# Fully-digital platform for local ultra-stable optical frequency distribution

Martina Matusko  Université de Franche-Comté, SUPMICROTECH, CNRS, institut
FEMTO-ST, F-25000 Besançon, France  Ivan Ryger  Université de Franche-Comté,
SUPMICROTECH, CNRS, institut FEMTO-ST, F-25000 Besançon, France  Gwenhaël
Goavec-Merou  Université de Franche-Comté, SUPMICROTECH, CNRS, institut FEMTO-
ST, F-25000 Besançon, France  Jacques Millo  Université de Franche-Comté,
SUPMICROTECH, CNRS, institut FEMTO-ST, F-25000 Besançon, France  Clément
Lacroûte  Université de Franche-Comté, SUPMICROTECH, CNRS, institut FEMTO-ST,
F-25000 Besançon, France  Émile Carry  Université de Franche-Comté,
SUPMICROTECH, CNRS, institut FEMTO-ST, F-25000 Besançon, France  Jean-Michel
Friedt  Université de Franche-Comté, SUPMICROTECH, CNRS, institut FEMTO-ST,
F-25000 Besançon, France  Marion Delehaye  Université de Franche-Comté,
SUPMICROTECH, CNRS, institut FEMTO-ST, F-25000 Besançon, France
<EMAIL_ADDRESS>
###### Abstract

This article reports on the use of an FPGA platform for local ultra-stable
optical frequency distribution through a 90 m-long fiber network. This
platform is used to implement a fully digital treatment of the Doppler-
cancellation scheme required by fiber links to be able to distribute ultra-
stable frequencies. We present a novel protocol that uses aliased images of a
digital synthesizer output to directly generate signals above the Nyquist
frequency. This approach significantly simplifies the set-up, making it easy
to duplicate within a local fiber network. We demonstrate performances
enabling the distribution of an optical signal with an instability below
$10^{-17}$ at one second at the receiver end. We also use the board to
implement an original characterization method. It leads to an efficient
characterization of the disturbance rejection of the system, that can be
realized without accessing the remote output of the fiber link.

††preprint: AIP/123-QED

##  I Introduction

Optical atomic oscillators can reach the $10^{-19}$ level of instability after
$10^{5}$ s averaging time McGrew  _et al._ (2018). Together with optical
clocks uncertainties that can be below the $10^{-18}$ level Brewer  _et al._
(2019), this paves the way to numerous applications, ranging from new GNSS
architecture Schuldt  _et al._ (2021) or improved geodesy Ohmae  _et al._
(2021) to advances in fundamental science, such as the measurement of the
relativistic redshift Pavlis and Weiss (2003); Chou  _et al._ (2010), or
searches for dark matter Safronova  _et al._ (2018).

Fibers links have proven to be the most robust Narbonneau  _et al._ (2006);
Cantin  _et al._ (2021) and efficient way to transfer ultra-stable signals
over distances up to thousands of kilometers Ma  _et al._ (1994); Predehl  _et
al._ (2012); Newbury, Williams, and Swann (2007); Grosche  _et al._ (2009);
Fujieda  _et al._ (2011); Lopez  _et al._ (2012); Calonico  _et al._ (2014);
Lisdat  _et al._ (2015); Husmann  _et al._ (2021). Long-distance fiber
networks now connect many European laboratories using dedicated
infrastructures Lisdat  _et al._ (2015); Roberts  _et al._ (2020). Here, we
focus on stable frequency distribution within an institute, where several
experiments distant by typically tens to hundred meters may require access to
ultra-stable optical signals. Fractional frequency instabilities that can be
reached in research facilities range from $10^{-15}$ to $10^{-17}$ at one
second ref ; Oelker  _et al._ (2019). This sets the targeted frequency
instability of the local distribution fiber network discussed in this paper.
Fibers can guide an optical signal over distances with limited power loss, but
the purity of the signals can be degraded while propagating because of thermal
and mechanical perturbations affecting the optical length of the fiber. Phase
noise induced in fiber links is usually tracked and compensated for by Doppler
cancellation techniques Williams, Swann, and Newbury (2008); Calosso  _et al._
(2015) in a phase-locked loop (PLL). These traditionally relied on analog
electronics for tracking and correcting the induced phase noise. Powerful
alternatives to analog devices are digital electronics. They are very suited
to ultra-stable frequency dissemination within an institute, as they are less
sensitive to environmental perturbations and possess the advantages of
versatility, reconfigurability, and easy duplication Ning  _et al._ (2013);
Baldwin  _et al._ (2012); Cárdenas Olaya  _et al._ (2016); Tourigny-Plante
_et al._ (2018).

In this paper, we demonstrate a fully-digital and open-source Doppler
cancellation platform based on a Red Pitaya $\mathrm{SDR^{lab}}122-16$ (RP16)
board for a 90 m fiber link at 1542 nm (194 THz) for frequency dissemination
within the FEMTO-ST institute. Longer links can also be compensated using this
setup, at the price of a reduced lock bandwidth. We make use of the
capabilities of the open-source FPGA platform _oscimpDigital_ , available on
github under the CeCILL licence osc (2022a). First, we assess the performances
of the newly available RP16 and compare them with the previous Red Pitaya
$\mathrm{STEM^{lab}125-14}$ (RP14). We then explore the effects of a fully-
digital signal treatment using the analog-to-digital (ADC) and digital-to-
analog (DAC) converters of RP16. In particular, we make a novel use of
aliasing at the output of a direct digital synthesizer (DDS) and characterize
the delays due to different blocks of the set-up and their contribution to the
PLL bandwidth. Lastly, we present an automatic method to measure the
disturbance rejection of the PLL in order to reach the best noise-cancellation
performances.

##  II Digital platforms

Ultra-stable frequency dissemination through fiber links has already been
performed using an RP14 board and showed promising results Mukherjee  _et al._
(2022); Cárdenas Olaya  _et al._ (2016); Cárdenas-Olaya  _et al._ (2019). An
RP16 board recently became available with a more powerful FPGA to embed more
logical functions and increase processing capabilities. It is explicitly
designed for radio frequency instrumentation, with an analog front-end design
targeted towards using higher Nyquist zones and a readily accessible external
reference input port. A schematic view of the constitutive blocks of RP16 is
shown Fig. 1. Extra key characteristics of RP16 and RP14 are indicated in
Table 1, as well as critical resource use for the schemes used in this
article.

