Loïc Amoudry
email: amoudry@lal.in2p3.fr

Huan Wang

Kevin Cassou

Ronic Chiche

Kevin Dupraz

Aurélien Martens

Daniele Nutarelli

Viktor Soskov

Fabian Zomer

Modal instabilities suppression in high average power and high finesse Fabry-Perot Cavity

An experimental method to remove modal instabilities induced by thermoelastic deformation in optical high finesse resonators is presented and experimentally investigated. The method is found suitable for multi-mirror folded monolithic and compact cavities as those used in particle accelerator environment. It is also suitable for very large stacked average power. Stable operation at 200 kW intracavity average power is experimentally demonstrated.

INTRODUCTION

External Fabry-Perot cavities (FPC) sustaining resonant highaverage laser-power are nowadays foreseen for various applications. In pulsed regime, Compton scattering of laser beams off charged particle beams allows to produce quasi-monochromatic X and γ ray beams [START_REF] Federici | Backward compton scattering of laser light against high-energy electrons: the ladon photon beam at frascati[END_REF][START_REF] Casano | Production of a beam of polarized and monochromatic gamma rays by compton scattering of laser light against high-energy electrons[END_REF][START_REF] Huang | Laser-Electron Storage Ring[END_REF]. Interaction of laser beams with gas provides high harmonics in the XUV range [START_REF] Pupeza | Compact high-repetition-rate source of coherent 100 eV radiation[END_REF]. In the context of the latter application an average power of 670 kW of a 10 picosecond laser beam was reached in a few thousand finesse cavity [START_REF] Carstens | Megawatt-scale average-power ultrashort pulses in an enhancement cavity[END_REF]. In continuous wave regime the LIGO FPCs have been operated with an intracavity average power around 100 kW [START_REF] Evans | Observation of Parametric Instability in Advanced LIGO[END_REF][START_REF] Barriga | Optical design of the proposed Australian International Gravitational Observatory[END_REF] and an upgrade to 800 kW is foreseen for the LIGO-VIRGO collaboration. Recently, a research and development program has been launched to reach the challenging goal of 3 MW intracavity power for the photo-neutralization of deuterium in the context of future fusion energy experiments [START_REF] Simonin | Negative ion source development for a photoneutralization based neutral beam system for future fusion reactors[END_REF].

However, all these applications face the same experimental issue related to thermal effects induced by high average power. One of the most limiting effect appears to be the transverse mode degeneracy [START_REF] Carstens | Megawatt-scale average-power ultrashort pulses in an enhancement cavity[END_REF][START_REF] Bullington | Modal frequency degeneracy in thermally loaded optical resonators[END_REF] induced by thermo-elastic deformation of the mirror surfaces [START_REF] Hello | Analytical models of thermal aberrations in massive mirrors heated by high power laser beams[END_REF][START_REF] Winkler | Light scattering described in the mode picture[END_REF]. Various methods have been proposed to circumvent this issue. Firstly, the higher order modes resonances can be mitigated by a specific design of the mirror shapes or by additionally heating the mirrors [START_REF] Day | Reduction of higher order mode generation in large scale gravitational wave interferometers by central heating residual aberration correction[END_REF][START_REF] Vajente | In situ correction of mirror surface to reduce round-trip losses in Fabry-Perot cavities[END_REF][START_REF] Matsko | Mitigating parametric instability in optical gravitational wave detectors[END_REF][START_REF] Pare | Custom laser resonators using gradedphase mirrors[END_REF]. Secondly, when interested in the FPC fundamental modes, losses can be induced to high order modes by means of an intracavity pinhole [START_REF] Klaassen | Transverse mode coupling in an optical resonator[END_REF]. Thirdly, effects of thermoelastic deformations can be compensated by inserting a Brewster plate in the FPC [START_REF] Lilienfein | Balancing of thermal lenses in enhancement cavities with transmissive elements[END_REF]. Lastly, for instance in the context of Compton experiments, FPCs have been operated in burst mode making use of the linear accelerator duty cycle which reduces the laser average power to the kW level at most [START_REF] Sakaue | Stabilization of burst laser pulse storage in an optical enhancement cavity using a counter propagating mode[END_REF][START_REF] Favier | Optimization of a Fabry-Perot cavity operated in burst mode for Compton scattering experiments[END_REF].

In this article, we concentrate on FPCs dedicated to Compton X-ray sources based on electron rings [START_REF] Huang | Laser-Electron Storage Ring[END_REF][START_REF] Walter | A new high quality X-ray source for Cultural Heritage[END_REF]. A few peculiarities concerning the design of these FPCs should be highlighted. They are only accessible with difficulties, and their footprint must be reduced since they are located inside the ultra-high vacuum (UHV) chamber of an electron ring, e.g. below 1 × 10 -9 mbar [START_REF] Walter | A new high quality X-ray source for Cultural Heritage[END_REF]. The use of the fundamental mode with a very small waist, typically few tens of microns, is requested to optimize the X-ray production rate. A very high finesse optical cavity, typically above 20000 is of major interest to reduce the average power delivered by the laser amplifier in a cost reduction strategy. The mechanical design must be stable while providing enough flexibility to adjust the cavity round-trip length, which leads to the choice of four-mirror geometries [START_REF] Zomer | Polarization induced instabilities in external four-mirror Fabry-Perot cavities[END_REF][START_REF] Dupraz | The ABCD matrix for parabolic reflectors and its application to astigmatism free four-mirror cavities[END_REF].

