text:
Clément Ferise

Philipp Del Hougne

Matthieu Davy

Optimal matrix-based spatio-temporal wave control for virtual perfect absorption, energy deposition and scattering invariant modes in disordered systems

We present and experimentally verify a matrix approach for determining how to optimally sculpt an input wavefront both in space and time for any desired wave-control functionality, irrespective of the complexity of the wave scattering. We leverage a singular value decomposition of the transport matrix that fully captures how both the spatial and temporal degrees of freedom available to shape the input wavefront impact the output wavefront's spatial and temporal form. In our experiments in the microwave domain, we use our formalism to successfully tackle three iconic wave-control tasks in a disordered cavity: (i) reflectionless transient excitation ("virtual perfect absorption"), (ii) optimal energy deposition, and (iii) scattering-invariant time-varying states.

I. INTRODUCTION

The ability to sculpt the wavefront incident on a scattering system allows for a judicious control of the outgoing wavefront's spatial and/or temporal shape, thanks to the linearity of the wave equation 1,2 . The crux lies in identifying the optimal input wavefront for a desired wave-control functionality. While iterative techniques optimize in situ the incident wavefront until the output converges to the desired one by relying on a feedback 3-6 , matrix-based techniques achieve provably optimal wave control using tools from linear algebra 1, [START_REF] Tanter | Time reversal and the inverse filter[END_REF][START_REF] Vellekoop | Universal optimal transmission of light through disordered materials[END_REF] . Different wave-engineering domains (acoustics, optics, etc.) have explored matrix approaches for optimal coherent wave control under various constraints (e.g., only spatial control over the input, only aiming at focusing in space and time). Here we present a general approach based on a multi-spectral transport matrix that can optimally leverage both spatial and temporal degrees of freedom (DoFs) on input and output sides.

For monochromatic waves, optimal wave-control functionalities are based on measuring the scattering matrix that fully encodes the linear relation between spatial input and spatial output DoFs. In optics, the transmission matrix was initially measured using deformable mirrors capable of spatially shaping the input wavefront to achieve maximal focusing of light traversing a multiple scattering medium [START_REF] Popoff | Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media[END_REF] . More complicated functionalities beyond focusing are then achieved using a singular value decomposition (SVD) of the transport matrix. This includes selective focusing inside the scattering medium [START_REF] Michel Popoff | Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis[END_REF] , perfect absorption of incident radiations [START_REF] Chong | Coherent perfect absorbers: time-reversed lasers[END_REF][START_REF] Wan | Time-reversed lasing and interferometric control of absorption[END_REF] , speckle engineering [START_REF] Devaud | Speckle engineering through singular value decomposition of the transmission matrix[END_REF] and access to open and closed transmission eigenchannels [START_REF] Vellekoop | Universal optimal transmission of light through disordered materials[END_REF][START_REF] Kim | Maximal energy transport through disordered media with the implementation of transmission eigenchannels[END_REF][START_REF] Benoît Gérardin | Full transmission and reflection of waves propagating through a maze of disorder[END_REF][START_REF] Sarma | Control of energy density inside a disordered medium by coupling to open or closed channels[END_REF] in diffusive media, e.g., for optimal energy deposition [START_REF] Jeong | Focusing of light energy inside a scattering medium by controlling the time-gated multiple light scattering[END_REF][START_REF] Bender | Depth-targeted energy delivery deep inside scattering media[END_REF] . New possibilities have recently emerged by applying an SVD or an eigendecomposition to other operators constructed from the transport matrix. This has led to proposals for schemes aimed at optimal energy storage [START_REF] Durand | Optimizing light storage in scattering media with the dwell-time operator[END_REF][START_REF] Del Hougne | Experimental realization of optimal energy storage in resonators embedded in scattering media[END_REF] , optimal micro-manipulation of targets embedded in scattering environments [START_REF] Horodynski | Optimal wave fields for micromanipulation in complex scattering environments[END_REF] and ultimate sensitivity of the output wavefield to perturbations [START_REF] Bouchet | Maximum information states for coherent scattering measurements[END_REF] -provided that the system's transport matrix can be rapidly characterized under low-noise conditions.

Manipulating spatially the incident wavefront also enables a degree of control on output signals in the time domain for incident pulses [START_REF] Aulbach | Control of light transmission through opaque scattering media in space and time[END_REF][START_REF] Katz | Focusing and compression of ultrashort pulses through scattering media[END_REF] . Focusing and maximal energy deposition at arbitrary times are based on measuring the multispectral TM [START_REF] Andreoli | Deterministic control of broadband light through a multiply scattering medium via the multispectral transmission matrix[END_REF][START_REF] Mounaix | Spatiotemporal coherent control of light through a multiple scattering medium with the multispectral transmission matrix[END_REF] or the time-gated TM [START_REF] Jeong | Focusing of light energy inside a scattering medium by controlling the time-gated multiple light scattering[END_REF][START_REF] Choi | Measurement of the timeresolved reflection matrix for enhancing light energy delivery into a scattering medium[END_REF][START_REF] Mounaix | Deterministic light focusing in space and time through multiple scattering media with a time-resolved transmission matrix approach[END_REF][START_REF] Devaud | Temporal light control in complex media through the singular-value decomposition of the timegated transmission matrix[END_REF] . More recently, spatio-temporal control over the output has also been achieved by engineering the scattering system as opposed to sculpting the input wavefront [START_REF] Del Hougne | Spatiotemporal wave front shaping in a microwave cavity[END_REF][START_REF] Mohammadreza | Metasurface-programmable wireless network-onchip[END_REF] , but this approach is conceptually very different because the parametrization of the transport matrix via tunable scattering parameters is in general non-linear [START_REF] Rabault | On the tacit linearity assumption in common cascaded models of ris-parametrized wireless channels[END_REF] .

While these approaches are still limited to spatial incident DoFs, existing optimal spatio-temporal control makes use of both spatial and temporal DoFs on the input side [START_REF] Allard P Mosk | Controlling waves in space and time for imaging and focusing in complex media[END_REF] . For the specific functionality of maximal focusing on an output channel, time reversal (TR) is an established technique pioneered in acoustics that captures the spatio-temporal input-output relation of a (usually complex) scattering system [START_REF] Fink | Time reversal of ultrasonic fields. i. basic principles[END_REF][START_REF] Derode | Robust acoustic time reversal with high-order multiple scattering[END_REF] . TR consists in emitting phase-conjugated broadband transmission spectra, and can be refined with iterative techniques [START_REF] Lemoult | Manipulating spatiotemporal degrees of freedom of waves in random media[END_REF] . In reverberating media, the spatio-temporal compression of long impulse responses shows that a control on output spatial DoFs is achieved using input temporal DoFs only for focusing [START_REF] Derode | Robust acoustic time reversal with high-order multiple scattering[END_REF][START_REF] Draeger | One-channel time re-versal of elastic waves in a chaotic 2d-silicon cavity[END_REF][START_REF] Lerosey | Focusing beyond the diffraction limit with far-field time reversal[END_REF] and perturbation sensing [START_REF] Bouchet | Temporal shaping of wave fields for optimally precise measurements in scattering environments[END_REF] . Corresponding results have also been achieved in the optical domain for a single-channel [START_REF] David J Mccabe | Spatio-temporal focusing of an ultrafast pulse through a multiply scattering medium[END_REF] and multi-channel 41 systems. However, the TR technique does not offer any insights into how to achieve more elaborate wave control.

