T Lambert

F Despetis

L Puech

P E Wolf

Evidence for a disorder driven phase transition in the condensation of 4 He in aerogels

Keywords: numbers: 64.60.-i, 64.70.Fx, 68.03.Cd, 67.70.+n

We report on thermodynamic and optical measurements of the condensation process of 4 He in two silica aerogels of same porosity 95%, but different microstructures resulting from different pH during synthesis. For a base-catalyzed aerogel, the temperature dependence of the shape of adsorption isotherms and of the morphology of the condensation process show evidence of a disorder driven transition, in agreement with recent theoretical predictions. This transition is not observed for a neutral-catalyzed aerogel, which we interpret as due to a larger disorder in this case.

Condensation of fluids in mesoporous media, i.e. with pores sizes falling in the nmµm range, has long been the subject of extended research, both fundamental and applied [1]. From the point of view of statistical physics, this phenomenon offers an experimental realization of a phase transition in presence of disorder and confinement. From the practical point of view, it is widely used as a way to characterize the pore distribution. The most common case is that of sponge-like porous media, where the fluid is confined in well-defined cavities, either highly connected as, e.g., in Vycor glass, or with little or no connection, as in MCM41 silica. In the case where the dense phase of the fluid wets the substrate, one observes the progressive adsorption of a thin film at low vapor pressure, followed by a rather abrupt filling at a pressure of order, but smaller, than the saturated vapor pressure. This phenomenon is coined capillary condensation, being driven by the surface energy between the liquid and the vapor. In the classical picture, the condensation pressure depends on this surface energy and the pore size through the so-called Kelvin equation [1]. The process is hysteretic, emptying occurring at a lower pressure than filling. While this hysteresis is well understood at the level of the unique pore, its origin remains debated in the case of disordered porous media, with connected pores of different sizes [1,2]. In particular, it has been recently shown by Kierlik et al [3] that it could result from the disordered nature of the porous media, rather than, as usually thought, of its geometry or topology ('pore-blocking').

In this new description, for a given pressure close to saturation, there exists a large number of metastable equilibrium states, the one being selected depending on the time history of the system, which gives rise to the observed hysteresis.

As such, this description should apply to porous media of very different topology, silica aerogels [4]. In these gels, the silica forms a complex arrangement of interconnected strands, resembling more a three-dimensional spider web than a sponge. Despite the positive curvature of the strands , which unfavors the film adsorption, aerogels fill below the saturated pressure, which, in a 'classical' picture, has been attributed to capillary condensation starting on the intersections between strands [5]. Experiments are however delicate to interpret as, due to their very low elastic modulus, the gels deform a lot upon adsorption of fluids with usual values of surface tension, such as nitrogen [6]. This leads to the idea of studying these gels using Helium, which has a much lower surface tension. Such an experiment, albeit differently motivated, was first performed by Wong and Chan, about 15 years ago [7]. Their results showed capillary condensation, but, quite surprisingly, at a well defined pressure rather than on some range, as one would expect considering the wide distribution of scales in the silica aerogels. Together with specific heat experiments, this lead these authors to propose the existence of a genuine phase transition of helium inside aerogels, with a slightly depressed critical temperature with respect to the bulk case (T c = 5.195K). However, latter experiments by other groups performed on similar aerogels (95% porosity, gelation process with basic pH), did not confirm this conclusion. The sorption isotherms are hysteretic, and condensation occurs over a finite range of pressures [8,9,10,11]. Still, this range remains unexpectedly narrow, especially at low porosity [8,12] or low temperature [11,12].

Both facts are in agreement with theoretical work [4,13] extending Kierlik et al's description to a 95% porosity aerogel, numerically synthesized using diffusion limited cluster aggregation (DLCA), a process known to account for the structure factor of base-synthesized aerogels. These studies show a disorder-induced transition similar to that predicted by the Random Field Ising Model at zero temperature [14]. At small porosity (or large disorder), filling takes place by a succession of small, microscopic avalanches, where a limited number of sites switch from gas to liquid at each pressure. The maximal size of these avalanches increases with the porosity, up to some critical porosity (corresponding to some critical disorder), where it diverges. Above this critical porosity, which increases with increasing temperature, filling involves a macroscopic avalanche, associated with a jump in the average density, at some well defined pressure. This implies a change of shape, from smooth to sharp, of the hysteresis loop, when the porosity is increased at a constant temperature or, alternatively, when the temperature is decreased below some critical value, at a constant porosity. In magnetic systems, an equivalent change of the hysteresis loop has been observed in Co/CoO disordered films [15]. Although the observed evolution of shape with temperature is an appealing argument in favor of the Kierlik et al vision of the condensation in disordered porous media, other explanations could exist as well. The aim of this paper is to further investigate the adequacy of the proposed mechanism, and its possible dependence on the aerogel structure, using combined optical and thermodynamic measurements.

