# Spatial Mode Multiplexing for Fiber-Coupled IM/DD Optical Wireless Links
with Misalignment

Jinzhe Che, Shenjie Huang , Majid Safari The authors are with the Institute
for Imaging, Data and Communications, The University of Edinburgh, Edinburgh
EH9 3JL, U.K. (e-mail: J.Che@sms.ed.ac.uk).

###### Abstract

Optical wireless communication (OWC) emerges as a pivotal solution for
achieving terabit-level aggregate throughput in next-generation wireless
networks. With the mature high-speed transceivers and advanced
(de)multiplexing techniques designed for fiber optics, fiber-coupled OWC can
be seamlessly integrated into existing ultra-high-speed networks such as data
centres. In particular, OWC leveraging spatial mode multiplexing (SMM) and
few-mode fiber (FMF) coupling can significantly increase capacity, though
misalignment may reduce performance. This paper presents a thorough
investigation into the SMM-enabled FMF coupling OWC systems affected by link
misalignment, specifically focusing on systems with intensity modulation with
direct detection (IM/DD) receivers. A theoretical analysis is conducted to
assess the fiber coupling efficiency of the considered system in the presence
of both pointing error and angle of arrival (AOA) fluctuations caused by
random device vibrations. Our model elucidates the dependence of coupling
efficiency to the order of the incident modes, highlighting the critical role
of beam properties in system performance. To mitigate the intermodal crosstalk
arising from link misalignment, we employ zero-forcing beamforming (ZFBF) to
enhance the overall aggregated data rate. Through extensive numerical results,
we identify optimal system configurations encompassing aperture design and
mode selection, leading to a capacity boost exceeding 200%.

###### Index Terms:

Optical wireless communication, spatial mode multiplexing, link misalignment,
data center network, few-mode fiber, fiber coupling.

††publicationid: pubid:

##  I Introduction

The prevalent radio frequency (RF) based wireless networks face significant
throughput limitations due to the increasingly congested RF spectrum, which
hinders the support for data-intensive applications like 4K and 8K ultra-high-
definition (UHD) video streaming [1]. Optical wireless communication (OWC),
leveraging the broad spectrum of light waves, presents a promising solution to
overcome this spectrum bottleneck. Additionally, OWC offers heightened
security levels as light waves cannot penetrate solid objects[2]. As a pivotal
technology within OWC, fiber-coupled OWC systems are gaining increasing
recognition for their potential in high-speed data transmission [3, 4, 5].
This rising prominence can be attributed to their potential for seamless
integration with mature high-speed devices designed for fiber optics,
including high-fidelity optical signal amplifiers [3], precise wavelength
filters [4], and high-bandwidth fast switching[6]. This integration is crucial
for incorporating OWC technology into the existing data communication
infrastructures, such as optical data center networks (ODCN) and passive
optical networks (PON) [7, 8, 9, 10].

The recent advancement in mode (de)multiplexing technology facilitates
efficient integration of the few-mode fiber (FMF) or the multi-mode fiber
(MMF) with the prevailing single-mode fiber (SMF) systems [11, 12, 13, 14].
FMF, with its larger core size, can accommodate several orthogonal higher-
order modes beyond the fundamental mode supported by SMF systems, thereby
alleviating the alignment issue in OWC systems. This capability not only
increases fiber coupling efficiency in conditions of turbulence or jitter but
also enables efficient mode division multiplexing [15, 16]. In parallel,
spatial mode multiplexing (SMM) in free space complements FMF-coupled OWC
systems, allowing for simultaneous transmission of multiple data streams
through orthogonal mode sets. The comparison of information capacity between
SMM systems and traditional multiple-input-multiple-output (MIMO) systems has
been discussed in [17]. It is concluded that SMM performs as the best
technique in realizing the capacity limit with the same space–bandwidth
product. Specifically, attributing to the smaller space-bandwidth product and
simpler multiplexing techniques, the orbital angular momentum (OAM) modes have
garnered significant attention, despite constituting only a subset of the
complete beam basis [18, 19, 20].

While SMM has the theoretical potential to increase aggregated channel
capacity, the performance of SMM-based OWC is significantly hindered by unique
OWC channel impairments, especially the misalignment loss. Generally, two
types of channel random variation can be introduced by the misalignment issue,
namely, pointing error and angle of arrival (AOA) fluctuations. Pointing error
is predominantly attributed to transmitter vibrations, which are commonly
introduced by the building sway and the device orientation for outdoor and
indoor OWC applications, respectively [21, 22]. The effect of pointing error
can be modeled as the radial displacement of the beam spot position at the
receiver[23, 16]. On the other hand, changes in the beam’s spatial phase,
driven by turbulence or receiver vibrations, can lead to AOA fluctuations.
Such angular deviation can lead to the shift of the diffracted pattern on the
focusing optics focal plane [24], resulting in a reduction of the detected
power due to the limited size of the fiber core, i.e., coupling loss. In
practical applications, the above-mentioned channel impairments induced by
misalignments are always coexistent[22], and their impacts on the intermodal
crosstalk and power penalty for the SMM system with free-space photodetector
(PD) receivers have been investigated experimentally [20, 25].

To mitigate the crosstalk introduced by mode coupling, traditional fiber
optics systems often resort to coherent detection-based MIMO digital signal
processing (MIMO-DSP) techniques [26], a practice that extends to the design
of receivers in OWC systems [27, 28, 17]. Despite their effectiveness, the
inherent complexity of MIMO-enabled coherent receivers impedes their large-
scale implementation in commercial networks. Consequently, intensity
modulation with direction detection (IM/DD) receivers, known for their
stability and cost-effectiveness, have been predominantly employed in
practical applications [1]. In such IM/DD-based multiplexing systems, some
works have assumed that the multiple channels are mutually incoherent, thus
simplifying the channel representation and enabling the traditional
interference mitigation method like singular value decomposition (SVD) [29].
However, this assumption of mutually incoherent channels may be only achieved
at the expense of losing spectral efficiency, where employing multiple laser
sources at different wavelengths or a single laser with a significantly wide
linewidth can diminish the beat noise generated at the output of the square-
law PD [30, 31]. In this paper, we consider the more practical and spectrally
efficient design, where multiple beams are generated by a single or multiple
narrow linewidth laser(s) as well as a spatial mode multiplexer [18, 32]. This
implies that the interfering optical beams form a coherent superposition at
the receiver, i.e., mutually coherent channels. Since the signals are mixed
non-linearly in such IM/DD systems, most conventional MIMO schemes are not
applicable, but techniques such as zero-forcing beamforming (ZFBF) and optical
phase retrieval can be employed to mitigate the intermodal crosstalk [33, 34].

This paper provides a comprehensive theoretical framework to evaluate the
communication performance of the SMM-enabled FMF coupling system in the
presence of both pointing error and AOA fluctuations. The main contributions
of this paper can be summarized as follows:

  * $\bullet$

We introduce a framework to analyze the performance of an FMF-coupled OWC
system characterized by mutually coherent multiplexing channels. By
considering the quadratic nature of the IM/DD receivers and misalignment-
induced modal crosstalk, we derive the non-linear channel transformation
matrix for the proposed system. This addresses a notable research gap,
shifting the focus from coupling efficiency models under ideal channel
conditions to a comprehensive evaluation of system performance impaired by
channel misalignments [16, 35].

  * $\bullet$

We propose an accurate theoretical model for indoor multi-mode fiber-coupling
channels, offering valuable insights into the joint effects of the pointing
error and AOA fluctuations. This model deviates from conventional approaches
that rely heavily on the far-field assumption and the transmission of the
fundamental Gaussian beam. Furthermore, we provide a theoretical coupling
model for evaluating the coupling efficiency of the FMF across various
incident beam modes, facilitating a deeper understanding of system behavior
under a broader range of operational scenarios.

  * $\bullet$

We employ a ZFBF scheme to mitigate the mode crosstalk and analytically derive
the aggregated data rate to assess the communication performance of the
considered mutually coherent system. We also present some simulation results
to provide insights into both the efficiency and communication performance of
FMF-coupled OWC under various conditions, indicating the variation of the
coupling performance for different mode groups and the significant capacity
enhancement introduced by the ZFBF.

This paper is organized as follows: Section II introduces the system schematic
with the channel matrix. Section III focuses on modeling fiber-coupled
channels and analytically deriving the aggregated data rate equations. Section
IV presents some simulation results and discusses the optimal system
geometrical configuration. Finally, Section V concludes this paper.

##  II Spatial mode multiplexing system with mutually coherent channels

###  II-A System Model

Figure 1: The considered FMF-coupled spatial mode multiplexing OWC system with
mutually coherent channels: EOM: electro-optic modulators; SMF: single-mode
fiber; SPP: spiral phase plate; (DE)MUX: modal (de)multiplexing; PD:
photodetector.

