Old Draft

# High-Speed Imaging Receiver Design for
6G Optical Wireless Communications:
A Rate-FOV Trade-Off

Mohammad Dehghani Soltani, Hossein Kazemi, Elham Sarbazi, Harald Haas and
Majid Safari Part of this work is accepted to be presented at the 7th IEEE ICC
Workshop on Optical Wireless Communications (OWC’22), 16–20 May 2022
[MDS2022ICC].

###### Abstract

The design of a compact high-speed and wide field of view (FOV) receiver is
challenging due to the presence of two well-known trade-offs. The first one is
the area-bandwidth trade-off of photodetectors (PDs) and the second one is the
gain-FOV due to the use of optics. The combined effects of these two trade-
offs imply that the achievable data rate of an imaging optical receiver is
limited by its FOV, i.e., a rate-FOV trade-off. To control the area-bandwidth
trade-off, an array of small PDs can be used instead of a single PD. Moreover,
in practice, a large-area lens is required to ensure sufficient power
collection, which in turn limits the receiver FOV (i.e., gain-FOV trade-off).
In this paper, we propose an imaging receiver design in the form of an array
of (PD) arrays to achieve a reasonable receiver FOV by limiting the size of
the focusing lens for each PD array. The proposed array of arrays structure
provides an effective method to control both gain-FOV trade-off (via an array
of lenses) and area-bandwidth trade-off (via arrays of PDs). We initially
derive a tractable analytical model for the signal-to-noise ratio (SNR) of an
array of PDs where the maximum ratio combining (MRC) has been employed. Then,
we extend the model for the proposed array of arrays structure. The
reliability of the analytical model is verified based on several Optic Studio-
based simulations. Next, we formulate an optimization problem to maximize the
achievable data rate of the imaging receiver while a minimum FOV is
guaranteed. The optimization problem is solved for two commonly used
modulation techniques, namely, on-off keying (OOK) and direct current (DC)
biased optical orthogonal frequency division multiplexing (DCO-OFDM) with
variable rate quadrature amplitude modulation (QAM). It is demonstrated that a
data rate of $\sim 21$ Gbps with a FOV of $15^{\circ}$ is achievable using
OFDM with a total receiver size of $2$ cm $\\!\times\\!$ $2$ cm.

###### Index Terms:

Photodetector, PIN photodiode, lens, array and optical wireless communication.

##  I Introduction

The global demand for high-speed data rate is increasing rapidly, where no
saturation trends have yet been observed [AIR_2020]. Three-dimensional (3D)
holographic displays with the ability to create virtual reality (VR)
environments, $8$K ultra-high-definition (UHD) video streaming and online
video games are just few examples of applications that require high-speed data
rate. It is forecasted there will be $5.7$ billion connected mobile users by
2023 [CiscoInternet_2023]. This yields an exponential growth in data traffic
of wireless communications. The radio frequency (RF) may struggle to handle
this high data traffic. In fact, RF spectrum is getting congested and wireless
devices are seriously interfering with each other. As a result, the data
throughput of the devices is affected severely, and the established connection
may be poor.

Optical wireless communication (OWC), which permits the availability of a huge
spectrum, is a propitious technology to alleviate the RF spectrum congestion
by offering solutions such as free space optical (FSO) communications
[FSOKhalighi2014], (ii) visible light communications (VLC)
[karunatilaka2015led], (iii) underwater OWC [UnderwaterZeng2017], (iv) optical
camera communications (OCC) [OCCShivani2017] and (v) wireless networking with
light, which is also referred to as LiFi[LiFi2016, OWCHaas2020,
Koonen2018JLT]. Moreover, OWC can create smaller communication cells in the
network, known as attocell [LiFi2016] which is a vital key to unlock the path
to exponential capacity growth for indoor scenarios [Elgala2011Mag]. OWC can
offload heavy traffic loads from congested RF wireless networks, therefore
making the RF spectrum to be available for emerging wireless applications such
as the Internet of Things (IoT).

High-speed aggregate Terabit per second indoor OWC systems are being realized
thanks to the large modulation bandwidth of laser diodes [Hong:21:Tbps,
OBrienTerabit2018, KoonenHighSpeed2020, Sarbazi2000PIMRC, Kazemi2000PIMRC].
Hong et al. demonstrated that a ten-channel wavelength division multiplexing
(WDM) intensity modulation/direct detection (IM/DD) system can achieve $>1$
Tb/s capacity at a perpendicular distance of $3.5$ m with a lateral coverage
up to $1.8$ m [Hong:21:Tbps]. A $1$ Tbit/s bi-directional free-space
connection has been presented in [OBrienTerabit2018], which employs ten WDM
channels with pulse amplitude modulation (PAM)-$4$. The coverage area of
$2.54$ m2 is reported. Beam-steering laser-based optical multiuser systems,
which is capable of supporting up to $128$ beams carrying up to $112$ Gbit/s
per beam has been developed in [KoonenHighSpeed2020].

One of the major challenges for the full adoption of OWC in 6G indoor wireless
networks is to design high-speed optical receivers that can operate reliably
in a mobile environment. Such receiver requires small detectors with large
bandwidth and a large FOV111It is noted that throughout this paper, FOV is
defined based on the full-cone angle. to support mobility. However, enabling a
high data rate and a large FOV at the same time is challenging due to the
following reasons. Firstly, there is a trade-off between photodetector (PD)
bandwidth and its sensitive area. The utilization of light-focusing optics
(lens or concentartor) can compensate for the small area of a high-bandwidth
PD and enhance the received signal-to-noise ratio (SNR). However, the use of
optics gives rise to a second trade-off between gain and the FOV of the
receiver in accordance with etendue law of conservation
[chaves2008introduction]. This indicates that the improved SNR is achieved in
expense of a narrow FOV receiver.

The choice of angle diversity receiver (ADR), which consist of multiple
narrow-FOV detectors facing different directions, is one way to achieve a
larger aperture and wider FOV [Kahn97WirelessInfrared]. In [Alkhazragi:21], a
wide-FOV receiver using fused fiber-optic tapers is proposed, which comprises
hundreds of thousands of tapered optical fibers. An overall FOV of
$30^{\circ}$ and optical gain of $~{}121.3$ are reported. However, receiver
designed based on these approaches are not as compact as typical imaging
receivers. Fluorescent optical concentrators are also introduced to enhance
the receiver FOV, which are mainly applicable for visible light
communications[CollinsFOV2014, Riaz2019Fluorescent, Manousiadis:WideFOV16]. A
fluorescent concentrator is shown to achieve a gain of $50$ times higher than
the conventional concentrators with the same FOV [CollinsFOV2014]. In
[Manousiadis:WideFOV16], a threefold increment in data rate is reported by
using a fluorescent antenna that can support FOV of $120^{\circ}$. However,
the maximum data rates reported using this approach are still limited since
color conversion restricts the bandwidth.

An array of PDs along with proper optics is another promising solution which
can offer an enhanced SNR and a wide FOV while ensuring a compact design,
which is essential for portable mobile devices. Detector arrays can be
designed using PIN or avalanche photodiodes (APD), which typically operate in
thermal noise limited and shot noise limited regimes, respectively
[StephenBAlexander1997]. For mobile scenarios, PIN photodetectors are more
preferable due to low biasing voltage [KHARRAZ20131493]. The output current of
positive-intrinsic-negative (PIN) photodetectors are typically so small that
this necessitates the use of an amplifier. Among various types of amplifiers,
transimpedance amplifiers are best suited to most optical wireless
applications, since they support a large dynamic range and have a wide
bandwidth without the need for an equalizer [personick1973receiver,
Kahn97WirelessInfrared]. High speed TIAs have been designed and fabricated in
many studies [Bashiri26GHzTIA2010, Kim40GHzTIA2019, Gracia60GHzTIA2017,
Garcia66GHzTIA2018, Vasilakopoulos96GHzTIA2015]. For instance, TIAs with $66$
GHz and $96$ GHz bandwidth are demonstrated in [Garcia66GHzTIA2018] and
[Vasilakopoulos96GHzTIA2015], respectively. The other consideration when
designing an detector array receiver is the choice of a combining technique,
e.g., equal gain combining (EGC), selection combining (SC), switched combining
(SWC) or the maximal ratio combining (MRC). It has been mathematically proven
that even though an array of PDs augments more noise, employing an efficient
combining algorithm can help us achieve a superior probability of error
performance compared to a single-detector receiver in practical channel
conditions [Bashir2020FSO, Tsai2021single].

Several experimental works in the literature have demonstrated the use of an
array of PDs to achieve high data rates. In [Koonen:2020:Receiver], a
broadband receiver with a large aperture is introduced. The authors
experimentally demonstrated a data rate of $1$ Gbps at a FOV of $10^{\circ}$.
Umezawa et al. have reported an array of $8\times 8$ pixels along with a $15$
mm diameter-lens, which manifests a FOV of $6^{\circ}$ in
[Umezawa2022LargeSubmil]. They have achieved data rate of $25$ Gbps at a
distance of $10$ m. Signal detection enhancement for PD arrays has been
discussed in [Vilnrotter2002Adaptive]. Vilnrotter et al. have proposed an
optimum array detection algorithm, which requires intensive computation and a
simpler suboptimum structure that can achieve performance improvement up to
$5$ dB. All these studies emphasize the realization of high-speed mobile
receiver by means of an array of PDs. However, to the best of our knowledge,
there is no tractable analytical framework in the literature enabling imaging
receiver designs that can optimally explore the trade-off between achievable
data rate and FOV. In this study, we initially derive a general SNR model for
a single array, where MRC technique is employed. Then, we extend the model for
an array of arrays. Afterwards, we develop an optimization problem to design
an imaging receiver with maximum achievable data rate supporting a required
FOV. In summary, the contributions of this paper are pointed out as follows:

  * $\bullet$

In-depth analysis of an imaging optical receiver by modeling beam spot size
considering both focused and defocused designs for the imaging receiver.

  * $\bullet$

Development of a tractable framework for SNR of a static single array as well
as the average SNR over different receiver tilt angles.

  * $\bullet$

Formulating an analytical optimization problem to achieve an optimum design
for receiver that can support required constraints such as a minimum FOV and
maximum bit error rate (BER). The optimization problem has been evaluated for
on-off keying (OOK) and direct current (DC) biased optical orthogonal
frequency division multiplexing (DCO-OFDM) with variable rate quadrature
amplitude modulation (QAM).

  * $\bullet$

Validating analytical models and derivations using realistic simulations
conducted in OpticStudio (Zemax) software.

  * $\bullet$

Presenting insightful results and detailed discussions about the impact of
various trade-offs on receiver design, in the presence of non-idealities and
practical aspects of the imaging receiver components.

The rest of this paper is organized as follows. In Section II, the area-
bandwidth and gain-FOV trade-offs are discussed. In Section III, we present
analytical framework of SNR for a single array as well as an array of arrays.
In Section IV, the optimization problem is formulated to obtain the optimum
design of a high-speed receiver. Simulation results are presented in Section
V. In Section VI, summary and concluding remarks of this study are drawn and a
number of possible directions are suggested for the future research.

