Y Battie
email: yann.battie@univ-lorraine.fr.

L Broch

A En Naciri

J.-S Lauret

M Gicquel-Guézot

A Loiseau

Diameter dependence of the optoelectronic properties of single walled carbon nanotubes determined by ellipsometry

We report ellipsometric measurement on single walled carbon nanotube (SWCNT) films performed in a large spectral range from 0.07eV to 4.97eV. The complex dielectric functions of SWCNTs are correlated to their diameter distribution extracted from transmission electron microscopy. Here we show that the transition energies between Van Hove singularities are directly related to the strong one dimensional confinement. In the infrared spectral range, the real part of the dielectric function becomes negative. The electronic properties of SWCNTs are extracted from ellipsometry by using a Drude model. The mobility and the mean free path of charge carriers are limited by the high number of SWCNT contacts. In accordance with tight binding simulation, the conductivity and the charge carrier concentration increase with the SWCNT diameter. Finally, we demonstrate that the -plasmon energy depends on the charge carrier concentration.

Introduction

Single walled carbon nanotube (SWCNT) is a one dimensional material which has remarkable optical and electronic properties. Indeed, depending on its chirality, a SWCNT can be metallic or semiconductor. Moreover, semiconducting SWCNTs (SC-SWCNTs) have direct-bandgap, so they can be used to emit and detect light. These properties make SWCNT an ideal candidate for new optoelectronic devices such as transparent conductive electrodes [START_REF] Fanchini | Optical anisotropy in single-walled carbon nanotube thin films: implications for transparent and conducting electrodes in organic photovoltaics[END_REF], highbit-rate all-optical signal regenerator [START_REF] Nong | A direct comparison of single-walled carbon nanotubes and quantum-wells based subpicosecond saturable absorbers for all optical signal regeneration at 1.55 m[END_REF], high-repetition mode-locked laser [START_REF] Valle | Passive mode locking by carbon nanotubes in a femtosecond laser written waveguide laser[END_REF] or bolometer [START_REF] Maine | Mid infrared detectors based on carbon nanotube films[END_REF]. However, the development and the optimization of such devices suffer to the lack of experimental reports on the optoelectronic properties of SWCNT film which are currently extracted from transport measurements. However, this technique requires electrical contacts and known current paths, which exclude the estimation of the intrinsic electronic properties of SWCNT assembled into percolated film in which Schottky barrier at the contacts are unambiguously observed [START_REF] Heinze | Carbon Nanotubes as Schottky Barrier Transistors[END_REF][START_REF] Battie | Thickness dependent sensing mechanism in sorted semi-conducting single walled nanotube based sensors[END_REF].

Ellipsometry is a non contact characterisation tool which allows determining the complex dielectric function of material. The dielectric function contains several insights on the optoelectronic properties of materials such as the optical band gap or the electronic conductivity. Thus, the dielectric function is crucial to design optoelectronic devices.

Simulations based on ab-initio calculation [START_REF] Guo | Linear and nonlinear optical properties of carbon nanotubes from first-principles calculations[END_REF] or tight binding approximation [START_REF] Lin | Optical spectra of single-wall carbon nanotube bundles[END_REF] show a strong dependence between the dielectric function and the chirality of SWCNTs. Ellipsometry was previously used to determine the chirality distribution and the porosity of SWCNT films [START_REF] Battie | Chirality distribution in single walled carbon nanotube films by spectroscopic ellipsometry[END_REF].

Using an effective medium approximation to analyse ellipsometric data, Fanchini et al. [START_REF] Fanchini | Optoelectronic properties of transparent and conducting single-wall carbon nanotube thin films[END_REF] claimed that for high SWCNT density, SWCNT film exhibits a metallic behaviour suggesting that the percolation threshold can be estimated by analysing the Drude contribution to the SWCNT dielectric function. In the particular case of oriented SWCNTs, ellipsometry have revealed that the optical anisotropy is related to anisotropic selection rule conditions [START_REF] Battie | Optical anisotropy of single walled carbon nanotubes investigated by spectroscopic ellipsometry[END_REF]. The optoelectronic properties of boron doped SWCNT was investigated by combining ellipsometric and transport measurements [START_REF] Fanchini | Modification of transparent and conducting single wall carbon nanotube thin films via bromine functionalization[END_REF]. In this reference, the authors claimed that boron functionalization shifts the Fermi level and induces an acceptor states in the density of state (DOS) of SWCNT. On the other hand, ellipsometric measurements performed on sorted SWCNTs have shown that the energy of the -plasmon band depends on the ratio between SC-SWCNTs and metallic SWCNTs (M-SWCNTs) [START_REF] Park | Investigation of ultraviolet optical properties of semiconducting-enriched and metal-enriched single-walled carbon nanotube networks using spectroscopic ellipsometry[END_REF]. Despite several advances that have been made to measure the dielectric function of SWCNT, the correlation between the SWCNT diameter and the dielectric function is still under debate. This paper focuses on the ellipsometric measurements of the complex dielectric function of SWCNT films in a large spectral range from 0.07eV to 4.97eV. The influence of SWCNT diameter on the optical transitions and -plasmon band is highlighted. Due to one dimensional confinement, we report a large dependence between the optical transition and the diameter.

By adjusting the infrared part of dielectric function by a Drude model, the electronic DC conductivity of SWCNT film, charge carrier concentration, mean free path and mobility are estimated. The variation of charge carrier concentration is compared to the simulated one from tight binding calculation. We demonstrate that the mean free path of charge carrier is limited by the large number of contacts between SWCNTs. Finally, we highlight a direct correlation between charge carrier density and the -plasmon energy of SWCNTs.

Experimental

Dispersion of SWCNTs:

Electrical arc discharge (EAD), high-pressure carbon monoxide (HiPCo), raw cobalt molybdenum catalyser (CoMoCat) SWCNTs and SWCNTs enriched in (6,5) chirality, purchased from Carbon Solution inc., Unidym and SouthWest Nanotechnologies, respectively, are investigated. 4 mg/ml of SWCNT aqueous suspension was prepared by mixing in water, SWCNTs with 2%wt of sodium cholate. The suspension was sonicated during 90 min with an ultrasonic tip. The temperature of the suspension was fixed at 5°C with a thermostatic bath. The suspension was ultra-centrifuged at 35000 rpm during 1h to separate non-dispersed carbon nanotubes from the well separated ones. The supernatant was then extracted.

(6,5) SWCNT chirality enrichment:

SWCNTs with a nearly single [START_REF] Battie | Thickness dependent sensing mechanism in sorted semi-conducting single walled nanotube based sensors[END_REF][START_REF] Heinze | Carbon Nanotubes as Schottky Barrier Transistors[END_REF] chirality are sorted by density gradient ultracentrifugation from CoMoCat SWCNTs [START_REF] Green | Nearly Single Chirality Single Walled Carbon Nanotubes Produced via Orthogonal Iterative Density Gradient Ultracentrifugation[END_REF][START_REF] Posseckardt | Improved sorting of carbon nanotubes according to electronic type by density gradient ultracentrifugation[END_REF]. Briefly a non-linear density gradient was made by juxtaposing 9 layers in an ultracentrifugation column. The layers consist in 0%, 15%, 17.5%, 20% 22.5%, 25%, 27.5%, 30% and 60% of iodixanol dissolved in water. These layers also contain 0.8%wt of sodium cholate. Then, a solution of CoMoCat SWCNTs with 25% of iodixanol and 0.8%wt of sodium cholate was introduced in the nonlinear density gradient. The gradient was then centrifuged during 14h at 35000 rpm. The SWCNT micelles must move down through the density gradient column during the ultracentrifugation in order to reach their isopycnic points. A pink layer enriched in [START_REF] Battie | Thickness dependent sensing mechanism in sorted semi-conducting single walled nanotube based sensors[END_REF][START_REF] Heinze | Carbon Nanotubes as Schottky Barrier Transistors[END_REF] SWCNT was observed at the top of the gradient. This layer was extracted with the gradient master device (Biocomp).

SWCNT film preparation and characterization

SWCNTs are deposited on nitrocellulose filter with 0.22µm pore size by filtering the SWCNT suspension with a Buchner filtration set up. The volume of the SWCNT suspension was adjusted to obtain a completely black and opaque thick film. The film was then washed thoroughly to remove the surfactant. Note that contrary to reference [START_REF] Battie | Optical anisotropy of single walled carbon nanotubes investigated by spectroscopic ellipsometry[END_REF], the SWCNT films are not transferred on a silicon substrate and nitrocellulose filter is used as substrate.

Electrical measurements are recorded at room temperature with an ECOPIA HMS-5500 set up. Briefly, four point probe measurements are performed in the Van der Paw geometry. The film resistivity is obtained by applying a current of 0.1 mA which flows along one edge of the sample and by measuring the voltage across the opposite edge. This set up can also be used to perform Hall measurements. A 0.51T magnetic field is created perpendicularly to the film surface. A current is applied along two opposite edges of the sample and a voltage induced by the magnetic field is recorded between the other edges. This allows estimating the Hall coefficient of the film. The carrier density and mobility are deduced from the film resistivity and Hall coefficient. Note that the SWCNTs are assumed to be p-doped by air.

Transmission electron microscopy (TEM) investigations were performed with a Technai CM20 electron microscope operating at 200 kV. The TEM grids were prepared by depositing a drop of SWCNT suspension on a copper grid. Scanning electron microscopy (SEM) was performed with a field emission gun scanning electron microscope Zeiss Gemini operating at 15 kV. The optical properties of SWCNT film are investigated by spectroscopic ellipsometry.

A home built rotating polariser ellipsometer and a photoelastic ellipsometer (Horiba Jobin Yvon) were used to cover a large spectral range from 0.07eV to 4.97eV. The ellipsometric measurements were performed at three angles of incidence (50°, 60°, 70°).

Results and discussion

SWCNT diameter distribution

Figure 1 depicts the diameter distributions of SWCNT extracted from a statistical analysis based on transmission electron microscopy (TEM) of more than 50 SWCNTs. Due to Van a Gaussian distribution. The mean radius of (6,5), CoMoCat, HiPCo and EAD SWCNTs are 0.75nm 0.08 nm, 0.75 nm 0.08 nm, 0.95nm 0.09 nm and 1.3 nm 0.12 nm, respectively (Fig. 1). Due to the limited TEM resolution few diameter differences are observed between [START_REF] Battie | Thickness dependent sensing mechanism in sorted semi-conducting single walled nanotube based sensors[END_REF][START_REF] Heinze | Carbon Nanotubes as Schottky Barrier Transistors[END_REF] and CoMoCat SWCNTs.