In this paper, we introduce an SVD-based approach enabling optimal spatio-temporal wavefront shaping for arbitrary desired wave-control functionalities. We illustrate the capabilities of our formalism by applying it (i) to achieve reflectionless transient excitation ("virtual perfect absorption", VPA), (ii) to optimally deliver energy at a targeted time, and (iii) to excite timevarying scattering-invariant states. Thereby, we generalize the VPA concept from the single-mode [START_REF] Denis | Coherent virtual absorption based on complex zero excitation for ideal light capturing[END_REF] to the multi-resonance regime, as well as scattering-invariant states from the harmonic [START_REF] Pai | Scattering invariant modes of light in complex media[END_REF] to the time-varying regime. Moreover, our formalism constitutes a unifying frame- work to understand to-date seemingly unrelated wavecontrol concepts.

y (t ) 1 1 = H y (t ) 2 1 y (t ) 1 N y (t ) 2 N y (t) 2 x (t ) 2 1 x (t ) 1 1 x (t ) 2 N x (t ) 1 N x (t) 2 y (t) 1 x (t) 1 † S(ν)] S(ν)

II. MATRIX FORMALISM

For monochromatic waves with frequency ν, the transport matrix H(ν) describes the linear relation between the N incoming channels x(ν) and M outgoing channels y(ν) as y(ν) = H(ν)x(ν). If all connected channels are consider for both input and output, H(ν) is the system's full scattering matrix S(ν).

To extend this monochromatic input-output relation to the time domain on both input and output sides, we consider that the N time signals x n (t m ) are sampled on N tx points within a time interval ∆t x = [0 t c ]. Their spectra x n (ν) are computed on N ν frequencies using the discrete-Fourier-transform (DFT) operator, x n (ν) = D x x n (t), where D x km = e -i2πν k tm / √ N tx . To obtain a single multi-channel spatio-temporal linear relation, we first concatenate these vectors into the single vectors x t and x ν of length N N tx and N N ν . The elements of x t are x tj = x n (t m ) with j = (m -1)N + n. The linear relation between vectors x t and x ν is described by the sparse multi-channel DFT operator Dx of dimensions N N ν × N N tx (see SM), x ν = Dx x t . For N tx = N ν and t c = t max = 1/δν, both D x and Dx are unitary operators. However, in the more general case of t c < t max , they are non-square matrices verifying D †

x D x = 1 Ntx and D †

x Dx = 1 N Ntx but D x D † x ̸ = 1 Nν and Dx D †

x ̸ = 1 N Nν . We apply the same procedure to the M outgoing signals y n (t) measured within the time window ∆t y .

Finally, we introduce the N N ν × N N ν multi-spectral block-diagonal matrix S D with the kth diagonal block given by S(ν k ), yielding the following linear relation,

y t = D † y S D Dx x t = Hx t , (1)
where we have defined the operator H = D † y S D Dx . This formulation relates the spatial and temporal DoFs of a coherent input wavefront to the spatial and temporal DoFs of the corresponding output wavefront.

Using Eq. ( 1), the spatio-temporal input wavefront x giving any arbitrary multi-channel time-varying output signal y T can be estimated using the pseudo-inverse H + of H as x = H + y T . The case of the well-established TR technique [START_REF] Fink | Time reversal of ultrasonic fields. i. basic principles[END_REF] can also be interpreted in light of Eq. (1). In a time-reversal experiment, the goal is to maximize the signal on the kth output channel at time t = t i by injecting a spatio-temporally sculpted wavefront in the time interval ∆t x = [0 t i ]. The desired output signal is y n (t) = δ(k -n)δ(t -t i ). Simple algebra shows that for a pseudo-inverse H + = H † , the corresponding input wavefront is x n (t) = Dx S * nk , corresponding as expected to the time-reversed impulse responses between the N input ports and the kth output port. TR is hence a special case of our more general technique for optimal spatio-temporal wave control.

Of particular interest is the SVD of H: H = U √ ΛV † . The diagonal matrix Λ directly yields the integral of output intensities over the time interval ∆t y and all ports λ n = ||HV n || 2 = ∆ty [Σ n |y n (t)| 2 ]dt upon injection through all ports within the time interval ∆t x of the spatio-temporal states x n (t). The incident signals x n (t) composing the vector V n are normalized such as ∆tx [Σ n |x n (t)| 2 ]dt = 1. If our transport matrix H is taken to be S, the highest and lowest values of λ n determine the complete range of achievable output reflection, and hence offer a simple method to identify the provably optimal input states for maximal or minimal output reflections. If instead H is a transmission matrix, then λ n corresponds to transmitted rather than reflected intensity instead.

III. EXPERIMENTAL SETUP

We consider the quasi-2D electromagnetic cavity shown in Fig. 1 metallic cylinders. This disordered system is coupled via matched coax-to-waveguide transitions to two channels at the left and right sides. Using a vector network analyzer, we can conveniently measure the 2 × 2 scattering matrix between 8 and 9 GHz (frequency step: δν = 1.5625 MHz). In order to inject spatiotemporally sculpted signals in situ, we generate baseband signals between 0 and 1 GHz on an FPGA board at a sampling rate of 4 GS/s and multiply them with a local oscillator at f = 8 GHz (see SM for additional details). We coherently record the spatio-temporal output signals with an ultra-wideband multi-channel oscilloscope using a sampling rate of 40 GS/s. For the particular choice of ∆t x = ∆t y = [0 t c ] with t c = 1/(2δν), we display in Fig. 1(d) the singular value (SV) spectrum of H for our disordered system shown in Fig. 1(a). The largest SV λ 1 = 0.84 remains below unity due to the inevitable presence of absorption in our experimental system. The presence of absorption is also evidenced by the unitarity deficit of S that is seen in Fig. 1(b). This unitarity deficit is frequency dependent and it is notable in Fig. 1(b) that for some frequencies the smaller one of the two eigenvalues of S † S even gets close to zero, indicating that a time-harmonic state yielding a rather small reflected intensity exists. For the transient regime, we observe in Fig. 1(d) that multiple of the smallest SVs of H are extremely close to zero, suggesting that in the transient regime (i.e., within the interval ∆t y ), the reflected signal intensity can be suppressed very efficiently if the corresponding spatio-temporal states are injected.

(f)| 1 2 |S (f)| 1 2 (c) (f) 0 1 -1 0 1 -1 0 -1 1 0 1 -1 FIG. 2.

IV. REFLECTIONLESS TRANSIENT EXCITATION.

We begin by considering reflectionless transient excitation (RTE) of a scattering system. As mentioned above and seen in Fig. 1(d), multiple λ n are extremely close to zero for ∆t x = ∆t y = [0 t c ]. We observe that two qualitatively different types of such RTEs exist in our system, and we display one experimentally measured example of each in the first and second rows of Fig. 2.

The first type, in the first row of Fig. 2, is an instance of a concept previously introduced as "coherent virtual absorption"(CVA) [START_REF] Denis | Coherent virtual absorption based on complex zero excitation for ideal light capturing[END_REF] . CVA is a promising concept for energy storage [START_REF] Trainiti | Coherent virtual absorption of elastodynamic waves[END_REF] , sensing and transient application of optical forces [START_REF] Lepeshov | Virtual optical pulling force[END_REF] where an isolated complex-valued zero of the scattering response is engaged in the transient regime to suppress reflection within the excitation interval. Any scattering response can be decomposed into its constituent scattering singularities, namely poles νn and zeros z n [START_REF] Grigoriev | Optimization of resonant effects in nanostructures via weierstrass factorization[END_REF][START_REF] Grigoriev | Singular analysis of fano resonances in plasmonic nanostructures[END_REF][START_REF] Krasnok | Anomalies in light scattering[END_REF] . If a zero is real-valued, reflectionless time-harmonic excitation of the scattering structure is possible [START_REF] Chong | Coherent perfect absorbers: time-reversed lasers[END_REF][START_REF] Pichler | Random anti-lasing through coherent perfect absorption in a disordered medium[END_REF][START_REF] Sweeney | Theory of reflectionless scattering modes[END_REF] . Within a limited frequency interval, a disordered system like ours usually does not have a real-valued zero unless the scattering system is purposefully perturbed to impose one [START_REF] Mohammadreza | Perfect absorption in a disordered medium with programmable meta-atom inclusions[END_REF][START_REF] Benjamin W Frazier | Wavefront shaping with a tunable metasurface: Creating cold spots and coherent perfect absorption at arbitrary frequencies[END_REF][START_REF] Del Hougne | On-demand coherent perfect absorption in complex scattering systems: Time delay divergence and enhanced sensitivity to perturbations[END_REF][START_REF] Del Hougne | Coherent wave control in complex media with arbitrary wavefronts[END_REF][START_REF] Sol | Meta-programmable analog differentiator[END_REF][START_REF] Sol | Reflectionless programmable signal routers[END_REF] . However, given only complex-valued zeros, reflectionless excitation can still be achieved, albeit only in the transient regime for a limited temporal interval, by shaping the incident signal in space and time. When the incident signal is interrupted, the energy accumulated within the medium leaks out through the channels.

