Nguyen Duy

Minh Phan

Yann Quinsat

Claire Lartigue
email: lartigue@lurpa.ens-cachan.fr

Simulation of laser-sensor digitizing for onmachine part inspection

Keywords: In situ measurement, Laser plane sensor, Digitizing quality, Trajectory simulation

Integrating measurement operations for on-machine inspection in a 5axis machine tool is a complex activity requiring a significant limitation of measurement time in order not to penalize the production time. When using a laserplane sensor, time optimization must be done while keeping the quality of the acquired data. In this paper, a simulation tool is proposed to assess a given digitizing trajectory. This tool is based on the analysis of sensor configurations relatively to the geometry of the studied part.

Introduction

Integrating inspection procedures within the production process involves rapid decision-making regarding the conformity of parts. In the case of on-machine inspection for instance, part geometry measurements are performed in the same phase as the machining operations without removing the part from its set-up, which facilitates comparing the machined part to its CAD model for the conformity analysis. This also contributes to reducing the time allocated to measurement. Within this context, a few recent studies have focused on the use of laser-plane sensors to carry out on-machine inspections as they have a great ability to measure deviations of machining parts within a time consistent with on-machine inspection [START_REF] Dubreuil | Multi-sensor approach for multi-scale machining defect detection[END_REF][START_REF] Poulhaon | Simulation-based adaptative toolpath generation in milling processes[END_REF]. In the particular case of the milling process, laser-plane sensors can be integrated in the machine tool, the sensor replacing the cutting tool. As the measurement operation is performed while the process is stopped, one of the main issues concerns sensor path-planning. In fact, the time allocated to measurement must be minimized to preserve global production time, but the quality of the acquired data must be sufficient to measure potential deviations. The integration of a laser-sensor in a machine-tool is an issue little addressed in the literature. The fact that the sensor accessibility is increased, considering the 5 degrees of freedom plus the spindle rotation, is an opportunity. Indeed, as for laser mounted on industrial robots, this gives the sensor the possibility to scan an object from any direction even along curved paths [START_REF] Larsson | Path planning for laser scanning with an industrial robot[END_REF]. Wu et al. [START_REF] Wu | Path planning for surface inspection on a robot-based scanning system[END_REF] propose a method of sensor path planning for surface inspection on a 6 degree-of-freedom robot that automatically adapts its trajectory to the complex shape of the object by continuously changing the viewing direction of the scanner mounted on the robot. Each viewpoint of the planned path must however satisfy several constraints: field of view, scanning distance, view angle and overlap. Within the context of sensor path planning for part inspection, it is quite classical to impose some constraints to the sensor relatively to the part to be measured. In [START_REF] Bernard | Visibility theory applied to automatic control of 3d complex parts using plane laser sensors[END_REF], authors introduce the concept of visibility (local and global) to generate a sensor trajectory well-adapted to the control of complex parts. Prieto et al. propose to keep the sensor as normal as possible to the surface, while obeying a criterion of quality depending on both the scanning distance and the sensor view angle [START_REF] Prieto | Range image accuracy improvement by acquisition planning[END_REF]. Son et al. include additional constraints such as the number of required scans and the checking of occlusions [START_REF] Son | Automated laser scanning system for reverse engineering and inspection[END_REF]. Yang and Ciarallo use a genetic algorithm to obtain a set of viewing domains and a list of observable entities for which the errors are within an acceptable tolerance [START_REF] Yang | Optimized sensor placement for active visual inspection[END_REF]. The approach developed in [START_REF] Lartigue | Voxelbased path planning for 3d scanning of mechanical parts[END_REF] relies on the representation of the part surface as a voxel map, for which the size of each voxel is defined according to the sensor field of view (fov). To each voxel, a unique point of view is associated in function of visibility and quality criteria leading to a set of admissible viewpoints to ensure the surface digitizing with a given quality. Mavrinac et al. [START_REF] Mavrinac | Semiautomatic model-based view planning for active triangulation 3-d inspection systems[END_REF] formalize the search of sensor viewpoints under constraints for 3D inspection using an active triangulation system. Each viewpoint (sensor configuration) is assessed thanks to a performance function that results from the combination of constraint functions to be respected. This interesting approach seems valuable to assess the validity of a sensor trajectory prior to its optimization.

This paper deals with laser-sensor trajectories well-adapted for on-machine inspection on a 5-axis milling machine-tool. First, we propose to define an original description format of the sensor trajectory directly interpretable by the CNC of the machine-tool. The laser-plane sensor takes the place of the cutting tool in the spindle. This original format takes advantage of the additional degree of freedom given by the spindle rotation. The sensor trajectory is a series of ordered viewpoints that must satisfy a set of constraints (visibility, quality, number of viewpoints, overlaps, etc.). Prior to the stage of trajectory optimization, it might be interesting to have a tool assessing given trajectories according to the constraints to be satisfied. In this direction, the second part of the paper presents a method for simulating the digitizing from a given trajectory.

Sensor trajectory in 5-axes

A description format must be defined that can be interpretable by the CNC of the machine-tool. The study can be applied for laser-plane sensor types for which the field of view (fov) is planar (2D) (figure 2). The sensor trajectory consists in a set of ordered sensor configurations (each configuration defining a viewpoint), i.e. a set of positions and orientations. The sensor orientation is given by the couple of vectors: c v  , the director vector of light-beam axis, and L v  , the director vector of the digitizing line (figure 2). By analogy of cutting tool trajectories for which the cutter location point (CL point) is the tool extremity [START_REF] Lavernhe | Model for the prediction of 3D surface topography in 5-axis milling[END_REF], the sensor position is defined through the point C E which positions the digitizing line:

c E v d C C    * 0 .
Therefore, in the part frame, the sensor trajectory is a set of configurations

) v ; v C ( L c ; E  
expressed in the part frame as a set of coordinates (X, Y, Z, I, J, K, I*, J*, K*) (figure 2). This trajectory is expressed in the machine-tool frame thanks to the Inverse Kinematics Transform, leading to (X, Y, Z, A, C, W) where A and C are the classical angles for a RRTTT machine tool and W allows the spindle indexation. This additional degree of freedom is particularly interesting to orient the laser beam relatively to the surface.



The width of the scanning line, which can be assimilated to the width of the cutting tool, varies in function of the scanning distance d* (see figure 2). The additional degree of freedom given by the spindle indexation W permits to orient L v  . It could thus be possible to plan a sensor trajectory so that the width of the scanning line is maximized by modifying the sensor indexation.

After describing the parameter setting for the definition of a digitizing trajectory on a 5-axis CNC machine-tool, a simulation tool is proposed with the aim of assessing the quality of the acquired data.

Digitizing simulation

Considering a given sensor trajectory, a simulator has been developed to assess this trajectory with regard to a set of constraints. As reported in the literature, the most usual constraints are visibility, and digitizing quality. The laser scanner is characterized by its actual fov, which corresponds to the area of the scanning plane which is visible by the camera. The fov is thus defined by its height, H, and its widths L min and L max , each width corresponding to the minimal and maximal height in the fov. Hence, a portion of part surface is visible if it belongs to the fov so defined (figure 4). However, as the sensor moves from one configuration

) v C ( ci ; Ei  to another ) v C ( 1 ci ; 1 Ei   
, a portion of surface is digitized if it belongs to the swept volume created by the displacement of the laser plane from the first configuration to the second, as displayed in figure 4.