Ellipsometric data analysis

Ellipsometric measurements were performed in the 0.07eV to 4.97eV spectral range on dense SWCNT films. Depending on the SWCNT film, the film thickness, estimated from mechanical profilometry, is in the 500nm-800nm range. These films are sufficiently thick, absorbent and reflective to be considered as an isotropic semi-infinite medium for the ellipsometric data analysis. In this context, the effective dielectric constant of SWCNT films can be analytically derived from the measured ellipsometric angles and not shown [START_REF] Azzam | Ellipsometry and polarized light[END_REF]

2 2 tan tan 1 tan 1 1 sin j j e e , (1)
where is the angle of incidence. In other words, the effective dielectric function of SWCNTs (Fig. 2) can be extracted from ellipsometric measurements by using equation (1) without fitting. Measurements performed at three angles of incidence (50°, 60°, 70°) reveal that the dielectric function does not depends on the angle of incidence (not shown). This unambiguously suggests that the surface roughness is sufficiently small to be neglected, confirming the correctness of our ellipsometric model.

Visible-near infrared optical properties of SWCNTs

Several bands are observed from 0.40 eV to 3.20 eV in the i spectra. These bands are associated to the transitions between the Van Hove singularities (VHS) in the SWCNT DOS induced by the one-dimensional nature of SWCNT. These bands are assigned to specific transitions between VHS according to references [START_REF] Bachilo | Structure-assigned optical spectra of single-walled carbon nanotubes[END_REF][START_REF] Pekker | Wide-range optical studies on various single-walled carbon nanotubes: Origin of the low-energy gap[END_REF][START_REF] Green | Colored semitransparent conductive coating consisting of manodisperse metallic single walled carbon nanotubes[END_REF][START_REF] Liu | One-step separation of high-purity (6,5) carbon nanotubes by multicolumn gel chromatography[END_REF][START_REF] Knupfer | Electron energyloss spectroscopy studies of single wall carbon nanotubes[END_REF][START_REF] Lauret | Ultrafast carrier dynamics in single wall carbon nanotubes[END_REF]. Indeed, the S 11 , S 22 , M 11 , S 33 bands of EAD SWCNTs are in the 0.5-1eV [START_REF] Pekker | Wide-range optical studies on various single-walled carbon nanotubes: Origin of the low-energy gap[END_REF], 1-1.5eV [START_REF] Green | Colored semitransparent conductive coating consisting of manodisperse metallic single walled carbon nanotubes[END_REF], 1.5-2.1eV [START_REF] Green | Colored semitransparent conductive coating consisting of manodisperse metallic single walled carbon nanotubes[END_REF] and 2.1-2.9eV [START_REF] Green | Colored semitransparent conductive coating consisting of manodisperse metallic single walled carbon nanotubes[END_REF] spectral ranges, respectively. The S 11 , S 22 , M 11 , bands of HiPCo SWCNTs are in the 0.6-1.4eV [START_REF] Pekker | Wide-range optical studies on various single-walled carbon nanotubes: Origin of the low-energy gap[END_REF], 1.4-2.6eV [START_REF] Green | Colored semitransparent conductive coating consisting of manodisperse metallic single walled carbon nanotubes[END_REF] and 2.6-3.5eV spectral ranges, respectively. The S 11 , S 22 , M 11 , [START_REF] Liu | One-step separation of high-purity (6,5) carbon nanotubes by multicolumn gel chromatography[END_REF], 1.5-2.5eV [START_REF] Liu | One-step separation of high-purity (6,5) carbon nanotubes by multicolumn gel chromatography[END_REF] and 2.6-3.1eV [START_REF] Liu | One-step separation of high-purity (6,5) carbon nanotubes by multicolumn gel chromatography[END_REF] spectral ranges, respectively. The transition energies associated to [START_REF] Battie | Thickness dependent sensing mechanism in sorted semi-conducting single walled nanotube based sensors[END_REF][START_REF] Heinze | Carbon Nanotubes as Schottky Barrier Transistors[END_REF] SWCNTs are given by Bachilo [START_REF] Bachilo | Structure-assigned optical spectra of single-walled carbon nanotubes[END_REF]. According to Knupfer et al. [START_REF] Knupfer | Electron energyloss spectroscopy studies of single wall carbon nanotubes[END_REF] and Lauret et al. [START_REF] Lauret | Ultrafast carrier dynamics in single wall carbon nanotubes[END_REF], the band located above 3.5 eV is the -plasmon band of SWCNTs.

In agreement with the Kramers-Kronig relation, large variations of r appear close to these transitions. The CoMoCat and HiPco SWCNT spectra are more structured than the EAD one.

These substructures are the optical signature of individual chiralities present in the film. Due to their large diameter, EAD SWCNT has broad chirality distribution. This induces an inhomogeneous broadening of the absorption bands.

Transitions between the first (S 11 ) and second (S 22 ) pairs of VHS of SC-SWCNTs appear in all SWCNT spectra while the transition between their third VHS (S 33 ) is only observed in the EAD spectra. In the other hand, the amplitude of the band associated to the transitions between the first pair (M 11 ) of VHS of M-SWCNTs is reduced as the SWCNT mean diameter decreases and completely disappears for sorted SWCNTs. Indeed, CoMoCat SWCNTs contain few metallic species [START_REF] Battie | Chirality distribution in single walled carbon nanotube films by spectroscopic ellipsometry[END_REF]. Moreover, the dielectric function of sorted SWCNTs is dominated by two wide bands located at 1.23 eV and 2.15 eV attributed to the S11 and S22 transitions of (6,5) SC-SWCNTs [START_REF] Bachilo | Structure-assigned optical spectra of single-walled carbon nanotubes[END_REF], respectively, confirming the high (6,5) chirality enrichment. As shown in Figure 2, the transition energies depend on the SWCNT source. As example, the S11 transition is centered at 1.23 eV, 1.11 eV, 0.99 eV and 0.67 eV for [START_REF] Battie | Thickness dependent sensing mechanism in sorted semi-conducting single walled nanotube based sensors[END_REF][START_REF] Heinze | Carbon Nanotubes as Schottky Barrier Transistors[END_REF] enriched, CoMoCat, HiPco, and arc discharge SWCNTs. By neglecting excitonic effects, these energies can be considered as the optical band gap of SC-SWCNTs [START_REF] Lin | Many-body effects in electronic bandgaps of carbon nanotubes measured by scanning tunnelling spectroscopy[END_REF]. The measured transitions energies are reported in a Kataura plot (Fig. 3) and are compared to the calculated ones from tight binding approximation [START_REF] Kataura | Optical properties of single-wall carbon nanotubes[END_REF]. Experimental data are in close agreement with the calculated ones. Small deviations with theoretical value can be attributed to the perturbations by the environment and excitonic effects. In agreement with tight binding and zone-folding calculations [START_REF] Samsonidze | The concept of cutting lines in carbon nanotube science[END_REF], the measured energies ESii and EM11 are inversely proportional to the SWCNT diameter d:

d ia E cc S ii 0 2 , (2) d a E cc M 0 6 11 , (3)
where ac-c and 0 are the carbon bond length and the transfer integral parameter used to describe the optoelectronic properties of SWCNTs, respectively. Thus, by fitting simultaneously all experimental transition energies by equations (2-3), the transfer integral can be estimated to 2.8 eV 0.1 eV. This value is in close agreement with the reported one by Dresselhaus et al. [START_REF] Samsonidze | The concept of cutting lines in carbon nanotube science[END_REF]. Thus, the redshift of the transition energies is due to the strong one dimensional confinement induced by the periodic boundary conditions that result from wrapping the graphene sheet into a tubular structure.

Electronic properties of SWCNTs:

In the 0.07eV-0.40eV range, the real part of the dielectric function become more and more negative while the imaginary parts abruptly increases as the photon energy decreases. This metallic behaviour was previously observed [START_REF] Maine | Complex optical index of single wall carbon nanotube films from the near-infrared to the terahertz spectral range[END_REF] and involves the free charge carrier motion in the SWCNT network. Note that SC-SWCNTs are known to be p-type in air [START_REF] Posseckardt | Improved sorting of carbon nanotubes according to electronic type by density gradient ultracentrifugation[END_REF]. Several works have previously reported the presence of an absorption band at 17meV [START_REF] Pekker | Wide-range optical studies on various single-walled carbon nanotubes: Origin of the low-energy gap[END_REF][START_REF] Maine | Complex optical index of single wall carbon nanotube films from the near-infrared to the terahertz spectral range[END_REF][START_REF] Kamaras | Wide range optical spectra of carbon nanotubes: a comparative study[END_REF][START_REF] Ugawa | Far-infrared gaps in single-wall carbon nanotubes[END_REF]. This band was attributed to the bandgap of M-SWCNTs induced by curvature effect. However, due to our limited spectral range, this band cannot be observed in Figure 2. As shown in figure 2, the SWCNT dielectric function in the infrared spectral range can be modelled by a Drude dispersion law [START_REF] Fanchini | Optoelectronic properties of transparent and conducting single-wall carbon nanotube thin films[END_REF][START_REF] Maine | Complex optical index of single wall carbon nanotube films from the near-infrared to the terahertz spectral range[END_REF]:

i p 2 , (4)
where and are the effective dielectric function of the SWCNT film and the photon energy, respectively. Three parameters are adjusted , p et which correspond to the dielectric function at high energy, the plasma energy and the damping constant of free charge carrier.

To give more insights on the electronic properties of SWCNT films, the free charge carrier concentration n and the conductivity of SWCNT film calculated from equations ( 5) and [START_REF] Battie | Thickness dependent sensing mechanism in sorted semi-conducting single walled nanotube based sensors[END_REF] are represented in Fig. 4a e and 0 are the elementary charge and vacuum permittivity. The effective mass m of charge carriers in carbon nanotubes is 0.9*10 -31 kg [START_REF] Marulanda | Current transport modeling of carbon nanotubes: Concepts, analysis, and design[END_REF]. The conductivity obtained from ellipsometry is compared to DC conductivity estimated from Van Der Paw electrical measurements (Figure 4a). Despite a limited ellipsometric spectral range, the film conductivity values obtain from ellipsometric measurements deviates by less than 6% from the estimated ones from Van Der Pauw electrical measurements confirming correctness of the ellipsometric data analysis. In other words, the additional low frequency band reported in references [START_REF] Pekker | Wide-range optical studies on various single-walled carbon nanotubes: Origin of the low-energy gap[END_REF][START_REF] Maine | Complex optical index of single wall carbon nanotube films from the near-infrared to the terahertz spectral range[END_REF][START_REF] Kamaras | Wide range optical spectra of carbon nanotubes: a comparative study[END_REF][START_REF] Ugawa | Far-infrared gaps in single-wall carbon nanotubes[END_REF] at 17 meV has a negligible effect on the DC conductivity. By considering that SWCNTs are p-doped in air [START_REF] Posseckardt | Improved sorting of carbon nanotubes according to electronic type by density gradient ultracentrifugation[END_REF], the free charge carrier concentration can be assimilated to the hole concentration. The hole concentration and the film conductivity increase with the SWCNT diameter (Fig. 4a). The charge carrier density of SWCNTs is close to the charge carrier density of graphite [START_REF] Dillon | Electronic energy band parameters of graphite and their dependence on pressure, temperature and acceptor concentration[END_REF].

