Guillaume Gabant

Martin Stekovic

Matej Nemcic

Justine Pinêtre

Martine Cadene
email: martine.cadene@cnrs-orleans.fr

A sDOE (simple design-of-experiment) approach for parameter optimization in mass spectrometry. Part 1. Parameter selection and interference effects in topdown ETD fragmentation of proteins in a UHR-QTOF instrument

Keywords: Design of Experiments, sDOE, mass spectrometry, UHR-QTOF, ETD, optimization

    A sDOE (simple design-of-experiment) approach for parameter optimization in mass spectrometry. Part 1. Parameter selection and interference effects in topdown ETD fragmentation of proteins in a UHR-QTOF instrument.

Introduction:

To carry out their functions, proteins are expressed in many flavors in vivo and in vitro. These flavors, called proteoforms, are isoforms resulting from sequence variability, post-translational modifications (PTMs), and/or modifications induced by exogenous agents such as radiations or chemicals. The precise localization of modifications and their quantification is becoming a major quest of biology. While the bottom-up proteomics approach based on proteolysis and peptide MS fragmentation can reliably be used to address this challenge [START_REF] Zhang | Protein analysis by shotgun/bottom-up proteomics[END_REF], it presents some drawbacks, such as the loss of filiation between a modified peptide and its proteoform of origin [START_REF] Cui | Top-down mass spectrometry: recent developments, applications and perspectives[END_REF]. The top-down approach based on the fragmentation of intact protein molecules leads not only to primary sequence information but also to positional identification of modifications [START_REF] Smith | The Human Proteoform Project: Defining the human proteome[END_REF][START_REF] Srzentić | Interlaboratory Study for Characterizing Monoclonal Antibodies by Top-Down and Middle-Down Mass Spectrometry[END_REF], and potentially, to their quantification [START_REF] Gabant | SSPaQ: A Subtractive segmentation approach for the exhaustive parallel quantification of the extent of protein modification at every possible site[END_REF]. Since sequence coverage is paramount to a successful top-down characterization of proteoforms, extending fragmentation coverage (FC) becomes central to top-down strategies.

In terms of sequencing, several fragmentation methods are readily available on commercial instruments such as collision-induced dissociation (CID), photodissociation (PD), electron-transfer dissociation (ETD), electron-capture dissociation (ECD), and (ExD). In CID, multiple collisions increase the internal energy of precursor ions, which can cause both internal fragmentation, a re-fragmentation of fragment ions, and loss of side-chain PTMs [START_REF] Sobott | Comparison of CID versus ETD based MS/MS fragmentation for the analysis of protein ubiquitination[END_REF]. In ETD [START_REF] Syka | Peptide and protein sequence analysis by electron transfer dissociation mass spectrometry[END_REF], a singly charged anion reagent known as the "reactant" transfers an electron to multiply protonated peptides/proteins, inducing fragmentation along the peptide backbone. In contrast to CID, this method achieves nonergodic fragmentation without prior redistribution of energy [START_REF] Sobott | Comparison of CID versus ETD based MS/MS fragmentation for the analysis of protein ubiquitination[END_REF] and leads to a rapid bond cleavage predominantly at the sites where the electron was trapped [START_REF] Li | The effect of fixed charge modifications on electron capture dissociation[END_REF], thereby promoting random N-Cα (alpha carbon) bond breakage while leaving side-chains and PTMs intact. Unlike CID, the outcome of ETD is affected by a number of parameters including but not limited to the amount of reagent and precursor ions, reaction time, and selected charge state [START_REF] Rose | A calibration routine for efficient ETD in large-scale proteomics[END_REF]. Top-down, middle-down and bottom-up ETD fragmentation in a Q-TOF instrument can be used to characterize the structural heterogeneity of antibodies [START_REF] Zhu | Integrating Intact Mass Analysis and Middle-Down Mass Spectrometry Approaches to Effectively Characterize Trastuzumab and Adalimumab Structural Heterogeneity[END_REF], to read out HDX results while minimizing scrambling [START_REF] Wollenberg | Avoiding H/D Scrambling with Minimal Ion Transmission Loss for HDX-MS/MS-ETD Analysis on a High-Resolution Q-TOF Mass Spectrometer[END_REF] and to discriminate leucine from isoleucine, in a dataindependent analysis on peptides [START_REF] Cramer | Electron Transfer Dissociation of All Ions at All Times, MS ETD , in a Quadrupole Time-of-Flight (Q-ToF) Mass Spectrometer[END_REF]. Top-down fragmentation presents a particular challenge as fragmentation of peptides bonds in the middle of the protein tends to be energetically unfavorable, resulting in a typical "middle-gap" coverage pattern, also known as the "dark zone" in fragmentation coverage along the sequence. Achieving the high fragmentation coverage necessary to localize modifications anywhere for any kind of proteoform thus requires the careful optimization of ETD fragmentation conditions [START_REF] Srzentić | Interlaboratory Study for Characterizing Monoclonal Antibodies by Top-Down and Middle-Down Mass Spectrometry[END_REF].

The principle of a rational design of experiments to give a sound statistical basis to experimental results interpretation was initially conceived by Fischer as exposed in his foundational book [START_REF] Fischer | The Design of Experiments[END_REF]. Design-ofexperiment (DOE) approaches per se were formalized by statisticians [START_REF] Cochran | Experimental design[END_REF] and saw exponential growth in the 1950s as a means to provide for fast, reliable, and objective optimization of production processes [START_REF] Taguchi | Taguchi Methods Orthogonal Arrays and Linear Graphs: Tools for Quality Engineering[END_REF]. These methods are regularly used in the pharmaceutical and food industry [START_REF] Myers | Response Surface Methodology[END_REF]. Complex production processes depend on multiple parameters, some of which may be interdependent. The motivation for DOE in this context is based on the simple observation that the parameters which are critical to the performance of a product cannot necessarily be intuited based on past experience in production.

To reach a certain level of performance to match a specific research aim, mass spectrometrists likewise need to optimize instrument parameters and quickly face a multiplicity of parameters, complexity, and interdependency. ETD fragmentation, alone or in combination with other fragmentation modes, is a good illustration of this type of situation. Figure 1 shows the main components a typical ETD fragmentation cycle.

Hereafter, we will refer to precursor intensity as PI, reagent intensity as RI, and reaction time as RT. Different strategies can be applied to optimize multiple parameters for a given desired outcome ("response" in DOE terms), such as maximum protein fragmentation coverage. The simplest and most frequent strategy is sequential optimization of parameters or "change one at a time", which can be described as the "manual" approach. This approach obviously consumes large amounts of resources, including sample quantities and time. After the first round of optimization, each parameter may need to be readjusted because its optimum has shifted in response to the change in at least one other parameter which interferes with it. One way to attempt circumvention of this problem is to plan a "co-optimization" experiment, by making a 2D table of values for two parameters to be tested jointly. Co-optimization has its own limits: first, the choice of parameters to cooptimize relies on assumptions about which co-dependencies are most relevant, and second, it is still timeconsuming and sample-intensive. Faced with this problem, some authors have proposed to use a "brute-force" approach to scan for all possible combinations of parameters in the instrument, or a large subset thereof [START_REF] Shliaha | Maximizing Sequence Coverage in Top-Down Proteomics By Automated Multimodal Gas-Phase Protein Fragmentation[END_REF].

Automatizing the optimization process requires scripting of instrument parameter setups. This approach, however, still requires both large amounts of time and sample.

In a research context, samples are often scarce, so we looked for an alternative to make the process less timeconsuming and sample-intensive. DOE, by cleverly combining a subset of conditions to test parameters effects, offers just such an alternative. Here we show how it can be used to optimize the efficiency of ETD fragmentation, in comparison with the "manual" and "co-optimization" parameter optimization approaches.

We give concrete examples of building matrices, performing experiments, and calculating effects of different parameters. This allows us to identify parameters that are critical to the outcomes of interest, mainly fragmentation coverage, and to show parameter interdependency. In this first paper we show how a selection of DOE tools, which we call sDOE, can offer the user a large part of the benefits of DOE, without necessarily delving into the more complex aspects of the method.