Figure 1: Schematic description of the relevant blocks of RP16.  | RP14 | RP16
---|---|---
ADC | 14 bits | 16 bits
| ADC effective
---
number of bits
11.8 | 12.5
| Measurement
---
bandwidth
52 MHz111The measurement bandwidth can be increased to reach the ADC track-and-hold bandwidth of 750 MHz by removing the anti-aliasing filter before the ADC. | 550 MHz
DAC | 14 bits | 14 bits
FPGA | Xilinx Zynq 7010 | Xilinx Zynq 7020
Resource use |  |
Dual PLL Mukherjee  _et al._ (2022) | 75 % of DSP222Digital Signal Processing | 27 % of DSP
| PLL with
---
characterization
(see Fig. 6)
94 % of BRAM333Blocks of Random Access Memory | 44 % of BRAM
| Quartz crystal
---
(TCXO)
FXO HC735-125 MHZ | ABLNO-122.88MHz
Clock frequency | 125 MHz | 122.88 MHz
Table 1: Key characteristics of RP14 and RP16 boards

To assess these boards for metrology purposes, we first measured their output
phase noise at 10 MHz with either their internal (temperature-compensated
crystal oscillator – TCXO) or an external (AD9915 DDS clocked by an active
hydrogen maser) clock reference. The results are shown in Fig. 2. In agreement
with the specified "low-noise" characteristics of RP16 TCXO, its phase noise
shows a 15 dB improvement close to 1 Hz with respect to RP14. However, below
0.1 Hz, the phase noise of $-10~{}\mathrm{dBrad^{2}/Hz}$ is similar for the
two boards. Such performances of the boards in a Doppler-cancellation scheme
for fiber links limit the relative instability of the frequency transfer to
the $10^{-15}$ range Rubiola and Vernotte (2022), similar to the degradation
of the signal stability we measured when propagating through 90 m-long optical
fibers in a noisy environment. Clocking the boards with a stable external
reference is therefore mandatory for most metrological applications. It should
be noted that both the RP14 and the RP16 present spurious signals at 80 and
170 kHz with an amplitude of about $-91~{}\mathrm{dBrad^{2}/Hz}$ (see squared
area Fig. 2). These are generated by the DAC voltage regulators and cannot be
removed easily444A low-noise version of Red Pitaya STEMlab 124-14 very
recently became available, that claims improved linear voltage regulators..
When the boards are used within a PLL, they may induce oscillations at these
frequencies.

Figure 2: Phase-noise of a 10 MHz signal generated by Red Pitaya boards when
clocked by their internal TCXO oscillator or by an external reference derived
from an active hydrogen maser. Green: RP16 with external reference. Red: RP 16
with internal TCXO. Purple: RP14 with external reference. Brown: RP14 with
internal TCXO. Yellow: hydrogen maser. The squared area indicates the spurious
signals coming from the DAC voltage regulators.

Regarding the physical board, the RP14 ADC has a low-pass, anti-aliasing
filter that is not present on the RP16. This makes possible for the RP16
direct undersampled detection of signals at a frequency higher than
$f_{\mathrm{clock}}/2$ (where $f_{\mathrm{clock}}$ is the clock frequency, see
Table 1): an incoming signal at $f_{\mathrm{in}}>f_{\mathrm{clock}}/2$ is
detected at a frequency $f^{\prime}_{\mathrm{in}}=|f_{\mathrm{in}}-n\times
f_{\mathrm{clock}}|$, where $n$ is an integer such that
$0<f^{\prime}_{\mathrm{in}}<f_{\mathrm{clock}}/2$.

Undersampling has already been used in the context of fiber links Mukherjee
_et al._ (2022) with an RP14 that had to undergo ADC input circuit
modification, while the use of an RP16 leads to a much simpler implementation.

##  III Fully-digital cancellation of fiber links noise

We performed Doppler cancellation of the phase noise arising from a 90 m-long
fiber due to thermal and mechanical fluctuations. We first measured the phase
noise of the system with and without the 90 m-long fiber and found that the
presence of the fiber leads to a noise increase in the
$1~{}\mathrm{Hz}-10~{}\mathrm{kHz}$ range by up to 40 dB (at 1 Hz). Our
Doppler-cancellation set-up is based on a heterodyne Michelson interferometer
as depicted in Fig. 3 and signal treatment is done with RP16. It is fully-
fibered with single-mode non polarization-maintaining fibers. One arm of the
interferometer is a short fiber ended by a Faraday mirror (FM1) and plays the
role of a reference arm. The second arm is composed of a fiber-coupled
acousto-optic modulator (AOM555Gooch & Housego T-M110-0.2C2J-3-F2S) driven at
$f_{\mathrm{AOM}}=110~{}\mathrm{MHz}$ followed by the fiber link and a second
fiber-coupled Faraday mirror (FM2). We measure the noise arising from the
fiber link by generating a beatnote between the signal reflected by FM1 and
the signal reflected by FM2 that is twice frequency shifted by the AOM at
$f_{\mathrm{beatnote}}=2\times f_{\mathrm{AOM}}=220~{}\mathrm{MHz}$. The
resulting beatnote is then detected on a photodiode (PD1) for the aim of the
external phase disturbance correction. For characterization purposes, the 90
m-long fiber does a round-trip, with the output of the link located within a
few centimeters from its input. Part of the link output interferes with the
signal coming directly from the master laser, and the resulting beatnote at
110 MHz is detected by a second photodiode (PD2) and carries the uncompensated
noise (see Fig. 3).

Figure 3: Doppler-cancellation set-up. OC: optical coupler; FM: Faraday
mirror; PD: photodiode; FC: frequency counter; input and output processing:
see text. PD1 detects the 220 MHz beatnote resulting from the return trip
through the AOM. PD2 detects a beatnote at 110 MHz that is used to
characterize the system.

The 220 MHz beatnote that is detected and the 110 MHz correction signal that
needs to be generated are outside the ADC and DAC baseband (between
$\mathrm{0}$ and $f_{\mathrm{{clock}}}/2$). For the ADC, one can perform an
analog treatment of the signal: using a 200 MHz signal from a reference, the
220 MHz beatnote can be downconverted to 20 MHz (see Fig. 3.a) so that it is
sampled by the ADC. Alternatively, it has been shown Mukherjee  _et al._
(2022) that the beatnote at $2\times f_{\mathrm{AOM}}$ can be sent directly to
the ADC and undersampled within the second Nyquist band at
$f^{\prime}_{\mathrm{in}}=|f_{\mathrm{beatnote}}-2f_{\mathrm{clock}}|=25.76~{}\mathrm{MHz}$
(see Fig. 3.b).

For the DAC, one can similarly perform an analog treatment of the signal: a 55
MHz signal within the Red Pitaya baseband can be output and frequency doubled
to drive the AOM at 110 MHz (see Fig. 3.a). Here, we propose to instead output
a signal at $f_{\mathrm{out}}=12.88~{}{\mathrm{MHz}}$, and to use the aliased
image of the fundamental at
$f_{\mathrm{clock}}-f_{\mathrm{out}}={\mathrm{110~{}MHz}}=f_{\mathrm{AOM}}$.
As the 3 dB bandwidth of the AOM is 30 MHz and the nearest spectral component
at the output is only $2\times f_{\mathrm{out}}=25.76$ MHz away, we use a
bandpass surface acoustic wave (SAW) filter666Golledge MP01327 to select the
110 MHz spectral component (see Fig. 3.b). The filter was chosen despite its
narrow bandwidth of 200 kHz for the very low thermal sensitivity of its group
delay (measured to be below $\mathrm{10^{-7}~{}K^{-1}}$). It is not expected
to impact the measurement in a laboratory environment, contrary to other
larger-bandwidth existing SAW filters. The phase-noise of the 110 MHz signals
obtained either by frequency doubling a 55 MHz signal or by selecting the
aliased image of a 12.88 MHz signal are indicated in Fig. 4. Due to the sinc
envelope of the DDS output, the amplitude of the signal at 110 MHz is a factor
6 smaller than at 55 MHz, but we see no degradation of the phase-noise by
using the aliased output configuration. The lower sensitivity to environmental
perturbations of digital electronics leads to an overall small improvement
($\lesssim 2$) of the link fractional frequency stability for integration
times below 2000 s. The lower phase-noise obtained for the aliased signal
above $200~{}\mathrm{kHz}$ results from the narrow bandwidth of the SAW
filter. We observe at $110~{}\mathrm{MHz}$ a larger influence of the
$80~{}\mathrm{kHz}$ and $170~{}\mathrm{kHz}$ spurious signals already observed
at $10~{}\mathrm{MHz}$. Noise peaks in the 5-100 Hz range are caused by high
current drawn by the RP16, inducing voltage noise on its power supply.