Figure 1: A common structure of an optical wireless receiver.

##  II Design Trade-offs for High-Speed OWC Receivers

A common architecture of a receiver for OWC is shown in Fig. 1, which includes
three main elements: a photodiode (PD), optics and a TIA. The proper selection
of these elements plays a crucial role in addressing the design trade-offs as
well as fulfilling the system requirements such as a desired data rate, FOV,
and bit error ratio (BER). High bandwidth PDs are more favorable for high-
speed applications, however, the higher the bandwidth, the smaller the
sensitive area. This is recognized as the area-bandwidth trade-off and states
that a high bandwidth PD may not be able to collect sufficient power,
particularly when the beam spot radius at the receiver is much larger than the
PD side length. Furthermore, choosing high bandwidth PDs in a power-limited
regime is not a good practice as it adds more noise to the system, resulting
in SNR degradation. The use of imaging optics in front of the PD can help to
collect more power and improve the SNR. However, the FOV of the receiver will
be limited due to the law of conservation of etendue [chaves2008introduction],
resulting in a gain-FOV trade-off. Next, we will discuss the area-bandwidth
and gain-FOV trade-offs in more details.

$0$$50$$100$$150$$200$$250$$300$$350$$400$$0$$20$$40$$60$$d\
{[\mu\rm{m}]}$Bandwidth [Gbps]Equation (2)Reference [Chen2019HighSpeed]LPD3040
[DataSheetLPD3040]Reference [Umezawa2018High]Reference
[Umezawa2018High]Newport: 818-BB-35 [DataSheet15GHzNewport]EOT: ET-5000
[DataSheetEOT]EOT: ET-3600 [DataSheetEOT]Do$007\\_100$um$\\_$PIN4
[DataSheetGCS]PD-LD: PDINBJ070 Series [DataSheetPDLD]Hamamatsu: G6854-01
[DataSheetHamamatsuG6854-01]

Figure 2: Bandwidth versus the side length of a PIN photodetector, presenting
the area-bandwidth trade-off. Several practical and commercial PIN PDs have
been included.

###  II-A Area-Bandwidth Trade-off

The bandwidth of PIN photodetectors is expressed as [StephenBAlexander1997]:

$B=\dfrac{1}{\sqrt{\left(2\pi R_{\rm s}\epsilon_{0}\epsilon_{\rm
r}\dfrac{A}{\ell}\right)^{2}+\left(\dfrac{\ell}{0.44v_{\rm
s}}\right)^{2}}},\vspace{-0.1cm}$ (1)

where $R_{\rm s}$ is the junction series resistance, $\epsilon_{0}$ is the
permittivity in vacuum ($8.85\times 10^{-12}$ F.m-1) and $\epsilon_{\rm r}$ is
the relative permittivity of the semiconductor. The area and the length of the
depletion region are denoted by $A$ and $\ell$, respectively. Also, $v_{\rm
s}$ is the carrier saturation velocity which is limited by the hole saturation
velocity. The junction series resistance, $R_{\rm s}$, typically varies from
$7~{}\Omega$ to $9~{}\Omega$ depending on the PD size [Dupuis2014Exploring].
If the PD is followed by a TIA with an input resistance of $R_{\rm L}$, that
is typically $50~{}\Omega$ for commercial products, the bandwidth is affected.
In this case, $R_{\rm s}$ in (1) should be replaced by the overall resistance
$R_{\rm s}+R_{\rm L}$ [StephenBAlexander1997].

The left and right terms under the radical sign in (1) represent the junction
capacitance and transit time of a PIN detector, respectively. Clearly, as
$\ell$ increases junction capacitance decreases, yielding a bandwidth
increment, while the transit time increases, causing a bandwidth decrements.
This trade-off results in an optimum length of depletion region which is
denoted by $\ell_{\rm opt}$ [StephenBAlexander1997]. Assuming a square shape
for the PD with a side length of $d$, after some manipulations of (1), the
optimal bandwidth at $\ell=\ell_{\rm opt}$ can be rewritten as:

$B=\dfrac{1}{C_{\rm t}d},\ \ \ \ \ \ {\rm where}\ \ \ \ C_{\rm
t}=\sqrt{\dfrac{4\pi R_{\rm s}\epsilon_{0}\epsilon_{\rm r}}{0.44v_{\rm
s}}},\vspace{-0.1cm}$ (2)

The proof is provided in Appendix VII. This equation is an upper bound for
bandwidth and other depletion regions where $\ell\neq\ell_{\rm opt}$ result in
a lower value for bandwidth.

Fig. 2 illustrates the bandwidth versus side length of a photodetector for
$\ell=\ell_{\rm opt}$, which provides a theoretical upper bound on the
performance of practical detectors. For comparison, we have included several
commercial PDs in Fig. 2, which shows the tightness of the theoretical bound.
Note that the area-bandwidth trade-off can be observed from these results,
where as $d$ (or equivalently $A$) increases, the bandwidth decreases. When
the received optical power is sufficiently high, the selection of high
bandwidth PDs are preferred to to acheive high data rates.

###  II-B Gain-FOV Trade-off

The use of optics boosts the collected optical power at the expense of a
reduced receiver FOV, i.e., the gain-FOV trade-off. In this paper, we focus on
using an imaging lens and the study of compound parabolic concentrator (CPC)
is kept for future investigations. Without loss of generality, in this study,
we employ an aspheric lens. We calculate the beam spot radius after the lens
and the FOV as a function of distance between the lens and the PD. These two
functions are essential in the SNR analysis and optimal design of receiver.
Aspheric lenses are known to produce a precise and small beam spots. As an
example, we consider a Thorlabs aspheric lens with the specifications given in
Table I and the geometry shown in Fig. 3. We note that the following
theoretical analyses can be extended to other imaging lenses.

(a)

(b)

Figure 3: Thorlabs $354140$-B aspheric lens: (a) unmounted, (b) simulated in OpticStudio [Zemax].  TABLE I: Thorlabs $354140$-B aspheric lens specifications [DataSheetAsphericLens]. | Effective
---
focal length
Outer diameter |  | Back
---
focal length
Clear aperture | Glass |  | refractive index
---
@ 850 nm
1.45 mm | 2.4 mm | 0.82 mm | 1.6 mm | D-ZK3 | 1.5809

(a)

(b)

Figure 4: (a) Beam spot radius versus distance from the lens obtained from
OpticStudio software and the fitted curve with $b_{1}=0.69$ and $b_{0}=1\
\mu$m, (b) beam spot at $L=820\ \mu$m in OpticStudio (dimensions are in mm).

Let $L$ denote the distance between the aspheric lens and the observing plane
and $W_{2}(L)$ denote the beam spot radius after the lens. The fraction of the
total beam power contained by $W_{2}(L)$ is denoted by $\eta$ in this study.
Fig. 4 illustrates $W_{2}(L)$ at different values of $L$ for $\eta=0.5$ the
Thorlabs $354140$-B aspheric lens. The values of $W_{2}(L)$ are obtained using
the commercial optical design software, OpticStudio [Zemax]. According to Fig.
4a, the beam spot radius can be well approximated by:

$W_{2}(L)=b_{1}(f_{\rm b}-L)+b_{0},\vspace{-0.2cm}$ (3)

where $f_{\rm b}$ is the back focal length of the aspheric lens. The two
constants $b_{0}$ and $b_{1}$ can be obtained from the clear aperture of the
lens denoted by $\rm{CA}$ and the spot radius in the diffraction limit.
Diffraction limit determines the maximum resolution that can be achieved by an
optical component. It is defined as $r_{\mathrm{DL}}=1.22\lambda f_{\rm
e}/\rm{CA}$, where $\lambda$ is the wavelength of the light, and $f_{\rm e}$
is the effective focal length of the lens [PrinciplesBorn1999]. Assuming that
the incident beam spot is larger than the lens clear aperture,
$W_{2}(L)=0.5\rm{CA}$ at $L=0$. Also, as shown in Fig. 4a, at $L=820~{}$nm,
$W_{2}(L)$ is equal to the diffraction limit. Therefore, $b_{0}$ and $b_{1}$
are obtained as:

$\displaystyle b_{0}=r_{\mathrm{DL}}\kappa,$ (4a) $\displaystyle
b_{1}=\frac{\kappa}{2f_{\rm b}}\left({\rm
CA}-2r_{\mathrm{DL}}\right),\vspace{-0.2cm}$ (4b)

where $f_{\rm e}$ and $\rm CA$ are the effective focal length and the clear
aperture of the lens, respectively, which are provided in the datasheet
[DataSheetAsphericLens]. In (4), $\kappa$ is a constant parameter, which is
$\kappa=\sqrt{\eta}$.

The thickness of the considered aspheric lens is $1$ mm, which is comparable
to its radius of curvature that is $0.84$ mm [DataSheetAsphericLens].
Therefore, the lens is not a thin lens222A thin lens is a lens with a
thickness that is negligible compared to the radii of curvature of the lens
surfaces [saleh2019fundamentals].. We therefore evaluate the relation between
FOV and $L$ using the OpticStudio simulator. We accentuate that the
traditional tangent equation that describes the relation between the FOV and
the effective focal length of the lens [EdmundOpticsFoV] cannot be directly
applied for all $0\leq L\leq f_{\rm b}$. There are several important remarks
to highlight before introducing the approximation of the FOV. Firstly, $L$ and
the back focal point, $f_{\rm b}$, are measured from the back surface of the
lens, while the effective focal length, $f_{\rm e}$, is measured from the
principal plane of the lens. Secondly, the traditional expression of FOV is
applicable for $L$ that ensure the condition of $W_{2}(L)\leq d/2$, i.e., the
beam spot is inscribed inside the PD. Thirdly, the FOV is defined based on the
half power metric. We can now present the approximation relation of FOV and
$L$ which is valid for $W_{2}(L)\leq d$ as:

${\rm{FOV}}=2\tan^{-1}\left(\dfrac{d}{2(L+f_{\rm e}-f_{\rm b})}\right),\ \ \ \
\ \ \ \ \dfrac{b_{0}-b_{1}f_{\rm b}-d/2}{b_{1}}\leq L\leq f_{\rm
b},\vspace{-0.1cm}$ (5)

where the lower bound of $L$ is obtained by solving $2W_{2}(L)=d$ and using
(3).