The orientation of the sensor relatively to the surface and the digitizing distance characterize the quality of the acquired data. Therefore, the digitizing quality is evaluated according to both the scanning distance d, and the view angle,  as we proposed in previous studies [START_REF] Bernard | Visibility theory applied to automatic control of 3d complex parts using plane laser sensors[END_REF][START_REF] Mehdi-Souzani | A knowledge database of qualified digitizing systems for the selection of the best system according to the application[END_REF]. Parameters defining the digitizing conditions are summarized in figure 4 andtable1. To develop our approach, we take advantage of the formalism proposed in [START_REF] Wu | Path planning for surface inspection on a robot-based scanning system[END_REF][START_REF] Mavrinac | Semiautomatic model-based view planning for active triangulation 3-d inspection systems[END_REF], in which the constraints to be verified are expressed as a combination of functions. Such formalism is rather flexible as it permits to add or remove constraints in function of the complexity we want to introduce in trajectory generation. The CAD model of the part is tessellated through a STL format. This gives a mesh defined by a set S T of n triangular facets T j . Each facet T j is defined by 3 vertices denoted i j V , and the normal vector to the facet is denoted j n  (see table 1). In a first approach only 2 functions are considered: the visibility function and the quality function. These functions are applied to the tessellated model.

Visibility function

The visibility function is used to determine the facets which we denote as seen by the laser sensor. As the trajectory is a set of sensor configurations, the visibility function is defined for each trajectory segment, i.e. between two successive configurations. For each facet T j of the CAD model, the function is defined as a combination of two functions as expressed in equation 1:

) T ( F ) T ( F ) T ( F j s j V j * V    (1)
The swept facet function F V (T j ) checks if the facet belongs to the volume swept by the laser beam between 2 configurations

1 Ei Ei C C V  .
A facet belongs to the swept volume if all its vertices belong to the swept volume:

            otherwise , 0 V V , 3 , 1 k if , 1 ) T ( F 1 Ei Ei C C k j j V (2)
Generally, the view angle is limited [START_REF] Mahmud | 3D part inspection path planning of a laser scanner with control on the uncertainty[END_REF]. If the angle between the normal vector to the facet and c v  exceeds the maximal view angle  max , the facet is not seen. This is expressed by:

      otherwise , 0 ) cos( v n , 1 ) T ( F max c j j s     (3)
At the end of this stage, when the whole sensor trajectory is considered (i.e. when all the trajectory segments defined by

) v C ( ci ; Ei  ) v C ( 1 ci ; 1 Ei  



are considered), all the facets verifying 1 ) T ( F j * V  are characterized as seen and define the set

  1 ) T ( F , S T S j * V T j s T    .

Quality function

The visibility of the facet does not ensure the digitizing quality. Indeed, numerous studies point out the importance of the digitizing distance and of the view angle on the digitizing noise, factors that strongly influence the digitizing quality [START_REF] Bernard | Visibility theory applied to automatic control of 3d complex parts using plane laser sensors[END_REF][START_REF] Lartigue | Voxelbased path planning for 3d scanning of mechanical parts[END_REF][START_REF] Mahmud | 3D part inspection path planning of a laser scanner with control on the uncertainty[END_REF][START_REF] Mehdi-Souzani | A knowledge database of qualified digitizing systems for the selection of the best system according to the application[END_REF]. Quality is ensured when the digitizing noise is lesser than a threshold, threshold generally given by the user in function of the considered application. This involves admissible ranges for both the digitizing distance and the view angle allowing the definition of the quality function as follows:

) T ( F ) T ( F ) T ( F ) T ( F j ws j wsd j * j ws V     (4)
In equation ( 4), F wsd and F ws account for the quality in terms of digitizing distance and view angle respectively. Therefore, a facet is said well-seen in terms of digitizing distance if all its vertices belong to the admissible range of digitizing distances I ad :

           otherwise , 0 I d , 3 , 1 k if , 1 ) T ( F ad V j wsd k j (5)
A facet is said well-seen in terms of view angle, if the angle between the normal vector to the facet and c v  , belongs to the admissible range of view angles defined by  1 and  2 :

       otherwise , 0 ) cos( v n ) cos( if , 1 ) T ( F 2 c j 1 j ws      (6)
At the end, all the facets verifying 1 ) T ( F j ws  are characterized as well-seen and define the set

Results and discussion

The objective is here to validate our simulator by comparing digitizing obtained using our simulator to the actual digitizing, and considering various trajectories. The simulator is tested using a case study, and for the laser sensor Zephyr KZ25 (www.kreon3d.com). Although most of the sensor parameters are given by the manufacturer, a protocol of sensor qualification is required to identify the actual sensor parameters such as the dimensions of the fov, or the limited view angle, but also to identify quality parameters that define the admissible ranges of digitizing distances and view angles.

Sensor parameters

First, the dimensions of the fov are identified by simply measuring a reference plane. As the intersection of the reference plane and the laser-beam is a line, the height of the fov is identified by observing if the line is visible in the CCD. The experiment gives H = 50mm. According to the protocol defined in [START_REF] Mehdi-Souzani | A knowledge database of qualified digitizing systems for the selection of the best system according to the application[END_REF], the evolution of the digitized noise, denoted , is identified in function of the digitizing distance and the view angle. The digitizing noise accounts for the dispersion of the measured points with respect to a reference element, and it is usually evaluated by measuring a reference plane surface for different digitizing distances and various view angles. The evolution of the digitizing noise in function of the digitizing distance exhibits a significant decrease of the noise from the bottom position to the top position in the fov (figure 5a). On the other hand, the evolution of the noise in function of the view angle does not show a significant trend (figure 5b). However, it can be pointed out that the maximal view angle is equal to   60 max 

, and that for the whole range     60 ; 0 , the noise remains lesser than 0.015mm. Considering that value as the threshold  ad for quality, the admissible range of digitizing distances I ad is defined by   50 ; 20 mm. Those two intervals guarantee a digitizing with a noise lesser than  ad = 0.015 mm.