Despite CoMoCat and (6,5) SWCNTs have similar diameter, their charge carrier concentration and conductivity differ. Indeed, the number of M-SWCNTs and so the concentration in free charge carriers is smaller in (6,5) SWCNTs. By considering the approach introduced by Marulada et al. [START_REF] Marulanda | Current transport modeling of carbon nanotubes: Concepts, analysis, and design[END_REF][START_REF] Marulanda | Carrier density and effective mass calculations in carbon nanotubes[END_REF], the hole carrier concentration of SWCNTs is defined by:

f E dE E f E D d h d n 1 25 . 0 1 2 2 , (7)
where f(E) is the Fermi Dirac distribution, h the effective wall thickness of SWCNTs equals to 0.617 Å and d the SWCNT diameter. The DOS (D(E)) of SWCNTs is given by [START_REF] Kamaras | Wide range optical spectra of carbon nanotubes: a comparative study[END_REF][START_REF] Marulanda | Carrier density and effective mass calculations in carbon nanotubes[END_REF]:

bands all i i cc E g a E D ) ( 3 8 0 , (8) with i c i c i c i c i E E si E si E E si E E E E g , , , 2 , 2 0 0 1 . (9)
E c,i is the minimum energy value of the i th conduction band. By considering a rigid band model whereby doping shifts the Fermi level without affecting the band structure, this approach can also be used to compute the charge carrier density of doped SWCNTs by air.

Thus, simulations have been made for undoped and p-doped carbon nanotubes by assuming the Fermi level in the mid-gap of SWCNTs and shifted by E g(6,5) /2~0.3eV toward the valence band [START_REF] Blackburn | Transparent conductive single-walled carbon nanotube networks with precisely tunable ratios of semiconducting and metallic nanotubes[END_REF][START_REF] Tchernatinsky | Adsorption of oxygen molecules on individual single-wall carbon nanotubes[END_REF], respectively. Although no accurate value of the Fermi level shift has been found in the literature, simulations performed in Figure 4b give qualitative information used to highlight the electronic properties of SWCNTs. In accordance with Marulada et al. [START_REF] Marulanda | Current transport modeling of carbon nanotubes: Concepts, analysis, and design[END_REF][START_REF] Marulanda | Carrier density and effective mass calculations in carbon nanotubes[END_REF], simulations presented in the Figure 4b show that the charge carrier concentration increases by three orders of magnitude for SC-SWCNTs while it slightly decreases for M-SWCNTs as the diameter increases. Due to the higher DOS of M-SWCNTs close to the Fermi energy, the charge carrier concentration in SC-SWCNTs is smaller than the metallic one. Since M-SCWNT has constant DOS near the Fermi level, the Fermi level shift induced by doping has little effect on the charge carrier concentration. On the contrary, doped SC-SWCNTs have a higher charge carrier concentration than undoped SC-SWCNTs. Thus, air doping may induce a drastic increase of the concentration of the SC-SWCNT hole charge carriers. The charge carrier concentration estimated from ellipsometry is smaller than the calculated charge carrier concentration of M-SWCNTs but slightly higher than the doped SC-SWCNT one. Indeed, the films are composed of a mixture of both kinds of SWCNTs. As the optical M 11 band of M-SWCNTs is more and more pronounced as the SWCNT mean diameter increases, we can conclude that the variations of the measured charge carrier concentration and so the DC conductivity can be attributed to the increase of both, the charge carrier concentration in doped SC-SWCNTs and the M-SWCNT concentration.

By considering that all holes which participate to the conduction have an energy close to the Fermi energy, the hole carrier mobility µ and mean free path l f are both related to the damping constant: m e [START_REF] Fanchini | Optoelectronic properties of transparent and conducting single-wall carbon nanotube thin films[END_REF] and

f f v l ( 11
)
where v f =8.8*10 5 m.s -1 is the Fermi velocity [START_REF] Marulanda | Current transport modeling of carbon nanotubes: Concepts, analysis, and design[END_REF]. Figure 4c shows that the variations of hole mobility µ and mean free path as a function of nanotubes diameter are similar. Large dispersion of mobility around the 75 cm 2 V -1 s -1 value is observed. The mobility is smaller than the measured one on individual nanotubes [START_REF] Dürkop | Extraordinary mobility in semiconducting carbon nanotubes[END_REF]. Despite a micrometer length ballistic transport was reported for individual SWCNT [START_REF] Bachtold | Scanned probe microscopy of electronic transport in carbon nanotubes[END_REF], the mean free path of charge carrier is in the nanometer scale. The hole mobility and mean free path are both related to the scattering events and potential barriers that occur in the conduction channel. As shown by scanning electron microscopy (inset of figure 4c), the films are composed of highly entangled SWCNTs. Schottky barrier between M-SWCNTs and SC-SWCNTs drastically reduces the mean free path of carriers. In addition, potential barriers are also present at the junction of two M-SWCNTs or two SC-SWCNTs. In other words, the small mean free path and mobility values come from the large number of SWCNT-SWCNT contacts in the film.

Diameter dependence of the -plasmon band

The broad band observed in the 3.95-4.50eV range (Fig. 2) is attributed to the -plasmon resonance induced by the collective oscillation of -electrons polarised along the tube axis which arises from the electronic transitions near the M-point of the Brillouin zone. In contrast with transitions between VHS, the position of the -plasmon band is blueshifted from 3.95eV to 4.50eV as the nanotubes diameter increase. In accordance with our results, a blueshift of the -plasmon band as the SWCNT diameter increases, was reported in recent ab-initio calculations which take into account local field effects [START_REF] Marinopoulos | Ab initio study of the dielectric response of crystalline ropes of metallic single-walled carbon nanotubes: Tube-diameter and helicity effects[END_REF]. As reported by Park et al. [START_REF] Park | Investigation of ultraviolet optical properties of semiconducting-enriched and metal-enriched single-walled carbon nanotube networks using spectroscopic ellipsometry[END_REF], the -plasmon band energy depends on the concentration ratio between M-SWCNTs and SC-SWCNTs i.e. the charge carrier concentration. Moreover, a large blue shift of the -plasmon band accompanied by an increase of SWCNT film reflectivity was reported for hydrogenated or fluorinated SWCNT [START_REF] Lee | Optical properties of hydrogen-and fluorine-doped single-walled carbon nanotubes[END_REF]. Hydrogenation and fluorination are known to dope SWCNTs. In other words, the variation of -plasmon band energy depends on the number of transferred charges. By assuming that all free charge carriers contribute to the collective surface oscillation of the electrons, a quadratic relation, derived from free electron gas model [START_REF] Pitarke | Electronic response of aligned multishell carbon nanotubes[END_REF], can be found after some calculations between the -plasmon energy ( ) and the charge carrier concentration:

n m e 0 2 2 0 2 2 , (12)
The first term of equation ( 12) is associated to the -* interband oscillator energy 0 , while the second term represents the surface plasmon energy of electrons. Note that due to the expansion of the wave function of -electrons over the SWCNT diameter, SWCNTs are considered, in first approximation, as a solid cylinder [START_REF] Pitarke | Electronic response of aligned multishell carbon nanotubes[END_REF]. In agreement with equation ( 12), a linear behaviour is obtained between 2 and n (Fig. 5), confirming the correlation between free charge carriers and the -plasmon band energy. Note that the slope corresponds to the calculated one from equation [START_REF] Fanchini | Modification of transparent and conducting single wall carbon nanotube thin films via bromine functionalization[END_REF]. The -* interband oscillator energy is estimated at 3.7 0.1 eV. Thus, we can conclude that the blue shift of the -plasmon band, observed by increasing the SWCNT diameter, is attributed to the increase of the free charge carrier concentration.

Conclusion

In summary, we have highlighted the dependences between the diameter distribution and the complex dielectric functions of SWCNTs. Due to the strong one dimensional confinement, the energy of optical interband transition between VHS is redshifted when the diameter increases. The DC electronic properties of SWCNT are evaluated by analysing the infrared part of the dielectric function by a Drude model. The mean free path of charge carrier is drastically reduced by the large number of SWCNT contact. Finally, we have demonstrated that the position of the -plasmon band is correlated to the concentration of charge carrier.

Fig. 1 -

 1 Fig. 1-Diameter distribution of (a) (6,5) enriched, (b) CoMoCat, (c) HiPCo and (d) EAD

Fig. 2 -

 2 Fig. 2-Real (red line) and imaginary (green line) parts of dielectric function of (a) (6,5)

  SWCNTs are located in the 0.6-1.5eV

Fig. 3 -

 3 Fig. 3-Comparison between the Kataura plot of experimental data (filled symbol), their

  .

Fig. 4 -

 4 Fig. 4-(a) Variation of the DC conductivity and charge carrier density measured by

Fig. 5 -

 5 Fig. 5-Square of the -plasmon energy versus the charge carrier density.


Daniel Finkelstein-Shapiro

Ines Urdaneta

Monica Calatayud

Osman Atabek

Vladimiro Mujica

Arne Keller

Fano-Liouville Spectral Signatures in Open Quantum Systems

The scattering amplitude from a set of discrete states coupled to a continuum became known as the Fano profile, characteristic for its asymmetric lineshape and originally investigated in the context of photoionization. The generality of the model, and the proliferation of engineered nanostructures with confined states gives immense success to the Fano lineshape, which is invoked whenever an asymmetric lineshape is encountered. However, many of these systems do not conform to the initial model worked out by Fano in that i) they are subject to dissipative processes and ii) the observables are not entirely analogous to the ones measured in the original photoionization experiments. In this letter, we work out the full optical response of a Fano model with dissipation. We find that the exact result for the excited population, Raman, Rayleigh and fluorescence emission is a modified Fano profile where the typical lineshape has an additional Lorentzian contribution. Expressions to extract model parameters from a set of relevant observables are given.

In a set of seminal papers spanning from 1935 to 1961, Beutler [1], Fano [2,3] and Friederichs [4] laid the basis of the theory to describe the absorption lineshapes of atomic photoionization experiments. These lineshapes present marked asymmetries which could not be explained by a simple Lorentzian resonance. The explanation was attributed to an interference between two photo-ionizaton pathways: one where the atom is ionized directly from its ground state and one where it is first excited to a higher discrete state which then ionizes (auto-ionizing states). The minimal Fano model consists in a discrete excited state coupled to a continuum set of excited states, both types of states being reachable by photo-excitation from the ground state. The resulting photo-fragmentation cross-section as a function of the excitation laser frequency ω L is known as the Beutler-Fano or Fano profile:

h( ; q) = (q + ) 2 2 + 1 , (1)
where q is the ratio of the transition dipole moment of the ground-discrete and ground-continuous transitions, and = ( ω L -E e )/γ where E e is the energy of the discrete state relative to the ground state and γ = nπV 2 is the linewidth of the excited state, induced by its coupling (per unit of energy) nV 2 to the continuum set of states, n being the density of states.