Fig. 1 presents the schematic diagram of the proposed SMM-based FMF-coupled
OWC system in the presence of link misalignment. For the considered indoor
scenarios, the misalignment is normally introduced by the device vibration,
which results in the orientation angle deviation for both transmitters and
receivers. At the transmitter, the single-wavelength laser sources with a
narrow linewidth are employed to realize the mutually coherent system, where
all the modulated signals are generated at the same wavelength, thereby
realizing a spectral-efficient system design. Electro-optic modulators (EOMs)
modulate $N$ input data streams onto the separated collimated Gaussian beams
coupled into $N$ SMFs. These modulated beams are then transformed into $N$
orthogonal spatial modes by the spiral phase plates (SPP), with the composite
multiplexed beam subsequently propagating over an indoor optical wireless
channel suffering from misalignment issues. At the receiver, the incoming
multiplexed optical beam, upon its arrival, is initially captured by the
aperture and promptly coupled into an FMF. Notably, the pointing error caused
by the transmitter vibration introduces the random spatial shift between the
beam center and the receiver aperture center. On the other hand, the receiver
vibration results in the unparalleled wavefront plane with respect to the
slant aperture, leading to the deviation in beam phase distribution over the
receiver telescope and the displacement of the diffracted pattern at the FMF
fiber end-face (AOA fluctuations). The combined effects of the aforementioned
impairments disrupt the orthogonality among free-space channels, and this
disruption leads to significant intermodal crosstalk in the light coupled into
the FMF. Subsequent to the fiber coupling, the adjacent demultiplexing
operation is employed to separate the different spatial modes for isolating
each mode before conversion back to the fundamental fiber modes, which are
then channelled into respective SMFs tailored for photodetection. Mode-
selective photonic lanterns are promising components that can be utilized in
the proposed system to realize the efficient transition from the FMF to SMFs
with minimal cost and power loss [11]. There are two prevalent types of
photonic lanterns: tapered and integrated. Tapered photonic lanterns combine
several SMFs into a multi-mode core, encapsulated by a cladding of low-index
medium to facilitate multi-mode propagation [12]. Conversely, integrated
photonic lanterns are fabricated by directly etching waveguides onto a chip,
enabling advanced stability for splitting light modes in sophisticated
devices, albeit with implications such as the increased cost [13]. Finally, an
array of $N$ photodetectors is deployed at the SMF ends to effectively detect
the optical power of each spatial mode, which are the superposition of the
signal and the interference from other modes generated due to channel
impairments.

###  II-B Multi-Mode Optical IM/DD Channel Model

In the proposed model, the field distribution of the $i$th spatial mode at the
transmitter side is denoted as $u^{\prime}_{i}(\vec{r^{\prime}},0)$, where
$\vec{r^{\prime}}$ indicates the radial vector in the transmitter cylindrical
coordinate system as shown in Fig. 1. The beam propagated for a distance $z$
through the misaligned link, denoted by $u^{\prime}_{i}(\vec{r^{\prime}},z)$,
can be projected into the receiver aperture plane defined by the radial vector
$\vec{r}$ as shown in Fig. 1. The projected field, denoted by
$\hat{u}_{i}(\vec{r},z)$, can be then decomposed into an orthogonal set of
fiber modes, backpropagated to the receiver aperture plane, i.e.,
$u_{k}(\vec{r},z)$, given by [18]:

$\hat{u}_{i}(\vec{r},z)=\sum_{k=-\infty}^{\infty}\mu_{ik}u_{k}(\vec{r},z),$
(1)

where $u_{k}(\vec{r},z)$ denotes the field distribution for $k$th fiber mode
back propagated to the receiver aperture plane, which satisfies the
orthogonality condition as

$\int
u_{k}(\vec{r},z)u_{k^{\prime}}^{*}(\vec{r},z)\mathrm{d}\vec{r}=\begin{cases}1,&\text{
if }k=k^{\prime}\\\ 0,&\text{ if }k\neq k^{\prime}\end{cases},$ (2)

and $\mu_{ik}$ denotes the channel coefficient for transmitting the $i$th mode
and receiving the $k$th mode, which can be expressed as [18]

$\mu_{ik}=\int_{\mathcal{A}}\hat{u}_{i}(\vec{r},z)u^{*}_{k}(\vec{r},z)\mathrm{d}\vec{r}.$
(3)

where $\mathcal{A}$ represents the two-dimensional space defined by the
receiver aperture. Assuming the use of an ideal demultiplexer, for the FMF-
coupled SMM system with the transmitted mode set $\mathcal{N}$, the total
received field at $k$th single-mode fiber can be expressed as

$U_{k}(\vec{r},z,t)=\sum_{i\in\mathcal{N}}\mu_{ik}u_{k}(\vec{r},z)s_{i}(t)e^{j\left(\nu
t+\phi_{i}\right)},$ (4)

where $s_{i}(t)$ denotes the modulated optical amplitude for the $i$th
transmitted mode, $\nu$ refers to the center frequency, and $\phi_{i}$ is the
phase difference between the output of the $i$th laser source and the 1st
laser source. Note that all lasers are assumed to be narrow linewidth
generating the same wavelength of light. In addition, if a single laser is
used to generate all transmitted modes, then $\phi_{i}=0$. Finally, the
received current at the square-law PD used to receive the signal of the $k$th
spatial mode can be represented as

$\displaystyle Y_{k}$
$\displaystyle=R\displaystyle\int\left|U_{k}(\vec{r},z,t)\right|^{2}\mathrm{d}\vec{r}$
$\displaystyle=R\displaystyle\int\left|\sum_{i\in\mathcal{N}}\mu_{ik}u_{k}(\vec{r},z)s_{i}(t)e^{j\left(\nu
t+\phi_{i}\right)}\right|^{2}\mathrm{d}\vec{r}$
$\displaystyle=R\left|\sum_{i\in\mathcal{N}}\mu_{ik}s_{i}(t)e^{j\phi_{i}}\right|^{2},$
(5)

with $R$ is the PD responsivity.

Notably, (5) indicates the coherent superimposition for the intended signal
$\mu_{kk}s_{k}(t)$ and the interference from other subchannels
$\mu_{ik}s_{i}(k)$. Note that the considered coherent superposition differs
from the incoherent superposition arising from assuming mutually incoherent
channels, which result in a simple intensity superposition as[33]:

$Y^{\prime}_{k}=R\sum_{i\in\mathcal{N}}\left|\mu_{ik}s_{i}(t)\right|^{2}.$ (6)

However, as elucidated in the literature, such mutually incoherent framework
tends to overly complicate the transmitter design and would lead to the lower
spectral efficiency [32, 18, 33, 34]. Consequently, the following analysis
will focus on mutually coherent channels, as defined in (5). Therefore, The
channel model for the considered mutually coherent SMM systems can be
expressed in the following matrix form

$\mathbf{Y}=R\left|\mathbf{H}\mathbf{s}\right|^{2}+\mathbf{Z},$ (7)

where $\mathbf{Y}=[Y_{1},\dots,Y_{N}]^{T}$ is the vector denoting the received
current with totally $N$ spatial modes in set $\mathcal{N}$,
$\mathbf{s}=[s_{1},...,s_{N}]^{T}$ indicates the signal amplitude vector, and
$\mathbf{Z}=[Z_{1},Z_{2},...,Z_{N}]^{T}$ denotes the noise vector. Normally
three main types of noise can affect the performance: thermal noise, shot
noise, and relative intensity noise. Considering the thermal noise as the
dominant noise source [36], elements in $\mathbf{Z}$ can be assumed to follow
a Gaussian distribution with zero mean and a variance of $\sigma^{2}_{n}$.
Note that $\mathbf{H}$ refers to the complex channel matrix, given by

$\displaystyle\mathbf{H}$
$\displaystyle=\left[\begin{array}[]{ccc}h_{11}&\ldots&h_{N1}\\\
\vdots&\ddots&\vdots\\\ h_{1N}&\ldots&h_{NN}\end{array}\right]$ (11)
$\displaystyle=\left[\begin{array}[]{ccc}\mu_{11}e^{j\phi_{1}}&\ldots&\mu_{N1}e^{j\phi_{N}}\\\
\vdots&\ddots&\vdots\\\
\mu_{1N}e^{j\phi_{1}}&\ldots&\mu_{NN}e^{j\phi_{N}}\end{array}\right],$ (15)

where the phase difference between the $\mu_{ik}$ and $h_{ik}$ stems from the
considered multiple laser sources and this relationship can be simplified to
$\mu_{ik}=h_{ik}$ for the single laser systems.