Simulator tests

The sensor trajectories used to test our simulator are classical pocket-type trajectories. For the first tests, the sensor orientation is constant, and the trajectory consists in a set of points C E defined at a constant altitude z (figure 6). To assess the simulator, the simulated digitizing is compared to the actual one. For this purpose, actual digitizing was carried out using a Coordinate Measuring Machine (CMM) equipped with a motorized indexing head, which enables the scanner to be oriented according to repeatable discrete orientations. We choose to assess our simulator using a CMM, because a 3-axis Cartesian CMM is a machine with less geometrical defaults than a machine-tool. But this does not change anything in the principle of our simulator. On the CMM, the orientations of the sensor are given by the two rotational angles A and B. Therefore, the trajectories expressed in the part coordinate system (for the simulation) must first be expressed in the CMM coordinate system (figure 6). Different trajectories for various digitizing distances and sensor orientations have been tested. Only results associated with one orientation (A = 0°; B = 90°) and two different distances (z = 0 and z = 30mm) are commented in this paper. The algorithm is applied to the tessellated CAD model of the part, and facets are classified in the corresponding set according to visibility and quality functions as proposed in section 3. To simplify the representation, a color code is adopted: well-seen facets are green, poorly-seen facets are orange, and not-seen facets are red (table 2). On the other hand, the actual digitizing gives a point cloud which is registered onto the mesh model. For each facet, a cylinder, whose basis is the triangle defining the facet and the height is the maximal measurement error, is created. The set of digitized points belonging to the cylinder so defined corresponds to the actual digitized facet. To compare actual digitizing to its simulation, we have to characterize each facet according to visibility and quality functions in the same way. In this direction, we consider that a facet is not-seen if the density of points associated with the facet is less than 5 points/mm 2 ; the facet color is red. For each facet, the geometrical deviations between the digitized points and the facet are calculated. The associated standard deviation accounts for the actual digitizing noise. If the noise is greater than the threshold  ad = 0.015mm, the facet is tagged as poorly-seen, and its color is set to orange. Conversely, if the noise is lesser than  ad , the facet is tagged as well-seen, and its color is green. Results displayed in table 2 bring out the good similarity between simulation and actual digitizing. This is particularly marked for the trajectory z = 0. However, some differences exist for which the simulator underestimates the digitizing. A whole area which appears red in the simulation is green in the actual digitizing (on the left of the part for the trajectory z = 30 mm for instance). This is likely due to the fact that the digitizing noise is evaluated using an artefact with a specific surface treatment which makes the surface very absorbing, whereas the part is coated with a white powder that matifies the surface. Digitizing is thus facilitated. Nevertheless, the simulator turns out to be an interesting predictive tool prior to sensor trajectory planning.

Conclusion

Within the context of on-machine inspection using laser-plane digitizing systems, sensor trajectory planning is a challenge. To ensure the efficiency of the measurement, it is necessary to minimize measurement time while ensuring the quality of the acquired data. The presented work proposes a description format of a sensor well-adapted to on-machine inspection on 5-axis machine-tools. Given a digitizing trajectory, a simulation tool of the acquired data quality is presented. After a real digitizing, a good similarity between simulation and actual digitizing can be observed. The simulator is thus an interesting predictive tool that can be used to assist in finding the best strategy to digitize the part with a quality consistent with geometrical deviations obtained in milling.

Fig. 1 .

 1 Fig. 1. Laser-sensor in the machine-tool

Fig. 2 .

 2 Fig. 2. Parameters defining the sensor trajectory for on-machine inspection

Fig. 4 .

 4 Fig. 4. Parameters of the laser sensor and digitizing volume between 2 configurations

  the facet Tj k j V d , distance from the bottom of fov to the vertex k j V 

  other seen facets are tagged as poorly-seen and in turn define a set

Fig. 5 .

 5 Fig. 5. Noise in function of the scanning distance (a) and the view angle (b).

Fig. 6 .

 6 Fig. 6. Scanning trajectories for test (A = 0°; B = 90°, z= 0 mm).

Table 1 .

 1 Parameters for surface digitizing

	Mesh Parameters						Digitizing parameters
	ST, set of n triangular facets	C V	Ei C	Ei	1  , swept volume between 2 configurations
	Tj, facet j ,	T	j		S T	,	j		  n , 1	dE, distance from the bottom to CEi in the fov
	Sv, set of vertices						H, distance from the bottom to the top in the fov
	k j V , vertex of a facet Tj,	k 	  3 , 1	max, limited view angle of the sensor

Table 2 .

 2 Results for actual and simulated digitizing.

	Simulation of the digitizing	Actual digitizing
	A=0°; B=90°;
	z = 0 mm
	A=0°; B=90°;
	z = 30mm


W Guerin

M T Rouabah

R Kaiser

Light interacting with atomic ensembles: collective, cooperative and mesoscopic effects

Light interacting with atomic ensembles: collective, cooperative and mesoscopic effects

W Guerin, M.

I. INTRODUCTION

'Cooperative scattering' of light by a collection of twolevel atoms has become a hot topic in the recent years, with a lot of experimental results on superradiance [1][2][3][4], subradiance [5] and 'cooperative' shifts [4,[6][7][8][9][10] in the 'single-photon superradiance' regime [11][12][13], or equivalently in the linear-optics regime. By contrast, the experiments performed in the 1970s-1980s were performed in a regime where a large number of atoms are excited [START_REF] Feld | Coherent Nonlinear Optics[END_REF][START_REF] Gross | [END_REF]. Related experiments with multi-level atoms [16] or artificial atoms [17] illustrate that the phenomena studied in two-level systems can find their application in other fields of research. Despite this recent interest, it seems that a common agreement on what should be called cooperativity is still lacking. It has recently been suggested that 'cooperativity' should be reserved to effects that could not be explained by 'the mean-field approach of traditional optics' [18], and that such effects would appear for dense atomic samples only [START_REF] Javanainen | Talk at the 46 th Winter Colloquium on the Physics of Quantum Electronics (PQE)[END_REF]. Here, 'dense' means that the atomic density ρ should not be very small compared to k 3 , where λ = 2π k is the wavelength of the atomic transition. However, our recent results on sub-and superradiance [START_REF] Dicke | [END_REF] show that these effects are controlled by the resonant optical depth of the sample and not by its density [3,5]. If one agrees that Dicke sub-and superradiance in extended samples [21,22] deserve being called 'cooperative' effects, the restriction to high spatial densities should clearly be dropped.

Although everyone certainly agrees on the physics, terminology confusion does not help in understanding this * william.guerin@inln.cnrs.fr topic. It can also be a real barrier when trying to communicate with experts from other, neighboring, fields. With this problematic in mind, in this article, we would like, first, to discuss the concept of 'cooperative scattering', and, second, to point out the possible link between the so-called 'mean-field approach of standard optics' used in [18] and what is called the 'independent scattering' approximation in mesoscopy [23][24][START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF].

Unfortunately, terminology confusion has always been present in this field. For instance, Bonifacio and Lugiato proposed the term 'superfluorescence' instead of superradiance when there is no initial macroscopic dipole in the atomic system but only uncorrelated excited atoms [START_REF] Bonifacio | [END_REF]. As soon as propagation effects are involved, superfluorescence can also be seen as the transient regime of 'amplified spontaneous emission' [START_REF] Feld | Coherent Nonlinear Optics[END_REF], a well-known, and somewhat trivial phenomenon in laser physics, although a difference does exist and has been studied in Ref. [27]. Dicke himself spoke of a mirrorless 'coherence brightened laser' [START_REF] Dicke | Proceedings of the third international congress on Quantum Electronics[END_REF]. Another example is the analogy or confusion between subradiance [5,[START_REF] Pavolini | [END_REF] and 'radiation trapping' [30][START_REF] Molisch | Radiation Trapping in Atomic Vapours[END_REF][START_REF] Labeyrie | [END_REF]. The difference and early confusion between the two phenomena was pointed out by Cummings [33].

Although we do not pretend to be able to solve all controversies, we feel that it may be instructive to tackle these questions from a broader perspective, which could be, in general, the study of all collective effects in light-atom interactions, or the transition from microscopic properties of light scattering by atoms to macroscopic, optical and transport properties of extended and/or dense atomic vapors. This subject is even older than Dicke's paper, as it was investigated by Rayleigh, Lorentz, Lorenz, Ewald, Oseen, and many others [START_REF] Born | Principle of Optics[END_REF].

