Emmanuel Mimoun

Luigi De Sarlo

Jean-Jacques Zondy

Jean Dalibard

Fabrice Gerbier

Solid-state laser system for laser cooling of Sodium

Keywords: De -42, 65, Ky

We demonstrate a frequency-stabilized, allsolid laser source at 589 nm with up to 800 mW output power. The laser relies on sum-frequency generation from two laser sources at 1064 nm and 1319 nm through a PPKTP crystal in a doubly-resonant cavity. We obtain conversion efficiency as high as 2 W/W 2 after careful optimization of the cavity parameters. The output wavelength is tunable over 60 GHz, which is sufficient to lock on the Sodium D 2 line. The robustness, beam quality, spectral narrowness and tunability of our source make it an alternative to dye lasers for atomic physics experiments with Sodium atoms. Key words 37.10

Introduction

Among the many atomic species that can be brought to quantum degeneracy, Sodium benefits from low inelastic losses and a relatively large elastic cross-section, allowing the production of large (> 10 8 atoms) Bose-Einstein condensates [1,2,3,4,5,6,7], or degenerate Fermi clouds by thermalization with the Sodium gas [8]. A current drawback of using Sodium is the necessity of using dye lasers to reach the resonant wavelength of 589.158 nm (Sodium D 2 transition). Although technically well-mastered, dye laser systems are expensive, hardly transportable and comparatively difficult to maintain and operate, justifying the need for alternatives as solid-state lasers.

In addition, new laser sources in the yellow spectral region find applications outside the domain of laser cooling. In fact, the generation of Sodium resonant radiation has been mainly driven by the astronomy community, with the development of high-power 589 nm lasers to create artificial "beacon" stars by exciting the mesospheric sodium layer [9,10,11,12,13,14,15,16,17]. Other possible applications for lasers in the yellow spectral region include Laser-induced detection in the atmospheric range (LIDAR) [18], eye surgery or dermatology [13].

In the literature, several methods for generating continuouswave (cw) laser light around 589 nm have been reported, including sum-frequency mixing of two infrared lasers around 1319 nm and 1064 nm [9,10,11,12,13,19], frequencydoubling of a Raman fiber laser [14,15,16], or sum-frequency mixing of two fiber lasers around 938 nm and 1535 nm [17]. Applications to laser cooling typically require powers of several hundred mW to 1 W, the possibility to tune the laser to the Sodium resonance, and a linewidth much smaller than the Γ = 2π × 9.8 MHz natural linewidth of the cooling line.

In a recent paper, we have reported on the realization of a laser source suitable for laser cooling of Sodium [20]. In the present article, we present an exhaustive account of our experimental approach. Our laser source is based on sum frequency generation (SFG) from 1064 nm and 1319 nm lasers. SFG is a second order non-linear optical process, in which two pump beams with frequencies ω 1 (λ 1 = 1064 nm) and ω 2 (λ 2 = 1319 nm) produce a signal beam with frequency ω 3 = ω 1 + ω 2 . We implement this sum frequency technique using commercial, solidstate infrared lasers. The 1064 nm and 1319 nm sources are monolithic solid-state lasers built upon an Yttrium Aluminium Garnet (YAG) Non-Planar Ring Oscillator (NPRO) crystal. We also tested another configuration in which the 1064 nm laser is replaced by an external cavity laser diode boosted by a single-mode fiber amplifier. This led to poorer performances attributed to misbehaviour of the amplifier, and this configuration was not pursued further in our work. The non-linear medium used is a periodically poled potassium titanyl phosphate crystal (pp-KTiOPO 4 or PPKTP), with a poling period chosen to achieve first-order quasi-phase matching (QPM) near room temperature [21,[START_REF] Boyd | Nonlinear optics[END_REF]. In single-pass configuration, the conversion efficiency is still too small to reach the output power required for laser cooling. To circumvent this problem, the crystal is enclosed in a doubly-resonant build-up cavity to enhance the conversion efficiency. Doing so, we reach an overall power conversion efficiency α cav ≈ 2 W/W 2 , where the conversion efficiency is defined through P 3 = αP 1 P 2 , with P 1,2,3 the power at each wavelength. This is to be compared to the value for single-pass conversion efficiency, α sp ≈ 0.022 W/W 2 . In terms of photon fluxes, about 92% of the photons of the weakest source which enter the cavity are converted [20].

The article is organized as follows. Section 2 gives an overview of our experimental setup. Section 3 recalls the main features of the process of sum frequency generation, and presents our results in a simple single-pass configuration. Section 4 discusses the experimental realization of a doubly resonant cavity and its optimization to achieve near-unit conversion efficiency. Section 5 characterizes the main properties of the laser source obtained at 589 nm.

Experimental setup

Our experimental setup is represented in Fig. 1. The pump laser sources at 1064 nm and 1319 nm are nonplanar ring oscillator YAG lasers (Innolight GmbH, Germany) with an instantaneous spectral linewidth narrower than 10 kHz and output power of 1.1 W and 500 mW, respectively. Both lasers are collimated to a 1/e 2 radius around 1 mm and combined on a dichroic mirror. After passing through an electro-optical phase modulator (EOM) operating near f mod ≈ 1 MHz, the beams are focused to match their spatial profiles with the fundamental spatial mode of the resonant cavity.

The cavity is built in a bow-tie planar configuration, with highly reflecting mirrors M 2 , M 3 , M 4 and an input coupler M 1 with lower reflectivities chosen such as to optimize the intracavity conversion (see section 4 below). Mirrors M 1 , M 2 are plane, while M 3 , M 4 are concave with radius of curvature R c = 100 mm. At the crystal location, the 1064 nm (1319 nm) beam is focused to a waist w 1 = 45 µm (w 2 = 47 µm). This corresponds to almost equal Rayleigh lengths in the crystal z R,i = π n i w 2 i /λ i ≃ 10 mm, with the refractive indices n 1 ≃ 1.83 and n 2 ≃ 1.82 for PPKTP. For our configuration, this choice offers a good trade-off between increasing the nonlinear conversion efficiency and avoiding detrimental effects such as thermal lensing. The cavity geometry is chosen to avoid transverse mode degeneracies, allowing to excite the fundamental Gaussian mode only and suppress higher order modes.

The PPKTP crystal used for SF mixing was manufactured at the Royal Institute of Technology of Stockholm (KTH). Its length is L = 20 mm, with a poling period Λ = 12.36 µm. The use of a periodically poled crystal allows to reach QPM conditions only with temperature tuning (see section 3.1). The crystal is mounted in a copper case with a Peltier thermo-electric cooler element. The case temperature is controlled by a standard Proportional-Integral-Derivative regulator with better than 10 mK stability. Using the phase-matching curve calculated in section 3.1, we estimate that this corresponds to output power drifts less than 1%.

Single-pass measurements

In this section, we discuss first our measurements in a single-pass configuration, i.e. without enhancement cavity. As shown later, this measurement is critical to optimize the resonant cavity parameters to reach maximal conversion efficiency. We first recall for completeness the theoretical results relevant to such measurements, first in the simple case where the pumps are plane waves and then in the more realistic situation where they are Gaussian beams. We then discuss our measurements -from which we derive a nonlinear coefficient d 33 ≃ 16pm/V for PPKTP.

Plane wave model

In this section, we recall the basic features of SFG using a simplified theoretical framework [START_REF] Boyd | Nonlinear optics[END_REF]. The starting point to describe the propagation in the nonlinear crystal are Helmholtz equations, including nonlinear polarization terms. We introduce the complex amplitudes a i , related to the electric field strengths by

E i = 2Z 0 ω i n i f i (ρ)a i (z)e i(kiz-ωit)
and to the powers by

P i = ω i |a i | 2 .
Here k i is the momentum of a photon with frequency ω i in a medium with index of refraction n i , z is the direction of propagation of light, ρ is the transverse coordinate, f i denotes the area-normalized transverse mode for each beam, and

Z 0 = µ 0 /ǫ 0 is the impedance of vacuum.
As a first approximation, we neglect the spatial profile of the laser beams and set f i (ρ) = S -1/2 , with a cross-section S identical for all beams. The Helmholtz equation for the harmonic wave a 3 then reads [START_REF] Boyd | Nonlinear optics[END_REF]

da 3 dz = -iγa 1 a 2 e -i ∆k z , (1)
where the non-linear coupling coefficient γ can be written as

γ = 2 ω 1 ω 2 ω 3 Z 3 0 ǫ 2 0 d 2 Sn 1 n 2 n 3 1/2 . ( 2
)
Here d denotes the nonlinear coefficient which characterizes the efficiency of the nonlinear process crystal, d can be treated as constant over the crystal length (neglecting possible defects and impurities). In contrast, a periodically poled crystal is characterized by an alternating permanent ferromagnetization [21]. As a result, d is a periodic function of the position z in the poling direction, with spatial period Λ. As such, it can be expanded as a Fourier series, d(z) = d 33 n c n e iqnz , where q n = 2πn/Λ. Significant conversion only takes place when the QPM condition q n = ∆k occurs for some integer n. Here we only consider the first term of the series, n = 1 (first order QPM). For a 50% poling dutycycle, the Fourier coefficient c 1 is 2/π and the nonlinear coefficient d in Eq.( 2) becomes an effective coefficient d pp = 2/π d 33 . Hence, the maximum efficiency is lower than for a bulk crystal with perfect phase matching by a factor 4/π 2 ≈ 0.4.

Assuming low conversion, we solve for a 3 in the undepleted pumps approximation, a i (z) ≈ a i (0), for i = 1, 2. For a crystal of length L, this gives the generated power at 589 nm as

P 3 = α sp P 1 P 2 sinc 2 ∆k -2π Λ L 2 , (3)
where sinc(x) = sin(x)/x and where the maximal singlepass conversion efficiency α sp is

α sp = γ 2 L 2 ω 3 ω 1 ω 2 . ( 4
)
As we will see, the single-pass efficiency α sp is the critical parameter to allow optimization of the resonant cavity. The argument of the sinc function in Eq.( 3) depends on temperature through the various refractive indices (the case of KTP has been studied experimentally in [START_REF] Fradkin | [END_REF]24]). Therefore, by adjusting the temperature one can reach the quasi-phase matching condition ∆k = 2π/Λ which maximizes the conversion efficiency. In practice, the period Λ is chosen so that this condition is fulfilled near room temperature.

