[1]Yiming Li

[1]Aston Institute of Photonic Technologies, Aston University, Birmingham, B4
7ET, UK

2]Instituto de Telecomunicações, University of Aveiro, Aveiro, 3810-193,
Portugal

3]Cailabs, 1 Rue Nicolas Joseph Cugnot, Rennes, 35000, France

# Beaconless Auto-Alignment for Single-Wavelength 5 Tbit/s Mode-Division
Multiplexing Free-Space Optical Communications

<EMAIL_ADDRESS>Gil Fernandes<EMAIL_ADDRESS>David Benton
<EMAIL_ADDRESS>Antonin Billaud<EMAIL_ADDRESS>Mohammed Patel
<EMAIL_ADDRESS>Andrew Ellis<EMAIL_ADDRESS>*  [  [

###### Abstract

Mode-division multiplexing has shown its ability to significantly increase the
capacity of free-space optical communications. An accurate alignment is
crucial to enable such links due to possible performance degradation induced
by mode crosstalk and narrow beam divergence. Conventionally, a beacon beam is
necessary for system alignment due to multiple local maximums in the mode-
division multiplexed beam profile. However, the beacon beam introduces excess
system complexity, power consumption, and alignment errors. Here we
demonstrate a beaconless system with significantly higher alignment accuracy
and faster acquisition. This system also excludes excess complexity, power
consumption, and alignment errors, facilitating simplified system calibration
and supporting a record-high 5.14 Tbit/s line rate in a single-wavelength
free-space optical link. We anticipate our paper to be a starting point for
more sophisticated alignment scenarios in future multi-Terabit mode-division
multiplexing free-space optical communications for long-distance applications
with a generalised mode basis.

###### keywords:

Free-space optical communication, mode-division multiplexing, multiple-input
multiple-output, alignment

##  1 Introduction

Mode-division multiplexing (MDM) is a promising technology to extend the
capacity limits in free-space optical (FSO) communication systems [1].
Orthogonal to other degrees of freedom such as amplitude [2], phase [3],
polarisation [4] and wavelength [5], MDM offers the spatial degree of freedom
for multiplexing, extending capacity beyond the conventional limit in both FSO
and fibre communications [1, 6]. Beyond conventional employment of orbital
angular momentum (OAM) modes [1], which constitute an incomplete basis, there
is a growing interest in employing complete mode basis, such as Hermite-
Gaussian (HG) or Laguerre-Gaussian (LG) modes [7, 8], to maximise potential
capacity gains in the spatial domain [9].

In contrast to fibre communications, FSO communication systems require
stringent alignment to maximise received power [10]. In MDM systems, there is
an additional requirement to minimise inter-mode crosstalk [11], thus
increasing the demand for precise alignment systems. Although single-input
single-output (SISO) FSO systems can naturally reuse the signal beam for
alignment [10], most MDM FSO systems employed a beacon beam for alignment due
to multiple local power extremums during the alignment [12, 11]. Although
adaptive optics can be employed for alignment through tip/tilt compensation,
it is also incompatible with generalised MDM systems due to the presence of
multiple power extremums, and a beacon beam is still required [13]. While
attempts have been made on beaconless alignment in OAM systems [14], it can
not support complete mode bases such as HG or LG modes because certain modes
have zero power on the x- and y-axis. Moreover, the employed quadrant-
detector-based alignment structure still introduced practical alignment errors
between the quadrant detector and the signal photodiodes. The errors were
mitigated by introducing order intervals, which significantly reduced the
number of applicable modes.

To minimise the inter-mode crosstalk in MDM system, it is important to reduce
the practical optical calibration error between the signal and beacon beams,
as well as between the quadrant detector and the signal photodiodes.
Therefore, we should eliminate any excessive FSO paths, including the beacon
beam and the 4-quadrant detector. This will also reduce the system cost and
the difficulties in FSO calibration.

In this work, we demonstrate a beaconless and quadrant-detector-free alignment
algorithm for HG-mode-based MDM system. By minimising power leakage to
unwanted modes, this algorithm enables precise alignment to reduce inter-mode
crosstalk. By eliminating redundant structures in alignment systems, it also
reduces the overall cost of MDM FSO systems. When employing 6 lowest-order
modes, this algorithm supported a 6 (modes)$\times$2 (polarisations)$\times$80
GBaud 16-QAM transmission, obtaining a record-high line rate of 3.84 Tbit/s in
single-wavelength FSO communication systems with a bit error rate (BER) below
the hard-decision forward error correction (HD-FEC) limit. By employing
probabilistic shaping (PS) 256-QAM in the same system, we achieved a record-
high generalized mutual information (GMI) of 5.14 Tbit/s in single-wavelength
FSO communication systems.