Figure 4: Phase-noise of the 110 MHz signal: (blue) obtained by selecting the
aliased image of a 12.88 MHz signal or (yellow) obtained by frequency doubling
a 55 MHz signal.

A summary of the frequency shifts we perform by aliasing at ADC and DAC is
shown in Fig. 5.

Figure 5: Digital ADC and DAC signal processing. ADC signal treatment relies
on undersampling, where the signal with a frequency
$f_{\mathrm{beatnote}}=220$ MHz outside the Nyquist baseband is sampled as a
signal inside the baseband with a frequency
${f^{\prime}_{\mathrm{in}}=2f_{\mathrm{clock}}-f_{\mathrm{beatnote}}}=25.76$
MHz. Regarding the ADC signal treatment, the DDS generates a signal at
$f_{\mathrm{{out}}}=12.88$ MHz that carries the noise correction. It is
aliased at $n{\times}f_{\mathrm{clock}}\pm f_{{out}}$, with $n$ an integer
number. We use a SAW filter to select the
$f_{\mathrm{AOM}}=f_{\mathrm{clock}}-f_{\mathrm{out}}=110$ MHz spectral
component that drives the AOM.

The beatnote detected by the ADC is numerically multiplied by a complex signal
at $f_{\mathrm{{demod}}}$ (with $f_{\mathrm{{demod}}}=20$ MHz for the analog
ADC treatment, and $f_{\mathrm{{demod}}}=25.76$ MHz for the digital ADC
treatment). The phase information is obtained by extracting the real part of
the product and filtering it by a finite impulse response (FIR) filter,
realizing overall a cosine demodulation. It is then provided to a PI
controller, the output of which drives a numerically controlled oscillator
($\mathrm{NCO_{out}}$) at $f_{\mathrm{out}}$, that carries the noise
correction.

The bandwidth of a PLL is ultimately limited by the delays of the system: for
a given delay $\tau$, perturbations with frequencies higher than $1/2\tau$ can
not be corrected. Therefore, we set the maximum bandwidth of the system to
$1/4\tau$ (such that $2\pi\tau f_{\mathrm{pert}}=\pi/2$, where
$f_{\mathrm{pert}}$ is the perturbation frequency; this may still be close to
the instability threshold, so a safer bandwidth of $1/8\tau$ is sometimes used
instead as an upper limit in the literature). The delays and resulting
cumulative bandwidth of our system are indicated in Table 2. It shows that the
ADC and DAC conversions induce an irreducible delay of 125 ns, which sets a
hard limit to 2 MHz for locks operated with Red Pitaya boards. Other digital
blocks (complex and real conversions, multiplications, adjustments of the
number of bits) increase the delay by about $345~{}\mathrm{ns}$, while analog
components play a dominant role, with 1.3 $\upmu$s delay due to the SAW filter
and 2.5 $\upmu$s due to the AOM. The propagation delay itself for a retro-
reflected 90 m-long fiber further reduces the $1/4\tau$ bandwidth by about
17%, down to 49 kHz. To emphasize the influence of each block, the phase shift
it induces at 49 kHz is also indicated in Table 2.

For longer links, the propagation delay would play a more important role: for
instance, a 500 m-long link would induce a delay of 2.5 $\upmu$s, making it
the dominant contribution to the total delay. Longer links are also noisier,
and would benefit from an arctan demodulation to correct large perturbations
inducing phase flips. Arctan demodulation is more demanding in terms of
digital processing and would further increase digital delays to 565 ns.

|  |  | Delay
---
$\tau$
| Cumulative
---
bandwidth
| Induced
---
dephasing
at 49 kHz
Digital electronics |  | ADC and DAC
---
conversions
125 ns | 2 MHz | 0.01 $\pi$
| FIR and other
---
digital blocks
345 ns | 532 kHz | 0.03 $\pi$
Analog electronics | SAW | 1.3 $\upmu$s | 141 kHz | 0.17 $\pi$
AOM | 2.5 $\upmu$s | 59 kHz | 0.21 $\pi$
Fiber | 2 $~{}\times$ 90 m | 880 ns | 49 kHz | 0.07 $\pi$
Table 2: Delays, cumulative bandwidth and induced dephasing at 49 kHz due to
different components of the system. For each component, the cumulative
bandwidth is obtained from $1/4\Sigma\tau$, where the sum is performed on the
delays of the component studied and of the components appearing above in the
table.

##  IV  Automatic characterization

To measure the noise rejection of an installed compensated fiber link, it is
usually necessary that actually two links are set up in a reverse
configuration, such that the output of the second link can be measured against
the input of the first link. We propose here a direct digital noise-rejection
evaluation method that does not require access to the remote end of the link.
It can be used on an existing system or efficiently deployed on multiple
installations within an institute. The idea is to measure the rejection of
sinusoidal perturbations for given proportional-integral (PI) controller
parameters. We use it to experimentally find PI parameters that lead to the
highest bandwidth (corresponding to the $1/4\tau$ bandwidth discussed
previously) without increasing the noise at high frequencies and then divide
the PI gains arbitrarily by a factor of two to avoid loop instabilities.

For this purpose, we induce a frequency modulation at $f_{\mathrm{pert}}$ of
the output signal: a sinusoidal perturbation at $f_{\mathrm{pert}}$ is
numerically added to the correction signal that drives $\mathrm{NCO_{out}}$.
This simulates the effect of a frequency perturbation arising inside the fiber
link. The disturbance rejection by the PLL is numerically measured right after
the PI block. A scheme of the modified Red Pitaya architecture is indicated in
Fig 6 and available at osc (2022b).

Figure 6: Red Pitaya design adapted for disturbance rejection measurements:
$\mathrm{NCO_{pert}}$ adds a perturbation to the correction signal, and the
cancellation of this perturbation is measured by $\mathrm{D2R{corr}}$.
$\mathrm{D2R}$: data to ram acquisition, $\mathrm{NCO}$: numerically
controlled oscillator. $\mathrm{FIR}$: finite impulse response filter.
$\mathrm{C2R}$: complex to real conversion.

