domain: sdv phys

text:
Melanie Derde
email: melanie.derde@agrocampus-ouest.fr

Francoise Nau

Valérie Lechevalier-Datin

Catherine Guérin-Dubiard

Gilles Paboeuf

Sophie Jan

Florence Baron

Michel Gautier

Françoise Nau

Valérie Lechevalier

Véronique Vié
email: veronique.vie@univ-rennes1.fr

Native lysozyme and dry-heated lysozyme interactions with membrane lipid monolayers: lateral reorganization of LPS monolayer, model of the E. coli outer membrane

Keywords: AFM, atomic force microscopy, AMP, antimicrobial peptide or protein, BAM, Brewster angle microscopy, DH-L, dry-heated lysozyme, HEPES, 4-(2-hydroxyethyl)piperazine-1

come

ABSTRACT: Lysozyme is mainly described active against Gram-positive bacteria, but is also efficient against some Gram-negative species. Especially, it was recently demonstrated that lysozyme disrupts E. coli membranes. Moreover, dry-heating changes the physicochemical properties of the protein and increases the membrane activity of lysozyme. In order to elucidate the mode of insertion of lysozyme into the bacterial membrane, the interaction between lysozyme and a LPS monolayer mimicking the E. coli outer membrane has been investigated by tensiometry, ellipsometry, Brewster Angle Microscopy and Atomic Force Microscopy. It was thus established that lysozyme has a high affinity for the LPS monolayer, and is able to insert into the latter as long as polysaccharide moieties are present, causing reorganization of the LPS monolayer. Dry-heating increases the lysozyme affinity for the LPS monolayer and its insertion capacity; the resulting reorganization of the LPS monolayer is different and more drastic than with the native protein.

KEYWORDS: BAM, AFM, Langmuir film, Dry-heated lysozyme, LPS monolayer

INTRODUCTION

Antibiotic resistance due to decades of misuse in human and veterinary medicine, is causing an enormous public health problem. Several pathogens, such as Staphylococcus aureus and Klebsiella pneumonia, have developed multiple antibiotic resistance mechanisms. The consequence is difficult and expensive treatments of several diseases. [START_REF]World Health Organization, Overcoming Antimicrobial Resistance[END_REF] The number of these multi-resistant strains is increasing, but only three new antibiotic molecules against Grampositive multiresistant strains were registered since 1970, and none for Gram-negative multiresistant strains. [START_REF] Rosbach | Report on the Microbial Challenge -Rising threats from Antimicrobial Resistance (2012/2041(INI)), European parlaiment, commitee on the environment, public health and food safety[END_REF] Research for novel antimicrobial compounds is thus needed, besides the measures of the European Union to limit the spread of antibiotic resistances. Preferably, novel compounds should decrease the development rate and spread of antibiotic resistance.

Several natural proteins and peptides can be considered as potential candidates, especially the antimicrobial proteins or peptides (AMP) which act on the bacterial membranes. Their target, i.e.

the bacterial cell membrane, is a generalized and vital target, and thus antimicrobial resistance development remains limited. [START_REF] Nguyen | The expanding scope of antimicrobial peptide structures and their modes of action[END_REF][START_REF] Peschel | The co-evolution of host cationic antimicrobial peptides and microbial resistance[END_REF] AMP are mostly positively charged molecules, amphiphilic, flexible, and contain several hydrophobic residues, suggesting electrostatic and hydrophobic interactions with the bacterial cell membrane. [START_REF] Nguyen | The expanding scope of antimicrobial peptide structures and their modes of action[END_REF] These interactions can then lead to the membrane disruption, causing bacterial cell death or translocation of the AMP into the cytoplasm where these can interact with several intracellular targets. [START_REF] Nguyen | The expanding scope of antimicrobial peptide structures and their modes of action[END_REF][START_REF] Jenssen | Peptide antimicrobial agents[END_REF] One of the natural antimicrobial proteins, widely studied, is hen egg white lysozyme. This small enzyme (129 amino acid residues) was for a long time only known for its antimicrobial activity against Gram-positive bacteria, due to its muramidase activity. [START_REF] Jolles | Whats New in Lysozyme Research -Always A Model System, Today As Yesterday[END_REF][START_REF] Masschalck | Antimicrobial properties of lysozyme in relation to foodborne vegetative bacteria[END_REF] However, several studies suggest other mechanisms of action against both Gram-positive and Gram-negative bacteria, such as perturbation of DNA or RNA synthesis, activation of autolysin production, and membrane permeabilisation. [START_REF] Masschalck | Antimicrobial properties of lysozyme in relation to foodborne vegetative bacteria[END_REF][START_REF] Pellegrini | Effect of lysozyme or modified lysozyme fragments on DNA and RNA synthesis and membrane permeability of _Escherichia coli[END_REF][START_REF] Derde | Hen egg white lysozyme permeabilizes the E. coli outer and inner membranes[END_REF][START_REF] During | The non-enzymatic microbicidal activity of lysozymes[END_REF] The disruption of both the outer and cytoplasmic membranes of E. coli by native lysozyme has been recently demonstrated in our laboratory. [START_REF] Derde | Hen egg white lysozyme permeabilizes the E. coli outer and inner membranes[END_REF] Especially, pore formation in the outer membrane of E. coli has been described. [START_REF] Derde | Dry-Heating of Lysozyme Increases Its Activity against Escherichia coli Membranes[END_REF] Moreover, pore size and/or quantity were higher with dry-heated lysozyme as compared to the native protein. [START_REF] Derde | Dry-Heating of Lysozyme Increases Its Activity against Escherichia coli Membranes[END_REF] The physicochemical modifications resulting from dry-heating are an increased positive charge at physiological pH as well as increased flexibility and hydrophobicity while preserving the secondary and tertiary structure; these modifications could be responsible for the enhanced antibacterial activity of dry-heated lysozyme, similarly to what has been described for lysozyme modification by enzyme hydrolysis, heat denaturation, or fusion with several chemical moieties. [START_REF] Masschalck | Antimicrobial properties of lysozyme in relation to foodborne vegetative bacteria[END_REF][START_REF] Mine | Antimicrobial peptides released by enzymatic hydrolysis of hen egg white lysozyme[END_REF][START_REF] Abdou | Antimicrobial peptides derived from hen egg lysozyme with inhibitory effect against Bacillus species[END_REF][START_REF] Ibrahim | A structural phase of heat-denatured lysozyme with novel antimicrobial action[END_REF][START_REF] Hoq | Triclosan-lysozyme complex as novel antimicrobial macromolecule: A new potential of lysozyme as phenolic drugtargeting molecule[END_REF][START_REF] Ibrahim | Antimicrobial effects of lysozyme against Gramnegative bacteria due to covalent binding of palmitic acid[END_REF][START_REF] Ibrahim | Bactericidal action of lysozyme against Gram-negative bacteria due to insertion of a hydrophobic pentapeptide into its C-terminus[END_REF] However, the interactions between the outer membrane lipids of Gram-negative bacteria and both types of lysozyme (native and dry-heated) remain to be investigated. In the presently reported study, lipopolysaccharide (LPS) monolayers of E. coli K12 have been used as a model for the bacterial outer membrane. [START_REF] Brockman | Lipid monolayers: why use half a membrane to characterize proteinmembrane interactions?[END_REF][START_REF] Roes | Probing the Properties of Lipopolysaccharide Monolayers and Their Interaction with the Antimicrobial Peptide Polymyxin B by Atomic Force Microscopy[END_REF] These LPS monolayers were formed in a Langmuir trough at a controlled initial surface pressure (π initial ). Multiscale analysis of the interfacial film using tensiometry, ellipsometry, Brewster angle microscopy (BAM) and atomic force microscopy (AFM), enabled to investigate the LPS-lysozyme interactions, to study the consequences of lysozyme interaction on the LPS monolayer

MATERIALS AND METHODS

Proteins and lipids.

Native lysozyme (N-L) powder (pH 3.2) was obtained from Liot (Annezin, 62-France). It was heated for 7 days at 80°C in hermetically closed glass tubes to obtain dry-heated lysozyme (DH-L). Lysozyme (N-L or DH-L) was solubilized (around 0.5 g/L) in 5 mM 4-(2hydroxyethyl)piperazine-1-ethanesulfonic acid (HEPES) buffer (Sigma Aldrich, Saint-Quentin, France), pH 7.0, 150 mM NaCl (Fluka, Saint-Quentin, France). The concentration of the lysozyme stock solution was precisely determined by absorbance at 280 nm (extinction coefficient = 2.6 L/g) [START_REF] Expasy | Chicken lysozyme)[END_REF]. The protein solution was then diluted in the HEPES buffer to obtain the desired concentration for used lysozyme solutions.

The lipopolysaccharides (LPS) of E. coli K12 were obtained from Invivogen (Toulouse, France). The LPS were solubilized in 2:1 chloroform/methanol mixture at 0.5 g/L. Lipid A-(KdO) 2 (KLA) were purchased by Avanti Polar Lipids (Alabaster, Alabama, USA) and were solubilized in a 2:1 chloroform/methanol mixture at 0.67 g/L.

Lipid/protein monolayers.

The experiments were performed in a homemade TEFLON ® trough of 8 mL at 20°C. Before each use, the trough was thoroughly cleaned with successively warm tap water, ethanol, demineralized water, and then boiled for 15 minutes in demineralized water. After cooling the TEFLON ® trough was then filled with 8 mL HEPES buffer. The LPS or KLA were spread with a high precision Hamilton microsyringe at the clean air / liquid interface to obtain an initial surface pressure between 18 and 30 mN/m. After 1h to allow the solvent to evaporate and the lipids to organize, 50 µL N-L or DH-L solution were injected in the subphase with a Hamilton syringe to obtain a final protein concentration of 0.02, 0.03, 0.05, 0.1, 0.2, 0.3 or 1 µM.

2.3

Surface pressure measurements.

The surface pressure was measured following a Wilhelmy method using a 10 mm x 22 mm filter paper as plate (Whatman, Velizy-Villacoublay, France) connected to a microelectronic feedback system (Nima PS4, Manchester, England). The surface pressure (π) was recorded every 4 s with a precision of ± 0.2 mN/m. The measured surface pressure is the result of the surface tension of water minus the surface tension of the lipid film.

Ellipsometry.

Measurements of the ellipsometric angle value were carried out with an in-house automated ellipsometer in a "null ellipsometer" configuration. [START_REF] Berge | Ellipsometry Study of 2D Crystallization of 1-Alcohol Monolayers at the Water Surface[END_REF][START_REF] Azzam | Ellipsometry and polarized light[END_REF] A polarised He-Ne laser beam (λ=632.8 nm, Melles Griot, Glan-Thompson polarizer) was reflected on the surface of the trough. The incidence angle was 52.12°, i.e. Brewster angle for the air/water interface minus 1°.