Previous experimental observations of RTE [START_REF] Trainiti | Coherent virtual absorption of elastodynamic waves[END_REF][START_REF] Delage | Experimental demonstration of virtual critical coupling to a single-mode microwave cavity[END_REF][START_REF] Delage | Reflectionless plasma ignition via high-power virtual perfect absorption[END_REF] considered systems with isolated resonances (similar to the first row in Fig. 2). In the simple picture of isolated resonances experiencing uniform absorption within the medium, poles νn and zeros z n are found at the same real frequency ν n but acquire different imaginary parts, νn = ν n -i(γ n + γ a ) and z n = ν n + i(γ n -γ a ) (γ n is the resonance width and γ a is the absorption strength). The incident signal corresponding to a zero z n is a monochromatic signal at frequency Re[z n ] modulated with an exponentially increasing envelope at the rate Im[z n ] 42 . Identi-fying poles and zeros however requires cumbersome protocols that becomes highly challenging in the regime of overlapping resonances [START_REF] Trainiti | Coherent virtual absorption of elastodynamic waves[END_REF][START_REF] Delage | Experimental demonstration of virtual critical coupling to a single-mode microwave cavity[END_REF][START_REF] Delage | Reflectionless plasma ignition via high-power virtual perfect absorption[END_REF][START_REF] Chen | Use of transmission and reflection complex time delays to reveal scattering matrix poles and zeros: Example of the ring graph[END_REF][START_REF] Vladimir | Harmonic inversion of time signals and its applications[END_REF][START_REF] Kuhl | Resonance widths in open microwave cavities studied by harmonic inversion[END_REF][START_REF] Davy | Selectively exciting quasi-normal modes in open disordered systems[END_REF] . In contrast, our technique solely requires the SVD of the transport matrix H and directly yields the non-intuitive spatio-temporal input signals. For overlapping resonances, these signals involve multiple zeros as seen in the second row of Fig. 2.

How many VPA states exist for a given transport matrix H? Theoretically, we expect this number to correspond to the number of complex zeros in the upper half of the complex frequency plane. We show in the SM that a VPA state indeed disappears as the corresponding zero crosses the real axis and acquires a negative imaginary part. The total number of poles and zeros can be estimated using Weyl's law [START_REF] Weyl | Ueber die asymptotische Verteilung der Eigenwerte[END_REF][START_REF] Arendt | Mathematical Analysis of Evolution, Information, and Complexity, Weyl's Law[END_REF][START_REF] Pierrat | Invariance property of wave scattering through disordered media[END_REF] . The average density of states ρ(ω) ∼ Aω/(2πc 2 0 ), where A is the area of the scattering region, gives a theoretical estimate of 25 modes between 8 and 9 GHz. This number can also be roughly confirmed by counting the peaks in Fig. 1(b). However, counting the number of zeros with positive imaginary part is more challenging. To do so, we study in Fig. 1(c) the number of positive peaks of the real part of the Wigner-Smith time delay τ

W (ν) = [Tr(Q(ν))]. Here, Q(ν) = -i [S(ν)] -1 [∂S(ν)/∂ν] is the Wigner-Smith op-
erator [START_REF] Wigner | Lower limit for the energy derivative of the scattering phase shift[END_REF][START_REF] Smith | Lifetime matrix in collision theory[END_REF] . The real part of the eigenvalues of Q is assumed to be proportional to a dwell time [START_REF] Durand | Optimizing light storage in scattering media with the dwell-time operator[END_REF][START_REF] Del Hougne | Experimental realization of optimal energy storage in resonators embedded in scattering media[END_REF][START_REF] Fan | Principal modes in multimode waveguides[END_REF][START_REF] Rotter | Generating particlelike scattering states in wave transport[END_REF][START_REF] Davy | Transmission eigenchannels and the densities of states of random media[END_REF][START_REF] Böhm | In situ realization of particlelike scattering states in a microwave cavity[END_REF][START_REF] Huang | Wave excitation and dynamics in non-hermitian disordered systems[END_REF] which is diverging as a zero crosses the real axis [START_REF] Del Hougne | On-demand coherent perfect absorption in complex scattering systems: Time delay divergence and enhanced sensitivity to perturbations[END_REF][START_REF] Asano | Anomalous time delays and quantum weak measurements in optical micro-resonators[END_REF][START_REF] Chen | Statistics of complex wigner time delays as a counter of s-matrix poles: Theory and experiment[END_REF][START_REF] Chen | Generalization of wigner time delay to subunitary scattering systems[END_REF] . A positive (negative) peak of Re[τ W ] is a clear signature of a zero with positive (negative) imaginary part. The number of positive peaks (16) in Re[τ W (ν)] is in good agreement with the number of λ n below λ = 10 -5 (15). This number increases with the number N of ports connected to the cavity as the channel decay rate scales as ⟨γ n ⟩ = N γ 0 + γ a , where γ 0 is the single-channel decay rate. In our system, if we increase N from 2 to 6, γ n ≫ γ a so that all the zeros have positive imaginary parts and the number of VPA states is equal to 25 (see SM).

V. OPTIMAL ENERGY DELIVERY

We now turn our attention to the optimal energy deposition, which consists in delivering the maximum possible amount of energy within the output time interval ∆t y given spatio-temporal coherent control in the input time interval ∆t x . In Fig. 3(a,b), we experimentally observe the input and output signals upon injecting the state corresponding to the largest singular value of H for the intervals considered in Fig. 1(d). During the transient excitation, a very large fraction (λ 1 = 0.84) of the injected energy exits the system within the same temporal interval (∆t x = ∆t y ). The remainder of the injected energy is absorbed in the system. Maximal-reflection states corresponding to other temporal intervals are presented in the SM.

By adjusting the definition of the output time interval ∆t y , the same approach can also yield optimal energy delivery at arbitrary times after the transient excitation interval. Two examples in the limit of a short output interval ∆t y = δ(t -t c ) are shown in Fig. 3(c,d). In all cases, a sharp increase of total reflection at the selected time is observed. Of course, the duration of the focused output signals is not arbitrarily short but is instead determined by the bandwidth of the spatio-temporal input signal.

The number of transmitting channels is here limited (N = 2) by our electronic hardware. To further demonstrate the interest of our approach, we consider a second 2D cavity of length L = 0.5 m and width W = 0.25 m with two arrays of N = 8 coax-to-waveguide transitions attached on the left and right interfaces (see Ref. [START_REF] Davy | Mean path length invariance in wavescattering beyond the diffusive regime[END_REF] for a description of the experimental setup). We measure the transmission matrix (TM) in the diffusive regime between between 10 and 15 GHz with a frequency step of 3.9 MHz. For a short incident pulse delivered at t = 0.5t max through a single channel, the temporal profile of the output intensity decays exponentially with a decay time equal to τ = 0.0186t max (t max = 0.26 µs), see Fig. 4(a). Using the linearity of the wave equation, we then reconstruct the transmitted intensity averaged over outgoing channels for an injection over the N = 8 incoming channels and an optimal energy delivery at time ∆t y = δ(t -t c ) with t c = 0.5t max , t c = 0.55t max and t c = 0.6t max (see Fig. 4(b-d)). In each case the optimal incoming signal is assumed to be injected within the interval ∆t y = [0 -0.5t max ]. A clear enhancement of the transmitted signals at selected temporal interval is now observed even at late times as a result of a larger number of controlled input ports.

