11-15 Avril 2022, Marseille 1 Experimental nonlinear acoustic techniques for visualizing damage in materials and structures have been intensively developing for more than 30 years. It is of interest to supplement them by relevant modeling methods capable of imitating acoustic propagation in damaged materials. The associated gains include the availability of all calculated mechanical fields instead of restrained experimental data, the possibility to predict false detections and false alarms, and, in perspective, to reconstruct real parameters of defects by comparing measured data to synthetic ones. In this work, we model mechanical processes corresponding to laser vibrometry experiments in which nonlinear components of generated acoustic standing waves in a sample approximately reveal the position of damage. To do so, we apply a previously developed numerical method that combines a physical model of frictional contacts representing damage and the finite element formulation for acoustic waves. This physical model accounts for the Coulomb friction law governing mechanics of contacting rough surfaces by using the analogy between rough profiles and axisymmetric contacts of Cattaneo-Mindlin type. The latter ones are successfully described with the help of the original method of memory diagrams capable of calculating contact response to an arbitrary acoustic excitation. In our numerical experiments, we form standing waves in a domain containing a crack with known geometric properties. Depending on structure geometry, excitation type and strength, as well as on damage size, position and depth, we obtain widely different responses containing nonlinear spectral components. We qualitatively compare them to available experimental data and formulate conclusions on theoretical sensitivity of nonlinear resonant methods in various situations.

Introduction

This paper is concerned a new method of acoustic modeling in materials with frictional contacts aimed at nondestructive testing (NDT) applications. Frictional contacts are frequency related to damage (cracks, delaminations) in solid structures. Nowadays, contact acoustic nonlinearity is used for detecting positions and extent of damage, at least in laboratory context. It is important to emphasize that nonlinear acoustic NDT is a family of essentially experimental techniques. Numerical models for underlying physical processes exist only in particular cases and on the basis on simplified description of contact nonlinearity. Description of an arbitrary frictional contact excited by an arbitrary acoustic field requires huge computational expenses. The main reason for that is in the fact that the account for friction engenders implicit calculations. The basic approximate model for frictional interaction (Coulomb friction law of T=µ N type) does not produce the required boundary conditions as a direct link between stresses and displacements. It just says that particular contact fragment can be found in the state of slick or of slip, and defines conditions for that in terms of stresses. The displacements in this case should be adjusted altogether and multiple times in order to satisfy those conditions. In addition, friction makes the problem hysteretic i.e. a current response of a frictional system is not instantaneous but depends on loading history.

Nevertheless, there exists a family of numerical contact mechanics methods [START_REF] Yastrebov | Numerical Methods in Contact Mechanics[END_REF][START_REF] Vollebregt | User Guide for CONTACT, Rolling and Sliding Contact with Friction[END_REF] that successfully address those issues. However, they are typically applied to simple loading histories (e.g. loading-unloading-reloading) whereas acoustic signal contain large number of oscillations. There also exist numerical acoustic methods [START_REF] Blanloeuil | Numerical study of nonlinear interaction between a crack and elastic waves under an oblique incidence[END_REF][START_REF] Blanloeuil | 2d finite element modeling of non-collinear mixing method for detection and characterization of closed cracks[END_REF][START_REF] Blanloeuil | Analytical extension of Finite Element solution for computing the nonlinear far field of ultrasonic wavesscattered by a closed crack[END_REF] applicable to ultrasound signals in the presence of inner frictional contacts but their use is associated with considerable computational expenses coming from an implicit nature of solution.

In a series of publications [START_REF] V. Aleshin | Method of memory diagrams for mechanical frictional contacts subject to arbitrary 2D loading[END_REF][START_REF] Popov | Method of Dimensionality Reduction in Contact Mechanics and Friction[END_REF][START_REF] Aleshin | Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction -Part I: Theoretical background[END_REF] it is suggested an alternative approach based on the Cattaneo-Mindlin [START_REF] Cattaneo | Sul contatto di due corpi elastici: distribuzione locale degli sforzi[END_REF] (also called Hertz-Mindlin [START_REF] Mindlin | Elastic spheres in contact under varying oblique forces[END_REF]) solution. This classical result links forces and displacements for contact of two spheres loaded by one of a certain set of particular protocols. More recently, two important advances have been suggested: account for non-spherical but axisymmetric shapes [START_REF] Jäger | Axisymmetric bodies of equal material in contact under torsion or shift[END_REF], and applicability to arbitrary loading histories/protocols [START_REF] V. Aleshin | Method of memory diagrams for mechanical frictional contacts subject to arbitrary 2D loading[END_REF][START_REF] Popov | Method of Dimensionality Reduction in Contact Mechanics and Friction[END_REF][START_REF] Aleshin | Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction -Part I: Theoretical background[END_REF]. These methods can be called semi-analytical since the solution remains cumbersome but analytical, and has coefficients determined by an algorithm. In addition, there exist arguments proving that fragments of isotropically rough surfaces in contact interact in the sense of load-displacement relationships in the same way as axisymmetric bodies of appropriate shapes. The semianalytical methods of this kind provide the sought-for loaddisplacement relationship in an explicit form that dramatically reduces the calculation effort in comparison with implicit schemes.

Our original method (Method of Memory Diagram, MMD [START_REF] V. Aleshin | Method of memory diagrams for mechanical frictional contacts subject to arbitrary 2D loading[END_REF][START_REF] Aleshin | Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction -Part I: Theoretical background[END_REF]) belonging to this group uses internal memory functions or diagrams in which the coefficients of the analytical solutions are encoded. Further, based on those principles, we have developed a so called MMD-FEM numerical tool [START_REF] Delrue | Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction -Part II: Numerical implementation[END_REF] that describes wave propagation in a material with frictional cracks including nonlinear effects coming from contact interactions. In this paper, we apply the corresponding code to solving a wave problem that underlies an existing nonlinear acoustic imaging technique that uses harmonic content in standing waves generated in a sample by a continuous sinewave excitation. Purely harmonic, the standing wave becomes slightly nonlinear if defects are present. Spatial distribution of the nonlinear wave component can approximately indicate positions and extents of hidden cracks. This technique, the nonlinear resonant laser vibrometry [START_REF] Jhang | Measurement of Nonlinear Ultrasonic Characteristics[END_REF], is relatively easy to imitate numerically since it does not require sharp focusing of acoustic waves as other methods do.

In this paper, we briefly explain how the MMD-FEM functions in the considered case and present a number of synthetic nonlinear acoustic images of hidden plane cracks of certain size. These images are shown in a variety of cases determined by geometric and physical parameters of the sample and of the excitation. In addition, the MMD algorithm has an important parameter related to roughness of crack faces. The influence of these factors is illustrated by numerous examples.

Method of memory diagrams

As mentioned above, the MMD is analytical with coefficients "ciphered" in internal memory functions (or diagrams) that are continuously being updated following changes in the loading protocol. The details can be found in [START_REF] V. Aleshin | Method of memory diagrams for mechanical frictional contacts subject to arbitrary 2D loading[END_REF][START_REF] Aleshin | Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction -Part I: Theoretical background[END_REF] and include the formulation of the method for partial slip of axisymmetric bodies in contact, and then the generalization of the method for the case of total sliding. As a result, for any sequence of contact states (partial slip, full sliding and contact loss), the solution is expressed in a closed form denoted here as

( ) ( ) ( ) ( ) , , MMD T T b t a t N N a = = . ( 1
)
Here a and b are normal and tangential contact displacements that depend on time, while N and T are normal and tangential forces. In the considerer 2D case the force and displacement vectors always stay in one plane.

Here the normal dependence N=N(a) is considered as known. Paper [START_REF] Aleshin | Nonlinear and hysteretic constitutive models for wave propagation in solid media with cracks and contacts[END_REF] explains why a model form

( ) 2 2 N a C a = (2)
is appropriate as an approximation arising from several assumption related to rough surface contact mechanics. The authors of [START_REF] Biwa | On the acoustic nonlinearity of solid-solid contact with pressure-dependent interface stiffness[END_REF] have arrived to the same conclusion on the basis of acoustic measurements of the load-force displacement. Their estimate for constant C obtained for two aluminum blocks is used here as an approximate value.

Inner cracks can have roughness largely different from the available case, therefore in this work we consider other possible values as well.

Figure 1 illustrates the MMD load-displacement response to an exemplar loading history in terms of the normal and tangential displacement. The automated calculations for hysteretic shapes of the similar kind provide boundary conditions for the wave propagation problem at each discretization point at crack faces (actually, pairs of points located at the opposite faces).

Geometry of imitated laser vibrometry experiment

It is important to note that numerical acoustics remains time consuming as far as the wave equation itself is concerned. In our examples, several milliseconds of modeled acoustic experiment correspond to hours of calculations. To keep this ratio reasonable, we had to accept two principal simplifications: restrained geometry consisting in a 2D rectangle containing crack, and high values of viscous losses in the material, realistic for composite materials but highly exaggerated for metals or ceramics. The increased viscosity considerably shortens time needed for formation of a stationary solution in the form of a standing wave those amplitude and shape do not depend on time anymore. In addition, the bulk and shear viscosities are considered equal, since their values have an approximate character anyway.

Geometry of the problem and boundary conditions are shown in Figure 2 together with information about excitation. Low reflecting boundaries at the left and right correspond to the case of a fragment of a realistic plate with high geometric aspect ratio. In addition, acoustic energy can effectively go away from the edges acting as additional damping that also shortens the standing wave formation time. Further, to comply with vibrometric experiments, the top surface is considered free; the corresponding displacements are measured by laser beam in reality and are used for the harmonic analysis. In contrast, the bottom surface is fixed. This is important since some boundary conditions should be of the first type (known displacement), otherwise an acoustic action will generate an unlimited motion of the entire plate in free space. Besides, fixing the bottom instead of the left and/or right edges helps prevent bending or cantilever motions that can be excited by low frequencies but are out of interest here. In real laboratory experiments, a plate can be suspended by elastic strings which are difficult to model here since we consider a small fragment only.

The sample is excited by a sinusoidal vertical displacement as shown in the figure. The excitation zone corresponds to an attached transducer and is limited by a supergaussian window.

The crack is located in top left corner. Since it represents a critical element of geometry, mesh is refined around it.