Together with the small waist requirement this later constraint implies that the cavity mode can be strongly elliptical [START_REF] Bonis | Non-planar four-mirror optical cavity for high intensity gamma ray flux production by pulsed laser beam Compton scattering off GeV-electrons[END_REF][START_REF] Labaye | Amplification passive d'un laser à fibre optique dans une cavité fabry-perot : application à la production de rayonnement gamma par diffusion compton inverse[END_REF] and operated close to instability region [START_REF] Kogelnik | Laser beams and resonators[END_REF]. Eventually, in order to maximize the X-ray flux produced by laser beam scattering off the electron beam one must reduce as much as possible the crossing angle between the electron and laser beams propagation axes. This means that small mirror diameters must be favored since they are imposing a minimum crossing angle value [START_REF] Bonis | Non-planar four-mirror optical cavity for high intensity gamma ray flux production by pulsed laser beam Compton scattering off GeV-electrons[END_REF].

The list of above technical solutions used to mitigate thermal effects would be extremely difficult to implement or would difficultly fit within the constraints of Compton X ray sources application summarized in the previous section. First, heating the cavity mirrors [START_REF] Day | Reduction of higher order mode generation in large scale gravitational wave interferometers by central heating residual aberration correction[END_REF][START_REF] Vajente | In situ correction of mirror surface to reduce round-trip losses in Fabry-Perot cavities[END_REF] would be very difficult to implement in an electron accelerator environment and not straightforward to apply to one-inch mirror diameters. Shaping the mirrors [START_REF] Matsko | Mitigating parametric instability in optical gravitational wave detectors[END_REF][START_REF] Pare | Custom laser resonators using gradedphase mirrors[END_REF] while keeping a very high reflectivity and thus the highest grade of polishing quality would require dedicated developments on mirror technology. Moreover, these methods would have to be adapted to a cavity regime close to the instability where thermoelastic deformations induce very large changes of the elliptical mode radii on mirror surfaces [START_REF] Carstens | Megawatt-scale average-power ultrashort pulses in an enhancement cavity[END_REF]. The pinhole solution [START_REF] Klaassen | Transverse mode coupling in an optical resonator[END_REF] does not fit with an elliptical beam shape and could also be damaged when high energy is stacked inside the cavity. Inserting a Brewster plate supplemented by a thermal control as in [START_REF] Lilienfein | Balancing of thermal lenses in enhancement cavities with transmissive elements[END_REF] would be difficult to implement in the UHV and confined electron accelerator environment. In this article we propose a simple solution adapted from [START_REF] Klaassen | Transverse mode coupling in an optical resonator[END_REF] that accounts for elliptical mode shapes and, high-finesse, high intracavity power, low loss FPCs. Instead of a pinhole, a pair of movable D-shape dielectric mirrors are used to induce large losses on high-order transverse modes while preserving the high FPC finesse for the fundamental mode. D-shape mirrors with metallic coatings have already been used successfully in [START_REF] Lilienfein | Enhancement cavities for few-cycle pulses[END_REF] to pick-up a small fraction of a short pulse intracavity laser beam in order to perform in situ spectrum measurement.

In this article, we report on the experimental results obtained with this technique. The article is organized as follow. The section 2 is devoted to description of the setup and the model used to describe the effects of thermal heating. The experimental observation of degenerate modes is given in section 3. Finally, significant improvements on the long term operation provided by the implemented solution are put forward in section 4.

EXPERIMENTAL SET-UP AND MODELING OF HEAT-ING EFFECTS

Planar bow-tie cavities with four high reflectivity mirrors {M i } i=1,..,4 are chosen for stability reasons [START_REF] Zomer | Polarization induced instabilities in external four-mirror Fabry-Perot cavities[END_REF][START_REF] Carstens | Large-mode enhancement cavities[END_REF]. A schematic of the research and development cavity used in the experiments described in this paper is given in Fig 1 . The injection mirror M 1 is planar and its substrate is made of Suprasil® [START_REF] Numata | Intrinsic losses in various kinds of fused silica[END_REF]. The three other mirror substrates are made of Ultra Low Expansion glass (ULE) which has been shown to be optimum in the high average power regime [START_REF] Carstens | Megawatt-scale average-power ultrashort pulses in an enhancement cavity[END_REF]. All the mirror coatings are made with alternate dielectric layers of Ta 2 O 5 and SiO 2 of λ/4 thickness for an incidence angle of 4°. Mirrors have been manufactured and characterized by "Laboratoire des Matériaux Avancés" (LMA) in Lyon (France). M 2 is planar and mirrors M 3 and M 4 are spherical with 50 cm of radius of curvature. The four-mirror cavity is located inside a vacuum chamber. A cavity finesse of 20000 has been measured [START_REF] Locke | A simple technique for accurate and complete characterisation of a Fabry-Perot cavity[END_REF] for the experiments presented in this article. The cavity modes exhibit two waists, one at mid-distance between M 1 and M 2 and the other one between M 3 and M 4 . The size of the latter can be easily changed by tuning the distance between the spherical mirrors M 3 and M 4 . The required very small waist and the large beam on the mirrors implies that the cavity is at the limit of the stability. The cavity is seeded by a pulsed laser oscillator at a 1031 nm wavelength and 133.33 MHz repetition frequency. The pulses are stretched to 250 ps in order to be amplified at a maximum of 40 W by a fiber amplifier and then compressed down to 68 ps [START_REF] Chaikovska | High flux circularly polarized gamma beam factory: coupling a Fabry-Perot optical cavity with an electron storage ring[END_REF]. The laser frequency is locked on the cavity round-trip frequency using the Pound-Drever-Hall method [START_REF] Drever | Laser phase and frequency stabilization using an optical resonator[END_REF][START_REF] Jones | Stabilization of the frequency, phase, and repetition rate of an ultra-short pulse train to a Fabry-Perot reference cavity[END_REF].