Our spatio-temporal matrix formulation takes full advantage of both spectral and spatial DoFs to achieve optimal total transmission at selected times. We show in Fig. 5 that the transmitted intensity at the focal time scales linearly with the number of incoming antennas N and the bandwidth of incident signals ∆ν. This is in agreement with theoretical predictions for time-reversal experiments in diffusive or chaotic systems, which demonstrates that the focused intensity increases linearly with the number of DoFs N ×N f . Here N f is the ratio between the bandwidth ∆ν and the spectral correlation length δν ∼ 1/τ . For N = 8 and a bandwidth of 5 GHz, we estimate that the total DoFs is equal to 190, which is of the same order of magnitude as the number of spatially controllable optical modes in Ref. [START_REF] Devaud | Temporal light control in complex media through the singular-value decomposition of the timegated transmission matrix[END_REF] .

VI. SCATTERING-INVARIANT TIME-VARYING STATES.

Finally, we explore the possibility of exciting states that have the same spatial and temporal input and output patterns, generalizing the recent concept of timeharmonic (monochromatic) scattering-invariant states that have the same spatial input and output patterns [START_REF] Pai | Scattering invariant modes of light in complex media[END_REF] . To that end, we solve the eigenvalue problem The blue line is the incident signal, the red line is the outgoing signal and the green dashed line is the output signal shifted by -∆τ = -8ns. The inset is a zoom of the field on a small interval represented by the dashed black lines in the figure .   short delay times. They are therefore peaked within frequency ranges in which the reflection coefficient presents a flattened shape [see Fig. 6(a,c)].

Hx n = α n x n (2)
We then consider scattering-invariant time-varying states in transmission through the larger cavity with N = 8 antennas on each side. The length of the input interval is fixed to 0.3t max and the output temporal interval is now shifted by dτ , ∆t y = ∆t x + ∆τ . We find a maximal similarity coefficient |α| = 0.77 for ∆τ = 0.08t max . The corresponding length c 0 ∆τ = 2.04 m is close to the mean path length of the cavity [START_REF] Davy | Mean path length invariance in wavescattering beyond the diffusive regime[END_REF] . The average temporal envelopes of input and output signals are shown in Fig. 7. The output signal shifted by -∆τ is very similar to the injected signal. Since input and output speckle patterns are correlated over a temporal interval, these states could be useful for broadband imaging through disordered systems.

VII. CONCLUSION

We have presented a matrix formalism enabling optimal spatio-temporal coherent control of waves in arbitrarily complex scattering systems by taking full advantage of both spatial and temporal DoFs. Our work presents a unifying perspective on many contemporary wave-control techniques (time reversal, wavefront shaping, virtual perfect absorption, scattering-invariant modes) and even generalizes them. Beyond providing fundamental insights, our technique may readily find applications in the microwave and acoustics regimes where the necessary hardware for coherent spatio-temporal wavefront generation is available. can be expressed in terms of the discrete Fourier transform (DFT) operator,

x n (ν) = D x x n (t) with elements [D x ] km = 1 √ N tx e -i2πν k tm . (A2)
The dimensions of D x is N ν × N tx . When we set δt = 1/(N ν dν) and t c = t max with t max = 1/δν, the operator D x is unitary, meaning that D x D † x = 1 Nν . In the more general case of control over smaller time invervals t c < t max (N tx < N ν ), the DFT operator is non-square. It however still verifies

D † x D x = 1 but D x D † x ̸ = 1.
We then seek for a single multichannel linear relation for the N input and output channels. We first concatenate the spectral and temporal matrices into the single vectors x t and x ν of length N N tx and N N ν , given below for N = 2:

x t =           x 1 (t 1 ) x 2 (t 1 ) x 1 (t 2 ) x 2 (t 2 ) . . . x 1 (t Ntx ) x 2 (t Ntx )           and x f =           x 1 (f 1 ) x 2 (f 1 ) x 1 (f 2 ) x 2 (f 2 ) . . . x 1 (f Nν ) x 2 (f Nν )           (A3)
The elements of x t are therefore x tj = x n (t m ) with j = (m-1)N +n. Using this formulation, we can find a linear relation between x t and x ν . We introduce the sparse multichannel DFT operator Dx of dimensions N N ν × N N tx expressed as (still for N = 2):

Dx =           (D x ) 00 0 • • • (D x ) 0Ntx 0 0 (D x ) 00 • • • 0 (D x ) 0Ntx (D x ) 10 0 • • • (D x ) 1Ntx 0 0 (D x ) 10 • • • 0 (D x ) 1Ntx . . . . . . . . . . . . . . . (D x ) Nν 0 0 • • • (D x ) Nν Ntx 0 0 (D x ) Nν 0 • • • 0 (D x ) Nν Ntx           (A4)
x ν = Dx x t (A5)
One can easily verify that Dx has the same properties as D x : it is unitary for t c = t max and that D †

x Dx = 1 for any temporal interval. However Dx D †

x ̸ = 1 for t c < t max . We apply the same procedure to the M outgoing signals y n (t) measured within the time window ∆t y . Finally, we define the block diagonal matrix S D with

S D =         S(ν 1 ) 0 0 • • • 0 0 S(ν 2 ) 0 • • • 0 . . . 0 . . . 0 . . . . . . . . . 0 . . . . . . 0 0 • • • 0 S(ν Nν )         (A6)
We finally get Dy y t = S D Dx x t and

y t = D † y S D Dx x t . (A7)
which is Eq. (1) of the main text.

Appendix B: Absorption and virtual perfect absorption

In this section, we demonstrate numerically that the VPA states disappear when a zero of the scattering matrix S crosses the real axis. We consider in the following a single channel N = 1 system but an extension to multichannel systems is straightforward. For a single resonance, the scattering matrix is expressed by a Breit-Wigner formula,

S(ν) = 1 - iγ n ν -ν n + i(γ n + γ a ) (B1)
Here γ n represents the coupling of the channel to the system and γ a is the absorption linewidth. The reflection coefficient |S(ν)| 2 is shown in Fig. 8(a) for γ a = γ n /2. The resonance is clearly visible at ν n = 8.5 GHz.

The pole of S is νn = ν n -i(γ n + γ a ) and the zero of S is z n = ν n + i(γ n -γ a ). When γ n > γ a , the imaginary part of z n is positive and the zero is located in the upper complex plane. This zero can then be accessed by sending a monchromatic signal at frequency ν n modulated by an exponentially growing envelope at rate γ n -γ a . This input temporal signal x(t) for an excitation between 0 and t c = t max /2 is directly extracted from the last subspace of the corresponding H = D † y S D Dx matrix. The outgoing energy in such state of virtual perfect absorption is suppressed between 0 and t c . Once the excitation is turned off, the energy stored within the cavity that has not been absorbed is released at rate γ n . The envelopes of input and output signals are illustrated in Fig. 8(b) for

γ a = γ n /2.
However, as γ a increases, zero reflection between 0 and t c becomes impossible for an excitation only within this range. For instance, for γ a = γ n , the zero of S is real, z n = ν n , and the excitation x(t) must be fully monochromatic to be perfectly absorbed with no reflection. This is the condition of coherent perfect absorption found when the absorption decay rate balances the excitation rate [START_REF] Chong | Coherent perfect absorbers: time-reversed lasers[END_REF] . The reflection between 0 and t c for any non-monochromatic signal cannot be suppressed. In Fig. 8(c), we observe that the minimal reflection state of the H matrix is still increasing in time but the reflection is not zero. For γ a > γ n , the state with minimal reflection presents a pronounced peak at t = t c /2 [see Fig. 8(d)].