Results and discussion

The MMD-FEM code allows one to compute acoustic fields in the sample depicted in Figure 2. The following characteristics are of special interest: wave displacements u x and u y at the top boundary that are measured by scanning laser in real experiments, and values of the first strain invariant ε xx +ε yy at points close to the transducer and to the crack. The magnitude close to the transducer characterizes the input level signal; generally, values of about 10 -7 are considered as weak, 10 -6 typical, and 10 -5 very strong. Here it is appropriate to cite a known result [START_REF] Sneddon | Fourier Series[END_REF] for the critical tensile stress at which crack starts growing:

( ) 2 0 2 2 1 EG R π σ ν = - , ( 3
)
where G is the surface energy, and E andν are respectively Young's modulus and Poisson's ratio of the material. Simple estimations show that for a 3 cm diameter crack in aluminum this critical stress corresponds to strain of about 3.4•10 -5 . To comply with the spirit of nondestructive testing, the strain in material should be less than that value with a margin. However, the strain invariant is weaker near the crack since the transducer is located at some distance. The ratio between strain invariants close to the crack and close to the transducer depends on excitation frequency f and ranges between a few percents to about 40%. Certainly, this penetration ratio should be high enough to generate a nonlinear signal of substantial level.

The nonlinear signal or criterion we are talking about is introduced as 2 , , , 1

x y n x y x y n A I A   =      (4)
where , x y n A denotes the n-th harmonic in the spectrum of u x and u y displacements at the top surface. These nonlinear criterions as functions of x-coordinate at the top boundary represent images of the crack. The definition Eq. ( 4) corresponds to the one introduced in experimental work [START_REF] Jhang | Measurement of Nonlinear Ultrasonic Characteristics[END_REF] in which its typical level is reported to be about 10 -3 . It is important to note that the fundamental harmonics , 1

x y A added in Eq. ( 4) as a denominator to compensate for the limited penetration effect can be close to zero at some points, as it frequently occur with a standing wave in general. Zeros in the first harmonics will produce huge parasite peaks in the images. To get rig of this, we applied running averaging to , 1

x y A in Eq. ( 4).

In Figure 3 we present results for a large crack of 3 cm size excited with the same randomly chosen frequency f=240 kHz and other parameters varied. Generally, even for weak excitation level in the vicinity of the crack its location and size are clearly visible in images. The signal level at the crack and in the excitation zone is characterized by ε cra and ε exc that represent the amplitudes of the first strain invariant ε xx +ε yy . This figure also illustrates the influence of parameter C of the normal force-displacement relationship Eq. ( 2). Its value measured in [START_REF] Biwa | On the acoustic nonlinearity of solid-solid contact with pressure-dependent interface stiffness[END_REF]

is 10 1/ 2 1 6 10 C Pa m - = ⋅
. Higher levels of C correspond to higher levels of nonlinearity as the nonlinear response is directly related to Eq. ( 2) and on the hysteretic tangential force also linked with the normal one [START_REF] V. Aleshin | Method of memory diagrams for mechanical frictional contacts subject to arbitrary 2D loading[END_REF][START_REF] Popov | Method of Dimensionality Reduction in Contact Mechanics and Friction[END_REF]. However, this increase or finally image contrast is not directly proportional to C 2 since apparently for rapidly increasing N(a) the actual displacement a becomes smaller.

The nonlinear response Eq (4) also depends on damping (highly exaggerated here); this dependence has to be clarified additionally. and for a crack size of 2 cm. In all these cases, the excitation displacement amplitude A (see Figure 2) was kept the same. First of all, it can be noted that damage is not necessarily seen in all figures. The higher harmonics are not always located near the crack; they have their own standing wave pattern. Certainly, for successful detection it is required to have strong amplitude in the vicinity of crack. It is expected that the idea of local defect resonance [START_REF] Jhang | Measurement of Nonlinear Ultrasonic Characteristics[END_REF] can help which consists in finding a specific frequency in the amplitude-frequency response of the sample which exists then damage is present and does not exist otherwise. But even without use of this concept, the choice of frequency is very important since the "penetration ratio" / cra exc ε ε greatly depends on frequency.

In fact, tendencies that show up in these assorted figures (and in those that remain unpublished) suggest the following strategy of obtaining quality images in which damage is clearly detected.

1. Consider intact sample and examine its amplitudefrequency response by making a series of timedomain calculations or (better) by reconsidering the problem in frequency domain.

2. Select frequencies maximizing the penetration ratio. Unfortunately, the problem of damping remains unsolved. Its highly exaggerated value essentially contributes in the nonlinear response. Decreasing damping to smaller and realistic for metals values results in higher times of standing wave formation, whereas the code is now exploited at maximum performance of modern PC.

Certainly, some factors are still not taken into account. For instance, faces of real cracks are typically prestressed. Indeed, once separated during cracking, the faces will not match each other anymore with the atomic precision.

At the same time, we hope that the implementation of the above strategy can help approach real experiments and then validate the entire nonlinear model experimentally. The validated MMD-FEM numerical tool can be an important advance towards modeling-based reconstruction of damage parameters in a better way than pure experimentation can do.

Conclusions

In this paper we present the results of numerical modeling for nonlinear laser vibrometry experiments. Using the previously developed MMD-FEM method, we have computed the nonlinear response (sum of higher harmonics amplitudes normalized by the first one) of a sample containing crack with friction. In some cases, damage is clearly seen in the obtained images and in the others it is not. Our preliminary results allowed one to formulate a possibly strategy for achieving robust damage detection. At the nest stage, comparison with experiments is envisaged.

Figure 1 -Figure 2 -

 12 Figure 1 -Hysteretic tangential load-displacement response (a) plotted for exemplar loading history in terms of displacements (b).

Figure 3 -

 3 Figure 3 -Nonlinear indicators I x and I y calculated for a sample with a crack of 3 cm width excited by a continuous wave of 240 kHz frequency. The next series of figures (Figure 4) illustrates the effect of frequency on image quality. The nonlinear responses I x,y are calculated for 10 1/ 2 1 100 , 10 Pa s C Pa m η

Figure 4 -

 4 Figure 4 -Nonlinear indicators I x and I y calculated for a sample with a crack of 2 cm width excited by a continuous waves of frequencies ranging from 120 to 330 kHz.

  ] since those measurements are done for globally plane aluminum blocks, and real crack faces are rougher. Rough surfaces correspond to smaller C.4. Make MMD-FEM computations for a crackedsample with selected parameters.

Acknowledgments

R. Verma acknowledges his PhD thesis support by the I-SITE foundation of the University of Lille.


Julien Fatome
email: julien.fatome@u-bourgogne.fr

Josselin Garnier

S Pitois

M Petit

Guy Millot

Mathilde Gay

Benoît Clouet

Laurent Bramerie

Jean-Claude Simon

J.-C Simon

All-optical measurements of background, amplitude, and timing jitters for high speed pulse trains or PRBS sequences using autocorrelation function

Keywords: High bit rate, Telecommunication system, Optical fiber, Timing and amplitude jitters, Autocorrelation function, 40 Gbit/s, 160 Gbit/s

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In future ultrafast telecommunication systems using optical time division multiplexing and dispersion management techniques, picosecond and even subpicosecond pulses will be transmitted through optical fibers at very high bit rates >80 Gbit/s [START_REF] Weber | Single channel 1.28 and 2.56 Tbit/s DQPSK transmission[END_REF][START_REF] Gnauck | 1-Tb/s(6× 170.6 Gb/s) transmission over 2000-km NZDF using OTDM and RZ-DPSK format[END_REF][START_REF] Weisser | Single-and alternating polarization 170-Gbit/s transmission up to 4000 km using dispersion-managed fiber and all-Raman amplification[END_REF]. Consequently, the ability to measure the quality and parameters such as extinction ratio, timing, and amplitude jitters of high speed clocks or transmitted pulse trains has become of a great interest for telecommunication engineers and transmission line designers. Unfortunately, for pulse trains beyond the bandwidth limit of electronic devices, no optical sampling oscilloscope are yet commercially available and we can assume that future devices will evolve in a very expensive trend. Consequently, checking the quality of ultrahigh speed pulse trains is still a recurring question. Frequency resolved optical gating (FROG) has been proposed a few years ago and was proved to be a powerful method to characterize such high speed pulse trains in intensity and phase even for pseudo ran-dom bit sequences (PRBS) [START_REF] Fatome | Sensitivity of SHG-FROG for the characterization of ultrahigh-repetition-rate telecommunication laser sources[END_REF][START_REF] Grelu | Pseudo-random pulse sequence characterization with frequency-resolved optical gating[END_REF][START_REF] Trebino | Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses[END_REF]. But, even though the FROG technique gives a large number of parameters (intensity and phase profiles, extinction ratio), it is generally a time consuming method (not suitable for real time monitoring) and does not permit timing and amplitude jitter measurements [START_REF] Trebino | Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses[END_REF]. The optical autocorrelator is often associated to the FROG technique [START_REF] Trebino | Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses[END_REF]. In contrast to the sampling oscilloscope, the time-averaging of the autocorrelation function allows the use of low-speed, high sensitivity detectors, does not require a local pulse source and can be used even on signals with closed eye diagrams [START_REF] Trebino | Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses[END_REF]. It is the reason why many people, even in the telecommunication field, use the autocorrelation function to control the pulse shape [START_REF] Weisser | Single-and alternating polarization 170-Gbit/s transmission up to 4000 km using dispersion-managed fiber and all-Raman amplification[END_REF]. But, even if the autocorrelation function has already been proved to be an efficient way to measure the timing jitter of 40-Gbit/s PRBS sequences [START_REF] Dörring | All-optical timing jitter measurements on 40 Gbit/s pseudorandom RZ data after long-haul transmission in dispersion-managed soliton system[END_REF] or the optical to noise ratio (OSNR) impairments at 10 Gbit/s [START_REF] Dinu | Characterization of signal quality using data stream optical autocorrelation[END_REF], generally speaking, they do not pull all the benefits out of their measurements [START_REF] Weisser | Single-and alternating polarization 170-Gbit/s transmission up to 4000 km using dispersion-managed fiber and all-Raman amplification[END_REF]. In this paper, we propose an extension of these methods and report an all optical technique for the simultaneous measurements of the intensity background, timing, and amplitude jitters of high speed repetition rate pulse trains or PRBS sequences. This technique is based on the distinct behavior of the auto-correlation and intercorrelation peaks of the signal under test and more specifically on the timing and amplitude jitter dependences of the intercorrelation peak amplitude and width. By means of numerical simulations followed by 42.66-Gbit/s PRBS and 160-GHz experiments, we successfully demonstrate that this method can be employed for the jitter characterization of periodic or PRBS pulse trains. We believe that such a simple, all optical and relatively cheap method could find a large area of applications in the field of ultrahigh repetition rate optical signals such as high speed mode lock lasers, optical clock or differential phase shift keying (DPSK) signal characterizations.

Theory and simulations

The first piece of work we carried out leads to the determination of the amplitude and timing jitters of a simulated 160-GHz pulse train. First of all, the intensity autocorrelation function A(τ ) of an optical signal I(t) is obtained from the interaction of two time-delayed (τ ) replicas of the signal under test and is defined as [START_REF] Trebino | Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses[END_REF] A(τ ) = I(t)I(tτ)dt.