Boyd-Kleinmann theory

Instead of collimated beams, experiments use focused Gaussian beams in order to reach high intensities, and hence efficient conversion. Non-linear processes with Gaussian waves were studied in a seminal paper by Boyd and Kleinmann [25], where explicit expressions were given for the conversion efficiency in the undepleted pumps approximation (see also [26]). The general expressions are rather complex, but they can be drastically simplified by assuming identical Rayleigh lengths z R for the three beams. Indeed, both infrared beams are resonant in the cavity, which implies that their confocal parameter is the same, essentially determined by the geometry of the cavity. Although the output beam generated by SFG is not resonant, it is generated only in the regions where both pump beams overlap significantly so that approximating its spatial mode by a Gaussian beam with the same confocal parameter as the infrared ones is a reasonable assumption [START_REF] Boyd | Nonlinear optics[END_REF]. With Gaussian beams, the coefficient γ defined in Eq.( 2) becomes a function of z proportional to the overlap integral

I(z) = d (2) ρf 1 f 2 f *
3 between the different modes f i . For Gaussian waves with waists w i at the crystal center, and Rayleigh length z R = π n i w 2 i /λ i , this can be calculated explicitly. After some rearrangement, the expression for the output power can be writtten as P 3 = α sp P 1 P 2 , where the single-pass conversion efficiency α sp reads

α sp = Z 1 d 2 pp L λ 3 3 h (a, b, c) . (5)
Here

Z 1 = 32πZ 0 λ 1 λ 2 n1 λ1 + n2 λ2 + n3 λ3 2 ≈ 2.15 kΩ (6)
has the dimension of an impedance and the dimension-

less function h h (a, b, c) = 1 4a a -a e ibτ (1 + iτ )(1 + icτ ) dτ 2 (7)
is the so-called Boyd-Kleinman factor. The latter depends on the reduced variables

a = L 2zR , b = ∆k -2π Λ z R , and c = ∆kw 2 eff /z R , with w -2 eff = (π/z R ) i n i /λ i . We can further write c = ∆k w 2 eff zR = b + 1 a πL Λ × w 2 eff zRL × a,
showing that the function h depends only on the variables a and b once the wavelengths, crystal length and crystal period are fixed2 . Since

w 2 eff zRL ∼ λ3
2πn3L ≪ 1, c = 0 can be assumed, and the integral h (a, b, c) is well approximated by

h (a, b, 0) = 1 4a a -a e ibτ 1 + iτ dτ 2 . ( 8
)
There are two limiting cases of interest :

1. Collimated beams, z R ≫ L or a ≪ 1: in this case we find the sinc function familiar from the plane wave case (see Eq.( 3)),

h (a, b, 0) ≈ a sinc 2 (b + 1)a . (9)
2. Focused beams, z R ≪ L or a ≫ 1: for tightly focused beams, the length L of the crystal naturally drops out of the problem. One finds that h tends to a limit function

h (a ≫ 1, b, 0) ≈ π 2 a e -2b , b > 0, 0, b < 0.
The experimental procedure of changing the temperature (which changes ∆k) corresponds to searching for the maximum h * (a) of h(a, b, 0) as a function of b for a fixed a [25]. The optimum phase mismatch is offset from the plane wave result (∆k = 2π/Λ) by a quantity on the order of z -1 R , a consequence of the Gouy phase accumulated as the beams pass through a focus in the crystal. The optimal focusing corresponds to the maximum of h * , which is found for a * = L/2z R ≈ 2.84 ( h * (2.84) ≈ 1.06). This optimum is quite loose, as h * > 1 for 1.5 a 5.

Results for single pass operation

Our experimental configuration corresponds to a configuration where z R ≈ 10 mm, or a ≈ 2 for a 20 mm long crystal. For this parameter, the shape of the function h is very close to the sinc function predicted by Figure 2 Power produced at 589 nm in a single-pass configuration when the temperature of the PPKTP crystal is varied. The plot is fitted for a value of the poling period Λ = 12.32 µm, for the Plane Wave Approximation (solid line, PWA) and the Boyd Kleinmann theory (dashed line, BK). The peak value measured for the power produced at the output of the cavity is 7 mW. Considering the losses due the optics used to separate yellow light from infrared, we infer a total produced power of 8.5 mW.

the plane wave model (see Fig. 2). To perform singlepass measurements, we use the same setup as in Fig. 1 but remove the mirror M 4 from the cavity. We find that the optimum temperature T (mes) QPM ≈ 50 • C for our crystal samples. For this temperature, the refractive indices are n 1 ≈ 1.83, n 2 ≈ 1.82, and n 3 ≈ 1.87. Using BK theory to fit the data, we measure an efficiency α mes sp ≈ 0.022 W -1 (P 3 = 9 mW), obtained for normal incidence on the crystal using P 2 = 440 mW and P 1 = 940 mW3 . The variation of the output power with the power of both infrared lasers are found to be linear, confirming the validity of the undepleted pumps approximation for a single pass operation. Boyd-Kleinmann theory predicts a value α BK sp ≈ 0.021 W/W 2 for d 33 ≈ 16 pm/V, which is in a good agreement with values for the non-linear coefficient found in the literature [27,28,29]. Note also that this is quite close to the optimal value α * ≈ 0.023 W/W 2 which would be obtained for slightly tighter focusing. Applying the equations (3-4) obtained in the plane wave approximation, one would expect α ≈ 0.022, taking for the cross section S the average of the waist of a gaussian beam over the length L of the crystal :

S = 1 L L/2
-L/2 πw 2 (z)/2 dz, with w(0) = 45 µm. This highlights the usefulness of these theories for quantitative predictions.

The PPKTP crystals used in our experiments was ordered with a poling period Λ = 12.36 µm. Using the val-ues given in Refs. [START_REF] Fradkin | [END_REF]24] for the temperature and wavelength dependence of the refractive indices, we calculate a QPM temperature T (calc) QPM ≈ 28 • C, apparently far from the measured T (mes) QPM ≈ 50 • C (see Fig. 2). We note however that the quasi-phase-matching temperature is rather sensitive to the exact value of the period. Using the same wavelength and temperature dependence for the refraction indices, we find that the measured T (mes) QPM corresponds to a spatial period Λ = 12.32 µm.

Intra-cavity conversion

Definition of the optimization problem

After having characterized the single-pass sum-frequency process, we turn to the cavity setup. The presence of the cavity enhances the infrared lasers intensities at the crystal location. For the geometry shown in Fig. 1, the intracavity power at resonance can be written for each infrared laser as [START_REF] Siegman | Lasers[END_REF]

P cav i = T i 1 -R i • (1 -δ i ) • C i 2 P inc i , i = 1, 2. (10)
In Eq.( 10), P cav i denotes the circulating intracavity power, P inc i the incident power coupled into the fundamental mode of the cavity, R i , T i denote the input coupler (mirror M 1 ) reflectance and transmittance (R i + T i + L i = 1, with L i a loss coefficient), δ i denote the passive losses after one round trip, excluding the input coupler (i.e. finite reflectances of the other mirrors, and losses in the crystal), and C i accounts for the nonlinear conversion. To evaluate C i , we use conservation laws for the photon fluxes which state that |a 1 | 2 + |a 3 | 2 and |a 2 | 2 + |a 3 | 2 are constant along the crystal length (in the absence of absorption). This corresponds to non-linear conversion factors given by

C i = 1 - λ 3 P 3 λ j P cav j , (j = i; i, j = 1, 2). ( 11
)
Assuming total transmission of the yellow light by the output mirror M4, the 589 nm power P 3 coupled out of the cavity is given by P 3 ≈ αP cav 1 P cav 2 , under the undepleted pump approximation. When both pumps have imbalanced powers, the output power at ω 3 is ultimately limited by the weakest one, since one photon from both pumps is required to create one at ω 3 . As it is the case in our experiment, we assume that the weakest source is the one at wavelength λ 2 . This translates into a maximum power P max 3 = (λ 2 /λ 3 )P inc 2 . Therefore, a figure of merit to characterize the conversion efficiency is the ratio

η = P 3 P max 3 = λ 2 λ 3 P 3 P inc 2 ( 12
)
between the actual power and the absolute maximum power that can be obtained from the available pump power coupled into the cavity P inc 2 . The problem at hand is thus to maximize η for given cavity parameters δ i , L i , α. This amounts to balancing the input coupler reflectances R 1 , R 2 with the total loss per round trip, including the non-linear conversion. This is usually termed impedance matching [START_REF] Kaneda | [END_REF]. In our case, finding the impedance matching point is a coupled problem, since one should maximize simultaneously both intensities in the cavity using (10) and (11). This last equation is critically dependent on the single-pass conversion coefficient α.

Total conversion in an idealized lossless cavity

Let us first study the case, where passive losses in the cavity and on the input coupler can be neglected (δ i , L i = 0 in the above equations). The question to be answered is whether it is possible to convert all photons at λ 2 into photons at the harmonic at λ 3 (cavity conversion efficiency η = 1). In [20], we showed that this is indeed the case for any value of the input coupler reflectance R 2 . We recall here the argument for completeness. We look for a solution where the output flux at λ 3 and the incident flux at λ 2 are equal, P 3 /( ω 3 ) = P inc 2 /( ω 2 ). According to Eq. (11), this corresponds to C 2 = 1 -

P inc 2 P cav 2 .
The cavity equation (10) becomes

P cav 2 = (1 -R 2 )P inc 2 1 -R 2 1 - P inc 2 P cav 2 2 . ( 13
)
This solves into the simple result

P cav 2 = P inc 2 1 -R 2 , (14)
valid for any R 2 . Thus we conclude that there is always a possibility to reach complete conversion in the ideal, lossless case, corresponding to the intracavity flux for the weak pump 2 as given above. The flux for the strong pump 1 is found from the relation P 3 = αP cav 1 P cav 2 ,

P cav 1 = (1 -R 2 )λ 2 α sp λ 3 . ( 15
)
The parameters of the cavity (R 1 , R 2 ) are linked via Eq. (10). For any R 2 , one can find a value of R 1 leading to the power P cav 1 given above, corresponding to complete conversion of the λ 2 photons.

Optimization of conversion for a realistic lossy cavity

In any practical situation, passive losses will be present. This modifies the conclusions of the last subsection, as these losses limit the enhancement factor that can be reached in the cavity. Unlike the lossless case, instead of a locus of optimal points in the (R 1 ,R 2 ) plane, one finds a unique value of (R 1 ,R 2 ) that maximizes P 3 , at a value smaller than P max 3 . However, this optimum is quite loose when non-linear conversion dominates over the passive losses (C i ≫ δ i ). This highlights the importance of a large single-pass efficiency, justifying the use of a highly nonlinear material such as PPKTP : the required power P 1 is reduced (see Eq.( 15)), making the cavity more tolerant to passive losses.

We rewrite C 2 = 1 -η

P inc 2 P cav 2
. The cavity equation for wave 2 then leads to two solutions for the intracavity power

P cav 2 = (1 + r)T 2 -(1 -r)rη (1 -r) 2 ± 2 T 2 r(T 2 -(1 -r)η) (1 -r) 2 ,
with r = R 2 (1 -δ 2 ) the total passive loss coefficient for the circulating waves. The existence of two solutions indicates a possible bistability. Such solutions are real provided

T 2 ≥ (1 -r)η. (16)
When this condition is not fulfilled, the cavity is unstable due to excessive passive or nonlinear losses. This condition sets a limit on the efficiency achievable for given cavity parameters T 2 , R 2 , δ 2 , η < η max = T 2 /(1 -r).