In Fig. 9, we show the variation of the minimal reflection value found from the singular value decomposition of H. It clearly shows that the reflection can indeed be completely suppressed for γ a ≪ γ n but this property is lost as γ a approaches γ n as expected. For γ a = 2γ n , the smallest reflection is even quite strong (min[λ n ] = 0.13). In Fig. 2 on the main text we have presented our new approach by displaying optimal energy deposition and its opposite: reflectionless transient excitation for ∆t x = ∆t y = [0 t c ] with t c = t max /2 and for N = 2 channels coupled to the scattering system. In this section we numerically explore the impact of the time interval on reflection coefficients.

We start by varying the interval length t c . We display in Fig. 11(a) the spectrum of the first (blue) and the last (red) singular value (SV) of H as a function of t c . The time axis is normalized by the maximum time which is inversely proportional to the frequency step: t max = 1/δν. These states correspond respectively to the maximum and minimum possible reflection within the output interval ∆t y . The first SV is zero for t c = 0, as expected, and then exponentially increases to rapidly reach a plateau at λ 1 ≈ 0.84 shortly after t c = t c2 = 0.05t max . Thus, for t > t c2 , the interval length does not impact the reflection coefficient. On the other hand, the last SV is minimum for t c2 < t < 0.8t max with an average value ⟨λ NΛ ⟩ ≈ 2 × 10 -11 . For very short and large values of t c and t c , it is however not possible to shape the incident wavefront temporally to modulate the reflection coefficient. This implies that VPA states presented in the main text do not exist when t c is close to 0 or t max . The temporal responses for four time intervals (t c1 = 0.004t max , t c2 = 0.05t max , t c3 = 0.35t max and t c4 = 0.9t max ) are shown in the four last rows of Fig. 11. These signals are reconstructed numerically from measurements of the scattering matrix. On the first column, input (blue) and output (red) signals upon injecting the state corresponding to the maximum possible amount of energy are shown for the four time intervals considered. RTE's input and output signals, i.e. signals corresponding to minimal reflection, are displayed in the same way on the second column. In order to better observe the signals on the first line (t c = 0.004t max ), the time axis is adjusted to [0 0.03]µs.

0 -1 1 0 -1 1 0 S (t) 1 S (t) 1 S (t) 1 (a) (d) (f) (h) t / t c max First SV Last SV SV of H -1 1 0 -1 1 0 -1 1 0 S (t) 1 S (t) 1 S (t) 1 (e) (g) (i)
As explained in the main text, the number of reflectionless states found from the spatio-temporal operator H is theoretically equal to the number of zeros located in the upper complex plane. The total number of zero within the frequency range [f 1 = ω 1 /(2π); f 2 = ω 2 /(2π)] is equal to the number of poles of the scattering matrix and can be estimated from Weyl's law : N ω = ω2 ω1 ρ(ω)dω, where ρ(ω) ∼ Aω/(2πc2 0 ) is the density of states. Here A is the area of the cavity.

In this section, we provide further evidence of the relation between the number of reflectionless transient states and the number of zeros with positive imaginary parts. To do so, we vary the number of channels coupled to the cavity, from N = 2 (corresponding to measurements presented in the main text) to N = 8. In each case, we measure the N × N scattering matrix and obtain τ W (ν).

The real part of the Wigner time delay is represented on the first column of Fig. 12 for different number of coupled channels N , N = 2 (a), N = 4 (c) and N = 6 (e). For each case, the corresponding spectrum of the last 40 singular values of H is shown on the second column.

For N = 2, the Wigner time delay comprises positive and negative peaks. We find 18 singular values of H bellow λ = 10 -5 . The small difference from the number presented in the main text is due to the slight changes in the system. This time, all the channels are matched with coax-to-waveguide transitions. For each measurements, N ports are connected to the Vector Network Analyser, the others being in open-circuit states. This pushes the reflective boundaries away from the center of the cavity and so virtually increases its size. For N = 4, the Wigner time delay comprises only one negative peak and 25 singular values below λ = 10 -5 are found. For N = 6, the peaks of τ W (ν) are all positive so that we expect that the number of reflectionless states to be equal to the number of resonances (or poles) with the bandwidth. We find 25 singular values of H smaller than λ = 10 -5 which is in good agreement with Weyl's estimate.

8 FIG

 8 FIG. 1. (a) Photographic image of our quasi-2D disordered cavity (L = W = 205.74 mm, h = 10.16 mm) with the top plate removed to show the inside. One coax-to-waveguide transition is attached at each side of the cavity, yielding a two-channel scattering system. The inset illustrates our spatio-temporal matrix formalism defined in Eq. (1). (b) Spectrum of the two eigenvalues of [S(ν)] † S(ν). (c) Spectrum of the real part of the trace of the Wigner-Smith operator Q(ν).(d) Singular value spectrum of H (see Eq. (1)).

8 FIG

 8 FIG. 3. (a,b) Optimal deposition of energy corresponding to the largest singular value of 0.84 for ∆tx = ∆ty = [0 0.5tmax] with tmax = 0.64 µs. (c,d) Optimal energy deposition at two chosen output times: (c) t = 0.5tmax, (d) t = 0.52tmax with tmax = 0.80 µs. The last two rows show the spatially averaged time-varying input (blue) and output (red) intensities upon injecting states for maximal reflection corresponding to the first singular value of H. The insets are a zoom around the focal time.

FIG. 5 .FIG. 6 .FIG. 7 .

 567 FIG.5. Enhancement of transmitted intensity with respect to the bandwidth (a) and number of antennas (b) used for optitransmitted energy. The markers represent experimental data for different selected times tc between tc = 0.5tmax and tc = 0.7tmax with tmax = 0.26 µs. The dashed lines are linear fits of the data. The intensity is normalized by its result found for N = 8, a bandwidth of 5 GHz and tc = 0.5tmax.

FIG. 8 .

 8 FIG. 8. (a) Reflection |S(ν)| 2 for a single resonance at νn = 8.5 GHz. The scattering matrix is described by the Breit-Wigner formula given in Eq. (B1). The channel decay rate is γn = 8 MHz and γa = γn/2. (b-d) Input and output signals for the minimal subspace of H corresponding to an excitation between 0 and tc = tmax/2 (tmax/2 = 1/δν) for γa = γn/2 (b), γa = γn (c) and γa = 2γn (c)

FIG. 9 .

 9 FIG. 9. Variation of the last eigenvalue of H † H as a function of the absorption rate γa/γn. VPA states with suppressed reflection are obtained only as γa ≪ γn.

  Appendix C: Details on the experimental setup In this section we present in details our experimental setup. A sketch is shown on Fig 10. A 4 × 4 multichannel square cavity (side L = 205.74 mm and height H = 10.16 mm) comprising only two coupled ports to the system is used. Uncoupled ports are obstructed by a metallic plate.The measurement of the 2 × 2 scattering matrix between 8 and 9 GHz is operated with a Vector Network Analyser (VNA) on the two coupled ports of the cavity via coax-to-waveguide transitions with one circulator (Aerocomm J80.160) connected to it. We use circulators on each port to inject and measure the signals with a 4-ports VNA. By applying the SVD-based approach and matrix formalism presented in the main text, we obtain baseband temporal signals with frequencies between 0 and 1 GHz. These signals are first up-converted in the frequency range between 0.2 and 1.2 GHz via a Fourier transform and an inverse Fourier transform successively. The two pseudo-baseband signals (s 1 and s 2 on Fig 10) are finally uploaded on a FPGA board (Xilinx ZYNQ-ZCU111 ). They are analogically up-converted in the frequency range [8 9] GHz using two wide band frequency mixers (Mini-Circuits ZX05-153LH-S+) connected to a local oscillator at 7.8 GHz. The same local oscillator is connected to the frequency mixers via a 4 ways power divider (AMD-GROUP PD4-2-18-10 ) to synchronize the signals.The RF output of the frequency mixers is connected to the two circulators matched with the two coaxto-waveguide transitions coupled to the cavity. To measure the signals reflected from the cavity, the third channel of each circulator is connected to an ultra-wideband oscilloscope with a sampling rate of 40GSamples/s (SDA 816Zi-B).Appendix D: Impact of the size of the input time interval

FIG. 10 .