Even if the mirror coatings absorption is low, about 2 ppm as measured after the coating process, the total absorbed power may be of the order of 1 W or more due to the high intracavity power of several hundreds of kW. As all the system is under vacuum and the mirror substrate made of ULE, the heat dissipation is not efficient and thermal effects may be very important depending on the cavity geometry. In order to model the effect of thermal deformation of the mirror surfaces, one uses the ABCD matrix formalism [START_REF] Kogelnik | Laser beams and resonators[END_REF] and a simplified thermo-mechanical model [START_REF] Winkler | Light scattering described in the mode picture[END_REF]. At the first order of the approximation, the local value of the radius of curvature R , given by the absorbed power in the coating at the center of the mirror, can be written as follows

1 R = 1 R - 2αA b P l 4πκw 2 , ( 1
)
where R is the initial radius of curvature of the mirror surface, α is the substrate's thermal expansion coefficient, respectively 0.01 × 10 -6 K -1 for ULE and 0.6 × 10 -6 K -1 for Suprasil, A b is the coating's absorption coefficient, κ is the substrate's thermal conductivity, respectively 1.31 W m -1 K -1 for ULE and 1.38 W m -1 K -1 for Suprasil, P l is the intracavity laser power and w is the radius of the laser-beam fundamental mode on the mirror surfaces. For numerical applications we shall use values of A b = 1.2 ppm coming from measurements made after the coating process. It is measured by a mirage effect set-up [START_REF] Boccara | Thermo-optical spectroscopy: Detection by the "mirage effect[END_REF] at factory by the LMA. Measurement of low absorption coefficient in the ppm range is very difficult, the accuracy of the measurement can be estimated to 10 % roughly. The value of w on the mirror surface, given by calculation, is of the order of 1.2 mm. As the elements of the ABCD cavity round-trip matrix depend on the radius of curvature of the mirrors surface, thermal effects (i.e. intracavity laser power) change their values and impact the stability condition of the cavity, -1 < (A + D)/2 < 1, where (A + D)/2 is the stability parameter. The resonance condition for higher order modes (q, n, m) is given in [START_REF] Gigan | Image transmission through a stable paraxial cavity[END_REF]. Due to thermal deformation of the mirror surfaces, higher order transverse modes TEM nm can be resonant at the same time as the fundamental TEM 00 mode. This resonance degeneracy has been computed as a function of the intracavity laser power and the results are shown in Fig. 2. It is observed that for the FPC geometry that we implemented, degenerated resonances are very dense and the order of the degenerated mode vary rapidly with the laser power. Since these high order modes are resonant with the fundamental mode, the energy can be transferred from one to the other and thus generate instabilities. This model has proved to be reliable and efficient for this purpose [START_REF] Bullington | Modal frequency degeneracy in thermally loaded optical resonators[END_REF][START_REF] Winkler | Light scattering described in the mode picture[END_REF].

Fig. 2. Stability parameter as function of the intracavity laser power

in the cavity. Each circle or square represents the occurrence of a degeneracy between the fundamental and a high order mode n and m, with n and m ranging from 1 to 30 for both x and y directions.

It is found that the degenerate mode order is quite high in both directions for an intracavity power corresponding to our operating condition as shown in Fig. 3. Furthermore, the extension of a Hermite-Gauss mode of order n in one of the transverse directions is asymptotically given by √ nw for large n [START_REF] Siegman | Lasers[END_REF]. It is then possible to introduce strong losses for these modes while preserving the fundamental one by simply using a mask [START_REF] Klaassen | Transverse mode coupling in an optical resonator[END_REF].

EXPERIMENTAL OBSERVATION OF MODAL INSTA-BILITIES

As shown in Fig. 4 high order modes have been observed in a range of power which in a good agreement with simulations. For instance, the mode (27.11) has been observed at 110 kW intracavity power instead of 136 kW, as predicted in Fig. 3. This difference can be explained by the fact that the model used for the simulations remains a first-order approximation and strongly relies on the measurement of the mirror coating absorption coefficient, which is subject to uncertainties given its smallness, or the radius of curvature of the mirrors. An increase of 170 µm of the value of the latter parameter allows to match the model to the experimental data, to be compared to a ±1 mm manufacturing uncertainty. We thus conclude that the observations are consistent with the model and thus allows the observations to be interpreted satisfactorily here. It must be pointed out that the size of the mirrors limits the maximum mode order. It has been observed on different types of FPC geometries that the energy between two resonating modes, in particular TEM nm and TEM 00 , induces instabilities on the TEM 00 [START_REF] Carstens | Megawatt-scale average-power ultrashort pulses in an enhancement cavity[END_REF][START_REF] Benedikter | Transverse-mode coupling and diffraction loss in tunable Fabry-Perot microcavities[END_REF]. We indeed observed this effect up to a loss of the lock, see Fig. 5 that shows the intracavity power as function of time. Up to 20 % average power losses are happening due to these degeneracies. The transition time between the fundamental mode (1) and a degenerated regime (2) observed in Fig. 5 is compatible with the transient thermoelastic deformations timescale as computed using [START_REF] Hello | Analytical models of thermal aberrations in massive mirrors heated by high power laser beams[END_REF][START_REF] Hello | Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams[END_REF]. Oscillations of the intracavity power due to a change of the thermal load on the mirrors are expected [START_REF] Bullington | Thermal loading and modal frequency degeneracy in optical resonators for the laser interferometer gravitational-wave observatory[END_REF][START_REF] Klaassen | Transverse mode coupling in an optical resonator[END_REF] and also observed here. They are due to a coupling between the dynamics of the thermal diffusion into the substrate and the properties of the electronic feedback loop acting on the laser. The growth of these oscillations irremediably induces a loss of the lock of the system and thus prevents its long term operation at average intracavity power above about 100 kW. Performance goals for X-ray sources based on Compton scattering [START_REF] Walter | A new high quality X-ray source for Cultural Heritage[END_REF] are thus not fulfilled.