##  2 Results

###  2.1 Hermite-Gaussian modes and the local maximums

As an orthogonal complete set to decompose a coherent paraxial beam, the
electric field of the HG laser modes which propagate along the z-axis, can be
described as [15]

$\displaystyle{E_{l,m}}\left({x,y,z}\right)$
$\displaystyle=C_{l,m}^{HG}\frac{1}{{{w_{z}}}}\exp\left({-\frac{{{x^{2}}+{y^{2}}}}{{w_{z}^{2}}}}\right)\exp\left[{-\frac{{ik\left({{x^{2}}+{y^{2}}}\right)}}{{2{R_{z}}}}}\right]$
(1)
$\displaystyle\times{H_{l}}\left({\frac{{\sqrt{2}x}}{{{w_{z}}}}}\right){H_{m}}\left({\frac{{\sqrt{2}y}}{{{w_{z}}}}}\right)\exp\left({-i{\psi_{z}}}\right)\exp\left({-ikz}\right),$

where $C_{l,m}^{HG}={\left({{2^{l+m-1}}\pi l!m!}\right)^{-1/2}}$ is the
normalisation factor, ${R_{z}}=\left({z_{R}^{2}+{z^{2}}}\right)/z$ is the
radius of curvature, ${z_{R}}=\pi w_{0}^{2}n/\lambda$ is the Rayleigh range,
$w_{0}$ is the radius of beam waist, $n$ is the refractive index, $\lambda$ is
the wavelength, ${w_{z}}={w_{0}}\sqrt{1+{{\left({z/{z_{R}}}\right)}^{2}}}$ is
the beam radius at $z$, $k=2\pi/\lambda$ is the wave number,
$\psi_{z}=\left({l+m+1}\right)\arctan\left({z/{z_{R}}}\right)$ is the Gouy
phase,
${H_{n}}\left(x\right)={\left({-1}\right)^{n}}\exp\left({{x^{2}}}\right)\frac{{{{\text{d}}^{n}}}}{{{\text{d}}{x^{n}}}}\exp\left({-{x^{2}}}\right)$
is the Hermite polynomial of order $n$.

To calculate the received power from the transmitted mode
$M_{t}=HG_{l_{t},m_{t}}$ to the received mode $M_{r}=HG_{l_{r},m_{r}}$, the
equation below applies:

$P_{M_{t},M_{r}}={\left({\iint{E_{{l_{t}},{m_{t}}}^{*}\left({x-\Delta
x,y-\Delta
y,z}\right)}{E_{{l_{r}},{m_{r}}}}\left({x,y,z}\right){\text{d}}x{\text{d}}y}\right)^{2}},$
(2)

where $\left({\Delta x,\Delta y}\right)$ is the misalignment of the
transmitted beam profile. As shown in Fig. 1(a), multiple local minimums occur
when either the transmitted or the received mode is not $HG_{0,0}$, requesting
the conventional beacon of an $HG_{0,0}$ (Gaussian) beam.

Figure 1: Concept of beaconless alignment. OF: objective function. (a) The
scanning profile matrix from transmit HG modes to receive HG modes. (b) The
received power summation of all transmitted modes at the receiver. (c) The
received power summation of unwanted modes at the receiver. (d) The objective
function.

###  2.2 Beaconless alignment

To align the MDM beam without a beacon, we need to find an objective function
with only one local maximum where the system is optimally aligned.

####  2.2.1 The requirement of transmit modes

Intuitively, the overall power leakage to unwanted modes will increase when
the misalignment is larger [16]. Although multiple local maximums exist in
Fig. 1(a), the power summation of all transmitted modes is a suitable
objective function with only one local maximum (Fig. 1(b)), which can be
written as:

${P_{Tx}}=\sum\nolimits_{{M_{r}}\in{\mathcal{X}_{T}}}{P\left({{M_{r}}}\right)},$
(3)

where $\mathcal{X}_{T}$ is the set of all transmitted modes,
$P\left({{M_{r}}}\right)=\sum\nolimits_{{M_{t}}\in{\mathcal{X}_{T}}}{{P_{{M_{t}},{M_{r}}}}}$
is the overall received power at receive mode $M_{r}$. Note the independence
between the x- and y-axis indicated by Eq. (1), we can set $y=0$ and analyse
the power distribution in the x-axis without loss of generality. As shown in
Fig. 3(a), Eq. (3) works for all the tested situations from a total of 1
($HG_{0,0}$) to 20 ($HG_{0,0}\sim HG_{19,0}$) transmitted modes. It is also
worth noting that if $HG_{L,M}$ is chosen as a transmitted mode, all
$HG_{l,m}$ modes that satisfy conditions $0\leq l\leq L$ and $0\leq m\leq M$
should also be chosen as transmitted modes to guarantee one single local
maximum in Eq. (3).

Although the transmit HG modes form an orthogonal basis, their projections on
the receive HG modes are not necessarily orthogonal to each other when
misalignment exists. When unmodulated laser beams with inherent slow phase
shifts are loaded into different transmit modes, the power collected by a
specific receive mode will exhibit fluctuations. These fluctuations are due to
the interference effect, which is influenced by the phase variations of
different transmit modes. To eliminate this fluctuation in beaconless FSO
systems, the phase of the laser source should be equally distributed in the
time domain. This can be naturally achieved in coherent systems with phase
modulation or quadrature amplitude modulation (QAM), or in direct detection
systems with a slight modification to the intensity modulators to introduce a
random phase shift of $\pm\pi/2$.

