# Capacity and Achievable Rates of Fading Few-mode MIMO IM/DD Optical Fiber
Channels
††thanks: The authors are supported by Danish National Research Foundation
Centre of Excellence SPOC, DNRF123, and the Villum YI, OPTIC-AI, 29344.

Metodi P. Yankov, Francesco Da Ros, Søren Forchhammer, and Lars Grüner-Nielsen
Department of Photonics Engineering
Technical University of Denmark
2800 Kgs. Lyngby, Denmark
{meya, fdro, sofo<EMAIL_ADDRESS>
###### Abstract

The optical fiber multiple-input multiple-output (MIMO) channel with intensity
modulation and direct detection (IM/DD) per spatial path is treated. The
spatial dimensions represent the multiple modes employed for transmission and
the cross-talk between them originates in the multiplexers and demultiplexers,
which are polarization dependent and thus time-varying. The upper bounds from
free-space IM/DD MIMO channels are adapted to the fiber case, and the
constellation constrained capacity is constructively estimated using the
Blahut-Arimoto algorithm. An autoencoder is then proposed to optimize a
practical MIMO transmission in terms of pre-coder and detector assuming
channel distribution knowledge at the transmitter. The pre-coders are shown to
be robust to changes in the channel.

###### Index Terms:

Few-mode fiber, MIMO, intensity modulation, direct detection, capacity,
autoencoder.

##  I Introduction

Few-mode transmission in optical fibers is typically considered in long-haul
communication links in order to go beyond the so-called nonlinear Shannon
limit of single-mode fibers and enhance the data rate by employing spatial
multiplicity [1, 2]. On the other hand, for short-reach connections as the
ones inside data centers (DCs), the problem is typically not capacity per
fiber, but rather the cost of the transceivers, which have to employ multiple
wavelengths to reach the target data rates. Such links typically operate with
intensity-modulation and direct detection (IM/DD), with digital signal
processing (DSP) functionalities kept at a minimum in order to reduce the cost
w.r.t. the standard long-haul coherent transmission, and also to reduce
latency which is critical for intra-DC communication.

The abundance and low cost of few-mode fibers (FMFs) and the lifted
requirement for frequency stabilization of the lasers make few-mode
multiplexing an attractive alternative to few-wavelength multiplexing. The
challenge in employing FMF transmission is multiplexing and demultiplexing the
transmission modes into and out of the same fiber. For that purpose, several
types of mode multiplexers may be employed, e.g. air clad photonic lanterns
[3], demonstrating 100G transmission over up to 2 km for 2-mode multiplexing.
Due to the cost requirement, multiple input multiple output (MIMO) processing
and synchronization of the modes at the receiver is typically prohibitive.
Instead, the multiplexer (MUX) and demultiplexer (DEMUX) are required to
produce negligible cross-talk (XT), which sets a harsh requirement on their
fabrication process.

Capacity and achievable rates for wireless IM/DD links have been studied[4],
more recently in [5, 6], including MIMO versions [7, 8], where the channel is
assumed fixed and known at the transmitter and/or receiver. In general, the
amplitude-constrained channel capacity study dates back to [9], where a linear
system was considered. Achievable rate regions for the MIMO multiple access
channel with IM/DD were derived in [8].

In this paper, the case of few-mode point-to-point optical fiber channel is
considered with joint or independent mode processing at the receiver. The
latter case falls into the category of MIMO broadcast channels with IM/DD, for
which to our knowledge, no closed-form expressions exist for the channel
capacity, especially in the case of fading. An upper bound on the capacity
with perfect channel knowledge at the transmitter and receiver is adapted from
the free-space case, and the Blahut-Arimoto algorithm (BAA) is used to
estimate a tightly-approaching achievable rate. An autoencoder is then
proposed to optimize a pre-coder and a detector for both cases of joint and
independent processing at the receiver, under the constraint of only channel
state distribution knowledge at the transmitter.