After reflection on the liquid surface, the laser light passed through a λ/4 retardation plate, a Glan-Thompson analyser, and a photomultiplier. The analyser angle, multiplied by two, yielded the value of the ellipsometric angle (Δ), i.e. the phase difference between parallel and perpendicular polarisation of the reflected light. The laser beam probed the 1 mm² surface with a depth in the order of 1 µm. Initial values of the ellipsometric angle (Δ 0 ) and surface pressure of buffer solutions were recorded for at least half an hour to assure that the interface is clean. Only in the case of a stable, minimal signal, experiments were performed. Values of Δ were recorded every 4 s with a precision of ± 0.5°. When measuring the pressure increase induced by lysozyme at the air/liquid interface a lysozyme solution at 0.1 µM is deposited in the trough. When the pressure increase induced by lysozyme is measured at the LPS/liquid interface a LPS monolayer is first created as formerly described.

For the detection of local ellipsometric angle values, an imaging ellipsometer EP3 (Nanofilm, Göttingen, Germany) in "null ellipsometer" configuration was used with a 10X objective. A solid-state laser (λ=532 nm) was used as a light source. Delta/Psi maps were recorded with the EP3 software for a 450 µm x 390 µm surface. For delta and psi maps, a polarizer and analyzer range of 20° was used. Delta/psi maps were based on 20 images taken at different polarizer and analyzer angles.

Brewster angle microscopy.

An ellipsometer EP3 (Nanofilm, Berlin, Germany) with a polarized incident laser (λ=532.0 nm) was used with a 10X objective in a Brewster angle configuration (angle of incidence was 53.1°). The images represented a 450 µm x 390 µm surface. Different zones of each sample were evaluated; the images here shown are representative of the whole samples.

AFM sample preparation and AFM imaging.

Experiments were performed with a computer-controlled and user-programmable Langmuir TEFLON ® -coated trough (type 601BAM) equipped with two movable barriers and of total surface 90 cm² (Nima Technology Ltd., England). Before starting the experiments, the trough was cleaned successively with ultrapure water (Nanopure-UV), ethanol, and finally ultrapure water. The trough was filled with 5 mM HEPES buffer pH 7 150 mM NaCl. LPS were spread over the clean air/liquid interface at a surface pressure of 25 mN/m or 30 mN/m. The solvent was then left to evaporate for 1 h. Then, a Langmuir-Blodgett transfer was performed onto freshly cleaved mica plates at constant surface pressure by vertically raising (1 mm/min) the mica through the air/liquid interface to obtain a sample of the initial LPS monolayer. The LPS monolayer stability was assured during the Langmuir-Blodgett transfer allowing the injection of lysozyme in the subphase.

Then, 0.1 µM lysozyme was injected in the subphase of the previously sampled LPS monolayer with a Hamilton syringe. Surface pressure variations were recorded until a stable surface pressure was reached (after ~1 h). Then, a second Langmuir-Blodgett transfer was performed onto freshly cleaved mica as described above to obtain the sample of the LPS monolayer after lysozyme interaction.

AFM imaging of Langmuir Blodgett films was performed in contact mode using a Pico-plus atomic force microscope (Agilent Technologies, Phoenix, AZ) under ambient conditions with a scanning area of 20x20 µm² and 5x5 µm². Topographic images were acquired using silicon nitride tips on integral cantilevers. The forces were controlled along the imaging process.

Different zones of each sample were scanned; the images here shown are representative of the whole samples.

RESULTS

Insertion capacity of lysozyme into a LPS monolayer.

The insertion capacity of N-L and DH-L into a LPS monolayer was determined by independent tensiometry experiments at different protein concentrations. Insertion can be detected by a surface pressure increase (Δπ = π final -π initial ). Here, lysozyme was injected under a LPS monolayer with an initial surface pressure (π initial ) of 18 mN/m.

In both cases (N-L and DH-L), a surface pressure increase is demonstrated indicating lysozyme insertion into the LPS monolayer for protein concentrations above 0.02 µM (figure 1A). Below 0.05 µM, no difference can be observed between both lysozymes, while above this protein concentration, DH-L induces a higher surface pressure increase than N-L (figure 1A). When increasing the lysozyme subphase concentration, a Δπ-plateau is obtained at a lysozyme concentration of 0.2 µM for both N-L and DH-L, indicating saturation of the interface in these conditions. However, the maximum Δπ value is higher for DH-L than for N-L (12 mN/m and 8 mN/m, respectively; figure 1A). For further investigation of the insertion capacity of lysozyme, 0.1 µM N-L or DH-L has been used. At this concentration, differences exist between both proteins, and lipid protein interactions can be observed, while minimizing protein-protein interactions in the bulk solution (aggregation) or at the lipid interface.

Affinity of lysozyme for LPS monolayers.

To evaluate the affinity of both N-L and DH-L for the LPS monolayer, Δπ was determined after lysozyme injection (0.1 µM) under LPS monolayers previously formed at different initial surface pressures (π initial ). Supplementary experiments demonstrated that no phase transition occurs in the π-range here used (supplementary data S3) comparisons are then valuable. Linear regression analysis of the Δπ values versus π initial allows calculation of three binding parameters of lysozyme: maximal insertion pressure (MIP), synergy factor, and Δπ 0 (figure 1B). [START_REF] Calvez | Parameters modulating the maximum insertion pressure of proteins and peptides in lipid monolayers[END_REF][START_REF] Boisselier | Influence of the Physical State of Phospholipid Monolayers on Protein Binding[END_REF][START_REF] Calvez | Analysis of the Contribution of Saturated and Polyunsaturated Phospholipid Monolayers to the Binding of Proteins[END_REF] MIP is the intercept of the straight line with x-axis after extrapolation; it is thus the initial surface pressure for which no surface pressure increase occurs when lysozyme is injected in the subphase. The synergy factor is determined as the slope of the linear regression +1. The synergy factor provides information on the affinity of the protein for the lipid monolayer. High positive synergy values indicate the existence of strong protein/lipid interactions, since it means that the protein is able to insert into the lipid film even when initial surface pressure is high. Δπ 0 is the intercept of the straight line with y-axis after extrapolation; it is thus the theoretical pressure increase in the absence of lipids (π initial = 0 mN/m).

Linear regression for N-L and DH-L resulted in equations 1 and 2, respectively, with respective determination coefficients (R²) of 0.96 and 0.91. y = -0.21 x + 8.75 (eq. 1) y = -0.15 x + 9.10 (eq. 2)

The MIP is higher with DH-L than with N-L (59.6 and 41.5 mN/m, respectively; table 1). The 2011) is also higher with DH-L than with N-L, and is positive for both proteins (0.89 and 0.79, respectively; table 1). [START_REF] Boisselier | Influence of the Physical State of Phospholipid Monolayers on Protein Binding[END_REF] Oppositely, the Δπ 0 are similar for N-L and DH-L (8.75 and 9.10 mN/m, respectively; table 1). It is noticeable that these latter values are smaller than the experimental surface pressure increase observed for N-L and DH-L lysozymes at the air/liquid interface at the same subphase concentration (10 and 11 mN/m, respectively).

The rate constant of adsorption k ads (M -1 •s -1 ) of a lysozyme solution with a concentration (c) of 0.1 µM at the air/liquid interface and the LPS/liquid interface can be evaluated by fitting the Langmuir equation (eq. 3) for adsorption to the surface pressure measurements. The rate constant of desorption k des (M -1 •s -1 ) can here be considered negligible.

π (t) = π final (1-exp (-σ t)) (eq. 3)
σ = k ads c + k des (eq. 4)

At the air/liquid interface, N-L and DH-L have a k ads value of 6.6•10² M -1 •s -1 and 6. conditions results in a surface pressure increase of 2.9 mN/m and 3.5 mN/m, respectively (figure 2A), and induces an increase of ellipsometric angle of 8 and 12°, respectively (figure 2B).

3.4

Changes of surface pressure (π) and ellipsometric angle (Δ) of KLA monolayer in the presence of lysozyme.

To estimate the influence of the polysaccharide moieties on lysozyme interactions with LPS monolayer, KLA lipids were used. KLA lipids are derivative forms of LPS from which the polysaccharide moiety besides two 3-deoxy-D-manno-octulosonic acid (KdO) groups are missing (figure 3). The use of KLA was also relevant to test the role of electrostatic interactions between lysozyme and the negative charge at the interface by making the access to the charge easier. KLA monolayers are homogeneous lipid films on the contrary to LPS monolayers. This was confirmed by AFM imaging (supplementary data S2).

Kinetics of the π and Δ changes after injection of N-L and DH-L in the subphase were recorded for a KLA monolayer with an initial surface pressure of 25 mN/m and 30 mN/m, respectively. For N-L, the surface pressure of the KLA monolayer is stable for the first half hour and then decreases after 3 h (-2.1 mN/m ) (figure 4A). Oppositely, DH-L injection induces an immediate and more intense decrease (-5 mN/m after 3 h) (figure 4A). Both N-L and DH-L interact with the KLA monolayer in such a way that the ellipsometric angle increases slightly after injection of both proteins: +0.65° and +1.5° after 3 h, respectively (figure 4B).

Microscopic observations of LPS monolayer in the presence of lysozyme.

Brewster angle microscopy (BAM) and ellipsometry were performed to visualize the LPS monolayer organization on a µm-scale before and after lysozyme injection in the subphase.

BAM-images give information on the thickness and refraction index of the LPS monolayer.

Thick and/or high refraction index zones will appear lighter (white) than thin and low index zones (black). Delta maps show the same information as the BAM images, but the differences in height and/or refraction index are more precisely measured. Blue is the baseline color of the delta maps and correspond to a small delta value. High delta zones will be represented from green till red.

Before lysozyme injection, the LPS monolayer is heterogeneous, with black and white zones, at both initial surface pressures (25 mN/m and 30 mN/m), as evidenced by BAM-imaging (figures 5A and 5E). In the absence of literature references, the black colored zones are assumed to correspond to LPS with short polysaccharide chains (low refractive index and low thickness), while the white regions are assumed to correspond to LPS with long polysaccharide chains (high refractive index and low thickness). Such domain-organization is likely considering the optimal thermodynamic configuration that suggests segregation of LPS with similar polysaccharide chain lengths. The same information is provided by the delta-maps (figures 5C and 5G).

One hour after injection of 0.1 µM N-L, the BAM-images and delta-maps do not show any significant change of the heterogeneity as compared to the initial LPS monolayer (figures 5B and 5D), despite a slight increase of the background Δ-value in the delta-map (figure 5D). On the contrary, after injection of DH-L, an unequivocal change of the LPS monolayer organization is observed in both BAM-images and delta-maps (figures 5F and 5H). Especially, the small high Δdomains make place for bigger ones, and the background Δ-value increases (figure 5H).