Assuming one chooses the input coupler to reach this maximum value, one finds for small losses (δ 2 , L 2 ≪ R 2 ) an intracavity power

P cav 2 ≈ P inc 2 1 -R 2 1 - L 2 + 2R 2 δ 2 1 -R 2 , (17)
close to the idealized case studied before. The solution of the coupled equations giving P cav 1 , P cav 2 [Eq. (10)] is performed numerically 4 , using as input the 4 A numerical algorithm maximizing the two dimensional function P cav 1 P cav 2 = f (R1, R2) was implemented. In practice, P cav 1 is first evaluated using an Euler secant method by substituting P cav 2 in C1 [Eq. (11)] by its expression given by Eq.( 10). Once P cav 1 solved, its value is used to derive P cav 2 available power in our infrared sources and the measured characteristics of the cavity. We carried out the optimization with respect to the input coupler transmittances at both infrared wavelengths [20].

Experimentally, we characterized carefully the transmition and reflection coefficients of the mirrors used for the cavity (see Table 1). The measured reflectances correspond to a passive (i.e. without non-linear conversion) amplification of the intra-cavity power by a factor around 22 at 1064 nm and 12 at 1319 nm. To find the powers coupled into the cavity fundamental mode, we sent the lasers independently into the cavity with a known incident power. Comparing the measured intracavity power (inferred from the power transmitted through M2 and the measured value of its transmittance) with the one expected from the reflectances gives the fraction of incident power effectively coupled to the fundamental mode, around 85% for both wavelengths. The measured reflectances as well as coupling efficiencies were taken into account in our numerical simulations, predicting a conversion efficiency η ≃ 0.9 for the photons at λ 2 coupled into the cavity (see [20]). The maximal measured output power of 800 mW, which corresponds to η = 0.92, is in fair agreement with this result.

Cavity setup and locking

An essential requirement to achieve a stable output with high efficiency is to ensure that both pump lasers are simultaneously resonant in the cavity. In order to maintain the cavity on resonance for both wavelengths, a double locking scheme using the stable 1319 nm source as a master laser is implemented (see Fig. 3a). Both lasers are routed together through an electro-optical modulator (EOM) placed before the cavity, and resonantly driven at a frequency f mod = 1 MHz. This dithers the laser frequencies and generate a dispersive signal from the cavity transmission. In our implementation, the weaker pump laser 2 is used as a master laser onto which the cavity length is locked using an integrating servo-loop. In a second step, the stronger pump 1 laser is locked onto the cavity, and consequently on the master laser, ensuring stable operation of the ensemble. In details, the small fraction of infrared light transmitted by the second mirror M 2 is collected by two separate photodiodes (see Fig. 1). Two piezoelectric transducers glued to the cavity mirrors M 2 and M 3 are used to tune the cavity length. The first one (M 3 ) allows fast response in the 30 kHz range, but has a limited travel of a few tens of nm. The second piezoelectric stack allows to correct for larger drifts of the cavity length, occuring over much longer timescales (from a few ms to a few hours). The and the function f . The optimal couple (R opt 1 ,R opt 2 ) is then tracked using an adaptative stepsize algorithm maximizing f . photodiode signal at 1319 nm (Ph.2) is demodulated by a lock-in amplifier operating at the modulation frequency f mod driving the EOM, producing a dispersive error signal subsequently fed back to both piezoelectric transducers (with appropriate filters in the feedback loop). This locks the length of the cavity on the stable 1319 nm source. The photodiode signal at 589 nm (Ph.3) is demodulated in a similar way, and the resulting signal can be used to react on laser 1 frequency using available piezoelectric and temperature control to ensure that it follows the cavity resonance.

EOM

In situations where the conversion efficiency is large, this standard locking scheme leads to serious stability problems with both IR lasers simultaneously present in the cavity. To see this, picture a situation where the cavity is on lock with the 1064 laser off-resonant. As the 1064 nm laser frequency is tuned to reach resonance, the power level of the 1319 nm drops due to conversion into 589 nm photons (see Fig. 3b). This large drop of the 1319 nm power level when both lasers resonate cannot be distinguished from a perturbation by the cavity lock. Hence, the cavity lock actually works against keeping both lasers on resonance simultaneously, and resists increasing the conversion efficiency above the level where the 1319 nm lineshape is distorted significantly. We have devised a simple solution to this problem [20,32]. First, instead of the bare 1319 nm photodiode transmission, the error signal for the cavity lock is derived from a linear combination of this transmission signal and of the yellow output of the laser. The combination is done electronically before the lock-in amplifier, with weights empirically chosen to minimize distortions of the cavity lineshape and to optimize the servo gain around the lock point. Our "fringe reshaping" method works for any level of conversion, and allows stable operation of the laser on a day time scale, even at the highest efficiencies. Second, choosing the 589 nm output as the error signal for the second servo-loop instead of the 1064 nm transmission ensures that the system locks to the maximal converted power. Our method relies on the fact that the SFG is a phase-coherent process : a modulation sideband present on the 1319 nm laser is automatically present on the output (with a different weight that depends on the 1064 nm power). Synchronous demodulation by the lock-in amplifier therefore preserves the linear combination.

When these two servos are in action, the cavity is doubly resonant, and the two lasers are frequency locked to each other. We have found that the lock of the second laser was somewhat sensitive to disturbances occurring near the optical table. This is a well-known features of integrating servo-loops, which typically encounter difficulties to recover from disturbances with large amplitude that cause the integrator to saturate [33]. Integrators are required to achieve zero DC errors in a servo loop, and replacing them with a simpler proportional control is not an option. We have implemented an electronic circuit that bypasses this problem and prevents the integrator from saturating after violent perturbations, while maintaining the laser locked at all times. This can be seen as an automatic gain control circuit that limits the DC gain when the input becomes too large. The circuit, shown schematically in Fig. 3c , uses the laser power level to detect such disturbances, and compares it to a preset threshold value (set to 80% of the nominal value in our case). A sample and hold amplifier (SH) samples the integrator voltage, with its output connected ("bootstrapped") to the integrator input through a differential amplifier. Regular operations correspond to the SH in "sample" mode, where the output closely tracks the input. The output V diff of the differential amplifier is zero and the integrator behaves normally. When the output of the laser falls below the threshold, the comparator triggers the SH circuit to switch to "hold" mode. The SH output is frozen at the value V th it had at threshold, so that the differential amplifier output becomes V diff = V out -V th . The output voltage at frequency ω then becomes

V out = - R2 R1 V in -V th 1 + iR 2 Cω , (18)
where V in is the incoming error signal and V out the output of circuit. The integrator is thus neutralized before saturating, and the circuit behaves as a proportional controller around the threshold value. When the perturbation is removed, the SH turns back to sample mode and restores regular integrator operation. We use this circuit on all servo controllers in the laser system. With this last improvement, the system can withstand severe mechanical perturbations without unlocking and requires very little maintenance compared to dye lasers. It is mostly insensitive to temperature fluctuations because the infrared lasers are thermally stabilized and the cavity length fluctuations are compensated by the servo. Alignement is left untouched over weeks, with a power drop below 10%, and when needed adjustments are only required on the injection path into the cavity. The cavity alignment itself has not been touched for six months. The laser stays locked for a day on its own, and for several hours when tracking an atomic line.

Finally, the frequency drift of the yellow laser is cancelled by locking laser 2 to the D 2 line of Na using standard saturated absorption spectroscopy, yielding a longterm frequency-stabilized laser source.

Yellow laser characterization

Beam quality

We have characterized the spatial mode of the laser emerging from the cavity. The output beam was focused through a converging lens and imaged on a charge-coupled device (CCD) camera at various distances from the lens. The beam profile for each distance was fitted to a gaussian with 1/e 2 radius w identified as the beam waist (see Fig. 4). We fitted this function to w 0 1 + θ(z/w 0 ) 2 , where w 0 is the waist of the beam near focus, θ is its divergence, and z the direction of propagation. This gives a M 2 parameter M 2 = πw 0 θ/λ = 1.02, indicating diffractionlimited performances. This shows the high-quality of the transverse mode of the output beam. Measurements in both transverse directions show no visible astigmatism. Intensity noise of the output was characterized by recording the beam power on a fast photodiode (bandwidth 10 MHz) followed by a 16 bits analog acquisition card (National Instruments NI-6259). From such samples, we determined the one-sided power spectral density S RIN of the instantaneous intensity normalized to the mean intensity,

Intensity noise measurements

S RIN (ν) = 1 T T 0 I(τ ) I e i2πντ dτ 2 , (19)
where • • • denotes statistical averaging and where T ≈ 100 ms is the measurement time. The results averaged over 100 samples are shown in Fig. 5. This corresponds to a noise δI/ I ≈ 4×10 -3 integrated over a 5 Hz-500 kHz bandwidth. Two broad noise peaks are visible near 190 kHz and 330 kHz, which probably reflect resonances in the cavity piezoelectric actuators. The noise level is sufficient for our application, but could be controlled actively to a lower level if needed, for instance by monitoring the instantaneous power and reacting on the incident power from the 1064 nm laser. Because the two pump sources are extremely narrow in frequency, one can expect similar spectral purity of the output. At present times, we have no means to measure such narrow linewidths, but we can place an upper bound on the laser linewidth from high-resolution spectroscopy measurements.

To this aim, we used cold atoms from a magnetooptical trap (MOT) formed using the SFG laser source. Sodium atoms were introduced in a high-vacuum cell using electrically-controlled dispenser sources (Alvatec GmbH). Repumping light was derived from the main laser using a high-frequency (1.7 GHz) acousto-optical modulator (Brimrose Corp.). A MOT was formed in the vacuum cell using a magnetic field gradient around 10 G/cm and approximately 10 mW optical power in each of the six MOT beams. The cloud typically contained a few 10 7 atoms, at a temperature T ∼ 110 µK. We measured the absorption of a weak probe beam (intensity ∼ 1 mW/cm 2 ) by the atomic cloud (with MOT beams turned off) as a function of the probe frequency. According to Beer-Lambert's law, this measures σ(δ), the optical density at a detuning δ = ω L -ω 0 , with ω L the laser frequency and ω 0 the atomic resonance frequency. Typical results are plotted in Fig. 6. These mea-surements have been fitted using the theoretical expression

σ(δ)/σ(0) = Γ 2 /4 δ 2 + Γ 2 /4 , ( 20
)
where Γ is the natural linewidth for the D 2 transition of sodium. We deduce a measured value Γ/(2π) ≈ 9.6 ± 0.5 MHz, compatible with the value found in the literature, Γ/(2π) = 9.8 MHz. Since no broadening of the absorption profile due to the linewidth of the laser could be observed within our measurement accuracy, the latter is small compared to the natural linewidth of the atoms. We conclude that the laser source fullfils all the requirements for laser cooling applications.