Methods

Reagents

Acetonitrile and formic acid were from Biosolve (Dieuze, France). 18 Mpurified water (MilliQ reagent grade system, Millipore) was used to prepare the ionization solution. Ubiquitin and apomyoglobin were purchased from Sigma (St. Louis, MO). Recombinant human Phosphatidyl Ethanolamine Binding Protein 1 (hPEBP1) was cloned and overexpressed in BL21DE3 E.coli as previously described by Schoentgen and coworkers [START_REF] Atmanene | Characterization of human and bovine phosphatidylethanolaminebinding protein (PEBP/RKIP) interactions with morphine and morphine-glucuronides determined by noncovalent mass spectrometry[END_REF]. The protein was purified without tag using the protocol originally developed by Bernier and Jolles [START_REF] Bernier | Purification and characterization of a basic 23 kDa cytosolic protein from bovine brain[END_REF]. HUalpha-beta protein was a kind gift from Dr Bertrand Castaing at CBM.

Top-down mass spectrometry fragmentation

Top-down MS sequencing of proteins was performed using ESI-QTOF ETD fragmentation method. Samples were diluted at 0.5 µM in a solution of 50% acetonitrile, 49.8% water and 0.2% formic acid and were analyzed by direct infusion in a 4-GHz MaXis ultra high resolution quadrupole-TOF mass spectrometer (Bruker Daltonics) equipped with an electrospray ion source. Spectra were acquired in positive ion MS mode over a 115-3000 m/z range and were externally calibrated with the ESI-L Low Concentration Tuning Mix (Agilent Technologies). Mass spectra were processed using DataAnalysis 3.1 software (Bruker Daltonics). When applied, the smoothing parameters were Savitzky Golay algorithm, width 0.05 Da and 1 cycle. Peaks were annotated using the SNAP algorithm with a quality factor of 0.5. Fragmentation coverage (FC) was obtained using Biotools software (Bruker). The c, c-1, c+1, z, z+1 and z+2 fragment ions were considered.

Design of experiments

The Design Of Experiments (DOE) method is an efficient and structured way to maximize information while minimizing resources. It is used to develop a process that will 1) identify the factors that have the largest effect on the studied output (also known as the response) and 2) adjust them to achieve an optimal response. In the present work, we focus on the first step and the output of interest is the fragmentation coverage.

In the first step, screening experiments are designed to gauge the relative effects of the factors studied. This allows for a first coarse selection of the most interesting factors, characterized as "crucial" or "critical" [START_REF] Hibbert | Experimental design in chromatography: a tutorial review[END_REF], in other words, factors which strongly affect the studied process. In mass spectrometry, factors could be instrumental parameters, sample conditions, or any other factor thought to influence the outcome of the experiment. To set up factor selection experiments, we can use a) a full factorial design or b) a fractional factorial design. A full factorial design also gives access to interference effects between parameters, while a fractional factorial design only gives access screening of individual factors, albeit with fewer experiments.

When measuring a response, the general principle of a factorial design is to plan to vary 2 or more factors at the same time. As shown in Table 1, a matrix is built by choosing for each factor two numerical or alphanumeric values called levels -1 and +1. Level -1 can be the "base level", i.e. a starting quantitative or qualitative value based on prior data. If the design is orthogonal, i.e. if factors are varied independently, then the effect of a factor can be calculated independently of the effect of other factors. Mathematically, two factors are orthogonal if the sum of their columns in standard format (+1, -1) equals 0. In the present work we used two full factorial designs of 4 and 5 factors to quickly highlight important parameters for FC. The building of experimental runs for a full 4-factor design is shown in Table 1.

Table 1. Full factorial matrix for 4 factors with 2 levels.

Experiment # X1 X2 X3 X4 Response y 1 -1 -1 -1 -1 y1 2 -1 -1 -1 1 y2 3 -1 -1 1 -1 y3 4 -1 -1 1 1 y4 5 -1 1 -1 -1 y5 6 -1 1 -1 1 y6 7 -1 1 1 -1 y7 8 -1 1 1 1 y8 9 1 -1 -1 -1 y9 10 1 -1 -1 1 y10 11 1 -1 1 -1 y11 12 1 -1 1 1 y12 13 1 1 -1 -1 y13 14 1 1 -1 1 y14 15 1 1 1 -1 y15 16 1 1 1 1 y16
In the 4 factors matrix, factors were protein identity, reaction time, isCID and charge state (CS), while in the 5 factors matrix, we picked PI, number of precursor ions (NPI), CS, RT, and number of accumulated spectra (NAS). The rationale for picking these factors is described in the results section.

For the DOE experiments presented herein, the effect of a parameter is calculated as the sum of the products of the level (-1 or +1) and the outcome (FC) of a given experiment, divided by the number of the experiments (n) as represented by the following equation:

𝐸𝑓𝑓𝑒𝑐𝑡 = ∑ (𝑙𝑒𝑣𝑒𝑙 𝑖 × 𝐹𝐶 𝑖 ) 𝑛 𝑖=1

𝑛

Once key factors have been identified, in the second step finer experiments focusing on these key factors are needed to find the set of factor values that will maximize the response, i.e. the optimal combination. These optimization experiment designs can rely on full factorial design if one the factors is qualitative, or on response surface design, if all the factors are quantitative. In these experiments, it may be advisable to test more than two levels for certain factors, for example when the response to a given factor is suspected of being non-linear.

Results

To get a useable ETD fragmentation reaction, one basically needs to optimize precursor intensity (PI), reagent intensity (RI) and reaction time (RT). PI is however directly dependent on the precursor accumulation time (AT), just as RI is dependent on the reactant reagent injection time (RgT). Since our instrument is not equipped with automatic ion gain control, the accumulation time needs to be adjusted manually to reach a given intensity.

We started by optimizing AT using two calibrant proteins, namely ubiquitin (Mmono = 8 560 Da) and apomyoglobin (Mmono = 16 941 Da) (Figure 2A). The dependency of PI upon AT values can be seen to be protein-dependent, which is the first clue that the ETD optimization process may have to be performed anew for each type of protein sample. Precursor isolation width is commonly optimized and shows a saturation effect (Figure 2B). After further optimization based on reagent time, a FC of 55.9% was obtained on the hPEBP1 protein. From this number, it is immediately apparent that in this example, "manual" optimization of parameters yields insufficient FC for most PTM localization applications. Apart from favorable cases such as ubiquitin and from thorough, time consuming manual optimization of the parameters, in ETD such manual approach may frequently yield a FC that is indeed too low for these applications. Thus, we looked into parameters that may benefit from a "cooptimization" approach.

A B

Using ubiquitin as an "expendable" model protein, we set up experiments varying both PI and RT. For this model system, the FC varies between 60 and 98%, while a co-optimum can be seen at PI = 74 000 and RT = 8 ms (Figure 3A).

A

To evaluate the repeatability of measurements, we performed a decaplicate experiment. The distributions of FC for ubiquitin and apomyoglobin are shown as a box plot (Figure 4). Standard deviations for FC were 3.9% for ubiquitin and 1.8% for apomyoglobin, respectively. These values can give an order of magnitude for the degree of variability for FC determination, which can give some context when evaluating the result of manual or co-optimization. We also noted that reagent accumulation time and reaction time cannot be completely independent because, as can be seen in Figure 1, the reaction of proteins/peptides with the ETD reactant can start as soon as the reactant is introduced into the fragmentation chamber. Co-optimization of these parameters proved useful as well. The plotted surface shows a co-optimum area from 70 000 to 76 000 for the reagent intensity and from 8 ms to 20 ms for the reaction time (Figure 3B).

To further our understanding of which parameters have the largest effect on the output and/or interfere with one another, we next set up a DOE experiment to start exploring this tool. A set of 4 parameters can be tested with a relatively simple full factorial matrix as shown in Table 1. For each parameter, some prior knowledge of the system is preferable to assist in choosing two different levels per factor, hereafter labeled as the "+1"

and "-1" conditions.

We included as parameters the identity of the protein, reaction time, the voltage for in-source collision-induced dissociation (isCID), and the charge state, all of which seemed to affect FC during manual optimizations (Table 2).

Table 2. Determination of the main effects and interactions. Full factorial matrix with two levels per factor. Reagent intensity was set at 70 000 and precursor intensity at 75 000. Proteins were HU alpha chain (M = 9535 Da) and apomyoglobin at 0.5 µM. For apomyoglobin the charge states used were 21+ and 25+, while for HU they were 11+ and 14+. For the value of z n , the charge state with the highest intensity in the charge state distribution of a previous experiment was chosen. Since by increasing the number of charges, we can expect to improve the yield of the ETD reaction, for the second charge state value we chose to test a value corresponding to a 20% increase in z, called z nx1.2 . The criterion for choosing 20% as the increase is a compromise between a higher charge state and a usable intensity. The 25+ and 14+ charges states of apomyoglobin and HU respectively, fit well with these specifications.