Atomic force microscopy (AFM) enables to investigate the LPS monolayer at a nanoscale with a high resolution. Thus, this technique was used to study more precisely the organization of the lipid monolayer observed in the background of BAM-images (black zones).

The resulting AFM-images show the heterogeneity of the initial LPS monolayer at a nanoscale at both initial surface pressures (25 mN/m and 30 mN/m; figures 6A, 6C, 6E and 6G). The height difference between the lower (zone 1) and higher (zone 2) lipid zones is 1.2 to 2.0 ± 0.2 nm. By grating the LPS (data not shown), the monolayer thickness could be measured and corresponds to 5 nm. The monolayer thickness is in coherence with the one found by Le Brun et al.

(2013). [START_REF] Le Brun | Structural Characterization of a Model Gram-Negative Bacterial Surface Using Lipopolysaccharides from Rough Strains of Escherichia coli[END_REF] The impact of N-L and DH-L on the lipid monolayer can also be studied by AFM, enabling to study more carefully the reorganization of the low Δ-domains present in the BAM-images.

AFM shows that the injection of 0.1 µM N-L or DH-L does not significantly modify the heterogeneous appearance of the LPS monolayer (figures 6B, 6D, 6F and 6H). However, the insertion and adsorption of 0.1 µM N-L gives rise to the formation of small domains (object 1)

with a height of 1.4 ± 0.4 nm (figures 6B and 6D). The height of these domains is equivalent to the height of the dense domains observed in absence of lysozyme (figures 6A and 6C). The adsorption and insertion of 0.1 µM DH-L induces the formation of two types of clusters (object 2 and 3 with a height of 25 ± 5 and 57 ± 12 nm, respectively) and small domains (object 4) (1.4 ± 0.3 nm height) (figures 6F and 6H).

Topographical information shown in the AFM images is representative for the whole sample.

However, the size and shape of the different domains is irregular and heterogeneously distributed over the sample, making it impossible to quantify the effect of lysozyme on the domain size and shape.

DISCUSSION

Native lysozyme (N-L) has been shown active against Gram-negative bacteria such as E.

coli. [START_REF] Derde | Dry-Heating of Lysozyme Increases Its Activity against Escherichia coli Membranes[END_REF][START_REF] Pellegrini | Bactericidal Activities of Lysozyme and Aprotinin Against Gram-Negative and Gram-Positive Bacteria Related to Their Basic Character[END_REF] Membrane permeabilization has been suggested as one of the mechanisms responsible for this activity. [START_REF] Pellegrini | Effect of lysozyme or modified lysozyme fragments on DNA and RNA synthesis and membrane permeability of _Escherichia coli[END_REF][START_REF] Wild | Reevaluation of the effect of lysoyzme on Escherichia coli employing ultrarapid freezing followed by cryoelectronmicroscopy or freeze substitution[END_REF] This assumption was recently confirmed by our group who demonstrated that N-L causes the formation of pores and ion channels in the outer and cytoplasmic membranes, respectively. [START_REF] Derde | Hen egg white lysozyme permeabilizes the E. coli outer and inner membranes[END_REF][START_REF] Derde | Dry-Heating of Lysozyme Increases Its Activity against Escherichia coli Membranes[END_REF] Pore formation due to N-L implies that interactions occur between the protein and the E. coli outer membrane. Nevertheless, the mode of insertion of lysozyme into the outer membrane remains unknown.

Moreover, dry-heated lysozyme (DH-L) has a higher antimicrobial activity and higher membrane disruption potential than N-L. [START_REF] Derde | Dry-Heating of Lysozyme Increases Its Activity against Escherichia coli Membranes[END_REF] This improved activity is supposed to be related to the modified physico-chemical properties of DH-L. DH-L is more hydrophobic, flexible and surface active than N-L, but its secondary and tertiary structures remain intact. [START_REF] Desfougères | Succinimidyl Residue Formation in Hen Egg-White Lysozyme Favors the Formation of Intermolecular Covalent Bonds without Affecting Its Tertiary Structure[END_REF][START_REF] Desfougères | Strong Improvement of Interfacial Properties Can Result from Slight Structural Modifications of Proteins: The Case of Native and Dry-Heated Lysozyme[END_REF] It is thus relevant to compare the interaction of native and dry-heated lysozymes with LPS, the lipid components of the outer leaflet of the outer membrane of Gram negative bacteria.

Interfacial monolayers are considered as good models to study interactions between antimicrobial peptides and bacterial membranes. [START_REF] Brockman | Lipid monolayers: why use half a membrane to characterize proteinmembrane interactions?[END_REF][START_REF] Roes | Probing the Properties of Lipopolysaccharide Monolayers and Their Interaction with the Antimicrobial Peptide Polymyxin B by Atomic Force Microscopy[END_REF] In the presently reported study, a LPS monolayer was used to mimic the outer leaflet of the E. coli K12 outer membrane, in order to explore the first step of lysozyme interaction with bacterial membrane. It is noticeable that wildtype LPS was here used for the first time to investigate protein-LPS interactions at a macroscopic and mesoscopic level, using biophysical tools such as tensiometry, ellipsometry, AFM and BAM.

The affinity of N-L for LPS is very high and makes possible the insertion of the protein into a LPS monolayer.

For the first time, protein insertion into a wild type LPS monolayer is here demonstrated. Until now, protein insertion was only recorded for LPS-derivative monolayers and lung surfactant protein D. [START_REF] Wang | Interaction of Recombinant Surfactant Protein D with Lipopolysaccharide: Conformation and Orientation of Bound Protein by IRRAS and Simulations[END_REF] . The surface pressure increase measured when N-L is injected into the subphase demonstrates that N-L is able to insert into a LPS monolayer (figure 1A). The ability of lysozyme to interact with LPS is consistent with the surface activity of the protein at the air/liquid interface. [START_REF] Desfougères | Strong Improvement of Interfacial Properties Can Result from Slight Structural Modifications of Proteins: The Case of Native and Dry-Heated Lysozyme[END_REF] However, insertion of N-L into the LPS monolayer remains lower than for antimicrobial peptides such as temporin L, as suggested by the lower surface pressure increase (table 2). [START_REF] Zhang | Interactions of Bacterial Cationic Peptide Antibiotics with Outer and Cytoplasmic Membranes of Pseudomonas aeruginosa[END_REF][START_REF] Giacometti | Interaction of Antimicrobial Peptide Temporin L with Lipopolysaccharide In Vitro and in Experimental Rat Models of Septic Shock Caused by Gram-Negative Bacteria[END_REF] The larger molecular size and higher rigidity of lysozyme, [START_REF] Canfield | The disulfide bonds of egg white lysozyme (muramidase)[END_REF] as compared to peptides, could be responsible for the lower efficiency of the protein.

The maximal insertion pressure (MIP), determined from measurements of N-L insertion at different initial pressures, is high (41.5 mN/m, table 1) and similar to MIP recorded for antimicrobial peptides and phospholipid monolayers (25-45 mN/m). [START_REF] Calvez | Parameters modulating the maximum insertion pressure of proteins and peptides in lipid monolayers[END_REF][START_REF] Giacometti | Interaction of Antimicrobial Peptide Temporin L with Lipopolysaccharide In Vitro and in Experimental Rat Models of Septic Shock Caused by Gram-Negative Bacteria[END_REF] Especially, it is remarkable that the MIP value is higher than the lateral pressure which is supposed to exist in natural membrane systems in eukaryotic cells (~ 30 mN/m). [START_REF] Marsh | Lateral pressure in membranes[END_REF] Unfortunately, no measurements or theoretical deductions of the lateral pressure in the outer and cytoplasmic membranes of prokaryotes are available in literature then no comparison is possible with the here observed MIP value. Moreover, the N-L synergy factor (0.79, table 1) is extremely high as compared to reported values for protein insertion into phospholipid monolayers (from 0.3 to 0.5). [START_REF] Boisselier | Influence of the Physical State of Phospholipid Monolayers on Protein Binding[END_REF] It can thus be concluded that the protein has a high affinity for the LPS interface between 18 and 30 mN/m and strikingly lysozyme insertion is almost not impacted by the lateral cohesion of the LPS molecules. These observations suggest a mode of action that is unusual compared to the interaction between protein and phospholipids.This could result from the LPS inherent molecular structure and from the specificities of a LPS monolayer compared to a phospholipid monolayer;

the LPS molecules have heterogeneous polysaccharide chains in length, thus the monolayer has a variable thickness induced by the auto-assemblage of similar LPS molecules observed by BAM and AFM microscopy (figures 5 and 6). Indeed, a LPS monolayer can be divided into two distinct zones, i.e. a polysaccharide zone and a phospholipid-like zone, on the contrary to a phospholipid monolayer which is composed of a unique zone.

BAM and AFM imaging were performed to evaluate the different hypotheses explaining the Δ increase. BAM and AFM imaging show the heterogeneity of the initial LPS monolayer at micrometer and nanometer scales, respectively (figures 4A, 4C, 5A and 5C), as a result of the variable lengths of the polysaccharides chains. After N-L injection and interaction with LPS, this heterogeneity is maintained as can be observed in the BAM (figures 5B and 5D Similarly to N-L, DH-L insertion into the LPS monolayer is enabled by the polysaccharides moieties, and DH-L reorganizes the LPS monolayer. However, differences in the behavior of DH-L versus N-L with the LPS monolayer can be noticed. This modified behavior could be related to its different physico-chemical properties such as increased hydrophobicity, surfaceactivity, positive charge and flexibility. [START_REF] Desfougères | Succinimidyl Residue Formation in Hen Egg-White Lysozyme Favors the Formation of Intermolecular Covalent Bonds without Affecting Its Tertiary Structure[END_REF][START_REF] Desfougères | Strong Improvement of Interfacial Properties Can Result from Slight Structural Modifications of Proteins: The Case of Native and Dry-Heated Lysozyme[END_REF] DH-L insertion into the LPS monolayer is more efficient than N-L at concentrations higher than 0.05 µM (figure 1A). This could be due to the higher flexibility of DH-L as compared to N-L, [START_REF] Desfougères | Succinimidyl Residue Formation in Hen Egg-White Lysozyme Favors the Formation of Intermolecular Covalent Bonds without Affecting Its Tertiary Structure[END_REF] which could allow more DH-L molecules to insert into the LPS monolayer, and/or to restructure more efficiently the interface. The increased insertion capacity of DH-L is consistent with its slight increased interfacial behavior (π final , table 1). Especially, it is noticeable that the surface pressure increase induced by DH-L insertion into the LPS-monolayer is similar to that measured with an antimicrobial peptide, i.e. temporin L, in equivalent conditions (table 2). DH-L has also more affinity for the LPS monolayer than N-L, demonstrated by its higher MIP and synergy factor (59.6 mN/m and 0.89, respectively; table 1). [START_REF] Calvez | Parameters modulating the maximum insertion pressure of proteins and peptides in lipid monolayers[END_REF][START_REF] Giacometti | Interaction of Antimicrobial Peptide Temporin L with Lipopolysaccharide In Vitro and in Experimental Rat Models of Septic Shock Caused by Gram-Negative Bacteria[END_REF] The drastically different reorganization of the LPS monolayer by DH-L is highlighted by BAM and AFM imaging (figure 5 and6). The BAM-images show that the many small domains with a high Δ-value visible in the

CONCLUSIONS

The presently reported study demonstrates the strong interaction between N-L and a LPS monolayer, usually considered as a relevant model of the outer membrane of Gram-negative bacteria. Even more, N-L is able to insert leading to a lateral reorganization of the LPS monolayer, which can explain pore formation into the E. coli outer membrane, previously observed in vivo. [START_REF] Derde | Dry-Heating of Lysozyme Increases Its Activity against Escherichia coli Membranes[END_REF] An original and unexpected result is that lysozyme insertion between the lipid A of LPS monolayers requires the presence of the polysaccharide moieties. This reveals specific interactions between lysozyme and the polysaccharide moieties leading to better insertion and decreased electrostatic attraction. Further experiments are needed in order to settle between the different hypotheses that could explain this finding.