Fig. 5a-5c represent the normalized collected power at the PD, $\bar{P_{\rm
r}}$, versus different tilt angles of transmitted beam about the $X$ axis.
These results are obtained in OpticStudio and the collected power is
normalized to the value at zero tilt angle. Three separate locations are
selected to exhibit the effect of tilt angle on the collected power. It can be
observed that at $L=400$ $\mu$m for the tilt angles between $-15.5^{\circ}$ to
$15.5^{\circ}$ about the $X$ axis, the normalized power is greater than $0.5$.
The range would be $[-10.5^{\circ},10.5^{\circ}]$ and $[-8^{\circ},8^{\circ}]$
at $L=600$ $\mu$m and $L=820$ $\mu$m, respectively. One typical definition of
the FOV is based on the half value of the collected power. Therefore, the FOV
at $L=400$ $\mu$m, $L=600$ $\mu$m and $L=820$ $\mu$m are $33^{\circ}$,
$21^{\circ}$ and $16^{\circ}$, respectively. Fig. 5d illustrates the results
of $L$ versus FOV acquired in OpticStudio. A square PD with side length of
$400$ $\mu$m is considered in these simulation results. The fitted curve to
these results is:

$L=a_{3}{\rm FOV}^{3}+a_{2}{\rm FOV}^{2}+a_{1}{\rm FOV}+a_{0},\vspace{-0.2cm}$
(6)

where $(a_{3},a_{2},a_{1},a_{0})=(-0.08506,6.142,-159.5,1720)$ with a
normalized root mean square error (NRMSE) of less than $10^{-4}$. The values
for $(a_{3},a_{2},a_{1},a_{0})$ are obtained using the curve fitting toolbox
of Matlab software. We have also plotted the approximation relation of FOV and
$L$, which provides a good estimate for the range $520\ \mu\rm{m}\leq L\leq
820\ \mu\rm{m}$. For comparison, the curve of $2\tan^{-1}\left(d/2L\right)$
has been included, where we can see there is a noticeable gap between this
curve and the OpticStudio simulations without applying the impact of
$f_{\rm{e}}-f_{\rm{b}}$.

(a)

(b)

(c)

(d)

Figure 5: Normalized received optical power versus transmitted beam tilt angle
at (a) $L=400$ $\mu$m, (b) $L=600$ $\mu$m and (c) $L=820$ $\mu$m. (d) $L$
versus FOV obtained from OpticStudio software, the fitted curve and the
conventional analytical model for FOV.

###  II-C Array of Arrays Receiver Structure

In an imaging optical receiver, the combined effect of the area-bandwidth and
gain-FOV trade-offs introduces a new trade-off between the achievable data
rate and FOV of the receiver, namely the rate-FOV trade-off. This is because
for achieving higher data rate, more bandwidth (i.e., smaller PD) and higher
collected power (i.e., larger collecting lens), which both imposing limits on
the receiver FOV. In this paper, we propose an array of array structure for
the imaging receiver that allows us to effectively control the area-bandwidth
and gain-FOV trade-offs in order to achieve the desired system requirement,
that is constrained by the rate-FOV trade-off.

(a)

(b)

Figure 6: (a) The proposed $\sqrt{\mathcal{N}_{\rm
a}}\times\sqrt{\mathcal{N}_{\rm a}}$ array of arrays structure for imaging
receiver, (b) geometry of beam spots and one array of PDs.

Fig. 6a illustrates the the proposed array of arrays structure, in which
$\mathcal{N}_{\rm a}$ arrays are assumed to be arranged on a square lattice.
It can be thus expressed as:

$\mathcal{N}_{\rm a}=\left(\frac{D_{\rm a}}{2r_{\rm
lns}}\right)^{2},\vspace{-0.1cm}$ (7)

where $D_{\rm a}$ is the side length of the array of array receiver. In this
structure, to produce a high received optical power, the combined output of
the array of $\mathcal{N}_{\rm a}$ imaging detectors are considered where each
imaging detector includes an $N_{\rm PD}$ photodetector array equipped with a
collecting lens placed at a distance of $L$ from it (see Fig. 6b). Note that
each array of PDs are assumed to be integrated into a single chip. Let the
side length of each PD and the array be denoted by $d$ and $D$, respectively.
The fill factor (FF) of an array of $N_{\rm PD}$ PDs, which are arranged in a
square lattice, can be described as:

${\rm{FF}}=\frac{N_{\rm
PD}d^{2}}{D^{2}}\,\mathpunct{\raisebox{1.72218pt}{.}}\vspace{-0.2cm}$ (8)

Assuming that the overall size of the receiver is fixed, the size of the outer
array (i.e., $\mathcal{N}_{\rm a}$) controls the gain-FOV trade-off while the
size of the inner arrays (i.e., $N_{\rm PD}$) controls the bandwidth-area
trade-off. Therefore the sizes of inner and outer arrays can be adjusted to
achieve desired receiver performance in terms of data rate, FOV, and BER.
Fixing the size of arrays, the key remaining control parameters are the side
length of each PD, $d$, and the distance between the lens and PD array, $L$.
Note that defocusing (i.e., reducing $L$ below the focal length) can further
increase the FOV of the receiver (see Fig. 6b). In the following sections, we
will initially present the SNR analysis of an array of arrays and then, the
problem formulation for an optimized design will be introduced.

##  III SNR Analysis of the Array of Arrays Receiver

We assume that a lens of radius $r_{\rm lns}$ is placed in front of the array,
as shown in Fig. 6b. It is also assumed that each PD in the array is
individually connected to a high bandwidth TIA, since the output current of
PIN photodetectors are typically small. Under this assumption, the receiver
bandwidth is limited by the PD’s bandwidth.

Three types of noise can affect the performance of this receiver; thermal,
shot and relative intensity noise (RIN) noise [Sarbazi2000PIMRC,
Kazemi2000PIMRC]. The dominant noise source is the TIA thermal noise whose
variance is given by [StephenBAlexander1997, Kahn97WirelessInfrared]:

$\sigma_{\rm n}^{2}=\dfrac{4k_{\rm b}TF_{\rm n}B}{R_{\rm
f}}\,\mathpunct{\raisebox{1.72218pt}{,}}$ (9)

where $k_{\rm b}$ is the Boltzmann constant, $R_{\rm f}$ is the feedback
resistor of the TIA, $T$ is temperature in Kelvin, $F_{\rm n}$ is the noise
figure of the TIA and $B$ is the bandwidth of the PIN photodetector.

Different combining methods such as maximum ratio combining (MRC), equal gain
combining (EGC) and selection combining (SC) can be used to combine the output
of PDs. Typically, MRC outperforms the other two techniques while requiring a
more complex hardware circuit. In this paper, we provide the analyses for MRC
and EGC, however, they can be extended to SC in a similar way. In the
following, we start by deriving the SNR of the MRC technique for a single
array shown in Fig. 6a. Next, we will present the SNR derivations of EGC
method. We will then extend our analysis to an array of
$\sqrt{\mathcal{N}_{\rm a}}\times\sqrt{\mathcal{N}_{\rm a}}$ PD arrays as
illustrated in Fig. 6b.

The SNR of MRC technique for a single array can be obtained as
[goldsmith2005wireless]:

$\gamma_{\rm MRC}=\sum_{i=1}^{N_{\rm PD}}\gamma_{i},\vspace{-0.1cm}$ (10)

where $\gamma_{i}$ is the SNR of the $i$th PD, which is:

$\gamma_{i}=\dfrac{\left(R_{\rm res}P_{{\rm r},i}\right)^{2}}{\sigma_{\rm
n}^{2}}\,\mathpunct{\raisebox{1.72218pt}{,}}\vspace{-0.1cm}$ (11)

where $R_{\rm res}$ is the PD responsivity, $\sigma_{\rm n}^{2}$ is the
variance of noise given in (9), and $P_{{\rm r},i}$ denotes the received
optical power of the $i$th PD. $P_{{\rm r},i}$ depends on the beam spot radius
after the lens, $W_{2}(L)$, which is expressed by (3) and is shown in Fig. 6b.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: Illustration of different conditions of beam spot radius for PD
arrays of $4\times 4$ and $3\times 3$.

The radius of beam spot at the receiver plane is denoted by $W(z)$, where it
is assumed that $W(z)\gg r_{\rm lns}$. Thus, we can assume almost a uniform
beam intensity over the whole area of the lens. Furthermore, ray-tracing
simulations conducted in OpticStudio [Zemax] confirm that the beam after the
lens has almost a uniform intensity profile. Accordingly, $P_{{\rm r},i}$ can
be readily calculated as:

$P_{{\rm r},i}=\iint\limits_{\mathcal{A}_{i}}\dfrac{\xi P_{{\rm r,lns}}}{\pi
W_{2}^{2}(L)}\,dx\,dy,\vspace{-0.1cm}$ (12)

where $\mathcal{A}_{i}$ denotes the overlapping area of beam after lens and
the $i$th PD. Also, $\xi=\xi_{\rm r}\xi_{\rm p}\xi_{\rm a}$; with $\xi_{\rm
r}$ representing the transmission coefficient of the lens; $\xi_{\rm p}=0.5$
due to definition of spot radius being the minimum radius that includes $50\%$
of total power and $\xi_{\rm a}={\rm CA}^{2}/(2r_{\rm lns})^{2}$, where ${\rm
CA}$ is the clear aperture. In (12), $P_{{\rm r,lns}}$ is the power collected
by the lens with the radius of $r_{\rm lns}$. Based on the relation between
$W_{2}(L)$, $D$ and $d$, as well as the number of PDs in the array, whether it
is even or odd, there are separate cases for $P_{{\rm r},i}$. As shown in Fig.
7a, small beam spot focused at the center of the PD array can result in zero
collected power when $N_{\rm PD}$ is even. In contrast, when $N_{\rm PD}$ is
odd as shown in Fig. 7d, a small beam spot radius can yield a high collected
power. When considering all these possible conditions, the received power can
be expressed as:

$P_{{\rm r},i}=\begin{cases}\xi P_{{\rm r,lns}}\rho,\ \ \ \ \
&W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}d,\\\ \xi P_{{\rm
r,lns}}\dfrac{\mathcal{A}_{i}}{\pi W_{2}^{2}(L)},\ \ \ \ \
&\dfrac{1}{\sqrt{\pi}}d<W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}D,\\\ \xi P_{{\rm
r,lns}}\dfrac{d^{2}}{\pi W_{2}^{2}(L)},\ \ \ \ \
&W_{2}(L)>\dfrac{1}{\sqrt{\pi}}D,\vspace{-0.1cm}\end{cases}$ (13)

where $\rho=1$ if $N_{\rm PD}$ is odd, and $\rho=0$ otherwise. Substituting
(13) into (11) and then into (10), the SNR of MRC technique for a single array
is obtained as:

$\gamma_{\rm MRC}=\dfrac{1}{\dfrac{4k_{\rm b}TF_{\rm n}B}{R_{\rm
f}}}\times\begin{cases}\rho\left(R_{\rm res}\xi P_{{\rm r,lns}}\right)^{2},\ \
\ \ \ &W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}d,\\\ \left(\dfrac{R_{\rm res}\xi
P_{{\rm r,lns}}}{\pi W_{2}^{2}(L)}\right)^{2}\sum\limits_{i=1}^{N_{\rm
PD}}\mathcal{A}_{i}^{2},\ \ \ \ \
&\dfrac{1}{\sqrt{\pi}}d<W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}D,\\\ N_{\rm
PD}\left(R_{\rm res}\xi P_{{\rm r,lns}}\dfrac{d^{2}}{\pi
W_{2}^{2}(L)}\right)^{2},\ \ \ \ \
&W_{2}(L)>\dfrac{1}{\sqrt{\pi}}D.\vspace{-0.2cm}\end{cases}$ (14)