Individually, the identity of the protein (which contains its sequence data and physicochemical profile) was found to have the largest effect, with over twice the effect observed for the next-most important factor, namely RT (Table 3). Factors that can make ETD fragmentation protein-dependent, may include protein size, sequence and residual gas phase structure. Protein sequence affects the distribution of polar, apolar, positively and negatively charged amino acids along the sequence, which may influence the transfer of electrons in the ETD process. A fully unfolded structure should allow full access to peptide bonds for electron transfer. However, the sequence of a protein affects its folding in solution, and unfolding kinetics in solvents and during transfer to the fragmentation cell is also protein-dependent. As a consequence, any structure that survives denaturing solvent conditions as well as transfer to the gas phase may influence ETD fragmentation, as was shown for example for ECD [START_REF] Skinner | How ubiquitin unfolds after transfer into the gas phase[END_REF].

Table 3. Effect of individual parameters and their interactions on fragmentation coverage.

Interestingly, the largest interference effects where seen for the charge state x RT effects and the protein identity x charge state effects, so that both of the largest observed interference effects involve the charge state of the precursor. While the interpretation of interference effects can be challenging, some hypotheses based on the distribution of positively charged sites on the protein can be formulated. For the interference of the charge state with RT, since the reactant brings an electron for transfer, electrostatic effects between positively charged sites and the reactant affect the transfer of electrons to the backbone, and the charge state affects the distribution of positively charged sites. As a consequence, for a given efficiency of transfer for a given charge state, reaction time may need to be adjusted to compensate for these electrostatic effects, and the adjustment would then depend on the charge state. From the point of view of the sequence, protein identity may also affect the efficiency of transfer of electrons to the backbone, as the identity of amino acid residues has a direct bearing on the ionization of residues. This may explain the interplay between charge state and protein identity.

Next we moved to optimizations for a more scarce protein, the hPEBP1 metastasis suppressor. For this single protein and 5 different parameters, a full factorial matrix consisting of 2 5 = 32 partially randomized top-down MS experiments was constructed (Table 4). After co-optimization, the best FC values were obtained with a RI would also have had an adverse impact on the length of the duty cycle. Thanks to the co-optimization results, the RI parameter was set at 70000 for subsequent experiments.

Table 4. Full factorial matrix with two levels per factor. Reagent intensity was set at 70 000. Protein was hPEBP1 (M = 20926 Da) at 0.5 µM.

The first 4 parameters are randomized, meaning fragmentation sequences are programmed in random order in the analysis. Upon closer examination of experimental data, we noticed that the fragmentation segments we programmed in the instrument had a variable number of spectra. This is due to the effect of variations in protein accumulation times and reaction time on duty cycle length. To test the effect of this parameter (number of averaged spectra), we constructed a 5-parameter matrix by picking either spectra, or the following 800 spectra within the same segment, which means that the NAS parameter is not randomized with respect to the others. In DOE, mixing randomized and non-randomized parameters is allowed. As can be seen in Table 5, PI and NAS are in a draw as the main parameter to optimize. In this representation, each X factor (parameter) interference with another X factor is shown. The major interference, PI with RT, is quite far ahead of all others.

Table 5. Effect of individual parameters and their interactions on hPEBP1 fragmentation coverage

Discussion

If we define fragmentation coverage (FC), as the percentage of amino acids bonds in the sequence where at least one fragmentation event is observed through fragment ions, the difficulty of achieving close to 100% FC in top-down sequencing increases with respect to protein size [START_REF] Pesavento | Quantitative Analysis of Modified Proteins and Their Positional Isomers by Tandem Mass Spectrometry: Human Histone H4[END_REF]. Fragmentation occurs predominantly at the ends of the protein, leaving an area at the center without corresponding fragments, which is termed "dark zone" to indicate the lack of sequence information. Dark zones can also occur outside the center area in a sequence.

For an exhaustive census of modified residues, dark zones are an obvious impediment to reaching the stated goal. When faced with incomplete fragmentation coverage, several responses are possible. The first is to abandon the study of a protein showing imperfect coverage and to focus on systems that behave perfectly on the first try. This "low-hanging fruit" approach has serious limitations, as the mass spectrometry community will quickly run out of such ideal systems. Moreover, this analysis ethics runs counter to the goal of science, which is to go beyond the tip of the iceberg to explore both emerged and immerged parts of the factual berg, and counter to the social compact, which is to provide information that is useful for human health, wellbeing, the environment, and economics. Therefore, to achieve high FC for any kind of protein and proteoform, ETD fragmentation requires careful optimization. In the present paper, we compare several strategies for optimization on an UHR-QTOF MS instrument with ETD fragmentation, with a final focus on a simple version of DOE. Ultimately, the reasoning could also apply to any number of MS techniques were multiple input parameters affect the outcome of the analysis. To improve the efficiency of top-down ETD fragmentation of proteins on a high-resolution Q-TOF mass spectrometer with the aim of maximizing FC, we started from single-parameter optimization, then moved to co-optimization of parameters. Through a peak-and-valley visualization of the optimization landscape, we could see the importance of interdependency (Figure 3). Yet co-optimization still only deals with 2 parameters at a time.

Other authors have made the case that sequential, manual optimization or co-optimization of MS/MS experiments is too slow and incomplete. Jensen et al had to deal with multiple parameters when using a combination of CID, HCD, ETD, ETciD, EThcD, and UVPD for top-down protein fragmentation on an

Orbitrap instrument [START_REF] Shliaha | Maximizing Sequence Coverage in Top-Down Proteomics By Automated Multimodal Gas-Phase Protein Fragmentation[END_REF]. The solution the authors proposed and successfully applied was to first narrow down the set of parameters they thought would most affect the outcome (FC), then to use a "brute-force" approach to scan for all possible combinations of these parameters in the instrument. This required the development of the "topdownr" R-package to automate multimodal MS/MS analyses as well as data treatment, which basically scripts instrument parameter setups to allow for automation of the data acquisition and filtering of thousands of MS/MS datasets [START_REF] Shliaha | Maximizing Sequence Coverage in Top-Down Proteomics By Automated Multimodal Gas-Phase Protein Fragmentation[END_REF]. This approach, however, requires both time and large amounts of sample, which are often scarce in a research context, prompting us to search for an alternative. DOE, by cleverly combining a subset of conditions to test parameters effects, offers just such an alternative.

Experimental plans are designed as matrices of conditions to run experiments. DOE can be used to maximize the efficiency of ETD fragmentation, or in principle any combination of MS/MS fragmentation methods. In addition to parameters that are selected and randomized prior to the experiment, parameters can be added to the analysis a posteriori as non-randomized parameters. This provides some flexibility in the exploitation of data, for a limited cost in data independence. Different sets of DOE tools have been successfully applied for optimization in an industrial [START_REF] Seto | Development of generic liquid chromatography-mass spectrometry methods using experimental design[END_REF] or academic [START_REF] Switzar | Protein digestion optimization for characterization of drug-protein adducts using response surface modeling[END_REF][START_REF] Zheng | Time-saving design of experiment protocol for optimization of LC-MS data processing in metabolomic approaches[END_REF][START_REF] Zhang | A designed experiments approach to optimization of automated data acquisition during characterization of bacteria with MALDI-TOF mass spectrometry[END_REF][START_REF] Zhang | Optimization of mass spectrometric parameters improve the identification performance of capillary zone electrophoresis for single-shot bottom-up proteomics analysis[END_REF][START_REF] Bheemanapally | Optimization of Ultra-High-Performance Liquid Chromatography-Electrospray Ionization-Mass Spectrometry Detection of Glutamine-FMOC Ad-Hoc Derivative by Central Composite Design[END_REF][START_REF] Serrano-Blesa | Robust optimization of SWATH-MS workflow for human blood serum proteome analysis using a quality by design approach[END_REF][START_REF] Pizzala | Design of Experiments for Matrix-Assisted Laser Desorption/Ionization of Amphiphilic Poly(Ethylene Oxide)-b-Polystyrene Block Copolymers[END_REF][START_REF] Afoullouss | Optimization of LC-MS 2 Data Acquisition Parameters for Molecular Networking Applied to Marine Natural Products[END_REF] context, and/or for method validation [START_REF] Frankenfield | Development and Comparative Evaluation of Endolysosomal Proximity Labeling-Based Proteomic Methods in Human iPSC-Derived Neurons[END_REF][START_REF] Tibon | Speciation analysis of organoarsenic species in marine samples: method optimization using fractional factorial design and method validation[END_REF].