##  III Fiber Coupling in the Presence of Misalignment

The communication performance of the proposed system is closely tied to the
characteristics of the channel matrix $\mathbf{H}$, which is directly
determined by the coupling efficiency of the considered fiber-coupled system.
With the perfectly aligned communication links, the orthogonality among
spatial modes can be well preserved, leading to the diagonalization of
$\mathbf{H}$. However, in the presence of misalignment, considerable crosstalk
between subchannels are introduced, thereby impairing the system stability
[20]. In order to provide insights into the communication performance, the
fiber coupling performance in the presence of the misalignment will be
comprehensively investigated in this section. In addition, to improve the
aggregated capacity, ZFBF is employed at the transmitter to mitigate the
interference between mutually coherent subchannels as conventional linear MIMO
techniques at the receiver side are not applicable.

Figure 2: The lens with diameter $D$ couples $i$th incident beam
${E^{(i)}_{\mathrm{I}}}$ to the $k$th fiber mode field
${E^{(k)}_{\mathrm{F}}}$. Plane A: aperture; plane B: fiber end face.

The magnitude of the complex channel coefficient $h_{ik}$ in $\mathbf{H}$ can
be estimated by measuring ${P^{(ik)}_{\,\mathrm{F}}}$, which denotes the
received optical power associated with the $k$th spatial mode, or equivalently
the power coupled into the $k$th mode of the fiber, with the condition that
only the $i$th mode is being transmitted with unit power. As illustrated in
Fig. 2, let’s denote ${E^{(i)}_{\mathrm{I,A}}}$ and ${E^{(i)}_{\mathrm{I,B}}}$
as the electric fields of the $i$th incident beam at the aperture plane
(referred to as plane A) and the fiber plane (referred to as plane B),
respectively. This fiber-coupled power ${P^{(ik)}_{\,\mathrm{F}}}$ can be then
expressed as [37]

$\left|h_{ik}\right|^{2}={P^{(ik)}_{\,\mathrm{F}}}={\left|\int_{\mathcal{B}}{E^{(i)}_{\mathrm{I,B}}}\left(\vec{r}\right)E^{(k)*}_{\mathrm{F,B}}\left(\vec{r}\right)\mathrm{d}\vec{r}\right|^{2}},$
(16)

where $E^{(k)*}_{\mathrm{F,B}}$ represent the complex conjugate of the $k$th
fiber mode at plane B. As it is more convenient to evaluate the overlap
integration on plane A, after yielding the back-propagated fiber mode
$E^{(k)}_{\mathrm{F,A}}$, (16) can be rewritten as [38]

$\displaystyle\left|h_{ik}\right|^{2}=\left|\int_{\mathcal{A}}{E^{(i)}_{\mathrm{I,A}}}(\vec{r})E^{(k)*}_{\mathrm{F,A}}(\vec{r})\mathrm{d}\vec{r}\right|^{2}.$
(17)

In addition to determine the magnitude of $h_{ik}$, its phase information can
also be acquired through different techniques, such as the method introduced
in [33].

###  III-A Fiber Mode Approximation

In a weakly guided step-index FMF, the distribution of the guided mode fields
$E^{(k)}_{\mathrm{F,B}}$ aligns with the linearly polarized (LP) modes, which
are derived from the solutions to the scalar Helmholtz equation. However, the
substantial mathematical complexity makes it challenging to accurately assess
the coupling loss directly from the LP modes, and previous works tend to adopt
Laguerre-Gaussian (LG) modes due to the aligned cylindrical symmetry [16].
Derived from the paraxial Helmholtz wave equation in cylindrical coordinates,
the LG modes provide a comprehensive representation of the field distribution
following free-space propagation. Specifically, at the beam waist position,
the existing literature has identified that the overlap relation $M_{lp}$
between $\mathrm{LP}_{l,p}$ and $\mathrm{LG}_{p-1,l}$ can reach 0.99 for the
FMF under the low radial order modes, where $M_{lp}=0$ indicates the
orthogonal mode fields and $M_{lp}=1$ indicates the perfectly matched fields
[39]. Thus, to simplify the following derivations, we can utilize LG modes to
accurately represent the $k$th fiber modes $E^{(k)}_{\mathrm{F,B}}$. To this
end, after applying the back-propagation theory, the fiber field over the
receiver aperture is given by

$\displaystyle
E^{(k)}_{\mathrm{F,A}}\left(\vec{r}\right)=\mathrm{LG}_{p_{\mathrm{F}},l_{\mathrm{F}}}\left(r,\theta\right)=$
$\displaystyle\frac{B^{\mathrm{F}}_{pl}}{\omega}\left(\sqrt{2}\frac{r}{\omega}\right)^{\left|l_{\mathrm{F}}\right|}L^{\left|l_{\mathrm{F}}\right|}_{p_{\mathrm{F}}}\left(\frac{2r^{2}}{\omega^{2}}\right)\exp{\left(-\frac{r^{2}}{\omega^{2}}\right)}e^{-jl_{\mathrm{F}}\theta}.$
(18)

In this context, $p_{\mathrm{F}}$ and $l_{\mathrm{F}}$ indicate the radial
index and azimuth index of the fiber mode,
$L^{\left|l_{\mathrm{F}}\right|}_{p_{\mathrm{F}}}$ are the associated Laguerre
polynomials, and
$B^{\mathrm{F}}_{pl}=\left(\frac{2p_{\mathrm{F}}!}{\pi(\left|l_{\mathrm{F}}\right|+_{\mathrm{F}}p)!}\right)^{\frac{1}{2}}$
is a normalization factor. The term $\omega=\frac{\lambda f}{\pi\omega_{a}}$
is the back-propagated mode radius [16], where $\lambda$ is the wavelength,
$f$ is the aperture focal length, and $\omega_{a}$ is the fiber mode field
radius. This fiber mode field, along with the incident beam field (as derived
later in Section III-B), can be substituted into (17) for the coupling power.

###  III-B Modeling the Incident Misaligned Beams

Figure 3: Illustration of the transmitter vibration-induced pointing error,
showing the displacement $\vec{d}$ between the beam center and the aperture
center.

Benefiting from the smaller space-bandwidth product and simpler multiplexing
techniques, the OAM multiplexed systems, which use a subset of LG modes, have
been notably highlighted in recent works [17, 18]. Moreover, LG modes feature
better compatibility with many optical subsystems due to their intrinsic
circular symmetry, especially for the considered fiber coupling systems [19,
20]. Therefore, in the following discussion, we focus on the transmission of
LG modes. Note that the analysis presented here can be readily expanded to the
transmission of other spatial modes. Therefore, after propagating over a
distance $z$ within perfectly aligned channels, the resultant field
distribution at the receiver side can be formally articulated as [40]

$\begin{split}&{E^{(i)}_{\mathrm{I}}}\left(\vec{r^{\prime}},z\right)={E^{(i)}_{\mathrm{I,A}}}\left(\vec{r},z\right)=\mathrm{LG}_{p_{\mathrm{I}},l_{\mathrm{I}}}\left(r,\theta,z\right)=\\\
&\frac{B^{\mathrm{I}}_{pl}}{\omega_{z}}\left(\sqrt{2}\frac{r}{\omega_{z}}\right)^{\left|l_{\mathrm{I}}\right|}L^{\left|l_{\mathrm{I}}\right|}_{p_{\mathrm{I}}}\left(\frac{2r^{2}}{\omega_{z}^{2}}\right)\exp{\left(-\frac{r^{2}}{\omega_{z}^{2}}-j\tilde{\nu}z\right)}\times\\\
&\exp{\left(-j\tilde{\nu}\frac{r^{2}}{2R_{z}}+j\left(2p_{\mathrm{I}}+\left|l_{\mathrm{I}}\right|+1\right)\zeta\left(z\right)-jl_{\mathrm{I}}\theta\right)},\end{split}$
(19)

where $\tilde{\nu}$ = $2\pi/\lambda$ stands as the wavenumber,
$p_{\mathrm{I}}$ and $l_{\mathrm{I}}$ represent the radial index and azimuth
index of the $i$th transmitted mode, respectively. The term $\omega_{z}$
indicates the radius of the beam spot, which can be calculated as
$\omega_{z}=\omega_{0}\sqrt{(1+2p_{\mathrm{I}}+|l_{\mathrm{I}}|)(1+z/z_{R})}$,
with $\omega_{0}$ and $z_{R}$ being the waist radius and Rayleigh range,
respectively. $R_{z}$ and $\zeta\left(z\right)$ define the curvature radius of
the beam’s wavefront and the Gouy phase at a distance of $z$, respectively.