HIGH ORDER MODES SUPPRESSION

In order to circumvent this limitation, several solutions [12-16, 18, 19] have been investigated but do not satisfy the requirement of operating a FPC at high average power with high-finesse under high vacuum and in an accelerator environment. We thus propose a new technical solution to comply with these requirements by installing two half-round mirrors with knife edge, so called D-shaped mirrors. As shown on the Figure 1, these mirrors D 1 and D 2 are installed on motorized translation stages between M 1 and M 2 , each on a separate transverse axis. This position has been chosen for practical reasons, indeed the beam is almost parallel between M 1 and M 2 and the higher order modes have a transverse size large enough so that they can be easily damped. Besides, it is not possible to put that kind of mirror system near to the other waist due to the proximity of the interaction point. We choose knife edge mirrors to allow placing them in the beam with an angle of about 10°. The reflected part of the beam thus hits the vacuum chamber wall to avoid any spurious reflection or additional heat load on the mirrors or their mounts. It is observed on Figure 6 that, as expected, power losses appear if D 1 and D 2 are moved too close to the fundamental mode beam. Several comments could be made from this test. First of all, as expected, there exist a position, in both transverse directions, for which losses are negligible on the fundamental mode. Secondly, it is noticeable that these positions are not the same for both axes and that they depend on the average intracavity power. This is due, at least partly, to the expected different increase of elliptical mode sizes along the two transverse axes induced by thermoelastic deformations [START_REF] Carstens | Megawatt-scale average-power ultrashort pulses in an enhancement cavity[END_REF]. Measured positions and their expected values from calculations are found to be in agreement within 10 %. Finally, for beam mode diameters of about 1.8 mm (horizontal) and 2 mm (vertical), degenerate high order modes were observed if D-shape mirrors are brought further than ≈ 400 µm from these positions. This ensures that there is a comfortable range of D-shape mirrors positions that ensure both damping of the high order modes and negligible losses on the fundamental mode. Experimentally, these positions are determined by requiring no noticeable variation of the transmitted power which is measured with a relative precision 5 %, by means of a power meter. One can thus conservatively estimate the maximum losses induced by the D-shape mirrors by assuming that they solely lead to 5 % decrease of the transmitted power. Taking our measured values for the mirror M 1 transmissivity (180 ppm) and of the total cavity round-trip loss (316 ppm), we obtain a maximum loss induced by the D-shape mirrors of about 8 ppm. A cavity optimally coupled with a total round-trip loss of 8 ppm corresponds to a cavity finesse of approximately 400000. The method proposed here could thus in principle be used for such very high cavity finesse values. As a result we have observed clear improvements on the intracavity power stability. In particular, we proceeded a 30 min run at about 200 kW for which the evolution of the intracavity power as a function of time is shown on Figure 7. This measurement has been done without any alignment or feedback corrections during the run. A drop of power from 205 kW to 197 kW is observed due to thermal stabilization of the vacuum vessel. The initial value of 205 kW is reached back after further alignment and feedback optimization. This level of intracavity power corresponds to a fluence of approximately 0.04 J/cm 2 on the cavity mirror coatings which is well below damage threshold. We further inspected mirror surfaces with an Atomic Force Microscope to check that no damage has effectively been induced.

CONCLUSION

In summary, an experimental method to remove high order modes degeneracy in a four-mirror Fabry-Perot cavity is described. A simple model was used to satisfactorily describe the degeneracy observed experimentally, as a function of the intracavity laser beam power, which leads to locking instabilities thus preventing nominal operation of the system. Introduction of D-shape dielectric coated mirrors which are adapted to handle high average power cavity modes because of their non-absorbing properties allowed to damp these high order modes while introducing negligible losses on the fundamental mode. This solution is particularly suitable for a particle accelerator environment typical of Compton back-scattering radiation sources. Furthermore, their implementation allows precise and convenient positioning, independent on the two main axes, thus especially suited for all fundamental mode shapes including elliptical beams. We estimated that losses induced by D-shape mirrors on the fundamental modes would fit with cavity finesse of the order of a few hundred thousand. In principle this simple method may thus be used to reach much higher intracavity power. Other processes such as mirror coating damage or degradation may however limit its increase. The former issue can be mitigated by increasing the spot sizes on the mirrors and using H f O 2 /SiO 2 instead of Ta 2 O 5 /SiO 2 multilayer coatings [START_REF] Gallais | Laser-induced damage thresholds of bulk and coating optical materials at 1030 nm, 500 fs[END_REF]. The latter issue can also be overcome by using for instance in-situ or ex-situ plasma cleaning as in intense laser beam and synchrotron beam line experiments (see e.g. [START_REF] Gubbini | Decontamination of cpa diffraction gratings[END_REF]). In our experiment we indeed observed a decrease of the cavity enhancement factor when increasing the incident laser beam power as it was also reported in [START_REF] Carstens | Megawatt-scale average-power ultrashort pulses in an enhancement cavity[END_REF]. We are presently investigating in detail the origin of such a behavior.