(1)

Practically, Fig. 1a shows how this optical autocorrelation function can be experimentally obtained and corresponds to the setup of our second-harmonic generation (SHG) backgroundfree autocorrelator device used in the following experiments [START_REF] Fatome | Sensitivity of SHG-FROG for the characterization of ultrahigh-repetition-rate telecommunication laser sources[END_REF].

The two time-delayed replicas of the input signal are obtained through a 50:50 beam splitter while the variable time delay is tuned by means of a step-motor controlled translation stage. Then, the two replicas are focused in a 2-mm-thick BBO crystal by a 5 cm focusing lens. Note that a polarization controller (PC) is also used to minimize the insertion loss of the polarizer placed at the autocorrelator input and to optimize the SHG signal. Finally, after injection in an optical fiber thanks to a micro-lens, the second harmonic autocorrelation signal is detected by means of a photo-multiplier and an oscilloscope. Figure 1b illustrates an example of an experimental recorded 160-GHz pulse train autocorrelation function. The autocorrelation peak A AC (τ ) is obtained from the correlation of the signal by its nondelayed replica (τ |T/2|, T is the signal period) and this is the reason why it represents the autocorrelation of an individual pulse and that its maximum is always equal to 1 even if neighboring pulses are quite different. At the opposite, the intercorrelation peak A XC (τ ) (τ |T/2|) results from the correlation between neighboring pulses and is consequently dramatically sensitive to neighboring pulse differences. The method we proposed in this paper is to characterize high speed pulse trains or PRBS sequences by comparing the autocorrelation and intercorrelation peaks of the signal under test.

During its generation or propagation, essentially because of nonlinear interactions between neighboring pulses or noise, the pulse train may get amplitude and timing jitters and can be expressed assuming Gaussian pulse shape as

I(t)= n I n (t -nT ), I n (t) = a n exp - (t + τ n ) 2 2t 2 0 , ( 2
)
where T is the pulse train period, n is the pulse index, a n is the amplitude jitter which is assumed to have Gaussian statistics with an average value equal to 1 and a standard deviation σ a . τ n is the timing jitter which is assumed to have Gaussian statistics with an average value of 0 and a standard deviation σ t .Note that this method can be applied only if jitters are sufficiently fast to consider that pulses act individually, i.e., no difference could be observed between auto-and inter-correlation peaks if a slow jitter is applied to the pulse train. Finally, t 0 corresponds to the RMS width and related to the full-width at half maximum (FWHM) by [START_REF]Nonlinear Fiber Optics[END_REF] FWHM = 2t 0 2ln(2).

(

) 3
In absence of jitters, the autocorrelation and intercorrelation peaks of I(t) are equal and are given by By comparing the initial (dashed line) and degraded (solid line) autocorrelation functions of the 160-GHz pulse train in Fig. 2b, we can clearly observe that the presence of amplitude and timing jitters is characterized by a dramatic attenuation and broadening of the intercorrelation peaks, translating the differences between neighboring pulses. This behavior could be greatly observed in Ref. [START_REF] Weisser | Single-and alternating polarization 170-Gbit/s transmission up to 4000 km using dispersion-managed fiber and all-Raman amplification[END_REF] where the autocorrelation function of an 85-Gbit/s DPSK signal has been recorded until 4320 km.

A AC (τ ) = A XC (τ ) = t 0 √ π exp - τ 2 4t 2 0 . (4)
In presence of timing and amplitude jitters, the autocorrelation and intercorrelation peaks (respectively A AC and A XC ) can be obtained by injecting expression (2) in Eq. ( 1) and are then given by

A AC (τ ) = a 2 0 exp - t 2 2t 2 0 - (τ -t) 2 2t 2 0 dt , A AC (τ ) = a 2 0 √ πt 0 exp - τ 2 4t 2 0 , A XC (τ ) = a 0 a 1 exp - t 2 2t 2 0 - (τ -t -Γ ) 2 2t 2 0 dt , A XC (τ ) = a 0 a 1 √ πt 0 exp - (τ -Γ ) 2 2t 2 0 , (5)
where Γ = τ 0 -τ 1 , τ 0 and τ 1 (respectively a 0 and a 1 )arethe timing jitters (respectively amplitude jitters) for two neighbor-ing pulses, andstands for statistical averaging. Assuming independent Gaussian statistics for the timing and amplitude jitters, the autocorrelation and intercorrelation peaks have now the following form:

A AC (τ ) = 1 + σ 2 a √ πt 0 exp - τ 2 4t 2 0 , A XC (τ ) = √ πt 0 1 + σ 2 t /t 2 0 exp - τ 2 4t 2 0 1 1 + σ 2 t /t 2 0 . (6)
From this analysis, we can express the timing and amplitude jitters in terms of the full-width at half maximum FWHM AC and FWHM XC of the autocorrelation and intercorrelation peaks, and in terms of their amplitudes P AC and P XC :

σ t = 1 4 √ ln 2 FWHM 2 XC -FWHM 2 AC 1/2 , σ a = P AC P XC 1 + σ 2 t t 2 0mes -1/2 -1 1/2 , ( 7
)
where t 0mes is given by t 0mes = FWHM AC /(4 √ ln 2). The expression for the timing jitter is consistent with the one derived in Ref. [START_REF] Dörring | All-optical timing jitter measurements on 40 Gbit/s pseudorandom RZ data after long-haul transmission in dispersion-managed soliton system[END_REF]. Here we show that it holds true even in the presence of amplitude jitter. In other words, because broadening of the intercorrelation peak was only due to the timing jitter, its measurement with the correlation setup is not sensitive to the amplitude jitter. Consequently, the amplitude and timing jitters can now be extracted simply from the experimental measurements of the autocorrelation and intercorrelation peak FWHMs and amplitudes.

Another interesting result is that we can compute the errors on the measured values of σ t and σ a by performing a linear stability analysis of expressions [START_REF] Dörring | All-optical timing jitter measurements on 40 Gbit/s pseudorandom RZ data after long-haul transmission in dispersion-managed soliton system[END_REF]. If t , respectively P , represents the temporal resolution, respectively the amplitude resolution, of the correlation setup, we obtain

σ t = 1 4 √ ln 2 FWHM 2 XC + FWHM 2 AC FWHM 2 XC -FWHM 2 AC 1/2 t, σ a = σ a P 2 P 2 AC + P 2 P 2 XC + t 2 FWHM 2 AC + t 2 FWHM 2 XC 1/2 . (8)
Note that the timing jitter measurement accuracy is inversely proportional to the square root of the timing jitter, which shows that this method has rather poor resolution when the timing jitter is small. At the opposite, the amplitude jitter measurement accuracy is, roughly speaking, proportional to the jitter itself, which shows that it is efficient even when the jitters are small.

For example, Table 1 gives a comparison between the simulated jitters defined as the standard deviation of the density functions plotted in Figs. 2c and2d and those retrieved from Eq. ( 7) by means of FWHMs and amplitudes of autocorrelation and intercorrelation peaks. We can clearly see an excellent agreement between the simulated and calculated results, proving the reliability of the method.

We would like to stress that this method is unlimited in frequency and thus could be used to characterize pulse sources with high repetition rates beyond the electronic sampling oscilloscope limitation. Recording the autocorrelation function of DPSK format during the propagation, as in Ref. [START_REF] Weisser | Single-and alternating polarization 170-Gbit/s transmission up to 4000 km using dispersion-managed fiber and all-Raman amplification[END_REF], could now become a powerful measurement which is presently underestimated in the telecommunication field.

PRBS pulse train

This method can be applied to a pseudo-random bit sequence (PRBS) as well, providing that the length of the sequence is sufficiently large to ensure a fair statistical distribution between "0" and "1." In this case, the maximum of the initial intercorrelation peak becomes R (ratio of "1" in the sequence) in contrast to 1 in the periodic case. Consequently, in the PRBS case, the expression of the timing jitter in Eq. ( 7) is still valid while that of the amplitude jitter becomes Note that in a true PRBS sequence, the number of "0" is equal to the number of "1" and R = 0.5. Figure 3a shows the eye diagram of a 160-Gbit/s PRBS sequence damaged by timing (0.2 ps) and amplitude (20%) jitters. Figure 3b represents the autocorrelation function of this initial (dashed line) and degraded (solid line) 160-Gbit/s sequence.

σ a = R P XC 1 + σ 2 t t 2 0mes -1/2 -1 1/2 . ( 9
) (a) (b)
By means of Eqs. ( 7) and ( 9), one can calculate the timing and amplitude jitters of the sequence. Table 2 sums up the results. As in the periodic case, we can see an excellent agreement between simulated and retrieved jitters.

Pulse train with background

Because of the intrinsic features of pulse sources (fiber ring lasers, nonlinear optical loop mirrors, optical modulators), or simply because of nonlinear interactions with noise or neighboring pulses in optical fiber, we have to consider the case where pulses are degraded or raised on an intensity background. As a result, the model for the nth pulse intensity becomes

I n (t) = f d + (1 -f d )a n exp - (t + τ n ) 2 2t 2 0 , (10)
where f d is the intensity background.

In the general case (R = 1), the autocorrelation peak of I(t) is now given by

A AC (τ ) = Tf 2 d + 2f d (1 -f d ) √ 2πt 0 + (1 -f d ) 2 √ πt 0 × exp - τ 2 4t 2 0 + R exp - (T -τ) 2 4t 2 0 × Tf 2 d + 2f d (1 -f d ) √ 2πt 0 + (1 -f d ) 2 √ πt 0 × 1 + R exp - T 2 4t 2 0 -1 , ( 11
)
where T is the pulse train period. The minimum of the autocorrelation function F d is approximately reached at

τ min =± T 2 1 - 4t 2 0 T 2 ln R . (12)
Note that τ min =±T/2 in the periodic case (R = 1). The value of F d is then given by

F d = Tf 2 d + 2f d (1 -f d ) √ 2πt 0 + 2 √ R(1 -f d ) 2 √ πt 0 exp - T 2 16t 2 0 × Tf 2 d + 2f d (1 -f d ) √ 2πt 0 + (1 -f d ) 2 √ πt 0 1 + R exp - T 2 4t 2 0 -1 . (13)
Measuring F d on the experimental autocorrelation function, we can get f d by solving a quadratic equation. If F d is small, then the solution can be expanded as

f d = F d -2 √ R exp(-T 2 /16t 2 0mes ) 2 √ 2 -4 √ 2R exp(-T 2 /16t 2 0mes ) ≈ F d 2 √ 2 , ( 14
)
where the last expression is obtained assuming T larger than 4t 0 (well separated pulses).