####  2.2.2 The requirement of receive modes

As shown in Fig. 3(a), the width of the “flat” top will increase when the
number of modes increases, leading to a worse alignment accuracy when noise
and practical imperfection exist. Therefore, a “sharp” metric is required when
the system is nearly aligned. Exploiting the intrinsic orthogonality in HG
modes, no power leakage to unwanted modes will be observed when the system is
properly aligned (refer to the pink area in Fig. 1(a) for details). However, a
small alignment error will introduce significant power leakage, especially to
adjacent HG modes (e.g. from $HG_{l,m-1}$ to $HG_{l,m}$ in Fig. 1(a)). As
shown in Fig. 1(c), when the system is properly aligned, a sharp minimum can
be observed in

${P_{Rx}}=\sum\nolimits_{{M_{r}}\notin{\mathcal{X}_{T}}}{P\left({{M_{r}}}\right)}.$
(4)

However, $P_{Rx}$ will also reduce when the beam goes beyond the aperture,
preventing Eq. (4) from being a stand-alone objective function. Fortunately, a
higher mode order leads to a larger effective area. Therefore, dividing Eq.
(3) by Eq. (4) will remove the problem of multiple local extremums when the
misalignment goes to infinity. On the other hand, several local extremums are
observed in Fig. 1(c) when the misalignment is a finite non-zero number. To
eliminate these local extremums in the objective function, the monotonicity of
Eq. (3) is exploited and a shaping coefficient $C\leq 1$ is introduced in Eq.
(5).

Given the x-axis misalignment insensitivity of the unwanted mode $HG_{0,m+1}$
and y-axis misalignment insensitivity of the unwanted mode $HG_{l+1,0}$, it is
suggested to incorporate at least two additional receive modes (e.g.
$HG_{0,m+1}$ and $HG_{l+1,0}$) to improve alignment accuracy and obtain
directional sensitivity, improving the acquisition speed in a quadrant-
detector-free system. Moreover, including extra receive modes (e.g.
$HG_{0,m+2}$ and $HG_{l+2,0}$) can further suppress practical errors and
enhance the performance of the objective function. Importantly, it should be
noted that in a quadrant-detector-free system, the directional sensitivity is
unattainable in a Gaussian-beacon-based alignment system.

####  2.2.3 The objective function

As shown in Fig. 1(d), invoking Eq. (3) and Eq. (4), one suitable objective
function can be written as

$OF=\frac{{{P_{Tx}}}}{{P_{Rx}^{C}}}=\frac{{\sum\nolimits_{{M_{r}}\in{\mathcal{X}_{T}}}{P\left({{M_{r}}}\right)}}}{{{{\left[{\sum\nolimits_{{M_{r}}\notin{\mathcal{X}_{T}}}{P\left({{M_{r}}}\right)}}\right]}^{C}}}},$
(5)

where $C\leq 1$ is a shaping coefficient. Multiple benefits can be obtained by
employing the proposed objective function, including:

  1. 1.

Better alignment accuracy: When the system is nearly aligned, utilising the
power leakage to the unwanted modes (Eq. (4)) yields a ’sharp’ metric (Fig.
1(d)) in the proposed objective function (Eq. (5)), thereby achieving
significantly higher accuracy compared to conventional methods. By excluding
the beacon beam, the practical error between the signal beam and the beacon
beam is also eliminated.

  2. 2.

Faster acquisition: The utilisation of transmitted higher-order modes (Eq.
(3)) in the proposed objective function (Eq. (5)) allows a larger power
profile (Fig. 3(a)). Without sacrificing alignment accuracy, a larger
acquisition step can be employed to reduce the number of iterations for faster
acquisition.

  3. 3.

Simplified system calibration: The system calibration process is significantly
simplified by excluding all the redundant optical structures such as beacon
beam or 4-quadrant detector.

  4. 4.

Better cost and energy efficiency: The proposed objective function can be
achieved by employing simple photodiodes, typically integrated at the input of
receiver erbium-doped fiber amplifiers (EDFAs) for amplification control. By
excluding all the redundant optical structures, the system cost can be
significantly reduced, with a higher energy efficiency by excluding power
absorption on the redundant devices (e.g. wavelength beam splitters).

###  2.3 Experimental setup

Figure 2: Experimental setup for beaconless MDM communication system. L: lens;
M: mirror; EDFA: erbium-doped fiber amplifier; MPLC: multi-plane light
conversion; OSA: optical spectrum analyser; FSO: free-space optics. (a)
Schematical diagram of experimental setup. (b) The structure of the MPLC
device. (c) The normalised transfer matrix when the system is properly
aligned.