Conclusion

In conclusion, we have demonstrated a single-frequency, tunable, compact and robust all-solid-state SFG yellow laser source for cooling and trapping sodium atoms. The long-term stability of the laser source, despite the complexity brought by the use of a doubly-resonant enhancement cavity, stems from an original electronic servo loop. This servo is designed both to bypass the large depletion dip observed on the weaker input laser resonance fringe under high conversion, and to avoid saturation due to disturbances of the various integrators used in the servo loops. In the current configuration, the maximum output power of 800 mW remains lower than what can be produced with a dye laser. However, based on our measurements, we calculate that increasing the powers of the infrared laser sources to P 1 = 2 W and P 2 = 800 mW (both commercially available) should allow output powers in excess of 1 W, ultimately limited by the possible occurrence of thermal effects in the crystal [29,34,35].

Figure 3 a:

 3 Figure3a: Locking scheme for the SFG cavity. Solid lines represent optical paths, while dashed lines indicate electronic connections. LA + I: Use of a lock-in amplifier and an integrator to lock the lasers or the cavity to the maximum of a signal produced by one of the photodiode (Ph.). b: Linear combination used to compensate for the dip in intra-cavity power due to conversion. The photodiode monitoring laser 2 sees a power drop while the photodiode monitoring laser 3 sees a peak. Summing them up allows one to always maintain a peak signal to lock to. c: Automatic gain control circuit to control integrator saturation. COMP: comparator, SH: sample-and-hold amplifier, INT: integrator, D: precision differential amplifier.

Figure 4

 4 Figure 4 Measurement of the M 2 coefficient for the 589 nm laser in the vertical (triangles) and horizontal (circles) directions. The laser is focused with a f = 100 mm converging lens at the output of the cavity.

Frequency 7 Figure 5

 75 Figure 5 Relative intensity noise spectral density of the laser source averaged over 100 samples.

5. 3 1 -Figure 6

 316 Figure 6 Optical density of a cold atom cloud in a magnetooptical trap. The probe beam is detuned from the resonance line by an amount δ.

  1 , and ∆k = k 3 -k 1 -k 2 is the phase mismatch parameter. In a bulk Overview of the laser system. Mirrors M1 (with reflectance Ri at wavelength λi) and M2 are flat, while mirrors M3 and M4 are concave with a radius of curvature of 10 cm to allow focusing in the crystal, in which a fraction Ci of the power at wavelength λi is converted. The incoming power at wavelength λi is noted P inc i , and the intra-cavity power P cav

	(1064 nm) Laser 1	Dichroic mirror	EOM	P i	inc	Input coupler (R i ) M 1	P i	cav	M 2	Dichroic Ph. 2 Ph. 1
		Laser 2	(1319 nm)				M 3	Crystal (C i )	M 4	P 3	mirror Ph. 3
	Figure 1 i	. Ph. stands
	for Photodiode and EOM for Electro-Optic phase Modulator.

Table 1

 1 Reflectances and transmittances of the optical elements inside the cavity, at both wavelengths. Reflectances are measured within 0.5% and transmittances within 0.2%. The values for the crystal are specifications by the manufacturer. R is the optimum given by the numerical simulation.

		1064 nm			1319 nm
	R	R	T	R	R	T
	M1 0.930 0.96 0.060 0.740 0.79 0.250
	M2, M3, M4 0.995	0.005 0.995	0.005
	Crystal	0.980		0.980

In principle, Z0 and c depend weakly on temperature as ∆k through the dependance of the indices. One finds that over a temperature range of 20 • -100 • the relative variations do not exceed a few 10 -4 . Thus, we can safely consider Z0 and c as constants for the rest of the calculations.

This is the highest value obtained so far and differs from the measurements presented in Fig.2, which we have taken with another laser source at 1064 nm of poorer quality, and another crystal.

Acknowledgements We would like to thank Pierre Lemonde, Wolfgang Ketterle and Aviv Keshet for useful discussions. We acknowledge financial support from ANR (contract Gascor), IFRAF (Microbec project), the European Union (MI-DAS project, Marie Curie Fellowship) and DARPA (OLE project).


Sebastián Basterrech

Jan Platoš

Gerardo Rubino

Michał Woźniak

Experimental analysis on dissimilarity metrics and sudden concept drift detection

Learning from non-stationary data presents several new challenges. Among them, a significant problem comes from the sudden changes in the incoming data distributions, the so-called concept drift. Several concept drift detection methods exist, generally based on distances between distributions, either arbitrarily selected or context-dependent. This paper presents a straightforward approach for detecting concept drift based on a weighted dissimilarity metric over posterior probabilities. We also evaluate the performance of three well-known dissimilarity metrics when used by the proposed approach. Experimental evaluation has been done over ten datasets with injected sudden drifts in a binary classification context. Our results first suggest choosing the Kullback-Leibler divergence, and second, they show that our drift detection procedure based on dissimilarity measures is pretty efficient.

Introduction

In many real-world problems, a data stream may suddenly change its distributions. This phenomenon is commonly called concept drift.

One of the most commonly used approaches, which is a reaction to the occurrence of this phenomenon, is its detection with the use of so-called drift detectors.

A drift detector signals that the change in data distribution is significant and requires reconstruction or upgrade of the used model [START_REF] Sobolewski | Concept drift detection and model selection with simulated recurrence and ensembles of statistical detectors[END_REF]. So far, a number of methods have been proposed on how to construct drift detectors. However, most of them require either access to labels or access to prediction metrics of the used prediction model to make a decision. [START_REF] Gama | A survey on concept drift adaptation[END_REF]. Other concept drift detectors are based on distances over the underlying data distributions [START_REF] Hinder | Suitability of Different Metric Choices for Concept Drift Detection[END_REF][START_REF] Basterrech | Tracking changes using Kullback-Leibler divergence for the continual learning[END_REF][START_REF] Ditzler | Hellinger distance based drift detection for nonstationary environments[END_REF][START_REF] Brzezinski | Reacting to different types of concept drift: The accuracy updated ensemble algorithm[END_REF]. A recent experimental framework for the drift detection evaluation can be found in [START_REF] Gonzalvez | A comparative study on concept drift detectors[END_REF]. To assess a concept drift detector's performance, among the metrics measuring how different two distributions are, some usually considered ones are the number of true positive drift detections, the number of false alarms, the drift detection delay, the confusion matrix, and so on. One difficulty here is that there is typically a cost-benefit trade-off to find between different metrics [START_REF] Gustafsson | Adaptive Filtering and Change Detection[END_REF]. The mentioned metrics are more often arbitrarily selected, or they are selected according to the characteristics of the data and the specific problem at hand. Nevertheless, the metric choice for identifying changes in the probability distributions is a crucial decision addressed in this article. The contributions of this brief paper are two-fold.

(i) First, we specify a universal drift detector method in a supervised context without considering any assumptions about data, and we explore the impact of the two main parameters of the proposed technique. (ii) Second, we empirically analyze the performance of our drift detector using three different and important dissimilarity metrics: KLdivergence, Hellinger distance and Wasserstein distance. The selection of these metrics is based on the fact that nowadays are often used in the learning area [START_REF] Ditzler | Hellinger distance based drift detection for nonstationary environments[END_REF][START_REF] Dasu | An information-theoretic approach to detecting changes in multidimensional data streams[END_REF][START_REF] Faber | WATCH: Wasserstein Change Point Detection for High-Dimensional Time Series Data[END_REF].

Our experimental results over ten simulated datasets with injected drifts show remarkable differences between Hellinger distance and the other two evaluated metrics. In addition, it seems to be also a difference between KL-divergence and Wasserstein distance that makes us provide insights about the advantages of relative entropy in cases of sudden drifts in binary multidimensional data. This paper also briefly describes the studied dissimilarity metrics. Then, we present the drift detector method and our general methodology. We report the results in Section 1.4. Finally, we conclude with some discussion on further studies.

Background

Drift detection problem

Streaming data processing is usually related to problems where data comes in regular data chunks (blocks). Because we focus on the supervised context, we receive a long sequence of (input, output) values organized in chunks of common size K.

Consider a system producing the output y ∈ Y when the input is u ∈ U. Formally, we receive a time series C 1 , C 2 , . . . where C i is the chunk i, composed of the K-length se-quence z

(1) i , . . . , z (K) i , with z (k) i = u (k) i , y (k) i (that is, z (k) i
is the (input,output) pair of the kth element in the ith chunk). We see the elements z

(1) i , . . . , z (K) i as a sample having size K of a random variable z over a discrete set U × Y. The concept drift idea refers to the phenomenon that the probability distribution of z changes over time, i.e. there exists a point t such that the underlying distribution of {. . . , z t-2 , z t-1 , z t } is different from the distribution of {z t+∆ , z t+∆+1 , . . .}. We refer to sudden (known also as abrupt) drift when ∆ = 1 [START_REF] Hinder | Suitability of Different Metric Choices for Concept Drift Detection[END_REF][START_REF] Igor | Survey of distance measures for quantifying concept drift and shift in numeric data[END_REF]. Observe that the change in the joint distribution can be provoked either by a change in the posterior distribution (Pr(y | u)) (referred as the real concept drift) or by a change in the independent variables collected in u (referred as the virtual concept drift) [START_REF] Igor | Survey of distance measures for quantifying concept drift and shift in numeric data[END_REF]. In this contribution, we are focusing only on the studies of abrupt real drifts.

Re-visiting the concepts on dissimilarities

Let us consider discrete probability distributions, our case of interest. The context is the following: we have two discrete probability distributions (two probability mass functions, pmfs) p and q, defined on some common space S, and we want to measure how different they are. We review three different ways proposed for this purpose in computer science applications, primarily used in data mining.

Kullback-Leibler divergence.

The Kullback-Leibler (KL) divergence (also abusively called distance) between p and q (better said from p to q) is [9]

KL(p q) = s∈S p(s) log p(s) q(s) . (1.1)
Observe that this is not strictly a distance as the other dissimilarities analyzed here.It is positive and its value is zero if and only if p and q are identical. In information theory, we know that KL(p q) is the quantity of information lost when we use q instead of p, or as an approximation of p. The KL divergence does not satisfy the triangular inequality, in general. Several variations of the canonical KL divergence have been introduced in the literature to reach the symmetry property. Also, observe that the expression defining this divergence needs the sum taken for all values s where p and q are not zero. This leads to some technical issues relevant to our work. In case of zero values, a correction is proposed, see [START_REF] Basterrech | Tracking changes using Kullback-Leibler divergence for the continual learning[END_REF][START_REF] Dasu | An information-theoretic approach to detecting changes in multidimensional data streams[END_REF]. Despite the mentioned inconveniences, the KL divergence also has several advantages, such that: there exists a relationship with the expected value of likelihood ratio, several hypothesis tests are equivalent to KL divergence, and in case of some specific distributions KL divergence computation can be performed very fast Pinsker's inequality, and so on. For more details, please see [START_REF] Basterrech | Tracking changes using Kullback-Leibler divergence for the continual learning[END_REF][START_REF] Dasu | An information-theoretic approach to detecting changes in multidimensional data streams[END_REF][START_REF] Thomas | Elements of information theory[END_REF].