Figure 4a shows the eye diagram of a 160-GHz pulse train with an intensity background of 4% and damaged by a timing (0.2 ps) and amplitude (20%) jitters. Figure 4b represents the autocorrelation function of the initial (dashed line) and degraded (solid line) pulse train.

As in Fig. 2b, we can see that the intercorrelation peak is dramatically reduced and broadened due to the presence of amplitude and timing jitters. By removing the contribution of neighboring pulse overlapping and by supposing that timing jitter does not participate to the autocorrelation background, that means that timing jitter does not exceed 10% of the bit slot, the timing jitter could be also evaluated thanks to Eq. ( 7) by calculating the FWHMs at 1 2

[P k + F d ], (15)
where P k is the inter-or autocorrelation peak and F d the measured autocorrelation background. Under the same assumptions, the decrease of the intercorrelation peak due to the jitters is now obtained by the following expression:

P AC -F d P XC -F d = 1 + σ 2 a 1 + σ 2 t t 2 0mes 1/2 . ( 16
)
For example, Table 3 gives the simulated and calculated jitters and background of the pulse train plotted in Fig. 4a. Thanks to Eqs. ( 7) and ( 14)-( 16), we have successfully derived all the desired features of the pulse train from its autocorrelation function.

Experimental results

42.66-Gbit/s results

The first series of experiments were carried out on the technological platform PERSYST (platform for test and research on optical telecommunication system) in Lannion, France, (persyst@persyst.fr). The purpose is to characterize in the time domain a RZ 42.66-Gbit/s PRBS sequence by means of a 50-GHz optical sampling oscilloscope and thanks to the second-harmonic generation background-free optical autocorrelator described in Fig. 1a as well. Figure 5 shows the experimental setup. The signal is coded thanks to a LiNbO 3 modulator driven by a 2 15 -1 PRBS sequence in NRZ format at 42.64 Gbit/s, generated by electrical multiplexing of 4 trains at 10.66 Gbit/s. The RZ 33% optical signal is obtained thanks toaLiNbO 3 modulator driven by a 21.3 GHz sinusoidal clock. Then, the sequence is injected into 100 km of nonzero dispersion shifted fiber (NZDSF, D = 4ps/nm/km). Chromatic dispersion was compensated with dispersion compensating fiber (DCF) and amplification is ensured by erbium-doped fiber amplifiers (EDFA). The 42.66-Gbit/s PRBS sequence was finally characterized at the receiver by a 50-GHz optical sampling oscilloscope and by the optical autocorrelator.

We first characterized the PRBS sequence in a back-to-back configuration. Figure 6a represents the corresponding eye diagram. The timing jitter was evaluated at 1 ps by the sampling oscilloscope (corresponding to the resolution limit of the equipment). Thanks to the experimental autocorrelation function, plotted in Fig. 6b (solid line), and to Eq. ( 7), we have calculated a timing jitter of 0.96 ps (±2%, t = 10 -2 ps) which is in good agreement with the first oscilloscope measurement. In both cases, the amplitude jitters were found negligible. Moreover, we have also plotted in Fig. 6b (circles), a simulated autocorrelation function of the 42.66-Gbit/s PRBS sequence calculated from the experimental features (FWHM, jitters, background). The excellent agreement between the two different curves stresses the accuracy of the method.

Figures 7a and7b show the experimental results obtained after 100 km of propagation. The NZDSF input power was fixed to 10 dB m in order to exacerbate the nonlinear effects so as to speed up the timing jitter creation. The timing jitter was measured to 2.5 ps by means of the sampling oscilloscope (Fig. 7a) and to 2.22 ps (±1%, t = 10 -2 ps) through the autocorrelation method (Fig. 7b, solid line). No significant decrease of the intercorrelation peak compared to the back-to-back mea- surement was observed after 100 km, indicating that no amplitude jitter was generated during the propagation, which is in good agreement with the oscilloscope conclusions. Finally, on Fig. 7b, we observe a good agreement between the experimental autocorrelation function (solid line) and that simulated from the calculated features (circles), proving the reliability of the measurement.

160-GHz results

The second series of measurements leads to the determination of the intensity background, timing and amplitude jitters of a 160-GHz pulse train during its propagation in a classical 100-km SMF/DCF recirculating loop setup. The experimental setup is shown in Fig. 8a. A high quality 160-GHz Gaussian pulse train, generated by multiple four wave mixing in a 1 kmlong NZDSF [START_REF] Pitois | Generation of a 160-GHz transform-limited pedestal-free pulse train through multiwave mixing compression of a dualfrequency beat signal[END_REF] is injected into a 100-km-long recirculating loop setup. The dispersion map is a symmetric map [START_REF] Fatome | Effectiveness of fiber lines with symmetric dispersion swing for 160-Gb/s terrestrial transmission systems[END_REF] made of 2 spans of 50 km of SMF fiber (D = 17 ps/km nm, S = 0.056 ps/km nm 2 ) separated by a dispersion compensating fiber (DCF) module which compensates for 99% of the total second-and third-order chromatic dispersion of the line. Two erbium-doped fiber amplifiers (EDFA) are used to compensate for the total losses of the line while a 1.8-THz optical band pass filter (OBPF) is inserted to avoid the accumulation of noise into the loop. The output signal is finally characterized by our background-free second-harmonic generation autocorrelator described in Fig 1a.

By supposing that pulses keep a Gaussian shape during their propagation but develop a growing intensity background and by using expressions ( 7) and ( 14)-( 16), we derived the FWHMs, the intensity backgrounds, the timing and amplitude jitters of the 160-GHz pulse train at 100, 400, 800, 1100, 1500, and 2000 km which experimental autocorrelation functions are plotted in Fig. 8b. Figure 9a shows the measured FWHMs and the calculated intensity backgrounds. We can observe that after an initial broadening, the pulse width stabilizes and remains constant during the propagation, indicating that the dispersion management is finely tuned. The intensity background increases during the whole propagation, indicating that strong nonlinear interactions occurred between neighboring pulses and noise and thus limited the propagation distance. Figure 9b shows the calculated timing σ t and amplitude σ a jitters as a function of propagation distance. We can notice that the 2 jitters dramatically increase to reach at 2000 km nearly the third of the pulse width for the timing jitter and the entire pulse peak for the amplitude jitter. Note that the error bars on Fig. 9b are calculated by means of expressions [START_REF] Dinu | Characterization of signal quality using data stream optical autocorrelation[END_REF]. Figure 9c shows the measured autocorrelation function at 800 km (solid line) and that simulated (circles) from the measured features (FWHM=1.44 ps, intensity background of 6%, σ t = 0.17 ps and σ a = 27%). We can observe that we found an excellent agreement between the two different curves indicating the reliability of our technique. Finally, for illustration, Fig. 9d shows the intensity profile of the 160-GHz pulse train corresponding to the simulated 800 km autocorrelation function plotted in circles in Fig. 9c. We can notice that the level of the amplitude jitter evolves in a dramatic behavior for a telecommunication system.

Conclusion

In this paper, we have reported a simple and all-optical method for the simultaneous measurements of the intensity background, timing, and amplitude jitters of high repetition rate pulse trains and PRBS sequences. This technique is based on the recording of the experimental optical autocorrelation function of the pulse train under test and on the timing and amplitude jitter dependences of the intercorrelation peak amplitude and width. We have derived simple expressions of the jitters as a function of the FWHMs and amplitudes of the autocorrelation and intercorrelation peaks both in the periodic and PRBS case. We have also completed the analysis by computing the errors on the measurements and by deriving an expression for the intensity background. Through numerical simulations, we have successfully demonstrated that this method could be employed for the characterization of PRBS or periodic high speed pulse trains. Then, we have experimentally demonstrated the reliability of our technique by successfully computing the timing jitter of a 42.66-Gbit/s PRBS RZ sequence. Finally, we applied our method to the characterization of a 160-GHz pulse train during its propagation in a classical 100-km SMF/DCF system and successfully derived the different features of the pulse train as a function of propagation distance (FWHM, intensity background, timing, and amplitude jitters). The main limitations of our method is that the measurement is under the assumption that pulses have a Gaussian shape, that jitters are sufficiently fast to consider that any bit acts individually, that the bandwidth of the autocorrelator (typically >THz) is sufficiently large toward the pulse train repetition rate and that in the case of an intensity background, timing jitter does not participate to the autocorrelation background. In conclusion, believe that such a simple, relatively cheap and all-optical method could find a large area of applications in the ultrahigh speed optical field; for example: characterizing fiber ring lasers, high speed DPSK signals or high speed optical sources and clocks.

Fig. 1 .

 1 Fig. 1. (a) Autocorrelation experimental setup. (b) Example of a 160-GHz pulse train experimental autocorrelation function.

Figure 2

 2 Figure2shows an illustration of the intensity profile of an ideal 160-GHz pulse train as well as its autocorrelation function, respectively Figs.2a and 2b, dashed line. At the same time, solid lines represent this pulse train numerically damaged by adding timing (0.4 ps) and amplitude (20%) jitters. The number of periods is chosen to be sufficiently large(2 12 ) in order to have an efficient statistical distribution. Density functions of timing and amplitude jitters are plotted in Figs.2c and 2dand show pretty well Gaussian shapes with an average of 0 ps for the arrival time and an average of 1 for the peak intensity.

Fig. 2 .

 2 Fig. 2. (a) Initial 160-GHz pulse train (dashed line) and degraded with timing (0.4 ps) and amplitude (20%) jitters (solid line). (b) Autocorrelation function of initial (dashed line) and degraded pulse train. (c) Probability density of the amplitude jitter. (d) Probability density of the timing jitter.

Fig. 3 .

 3 Fig. 3. (a) Eye diagram of the 160-Gbit/s PRBS sequence degraded with timing (0.2 ps) and amplitude (20%) jitters. (b) Autocorrelation function of the initial (dashed line) and degraded 160-Gbit/s PRBS sequence.

Fig. 4 .

 4 Fig. 4. (a) Eye diagram of the 160-GHz pulse train with a 4% intensity background and degraded with timing (0.2 ps) and amplitude (20%) jitters. (b) Autocorrelation function of the initial (dashed line) and degraded pulse train.

Fig. 5 .

 5 Fig. 5. 42.66-Gbit/s transmission experiment, optical band pass filter (OBPF).

Fig. 6 .

 6 Fig. 6. (a) Eye diagram of the initial 42.66-Gbit/s PRBS sequence. (b) Corresponding autocorrelation function: experimental (solid line) and simulated (circles).

Fig. 7 .

 7 Fig. 7. (a) Eye diagram of the 42.66-Gbit/s PRBS sequence after 100 km of propagation. (b) Corresponding autocorrelation function: experimental (solid line) and simulated (circles).

Fig. 8 .

