Denis Château

Sylvain David

Gérard Berginc

Cesar Lopes

Frédéric Chaput

Frédéric Lerouge

Anthony Désert

Chantal Andraud
email: chantal.andraud@ens-lyon.fr

Stephane Parola
email: stephane.parola@ens-lyon.fr

Plasmonic Bipyramidal Au Nanoparticles Enhance Near-Infrared Nonlinear Absorption of Dyes Confined in Sol-Gel Materials: Implications for the Safe Utilization of Lasers

Keywords: (Nonlinear optics, Optical limiting, sol-gel, plasmonic, azabodipy, SWIR)

versatile and efficient one. Earlier research has focused on optical limiting materials for visible wavelengths. NIR wavelengths used in telecommunication or optical sensors such as for autonomous vehicles, represent a highly challenging area with very few reports in particular in the solid-state. This work discusses the preparation of efficient NIR optical filter combining azabodipy dyes with plasmonic nanoparticles in sol-gel monoliths. The strategy allows to strongly enhance the nonlinear performance of the dyes, creating a synergy effect never observed on such system in the NIR.

INTRODUCTION.

The development of lasers in the 60s gave rise to intense research programs in optics and nonlinear optics. [START_REF] Maiman | Stimulated Optical Radiation in Ruby[END_REF] This has led to a broad range of available continuous or pulsed lasers, ranging from the UV to the NIR wavelengths. They are used in many different fields, from data storage to telecommunications, and for instance in telemetry for airplanes or cars to detect other vehicles or obstacles. One drawback is the requirement for optical protection to ensure a safe utilization of the commercially available lasers. An efficient protection technology emerged which is based on nonlinear optical limiters. [START_REF] Hollins | Materials for optical limiters[END_REF][START_REF] Spangler | Recent development in the design of organic materials for optical power limiting[END_REF] Nonlinear optical limiters can possess extremely fast activation times, low laser clamping values and high damage thresholds. Reverse Saturable Absorption and Multiphoton Absorption are among the most efficient processes to achieve nonlinear optical limiting. 4- 12 During the past 20 years, numerous publications report efficient nonlinear optical limiting dyes in the visible and in the near infrared wavelengths (NIR). [START_REF] He | Multiphoton Absorbing Materials: Molecular Designs, Characterizations, and Applications[END_REF] Even if some of them are efficient in solution, there is a requirement for integration into solid materials to fulfill the field application criteria. [START_REF] Miller | General Optical Limiting Requirements[END_REF][START_REF] Justus | Broadband thermal optical limiter[END_REF] Regarding the design of solid-state materials, publications are remaining scarce and essentially limited to visible wavelengths. [START_REF] Bouit | Multiphoton processes in organic materials and their applications[END_REF][START_REF] He | Multiphoton Absorbing Materials: Molecular Designs, Characterizations, and Applications[END_REF][START_REF] Parola | Optical Properties of Hybrid Organic-Inorganic Materials and their Applications[END_REF] While some approaches report thin film materials, [START_REF] Doyle | Nonlinear optical performance of chemically tailored phthalocyanine-polymer films as solid-state optical limiting devices[END_REF][START_REF] Su | Preparation and optical limiting properties of a POSS-containing organic-inorganic hybrid nanocomposite[END_REF][START_REF] Su | The preparation and optical limiting properties of POSS-based molecular hybrid functional materials[END_REF][START_REF] Su | Molecular hybrid optical limiting materials from polyhedral oligomer silsequioxane: Preparation and relationship between molecular structure and properties[END_REF][START_REF] Manthrammel | Optical analysis of nanostructured rose bengal thin films using Kramers-Kronig approach: New trend in laser power attenuation[END_REF] the use of 3D monolithic materials appears as a credible alternative to thin films since they allow better localisation of the focal point in the core of the active material. Most of the reported monolithic materials, for visible wavelengths applications are based on organic polymers, [START_REF] Bouit | Multiphoton processes in organic materials and their applications[END_REF][START_REF] He | Two-photon absorption based optical limiting and stabilization in organic molecule-doped solid materials[END_REF][START_REF] Jiang | Polymer host materials for optical limiting[END_REF][23][START_REF] Price | Polymer Monoliths Containing Two-Photon Absorbing Phenylenevinylene Platinum(II) Acetylide Chromophores for Optical Power Limiting[END_REF][START_REF] Ouyang | Enhanced nonlinear optical and optical limiting properties of graphene/ZnO hybrid organic glasses[END_REF][START_REF] Ouyang | Synthesis and enhanced nonlinear optical properties of graphene/CdS organic glass[END_REF] sol-gel hybrid glasses [START_REF] Fuqua | Synthesis and nonlinear optical properties of sol-gel materials containing phthalocyanines[END_REF][START_REF] Bentivegna | Reverse saturable absorption in solid xerogel matrices[END_REF][START_REF] Innocenzi | 3-(Glycidoxypropyl)-trimethoxysilane-TiO2 hybrid organic-inorganic materials for optical limiting[END_REF][START_REF] Kopitkovas | Reverse saturable absorption of fullerodendrimers in porous SiO2 sol-gel matrices[END_REF][START_REF] Sanz | Organic nanocrystals grown in gel glasses for optical-power-limiting applications[END_REF][START_REF] Morales-Saavedra | Linear and nonlinear optical properties of trans-and cis-poly(1-ethynylpyrene) based sonogel hybrid materials[END_REF][START_REF] Desroches | Design and synthesis of multifunctional thiacalixarenes and related metal derivatives for the preparation of sol-gel hybrid materials with non-linear optical properties[END_REF][START_REF] Zieba | Preparation of Functional Hybrid Glass Material from Platinum (II) Complexes for Broadband Nonlinear Absorption of Light[END_REF][START_REF] Chateau | Silica hybrid sol-gel materials with unusually high concentration of Pt-organic molecular guests: studies of luminescence and nonlinear absorption of light[END_REF][START_REF] Zheng | Giant optical limiting effect in Ormosil gel glasses doped with graphene oxide materials[END_REF][37][START_REF] Kumar | Optical limiting applications of resonating plasmonic Au nanoparticles in a dielectric glass medium[END_REF] or sodium-zinc borate glass doped with gold nanoparticles.To the best of our knowledge the two only reports on 3D monolithic solid-state nonlinear optical limiters, operating in the short-wave infrared (SWIR), which is a promising range for the development light detection and ranging (LIDAR) systems working at the eye safe wavelength of 1550 nm, are by our group, on sol-gel derived materials. [START_REF] Château | Efficient hybrid materials for optical power limiting at telecommunication wavelengths[END_REF][START_REF] David | High-Performance Optical Power Limiting Filters at Telecommunication Wavelengths: When Aza-BODIPY Dyes Bond to Sol-Gel Materials[END_REF] In most cases, silica based matrices are favoured over organic polymer matrices due to higher damage threshold and nonlinear parameters (n2, β, Re[χ [START_REF] Spangler | Recent development in the design of organic materials for optical power limiting[END_REF] ], Im[χ [START_REF] Spangler | Recent development in the design of organic materials for optical power limiting[END_REF] ]). [START_REF] Parola | Optical Properties of Hybrid Organic-Inorganic Materials and their Applications[END_REF][START_REF] Sharma | Measurement of nonlinear properties and optical limiting ability of Rhodamine6G doped silica and polymeric samples[END_REF] Low compatibility and solubility of dyes in the sol-gel host matrices is often a limitation since the optical performances are concentration dependent. This limitation can be overcome by silanization of dopants to highly improve their concentration through strong covalent bonding between the dyes and the matrix. [START_REF] Kopitkovas | Reverse saturable absorption of fullerodendrimers in porous SiO2 sol-gel matrices[END_REF][START_REF] Morales-Saavedra | Linear and nonlinear optical properties of trans-and cis-poly(1-ethynylpyrene) based sonogel hybrid materials[END_REF][START_REF] Desroches | Design and synthesis of multifunctional thiacalixarenes and related metal derivatives for the preparation of sol-gel hybrid materials with non-linear optical properties[END_REF][START_REF] David | High-Performance Optical Power Limiting Filters at Telecommunication Wavelengths: When Aza-BODIPY Dyes Bond to Sol-Gel Materials[END_REF][START_REF] Signorini | Optical limiting and non linear optical properties of fullerene derivatives embedded in hybrid sol-gel glasses[END_REF][START_REF] Brusatin | Materials for Photonic Applications From Sol-Gel[END_REF][START_REF] Tao | Fabrication of covalently functionalized graphene oxide incorporated solid-state hybrid silica gel glasses and their improved nonlinear optical response[END_REF] Another alternative to improve the optical performance is to use plasmon enhanced optical limiting. This strategy have already proven to be of high efficiency in the visible wavelength but has never been demonstrated so far in the SWIR. 37 Based on these earlier findings, the intention was set on combining a dye and a plasmonic nanostructure in a solid host for SWIR applications.