Our experimental MDM multiple-input multiple-output (MIMO) FSO communication
system is depicted in Fig. 2. At the transmitter, a 1550.12 nm laser was
modulated by a dual-polarisation (DP) in-phase and quadrature (IQ) modulator
with a nominal 6 dB bandwidth of 45 GHz using a 4-channel 120 GSa/s arbitrary
waveform generator (AWG) with a 3 dB bandwidth of 50 GHz, and a 80 GBaud
signal was generated. The signal was root-raised cosine (RRC) shaped with a
roll-off factor of 0.1. The signal had a frame structure of 20,000 symbols
with 1,920 quadrature phase-shift keying (QPSK) symbols in the training
sequence, and 1 random QPSK pilot symbol for every 9 data symbols, the data
symbols were either 16-QAM symbols generated from a $2^{15}-1$ pseudo random
binary sequence (PRBS) or PS-256-QAM symbols generated from a mt19937ar
Mersenne Twister [17]. After amplified by an EDFA, the signals were passed
through an acousto-optic modulator (AOM) to generate a 20 µs burst signal for
each 160 µs period (Fig. 2(a1)) to enable the time-division multiplexing (TDM)
receiver setup which will be detailed later. To emulate independent
transmitters, 6 copies were split from the burst signal and delayed by
variable fibre delay lines (FDLs) with lengths of 0, 320, 640, 960, 1280, and
1600 symbols. The delayed signals were passed through variable optical
attenuators (VOAs) to compensate for practical power differences before the
VOAs. The compensated signals were then converted to corresponding HG modes
using MPLC technology (Fig. 2(b), the output beam waist diameter of $HG_{0,0}$
mode is approximately 0.3 mm) [18].

In the FSO channel, the transmitted HG modes were passed through a beam
expander consisting of a convex lens with a focal length of 30 mm (L1)
followed by another convex lens with a focal length of 100 mm (L2). The
expanded beam was passed through a steering mirror (M1) with two high-
precision motorised actuators for automatic alignment. The stepper motors of
the actuators support an incremental motion of approximately 1 µm in standard
mode or approximately 0.5 nm using micro-steps in high-precision mode,
providing adequate precision for our experiments. The beam was then passed
through two more mirrors (M2, M3), and another symmetric beam expander
consisting of a convex lens with a focal length of 100 mm (L3) followed by
another convex lens with a focal length of 30 mm (L4). The beam was then
coupled into the receiver MPLC device after a total FSO length of ~1.6 m. As
shown in Fig. 2(c), the normalised transfer matrix can achieve $<$1.7 dB mode-
dependent loss, and $>$25 dB suppression for $6\times 6$ MIMO when the system
is properly aligned using the proposed algorithm.

At the receiver, the optical signals split by the receiver MPLC device were
delayed by the FDLs for the 6 lowest order modes to realise the TDM receiver.
Here, a 5 km difference between adjacent modes introduced a ~24.5 µs delay
(Fig. 2(a2)), slightly longer than the signal burst (Fig. 2(a1)). Each delayed
signal was then split by a 99:1 fibre coupler so that the overall power
received by each mode could be measured by a corresponding power meter. Here,
we also connected two more unused higher-order modes (i.e. $HG_{0,3}$ and
$HG_{3,0}$) to calculate the denominator term in Eq. (5). The 6 lowest order
modes were then amplified by 6 EDFAs, coupled into one single-mode fibre (SMF)
for TDM combining (Fig. 2(a3)), and amplified by another EDFA. Afterwards, the
variable optical noise was loaded by a sequence of devices involving an EDFA,
an optical bandpass filter, another EDFA, a VOA, and a 50:50 coupler. The
average optical signal-to-noise ratio (OSNR) was then accessed using an
optical spectrum analyser (OSA) to facilitate OSNR scanning. Finally, the TDM
signals were received by a coherent receiver with a 70 GHz, 200 GSa/s
oscilloscope and demodulated by offline digital signal processing (DSP).

###  2.4 Experimental results on alignments

Figure 3: Summation of received power versus misalignment. (a) Received power
profile versus x-axis misalignment. (b) Traces of alignments from different
initial points. (c) Mode-wise received power versus alignment steps for the
y-axis and diagonal misalignments, respectively. (d) Simulational objective
function versus misalignment for $1\times 3$, $3\times 5$, and $6\times 8$
systems, respectively. (e) Experimental objective function versus misalignment
for $1\times 3$, $3\times 5$, and $6\times 8$ systems, respectively.

We transmitted the 6 lowest order modes and employed $HG_{0,3}$ and $HG_{0,3}$
as unwanted receive modes for better alignment accuracy. Equipped with the
proposed objective function, we’re ready to align the system by employing the
greedy algorithm [19]. Our strategy was to sequentially move towards nine
neighbouring points including $\left({-\Delta x,-\Delta x}\right)$,
$\left({-\Delta x,0}\right)$, $\left({-\Delta x,\Delta x}\right)$,
$\left({0,-\Delta x}\right)$, $\left({0,0}\right)$, $\left({0,\Delta
x}\right)$, $\left({\Delta x,-\Delta x}\right)$, $\left({\Delta x,0}\right)$,
and $\left({\Delta x,\Delta x}\right)$, where $\Delta x$ is the step size. We
recorded the objective function for all nine points and specifically targeted
the point with the maximum value of the objective function. Repeating the
greedy algorithm ensures precise alignment by focusing on the most favourable
point within the immediate vicinity. The alignment traces of a $6\times 8$ MDM
system from different starting points are depicted in Fig. 3(b). A beam
profile including 6 transmitted modes is also depicted in Fig. 3(b) as a
reference. Here the HG modes are rotated by $45^{\circ}$ for symmetric
considerations when designing the MPLC device [18]. Despite the choice of
different initialisation points, the system could always converge to the
centre of the beam profile. Moreover, the alignment traces tend to follow
diagonal directions, which aligns with the “diagonal ridge” effect in the
objective function as shown in Fig. 3(d3) and Fig. 3(e3).