 8 Fig. 8. (a) Experimental setup, optical switch (OS), polarization controller (PC). (b) Autocorrelation functions of the 160-GHz pulse train at (bottom to top) 100, 400, 800, 1100, 1500 and 2000 km.

Fig. 9 .

 9 Fig. 9. (a) Measurements from experimental autocorrelation functions of FWHM (circles) and intensity background (triangle) of the 160-GHz pulse train as a function of propagation distance. (b) Measurements from experimental autocorrelation functions of amplitude σ a (bottom) and timing jitters σ t (top) as a function of propagation distance. (c) Autocorrelation function of the 160-GHz pulse train: recorded at 800 km (solid line) and simulated from calculated features (circles). (d) Corresponding intensity profile of the autocorrelation function simulated in (c).

Table 1

 1 Comparison between the simulated and retrieved jitters of a 160-GHz pulse train

		FWHM (ps)	Peak		σ t (ps)	σ a
	A AC	1.81	1	Simulation	0.40	0.20
	A XC	2.25	0.78	Retrieved	0.40	0.19

Table 2 Comparison

 2

	σ a

between the simulated and retrieved jitters of a 160-Gbit/s PRBS sequence σ t (ps)

Table 3

 3 Comparison between the simulated and retrieved jitters of a 160-GHz pulse train in presence of an intensity background

		Background	σ t (ps)	σ a
	Simulation	0.04	0.40	0.20
	Retrieved	0.04	0.40	0.19


Guillaume Alombert Goget

Christina Armellini

Simone Berneschi

Brigitte Boulard

Andrea Chiappini

Alessandro Chiasera

Claire Duverger-Arfuso

Maurizio Patrice Féron

G Alombert-Goget

M Ferrari
email: mferrari@science.unitn.it

S Guddala

E Moser

S Pelli

D N Rao

G C Righini

G Speranza

Fabrication, assessment, and application of confined structures in photonic glasses

Keywords: silica micro-nano spheres, microresonators, inverse opals, sol-gel, erbium, quantum efficiency

INTRODUCTION

Silica-based glasses take up a consolidated place as photonic materials [START_REF] Ferrari | Rare-earth-doped glasses for integrated optical amplifiers[END_REF]. The fruitful exploitation of silicate glasses is not restricted only to the ICT area. Many other photonic devices, with a large spectrum of applications covering Health and Biology, Structural Engineering, and Environment Monitoring Systems, have been developed during the last years. Even if several of these devices are actually available on the market, at the state of the art the research is focused on optimising chemical composition and developing innovative fabrication processes and structures, in order to reduce the costs, increase the performances, and discover new functionalities. In this context, a significant role is played by confined structures such as amorphous and transparent glass-ceramic waveguides [START_REF] Alombert-Goget | Glass-based erbium activated micro-nano photonic structures[END_REF], spherical microresonators [START_REF] Chiasera | Spherical whispering-gallery-mode microresonators[END_REF], colloidal-based system [START_REF] Gonçalves | 3-D rare earth-doped colloidal photonic crystals[END_REF], and photonic crystals [START_REF] Jasieniak | Sol-gel based vertical optical microcavities with quantum dot defect layers[END_REF]. Aim of this lecture is to highlights recent results obtained in some of the above-mentioned systems.

NANO SCALE PHOTONIC STRUCTURES

The research on nano-scale structures can find application in different fields such as bioimaging, biosensing and photonic. One of the promising system for application in the above mentioned areas are the Er 3+ -activated coreshell like structures. Typically, monosize silica particles are synthesized via base-catalyzed hydrolysis of TEOS. However, the incorporation of rare earth ions, into the silica spheres, by dissolving a rare earth salt in ethanol fails for the base-catalyzed reaction because the rare earth ion immediately forms an insoluble rare earth hydroxide. Incorporation of lanthanide ions during the growth of silica particles is possible using an acid catalysis. On the other hand acid-catalyzed hydrolysis of TEOS results in particles with broad size distribution. Using a seeded growth method on the previously fabricated monosize silica spheres it is possible to grow monosize larger particles and incorporate Er 3+ ions into the shell, obtaining a core-shell-like Er 3+ -activated structure [START_REF] Chiappini | Fabrication and optical assessment of sol-gel-derived photonic bandgap dielectric structures[END_REF]. Core-shell-like Er 3+ -activated silica spheres, where the core is the silica sphere and the shell is an Er 2 O 3 -SiO 2 coating, were prepared using the following protocol. 150 mg of silica spheres of 270 nm, obtained by Stober method, were added to a solution with the molar ratio TEOS:CH 3 COOH:H 2 O of 1:8:8, plus 0.2 wt% ErCl 3 . The mixture was agitated for 45 min using a magnetic stirrer. After synthesis, the SiO 2 particles were separated from the solution by centrifuging at 1000 rpm and washed at least twice with pure ethanol. Subsequently core-shell-like Er 3+ -activated silica spheres were annealed at 950 °C for 30 min. Figure 1 shows a SEM image of the core-shell-like Er 3+ -activated silica spheres after annealing at 950 °C for 30 min. From Fig. 1 it is evident that seeded growth in the Erbium solution occurs on individual particles without clustering.

The photoluminescence in the 1.5 µm region of core-shell-like Er 3+ -activated silica spheres, were measured using the 514.5 line of an Ar + laser as excitation source. The shape of the emission band is typical of Er 3+activated silicate glasses and the spectral bandwidth, measured at 3 dB from the maximum intensity, is 27 ± 2 nm [START_REF] Chiappini | Design of photonic structures by sol gel-derived silica nanospheres[END_REF]. The decay curve exhibits a single-exponential behaviour with a measured lifetime (τ meas ) of 12.8 ± 0.1 ms. We can compare this result with the lifetime of 18 ± 3 ms reported by Dood et al. [START_REF] De Dood | Acid-based synthesis of monodisperse rare-earth-doped colloidal SiO 2 spheres[END_REF], which can be considered close to the radiative lifetime (τ rad ) for erbium in silica. From the ratio: η = τ meas / τ rad we estimate a quantum efficiency of 71% for the sol-gel-based core-shell silica spheres. Er 3+ -activated silica inverse opals are another example of nano-scale structures. The opal templates were assembled using a commercially available monodispersed polystyrene (PS) microsphere colloidal suspension (Duke Company -1%wt) with an average diameter of 270 nm. The PS template was prepared using a vertical deposition technique. The silicon substrate was placed in a vial with deionized water containing 0.036 wt % PS in suspension. The water was then slowly evaporated at 50 °C in an oven. The silica sol activated by erbium ions (SsEi) was obtained after stirring a solution of the following precursors for 2 h at room temperature: 0.428cc Tetraethyl Orthosilicate (TEOS), 0.001cc HCl, 3.6cc Ethanol, 0.069cc H 2 O and 1.68mg ErCl3. The SsEi was filtered using a syringe filter with a 0.2 µm pore size. The void spaces of the opal template were then infiltrated by dropping ~10 µl of SeEi over the samples and were left to dry at room temperature. The template was removed by heat-treatment: the samples were slowly heated to 450 °C (at 0.1 °C/min) and then left at this temperature for 30 min. Furthermore a rapid thermal annealing at 900 °C for 10 min was carried out in order to promote a more complete condensation densification of the structure and elimination of adsorbed water. An SEM image of Er 3+ -activated silica (ErSiO 2 ) inverse opals is shown in Figure 2. Figure 2 reveals that the hollow regions of the air spheres are well ordered in a triangular lattice corresponding to the (111) planes of a fcc crystalline structure. The average dimension of the air-hollows is ~210 nm. This value is smaller than the diameter of the latex spheres (270 nm) used to form the template, which demonstrates that shrinkage occurs during the sintering process. This fact is not surprising given that the latex spheres are mesoporous and, in the early stages of calcination, probably decrease in size as water vapour is released from the pores. The ErSiO 2 inverse opal sample exhibited violet regions, and the reflected light was easily observed by the naked eye, because the bright colour resulted from optical Bragg diffraction from the crystal planes. Figure 3 From Figure 5 we can observe that with an increase of the incident angle, the stop band position shifts to shorter wavelengths. This behaviour can be express by the modified form of the Bragg's law:

( ) 2 2 2 0.816 sin eff dn λ θ =⋅ ⋅⋅ - , ( 1
)
where λ is the free-space wavelength of light, d is the dimension size of the hollow spheres, n eff is the effective refractive index and θ is the angle measured from the normal to the planes. Moreover by using equation ( 1) and taking in count the following formula [START_REF] Chiappini | Design of photonic structures by sol gel-derived silica nanospheres[END_REF]:

( ) 22 2 1 eff air silica nn f n f =⋅ + ⋅ -. (2)
It is possible to estimate the volume fraction of ErSiO 2 in the inverse opal sample. In fact, assuming an effective refractive index of n eff = 1.1268 (obtained, by fitting formula (1) with the incidence angles θ versus the stop band positions (free-space wavelength) λ) and considering the following values n silica = 1.4742 and n air = 1, we can calculate a volume fraction of about 23%, that it's very close to a maximal infiltration fraction for a inverse opal structure (26%). These results confirm the very good quality of the inverse structure obtained.

As far as concerns spectroscopic assessment the photoluminescence of ErSiO 2 inverse opal exhibits a main emission peak at 1540 nm. The spectral width of the emission band measured at 3 dB from the maximum of the intensity is 21 ± 1 nm, and the Stark structures at 1490, 1567 and 1617 nm appear well defined. The decay curve exhibit a single exponential profile with a lifetime τ 1/e of 16.8 ± 0.1 ms, leading to a quantum efficiency of about 90%.

SPHERICAL MICRORESONATORS

Dielectric microspheres are an important scientific and technological tool for manipulation of light on micrometric and sub-micrometric scale. The following example focuses onto the spectroscopic properties of Er 3+ -activated silica spheres. Silica microspheres serving as a base resonator structure were made by melting the end of a stripped standard telecommunication fiber [START_REF] Righini | Photonic properties and applications of glass micro-and nano-spheres[END_REF]. Typical diameters were in the range 125 µm (the fiber diameter) to 450 µm. In order to functionalise the surface, a thin film of Er 3+ -activated 70SiO 2 -30HfO 2 was solgel deposited by dip coating technique [START_REF] Jestin | Micro resonator stabilization by thin film coating[END_REF]. Before further coating, each layer was annealed in air for 50 s at 900 °C. Final films, resulting from 30 to 40 coatings, were stabilized by a last treatment for 5 min in air at 900 °C. As a result of this procedure, crack-free films surrounding the microspheres were obtained. The main characteristics of two microspheres are summarized in Table 1.

Table 1. Diameter D of the silica spheres coated by Er 3+ -activated 70SiO 2 -30HfO 2 film of thickness t and erbium concentration C(Er).