Hellinger The definition is as follows [START_REF] Gibbs | On choosing and bounding probability metrics[END_REF]:

H(p, q) = s∈S p(s) -q(s) 2 1/2 . (1.2)
This is a distance, so it is equal to zero if both distributions are the same [START_REF] Ditzler | Hellinger distance based drift detection for nonstationary environments[END_REF]. A particularly interesting property of the Hellinger distance is that it is bounded, the H(p, q) values are in

[0, √ 2].
Wasserstein distance. Let Γ = Γ (p, q) stands for the set of pmfs on S 2 having p and q as marginals. Then, given some real ν ≥ 1, the ν-Wasserstein distance W ν (p, q) between the two distributions is

W ν (p, q) = inf f ∈Γ Exp f dist(X, Y ) ν 1/ν
, where dist(.,.) denotes the Euclidean distance and (X, Y ) is a pair of random variables having distribution f ∈ Γ . The implementation of this distance has technical issues [START_REF] Faber | WATCH: Wasserstein Change Point Detection for High-Dimensional Time Series Data[END_REF], and the usual approach is to get approximations of the theoretical value. This is provided by available packages, like the one used in this paper (see below).

Methodology

Computation of dissimilarity scores. For computing the dissimilarities between two distributions, first we need to build a descriptor of the distribution of the data [START_REF] Hinder | Suitability of Different Metric Choices for Concept Drift Detection[END_REF]. Here, we use the standard estimator as a descriptor based on the binning strategy. Following the previous notation, we receive in the ith chunk a Klength sequence z

(1) i , . . . , z (K) i , with z (k) i = u (k) i , y (k) i
. Then, for the latter, that is, for the output values of the system, the number Pr(y i = ), for any ∈ Y, is naturally estimated by its standard estimator

Pr(y i = ) = 1 K K k=1 1(y (k) i = ),
where we denote by 1(P ) the indicator function.For an input u to the system, we apply the binning strategy decomposing the input space U into J disjoint "bins" b (1) , b (2) , . . . , b (J) . For the the C i th chunk the conditional probability of having a class in a specific bin b j i is estimated by

Pr y i = | {u i : u k i ∈ b j i , ∀k} = K k=1 1((u k i ∈ b j i ) ∩ (y k i = l)) K k=1 1(u k i ∈ b j i )
.

(1.3)

Then, by applying expression (1.3) for each output class we may compute the probability mass function Pr(y i | {u i : u k i ∈ b j i , ∀k} . Hence, each bin has associated a pmf, and then we evaluate a change through any two chunks C i and C t as follows

d j i,t = φ Pr(y i | {u i : u k i ∈ b j i , ∀k}), Pr(y t | {u t : u k t ∈ b j t , ∀k}) , (1.4)
where φ(•) is any selected function for estimating the distribution dissimilarity. Finally, we aggregate the estimated dissimilarity for covering the whole input space (for all the bins)

Φ(C i , C t ) = 1 J J j=1 d j i,t . (1.5)
Finally, we modify the previous aggregation form using a weighted sum. We consider the chance of sampling in a specific bin

Φ(C i , C t ) = 1 J J j=1 γ j i d j i,t , (1.6)
where the weight γ j i is the probability estimation of sampling in a specific region b j i of the reference chunk S i

γ j i = 1 K K k=1 1(u k i ∈ b j i ).
Decision rule using a variancebased threshold. Now, let us consider windows of chunks W 1 , W 2 , . . . where W i = C i , C i+1 , . . . , C i+N -1 . We employ the previous approach again to look for changes in the data distributions but see a block of chunks as a sliding window on the series of chunks, having KN instances. We proceed as before, except that instead of comparing two windows starting at chunks C i and C i+1 , we shift the blocks by N individual chunks, which is, we compare the window starting at chunk i with the one starting at chunk i + N . For each N individual chunk is possible to compute a new dissimilarity score by collecting the chunks in batches (windows) and computing a dissimilarity score Φ(W j , W j+1 ) applying expression (1.6). Therefore, a sequence of dissimilarity scores is generated Φ(W 1 , W 2 ), Φ(W 2 , W 3 ), . . . , Φ(W j , W j+1 ). It is necessary to define a procedure for identifying locations where critical points occur to make an automatic decision. Let m j be the mean of the dissimilarity scores until the last processed window W j , and σ k the standard deviation of this sequence. Given a new dissimilarity score value Φ(W j , W j+1 ), we decide that a drift occurs when

Φ(W j , W j+1 ) / ∈ [m k -ασ k , m k + ασ k ], (1.7)
where α is a threshold parameter. This specific decision rule is inspired by techniques for artifact and outliers detection [START_REF] Basterrech | A natureinspired biomarker for mental concentration using a single-channel eeg[END_REF][START_REF] Basterrech | Nature-inspired Algorithms for Selecting EEG Sources for Motor Imagery Based BCI[END_REF]. Window length and α value are the main parameters of the method. A larger α may increase the chance of false negatives. When α is too small, then the chances of false positives increase. Here, we analyze only scenarios where the windows are disjoint, and the α values are static (we don't modify them according to changes in the data). Another parameter that has an impact on the results is the number of bins. It impacts the pmf estimation. A large number also increases computational costs. After a preliminary evaluation, we decided to present results using J = 5×dim(U) homogeneous bins, where dim(U) denotes the dimensionality of the input space.

Methodological approach overview. The concept drift detector method analyzed here is summarized in the high-level workflow presented in Figure 1.1. It has the following main steps:

(i) Homogeneous partition of the input space. We decompose the input space into disjoint bins using parameterized range constraints.

The search for the best splitting hyperplanes in U is out of the scope of this paper.

Here, we decided to create homogeneous partitions following the standard binning strategy. (ii) Posterior probability estimation. The probability mass function is estimated by applying the expression (1.3). Note that the conditional distribution is made for each partition of the input space.

Experimental study

In the previous section, we provided a framework for explicitly monitoring the data stream and detecting if a drift occurs. We hypothesize that proposed measures could be used as the base for a decision about drift. We designed experiments to compare the performance of the three mentioned earlier dissimilarity metrics. We used simulated data streams where the drift appearances are marked. In this ongoing work, we study only binary datasets with injected sudden concept drifts. The analyzed window lengths are {250, 500, 1000, 2000, 4000}, and we studied α values over a large domain (the specific range depended on the metric).

Benchmark data streams. We employed 10 datasets in our performance evaluation studies. We generated 5 binary datasets with 3 features and 5 datasets with 5 features, all of them were created using the stream-learn library [START_REF] Ksieniewicz | stream-learnopen-source python library for difficult data stream batch analysis[END_REF].

Each data stream has 10000 chunks with 250 instances and 20 induced sudden concept drifts. The stream-learn library is useful for generating a wide range of datasets with injected drifts, it has the additional advantage that provides the time-stamp where the drifts were injected. The stream-learn simulator has a parameter that determines how sudden the change of drift concept is. We used the maximum allowed value for this parameter. More details about the simulation of data streams with sudden drift concepts are specified in [START_REF] Ksieniewicz | stream-learnopen-source python library for difficult data stream batch analysis[END_REF].

Performance evaluation. We chose the standard metrics: sensitivity, precision, balance accuracy score (BAC) and F1-score [START_REF] Gonzalvez | A comparative study on concept drift detectors[END_REF][START_REF] Gustafsson | Adaptive Filtering and Change Detection[END_REF].

Results. According to our empirical results, we do not appreciate notable differences between results over data with 3 and 5 features. However, we obviously cannot affirm similar behavior in larger input space dimensions. Results obtained by the KL-divergence dissimilarity, the Hellinger distance, and the Wasserstein distance are presented in the figures 1.2, 1.2, and 1.4, respectively. Each of these figures has two graphics, in the left graphic is presented the specificity according to the window length, and the right side is shown the precision according to the window length. We present the results of the specificity metric over datasets with 5 features and the precision obtained over data with 3 features. Each graphic has several curves resulting from different experiments over five datasets. A common behavior in the figures is that the window length is a relevant parameter, which is intuitive because it directly affects the distribution estimation. Another characteristic is that the specificity decreases when the window length is large. On the other hand, the precision also is impacted by the window length, but it seems more stable in the case of KL and Hellinger metrics than in the case of Wasserstein distance.

Let us note that from previously described figures, Hellinger distance seems less competitive than the other two metrics. For illustrative reasons, the α threshold used for creating the mentioned curves was empirically tuned to obtain 20 drifts during the whole stream. Figures 1.5 and 1.6 show results over different threshold values α. From Fig. 1.2 and Fig. 1.4 we see a minor difference between KL and Wasserstein dissimilarities. Then, we also present a specific comparison between KL-divergence and Wasserstein distance for different α thresholds using BAC and F1-score values. We fixed the window length to 500 instances (a value that both metrics perform "pretty well" according to Fig. 1.2 and Fig. 1.4). Fig. 1.5 presents two graphics with BAC results, and Fig. 1.6 has two figures with a comparison between KL and Wasserstein using F1-scores. We appreciate a slight difference between both metrics from the obtained results, indicating that KL-divergence has a better global performance.

It also seems that KL is more robust, i.e., it is less sensitive to the window length and the α value. In addition, KL-divergence is faster than the computation of Wasserstein distance.

Experimental

protocol and implementation. We used python v3.9, the libraries numpy v1.19.5, stream-learn v0.8.16 and scipy.stats v1.5.4. We used the scipy.stats.wasserstein_distance function for computing the Wasserstein distance; KL-divergence and Hellinger distance were implemented by us based on numpy functions.

Conclusions and future work

We presented a drift detection method based on the evaluation of changes over the empirical statistical distributions of the data. The method does not require any assumptions about the data. We show its performances using three well-known dissimilarity metrics over binary data with sudden drifts. We compare the behavior of each of the metrics. It is interesting to note that the KL divergence obtains better results globally, and with it, our proposed detector achieves good performance. Further work needs to be done exploring real data to analyze statistical differences between the results and other types of concept drifts. Note that the number of bins grows exponentially with the number of dimensions. Then, the binning strategy has well-known limitations in highdimensional data. For this reason, we also plan to explore other data descriptors.