A number of reviews [START_REF] Hibbert | Experimental design in chromatography: a tutorial review[END_REF][START_REF] Hecht | Optimizing mass spectrometry analyses: A tailored review on the utility of design of experiments[END_REF] have sought to make full sets of DOE tools accessible and attractive to the analytical sciences community. Hibbert produced a detailed and instructive guide on how to choose and apply tools in DOE [START_REF] Skinner | How ubiquitin unfolds after transfer into the gas phase[END_REF]. Hecht et al. wrote an extensive review with case studies to "enable a post-doctorate mass spectrometrist to utilize DOE to design and execute an optimization study independently and without a statistician consultant" [START_REF] Hecht | Optimizing mass spectrometry analyses: A tailored review on the utility of design of experiments[END_REF]. Despite the clarity and pedagogy displayed in these reviews, DOE as a general set of tools may still seems quite complex and daunting to the everyday user. Yet, DOE should attract a much larger audience since a subset of these tools potentially yields a large portion of useful information for a fraction of the time and sample costs.

The present work furnishes concrete examples of building matrices, performing experiments, and calculating the relative effects of different parameters. This allows us to identify parameters that are critical to FC, the outcome of interest, and to show parameter interdependency for the purpose of adjusting critical parameters for the best possible outcome.

In comparison to manual optimization methods, DOE enables a relatively fast screening of crucial parameters and interactions thereof. Although DOE can be made into quite a complex and powerful tool, in its simplest form, shown here as simpleDOE (sDOE), it can be leveraged quickly, with a lower learning curve, to solve the type of problems faced in the mass spectrometry lab.

In its simplest form, sDOE consists in creating a full-factorial matrix to determine the most relevant parameters and interference between them. In this work for example, some general knowledge of relevant parameters was acquired through 16 experiments, followed by a protein-specific optimization using 10 experiments.

In principle, this sDOE step can be followed by co-optimization of the top two parameters, without necessarily going to the modeling step. For experiments considered as critical or in the context of regulation, additional statistical tests on replicates may be needed to evaluate the significance of the result, although replication on a single set of conditions can be enough to derive measurement errors [START_REF] Hecht | Optimizing mass spectrometry analyses: A tailored review on the utility of design of experiments[END_REF]. Where necessary, further FC optimization towards maximal coverage could be reached by multi-level optimization of relevant parameters followed by modeling of the equation for each parameter's influence. These sDOE-2 steps will be covered in a future publication.

In many cases, sDOE shows great potential for rapid and efficient optimization of experimental conditions and instrumental setup in mass spectrometry and associated techniques.

Figure 1 .

 1 Figure 1. Overview of input and output parameters. The ETD cycle includes the 4 following steps: 1) Accumulation of precursor ions, 2) Accumulation of reagent ions, 3) ETD Reaction, 4) Fragment transfer and 5) Detection of fragment ions. An additional step was considered: 6) Data processing. Image inspired from Bruker Daltonics illustration, with permission.

Figure 2 .

 2 Figure 2. Manual optimization of precursor accumulation time and isolation width. A. Precursor intensity as a function of accumulation time for ubiquitin and apomyoglobin, B. Precursor intensity as a function of isolation width for ubiquitin. The accumulation time used in the isolation width study was 300 ms.

Figure 3 .

 3 Figure 3. 2D plots of co-optimization of parameters for maximizing FC. A. Fragmentation coverage as a function of pairs of (PI,RT) values. B. Fragmentation coverage as a function of pairs (RI,RT) of values.

Figure 4 .

 4 Figure 4. Distribution of FC for ETD decaplicate experiments on ubiquitin and apomyoglobin.

Factor

  of 70000-76000 which correspond to a reagent accumulation time of approximately 10 ms. Since the maximum reagent intensity and the length of reagent injection time are dependent on the amount of reagent in the cartridge, and the reagent gets consumed over time, keeping reagent intensity at 70000 makes it possible to do multiple experiments without being limited in reachable reagent intensity. Increasing the reagent injection time

Figure 5 Figure 5 .

 55 Figure5shows the extent of progress that could be achieved with manual optimization (upper panel) versus

Acknowledgements

Financial support of European FEDER funds (grant #2699-33931, SyMBioMS), Région Centre Val de Loire (Biocontrol), Université d'Orléans and CNRS for high resolution mass spectrometry is gratefully acknowledged. We thank Dr Bertrand Castaing for the kind gift of HU alpha protein and Dr Kathrin Breuker for stimulating discussion.


C Finot

T N Nguyen

J Fatome

T Chartier

S Pitois

L Bramerie

M Gay

J.-C Simon

Numerical study of an optical regenerator exploiting self-phase modulation and spectral offset filtering at 40 Gbit/s

Keywords:

In this work, we numerically investigate the performances of optical regenerators based on self-phase modulation and spectral offset filtering at 40 Gbit/s. We outline the different effects affecting the device performances and explain the choice of the optimal working power. The impact of the regenerator on the output signal is also analysed through a statistical approach. Both single and double stage configurations are investigated.

Introduction

With the development of long haul optical telecommunication systems working at high repetition rates (40 Gbit/s and beyond), performing an all-optical regeneration of the transmission signal has become of a great interest to combat the cumulative impairments occurring during the signal propagation and overcoming the electronics bandwidth limitations of the current devices [1]. Indeed, during its propagation, the signal undergoes various degradations such as amplified spontaneous emission (ASE) noise accumulation, chromatic dispersion as well as intra-channel non-linear effects [START_REF]Fiber-Optic Communication Systems[END_REF]. Non-linear effects lead to jitter degradations (in time an amplitude) and ghost-pulse generation in the zero bit-slot [START_REF] Essiambre | [END_REF].

Consequently, it is required to limit the amplitude jitter and the timing jitter of the pulses and to enhance the extinction ratio (ER) of the signal. To achieve simultaneously such operations, different optical methods based on non-linear effects in optical fibers have been proposed; for example: the use of non-linear optical loop mirrors (NOLMs) [4], four-wave mixing [5], or self-phase modulation (SPM) [START_REF] Mamyshev | All-optical data regeneration based on self-phase modulation effect[END_REF]. The latest method is based on spectral broadening followed by an offset spectral filtering and has been proposed in 1998 by P.V. Mamyshev. Many works, both experimental [START_REF] Matsumoto | [END_REF][START_REF] Provost | Self-Phase Modulation-based 2R optical regenerator for the simultaneous processing of two WDM channels[END_REF][START_REF] Provost | Generalisation and experimental validation of design rules for self-phase modulation-based 2Rregenerators[END_REF][START_REF] Finot | [END_REF][11][12][13] and theoretical [START_REF] Provost | Generalisation and experimental validation of design rules for self-phase modulation-based 2Rregenerators[END_REF]11,[14][15][16][17][START_REF] Lakoba | Multicanonical Monte-Carlo simulations of the dynamix power transfer characteristic of an all-optical 2R regenerator[END_REF][START_REF] Pitois | [END_REF], have dealt with the possibilities of such a device known as the Mamyshev regenerator. For example, its capacity of multi-wavelength regeneration [START_REF] Provost | Self-Phase Modulation-based 2R optical regenerator for the simultaneous processing of two WDM channels[END_REF]17], the existence of eigen-pulses in concatenated regenerators [START_REF] Pitois | [END_REF] or its inclusion in a set-up compatible with DPSK signals [20] have been recently reported.

In Ref. [14], general theoretical guidelines have been proposed to design a highperformances single-stage regenerator with a transfer function exhibiting both a high extinction ratio improvement and a locally flat region on the one-level called the plateau. In an other numerical study, the authors have tried to link the properties of the transfer function with the performances of the regenerator in terms of Q-factor improvement [16]. The authors have noticed that the transfer function itself is not sufficient to fix the working power giving the best performance and that a trade-off between the improvements and impairments raised by the regenerator is then necessary. But to date, no theoretical argument has been suggested to clearly explain this observation.

More importantly, in order to restore the signal at its initial wavelength, a double-stage configuration is required and once again, even if some experimental devices have already been demonstrated [START_REF] Matsumoto | [END_REF], no detailed theoretical work has been reported in the literature.

In this article, we propose a numerical study of the Mamyshev regenerators both in single-stage and double-stage configurations. Our study combines both the transfer function and the Q-factor improvement approaches. Based on the statistical study of the initial and regenerated signals, we present an interpretation of the impact of the regenerator on the regenerating signal and propose an explanation regarding the choice of the working power.