Microsphere labelling

D [± 5 µm ] T [± 0.05 µm ] C(Er) [mol%] SP1 250 0.8 1 SP2 430 1.1 0.1
Room temperature photoluminescence measurements in the region of the 4 I 13/2 → 4 I 15/2 transition of the Er 3+ ions were performed using a 980 nm diode laser as excitation source, detecting the scattered light from the microsphere in free space configuration and dispersing the luminescence light with a 320 mm single grating. The light was detected using a InGaAs photodiode and a lock-in technique. Decay curves were obtained by chopping the CW exciting beam with a mechanical chopper and by recording the signal with a digital oscilloscope. The photoluminescence spectra as well as the decay curves obtained for SP1 and SP2 microspheres exhibit the typical spectroscopic features already observed for planar waveguides of same composition [START_REF] Alombert-Goget | Erbium-activated silica-hafnia: a reliable photonic system[END_REF].

In particular a main emission peak at 1.53 µm with a shoulder at about 1.55 µm and a spectral bandwidth of about 48 nm for both Er 3+ concentrations, are observed. The 4 I 13/2 decay exhibits a single exponential profile with a lifetime of 4.4 ms and 6.2 ms for the microspheres doped with 1 mol% and 0.1 mol%, respectively. The decrease of the lifetime with the increase of the erbium concentration indicates that energy transfer processes, including cross-relaxation and up-conversion, are effective. In fact, in the case of the 1 mol% doped SP1 microspheres, an intense infrared-to-green up-conversion luminescence was observed upon 980 nm excitation. These lifetimes values are in perfect agreement with those obtained in [START_REF] Alombert-Goget | Erbium-activated silica-hafnia: a reliable photonic system[END_REF][START_REF] Gonçalves | Erbium-activated HfO 2 -based waveguides for photonics[END_REF], where a detailed discussion regarding concentration quenching effect is reported. On the basis of these results only the SP2 microsphere was considered for whispering gallery mode (WGM) analysis. To excite WGMs, light was launched by a tapered single-mode fiber. The pumping wavelength was chosen at 1480 nm in order to obtain a good overlap between the pump and laser mode volumes in the micro-sphere, and to have efficient microsphere in/out coupling for both pump and laser signals. The tapered fiber was obtained by heating and stretching a standard telecommunication fiber, single mode at 1.55 µm. Further details on the fiber coupling experimental setup can be found in [START_REF] Chiasera | Spherical whispering-gallery-mode microresonators[END_REF]. Figure 4 shows the fluorescence spectrum of the microsphere SP2 after excitation of the WGMs; the free spectral range (FSR) of 1.22 nm is highlighted in the inset. The spectrum shown in the inset also makes evident the presence of the two TE and TM polarizations and a degenerating effect on the azimuthal index finding his origin in the ellipticity of the sphere. Indeed, these spheres present an average ellipticity of 0.12%, due to inhomogeneities in the thickness of the coating. This fact is responsible of losses which reduce the quality factor and hamper laser. action. A positive outcome of this analysis, however, is that these microspheres exhibit a behaviour of the fluorescence spectra which is quasi independent on the thermal effect linked to injected power. This result indicates that the silica-hafnia coating strongly reduces the typical red-shift observed in spherical microresonators fabricated by other glass systems [START_REF] Jestin | Micro resonator stabilization by thin film coating[END_REF] 1550

CONCLUSIONS

Three examples of confined structures are discussed: i) core-shell Er 3+ -activated silica particles; ii) Er 3+ -activated silica inverse opal; iii) spherical microresonators coated by Er 3+ -activated silica-hafnia film. Core-shell-like Er 3+ -activated silica spheres, where the core is a 270 nm silica sphere and the shell is an Er 2 O 3 -SiO 2 coating, exhibit quantum efficiency of 71%. Large dimension and well ordered Er 3+ -activated silica inverse opals were fabricated using colloidal route. Reflectance measurements showed the presence a pseudo photonic stop band in the visible region. Inverse opal exhibits a main emission peak at 1540 nm, with a spectral width of 21 ± 1 nm, and a 4 I 13/2 lifetime of 16.8 ± 0.1 ms corresponding to a very high quantum efficiency of the system. The coated microspheres have exhibited a WGM fluorescence behaviour quasi independent on the thermal effects related to the injected power. This result confirms the suitability of these microspheres as microlasers, once the inhomogeneity in the thickness of the coating is reduced. It is also confirmed that sol-gel coating is compatible with functionalization or chemical modification of the microsphere surface without significant de-crease of the resonator's quality factor: this characteristic is important for the use of microspheres as potentially highsensitivity biosensors. The possibility to develop appropriate protocols for fabrication of optimized confined structures in photonic glasses on nano-scale domain represents a breakthrough to higher performance and low cost products as well as to innovative physics.

Figure 1 .

 1 Figure 1. SEM images of core-shell Er 3+ -activated silica particles prepared by seeded growth procedure.

Figure 2 .

 2 Figure 2. Scanning Electron Microscopy Image related to (111) facet of Er3+-activated silica inverse opal.

Figure 3 .

 3 Figure 3. Reflectance spectra at different incidence angles performed on the of Er 3+ -activated silica inverse opal along the normal of the (111) plane of the sample.


Y 1 ; Thiery

G Grandjean

; Cottereau

Y 1 ; Colas

B 1 ; Maquaire

; Fressard

; Premaillon

M 1 ; Bernardie

; Gourdier

From landslide susceptibility to risk mapping for France: improvement and application at 1:500,000 scale of work

Keywords: landslide susceptibility, landslide risk, national scale, mapping, quantitative method

Introduction

Assessing landslide risk at national scale has become a major issue for countries, as these maps provide a first overview and qualitative information of the areas potentially susceptible to landslides. In Europe, for more than ten years, several maps have been proposed at the initiative of the JRC (https://esdac.jrc.ec.europa.eu/themes/european-landslide-expert-group). Thus, several maps were produced whether at supranational scale (Gunther et al., 2013;Wide et al., 2018) or at national scale (e.g. for Italy, Triglia et al., 2013;Romania, Bãlteanu et al., 2020;Austria, Lima et al., 2021). In France, some tests for metropolitan territory and Corsica were proposed based on expert knowledge as in 1984 (Delauney, 1984) and later on the basis of the approach proposed by [START_REF] Hervás | Guidelines for Mapping Areas at Risk of Landslides in Europe[END_REF][START_REF] Hervás | Harmonised approaches for landslide susceptibility mapping in Europe[END_REF]Tier 1 approach;[START_REF] Malet | Landslide susceptibility mapping at 1:1M scale over France: exploratory results with a heuristic model[END_REF]. The first results of the Tier 1 approach at the million scale were improved in 2013 [START_REF] Malet | Integrating spatial multi-criteria evaluation and expert knowledge for national-scale landslide susceptibility analysis: application to France[END_REF]. However, while this map presents a first result for landslide susceptibility assessment, limitations can be highlighted. These limitations are: i.

Methodological, for instance lithology reclassification and weightings are subjective and based on the expertise of the person in charge of the map. At this scale of work, it is complex to know the influence of each class of variables on the susceptibility to landslides; this is why Hervàs et al. ( 2010) also proposes a Tier 2 approach based on a more objective calculation and quantification of the susceptibility or hazard. ii.

Geographical, with areas underestimated in term of susceptibility class in particular slopes of plateau areas knowing landslides. iii.

Linked to the improvement of some spatial variables in term of resolution and nomenclature such as the DTM, the national landslide database (BD-MVT), or lithological map.

Thus, this work proposes a new approach for national landslide susceptibility mapping (NLSM) for France. Furthermore, it is proposed to go further and realize landslide risk maps at the national scale. The principles are based on the guidance given by [START_REF] Hervás | Harmonised approaches for landslide susceptibility mapping in Europe[END_REF], [START_REF] Hervás | Harmonised approaches for landslide susceptibility mapping in Europe[END_REF][START_REF] Günther | Climatephysiographically differentiated Pan-European landslide susceptibility assessment using spatial multi-criteria evaluation and transnational landslide information[END_REF], Sedan et al. (2012), andWide et al. (2018), which are:

i.

The computations are performed using a data driven method (quantitative method) and not a heuristic method limiting the subjectivity of the weightings. ii.

New spatial variables are tested in order to spatially improve the unidentified areas.

Finally, the risk is assessed on the principles of Sedan et al (2012) for each municipality with a selection of the stakes based on land use. These issues at this scale are the built-up area and the linear roads of each municipality.

Methodology

To achieve these objectives, the strategy was divided into two phases with phase 1 focusing on susceptibility analysis, and phase 2 performing risk computations. The first phase was carried out in four steps:

i. Selection of the best spatial variables (i.e. topography, geomorphology, geology, land use, physio-geographic area ...) according to the aptitude to spatial variables to build landslide susceptibility maps recommended by [START_REF] Thiery | Landslide susceptibility assessment by bivariate methods at large scales: Application to a complex mountainous environment[END_REF], the computations were performed with a cell size of 250 m in order to optimize the calculation time. ii. Once the best set of spatial variables were defined, in order to meet the objectives set for the working scale, the calculations were carried out with a cell size of 50m and on a partition of the country in 18 physio-geographical units based on topographic and climatic units.

iii. Each unit was subjected to a statistical validation procedure with four tests before reclassification into 6 classes following the limits given by the JTC-1. The limits are given for guidance purposes. The reclassified maps are then merged to obtain a single national landslide susceptibility map. iv. Finally, the final landslide susceptibility map (Fig. 1) was compared to different local landslide databases not implemented in the national landslide database (BD-MVT) and to different national, regional, and local landslide susceptibility maps for different areas.

The second steps consisted in crossing the national landslide susceptibility map with a map of the stakes represented by the built-up areas and the roads for each commune (Sedan et al., 2012). Four types of maps were obtained with:

i. Absolute risk maps representing the surface (in m²) of built areas or road length (in m) affected by moderate, high and very high susceptibility by municipality ii.

Relative risk maps representing dimensionless maps. These maps are the result of a ratio between the built-up surfaces or the road surfaces and the total surface of the municipality.

Results

The results show that the national landslide susceptibility map (NLSM, Fig. 1) obtaining by the procedure based on the principle of selection of the best variables and computation by physiogeographical units significantly improves the map. For the whole territory, the relative error is 0.27, the success rate is 72.8% and the prediction rate is 73.7%. The non-recognition rate is low, meaning that few phenomena are in the negligible class (9%). If we retain the edge areas (e.g. plateau edges) known to produce landslides, they are very well identified with recognition rates higher than 80% (Fig. 1B &1C). We can add that 80-97%of the landslides from local databases not introduced in the computation process are well identified in, very high, high and moderate susceptibility. There are strong similarities between the landslides susceptibility maps obtained at other scales and by other techniques, and despite a different number of classes, the very high, high and moderate classes retained for the NLSM overlap well with the high and moderate classes of the other maps without large overlapping. Finally, the cross-referencing with simplified maps of the stakes (built-up areas and linear roads) makes it possible to identify the communes that potentially have the highest landslide risks.