###  I-A A note on notation

Variables are denoted with capital letters, e.g. $X$, their distributions by
$p_{X}(X)$, their support set by ${\cal{X}}$, with ${\cal{X}}_{i}$ being the
$i-$th element, and their realizations by small letters, e.g. $x$ for scalars
and $\mathbf{x}$ for vectors, $x_{i}$ being the $i-$th vector element.
Matrices are denoted $\mathbf{H}$, to be distinguished from vector random
variables $\boldsymbol{X}$.

##  II Channel model

A transmission system with independent data on each spatial path of the FMF is
considered. The data bits modulate the waveform using pulse amplitude
modulation (PAM), where the modulation alphabet is ${\cal{X}}={0,1,...M-1}$ of
size $|{\cal{X}}|=M$. The symbols are independent in time and the signal
vector is denoted $\mathbf{x}=\left[x_{1},x_{2},...x_{N}\right]^{T}$, where
$N$ is the number of employed modes. The signal may then be pre-coded by a
linear or nonlinear function $f$, $\mathbf{v}=f(\mathbf{x})$. A Mach-Zehnder
modulator (MZM) is employed for modulating the intensity of optical carriers
of wavelengths $\mathbf{\lambda}=\\{\lambda_{1},\lambda_{2},...\lambda_{N}\\}$
with the corresponding pre-coded symbols. The MZM is biased in quadrature, and
its output intensity $s_{i}$ as a function of the modulating voltage $v_{i}$
can be modeled as

$\displaystyle
s_{i}=P_{i}\cos^{2}\left(\frac{\pi}{2}\frac{v_{i}}{V_{\pi}}\right),$ (1)

where $V_{\pi}$ is the half-wave voltage of the modulator, and $P_{i}$ is the
optical power of the $i-$th carrier. The multiple fibers leading out of each
MZM are combined in a mode MUX. An example of a MUX is the air clad photonic
lantern made of single mode fibers, which are fused together and down tapered.
The end of the taper is spliced to a few-mode fiber. After down tapering the
surrounding air will act as the cladding. By use of single mode fibers with
different core sizes mode selectivity is obtained [3, 10]. At the end, a DEMUX
is used to demultiplex the modes into independent spatial paths. Each mode is
detected by a photodetector (PD), which translates light intensity to
electrical current. The signal at the receiver is denoted
$\mathbf{y}=\left[y_{1},y_{2},...y_{N}\right]^{T}$. A block diagram of the
system is given in Fig. 1. Relatively short fiber distance is assumed so that
dispersion, loss and distributed XT along the fiber span can be neglected
[10]. The channel is thus memoryless. Without loss of generality and for
easier notation, the case of 2-mode transmission is assumed where the spatial
modes $LP_{01}$ and $LP_{11}$ are excited for propagation. This case
corresponds to the lanterns demonstrated in [3].

Figure 1: Considered MIMO IM/DD channel model of a few-mode fiber
transmission.

Each of the MUX, DEMUX and splices can then be modeled by a MIMO matrix
$\mathbf{H}_{MUX}$, $\mathbf{H}_{DEMUX}$ and $\mathbf{H}_{SPL}$, respectively
and has the form (for example for $\mathbf{H}_{MUX}$)

$\mathbf{H}_{MUX}=\begin{bmatrix}(1-xt_{1})/\alpha_{1}&xt_{2}/\alpha_{2}\\\
xt_{1}/\alpha_{1}&(1-xt_{2})/\alpha_{2}\end{bmatrix},$ (2)

where $xt_{1}$ and $xt_{2}$ are the XTs from mode $LP_{01}$ to mode $LP_{11}$
($LP_{11}$ to $LP_{01}$, respectively), and $\alpha_{1}$ and $\alpha_{2}$ are
the losses of the corresponding modes in the component. The transfer matrices
are subject to polarization-dependent XT. In this work, without loss of
generality, this dependence is modeled by drawing the realization of each XT
value in (2) for each time instant from a uniform distribution with a given
mean ${\mathbb{E}}\left[XT\right]$ and a range of $6$ dB. The losses are
assumed constant. Assuming the carriers are not coherent at the receiver
(which is reasonable for cheap, independent lasers with significant phase
noise and assuming large enough differential group delay, as well as proper
wavelength selection for the carriers [11]), the beating between modes can be
minimized, and the channel from transmitter to receiver can be modeled as