Since the original photoionization experiments, the Beutler-Fano profile has been observed in an ever increasing variety of physical systems, and in particular in nanoscale structures [5,6]. These include plasmonic nanostructures [7][8][9], quantum dots, decorated nanoparticles [10] and spin filters [11], to name a few. Although the Fano theory was built on a scattering framework where the observable was the population on the contin-uum (i.e. the ionized electrons), the result continued to be applied (with remarkable success) to dissipative, nonscattering systems where the observable was not always the population in the continuum of states. As noted by A. E. Miroshnichenko in 2010, "a suitable theory for a quantitative description of these cases is still lacking" [5]. Some attempts to supplement the Fano model with dissipation processes have already been made. First by Fano himself in 1963 [12] in a less-known paper, where the objective was to model pressure lineshape broadening by atomic collisions. In this paper, only pure dephasing was considered which amounts to considering solely elastic collisions. In the eighties, Agarwal et al. [13] included population relaxation from the discrete excited state to an additional discrete state to model the competition between atomic photo-ionization and spontaneous emission. K. Rzyzewski and J. H. Eberly [14] considered pure dephasing using a Wiener-Levy stochastic process to describe phase fluctuations. More recently Kroner et al. [15] and Zhang et al. [16] included a general dissipation process for a system of semiconductor quantum dots but the equations were solved approximatively by neglecting the population on the continuum set of states. It is worth noting that these three last works focus on intense field effects, which will not be adressed in the present letter and will be the subject of a future work.

In this letter, we solve exactly the Fano dissipation problem for weak field, fulfilling the conditions for absorption, Raman, Rayleigh and fluorescence spectroscopies. Within the wideband approximation to describe the coupling to the continuum, we obtain explicit and simple expressions for all optical observables, such that they can be used explicitly by experimentalists, notably to directly obtain the system parameters. For this, we solve the dynamics of the quantum system with energy levels in a Fano like configuration and coupled to a bath which induces excited states relaxation and pure dephasing. The system density matrix evolution is described in the Born-Markov approximation, by the Liouville equation in Lindblad form [17].

Although our formalism is general and can be applied to a variety of systems, an important motivation is the type of architectures encountered in light harvesting systems called Grätzel or dye-sensitized solar cells [18,19], where a molecule is adsorbed to a semiconductor surface. In most important physical realizations, the electronic ground state of this hybrid system is an isolated quantum state located in the semiconductor energy gap while the excited states can be considered as superpositions of localized molecular excited states and delocalized semiconductor conduction band states [19,20]. Evidence suggests that such a model with minor modifications could also account for molecules on metal nanoparticles [10,21,22]. Be it for solar energy applications, electronics or sensors, there is strong evidence that the details of the interface distinguish functioning from nonfunctioning devices [23].

The energy levels of our model, along with the possible transitions, are shown in Figure 1. A discrete excited state |e with energy E e is coupled to a continuum of states |k with energy E k . These states can be reached from a ground state |g through laser excitation. A submanifold (typically vibrational) |ν with energy E ν is included in the electronic ground state to open inelastic scattering channels as |ν → |ν . The Hamiltonian is:

|ν |e |k µνe µ νk V (k) Γ ( k , ν ) Γ vib
FIG. 1: Energy levels and transitions of a Fano model with dissipation, including a vibrational ground state manifold, a discrete excited state and a continuum (bottom). A particular realization of a Fano system with dissipation is a molecule adsorbed on a metal oxide semiconductor, here two catechol molecules on a (101) anatase TiO2 surface (top)

H = H 0 + H V + H F (2)
H 0 = ν E ν |ν ν| + E e |e e| + dkE k |k k| H V = dk [V (k)|e k| + V * (k)|k e|] H F = F ν [µ νe cos(ω L t)|ν e| + µ * νe cos(ω L t)|e ν|] + F ν dk [µ νk cos(ω L t)|ν k| + µ * νk cos(ω L t)|k ν|] (3)
where H 0 is the site Hamiltonian, H V is the coupling of the excited state to the continuum, for simplicity, in the following, we will consider that

V (k) = e|H V |k is real.
H F is the interaction with the incident radiation field of frequency ω L , allowing transitions from the ground state to the discrete excited state ν ↔ e and to the continuum of states ν ↔ k, µ ij = i|µ|j is the transition dipole moment between states i and j and F is the field amplitude.

The originality of our model consists in taking into account a possible population relaxation from the continuum to the ground state at a rate Γ(k, ν), and pure dephasings. When the photo-excitation involves an electron, this relaxation is mainly the result of the electronhole Coulombic attraction followed by the thermal relaxation through the phonons of the environment. We phenomenologically capture this dissipation process using a superoperator of Lindblad [17] form, which ensures the trace-preserving and complete positivity of the dynamical map, and solve the evolution of the density matrix with Liouville's equation:

∂ρ ∂t = L(t)ρ (4)
where

L(t) = L H (t) + L D , with L H = -i(1 1 ⊗ H T (t) - H(t) ⊗ 1 1), L D = L D k + L D vib + L D pure L D k = ν dkΓ(k, ν) A(k, ν) ⊗ A(k, ν) -1 2 1 ⊗ A † (k, ν)A(k, ν) + A † (k, ν)A(k, ν) ⊗ 1 , (5)
L D vib = ν =0 Γ vib A(ν, 0) ⊗ A(ν, 0) -1 2 1 ⊗ A † (ν, 0)A(ν, 0) + A † (ν, 0)A(ν, 0) ⊗ 1 (6)
and

L D pure = -ν γ eν |e e| ⊗ |ν ν| + |ν ν| ⊗ |e e| -ν dkγ kν |k k| ⊗ |ν ν| + |ν ν| ⊗ |k k| -dkγ ke |k k| ⊗ |e e| + |e e| ⊗ |k k| , (7)
where H T denotes the transpose of H, A(i, j) = |j i| are the jump operators and Γ(k, ν) is the population relaxation rate from state |k to |ν . The superoperator L D vib relaxes the ground states vibrational manifold and Γ ν0 is the population relaxation rate within the electronic ground state manifold. γ ij are the pure dephasing rates for the ij coherences. We have used the isomorphism LρR → L ⊗ R T ρ, where ρ is the column form of the matrix ρ through the correspondence: |l m| ↔ |l ⊗ |m ≡ ||lm [24].

Non-radiative and radiative transitions from the electronic excited state |e to the ground states |ν exist. They have been left out for simplicity, although their inclusion can be done straigthforwardly. Furthermore, in most cases of interest, the electronic coupling V induces an effective population relaxation rate 2πnV 2 / associated to electron injection into the semiconductor band which is much larger than the e → ν rate corresponding to the direct relaxation mechanism, hence justifying this simplification [25].

The optical response (absorption and emission spectra) is obtained through the Fourier transform of the field two-times correlation function which in the far field is proportional to the dipole two-times correlation function. Using the quantum regression theorem [26][START_REF]Quantum Optics[END_REF][START_REF] Cohen-Tannoudji | Processus d'interaction entre photons et atomes[END_REF], the emitted light differential scattering cross-section can be written in the steady-state of the system as [START_REF] Johansson | [END_REF]30]:

d 2 σ dΩd( ω) = A(θ)× × a∈e,k b∈ν µ 2 ab r∈ν,e,k Re[ρ ra G ab,rb -i(ω -ω L ) ], (8)
with A(θ) = ω 4 sin 2 θ Iin8π 3 c 3 0 , dΩ the element of solid angle, I in the incident laser intensity, θ the polar angle in spherical coordinates, ρ the steady state density matrix and G(z) = (z1 1 -iΩ L -L) -1 the resolvent, which is the Laplace transform of the evolution superoperator, corresponding to the time-independent Liouvillian L = e iΩ L t L(t)e -iΩ L t in the rotating-wave approximation (RWA), which consists in removing resonant oscillating prefactors and discarding non resonant terms. The Ω L matrix is a diagonal matrix with ±ω L for excited(ground)-ground(excited) coherences, and zero elsewhere. In summary, in order to obtain the optical emission we need to calculate the resolvent and the steady-state density matrix, which we do next. An explicit expression for G(z) is obtained by separating L into its diagonal and non-diagonal parts. While the diagonal contribution leads to a trivial calculation of its resolvent, the non-diagonal contribution is worked out through a Dyson equation to all orders in the field-free coupling and to second order in the field interaction. As for the steady-state density matrix ρ, it appears as the kernel of the Liouvillian within the RWA, that is (-iΩ L -L)ρ = 0, which can in turn also be expanded to second order in the field, with the help of the previously calculated resolvent G(z) [25].

In principle, the above method allows to calculate the optical response for arbitrary couplings and density of states. Nevertheless, for many materials, the wideband approximation, which considers k-independent couplings and a constant density of states, can faithfully reproduce experimental measurements. This was the same approximation which allowed a closed analytic form in the original Fano model [3] and which has been used in all previous partial attempts to solve the Fano-dissipative problem [12][13][14][15][16]. We introduce the notation Γ cν = Γ(k, ν) and µ νc = µ νk for the k-independent parameters.

A straightforward but tedious calculation [25] yields all the terms needed to calculate the absorption spectrum and the emission differential cross-section (Eq (8)). The absorption and emission profiles, which carry the ω L dependence, can be expressed in a compact and simple form in terms of the following function:

f ( , q, η, α) = α (q + ) 2 2 + 1 + η q + 1 2 + 1 (9)
which is a linear combination of a Fano profile (first term) and a Lorentzian (second term); q being the Fano asymmetry parameter and the normalized incident laser energy. α (which can only take the values 0 and 1) and η are weighting coefficients for the Fano and Lorentzian components, respectively. The sum of excited state populations (ρ ee + dkρ kk ) can be written as:

N excited = B abs f ( , q, η, α) (10)
where B abs is a constant [25], and q are the standard Fano parameters ( = (E e -ω L )/γ, γ = nπV 2 , q = µ 0e /(nπV µ 0c )), α = 1, and η = ( ν Γ cν )/(4πnV 2 ). We can obtain the extinction coefficient β measured in optical absorption measurements using arguments of detailed balance [25]:

β = nπµ 2 0c c 0 ω L f ( , q, η = 0, α = 1) ( 11
)
where c is the speed of light and 0 the vacuum permittivity. The differential cross sections for the three processes Rayleigh, Raman and fluorescence can all be expressed referring to the same functional form:

d 2 σ i dΩd( ω) = A(θ)B i ν R(ω, ω i 0 , ∆ i )f ( i , q i , η i , α i ) ( 12
)
where i = Rayleigh, Raman or fluorescence. The expression for emission has 3 parts:

A(θ)B i ν is a prefac- tor [25], R(ω, ω 0 , ∆) = ∆ π [(ω-ω 0 ) 2 +(∆) 2 ]
-1 is a normalized Lorentzian of central frequency ω 0 and half-width ∆ which gives the emission lineshape (for a fixed incident laser frequency), and f ( , q, η, α) which gives the profile -the signal integrated over the emitted frequency -and carries all the dependence of the incident laser frequency ω L .