It can be shown that $\sum\limits_{i=1}^{N_{\rm PD}}\mathcal{A}_{i}^{2}$ can
be approximated as (this can be interpreted as a beam spot of radius
$W_{2}(L)$ that includes $\pi W_{2}^{2}(L){\rm{FF}}/d^{2}$ active PDs out of
all $N_{\rm PD}$ PDs of the array with $\mathcal{A}_{i}=d^{2}$):

$\sum\limits_{i=1}^{N_{\rm PD}}\mathcal{A}_{i}^{2}\approx\pi
W_{2}^{2}(L){\rm{FF}}d^{2}.\vspace{-0.1cm}$ (15)

Therefore, when $N_{\rm PD}$ is odd, (14) can be simplified as:

$\gamma_{\rm MRC}=\dfrac{1}{\dfrac{4k_{\rm b}TF_{\rm n}B}{R_{\rm
f}}}\times\begin{cases}\left(R_{\rm res}\xi P_{{\rm r,lns}}\right)^{2},\ \ \ \
\ &W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}d,\\\ (R_{\rm res}\xi P_{\rm
r,lns})^{2}\dfrac{d^{2}}{\pi W_{2}^{2}(L)}{\rm{FF}},\ \ \ \ \
&\dfrac{1}{\sqrt{\pi}}d<W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}D,\\\ N_{\rm
PD}\left(R_{\rm res}\xi P_{{\rm r,lns}}\dfrac{d^{2}}{\pi
W_{2}^{2}(L)}\right)^{2},\ \ \ \ \
&W_{2}(L)>\dfrac{1}{\sqrt{\pi}}D.\vspace{-0.1cm}\end{cases}$ (16)

When $N_{\rm PD}$ is even, we can present the approximation of SNR by
considering the impact of $d_{\sigma}$ as well, which is the gap distance
between the PDs. As shown in Fig. 7a, when
$W_{2}(L)\leq\frac{\sqrt{2}}{2}d_{\sigma}$, the collected power and the SNR is
zero. If $\frac{\sqrt{2}}{2}d_{\sigma}<W_{2}(L)\leq
d+\frac{\sqrt{2}}{2}d_{\sigma}$, then, $\sum\limits_{i=1}^{N_{\rm
PD}}\mathcal{A}_{i}^{2}\approx\pi
W_{2}^{2}(L){\rm{FF}}\left(W_{2}(L)-\frac{\sqrt{2}}{2}d_{\sigma}\right)^{2}$.
When beam radius becomes larger, i.e.,
$d+\frac{\sqrt{2}}{2}d_{\sigma}<W_{2}(L)\leq\frac{1}{\sqrt{\pi}}D$,
$\sum\limits_{i=1}^{N_{\rm PD}}\mathcal{A}_{i}^{2}$ can be obtained from (15).
For beam radii greater than $\frac{1}{\sqrt{\pi}}D$, the SNR is the same as
the one for odd PD array configuration. Hence, when $N_{\rm PD}$ is even, (14)
can be expressed as:

$\gamma_{\rm MRC}\\!=\\!\dfrac{1}{\dfrac{4k_{\rm b}TF_{\rm n}B}{R_{\rm
f}}}\\!\times\\!\begin{cases}0,\ \ \
&W_{2}(L)\leq\frac{\sqrt{2}}{2}d_{\sigma},\vspace{-0.1cm}\\\ (R_{\rm res}\xi
P_{\rm
r,lns})^{2}\dfrac{\\!\left(\\!W_{2}(L)\\!-\frac{\sqrt{2}}{2}d_{\sigma}\\!\right)^{2}}{\pi
W_{2}^{2}(L)}{\rm{FF}},\ \ \ &\frac{\sqrt{2}}{2}d_{\sigma}<W_{2}(L)\leq
d+\frac{\sqrt{2}}{2}d_{\sigma},\vspace{-0.1cm}\\\ (R_{\rm res}\xi P_{\rm
r,lns})^{2}\dfrac{d^{2}}{\pi W_{2}^{2}(L)}{\rm{FF}},\ \ \
&d+\frac{\sqrt{2}}{2}d_{\sigma}<W_{2}(L)\leq\frac{1}{\sqrt{\pi}}D,\\\ N_{\rm
PD}\left(R_{\rm res}\xi P_{{\rm r,lns}}\dfrac{d^{2}}{\pi
W_{2}^{2}(L)}\right)^{2},\ \ \
&W_{2}(L)>\frac{1}{\sqrt{\pi}}D,\vspace{-0.3cm}\end{cases}$ (17)

where $d_{\sigma}$ is the gap distance between PDs as shown in Fig. 7a.

The SNR of EGC is given as [ZheAngle2014]:

$\gamma_{\rm EGC}=\dfrac{\left(\sum\limits_{i=1}^{N_{\rm PD}}{R_{\rm
res}P_{{\rm r},i}}\right)^{2}}{N_{\rm PD}\sigma_{\rm n}^{2}}.\vspace{-0.1cm}$
(18)

Following the same approach described for MRC, we can readily obtain the SNR
of EGC technique, where for odd values of $N_{\rm PD}$, it is given as:

$\gamma_{\rm EGC}=\dfrac{1}{\dfrac{4k_{\rm b}TF_{\rm n}B}{R_{\rm
f}}}\times\begin{cases}\dfrac{1}{N_{\rm PD}}\left(R_{\rm res}\xi P_{{\rm
r,lns}}\right)^{2},\ \ \ \ \ &W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}d,\\\
\dfrac{1}{N_{\rm PD}}(R_{\rm res}\xi P_{\rm r,lns}{\rm{FF}})^{2},\ \ \ \ \
&\dfrac{1}{\sqrt{\pi}}d<W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}D,\vspace{-0.1cm}\\\
N_{\rm PD}\left(R_{\rm res}\xi P_{{\rm r,lns}}\dfrac{d^{2}}{\pi
W_{2}^{2}(L)}\right)^{2},\ \ \ \ \
&W_{2}(L)>\dfrac{1}{\sqrt{\pi}}D.\vspace{-0.2cm}\end{cases}$ (19)

Moreover, when $N_{\rm PD}$ is even, the SNR of EGC is expressed as:

$\gamma_{\rm EGC}\\!=\\!\dfrac{1}{\dfrac{4k_{\rm b}TF_{\rm n}B}{R_{\rm
f}}}\\!\times\\!\begin{cases}0,\ \ \
&W_{2}(L)\leq\frac{\sqrt{2}}{2}d_{\sigma},\vspace{-0.1cm}\\\ \dfrac{1}{N_{\rm
PD}}\\!\\!\left(\\!\\!R_{\rm res}\xi P_{\rm
r,lns}\dfrac{\\!\\!\left(\\!W_{2}(L)\\!-\frac{\sqrt{2}}{2}d_{\sigma}\right)^{2}{\rm{FF}}\\!}{W_{2}^{2}(L)}\right)^{2},\
\ \ &\frac{\sqrt{2}}{2}d_{\sigma}\\!<\\!W_{2}(L)\\!\leq
d+\frac{\sqrt{2}}{2}d_{\sigma},\vspace{-0.1cm}\\\ \dfrac{1}{N_{\rm
PD}}\left(R_{\rm res}\xi P_{\rm r,lns}{\rm{FF}}\right)^{2},\ \ \
&d+\frac{\sqrt{2}}{2}d_{\sigma}\\!<\\!W_{2}(L)\\!\leq\frac{1}{\sqrt{\pi}}D,\\\
N_{\rm PD}\left(R_{\rm res}\xi P_{{\rm r,lns}}\dfrac{d^{2}}{\pi
W_{2}^{2}(L)}\right)^{2},\ \ \
&W_{2}(L)>\frac{1}{\sqrt{\pi}}D,\vspace{-0.1cm}\end{cases}$ (20)

(a)

(b)

Figure 8: SNR comparison of EGC, MRC and analytical model considering a single
array of (a) $8\times 8$ and, (b) $9\times 9$.

Fig. 8 presents a comparison between the SNRs of EGC and MRC obtained from
OpticStudio as well as the analytical model given in (16). The distance
between transmitter and receiver is set to $3$ m and a lens is placed in front
of the laser source at the transmitter side to ensure a beam spot radius of
$10$ cm at the receiver side. The eye-safe transmit power is calculated as
$10$ mW considering the lens impact [MDSoltani2021safety]. The valley in the
SNR results of array of $8\times 8$ is due to the small beam spot radius after
the lens, where it happens right in the gap between PDs. On the other hand,
for the array of $9\times 9$, the SNR has its maximum when
$W_{2}(L)\leq\frac{1}{\sqrt{\pi}}d$. The results shown in Fig. 8 confirms the
accuracy of our analytical model. There is a slight difference between MRC
simulations of OpticStudio and analytical model for lower values of $L$. This
is mainly due to non-uniformity of the beam profile over the considered beam
spot. However, we can observe from Fig. 8 that the SNR at this regime is low
thereby not being the regime of interest.

Let us now investigate the mean SNR at the output of the proposed array of
arrays receiver illustrated in Fig. 6a by averaging over different transmit
beam angles within the FOV of the receiver. The change of transmit beam angle
will be equivalent to shift of beam spot on the PD array plane. By considering
(14), when $W_{2}(L)>\frac{1}{\sqrt{\pi}}d$, the shift of the beam spot on the
PD array plane will not affect the SNR. However, when
$W_{2}(L)\leq\frac{1}{\sqrt{\pi}}d$, the SNR can be zero with the probability
of $(1-{\rm FF})$ or non-zero with the probability of ${\rm FF}$. Thus, the
average value is $(1-{\rm FF})\times 0+{\rm FF}\times(\frac{4k_{\rm b}TF_{\rm
n}B}{R_{\rm f}})^{-1}(R_{\rm res}\xi P_{{\rm r,lns}})$. Therefore, the average
SNR of MRC can be approximated as follows:

$\displaystyle\overline{\gamma}_{\rm MRC}$
$\displaystyle=\dfrac{\sqrt{\mathcal{N}_{\rm a}}}{\dfrac{4k_{\rm b}TF_{\rm
n}B}{R_{\rm f}}}\times\begin{cases}(R_{\rm res}\xi P_{\rm
r,lns})^{2}{\rm{FF}},&W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}d,\\\ (R_{\rm res}\xi
P_{\rm r,lns})^{2}\dfrac{d^{2}}{\pi
W_{2}^{2}(L)}{\rm{FF}},&\dfrac{1}{\sqrt{\pi}}d<W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}D,\\\
N_{\rm PD}\left(R_{\rm res}\xi P_{\rm r,lns}\dfrac{d^{2}}{\pi
W_{2}^{2}(L)}\right)^{2},&W_{2}(L)>\dfrac{1}{\sqrt{\pi}}D.\vspace{-0.1cm}\end{cases}$
(21)