ACKNOWLEDGMENTS

We are grateful to Dr Jérôme Degallaix for fruitful discussions and his comments on this article.

Fig. 1 .

 1 Fig. 1. 3D view of the Fabry-Perot cavity. M 1 input planar mirror, M 2 high reflectivity planar mirror, M 3 and M 4 high reflectivity spherical mirrors. D 1 and D 2 are two half round mirrors, so called D-shaped pickoff mirrors.

Fig. 3 .

 3 Fig. 3. Zoom on Figure 2 around 135 kW intracavity power for x direction. The transverse mode numbers are specified on the figure.

Fig. 4 .

 4 Fig. 4. Superimposition of the resonant high order mode (27.11) and the fundamental mode observed in transmission of M 2 at an intracavity power of 110 kW. The fundamental mode is saturated to clearly distinguish the larger mode.

Fig. 5 .

 5 Fig. 5. The curve shows the intracavity power for 1.7 s, measured through M 4 . (1) only fundamental mode, (2) switch to a degenerated mode, (3) stable degenerated region and (4) oscillations due to thermal effects.

Fig. 6 .

 6 Fig. 6. Experimental measurement of the intracavity power at different D-shaped mirror positions for both vertical and horizontal axes and at several input power values. The numerical value of the displacement increases while the mirrors are getting closer to the optical axis.

Fig. 7 .

 7 Fig. 7. Laser intracavity power for 30 min, measured by transmission of a cavity mirror.

FUNDING

This work is supported by the French "Agence Nationale de la Recherche" as part of the program "investing in the future" under reference ANR-10-EQPX-51.

DISCLOSURES

The authors declare no conflicts of interest.


Loren Méar
email: loren.mear@gmail.com.

Charles Pineau

François Vialard

Juan-Felipe Velez De La Calle

Endometriosis screening in patients attending an IVF clinic: a proof-of-concept 5 retrospective cohort study. 6 7 8

Keywords: Endometriosis screening, IVF, high-risk population 43

25 Given that (i) endometriosis affects approximately 40% of women diagnosed with 26 fertility problems and (ii) this condition may be an underestimated cause of idiopathic 27 infertility, it is essential to identify high-risk patients for laparoscopic screening and 28 reduce the diagnostic delay. We performed a retrospective analysis of 312 women (208 29 diagnosed with endometriosis and 104 controls) admitted to an in vitro fertilization 30 (IVF) unit in the city of Brest (France) between June 2007 and July 2014. As part of the 31 women's infertility treatment, levels of cancer antigen 125 (CA-125) were assayed in 32 blood samples collected on the day of oocyte retrieval. Surplus serum was used to set up 33 a new sperm agglutination test. It was observed that sperm agglutination was 34 significantly correlated with endometriosis and CA-125 levels (p<0.01 for both). By 35 building a decision tree, we identified a subpopulation of patients with low CA-125 36 levels and a high risk of endometriosis. This proof-of-concept study constitutes a first 37 step towards a high-quality, controlled, multi-centre trial. If our preliminary results are 38 confirmed, the decision tree should improve the medical care given to women in IVF 39 programmes by identifying potential endometriosis sufferers for laparoscopic 40 examination and enabling them to be counselled about precautionary measures.

119 July 2014 were included in the study. The sole inclusion criterion was admission to an 120 IVF programme. Two groups of participants were defined: (i) women with 121 laparoscopically and histologically confirmed endometriosis (hereafter referred to as the 122 endometriosis group), and (ii) women without a documented history of endometriosis 123 (the control group). For ethical reasons, the absence of endometriosis was not confirmed 124 laparoscopically in the control participants; there was no justification for an invasive 125 surgical procedure in these patients.

126

All women received standard treatment for IVF, namely the administration of a

  x 10 -9 ) and between age and endometriosis (p = 0.03595). 565 566 S2. Fig. A) Variable importance in the model, tested in a random forest analysis. 567 Agglutination 1:4 was the most important variable in this model, with regard to both 568 accuracy and the Gini index. B) The random forest plot for 200 trees, mtry= 3, and an 569 estimated OOB error rate of 23.72%.

Table 1 . A qualitative description of the study population.

 1 The results of the chi-squared test in a bivariate analysis of each of the two dilutions showed that the CA-125 level and the agglutination level were significantly associated with the origin of the IVF sample (endometriosis or not).

	Variable ≤ 35 years > 35 years	Control group n % 50 48.08 54 51.92	Endometriosis group n % Age class 124 59.62 84 40.38 CA-125 level m a n u s c r i p t Total n % 𝟀2 p-value 174 55.77 138 44.23 3.289 0.07
	≤ 30 U/mL > 30 U/mL -+	87 17 21 28 A c c e p t e d 83.65 108 51.92 16.35 100 48.08 Agglutination with a 1:2 dilution 195 117 20.19 18 8.65 39 26.92 34 16.35 62	62.5 37.5 12.5 19.87	28.446 16.302	<0.0001*** <0.0005**

Table 2 . The training and test sets.

 2

	Training set

Table 3 . Multivariate model (obtain by logistic regression) for discriminating between the endometriosis group and the control group.