Moreover, dry-heating modifies lysozyme properties in such a way that its affinity for LPS, its insertion capacity, and its ability for LPS monolayer reorganization are emphasized. These results are thus consistent with in vivo experiments that demonstrated larger and/or more numerous pores induced by DH-L into the E. coli outer membrane, as compared to N-L. [START_REF] Derde | Dry-Heating of Lysozyme Increases Its Activity against Escherichia coli Membranes[END_REF] The interaction of N-L and DH-L with the outer membrane lipids is now well established and consistent with the pore formation previously demonstrated in vivo. Self-uptake mechanism is then imaginable meaning that lysozyme molecules involved in pore formation and stabilization could enable the entrance of free lysozyme in the bacterial cell. Then, it is relevant to further study the interaction of lysozyme with the cytoplasmic membrane, the final hurdle before access to the cytoplasm. The findings resulting from this study are currently analyzed and will soon be published.

FIGURES

  synergy factor as introduced by Boisselier et al. (2012) and Calvez et al. (

  pressure (π) and ellipsometric angle (Δ) of LPS monolayer in the presence of lysozyme Kinetics of the π and Δ changes after injection of N-L and DH-L in the subphase were recorded using a LPS monolayer with an initial surface pressure of 25 mN/m and 30 mN/m, respectively. Different initial surface pressures of the LPS monolayers were chosen because of the different insertion capacities of N-L and DH-L for this experiment.The aim of this study is to evaluate the effects of N-L or DH-L on the LPS monolayer after a similar insertion of proteins, i.e. a similar Δπ. The initial surface pressures which correspond to this prerequisite is 25 mN/m and 30 mN/m for a concentration of 0.1 µM N-L and DH-L, respectively (figure 1B); more so, LPS monolayers with an initial surface pressure of 25 mN/m and 30 mN/m have similar Δ values (supplementary data S1). The injection of N-L and DH-L under the LPS monolayer in these

  ) and AFM images (figures 6B and 6D). But N-L injection also results in a slight increase of the background Δ-value in the delta-map (figure5D), and in the formation of small domains on the background zones in AFM-imaging (figures 6B and 6D). It can thus be concluded that N-L reorganizes the LPS monolayer, even if this reorganization remains limited. The reorganization of the LPS monolayer and the LPS/lysozyme complex formation could possibly be the preliminary steps for pore formation by N-L as observed in vivo by[START_REF] Derde | Hen egg white lysozyme permeabilizes the E. coli outer and inner membranes[END_REF].[START_REF] Derde | Hen egg white lysozyme permeabilizes the E. coli outer and inner membranes[END_REF] has a stronger affinity for LPS than N-L, and causes more radical reorganization of the LPS monolayer.

  presence of N-L (figures 5B and 5D) are replaced by larger and fewer high Δ-value domains in the presence of DH-L (figures 5F and 5H). Concurrently, more or less thick, and more or less large clusters appear in the presence of DH-L, as evidenced by AFM images (figures 6F and 6H). These clusters could be protein aggregates caused by high local concentration of DH-L at the LPS-monolayer, consistently with the higher sensitivity to aggregation of DH-L as compared to N-L, previously established byDesfougères et al (2011).[START_REF] Desfougères | Strong Improvement of Interfacial Properties Can Result from Slight Structural Modifications of Proteins: The Case of Native and Dry-Heated Lysozyme[END_REF]

Figure 1 Figure 1 .Figure 2 .Figure 3 :Figure 3 .

 11233 Figure 1:

Figure 4 Figure 4 .Figure 5 :Figure 5 . 1 .Table 2 .

 445512 Figure 4:

4.2

The polysaccharide moieties of LPS are needed for N-L insertion.

When LPS molecules are depleted from their polysaccharide moieties (KLA), lysozyme is no longer able to insert into the lipid monolayer, since no increase of the surface pressure occurs (figure 4A). However, lysozyme adsorption is evidenced by the increase of the ellipsometric angle (Δ) (figure 4B). Lysozyme adsorption could involve hydrogen bonds between the protein and the remaining two sugar moieties, or electrostatic interactions between the positive lysozyme and the negative KLA. The latter assumption is reinforced by the immediate and higher adsorption of DH-L which is more positively charged than N-L (figure 2B). It is also in accordance with [START_REF] Brandenburg | Biophysical characterisation of lysozyme binding to LPS Re and lipid A[END_REF] who reported electrostatic interactions between S.

minnesota Re LPS and lysozyme in solution. [START_REF] Brandenburg | Biophysical characterisation of lysozyme binding to LPS Re and lipid A[END_REF] Actually, while N-L adsorption is proceeding for 3 h (figure 4B), the surface pressure of the lipid monolayer is decreasing (figure 4A). This could be due to a destabilization and partial solubilization of the lipid monolayer as has been previously described for with the antimicrobial peptide protegrin-1 at a lipid A monolayer [START_REF] Ishitsuka | Insertion selectivity of antimicrobial peptide protegrin-1 into lipid monolayers: Effect of head group electrostatics and tail group packing[END_REF]; another hypothesis is the reorganization or reorientation of the lipid headgroups induced by lysozyme presence beneath the monolayer,

similarly to what has been previously reported for a dystrophin subdomain R20-24 at a DOPC/DOPS monolayer. [START_REF] Vié | Specific anchoring modes of two distinct dystrophin rod sub-domains interacting in phospholipid Langmuir films studied by atomic force microscopy and PM-IRRAS[END_REF] If a partial solubilization of the KLA occurs, this should be reflected in a decrease of the ellipsometric angle (Δ), due to the loss of matter at the interface.

Here, the ellipsometric angle increases (figure 4), meaning that rather than a solubilization of the KLA monolayer, a reorganization of the KLA head groups takes place leading to a relaxation of the lipid film. Lysozyme molecules are trapped beneath the KLA monolayer caused by strong electrostatic attractive forces between lysozyme and the KLA lipids.

On the contrary, when N-L interacts with a LPS monolayer, i.e. including polysaccharide moieties, surface pressure and ellipsometric angle simultaneously increase (figures 2A and 2B).

Undoubtedly, N-L is thus able to insert deeply in the interface, up to the hydrophobic zone of the LPS monolayer. A hypothesis can be the effect of steric hindrance of the polysaccharides which prevents total coverage of the interface by the lipid headgroups, thus leaving free space for lysozyme insertion. Moreover, the polysaccharide chains can also cause simultaneously partial shielding of the negative charges on the headgroups and therefore prevent the entrapment of the positive lysozyme molecules at the level of these negative charges as is the case for KLA lipids.

The decreased interaction of the negative charges with positive lysozyme could enable insertion of the protein between the LPS headgroups. At last, lysozyme and the polysaccharides moieties could interact and create compact zones as LPS/lysozyme domains and complexes (figure 6)

resulting in lesser density in other areas enabling the remaining free lysozyme to attain the interface. Such strong hydrophobic interactions have already been reported between LPS and lysozyme in solution, [START_REF] Ohno | Lipopolysaccharide interaction with lysozyme. Binding of lipopolysaccharide to lysozyme and inhibition of lysozyme enzymatic activity[END_REF] and LPS/lysozyme complexes have been observed. [START_REF] Brandenburg | Biophysical characterisation of lysozyme binding to LPS Re and lipid A[END_REF][START_REF] Ohno | Characterization of Complex-Formation Between Lipopolysaccharide and Lysozyme[END_REF]

N-L interaction with LPS causes a slight reorganization of the LPS monolayer.

At the same time as the surface pressure increases when N-L is injected under a LPS monolayer, a strong increase of the ellipsometric angle Δ (+7°, figure 2B) is observed, which is higher than the ellipsometric angle increase for protein/phospholipid monolayers. [START_REF] Sarkis | Spectrin-like Repeats 11-15 of Human Dystrophin Show Adaptations to a Lipidic Environment[END_REF] This unusually high Δ increase can be explained by the LPS/lysozyme complex formation, polysaccharide reorganization, and/or the presence of N-L at the interface, since the ellipsometric angle depends on the refraction index and the film thickness.

ASSOCIATED CONTENT

Supporting information. Additional experimental data on the ellipsometric angle of a LPS monolayer (S1),Atomic Force images of LPS or KLA monolayers (S2) and isothermal compression of LPS and KLA monolayers (S3). This material is available free of charge via the Internet at http://pubs.acs.org.

ACKNOWLEDGMENT

The authors thank "Conseil Regional de Bretagne" for the funding of this work.


Fabio Stefani

Olivier Lopez

Anthony Bercy

Won-Kyu Lee

Christian Chardonnet

Giorgio Santarelli

Paul-Eric Pottie
email: paul-eric.pottie@obspm.fr.

Anne Amy-Klein

Tackling the Limits of Optical Fiber Links

Keywords: numbers: 06.20.fb, 06.30.Ft, 42.62.Eh

We theoretically and experimentally investigate relevant noise processes arising in optical fiber links, which fundamentally limit their relative stability. We derive the unsuppressed delay noise for three configurations of optical links : two-way method, Sagnac interferometry, and actively compensated link, respectively designed for frequency comparison, rotation sensing, and frequency transfer. We also consider an alternative two-way setup allowing real-time frequency comparison and demonstrate its effectiveness on a proof-of-principle experiment with a 25-km fiber spool. For these three configurations, we analyze the noise arising from uncommon fiber paths in the interferometric ensemble and design optimized interferometers. We demonstrate interferometers with very low temperature sensitivity of respectively -2.2, -0.03 and 1 fs/K. We use one of these optimized interferometers on a long haul compensated fiber link of 540 km. We obtain a relative frequency stability of 3 • 10 -20 after 10,000 s of integration time.