Finally, we highlight several key points which affect the regenerator performances. This paper will thus be organised as follows. After having recalled the principle of the Mamyshev regenerators, we will investigate the performance of a single-stage regenerator both in terms of Q-factor improvement and statistical impact. In the fourth part of the paper, we will propose a set of explanations about the working power influence and in the final part, we will numerically study a double-stage device.

Principle and set-up

The set-up of the single stage Mamyshev regenerator under study is described in Fig 1 . The signal to be regenerated is first amplified in order to reach an optimum average power P m (corresponding to a peak power P 0 for '1' pulses). An optical bandpass filter (OBPF), centred at the signal wavelength λ 0 and characterized by its spectral full-width at half maximum (FWHM) δ f 1 is also used to reduce the ASE noise power [16]. A spectral broadening is then achieved through the SPM effect occuring during the propagation in a highly non-linear optical fiber (HNLF) with a length L. The broadened spectrum is finally partially sliced by a second OBPF shifted by ∆λ. The spectral bandwidth of this second filter δ f 0 , which determines the output pulse width, is then fixed according to the initial pulse properties. Fig. 1. Set-up of the single stage Mamyshev regenerator. Erbium Doped Fiber Amplifier (EDFA), Optical BandPass Filter (OBPF), Highly Non-Linear Fiber (HNLF).

In Ref. [14] the authors have shown that, according to the system parameters (L, P 0 and ∆λ ), three different regimes of transfer function ( P out = f ( P in ) ) can be observed : a transfer function (TF) characterised by a plateau (B regime), by a continuously increasing behaviour (C regime) or by a non-monotonic regime (A regime). The transfer function associated with the plateau zone of the B regime (and corresponding to an input peak-power P theo ) has been considered as the most suitable to achieve an efficient regeneration process, mainly due to its capacity to equalize pulse-to-pulse peak-power fluctuations.

Our regenerator is based on a HNLF with a length L of 3500 m, a non-linear coefficient γ of 8.4 W -1 .km -1 , a dispersion D of -0.7 ps.km -1 .nm -1 and a linear loss coefficient α of 0.6 dB.km -1 . We have included in our simulations the ASE noise introduced by the erbium doped fiber amplifier (EDFA) by taking into account a noise figure N f = 4 dB.

Following the method detailed in [14], we have calculated the TF thanks to the propagation of a single Gaussian pulse having a temporal FWHM of T 0 = 6.25 ps, which corresponds to the pulses used in a classical 40-Gbit/s return-to-zero-25% (RZ-25) optical telecommunication system.

Our simulations have been computed by means of the software VPI WDM Transmission Maker and rely on the numerical integration of the non-linear Schrödinger equation by a split-step Fourier method. Note that polarisation mode dispersion (PMD) as well as other higher-order non-linear effects (Raman, Brillouin or self-steepening contributions) have been neglected in our simulations.

OBPF 2 is a Gaussian filter with a spectral width δf 0 = 70 GHz and a spectral offset ∆λ = 160 GHz. For an initial 4-th order supergaussian OBPF with a spectral width δf 1 = 100 GHz (OBPF 1), we obtain a B-like TF (Fig. 2, solid line), with a plateau around P theo = 1.05 W. Let us note that we can easily obtain other TF types by simply adjusting δf 1 . Indeed, by changing the spectral width of the initial filter, we also modify the initial temporal width of the pulses entering the HNLF. Given the dependence of TF with the initial pulse properties, this leads to noticeable changes of the TF. For example, if δf 1 = 160 GHz (mixed line), we obtain a C-like TF and by using narrower filters, an A-like TF could also be achieved.

Performance of the single-stage regenerator

In this part, we aim to quantify the regenerative capacity of the B-like single-stage Mamyshev regenerator described in the above section. The transfer function is usually built by computing the response of the regenerator to an input single pulse. However, in order to fully characterize the performance of a regenerator, a stream of pulses (namely a pseudo-random binary sequence (PRBS)) has to be taken into account. In that context, the usual parameter widely used to evaluate the signal quality is the Q-factor [START_REF]Fiber-Optic Communication Systems[END_REF] :

Q = (V 1 -V 0 ) / (σ 1 -σ 0 ), with
V 1 and V 0 the average electrical voltage of the ones and zeros respectively at the detection instant, and σ 1 and σ 0 the corresponding standard deviations. A good estimation of the performance of our regenerator will be the Q-factor improvement defined as the ratio between the input and output signal Q-factor (or equivalently the difference when expressed in dBs).

On the other hand, the Q-factor parameter, obtained in the electrical domain, often hides the physical meaning of the signal degradation and consequently, this data itself is not enough to get a precise understanding of the degradation affecting the pulse train. So in order to complete the analysis, we will also get interested in the statistical properties of the optical signal, i.e. its temporal pulse width, its peak-power, its temporal position and its extinctionratio.

Initial input pulse properties

In this section, we shall first study the features of the degraded 40-Gbit/s PRBS signal used at the input of the regenerator.

In order to model realistically the signal degradation [21] and to maintain an affordable computation time, we have numerically propagated a (2 11 -1)-long 40-Gbit/s RZ-25 PRBS signal in the recirculating loop system represented in Fig. 3(a). The FWHM of the pulses is 6.25 ps, the initial extinction ratio (defined by the ratio between the '1' peak-power and the '0' level) is 20 dB and the initial signal average power launched into the loop is 1 dBm. The loop is made of 100 km of standard single mode fiber (SMF) with an anomalous dispersion of 16 ps/km/nm, a dispersion slope of 0.08 ps/km/nm 2 , a non-linear coefficient of 1.3 W -1 .km -1 and an attenuation of 0.2 dB/km. Dispersive effects are compensated by 16 km of dispersion compensating fiber (DCF) with a normal dispersion of -100 ps/km/nm, a dispersion slope of -0.5 ps/km/nm 2 , a non-linear coefficient of 5.2 W -1 .km -1 and an attenuation of 0.6 dB/km. This compensating fiber is preceded and followed by erbium doped fiber amplifiers (EDFA) with gains G A = 14 dB and G B = 15.6 dB, respectively. Both amplifiers are followed by a 4-th order supergaussian OBPF with a spectral width of 400 GHz in order to partly remove the ASE noise introduced by the amplifiers. Finally, the 40-Gbit/s receiver module is preceded by a supergaussian OBPF with a FWHM of 160 GHz. of the number of turns completed into the loop. As expected, the Q-factor progressively decreases due to the cumulative impairments occurring during the propagation, namely the accumulation of in-band ASE noise and combined effects of chromatic dispersion and nonlinear effects (intrachannel cross-phase modulation and four-wave mixing [START_REF] Essiambre | [END_REF]). Consequently, without any regeneration, only 20 round trips (2000 km) could be covered before falling down to the transmission limit corresponding to Q in = 6. Let us note that the present loop is not fully-optimized to its best performances and that some pre-chirp/post-chirp could still enhance the transmission distance.

In the following of the paper, we will particularly investigate the evolution in the regenerator of three different levels of Q-factor at the input, i.e. Q in = 24 (signal of excellent quality corresponding to a 200 km propagation distance), Q in = 6 (limit transmission signal quality [START_REF]Fiber-Optic Communication Systems[END_REF] ) and Q in = 3.3 (signal of extremely poor quality after 3500 km of propagation which can only be used in presence of forward error correction (FEC)). Eye-diagrams corresponding to these three levels of degradation are presented in Figs. 3(c) and clearly reveal that the degradation of the input Q-factor has multiple sources: amplitude jitter, timingjitter as well as an extinction ratio decrease.

In order to get a more precise quantitative description of the degraded 40-Gbit/s PRBS signal, we now consider in Fig. 4 the average values and root-mean square (rms) deviations of several signal key parameters.

As can be seen in Fig. 4(a-b), the amplitude jitter σ Ain (rms deviation of distribution of the peak power of the pulses) as well as the temporal jitter σ Tin (rms deviation of distribution of the central pulse position as defined by its highest peak-power) strongly increase during the propagation to reach after 4000 km, 20% of the pulse width and peak power, respectively.

The effects of the ASE accumulation and the ghost-pulses generation are also apparent through the significant decrease of the extinction ratio (Fig 4(d)) which drops from 10 dB after 40 round trips. We can finally note in Fig. 4(b) that during the propagation, the temporal pulse width increases, with a rise as high as 20 % of its initial value after 4000 km, leading to a decrease of the average pulse peak-power. We will discuss in more details the implications of those fluctuations in section 4.3.