Discussion -conclusion

If the results of the NLSM are very satisfactory from a statistical and expert point of view, the results for the risk maps are less conclusive. Indeed, the treatment of simplified issues for the built-up environment was carried out from the Corine Land Cover database. Even though it was improved over time, its resolution does not allow obtaining all built-up areas, especially when the habitat is loose. Thus, some municipalities appear without buildings and consequently are not integrated in the maps of absolute and relative risks. It is therefore necessary to re-process the numerical computations by municipality with a more spatially precise database such as the landuse database from THEIA (https://www.theia-land.fr/product/carte-doccupation-des-solsde-la-france-metropolitaine/ ).

Finally, the landslide susceptibility and risk maps for a scale of 1:500,000 is a new spatialization tool that can be integrated into the global assessment scheme of landslide hazard in France with a nested scale approach proposed by Thiery et al (2020). These two tools could allow to better target the municipalities not yet been integrated in the global scheme of PPRs elaboration or that are not a priority although they could have landsliding probability. Thus, it would be necessary, after reprocessing the risk maps, to compare the results with the GeoHazards database (http://www.georisques.gouv.fr/), which provides information on the PPRs planned, in progress or approved.

Figure 1 .

 1 Figure 1. National landslide susceptibility map (NLSM) obtained with quantitative method (WoE method). A. LSM for metropolitan territory and Corsica. B Zoom of the NLSM for the Pays de Caux (with BD-Mvt and independent landslide inventory; Fressard, 2013). C. Zoom for the Touques area (with independent landslide inventory; Fressard, 2013).


André Nel

Corentin Jouault

Guilherme Cunha

The third aeschnidiid dragonfly genus and species from the Lower Cretaceous Crato Formation (Odonata, Anisoptera)

Keywords: Insecta, Aeschnidiidae, Aptian, Early Cretaceous, gen. et sp. nov., Brazil

Cratoaeschnidium martinsnetoi gen. et sp. nov., third genus and species of Aeschnidiidae from the Crato Formation (Brazil), is described and illustrated based on one well-preserved forewing.

It can be differentiated from the other genera in the family, inter alia, because of its discoidal triangle very narrow with only one row of cells; three rows of cells between Mspl and MAa, and between Rspl and IR2; one row of cells between CuAa and MP in basal part; three rows of cells between RP3/4 and MAa. This new genus and species confirms that much can be expected in regard to discovering the Odonata in the Crato Formation.

Introduction

The family Aeschnidiidae is a speciose fossil dragonfly clade currently encompassing 33 genera.

Representatives of this family were distributed worldwide between the Late Jurassic and the latest Cretaceous [START_REF] Fleck | Revision of the Mesozoic family Aeschnidiidae (Odonata: Anisoptera)[END_REF][START_REF] Nel | Maastrichtian representatives of the dragonfly family Aeschnidiidae question the entomofaunal turnover of the early Late Cretaceous[END_REF]. Recent studies indicate that the family seems to die out during the Cretaceous-Cenozoic (K-Pg) crisis, and not during the mid-Cretaceous floristic and faunal turnover as previously supposed by [START_REF] Fleck | The oldest record of the Libellulidae from the Upper Cretaceous of Kazakhstan (Odonata, Anisoptera)[END_REF]. However, the effect of the K-Pg crisis on insects appears to be soft and poorly quantified [START_REF] Condamine | Global patterns of insect diversification: towards a reconciliation of fossil and molecular evidence?[END_REF][START_REF] Schachat | Are insects heading toward their first mass extinction? Distinguishing turnover from crises in their fossil record[END_REF] suggesting that the family may be present during the Paleocene, even if not recorded from Paleocene deposits (e.g., [START_REF] Nel | The odonatan insects from the Paleocene of Menat, central France[END_REF]. The Aeschnidiidae were apparently extremely diversified in the Lower Weald Clay (Hauterivian), in the UK, but less diverse in the Lower Cretaceous of Spain (Las Hoyas and Montsec) (e.g., [START_REF] Nel | Essai de révision des Aeschnidioidea (Insecta, Odonata, Anisoptera)[END_REF][START_REF] Fleck | Revision of the Mesozoic family Aeschnidiidae (Odonata: Anisoptera)[END_REF]. In China, the family is known by a few genera, some of them being very frequent in the Lower Cretaceous of Liaoning (e.g., [START_REF] Huang | Protoliupanshania wangi, a new genus and species from the Chinese Early Cretaceous (Odonata: Aeshnoptera: Liupanshaniidae)[END_REF].

Interestingly, several species are known from the Southern Hemisphere with specimens known from Australia, Brazil, and Colombia (e.g. [START_REF] Martill | A new dragonfly from the Crato Formation (Lower Cretaceous, Aptian) of N.E. Brazil[END_REF][START_REF] Fleck | Revision of the Mesozoic family Aeschnidiidae (Odonata: Anisoptera)[END_REF][START_REF] Adj | Fósiles de Insecta (Odonata) del Aptiano tardío en la Formación Paja[END_REF]; https://paleobiodb.org/). This 'worldwide' distribution during the Cretaceous suggests that the family was perhaps present on the Gondwana masses during the Jurassic and not only present in the northern hemisphere, as evidenced by their current fossil record. They were possibly not found due to the scarcity of Gondwanan Jurassic insect assemblages. The family could also have rapidly dispersed from Laurasia to Gondwana during the earliest Cretaceous. The Brazilian Lagerstätte of the Crato Formation is worldwide known for its numerous perfectly preserved fossils of insects and odonatan specimens are not an exception. This rich entomofauna encompasses two aeschnidiid species, viz. Santanoptera gabbotti Martill and[START_REF] Martill | A new dragonfly from the Crato Formation (Lower Cretaceous, Aptian) of N.E. Brazil[END_REF]Leptaeschnidium araripina (Carle and[START_REF] Carle | Insects from the Santana Formation, Lower Cretaceous, of Brazil[END_REF]) (originally in the genus Wightonia [START_REF] Carle | Insects from the Santana Formation, Lower Cretaceous, of Brazil[END_REF]. The latter genus is known from numerous specimens while the former is known by only one specimen.

Here we describe a new genus and species from this formation, superficially looking like Leptaeschnidium araripina, but showing a unique combination of characters. This discovery confirms that the odonatan diversity of the Crato Formation is really impressive [START_REF] Bechly | Chapter 11.5 Odonata: damselflies and dragonflies[END_REF].

Material and methods

The specimen studied herein comes from the Crato Formation, which is one of the stratigraphic units that constitute the Santana Group of the Araripe Basin. From the base to the top, the Santana Group consists of the Barbalha, Crato, Ipubi, and Romualdo formations. The specimen was collected at 'Pedreira Três Irmão', one of the many local mining queries near the city of Nova Olinda (State of Ceará, Northeastern Brazil), where the Crato Formation outcrops are explored for the extraction of paving stones.

The geology and paleontology of the Crato Formation were revised in detail by Martill et al. (2007a,b) and [START_REF] Ribeiro | Towards an actualistic view of the Crato Konservat-Lagerstätte paleoenvironment: a new hypothesis as an Early Cretaceous (Aptian) equatorial and semi-arid wetland[END_REF]. The age of the Crato Formation is considered to be upper Aptian (early Cretaceous) [START_REF] Pons | Quelques observations sur la palynologie de l'Aptien supérieur et de l'Albien du bassin d'Araripe[END_REF][START_REF] Heimhofer | Early Cretaceous angiosperm pollen from a lowlatitude succession (Araripe Basin, NE Brazil)[END_REF][START_REF] Varejão | Mixed siliciclastic-carbonate sedimentation in an evolving epicontinental sea: aptian record of marginal marine settings in the interior basins of north-eastern Brazil[END_REF]. [START_REF] Dias | The role of microbial mats in the exquisite preservation of Aptian insect fossils from the Crato Lagerstätte, Brazil[END_REF] recently demonstrated the role of the microbial mats in the preservation of the insects in the Crato Formation.

According to [START_REF] Ribeiro | Towards an actualistic view of the Crato Konservat-Lagerstätte paleoenvironment: a new hypothesis as an Early Cretaceous (Aptian) equatorial and semi-arid wetland[END_REF], the paleoenvironment of the fossil-rich interval of the formation -the 'Crato Konservat-Lagerstätte' or CKL -consisted of a semi-arid seasonal lacustrine wetland with a shallow water body. A diverse aquatic fauna and flora were present as an integral part of this biome, and was succeeded up landward by neighboring mesophytic ecotones, which were periodically flooded. This system was surrounded by outer xeric habitats.

Trophic structure analysis detailing the putative food web that took place within the Crato Ecosystem was studied by [START_REF] Mendes | Ecosystem structure and trophic network in the Late Early Cretaceous Crato Biome[END_REF].

The fossils were preserved with the cuticle being replaced by dark brown iron hydroxide (goethite) [START_REF] Barling | High fidelity preservation of fossil insects from the Crato Formation (Lower Cretaceous) of Brazil[END_REF][START_REF] Barling | A geochemical model for the preservation of insects in the Crato Formation (Lower Cretaceous) of Brazil[END_REF]. The holotype of Cratoaeschnidium martinsnetoi gen. et Remark: The coloration on wings or bodies of fossil insects from the Crato Formation is often well preserved [START_REF] Nel | New grasshoppers (Orthoptera: Elcanidae, Locustopsidae) from the Lower Cretaceous Crato Formation suggest a biome homogeneity in Central Gondwana[END_REF] allowing species delineation in some cases. However, it is impossible to ensure that the color pattern recorded in the present specimen is that of the original wing or results from taphonomic processes.

Discussion

This fossil has all the characteristics typical of a forewing of a dragonfly belonging to the family Aeschnidiidae, especially the presence of two rows of cells between C and RA distal of the nodus, a strongly transverse discoidal triangle, a special shape of the anal area with a series of large areas limited by branches of AA [START_REF] Fleck | Revision of the Mesozoic family Aeschnidiidae (Odonata: Anisoptera)[END_REF].