 3

	Page 29 of 33
	All the variables considered here were qualitative.
	The Akaike information criterion for the model was 342.33.
		Odds ratio	95%CI	p value
	(Intercept)	0.5513037	0.3091266 -0.9630029	>0.01	*
	CA-125 > 30 U/mL	4.3015392	2.3748690 -8.1532708	<0.0001	***
	Agglutination1:4 +	2.2483028	1.0959224 -4.6887153	>0.01	*
	Agglutination1:4 ++	5.3711769	2.9005988 -10.1558972	<0.0001	***
	Age > 35	0.6644742	0.3906826 -1.1264008	>0.1
	573


D Pelati

G Patriarche

L Largeau

O Mauguin

L Travers

F Brisset

F Glas

F Oehler

Microstructure of GaAs thin films grown on glass using Ge seed layers fabricated by Aluminium Induced Crystallization

Keywords: Metal Induced Crystallization, ALILE, Germanium, Aluminium, Gallium Arsenide, Molecular Beam Epitaxy remains lower than commercial mono-crystalline Ge

We perform the growth of GaAs epilayers by molecular beam epitaxy (MBE) on Ge pseudo-substrates obtained by the Aluminium Induced Crystallisation (AIC) of thin amorphous Ge layers deposited on silica. Despite the apparent uniformity of the AIC-Ge layer, large domains (more than 50 µm wide) previously thought to be monocrystalline are found to actually consist in smaller grains (500 to 1000 nm wide), separated by low angle grain boundaries. These defects are transferred during epitaxy to the GaAs layer and degrade the quality of the III-V material. In our growth conditions, the MBE results in the selective deposition of thin GaAs layers on Ge with respect to the silica support, but selectivity is progressively lost with increasing layer thickness.

The low cost of the ALILE technique has primarily prompted solar applications [5,[START_REF] Gordon | Thin-film polycrystalline silicon 447 solar cells on ceramic substrates by aluminium-induced crys-448 tallization[END_REF][START_REF] Gestel | Aluminum-induced 451 crystallization for thin-film polycrystalline silicon so-452 lar cells: Achievements and perspective[END_REF][START_REF] Shekoofa | P-silicon thin film 457 fabricated by magnetron sputtering and aluminium induced 458 crystallization for schottky silicon solar cells[END_REF], although there have been other outcomes such as transistors [START_REF] Chen | High-462 performance polycrystalline silicon thin-film transistors based 463 on metal-induced crystallization in an oxidizing atmosphere[END_REF][START_REF] Moto | Polycrystalline thin-film transistors fabricated on high-468 mobility solid-phase-crystallized Ge on glass[END_REF].

Interestingly, the control of the diffusion of Ge in the metal can lead to the crystallization of oriented grains with normals mainly along the [111] or [100] directions, depending on the nature of diffusion barrier and substrate [START_REF] Okada | Dependence of crystal orientation in Al-induced 472 crystallized poly-Si layers on SiO2 insertion layer thick-473 ness[END_REF][START_REF] Numata | Orientation control of large-grained Si films on 477 insulators by thickness-modulated Al-induced crystallization[END_REF][START_REF] Nakazawa | Large-grained polycrystalline (111) Ge films on in-482 sulators by thickness-controlled Al-induced crystallization[END_REF][START_REF] Toko | Selective formation of large-grained, (100)-or 487 (111)-oriented Si on glass by Al-induced layer exchange[END_REF][START_REF] Toko | Improved surface quality of the metal-induced crystal-492 lized Ge seed layer and its influence on subsequent epitaxy[END_REF][START_REF] Park | 496 Nucleation-controlled gold-induced-crystallization for selective 497 formation of Ge(100) and (111) on insulator at low-temperature 498 (250C)[END_REF][START_REF] Oya | Effects 501 of flexible substrate thickness on Al-induced crystallization of 502 amorphous Ge thin films[END_REF]. The fabrication of flat oriented crystals with large grains * Email address: fabrice.oehler@c2n.upsaclay.fr (F. Oehler) (wider than 50µm) is interesting and can compete 21 with other approaches such as localized liquid phase 22 crystallization [START_REF] Hu | Single-505 crystal germanium layers grown on silicon by nanowire seed-506 ing[END_REF] or laser annealing [START_REF] Kühnapfel | Towards 509 monocrystalline silicon thin films grown on glass by liquid 510 phase crystallization[END_REF][START_REF] Toko | Formation of single-513 crystalline Ge stripes on quartz substrates by SiGe mixing-514 triggered liquid-phase epitaxy[END_REF]. [START_REF] Toko | Vertically aligned Ge nanowires on flexible plastic films syn-530 thesized by (111)-oriented Ge seeded vapor-liquid-solid growth[END_REF] Yet, the use of MIC layers as oriented crystalline 24 pseudo-substrates for subsequent epitaxy has been [START_REF] Mauk | Low-cost III-V compound semiconductor so-538 lar cells[END_REF] rather limited. Thin MIC-Si or MIC-Ge layers on silica 26 have been used as substrates for the growth of GaAs [21, 27 22] or Ge [START_REF] Toko | Vertically aligned Ge nanowires on flexible plastic films syn-530 thesized by (111)-oriented Ge seeded vapor-liquid-solid growth[END_REF] nanowires, but nanostructures with small 28 footprints are not very demanding in terms of substrate 29 crystallinity [START_REF] Dhaka | High quality GaAs 535 nanowires grown on glass substrates[END_REF]. In an attempt to reduce defect density (Fig. 2.d). Fortunately, the EBSD data (Fig. 3 and4) show that the main orientation of our MIC-Ge layers is (111), similarly to most layers fabricated by ALILE on silica [START_REF] Nakazawa | Large-grained polycrystalline (111) Ge films on in-482 sulators by thickness-controlled Al-induced crystallization[END_REF][START_REF] Numata | Growth promotion of Al-induced crystallized Ge films on insulators by insertion of a Ge membrane below the Al layer[END_REF][START_REF] Toko | Improved surface quality of the metal-induced crystal-492 lized Ge seed layer and its influence on subsequent epitaxy[END_REF]. On commercial 'epiready' Ge(111)