We used a one step process [START_REF] See E | Handbook of Sol Gel Science and Technology, Processing Characterization and Applications[END_REF] to synthesize two aerogels of same porosity (95%), but different microstructures, by changing the pH from basic to neutral. Neutron [START_REF] Vacher | [END_REF] and X-ray [18] scattering experiments previously performed on similar samples reveal the sensitivity of the structure to the different kinetics of the sol-gel process in both cases. The size of the building silica units is larger for the base-catalyzed aerogel (B100, i.e. of density 100 kg/m 3 ) than for the neutral aerogel (N102), whereas the correlation length is larger for N102 than for B100. This results in a fractal structure over a much larger range for N102 than for B100, and, accordingly, a wider distribution of pores sizes. The correlation lengths of our own samples were deduced from their structure factor S(q) at zero q (as measured by light scattering) and their known porosities and fractal dimensions, using a theoretical expression of S(0) which has been experimentally checked [19]. This gives values of order 10 nm for B100 and 20 nm for N102. Sorption isotherms were performed on both samples at a number of temperatures, and, at the same time, the way they scattered light was studied, so as to gain information on the distribution of helium inside the aerogels, both at the microscopic and macroscopic levels.

The experimental cell is a 4 mm thick copper block, traversed by a 20 mm diameter hole, and closed by two sapphire windows. Aerogels samples were sliced from 14 mm diameter cylinders. In the first experiment, we used a half disk of B100, about 2.7 mm thick. In the second experiment, we studied a 3.8 mm thick full disk of N102. The cell is mounted in a cryostat with 8 optical ports 45 • apart, and is connected to the room temperature gas-handling system by two lines, one of which is used to fill or empty the cell through a regulated flowmeter, and the other is connected to a room temperature pressure sensor, with a resolution of 0.1 mbar. The flowrate is adjusted so that the stage of capillary condensation takes from 15 to 30 hours, which we checked to be long enough to get rate independent results. The helium mass in the aerogel is obtained by integrating the flowrate over time, and correcting for dead volumes as discussed in ref. [11]. It is converted into a fraction Φ of dense phase ('liquid'), assuming that the vapor inside the aerogel has its bulk value and that the density of the dense phase is that measured for the fully filled aerogel [11].

Figure 1(a) shows the isotherms obtained at four temperatures below the bulk critical temperature T c for the two aerogel samples. The general trends are in qualitative agreement with the usual picture of capillary condensation; the hysteresis loop gets narrower and closer to the bulk saturation pressure P sat as the surface tension decreases with increasing temperature. Also, it is smoother, and closer to P sat for N102, consistent with the wider distribution of pores extending up to larger sizes. Quantitatively, we measure the position of the adsorption isotherm by ∆P max = P -P sat , at the point of largest slope of the adsorption branch. For a cylindrical pore of radius R, filling occurs at the stability limit of the adsorbed film, which, for R much larger than the film thickness, is given by [20,21] ∆P max = σ/R.ρ V /(ρ L -ρ V ), where σ is the helium surface tension, and ρ L and ρ V are the bulk liquid and vapor densities at saturation [22]. For a collection of independent pores, one expects ∆P max to correspond to some typical value of the pores size. Although, a priori , aerogels cannot be so simply described, figure 2 shows that the equation above accounts for the temperature dependence of ∆P max for N102 in the whole temperature range, and for B100 above 4.95 K, with characteristic 'pores' sizes R ≃ 35 nm and 20 nm. Both values are about twice the measured correlation lengths, a difference which is not surprizing, considering the complex microstructure of the samples. However, this simple approach fails to describe the position of the pressure plateau for B100 at the two lower temperatures, which occurs at a smaller ∆P max than expected. Furthermore, in such an approach, the small width of the corresponding isotherms would correspond to a much narrower distribution of pores sizes than that could be deduced from the isotherms at 4.95 K and above.