We first discuss the characteristic function f of the population given by equation 10. The result is the combination of a standard Fano profile, and a Lorentzian. The signature of the relaxation process is embodied in the parameter η. Its physical meaning is the ratio of the relaxation rate to the injection rate into the continuum. When η 1 the relaxation quenches the population of the continuum and therefore suppresses the Fano interference signal giving rise to a pure Lorentzian lineshape. When η 1, we recover the original Fano profile. Figure 2 shows normalized population profiles as given by Eq. ( 10), for several values of q and η. As η → 0, the curve approaches the standard Fano profile while η = 1 shows the Fano with a Lorentzian contribution for different values of the q parameter. q =1

,QWHQVLW\ q =0 q =100

FIG. 2: Characteristic function for absorption or Raman emission for η = 0 (solid black) and η = 1 (red dashed) for q = 0, 1, 100 (intensities are normalized).

For the emission processes, we separate the full optical response given by Eq. ( 8) into several components according to their central emission frequency ω i 0 of the lineshape functions R(ω, ω i 0 , ∆ i ). The Rayleigh process corresponds to ω Ray 0 = ω L and the Raman process to ω Ram 0 = ω L -ω vib , ( ω vib = E 1 -E 0 ). Contrary to the Rayleigh and Raman scattering which are coherent processes, fluorescence corresponding to a radiative transition from excited population is incoherent. This stationary population can only exist if in addition to population relaxation, pure dephasing processes are taken into account (γ ij = 0 in Eq. (7)). The population of the discrete excited state leads to an emission at E e / and at E e / -ω vib central frequencies. The continuum states are populated at the laser frequency ω L and therefore give a fluorescence emission with central frequencies at ω L and ω L -ω vib . We note that the fluorescence from the continuum states occurs at the same central frequencies as the Rayleigh and Raman scattering, but can be distinguished by their respective lineshape widths ∆ i . For Rayleigh, the width comes from the laser linewidth, ∆ Ray = δ, and for the Raman the width is given by the inverse lifetime of the vibrationally excited state ν = 1, ∆ Ram = Γ vib /2. The width of the fluorescence lineshape is dominated by the excited state population lifetime: for the emission at ω L the width is ∆ fluor,c0 = ν Γ cν and for the emission at ω L -ω vib the width is ∆ fluor,c1 = ν Γ cν + Γ vib /2. All the parame-

,QWHQVLW\DX q =4 D q =1 E η (q eff -q)/q q =4 F η q =1
G FIG. 3: Comparison between different Fano models. Top: fits of the profile presented in this paper Eq. ( 9) with a) q = 4 and b) q = 1, with η = 1 by a standard (blue dashed) or a shifted Fano profiles (solid green). Bottom: relative differences between the extracted q eff and the actual q value for c) q = 4 and d) q = 1 for the standard (blue dashed) or the shifted Fano profiles (solid green).

ters for each emission component given by Eq. ( 12) are collected in Table I.

The parameters of the model can be extracted from spectroscopic measurements. By fitting the profiles, we can get γ, q, B abs , B Ram , B Ray , F (experimental parameter), η Ray , and η Ram (Table I and S.M. [25]). From these, there are various ways to obtain the model parameters, for instance:

1 ν=0 Γ cν = B Ray (B abs ) 2 F 2 2 , √ nV = γ π √ nµ 0c = B Ray πB abs , µ 0e = B Ray γ B abs q √ nµ 1c = B Ram πB abs , µ 1e = 8η Ram γB abs B Ram πB Ray F 2 (13)
We now discuss how important is to include the Lorentzian term arising from dissipation in addition to the standard Fano profile. In Fig. 3, we present a profile with our model Eq. (10) with a given set of parameters and fit it both with a standard Fano model h( ; q eff ) (Eq. ( 1)) and with a shifted Fano model 1 N [h( ; q eff )+D]. The shifted Fano fitting is usually performed in optical measurements and corresponds to a constant background substraction. Below, we calculate the relative error between the extracted value of the Fano asymmetry parameter, which we call q eff , and the original q as a function of the Lorentzian weight factor η. We show that even when the fitting is very good, the extracted parameters can be off by a factor of two for η = 1. This definitely shows the importance of including properly the dissipation in the Fano model.

In conclusion, we expect that the equations will motivate the experimentalists to extract system parameters TABLE I: Parameters for all optical processes depicted in Eqs. (10) (12). The parameters are expressed as a function of = ( ωL -Ee)/(nπV 2 ), q = µ0e/(nπV µ0c) and η = ν Γcν /(4nπV 2 ).

which were not accessible previously, from routine optical spectroscopies, characterizing with increasing precision the discrete-continuum interface both relevant in devices and interesting from a fundamental standpoint. Furthermore, our model serves as a stepping stone for further theoretical studies: optical response beyond the wide-band approximation (near band edges for example), strong field effects relevant for plasmonics, generalization to time-dependent laser pulse sequences relevant to nonlinear 2D spectroscopy and application to real systems with the help of computational tools in order to obtain ab-initio the parameters of the model.

Acknowledgments We thank Prof. J.H. Eberly for an overview of relevant literature. D.F.S. acknowledges the Research in Paris program for a fellowship. This work was financially supported by project NSF-ANR (ANR-11-NS04-0001 FRAMOLSENT program, NSFCHE-112489). This work was performed using HPC resources from GENCI-CINES/IDRIS (Grant 2015-x2015082131, 2014-x2014082131) and the CCRE-DSI of Université P. M. Curie.


Daniel Finkelstein-Shapiro

Monica Calatayud

Osman Atabek

Vladimiro Mujica

Arne Keller

Non-linear Fano Interferences in Open Quantum Systems: an Exact Solution

We obtain an explicit solution for the stationary state populations of a dissipative Fano model, where a discrete excited state is coupled to a continumm set of states; both excited set of states are reachable by photo-excitation from the ground state. The dissipative dynamic is described by a Liouville equation in Lindblad form and the field intensity can take arbitrary values. We show that the continuum states population as a function of laser frequency can always be expressed as a Fano profile plus a Lorentzian function with effective parameters whose explicit expressions are given in the case of a closed system coupled to a bath as well as for the original Fano scattering framework. Although the solution is intricate, it can be elegantly expressed as a linear transformation of the kernel of a 4 × 4 matrix which has the meaning of an effective Liouvillian. We unveil key notable processes related to the optical nonlinearity and which had not been reported to date: electromagnetic induced-transparency, population inversions, power narrowing and broadening, as well as an effective reduction of the Fano asymmetry parameter.

The original Fano model was introduced by U. Fano in 1961 [1] to explain the asymmetric profile that was obtained when monitoring the photo-current as a function of the laser frequency used to ionize a gaz of Helium-like atoms. In parallel, Friederichs [2] developed the mathematical formalism to describe the essential feature of the Fano model: a discrete state coupled to a continuum set of states; both sets of states being reachable by photoexcitation from the ground state. The resulting photocurrent, which is proportional to the population of the continuum set of states, as a function of the excitation laser frequency ω L is known as the Beutler-Fano or Fano profile: f ( ; q) = (q + ) 2 /( 2 + 1), where q is the ratio of the transition dipole moment of the ground-discrete and ground-continuum transitions, and = (ω L -ω e )/γ where ω e is the energy of the discrete state relative to the ground state, ω L is te radiation field frequency, and γ = nπV 2 is the linewidth of the excited state, induced by its coupling (per unit of energy) nV 2 to the continuum set of states, n being the density of states.

Two extensions of the original model have been considered: inclusion of incoherent relaxation and dephasing processes and high field intensities. The motivation to include incoherent processes was first to describe the pressure broadening [3] due to elastic collisions, laser phase fluctuations [4] and spontaneous emission [5,6]. Nowadays, Fano profiles in nanoscale structures are standard [7,8], for example in plasmonic nanostructures [9][10][11], quantum dots, decorated nanoparticles [12] and spin filters [13]. The coherent coupling with the incident light induces large Rabi frequencies which in turn compete with the relaxation rates in order to modify the stationary state. The ability to predict the lineshape, and in general to investigate non linear optical phenomena in the presence of a continuum set of states are the main motivations to consider an arbitrary incident field intensity.

In spite of the ubiquity of the Fano interferences, to the best of our knowledge, explicit solutions for arbitrary laser intensities and including general relaxation processes have not been obtained. Even in Refs. [14,15], dealing with quantum dots, despite some approximations no analytical expressions are derived that afford a simple physical interpretation of the results.

In a recent work [16], we investigated the Fano interferences present in the emitted spectrum of a Fano system with a vibrational manifold. We obtained explicit expressions of spectroscopic observables like Rayleigh, Raman and fluorescence emission but restricted to the low intensity field limit where the lowest order of perturbation was enough to describe the laser-matter interaction. In this letter, we focus on the description of the non-linear Fano effect on the total population of the continuum excited state, the observable in the original Fano model, for the general case of arbitrary large intensities of the laser field and including dissipation process. Here, unlike Ref. [16], obtaining explicit expressions requires non perturbative calculations. We present a method that allows us to obtain such an explicit formulation for arbitrary field strengths and Markovian baths in an elegant and intuitive framework where the entire solution is formulated in terms of a 4 × 4 matrix corresponding to an effective Liouvillian acting in the space of the discrete states only. We calculate the usual Fano observable, that is, the total population of the continuum set of states, which is related to the optical absorption or to the photo-current, as a function of the incident laser frequency. The striking result is that such a function can be written exactly as a linear combination of a Fano profile and a Lorentzian function like in Ref. [16], but where the Fano q and coefficients become effective parameters that are functions of the field intensity and the decay rates.

Although the objectives and results of the present work differ from our previous one [16], the model is similar. As there are some differences and for the sake of introducing the notation in a self-contained way, we summarize it below. The ingredients of the model are schematically presented in Fig. 1. The Hamiltonian H = H 0 +H V +H F is exactly the same as in the original Fano model [1]:

H 0 = E 0 |g g| + E e |e e| + dk k |k k| H V = dkV (k)|e k| + V * (k)|k e| H F = F [µ e cos(ω L t)|g e| + µ * e cos(ω L t)|e g|] + F dk [µ k cos(ω L t)|g k| + µ * k cos(ω L t)|k g|] , (1)
where H 0 is the site Hamiltonian, H V is the coupling of the excited state to the continuum. For simplicity, in the following, we will consider that

V (k) = e|H V |k is real.
H F is the interaction with the incident radiation field, allowing transitions from the ground state to the discrete excited state g ↔ e and to the continuum of states g ↔ k, µ ij = i|µ|j is the transition dipole moment between states i and j and F is the field amplitude. The relaxation and dephasing processes are introduced in an analogue way as in Ref. [16]. It consists in a Liouville equation in Lindblad form to insure complete positivity of the density matrix describing the quantum system. The dynamics of the system is given by:

∂ρ ∂t = L(t)ρ, where L(t) = L H (t) + L D , with L H = -i(1 1 ⊗ H T (t) -H(t) ⊗ 1 1) is the Hamiltonian conser- vative part and L D = L D pop + L D pure is the generator of dissipative dynamics. L D pop = dkΓ(k) A(k, g) ⊗ A(k, g) - 1 2 1 ⊗ A † (k, g)A(k, g) + A † (k, g)A(k, g) ⊗ 1
+ Γ e A(e, g) ⊗ A(e, g)

- 1 2 1 ⊗ A † (e, g)A(e, g) + A † (e, g)A(e, g) ⊗ 1 , (2)
describes the population relaxation from the |k manifold and from the |e state to the ground state.