wafers, the polarity of GaAs epilayers grown by MBE is uniquely determined and insensitive to the surface roughness, so that no antiphase domain form during the growth [START_REF] Koh | GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices[END_REF][START_REF] Pelati | GaAs (111) epilayers grown by MBE on Ge (111): Twin reduction and polarity[END_REF]. Our TEM analysis (Fig. 5) did not reveal any APB in our GaAs layers, despite the lower surface quality of our MIC-Ge pseudo-substrates compared to commercial Ge(111) wafers. The growth of GaAs on (111)-oriented MIC-Ge templates remains ABP-free and it is thus a promising combination for the fabrication of low-cost III-V devices on silica.

Selective growth of GaAs on MIC-Ge(111)/SiO 2

The use of a silica support for the MIC-Ge layers is particularly relevant since SiO 2 masks are already used in the selective area growth of Ge [START_REF] Li | Probing interactions of Ge with chemical and thermal SiO2 to understand selective growth of Ge on Si during molecular beam epitaxy[END_REF] or GaAs [START_REF] Plissard | High yield of selfcatalyzed GaAs nanowire arrays grown on silicon via gallium droplet positioning[END_REF][START_REF] Munshi | Positioncontrolled uniform GaAs nanowires on silicon using nanoimprint lithography[END_REF] by MBE. Despite the relatively high substrate temperature (∼630 • C), we only observe a partial selectivity for the GaAs growth between the MIC-Ge islands and the SiO 2 support, which is progressively lost as the deposition duration increases from 4.5 to 40 min (Fig. 2). It is possible that the exposed silica still host some residual Al or Ge contaminations from the deposition of the initial MIC structure, despite the thorough chemical cleaning of the MIC-Ge layers (see methods). A solution to this partial selectivity issue is to fabricate MIC-Ge layers with a high surface coverage [START_REF] Nakazawa | Large-grained polycrystalline (111) Ge films on in-482 sulators by thickness-controlled Al-induced crystallization[END_REF][START_REF] Toko | Improved surface quality of the metal-induced crystal-492 lized Ge seed layer and its influence on subsequent epitaxy[END_REF] so that no SiO 2 is exposed to GaAs during the MBE growth [START_REF] Nishida | High photoresponsivity in a GaAs film synthesized on glass using a pseudo-single-crystal Ge seed layer[END_REF].

308

From previous work [START_REF] Pelati | GaAs (111) epilayers grown by MBE on Ge (111): Twin reduction and polarity[END_REF] and literature [START_REF] Kajikawa | Suppression of twin generation in the growth 618 of GaAs on Ge (111) substrates[END_REF]

  Metal induced crystallization (MIC) is a promising technique for the fabrication of crystalline Si or Ge layers on low cost supports. Depending on the nature of the metal, the crystallization of amorphous Ge can occur at very low temperatures, down to 177 • C for Au or 211 • C for Al, below the respective eutectic temperature of the binary mixtures (Ge-Al 420 • C, Ge-Au 361 • C)[1, 2, 3]. The aluminium-induced crystallization (AIC) of Ge has attracted much interest due to the peculiar layer inversion process (ALILE) during the crystallization[4].

30 presentFigure 1 :

 301 Figure 1: Simplified processing scheme of the samples. (a) Fabrication by MIC (ALILE) of the crystalline (c-Ge) pseudo-substrates starting from 20 nm thick amorphous Ge (a-Ge) and aluminium layers deposited on silica. (b) Growth of GaAs on the Ge pseudo-substrates by molecular beam epitaxy (MBE).

Figure 2 Figure 2 :Figure 3 :

 223 Figure 2 presents the surface morphology before and

Figure 3 Fig. 3 .

 33 Figure 3 presents SEM and EBSD characterizations

Figure 4 :

 4 Figure 4: SEM and EBSD characterizations of an area with uniform in plane orientation. The yellow dashed line marks the position of the FIB cut used to extract the TEM foil. (a) Top view SEM image showing the zone selected for EBSD analysis (white rectangle). (b) Inverse pole figure (IPF) color scheme used in the EBSD maps.(c) EBSD map of the in plane crystal orientation. (d) Out of plane EBSD map showing grain boundaries (0-10 • ) in green. (e) Out of plane crystal orientation, twin boundaries are marked in white.

  147

Figure 3 .

 3 Figure 3.b presents the morphology after the growth

176Figure 5 :

 5 Figure 5: Dark field cross-sectional g 002 TEM image of the GaAs / MIC-Ge pseudo-substrate near the [110] zone axis. (a) Large scale view showing the zones selected for further analysis. The lateral size of the sub-grains is indicated above the image. (b) Details of a smooth area showing the thickness of each layer on the SiO 2 support. (c) Details of a pyramidal hillock. A defect from the MIC-Ge pseudo-substrate (orange arrow) extends to the GaAs layer (green arrows).

Figure 6 :Figure 6 Fig. 6 .