Practical transceiver misalignment can result in a dual set of impairments:
pointing error and AOA fluctuations. The fluctuations in the transmitter’s
orientation angle lead to the displacement between the optical fields and the
receiver’s aperture, consequently causing pointing error. Fig. 3 depicts the
pointing error induced by the transmitter vibration, highlighting a
displacement $\vec{d}$ between the beam center and the aperture center.
Without loss of generality, we assume $\vec{d}$ aligns with the $x$-axis. In
this context, the radial vector $\vec{r}$ in the receiver plane essentially
indicates the position of $\vec{r^{\prime}}$ relative to the transmitter
coordinate system. After the coordinate transformation, the $\vec{r^{\prime}}$
is with a magnitude of $g(r,\theta)=\sqrt{r^{2}+d^{2}-2rd\cos{\theta}}$ and
the azimuth angle of $\vartheta$. The relationship between $\vartheta$ and
$\theta$ is given by

$\vartheta=\begin{cases}\pi-\arccos{\left(\frac{d-r\cos{\theta}}{\sqrt{r^{2}+d^{2}-2rd\cos{\theta}}}\right)},&{\theta\in[0,\pi]},\\\
\pi+\arccos{\left(\frac{d-r\cos{\theta}}{\sqrt{r^{2}+d^{2}-2rd\cos{\theta}}}\right)},&{\theta\in[\pi,2\pi]}.\end{cases}$
(20)

While the existing work has emphasized the misaligned field impaired by the
lantern displacement [41, 16], they generally neglect the potential impacts on
the azimuthal dependence which is fundamental to the functioning of SMM
systems. Therefore, after accounting for the transformation of both radial
distance $|\vec{r^{\prime}}|$ and the azimuth angle $\vartheta$ induced by the
pointing error $\vec{d}$, the field distribution of the incident LG mode over
the receiver lens becomes

$\displaystyle{E^{(i)}_{\mathrm{I}}}\left(\vec{r^{\prime}},z\right)={E^{(i)}_{\mathrm{I,A}}}\left(\vec{r},z,\vec{d}\right)=\mathrm{LG}_{p_{\mathrm{I}},l_{\mathrm{I}}}\left(r,\theta,z,d\right)=$
$\displaystyle\frac{B^{\mathrm{I}}_{pl}}{\omega_{z}}\left(\sqrt{2}\
\frac{g(r,\theta)}{\omega_{z}}\right)^{\left|l_{\mathrm{I}}\right|}L^{\left|l_{\mathrm{I}}\right|}_{p_{\mathrm{I}}}\left(2\
\frac{g^{2}(r,\theta)}{\omega_{z}^{2}}\right)\exp{\left(-\frac{g^{2}(r,\theta)}{\omega_{z}^{2}}\right)}\times$
$\displaystyle\exp{\left(-j\tilde{\nu}\frac{g^{2}(r,\theta)}{2R_{z}}-j\tilde{\nu}z+j\left(2p_{\mathrm{I}}+\left|l_{\mathrm{I}}\right|+1\right)\zeta\left(z\right)-j\left|l_{\mathrm{I}}\right|\vartheta\right)}.$
(21)

Figure 4: Illustration of the receiver vibration-induced AOA fluctuation,
showing the cross-section of the wavefront intersecting the aperture at an
angle $\varepsilon$ at distance $z$.

On the other hand, receiver vibrations can give rise to fluctuations in the
spatial phase of the near-field spherical beam across the aperture, commonly
referred to as the AOA fluctuations [42, 24]. For a receiver aperture which is
significantly smaller than the beam’s coherence spot, the wavefront is
essentially a plane over the aperture [24]. As described in (19), when the
incident wavefront is perfectly parallel with the receiver aperture, i.e.,
signifying no AOA deviations, the spatial phase of the $i$th beam across the
aperture can be expressed as

$\varphi_{i}\left(r^{\prime},z\right)=\varphi_{i}\left(r,z\right)=\frac{r^{2}}{2R_{z}}+\tilde{\nu}z-\left(2p_{\mathrm{I}}+\left|l_{\mathrm{I}}\right|+1\right)\zeta\left(z\right)+l_{\mathrm{I}}\theta.$
(22)

However, given that the incident wavefront has an angle of $\varepsilon$ with
the aperture plane due to the AOA fluctuations, as depicted in Fig. 4, the
phase distribution across the aperture becomes

$\displaystyle\varphi_{i}\left(r^{\prime},z\right)=\varphi_{i}\left(r,z,\varepsilon\right)=\frac{\left(r\cos{\varepsilon}\right)^{2}}{2R_{z+r\sin{\varepsilon}}}+$
$\displaystyle\tilde{\nu}(z+r\sin{\varepsilon})-\left(2p_{\mathrm{I}}+\left|l_{\mathrm{I}}\right|+1\right)\zeta\left(z+r\sin{\varepsilon}\right)+l_{\mathrm{I}}\theta.$
(23)

Assuming the typically small deviation angle $\varepsilon$, as observed in
measurements from real-world vibrations [22], the small angle approximation is
justified. This permits the assumptions that $\cos{\varepsilon}\approx 1$ and
$\sin{\varepsilon}\approx\varepsilon$. Additionally, taking into account the
slow variation of both the wavefront curvature radius, $R_{z}$, and the Gouy
phase, $\zeta(z)$, with respect to the distance $z$ [40], (23) can be
simplified to

$\varphi_{i}\left(r,z,\varepsilon\right)=\frac{r^{2}}{2R_{z}}+\tilde{\nu}z+\tilde{\nu}r\varepsilon-\left(2p_{\mathrm{I}}+\left|l_{\mathrm{I}}\right|+1\right)\zeta\left(z\right)+l_{\mathrm{I}}\theta.$
(24)

Subsequently, by substituting (24) into (21), the incident field distribution
across the aperture, in the presence of both pointing error denoted by $d$,
and AOA fluctuations represented by $\varepsilon$, can be articulated as

$\displaystyle{E^{(i)}_{\mathrm{I}}}\left(\vec{r^{\prime}},z\right)={E^{(i)}_{\mathrm{I,A}}}\left(\vec{r},z,\vec{d},\varepsilon\right)=\mathrm{LG}_{p_{\mathrm{I}},l_{\mathrm{I}}}\left(r,\theta,z,d,\varepsilon\right)=$
$\displaystyle\frac{B^{\mathrm{I}}_{pl}}{\omega_{z}}\left(\sqrt{2}\
\frac{g(r,\theta)}{\omega_{z}}\right)^{\left|l_{\mathrm{I}}\right|}L^{\left|l_{\mathrm{I}}\right|}_{p_{\mathrm{I}}}\left(2\
\frac{g^{2}(r,\theta)}{\omega_{z}^{2}}\right)\times$
$\displaystyle\exp{\left(-\frac{g^{2}(r,\theta)}{\omega_{z}^{2}}-j\tilde{\nu}\frac{g^{2}(r,\theta)}{2R_{z}}-j\tilde{\nu}\left(z+\varepsilon
g(r,\theta)\right)\right)}\times$
$\displaystyle\exp{\left(j\left(\left(2p_{\mathrm{I}}+\left|l_{\mathrm{I}}\right|+1\right)\zeta\left(z\right)-l_{\mathrm{I}}\vartheta\right)\right)}.$
(25)

By utilizing the derived incident field across the aperture, the fiber-coupled
power across different modes in the presence of misalignment can be
calculated. Specifically, by substituting (18) and (25) into (17), and
following a series of algebraic manipulations, the final coupling power
between the $i$th transmitted $\mathrm{LG}_{p_{\mathrm{I}},l_{\mathrm{I}}}$
mode and the $k$th fiber mode
($\mathrm{LP}_{l_{\mathrm{F}},p_{\mathrm{F}+1}}$), incorporating the impact of
transceiver vibrations, can be expressed as