L D pure = -γ eg |e e| ⊗ |g g| + |g g| ⊗ |e e| -dkγ kg |k k| ⊗ |g g| + |g g| ⊗ |k k| -dkγ ke |k k| ⊗ |e e| + |e e| ⊗ |k k| , (3)
describes pure dephasing, that is the dynamics of the non diagonal matrix elements of ρ. H T denotes the transpose of H, A(i, j) = |j i| are the jump operators and Γ(k) is the population relaxation rate from state |k to |g as is Γ e for the |e population. γ ij are the pure dephasing rates for the ij coherence relaxations. As in Ref. [16], we have used the correspondance: [17]. We use the rotatingwave approximation (RWA) on L = e iΩ L t L(t)e -iΩ L t and remove non-resonant terms such that L can be considered time independent. Ω L is a diagonal matrix whose matrix elements are equal to ±ω L for excited(ground)ground(excited) coherences, and zero elsewhere. The continuum states population, dkρ kk , where ρ is the full steady state density matrix can be obtained by finding the kernel of the time-independent operator (Ω L -L), that is (Ω L -L)ρ = 0. To solve this equation, we split L in two terms, L = L 0 + V where L 0 is diagonal and V is purely non diagonal, and proceed to Feshbach partitioning. For that, we introduce the projectors P = |g g| + |e e| Q = dk|k k|, with P + Q = 1. The corresponding projectors for the discrete and continuum parts in Liouville space are given by:

|l m| ↔ |l ⊗ |m ≡ ||lm
P = P ⊗ P ; Q = P ⊗ Q + Q ⊗ P + Q ⊗ Q. ( 4
)
This allows us to rewrite the kernel equation as (Ω L -L)(P + Q)ρ = 0. By projecting on both P and Q and after some algebra, we obtain

P(Ω L -L eff )Pρ = 0, (5)
with an effective Liouville operator,

L eff = PLP + PVQG Q QVP, (6)
and

G Q = (QΩ L Q -QLQ) -1 .
G Q , which is the resolvent of QLQ, is not straightforward to calculate unless QLQ is diagonal. To achieve the calculation of G Q , we proceed to a sub-partition of the Q subspace until the projected liouvillian be diagonal [18]. PΩ L P -L eff acts on the P space only, but its kernel is equal to the exact stationary density matrix Pρ describing the discrete states coupled to the continuum. Once the density matrix Pρ on the discrete space has been obtained, the density matrix Qρ in the continuum subspace can be computed through the following equation:

Qρ = QG Q QVPρ. (7)
L eff can be thought as a 4 × 4 matrix when Pρ is considered as a column vector with 4 elements. To obtain an explicit expression for L eff , the usual wide-band approximation is employed. It is also in this same approximation that an explicit expression was obtained in the original Fano problem [1]. It consists in assuming that the parameters of the model do not depend upon k. From now on, we consider this approximation and define Γ c ≡ Γ(k), µ c ≡ µ k as constants.

After tedious but straightforward calculations, the effective Liouvillian L eff defined in Eq. ( 5) can be written in a surprisingly simple form:

L eff = -i(1 1 ⊗ H eff -Heff ⊗ 1 1) + L D eff , (8)
where H eff is an effective Hamiltonian ( Heff its complex conjugate) and L D eff is the dissipative part of the effective Liouvillian. The effective Hamiltonian can be written as:

H eff = P H 0 P + H field , with (9)
H field = F 2 µ e -inπV µ c |g e| + |e g| .
It contains an Hermitian and an anti-Hermitian part arising from the imaginary component of the transition dipole moment which corresponds to the groundcontinuum transition.

Hamiltonians with an anti-Hermitian component are commonly used for the simulation of decay processes in which the population is not conserved [19,[START_REF] Moiseyev | Non-Hermitian Quantum Mechanics[END_REF]. The dissipative part of the effective Liouvillian is:

L eff = L D e + L D c + L D pure , with: L D e = (2nπV 2 + Γ e ) A(e, g) ⊗ A † (e, g) - 1 2 A † (e, g)A(e, g) ⊗ 1 + 1 ⊗ A † (e, g)A(e, g) L D pure = -γ eg |e e| ⊗ |g g| + |g g| ⊗ |e e| L D c = 2nπV µ c (||gg eg|| + ||gg ge||) -nπµ 2 c (||ge ge|| + ||eg eg||). (10)
L D e is a dissipation superoperator in Lindblad form that describes the population decay with rate 2nπV 2 / + Γ e due to the coupling to the continuum and the natural decay rate Γ e , and L D pure is a pure dephasing superoperator. L D c is an additional part of the relaxation superoperator which cannot be put into a Lindblad form.

Finally, solving Eqs. ( 5) amounts to finding the kernel of a 4 × 4 matrix, which can be explicitely done with the help of a symbolic calculation software. Then, applying Eq. ( 7) along with the normalization condition ρ gg + ρ ee + dkρ kk = 1 gives us all of the populations and coherences.

The results will be given in terms of dimensionless quantities and γ = nπV 2 is taken as the unit of energy. In addition to the original dimensionless Fano parameters = (ω L -ω e )/γ and q = µ e /nπV µ c , we introduce the new parameter Ω = µeF 2q γ = µ c F/2V , which corresponds to a dimensionless Rabi frequency. Also, all relaxation rates will be given in units of γ, this amounts to perform the following replacement:

Γ c → Γ c /γ, Γ e → Γ e /γ, γ eg → γ eg /γ.
The main result of the paper is that the population of the excited state n c = dkρ kk can always be brought back to a Fano profile f , plus a Lorentzian term:

n c ( eff ; q eff ) = C f ( eff ; q eff ) + D 2 eff + 1 , ( 11
)
where the dependence upon ω L is solely contained in eff = ω L -ω eff γ eff . ω eff , γ eff and q eff are effective Fano parameters that depend on all the parameters of the model but ω L . C is a measure of the total population and D indicates the relative weight of the Lorentzian term in comparison to the Fano profile.

Simple explicit expressions for the effective Fano parameters can be given when the relaxation and the dephasing rates concerning the |e state can be neglected, that is when Γ e = 0 and γ eg = 0 (see Eq. ( 12)). This is often a very good approximation in the context of semiconductor quantum dots or in hybrids consisting of an organic molecule adsorbed on metallic or semiconductor nanoparticles [START_REF] Raymond | [END_REF][22][START_REF] Turro | Modern Molecular Photochemistry[END_REF][START_REF] Piotrowiak | [END_REF][25] (1/Γ e ≈ nanoseconds, /nπV 2 ≈ 10 femtoseconds). In that case, the only relaxation process consists in the continuum states population decay to the ground state and the profile is given by a pure Fano profile and the Lorentzian term is absent, D = 0 and C = 2Ω 2 2Ω 2 +Γc . We have obtained:

γ eff γ = Γ c 1 + (q 2 + 1)Ω 2 (Ω 2 [(2Ω 2 + 4)/Γ c + 1] + 2/Γ c + 2) 1/2 2Ω 2 + Γ c ω eff γ = ω e γ + qΩ 2 1 - 2 2Ω 2 + Γ c ; q eff q = Γ c 2Ω 2 + Γ c 1 γ eff . (12)
We now discuss each one of the parameters as a function of the Rabi frequency and illustrate them in Figure 2. The prefactor C which is proportional to the intensity of the field for weak fields (linear regime) saturates when 2Ω 2 Γ c . In Fig. 2a, we show the normalized Fano profiles at two intensities of the field (Ω = 0.001 and Ω = 0.1 for q = 5). As the field intensity increases, we see changes in all of the Fano parameters. The effective width γ eff increases or decreases (power narrowing or power broadening) depending on the value of the relaxation (see Fig. 2b and Eq. ( 12)). The effective q eff de-creases monotonically with q (see Figure 2c and Eq. ( 12)). As shown in the inset, the population of the continuum, even for modest values of Ω is significant, underlying the importance of a theory which can handle non-negligible population in the continuum set of state, contrary to the approximations in Ref. [15]. The decreases of q eff can be thought as a consequence of the saturation of the discrete excited state population. The energy shift (ω eff -ω e ) of the discrete state |e has an interesting behavior. For Γ c < 2, the shift is negative if 0 < Ω < (2 -Γ c )/2 and it is positive if Ω > (2 -Γ c )/2. Therefore, Γ c < 2, Ω = (2 -Γ c )/2 is a null point. On the contrary, for large relaxation rates such that Γ c > 2, the shift will be positive for all values of the field (see Fig. 2d).

q eff /q q =5 q =10 q =50 c Ω q 0 10
Inclusion of population relaxation (Γ e = 0) or pure dephasing processes (γ eg = 0) results into heavy expressions of the states populations that we provide in [18]. The main qualitative features remain unchanged except for the appearance of a Lorentzian term.

Until now, we have focused on the stationary population of the continuum set of states dkρ kk . Another quantity that can be measured is the photocurrent, that is the total flow of electrons in the continuum. Assuming that all the electrons emitted in the continuum |k states are collected by an electrode, in the limit of low bias voltage, the current intensity I can be obtained from the stationary populations as I = lim Γc→0 |e| Γc dkρ kk ρgg [26], where e is the electron charge. It turns out that I can also be brought into the form of a Fano factor and a Lorentzian factor as in equation (11). Explicit expressions for the Fano and Lorentzian parameters can be given with both the population relaxation and dephasing included:

γ tr eff = (1 + Γ e ) -1 Ω 4 Γ e (q 2 + Γ e + 1) + Ω 2 (1 + Γ e )(q 2 + 2Γ e + 1)(Γ e + γ eg + 1) + (1 + Γ e ) 2 (Γ e + γ eg + 1) 2 1/2 (13) D tr = (1 + Γ e ) -2 Ω 4 Γ e (q 2 + Γ e + 1) + Ω 2 (1 + Γ e )(q 2 + 2Γ e + 1)(Γ e + γ eg ) + (1 + Γ e ) -1 Γ 3 e + Γ 2 e + γ eg (2Γ 2 e + Γ e γ eg + q 2 + 2Γ e + γ eg + 1) , (14)
ω tr eff γ = ω e γ + qΩ 2 1 + Γ e ; q tr eff = 1 γ tr eff ; C tr = 2Ω 2 . ( 15
)
These expressions give an exact description of the scattering problem, as formulated by Fano in its original work [1], but extended to arbitrary field intensities and dissipation processes.