We will focus on experiments in which near-resonant light from a laser is exciting a sample of N two-level atoms at rest. The experimental observables that can be measured include extensive quantities that increase with the atom number, such as the total amount of scattered light, or some intensive quantity that could a priori be independent of the atom number. If an extensive quantity is not simply given by the atom number times the same quantity for a single-atom, or if an intensive quantity does depend on the atom number, nontrivial collective effects probably are at work. A first example of an extensive quantity is the total amount of scattered light, which, in some limit, is proportional to the atom number. Another example of an extensive quantity is the dephasing accumulated by the wave crossing the sample, which is also increasing with the atom number, and the notion of refractive index, which is a macroscopic (collective) quantity provides an efficient description. Collective effects thus take place in most situations, with some effects depending on the atomic density ρ, and others on the size L of the medium, as detailed in the following. Note that as soon as propagation effects or quantities associated to transient phenomena become relevant, the size of the sample will play an important role, and the important parameter turns out to be the optical depth. On the other side, for quantities that can be defined in steady state for a bulk medium, the density is the relevant parameter.

II. COLLECTIVE EFFECTS THAT DEPEND ON THE DENSITY

Obviously, all effects or quantities that can be defined for an infinite medium depend on the atomic density ρ, which is the only parameter characterizing the medium.

A. At low atomic density

The most simple example is the medium susceptibility χ. At low density, it is related to the atomic polarizability α by χ = ρα. For simple two-level atoms and in the linearoptics regime (low intensity), the polarizability is

α = 6π k 3 × -1 2∆ Γ + i = 6π k 3 × -2∆ Γ + i 1 + 4∆ 2 Γ 2 . (1)
The complex refractive index n is defined from the susceptibility by n 2 = 1 + χ. At low density, χ ≪ 1 and n ≃ 1+χ 2. Inside the medium, the wavevector is changed from k to nk. The real part of the refractive index is thus responsible for the dispersive properties of the material and its imaginary part for the attenuation of the wave. One can thus define a characteristic extinction length (or linear extinction coefficient) for the intensity of the light propagating in the incident optical mode,

ℓ -1 ex = ρkIm(α) = ρσ 0 1 + 4∆ 2 Γ 2 , (2)
where σ 0 = 6π k 2 = 3λ 2 (2π) is the resonant scattering cross-section.

The susceptibility is not sufficient to describe the behavior of the light inside the medium. In particular, it does not make a difference if the attenuation is due to absorption, when the electromagnetic energy is transferred to the medium, or to scattering, when the energy is just removed from the incident electromagnetic mode to feed other spatial modes. Scattering is often seen as being due to impurity, or granularity of matter, and as such, it is neglected in the susceptibility, which corresponds to a continuous-medium approximation. It is often stated, in particular in the recent literature about cooperative scattering (see, e.g., Refs. [START_REF] Svidzinsky | [END_REF][36][37]), that the continuousmedium approximation is only valid when the density is high, ρk -3 ≳ 1. However, the continuous-medium approximation does actually neglect the scattered light and treats it as if it were absorbed. This can indeed be a very good approximation when one is interested in the coherent transmission and not in the scattered light. It is important to note that this approximation is not restricted to high spatial densities (as in condensed matter or very high density vapors) but also holds in the dilute limit where the interatomic distances are larger than the optical wavelength. The refractive index is thus very useful for describing light propagation in dilute atomic clouds. It has been successfully used to describe subtle experiments involving nonlinear effects [38] or photonic band gaps [39,40]. As discussed below, it turns out that the continuous-medium approximation for atomic vapors is in fact more accurate for dilute than for dense vapors [18,41].

In order to describe the scattered light, another approach is necessary, in which one basic quantity is the mean-free path ℓ sc = 1 (ρσ sc ). If we only consider elastic scattering, which is valid if the incoming light has a weak intensity [START_REF] Cohen-Tannoudji | Atom-Photon Interactions: Basic Processes and Applications[END_REF], the (elastic) scattering cross-section is related to the polarizability by [23]

σ sc = k 4 6π α 2 . (3)
In a large medium of size much larger than the mean-free path, light will be scattered many times before escaping. In this case, many observables can be very well described by a diffusion equation for the electromagnetic energy density, at the condition to perform an average over the disorder configurations. This fact is often surprising for quantum opticians, who expect that light scattered off a disordered medium produces a speckle pattern. However, it is well known to people from mesoscopic physics that after averaging over the disorder, the remaining intensity distribution, or its temporal dynamics in a pulsed experiment, fits perfectly well the prediction of the diffusion equation, although this approach neglects all interference effects [43] [44]. At low density, there is one well-known exception, which corresponds to an enhanced backscattering [45][46][47]. The diffusion coefficient is thus another, important macroscopic quantity, which is governed by the atomic density, and which allows describing the dif-fuse light. In three-dimensional space it reads [START_REF]A more general expression is D0 = ⟨z 2 ⟩ (2dτ ), where d is the number of dimensions and ⟨z 2 ⟩ is the second moment of the step-length distribution P (z). For a random walk with constant step ℓ, ⟨z 2 ⟩ = ℓ 2 , but for the exponential distribution due to Beer's law (Sec. III A), P (z) = exp(-z ℓ) ℓ and ⟨z 2 ⟩ = 2ℓ 2 , which gives Eq. (4)[END_REF]

D 0 = v E ℓ sc 3 = ℓ 2 sc 3τ tr , (4)
where v E = ℓ sc τ tr is the energy transport velocity inside the medium and τ tr the transport time [23,24]. Here, we considered only isotropic scattering such that the transport length equals the mean-free path. For near-resonant light, a remarkable property of cold atomic vapor is that τ tr = τ at , the lifetime of the excited state, independently of the detuning [23,[START_REF] Labeyrie | [END_REF]. In a nonabsorbing passive medium like an atomic vapor probed by a weak intensity laser, the attenuation of the coherent propagating wave is only due to elastic scattering. The mean-free path and the extinction length are thus equal, which leads to the relation

k 3 6π α 2 = Im(α) , (5)
closely related to the so-called optical theorem [23,[START_REF] Newton | [END_REF].

There is thus a close link between the diffusion coefficient, or the mean-free path, governing the transport of the diffuse light, and the susceptibility, governing the transmission of the coherent light.

B. At high atomic density

When the typical distance between atoms becomes on the order of the wavelength, i.e., at large density ρk -3 ≳ 1, several nontrivial effects must be taken into account to describe the coherent propagation and the transport of the diffuse light.

First, the relation between the susceptibility and the polarizability has to be modified to include the Lorentz local field [START_REF] Born | Principle of Optics[END_REF],

χ(ω) = ρα(ω) 1 -ρα(ω) 3 . (6)
This leads to the famous Lorentz-Lorenz shift [50],

∆ LL = -πρk -3 Γ , (7)
which depends on the density. Here Γ = τ -1 at is the linewidth of the transition. Although the derivation of Eq. ( 6) is based on mean-field arguments and uncontrolled approximations, it yields correct results in many situations [51]. With hot atomic vapors, a densitydependant shift of the resonance line has indeed been observed [7,52]. With cold atoms, however, recent results suggest that correlations between scatterers at high density are not negligible and the mean-field approximation breaks down [18,41]. The difference with the hot vapor case is probably due to the absence of inhomogeneous broadening, which may suppress the correlations for hot atoms [10,53]. This argument is also consistent with the recent experiment reported in Ref. [2].