$f_{\mathrm{pert}}$ is swept between 3 kHz and 10 MHz for both open loop
configuration (no PI gain) and closed loop configuration, with adjusted PI
gains. The disturbance rejection (ratio of the perturbation amplitudes in the
closed and open loop configurations) measured with a 90 m-long link is shown
in Fig. 7 with dashed lines for four sets of PI gains. It is modelled with
solid lines using optical system transfer functions that were first described
in works of Jiang (2010); Bercy (2015). We chose an equivalent approach in the
discrete time domain and used $z$-transforms formalism that takes into account
analog, digital and optical propagation delays, with the global gain as the
only free parameter. In the case of cosine demodulator the mixer conversion
gain (rad/V) is dependent on the magnitude of the input signal. Thus it
creates a scaling of the total loop gain, which is then dependent on the
photodiode beatnote amplitude. The model does not take into account the
spurious signals observed around 80 and 170 kHz. At low frequencies there is a
good agreement between the model and experimental disturbance rejection
curves. For perturbation frequencies above 200 kHz, the SAW filter acts as a
very efficient low-pass filter, regardless of the PI gains. As it is then the
ratio of two small quantities, the experimental disturbance rejection curve
becomes noisy, but its value consistently averages to 0 dB. For all
experimental curves, we observe a shift of the overshoot towards 80 kHz with
respect to the model. We attribute this to the proximity of the 80 kHz
instability mentioned previously (see Fig. 4). The highest bandwidth that can
be reached with this system is about 40 kHz (see blue curve Fig. 7),
consistent with Table 2. This configuration is close to the instability
threshold (where oscillations induce excess noise above 200 kHz), so we rather
favored the global system stability and reduced these PI gains by a factor
two, leading to a 3 dB bandwidth around 30 kHz (see green curve Fig. 7).

Figure 7: Disturbance rejection of the system with a 90 m-long fiber link.
Dashed lines correspond to the experimental disturbance rejection. Solid lines
correspond to the modeled disturbance rejection. Red curves correspond to the
lowest PI gains; purple curves correspond to PI gains 5 times larger, green
curves to PI gains 10 times larger and blue curves to PI gains 20 times larger
than the ones of the red curve.

We used this method to choose PI gains and then measured the resulting signal
frequency instability of the monitoring 110 MHz signal (see Fig. 3) for both
open and closed loop in three situations: first, with a 1 m-long fiber link,
then with a 90 m-long fiber link, and last, for reference, without the
physical system constituted of the AOM and fibers. Results are indicated in
Fig. 8 by open symbols for the open loop and full symbols for the closed loop.

For the 1 m-long and the 90 m-long links (blue circles in Fig. 8 and red
squares in Fig. 8, respectively), the plateau in stability for the 1 - 40 s
range points towards flicker frequency noise, consistent with the $1/f$ slope
we observe in their power spectrum for frequencies higher than $10^{-2}$ Hz
(not shown).

For the 1 m-long link, the open and closed loop curves overlap for the first
40 s of integration, indicating that for short integration times the stability
of the system is limited by the reduction of the lock bandwidth due to the
optical components (AOM, couplers, Faraday mirrors, photodiodes). For closed
loop configuration, the fractional frequency instability is below $10^{-17}$
for all integration times and reaches $1\times 10^{-18}$ for integration times
longer than 500 s.

For the 90 m-long link, the open loop signal is degraded to the $10^{-16}$
range, illustrating the reduction of the signal stability due to the 90 m
fiber. When the loop is closed, the signal shows the same instability as that
of the 1 m-long fiber link, showing that we reached the limit of the system.
With an instability at most in the $\mathrm{10^{-18}}$ range for all
integration times and that averages down to $6\times 10^{-19}$ for 2000 s
integration time, the set-up fully meets the requirements for frequency
transfer within an institute.

For reference, we removed the physical system containing the AOM and the
optical fibers. The 110 MHz component selected by the SAW filter is split in
two. One part is directly sampled by the ADC of RP16 (with
$\mathrm{NCO_{demod}}$ set to 12.88 MHz). The other part is sent to a
frequency counter to measure the in-loop fractional frequency instability of
the 110 MHz signal. It is then scaled to optical frequencies and shown with
green triangles in Fig. 8. The open and closed loop curves overlap for all
integration times, indicating the limitations due to the RP16 board.

Figure 8: Fractional frequency instabilities of the optical signal at PD2 for
different experimental set-ups. The instability is measured on the 110 MHz
beatnote and scaled to the optical frequency of 194 THz with a leverage factor
of $5.7\times\mathrm{10^{-7}}$. Open symbols: free running system. Full
symbols: closed loop measurement. Green triangles: no physical system (see
text). Blue circles: with 1 m link. Red squares: with 90 m link. Purple line:
Lower limit set by the hydrogen maser, given by the maser fractional frequency
instability multiplied by the leverage factor. Lines are a guide to the eye.
All measurements were taken without any specific shielding of the system.

##  V Conclusion

We compared RP16 and RP14 boards for Doppler cancellation of fiber links using
the fully open-source FPGA framework OscimpDigitalosc (2022a). With its design
explicitly targeting radio frequency instrumentation, RP16 proved more adapted
for metrology applications. Here, we focused on the development of a local
fiber network for ultra-stable signal distribution.

The noise arising from the fiber links is compensated by a digital PLL hosted
on the RP16 board with a novel signal processing based on aliasing at the RP16
DAC and undersampling at the RP16 ADC. This alleviated the need for many
environment-sensitive analog components and resulted in a compact set-up easy
to duplicate. We have measured the delays of the system and estimated the
resulting PLL bandwidth. The delay due to ADC and DAC conversions set an upper
limit of the bandwidth for any Red Pitaya-based system at 2 MHz. Taking all
the delays of the system into account brings the upper limitation of the PLL
bandwidth to 49 kHz. For a 90 m-long link, propagation delay contributes to
about 17% of the total delay, therefore the total bandwidth is significantly
reduced for longer links.

Lastly, we have developed an automatic measurement of the disturbance
rejection for given PI gains of the PLL, therefore providing a rigorous method
to measure the bandwidth. The maximum bandwidth we measured was in agreement
with the calculated one, but the system was close to the instability
threshold. For a 30% lower bandwidth, the system was stable and we measured
fractional frequency instabilities in the $10^{-18}$ range for all timescales.
Performances could in principle be improved by the use of a second AOM at the
remote end of the system to discriminate unwanted reflections Manamanni  _et
al._ (2022), but the current system is already perfectly suitable for local
ultra-stable frequency distribution within a metrology institute.

## Acknowledgments

This work was supported by Agence Nationale de la Recherche
(ANR-21-CE47-0006-01 CONSULA, ANR-10-LABX-48-01 First-TF, and ANR-11-EQPX-0033
Oscillator-IMP), the Région Bourgogne Franche-Comté and the EIPHI Graduate
School (contract ANR-17-EURE-0002). The authors would like to thank Gonzalo
Cabodevila for discussions on PLL characterization, Vincent Giordano for
fruitful discussions and Claudio Calosso for careful reading of the
manuscript.