Optical measurements also show a markedly different behavior below 4.71 K, and above 4.95 K, respectively. We illuminate the aerogel by a thin (100 µm wide), vertically polarized, He-Ne laser sheet under a 45 • incidence with respect to its faces, and image it at 45 • , 90 • , and (for N102) 135 • using CCD cameras. At any point of the intercept of the aerogel by the laser sheet, the brightness of the image is proportional to the intensity scattered by the aerogel in the direction of observation, and depends on the local strength and scale of the spatial fluctuations of the helium and silica densities. Figure 3 shows (a) images of B100 for increasing helium fractions along the previous isotherms, and (b) the scattered intensity by three selected spots in the images. Quantitative analysis of these curves show that, below Φ ≈0.3, the scattered intensity is that expected for a thin film of helium covering uniformly the silica strands, whereas, beyond this value, it becomes much larger, implying the formation of liquid clusters correlated over distances larger than the correlation length of the silica [11]. The striking fact is that, while this increase of the scattered intensity occurs uniformly over the sample for the two larger temperatures, for the two lower ones, it is only so for fractions less than about 0.5. Beyond, liquid progressively invades the aerogel, resulting in the growth of dark regions. Comparison of figures 1 and 3(b) shows that this invasion occurs along the vertical part of the isotherm. This could suggest that the bright and dark regions correspond to two coexisting phases at thermodynamic equilibrium. However, this is ruled out by the fact that the intensity scattered by the bright regions evolves with Φ, as shown by figure 3(b) at 4.71 K. On the other hand, the observations exhibit the behavior expected for macroscopic avalanches ; at any given spot, the transition to the dark state occurs discontinuously, when this spot is swept by the boundary between the bright and the dark regions. This is consistent with the occurrence of a macroscopic (but local) avalanche from a partly filled to a fully filled state. In this scenario, the macroscopic morphology, which is identical for 4.47 K and 4.71 K, and for different rates of condensation as well, must originate in macroscopic, heterogeneities of the average density of silica, which make some regions favor slightly more the liquid. This will make the pressure at which the avalanche takes place to vary continuously in space, so that the transition should occur successively at different locations in the aerogel, in qualitative agreement with the observations. In summary, for B100 at 4.71 K and below, the isotherms exhibit a nearly vertical portion, their position is anomalous, and a significant fraction of the condensation takes place in an heterogeneous way. These facts are consistent with the occurrence of a disorder induced transition for B100 between 4.95 K and 4.71 K. In contrast, for N102 at all temperatures, the isotherms are smooth, and the scattering is homogeneous, as shown in figure 4(a) : as a function of the condensed fraction Φ, the intensity scattered at 45 • is the same along any vertical line of the picture, corresponding to a given depth, hence path length, inside the sample. There is a difference between points at different depths, but this (as well as the bump in the signal at Φ ≈ 0.5) can be explained by multiple scattering effects, which are stronger for N102 than for B100, due to the larger thickness and silica correlation length.

Hence, no transition is observed for N102 in the temperature range where it is for B100. In the disorder induced transition scenario, this implies that the disorder does not only depend on porosity and is larger for N102 than for B100. We speculate that this is due to the wider distribution of pores sizes in this case. It would be interesting to see whether calculations performed on 95% aerogels numerically synthesized using the Diffusion Limited Aggregation algorithm appropriate for neutral aerogels would indeed confirm the absence of transition in this case.

Finally, as shown by figure 4 (b), the intensity scattered at 135 • close to the entrance of the laser sheet, once corrected by the contrast factor( (∆n) 2 , where ∆n is the difference in refractive index between the liquid and the vapor) behaves roughly the same, whatever the temperature. This suggests that, at a given Φ, the distribution of liquid does not depend on temperature. This is not the case for B100 below 4.95 K. As shown by figures 3(a) and (b), the scattered intensity, at a given Φ, is smaller at 4.46 K than at 4.71 K, although the contrast is larger. This implies that the liquid must be distributed in smaller clusters. Here also, theoretical calculations would be needed in order to see whether such a temperature dependence follows from the disorder induced transition scenario.

In conclusion, our results provide a strong evidence for the occurrence of a disorder induced transition in the condensation of 4 He in base-catalyzed aerogels. Planned experiments using lighter aerogels should allow to check whether the transition moves to larger temperatures, as the theory would predict. Here again, the very low index of refraction of helium, which allows quantitative single scattering measurements to be performed over a much wider range of conditions than for any other simple fluid, will be a unique advantage. This ability to perform optical studies in transparent porous media could be used to study other problems as well, such as the origin of the hysteresis between adsorption and desorption.

FIG. 2 :

 2 FIG. 2: Position ∆P max of the point of maximal slope in the adsorption branch versus the temperature dependent ratio σρ V /(ρ L -ρ V ). The straight lines correspond to pore diameters (see text) of 18 and 36 nm.

9

 9

FIG. 4 :

 4 FIG. 4: Intensity scattered by N102 as a function of the condensed fraction Φ : (a) at 4.47 K and 45 • . The signal, normalized by the silica contribution, corresponds to five regions at three different depths inside the aerogel disk. Differences between curves for different depths (and path lengths) are consistent with multiple scattering effects combined with an homogeneous distribution of the liquid clusters within the sample, in contrast to B100 at the same temperature; (b) at 135 • , close to the entrance of the laser sheet. The silica contribution I Si has been subtracted, and the result normalized by I Si and the optical contrast squared. The signal is similar at all temperatures, suggesting that the spatial organization of the liquid only depends on Φ, not on temperature.

We acknowledge T. Herman, J. Beamish, F. Detcheverry, E. Kierlik, M.L. Rosinberg, and J. Phalippou for useful suggestions or discussions.


Yann Lepoittevin
email: yann.lepoittevin@inria.fr

Isabelle Herlin
email: isabelle.herlin@inria.fr

Modeling high rainfall regions for flash flood nowcasting

The paper concerns the description of a nowcasting system used for flash floods mitigation, which is based on ground radar images acquired at local scale. The forecast of rain quantity is usually obtained either with numerical weather prediction models or with image-based methods. The first solution is suitable for temporal horizons above three hours due to the computational time required for applying data assimilation techniques to these high dimension models. The second solution is therefore applied for nowcasting up to two hours. Such nowcasting system involves two main parts. The first one estimates the motion field and the second one forecasts the rainfall location and quantity using the radar images and the estimated motion field. The paper focuses on the estimation process, which is obtained by assimilating radar images with an image model describing the temporal evolution of the studied system. The state vector of this model includes the motion field, the image brightness and a description of the objects displayed on the image data. These objects correspond to the rainy regions, whose displacement must be correctly assessed if the estimated motion field is accurate. Results demonstrate that this extended description of the image data allows to produce an improved forecast.