An interesting consequence of the nonlinear Fano effect, is electromagnetic-induced transparency [27]. Indeed, in the absence of discrete state relaxation and dephasing (Γ e = γ eg = 0), the continuum population goes through zero when eff = -q eff (see Eq. ( 11)). It can be shown that this is equivalent to Ω 2 = 1 + q either for the light absorption or the photocurrent intensity. This phenomenon is shown in Fig. 3.a for the case = 0, q = 15 for different values of Γ e , and q = 15, and Fig. 3.b Γ e = 0 for different values of . This phenomenon provides an interesting tool for devices as well as a means for determining the system parameters. For example, irradiating at the discrete level resonance ( = 0) in weak field and increasing the intensity until the induced transparency is found determines the ratio of transition dipole moment µ c to the coupling V such that µ c /2V = 1/F . In the presence of pure dephasing or of relaxation from the discrete state, the zero becomes a minimum but its position does not change appreciably (see Figure 3 (left)).

We have no doubt that the generality of the model, and the derived expressions given here will open new horizons in the analysis of Fano profiles under intense fields as well as in applications and devices that exploit the new phenomena we have reported in this letter.

FIG. 1 :

 1 FIG. 1: Energy levels and transitions of a Fano-type model with dissipation (see main text for notations)

FIG. 2 :

 2 FIG.2: Effect of the field on the Fano profiles and parameters: (a) Fano profiles for q = 5 for Ω = 0.001 and Ω = 0.1 (b) γ eff for different values of Γc (c) q eff /q for different values of q. Inset show nc as a function of q and Ω. Upper end (red) corresponds to nc = 1 and lower end (blue) corresponds to nc = 0 (d) ω eff for different values of Γc and ωe = 10

FIG. 3 :

 3 FIG. 3: Population in the continuum showing electromagnetically induced transparency (EIT) for a) = 0, q = 15 with different values of Γe and b) for q = 15, Γe = 0 different values of

Acknowledgements D.F.S. acknowledges the Research in Paris program for a fellowship. This work was financially supported by project NSF-ANR (ANR-11-NS04-0001 FRAMOLSENT program, NSFCHE-112489). This work was performed using HPC resources from GENCI-CINES/IDRIS (Grant 2015-x2015082131, 2014-x2014082131) and the CCRE-DSI of Université P. M. Curie.


Phone Guillaume Chanel
email: guillaume.chanel@unige.ch

Denis Lalanne
email: denis.lalanne@unifr.ch

Elise Lavoué
email: elise.lavoue@univ-lyon3.fr

Kristine Lund
email: kristine.lund@univ-lyon2.fr

Gaëlle Molinari
email: gaelle.molinari@unidistance.ch

Fabien Ringeval
email: fabien.ringeval@uni-passau.de

Armin Weinberger
email: a.weinberger@mx.uni-saarland.de

Grand Challenge Problem 2: Adaptive awareness for social regulation of emotions in online collaborative learning environments

Keywords: Computer-supported collaborative learning, affective computing, emotion awareness, emotino management, multi-modal interaction

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

Students' ability to understand and manage emotions in self and others plays an important role in the success of collaborative learning [START_REF] Järvenoja | Emotion control in collaborative learning situations: Do students regulate emotions evoked by social challenges[END_REF][START_REF] Lajoie | The role of regulation in medical student learning in small groups: Regulating oneself and others' learning and emotions[END_REF][START_REF] Lavoué | Reflection-in-action markers for reflectionon-action in computer-supported collaborative learning settings[END_REF]. In online learning environments, the use of new social media technologies (blogs, forums, wikis, social networking sites, etc.) facilitates connections between learners, but may also introduce new communication challenges. Moreover, the access of emotional cues (e.g., interest, workload, stress, anxiety) is reduced in such environments. Online collaborative learners may therefore experience difficulties in evaluating their partners' emotions and adjusting their behavior in response. A lack of emotion awareness among online learners may have a detrimental effect on collaboration and learning [START_REF] Eligio | Emotion understanding and performance during computer-supported collaboration[END_REF]. It may also lead to inappropriate or even dangerous social behaviors such as cyberbullying. This project aims at substantially improving learning effects with social media through the use of adaptive emotion awareness technology [START_REF] Chanel | Assessment of computersupported collaborative processes using interpersonal physiological and eye-movement coupling[END_REF][START_REF] Ringeval | On the influence of emotional feedback on emotion awareness and gaze behavior[END_REF] designed to support students' emotional regulation in online groups.

Problems of the European Education System Addressed, and Long Term Benefits for Society

The increasing use of social networking sites introduces new social problems such as cyberdisinhibition and cyber-bullying that interfere with school and learning. Such problems are based on failure to address the development of socio-emotional skills in school curricula. A new pedagogy is needed that is based on self-regulated, experiential learning in groups where learners are supported to achieve a deeper understanding of self in relation to others. In order to contribute to the goal of building emotion-centered learning programs, research should focus on the investigation of how awareness tools can be adapted to support the social regulation of emotions in online learning contexts [START_REF] Molinari | Emotion feedback during computer-mediated collaboration: Effects on self-reported emotions and perceived interaction[END_REF].

Main Activities to Address the Grand Challenge Problem

There will be five main activities:

1. Selection of a population of learners involved in online collaborative learning experiences. An assessment will be conducted targeting learners and groups of learners with emotional regulation problems.

2. Investigation into the role of emotions in online social learning environments. The output will be a range of emotions identified as playing a significant role in the success of collaborative learning.

3. Development of methods to identify learners' emotions in real time during online collaborative learning. These methods will be based on a multimodal dynamic analysis approach including signals from users (e.g., facial expression, speech, eye movements, physiological data) and contextual information. A focus will also be on how co-learners' emotions evolve over the course of their interactions.

4. Development and implementation of adaptive emotion awareness systems. To be efficient, they should promote the sharing of emotions identified as beneficial to collaborative learning at the proper time during interaction.

5. Evaluation process of systems developed in (4) and with participants selected in (1), using the following criteria: users' acceptance; capacity to enhance mutual emotion awareness and collaborative learning; capacity to help students deal with emotional issues in online social learning environments.

Timeframe for the Grand Challenge Problem

The proposed Grand Challenge Problem could be solved in a timeframe of 4 years: 6-12 months for Activity 1 (Year 1), 24 months for Activity 2 (Years 1-2), Activity 3 (Years 2-3), Activity 4 (Years 3-4), and 12 months for Activity 5 (Year 4).

Measurable Progress and Success Indicators:

1. Development and validation of models for the investigation of the dynamics of emotions and their regulation in computer-supported collaborative learning (CSCL) contexts.

2. Development of adaptive emotion awareness systems for various types of CSCL environments, implementation of studies to assess their acceptance and usefulness in laboratory and authentic settings.

3. Higher emotion awareness, better collaborative processes, and learning outcomes in CSCL settings.

4. Development of a methodology for the assessment of skills for social regulation of emotions in online learning contexts, development of emotion-centered learning programs.

5. Development of an interdisciplinary research community working on awareness tools for regulation of emotions in CSCL.

Attraction of Funding

  Interdisciplinary European projects funded under the European Commission (e.g., H2020-ICT, Lifelong Learning Programme such Comenius, Erasmus, or Grundtvig); COST (European Cooperation in Science and Technology) fund for covering the cost of networking activities; National funding, e.g., Swiss National Science Foundation (SNSF), French National Research Agency (ANR); Funds for applied R&D, e.g., Hasler Foundation in Switzerland which supports educational, research, and innovation projects where the aim is to promote the development of ICT or innovative applications of ICT.


Florent Berthaut
email: florent@hitmuri.net

Deepak Ranjan Sahoo

Jess Mcintosh

Diptesh Das
email: diptesh.das09@gmail.com

Sriram Subramanian
email: sriram@sussex.ac.uk

Dynamir : Optical Manipulations using Dynamic Mirror Brushes

Keywords: Dynamir, Shape Changing Mirrors, Mixed Reality, ACM Classification H.5.m. Information Interfaces and Presentation (e.g. HCI): Miscellaneous

. Dynamic mirror brushes allow for optical manipulations by deforming the mirror surface (displayed below). A) Applying a controlled stretching effect using a vertical brush with a 1D curvature. B) Contracting the user's hand for the selection of a small sphere on a stereoscopic screen by changing the curvature and position of a small brush interactively. C) Moving the reflection of physical objects with a finger by changing the orientation of a large brush.

INTRODUCTION

Curved mirrors are extensively used in everyday life, from theme parks to street signs. These surfaces create a reflection of the physical space that can be altered while preserving some of its physical qualities. For example, convex traffic mirrors are often used at road intersections so on-coming vehicles can see around corners. Large field of view convex mirrors are also embedded in side-view mirrors of vehicles to minimise blind spots. It is common to find concave mirrors in hotel bathrooms that provide a magnified image of the face for shaving. Curved mirrors have also been used in the context of imaging and displays. For instance, in [START_REF] Kiyokawa | A wide field-of-view head mounted projective display using hyperbolic half-silvered mirrors[END_REF], the authors increase the field-of-view (FOV) of a head mounted projected display using a hyperbolic half-silvered mirror. Continuous deformable mirrors are commonly used to correct optical aberration in telescopes, microscopes and laser optical systems. Discrete deforming mirror array are used in projectors [START_REF] Nayar | Programmable imaging: Towards a flexible camera[END_REF], as well as solar cells, camera and interactive art. Planar mirrors have been used extensively in mixed-reality displays, however these mirrors remain static. For example, a semi-transparent flat-mirror enables the reflection of a virtual screen to be projected in a physical space allowing augmentation of an interactive surface [START_REF] Hachet | Multitouch and stereo combined in a seamless workspace[END_REF]. Another use is in [START_REF] Martinez Plasencia | Through the combining glass[END_REF], where the authors show how the shared space created by semi-transparent mirrors can be used for bidirectional augmentations, e.g., of physical objects on both sides. However, the mirrors remain flat and static, and manipulations of the physical space are done through the virtual augmentations.

Dynamir is a shape changing mirror system that enables a novel approach to optical manipulations based on dynamic mirror brushes. These brushes allow for controlling local deformation of the surface of a flexible mirror with the purpose of enabling manipulations of the physical space, resulting in novel interaction possibilities. Dynamic street mirrors could adapt to the traffic, giving a wider field of view when more and/or faster vehicles are detected. Bathroom mirrors could allow for different levels of zoom when shaving. Individual mirrors could emerge from a collective one when practicing sports or arts and provide private views on movements of users. Users could also be augmented with physical content in novel ways. For example their reflection could be adjusted so that it fits specific clothes or accessories in shopping windows, or they could be made smaller with a convex mirror and integrated in miniature scenes during storytelling performances. On the contrary to augmented mirrors based on cameras and displays, which might limit resolution and contrast of the mirrored scene, Dynamir preserves the optical scene, including the visual cues. Finally, Dynamir can also be used for 3D user interfaces similar to the HoloCubtile [START_REF] De La Rivière | Holocubtile: 3d multitouch brings the virtual world into the user's hands[END_REF], adding the possibility of optically manipulating the physical part of interaction techniques, e.g. hands and props, in addition to the virtual part.