$\displaystyle\left|h_{ik}\right|^{2}={P^{(ik)}_{\,\mathrm{F}}}=\frac{\left({B^{\mathrm{F}}_{pl}}{B^{\mathrm{I}}_{pl}}\alpha\right)^{2}}{2}\exp{\left(-\frac{2d^{2}}{\omega^{2}(z)}\right)}\left(2\beta^{2}\right)^{\left|l_{\mathrm{F}}\right|+1}\times$
$\displaystyle\left|\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\left(\frac{\sqrt{2}f(x,\theta)}{\omega_{z}}\right)^{\left|l_{\mathrm{I}}\right|}L_{P_{\mathrm{I}}}^{\left|l_{\mathrm{I}}\right|}\left(\frac{2f^{2}(x,\theta)}{\omega_{z}^{2}}\right)L_{P_{\mathrm{F}}}^{\left|l_{\mathrm{F}}\right|}\left(2\beta^{2}x^{2}\right)\right.\times$
$\displaystyle\left.\exp{\left(-j\tilde{\nu}\frac{xDd\cos{\theta}}{2R_{z}}+j\tilde{\nu}\varepsilon
f\left(x,\theta\right)-\gamma
x^{2}\right)}x^{\left|l_{\mathrm{F}}\right|+1}\right.\times$
$\displaystyle\left.\exp{\left(\frac{xDd\cos{\theta}}{\omega^{2}_{z}}+jl_{\mathrm{I}}\vartheta-
jl_{\mathrm{F}}\theta\right)}\mathrm{d}x\mathrm{d}\theta\right|^{2},$ (26)

where the function
$f(x,\theta)=(\frac{D^{2}}{4}x^{2}+d^{2}-xDd\cos{\theta})^{\frac{1}{2}}$, and
$\gamma$ is defined as
$\gamma=\alpha^{2}+\beta^{2}-j\frac{\tilde{\nu}D^{2}}{8R_{Z}}$, with
$\alpha=\frac{D}{2\omega_{Z}}$ representing the ratio of the aperture radius
to the incident beam radius. Moreover, $\beta=\frac{D}{2\omega}$ signifies the
ratio of the aperture radius to the back-propagated waist radius, which is
commonly referred to as the coupling geometry parameter.

###  III-C Coupling Efficiency in the Presence of Misalignment

The coupling efficiency denotes the ratio between the coupled power into the
fiber and the collected power by the aperture, which can be expressed as

$\eta_{ik}=\frac{{P^{(ik)}_{\,\mathrm{F}}}}{{P^{(i)}_{\,\mathrm{A}}}}=\frac{|h_{ik}|^{2}}{\int_{A}|{E^{(i)}_{\mathrm{I,A}}}\left(\vec{r}\right)|^{2}\mathrm{d}\vec{r}},$
(27)

where the term ${P^{(i)}_{\,\mathrm{A}}}$ denotes the collected power by the
receive aperture from $i$th incident mode. Substituting (25) and (26) into
(27) and following a series of algebraic manipulations, the final coupling
efficiency expression for the $k$th fiber mode with the $i$th incident mode
can be expressed as

$\displaystyle\eta_{ik}={B^{\mathrm{F}}_{pl}}^{2}\left(2\beta^{2}\right)^{\left|l_{\mathrm{F}}\right|+1}\left|\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\exp{\left(-\gamma
x^{2}+j\tilde{\nu}\varepsilon f\left(x,\theta\right)\right)}\right.\times$
$\displaystyle\left.\exp{\left(xDd\cos{\theta}\left(\frac{2R_{z}-j\tilde{\nu}\omega^{2}_{z}}{2\omega^{2}_{z}R_{z}}\right)+j\left(l_{\mathrm{I}}\vartheta-
l_{\mathrm{F}}\theta\right)\right)}x^{\left|l_{\mathrm{F}}\right|+1}\right.\times$
$\displaystyle\left.\left(\frac{\sqrt{2}f(x,\theta)}{\omega_{z}}\right)^{\left|l_{\mathrm{I}}\right|}L_{P_{\mathrm{I}}}^{\left|l_{\mathrm{I}}\right|}\left(\frac{2f^{2}(x,\theta)}{\omega_{z}^{2}}\right)L_{P_{\mathrm{F}}}^{\left|l_{\mathrm{F}}\right|}\left(2\beta^{2}x^{2}\right)\mathrm{d}x\mathrm{d}\theta\right|^{2}\times$
$\displaystyle\left(2\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\left(\frac{2f^{2}(x,\theta)}{\omega_{z}}\right)^{\left|l_{\mathrm{I}}\right|}\left(L_{P_{\mathrm{I}}}^{\left|l_{\mathrm{I}}\right|}\left(\frac{2f^{2}(x,\theta)}{\omega_{z}^{2}}\right)\right)^{2}\right.\times$
$\displaystyle\left.\exp{\left(2\left(\frac{xDd\cos{\theta}}{\omega^{2}_{z}}-\alpha^{2}x^{2}\right)\right)}x\mathrm{d}x\mathrm{d}\theta\right)^{-1}.$
(28)

Under the far-field assumption, considering that the input optical intensity
is independent of $\vec{r}$ across the receiver aperture, the coupling
efficiency of the incident Gaussian beam with SMF in the absence of link
misalignment is reduced to

$\eta=8\beta^{2}\left|\int_{0}^{1}x\exp\left(-\beta^{2}x^{2}\right)\mathrm{d}x\right|^{2},$
(29)

which aligns with the literature with negligible speckle size as depicted in
[37, 38].

Assuming that both the vertical and horizontal AOA deviations, denoted by
$\varepsilon_{v}$ and $\varepsilon_{h}$ respectively, follow independent and
identical zero-mean Gaussian distribution, the combined AOA deviation under
the small angle approximation is
$\varepsilon=\sqrt{\varepsilon_{v}^{2}+\varepsilon_{h}^{2}}$. This leads to a
Rayleigh distribution characterized by a scale parameter of
$\sigma_{\varepsilon}$ with the probability density function (PDF)

$p_{\varepsilon}(\varepsilon)=\frac{\varepsilon}{\sigma_{\varepsilon}^{2}}\exp\left(-\frac{\varepsilon^{2}}{2\sigma_{\varepsilon}^{2}}\right).$
(30)

Similarly, by considering identical distributions for vertical and horizontal
transmitter orientation angles, denoted as $\epsilon_{v}$ and $\epsilon_{h}$
respectively, the resultant vertical and horizontal displacements can be
modeled as $d_{v}$ = $z\epsilon_{v}$ and $d_{h}$ = $z\epsilon_{h}$.
Consequently, the aggregate displacement
$d=z\sqrt{\epsilon_{v}^{2}+\epsilon_{h}^{2}}$ is also subject to a Rayleigh
distribution with a scale parameter $\sigma_{d}=z\sigma_{\epsilon}$, and can
be represented as $p_{d}(d)$. Therefore, the expected coupling efficiency
$\left\langle\eta_{ik}\right\rangle$ can be calculated by integrating (28)
over the effects of both pointing error and AOA deviations, which is given by

$\left\langle\eta_{ik}\right\rangle=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\eta_{ik}(d,\varepsilon)p_{d}(d)p_{\varepsilon}(\varepsilon)\mathrm{d}\varepsilon\mathrm{d}d,$
(31)

where the angle brackets represent the ensemble-average operator.

###  III-D Achievable Data Rate

Building upon the channel coefficients derived in (26), the estimated complex
channel matrix for the considered FMF-coupled SMM system can be expressed as

$\mathbf{\tilde{H}}=\left[\begin{array}[]{ccc}\left|h_{11}\right|&\ldots&\left|h_{N1}\right|e^{j\Delta\varphi^{1}_{N-1}}\\\
\vdots&\ddots&\vdots\\\
\left|h_{1N}\right|e^{j\Delta\varphi^{N}_{1-N}}&\ldots&\left|h_{NN}\right|\end{array}\right],$
(32)

where $\Delta\varphi^{k}_{n-i}$ denotes the phase difference between $h_{ik}$
and $h_{nk}$. In [43], it was shown that if ${s}_{k}$ obeys the Rayleigh
distribution and ${s}_{k}^{2}$ obeys the exponential distribution, a capacity
lower bound for optical IM/DD channel can be derived. Using this result, an
achievable data rate for $k$th subchannel can be expressed as

$r_{k}=\frac{1}{2}B\log{\left(1+\frac{\varsigma_{k}e}{2\pi}\right)},$ (33)

where $B$ stands for the bandwidth, and $\varsigma_{k}$ indicates the signal-
to-interference-plus-noise ratio (SINR) for channel $k$. In real-world
demonstrations, a total power constraint needs to be considered due to issues
such as eye-safety requirement, which can be expressed as
$\sum\xi_{k}\leq\xi_{\mathrm{t}}$ [44], where
$\xi_{k}=\langle{s}_{k}^{2}\rangle$ denotes the average optical power for
$k$th channel and $\xi_{\mathrm{t}}$ is the total power limit. Considering
that significant interference is introduced by the channel misalignment,
according to the non-linear channel transformation for the mutually coherent
system as depicted in (7), the average received current for $k$th PD, $\langle
Y_{k}\rangle$, can be denoted as