I. INTRODUCTION

The paper concerns the nowcasting of flash floods from ground-based radar images acquired at the city scale. Illustration is obtained with data acquired for the city of Clermont-Ferrand in France, which is strongly affected by flood events in spring and summer. Rainfalls forecast is usually obtained in two ways: either from numerical weather prediction models or from image-based methods. The first solution is used for temporal horizons above three hours according to the computational time required for applying data assimilation techniques to these high dimension models. The second solution is usually applied for nowcasting up to two hours, for instance in Mecklenburg et al. [START_REF] Mecklenburg | Quantitative precipitation forecasts (QPF) based on radar data for hydrological models[END_REF] and Ebert et al. [START_REF] Ebert | Verification of nowcasts from the WWRP Sydney 2000 forecast demonstration project[END_REF], as modeling the long term appearance or disappearance of clouds requires involving physical laws. A nowcasting system involves two main parts, as described in Lepoittevin et al. [START_REF] Lepoittevin | Assimilation of radar reflectivity for rainfall nowcasting[END_REF]. The first one estimates the current motion field and the second one forecasts the rainfall location and quantity, using as input the motion field and the radar acquisitions. The core of the paper concerns the first part, which is obtained by assimilating radar images with an image model. The innovation concerns the definition of the state vector associated to the image model and of its evolution law. As the clouds displayed on the image data are transported by motion accordingly with the image brightness values and provide additional information on the observed dynamics, the state vector, which describes the studied system, includes both the brightness function and a representation of the objects. Such extended description is required in order to better assess the rainfall displacement and their future location.

In this paper, motion estimation relies on three consecutive radar acquisitions that are named A e1 to A e3 . I fi denotes the ith forecasted image and A fi the radar image acquired at the same date. As no ground-truth on motion is available, the only validation of the method comes from the comparison of I fi and A fi . However, their discrepancy corresponds not only to the inaccuracy of the estimated motion field but also of physical processes that create appearance or disappearance of clouds. The estimation and forecast parts are illustrated by Fig 1.

The input data are X band images from ground radars, acquired every five minutes, with an angular resolution of one degree and a radial resolution of 150 meters. The reflectivity measures are first converted into precipitations rates with the Z-R formula from Marshal-Palmer [START_REF] Marshall | The distribution of raindrops with size[END_REF] and second transformed from the polar grid to the Cartesian one, as displayed on the left column of Fig. 2. Motion is then estimated (see Section II) from acquisitions A e1 to A e3 and used for the nowcast (see Section III) with A e3 as initial value (see Fig. 1).

II. ESTIMATION COMPONENT

Most estimation methods of the literature on nowcasting are either based on optical flow computation, which was first described by Horn and Schunk [START_REF] Horn | Determining optical flow[END_REF], or on clouds tracking. They usually rely on only two consecutive image acquisitions. Our approach is alternatively based on an image evolution model and an image assimilation approach, as the one described by Béréziat et al. [START_REF] Béréziat | Solving ill-posed image processing problems using data assimilation[END_REF]. The temporal evolution of motion is ruled either by a stationarity equation or by advection laws, depending on the image acquisition period and on the duration of the time interval used for the estimation. This paper illustrates the approach with the stationarity option, which is valid in the applicative context because the acquisition period is equal to 5 minutes and only three acquisitions are used for the estimation on a 10 minutes period.

The studied image domain is denoted Ω. A state vector X is defined on the space-time domain D, depending on the location x and date t. The vector X has three components: it Observations : P seudo -images :

Estimation F orecast A e1 A e2 A e3 A f1 A f2 . . . A f K I e1 I e2 I e3 I f1 I f2 . . . I f K x x x x x x
             ∂w ∂t (x, t) = 0 ∂I ∂t (x, t) = -w(x, t) • ∇I (x, t) ∂φ ∂t (x, t) = -w(x, t) • ∇φ(x, t) (1)
Motion is then estimated by an iterative process so that its final value allows: 1-the image function I to be close to the effective acquisition A, 2-the object component φ to be close The estimation relies on a 4D-Var data assimilation approach, which was initially described by Le Dimet and Talagrand [START_REF] Dimet | Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects[END_REF]. This method computes the minimum of a cost function J, depending on the initial value of the state vector X(0), with a gradient descent technique. The cost function value is obtained from a forward integration of the evolution model described by System (1) and its gradient value from the backward integration of the adjoint model. J is defined by:

J(X(0)) = D ε I (x, t) 2 R I (x, t) dxdt + D ε φ (x, t) 2 R φ (x, t) dxdt + Ω ε b (x) T B(x) -1 ε b (x)dx (2)
The first term is the integral, on the space-time domain D, of a function of the error term ε I , which is defined as the difference between the image function and the radar acquisition:

ε I (x, t) = I (x, t) -A(x, t) (3)
R I is the variance of the Gaussian variable ε I . It is given the same value than the variance of the acquisition error.