I. INTRODUCTION

Optical fiber links for frequency and time dissemination have been developed fast for the last 5 years demonstrating optical frequency transfer below 10 -13 at one second integration time over typical distance of 1 000 km [START_REF] Newbury | Coherent transfer of an optical carrier over 251 km[END_REF][START_REF] Fujieda | All-optical link for direct comparison of distant optical clocks[END_REF][START_REF] Lopez | Ultra-stable long distance optical frequency distribution using the internet fiber network[END_REF][START_REF] Droste | Optical-frequency transfer over a single-span 1840km fiber link[END_REF][START_REF] Marra | Dissemination of an optical frequency comb over fiber with 3 • 10 -18 fractional accuracy[END_REF][START_REF] Calosso | A frequency transfer via a two-way optical phase comparison on a multiplexed fiber network[END_REF][START_REF] Krehlik | Fiber-optic joint time and frequency transfer with active stabilization of the propagation delay. Instrumentation and Measurement[END_REF][START_REF] Calonico | High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link[END_REF]. The most demanding application in terms of stability and uncertainty is the direct comparison of distant optical clocks (see for instance [START_REF] Giorgetta | Optical twoway time and frequency transfer over free space[END_REF] and references herein). Their record relative frequency stability of 10 -18 in less than 1 000 s is out of reach for satellite techniques [START_REF] Fujieda | Carrier-phase two-way satellite frequency transfer over a very long baseline[END_REF]. Fiber links are now extending towards a much wider field of application with the development of national or transnational fiber networks as for instance REFIMEVE+ in France and LIFT in Italy. Frequency transfer with fiber links is expected to play a key role for fundamental physics and high resolution atomic and molecular physics [START_REF] Matveev | Precision measurement of the hydrogen 1s-2s frequency via a 920km fiber link[END_REF][START_REF] Chanteau | Mid-infrared laser phase-locking to a remote nearinfrared frequency reference for high-precision molecular spectroscopy[END_REF] and for syncing large area networks as for astrophysics applications.

Fiber links operated in the optical frequency domain use an ultra-stable laser as a frequency signal. This signal is degraded by the propagation delay fluctuations, due to thermal and mechanical perturbations that affect the optical length of the link setup. Although these perturbation are passively or actively compensated in usual fiber links configurations, a fine understanding of the phase noise contributions of a fiber link will enhance the resolution of the frequency comparison and transfer and reduce the integration time. Our aim is also to improve further the performances of fiber links for new applications, such as Sagnac interferometry on giant loops [START_REF] Schiller | Feasibility of giant fiber-optic gyroscopes[END_REF][START_REF] Clivati | Large-area fiber-optic gyroscope on a multiplexed fiber network[END_REF].

In this paper we first describe three configurations of fiber links used for frequency comparison, rotation sensing of the inertial frame, and frequency transfer. We derive the rejection of the propagation noise in the case of a noise process with spatial correlation. Then we study the so-called interferometer noise arising from uncommon optical paths for both directions and its contribution to the overall budget. Complementary to the work reported in [START_REF] Williams | Highstability transfer of an optical frequency over long fiberoptic links[END_REF], we investigate interferometric noise of fiber-based interferometers, which is related to the temperature sensitivity of the interferometers. We realize three interferometers, one for each link configurations, and demonstrate their very low temperature sensitivity. We finally demonstrate a 540-km optical link with such an ultra-low noise interferometer and show that the interferometric noise contribution at long integration time is at 10 -20 level.

II. LINK TOPOLOGY

We consider three configurations of optical links, all of them enabling the compensation of the propagation noise. These three configurations are respectively designed for frequency comparison, rotation sensing and frequency transfer. Their setups are depicted on Fig. 1. For frequency comparison of remote clocks, the most simple method consists in implementing a two-way link. This method was introduced by [START_REF] Hanson | Fundamentals of two-way time transfer by satellite[END_REF] and extended to optical links [START_REF] Calosso | A frequency transfer via a two-way optical phase comparison on a multiplexed fiber network[END_REF][START_REF] Williams | Highstability transfer of an optical frequency over long fiberoptic links[END_REF][START_REF] Bercy | Two-way optical frequency comparisons at 5 • 10 -21 relative stability over 100-km telecommunication network fibers[END_REF]. At each end of the link, a laser is sent to the other end. One detects the frequency difference between the local laser signal and the remote laser signal by measuring their beat note signals using a Michelsontype interferometer (see Fig. 1a). At each end, the beat note interferometric detection setup is made of an optical coupler, a short fiber arm terminated by a Faraday mirror and a photodiode. The fiber link itself is the second arm of the interferometer. Assuming that the long arm's fiber noise is equal for the two directions of propagation, one can efficiently reject the delay fluctuations by synchronizing and then post processing the data, simply by dividing by two the subtraction of the data sets that were recorded at the two ends. Note that accurate frequency comparisons require an accurate synchronization of the frequency data. This requirement is about the microsecond in most practical cases, which is not that stringent, and depends on the frequency drift of the two lasers and on the length of the link [START_REF] Raupach | Measuring the relative synchronization error of frequency counters with sub-µs accuracy via a phasestabilized fiber link[END_REF].

When the fibre link follows a loop as depicted in Fig. 1b, starting and ending in the same lab and enclosing a non-zero effective area, an optical link can be turned into a giant Sagnac interferometer [START_REF] Schiller | Feasibility of giant fiber-optic gyroscopes[END_REF][START_REF] Clivati | Large-area fiber-optic gyroscope on a multiplexed fiber network[END_REF][START_REF] Sagnac | Les systèmes optiques en mouvement et la translation de la Terre[END_REF][START_REF] Arditty | Sagnac effect in fiber gyroscopes[END_REF]. A Sagnac phase can be detected by splitting a single laser in two optical signals, injecting them clockwise and counterclockwise in the loop, and by applying the two-way compensation method described previously [START_REF] Ezekiel | Closed-loop, low-noise fiber-optic rotation sensor[END_REF]. The phase signal arises from the phase comparison between the long and the short (reference) loop. This approach was revisited recently in [START_REF] Clivati | Large-area fiber-optic gyroscope on a multiplexed fiber network[END_REF]. Compared to the two-way link, the experimental setup is even simpler, because the data sampling synchronization is made trivial by the colocation of the measurements. Even more, an unstabilized laser source can be used as a reference laser, since all the laser-related phase and frequency noises are rejected in common mode.

The two-way method for frequency comparison is ineffective when the transfer of an ultra-stable frequency from a reference laboratory to a distant user is required, as when an end user has no stable clock in the laboratory. Thus, an active method should be used to compensate in real-time for the propagation noise along the fiber [START_REF] Ma | Delivering the same optical frequency at two places: accurate cancellation of phase noise introduced by an optical fiber or other time-varying path[END_REF]. The principle of the method is sketched on the right part of Fig. 1c. The optical signal sent at one end travels forth and back through the same fiber, to allow the detection of a beat note of the local laser with the round-trip signal. AOMs are used at the two ends to distinguish the round-trip signal from parasitic back reflections. The beat note signal is a measurement of the round-trip phase noise. Assuming again that the accumulated fiber noise over the link is equal forth and back, one can then actively compensate the propagation noise by correcting the input signal phase with half of the opposite of the measured phase noise. This is done with the acousto-optic modulator at the link input end.

For all these setups the propagation noise is detected through a time delayed interferometer with strongly length-unbalanced arms. One arm consists of the fiber link, the other one is a local short reference arm. In order to improve the signal to noise ratio, an heterodyne detection of the phase difference between the two arms is im-plemented, by shifting the frequency in the long-arm with an AOM. The optical propagation delay in the long-haul link is a limiting factor for the noise correction. Actually both the correction bandwidth and the noise amplitude rejection are limited by the round-trip propagation time [START_REF] Williams | Highstability transfer of an optical frequency over long fiberoptic links[END_REF]. For instance a link of 1840 km leaves the acoustic noise almost uncorrected as the bandwidth is below 20 Hz [START_REF] Droste | Optical-frequency transfer over a single-span 1840km fiber link[END_REF]. In order to circumvent this issue and to face the large losses of public network, our group introduced cascaded link, where the propagation noise is compensated by successive and consecutive fiber spans [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF][START_REF] Lopez | Cascaded multiplexed optical link on a telecommunication network for frequency dissemination[END_REF]. With this approach, the optical signal has to be repeated from one span to the other using a so-called repeater station. In this setup, a local laser is phase-locked to the incoming signal of the previous span in order to clean up and amplify the metrological signal, as depicted on Fig. 1c.

Part of the fiber noise cannot be rejected by the noise compensation system. This noise arises from the interferometric detection of the fiber noise [START_REF] Williams | Highstability transfer of an optical frequency over long fiberoptic links[END_REF][START_REF] Grosche | Optical frequency transfer via 146 km fiber link with 10 -19 relative accuracy[END_REF]. It can be due for instance to propagation noise in the reference arm. The main noise sources of fiber-based interferometers are thermal effect. Addressing this noise is crucial for the design of the optical part of a station, in which the laser phase should be well defined and should not suffer any thermal fluctuations, as explained below.

Apart from this interferometric noise and unsuppressed delay noise that will be addressed successively in the next two parts, the fiber noise that is accumulated from the output of the optical oscillators to the input of the interferometric setup is also contributing to the noise budget. It is indistinguishable from the intrinsic noise of the optical oscillators itself. It remains even if the unsuppressed delay noise and the interferometric noise are zero. To get rid of that, one should consider very short fiber paths from the oscillator to the interferometer, very good thermal stabilization and low acoustical noise levels, or a short compensated link. This noise can be neglected in a case of a demonstration setup for frequency comparison or dissemination (where an optical oscillator is compared to itself), and in any case for a giant Sagnac interferometer. It will not be considered below.

III. UNSUPPRESSED DELAY NOISE

A. Free fiber noise Following Newbury and coworkers, we will consider a phase noise density per unit of length δϕ(z, t), that depends of time t and position z along the link [START_REF] Newbury | Coherent transfer of an optical carrier over 251 km[END_REF]. δϕ(z, t) dz gives the phase noise of the fiber elementary length dz. Thus the fiber noise at time t arising from the propagation from the end 1 at position z = 0 to the end 2 at position z = L is given by φ 12 (t) = L 0 δϕ(z, t -T + z/v) dz, where v is the light velocity in the fiber and T = L/v the one-way propagation delay.