$\displaystyle\mathbf{y}=\mathbf{H}_{DEMUX}\cdot\mathbf{H}_{SPL}^{2}\cdot\mathbf{H}_{MUX}\cdot\mathbf{s}+\mathbf{w}=\mathbf{H}\cdot\mathbf{s}+\mathbf{w},$
(3)

where $\mathbf{s}=\left[s_{1},s_{2},...s_{N}\right]^{T}$. The noise terms
$\mathbf{w}=\left[w_{1},w_{2},...w_{N}\right]^{T}$ are independent,
identically distributed samples from a Gaussian distribution with a variance
$\sigma_{w}^{2}$. The noise is assumed dominated by the receiver thermal and
quantization noise. At the receiver, independent or joint detection of the
signal $\mathbf{y}$ may be performed with the function $g(\mathbf{y})$.

The MZM transfer function imposes an amplitude constraint on the signal $0\leq
v_{i}\leq V_{\pi}$, since outside of these boundaries the cosine transfer
function becomes non-bijective. Equivalently, the channel in (3) can be
considered the standard MIMO IM/DD channel [7, 8] with input $\mathbf{s}$ and
amplitude constraint $0\leq s_{i}\leq P_{i}$.

In the case of fiber communications and for IM/DD links operating at short
reach, there is no practical constraint (within reason) associated with the
average power, unlike for example free-space optical wireless links, where
safety and interference may be factors.

The signal to noise ratio (SNR) is defined as
$SNR=10\log_{10}(\frac{\sum_{i}P_{i}}{2N\cdot\sigma_{w}})$, motivated by the
fact that ${\mathbb{E}}\left[|Y_{i}-S_{i}|\right]=\sigma_{w}$ and that for
uniformly distributed $s_{i}$ we have
${\mathbb{E}}\left[|S_{i}|\right]=P_{i}/2$. A total power constraint may be
imposed in the form of $\sum_{i}P_{i}=P_{tot}$ so that the SNR can be swept.

##  III Capacity and capacity achieving distributions

Assuming full-rank, positive $\mathbf{H}$ with well-behaved distribution, the
ergodic capacity ${\cal{C}}$ of the MIMO fading channel with perfect channel
knowledge at the transmitter and receiver is defined as

$\displaystyle{\cal{C}}=\max_{p_{\boldsymbol{X}}(\boldsymbol{X}),f({\boldsymbol{X}}),P_{\\{1\dots
N\\}}}{\mathbb{E}}_{H}\left[{\cal{I}}(\boldsymbol{Y};\boldsymbol{X}|\mathbf{H},p_{\boldsymbol{X}}(\boldsymbol{X}))\right],$
(4)

where the mutual information (MI) ${\cal{I}}(\cdot;\cdot)$ between the input
and output is maximized over the input distribution
$p_{\boldsymbol{X}}(\boldsymbol{X})$, the pre-coder $f({\boldsymbol{X}})$ and
the power allocation. Assuming that the optimal pre-coder is applied, we have