Closely related to the Lorentz-Lorenz shift is the 'cooperative' Lamb shift [4,6,7,9,10,13,50,53]. It is also a density effect but it is influenced by the finite size and the geometry of the medium. We will discuss this effect in more detail in Section V.

On the mesoscopic side, the diffusion coefficient should also be corrected at high density, this is the so-called "weak localization" (WL) correction [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF],

D ∼ D 0 1 - 1 (kℓ sc ) 2 . ( 8
)
Note that short-path diagrams not captured by the diffusion approximation are expected to yield supplementary 1 kℓ sc corrections to the transport [54,55]. Weak localization corresponds to a slowing down of diffusion due to the constructive interference between reversed-paths of closed loops, which increases the probability of returning to the original point. The parameter 1 (kℓ sc ) quantifies the amount of 'disorder'. On resonance, since

σ sc = σ 0 = 6π k 2 , 1 (kℓ sc ) ∼ 6πρk -3 .
Weak localization is thought to be a precursor of Anderson (or strong) localization [56,57], in particular in the self-consistent theory of localization [START_REF] Vollhardt | Electronic Phase Transition[END_REF]. Anderson localization is expected when kℓ sc ∼ 1 (Ioffe-Regel criterion [START_REF] Ioffe | Progress in Semiconductors[END_REF]) and corresponds to the complete absence of diffusion. Although weak localization has been observed for light in different systems, including cold atoms using the coherent back-scattering effect [START_REF] Labeyrie | [END_REF][61][62][63], Anderson localization of light in 3D has still not been observed in any experiment [64,65], and cold atoms may provide a route towards this goal [66].

Beyond transport properties, other subtle effects appear in the correlations and fluctuations of scattered light, such as universal conductance fluctuations [24,[START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]. These correlations come from common scatterers involved in different paths. If the so-called C 1 correlation is short-range and corresponds to the standard speckle grain of traditional statistical optics [START_REF] Goodman | Speckle Phenomena in Optics: Theory and applications[END_REF], the long-range C 2 and C 3 correlations are highly nontrivial effects, which become nonnegligible when the Thouless conductance g is small, i.e., near the strong localization regime. These effects have not been studied yet using multiple scattering of light in cold atoms.

III. COLLECTIVE EFFECTS THAT DEPEND ON THE OPTICAL DEPTH

As soon as propagation effects are important, the macroscopic local quantities (susceptibility, diffusion coefficient) are not sufficient to describe experimental observables, but should be combined with the size of the medium. The relevant quantity then turns out to be the optical depth.

A. Coherent and diffuse transmission

The complex susceptibility of an atomic medium is measured by the attenuation or the dephasing of a probe wave crossing the sample. It is then obvious that the finite size L of the medium enters the problem. The phase shift induced by the sample will be

ϕ = [Re(n) -1]kL = Re(χ)kL 2 , (9)
and the transmitted intensity will be T

= exp[-Im(χ)kL].
The above relation is called the Beer's or Beer-Lambert law. The argument of the exponential is called the optical depth (or thickness) b,

i.e., b = -ln(T ) = Im(χ)kL = ρσ sc L = L ℓ sc . ( 10
)
Although this seems completely trivial, it is interesting to note that a consequence of this exponential attenuation is the nonlinear evolution of the total fluorescence (or 'offaxis scattering') with the atom number. The total fluorescence F corresponds indeed to what is not transmitted in the incident mode. We thus get F ∝ 1e -b , which is linear only for low b and saturates for high b. This saturation can be intuitively understood as a 'shadow effect': atoms at the back of the sample do not radiate because they are not illuminated by the incoming laser. For the same reason, if we sweep the detuning of the probe beam and measure the fluorescence spectrum, it will be a Lorentzian of width Γ at low optical thickness only, and deviate from a Lorentzian line shape at larger optical thickness, even though the scattering cross-section for single-atoms

σ sc = σ 0 (1 + 4∆ 2 Γ 2 ) is still Lorentzian.
More precisely, it induces an effective broadening and a strong directional dependence of the fluorescence, as shown experimentally in [START_REF] Labeyrie | [END_REF]. As a consequence, such behaviors are not unique to cooperative effects, even if they are also consistent with a full coupled-dipole model including those effects [69,70]. The shadow effect can also explain the reduction of the radiation pressure force reported in [71,72], first interpreted as a signature of cooperativity [71]. The different collective effects contributing to a modification of the radiation pressure force will be discussed in detail in a forthcoming publication [73].

To illustrate the broadening and saturation effects induced by Beer's law, we show in Fig. 1(a) the behavior of the fluorescence of an atomic cloud as a function of the detuning and the optical depth. Considering a Gaussian atomic density distribution profile, the total scattering cross-section of the cloud according to Beer-Lambert law, reads [72,73]

Σ sc = N σ sc × Ein(b) b , ( 11
)
where Ein is the integer function [START_REF] Weisstein | Ein Function, From MathWorld-A Wolfram Web Resource[END_REF] Ein

(b) = b 0 1 -e -x x dx = b 1 + ∞ n=1 (-b) n (n + 1)(n + 1)! . (12)
Here, b = √ 2πρ 0 σ sc R z is the optical depth that would be measured using a small beam crossing the cloud through its center, ρ 0 is the peak density and R z the rms radius along the propagation axis. Although Beer's law neglects diffraction and refraction effects inside the sample, which are a priori not negligible at high b and small nonzero detuning or for very small sample, it appears that such a simplified model is sufficient to qualitatively explain the results of [69].

The angular dependence is, however, not possible to compute analytically, and numerical simulations based on a random walk process can be used. For an isotropic cloud illuminated by a plane wave, the anisotropy of the emission diagram results from the multiple scattering of light before escaping the sample [73]. We illustrate this anisotropy in Fig. 1(b). It should be noted that it can lead to nonintuitive behaviors. For instance, if light is detected near the forward direction, the measured fluorescence can decrease as the atom number increases or as the detuning is reduced ('self-absorption') [START_REF] Labeyrie | [END_REF]. Although a simple random walk model completely neglects all wave or coherence effects, it has been shown in Ref. [72] that, for a large and dilute cloud of two-level systems, the corresponding emission diagram agrees very well with the emission diagram computed from a full coupled-dipole model after averaging over the disorder configurations, except in narrow angular ranges around the forward and backward directions. The forward lobe [2,11,[START_REF] Dicke | [END_REF]22], absent in an incoherent random walk model, can be explained by diffraction/refraction of light by a continuous index distribution [START_REF] Bachelard | [END_REF]. Around backward direction, a narrow cone is visible, which can be explained neither by a random walk model nor by a homogeneous index of refraction. This is a signature of interference in a disordered systems, robust against configuration average and well studied in mesoscopic physics (coherent backscattering) [45][46][47][START_REF] Labeyrie | [END_REF][61][62][63]. For a Gaussian cloud geometry, analytical expressions for the lobe and cone shapes have been obtained in the double scattering limit [76].