Considering standard random signal processing, the fiber phase noise Power Spectral Density (PSD) is given by the Fourier transform of the autocorrelation function of its phase noise. This approach allows to take into account the statistical properties of the phase noise, as it depends on the model established for the random signal δϕ(z, t). The full calculations are detailed in Appendix A. Since it originates from thermal and acoustical fluctuations, the phase noise density δϕ(z, t) has a very short spatial correlation along the fiber. Here we assume that its spatial autocorrelation function has a narrow gaussian profile of deviation ∆L, with ∆L ≪ L. δϕ(z, t) is thus practically uncorrelated for two different positions z and z' separated by more than a few ∆L. It extends the previous studies in which the spatial autocorrelation of the noise was considered as an impulse function [START_REF] Williams | Highstability transfer of an optical frequency over long fiberoptic links[END_REF][START_REF] Bercy | In-line extraction of an ultrastable frequency signal over an optical fiber link[END_REF]. This calculation method can be extended to arbitrary correlation functions, in time or in space.

For sake of simplicity, we also consider here that the fiber noise per unit length δϕ(z, t) has constant statistical properties over z. We obtain the following PSD for the free fiber phase noise φ 12 (t)

S 12 (f ) = ∆LL √ 2πe -( √ 2πf ∆L) 2 v 2 S δϕ (f ) (1)
where S δϕ (f ) is the PSD of the fiber noise per unit of length δϕ(z, t).

B. Two-way link for frequency comparison and Sagnac interferometry for rotation sensing

In the case of a two-way frequency comparison, we consider the one way fiber noise from the end 1 to the end 2, φ 12 (t), and from the end 2 to the end 1, φ 21 (t). The laser phases φ 1 (t) of the Laser 1 and φ 2 (t) of the Laser 2 are defined as the lasers enter the interferometric setup. From the frequency measurements made on the photodiodes PD1 and PD2, one obtains in post-processing the two-way phase signals (-φ PD1 + φ PD2 )/2 = (φ 1φ 2 ) + (φ 12 -φ 21 )/2, where the time dependency has been omitted for sake of simplicity. The ideal situation for a frequency comparison is when φ 1 -φ 2 is the dominant term. The performance of the comparison is asset when the two lasers are perfectly coherent, i.e. when φ 1 = φ 2 . For this study focused on the fiber noise contribution we conveniently defined the two-way phase signal as φ tw (t) = (φ 12 (t) -φ 21 (t))/2. Considering that 2πf T ≪ 1, and for a first order expansion, one has :

φ tw (t) = L 0 z v - T 2 δϕ ′ (z, t)dz (2)
One deduces that the two-way phase PSD is then :

S tw (f ) = 1 12 (2πf T ) 2 • e + ( √ 2πf ∆L) 2 v 2 • S 12 (f ) (3)
The full calculation is detailed in Appendix A.

It is convenient to limit our observations to a maximum frequency f H for which 1 12 (2πf H T ) 2 = 1, in order to avoid that the post-processed two-way noise overtakes the original one way noise. In that case, it can be easily demonstrated that the exponential term in Eq. 3 is bounded to e +6•(∆L/L) 2 , making it negligible for every practical realization. The same formulas are found for a Sagnac configuration.

C. Actively noise compensated link for frequency dissemination

In the case of an actively noise compensated (ANC) link, and for a first order expansion, the compensated fiber noise at output end 2 is φ anc (t) = L 0 z/vδϕ ′ (z, t) dz [START_REF] Williams | Highstability transfer of an optical frequency over long fiberoptic links[END_REF][START_REF] Bercy | In-line extraction of an ultrastable frequency signal over an optical fiber link[END_REF]. One can demonstrate that the PSD of φ anc (t) is :

S anc (f ) = 1 3 (2πf T ) 2 • e +( √ 2πf ∆L v ) 2 • S 12 (f ) ( 4
)
The calculation is detailed in Appendix A. That's almost the same formula that the one derives in [START_REF] Newbury | Coherent transfer of an optical carrier over 251 km[END_REF], with an additional exponential coefficient arising from the spatial correlation of the noise. According to the theory of servo-loops, the active noise compensation system can operate up to a frequency approximatively equal to 1/(4T ), setting also an upper limit for the validity of Eq. 4 [START_REF] Williams | Highstability transfer of an optical frequency over long fiberoptic links[END_REF]. Beyond this limit, the fiber noise remains uncorrected. Considering this, we can derive that the exponential coefficient in Eq. 4 can reach a maximum value of e (π 2 /8)(∆L/L) 2 , which is negligible in practice to the best of our knowledge. From Eq.4 and following considerations, one expects that the amplitude and the bandwidth of the noise correction are decreasing linearly with L. The performance of the link can then be improved by dividing a link into smaller subsections. This is the cascaded link approach [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF][START_REF] Lopez | Cascaded multiplexed optical link on a telecommunication network for frequency dissemination[END_REF]. For instance, dividing a link in N independent subsections of equal length, and considering that the power of the free fiber noise scales as the length of each subsection, both the amplitude and the bandwidth of the noise rejection are increased by a factor N .

D. Two-way link with noise compensation

A question that arises therefore is whether a composite system can improve further the fiber noise rejection. Several configurations are considered here. We first consider the case of an active compensation system set at one end of a two-way setup. We demonstrate in Appendix B that it does not lead to a higher noise rejection. On the contrary, one finds that the two-way phase noise PSD is indeed 4 times higher than that of a post-processed twoway without active compensation. We also consider the case of a double round-trip noise detection system at each end, and a post-processing correction of each end phase signal. We demonstrate in Appendix B that the two postcorrections at each ends canceled each other and that this configuration does not lead to higher noise rejection.

E. Real-time two-way link

In this subsection we consider an alternative two-way setup which takes advantage of the round-trip noise detection performed with the ANC configuration. This technique was introduced in [START_REF] Bercy | Two-way optical frequency comparisons at 5 • 10 -21 relative stability over 100-km telecommunication network fibers[END_REF] for a two-fiber setup where the light travels the fibers with uni-directional propagation. We introduce here a bi-directional configuration with a single fiber. If one flips at both ends the play role of the Faraday mirrors and the photodiodes of a two-way setup (see Fig. 1a and Fig. 2a), one has the possibility to record simultaneously on each photodiode two signals, that we labelled A and B. One can write the beat notes detected by the photodiodes PD1 and PD2 as follows :

         φ PD1 A = (φ 2 + φ 21 ) -φ 1 at frequency F φ PD1 B = (φ 12 + φ 21 ) at frequency 2F φ PD2 A = (φ 1 + φ 12 ) -φ 2 at frequency F φ PD2 B = (φ 12 + φ 21 ) at frequency 2F (5)
where φ 1 , φ 2 , φ 21 , and φ 12 are defined in section III B. We set

F = f 1 + f 2 .
(-φ PD1 A + φ PD2 A )/2 = (φ 1 -φ 2 ) + (φ 12 -φ 21 )/2 (6)
Assuming again that φ 12 = φ 21 , one finds the regular two-way noise rejection, as the fiber noise cancels out and only the laser's phase difference remains. Note that the power of the optical beat note signals in post processing is the same as for the setup of Fig. 1a.

In addition, the same results can be obtained by combining the 2F and F beat notes signals detected on the same photodiode. For instance at the end 1 on PD 1 one has : [START_REF] Krehlik | Fiber-optic joint time and frequency transfer with active stabilization of the propagation delay. Instrumentation and Measurement[END_REF] the same phase difference can be obtained at end 2 on PD 2. This approach, that we called "local two-way", allows to process in real-time the noise rejection in both distant laboratories, relying only on data acquired locally. It avoids the necessity to exchange and synchronize data between distant sites to compute the stability of the link and diagnose any malfunction. The real-time processing can be done by a simple tracking oscillator and do not need high-speed digital electronics.

-φ PD1 A + φ PD1 B /2 = (φ 1 -φ 2 ) + (φ 12 -φ 21 )/2
The main difference between the real-time signal and the post-processed one is that the limit set by the propagation delay is twice for the real-time data, due to the round trip. Following the same approach as for the previous setups, we demonstrate that the noise rejection for the real-time two-way is given by Eq. 4, as for the actively compensated noise link. The post-processed twoway obeys Eq.3, and the noise rejection is expected to be four times higher. Details are given in Appendix B. One expects then a factor of 2 between the relative frequency stability in real-time and in post processing.

In order to check the validity of the approach, we have experimented this scheme on a fiber spool of 25 km, with a single laser split in two with an optical fiber coupler. We used on-the-shelf fiber components with approximately matched lengths. The interferometric setup was set in a box with an insulating foam. The signals of each photodiodes are separated with RF filters, and then amplified, and filtered. The signal's frequencies are recorded with a dead-time free frequency counter, operated in Λ-mode with 1-s gate time [START_REF] Kramer | Multi-channel synchronous digital phase recorder[END_REF]. Less than 60 points are removed from the two-way data set in post processing and from the real-time data set recorded on PD1. The second real-time data set recorded on PD2 suffers from experimental imperfections (as insufficient fil-tering) and has a poorer signal-to-noise ratio. 120 points are removed from the data set. The modified Allan deviation (MDEV) of the relative frequency fluctuations of the beat notes are displayed on Fig. 2. We found a nice agreement to theory, with as expected, a factor of two between real-time two-way and the post-processed twoway. As the proof of principle is obtained with a 25-km fiber spool, the MDEV are as low as 1•10 -17 and 5•10 -18 at 1-s integration time for the real-time data and postprocessed data respectively. The measurements reaches the interferometric noise floor between 50 and 100 s of integration time, and the three curves perfectly merge themselves at longer integration time. This limitation can be overcome with a proper design and a careful realization of the interferometric setup, as it is discussed in the next section.

IV. INTERFEROMETRIC NOISE

The noise compensation system relies on the assumption that φ 12 = φ 21 . This assumption can be violated by non-reciprocal noise, non reciprocal path, and non reciprocal effect (as Sagnac effect).

In practical implementation, the measurement of these quantities (φ 12 and φ 21 ) contains noise sources inherent to the detection system and that cannot be rejected by constitution. Actually the main noise source may arise from the instability of the interferometric setup. The case of free-space optics interferometer was addressed in [START_REF] Williams | Highstability transfer of an optical frequency over long fiberoptic links[END_REF]. We will focus here on fiber-based interferometric setup.

The detected phase noise is obviously sensitive to the propagation noise in the reference arms [START_REF] Grosche | Optical frequency transfer via 146 km fiber link with 10 -19 relative accuracy[END_REF]. The interferometric noise can be asset by carefully writing the propagation's equations through the interferometric setup. It will be demonstrated below that the interferometric noise arises from the uncommon paths and their sensitivity to external perturbations. Acoustic and mechanical noises can be made negligible thanks to the very short lengths of the reference arms and a well designed isolation box. The temperature is therefore homogeneous inside the interferometer enclosure. Thus the so-called interferometric noise is mainly due to long-term thermal effects.