 666 Figure 6: HR-TEM images of the zones analyzed in Fig. 5. (a) Atomically resolved image of the smooth area (see Fig. 5.b). (b) Image of the defective zone with two sub-grains (see Fig. 5.c). The left and right squares mark the zones used to perform FFT. (c) FFT patterns of the left and right sub-grain. Both GaAs crystals are cubic with a relative angular misalignment of ∆ϕ=2.4 • .

  reports on nanowire growth[START_REF] Cohin | Growth of vertical GaAs 518 nanowires on an amorphous substrate via a fiber-textured 519 Si platform[END_REF][START_REF] Ren | Growth op-523 timization for self-catalyzed GaAs-based nanowires on metal-524 induced crystallized amorphous substrate[END_REF][START_REF] Toko | Vertically aligned Ge nanowires on flexible plastic films syn-530 thesized by (111)-oriented Ge seeded vapor-liquid-solid growth[END_REF], 215 we could only find an two attempts at planar layer 216 epitaxy on MIC substrates: homoepitaxy of Ge on 217 MIC-Ge(111)[15] and hetero-epitaxy of GaAs on MIC-218 Ge(111)[28]. Contrary to Toko et al.[28], who report 219 good photoresponse from GaAs layers grown on MIC-220 Ge, the disrupted morphology of our thick GaAs layers 221 (Fig. 2) indicates that defects degrades the III-V mate-222 rial quality, at length scales much smaller to the appar-223 ent domain size (over 20µm) determined by large scale 224 EBSD (Fig. 3).

2254. 1 .

 1 Absence of antiphase boundariesCompared to commercial 'epiready' monocrystalline Ge wafers, the surface of our MIC-Ge pseudo-substrates is very rough: nanometer sized variations of the MIC-Ge thickness can be seen directly by cross-sectional TEM in Fig.5, even in the defect-free area. Due to the small thickness of the MIC-Ge layer (20 nm) further chemical or mechanical polishing is proscribed. It is thus critical that the GaAs epitaxy is not sensitive to the Ge surface roughness. Yet, even the standard epitaxy case of polar GaAs on non-polar Ge is known to require a carefully reconstructed double-step Ge(100) surface and a specific wafer miscut[START_REF] Pukite | Suppression of antiphase domains in the growth of GaAs on Ge(100) by molecular beam epitaxy[END_REF][START_REF] Sieg | Anti-phase domain-free growth of GaAs on offcut (001) Ge wafers by molecular beam epitaxy with suppressed Ge outdiffusion[END_REF]. The direct GaAs epitaxy on (100)-oriented MIC-Ge layers is thus likely to hold high densities of antiphase boundaries (APB).The identification of APB from EBSD maps is a difficult task, as the standard IPF color scheme (Fig. 4.b) is insensitive to 180 • rotations, so that it cannot discriminate grains with opposite orientations. Note that this applies to both out of plane (Fig. 4.d) and in plane (Fig. 4.c) EBSD maps.

4. 3 .

 3 Epitaxial relationship GaAs/MIC-Ge(111) 277 By focusing on the thin GaAs layer (∼40 nm) for 278 which the growth is still fully selective (Fig. 2), we 279 can learn more about the epitaxial relationship between 280 the GaAs layer and the MIC-Ge pseudo-substrate. The 281 transfer of the large scale (111) crystal texture from 282 the MIC-Ge template to the GaAs epilayer (Fig. 3) is 283 a good indication of the epitaxy relationship. This hy-284 pothesis is confirmed by cross-sectional HR-TEM im-285 ages (Fig. 6), where we observe the continuation of the 286 crystal lattice from the cubic MIC-Ge grain to the GaAs 287 cubic crystal. The growth of GaAs (111) on the MIC-288 Ge(111) thus proceeds as on standard monocrystalline 289 Ge substrates[37, 41, 34] and preserve in GaAs the same 290 in plane and out of plane orientations as the Ge sub-291 strate. The absence of visible defect by TEM at the 292 GaAs/Ge(111) interface (Fig. 5) demonstrates that our 293 substrate fabrication and cleaning procedures enable the 294 defect-free epitaxy of GaAs(111) on MIC-Ge(111).

2954. 4 .

 4 Twinning in GaAs(111) epilayers 296 As the GaAs growth proceeds in epitaxy with the 297 MIC-Ge substrate, it seems likely that the crystal ori-298 entation of MIC-Ge grains is exactly that of GaAs epi-299 layer. Yet it is possible that extended defects in the 300 GaAs layer alter this initial epitaxy relationship. 301 The color scheme of the EBSD maps stems from a 302 standard IPF (Fig. 4.b). As stated before, this color cod-303 ing does not differentiate between 180 • rotations, so that 304 it is not only insensitive to antiphase domains but also 305 to twin defects. However twin boundaries can be re-306 vealed by showing the local 60 • in plane misorienta-307 tions in (111)-oriented domains (Fig. 4.e).

3224. 5 .

 5 Fig. 5) matches the length scale of the small tonal changes in the EBSD maps (Fig. 4.c) and the sub-grain width of Fig. 4.d. The inner structure of a LAGB consisting in a dense 1D array of dislocations[42], the high density of LAGBs also explains the poor optical quality of our GaAs epilayers (not shown).The Fig.5.a and Fig.6.b clearly show that the LAGB originates from the MIC-Ge layer. This implies that the epitaxial growth merely extends the defective microstructure of the MIC-Ge pseudo-substrate to the GaAs layer. Compared to other attempt at planar epitaxy using MIC-Ge pseudo-substrates[START_REF] Toko | Improved surface quality of the metal-induced crystal-492 lized Ge seed layer and its influence on subsequent epitaxy[END_REF][START_REF] Nishida | High photoresponsivity in a GaAs film synthesized on glass using a pseudo-single-crystal Ge seed layer[END_REF], we ob-

nium, Journal of Vacuum Science & Technology A 24(3) (2006)

nal of Applied Physics 105 (8) (2009) 083532 (2009).