Similarly, the average SNR of EGC over various tilt angles within the FOV of
the receiver for an array of arrays of PIN detectors when thermal noise is
dominant can be obtained as:

$\overline{\gamma}_{\rm EGC}=\dfrac{\sqrt{\mathcal{N}_{\rm a}}}{\dfrac{4k_{\rm
b}TF_{\rm n}B}{R_{\rm f}}}\times\begin{cases}\dfrac{1}{N_{\rm PD}}(R_{\rm
res}\xi P_{\rm r,lns})^{2}{\rm{FF}},&W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}d,\\\
\dfrac{1}{N_{\rm PD}}(R_{\rm res}\xi P_{\rm
r,lns}{\rm{FF}})^{2},&\dfrac{1}{\sqrt{\pi}}d<W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}D,\\\
N_{\rm PD}\left(R_{\rm res}\xi P_{\rm r,lns}\dfrac{d^{2}}{\pi
W_{2}^{2}(L)}\right)^{2},&W_{2}(L)>\dfrac{1}{\sqrt{\pi}}D.\vspace{-0.1cm}\end{cases}$
(22)

By comparing (21) and (22), it can be concluded that both MRC and EGC have a
similar performance for $W_{2}(L)\geq\frac{1}{\sqrt{\pi}}D$. However, when
$W_{2}(L)\leq\frac{1}{\sqrt{\pi}}d$, MRC outperforms EGC around:

$G=10\log_{10}\left(\frac{\overline{\gamma}_{\rm MRC}}{\overline{\gamma}_{\rm
EGC}}\right)=10\log_{10}(N_{\rm PD})\ \ \ {\rm[dB]}.\vspace{-0.1cm}$ (23)

Larger values of $N_{\rm PD}$ can result in a significant gain difference
between MRC and EGC. Due to this notable difference, we will consider MRC for
the rest of our analyses.

##  IV Optimum Geometric Design of High-Speed Receiver

In this section, assuming a fixed size for inner arrays, we aim to optimize
the side length of each PDs in the array, $d$, as well as the distance between
the lens and the array, $L$, to achieve the maximum data rate while satisfying
a desired receiver FOV. In order to develop a compact receiver, $L$ is assumed
to be limited to the range $0\leq L\leq f_{\rm b}$. That is, the lens is not
placed at a distance more than its focal length from the PD array. This allows
for both focused ($L=f_{\rm b}$) and defocused ($L<f_{\rm b}$) designs while
the latter can relax the FOV constraint to some degree. It will be shown that
the optimum value of $L$ can be specified according to the desired FOV. In
addition, the optimum value of $d$ is selected in such a way that it achieves
the maximum data rate while fulfilling the required BER (or SNR). We note that
although small PDs have a higher bandwidth, they add more noise to the system
thereby degrading the SNR. On the other hand, large area PDs decrease the
system bandwidth and consequently the achievable data rate. Such a behavior in
a power-limited regime yields an optimum $d$ that maximizes the data rate. The
optimization problem can be formulated as:

$\displaystyle\max\limits_{L,d}\ \ \ $ $\displaystyle R$ (24a)
$\displaystyle\rm{s.t.}\ \ \ $ $\displaystyle{\rm{FOV}}(L)\geq\rm{FOV}_{\rm
req},$ (24b) $\displaystyle\overline{\gamma}_{\rm MRC}(L,d)\geq\gamma_{\rm
req},$ (24c) $\displaystyle d_{\rm min}\leq d\leq
D\sqrt{\frac{{\rm{FF}}}{N_{\rm PD}}}.\vspace{-0.2cm}$ (24d)

The first constraint guarantees the FOV requirement. The second constraint
enforces limitations on both the size of PDs and the distance between the lens
and the array. In fact, SNR is directly related to $d$ and $L$, where small
values of $d$ and $L$ result in lower SNRs. By substituting (2) and (8) into
(21), we can express this relationship as

$\displaystyle\overline{\gamma}_{\rm
MRC}=\frac{1}{A_{x}}\times\begin{cases}d^{3},&W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}d,\\\
\dfrac{d^{5}}{\pi
W_{2}^{2}(L)},&\dfrac{1}{\sqrt{\pi}}d<W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}D,\\\
D^{2}\dfrac{d^{5}}{\left(\pi
W_{2}^{2}(L)\right)^{2}},&W_{2}(L)>\dfrac{1}{\sqrt{\pi}}D,\vspace{-0.1cm}\end{cases}$
(25)

where

$A_{x}=\dfrac{4k_{\rm b}TF_{\rm n}D^{2}}{N_{\rm PD}\sqrt{\mathcal{N}_{\rm
a}}C_{\rm t}R_{\rm f}(R_{\rm res}\xi P_{\rm
r,lns})^{2}}\,\mathpunct{\raisebox{1.72218pt}{.}}\vspace{-0.1cm}$ (26)

In constraint (24c), $\gamma_{\rm req}$ is the required SNR to ensure that BER
is less than a target BER, i.e., $\rm{BER}_{\rm req}$. The threshold for SNR
can be represented based on the BER requirement as [ghassemlooy2019optical]:

$\gamma_{\rm req}=\left(Q^{-1}(\rm BER_{req})\right)^{2},\vspace{-0.2cm}$ (27)

where $Q^{-1}(\cdot)$ is the inverse of $Q$-function. The size of each PD on
the array is limited to $d_{\rm min}$ and $D\sqrt{\frac{{\rm{FF}}}{N_{\rm
PD}}}$. The right side of the third constraint is obtained based on (8) and
ensures the fill factor of the array is limited to ${\rm{FF}}$. To guarantee
that the designed inner arrays follow a square lattice configuration, the
number of PDs on each array are selected from the set of square numbers. The
transmit power in this optimization problem is set to the maximum permissible
value according to the eye safety regulations, i.e., $P_{\rm t}=P_{\rm
t,max}$. The details of the eye safety considerations for obtaining $P_{\rm
t,max}$ can be found in [MDSoltani2021safety]. In what follows, we solve the
optimization problem given in (24) for two well-known and practical modulation
schemes; on-off keying (OOK) and direct current-biased optical orthogonal
frequency division multiplexing (DCO-OFDM) with adaptive quadrature amplitude
modulation (QAM).

###  IV-A OOK Modulation

The data rate of OOK modulation is limited by the Nyquist’s sampling rate,
i.e., $R\leq 2B$. Although this requires employing efficient pulse shaping
techniques, experimental studies have shown that data rates close to the
Nyquist limit are achievable for OOK modulation [Minh2009OOK, Watson:13-OOK,
Lee:15-OOK]. The linear relationship between $R$ and $B$, indicates that $B$
can replace $R$ as the objective function of the optimization problem in (24).
Note that the objective function $B$ is a monotonically decreasing function of
$d$ and does not depend on $L$ (see (2)); hence the maximized solution should
happen at the boundary of the feasible set, i.e., at the minimum value of $d$
that satisfies the constraints of (24). To ensure the first constraint in
(24), $L\leq L_{\rm max}$, where $L_{\rm max}$ can be obtained by substituting
$\rm{FOV}_{\rm req}$ in (6) as:

$L_{\rm max}=a_{3}{\rm FOV}_{\rm req}^{3}+a_{2}{\rm FOV}_{\rm
req}^{2}+a_{1}{\rm FOV}_{\rm req}+a_{0}.\vspace{-0.2cm}$ (28)

The second and third constraints define the feasible set of $d$. The second
constraint depends on both $d$ and $L$ based on the three different conditions
defined in (25). Next, we find the optimum (i.e., minimum) $d$ within the
feasible set of (24). The optimization problem in (24) along with the
piecewise SNR function in (25) can be broken down into three separate
optimization problems, where the global solution will be the minimum solution
of $d$ among them.

Optimization problem 1:

$\displaystyle\max\limits_{d}\ \ \ $ $\displaystyle\frac{2}{C_{\rm t}d}$ (29a)
$\displaystyle\vspace{-0.1cm}\rm{s.t.}\ \ \ $
$\displaystyle{\rm{FOV}}(L)\geq\rm{FOV}_{\rm req},$ (29b)
$\displaystyle\frac{d^{3}}{A_{x}}\geq\gamma_{\rm req},$ (29c) $\displaystyle
d_{\rm min}\leq d\leq D\sqrt{\frac{{\rm{FF}}}{N_{\rm PD}}},$ (29d)
$\displaystyle d\geq\sqrt{\pi}W_{2}(L).\vspace{-0.3cm}$ (29e)

The second constraint of (29) implies

$d\geq d_{\Delta}\triangleq\left({A_{x}\gamma_{\rm
req}}\right)^{\frac{1}{3}}.\vspace{-0.2cm}$ (30)

The first constraint of (29) enforces $L<L_{\rm max}$. Since $W_{2}(L)$ is a
decreasing function of $L$, $L=L_{\rm max}$ ensures the minimum value of $d$
in (29e). Therefore, by applying the third and fourth constraints in (29), the
optimum (i.e., minimum) value of $d$ can be expressed as

$d_{\rm opt,1}=\max\left(d_{\rm min},d_{\Delta},\sqrt{\pi}W_{2}(L_{\rm
max})\right),\vspace{-0.2cm}$ (31)

for $\max\left(d_{\Delta},\sqrt{\pi}W_{2}(L_{\rm max})\right)\leq
D\sqrt{\frac{{\rm{FF}}}{N_{\rm PD}}}$; otherwise this condition does not
provide a solution. Also note that $L$ should be taken from
$[L_{\Delta},L_{\rm max}]$, where $L_{\Delta}=f_{\rm
b}-\frac{1}{b_{1}}\left(\frac{1}{\sqrt{\pi}}d_{\rm opt,1}-b_{0}\right)$ to
satisfy the constraints of the optimisation problem.