Multiple scattering obviously depends on the optical thickness. Although the diffusion coefficient depends on the density, light should first escape the sample before being detected. In steady state, the main consequence of multiple scattering is the complex emission diagram discussed above. Note that for a slab geometry (homogeneous medium of infinite transverse size and finite width L), space can be divided into two parts and we can speak of diffuse transmission and diffuse reflection. A wellknown result is the asymptotic decrease of the diffuse transmission scaling as 1 L, which corresponds to Ohm's law for photons [Fig. 2],

T diff = 1 + ξ b + 2ξ , ( 13
)
where ξ ≃ 0.7104 [24,77]. This result can be obtained from the diffusion equation or random walk simulations [Fig. 2(a)], i.e., a model of light propagation that ignores coherence and interference effects. We have also per- formed numerical simulations of the diffuse transmission through a slab of atoms, using the coupled-dipole model, in the scalar approximation, where only the far-field term (∝ 1 r) of the dipole-dipole interaction is present. We clearly observe a departure from Ohm's law at large density [Fig. 2(b)]. The precise origin of the deviation is still under investigation and might be due to cooperativity. Note that the observed deviation corresponds to a relative increase compared to Ohm's law for photons and is thus not consistent with a decrease of the diffusion coefficient due to weak localization. However, it resembles recent observations made with classical scatterers [78], in which the increased transport is attributed to near-field scattering, where evanescent waves open new channels of transmission. In our model though, the near-field terms are neglected, and the physical origin of the increased transmission remains to be explained. Recent numerical studies on the distribution of excitation inside atomic samples of different geometries [70] may also be related to our observation. The temporal dynamics of the diffuse light is also governed by the optical thickness. After some time of illumination, a sudden switch-off of the exciting laser leads to a slow decrease of the fluorescence due to multiple scattering. This 'imprisonment of radiation' [30], or 'radiation trapping' [START_REF] Molisch | Radiation Trapping in Atomic Vapours[END_REF], has been studied in cold atoms [START_REF] Labeyrie | [END_REF]79], taking also into account subtle effects like the frequency redistribution induced by the Doppler shift [80,81] or the multilevel structure [82]. Neglecting those effects, one can easily find the scaling of the radiation trapping time with the optical depth. For a Gaussian random walk in 3D, we have ⟨r 2 ⟩ = 6Dt. The average number of scattering events for escaping photons is the ratio between the time spent in the system and the scattering time τ at ,

⟨N sc ⟩ = t τ at ∼ ⟨r 2 ⟩ 6Dτ at , (14)
with D = ℓ 2 sc (3τ at ). When ⟨r 2 ⟩ ∼ R = bℓ sc 2, the radiation can escape the system, leading to ⟨N sc ⟩ ∼ b 2 8. Radiation trapping times are thus expected to scale as b 2 , with a precise numerical prefactor that depends on the geometry [START_REF] Labeyrie | [END_REF].

B. Superradiance and subradiance in extended and dilute samples

The previous effects all depend on the detuningdependent optical depth,

b(∆) = b 0 1 + 4∆ 2 Γ 2 , ( 15
)
where b 0 is the on-resonance optical depth. For a cloud of size R, and using σ 0 ∼ 1 k 2 , we have b 0 ∼ N (kR) 2 , with a numerical prefactor that depends on the geometry. We now discuss why the super-and subradiant decay rates, measured at large detuning such that attenuation or incoherent multiple scattering are negligible, depend on b 0 , independently of the detuning, in the case of an extended R ≫ λ and dilute (ρk -3 ≪ 1) sample.

The physical argument to understand why the cooperativity parameter for super and subradiant decay is b 0 , is the following. Consider a sample of finite size ∼ R radiating in free space. Although the number of modes in free space is infinite, the boundary condition due to the sample surface (∝ R 2 ) sets limitations on the modes that are efficiently coupled to the sample. In particular, the diffraction limit ensures that no mode with a divergence smaller than θ ∼ 1 (kR) can be emitted from the sample and, in 3D, the total number of modes M efficiently coupled to the sample is related to the sample surface, M ∼ (kR) 2 . It means that if we choose an arbitrary, infinite basis to express the modes in free space, a number M of those modes are enough to describe the radiation pattern of the sample. Then, if the number N of atoms in the sample is larger than the number M of modes, the emission will be cooperative because, in average, N M atoms emit in the same modes: they are thus coupled to each other via their common coupling to the electromagnetic mode (Fano coupling). We conclude that the 'cooperativity parameter' in this problem is the ratio N M , which turns out to be the on-resonance optical depth b 0 , up to a numerical prefactor. This argument is consistent with the Dicke limit R ≪ λ, for which the only possible outgoing mode is a spherical wave, i.e., M = 1, and the superradiant enhancement factor is N . This reasoning was given in brief in [5,83,84] and is consistent with the classification of superradiance given in [85].

Let us note that super-and subradiance for N = 2 has a marked difference with the case of N ≫ 2. First, in order to obtain large dipole-dipole coupling of two atoms in free space, these atoms need to be at a distance comparable to the optical wavelength. When an atomic pair is separated by many wavelengths, interferences between light emitted by these atoms can still occur [86]. However a modification of the atomic lifetime requires the interatomic distance to be smaller than or comparable to the wavelength [87,88]. For small atomic distances, near-field dipole-dipole coupling, scaling as 1 r 3 , becomes dominant, while super-and subradiance obtained for many atoms in the dilute limit is relying on the far-field dipole coupling, scaling as 1 r. This long-range behaviour of the far-field dipole-dipole coupling is essential for the collective scaling of the Dicke subradiance observed in [5]. Let us also note that the competition between the near and far-field dipole-dipole coupling has been discussed in [89,90] and has been called 'van der Waals dephasing' in [START_REF] Gross | [END_REF]. It is interesting to note that while for N = 2, the eigenstates of the near-field part of the interaction Hamiltonian H NF are also eigenstates of the far-field interaction H FF (both being proportional to the same σ 1 Pauli matrix), for N > 2, H NF and H FF are not proportional if the distances between all atoms are not equivalent (which can be obtained in the particular case of 3 atoms equally spaced on a ring or four atoms equally spaced on a sphere). In this case H NF and H FF do not commute and have different eigenvectors (see Sec. 4 of [START_REF] Gross | [END_REF]). This particular situation does not prevent the total Hamiltonian to have eigenvalues corresponding to long-lived (subradiant) and short-lived (superradiant) modes. However the symmetries of the eigenstates (and potentially their sensitivity to perturbations) will differ from those of H FF .