## References

  * McGrew  _et al._ (2018) W. McGrew, X. Zhang, R. Fasano, S. Schäffer, K. Beloy, D. Nicolodi, R. Brown, N. Hinkley, G. Milani, M. Schioppo, T. Yoon, and A. Ludlow, “Atomic clock performance enabling geodesy below the centimeter level,” Nature 564, 87—90 (2018).
  * Brewer  _et al._ (2019) S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt, “27Al+ quantum-logic clock with a systematic uncertainty below ${10}^{-18}$,” Phys. Rev. Lett. 123 (2019), 10.1103/PhysRevLett.123.033201.
  * Schuldt  _et al._ (2021) T. Schuldt, M. Gohlke, M. Oswald, J. Wüst, T. Blomberg, K. Döringshoff, A. Bawamia, A. Wicht, M. Lezius, K. Voss, M. Krutzik, S. Herrmann, E. Kovalchuk, A. Peters, and C. Braxmaier, “Optical clock technologies for global navigation satellite systems,” GPS Solutions 25 (2021), 10.1007/s10291-021-01113-2.
  * Ohmae  _et al._ (2021) N. Ohmae, M. Takamoto, Y. Takahashi, M. Kokubun, K. Araki, A. Hinton, I. Ushijima, T. Muramatsu, T. Furumiya, Y. Sakai, N. Moriya, N. Kamiya, K. Fujii, R. Muramatsu, T. Shiimado, and H. Katori, “Transportable strontium optical lattice clocks operated outside laboratory at the level of $10^{-18}$ uncertainty,” Advanced Quantum Technologies 4, 2170081 (2021).
  * Pavlis and Weiss (2003) N. K. Pavlis and M. A. Weiss, “The relativistic redshift with $3\times 10^{-17}$ uncertainty at NIST, Boulder, Colorado, USA,” Metrologia 40, 66–73 (2003).
  * Chou  _et al._ (2010) C. W. Chou, D. B. Hume, T. Rosenband, and D. J. Wineland, “Optical clocks and relativity,” Science 329, 1630–1633 (2010).
  * Safronova  _et al._ (2018) M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, “Search for new physics with atoms and molecules,” Rev. Mod. Phys. 90, 025008 (2018).
  * Narbonneau  _et al._ (2006) F. Narbonneau, M. Lours, S. Bize, A. Clairon, G. Santarelli, O. Lopez, C. Daussy, A. Amy-Klein, and C. Chardonnet, “High resolution frequency standard dissemination via optical fiber metropolitan network,” Review of Scientific Instruments 77, 064701 (2006), https://doi.org/10.1063/1.2205155 .
  * Cantin  _et al._ (2021) E. Cantin, M. Tønnes, R. Le Targat, A. Amy-Klein, O. Lopez, and P.-E. Pottie, “An accurate and robust metrological network for coherent optical frequency dissemination,” New Journal of Physics 23, 053027 (2021).
  * Ma  _et al._ (1994) L. Ma, P. Jungner, J. Ye, and J. Hall, “Delivering the same optical frequency at two places: accurate cancellation of phase noise introduced by an optical fiber or other time-varying path,” Optics letters 19, 1777–9 (1994).
  * Predehl  _et al._ (2012) K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, “A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place,” Science 336, 441 – 444 (2012).
  * Newbury, Williams, and Swann (2007) N. R. Newbury, P. A. Williams, and W. C. Swann, “Coherent transfer of an optical carrier over 251 km,” Opt. Lett. 32, 3056–3058 (2007).
  * Grosche  _et al._ (2009) G. Grosche, O. Terra, K. Predehl, R. Holzwarth, B. Lipphardt, F. Vogt, U. Sterr, and H. Schnatz, “Optical frequency transfer via 146 km fiber link with $\mathrm{10^{-19}}$ relative accuracy,” Opt. Lett. 34, 2270–2272 (2009).
  * Fujieda  _et al._ (2011) M. Fujieda, M. Kumagai, S. Nagano, A. Yamaguchi, H. Hachisu, and T. Ido, “All-optical link for direct comparison of distant optical clocks,” Opt. Express 19, 16498–16507 (2011).
  * Lopez  _et al._ (2012) O. Lopez, A. Haboucha, B. Chanteau, C. Chardonnet, A. Amy-Klein, and G. Santarelli, “Ultra-stable long distance optical frequency distribution using the internet fiber network,” Opt. Express 20, 23518–23526 (2012).
  * Calonico  _et al._ (2014) D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, “High-accuracy coherent optical frequency transfer over a doubled 642 km fiber link,” Applied Physics B 117, 979–986 (2014).
  * Lisdat  _et al._ (2015) C. Lisdat, G. Grosche, N. Quintin, C. Shi, S. Raupach, C. Grebing, D. Nicolodi, F. Stefani, A. Al-Masoudi, S. Dörscher, S. Häfner, J.-L. Robyr, N. Chiodo, S. Bilicki, E. Bookjans, A. Koczwara, S. Koke, A. Kuhl, F. Wiotte, and P.-E. Pottie, “A clock network for geodesy and fundamental science,” Nature Communications 7, 12443 (2015).
  * Husmann  _et al._ (2021) D. Husmann, L.-G. Bernier, M. Bertrand, D. Calonico, K. Chaloulos, G. Clausen, C. Clivati, J. Faist, E. Heiri, U. Hollenstein, A. Johnson, F. Mauchle, Z. Meir, F. Merkt, A. Mura, G. Scalari, S. Scheidegger, H. Schmutz, M. Sinhal, S. Willitsch, and J. Morel, “SI-traceable frequency dissemination at 1572.06 nm in a stabilized fiber network with ring topology,” Opt. Express 29, 24592–24605 (2021).
  * Roberts  _et al._ (2020) B. M. Roberts, P. Delva, A. Al-Masoudi, A. Amy-Klein, C. Bærentsen, C. F. A. Baynham, E. Benkler, S. Bilicki, S. Bize, W. Bowden, J. Calvert, V. Cambier, E. Cantin, E. A. Curtis, S. Dörscher, M. Favier, F. Frank, P. Gill, R. M. Godun, G. Grosche, C. Guo, A. Hees, I. R. Hill, R. Hobson, N. Huntemann, J. Kronjäger, S. Koke, A. Kuhl, R. Lange, T. Legero, B. Lipphardt, C. Lisdat, J. Lodewyck, O. Lopez, H. S. Margolis, H. Álvarez-Martínez, F. Meynadier, F. Ozimek, E. Peik, P.-E. Pottie, N. Quintin, C. Sanner, L. D. Sarlo, M. Schioppo, R. Schwarz, A. Silva, U. Sterr, C. Tamm, R. L. Targat, P. Tuckey, G. Vallet, T. Waterholter, D. Xu, and P. Wolf, “Search for transient variations of the fine structure constant and dark matter using fiber-linked optical atomic clocks,” New Journal of Physics 22, 093010 (2020).
  * (20) “REFIMEVE project,” .
  * Oelker  _et al._ (2019) E. Oelker, R. B. Hutson, C. J. Kennedy, L. Sonderhouse, T. Bothwell, A. Goban, D. Kedar, C. Sanner, J. M. Robinson, G. E. Marti, D. G. Matei, T. Legero, M. Giunta, R. Holzwarth, F. Riehle, U. Sterr, and J. Ye, “Demonstration of $4.8\times 10^{-17}$ stability at 1 s for two independent optical clocks,” Nature Photonics 13, 714–719 (2019).
  * Williams, Swann, and Newbury (2008) P. A. Williams, W. C. Swann, and N. R. Newbury, “High-stability transfer of an optical frequency over long fiber-optic links,” J. Opt. Soc. Am. B 25, 1284–1293 (2008).
  * Calosso  _et al._ (2015) C. E. Calosso, E. K. Bertacco, D. Calonico, C. Clivati, G. A. Costanzo, M. Frittelli, F. Levi, S. Micalizio, A. Mura, and A. Godone, “Doppler-stabilized fiber link with 6 dB noise improvement below the classical limit,” Opt. Lett. 40, 131–134 (2015).
  * Ning  _et al._ (2013) B. Ning, D. Hou, T. Zheng, and J. Zhao, “Hybrid analog-digital fiber-based radio-frequency signal distribution,” IEEE Photonics Technology Letters 25, 1551–1554 (2013).
  * Baldwin  _et al._ (2012) K. G. H. Baldwin, Y. He, M. T. L. Hsu, M. J. Wouters, M. B. Gray, B. J. Orr, A. Luiten, S. Schediwy, J. Chow, D. A. Shaddock, G. Aben, P. T. H. Fisk, and R. B. Warrington, “Analog and all-digital frequency distribution via optical fiber links,” in  _Conference on Lasers and Electro-Optics 2012_ (Optica Publishing Group, 2012) p. CTh4A.2.
  * Cárdenas Olaya  _et al._ (2016) A. C. Cárdenas Olaya, S. Micalizio, M. Ortolano, C. E. Calosso, E. Rubiola, and J.-M. Friedt, “Digital electronics based on Red Pitaya platform for coherent fiber links,” in  _2016 European Frequency and Time Forum (EFTF)_ (2016) pp. 1–4.
  * Tourigny-Plante  _et al._ (2018) A. Tourigny-Plante, V. Michaud-Belleau, N. Bourbeau Hébert, H. Bergeron, J. Genest, and J.-D. Deschênes, “An open and flexible digital phase locked loop for optical metrology,” Review of Scientific Instruments 89 (2018), 10.1063/1.5039344.
  * osc (2022a) “Oscillator IMP project: https://github.com/oscimp/app, see directory redpitaya/double_iq_pid_vco,”  (2022a).
  * Mukherjee  _et al._ (2022) S. Mukherjee, J. Millo, B. Marechal, S. Denis, G. Goavec-Mérou, J.-M. Friedt, Y. Kersalé, and C. Lacroûte, “Digital doppler-cancellation servo for ultrastable optical frequency dissemination over fiber,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 69, 878–885 (2022).
  * Cárdenas-Olaya  _et al._ (2019) A. C. Cárdenas-Olaya, A. Tampellini, E. Bertacco, C. Clivati, A. Mura, D. Calonico, S. Micalizio, and C. E. Calosso, “Digital instrumentation for phase-coherent frequency transfer over 300 km fiber link,” in  _2019 Joint Conference of the IEEE International Frequency Control Symposium and European Frequency and Time Forum (EFTF/IFC)_ (2019) pp. 1–2.
  * Rubiola and Vernotte (2022) E. Rubiola and F. Vernotte, “The companion of the Enrico’s chart for phase noise and two-sample variances,”  (2022).
  * Note (1) A low-noise version of Red Pitaya STEMlab 124-14 very recently became available, that claims improved linear voltage regulators.
  * Note (2) Gooch & Housego T-M110-0.2C2J-3-F2S.
  * Note (3) Golledge MP01327.
  * osc (2022b) “Oscillator IMP project, Digital PLL characterization: https://github.com/oscimp/digitalpllcharacterization,”  (2022b).
  * Jiang (2010) H. Jiang,  _Development of ultra-stable laser sources and long-distance optical link via telecommunication networks_ , Theses, Université Paris-Nord - Paris XIII (2010).
  * Bercy (2015) A. Bercy,  _Liens ultra-stables par fibres optiques : déploiement vers plusieurs utilisateurs distants, étude des limites fondamentales et technologiques et nouvelles applications_ , Theses, Université Paris 13 - Sorbonne Paris Cité (2015).
  * Manamanni  _et al._ (2022) K. Manamanni, T. Steshchenko, F. Wiotte, R. L. Targat, M. Abgrall, O. Lopez, Étienne Cantin, P. Éric Pottie, A. Amy-Klein, V. Roncin, and F. Du-Burck, “Limitations due to residual interference in a fiber-based optical frequency reference at 1.55 $\upmu$m,”  (2022).


