep
learning [46]. The scientists leveraged a well-known object detection system,
namely YOLO (You Only Look Once), which was developed earlier by Redmon and
_et al_ in [47] and assessed its performance on real-time data.

####  3.2.4 Learning process and emerging new type of data problems

In this section, we address possible challenges and solutions in the world of
data analysis. The first and foremost problem that researchers tackle is lack
of data for rare classes within the dataset that are used to make a model. The
less number of samples we have in the dataset, the higher chance of ignoring
that sample while we train and make the model. To handle this problem,
Metalearning has come to play an essential role to make a model only using few
samples. It has three important promises: zero shot learning, one shot
learning, and few shot learning [7]. Zero shot learning is a certain type of
Metalearning when a training dataset does not have any samples for classes,
and we still can predict them during a test process. For instance, a research
work [44] was conducted to not use any annotation for processing vehicles-
tracklets. This study established a route understanding system based on zero-
shot theory for intelligent transportation, namely zero-virus, which obtains
high effectiveness with zero samples of annotation of vehicle tracklets.
Further, another research work [40] established a new technique, namely
Metasense, which is the process of learning to sense rather than sensing to
learn. This process takes advantage of the lack of samples of classes by
learning from learning rather than learning from samples.

The process of non-annotation helps to work with data that lacks classes
because typical machine learning algorithms fail to detect non-annotation
classes. Furthermore, sensing to learn helps new algorithms predict
information that is completely lacking from the dataset itself. These
advancements lead us closer to (or are part of) zero shot learning, which is
critical for the advancement of Smart Cities.

The second promise is one shot learning, in which each epoch in a training
phase has only one sample per each class which is taken by a deep learning
algorithm or a combination of neural networks [7].

The third promises but not the least one, is few (k) shot learning, in which
each epoch in a training phase has only few (k) samples per each class which
are taken by a deep learning algorithm or a combination of neural networks
[7].

###  3.3 Decision Making Problems in smart cities

Decision making problems are becoming challenging issueS in smart cities where
not only the problem itself but also other relevant problems in other aspects
of smart cities need to be analyzed. Additionally, decision makers must depict
the consequence of the decision they are going to make. Thus, decision making
systems are needed in smart cities in which the systems take care of all
issues within the connected networks and only some limited information is
taken which would be enough make a optimum decision. In this section, we
highlight the challenging decision making problems and solutions.

####  3.3.1 Traffic decision making system

Traffic decision making system, as known as intelligent transportation system
(ITS), aims to detect traffic flow within a smart city and offer optimal
solutions using proper big data analytics [48]. Assessing and analysing this
big data plays a pivot role in decision making systems that makes the process
time- and cost-efficient [37]. Researchers addressed some case studies withing
traffic decision making systems, such as road traffic accidents analysis [49,
50], public transportation management and control, and road traffic flow
prediction.

$\noindent\diamond$ Challenges: In these such systems, users feed data into
them by gathering data from heterogeneous resources, such as high-performance
IOT devices, video detectors, and GPS. The systems must use big data analysis
approaches to evaluate the data online [51] in order to provide efficient and
intimate knowledge for decision making. To be more certain about the amount of
data these systems handle, consider Petabyte level data which are beyond the
utmost of traditional machine learning analytic abilities [51]. This is due to
two important issues: these traditional data processing algorithms are not yet
developed for online and real-time monitoring systems, and additionally, they
fail to learn from these data due to disorganized and nonstandard structures.

$\noindent\diamond$ Solutions: Smart cities, particularly in this domain (i.e
one of their important features, smart mobility, discussed in figure 1), need
optimum traffic decision making systems which address the aforementioned
challenges and provide efficient solutions to them. One of the solutions
discussed in [51] is the use of big data analytics approaches. They solved the
challenges by providing enough data storage, analysis and management tools.
The most important and promising frameworks, libraries or technologies are
used to analyze the big data, such as Hadoop and Spark, but not limited to
them. For instance, researchers in [52] established a time-efficient and
scalable distributed technique, namely Partition-Aware Connected Components
(PACC), for connected component computation that relies on main approaches:
edge filtering, two-step processing of partitioning and computation, and
sketching. PACC outperforms MapReduce and Spark frameworks and is time-
efficient on real-world graphs [52].

Additionally, Big Data analysis algorithms facilitate and boost the process of
handling large amount of data to provide enough information for making
decisions. Traffic decision making systems using Big Data algorithms and
technologies help associate offices and the people in charge to get to learn
drivers journey patterns within the transportation network where reports whole
networks trends or better understanding of similar drivers [37] . Considering
this feature, the systems provide the best path to drivers to reach their
destination through the minimum time possible. Furthermore, the systems help
the city to relay the traffic by controlling traffic light. The best timing to
make which light should be on or off and for what period.

Furthermore, Traffic decision making systems predicts the the probability of
traffic accident occurrence using Big Data algorithms [37]. This requires to
have smart healhthcare systems, which is nominated as one of smart cities
features discussed in figure 1, to help emergency centers to facilitate the
process of emergency rescue.

####  3.3.2  Safe and Smart Environment

Researchers in [5] leveraged deep learning algorithms and advanced
communication technologies to link vehicles, roads and drivers in order to
facilitate and enhance various traffic-related tasks and improve air pollution
[6]. The scientists focused on different initiatives with the goal of creating
a safe and smart environment and transportation. Further, Zhu _et al_ in [37],
developed an approach to use two algorithms: Bayesian inference and Random
forest to execute real-time tp predict the probability of crash occurrence to
decrease crash risks in smart cities. Moreover, researchers [38] have
established a combination of supervised and regression algorithms such as
multivariate adaptive regression splines, and classification and regression
trees, logistic regression and to experiment analytical study using of a
dataset of motor vehicle accident injury.

To have a safe environment, air pollution prediction in smart cities plays
imperative role. There have been a significant amount of work have tied to
enhance the prediction using several machine learning algorithms. The usage of
machine learning algorithms to make environment safe has increased
consistently, stating that how important this prediction would be for smart
cities [6]. We examined the most relevant research studies [6, 4, 23, 38] but
not limited to these, applying different evaluation based on several metrics.
The evaluation of those research lead us to following common observations:
first, the rate of applying of the advanced (deep learning) machine learning
algorithms have proliferated rather than typical machine learning algorithms;
second, among the prediction elements for air pollution prediction, PM2.5 is
considered as the most popular element; third, the data used for air pollution
prediction had already generated hourly rather than daily; and final
observation, efficient prediction occurs when air-quality captured data merged
with other relevant data of other networks like healthcare network within the
smart cities.

##  4 Conclusion

We highlight smart cities’ research branches and technology advancement
regarding different complex domains. We propose a solution namely, universal
smart cities decision making, which has three main section: data capturing,
data analysis and making decision. We provide an abstract review of the
fundamental concepts of Big Data, ML, and DL algorithms have been applied to
smart cities. We explore the essential role of the aforementioned algorithms
on making decision within smart cities. The goal of this study is to provide a
comprehensive survey of data analytics in smart cities, more specifically, the
role of Big Data algorithms and other advanced technologies like ML and DL for
making decision in smart mobility within smart cities.

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