$\displaystyle\langle
Y_{k}\rangle=R\left\langle\left|\sum_{i=1}^{N}h_{ik}{s_{i}}\right|^{2}\right\rangle=R\sum_{i=1}^{N}\sum_{n=1}^{N}h_{ik}h_{nk}^{*}\left\langle{s_{i}}{s_{n}}\right\rangle$
$\displaystyle=R\left(\left|h_{kk}\right|^{2}\xi_{k}+\sum_{i=1\atop i\neq
k}^{N}\left|h_{ik}\right|^{2}\xi_{i}+\sum_{i=1}^{N}\sum_{n=1\atop n\neq
i}^{N}\frac{\pi}{4}h_{ik}h_{nk}^{*}\sqrt{\xi_{i}\xi_{k}}\right),$ (34)

where the first element illustrates the desired signal emitted by the $k$th
transmitter, and the second and third elements indicate the interference and
beat noise, respectively. Note that since ${s}_{k}^{2}$ is assumed to be
exponentially distributed, we have $\langle{s}_{k}^{2}\rangle=\xi_{k}$ and
$\langle{s}_{k}\rangle=\sqrt{\xi_{k}\pi}/2$. With the uniform power allocation
among total $N$ transmitters, i.e.,
$\xi_{1}=\dots=\xi_{N}=\frac{\xi_{\mathrm{t}}}{N}$, after using $X_{k}$ to
collectively denote the interference and beat noise, the $\varsigma_{k}$ is
given by

$\varsigma_{k}=\frac{R^{2}\xi_{\mathrm{t}}^{2}\left|h_{kk}\right|^{4}}{N^{2}\left(\sigma^{2}_{n}+X_{k}^{2}\right)}=\frac{R_{f}R^{2}\xi_{\mathrm{t}}^{2}\left|h_{kk}\right|^{4}}{N^{2}\left(4k_{b}TF_{n}B+R_{f}X_{k}^{2}\right)},$
(35)

with $k_{b}$ being the Boltzmann constant, $R_{f}$ being the feedback resistor
for the receiver transimpedance amplifier (TIA), $F_{n}$ being the noise
figure and $T$ being the temperature in Kelvin.

Following the substitution of (35) into (33), and taking into account the sum
of achievable data rates across all subchannels as $C=\sum_{k}r_{k}$, the
aggregated channel capacity in the presence of interference can be described
as

$C_{\mathrm{I}}=\sum_{k=1}^{N}\frac{1}{2}B\log{\left(1+\frac{eR_{f}R^{2}\xi_{t}^{2}\left|h_{kk}\right|^{4}}{2\pi
N^{2}\left(4k_{b}TF_{n}B\pi+R_{f}X_{k}^{2}\right)}\right)}.$ (36)

As depicted in (36), the presence of interference and beat noise precipitates
significant performance degradation, underscoring the imperative need for a
robust interference mitigation strategy to ensure optimal signal integrity and
system efficiency. The investigation into the non-linear transformation has
demonstrated that traditional postcoding techniques are inadequate for
mitigating interference arising from subchannels [32, 33, 34]. Nonetheless,
given the availability of channel state information at the transmitter end,
ZFBF can be used as a viable solution to achieve crosstalk-free parallel
transmission. This is particularly relevant under high-speed transmission,
which suggests the channel matrix $\mathbf{H}$ is assumed to be quasi-static
during the transmission of a symbol block. Observing from (32), the
relationship between the channel matrix $\mathbf{H}$ and $\mathbf{\tilde{H}}$
can be revealed as $\mathbf{H}=\mathbf{D}\mathbf{\tilde{H}}$, with
$\mathbf{D}$ being the diagonal matrix with only phase elements. Therefore, if
we choose the inverse of the estimated channel matrix as the precoder, the
received electrical current matrix $\mathbf{Y}$ is given by

$\mathbf{Y}=R\left|\mathbf{H}\mathbf{\tilde{H}}^{-1}\mathbf{\tilde{s}}\right|^{2}+\mathbf{Z}=R\left|\mathbf{\tilde{s}}\right|^{2}+\mathbf{Z},$
(37)

where $\mathbf{\tilde{s}}=[\tilde{s_{1}},\tilde{s_{2}},...,\tilde{s_{N}}]$
indicates the information bearing signal. Assuming that $\tilde{s}_{k}$ also
obeys the Rayleigh distribution, after the crosstalk compensation introduced
by the ZFBF, the signal-to-noise ratio (SNR) for $k$th subchannel is given by

$\varsigma^{\prime}_{k}=\frac{\left(R\tilde{\xi_{k}}\right)^{2}}{\sigma^{2}_{n}}=\frac{R_{f}R^{2}\tilde{\xi_{k}}^{2}}{4k_{b}TF_{n}B}.$
(38)

With the non-linear power allocation induced by the ZFBF, the average total
transmitted power, $\langle||\mathbf{\tilde{s}}||^{2}\rangle$, which satisfies
$\langle||\mathbf{\tilde{s}}||^{2}\rangle\leq\xi_{\mathrm{t}}$, can be derived
as

$\displaystyle\langle||\mathbf{\tilde{s}}||^{2}\rangle=\left\langle\sum_{k=1}^{N}\left|\sum_{i=1}^{N}I_{ik}\tilde{s_{i}}\right|^{2}\right\rangle=\sum_{k=1}^{N}\sum_{i=1}^{N}\sum_{n=1}^{N}I_{ik}I_{nk}^{*}\left\langle\tilde{s_{i}}\tilde{s_{n}}\right\rangle$
$\displaystyle=\sum_{k=1}^{N}\sum_{i=1}^{N}\left(\left|I_{ik}\right|^{2}\tilde{\xi}_{i}+\frac{\pi}{4}\sum_{n=1\atop
n\neq i}^{N}{I_{ik}I_{nk}^{*}\sqrt{\tilde{\xi_{i}}\tilde{\xi_{n}}}}\right),$
(39)

where $I_{ik}$ stands for the elements in the inverse of estimated channel
matrix $\mathbf{\tilde{H}}$. Therefore, considering the uniform power
allocation among the subchannels, i.e.,
$\tilde{\xi_{1}}=\dots=\tilde{\xi_{N}}=\tilde{\xi_{\mathrm{un}}}$, and that
capacity lower bound is achieved at the boundary, i.e.,
$\langle||\mathbf{\tilde{s}}||^{2}\rangle=\xi_{\mathrm{t}}$, we can obtain
$\tilde{\xi_{\mathrm{un}}}$ from (39) as

$\tilde{\xi_{\mathrm{un}}}=\frac{\xi_{\mathrm{t}}}{\sum_{k=1}^{N}\sum_{i=1}^{N}\left(\left|I_{ik}\right|^{2}+\frac{\pi}{4}\sum_{n=1,n\neq
i}^{N}{I_{ik}I_{nk}^{*}}\right)}.$ (40)

Consequently, by following the substitution of (40) and (38) into (33), the
total channel capacity with the ZFBF can be revealed as

$C_{\mathrm{ZF}}=\frac{N}{2}B\log{\left(1+\frac{eR_{f}R^{2}\tilde{\xi_{\mathrm{un}}}^{2}}{8k_{b}TF_{n}B\pi}\right)}.$
(41)

##  IV Simulation results and discussion

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: The coupling efficiency for a six-mode FMF as the function of
coupling parameter $\beta$ in the absence of misalignment for an incident (a):
$\mathrm{LG}_{00}$ mode, (b): $\mathrm{LG}_{01}$ mode, (c): $\mathrm{LG}_{02}$
mode; and in the presence of misalignment for an incident (d):
$\mathrm{LG}_{00}$ mode, (e): $\mathrm{LG}_{01}$ mode, (f): $\mathrm{LG}_{02}$
mode.

In this section, we will evaluate the performance of the SMM-enabled FMF
coupling OWC system in the presence of misalignment, including both coupling
efficiency and communication performance. In the simulation, the transmitted
beam is considered with a wavelength of $1550$ $\mathrm{nm}$ and a waist
radius of $\omega_{o}=800$ $\mathrm{\mu m}$. A link distance of $10$
$\mathrm{m}$ is assumed to emulate a typical indoor laser-based OWC link such
as in data centers [22]. The total transmission power constraint $\xi_{t}$ is
set as 10 dBm [44], unless otherwise indicated. The other employed parameters
include a feedback resistor value of $R_{f}=500$ $\Omega$, a noise figure of
$F_{n}=5$ dB, a stable ambient temperature of $T=300^{\circ}$ K, and a
receiver responsivity of $R=0.7$ A/W [36].