 10 FIG. 10. Detailed sketch of the experimental setup. The inset explicitly shows the role of the circulators.

FIG. 11 .FIG. 12 .

 1112 FIG. 11. (a) Spectrum of the first (blue) and last (red) singular value of H as a function of the variation of tc. (b-i) Temporal representation from numerical simulations upon coherent spatio-temporal injection of the first (b,d,f,h) and the last (c,e,g,i) right singular vectors of H for four different input time intervals: (b,c) tc1 = 0.004tmax, (d,e) tc2 = 0.05tmax, (f,g) tc3 = 0.35tmax and tc4 = 0.9tmax.

  Experimentally measured output signals upon coherent spatio-temporal in situ injection of various right singular vectors of H. For each considered state, we display the injected (blue) and outcoming (red) time-varying signals on the two channels (middle and right columns), as well as the corresponding signal spectra averaged over both channels (left column). The first and second rows correspond to two representative choices of the smallest near-zero singular values that yield qualitatively distinct results.

Stefan Rotter and Sylvain Gigan, "Light fields in complex media: Mesoscopic scattering meets wave control," Rev. Mod. Phys. 89, 015005 (2017).

Hui Cao, Allard Pieter Mosk, and Stefan Rotter, "Shaping the propagation of light in complex media," Nature Physics 18, 994-1007 (2022).

I. M. Vellekoop and A. P. Mosk, "Focusing coherent light through opaque strongly scattering media," Opt. Lett.32, 2309-2311 (2007).

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, "Exploiting disorder for perfect focusing," Nature Photon. 4, 320-322 (2010), 10.1038/nphoton.2010.3

.5 Cheng Ma, Xiao Xu, Yan Liu, and Lihong V Wang, "Time-reversed adapted-perturbation (trap) optical focusing onto dynamic objects inside scattering media," Nature Photonics 8, 931 (

2014).6 Roarke Horstmeyer, Haowen Ruan, and Changhuei Yang, "Guidestar-assisted wavefront-shaping methods for focus-

Acknowledgements

We acknowledge Clara Moy for assistance with the work presented in Sec. IV. This publication was supported by the European Union through the European Regional Development Fund (ERDF), by the French region of Brittany and Rennes Métropole through the CPER Project SOPHIE/STIC & Ondes. M. D. acknowledges the Institut Universitaire de France. C. F. acknowledges funding from the French "Ministère de la Défense, Direction Générale de l'Armement".

Appendix A: Theory

We provide here details on the multichannel temporal operator given in the main text. We consider N time signals x n (t m ) sampled on N tx points between 0 and t c . The Fourier transform on N ν frequencies


Vincent Placet

Pauline Butaud

Fanny Pelisson

Taiqu Liu

Yves Gaillard

Gilles Bourbon

Fabien Amiot

Morvan Ouisse

Viscoelastic properties of plant fibers -Dynamic analysis and nanoindentation tests

come   L'archive ouverte pluridisciplinaire

Viscoelastic properties of plant fibers: dynamic analysis and nanoindentation tests.

V. Placet, P. Butaud, F. Pelisson, T. Liu, Y. Gaillard, G. Bourbon, F. Amiot, M. Ouisse Univ. Bourgogne Franche Comté, FEMTO-ST Institute, CNRS/UFC/ENSMM/UTBM, Department of Applied Mechanics, 24 rue de l'épitaphe, 25000, Besançon, France pauline.butaud@univ-fcomte.fr

It is now established that plant fibers are suitable reinforcements to replace or to complete synthetic fibers for composite applications. They have many well-known advantages such as being renewable, biodegradable and having high specific mechanical properties and intrinsic damping properties that can impart attractive benefits to composite materials. When compared to glass and carbon fibers reinforced polymer composites, PFCs exhibit generally higher damping properties. This is commonly attributed to the polymeric nature of the fibers. It could also result from their hierarchical structure and microstructural specificities. Indeed, damping in composite materials is induced by several microscopic level mechanisms related to the viscoelasticity of both matrix and fibers, but also by the fiber/matrix interface and the friction between fibers inside bundles or between layers. The presence of lumen could also play a role. At this stage, the damping source is not well identified, and the design and the optimization of composites, for structural applications, are then difficult. A better knowledge of the physical phenomena involved is therefore necessary.

A first approach is proposed using dynamic nanoindentation. Different dynamic loading paths are applied to plant fiber composites. Flax fibers are tested in situ in the composite. Indents were performed using predefined grids.

The sinusoidal signal of force and penetration depth as a function of time are post-processed to ensure a reliable determination of the loss factor. Finally damping maps are plotted. Results allow the variation of elastic and damping properties in the fiber/fiber and fiber/matrix interphases to be quantified and thus to shed new light on the origins of damping in PFCs.

In a second time, a specific experimental set-up is developed based on the analysis of the transverse (bending) vibratory response of a fiber clamped at one end, free at the other one. Both forced (using a piezoelectric actuator) or free (using a release of a static constraint) movements can be generated. The loss factor is then identified from the vibration displacement by tracking the fiber free-end recorded with a high-speed camera. Conclusions are drawn on the damping properties of plant fibers compared to synthetic fibers.

ACKNOWLEDGMENT

This project has received funding from the BQR ENSMM 2020 and from the bio-Based Industries Joint undertaking under the European Union's Horizon 2020 research and innovation program under grant agreement No 744349 -SSUCHY.


Bettina Horsch

Learning from Spatial Capital in Architectural Education -Tools and Perspectives

Keywords: Satial capital, students, architectural education

This article aims to seize the "spatial capital" as a notion to question the way in which space is taught in Schools of Architecture, both "in" and "outside" the walls. It being understood that spatial capital, following Pierre Bourdieu's capital theory [START_REF] Bourdieu | Distinction: A Social Critique of the Judgement of Taste, translated by R. Nice[END_REF] , "is the sum of the skills acquired by an individual or a group of individuals in the field of space mastery.

[…] It describes the way in which space, in particular its use, its knowledge and its control, are the subject of learning which is a social and cultural construct. Spatial capital covers the ability to move around, to control, to dominate and to transform a space" 2 . Thus, we start from the premise that the Schools of Architecture are institutions whose vocation is to transmit a specific capital to future architects -and it is this "specificity" that we intend to question and criticize. From the crossed trajectories of architecture students, we will discuss the way in which the spatial capital of future architects is built "in" but also "outside" the School, by particular space-times which are spread out over the long term. We will focus our attention on what the institution does to students' spatial capital, and more particularly how it transmits, compensates, distinguishes, but also reproduces specific spatial skills through teaching methods, theories and doctrines provided.

EXPLORING THE CONCEPT OF SPATIAL CAPITAL TO UNDERSTAND THE SOCIALIZATION PROCESS DIFFERENTLY

During our research, a line of work emerged on spatial capital as a possible key to understanding how students socialize with architecture. It gradually appeared that the concept of spatial capital would be a good way to investigate to better understand how students' relationship to space evolves and is shaped during their studies. It also informs us how previously constituted spatial capital determines their way of socializing according to their representations of space and architecture. To make the concept of spatial capital operational, we start from the postulate that space is at the heart of the training of architects.