Optimization problem 2:

$\displaystyle\max\limits_{d}\ \ \ $ $\displaystyle\frac{2}{C_{\rm t}d}$ (32a)
$\displaystyle\rm{s.t.}\ \ \ $ $\displaystyle{\rm{FOV}}(L)\geq\rm{FOV}_{\rm
req},$ (32b) $\displaystyle\frac{d^{5}}{A_{x}\pi W_{2}^{2}(L)}\geq\gamma_{\rm
req},$ (32c) $\displaystyle d_{\rm min}\leq d\leq
D\sqrt{\frac{{\rm{FF}}}{N_{\rm PD}}},$ (32d) $\displaystyle
W_{2}(L)>\frac{d}{\sqrt{\pi}},$ (32e) $\displaystyle
W_{2}(L)\leq\dfrac{1}{\sqrt{\pi}}D,\vspace{-0.2cm}$ (32f)

The second constraint of (32) implies

$d\geq\left(\pi W_{2}^{2}(L)A_{x}\gamma_{\rm
req}\right)^{\frac{1}{5}}.\vspace{-0.2cm}$ (33)

Let us denote the minimum of $d$ given in (33) as
$d_{\lambda}(L)\triangleq\left(\pi W_{2}^{2}(L)A_{x}\gamma_{\rm
req}\right)^{\frac{1}{5}}$. Noting that $d_{\lambda}(L)$ is a decreasing
function of $L$, the minimum value of $d$ (to ensure maximum data rate) can be
obtained on the boundary of feasible set at the maximum possible value of $L$,
i.e., by substituting the fourth constraint of (32) into (33), which is
$d_{\lambda,\min}=\left(A_{x}\gamma_{\rm req}\right)^{\frac{1}{3}}$. Applying
the third constraint in (32), the solution for $d$ is given by:

$d_{\rm opt,2}=\max\left(d_{\rm min},d_{\lambda,\min}\right),\vspace{-0.2cm}$
(34)

for $d_{\lambda,\min}\leq D\sqrt{\frac{{\rm{FF}}}{N_{\rm PD}}}$; otherwise
this condition does not provide a solution. The first constraint of (32)
enforces $L<L_{\rm max}$. Also, (34) and (32f) imposes
$L<L_{\lambda}\triangleq f_{\rm b}-\frac{1}{b_{1}}\left(\max\left(d_{\rm
min},d_{\lambda,\min}\right)-b_{0}\right)$. The fifth constraint imposes
$L\geq L_{\rm D}$, where $L_{\rm D}=f_{\rm
b}-\frac{1}{b_{1}}\left(\frac{1}{\sqrt{\pi}}D-b_{0}\right)$. Hence, $L_{\rm
D}\leq L\leq\min(L_{\lambda},L_{\rm max})$, where the optimum value of $L$ is
$\min(L_{\lambda},L_{\rm max})$.

Optimization problem 3:

$\displaystyle\max\limits_{d}\ \ \ $ $\displaystyle\frac{2}{C_{\rm t}d}$ (35a)
$\displaystyle\rm{s.t.}\ \ \ $ $\displaystyle{\rm{FOV}}(L)\geq\rm{FOV}_{\rm
req},$ (35b) $\displaystyle\frac{d^{5}}{A_{x}\left(\pi
W_{2}^{2}(L)\right)^{2}}D^{2}\geq\gamma_{\rm req},$ (35c) $\displaystyle
d_{\rm min}\leq d\leq D\sqrt{\frac{{\rm{FF}}}{N_{\rm PD}}},$ (35d)
$\displaystyle W_{2}(L)>\dfrac{1}{\sqrt{\pi}}D,\vspace{-0.2cm}$ (35e)

The second constraint of the optimization problem gives

$d\geq\left(\pi^{2}W_{2}^{4}(L)\dfrac{A_{x}}{D^{2}}\gamma_{\rm
req}\right)^{\frac{1}{5}}.\vspace{-0.2cm}$ (36)

Let us define $d_{\rm
g}(L)\triangleq\left(\pi^{2}W_{2}^{4}(L)\dfrac{A_{x}}{D^{2}}\gamma_{\rm
req}\right)^{\frac{1}{5}}$. Based on the first and fourth constraints of (35),
$L\leq\min(L_{\rm D},L_{\rm max})$. Similarly, $d_{\rm g}(L)$ is a decreasing
function of $L$; hence, the minimum value of $d$ is achieved on the boundary
of feasible set at the maximum value of $L$. Similarly, by applying the third
constraint in (35), the solution for $d$ is obtained as:

$d_{\rm opt,3}=\max\left(d_{\rm min},d_{\rm g}(\min(L_{\rm D},L_{\rm
max}))\right).\vspace{-0.2cm}$ (37)

for $d_{\rm g}(\min(L_{\rm D},L_{\rm max}))\leq D\sqrt{\frac{{\rm{FF}}}{N_{\rm
PD}}}$; otherwise this condition does not provide a solution.

The possible solutions of the three conditions above are summerized in Table
II. The global optimal solution of OOK can be then written based on the
minimum value of (31), (34) and (37).

TABLE II: The possible solutions of the optimization problem for OOK. Optimum $L$ | Optimum $d$ | condition
---|---|---
$[L_{\Delta},L_{\rm max}]$ | $d_{\rm opt,1}=\max\left(d_{\rm min},d_{\Delta},\sqrt{\pi}W_{2}(L)\right)$ | $\max\left(d_{\Delta},\sqrt{\pi}W_{2}(L)\right)\leq D\sqrt{\frac{{\rm{FF}}}{N_{\rm PD}}}$
$\min(L_{\lambda},L_{\rm max})$ | $d_{\rm opt,2}=\max\left(d_{\rm min},d_{\lambda}(\min(L_{\lambda},L_{\rm max}))\right)$ | $d_{\lambda}(\min(L_{\lambda},L_{\rm max}))\leq D\sqrt{\frac{{\rm{FF}}}{N_{\rm PD}}}$
$\min(L_{\rm g},L_{\rm max})$ | $d_{\rm opt,3}=\max\left(d_{\rm min},d_{\rm g}(\min(L_{\rm g},L_{\rm max}))\right)$ | $d_{\rm g}(\min(L_{\rm g},L_{\rm max}))\leq D\sqrt{\frac{{\rm{FF}}}{N_{\rm PD}}}$

###  IV-B DCO-OFDM

To ensure a high spectral efficiency with intensity modulation and direct
detection, we here implement the DC-biased optical OFDM with adaptive QAM. By
properly choosing the variance of the OFDM signal and the DC bias, a tight
upper bound for the achievable data rate is given by [Sarbazi2000PIMRC,
Kazemi2000PIMRC]:

$R=\nu B\log_{2}\left(1+\frac{\overline{\gamma}_{\rm
MRC}}{\Gamma}\right),\vspace{-0.2cm}$ (38)

where $\nu=\frac{N_{\rm sc}-2}{N_{\rm sc}}$ with $N_{\rm sc}$ denoting the
number of subcarriers. In (38), $\Gamma$ represents the SNR gap due to the
required BER, which is given by:

$\Gamma=\frac{-\ln(5{\rm BER_{\rm
req}})}{1.5}\,\mathpunct{\raisebox{1.72218pt}{.}}\vspace{-0.1cm}$ (39)

The optimization problem defined in (24) can be applied here to find the
optimum side length of PDs and the distance between the lens and the inner
array to maximize the data rate. By considering the first constraint of (24),
we have $L\leq L_{\rm max}$. Since $R$ is a monotonically increasing function
with respect to $L$, therefore, $L=L_{\rm max}$ is the optimum value for $L$
when $W_{2}(L)>\dfrac{1}{2}d$. When $W_{2}(L)\leq\dfrac{1}{2}d$, the SNR and
$R$ are maximum for $L\in[L_{0},L_{\rm max}]$. Therefore, the optimum value of
$L$ is the same for both OOK and DCO-OFDM. We now need to obtain the optimum
$d$. The constraints obtained in (30), (33) and (36) are required to ensure
that $\gamma\geq\gamma_{\rm req}$. For OOK modulation, the optimization
problem was fairly simple as $R$ was a decreasing function of $d$. As
highlighted $L=L_{\rm max}$ will result in the maximum SNR and data rate. By
substituting, $L=L_{\rm max}$ in (38), $R$ can be treated as a function of $d$
only. Here, $R$, which is a differentiable function with respect to $d$, has
different behaviors in the range of $[d_{\rm min},d_{\rm
max}=D\sqrt{\frac{{\rm{FF}}}{N_{\rm PD}}}]$. Therefore, the derivative of $R$
should be evaluated to check whether the peak data rate occurs at any of the
extremum points. Let us start by substituting (21) into (38) and analyzing
each case as follows.

1) if $W_{2}(L_{\rm max})\leq\dfrac{1}{2}d$,

$R=\dfrac{\nu}{C_{\rm t}d}\log_{2}\left(1+\frac{d^{3}}{\Gamma
A_{x}}\right).\vspace{-0.1cm}$ (40)

The derivative of (40) can be readily found and the extremum point is given
as:

$d_{*}=\left({15.8\Gamma A_{x}}\right)^{\frac{1}{3}}.\vspace{-0.2cm}$ (41)

Based on the constraints on $d$, i.e., $d_{\rm min}<d<d_{\rm max}$ and
$d>d_{\Delta}$, and $d_{*}$, twelve possible cases can be realized. These
cases are summarized in Table III. It is noted that the behaviour of $R$,
whether it is monotonically increasing/decreasing or it increases then
decreases, specifies the optimum $d$. The behavior of $R$ is also added to the
Table for the sake of better understanding.

TABLE III: Different relations between $d_{*}$, $d_{\Delta}$, $d_{\rm min}$ and $d_{\rm max}=D\sqrt{\frac{{\rm{FF}}}{N_{\rm PD}}}$ for first subfunction of SNR using DCO-OFDM. Case No. | Condition | Optimum $d$ |  | Behavior of $R(d)$
---
@ $d\in\left[d_{\rm min},d_{\rm max}\right]$
1 | $d_{*}<d_{\Delta}<d_{\rm min}$ | $\max(d_{\rm min},2W_{2}(L_{\rm max}))$ | decreasing
2 | $d_{\Delta}<d_{*}<d_{\rm min}$ | $\max(d_{\rm min},2W_{2}(L_{\rm max}))$ | decreasing
3 | $d_{\Delta}<d_{\rm min}<d_{*}<d_{\rm max}$ | $\max(d_{*},2W_{2}(L_{\rm max}))$ |  | increasing till $d_{*}$
---
then decreasing
4 | $d_{\Delta}<d_{\rm min}<d_{\rm max}<d_{*}$ | $d_{\rm max}$ | increasing
5 | $d_{*}<d_{\rm min}<d_{\Delta}<d_{\rm max}$ | $\max(d_{\Delta},2W_{2}(L_{\rm max}))$ | decreasing
6 | $d_{\rm min}<d_{*}<d_{\Delta}<d_{\rm max}$ | $\max(d_{\Delta},2W_{2}(L_{\rm max}))$ |  | increasing till $d_{*}$
---
then decreasing
7 | $d_{\rm min}<d_{\Delta}<d_{*}<d_{\rm max}$ | $\max(d_{*},2W_{2}(L_{\rm max}))$ |  | increasing till $d_{*}$
---
then decreasing
8 | $d_{\rm min}<d_{\Delta}<d_{\rm max}<d_{*}$ | $d_{\rm max}$ | increasing
9 | $d_{*}<d_{\rm min}<d_{\rm max}<d_{\Delta}$ | No solution | –
10 | $d_{\rm min}<d_{*}<d_{\rm max}<d_{\Delta}$ | No solution | –
11 | $d_{\rm min}<d_{\rm max}<d_{*}<d_{\Delta}$ | No solution | –
12 | $d_{\rm min}<d_{\rm max}<d_{\Delta}<d_{*}$ | No solution | –

2) if $\dfrac{1}{2}d<W_{2}(L_{\rm max})\leq\dfrac{1}{\sqrt{\pi}}D$,

$R=\dfrac{\nu}{C_{\rm t}d}\log_{2}\left(1+\frac{d^{5}}{\pi W_{2}^{2}(L_{\rm
max})\Gamma A_{x}}\right).\vspace{-0.2cm}$ (42)

The extremum point for this case can be simply obtained as:

$d_{**}=\left({142.32\pi W_{2}^{2}(L_{\rm max})\Gamma
A_{x}}\right)^{\frac{1}{5}}.\vspace{-0.2cm}$ (43)

Similarly, based on the constraint $d_{\rm min}<d<d_{\rm max}$ and
$d>d_{\lambda}$, and $d_{**}$, twelve different cases should be assessed for
finding the optimum value of $d$. These cases are presented in Table IV. The
last column providing the bahaviour of $R$ is removed in this Table as it is
similar to the one shown in Table III.