* Fill and Pemantle [1993] J. A. Fill and R. Pemantle, Percolation, first-passage percolation and covering times for Richardson’s model on the $n$-cube, _Ann. Appl. Probab._ 3 (1993), 593–629.
  * Frieze [2014] A. Frieze, Random structures and algorithms, _Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. 1_ , 311–340, Kyung Moon Sa, Seoul (2014).
  * Frieze [2019] A. Frieze, Hamilton cycles in random graphs: a bibliography, _arXiv e-prints_ (2019), 1901.07139.
  * Gregor and Dvořák [2008] P. Gregor and T. Dvořák, Path partitions of hypercubes, _Inform. Process. Lett._ 108 (2008), 402–406.
  * Hahn-Klimroth, Maesaka, Mogge, Mohr and Parczyk [2020] M. Hahn-Klimroth, G. S. Maesaka, Y. Mogge, S. Mohr and O. Parczyk, Random perturbation of sparse graphs, _arXiv e-prints_ (2020), 2004.04672.
  * Hoeffding [1963] W. Hoeffding, Probability inequalities for sums of bounded random variables, _J. Amer. Statist. Assoc._ 58 (1963), 13–30.
  * van der Hofstad and Nachmias [2017] R. van der Hofstad and A. Nachmias, Hypercube percolation, _J. Eur. Math. Soc. (JEMS)_ 19 (2017), 725–814.
  * Janson, Łuczak and Ruciński [2000] S. Janson, T. Łuczak and A. Ruciński, _Random graphs_ , Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York (2000).
  * Johansson [2018] T. Johansson, On Hamilton cycles in Erdős-Rényi subgraphs of large graphs, _arXiv e-prints_ (2018), 1811.03501.
  * Knox, Kühn and Osthus [2015] F. Knox, D. Kühn and D. Osthus, Edge-disjoint Hamilton cycles in random graphs, _Random Structures Algorithms_ 46 (2015), 397–445.
  * Knuth [2005] D. E. Knuth, _The Art of Computer Programming, Volume 4, Fascicle 2: Generating All Tuples and Permutations (Art of Computer Programming)_ , Addison-Wesley Professional (2005).
  * Kohayakawa, Kreuter and Osthus [2000] Y. Kohayakawa, B. Kreuter and D. Osthus, The length of random subsets of Boolean lattices, _Random Structures Algorithms_ 16 (2000), 177–194.
  * Komlós and Szemerédi [1983] J. Komlós and E. Szemerédi, Limit distribution for the existence of Hamiltonian cycles in a random graph, _Discrete Math._ 43 (1983), 55–63.
  * Koršunov [1976] A. D. Koršunov, Solution of a problem of P. Erdős and A. Rényi on Hamiltonian cycles in undirected graphs, _Dokl. Akad. Nauk SSSR_ 228 (1976), 529–532.
  * Krivelevich, Kwan and Sudakov [2017] M. Krivelevich, M. Kwan and B. Sudakov, Bounded-degree spanning trees in randomly perturbed graphs, _SIAM J. Discrete Math._ 31 (2017), 155–171.
  * Krivelevich, Lee and Sudakov [2014] M. Krivelevich, C. Lee and B. Sudakov, Robust Hamiltonicity of Dirac graphs, _Trans. Amer. Math. Soc._ 366 (2014), 3095–3130.
  * Krivelevich and Samotij [2012] M. Krivelevich and W. Samotij, Optimal packings of Hamilton cycles in sparse random graphs, _SIAM J. Discrete Math._ 26 (2012), 964–982.
  * Kühn and Osthus [2014a] D. Kühn and D. Osthus, Hamilton cycles in graphs and hypergraphs: an extremal perspective, _Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. IV_ , 381–406, Kyung Moon Sa, Seoul (2014a).
  * Kühn and Osthus [2014b] D. Kühn and D. Osthus, Hamilton decompositions of regular expanders: applications, _J. Combin. Theory Ser. B_ 104 (2014b), 1–27.
  * Leighton [1992] F. T. Leighton, _Introduction to parallel algorithms and architectures_ , Morgan Kaufmann, San Mateo, CA (1992), arrays, trees, hypercubes.
  * McDiarmid, Scott and Withers [2018] C. McDiarmid, A. Scott and P. Withers, The component structure of dense random subgraphs of the hypercube, _arXiv e-prints_ (2018), 1806.06433.
  * Pósa [1976] L. Pósa, Hamiltonian circuits in random graphs, _Discrete Math._ 14 (1976), 359–364.
  * Savage [1997] C. Savage, A survey of combinatorial Gray codes, _SIAM Rev._ 39 (1997), 605–629.