The second term is the integral, on the space-time domain D, of a function of the error term ε φ , which is defined as the difference between the distance map φ, computed on the image function I , and the distance map D A , computed on the radar acquisitions:

ε φ (x, t) = φ(x, t) -D A (x, t) (4)
R φ is the variance of the Gaussian variable ε φ .

Let IP be the projection of the state vector X on the vector with two components I and φ. The third term is the integral, on the image domain Ω, of a function of the difference ε b between the estimated value of IP(X(0)) and an initial value IP(X b ):

ε b = IP(X(0)) -IP(X b ) (5)
X b is the background of the data assimilation process. It defined in the following way. I b is the first image A e1 of the estimation process and φ b is the distance map to the boundary of the rainy regions computed on A e1 . The operational system is implemented with a sliding-window approach that estimates the state vector X on temporal windows of three acquisitions with one frame shift between two consecutive windows. w b is then defined as the result of motion estimation on the previous window. The covariance B(x), which is involved in Eq. ( 2), is diagonal with values equal to R I (x, t) and R φ (x, t).

As no ground-truth is available, validation of the estimated motion field is postponed to the forecast process. However, a first analysis is at hand by comparing the estimated image function I and the radar images A. The left column of Fig. 2 displays three consecutive acquisitions, A e1 , A e2 and A e3 , used as input for the motion estimation process, while the right column gives the estimations I e1 , I e2 and I e3 . An additional analysis comes from the comparison between the estimated distance map φ and the distance map D A that is computed on the radar image acquired at the same date. During the data assimilation process, φ (and therefore B) is iteratively modified from its background value, in order to correctly describe the boundary B A of the rainy regions on the radar acquisitions. Modifying the position of a given point c of the boundary B has a strong influence on all pixels x, which have c as closest point. The estimation of the state vector X(x) and of motion w(x) at these pixels is then linked to the estimation at point c. If c is moved for being closer to one point c A of the boundary B A , this impacts the estimation of the state vector (and consequently of motion) on all pixels x that are linked to c. This explains why motion is estimated even far away of the rainy regions as proven on Fig. 6. An influence value is defined for each point c as:

ξ(x) = 1 |d(x, c) -φ(x)| (6)
with d(x, c) being the Euclidean distance between pixel x and point c. Fig. 7 gives the influence map related to the point c displayed by the gray circle on the black boundary.

III. FORECAST COMPONENT

In a sliding-window framework, as soon as the third observation A e3 is available, a motion field w is estimated and used for nowcasting from A e3 , before the next acquisition A f1 , as displayed on Fig. 1. As the time horizon ranges from 30 minutes to 2 hours, in order to have enough time for applying preventive measures, the temporal duration of nowcasting is higher than the one of the estimation. The advection option is then applied for describing the temporal evolution of the motion field:

∂w ∂t (x, t) = -w(x, t) • ∇w(x, t) (7)
Temporal integration of Eq. 7 follows a three step process:

• Motion is integrated from one acquisition date i to the next one i + 1 thanks to a Godunov scheme with a one second time step. • A semi-Lagrangian scheme is then applied at date i + 1

to compute I and φ from their values at date i. • The two previous points are iterated from i + 1 to i + 2. The use of these two integration schemes allows to get benefit of both the stability of the semi-Lagrangian scheme and the accuracy of the Godunov scheme when applied to the non linear advection.

From an operational point of view, comparison of the forecasted images with the effective acquisitions has no interest at the pixel level. The required output is the location and approximate date of high cumulative rain quantities that can result in floods. The cumulative rain quantity is obtained from the time integration of the instantaneous rainfall rates over a given period that is set to 1 hour for illustrating the next discussion. The performance of the method is measured by the Root Mean Square Error (RMSE) between these forecasted rain quantities and the observed ones. For the data set corresponding to the the studied meteorological events (more than 150 frames), the value of the RMSE is around 2 millimeters, which is accurate enough for operational use.

Another interesting output is the detection of threshold exceedence, when the cumulative rain quantity is above a given value that requires emergency measures. The percentage of a correct forecast at 1 hour of these threshold exceedence events (the forecast corresponds to a real event) is around 90%.

The last performance criteria concerns the detection of such events. The involvement of an object representation φ in the state vector has a positive or neutral impact on that criteria. In average, the percentage of detection at 1 hour is close to 60% with the object component and decreases to 50% without, as displayed in Fig. 8.

IV. CONCLUSION AND PERSPECTIVES

This paper describes an operational nowcasting system for flash floods associated to high rainfalls. The short horizon of nowcasting is obtained thanks to an image-based method applied on ground radar acquisitions. In a similar way than usual image-based systems, our approach relies on two main parts: one related to wind fields estimation and one to the forecast. The innovation comes from the estimation part that relies on a data assimilation technique, with an image evolution model including a description of the high rainfall regions. This object description allows a better monitoring of the rainfall displacement and results in an improved nowcast.