Fig. 1 . 1 :

 11 Fig. 1.1: High-level flowchart of investigated algorithm.

Fig. 1 . 2 :

 12 Fig. 1.2: KL-divergence: specificity and precision according to window length.

Fig. 1 . 3 :

 13 Fig. 1.3: Hellinger distance: specificity and precision according to window length.

Fig. 1 . 4 :

 14 Fig. 1.4: Wasserstein: specificity and precision according to window length.

Fig. 1 . 5 :

 15 Fig. 1.5: Comparison using BAC between KLdivergence and Wasserstein distance.

Fig. 1 . 6 :

 16 Fig. 1.6: Comparison using f1-score between KL-divergence and Wasserstein distance.

Acknowledgements This work was supported by the CEUS-UNISONO programme, which has received funding from the National Science Centre, Poland under grant agreement No. 2020/02/Y/ST6/00037, and the GACR-Czech Science Foundation project No. 21-33574K "Lifelong Machine Learning on Data Streams".

It was also supported by the ClimateDL project (code 22-CLIMAT-02) belonging to the Climate Am-Sud programme, where the central problem is forecasting extreme temperatures in future periods such as in the following summer.


Emmanuel Mimoun
email: emmanuel.mimoun@ens.fr

Luigi De Sarlo

Jean-Jacques Zondy

Jean Dalibard

Fabrice Gerbier

Sum-frequency generation of 589 nm light with near-unit efficiency

We report on a laser source at 589 nm based on sum-frequency generation of two infrared laser at 1064 nm and 1319 nm. Output power as high as 800 mW is achieved starting from 370 mW at 1319 nm and 770 mW at 1064 nm, corresponding to converting roughly 90% of the 1319 nm photons entering the cavity. The power and frequency stability of this source are ideally suited for cooling and trapping of sodium atoms.

Introduction

Among all elements of the periodic table, Sodium historically played a key role in the development of atomic physics and spectroscopy, and still retains a considerable importance for fundamental research or applications such as artificial beacon stars, Laser-induced detection in the atmospheric range (LIDAR) [START_REF] Fugate | Measurement of atmospheric wavefront distortion using scattering light form a laser guide-star[END_REF], and quantum degenerate gases. However, reaching the yellow resonance ("Sodium doublet") near 589 nm requires to use dye lasers, which are expensive and difficult to maintain and operate. For this reason, many alternative methods based on non-linear freqency conversion of solid-state infrared lasers have been explored [START_REF] Jeys | Sum frequency generation of sodium resonance radiation[END_REF][START_REF] Moosmüller | Sum-frequency generation of continuous-wave sodium d2 resonance radiation[END_REF][START_REF] Vance | Continuous-wave, all-solid-state, single-frequency 400-mw source at 589 nm based on doubly resonant sum-frequency mixing in a monolithic lithium niobate resonator[END_REF][START_REF] Bienfang | 20w of continuouswave sodium d2 resonance radiation from sum-frequency generation with injection-locked lasers[END_REF][START_REF] Denman | 20 w cw 589 nm sodium beacon excitation source for adaptative optical telescope applications[END_REF][START_REF] Mildren | Efficient, all-solid-state, raman laser in the yellow, orange and red[END_REF][START_REF] Feng | 589 nm light source based on raman fiber laser[END_REF][START_REF] Janousek | Efficient all solid-state continuous-wave yellow-orange light source[END_REF][START_REF] Georgiev | Wattslevel frequency doubling of a narrow line linearly polarized raman fiber laser to 589nm[END_REF][START_REF] Dawson | Multi-watt 589nm fiber laser source[END_REF]. Second harmonic generation from a Raman fiber laser [START_REF] Mildren | Efficient, all-solid-state, raman laser in the yellow, orange and red[END_REF][START_REF] Feng | 589 nm light source based on raman fiber laser[END_REF][START_REF] Georgiev | Wattslevel frequency doubling of a narrow line linearly polarized raman fiber laser to 589nm[END_REF] and sum frequency generation (SFG) from two lasers around 938 nm and 1583 nm [START_REF] Dawson | Multi-watt 589nm fiber laser source[END_REF], or around λ 1 = 1064 nm and λ 2 = 1319 nm, have been demonstrated. This last solution seems particularly appealing since both wavelengths are accessible using YAG lasers. Such infrared sources in cavity-enhanced configurations have been demonstrated [START_REF] Jeys | Sum frequency generation of sodium resonance radiation[END_REF][START_REF] Vance | Continuous-wave, all-solid-state, single-frequency 400-mw source at 589 nm based on doubly resonant sum-frequency mixing in a monolithic lithium niobate resonator[END_REF][START_REF] Bienfang | 20w of continuouswave sodium d2 resonance radiation from sum-frequency generation with injection-locked lasers[END_REF][START_REF] Denman | 20 w cw 589 nm sodium beacon excitation source for adaptative optical telescope applications[END_REF][START_REF] Janousek | Efficient all solid-state continuous-wave yellow-orange light source[END_REF]. Up to P 3 = 20 W output power at λ 3 = 589 nm has been reported [5, 6], based on two custom high-power (P 1 =20 W and P 2 =15 W) infrared lasers.

Here we report on the experimental realization of a 589 nm source with up to P 3 ≈ 800 mW output power at λ 3 = 589 nm, using moderate infrared powers from commercial laser sources. Our system operates in a highly efficient regime, where roughly 90% of the photons of the weakest (1319 nm) source effectively coupled into the cavity are converted. We show below that the resulting depletion of the 1319 nm pump source strongly distorts the cavity fringe pattern. We have therefore designed and implemented an original fringe reshaping method to efficiently and robustly lock the lasers to the cavity. Using the resulting 589 nm laser source, a magneto-optical trap of sodium is obtained, confirming a linewidth of the laser below the natural linewidth (10 MHz) of the atomic transition. To our knowledge, this had only been achieved previously using dye lasers (see e.g. [START_REF] Davis | Bose-einstein condensation in a gas of sodium atoms[END_REF][START_REF] Streed | Large atom number bose-einstein condensate machines[END_REF]).

The paper is organized as follows. In Section 2, we present the main elements in our experimental setup. In Section 3, the theory of cavity-enhanced SFG is recalled, with particular emphasis on how to reach an optimal regime where almost all incoming photons from the weakest source are converted. In Section 4, we discuss experimental issues associated with large conversion efficiencies, and how to resolve them. Experimental results are presented in Section 5, and conclusions are exposed in Section 6.

Experimental setup

Our system is described schematically in Fig. 1. The lasers operating around λ 1 = 1064 nm and λ 2 = 1319 nm respectively are coupled into a bow-tie cavity, which is resonant for both of them and transparent for the output laser at λ 3 = 589 nm. For these lasers we use commercial sources (manufactured by Innolight GmbH, Germany) delivering 1 W at 1064 nm and 500 mW at 1319 nm. The useful powers effectively coupled into the cavity are lower, P 1 ≈ 770 mW at 1064 nm and P 2 ≈ 370 mW at 1319 nm, due to losses through the optical path and a measured 85% coupling efficiency into the fundamental mode of the cavity. Inside the cavity we place a

Crystal (C i )

Input coupler (R i ) For the pump lasers i (i = 1, 2), we note P (cav) i the intra-cavity power, P i the incident power,

P i P i (cav) P i (ref) M 1 M 3 M 4 M 2 P 3
P (ref) i
the reflected power, C i the power fraction transmitted through the crystal, ad R i the reflectivity of the input coupler. P 3 is the output power produced at λ 3 .

periodically poled KTiOPO 4 (ppKTP) crystal (manufactured by KTH, Sweden), with a poling period ≈ 12.7 µm. The choice of ppKTP was motivated by its relatively high non-linear coefficient, by its tolerance to large input powers, and by its negligible absorption in the visible range. The crystal is enclosed in a copper mount which is temperature-regulated within a few 10 mK. We found an optimal single-pass operating temperature around 50 • C, with single-pass efficiency α = P 3 /P 1 P 2 ≈ 0.022 W/W 2 , in reasonable agreement with the previously measured values (e.g. [START_REF] Janousek | Efficient all solid-state continuous-wave yellow-orange light source[END_REF]).

Sum-frequency generation in cavity-enhanced configuration

In this Section, we first recall the theory of SFG and discuss how it can be used to optimize the desired 589 nm output power. In our theoretical model, we assume that only a small fraction of the intra-cavity powers is consumed in the SFG process. In this weakly-depleted pumps approximation, the output power P 3 at wavelength λ 3 is simply given by [START_REF] Boyd | Nonlinear optics[END_REF][START_REF] Boyd | Parametric interaction of focused gaussian light beams[END_REF][START_REF] Zondy | Absolute value of the d 36 nonlinear coefficient of aggas2: prospect for a low-threshold doubly resonant oscillator-based 3:1 frequency divider[END_REF]:

P 3 = α P (cav) 1 P (cav) 2
.

(

) 1
where

P (cav) i
is the intra-cavity power at wavelength λ i . On resonance, the intra-cavity power P (cav) i is linked to the incident power P i by [START_REF] Siegman | Lasers[END_REF]

P (cav) i P i = 1 -R i -L i 1 -R i (1 -δ i )C i 2 . (2)
Here, R i and L i denote reflection and loss coefficients of the input coupler M 1 for lasers i (i = 1, 2), and δ i denote the total passive fractional loss per cavity round trip. The coeffi- cient C i is the ratio between the photon flux before and after the crystal, and accounts for the depletion of laser i due to SFG. Neglecting absorption in the crystal, energy conservation in the conversion process implies

C i = 1 -λ 3 P 3 /λ i P (cav) i
. As long as C i ≃ 1, the weakly depleted pumps approximation is valid.

Since SFG removes one photon from each infrared pump for each output photon, the efficiency of SFG is ultimately limited by the weakest pump 2. Neglecting losses, the maximal output flux at λ 3 equals the input flux at λ 2 , corresponding to P (max) 3 = (λ 2 /λ 3 )P 2 . In the following, we take the ratio η = P 3 /P as a figure-of-merit for the conversion efficiency. This ratio compares the outcoming flux at λ 3 to the limiting incoming flux effectively coupled into the cavity at λ 2 . With this definition, the conversion efficiencies reported in [START_REF] Jeys | Sum frequency generation of sodium resonance radiation[END_REF][START_REF] Moosmüller | Sum-frequency generation of continuous-wave sodium d2 resonance radiation[END_REF][START_REF] Vance | Continuous-wave, all-solid-state, single-frequency 400-mw source at 589 nm based on doubly resonant sum-frequency mixing in a monolithic lithium niobate resonator[END_REF][START_REF] Bienfang | 20w of continuouswave sodium d2 resonance radiation from sum-frequency generation with injection-locked lasers[END_REF][START_REF] Mildren | Efficient, all-solid-state, raman laser in the yellow, orange and red[END_REF][START_REF] Feng | 589 nm light source based on raman fiber laser[END_REF][START_REF] Janousek | Efficient all solid-state continuous-wave yellow-orange light source[END_REF][START_REF] Georgiev | Wattslevel frequency doubling of a narrow line linearly polarized raman fiber laser to 589nm[END_REF][START_REF] Dawson | Multi-watt 589nm fiber laser source[END_REF] are generally around 50 % (up to 60% for [START_REF] Bienfang | 20w of continuouswave sodium d2 resonance radiation from sum-frequency generation with injection-locked lasers[END_REF]).