##  Appendix A Proof of Lemma 8.9

###### Proof of Lemma 8.9.

The proof is similar (but easier) to that of Lemma 8.8. By relabelling the
atoms, we may assume that
$\mathcal{M^{*}}=\mathcal{A}_{1}\cup\dots\cup\mathcal{A}_{t}$. Without loss of
generality, we may assume that, for each $r\in[2]$, if $u_{r}\in R$, then
$u_{r}=z_{1}$, and if $v_{r}\in R$, then $v_{r}=w_{t}$. Moreover, we may
assume that $x\neq_{\mathrm{p}}u_{1}$. Let
$S\coloneqq\\{u_{1},v_{1},u_{2},v_{2}\\}$. Let $I_{R}\coloneqq\\{k\in[t]:R\cap
V(\mathcal{A}_{k})\cap S\neq\varnothing\\}$, $R^{*}\coloneqq
R\setminus\bigcup_{k\in I_{R}}V(\mathcal{A}_{k})$ and
$I_{R^{*}}\coloneqq\\{k\in[t]:R^{*}\cap V(\mathcal{A}_{k})\neq\varnothing\\}$.
For each $r\in[2]$, let $I_{R}^{r}\subseteq\\{1,t\\}$ be such that $1\in
I_{R}^{r}$ if and only if $u_{r}\in R$ and $t\in I_{R}^{r}$ if and only if
$v_{r}\in R$. Note that $I_{R}=I_{R}^{1}\cup I_{R}^{2}$. Fix an index
$t^{*}\in[t-1]\setminus(I_{R^{*}}\cup\\{1\\})$. If $|L|=2$, let
$I_{L}^{1}\coloneqq\\{i\\}$, $I_{L}^{2}\coloneqq\\{j\\}$ and
$I_{L}\coloneqq\\{i,j\\}$; otherwise, let $I_{L}^{1}\coloneqq
I_{L}^{2}\coloneqq I_{L}\coloneqq\\{t^{*}\\}$.

For each $r\in[2]$, we create an ordered list $\mathcal{L}_{r}$ of vertices,
which will be used to construct the vertex-disjoint paths $\mathcal{P}_{r}$.
Given any list of vertices $\mathcal{L}_{r}$, we write $L^{*}_{r}$ to denote
the (unordered) set of vertices in $\mathcal{L}_{r}$, and whenever
$\mathcal{L}_{r}$ is updated, we implicitly update $L_{r}^{*}$. In the end,
for each $r\in[2]$ we will have a list of vertices
$\mathcal{L}_{r}=(x_{1}^{r},\dots,x_{\ell_{r}}^{r})$. For each $r\in[2]$ and
$k\in[t]$, let $I_{r}(k)\coloneqq\\{h\in[\ell_{r}-1]:2\nmid h\text{ and
}x^{r}_{h},x^{r}_{h+1}\in V(\mathcal{A}_{k})\\}$. We will require
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ to be vertex-disjoint and to satisfy
the following properties:

  1. $(\mathcal{L}^{\prime}1)$

$\ell_{1}$ and $\ell_{2}$ are even.

  2. $(\mathcal{L}^{\prime}2)$

For each $r\in[2]$, for all $h\in[\ell_{r}-1]$, if $h$ is odd, then
$x^{r}_{h},x^{r}_{h+1}\in V(\mathcal{A}_{k})$, for some $k\in[t]$; if $h$ is
even, then $x^{r}_{h}x^{r}_{h+1}\in E(G\cup\mathcal{M}^{*})$.

  3. $(\mathcal{L}^{\prime}3)$

For all $k\in[t]$ we have that $|I_{1}(k)|,|I_{2}(k)|\geq 1$ and
$2\leq|I_{1}(k)|+|I_{2}(k)|\leq 3$.

  4. $(\mathcal{L}^{\prime}4)$

For each $r\in[2]$, the following holds. For each
$k\in[t]\setminus(I_{L}^{r}\cup I_{R}^{r})$ and each $h\in I_{r}(k)$, we have
$x_{h}^{r}\neq_{\mathrm{p}}x_{h+1}^{r}$. For each $k\in I_{L}^{r}\cup
I_{R}^{r}$, we have that $|I_{r}(k)|=1$ and for the unique index $h\in
I_{r}(k)$ we have $x_{h}^{r}=_{\mathrm{p}}x_{h+1}^{r}$, with the same parity
as $u_{r}$ in the case when $k\in I_{L}^{r}$, and with parity opposite to that
of the unique vertex in $\\{w_{k},z_{k}\\}\cap\\{u_{r},v_{r}\\}$ in the case
when $k\in I_{R}^{r}$.

  5. $(\mathcal{L}^{\prime}5)$

For each $r\in[2]$, we have the following. If $u_{r}\notin R$, then
$u_{r}=x_{1}^{r}$. If $v_{r}\notin R$, then $v_{r}=x^{r}_{\ell_{r}}$. If
$u_{r}\in R$ (and thus $u_{r}=z_{1}$), then $w_{1}=x_{1}^{r}$ and $u_{r}\notin
L^{*}_{1}\cup L^{*}_{2}$. If $v_{r}\in R$ (and thus $v_{r}=w_{t}$), then
$z_{t}=x_{\ell_{r}}^{r}$ and $v_{r}\notin L^{*}_{1}\cup L^{*}_{2}$.