A. Fiber thermal sensitivity and first-order thermal noise coefficient

We estimate the long term thermal effects on the fiber phase noise density. Temperature variation acts both on the fiber physical length and on the fiber index. Around the working temperature T 0 , the thermal fibre noise density can be expressed as a deterministic function of the temperature variation [START_REF] Cohen | Effect of temperature on transmission in lightguides[END_REF] : where n(T 0 ) = 1.468 is the fibre refraction index, α SiO2 = 0.55 • 10 -6 K -1 is the fiber linear expansion coefficient, c the speed of light, and ∆T the variation from the reference temperature T 0 . ω is the optical carrier angular frequency. The second term inside the parenthesis is the so-called thermo-optical coefficient and is equal to 7 • 10 -6 K -1 [START_REF] Chen | Fiber Bragg gratings and their applications as temperature and humidity sensors[END_REF].

δϕ(z, t) = ω n(T 0 ) c α SiO2 + 1 n(T 0 ) ∂n ∂T (T 0 ) • ∆T (z, t) (8)
In order to be independent of the angular carrier frequency, we conveniently expressed the phase error as a time error in fs. Eq. 8 can be re-written as δϕ(z, t) = γ • ∆T (z, t), were γ is a phase-temperature coefficient at first order. For the SMF28 fiber specifications, we obtain γ = 37 f s/(K • m) for an optical carrier at 194.4 THz and at T 0 = 298 K.

Note that these thermal variations which affect the interferometers are quasi-static compared to the light round-trip propagation delay in a long haul fiber link. Thus the total thermal phase variation along a fiber section of length L can be simply expressed as :

∆φ(t) = γω L 0 ∆T (z, t) dz (9)
When one considers a periodic modulation affecting the phase measurements with a period T m , and using the γ coefficient introduced above, one can show that the Allan deviation of the relative frequency modulation is as high as 1.7•10 -13 s/(K•m•T m ). For a typical perturbation due to an air conditioning system, with T m = 1 000 s, 0.1 K of peak-to peak modulation depth, and 10 cm of optical length mismatch, one expects a bump of the Allan deviation as high as 10 -18 at approximately 270 s.

In order to minimize the interferometric noise, the actual strategy is to adapt as precisely as possible the relevant optical lengths, and keep those arms at the same controlled temperature as well as possible [START_REF] Bercy | In-line extraction of an ultrastable frequency signal over an optical fiber link[END_REF][START_REF] Grosche | Eavesdropping time and frequency: phase noise cancellation along a time-varying path, such as an optical fiber[END_REF]. Such a passive compensation is quite difficult to implement since the contributions of all the optical components, as for instance the Faraday mirrors, optical isolators, couplers, and AOMs, have to be taken into account. Our approach is then to measure the net phase sensitivity of the constituted interferometer.

B. Experimental setup

In order to determine the temperature sensitivity of the interferometric setups depicted in Fig. 1, we have built a large aluminum box. The box is isolated from the experimental breadboard with insulating foams, and shielded with a Mylar insulating film. The temperature of the interferometer is sensed by one thermistor, set in thermal contact with the interferometer. The temperature is recorded by a data logger Agilent 34970A. The box is heated by a Peltier element in thermal contact with one wall of the box. The temperature of the hot face of the Peltier element is sensed by a second thermal sensor. We applied a square modulation to the set-point temperature of the temperature stabilization loop with a period of about 15 000 s.

C. Two-way link for frequency comparison

We are considering first a two-way setup as depicted on Fig. 1a, designed for an optical frequency comparison. The uncommon paths are constituted by the two short arms of the two interferometers at each end. Thus the two-way phase error ∆φ is :

∆φ(t) = 2γω FM1 C1 ∆T (z, t) dz - FM2 C2
∆T (z, t) dz [START_REF] Fujieda | Carrier-phase two-way satellite frequency transfer over a very long baseline[END_REF] For each end of the link, the critical optical length is thus given by twice the length of the short reference arm. For an homogeneous temperature inside the interferometer, the response of the interferometer to a temperature step will be minimized when these optical lengths are minimized and made equal at each end. Experimentally, we have measured the temperature sensitivity of such interferometer, by placing two independent interferometers on the same heating device, exciting the interferometer with temperature step of 4 K. The phase error is simultaneously measured and converted to time error and expressed in fs. We plot the results of the experiment in Fig. 3a. Neglecting one glitch that occurs around 30 000 s, we found from a multiple linear regression fit a temperature sensitivity of -2.2 fs/K, a sensitivity to the temperature derivative of +0.8ps/(K/s), and a time error drift of +1 • 10 -4 fs/s. This temperature sensitivity is the best value reported so far for a two-way link with such an interferometer, to the best of our knowledge.

This two-way interferometers were tested on a 100-km fiber loop surrounding Paris. The full experimental description and the frequency comparisons performances are reported in [START_REF] Bercy | Two-way optical frequency comparisons at 5 • 10 -21 relative stability over 100-km telecommunication network fibers[END_REF]. A flicker noise floor as low as 5.10 -21 at 30 000 s integration time was reached with this interferometric setup.

We note also that a real-time two-way interferometer as introduced in Section III E may have exactly the same performances as this one, as the difference consists only in the respective positions of the Faraday mirrors and the photodiodes.

D. Two-way closed loop link for Sagnac measurement

Sagnac interferometers are that special they are made to measure phase shift in one laboratory, as sketched in Fig. 1b. By contrast with a two-way setup, there is only one interferometer and two beat-notes. Thus there is no uncommon fiber paths.

We realized experimentally such an interferometer, and we have measured its temperature sensitivity with the same experimental apparatus. We excited the interferometer with temperature step of 2 K. We plot the results of the experiment Fig. 3b. We apply again a multiple regression fit to the data, and we found a very low temperature sensitivity of -0.03 fs/K, a sensitivity to the temperature derivative of +1.4 fs/(K/s), and a time error drift of +1.3 • 10 -6 fs/s, as expected from this kind of interferometers.

E. Compensated link for frequency dissemination

We are considering the interferometer of a repeater laser station of a cascaded link [START_REF] Lopez | Ultra-stable long distance optical frequency distribution using the internet fiber network[END_REF]. The experimental setup is depicted on Fig. 1c. Detailed descriptions of the experimental setup are given in [START_REF] Lopez | Ultra-stable long distance optical frequency distribution using the internet fiber network[END_REF][START_REF] Lopez | Cascaded multiplexed optical link on a telecommunication network for frequency dissemination[END_REF]. For sake of clarity, we depicted only the main optical elements. The principle of operation is to successively phase lock a repeater laser (RL) to the upcoming signal arriving from an ultra-stable laser after propagation on the span N , using their beat note detected on PD2. The repeater laser copy the phase of the ultra-stable laser, when the span N is actively stabilized and within the bandwidth of the loops. The repeater laser is used as a seed for the next span, N + 1. Then at Lab 2 or at the next repeater laser station, part of the light is reflected back by a Faraday mirror, and a beat note between the round-trip signal and the RL is detected on the photodiode PD3. After filtering, amplification and processing, the span N + 1 can be stabilized by acting on a AOM for instance. That way, one can repeat step-by-step the ultra-stable laser.

For sake of simplicity, lets assumes that the servo loop and the link compensation are ideal for the first fiber span, denoted N . One can derive that the phase of the remote laser signal on the Faraday mirror FM R1 is equal to the phase of the ultra-stable laser on FM 1. Considering the next fiber span denoted N + 1, one finds similarly that the remote laser phase on the Faraday mirror FM 2 at the link output is equal to the phase of the remote laser on the Faraday mirror FM R2. Finally, the end-to-end phase difference ∆φ is given by : [START_REF] Matveev | Precision measurement of the hydrogen 1s-2s frequency via a 920km fiber link[END_REF] where A corresponds to the position of the output coupler OC R1, and given the assumption that the input phase is defined on the Faraday mirror FM 1. For an homogeneous temperature inside the interferometer, the response of the interferometer to a temperature step will be minimized when the equivalent fiber lengths between the local laser and the two Faraday mirrors of the station are made equal, i.e. L R1 = L R2 .

∆φ(t) = γω• FMR1 A ∆T (z, t) dz - FMR2 A ∆T (z, t) dz
We have checked experimentally this equation by exciting such an interferometer with a temperature step of 4 K, and with the same experimental apparatus as described above. The experimental realization of such a well-balanced interferometer setup is quite difficult because the fiber paths include an AOM and a coupler, as detailed in [START_REF] Lopez | Ultra-stable long distance optical frequency distribution using the internet fiber network[END_REF].We plot the results of the experiment in Fig. 3c. From a multiple linear regression on the data, we found a temperature sensitivity of +1 fs/K, a sensitivity to the time derivative of the temperature of -6.2 ps/(K/s), and a time error drift of -1 • 10 -4 fs/s. By comparison with the design we used before and reported in [START_REF] Lopez | Ultra-stable long distance optical frequency distribution using the internet fiber network[END_REF], there is a 7-times improvement for the temperature sensitivity.

Finally, we tested the 1 fs/K optimized interferometer of Fig. 3(c) on our 540-km long-haul fiber link set between Paris and Reims [START_REF] Lopez | Ultra-stable long distance optical frequency distribution using the internet fiber network[END_REF], in order to check the impact on the long-term stability of the link. We compare the stability of the transferred phase obtained with our first generation interferometer, that has a measured sensitivity of 7 fs/K, and with the new interferometer. The experimental setup is otherwise exactly the same and is reported in [START_REF] Lopez | Ultra-stable long distance optical frequency distribution using the internet fiber network[END_REF]. After adequate filtering and amplification, frequency data are recorded with a dead-time free counter [START_REF] Kramer | Multi-channel synchronous digital phase recorder[END_REF], operated in Λ-mode and with a 1-s gate time. The time error and the corresponding MDEV are plotted in Fig. 4. The time error plot shows a 3-times narrower rms-trace for the new interferometer at short timescale (1-10 s). The phase drift that we observed previously is also reduced, from 2 • 10 -4 fs/s for the 7 fs/K interferometer, to 2.5 • 10 -5 fs/s for the low sensitivity interferometer. This is a reduction by a factor 8. Phase oscillations due to the air conditioning system are also much smaller and their impact on the stability plot is significantly reduced. Note that the oscillation frequencies are different, as the data were not taken at the same epoch of the year. With the low sensitivity interferometer, the MDEV reaches a floor at 3 • 10 -20 after 10 000 s of integration time, which is 10-times better than with the previous setup. We also observe a factor 3 improvement on the short-term relative frequency stability. It could be an indication that the design of the interferometer matters at short time and leads to a better rejection of fast frequency fluctuations. However, the short-term stability strongly depends on the fiber phase noise PSD, which can vary significantly along time. Therefore further investigations are needed to corroborate this result.

V. CONCLUSION

In this article, we have presented investigations on noise processes occurring in bi-directional optical fiber links. We described three setups, for frequency comparison, Sagnac interferometry, and frequency transfer respectively, and introduce a novel setup for two-way frequency comparisons in real-time. For all these setups, we addressed both the unsuppressed delay noise and the interferometric noise for fiber-based interferometers. We derived the unsuppressed delay noise for each setup in the case of spatial correlation that is not an impulse function.