Let us first consider an ideal situation with no passive losses in the cavity (L i =0 and δ i = 0). In this case complete conversion of the weakest pump 2 is possible (P 3 = P (max) 3

). From Eq. ( 2), we find that this occurs when C opt 2 = R 2 , corresponding to the impedance matching between the power transmitted through the input coupler and that consumed in the crystal due to conversion [START_REF] Kaneda | Theoretical treatment, simulation, and experiments of doubly resonant sum-frequency mixing in an external resonator[END_REF]. The corresponding cavity enhancement factor is

P (cav) 2 /P 2 = (1 -R 2 ) -1 .
Eq. ( 1) then fixes the corresponding value for P

(cav) 1 = (1 -R 2 )λ 2 /
αλ 3 , which is achieved for a reflectivity R 1 that can be computed from Eq. [START_REF] Jeys | Sum frequency generation of sodium resonance radiation[END_REF]. Without losses, one can always find a solution, implying that total conversion can be obtained in the lossless case for any value of R 2 (see Fig. 2(a-b)). Consequently, the power reflected at mirror M 1

P (ref) 2 P 2 = ( √ C 2 - √ R 2 ) 2 1 - √ R 2 C 2 2 . ( 4
)
vanishes under these optimum conditions: all the incoming photons are converted in the crystal and no photon at λ 2 comes out of the cavity. This can be interpreted as a destructive interfer- ence on M 1 between the reflected field and the transmitted one. Note that the assumption of complete conversion does not contradict the weakly-depleted pumps approximation underlying the calculation. Indeed, because of the cavity power enhancement, the incident flux is a small fraction of the power inside the cavity. This lossless model is already sufficient to interpret qualitatively our experimental findings. In Fig. 3 we show the transmitted (b,d) and reflected (c,e) power while scanning the cavity length around the position where both lasers are simultaneously resonant. When only one laser is present (either 1 or 2) we observe the expected Lorentzian profile. However when both lasers are present a pronounced dip appears in the resonance profile for laser 2, corresponding to efficient conversion into 589 nm photons. A dramatic decrease of the reflected power is simultaneously observed, corresponding to the destructive interference previously mentioned.

The more realistic model including passive losses in Eq. ( 2) can be solved numerically for a given single-pass efficiency and cavity parameters. The passive loss coefficients (δ 1 = 2.4%, δ 2 = 1.6%) have been determined by injecting only one laser at a time in the cavity, and comparing the measured transmitted and reflected powers to the theoretically expected values. The calculated conversion efficiency in this situation is shown in Fig. 4b, as a function of the input coupler reflectivities. While total conversion cannot be achieved as in the lossless case (Fig. 4a), a locus of points with a maximal efficiency close to 1 can still be identified. Among them, a sensible choice is to select R 1 and R 2 close to each other to minimize the total intracavity power and thermal effects in the crystal. Note also that the optimum is quite loose and the reflectivities relatively low, making the input coupler tolerant to small fabrication imperfections.

Locking scheme in the regime of large conversion

Experimentally, working in a regime with such large conversion efficiencies leads to serious stability problems for a conventional locking system. The key to efficient SFG operation is to ensure that the cavity is simultaneously resonant with both IR lasers at all times. This is usually enforced by two servo-loops maximizing the intra-cavity powers independently. The locking scheme is as follows in our experiment. The 1319 nm laser is used as a master laser onto which the cavity length is locked using an integrating servo-loop. Then, the 1064 laser is locked onto the cavity, and therefore on the master laser, ensuring stable operation of the ensemble. In our experiment each servo loop uses an error signal generated from the power leaking through one of the cavity mirror by a modulation/demodulation technique. However what follows would still be valid for other locking techniques.

In the regime of large conversion, methods relying on such a master/slave scheme fail due to the above mentioned dip in the transmission of laser 2 (see Fig. 3d). Indeed, the cavity servo cannot distinguish this power reduction from that caused by an external perturbation, and actually works against keeping both lasers on resonance simultaneously. To circumvent this problem, we have designed an original analog processing of our error signals. Instead of the "bare" error signal produced by laser 2, the cavity lock uses a linear combination of this . Contour map of the efficiency η of the conversion process as a function of the reflectivities R 1 and R 2 of the input coupler M 1 for laser 1 and 2 (λ 1 = 1064 nm and λ 2 = 1319 nm). a. Lossless case. The dashed line (same as in Fig. 2b) corresponds to total conversion (η = 1). b. Passive losses are taken into account using (δ 1 = 2.4%, δ 2 = 1.6%). While η = 1 cannot be obtained anymore experimentally, efficiencies higher than 90% can still be reached.

signal and of the output at λ 3 . The combination is done electronically, with weights chosen empirically to restore a lineshape with a single maximum and optimize the slope around the lock-point. This fringe reshaping method works for any level of conversion, and allows stable operation of the laser on a day timescale, even at the highest efficiencies. Finally, choosing the 589 nm output as the error signal for the second servo-loop locking laser 1 to the cavity ensures that the system locks to the maximal converted power. This fringe reshaping method is protected by French patent INPI 0803153 (international patent pending), and further details will be given in a future publication [19].

Experimental results

We have performed a systematic study of the dependance of the output power on the infrared pump powers. Such a measurement is shown in Fig. 5a. The model including passive losses (δ 1 = 2.4%, δ 2 = 1.6%) compares favorably to our experimental findings. Both the calculated steady state values (dashed lines in Fig. 3(b-e)) and the variation of output power P 3 with pump power P 1 (shown in Fig. 5a) are well-reproduced by the model. Furthermore, Fig. 5b shows no variation of the coefficient α deduced from Eq. ( 1) even at the highest powers, thus validating the weakly depleted pump approximation. This also rules out additional effects (such as thermal effects related to absorption of the infrared beams) which would reduce α at higher powers. Overall, we find that the model gives a reliable description of the SFG process for our experimental configuration, and allows one to optimize the parameters of the cavity to ensure maximum efficiency. A key parameter to achieve η ≃ 0.9 is the choice of a highly nonlinear crystal, resulting in a nonlinear loss αP

(cav) 1
exceeding by far the roundtrip passive loss. With our set of parameters, we were able to reach output powers as high as P 3 = 800 mW or η ≈ 90%. This implies that our apparatus works very close to the theoretical ideal limit studied in the first part of the paper.

In our experiment, the laser is locked on the Sodium D 2 resonance line by reacting on the frequency of laser 2 using an error signal obtained from saturated absorption spectroscopy, as shown in Fig. 6. Using this laser source, we have obtained a magneto-optical trap containing roughly 10 7 atoms in a ultra-high vacuum cell. Repumping light could be derived from the same source as the trapping laser itself by using a high-frequency (1.7 GHz) acousto-optic modulator (manufactured by Brimrose Corporation of America). This confirms the viability of our approach for demanding applications such as laser cooling and trapping, or high resolution spectroscopy.

Conclusion

In conclusion, we presented an efficient all-solid state laser source based on SFG at 589 nm with an output power of 800 mW. The source acts as a wavelength converter for the weakest source, ensuring a conversion efficiency around 90% while keeping input powers in the watt-level range. Such a setup can be used to produce other wavelengths in the visible range, provided the existence of input lasers at the right wavelengths. This provides a cost-effective solution for atomic physics experiment, free from the drawbacks of dye lasers.

Figure 1 .

 1 Figure 1. Setup for SFG in a doubly resonant bow-tie cavity formed by four mirrors M 1,..,4 .

1 Figure 2 .

 12 Figure 2. Lossless model : a. Intracavity powers for laser 1 and 2 (λ 1 = 1064 nm and λ 2 = 1319 nm) required to reach total conversion, plotted against the reflectivity R 2 of the input coupler; b. Reflectivity R 1 of the input coupler plotted againt R 2 : Each couple (R 1 ,R 2 ) on this curve ensures total conversion.

Figure 3 .

 3 Figure 3. b-d. Intra-cavity powers for lasers 1 (b) and 2 (d) (λ 1 = 1064 nm, λ 2 = 1319 nm), plotted against the cavity resonance frequency tuned with a piezoelectric transducer; c-e. Powers reflected out of the cavity for lasers 1 (c) and 2 (e). The dashed lines show the predictions from the model described in the text, including both conversion and passive losses.

Figure 4

 4 Figure 4. Contour map of the efficiency η of the conversion process as a function of the reflectivities R 1 and R 2 of the input coupler M 1 for laser 1 and 2 (λ 1 = 1064 nm and λ 2 = 1319 nm). a. Lossless case. The dashed line (same as in Fig.2b) corresponds to total

Intra-cavity power P 1

 1

Figure 5 .

 5 Figure 5. a. Output power at 589 mn plotted against intra-cavity power at 1064 nm (varied by changing the incoming power into the cavity). The solid line is the result of the numerical calculations as described in text. b. Conversion coefficient α (see Eq. (1)), varying the incoming power of one pump laser while leaving the other fixed. The conversion coefficient is constant and equal to that measured in the single pass configuration (dashed line), irrespective of laser power.

Figure 6 .

 6 Figure 6. Saturated absorption signal (solid line) while scanning the frequency of the 1319nm laser, thus the one of the 589nm one. The D2 transitions corresponding to atoms in the ground F=2 (a) and F=1 (b) electronic states are represented. (ij) represents the level crossing line between transitions i and j.

Acknowledgements

We thank Pierre Lemonde, Antoine Browaeys, Wolfgang Ketterle, Aviv Keshet and Volker Leonhardt for discussions. This work was supported by ANR (Gascor contract), IFRAF, DARPA (OLE project), and EU (MIDAS network). LdS acknowledges financial support from IFRAF. JJZ acknowledges funding from Laboratoire National de Métrologie et d'Essais (LNE).


Ahmeda Assann Ouédraogo
email: ahmeda.ouedraogo@cerema.fr

Emmanuel Berthier

David Ramier

Tan Yanlin

Marie-Christine Gromaire

Marie-Christine Gro- Maire Evapotranspiration

Evapotranspiration d'une toiture végétalisée expérimentale : observations et modélisation

Keywords: évapotranspiration, observation, modélisation, toitures-végétalisées, milieu-urbain

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Les toitures végétalisées (TV) s'intègrent dans la grande famille des ouvrages végétalisés (noues d'infiltration, jardins de pluies, parkings perméables, etc.) de gestion à la source des eaux pluviales. Dans un contexte de changement climatique et de forte imperméabilisation du milieu urbain qui favorisent le ruissellement et l'augmentation de la température ambiante, ces nouveaux outils de gestion des eaux pluviales urbaines viennent compléter les outils conventionnels (réservoirs, réseaux enterrés d'eaux pluviales, etc.) existants en limitant le ruissellement au profit de l'infiltration et de l'évapotranspiration (ET). Ces ouvrages permettent l'abattement du ruissellement pour les pluies courantes, contribuent au rafraîchissement du milieu pendant les périodes chaudes et à la restauration d'un bilan hydrologique urbain plus proche du bilan hydrologique naturel.