The mode-wise received power of two typical alignment curves (along the y-axis
and diagonal direction, respectively) is depicted in Fig. 3(c1)-(c2). As
anticipated, the power of transmitted modes reached its maximum, while the
power of unwanted modes was minimised when the system was properly aligned.
After step 15, the unwanted modes made substantial contributions when the
power of the transmitted mode reached a near-consistent level, indicating the
importance of the “sharp” metric in the proposed objective function for
precise alignment. Furthermore, the directional sensitivity between $HG_{l,0}$
and $HG_{0,m}$ is evident in Fig. 3(c2) when the path follows the same
direction as $HG_{l,0}$ modes.

By deliberately misaligning the system away from its aligned central point, we
can compare the experimental objective function against the simulation
results. As shown in Fig. 3(d1)-(d3), the objective function works for
$1\times 3$, $3\times 5$, and $6\times 8$ MIMO systems when the lowest order
modes (i.e. smallest $l+m$) were employed. The experimental results in Fig.
3(e1)-(e3) align well with the corresponding simulation predictions. Moreover,
the symmetry in the experimental results also indicated the high precision of
the proposed objective function.

###  2.5 Achieved BER and data throughput

Figure 4: Experimental performance of communication system. EVM: error vector
magnitude; BER: bit error rate; OSNR: optical signal-to-noise ratio; HD-FEC:
hard-decision forward error correction; GMI: generalized mutual information.
(a) Constellation diagram. (b) BER versus average OSNR. (c) BER versus
misalignment. (d) GMI versus misalignment.

Fig. 4 shows the experimental performance of the proposed communication
system, including the constellation diagram and the BER for 16-QAM signals,
and the achievable data rate estimated from GMI for 16-QAM and PS-256-QAM
signals, respectively. Fig. 4(a) depicts the channel-wise constellation
diagrams for 16-QAM signals. Benefit from the proposed alignment algorithm,
the constellations perform similarly in all channels, while some differences
in error vector magnitude (EVM) are observed due to the mode-dependent loss
introduced by the MPLC devices. Fig. 4(b) shows the mode-wise BER versus OSNR
for 16-QAM signals. Here we observe an implementation penalty of approximately
2.9 dB from the theoretical limit at the HD-FEC limit of $4.7\times 10^{-3}$
[20]. This is mainly due to practical implementation loss including shot noise
and quantisation noise introduced by the digital coherent receiver. As
illustrated by the pink area, the channel-wise BER shows slight variations.
Consistent with the EVM difference shown in Fig. 4(a), these variations are
primarily attributed to the mode-dependent loss in the MPLC devices.

Fig. 4(c) depicts the BER performance versus the misalignment when the OSNR
loading was disabled. When the system was properly aligned, we obtained a BER
of $4.52\times 10^{-4}$, an order of magnitude smaller than the HD-FEC limit.
Moreover, an approximately 34% BER degradation was observed when the beam was
deliberately misaligned by 30 µm, which was 1/10 of the $HG_{0,0}$ beam
diameter. Furthermore, the degradation significantly increased as the
misalignment increased. This phenomenon indicated the importance of accurate
alignment provided by our proposed algorithm. Fig. 4(d) depicts the GMI
performance versus misalignment when the OSNR loading was disabled. The
maximum GMI of 16-QAM signals are 47.88 bit/symbol, slightly smaller than the
theoretical limit of 6 (modes)$\times$2 (polarisations)$\times$4 bit/symbol.
By loading the PS-256-QAM signals, we obtained a maximum GMI of 64.33
bit/symbol, equivalent to an achievable line rate of 5.146 Tbit/s when
transmitting 80 GBaud signals.

##  3 Methods

###  3.1 DSP details

A nonlinear Volterra predistorter was employed to compensate for the
nonlinearities and imperfections in the 80 GBaud system [21, 22]. To perform
the system identification, an 80 GBaud DP-16QAM probe signal with a roll-off
factor of 0.1 was employed and a nonlinear filter with kernel memory lengths
of (201,9,9) taps was created.

The offline DSP includes a sequential combination of the Gram-Schmidt
orthogonalisation [23], polarisation demultiplexing in Stokes space [24],
frequency-domain frequency offset estimation, frequency-domain chromatic
dispersion (CD) compensation for the receiver FDLs [25], Godard timing
recovery [26], phase-asynchronous phase and channel estimation [27], and
3-dimensional MIMO equalisation [28].