$\displaystyle{\cal{C}}=\max_{p_{\boldsymbol{S}}(\boldsymbol{S}),P_{\\{1\dots
N\\}}}{\mathbb{E}}_{H}\left[{\cal{I}}(\boldsymbol{Y};\boldsymbol{S}|\mathbf{H},p_{\boldsymbol{S}}(\boldsymbol{S}))\right],$
(5)

which is the ergodic capacity of the channel in (3) with ideal knowledge at
the receiver. The latter can be upper bounded by splitting the channel into
parallel IM/DD channels by using the QR decomposition of $\mathbf{H=QR}$,
where $\mathbf{Q}$ is unitary and $\mathbf{R}$ is upper triangular. The
receiver then pre-processes $\mathbf{y}$ to
$\hat{\mathbf{y}}=\mathbf{Q}^{T}\mathbf{y}=\mathbf{R}\mathbf{s}+\mathbf{w}$,
the interference from previously decoded channels is canceled on sub-sequent
channels and the MIMO channel is equivalent to the set of parallel channels
$\hat{y}_{i}=\mathbf{R}_{i,i}s_{i}+w_{i}$, where $\mathbf{R}_{i,i}$ is the
element on the $i-$th row and $i-$th column on $\mathbf{R}$ [7]. The capacity
may then be upper-bounded by

$\displaystyle{\cal{C}}\leq\max_{P_{\\{1\dots
N\\}}}{\mathbb{E}}_{H}\left[\sum_{i}\bar{{\cal{C}}}_{i}(\mathbf{R}_{i,i},P_{i},\sigma)\right],$
(6)

where $\bar{{\cal{C}}}_{i}(\mathbf{R}_{i,i},P_{i},\sigma)$ is the upper bound
(UB) on the scalar IM/DD channel with power $P_{i}$ and a gain coefficient
$\mathbf{R}_{i,i}$. The maximization over the power allocation can be done by
a grid search for reasonable number of spatial modes with the above mentioned
constraint of $\sum_{i}P_{i}=P_{tot}$. Two upper bounds are considered, UB1
(obtained from [6, Theorem 1]) and UB2 (obtaiend from [4, Eq. $19$]), where
UB1 is tight at high SNR, and UB2 is superior at low SNR. The limitation of
these UBs is that they do not provide the pre-coder function, since they are
derived in the already pre-coded space.

On the other hand, practically achievable capacity estimates can be derived
numerically using the classical BAA. If the optimal distribution for the
channel in (3) is defined as
$p_{\boldsymbol{S}}^{opt}(\boldsymbol{S})=\sum_{k=1:|{\cal{S}}|}a_{k}\delta(S-{\cal{S}}_{k})$,
where $a_{k}$ are the weights of the $k-$th mass point ${\cal{S}}_{k}$ and
$\delta(\cdot)$ is the Dirac delta function, the following can be proven.

###### Theorem 1

For a given power allocation $P_{\\{1\dots N\\}}$, an initialization of the
BAA with a discrete uniform distribution of $M\rightarrow\infty$ on the
constrained hypercube produces $p_{\boldsymbol{S}}^{opt}(\boldsymbol{S})$ for
which the MI
${\mathbb{E}}_{H}\left[{\cal{I}}(\boldsymbol{Y};\boldsymbol{S}|\mathbf{H},p_{\boldsymbol{S}}^{opt}(\boldsymbol{S}))\right]$
is the capacity of the channel in (3), and a lower bound on ${\cal{C}}$.

###### Proof:

The following observations can be made:

  1. 1.

The capacity-achieving distribution of channel (3) is discrete and unique
[12][13, Chapter 11], and the MI is concave in
$p_{\boldsymbol{S}}(\boldsymbol{S})$ [14, 4].

  2. 2.

For discrete, memoryless channels the BAA achieves improvement on every
iteration [15] until convergence (i.e., it always achieves a local optimum).

  3. 3.

The BAA is allowed to approach zero probability to individual mass points,
effectively removing points from the support set of $\boldsymbol{S}$.

  4. 4.

Since the power constraint is ensured by the MZM transfer function (1) and the
initialization of the mass points, the BAA is unconstrained (except for
$\sum_{i}a_{i}=1$).