TABLE IV: Different relations between $d_{**}$, $d_{\lambda}$, $d_{\rm min}$ and $d_{\rm max}$ for second subfunction of SNR using DCO-OFDM. Case No. | Condition | Optimum $d$
---|---|---
1 | $d_{**}<d_{\lambda}<d_{\rm min}$ | $d_{\rm min}$
2 | $d_{\lambda}<d_{**}<d_{\rm min}$ | $d_{\rm min}$
3 | $d_{\lambda}<d_{\rm min}<d_{**}<d_{\rm max}$ | $d_{**}$
4 | $d_{\lambda}<d_{\rm min}<d_{\rm max}<d_{**}$ | $d_{\rm max}$
5 | $d_{**}<d_{\rm min}<d_{\lambda}<d_{\rm max}$ | $d_{\lambda}$
6 | $d_{\rm min}<d_{**}<d_{\lambda}<d_{\rm max}$ | $d_{\lambda}$
7 | $d_{\rm min}<d_{\lambda}<d_{**}<d_{\rm max}$ | $d_{**}$
8 | $d_{\rm min}<d_{\lambda}<d_{\rm max}<d_{**}$ | $d_{\rm max}$
9 | $d_{**}<d_{\rm min}<d_{\rm max}<d_{\lambda}$ | No solution
10 | $d_{\rm min}<d_{**}<d_{\rm max}<d_{\lambda}$ | No solution
11 | $d_{\rm min}<d_{\rm max}<d_{**}<d_{\lambda}$ | No solution
12 | $d_{\rm min}<d_{\rm max}<d_{\lambda}<d_{**}$ | No solution

3) if $W_{2}(L_{\rm max})>\dfrac{1}{\sqrt{\pi}}D$,

$R=\dfrac{\nu}{C_{\rm
t}d}\log_{2}\left(1+\frac{D^{2}d^{5}}{\pi^{2}W_{4}^{2}(L_{\rm max})\Gamma
A_{x}}\right),\vspace{-0.1cm}$ (44)

The extremum point of $R$ is:

$d_{***}=\left(\frac{142.32\pi^{2}W_{2}^{4}(L_{\rm max})\Gamma
A_{x}}{D^{2}}\right)^{\frac{1}{5}}.\vspace{-0.1cm}$ (45)

For this condition also, twelve different cases should be checked to obtain
the optimum $d$. These cases can be simply denoted by replacing $d_{**}$ and
$d_{\lambda}$ in Table IV with $d_{***}$ and $d_{\rm g}$, respectively.

##  V Simulation Results

A single mode vertical cavity surface emitting laser (VCSEL) transmitter is
considered which operates at $850$ nm with a beam waist of $5~{}\mu$m. A lens
is placed in front of the VCSEL source to produce a $20$ cm beam spot diameter
at $2$ m. The transmit power of $10$ mW is obtained based on the eye safety
considerations [MDSoltani2021safety], which enforces a power-limited regime.
The maximum size of receiver is set to be $2$ cm $\times$ $2$ cm. The FF of
the array is set to ${\rm{FF}}=0.64$ similar to the fabricated array in
[Umezawa2022LargeSubmil]. We have fixed the size of each inner array to $400$
$\mu$m $\times$ $400$ $\mu$m (similar to [Umezawa2022LargeSubmil]). Each inner
array is equipped with the aspheric lens $354140-$B from Thorlabs333This lens
is just an example and other lenses can be also used.. The lens parameters are
provided in Table I. We have also evaluated the transmission coefficient of
the considered aspheric lens using OpticStudio by calculating the ratio
between the transmission power through the lens and the incident power to it.
The results show that the transmission coefficient is $\xi_{\rm r}=0.88$. The
rest of the simulation parameters are provided in Table V.

TABLE V: Simulation Parameters. Parameter | Symbol | Value
---|---|---
Feedback resistor | $R_{\rm f}$ |  $500$ $\Omega$
Load resistor | $R_{\rm L}$ |  $50$ $\Omega$
Junction series resistor | $R_{\rm s}$ |  $7$ $\Omega$ [Dupuis2014Exploring]
Carrier saturation velocity | $v_{\rm s}$ |  $4.8\times 10^{4}$ m/s
TIA noise figure | $F_{\rm n}$ |  $5$ dB
Temperature | $T$ |  $300^{\circ}$ K
Side length of inner array | $D$ |  $400\ \mu$m
Transmission coefficient | $\xi_{\rm r}$ | $0.88$
Transmit power | $P_{\rm t}$ |  $10$ mW
Receiver responsivity | $R_{\rm res}$ |  $0.5$ A$/$W
Figure 9: SNR versus $L$ for EGC, MRC and the analytical model based on (21)
for $\mathcal{N}_{\rm a}=64$ and $N_{\rm PD}=64$.

Fig. 9 show the SNR results of MRC and EGC obtained from OpticStudio. The
results are shown for $\mathcal{N}_{\rm a}=64$ and $N_{\rm PD}=49$. A
remarkable gap of $\sim 18$ dB can be observed between MRC and EGC techniques
(this is compatible with $10\log(N_{\rm PD})=18$ dB discussed in Section III).
This means that in order to attain the SNR performance of MRC by EGC, more
number of outer arrays (i.e. a larger receiver) is required. However, this may
not be applicable for mobile devices. In Fig. 9, the analytical expression for
the SNR of MRC given in (25) is compared with the simulation results, which
proves the accuracy of our analytical model. Hence one can rely on the
proposed analytical model instead of OpticStudio ray-tracing simulations to
save computation time.

Fig. 10 illustrates the optimized design for the inner array and the array of
arrays based on the analytical derivations given in Section IV-A. A maximum
data rate of $23.82$ Gbps can be achieved for a given BER of $10^{-3}$ and a
FOV of $15^{\circ}$ using OOK modulation. It is assumed that the maximum
length of the receiver is $2$ cm in these results. As shown in Fig. 10a, the
inner array that provides the maximum data rate includes $7\times 7$ PDs each
with a side length of $44.81\ \mu$m. This size of a PD provides a bandwidth of
$11.91$ GHz and it ensures the BER is less than $10^{-3}$. The achieved FF of
the array is $0.61$ which is very close to our target FF of $0.64$. Fig. 10c
presents the achievable data rate versus the number of PDs in each inner array
and the number of arrays. A minimum $3\times 3$ array of arrays is required to
fulfill the required BER of $10^{-3}$ and the required FOV of $15^{\circ}$,
which achieves data rate of $3.38$ Gbps. It can be also observed that when the
number of PDs ($N_{\rm PD}$) increases, bigger size of receiver (i.e., larger
$\mathcal{N}_{\rm a}$) is also required to satisfy the BER and FOV
requirements. This is due the higher thermal noise accumulated in a larger PD
array.

(a) Optimized single PD array

(b) Array of arrays receiver

(c) Data rate

Figure 10: Optimized inner array and array of arrays configurations that
provide the maximum data rate of $23.82$ Gbps with OOK for a given FOV of
$15^{\circ}$.

The constraints given in (24) and the piecewise SNR function forms different
feasible regions of the optimization problem. Fig. 11 illustrates these
regions for $\mathcal{N}_{\rm a}=64$, $N_{\rm PD}=49$, ${\rm{FF}}=0.64$,
$\gamma\geq\gamma_{\rm req}=9.55$ and ${\rm FOV}\geq{\rm
FOV_{req}}=15^{\circ}$. The constraint $\gamma\geq\gamma_{\rm req}=9.55$
ensures ${\rm BER}\leq{\rm BER_{req}}=10^{-3}$ for the OOK modulation
[Kahn97WirelessInfrared], and according to (6), the constraint ${\rm
FOV}\geq{\rm FOV_{req}}=15^{\circ}$ is equivalent to $L\leq 820\ \mu$m. The
cyan areas correspond to each sub-function of (25). The magenta areas
represent the conditions given in (30), (33) and (36). These conditions
guarantee the required SNR. The yellow areas indicate the third constraint in
(24), which is $10\ \mu$m $\leq d\leq 400\sqrt{\frac{0.64}{{49}}}=45.7\ \mu$m.
The feasible region for the optimization problem can be found by looking into
the intersection of these areas. The subsets in Fig. 11a and Fig. 11b depict
the zoomed intersection areas for convenience. The optimum value of $d$ is the
smallest in the intersection area to ensure the maximum data rate since
$R=\frac{1}{C_{\rm t}d}$. Therefore, $d_{\rm opt}=44.81$ $\mu$m and $L_{\rm
max}\in[785,820]$ $\mu$m, which are in agreement with the analytical solutions
given in first row of the Table II for OOK modulation. We note that the choice
of $L=785$ $\mu$m results in the FOV of $16.2^{\circ}$.

(a)

(b)

(c)

Figure 11: Feasible region based on (a) first, (b) second and (c) third
equations of SNR given in (25).

It is noted that by setting ${\rm{FF}}=0.64$, the optimization problem will
only find solution with ${\rm{FF}}\\!\leq\\!0.64$. However, one can simply set
${\rm{FF}}=1$ and observe solutions with FF less than $1$. If we choose
${\rm{FF}}=1$ and solving the optimization problem for OOK modulation, the
maximum data rate of $15.11$ Gbps can be achieved. This is related to the
receiver with $N_{\rm PD}=100$, $d_{\rm opt}=35.33\ \mu$m, ${\rm{FF}}=0.78$
and $\mathcal{N}_{\rm a}=64$.

(a) $\mathcal{N}_{\rm a}=25$

(b) $\mathcal{N}_{\rm a}=36$

(c) $\mathcal{N}_{\rm a}=49$

(d) $\mathcal{N}_{\rm a}=64$

Figure 12: Maximum data rate versus $\rm{FOV_{req}}$ for different number of
PDs on the inner array and different sizes of array of arrays. OOK modulation
and FF of $0.64$ are assumed for these results.