  6. $(\mathcal{L}^{\prime}6)$

Every pair $(w_{k},z_{k})$ with $\\{w_{k},z_{k}\\}\subseteq R^{*}$ is
contained in $\mathcal{L}_{1}$ and $z_{k}$ directly succeeds $w_{k}$ or vice
versa.

If $R^{*}\cap V(\mathcal{A}_{1})=\\{w_{1},z_{1}\\}$, then let
$\mathcal{L}_{1}\coloneqq(u_{1},w_{1},z_{1})$, where we assume that
$w_{1}\neq_{\mathrm{p}}u_{1}$; otherwise, let
$\mathcal{L}_{1}\coloneqq(u_{1})$. Observe once more that, in what follows,
the existence of each alternating parity sequence follows from the bondedness
of $\mathcal{M}$.

Let $F_{1}\coloneqq L\cup R\cup S$ and let $t^{\bullet}_{1}\in I_{L}^{1}$. Let
$\mathcal{S}_{1}$ be a $(u_{1},t^{\bullet}_{1},F_{1},R)$-alternating parity
sequence. If $u_{1}\in R$, update $\mathcal{L}_{1}\coloneqq\mathcal{S}_{1}$;
otherwise, update
$\mathcal{L}_{1}\coloneqq\mathcal{L}_{1}\mathcal{S}_{1}^{-}$. Choose any
vertex $u_{t^{\bullet}_{1}}\in V(\mathcal{A}_{t^{\bullet}_{1}})$ with
$u_{t^{\bullet}_{1}}\neq_{\mathrm{p}}u_{1}$, and let $\mathcal{S}_{2}$ be a
$(u_{t^{\bullet}_{1}},t,F_{1},R^{*})$-alternating parity sequence. Update
$\mathcal{L}_{1}\coloneqq\mathcal{L}_{1}\mathcal{S}_{2}^{-}$. If $v_{1}\in R$,
update $\mathcal{L}_{1}\coloneqq\mathcal{L}_{1}(z_{t})$. Otherwise, update
$\mathcal{L}_{1}\coloneqq\mathcal{L}_{1}(v_{1})$.

Next, let $F_{2}\coloneqq F_{1}\cup L_{1}^{*}$ and let $t^{\bullet}_{2}\in
I_{L}^{2}$. Let $\mathcal{S}_{3}$ be a $(u_{2},t^{\bullet}_{2},F_{2},R\cap
V(\mathcal{A}_{1}))$-alternating parity sequence, and let
$\mathcal{L}_{2}\coloneqq\mathcal{S}_{3}$. Choose any vertex
$u^{\prime}_{t^{\bullet}_{2}}\in V(\mathcal{A}_{t^{\bullet}_{2}})$ with
$u^{\prime}_{t^{\bullet}_{2}}\neq_{\mathrm{p}}u_{2}$, and let
$\mathcal{S}_{4}$ be a
$(u^{\prime}_{t^{\bullet}_{2}},t,F_{2},\varnothing)$-alternating parity
sequence. Update $\mathcal{L}_{2}\coloneqq\mathcal{L}_{2}\mathcal{S}_{4}^{-}$.
Finally, if $v_{2}\in R$, update
$\mathcal{L}_{2}\coloneqq\mathcal{L}_{2}(z_{t})$. Otherwise, update
$\mathcal{L}_{2}\coloneqq\mathcal{L}_{2}(v_{2})$.

Observe that $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ satisfy
$(\mathcal{L}^{\prime}1)$–$(\mathcal{L}^{\prime}6)$. We are now in a position
to apply Lemma 8.6. For each $k\in[t]$, let
$t_{k}\coloneqq|I_{1}(k)|+|I_{2}(k)|$. Again, for any $r\in[2]$ and $k\in[t]$,
for each $h\in I_{r}(k)$, we refer to the pair $x^{r}_{h},x^{r}_{h+1}$ as a
matchable pair. By $(\mathcal{L}^{\prime}3)$, $(\mathcal{L}^{\prime}4)$ and
Lemma 8.6$(\mathrm{i})$, each $\mathcal{A}_{k}$ with
$k\in[t]\setminus(I_{L}\cup I_{R})$ can be covered by $t_{k}$ vertex-disjoint
paths, each of whose endpoints are a matchable pair contained in
$\mathcal{A}_{k}$. Similarly, by $(\mathcal{L}^{\prime}3)$,
$(\mathcal{L}^{\prime}4)$ and Lemma 8.6$(\mathrm{ii})$, each $\mathcal{A}_{k}$
with $k\in I_{R}$ contains $t_{k}$ vertex-disjoint paths, each of whose
endpoints are a matchable pair in $\mathcal{A}_{k}$, such that the union of
these $t_{k}$ paths covers precisely $V(\mathcal{A}_{k})\setminus(S\cap R)$.
Similarly, if $L\neq\varnothing$ and $k\in I_{L}$, then $\mathcal{A}_{k}$
contains $t_{k}$ paths, each of whose endpoints are a matchable pair in
$\mathcal{A}_{k}$, such that the union of these $t_{k}$ paths covers precisely
$V(\mathcal{A}_{k})\setminus L$. Finally, by $(\mathcal{L}^{\prime}3)$,
$(\mathcal{L}^{\prime}4)$ and Lemma 8.6$(\mathrm{iii})$, if $L=\varnothing$
and $k\in I_{L}$ (that is, $k=t^{*}$), then $\mathcal{A}_{k}$ can be covered
by $t_{k}$ paths, each of whose endpoints are a matchable pair in
$\mathcal{A}_{k}$. For each matchable pair $x^{r}_{h},x^{r}_{h+1}$ in
$\mathcal{A}_{k}$, let us denote the corresponding path by
$\mathcal{P}_{x^{r}_{h},x^{r}_{h+1}}$.

The paths $\mathcal{P}_{r}$ required for Lemma 8.9 can now be constructed as
follows. For each $r\in[2]$, let $\mathcal{P}_{r}$ be the path obtained from
the concatenation of the paths $\mathcal{P}_{x^{r}_{h},x^{r}_{h+1}}$, for each
odd $h\in[\ell_{r}]$, via the edges $x^{r}_{h}x^{r}_{h+1}$ for
$h\in[\ell_{r}-1]$ even. By $(\mathcal{L}^{\prime}5)$, if $\mathcal{P}_{r}$
does not contain $u_{r}$, then $\mathcal{P}_{r}$ starts in $w_{1}$, and
$u_{r}$ does not lie in any other path; therefore, we can update
$\mathcal{P}_{r}$ as $\mathcal{P}_{r}\coloneqq u_{r}\mathcal{P}_{r}$.
Similarly, if $\mathcal{P}_{r}$ does not contain $v_{r}$, then
$\mathcal{P}_{r}$ ends in $z_{t}$ and $v_{r}$ does not lie in any other path,
hence we can update $\mathcal{P}_{r}$ as
$\mathcal{P}_{r}\coloneqq\mathcal{P}_{r}v_{r}$. It follows that
$V(\mathcal{P}_{1}\cup\mathcal{P}_{2})=V(\mathcal{M}^{*})\setminus L$, and
thus the paths $\mathcal{P}_{r}$ are as required for Lemma 8.9. ∎