Performance of the regenerator and optimum working point

We first test our regenerator on a degraded signal with Q in = 6. Eye-diagrams after regeneration are plotted in Figs. 5 for two regenerator input values of the average powers P m (called the working power (WP)). As can be seen, in both cases, the regenerator improves very efficiently the extinction ratio of the pulse train and also tends to limits the amplitude jitter. The eye is then clearly more opened than in Fig. 3(c2), despite the presence of some additional timing jitter. We can also make out that the WP largely influences the quality of the output pulse, with better results obtained at a WP of 180 mW. In order to quantify the impact of the regenerator, we measure the improvement of the signal in terms of Q-factor enhancement [16], i.e. the ratio Q out / Q in where Q out is the quality factor at the output of the regenerator. We present in Fig. 6(a) the Q-factor improvement as a function of the WP for three different levels of degradation. Let us notice that studying the Q factor improvement is not sufficient to quantify the performances of a regenerator [START_REF] Gay | 2R and 3R optical regeneration : from device to system characterization[END_REF].

However, in the context of our study, where we focus on the physical mechanism of the regeneration process, the Q-factor is a convenient parameter to quantify the quality of the signal after regeneration. First and in disagreement with the TF approach described in Fig. 2, we can observe, for Q in = 24 and Q in = 6, that the optimal regeneration is not obtained on the plateau (around P theo ), but for a higher WP. This observation is quite surprising but consistent with the behaviour previously highlighted in Ref. [12,16]. We can also make out that the signal quality improvement strongly depends on its initial degradation. Indeed, for a high Q in (Q in = 24), the optimum regeneration is achieved on a relatively narrow input power range and reaches 3.5 dB for an input average power of 170 mW. On the contrary, for a more degraded signal (Q in = 6), the performances of the regenerator fortunately do not exhibit such a strong power dependence. (c) Q-factor improvement versus Q in for different WP : optimum WP (solid black line), WP of 130 mW, 158 mW and 183 mW (solid grey line, dotted grey line and mixed grey line, respectively), corresponding to input peak-powers of 1 W, 1.2 W and 1.4 W, respectively. The shaded area denotes the zone where the regenerated Q-factor is below 6.

We have also plotted in Fig. 6(b) the evolution of the optimal average WP as a function of Q in . We can see that the optimum power has to be tuned above the plateau area between 140 and 190 mW and depends on the level of degradation, with a decrease of the power for low Q in . We will give some explanations about such a dependence in the next section.

The next plot, Fig. 6(c), shows the Q-factor improvement as a function of the input Q-factor Q in for several values of WP. For comparison, we have plotted in solid black line the maximum feasible improvement; that is to say for each value of Q in , the maximum of Q-factor improvement at the optimum WP. But in practice, it could be easier to fix the amplifier to a given WP, no matter what Q in is. This is the reason why we have also plotted the results corresponding to a WP of 130, 158 and 183 mW in solid grey line, dotted grey line and mixed grey line, respectively. It is then important to notice that in all cases, we have improved the signal quality, but the value of this improvement could be significantly different. For example, the choice of a working power of 130 mW (corresponding to the value of the plateau P theo ) clearly appears to lead to a non-optimum regeneration whereas a working power of 183 mW is closer to the all-cases optimum regeneration.

In order to better understand the behaviours related above, we now quantify the action of the 2R Mamyshev regenerator in terms of statistics. In Figs. 7, we study the modification of the statistics induced by the regenerator on the 40-Gbit/s PRBS signal as a function of the input average power for three different levels of degradation (Q in = 24, 6 and 3.3, solid black line, mixed line and grey dotted line, respectively). For memory, the optimal WP giving the highest Q-factor improvement found in part 3.2, Fig. 6 are indicated by means of a black circle, a grey triangle and a grey square for Q in = 24, 6 and 3.3, respectively.

Let us first study the benefits provided by our regenerator in terms of amplitude jitter reduction. We define the amplitude jitter improvement as the ratio between the input and output amplitude jitter σ Ain / σ Aout (or the difference when expressed in dBs). We can see from Fig. 7(a) that for a WP above 125 mW, the regenerator always leads to an amplitude jitter improvement, with a maximum as high as 2.6 dB of improvement for weakly degraded signals ( Q in = 24).

We have also calculated the action of the regenerator on the temporal width fluctuations by defining the temporal width fluctuations improvement as the ratio between the input and output rms deviation of the pulse width distribution σ ∆Tin / σ ∆Tout . We can see in Fig. 7(b) the beneficial impact of the regenerator [13]. The output pulse width fluctuations strongly decreases whatever WP for Q in =6 and Q in =3.3 and decreases if WP > 130 mW for Q in = 24. Note that this regeneration of the temporal pulse width is a specific benefit of the Mamyshev regenerators compared with other usual 2R techniques. This is due to the presence of the output filter whose role is to shape the pulse profile at the output of the regenerator.

If we now have a look at the regenerator impact in terms of timing jitter, we can calculate σ Tin / σ Tout . We then observe in Fig. 7(c) that the regenerator introduces an extra temporal jitter [15] which always degrades the input signal by inducing an average of 1 dB penalty whatever the WP is.

Finally, we have plotted the ER enhancement defined as ER out / ER in in Fig. 7(d).

These results show that our regenerator leads to a significant ER improvement for any input average power between 100 and 250 mW.

Explanations for the choice of the working power

In the last subsection, thanks to the statistical analysis completed on the 40-Gbit/s PRBS signal, we now better understand the origin of the Q-factor improvement provided by the Mamyshev regenerator as a function of the input average power. However and in agreement with reference [16], we observed the non intuitive phenomenon that the effective optimal WP does not necessary correspond to the range of powers at which the TF exhibits a plateau area around P theo . It is the reason why, in the present section, we will try to link the 40-Gbit/s PRBS Q-factor improvement approach and its statistical analysis with the TF approach computed by the propagation of a single Gaussian pulse. We will then propose and discuss explanations about the difference between the usual theoretical WP P theo and the effective optimal WP where the Q-factor improvement is maximal as well as an explanation to the dependence of Q factor improvement over Q in .

Induced timing jitter and output temporal width fluctuations

In reference [14], the aim of the paper was mainly focus on both the peak-power equalisation and the extinction ratio-improvement by means of the single-pulse TF optimisation.

Following a similar approach, we can calculate the temporal changes affecting the output Gaussian pulse properties as a function the input power. Let us note that this study is different from the results dealing with the nominal soliton number parameter presented in [14] (cf. Fig. 5 of this reference) where the regenerator properties (fiber length and filter offset) where constantly adjusted in order to operate on the plateau area.

Results plotted in Fig. 8(a) show that, in the case of a single pulse propagation, the fluctuations of the output temporal width are maximum in the range of power corresponding to the plateau area, which has been recently confirmed by experimental FROG measurements [START_REF] Provost | Generalisation and experimental validation of design rules for self-phase modulation-based 2Rregenerators[END_REF]. Let us however relativize the impact of those fluctuations: as can been seen in Fig. 7 Note that the sign of the delay depends on the spectral offset sign. Changing the sign of the spectral offset in -∆λ will lead to an opposite time delay. Once again, we notice that the temporal delay exhibits maximum variations in the plateau zone. But here, the most interesting point does not lie in the absolute value of the delay in itself, but rather in its fluctuations, which directly induce timing jitter. Figure 8(c) shows the amplitude-derivative of the time-delay as a function of input power. We can clearly observe that this function is remarkably maximum at the plateau zone and that the time delay fluctuations and thus the extra induced timing jitter could be minimised by at least a factor four for a higher WP (for example 180 mW) which could partially explain the behaviour observed in Fig. 7 for Q in = 24 and Q in = 6 where, as the input signal is of a pretty good quality, the optimum WP corresponds to the minimum of timing jitter degradation provided by the regenerator itself.