From Observations 3) and 4), for $M\rightarrow\infty$ the BAA can ’reach’ any
discrete support set within the hypercube, making the corresponding local
optimum from Observation 2) the global optimum due to the concavity in
Observation 1), proving the first part of the Theorem. The MI achieved with
the BAA is an achievable rate on the original channel with the trivial pre-
coder $\mathbf{V}=f(X)=\mathbf{X}$ and support set ${\cal{V}}$ obtained by
inverting (1), making it by definition a lower bound on the capacity in (4),
proving the second part of the Theorem ∎

Theorem 1 has been used implicitly to estimate the capacity of the scalar
IM/DD channel [13], and is here formalized and proven. Furthermore, we point
out that initialization with a distribution of any size $M$ and any selection
of mass points ${\cal{S}}_{k}$ also provides an achievable rate, albeit
potentially lower. A particularly convenient and relevant initialization is
for $\boldsymbol{V}$ to be uniformly distributed on the hypercube confined by
$\left[0;V_{\pi}\right]^{N}$ and $\boldsymbol{S}$ obtained by applying (1),
which allows for the potential shaping gain to be estimated w.r.t. classical
equidistant PAMN schemes (as shown below).

Some capacity estimates are provided in Fig. 2 for initializations as
described above of different size, together with the two considered UBs. The
expected channel parameters in this case are given in Table I, and correspond
to the measurements done in [10]. For the rest of the paper, the total power
constraint is arbitrarily chosen to be $\sum_{i}P_{i}=3$ dBm. In certain SNR
regions of interest (between $\approx$ 15 dB and $\approx$ 20 dB), the BAA
constrained to 32PAM per dimension closes the gap to the upper bound,
effectively achieving the channel capacity with the trivial pre-coder
described above and ideal channel knowledge at the receiver. The shaping gain
for this constellation size is limited to $\approx$ 0.5 dB w.r.t. the MI
achieved with uniformly distributed, equidistant points. As shall be seen in
the Results section, when the constellation size remains fixed and the SNR is
increased above a certain modulation-size dependent threshold, pre-coding
plays a more important role and will influence the tightness of the bound
provided by the rate, achievable with the BAA.

TABLE I: Expected parameters for the MIMO components. |  ${\mathbb{E}}\left[XT_{1}\right]$, dB |  ${\mathbb{E}}\left[XT_{2}\right]$, dB |  $\alpha_{1}$, dB |  $\alpha_{2}$, dB
---|---|---|---|---
MUX | -18 | -15 | 0.7 | 1.4
DEMUX | -11 | -11 | 1.5 | 3
SPL | -25 | -25 | 0.04 | 0.04
Figure 2: Capacity results for the parameters given in Table I.  Figure 3: An
example optimal distribution $p_{\boldsymbol{V}}(\boldsymbol{V})$ for $M=32$
and $SNR=15$ dB.

An example of the capacity achieving distribution for $M=32$ and $SNR=15$ dB
is given in Fig. 3. We see a slight asymmetry in the distribution
corresponding to the asymmetric XT. The optimal powers found by grid search
(in this case a line search, since $N=2$ and $P_{1}+P_{2}=3$ dBm) were found
to be $P_{1}=0.41$ dBm and $P_{2}=-0.46$ dBm.

##  IV Autoencoder based communication

As mentioned in the introduction, full MIMO processing may be too complex for
the target communication system because of both hardware (in terms of
synchronizing the receivers for each mode) and software (in terms of
processing for channel estimation and multi-dimensional detection)
requirements. However, partial channel knowledge may still be available in
terms of the distribution of $\mathbf{H}$ at the transmitter and receiver,
which can be obtained e.g. from measurements statistics. For such cases, an
auxiliary channel likelihood function must be employed for detection and MI
estimation, which is subject to optimization. The likelihood function needs to
be robust to the distribution of $\mathbf{H}$. At the same time, the knowledge
of $\mathbf{H}$ may allow to design a pre-coder which at least partially
mitigates the mode interference. Both of these optimizations can be
efficiently approached with an autoencoder (AE) [16]. The AE employs a neural
network (NN) for pre-coding, another NN for detection, and wraps them around
the channel model at hand. The most closely related to this work AE
application is for the Rayleigh fading MIMO channel [17], where the channel is
linear. In this paper, both the MZM and the photodetection induce
nonlinearities. Furthermore, in this paper, the AE is employed to also
optimize the power allocation (instead of the grid-search performed in the
previous Section). A basic schematic of the AE is given in Fig. 4, and the
following optimization scenarios are considered for the pre-coder:

Figure 4: Autoencoder model for optimization. Showcased with independent
detector NN per layer.