Two main perspectives will be studied. First, the whole system will be rewritten in polar coordinates, according to the acquisition sensor. This will suppress unnecessary preprocessing. Second, the nowcast system will be coupled with a Numerical Weather Prediction model (NWP), in order to slowly switch from the image-based nowcast to the NWP output and provide a forecast on the whole temporal horizon as done in Ridal et al. [START_REF] Ridal | Optimized advection of radar reflectivities[END_REF].

Fig. 1 :Fig. 2 :

 12 Fig. 1: Estimation and forecast.

Fig. 3 :

 3 Fig. 3: Left: threshold of the precipitation rate. Right: distance map to the boundary B of the rainy regions.

Fig. 4 :

 4 Fig. 4: Left: estimated distance map φ e3 . Right: distance map to the contours on the radar image A e3 .The estimated motion field is displayed on the left column of Fig.5and demonstrates the impact on φ. A same display

Fig. 6 :

 6 Fig. 6: Left: estimation obtained with φ on another time window. Right: without.

Fig. 7 :

 7 Fig. 7: Influence map of the circled pixel.

Fig. 8 :

 8 Fig. 8: Percentage of detection with and without φ.

ACKNOWLEDGEMENTS

This research has been partially funded by the DGA. The authors acknowledge the company Weather Measures for providing data from its radars and the company Numtech for the implementation of the method in an operational context.


P Sévillano
email: sevillano@celia.u-bordeaux1.fr

Patrick Georges

Frédéric Druon

D Descamps

E Cormier

Kerr-lens mode-locked Yb:CaGdAlO 4 oscillator optically pumped by a bright fiber laser

Keywords:

32-fs

Kerr-lens mode-locked Yb:CaGdAlO 4 oscillator optically pumped by a bright fiber laser

During the past decade, Yb-doped crystals have raised a great interest for the generation of high average power ultra-short pulses due to their excellent spectroscopic and thermal properties [1]. Among them, Yb:CaGdAlO 4 (Yb:CALGO) crystal is one of the most promising candidates for sub-50 fs oscillator thanks to its broad and flat emission band [2]. Recent works reported pulse generation as short as 40 fs in this material [3] which is very close to the shortest (35 fs) pulse duration ever obtained from an Yb doped bulk material based oscillator [4,5]. However, these ultra-short pulse oscillators generate average powers never exceeding a few tens of milliwatts. This power limitation results from the unavailability of bright powerful (diffraction limited) pump sources as well as the typical response time of saturable absorbers. The thin-disk technology provides an efficient alternative to significantly increase the average power [6]. Although very impressive powers are demonstrated, the shortest pulse duration reported in Yb:CALGO thin-disk oscillator is limited to 62 fs [7]. Improving the average power while maintaining the pulse duration as short as possible is a challenging goal we aim at achieving making use of pure Kerr-lens mode-locking (KLM) technique combined with high-brightness pumping. By this way, recent works on Yb:CaF 2 reported significant improvements in average power of bulk oscillators with the generation of sub 70 fs with several watts [8]. In this letter, we report on a pure Kerr-lens mode-locked Yb:CALGO oscillator delivering pulse duration of 40 fs at the watt level by means of a powerful single mode, polarized Yb-doped fiber pump emitting around 979.5 nm. After optimization of the output coupler transmission, the oscillator can provide the shortest pulse ever produced in Ybdoped bulk material with pulses as short as 32 fs. To the best of our knowledge, it is the first realization of a pure Kerr-lens mode-locking in Yb:CALGO crystal. For this we use a highbrightness optical pumping with a single-mode fiber laser source [9]. After describing the experimental setup, we present the performance of the KLM Yb:CALGO oscillator and discuss the optimization to provide the shortest pulse durations.

The experimental setup of the laser cavity and pumping geometry is sketched in figure 1. We use a 5-mm-long, 2 x 4 mm² Brewsterangle cut, Yb:CALGO crystal doped at 5 at. %. It is mounted in a water-cooled copper holder. We have chosen the c-cut orientation to avoid the anisotropy of temperature-dependent refractive index which could appear at high average pump power [10] and to eliminate the polarization state sensitivity in cross-section emission [2]. The Yb:CALGO crystal is positioned between two concave mirrors (M 1 , M 2 ,100-mm-radius-of-curvature) in a standard X-fold cavity configuration. To compensate astigmatism due to the Brewster-angle incidence, these mirrors are tilted with an angle of 8°. On one side, the cavity ends by a high reflection (HR) mirror and closed on the other side by a 35% output-coupler (OC). A pair of SF10 prisms separated by a distance of 450 mm is used for fine intra-cavity dispersion control. The net intracavity group delay dispersion (GDD) is calculated to be close to -2200 fs² per round trip. All the mirrors used are specified with low group delay dispersion. The cavity is asymmetric with a short arm of 600 mm and a long one of 900 mm. The Yb:CALGO crystal is longitudinally pumped by the highbrightness fiber laser through the M 1 dichroic mirror (HT for a wavelength below 980 nm and HR above 1020 nm). We use a commercially available diffraction limited polarized pump source (M²=1.1) emitting 10 Watts at 979.5 nm (from Azur Light System).