The shaping factor of PS-256-QAM is set to 2 to obtain $<$0.1 dB signal-to-
noise ratio (SNR) penalty when estimating data rate from GMI for up to 6
bit/symbol for the QAM constellations [29].

###  3.2 Measurement of transfer matrix

The normalised transfer matrix given in Fig. 2(c) was measured by sequentially
connecting each mode of the transmitter MPLC directly to the transmitted
signal which was amplified to 10 dBm by an EDFA. The power attenuation
introduced by receiver FDLs were compensated to reflect proper power readings
directly after the receiver MPLC. The power measurement of $\leq$-35 dBm in
Fig. 2(c) was due to the limited sensitivity of the power meters.

##  4 Discussion

We proposed a novel objective function for beaconless and quadrant-detector-
free MDM FSO auto-alignment. This method significantly improved the alignment
accuracy and acquisition speed. It also eliminated redundant hardware,
reducing system cost, improving energy efficiency and facilitating system
calibration. Moreover, this method supports a complete modal basis of Hermite-
Gaussian modes. With identical beam divergence, this will enable the
transmission of a significantly larger number of spatial channels compared to
the incomplete OAM mode basis.

Our MDM FSO communication experiments demonstrated that the proposed alignment
algorithm enables an inter-mode crosstalk suppression of $>$25 dB in our
$6\times 6$ MDM FSO communication system. By employing Volterra-based system
identification, we achieved a record-high 80 GBaud in MDM transmissions,
supporting up to a record-high 3.84 Tbit/s and 5.14 Tbit/s line rate when
employing 16-QAM and PS-256-QAM signals, respectively.

Our proposed algorithm has shown superb alignment accuracy, inter-mode
crosstalk suppression, and hardware efficiency. By exploiting spatial
diversity, our experimental demonstration underpins the potential of ultra-
high data rate MDM transmission. We anticipate this paper to be a starting
point for more sophisticated alignment scenarios in future MDM FSO
communications for long-distance applications with generalised mode basis such
as LG modes, Ince-Gaussian modes, Bessel modes, etc.

Acknowledgements We acknowledge the support by EPSRC under Grants
EP/T009047/1, EP/S003436/1, and EP/S016171/1, and by the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie Grant 713694.

Author contributions Y.L. and G.F. designed and carried out the experiment and
analysed the data. A.B. designed and assembled the multi-plane light
conversion devices. Y.L. and D.B. designed and calibrated the optical system.
Y.L. designed and calibrated the DSP. M.P. and A.E. provided the technical
support. Y.L. drafted the manuscript with support from all co-authors. A.E.
supervised the project.

Competing interests The authors declare no competing interests.

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# Search for Super Earths by Timing of Transits with CoRoT

Juan Cabrera  Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstrasse 2,
12489 Berlin, Germany
LUTH, Observatoire de Paris, CNRS, Université Paris Diderot ; 5 Place Jules
Janssen, 92190 Meudon,France

###### Abstract

We explore the possibility of detecting Super Earths via transit timing
variations with the satellite CoRoT.

##  1 Introduction

The satellite CoRoT (Baglin et al., 2006) was launched on 27th December 2006
with a double scientific purpose: the analysis of stellar seismology and the
detection of extrasolar planets by the method of transits. So far, 7 transits
have been published (see Jean Schneider’s Extrasolar Planets
Encyclopaedia111http://exoplanet.eu), but more will come in the near future.
Table 1 gathers the data from these six planets and one brown dwarf. CoRoT’s
photometric precision is below $8\cdot 10^{-4}$ in 2h at $R=15$ (Aigrain et
al., 2009) and candidates are found with transit depths of $0.034\%$ (Leger et
al., 2009). Neptune size planets seem to be common (Gould et al., 2006) and,
there is something even more interesting: they don’t come alone (Mayor et al.,
2009). Super Earths, which should also be numerous, are in the range of
detectability of CoRoT(Leger et al., 2009; Queloz et al., 2009). In the near
future the number of candidates will increase and we will find ourselves with
a collection of planets whose diversity we can only start to imagine.

Section 2 is a short introduction to the CoRoT mission. Section 3 gives a
short overview of different sources of transit timing variations which could
allow the detection of Super Earths with the satellite CoRoT.

##  2 CoRoT

CoRoT is an afocal telescope with a 27 cm diameter pupil, equipped with 4 CCDs
($2048\times 2048$ pixels each); the pixel scale is $2.32^{\prime\prime}$ and
the field of view is $3.05^{\circ}\times 2.8^{\circ}.$ The selection of
observational targets follows two different strategies: the seismology channel
observes a small number (10) of bright stars ($6<m_{v}<9$) with a cadence of
32s whereas the exoplanet channel observes a large number ($\sim 11\,000$) of
faint stars ($12<m_{v}<16$) every 512s (although a limited number of targets
is measured every 32s). CoRoT is placed in a polar low Earth orbit which
determines the observational scheme. The satellite continuously monitors the
same region of the sky during 150 days; but then it has to turn around to
avoid the Sun entering the field of view: these are the _long runs_.
Immediately before or after the turnaround, the satellite is pointed during
roughly 20 days to perform a _short run_ in a different direction. Every year,
CoRoT observes 2 long runs and 2 short runs providing roughly $40\,000$ light
curves. These light curves are narrowed down in the search for transits and a
list of candidates is built. The most promising candidates are followed up
photometrically and spectroscopically from the ground.