###  IV-A Coupling Efficiency

We first assess the coupling efficiency as the function of $\beta$ considering
different incident OAM modes ($\mathrm{LG}_{00}$, $\mathrm{LG}_{01}$, and
$\mathrm{LG}_{02}$) without link misalignment, where $\beta$ is the coupling
geometry parameter which denotes the ratio of the aperture radius to the back-
propagated waist radius. This assessment is conducted in a six-mode FMF
supporting $\mathrm{LP}_{01}$, $\mathrm{LP}_{02}$, $\mathrm{LP}_{11}$, and
$\mathrm{LP}_{21}$ modes. As illustrated in Fig. 5a Fig. 5b and Fig. 5c, the
coupled power between the fiber mode
$\mathrm{LP}_{l_{\mathrm{F}},p_{\mathrm{F}}+1}$ (approximated by
$\mathrm{LG}_{p_{\mathrm{F}},l_{\mathrm{F}}}$), and an incident beam,
$\mathrm{LG}_{p_{\mathrm{I}},l_{\mathrm{I}}}$, with different azimuth order
(i.e., $l_{\mathrm{I}}\neq l_{\mathrm{F}}$) is found to be zero. This result
underscores that the orthogonality between modes with distinct azimuthal phase
dependencies has been well maintained. It is also observed that, in Fig. 5a,
the coupling efficiency of $\mathrm{LP}_{01}$ attains its maximal value of
0.74 when $\beta$ is at 1.01. Under the far-field assumption, the coupling
efficiency can increase to 0.81 when $\beta$ is 1.12, a result that aligns
with the findings in the existing literature [37, 38]. On the other hand,
$\mathrm{LP}_{02}$ mode is less excited compared with $\mathrm{LP}_{01}$ mode
due to the field vectors reverse [16], with a maximum coupling efficiency of
0.166 when $\beta$ is 2.2. A notable fluctuation in coupling efficiency for
the mode $\mathrm{LP}_{02}$ is observed with increasing $\beta$, which is
linked to the intensity profile of $\mathrm{LP}_{0p}$ modes with a central
spot surrounded by $p-1$ concentric rings when $p\geq 2$ [40]. Given that
$\omega$ is held constant in our simulation, an increase in the coupling
parameter $\beta$ leads to a larger aperture radius. This enlargement of the
aperture radius affects the spatial separation between the central spot and
the surrounding ring in the $\mathrm{LP}_{02}$ mode, leading to a fluctuated
coupling performance. As illustrated in Fig. 5b, the calculated optimal
$\beta$ stands at 1.33, which is associated with a maximum coupling efficiency
of 0.71 for the incident $\mathrm{LG}_{01}$ mode. In the case of the incident
$\mathrm{LG}_{02}$ beam in Fig. 5c, a $\beta$ setting of 0.66 is found to
facilitate the highest coupling efficiency which is recorded at 1.54.

Subsequently, we investigate the coupling performance under conditions of both
pointing error and AOA fluctuations, focusing on the previously mentioned
transmitted modes set and the FMF. Notably, we adopt a parameter of 0.125
$\mathrm{mrad}$ for the Rayleigh distribution ($\sigma_{\epsilon}$ and
$\sigma_{\varepsilon}$), aligning with real-world scenarios [22]. As depicted
in Fig. 5d Fig. 5e and Fig. 5f, the deleterious effects of transceiver
vibrations significantly degrade the coupling efficiency, and the performance
degradation escalates with the increase of the transmitted mode orders. For
example, with the incident $\mathrm{LG}_{00}$ beam, the ideal $\beta$ has
shifted to 0.77 to attain the highest coupling efficiency for the
$\mathrm{LP}_{01}$, which is 0.29 and with a 61% decrease. As for the incident
$\mathrm{LG}_{01}$ mode, the maximum coupling efficiency has been diminished
by 72%, to a value of 0.20 with the optimal $\beta$ being 1.28. For the
incident $\mathrm{LG}_{02}$ beam, the optimal $\beta$ is 1.63 now, leading to
the maximum coupling efficiency of only 0.15 after a notable 77% drop.
Moreover, in the presence of the misalignment, the orthogonality among modes
with different azimuthal dependence can not be conserved, aligning with the
power leakage observed in previous research[25, 20]. This phenomenon occurs
because, in the absence of misalignment, the transmitted mode is perfectly out
of phase with half of the modes with a differing azimuthal order. Such a
perfect phase mismatch leads to a cancellation effect, which prevents the
excitation of the remaining modes in the set. Considering the link
misalignment introduces a phase change to the incident beam before being
coupled into the fiber, and the induced imperfect phase mismatch results in
the mode excitation [16]. In addition, it is observed that the power of the
transmitted mode is more likely to be coupled into the supported mode with the
nearest mode order under the considered misalignment. Specifically, Fig. 5d
depicts that with the $\beta$ of 0.77, after being normalized by the aperture
collected power, 11$\%$ of the power is leaked into the $\mathrm{LP}_{11}$
mode and 5$\%$ power is coupled into $\mathrm{LP}_{21}$ mode. Similar trends
can also be observed in Fig. 5e and Fig. 5f, namely, 10$\%$ of the power is
leaked into the $\mathrm{LP}_{01}$ mode and 8$\%$ power is coupled into
$\mathrm{LP}_{21}$ mode with the incident $\mathrm{LG}_{01}$ beam; 15$\%$ of
the power is leaked into the $\mathrm{LP}_{01}$ mode and 3$\%$ power is
coupled into $\mathrm{LP}_{21}$ mode with the incident $\mathrm{LG}_{02}$
beam. Consequently, it is deduced that transceiver vibrations not only
substantially diminish coupling efficiency but also severely disrupt the
orthogonality among distinct spatial modes, and the severity of the
deleterious impact is amplified with higher-order LG beams.

###  IV-B SMM System Communication Performance

In this section, the communication performance of the ZFBF-based SMM system
will be evaluated. It should be noted that the achievable data rate for each
channel within the selected transmitted mode set can be quantitatively
determined by applying (36) and (41), corresponding to systems without and
with ZFBF, respectively. Fig. 6 illustrates the channel capacity after ZFBF as
a function of the receiver aperture configuration, varying with the number of
channels denoted by $N$. In this work, the available transmitted OAM modes
$\mathcal{N}$ span from 0 to 5, to match with the FMF-supported modes ranges
from $\mathrm{LP}_{01}$ to $\mathrm{LP}_{51}$ [45]. In addition, we make the
common assumption for thin lens systems that the aperture diameter $D$ is
equal to the focal length $f$, thereby simplifying the optical design and
aligning with standard optical system assumptions [24]. As we can see, for the
system with a single channel, the achievable data rate commences at 0.18 Tb/s,
which is in contrast to systems with $N\geq 1$ channels. Moreover, under a
small telescope regime, with the increase of aperture size, the data rate
usually grows much faster for the systems with smaller $N$. The aforementioned
results can be attributed to two main reasons. First, with more concentrated
intensity distribution, the single-channel system with incident fundamental
Gaussian mode features the most resistance to pointing error, compared with
other incident higher-order modes with the donuts-shaped intensity profile in
multiple-channel systems. The $\mathrm{LP}_{01}$ fiber mode is more
susceptible to angular deviations attributed to the smaller field radius
compared with other higher-order $\mathrm{LP}$ modes. However, the channel
fluctuations introduced by AOA error can be regarded as a diffracted image
jitter on the fiber plane where $d_{i}=\varepsilon f$, and the performance
degradation due to angular deviations can be significantly mitigated in a
small aperture regime [24].

Figure 6:  The achievable data rate as the function of aperture radius under
different $N$ where the focal length $f$ equals to the aperture diameter $D$.

On the other hand, the spot radius for the higher-order incident beam mode is
positively correlated with the order number. This correlation results in
limited power being captured by the small aperture, causing a lag in
performance enhancement as the size of the aperture is increased. Moreover,
the overall trend initially exhibits an increase followed by a subsequent
decrease. This pattern can be attributed to the fact that when the aperture is
small, increasing the lens size allows for the collection of more optical
power, which in turn makes the system more tolerant to pointing error, as also
elucidated in prior research [24]. This increased tolerance is due to the
larger lens’s ability to capture a greater portion of the diffused beam, which
is particularly advantageous when precise alignment cannot be maintained, thus
enhancing the channel capacity up to a certain point. However, when the
aperture is big enough, the resulting larger focal length results in a smaller
field of view (FOV), which makes the system more vulnerable to AOA deviation
[24]. Therefore, the trade-off between aperture diameter $D$ and focal length
$f$ results in the optimal telescope configuration to maximize the system
performance. Specifically, for $N$ equals 1 to 6, the optimal aperture
diameter can be determined as 6 mm, 12 mm, 15 mm, 19 mm, 22 mm, and 24 mm,
respectively.

(a)

(b)

Figure 7: The optimal mode set $\mathcal{N}$ to maximize the channel capacity
under different number of multiplexing channels for (a): mutually coherent
system without ZFBF, and (b): mutually coherent system with ZFBF.