Prec1

Power allocation only. The pre-coder is
$\mathbf{v}=\frac{V_{\pi}}{M-1}\mathbf{x}$.

Prec2

Power allocation + linear pre-coder
$\mathbf{v}=f(\mathbf{x})=\mathbf{A}\mathbf{x}$, where $A$ is an NxN matrix.

Prec3

Power allocation + nonlinear pre-coder, where $f(\mathbf{x})$ is a multi-layer
NN.

and for the detector:

Det1

Each mode is detected by an independent NN.

Det2

All modes are detected jointly by a common NN.

The outputs of the detector NNs are auxiliary distributions
$q(\mathbf{x}|\mathbf{y})$ for the conditional distribution
$p(\mathbf{x}|\mathbf{y})$, and the detectors thus provide an achievable rate
[18]. Det1 represents a more practical scenario. In order to simplify it even
further, its performance is reported also for the case of a standard Gaussian
auxiliary channel receiver, for which the auxiliary distribution is
$q({\cal{X}}_{i}|y_{k})\propto
p_{X_{i}}({\cal{X}}_{i})\cdot\exp\left[-0.5/\hat{\sigma}_{w_{i}}\left(y_{k}-\mu_{i}\right)^{2}\right]$.
The parameters $\left(\mu_{i},\hat{\sigma}_{w_{i}}\right)$ of the auxiliary
Gaussian function are estimated using training data [19]. The scenario with
Prec1 and independent Gaussian detection per mode can be thought of as the
current state of the art in few-mode, short reach IM/DD fiber links, and is
our point of performance reference. In the case of Prec2 and Prec3 and
Gaussian detection, a NN is still employed at the receiver but only in order
to facilitate training of the pre-coder using the cross-entropy (CE) cost
function.

In the case of Det1, the AE is trained using the cost function
$Cost=\max_{i=1\dots N}CE_{i}(X_{i},g(Y_{i}))$, where
$CE_{i}(X_{i}|Y_{i})={\cal{H}}(X_{i}|Y_{i})$ is the CE, which in this case is
equivalent to the conditional entropy ${\cal{H}}(X|Y)$. Since the MI
${\cal{I}}(X;Y)={\cal{H}}(X)-{\cal{H}}(X|Y)$, this cost function targets equal
performance on all modes, which is the desired mode of operation in simple,
robust systems which do not allow rate adaptivity. The case of Det2 is
considered as a reference performance and for comparison to the bounds and
rates of Section III. In this case, the cost function is
$Cost=CE(\boldsymbol{X},g(\boldsymbol{Y}))$. For all detectors and pre-coders,
the NN topologies were optimized by increasing their depth and width by
factors of 2 until convergence of the cost function. The resulting topologies
for $N=2$ are given in Table II. The last layer of the detectors is a
classifier, and thus has output size of $M$ and $M^{2}$ for the cases of
independent and joint processing, respectively. All NNs employ the ReLU
activation function and all NNs are biased. The Adam optimizer was employed
for stochastic gradient descent optimization of all parameters. The batch size
was 200, and a total of 1.000.000 symbols are used for training and
independent 100.000 for testing. The channel is drawn from a uniform
distribution as described above for every sample during training and testing.
The result is an AE robust to the desired channel distribution. Observe, no
explicit constraint is imposed on the encoder part. The AE is naturally forced
to operate within the boundaries of the cosine transfer function of the MZM
(1).