Our contribution is three-fold: 1) We propose dynamic mirror brushes that allow for optical manipulations with shapechanging mirrors; 2) We propose an implementation of the brushes with a flexible mirror sheet that also allows for augmented reality scenarios; 3) We demonstrate three novel interactive scenarios.

DYNAMIC MIRROR BRUSHES

Mirrors produce specular reflection of light, i.e., the angle of incidence θ i at a point P on the mirror equals the angle of reflection θ r . When an object is placed in front of the mirror, the reflections of the rays coming from the object produce a reflected image of this object. The perception of this reflected image by an observer will depend on the positions of both the observer and the object relative to the mirror. It will also depend on the normals on the mirror surface at all intersections with the light rays coming from the object. Manipulating the reflections therefore amounts to controlling the normals at all points of the surface of the mirror. In this paper, we propose to simplify this manipulation in the context of interactive applications by introducing the concept of dynamic mirror brushes. These brushes are local deformations of the mirror surface which: 1) are aimed at a group of observers, 2) affect a certain region of interest (ROI), and 3) have several attributes that define the effect that they create. The ROI can be the observers themselves or any group of objects in the physical space. As depicted on Figure 2, several brushes can be created and manipulated on a single mirror surface, allow- ing for different optical effects for different groups of users. Next, we describe the attributes of the optical brushes, i.e., how various optical effects can be specified using them, and how they can be translated into the description of the curvature of the mirror.

Size and shape attributes

The first group of attributes, size and shape, allow for controlling both the number of observers and the size of the ROI. Changing the size allows, for example, to control how much of a group of people an observer can watch. By varying the shape of the brush, a particular selection of physical objects can be defined from the observer's point of view with any contour, as shown on Figure 2.B. Given the extremities of a ROI, we can calculate the brush size from the lines forming the frustum which define the view of the observer. Finding the intersection points between these lines and the mirror plane give us the coordinates for the corners of the brush.

Position and orientation attributes

The second group of attributes correspond to the position and orientation of the brushes relative to the mirror surface. They can be seen as allowing for three different effects. The first is the selection of a region of interest from the observer's point of view. In this case, a brush with a fixed position is rotated according to the observer and ROI positions. The second effect is the manipulation of the position of the reflected image of the ROI. The brush can be moved and rotated in order to move the reflection relative to the observer. Finally, these attributes can be used to inspect the ROI from different angles.

Curvature attribute

The curvature attribute enables two main effects, contraction and expansion (i.e., zoom in and zoom out) of the ROI. These effects can be applied either horizontally or vertically as 1D effects, or by combining both as 2D effects. A possible use is the modification of the field of view so that observers can perceive more of a scene while preserving the size of the brush, as demonstrated in Figure 2 Left. It can also be used as a way of enriching 3D interaction done through the observer's reflection, by manipulation the observer himself. The contraction/expansion and hence, zooming out/in of the ROI is achieved by decreasing/increasing the radius of curvature, making the brush respectively convex or concave.

IMPLEMENTATION

Hardware

We used a silvered high impact polystyrene (HIPS) sheet with area 420 mm × 297 mm (A3 size) and thickness t = 1 mm as the shape changing mirror surface in our implementation. The HIPS mirror is highly flexible with flexural modulus E ≈ 2.4×10 5 psi, and highly stress resistant with flexural strength σ m ≈ 4.21 × 10 3 psi. This corresponds to yield strain m = σ m /E = 1.75 × 10 -2 , which means that a sharp edge can be approximated with a round edge with minimum radius of curvature ρ m ≈ t/2 m = 2.85 cm. Using the HIPS sheet with t = 0.1 mm, very sharp edges with ρ m = 2.85 mm can be achieved. Large HIPS mirror with size 2500 mm × 1250 mm are available off-the-shelf. HIPS is a copolymer of polybutadiene rubber, which has high Poisson ratio ν ≈ 0.5. In other words, the HIPS mirror shows good flatness for undistorted reflection, and at the same time good bending, stretching and twisting performance for local shape control.

To change the shape of the mirror, we used a new linear actuator, ShapeClips [START_REF] Hardy | Shapeclip: Towards rapid prototyping with shape-changing displays for designers[END_REF], which has a footprint W 2 of 2 cm × 2 cm and travel range R = 6 cm. The ShapeClips were mounted vertically to support the weight of the mirror for simplicity; i.e., we kept the actuated mirror surface horizontal. A spring mechanism was used to attach the HIPS sheet that allowed the desired deformation of the mirror. In our implementation, the maximum magnitude of shapes is 6 cm, and the maximum angle of sharp edge is tan -1 (R/W ) = 71.56 • . Due to the inherent solid mechanics of the HIPS sheet, the shapes generated comprise of hyperbolic functions; the radius of curvature can be calculated from the solid mechanics analysis [START_REF] Ventsel | Thin Plates and Shells: Theory: Analysis, and Applications[END_REF]. The maximum speed of the ShapeClips is 8 cm/s, which dictates the speed of our implementation. We used nine ShapeClips in our implementation. The ShapeClips are compact and modular, and more of them can be incorporated in our implementation seamlessly to create shapes with higher complexity [START_REF] Follmer | inform: Dynamic physical affordances and constraints through shape and object actuation[END_REF], and to alleviate the trade-off between achieving flat surface and sharp edges. For example, more ShapeClips and thinner HIPS mirror are required to effectively demonstrate the application described in Figure 2.B that uses both flat surfaces and sharp edges. Our implementation with nine ShapeClips and a thicker HIPS mirror allows for curved brushes, either two dimensional (bumps), or one dimensional vertical or horizontal curved ridges, as well as flat brushes when rotating the the entire mirror sheet. As depicted on Figure 3, we placed a half-silvered mirror at 45 • angle above the shape changing mirror. The space in front of this half-silvered mirror is therefore reflected on to the shape changing mirror, and back to the front space towards the user. In addition, a polarized stereoscopic screen is placed behind the half-silvered mirror at the virtual image of the shape changing mirror, making it possible to overlap virtual content with the manipulated/deformed reflection space.

Software

The software part of our implementation is mostly written in Javascript and runs in a browser. 3D shapes are rendered with Three.js, Leap Motion data is gathered with the leap Javascript sdk and position of the head computed from the depth camera is sent from a C++ application to the browser via a node.js bridge and OpenSoundControl messages. We want to be able to manipulate the optical brushes by simply generating and controlling corresponding virtual brushes in the form of meshes. Because ShapeClips translate the light field below them to a vertical displacement, we render the virtual brushes in a 3D scene below them. The frustum is defined so that near and far clipping planes match the minimum and maximum physical heights, and its dimensions match the dimensions of the screen. We then use a GLSL fragment shader that renders the depth of the meshes instead of their color. In order to compensate for variations in the ShapeClips heights, we also integrate a calibration texture in our fragment shader, which offsets the rendered depth separately for each ShapeClip. The virtual brushes themselves can be created in different ways depending on which attributes need to be controlled. For example, spheres with changes in position and scale can serve as brushes with a controllable 2D curvature, cylinders as brushes with a controllable 1D curvature, simple planes for flat brushes with changes in orientation. A more generic method consists in building the brush mesh from a Non-Uniform Rational B-spline surface, allowing for changes in size, shape and curvature on both axes, following the guidelines on shape-changing displays given in [START_REF] Roudaut | Toward high "shape resolution" in self-actuated flexible mobile devices[END_REF].

SCENARIOS

We propose three interactive scenarios enabled by Dynamir which demonstrate the effectiveness of our implementation.

The first scenario, depicted on Figure 1.A, and demonstrates the use of the position and curvature attributes. It allows an observer to apply a vertical contraction effect to himself, which remain applied to him as he moves, contrary to what would be possible with a static mirror. The magnitude of the contraction, is increased and decreased by moving the hand respectively away from and towards the mirror. With a larger scale implementation, several brushes could be created and applied to different regions of interest of the scene reflected in the mirror. Being defined relative to the user, these brushes would move with him as the single brush does in our scenario, keeping the region of interest of each brush constant. Each of their attributes could be set independently. In effect, this scenario corresponds to an optical version of an image manipulation application.

The second scenario, depicted on Figure 1.B, demonstrates how the curvature attribute can be controlled to expand and contract a region of interest in two dimensions. It draws inspiration from augmented mirrors such as the ones presented in [START_REF] Martinez Plasencia | Through the combining glass[END_REF], which allow users to manipulate virtual content through their reflection. Using our dynamic mirror brushes, their reflection can be altered to enrich the interaction techniques. In our case, a small sphere and a large cube are displayed on the stereoscopic screen placed behind the half-silvered mirror. By closing / opening their hand, the user controls the curvature of the brush, which in turns makes their hand smaller or larger, allowing them to select and manipulate the object of corresponding size at an appropriate scale. Other 3D manipulation and navigation techniques could be enriched by manipulating the user's image, for example to optically modify the control-display ratio.

With our third scenario, depicted on Figure 1.C, we demonstrate the use of the orientation attribute to manipulate the physical world. The user is able to move the reflection of a pre-defined physical object, here a mug. From the point of view of the user, the mug remains at the tip of his finger. The optical brush, which covers the whole mirror, changes orientation in order to align the virtual image of the object with the finger. With multiple brushes, multiple objects from the physical space could be manipulated at the same time.

CONCLUSION

In this paper, we proposed a novel approach for optical manipulations using shape changing mirrors. We described dynamic mirror brushes that allow for describing these manipulations with controllable attributes. We described an implementation of a shape-changing mirror and three interactive scenarios demonstrating the use of the brushes. We believe that our approach opens opportunities for research in public displays, augmented reality and 3D interaction. It could also benefit many everyday life applications. Finally, it leads to more interesting research questions. A large scale implementation will for example require studying materials with different stretchability and actuation mechanisms, in order to enable large changes in position and size attributes and sharp changes of orientation anywhere on the surface.

Figure 2 .

 2 Figure 2. Mockup of three brushes controlled on a large mirror: Left) a group of observers changes the curvature of the Brush A to perceive more of the physical space, Center) the blue character controls the orientation, size and position of the Brush B to select the head of the green character, Right) the yellow character scales his body by controlling the curvature of Brush C.

Figure 3 .

 3 Figure 3. Implementation of dynamic mirror brushes: The shape of a horizontal HIPS mirror sheet (a) is controlled using a LCD monitor (b) displaying a depth-map of the mirror surface and the Shapeclips (c).The reflection image of the mirror is formed as a vertical surface (d) using a half-silvered semi-transparent mirror (e) and is overlapped with a stereoscopic screen (f). A Leap Motion (g) and an Asus Xtion depth camera (h) are used for tracking the positions of the user's head and hands to interact with the mirror.

ACKNOWLEDGEMENTS

This project was funded by the EC FP7 through the Marie Curie Fellowship (PIEF-GA-2012-330770), ERC Starting Grant (278576) and FET Open Scheme (309191).