Notably, in SMM-enabled OWC systems, the choice of the transmitted spatial
mode set $\mathcal{N}$ is of paramount importance due to the varying crosstalk
attributes exhibited by each mode during misaligned propagation. Fig. 7
depicts the optimal mode selection for both systems with and without ZFBF
through exhaustive search. As depicted in Fig. 7a, mode 0 is invariably
selected across all considered mode sets $\mathcal{N}$ owing to the
fundamental Gaussian mode’s robust resistance to vibrations as evidenced in
Fig. 5d. For cases where $N\geq 2$, the mode set $\mathcal{N}=\\{0,1\\}$
consistently emerges as the preferred choice. This preference is largely due
to the effective mitigation of inter-channel interference by ZFBF, which in
turn accentuates the importance of signal power when calculating the channel
capacity. Additionally, it should be noted that in configurations requiring
the transmission of more than a binary set of modes, the selection of adjacent
modes results in substantial power consumption for interference compensation.
As observed in (39), unlike the utilization of SVD as a linear coding strategy
to preserve the signal power, the ZFBF may require substantial power
allocation to the transmitted signals to compensate for the interference. Such
compensation can lead to the non-uniform and occasionally excessive
amplification of power in specific data streams, as it strives to attain the
required level of interference mitigation. This outcome brings the trade-off
between allocating power for signal transmission and power for compensating
interference, an essential consideration in the design and optimization of
such communication systems. For example, for $N=3$, the optimal mode set is
$\\{0,1,3\\}$ instead of $\\{0,1,2\\}$, and for $N=5$, the optimal mode set is
$\\{0,1,2,3,5\\}$ instead of $\\{0,1,2,3,4\\}$.

On the other hand, as demonstrated in Fig. 7b, the absence of ZFBF
significantly alters the mode selection strategy, where the importance of
maximizing the relative separations between transmitted mode states to
mitigate modal crosstalk has been underscored as elucidated in previous works
[20, 18]. This shift can be attributed to the fact that power is more likely
to leak into adjacent modes as discussed in Fig. 5, and the presence of inter-
channel interference accentuates the importance of mitigating interference
power, which is in contrast with systems where ZFBF is employed. Additionally,
in configurations where transmission involves more than a binary set of modes,
it has been observed that there is a tendency for power from higher-order
modes to preferentially couple into modes that are more distantly located
within the spectrum. This behavior calls for a deliberate strategy in
selecting modes that are adjacent to the highest-ordered mode in the set, to
facilitate efficient transmission and to mitigate the effects of power
coupling into the unintended modes. For example, when $N=3$, the optimal mode
set is $\mathcal{N}=\\{0,4,5\\}$, and when $N=4$, the optimal mode set is
$\mathcal{N}=\\{0,3,4,5\\}$.

Figure 8: The optimal mode set to maximize the channel capacity under
different numbers of multiplexing channels $N$

Fig. 8 illustrates the aggregated achievable data rate as the function of the
number of subchannels for SMM systems with and without ZFBF. Note that the
optimal aperture configurations and mode sets shown in Fig. 6 and Fig. 7,
respectively, are utilized to attain this result. It is observable that as the
size of the available mode set $N$ expands, the channel capacity with ZFBF
exhibits an initial increase followed by a subsequent decrease. This pattern
can be attributed to the fact that, with a constrained array of subchannels,
ZFBF expends only a minor fraction of the total power to counteract
interference. Consequently, the system is able to benefit from the augmented
spatial degree of freedom, leading to a marked elevation in the channel
capacity. However, adding more transmitted modes significantly decreases the
allocated power to each channel under the constant power constraint. On the
other hand, more subchannels require increased power to mitigate the crosstalk
between each mode. As a result, when the number of modes $N$ is increased,
only limited power can be applied to the information-bearing signals, which in
turn results in a decreased received power for each channel, leading to a
significant drop in the achievable data rate. Therefore, a trade-off emerges
between the power used for interference elimination and that used for signal
transmission. This leads to the identification of an optimal mode number for
the ZFBF-based system as illustrated in Fig. 8, which is realized when $N=4$,
corresponding to the mode set $\mathcal{N}=\\{0,1,2,4\\}$. Based on the
previous discussion, the transmission of different OAM modes through the
misaligned channels is known to exhibit variability in maintaining the
transmitted power within the original mode state. Concurrently, the degree of
power leakage from one channel to another is dependent on the specific states
of the modes involved. This leads to an inconsistency in the average signal
and interference power across channels, resulting in the multiplexed channels
within the SMM system with inherently non-uniform characteristics.

Regarding system performance in the absence of ZFBF, the selection of higher-
order OAM modes results in a pronounced increase in crosstalk under the real-
world vibration parameter. Coupled with the uniform distribution of signal
power across all transmission modes, there is a consequential reduction in the
SINR within individual subchannels, which in turn precipitates a decline in
system performance when $N\geq 3$. Consequently, the data presented in Fig. 8
show that implementing ZFBF can facilitate a significant enhancement in system
performance under, namely, 43% improvement when $N=3$ and 236% improvement
when $N=6$. Notably, the findings presented in Fig. 8 hold significant
implications for the design of practical OWC-based SMM systems. The
limitations set by the physical dimensions of transceivers in actual
deployments are crucial factors that restrict the operational range, thereby
highlighting the necessity of choosing the optimal combination of modes to
realize the maximum channel capacity within these constraints.

Figure 9:  The achievable data rate with various $N$ under different
transmission power $\xi_{t}$ (from -15 dBm to 30 dBm), with arrows indicating
the optimal mode number under corresponding power budgets.

The achievable data rate with respect to $\xi_{t}$ for various $N$ under the
optimal mode set is illustrated in Fig. 9. The depicted trends reveal that,
for the ZFBF-based system operating under low-power conditions, a
configuration with a smaller number of subchannels can exhibit superior
performance relative to a system with a larger $N$. Specifically, when the
transmission power falls below 4 dBm, the single-channel system surpasses the
performance of all multiplexed configurations, and the 5-mode system features
the best performance as the $\xi_{t}$ is larger than 11 dBm. This is because
under the power-limited scenario, as more power is used to compensate for the
interference among subchannels when increasing $N$, less signal power will
result in the degradation in channel capacity as the constant noise variance.
However, under the high-power situation, since more power can be applied to
the information-bearing signal, the SNR can experience a huge enhancement
under such a constant noise power. Therefore, with the larger power budget,
the larger $N$ can benefit more from the extra spatial DOF with the
elimination of interference. For example, with $\xi_{t}$ is 11 dBm, the
5-channel can feature 35.6$\%$ and 4.8$\%$ improvement when compared with the
single-channel and 3-channel system, while the enhancement can be increased to
117.2$\%$ and 22.5$\%$ as the power budget is extended to 20 dBm. Hence, the
selection of a specific number of subchannels should be tailored to different
power budgets. As concluded by the arrows, after applying the ZFBF, $N=1$ is
the optimal option when the transmission power is less than 4 dBm. As further
expanding the power budget, a system with two subchannels outperforms others
up to a transmission power $\xi_{t}\leq 9$ dBm. In addition, as the power
ranges between 9 dBm and 11 dBm, $N=4$ maximizes the system capacity. Finally,
for a power budget exceeding 11 dBm, a configuration with five subchannels
($N=5$) optimizes system performance. On the other hand, for the system
without ZFBF, one can see that the performance shows a tendency toward
saturation in high-power regimes. This saturation occurs because an increase
in transmission power concurrently amplifies the interference power.
Therefore, as evidenced by Fig. 9, the ZFBF not only allows the system to
capitalize on the spatial DOF but also facilitates the system performance
enhancement through the escalation of transmission power.

##  V Conclusion

In this paper, we investigate the SMM-enabled FMF coupling system based on the
IM/DD receiver with mutually coherent channels impaired by link misalignment.
We introduce a theoretical method that provides more accurate and compact
misalignment-impaired channel models compared with the prior work in the
literature. Building upon this, we develop a theoretical coupling efficiency
model from the multi-mode OWC link to the FMF. It is found that in the
presence of misalignment, the coupling efficiency varies significantly for the
transmission of different modes. Our research also observes significant
intermodal power leakage due to device vibrations, for which we employ the
ZFBF to mitigate subchannel crosstalk and accordingly derive the analytical
aggregated data rate equations. Through simulations, we identify optimal
system configurations that maximize performance, including aperture design,
number of transmitting modes, and mode set selection. These optimal
configurations result in a performance enhancement of over 200% when using
ZFBF. Considering the system performance is significantly altered by the
received power, this paper finally presents the most effective mode
exploitation strategies for varying power budgets.

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