TABLE II: Optimized topologies in terms of nodes per layer of the AE components for $N=2$. All NNs are biased. | 8PAM | 16PAM
---|---|---
Prec2 | [2x2] | [2x2]
Prec3 | [2x16x16x2] | [2x16x16x2]
Det1 | [1x64x64x8] | [1x256x256x16]
Det2 | [2x128x128x64] | [2x512x512x256]

##  V Results

The practically-relevant system with Det1 is first treated where independent
mode processing is employed at the receiver in Fig. 5 for $M=8$. The reported
rate corresponds to
$N\cdot\min_{i}{\mathbb{E}}_{H}\left[{\cal{I}}(Y_{i};X_{i}|p_{\mathbf{H}}(\mathbf{H}))\right]$,
i.e., only the distribution of $\mathbf{H}$ is known, and the performance of
the worst mode is considered. The standard Gaussian receiver without
additional pre-coding is limited in rate to $\approx 4.8$ bits/symbol at
$SNR=20$ dB. The rate is substantially improved by the AEs for $SNR>10$ dB by
up to $1$ bits/symbol at the high end. Alternatively, the SNR gain is $\approx
5$ dB at the maximum rate for the conventional system. The gain of nonlinear
pre-coding is up to $\approx 0.25$ bits/symbol w.r.t. linear pre-coding, both
at low and high SNR. As expected from the Gaussian statistics of the noise,
there is no extra benefit from nonlinear detection w.r.t. standard Gaussian
receiver. The BAA rate is also shown for comparison. In contrast, its rate is
${\mathbb{E}}_{H}\left[{\cal{I}}(\boldsymbol{Y};\boldsymbol{X}|\mathbf{H})\right]$.
We see that nonlinear pre-coding operates near the BAA rate, demonstrating
that the penalty from independent processing and lack of ideal knowledge of
$\mathbf{H}$ at the receiver is nearly recovered by simple, time invariant
pre-coding. The pre-coded constellations are shown in Fig. 6 for $M=8$ and
$SNR=15$ dB. The linear pre-coder can only skew the constellation, whereas the
nonlinear one can shape the constellation so that the distribution of energy
is similar to that of Fig. 3. The optimal distribution obtains a shape, which
takes advantage of the full-range of the modulation of
$\left[0;V_{\pi}\right]$, while at the same time concentrates the mass around
the center of the modulation support $\left(V_{\pi}/2;V_{\pi}/2\right)$ in
order to increase the robustness to the Gaussian noise. Similarly to the
linear case, the nonlinear edges and skewing are attributed to the asymmetric
expectation of the cross-talk. For reference, the optimal powers found by the
AE are also indicated in Fig. 6 for each case. The powers are now different
from the ones obtained for the BAA because the target is equalizing the
performance per mode, instead of the total sum-rate.

Figure 5: AE performance for the channel with expected parameters from Table I
and uniform distribution, at $M=8$. The rate is estimated both with the NN
detector and with a Gaussian receiver, both with independent processing per
mode (Det1). Figure 6: Precoded constellations for the channel with expected
parameters from Table I and uniform distribution, at $M=8$ and $SNR=15$ dB.
a): Prec2 (linear); b) Prec3 (nonlinear).

The benefits of the AE w.r.t. the standard uncoded approach can also be
exemplified in terms of XT tolerance. To that end, without loss of generality
the expected $XT_{2}$ of the DEMUX is varied between -20 and -5 dB, and the
rest of the parameters remain the same. The results at the $SNR=20$ dB are
given in Fig. 7. In order to achieve an example target rate of 5.094, which
corresponds to a code rate of 0.849 of a typical passive optical network
forward error correction low-density parity check code [20], the standard
system requires that the XT is reduced to $\approx-13$ dB. The proposed pre-
coders achieve the target rate with XT tolerances of $4$ dB and $5$ dB for the
linear and nonlinear pre-coder, respectively. This tolerance is directly
translated to loosened fabrication requirements for the MUX and DEMUX. At the
extreme end of the XT, the nonlinear detectors also exhibit gain w.r.t. the
Gaussian receiver, potentially due to their capability to achieve an
asymmetric likelihood function.