The pump radiation is focused into the Yb:CALGO crystal by a 60 mm focal-length lens. The high spatial quality of the pump beam allows us to obtain a spot radius at the focal plane of 25 µm (at 1/e²) leading to a confocal parameter of 3.7 mm and a maximum pump intensity of 1 MW/cm 2 . When the pump focus is positioned in the middle of the crystal, absorption saturates leading to an unabsorbed power of 4 W without laser effect. First of all, the Yb:CALGO oscillator is characterized in the CW regime. For 10 W of incident pump power, a maximum output power of 4.5 W is obtained at the central wavelength of 1044 nm corresponding to an optical to optical efficiency of 45%. In these conditions, the unabsorbed pump power is 2W. Since the pump source used in this setup is characterized by a low M 2 value, a very good spatial overlap between the laser beam and the gain channel is obtained over the 5 mm crystal length and leads to a high optical-tooptical efficiency. In order to discriminate between CW and ML regimes in the oscillator, the cavity is set at the edge of the stability domain by translating the M 2 mirror by 400 µm toward the crystal. At this point, the CW output power drops down to 0.9 W. Stable Kerr-lens mode-locking is initiated by vibrating a cavity mirror (starter). When locked, the Yb:CALGO oscillator delivers a pulse train at a repetition rate of 96 MHz with an average power of 1.1 W. Assuming a sech 2 pulse shape, an autocorrelation measurement gives a pulse duration of 94 fs (FWHM). The spectrum of the femtosecond pulses is characterized with an optical spectrum analyzer (ANDO AQ6315). It is centered at 1044 nm with 34 nm bandwidth (see grey curve in figure 2.a) corresponding to a Fourier-Transform limited duration of 40 fs (FWHM). The slightly chirped output pulses (mainly quadratic phase) are externally compressed with 10 bounces on -100 fs² chirped mirrors (total transmission of 98%) and further characterized with a SHG-FROG device. The retrieved temporal pulse profile (see figure 2.b)

indicates the generation of pulse as short as 40 fs (FWHM) at the average power of 1.08 W. The computed autocorrelation width (58 fs) is in very good agreement with the independently measured autocorrelation width of 58 fs. At the external compressor output, the time-bandwidth product of the pulses equals 0.373. In order to optimize the pulse duration, we investigate the influence of the output coupler transmission. The KLM stability criteria and the soliton formation to provide stable single-pulse regime [11,12] impose a rather constant intracavity peak power irrespective of the OC. We therefore adjust the pump power to fulfill this condition and achieve single-pulse mode-locking. The table 1 summarizes the parameters recorded for various OC transmission values. Increasing the OC transmission clearly allows to extract more average power by simply cranking up the pump power accordingly to maintain the intracavity peak power. However, the laser dynamics varies significantly with significant changes on the output pulse spectrum. Similar to other Yb-doped materials, the gain cross section is a function of both the wavelength and the population inversion in the crystal. In our case, we observe a red shift of the central wavelength for decreasing OC transmission. This observation is confirmed by our dynamical gain calculations based on absorption and emission cross-sections (from [13]) weighted by the population inversion and where we estimate that the cavity losses are mainly attributed to the OC. We estimated for a 35% OC transmission an inverted population ratio (β) of 10.7 % with a maximum gain cross-section centered at 1045 nm. At contrast, for a 3% OC transmission, we calculated a β of 3.9% with a gain cross-section maximum at 1063 nm. Additionally to the spectral shift, we noticed an enhancement of the spectral bandwidth when the OC transmission is reduced. The above gain model also agrees with the latter tendency. However, the increase of the spectral bandwidth cannot be explained solely by the gain cross-section evolution. In fact, it is known that the GDD of dichroic mirror is not well controlled in the spectral range where the coating reflectivity varies significantly. To evaluate this effect on the output spectrum, we replaced M 2 (sub-cavity mirror with a flat reflectivity and constant GDD) by the same dichroic concave mirror as M 1 thus highlighting the contribution of the net GDD. In figure 3, we compare spectra recorded with one and two dichroic mirrors for a 10% OC transmission. Also displayed is the net intracavity GDD for each case. It is clearly shown that the femtosecond pulse tends to red shift in order to overlap with a spectral region where GDD is as constant as possible. However, the addition of the second dichroic mirror deteriorates the intracavity GDD flatness and leads to a reduction of the spectral bandwidth from 46 nm (1 dichroic mirror) to 40 nm (2 dichroic mirrors). Moreover, in the case of 2 dichroic mirrors, a spectral peak raises around 1 µm. This phenomenon has already been studied by J. Hermann and al. [14], already observed in Yb:CALGO [15], and originates from uncompensated group delay dispersion. This structure is also observed with a single dichroic mirror however at a much lower level. It is therefore the combined contribution of population inversion and uncontrolled GDD of dichroic mirror coating that explains the reduction of the spectral bandwidth with increasing OC. Further decreasing the pulse duration with higher OC to improve average power would require designing custom dichroic mirror with accurate GDD control.