Pulse-to-pulse overlapping induced amplitude jitter

Besides this first aspect, another key point we got interested in is the effect provided by the pulse-to-pulse overlapping occurring inside the regenerator due to the chromaticdispersion induced pulse broadening [START_REF] Rothenberg | [END_REF]. To this aim, we study, in a similar manner as in references [14,17], the amplitude jitter induced by the cross-phase modulation occurring during the adjacent pulse-to-pulse overlapping (what is called "collision" in ref. [14]) in a sequence of 2 or 3 pulses propagating in the regenerator. We have also considered a PRBS of 1024 pulses which includes all the possible combinations up to tens consecutive pulses. Using longer sequences has not lead to noticeable changes. In order to be solely sensitive to the impact of this overlapping, we have considered an initial signal with an infinite ER and we have neglected the ASE introduced by the EDFA. For each input peak-power, we have computed the difference ∆A (see Fig. 9 As in the previous timing-jitter study, the degradation due to overlapping is more pronounced on the plateau zone than at higher working power: the minimum of induced amplitude jitter is observed around 1.3 W, corresponding to a 40-Gbit/s PRBS average power of 170 mW. These conclusions then clearly explain the WP offset observed in Fig. 7 for the Q in = 24 initial pulse train where the optimal input average power mostly corresponds to the minimum of amplitude jitter degradation introduced by the regenerator itself which, with the intrinsic enhancement provided by the Mamyshev method, participates to reach a maximum amplitude jitter improvement. In other words, for input signals of good quality, the main factor affecting the 2R performance is the pulse-to-pulse overlapping induced amplitude jitter.

For a poor quality signal (Q in = 3.3), the interpretation is still more complex.

Let us finally note that it could be possible to significantly reduce the impact of pulseto-pulse overlapping by an appropriate choice of the regenerator parameters (L, ∆f), as well as the initial OBPF 1 (as it determines the duty-cycle of the signal). More precisely, using a lower nominal soliton number parameter (as defined in ref [14]) would allow us to partly eliminate this deleterious pulse overlapping [14]. However such a change in the regenerator features is usually at the cost of the ER. So, a trade-off between the ER improvement and the collision-induced amplitude jitter reduction has to be found. Such a study is however beyond the scope of the present paper.

Impact of temporal width change

In summary, from the conclusions outlined in the last subsections 1 and 2, it could be efficient to use a WP higher than the plateau power in order to minimise the timing jitter intrinsically induced by the regenerator, as well as the amplitude jitter induced by the pulse-to-pulse overlapping.

In this subsection, we highlight the strong impact of the initial temporal pulse fluctuations, leading to an additional source of amplitude jitter. Statistics of the input signal presented in Fig. 4 have revealed that the temporal width undergoes some non-negligible fluctuations (both in terms of average value and standard deviation). This can have a large impact on the regenerator behaviour. Indeed, it has been previously demonstrated [11,14,[START_REF] Pitois | [END_REF] that, contrary to NOLMs or other 2R regenerating devices, temporal width of the incoming pulses is a crucial parameter in the sense that the transfer function strongly depends on this parameter. In this context, considering a single TF relying on well defined 6.25 ps Gaussian pulses can be viewed as an inaccurate approximation.

In order to better apprehend the impact of this temporal width distribution, we have plotted in Figs. 10(a We have plotted in Figs. 10(b) the transfer functions obtained by taking into account the whole distribution of temporal pulse width. In order to do that, we have taken an input series of 256 Gaussian pulses with a temporal width distribution similar to that exposed in Figs. 10(a) (same average, same deviation). Note that the input pulses have been separated by an empty time slot in order to remove pulse-to-pulse overlapping induced effects. All the initial pulses have the same initial energy. In other words, the fluctuations of temporal width are directly associated with fluctuations of the input peak power. The output peak power (black solid line) represents the output pulse train average peak-power. It can be seen that in all cases, a TF exhibiting a plateau is achieved. However, the change in the average temporal width leads to a slight shift in the value of the plateau which increases from an input average power of 140 mW for Q in = 24 up to a 160 mW for Q in = 3.3. But, the most striking point is undoubtedly the significant fluctuations in the TF. We have plotted as shaded area the whole values of the output peak-powers. For high Q in , as the deviation of the temporal width is not high, the fluctuations are not severe. But for more degraded signals, the temporal width fluctuations may lead to a non-negligible additional source of amplitude jitter in the regenerated signal. It is then remarkable that the output amplitude fluctuations are reduced when increasing the input average power, in agreement with the trends observed in Fig. 7(a). This phenomenon underlines again the key role played by the amplitude jitter improvement in the choice of the optimal input power and thus the trend to operate with a WP higher than the plateau area.

Let us however note that the choice of taking all the initial pulses with the same energy represents an approximation, since in reality temporal fluctuations are also not strictly correlated with peak-power fluctuations. More realistic simulations could be carried out by use of more sophisticated methods such as multicanonal Monte-Carlo approach [START_REF] Lakoba | Multicanonical Monte-Carlo simulations of the dynamix power transfer characteristic of an all-optical 2R regenerator[END_REF], but such an analysis is beyond the scope of the present paper.

Impact of Extinction Ratio improvement

The three previous subsections have provided us several trends which help us to better understand the choice of the optimal WP for high or intermediate qualities of initial signals (Q in = 24 or Q in = 6). We have indeed shown that amplitude-and timing-jitter reductions are the key issues that govern the choice of the WP.

On the contrary, for a highly degraded signal (Q in = 3.3), the choice of the WP becomes a trade-off between the ER improvement and the amplitude jitter reduction. Indeed, in Fig. 7(d), as the evolution of the ER improvement is quite flat for Q in = 6 between 125 and 225 mW, the performances of the regenerator is weakly influenced by the ER improvement but rather governed by the amplitude-and timing-jitters reductions. On the opposite, variations as high as 4 dB can be observed for Q in = 3.3 in the same range of power. The Q in = 3.3 ER enhancement being a decreasing function with respect to WP, a large improvement of the ER will consequently lead to a lower WP with a minimum of 130 mW corresponding to the limit of a non degradation of the initial amplitude jitter (σ Ain / σ Aout = 0 dB). This behaviour which explains the decrease of the optimal WP observed for Q ini < 6 in Fig. 6(b).

Performance of the double-stage regenerator

Let us first recall that in order to restore the initial signal wavelength, a second wavelengthconverter stage is required. We now study the performance of such a double stage regenerator.

Principle and set-up

The set-up based on a two-stage Mamyshev regenerator is presented in Fig. 11. Two single stage regenerators are then concatenated. In order to have a device compatible with a bidirectional use [START_REF] Matsumoto | [END_REF][START_REF] Provost | Self-Phase Modulation-based 2R optical regenerator for the simultaneous processing of two WDM channels[END_REF], we have used the same HNLF properties in the two stages of regeneration.

Let us outline that the OBPF 1 and 3 can be identical, in which case a configuration B + B is obtained. But they can also be different, leading to various arrangements such as C + B or A + B, or any other type of association. This opens many possibilities, but a complete study of those combinations is out of the scope of the present paper. We will then only focus on the B + B arrangement. The TF obtained by this concatenation is plotted in Fig. 12 for several values of the intermediate amplifier gain G 2 (noise figure of the amplifier N f = 4 dB). We have plotted the results for three different gains of 11.3 dB, 12.8 dB and 14.3 dB (dotted line, solid black line and mixed line, respectively). We can see that the 12.8 dB gain leads to a good power equalisation (Fig. 12(a)) and limits the impact of the extra-temporal jitter (Fig. 12(c)). Indeed, temporal jitter introduced by the first regenerator is partly compensated by the second one.

Let us however outline that the time delay introduced by each regenerator do not add and that the concatenation of two regenerator leads to a more complex dynamics [START_REF] Pitois | [END_REF]. Regarding the output pulse temporal width fluctuations (Fig. 12(b)), they are quite limited.

We can also see from Fig. 12(a) that the TF is now steeper with a larger plateau and an enhanced ER compared to a single stage B (black circles).

Q-factor improvement

We have plotted in Fig. 13 the performances of the double stage regenerator in terms of Qfactor improvements. Compared to the results described in section 3.2 Fig. 6, several differences can be outlined. First, we can see in Fig. 13(a) that the regeneration input power range is broader than for a single stage regenerator. We can also observe that the dependence (Fig. 13(b)) of the optimal WP versus the initial degradation is less pronounced than in the single stage configuration, which is beneficial for system implementation.

We can also make out from the comparison of Figures 6(c) and 13(c) that for a low degraded signal, a double stage device is slightly less efficient than a single-stage configuration. But for lower Q ini (Q ini < 10), the double stage leads to regeneration enhancements. For example, the regeneration (Q out > 6) of a degraded signal with a quality factor as low as 3.3 becomes possible. Let us however here recall that the Q-factor data must be handled with special care in the case of optical regenerators [START_REF] Gay | 2R and 3R optical regeneration : from device to system characterization[END_REF] and that a link with bit error rates is not straightforward in that context.