As explained from the beginning, it is not surprising that the size of the sample is an important parameter to take into account, since the superradiant and subradiant decays are transient phenomena, which means that light must escape the sample. However, from a mathematical point of view, when looking at the coupled-dipole model [91][92][93][94][95], it is not obvious which combination of the parameters yields to correct scaling laws for the different features that can be observed. In this model, once the geometry of the cloud is fixed (and spherical), there are three independent parameters: the atom number N , the size R and the detuning of the driving laser. In the limit of a very large detuning, after a long enough illumination, the system reaches a steady state given by the 'timed-Dicke state' [11,93,95]. Once the laser is switched off, the decay of this state could be governed by any combination of N and R. In particular, it might be considered more intuitive that the density ρ ∼ N R 3 is the important parameter, in particular because the dipole-dipole coupling terms, which decay as 1 (kr) with the interparticle distance r, become very small as the density decreases. But even if the coupling terms are small, each N = 4000 FIG. 3. Decay of the total scattered power after the switch-off of the driving laser at t = 0 computed from the coupled-dipole model in the scalar approximation, as in [5], averaged over only two configurations. The time axis is normalized to the lifetime of the excited state τat and the vertical axis to the steady-state scattered power. The driving laser has a large detuning, ∆ = 50Γ, and has been applied for a duration of 30τat so that the system reaches its steady state. In panel (a), the atom number N = 4000 is kept constant and the size is varied, b0 = 3N (kR) 2 is varied from 30 to 2.3. In panel (b), both N and R are varied such that N kR is kept constant, b0 is varied from 15 to 3.75. In panel (c), N and R are varied such that b0 = 15, and in panel (d) N and R are varied such that the density is kept constant, n0k -3 = 0.01, and b0 is varied from 11 to 17.5.

atom is coupled to N -1 other atoms, and N can be huge, which, somehow, balances the low density and makes b 0 ∼ N (kR) 2 to be the scaling parameter. Note that the 1 r term in the dipole-dipole interaction bears all the relevant ingredient for 'long-range' interaction, which, as in gravitational or Coulomb interactions, yields to wellknown size-dependent effects [START_REF]Long-Range Interacting Systems[END_REF]. This parameter also appears in the statistical properties of the eigenvalues of non-Hermitian Euclidean random matrices such as the one involved in the couple-dipole model [START_REF] Skipetrov | [END_REF]98] and the early (superradiant) decay of the timed-Dicke state [11], or 'phased state', has been computed analytically by many authors, Γ sup ∝ b 0 [21, 22, 35-37, 93, 99, 100]. Numerical and experimental evidence have been given in [5,84] that the long time decay is also governed by b 0 , Γ sub ∝ 1 b 0 . Here, we show in Fig. 3 that not only the short and long time limits are both depending on b 0 , but the full decay curve is in fact only dependent on b 0 . An analytical function for describing this decay curve remains to be found. We have checked that this is neither a power law, nor a stretched exponential.

IV. THE MEAN-FIELD APPROACH OF TRADITIONAL OPTICS AND THE INDEPENDENT SCATTERING APPROXIMATION OF MESOSCOPY

It was pointed out in recent theoretical [18] and experimental studies [41] that Eq. ( 6), which includes Lorentz local field correction, is inaccurate for cold atoms at high density. Indeed, another term, usually not considered, has the same order of magnitude. Following [41], the more precise expression is given by

χ(ω) = ρα(ω) 1 -ρα(ω)(1 3 + β(ω)) , (16)
where β(ω) ∼ α(ω)k 3 and specifically depends on the geometry. This term is neglected in the mean-field approach leading to Eq. ( 6) even though on resonance, α(ω) ∼ k -3 , and thus β(ω) has the same order of magnitude than the 1 3 corresponding to the densitydependent Lorentz local-field correction. To our knowledge, this result has been first obtained by Saunders and Bullough [101,102] (see Eq. ( 22) of [102]) and has been later rediscovered several times [91,103]. One hypothe-sis for the success of Eq. ( 6) could be that the mean-field result is recovered if some mechanism breaks the correlation between scatterers, as it could be the case, e.g., for hot vapors with the Doppler broadening [10,53]. At this point, it is interesting to point out the close link between Eq. ( 16) and the weak localization correction of the diffusion coefficient (Eq. 8). First, we note that Eq. ( 16) can be rewritten

χ(ω) = ρα(ω) 1 -ρα(ω) 3 -ρα(ω)β(ω) , (17)
which isolates a correcting factor ρα(ω)β(ω) in the usual Lorentz-Lorenz formula (6). Then, if β(ω) ∼ α(ω)k 3 , this correcting factor is ∼ ρk 3 α(ω) 2 , which is almost identical to

1 kℓ sc = ρσ sc k = ρk 3 α(ω) 2 6π , (18)
the disorder parameter used in mesoscopic optics. Given the close link between the diffusion coefficient and the susceptibility, there is a strong similitude between the correction of Eq. ( 16) and the weak localization correction to the diffusion coefficient Eq. ( 8). This similitude is not surprising. The mean field approach of traditional optics neglects correlation between scatterers. When they exist, these correlations are due to the nonnegligible probability that a photon goes back to a previous scatterer, a process sometimes called recurrent scattering. In the mesoscopy community, neglecting this recurrent scattering and the subsequent correlations is called the 'independent scattering approximation' (ISA) [23][24][START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]. Weak localization effects are typical signatures of 'dependent scattering'. The link between the ISA and the Lorentz-Lorenz local field theory has been already briefly discussed in Ref. [51] but is often not discussed in more recent work.

V. WHAT SHOULD BE CALLED COOPERATIVE?

Based on the previous discussion, we consider that the term 'cooperative scattering' should not be kept for effects related to recurrent scattering only, because it is already called 'dependent scattering' in the mesoscopy community, and several such effects have well-established names, like weak or strong localization. Moreover, effects related to Dicke superradiance and subradiance can appear at low density, as shown in Fig. 3 and in [3][4][5], without recurrent scattering, and have been called 'cooperative' for a very long time (see, e.g., Refs. [21,22,[START_REF] Bonifacio | [END_REF]89,90,102,104,105]). Changing the semantics without a clear distinction of previously studied effects in different communities is therefore probably not recommendable.

However, we agree that it is questionable to call collective effects 'cooperative' if these effects can also be explained by conceptually more simple models, like wave propagation in a medium characterized by its susceptibility or standard multiple scattering.

In addition to the examples already mentioned [69,71], the coherent forward emission [2,11,[START_REF] Dicke | [END_REF]22] is certainly such an effect. Even if using a single-photon Gedankenexperiment requires a quantum formalism to properly demonstrate that the absorbed photon will be preferentially emitted in the forward direction [11], this effect is a simple consequence of an N -wave interference, like in a multiple slit experiment, and it is well-known that slit experiments produce interference patterns even with single particles [106]. The interference pattern from two atoms radiating a single photon has also been observed in [86,107]. The analogy between N -slit interference and the directionality character of superradiance has been further developed, including higher-order correlations, in Ref. [108]. In the CW regime, considering the phase-matching in the forward direction is also a way to understand the index of refraction of a polarizable medium. Thus, the resulting forward lobe is actually the diffraction/refraction pattern of the sample, like a Mie scatterer [START_REF] Bachelard | [END_REF].

The 'cooperative' Lamb shift is another questionable example. In the case of macroscopic samples, for which a susceptibility can be defined, Javanainen and Ruostekoski have shown [18], at least for the slab geometry that applies to the experiment of Ref. [7], that the 'cooperative' Lamb shift was the result of 'standard optics', i.e., wave propagation in a dielectric medium following the Helmholtz equation. It would be interesting to know, using wave propagation simulations, if this also holds for the cigar-shaped sample used in Ref. [4] or any other geometry [109]. For example, one of the geometries studied by Manassah [109] consists of a density-modulated slab of atoms and a 'giant cooperative Lamb shift' was predicted when the density is modulated at the Bragg condition [110]. Such an experiment has already been performed [39], and very asymmetric and shifted Braggreflection spectra have been indeed reported (see, e.g., Fig. 2 of [39]). These feature can be explained by the coupled-dipole model [111], but they can also be very well described by a standard-optics wave-propagation equation based on a periodic susceptibility (in that case using transfer-matrices to exploit the periodicity). Thus, it seems that in many (or most?) situations, the 'cooperative Lamb shift' is in fact a collective shift related to the shape and finite size of the medium, inducing refraction, reflection, lensing and waveguide effects [112], etc., which can be simulated by wave-propagation simulations.