Thus, the School of Architecture would be an institution that trains students in the observation, analysis and transformation of space. This training would endow them with a specific culture of space through the transmission of arts of thinking and projecting architecture and space. How do students develop spatial capital prior to their studies? How does this prior spatial capital impact the architectural socialization of the students? How does the School equip students with specific spatial dispositions and what effects do these dispositions have on their prior spatial capital? How does the School encourage students to question, shift and enrich their initial spatial references? Spatial capital would depend on their socio-economic and cultural origins. The prior spatial capital would guide the way in which they represent themselves, apprehend and invest their studies (through their ability to mobilize initial spatial references). The prior spatial capital (and the spatial repertoire that accompanies it) would endow them with a more or less legitimate culture of space (or more or less recognized by Schools) which would directly impact their socialization to architecture. The cross-analysis of the trajectories of the respondents confirms the impact of primary (family) socialization on prior spatial capital (spatial practices including modes of living, mobility, culture of specific space through extra-curricular training, etc.). Secondary socialization (high school in particular) also impacts prior spatial capital (culture of space "inside" and "outside" the walls, through class trips, literally and figuratively) and allows some respondents to "compensate" for spatial skills valued by the School of Architecture. Discussions with teachers or internship supervisors during high school is cited by some as part of an opening to other spatial references, or even as opening the way for them to enter architectural studies.

Architectural studies would provide students with specific spatial aptitudes by what is transmitted "in" and "outside" the walls. These spatial dispositions would enrich, reinforce and/or displace their prior spatial capital. The arts of doing and thinking provided by Schools through disciplinary teaching would promote a certain type of spatial culture and related spatial repertoire (legitimate/illegitimate) and therefore promote forms of socialization.

In order to illustrate our point, we will retrace what contributes, during the studies, to the acquisition of a specific repertoire of spatial action, whether short or long so-called "field" immersions during project studios or study trips, as well as so-called "professionalizing" experiences such as internships, or other educational devices. A more prospective part will open on what prior and transmitted spatial capital augurs as pedagogical horizons.

Through this research, we discuss the updating of the determinisms at work, but also what makes it possible to counter them through the figure of the "space defector" (in reference to the class defector). To support our argument, we will also rely on a more ethnographic corpus of pedagogical experiences lived by the two authors, including study trips to Rome, a teaching partnership with École Supérieure du Bois (Wood Engineering School), the Solar Decathlon Europe 2014, etc.

The corpus is based on two qualitative surveys aimed at documenting the students' trajectories of the Nantes School of Architecture over the past ten years [START_REF] Horsch | Architecture d'un métier, les étudiants architectes entre orientation, socialisation et insertion professionnelle[END_REF] , part of which focuses more particularly on the way in which the spatial capital of students is shaped over time, of their trajectory, from childhood to professional integration, via architectural studies. Approximately 250 semi-structured interviews have been conducted several years after graduation ("trajectory survey"). To counter the bias of the narrative told a posteriori, a more recent ongoing survey completes the research device with filmed interviews of a cohort of ten students who entered School in 2021 and who are interviewed once a year until their professional integration around 2026 ("documentary survey"). [START_REF] Horsch | Devenir architecte(s) : des espaces de socialisation au capital spatial. Enquête par entretiens semi-directifs filmés et récurrents, le cas des étudiant.e.s de l'ENSA Nantes[END_REF]

FIRST RESULTS ON THE PROCESS OF ARCHITECTURAL SOCIALIZATION THROUGH THE PRISM OF SPATIAL CAPITAL SOCIALIZATION TO ARCHITECTURE AND SOCIAL CLASSES: BETWEEN REPRODUCTION AND DISTINCTION

The "trajectory survey" revealed a close link between social classes and professional integration, confirming social determinism. Thus, social inequalities are replayed both at the entrance and at the end of architectural studies. We also analyzed the trajectories of students who contradict social determinisms in order to understand what role plays the school. Thus, the "documentary" survey encouraged us to track, from the filmed interviews, the social markers both through the descriptions that the respondents make of their backgrounds and practices, and through the way in which they recount and arrange these descriptions. It is a question of understanding how this way of telling evolves (or not) over the interviews, in parallel to their acculturation to architecture.

The first two series of interviews have thus brought to light the importance of the social, cultural and economic background of the students in order to fully understand the way in which they apprehend their studies and get involved, and how they socialize. The social status, the level of education and the cultural background of the parents are more or less close to that of the architectural milieu. While some will have pre-socialized in the midst of architecture before studying, others have little or no prior socialization. The trajectories analyzed testify to the way in which the social background and primary (family), then secondary (school) socialization shape the ability of students to socialize more or less easily and quickly with architecture. These gaps are depending on the position of the respondents on the social scale. To claim that the student status would reshuffle the cards and level out inequalities would not be honest. Indeed, the importance of the social, economic and cultural background immediately impacts the first steps at the School of Architecture. Some are very comfortable, where others will be confused, even distant from the concerns of studies because too busy trying to earn a living, to find accommodation.

SOCIALIZATION "IN" AND "OUTSIDE" THE SCHOOL

The School offers a framework, a frame of reference and opportunities for socialization through its decorum (its spaces and their capacities) [START_REF] Lemaître | Architecture et culture institutionnelle : le décorum des grandes écoles en France[END_REF] , the training provided, the activities and the student life that unfold there. The School offers a potential for socialization which the students discover as they become familiar with the School as a place and an institution. It is a potential which they more or less manage to seize and make fruitful according to their prior path. This potential also unfolds "outside the walls", and sometimes reinforces, but also collides with, or displaces (pre)established representations and requires adjustments.

Over the course of the interviews, the respondents testify to their discovery and their progressive appropriation of the establishment. The School, from its surroundings to its amphitheaters, passing through its project studios, its workshop, its library or its corridors, gradually becomes a familiar space, following the successive appropriations by the students, between formal and informal uses. Student life, through friendship or associative activities, as well as the classes allow the students to invest the different spaces. Group work, the production of models, intensive work formats, are all situations that promote dialogue between peers "within" the walls, on the periphery of the lessons. Several testimonies converge on an attenuation of the limits between studies and private life which gradually spread around the School, concentrating most of the activities and the time of the students.

If for some School attendance is above all a place of learning where they only pass through to take lessons, for others it quickly becomes "more than a School", a "second home", an extension of cramped student accommodation, especially in the first years when students have not yet grouped together in shared accommodation. Also, the School seems to be more appropriated by those whose families are geographically distant and those whose accommodation is close to the institution. Similarly, affinities

P A P E R

and friendships seem to have spatial reasons and resonances. Respondents describe the importance of the group of friends, formed almost immediately "from the first day, on the lawns [in front of the school building]", or during the semester when group work contributes to socialize between peers that goes far beyond the framework of university work. Thus, socialization also takes place, by mimicry and distinction, between groups of friends discovering together the milieu of architecture. It can be in an individual way (depending on life stories and prior social positioning) but also largely as a collective experience. Beyond the group of friends of the same school year, students build relations with peers of the upper years who seem to play a spurring role, particularly a form of mentoring.

"Outside the walls" offers specific space-times of socialization which tend to unframe the relationship between students and teachers, but also between peers. Study trips in full promotion (to Paris in the first year, to Rome in the second year) or partial (to a European capital in the third year), field surveys during tutorials in sociology or on-site immersion during project workshops, but also more individual experiences such as internships, are all opportunities to socialize with architecture outside of school. Thus, we can question the way in which experiences "outside the walls" -whether they are part of an educational framework or "all against" -shape the trajectories of architecture students and endow them with a specific spatial capital. The figure of the "pass-through" would make it possible to explore the type of borders (disciplinary, scalar, social but also more symbolic, etc.) that the students collide with, cross or from which they free themselves during these experiences. It is a question of understanding how the students foster the acquisition of knowledge, know-how and interpersonal skills, in particular between peers. The concept of "space defector" (like the "class defector") might help as a possible characterization of these future architects.