Fig. 12 show the maximum data rate versus the required FOV. To have a fair
comparison, the overall sensitive area of each configuration is almost equal
so that the FF of $0.64$ is achieved. Each subfigure represents a specific
size of receiver (indicated by the number of outer arrays, $\mathcal{N}_{\rm
a}$). We note that neither $\mathcal{N}_{\rm a}=1$ nor $\mathcal{N}_{\rm a}=4$
are able to fulfill the BER and FOV requirement. Both sizes of
$\mathcal{N}_{\rm a}=9$ and $\mathcal{N}_{\rm a}=16$ can ensure the design
requirements, only if $N_{\rm PD}=1$. A receiver with $\mathcal{N}_{\rm a}=25$
with either $N_{\rm PD}=1$ or $N_{\rm PD}=4$ configuration is able to
guarantee BER of less than $0.001$ and required FOV of $15^{\circ}$ as shown
in Fig. 12a. The rate-FOV trade-off can be observed in these results, where
higher data rate are achievable in cost of lower FOV. For $\mathcal{N}_{\rm
a}=25$, the PD configuration with $N_{\rm PD}=4$ is able to fulfill
$\rm{FOV_{req}}\leq 30^{\circ}$ while the PD configuration with $N_{\rm PD}=1$
can support $\rm{FOV_{req}}\leq 35^{\circ}$. As shown in Fig. 12d, when
$\mathcal{N}_{\rm a}$ increases to $64$, inner arrays with $N_{\rm PD}$ upto
$49$ detectors are able to support the desired BER and FOV. The maximum data
rate is related to the receiver structure with $\mathcal{N}_{\rm a}=64$ and
$N_{\rm PD}=49$, which is $23.82$ Gbps. These maximum data rate can be
achieved for $785\leq L\leq 820$ $\mu$m. It worth mentioning that inner arrays
with $N_{\rm PD}\geq 64$ are not able to satisfy the BER requirement. The
reason is that the high bandwidth of the PDs adds more noise to the system.
This is in accordance with the early statement that choosing high bandwidth
PDs in a power-limited regime is not helpful as more noise will be added to
the system. It is also noted that $\rm{FOV}\geq 30^{\circ}$ can be achieved
only with single-PD arrays at very close distances to the lens. Arrays with
other PD configurations fail to assure $\rm{BER}\leq 0.001$. For a single PD
equal to the size of the array ($400\times 400$ $\mu$m2), we can attain
$70^{\circ}$ FOV at $L\approx 0$ (see Fig. 5d) and a maximum data rate of
$2.68$ Gbps with $\mathcal{N}_{\rm a}=64$.

Next, we present the results for DCO-OFDM. The number of subcarriers is $512$
for the following results. Fig. 13 illustrates the optimized design for each
single array and the array of arrays based on the analytical derivatives given
in Section IV-B. The maximum data rate of $21.14$ Gbps can be achieved for the
given BER of $0.001$ and the FOV of $15^{\circ}$ using DCO-OFDM. Similarly, it
is assumed that the maximum size of the receiver is $2$ cm $\times$ $2$ cm in
these results. As shown in Fig. 13a, the inner array that provides the maximum
data rate includes $6\times 6$ PDs ($N_{\rm PD}=36$) with side length of
$53.33\ \mu$m. This size of a PD provides a bandwidth of $10$ GHz and it
ensures the BER is less than $0.001$. The structure of the optimized inner and
outer arrays and its dimensions are depicted in Fig. 13a and Fig. 13b,
respectively. Fig. 13c presents the achievable data rate versus $N_{\rm PD}$
in each inner array and $\mathcal{N}_{\rm a}$. A minimum of $\mathcal{N}_{\rm
a}=16$ arrays is required to guarantee the desired BER of $0.001$ and the
required FOV of $15^{\circ}$, which achieves data rate of $4.3$ Gbps.

(a) Optimized single array

(b) Array of arrays

(c) Data rate

Figure 13: Optimized single array and array of arrays configurations that
provide the maximum data rate of $21.14$ Gbps with DCO-OFDM for the given FOV
of $15^{\circ}$.

Fig. 14 illustrates the regions for $N_{\rm PD}=36$, $\mathcal{N}_{\rm a}=64$,
${\rm{FF}}=0.64$, $\gamma\geq\gamma_{\rm req}=10.6$ (this ensures ${\rm
BER}\leq{\rm BER_{req}}=0.001$ for DCO-OFDM [Kazemi2000PIMRC]) and ${\rm
FOV}\geq{\rm FOV_{req}}=15^{\circ}$ (or equivalently for $L\leq 820\ \mu$m
based on (6)). The feasible region for the optimization problem can be found
by looking into the intersection of these areas. It can be seen that there is
no intersections between the regions in Fig. 14c. However, the first equation
of SNR in (25) yields to a feasible region as shown in Fig. 14a. There are
plenty of combinations for the pair of $(d,L)$ that fulfil ${\rm BER}\leq
0.001$ and ${\rm FOV}\geq 15$. The optimum one that maximizes $R$ can be
determined based on the eighth row of Table III. Since $R(d)$ is a
monotonically increasing function in the range of $d\in[d_{\rm min},d_{\rm
max}]$, $d_{\rm opt}=d_{\rm max}=53.33$ $\mu$m. Furthermore, $L_{\rm
max}\in[777,820]$ $\mu$m, where the choice of $L=777$ $\mu$m yields the FOV of
$16.4^{\circ}$.

(a)

(b)

(c)

Figure 14: Feasible region based on (a) first, (b) second and (c) third
equations of SNR given in (25).

(a) $\mathcal{N}_{\rm a}=25$

(b) $\mathcal{N}_{\rm a}=36$

(c) $\mathcal{N}_{\rm a}=49$

(d) $\mathcal{N}_{\rm a}=64$

Figure 15: Maximum data rate versus $\rm{FOV_{req}}$ for different number of
PDs on a single array and different sizes of array of arrays. DCO-OFDM and FF
of $0.64$ are assumed for these results.

Fig. 15 indicates the maximum data rate versus the required FOV for DCO-OFDM.
Each subfigure represents a specific $\mathcal{N}_{\rm a}$. It should be noted
that none of the receiver structures with $\mathcal{N}_{\rm a}=1$,
$\mathcal{N}_{\rm a}=4$ and $\mathcal{N}_{\rm a}=9$ are able to fulfill the
BER and FOV requirements. A receiver with $\mathcal{N}_{\rm a}=16$ arrays can
ensure the design requirements, only if each inner array includes $1$ PD. As
shown in Fig. 15a, a receiver that includes $\mathcal{N}_{\rm a}=25$ arrays
with either $N_{\rm PD}=1$ or $N_{\rm PD}=4$ PD configuration for the inner
arrays can ensure the desired BER and FOV. The inner array with $N_{\rm PD}=4$
PDs support higher data rate and lower FOV compared to the one with $N_{\rm
PD}=1$ structure. When the size of the receiver increases to $\mathcal{N}_{\rm
a}=64$ arrays, each inner array with $1$ PD upto $36$ PD arrangement are able
to support BER of less than $0.001$ and the required FOV. The maximum data
rate is related to $\mathcal{N}_{\rm a}=64$ arrays with $N_{\rm PD}=36$ PDs on
each inner array, which is $21.14$ Gbps. These maximum data rate can be
achieved for $777\leq L\leq 820$ $\mu$m. It should be highlighted that we are
in a power-limited regime due to eye safety considerations, where lower
modulations order are preferable. That is the reason, OOK outperforms DCO-
OFDM. However, in a high SNR regime (which can be realized with APD), DCO-OFDM
outperforms OOK.

##  VI Summary and Concluding Remarks

In this study, we proposed an imaging receiver which consists of an array of
arrays structure to deliberately address the two well-known tradeoffs, i.e.,
area-bandwidth and gain-FOV. The proposed receiver is able to control the
combined impacts of both tradeoffs effectively, which was introduced as the
rate-FOV tradeoff. We derived the analytical models of SNR for both EGC and
MRC techniques for an array of arrays structure. The analytical models are
verified with realistic simulations conducted in OpticStudio (Zemax) software.
We then formulated an optimization problem to assure design requirements such
as FOV and BER for OOK modulation and DCO-OFDM. Furthermore, we derived the
optimum side length of PDs and the distance between an array and the lens. In
order to have realistic simulations, we considered practical aspects of
receiver elements, such as the transmission coefficient of the lens, and so
on. The simulations of beam intensity profile and SNR are carried out using
the OpticStudio software, which ensures the reliability of our results.
Insightful results and in-depth discussions are demonstrated for both OOK and
DCO-OFDM. Simulation results confirm that with a square lattice arrangement of
$8\times 8$ array of arrays, we are able to achieve a maximum data rate of
$23.82$ Gbps and $21.14$ Gbps data rate with a FOV of $15^{\circ}$ using OOK
and DCO-OFDM, respectively. Some possible directions for future research can
be the use of a non-imaging structure, angle diversity receiver formation and
selection combining technique.

##  VII Acknowledgement

The authors acknowledge financial support from the EPSRC under program grant
EP/S016570/1 “Terabit Bidirectional Multi-User Optical Wireless System (TOWS)
for 6G LiFi”.

### Proof of (2)

For ease of notations, we write (1) as below:

$B=\dfrac{1}{\sqrt{\left(\dfrac{p_{1}}{\ell}\right)^{2}+\left(p_{2}\ell\right)^{2}}},$
(46)

where $p_{1}=2\pi R_{\rm s}\epsilon_{0}\epsilon_{\rm r}d^{2}$ and
$p_{2}=1/{0.44v_{\rm s}}$ are independent of $\ell$. The derivative of (46)
with respect to $\ell$ is given as:

$\dfrac{{\rm d}B}{{\rm
d}\ell}=-\dfrac{p_{2}^{2}\ell^{4}-p_{1}^{2}}{2\ell^{3}\left(\left(\dfrac{p_{1}}{\ell}\right)^{2}+p_{2}^{2}\ell^{2}\right)^{\frac{3}{2}}}.$
(47)

Finally, the optimum depletion region can be found by solving,
$p_{2}^{2}\ell^{4}-p_{1}^{2}=0$, which is given as:

$\ell_{\rm opt}=\sqrt{\dfrac{p_{1}}{p_{2}}}=d\sqrt{0.88\pi R_{\rm
s}\epsilon_{0}\epsilon_{\rm r}v_{\rm s}}.$ (48)

By substituting the $\ell_{\rm opt}$ in (46), we have:

$B=\dfrac{1}{C_{\rm t}d},\ \ \ \ \ \ {\rm where}\ \ \ \ C_{\rm
t}=\sqrt{\dfrac{4\pi R_{\rm s}\epsilon_{0}\epsilon_{\rm r}}{0.44v_{\rm s}}}.$
(49)

This completes the proof of (2).

  *[FOV]: field of view
  *[SNR]: signal-to-noise ratio
  *[PIN]: positive-intrinsic-negative
  *[DC]: direct current
  *[PD]: photodiode
  *[TIA]: transimpedance amplifier
  *[BER]: bit error ratio
  *[CPC]: compound parabolic concentrator
  *[FF]: fill factor
  *[RIN]: relative intensity noise
  *[MRC]: maximum ratio combining
  *[EGC]: equal gain combining
  *[SC]: selection combining
  *[SNRs]: signal-to-noise ratio
  *[PDs]: photodiode
  *[OOK]: on-off keying
  *[DCO-OFDM]: direct current-biased optical orthogonal frequency division multiplexing
  *[QAM]: quadrature amplitude modulation
  *[VCSEL]: vertical cavity surface emitting laser