The problem is that such wave-propagation simulations are not simple to perform. In the recent years, the coupled-dipole model has been widely used [3,5,9,36,69,70,91,[93][94][95][113][114][115]. It is computationally limited to a few thousand atoms, but it is otherwise very simple to use. Another strength is its completeness: it includes all collective effects we have discussed so far. But this is also a drawback: it is sometimes hard to give simple interpretation of the numerical results and, when com-paring to experimental results, a good agreement does not help identifying the relevant physical ingredients of the experiment. As a consequence, it would be wise, although tedious, to systematically compare the results with simulations based on a standard-optics calculation (wave propagation in a dielectric) or with a standard multiple-scattering computation (random walk), to check if the observations do not have simple explanations. This methodology has been used in [5,41].

It seems to us that less ambiguous signatures of cooperativity can be found in the transient response of the system, as initially envisioned by Dicke for the superradiant emission of a fully inverted system [START_REF] Feld | Coherent Nonlinear Optics[END_REF][START_REF] Gross | [END_REF][START_REF] Dicke | [END_REF] and its subradiant counterpart [START_REF] Pavolini | [END_REF]. In the single-photon or linear-optics regime, superradiant decay rates have been observed recently in the forward direction [4] and also off-axis [3].

In the forward-direction case, the measured light is the diffracted/refracted light by the cloud. Its dynamics is thus related to the dynamics of the refractive index. The imaginary part of the refractive index is also what governs the transmission, and its dynamics at the switchon or -off of the incident laser is what gives rise to optical precursors [116,117] or 'flashs' [118][119][120]. It has been shown experimentally and theoretically, using only Beer-Lambert law including the dephasing, that dynamics faster than Γ can appear at large on-resonance optical thickness [120]. This is somehow the counterpart of the spectral broadening induced by Beer's law at high b 0 .

Since this broadening is also present for the light scattered off-axis (see Fig. 1a), one can wonder if similar arguments also explains off-axis superradiance. Off axis, though, the field arriving on the detector emitted by different atoms have random phases (no phase-matching). There is thus no straightforward calculation similar to the one of [120], at least to our knowledge, that could relate the spectral broadening of the system response to the temporal dynamics observed on the fluorescence. This is, we think, the nontrivial feature of cooperativity and superradiance in the linear regime that this accelerated decay is still preserved despite the absence of any obvious phase-matching condition. Thus, the most efficient way to understand this result is to invoke the collective (cooperative) modes of the atomic excitation in the cloud, some of them decaying fast (superradiant modes), and some decaying slowly (subradiant modes). In other words, we trace over the photon degrees of freedom and look at the collective (cooperative) atomic behavior. The reverse approach, looking at the light, scattered by all atoms and interfering, is certainly possible and might provide an alternative view on the light-matter interaction for many atoms and at low intensities.

Subradiant decay, as the counterpart of superradiance, is also a good signature of cooperativity. However, a particular care is necessary to distinguish between different effects that could lead to a slow decay of the scattered light. In a dense sample with sharp boundaries, for example, long-lived cavity-like modes (or 'polaritonic modes' [121]) may exist, and may also give rise to high-Q Mie-like resonances [START_REF] Bachelard | [END_REF]. If the optical thickness b(∆) is large (near resonance), multiple scattering may trap the light in the sample for a long time (radiation trapping, see Sec. III A and ref. [START_REF] Labeyrie | [END_REF]), and even strong localization might need to be considered [122]. In that case, the qualitative difference between those various effects can be seen in their different scalings with the experimental parameters. The radiation trapping time scales as b(∆) 2 whereas the subradiant time scales as b 0 [5], independently of the detuning. Here again, Mie resonances or radiation trapping are well understood by looking at the light, whereas subradiance is well understood by looking at the collective modes of the atomic excitation. If we want to explain subradiance with a physical picture using light, we can say that coherence and interference effect modifies multiple scattering such that some 'lucky photons' are trapped in the sample, even at large detuning. If we neglect coherence and interference effects (diffusion or random walk model), we find, on the contrary, that multiple scattering vanishes far from resonance. The interplay between this incoherent radiation trapping and subradiance for intermediate detuning, and the way to distinguish between them, is the subject of our current investigation.

VI. CONCLUSION

In this article, we have tried to discuss the meaning of 'cooperative scattering' from a broad perspective, by studying various collective effects in atom-light interaction. We conclude that what should be called cooperative is often mainly a question of perspective. When we describe collective effects by looking at the atoms, with a microscopic modeling (based on Dicke states, coupled dipoles, etc.), most features appear to be of cooperative nature. Nevertheless, one can also understand many of these effects using the point of view of light, undergoing multiple scattering or propagating through a macroscopic sample, without using the concept of cooperativity at all.

The temporal decay of the light emitted off-axis seems to be the best example so far of a situation where a cooperative-scattering approach is the most efficient, numerically as well as conceptually, to describe the physics. We have shown that, at large detuning, the full decay curve only depends on the on-resonance optical thickness of the sample (and of its precise shape), but an analytical description of this result is still missing.

Finally, we emphasize that most, if not all, of the effects that we discussed in this article, are already known, but not always in the same fields. Superradiance and subradiance are hardly known in the mesoscopy community, and the same can be said for effects related to multiple scattering with the quantum-optics community. Establishing a bridge between those communities and their concepts seems very important to us. In particular, we have pointed out the similarity between the recurrent-scattering corrections to the susceptibility recently discussed in the quantum-optics community [18,41] and the weak-localization corrections, which are well-known in the mesoscopy community. We have also presented new results on the deviation from Ohm's law predicted by the coupled-dipole model. The physical origin of this deviation remains to be understood, but this is an attempt to relate cooperativity to transport properties [123]. This connection may be fruitful for the question of Ander-son localization of light in 3D [66,122] or in relation with other topics, like superconductivity [124] or lightharvesting systems [125,126].

100 FIG. 1 . 40 FIG. 2 .

 1001402 FIG. 1. Collective effects in the fluorescence, which can be explained by Beer-Lambert law or by multiple scattering. (a) Total fluorescence as a function of the detuning for different atom numbers (N = 1, 5, 20, 50, 500, 325, 450 from the bottom to the top) and a Gaussian cloud of rms transverse size R = 0.6λ [69]. Broadening and saturation effects are well visible. (b) Emission diagram of a spherical Gaussian cloud illuminated by a plane wave, computed from random walk simulations, for several optical depths from 0.5 to 100. The emission diagram becomes more backward directed as the optical depth increases.

We thank Antoine Browaeys, Yvan Sortais and Mark Havey for fruitful discussions at the PQE conference and Joachim von Zanthier for his invitation. This work was supported by the Agence National pour la Recherche under grant No. ANR-14-CE26-0032 (project LOVE). M.T.R. was supported by an Averroès exchange program.