Because of the configuration of the discoidal triangle recorded in the new fossil, we restrain our comparison with genera possessing a very narrow discoidal triangle with only one (at most two) row(s) of cells [START_REF] Fleck | Revision of the Mesozoic family Aeschnidiidae (Odonata: Anisoptera)[END_REF][START_REF] Huang | A new aeschnidiid genus from the Early Cretaceous of China (Odonata: Anisoptera)[END_REF][START_REF] Nel | Maastrichtian representatives of the dragonfly family Aeschnidiidae question the entomofaunal turnover of the early Late Cretaceous[END_REF]. Therefore, the new fossil differs from the genus Aeschnidiella because of the presence of only one row of cells between CuAa and MP, and crossveins between IR2 and RP2 perpendicular to these veins, instead of being oblique [START_REF] Zalessky | Novye mestonakhozhdeniya Melovykh nasekomykh v Povolzh'e, Kazakhstane i Zabaykal'e[END_REF]. It differs from Aeschnidiopsis owing to the presence of one row of cells between IR2 and RP2 in their basal parts, and two rows of cells between IR2 and RP3/4 below subnodus, vs. three to four [START_REF] Woodward | On the wing of a neuropterous insect from the Cretaceous limestone of Flinders River, north Queensland, Australia[END_REF]. Note that the pterostigmal zone is unknown in these two genera. Gigantoaeschnidium has no pterostigma and no pterostigmal brace, as in the new fossil, but it also has much more rows of cells between Mspl and MAa and between Rspl and IR2 than the new fossil [START_REF] Nel | Essai de révision des Aeschnidioidea (Insecta, Odonata, Anisoptera)[END_REF].

Iberoaeschnidium has a widened pterostigmal zone with a curve of RA and RP1, unlike the new fossil [START_REF] Nel | Essai de révision des Aeschnidioidea (Insecta, Odonata, Anisoptera)[END_REF]. Leptaeschnidium araripina (from the Crato Fm.)

and Lleidoaeschnidium maculatum have a weak but distinctly oblique pterostigmal brace, and a curvature of RA in the pterostigmal zone, unlike the new fossil [START_REF] Carle | Insects from the Santana Formation, Lower Cretaceous, of Brazil[END_REF][START_REF] Fleck | Revision of the Mesozoic family Aeschnidiidae (Odonata: Anisoptera)[END_REF]. Leptaeschnidium latum (based on a hind wing) seems to have no defined pterostigmal brace and curvature of RA in the pterostigmal zone, but it has much more rows of cells between RP3/4 and MAa and between RP1 and RP2 [START_REF] Fleck | Revision of the Mesozoic family Aeschnidiidae (Odonata: Anisoptera)[END_REF].

Leptaeschnidium araripina, Lleidoaeschnidium valloryi, Lleidoaeschnidium maculatum, and

Coramaeschnidium minimum have distinctly narrower basal and distal parts of the subdiscoidal area, with a vein AA1a angled between these two parts of the subdiscoidal area, vs. straight in the new fossil [START_REF] Fleck | Revision of the Mesozoic family Aeschnidiidae (Odonata: Anisoptera)[END_REF][START_REF] Nel | Essai de révision des Aeschnidioidea (Insecta, Odonata, Anisoptera)[END_REF]. Nannoaeschnidium pumilio has also a distinct angle in the course of AA1. Angloaeschnidium toyei, Angloaeschnidium montreuili, Angloaeschnidium lacaui, Cooperaeschnidium durandi, and Linaeschnidium sinensis also have a well-defined pterostigmal brace and a curvature of RA in the pterostigmal zone [START_REF] Fleck | Revision of the Mesozoic family Aeschnidiidae (Odonata: Anisoptera)[END_REF][START_REF] Huang | A new aeschnidiid genus from the Early Cretaceous of China (Odonata: Anisoptera)[END_REF]. Delclosaeschnidium magnum and Jarzembowskiaeschnidium polandi have much more rows of cells between all main veins [START_REF] Fleck | Revision of the Mesozoic family Aeschnidiidae (Odonata: Anisoptera)[END_REF]. The latter genus can also be differentiated from our specimen owing to its oblique pterostigma brace. Diastatopsaeschnidium reneeheiko shares with the new fossil a vein AA1 without angle, but it has a pterostigmal brace and a curvature of RA in the pterostigmal zone and basal part of the subdiscoidal area distinctly larger [START_REF] Fleck | Revision of the Mesozoic family Aeschnidiidae (Odonata: Anisoptera)[END_REF].

Similarly, affinities with Kessleraeschnidium simonae are excluded [START_REF] Fleck | Revision of the Mesozoic family Aeschnidiidae (Odonata: Anisoptera)[END_REF].

Santanoptera gabbotti, second described Aeschnidiidae from the Crato Fm., strongly differs from the new fossil in having a very broad discoidal triangle and much more rows of cells between all main veins [START_REF] Martill | A new dragonfly from the Crato Formation (Lower Cretaceous, Aptian) of N.E. Brazil[END_REF].

As aforementioned, the new fossil does not fit with any described genus and species of Aeschnidiidae. We erect a new genus and species to accommodate this new fossil, which is the third aeschnidiid from the Crato Fm.

Conclusion

Cratoaeschnidium martinsnetoi gen. et sp. nov. is the third genus and species of the family Aeschnidiidae from the Lower Cretaceous Crato Fm. of Brazil. Cratoaeschnidium gen. nov. resembles the Early Cretaceous genera Leptaeschnidium and Lleidoaeschnidium, and shares with the early Cretaceous clade (Lieidoaeschnidium, Angloaeschnidium, Leptaeschnidium, Kessleraeschnidium, Iberoaeschnidium, Aeschnidiopsis, Aegyptidium, Santanoptera, Diastatopsaeschnidium) the putative synapomorphy 'presence of basal concave supplementary vein between RP1 and RP2'. Even if this clade seems to be stable after Fleck and Nel (2003: fig.147-148), a new phylogenetic analysis of the Aeschnidiidae, including the newly described taxa, is necessary to clarify the relationships within the family.

Figures

  Figures 1-2 Zoobank xxxx Etymology. Named after Dr Rafael Martins-Neto, who extensively studied the entomofauna of the Crato Formation. The species epithet is to be treated as a noun in a genitive case. Type material. Holotype specimen CCNH 445, a nearly complete forewing, with only costobasal part missing, stored at the paleontological collection of Centro de Ciências Naturais e Humans, Universidade Federal do ABC, Santo André, São Paulo, Brazil. Age and outcrop. Lower Cretaceous, late Aptian, Crato Formation, Araripe Basin, Brazil.

Figures

  Figures

Figure 1 .

 1 Figure 1. Cratoaeschnidium martinsnetoi gen. et sp. nov., holotype CCNH 445, forewing. (A) Composite image; (B) Composite image of forewing made in Photoshop using 'photomerge' function; scale bars = 2 mm.

Figure 2 .

 2 Figure 2. Cratoaeschnidium martinsnetoi gen. et sp. nov., holotype CCNH 445, forewing. Photographs. (A) Detail of basal portion of wing; (B) Detail of mid-portion of wing; (C) Detail of distal portion of wing; scale bars = 2 mm.

Acknowledgments

We thank three anonymous referees for their useful remarks on the first version of the paper.

We also thank the Brazilian National Mining Agency (ANM) for collecting permits for GCR.

This study was supported by the São Paulo State Research Agency (FAPESP) (grant no. 2020/02844-5 to GCR). This paper was contributed by CJ during his PhD project.

Disclosure statement

No potential conflict of interest was reported by the author(s).


Fluid properties probed by vibrating microcantilevers

The aim of this paper is to demonstrate that vibrating microcantilevers can be used to quantify fluid properties as both density and viscosity. For this, an analytical model is exploited and the preliminary measurements, allowing the determination of the viscosity as a function of frequency, are encouraging for the development of a useful microrheometer on a silicium chip for microfluidic applications.

Introduction

Contrary to classical rheological measurements using microcantilever sensors 1,[START_REF] Boskovic | Rheological measurements using microcantilevers[END_REF] , the development of the proposed microrheometer is based on the measurement of fluid properties at each vibration frequency, without necessarily being bound to resonance phenomenon. Simultaneous measurements of both phase and amplitude lead to the fast determination of both unknown fluid parameters (viscosity and density.

Beam Vibrations in Viscous Fluid

Viscosity, commonly perceived as the resistance to flow is an intrinsic constant which characterizes newtonian fluid. In Fourier space, a viscous fluid exerts on the beam surface the drag force 3,[START_REF] Rosenhead | Laminar Boundary Layers[END_REF] :

(1)

) ( ) ( 2 2 1 ω ω ω ω w g jw g F fluid + - =
with g 1 and g 2 terms due respectively to the dissipative and inertial parts, w the deflection at the end of the beam, ω the angular frequency.

In Fig. 1, the analytical resolution of the equation governing the deflection of a microcantilever in a viscous fluid demonstrates that the evolution of both, the amplitude and phase, as a function of the frequency depends on the fluid properties. An analogy between beam dynamic in viscous fluid and second order low-pass band allows the simplification of the mechanical transfer function. Then, by identification, knowing the amplitude and phase we can retrieve the drag force components and see their variations according to the frequency:

φ π sin 2 2 , 0 0 1 f f H H m g vac L - = (2) 1 cos 1 2 , 0 0 2 - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = f f H H m g vac L φ ( 3
)
with m L the cantilever mass per unit length, f 0 , vac the resonant frequency in vacuum, φ the phase, f the frequency, H the transfer function, H 0 its static value.

Thus, this determination of the two parts of the drag force allows to extract analytically at each frequency viscosity and density in order to see the rheological behaviour of the studied fluid.

Measurements and Discussion

Using microcantilevers of various geometrical properties, measurements were made in silicon oils covering a wide viscosity range (Fig. 2). The measurements confirm that, even without resonant phenomenon, fluid properties govern the vibrational characteristics of the beam oscillation. Viscosity, density and the rheological behaviour can be deduced after drag force components extraction from beam spectrum. First measurements (Fig. 3) are in good agreement with the Newtonian behaviour of the silicon oils for this frequency range.

CONCLUSION

The results are very encouraging as they demonstrate the potential for beam-based microstructures to serve as useful tools for rheological measurements at fixed frequency over a wide frequency-range.

Fig. 1 :

 1 Fig. 1: Theoretical phase and amplitude of a silicon cantilever (23µm, 600µm, 4mm) oscillating in fluids with different viscosities.

Fig. 2 :

 2 Fig. 2: Measurements corresponding to the simulation presented in Fig.1 for a silicon cantilever (23μm x 600μm x 4000μm) immersed in various silicon oils: 9.4cP, 19cP, 96.5cP, 288cP, 485cP and 1000cP.

Fig. 3 :

 3 Fig. 3: Determination of viscosity and density with measurements of a silicon cantilever (70μm x 200μm x 3000μm) immersed in silicon oil whose viscosity and density are respectively equal to 96cP and 1000kg.m -3 . 3000 3500 4000 4500 5000 10 2

ACKNOWLEDGMENTS

We thank the "Région Aquitaine" for its financial sponsorship.