Figure 7: AE performance for the channel with expected parameters from Table I
and uniform distribution, except for variable $XT_{2}$ of the DEMUX, at $M=8$,
$SNR=20$ dB. The rate is estimated both with the NN detector and with a
Gaussian receiver, both with independent processing per mode (Det1).

To further study the effect of the pre-coder, the rate of the AE is estimated
and compared to that of the BAA for the case of 0 uncertainty in the channel
in Fig. 8. This case still has relevance in practice if e.g. polarization
controllers are employed at the transmitter and receiver to stabilize the
polarization and remove the XT drift. The AE requirements for robustness to
channel uncertainty are now relaxed, which means that it can find a better
encoder targeted at the known channel. No such benefit can be exploited by the
BAA assignment of probabilities since as mentioned above, it already operates
in the pre-coded space. At the moderate SNR, the BAA is near-capacity
achieving, as is the AE. However, at higher SNR and at rates above $\approx 5$
bits/symbol, the impact of the mode interference becomes dominant to that of
the noise, and pre-coding becomes beneficial. The simplest AE with linear pre-
coding and independent processing at the receiver already outperforms the BAA.
Observe, the latter has access to the channel at the receiver. The full AE
with nonlinear pre-coding and joint mode detector outperforms the BAA on the
entire SNR range and closes the gap to capacity further.

Figure 8: AE performance for the channel with expected parameters from Table I
with 0 uncertainty, at $M=8$.

##  VI Discussion

The BAA optimization can be thought of as probabilistic shaping, while the
pre-coding can be thought of as geometric shaping. An obvious direction for
future research is to explore joint optimization of both, since as it was seen
above, each has its benefits.

In this work, uniform distribution of the cross-talk was arbitrarily chosen.
The described methods directly support other types of distributions, e.g.
Gaussian and or empirical, depending on the fit to the measured polarization
drift.

The case of training independent detectors per mode with the target of
maximizing the sum-rate instead of equalizing the rate per mode is well-within
the realm of capabilities of the AE. It is relevant for cases where rate-
adaptivity is permitted per mode. It was omitted here for space
considerations.

In this paper, $N=2$ in order to present some practically relevant and
motivated results, directly applicable to the system of [10]. Higher-order
mode multiplexing with $N>2$ is well-supported by the presented concepts.

As mentioned in the introduction, the capacity of the MIMO IM/DD multiple
access channel has been studied before [8]. The channel presented in this work
falls instead in the category of MIMO broadcast channels, since the receivers
are independent. Tapping into the general broadcast MIMO channel capacity
results, e.g. from [21] in order to improve the estimates and develop novel
pre-coding techniques for the case of IM/DD transmission is of interest for
future work.

The air clad lanterns were only used as an example in order to provide
practical cross-talk numbers. The bounds and AE method are general to other
types of mode multiplexers, e.g. integrated ones [22].

##  VII Conclusion

The capacity of the optical FMF MIMO channel was lower-bounded by means of an
unconstrained BAA, and tightness was demonstrated to previous upper bounds,
resulting in accurate capacity estimates. The lower bounds are shown to be
achievable by means of trivial pre-coders. Non-trivial pre-coders and
detectors were then designed based on an AE. The AE with independent mode
processing at the receiver was shown to approach the BAA lower bound and at
the same time to be robust against channel uncertainties and drifts. For a
fixed channel, the pre-coders were shown to be superior to the BAA and further
close the gap to capacity for constellations of practical size. Furthermore,
up to $5$ dB of cross-talk tolerance was reported by the AE w.r.t. standard
techniques, which will enable cheaper multiplexers and demultiplexers to be
employed for FMF transmission alternatives to standard wavelength division
multiplexing of short reach optical links.

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