LEARNING TO BE "PART OF THE BODY" : BETWEEN INCORPORATION AND CORPORATE EFFECTS

Over the course of the survey, the notion of "being part of a body" emerged to explain the way in which the process of socialization affects both the body of the respondents and the social body they constitute. Thus, the architectural acculturation "in" and "outside" the walls affects the bodies and modifies them through the ethos, the posture, the clothes, etc. The voice of respondents is also impacted, as well as by phrasing and diction. The first series of interviews reveal the first signs of these transformations. From one interview to another, the respondents no longer occupy the space in the same way: the posture of the body, the posture of the head, the way of putting the voice change. The voice is louder, the movements fuller. The regional or foreign accent is attenuated, has almost disappeared. New words or phrases appear. The bodies are accessorized, such as the bracelet of Archiculture (annual festival organized by the Schools' associations) that some have kept once the event has passed in the manner of nostalgic festival-goers.

How is architectural socialization (as a milieu and as a discipline) incorporated? The premises of this long-term research provide the first elements of understanding the way in which the bodies of the students are put "on trial", from the first week of their first study year, at a dance workshop intended for the entire promotion. The newcomers are invited to come into contact with architecture through the school building and the bodies of others. This introductive workshop, which elicited many reactions from respondents during the interviews, testifies to the first forms of socialization by the body and describes forms of learning by the body or by putting the body "on trial".

P A P E R

Acculturation to architecture is incorporated through the confrontation of the body with spatialities, through surveying and manipulation. Students practice "taking the measure" of the spaces crossed, to grasp their atmospheres by summoning their five senses. These trials of the body, in and through space, would contribute to the acquisition of a specific spatial repertoire. The eye and the hand are particularly solicited, by learning observation and drawing. To learn to look at space, to understand it, to decipher it, to describe it, but also to represent it, is one of the major activities of architects apprentices, accompanied by teachers, but also by practicing with peers. The confrontation of the body with the full-scale model during the project studio, drawing classes, but also visits of buildings previously studied in books are cited by respondents as highlights, where architecture is incorporated, where students are learning by direct confrontation of the body with spaces. The acquisition of the arts of doing and thinking would then take place in different stages and by iteration "between" and "outside" the walls of the School. This iteration would promote the successive incorporation and adjustment of representations and practices. The body is also put to trial by the rhythm of studies and the rite of the "charrette" [french : period of intensive work] . If one of them proudly recounts his first sleepless night, others explain that they were made aware of the risks of lack of sleep and the impact on their health from the start of the school year.

Another possible mobilization of the notion of "being part of the body" is that of the "social body" of the School of Architecture. It is a group of belonging in which the students will gradually immerse themselves, from the animated discussions between peers on the lawns in front of the building to the classrooms or project studios, via the library, the university restaurant, etc. Students socialize through their peers (between so-called circumstantial socialization, friendships, love stories, etc.), concomitantly but at their own pace, between mimicry and distinction. Socialization takes place "in reference to... ": one or more teachers, or students of higher classes, between identification, ripple effect, and reproduction. The inculcation of the architect's habitus is punctuated by repetitive rites of passage (the rendering of the project and its oral presentation, the sometimes mythologized charrette). The teacher-student relationship, of which the teaching of the project would be the most emblematic, is at the heart of the professional socialization of architecture students. They are confronted with a formal curriculum, whose assessment methods are poorly explained, and a hidden curriculum that requires them to appropriate themselves the values, attitudes and implicit standards that govern the profession. Several respondents report how studies and the School of Architecture have gradually become a primary concern -a total social space-time or topoï [START_REF] Darmon | Classes préparatoires : la fabrique d'une jeunesse dominante[END_REF] -which occupy their daily thoughts and activities without interruption. They evoke the project teaching that monopolize their minds. Study trips are very important: Living together and sharing common experiences and references would seem to be a major part of socialization, recognition between peers and corporate effects.

SOCIALIZATION, BETWEEN MANUFACTURING REPRESENTATIONS AND DISCOURSE

Socialization in architecture is a long, non-linear process. Initiated before studies, it shapes their representations and guides their ways of entering them. Socialization takes place in successive stages and involves invisible work of the respondents, which consists of adjusting their representations and their interpretations of the context as they progress. By encountering different situations (crucibles, rites of passage, etc.) they seem to comply by acquiring the arts of doing, thinking and behaving (sometimes involving forms of trial and error, questioning, resistance). Students act as interpreters whose activities would be a concomitant narration of the emergence of an individual and collective reflexivity. The socialization process would be accompanied by a discursive dimension where students, in order to produce meaning and appropriate themselves what they do, summon several registers of discourse and narrative forms [START_REF] Thompson | Narratives of architectural education, from student to architect[END_REF] . This individual and collective narrative thus participates in an "invisible work" which consists of shaping and adjusting their representations, by trial and error, successive successes and failures before managing to produce meaning.

The description of a typical week, a recurring question asked in each series of interviews, highlights the blurring between the private and the academic sphere. It also shows how the respondents seem to initially socialize in a disciplinary, partial and fragmentary way, by juxtaposition of the activities carried out. The testimonies collected attest to a difficulty in making the link between the different teachings provided and a vagueness on what architecture covers as a discipline. It makes it difficult for the students to get a better idea of the studies and the professional horizons that open up to them. They seem to be groping and a whole part of their activities is dedicated to understanding what they are doing, and what their teachers, the school, and the professional milieu expect of them. Some respondents refer more specifically to their project teachers, whom they often see and identify as referents. Socialization in architecture is therefore accompanied by a permanent activity of interpretation and translation, carried out individually, but also by successive adjustments between peers, with reference to their elders.

PROSPECTIVE/ VISION TOWARDS "SPATIAL CAPITAL" TO TEACH ARCHITECTURE

What does prior and transmitted spatial capital augurs as pedagogical horizons? How could it be implemented into the curriculum? What could be a cosmopolitan spatial capital for architects delivered by a School of Architecture? Such a curriculum would imply to criticize and objectivize the type of space culture and practice Schools transmit and reproduce. We need to quantify and qualify the project supports, typologies and theoretical backgrounds of space production proposed to the students. This analysis should orient, increase and improve the repertoire of spatial action and spatial culture in order to compensate, or even counter for prior social determinisms.

We should also offer teaching modules that provide students with an understanding of what is at stake during their studies, and an understanding of how work the "architecture worlds" [START_REF]According to the title of Howard Becker's sociology work on artists : Art worlds[END_REF] in which they will circulate, exercise and transform. Thus, we propose since recently a module which offers students the opportunity to grasp the functioning of the socio-professional environment of the production of space through the places, roles and positions occupied by architects, multi-positioned actors of the "architectural and urban factory". During the semester, the students carry out an interview survey entitled "Portraits of architects". This study contributes both to helping students to confront their representations with those of "real" architects and to participating in research work carried out for a better knowledge of professional trajectories.

Ultimately, acquiring spatial capital should enhance students to become experts of transdimensions of space, crossing time, scales, uses and ambiances to transform and perform the social and environmental challenges.

Figure 1 .

 1 Figure 1. Study trip to Rome of second year students. Credit: ENSA Nantes.

Figure 2 .

 2 Figure 2. Solar Decathlon Europe 2014. Team Atlantic Challenge with students of the School of Architecture and two Schools of Engineeing. Credit: ENSA Nantes.

Figure 3 .

 3 Figure 3. Archiculture, dinner of the annual festival organized by the students. Credit: ENSA Nantes.

Figure 4 .

 4 Figure 4. Dance workshop during the first week of the first study year. Credit: ENSA Nantes.