The photometric follow-up attempts to discover if the transit is on target or,
on the contrary, if it is produced by a background binary. The PSF of CoRoT is
quite large: the flux for each star is calculated on-board in masks of size 60
pixels on average. In front of the CCD there is a prism used to produce
chromatic light curves with the aim of distinguishing between stellar
(coloured) activity and (achromatic) transits. However, for faint stars it is
not possible to make this distinction. Large masks raise the probability of
observing background binaries, which are a major source of confusion (Pont et
al., 2005).

Spectroscopic follow-up measures the mass of the transiting object. There is a
degeneracy between the mass and the radius of low mass stellar objects, brown
dwarfs and planets (see Fig. 1); to confirm the nature of a transiting object
it is mandatory to perform radial velocity measurements and calculate the
object’s mass. This can become a bottleneck for the characterization because
the measurement of faint candidates is challenging.

Deleuil and Baglin (2008) is a very interesting short summary of CoRoT and its
achievements. Fully detailed recent information about the technical
characteristics of the mission can be found in Fridlund et al. (2006); Barge
et al. (2008b) and Drummond et al. (2008).

Table 1: 5 transits found by CoRoT. | 1b222Barge et al. (2008a). | 2b333Alonso et al. (2008); Bouchy et al. (2008); Alonso et al. (2009). | 4b444Aigrain et al. (2008); Moutou et al. (2008). | 5b555Rauer et al. (2009). | 6b666Fridlund (2009). | 7b777Leger et al. (2009); Queloz et al. (2009). | 3b888Deleuil et al. (2008).
---|---|---|---|---|---|---|---
radius | 1.49 | 1.47 | 1.19 | 1.39 | 1.15 | 0.15 | 1.01
(Jupiter’s radii) |  |  |  |  |  |  |
mass | 1.03 | 3.31 | 0.72 | 0.47 | 3.3 | 0.015 | 21.66
(Jupiter’s masses) |  |  |  |  |  |  |
period | 1.51 | 1.74 | 9.20 | 4.04 | 8.89 | 0.85 | 4.25
(days) |  |  |  |  |  |  |
Figure 1: Figure from Deleuil et al. (2008) showing the mass–radius diagram
for planets and low mass M stars. CoRoT-3b is highlighted. The theoretical
isochrones at 10 and 1 Gyr are from Baraffe et al. (2003).

##  3 Transit Timing Variations

Kepler’s laws of motion assign periodical orbits to planets. However, there
are numerous sources of perturbations which produce deviations from the
periodicity. Not only are there differences in the observed minus calculated
(O$-$C) epochs of transits, but also in their durations and depths. Some
possible sources are general relativity effects, the quadrupolar moment of the
gravitational potential of the star, tidal interaction or even the proper
motion of the star (Miralda-Escudé, 2002; Jordán and Bakos, 2008; Pál and
Kocsis, 2008; Rafikov, 2009). But all these perturbations act on timescales
much longer than the baseline of CoRoT observations and so, in spite of their
interest, are beyond the scope of this study.

However, there are still several other sources of perturbations acting on
shorter timescales, such us those produced by other planets (Schneider, 2004;
Holman and Murray, 2005; Agol et al., 2005; Nesvorný and Morbidelli, 2008),
Trojan planets (Laughlin and Chambers, 2002; Dvorak et al., 2004; Ford and
Holman, 2007), moons (Doyle and Deeg, 2004; Kipping, 2009a, b), orbital
eccentricity (Kipping, 2008) and the light time effect, the so-called LITE
(widely studied in binary systems, see Irwin 1952; Mayer 1990; Borkovits et
al. 2003). In addition to these, Winn (2009) contains a very interesting list
of the information that can be obtained from transits.

###  3.1 Photometric Precision

From Doyle and Deeg (2004), we can calculate the maximal accuracy of $\delta
t_{0}$ that one can achieve when determining the position of a transit of
length $T_{tr}$ and depth $\Delta L$; the photometric accuracy is $\delta L$
and the number of observations is $N$. This accuracy is:

$\delta t_{0}=\delta_{L}\frac{T_{tr}}{2\Delta L\sqrt{N}}.$

In CoRoT, with a photometric accuracy of $0.1\%$, measuring a transit of depth
$1\%$ at the observing cadence of 32s, we can achieve a timing accuracy on the
order of seconds.

###  3.2 Multiple Systems

On the day this manuscript was submitted, there were 374 extrasolar planets
known, among which most are isolated. But there are already 40 known multiple
planet systems and in the future, as Dr. Udry pointed out in this conference,
probably more and more planets will be found in multiple systems. See also the
work by Dr. Wright in this volume.