Une étude de la performance des TV dans 21 pays synthétisant 75 études internationales publiées entre 2005 et 2020 montre que les taux de rétention événementiels (c'est-à-dire la proportion de pluie retenue et abattue par évènement) sont très variables (0-100%), avec une moyenne autour de 62% [START_REF] Zheng | Green roofs for stormwater runoff retention: A global quantitative synthesis of the performance[END_REF]). Ces variabilités s'expliquent par des facteurs tels que l'intensité des pluies, la profondeur du substrat, le type de climat, le type de végétation et le type de saison. Si la capacité des TV à réduire le ruissellement est largement étudiée, les processus qui permettent à ces dernières d'être performantes sont insuffisamment documentés. En effet, lorsqu'il pleut, une partie de l'eau va être interceptée par la végétation (processus d'interception) et l'autre partie par le processus d'infiltration arrive dans les pores du substrat de la toiture, et c'est le processus d'ET (transpiration et évaporation) qui permet de restaurer la capacité de stockage de ces pores.

Dans ce travail, une attention particulière est portée sur l'ET et les principaux objectifs poursuivis se situent à trois niveaux : i) évaluation de l'ET à partir de la base de données mesurées disponibles ; ii) compréhension des facteurs de l'ET sur une toiture végétalisée et iii) modélisation de l'ET avec un modèle hydrologique.

Matériels et méthodes

Le dispositif expérimental mis en place est une toiture végétalisée extensive installée à Trappes (Ile de France) sur le site du Cerema. Il est composé de différentes structures de végétalisation extensive, en faisant varier un seul paramètre entre chaque structure. Le toit a donc été divisé en six compartiments végétalisés de 35 m² (7x5 m) et deux compartiments de référence de 21 m² (7x3 m) dont un a été laissé avec seulement l'étanchéité et un autre avec des graviers. Les six compartiments végétalisés ont des compositions différentes. Le couvert végétal (sedum et/ou graminée), l'épaisseur (3 cm ou 15 cm), la nature du substrat (intensif ou extensif), et le type de couche de drainage (polystyrène expansée ou de pouzzolane) ont été modifiés d'un compartiment à l'autre.

Un pluviomètre et des augets permettent de mesurer respectivement la pluie et le débit de drainage à la sortie de la gouttière de chaque compartiment. Une station météorologique permet de mesurer la température et l'humidité relative de l'air à 2 m au-dessus de la végétation, la vitesse et la direction du vent à 3 m au-dessus de la végétation et le rayonnement net à 1 m au-dessus de la végétation. Pour chaque compartiment, des capteurs ont permis le suivi de la teneur en eau et de la température à différentes profondeurs dans le substrat. On dispose aussi de mesures directes d'évapotranspiration ponctuelles, et temporellement, à l'aide d'une chambre [START_REF] Loustau | Utilisation d'une chambre de transpiration portable pour l'estimation de l'évapotranspiration d'un sous-bois de pin maritime à molinie (Molinia coerulea (L) Moench)[END_REF]. L'ensemble des données a été acquis à un pas de temps fin sur une période de plusieurs années de juillet 2011 à juillet 2018.

ET = P -Q -dθ (en mm) (1)
ii) par bilan énergétique de surface, grâce aux mesures du rayonnement net (Q * ) par un radiomètre et des profils de température mesurés par des thermocouples dans les toitures végétalisées :

ET = Q * -Q G -Q H (en w. m -2 ) (2)
avec Q G le flux de chaleur entrant dans le substrat de chaque compartiment et Q H le flux de chaleur sensible.

Ces flux sont estimés à partir des températures mesurées à la surface des différents compartiments végétalisés et de l'air.

iii) par une chambre à évapotranspiration développée au Cerema : il s'agit de mesurer les variations d'humidité à l'intérieur d'une enceinte fermée [START_REF] Loustau | Utilisation d'une chambre de transpiration portable pour l'estimation de l'évapotranspiration d'un sous-bois de pin maritime à molinie (Molinia coerulea (L) Moench)[END_REF] avec ρv l'humidité absolue (en g. m -3 ), L la chaleur latente de vaporisation, h la hauteur de l'enceinte et t le temps (s).

Le modèle Hydrus 1D [START_REF] Simunek | HYDRUS-1D. Simulating the one-dimensional movement of water, heat, and multiple solutes in variably-saturated media[END_REF], qui résout l'équation de Richards pour des milieux variablement saturés en eau, seront utilisé pour modéliser l'ET de chaque compartiment de la toiture. Les profils de teneur en eau et les données de ruissellement seront mobilisés pour le calage du modèle. L'évapotranspiration potentielle constitue une entrée dans ce modèle hydrologique et est prévu dans un premier temps d'utiliser l'ET potentielle (ET p ) paramétrée selon Penman- [START_REF] Monteith | Evaporation and environment[END_REF].

Premiers résultats

La Figure 2

Conclusion

L'ET d'une toiture végétalisée peut atteindre 10 mm/j et reste non négligeable même en l'absence de végétation (max d'ET de 7.4 mm/j). Si l'épaisseur du substrat est un facteur prépondérant qui augmente l'ET pour une même végétation, le type de végétation reste un facteur influent : les sedums ont montré une ET totale plus importante que les graminées durant les périodes sèches (293 mm vs 265.8 mm). Lors de la présentation en octobre, il est prévu des résultats d'estimation de l'ET par toutes les différentes méthodes précitées et une modélisation de l'ET sur un des compartiments de la TV avec le modèle hydrologique Hydrus.

Figure 1 :

 1 Figure 1: Toiture végétalisée de Trappes. À droite du toit se trouve la station météo, à gauche de cette station, le pluviomètre. Les bacs verts au pied du bâtiment contiennent les augets pour la mesure des débits de drainage issus des gouttières. Les symboles sont définis selon : G : Graminées, S : Sedum, E : Extensive, I : Intensive, 15 cm ou 3 cm pour l'épaisseur de substrat, polystyrène expansé (Y) ou pouzzolane (Z) pour la nature de la couche de drainage et N pour sol nu. L'ET est estimée pour chaque compartiment : i) par bilan hydrologique, qui s'exprime avec la pluie (P), le débit de drainage sortant (Q), et les variations de stock d'eau (dθ) dans les structures des toitures végétalisées :

  représente les ETs réelles estimées par bilan hydrologique. Sur les sept (7) années (2563 jours) de données traitées, seules 42 à 55% des jours selon le compartiment (SE3Y : 44%, SE15Y : 44%, SE3Z : 55%, GE15Y : 44%, GI15Y : 42% et NE3Y : 50%) ont pu être validés et utilisés pour estimer les ETs. La dynamique annuelle de l'ET y est illustrée avec des valeurs journalières pouvant atteindre 10 mm au printemps et en été, et des valeurs faibles en hiver et en automne. Les études pilotes (lysimètres imitant les toitures végétalisées) menées en Europe et aux USA confirment ces résultats indiquant des ETs journalières généralement estimées entre 0,5 et 10 mm (Ebrahimian et al. 2019) pour des climats tempérés. Quant à la saisonnalité du flux d'ET, elle est liée aux facteurs atmosphériques (le rayonnement solaire, la vitesse du vent, l'humidité de l'air et la température de l'air).

Figure 2 :

 2 Figure 2:Evapotranspiration estimée par bilan hydrologique au pas journalier pour les six compartiments de la toiture végétalisée du Trappes du 8 juillet 2011 au 14 juillet 2018. Les compartiments suivants sont représentés : (a) Graminées, Extensive, 15 cm de substrat et polystyrène expansé (Y), (b) Sedum, Extensive, 15 cm de substrat et polystyrène expansé (Y), (c) Graminées, Intensive, 15 cm et polystyrène expansé, (d) Sedum, Extensive, 3 cm de substrat et polystyrène expansé (Y), (e) Sedum, Extensive, 3 cm de substrat, pouzzolane (Z) et (f) Sol Nu (N), Extensive, 3cm de substrat et du polystyrène expansé. Les facteurs liés au substrat (épaisseur et type) et aux végétaux (sedum ou graminées) ont été testés dans cette étude. Sur toute la période d'analyse, on a 966 jours secs (c'est-à-dire sans pluie et sans drainage). Ces périodes sont propices à l'étude d'impact sur l'ET du fait de la réduction du nombre de variables impliquées dans l'estimation du flux (donc moins d'incertitude). Le tableau 1 illustre une comparaison entre les différents compartiments de la TV sur l'ensemble des périodes sèches valides (415 à 486 jours). D'abord, on note un faible impact de la végétation pour les substrats de 3 cm (SE3Y et le NE3Y ont respectivement des cumuls

  ). Ensuite, pour un même type de végétation (le sedum dans notre cas), plus le substrat est épais, et plus on a d'évapotranspiration (SE3Y et le SE15Y ont respectivement des cumuls totaux de 117 mm et 315 mm). Ces observations indiquent l'importance du stock d'eau disponible pour l'ET qui augmente lorsque pour un même type de substrat, l'épaisseur augmente. Aussi, le substrat intensif (GI15Y, 331 mm) qui retient mieux l'eau évapo-transpire plus par rapport au substrat extensif plus grossier (GE15Y, 266.7 mm). Enfin, on remarque que l'ET du sedum (293.6 mm) est plus importante que celle des graminées (265.8 mm) en cumuls. Ces premières comparaisons de cumuls permettent d'illustrer l'impact de chaque configuration sur le flux d'ET. Cependant, une plus grande fiabilité de ces résultats passe principalement par l'estimation des incertitudes de mesures associées et aussi par la comparaison avec d'autres méthodes d'estimation du flux d'ET.Tableau 1: Comparaison des différentes configurations de toiture végétalisée à partir de toutes les périodes sèche (c'est-à-dire sans pluie et sans ruissellement) de juillet 2011 à juillet 2018.

			Nombre de	ET journalière (mm/j)	Cumuls d'ET sur
	Determinants		données communes/valides	(Statistique descriptive)	les jours valides comunes (mm)
			(2011-2018)	Min -max Ecartype	Moyenne
	Epaisseur du	SE15Y	282/420	0 -9.26	1.26	1.11	315
	Substrat	SE3Y	282/415	0 -2.38	0.38	0.42	117
	Type de substrat	GE15Y	264/431	0 -7.11	0.92	1.01	266.7
		GI15Y	264/359	0 -8.06	1.32	1.25	331.1
	Type de vegetation	SE15Y	310/420	0 -9.26	1.14	0.94	293.6
		GE15Y	310/431	0 -7.11	0.85	0.85	265.8
	Presence ou	SE3Y	338/415	0 -3	0.44	0.47	158.4
	abscence de vegetation	NE3Y	338/486	0 -3.2	0.66	0.5	171.5