From the above analysis, it is not surprising to achieve the shortest pulse duration with a 3% OC. The spectrum is centered at 1063 nm with a spectral bandwidth of 51 nm and an average power of 90 mW with only 3.6 W of incident pump power. The spectrum displayed in figure 4.a is very smooth and exhibits a Gaussian shape. A tiny peak is visible at 1000 nm attributed to the remaining dichroic mirror. the experimental spectrum (grey curve) and spectral phase (dashed red curve). b) Retrieved intensity profile (black curve) and temporal phase (dashed red curve) of the output pulses.

After external compression (-1000 fs²), the pulse is retrieved from a SHG-FROG measurement and leads to a Fourier transform limited pulse duration of 32 fs and a time bandwidth product of 0.433 (see figure 4.b). Experimental and retrieved spectra are superimposed in figure 4.a with an excellent agreement.

In addition further characterizations (beam profile and pulse train stability) have been performed. We measured an output beam quality characterized by a factor M² x .M² y of 1.15x1.2 with no evidence of astigmatism. The mode-locking regime is stable during several hours. In the aim of decreasing the oscillator footprint, reducing the intracavity losses and improving its robustness, we discuss here the potential of using intracavity chirped mirror for ultra-short pulse generation. Chirped mirrors (-100 fs² and -250 fs² per bounce from Layertech) are inserted in each arm of the cavity and SF10 prisms are replaced by silica prisms separated by 120 mm to provide a fine intracavity GDD control. For 10 W of pump power, the oscillator delivers an average power of 1.5 W and pulses with a spectral bandwidth of 34 nm as shown in figure 6 (black curve).A pulse duration of 37 fs is obtained after SHG-FROG measurement. coupler of 40 %. In this configuration, the oscillator provides an average power of 1 W and pulse duration of 43 fs after external compression with a spectrum of 28 nm (see figure 6 (grey curve)).

In the compact configuration with intracavity chirped mirrors, we obtained results very closed to the configuration with SF10 prisms and large OC values. We believe that shorter pulses could be obtained with custom broadband chirped mirrors where the GDD is controlled over the complete spectral bandwidth of Yb:CALGO.

In conclusion, we have presented the first pure Kerr-lens modelocked Yb:CALGO oscillator. Based on a powerful diffraction limited fiber pump laser, we have built a pure KLM oscillator at 96 MHz delivering sub-40 fs pulses at the watt level. The lack of optics with constant GDD over large spectral ranges forces us to use SF10 prisms only. In this configuration, we have investigated different OC to shorten the pulse duration. With an output coupler value of 3% , our oscillator produced pulses with a duration of 32 fs and 90 mW of output power after external recompression. Furthermore, these pulses exhibit a structure-free spectrum with a bandwidth of 51 nm centered at 1063 nm. It is, to our best knowledge, the shortest pulse duration ever reported for Yb-doped bulk material. This result is the combination of the excellent spectral properties of Yb:CALGO and the use of a high brightness pumping scheme allowing pure KLM operation. By designing specific ultra-large band GDD controlled chirped mirrors, the same characteristics could be attained in a prism free configuration. Similarly, custom designed dichroic mirror should produce ultimate pulse duration together with high average power. We acknowledge the support from the European Community through the I3 LASERLAB III JRA Eurolite, the Conseil Régional d'Aquitaine with the "Lougre Brillant" program, Aquitaine Valorisation, the CNRS national networks FEMTO and CMDO+ through the entitled HABRY program, and the Agence Nationale de la Recherche (ANR) through the PAMPERO (ANR-12-B509-0033) projects

Fig. 1 .

 1 Fig. 1. KLM Yb:CALGO oscillator experimental setup.

Fig. 2 .

 2 Fig.2. a) Retrieved SHG-FROG spectrum (black curve) superimposed with the experimental spectrum (grey curve) and spectral phase (dashed red curve). b) Retrieved intensity profile (black curve) and temporal phase (dashed red curve) of the output pulses.

Fig. 3 .

 3 Fig.3. Experimental output spectra obtained with 10% OC transmission and calculated intracavity GDD for one and two dichroic mirrors configuration

Fig. 4 .

 4 Fig. 4. a) Retrieved SHG-FROG spectrum (black curve) superimposed with

Figure 5 Fig. 5 .

 55 Fig. 5. Radio-frequency spectrum.

Fig. 6 .

 6 Fig.6. Experimental spectra for OC =30 % (black curve) and OC =40 % (grey curve), calculated intracavity GDD (dashed red curve) Sharp spectral structures appear at 1 µm and 1.1 µm caused by additional uncompensated high-order dispersion introduced by the chirped and dichroic mirrors (red curve). The structures are dissipated by reducing the intra-cavity peak power with an output

Table 1 : Spectral and temporal characterization as a function of OC transmission

 1

	OC	Ppump	Pout	λc	Δλ	Δτ
	[%]	[W]	[mW]	[nm]	[nm]	[fs]
	3	3,6	90	1063	51	32
	10	4,3	280	1056	46	35
	20	6,3	640	1052	41	37
	35	8,9	1100	1045	34	40