We can calculate the perturbations in the time of arrival of the transit
signal of a planet if there is another planet in the system in an interior
orbit. Rigorous calculations are done in Agol et al. (2005), but we can
estimate $\delta t$, the amplitude of this perturbation, with the expression:

$\delta
t={\displaystyle\frac{P_{e}}{2\pi}}{\displaystyle\frac{m_{i}}{m_{*}+m_{i}}}{\displaystyle\frac{a_{i}}{a_{e}}};$
(1)

where $P$ stands for period, $m$ for masses, $a$ for the semi-major axis of
the orbits and the subscripts $i$ and $e$ refer to the inner and outer
(exterior) planet respectively. For a Jupiter outer planet with a period of 20
days around a star of one solar mass, an interior Super Earth of 11
terrestrial masses would produce a perturbation of 3 seconds, which is within
the limits of CoRoT.

Dynamics in multiple planets systems is a complicated matter (for example see,
in this volume, the work by Dr. Michtchenko) and resonances are one of the
most important features because they enhance the amplitude of these
perturbations and could open the door to the discovery of low mass planets
(Holman and Murray, 2005; Haghighipour et al., 2007).

In 2008 alone, at least 8 publications have seen the light on the detection of
this kind of perturbation: Agol et al. (2009); Alonso et al. (2009); Díaz et
al. (2008); Hrudková et al. (2009); Irwin et al. (2008); Miller-Ricci et al.
(2008a, b); Shporer et al. (2009).

###  3.3 Trojan planets

In our Solar System, Trojan satellites are a group of asteroids moving close
to the Lagrange points L4 and L5 in 1:1 mean-motion resonance with Jupiter’s
orbit. Many efforts have been done in the search for these kind of objects in
extrasolar systems (see for example Moldovan and Matthews 2008 and Madhusudhan
and Winn 2009). Bodies in these orbits are stable (Ford and Gaudi, 2006;
Dvorak et al., 2004) and can be found not only photometrically or by radial
velocity, but also by the timing variations that they produce in the transits
of the planet whose orbit they share. We can estimate the amplitude of this
perturbation:

$\delta t=\frac{M_{Trojan}}{M_{planet}}\frac{{\alpha}}{2\pi}P_{planet};$ (2)

for a Trojan object with the mass of the Moon and a transiting planet with the
mass of Jupiter in a 20 day orbit, $\alpha$ being the typical angle involved
in the calculation, with $\alpha\sim 30$ degrees (see the references given
above for justification), the amplitude of the perturbation is about 5
seconds. Needless to say, if the transiting planet is a Super Earth, this
perturbation is far more important. Another speculative hypothesis is the
existence of massive Trojan planets. If the ratio between the mass of the
transiting planet plus the mass of the Trojan over the mass of the star is
below $\sim 1/27,$ the system can be stable; this opens the possibility of
Trojan Super Earths (see Nauenberg 2002; Schwarz et al. 2007 and references
therein).

###  3.4 LITE

LITE was first used by the astronomer Ole Römer, working in Paris Observatory
with Jean-Dominique Cassini, to measure the speed of light (Römer, 1676).
Nowadays it is used to find hidden companions to binary systems, even those of
planetary mass (Deeg et al., 2008; Lee et al., 2008). But we can find the same
effect in multiple planet systems (Schneider, 2005). The reflex motion induced
in our Sun by Jupiter has an amplitude of one solar radius, which light covers
in 2 seconds. If we observe the transits of an inner planet and there is an
outer planet in the system, the amplitude of the LITE perturbation is:

$\delta t=2\frac{m_{e}}{M_{*}}\frac{a_{e}}{c};$ (3)

which is of the order of 0.1 seconds for the time baselines of CoRoT (and in
this case, we must concede that this favors the detection of high mass
companions and not of Super Earths).

###  3.5 Moons

No moon has yet been detected around any extrasolar planet, although their
existence is expected (Sartoretti and Schneider, 1999). However, their
detection is difficult (Brown et al., 2001). In general, it is not an easy
task to estimate the magnitude of the perturbation because it depends on the
orbit of the satellite around the planet; and for planets within the specific
period range detectable by CoRoT, we don’t yet have any clue as to how much
this estimate may be. Nevertheless, reasonable assumptions in the general case
give perturbations under 1 s, which is below CoRoT limits.

However, we point out the possibility of finding binary planets (Cabrera and
Schneider, 2007). Binarity is common among stars, from bright massive objects
down to brown dwarfs; and we can also find binary objects from the size of
trans-neptunian objects down to asteroids. Binarity should be possible among
planets and those systems will produce peculiar transit signals.

##  4 Conclusions

The photometric precision achieved by CoRoT allows the detection of Super
Earth planets in transit; but here we have shown that also non-transiting
Super Earths could be detected in multiple systems by the perturbations they
might produce in transiting planets. We have shown several possible scenarios
and discussed their limitations.

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