We perform theoretical and experimental investigations on the interferometric noise, for which we introduced a linearization of the thermal induced fibre noise. This work provides handful analysis for designing low noise fiber-based interferometers for the three kinds of interferometers we addressed. We have shown that a well designed Sagnac interferometer, which is intrinsically symmetric, is therefore the easiest to realize and the less sensitive to temperature fluctuations in practice. We reported outstanding temperature sensitivity performance of .03 fs/K, which is the lowest experimental value reported so far to the best of our knowledge. We reported the temperature sensitivity of a two-way setup to be 2.2 fs/K.

We realized a proof-of-principle experiment for realtime two-way frequency comparison and show excellent agreement of the experimental results with theory. We demonstrate finally an interferometer for cascaded fiber links with sensitivity as low as 1 fs/K. We compare the phase measurement recorded with our previous interferometer and with the improved one. We showed that the noise floor of a 540-km link is as low as 3 • 10 -20 at 4 000 s of integration time, which is the best relative instability value reported so far to our knowledge for an actively noise cancelled link, over such distance of many hundreds of km and in such short integration time.

the R function into one spatial-dependent term and one time-dependent term, R(τ ) = R t (τ ) • R s (zz ′ ). We describe again the spatial dependency with a gaussian function written as Eq. A7.

One obtains :

R tw (τ ) = - 1 4 L 0 (2z/v -T ) (2z ′ /v -T ) × (A15) R ′′ t (τ )e -1 2 z-z ′ ∆L 2 dzdz ′
where R ′′ t (τ ) denotes the second derivative of R t (τ ). After some mathematics one obtains :

R tw (τ ) = - R ′′ t (τ ) 4 L 0 (2z ′ /v -T ) 2 +∞ -∞ e -1 2 ( x ∆L )
where the terms φ c,2 and φ c,1 are the two fiber noise corrections calculated in L and 0 and have similar expression as the one given by Eq B2. The corrections terms are applied in post processing to the signals registered at the opposite end of the link. Thus the two-way phase noise with double compensation φ tw+2c (t) = 1/2(φ L (t)-φ 0 (t)) at time t, is :

φ tw+2c (t) = 1 2 [φ 0 (t) -φ L (t)] = φ tw (t)+ + 1 2 [φ c,2 (t -T ) -φ c,1 (t -T )] (B7)
The first term derives from the subtraction of the two free fiber noises integrated from 0 to L and from L to 0 and corresponds to the standard two-way signal of Eq A12. The second term derives from the interaction of the two ANC corrections and can be developed as follows : We can easily calculate that this term is zero with a first order development. We conclude that implementing an additional double ANC-like compensation to a twoway link does not improve the overall noise rejection.

Two-way Real-Time PSD

In a real-time two-way implementation, the one way fiber noise measured at one end is corrected with half of the round trip fiber noise measured at the same end of the link. For example, the round trip fiber noise measured at the end 1 (coordinate 0) and time t is : (B11) Following the same steps from Eq. A6, A14-A17 we obtain the following result :

R rt-tw (τ ) = -R ′′ t (τ ) √ 2π∆Lv T 3 3 (B12)
that is exactly the same obtained in Eq A22. We can conclude that the real-time two-way noise compensation system leads to the same reduction on the free fiber noise of the ANC frequency dissemination setup :

S rt-tw (f ) = (2πf T ) 2 3 e √ 2πf ∆L v 2 S 12 (f ) (B13)

FIG. 1 .

 1 FIG. 1. Experimental setups for (a) Two-way frequency comparison (b) Sagnac interferometry (c) Actively Noise Compensated link and cascaded link. FM: Faraday mirror. OC: optical coupler. AOM: acousto-optic modulator. PLL: phase-lock loop. PC: polarization controller. Ω Earth is the rotation rate of the Earth.

  The signals labelled as A are the beat note signals of the local laser with the remote laser that propagated in single trip over the fiber. Its mean value is the sum of the AOM's frequencies, F . The signals labelled as B are the beat notes of the local laser with itself after a round trip. Its mean frequency is twice the sum of the AOM's frequencies, 2F . The signals labelled as A in Eq. 5 from the two photodiodes at each end can be post-processed, and gives the usual two-way phase signal :

FIG. 2 .

 2 FIG. 2. (a) Experimental sketch of the real-time two-way setup. FM: Faraday mirror. OC: optical coupler. AOM: acousto-optic modulator. PC: polarization controller. (b) Modified Allan Deviation of the recorded data in real-time on PD1 and PD2 (red filled-up triangles and red open downtrainagles), in post-processing from signals on PD1 and PD2 (black circles), and free-running fiber noise (gray circles).

FIG. 3 .

 3 FIG. 3. Temperature modulation of the interferometer and the induced phase error at the output of the interferometer expressed in fs as a function of time. (a) Two-way interferometer (b) Sagnac interferometer (c) Cascaded link.

FIG. 4 .

 4 FIG. 4. Comparison of the transferred phase with an ANC setup with two interferometers, with a sensitivity of 7 fs/K (black trace) and 1 fs/K (red trace). (a) Time error data as a function of the measurement time. (b) Modified Allan deviation as a function of the integration time.

1 2 [ 2 L 0 [ 2 L 0 [

 122020 φ c,2 (t -T ) -φ c,1 (t -T )] = = 1 δϕ(z, tz/v) + δϕ(z, t -2T + z/v)] dz+ -1 δϕ(z, t -T + z/v) + δϕ(z, t -Tz/v)] dz(B8)

  , tz/v)dz + L 0 δϕ(z, t -2T + z/v)dz (B9)So, the real-time two-way signal is, with a first order approximation : φ rt-tw (t) = φ 21 (t) -We can proceed to calculate the autocorrelation of φ real (t) with the method described in the previous paragraphs :R rt-tw (τ ) = L 0 Lz v Lz ′ v δϕ ′ (z, t)δϕ ′ (z ′ , tτ )dzdz ′

ACKNOWLEDGMENTS

The authors thank Emilie Camisard, Thierry Bono and Patrick Donath at the GIP Renater, for giving the opportunity to use the RENATER network.

This work is supported by the European Metrology Research Programme (EMRP) under SIB-02 NEAT-FT. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. This work is supported by the Labex FIRST-TF, the french spatial agency CNES, IFRAF-Conseil Régional Ile-de-France and Agence Nationale de la Recherche (ANR BLANC 2011-BS04-009-01 and REFIMEVE+).

The phase noise accumulated by an optical carrier propagating along a fiber link from an end 1 at coordinate 0 to an end 2 at coordinate L and measured in 2 at time t is :

where z is the position coordinate along the fiber, T = L/v is the propagation delay from 0 to L, v = c/n is the speed of light in the fiber and δϕ(z, t) is the fiber noise per unit of length at position z and time t. Similarly the phase noise accumulated by the optical carrier propagating in the backward direction from the position 2 to the position 1 and measured at position 1 at time t is :

The general approach followed here consists on calculating the phase noise PSD S x of a random signal by calculating the Fourier transform of the autocorrelation function of its phase noise R x (τ ) = x(t)x(tτ ), where the bar denotes a time average :

Free Fiber Noise PSD

We consider the autocorrelation function R 12 (τ ) of φ 12 (t) :

The correlation function R inside the double integral can be expressed as a function of the difference of the linear coordinates and of the relative delay :

We assume here that the space and time dependencies can be separated. We split R into two different functions R t and R s .

We write R s as a gaussian with a deviation ∆L :

We deduce the phase noise PSD at the end 2 as :

We have S δϕ (f ) = F(R t (τ )), which is the PSD of the phase noise per unit of length δϕ(z, t). Note that starting the calculations from the phase noise φ 21 (t) leads exactly to the same result, so S 12 (f ) = S 21 (f ).

We can approximate the limits of the inner integral of Eq. A8 with ±∞ on the assumption that ∆L << L and by neglecting the boundary effects. We set x = zz ′ and one obtains :

We obtain finally :

The two-way phase noise at time t is given by φ tw (t) = 1 2 (φ 12 (t) -φ 21 (t)). From Eq. A2 one has :

A11) A first order (and even a second order) Taylor expansion gives :

where δϕ ′ (z, t) is the time derivative of δϕ(z, t).

We calculate the autocorrelation function R tw (t) of the two-way fiber noise as :

Considering that [START_REF] Bendat | Stationary Random Processes[END_REF] :

) we can use once again the R function introduced in Eq. A6 with R(zz ′ , τ ) = δϕ(z, t)δϕ(z ′ , tτ ). We split dxdz ′ (A16) which gives finally :

The PSD of the two-way fiber noise is calculated as the Fourier transform of R tw (τ ) :

The two-way PSD can be expressed as a function of the one-way PSD :

Actively Noise Compensated PSD

We can approximate at first order the compensated fiber noise [START_REF] Williams | Highstability transfer of an optical frequency over long fiberoptic links[END_REF] as :

The corresponding auto-correlation function is :

Following the same steps of Eq. A6, A14-A17 we obtain :

(A22) which leads to the expression :

which is four times the expression of R tw (τ ) given by Eq. A17.

The noise PSD of the ANC setup can be expressed as a function of the one way free fiber noise PSD S 12 (f ) :

Appendix B: Mixed Two-way and ANC noise rejection

Two-way + Active Noise Compensation

We are now calculating the effects on the fiber noise of a composite two-way + Actively Noise Compensated setup.

We consider first a two-way scheme combined with one single active noise compensation set at one end, that we choose to be at end 1 for instance. The phase noise accumulated by the optical carrier propagating along the link in the forward direction from the end 1 and exiting the fiber at output end 2 at time t is :

where [START_REF] Bercy | In-line extraction of an ultrastable frequency signal over an optical fiber link[END_REF] :

The phase noise accumulated by the optical carrier propagating in the backward direction from the end 2 and exiting the fiber at the end 1 at time t is :

Thus the two-way phase noise with compensation φ tw+c (t) = 1/2(φ L (t) -φ 0 (t)) at time t is given by :

Assuming a negligible correlation length for the sake of simplicity, the phase noise PSD at the output end 2 is again calculated as the Fourier transform of the φ tw+c (t) auto-correlation function, and one finds :

The power spectral density of noise is therefore four times more than that of a post-processed two-way without active compensation.

Two-way + Double Noise Compensation

The second approach is as follow : in a first step the fiber noise is removed by post-processing at both ends of the fiber, and in a second step the two resulting phases are subtracted with a two-way technique. We can write the phases recorded at the positions 1 and 2 (at coordinates 0 and L respectively) at time t as [START_REF] Williams | Highstability transfer of an optical frequency over long fiberoptic links[END_REF] :