Statistical impact of the double stage regenerator

We carry out in this sub-section similar study as the one presented in part 3.2. Results are plotted in Fig. 14 and the general trends are similar to the results obtained for a single stage configuration. As compared with the quantitative results obtained for a single stage, we can note the followings points. First, the amplitude jitter reduction capacity is improved, which is consistent with the enhanced plateau observed Fig. 12(a). Then, improvement of the temporal width fluctuations is also observed, in agreement with Fig. 12(b). The extinction ratio improvement is also significantly enhanced, in agreement with the stepper TF observed in Fig. 12(a). Regarding the timing jitter introduced during the regeneration, the double stage scheme does not seem to significantly further degrade this data. We could have expected from Fig. 12(c) a reduction of the timing jitter, but one has also to take into account the other sources of timing jitter such as intra channel pulse-to-pulse overlapping. As a consequence, for a low degradated initial signal, the extra timing jitter is slightly higher in the double stage configuration than in the single stage, which can partly explain the performance degradation of the double stage regenerator for Q in above 10.

Conclusion

We have numerically studied the performances of a 2R Mamyshev optical regenerator used in a 40-Gbit/s RZ-25 system. Based on this given configuration, we have outlined several key elements of Mamyshev regenerators which have to be taken into account into the set-up design : the induced timing jitter, the temporal width fluctuations and the pulse-to-pulse overlapping. Impact of the fluctuations of the temporal width of the initial pulses has also been outlined. Those generic elements enable us to physically explain the differences which can be observed between the general predictions of Provost et al. in Ref. [14] and the working power which is required to achieve optimum eye opening for our given configuration.

We have also used a statistical approach to describe quantitatively the role of the regenerator and to explain the dependence of the performance on the degradation of input signal.

We have finally proposed and analyzed a two-stage configuration where the deleterious impact of the timing jitter can be partly minimized. Improvement of amplitude jitter and extinction-ratio is demonstrated and the distance of propagation can be improved by using such a double-stage in front of the receiver. A further step will be the investigation of the regenerator capacity in a loop configuration but let us note that the requirements of such a regenerator will be quite different : due to the periodic use, improvements of the ER and amplitude jitter will not be so severe as amplitude jitter and noise will not develop so heavily.

On the contrary, in a loop configuration, issues such as timing-jitter reduction and collisions induced-effects [14] will become even more crucial. Given those given specific requirements, we anticipate that further optimization process adapted to in-line regeneration will be necessary.

Fig. 2 .

 2 Fig. 2. Transfer function of a single-stage regenerator. Evolution of the output peak-power versus the input power for two different spectral widths of OBPF 1 (δf 1 = 100 GHz and δf 1 = 160 GHz, solid black line and mixed line, respectively). The input average power corresponds to the average power of a 40-Gbit/s pseudo-random binary sequence (PRBS) signal.

Fig. 3 .

 3 Fig. 3. (a) Scheme of the loop system used to model the degradation of the 40-Gbit/s PRBS signal during the propagation. (b) Evolution of Q in as a function of the number of round trips. (c) Optical eye-diagrams for three different levels of degradation ( Q in = 24, 6 and 3.3, subplots 1, 2 and 3 respectively )

Fig. 4 .

 4 Fig. 4. Statistics of the 40-Gbit/s PRBS signal as a function of propagation distance (or equivalently number of turns n). (a) Evolution of the average peak-power (solid black line, left, normalized by the initial peak-power P in ) and the amplitude jitter (grey dotted line, right, defined as the standard deviation of the peak-power distribution, normalized by the average peak-power) (b) Evolution of the temporal FWHM T n (left, solid black line) and its standard deviation (right, normalized by the temporal width T n , grey dotted line). (c) Temporal jitter (defined as the standard deviation of the distribution of the central position of the pulse, normalized by the FWHM average temporal width T n at turn n) (d) Evolution of the extinction ratio.

Fig. 5 .

 5 Fig. 5. Optical eye-diagrams after regenerator for Q in = 6 for two different working powers. (a) WP = 130 mW (b) WP = 180 mW.

Fig. 6 .

 6 Fig. 6. Regenerator performances. (a) Q-factor improvement as a function of the WP for three different levels of degradation ( Q in = 24, 6 and 3.3, solid black line, mixed line and grey dotted line, respectively). (b) Optimal WP versus the Q in . In Figs (a) and (b), the shaded areas correspond to the plateau zone (see Fig. 2(a)).(c) Q-factor improvement versus Q in for different WP : optimum WP (solid black line), WP of 130 mW, 158 mW and 183 mW (solid grey line, dotted grey line and mixed grey line, respectively), corresponding to input peak-powers of 1 W, 1.2 W and 1.4 W, respectively. The shaded area denotes the zone where the regenerated Q-factor is below 6.

Fig. 7 .

 7 Fig. 7. Statistics regarding the single-stage regenerator impact on a 40-Gbit/s degraded signal for three levels of degradation (input Q-factor of 24, 6 and 3.3, solid black line, mixed line and grey dotted line, respectively) as a function of the regenerator input average power (a) Amplitude jitter improvement (b) Temporal-width fluctuations improvement (c) Timing jitter improvement and (d) Extinction-ratio improvement. The optimal WP found in part III.B, Fig. 5 are indicated by means of a black circle, a grey triangle and a grey square for Q in = 24, 6 and 3.3, respectively. The shaded areas correspond to the plateau zone (see Fig. 2).

  (b), the 40-Gbit/s statistical study outlines a clear improvement of the regenerated pulse train in terms of temporal width fluctuation. In other words, the fluctuations induced by the regenerator are largely compensated for by the intrinsic enhancement provided by the Mamyshev method [13] and are consequently not sufficient to explain the WP offset.

Fig 8 .

 8 Fig 8. (a) Evolution of the temporal width ∆T out of the output pulse as a function of the input peak-power

  (a)) between the two peak-power extrema obtained in the output pulse sequence. The quantity ∆A qualifies the amplitude jitter induced by the intrachannel interactions. Results are normalized by the average output pulse peak-power and are plotted in Fig. 9(b).

Fig. 9 .

 9 Fig. 9. Effects of pulse-to-pulse overlapping in the optical regenerator. (a) Principle and notations (b) Peak-power fluctuations at the output of the regenerator (normalized by the average output peak-power) for a sequence of 2 (black solid line), 3 (mixed line) and 1024 PRBS pulses (dotted grey line). The grey area represents the plateau zone

  ) the distribution of the temporal width of the 40-Gbit/s PRBS signal for three levels of degradation (Q in = 24, 6 and 3.3, (a1), (a2) and (a3) respectively).

Fig. 10 .

 10 Fig. 10. Impact of the statistical evolution of the temporal width for three different values of Q in of 24, 6 and 3.3 (subplots 1, 2 and 3 respectively). (a) Statistical distribution of the temporal pulse width of the 40-Gbit/s signal (b) statistical TF for input pulses with a temporal width distribution corresponding to Figs. (a). Black line corresponds to the average peak power of the pulse train and the shaded area corresponds to the whole output peak powers possibilities.

Fig. 11 .

 11 Fig. 11. Set-up of the Mamyshev double stage regenerator

Fig. 12 .

 12 Fig. 12. Characteristic of a double-stage regenerator. Evolution of the output pulse properties according to the input power for three different gains G 2 = 11.3, 12.8 and 14.3 dB (dotted line, solid black line and mixed grey line respectively). Results are compared with results of the single-stage regenerator (black circles) (a) Evolution of the output peak-power. (b) Evolution of the temporal width ∆T out of the output pulses (c) Derivative of the time delay versus the input peak-power. Results (b) and (c) are normalised by the input pulse width T 0 .

Fig. 13 .

 13 Fig. 13. Double-stage regenerator performances. (a) Q-factor improvement according to the WP for three different levels of degradation (same convention as Fig. 4(a)) (b) Optimal WP versus the input Q-factor. (c) Q-factor improvement relative to the input Q-factor for different WP : optimum working power (solid black line), working power of 183 mW (mixed grey line), Results of the two-stage regenerator are compared with the results of a single-stage configuration (dotted grey line).

Fig. 14 .

 14 Fig. 14. Statistics regarding the double-stage regenerator impact on a degraded signal for three levels of degradation (input Q-factor of 24, 6 and 3.3, solid black line, mixed line and grey dotted line respectively). Same conventions as Fig. 7. Optimal working points are the WP found Fig. 13.

Acknowledgments

This research was supported by the Agence Nationale de la Recherche (FUTUR project). We acknowledge fruitful discussions with B. Patin (Institut Carnot Bourgogne, France). We are very grateful to I. Joindot and M. Joindot (FOTON, France) for their comments on the manuscript.