Silica based sol-gel materials are therefore selected as hosting matrices as they have proven to give highly efficient optical power limiting materials, both in the visible and SWIR. [START_REF] Zieba | Preparation of Functional Hybrid Glass Material from Platinum (II) Complexes for Broadband Nonlinear Absorption of Light[END_REF][START_REF] Château | Efficient hybrid materials for optical power limiting at telecommunication wavelengths[END_REF][START_REF] David | High-Performance Optical Power Limiting Filters at Telecommunication Wavelengths: When Aza-BODIPY Dyes Bond to Sol-Gel Materials[END_REF] Combining plasmonic nanostructures with the dyes, in the glass material, previously allowed an enhancement of two-photon absorption especially at the wavelengths where it is weak. 37 Similar strategy transfered to the SWIR is not trivial. It requires the use of stable metal nanoparticules with longitudinal plasmon resonances above 1.5 µm, dispersability and stabilisation in a hybrid glass together with an adapted SWIR chromophore.

Herein, we investigate the synergy between conjugated aza-bodipy dyes and plasmonic nanostructures for optical power limiting application in the SWIR. Aza-bodipy dyes, 1 (Figure 1) functionalized by the dialkylaminophenylethynyl in the 1,8 positions is known to present good nonlinear optical properties (nonlinear absorption induced by two-photon absorption and followed by excited state absorption) in the SWIR and particularly at 1.5µm. [START_REF] Bouit | Two-Photon Absorption-Related Properties of Functionalized BODIPY Dyes in the Infrared Range up to Telecommunication Wavelengths[END_REF] These dyes also present the chemical inertness to be successfully incorporated in a bulk xerogel, which gave rise to the first solid-state optical power limiting material at 1.5µm reported in the literature so far. [START_REF] Château | Efficient hybrid materials for optical power limiting at telecommunication wavelengths[END_REF][START_REF] David | High-Performance Optical Power Limiting Filters at Telecommunication Wavelengths: When Aza-BODIPY Dyes Bond to Sol-Gel Materials[END_REF] Nanobipyramidal gold structures (AuBPs, figure 1) are considered in this work for the synergy enhancement of optical power limiting of the Aza-bodipy dyes, since they allow very precise tuning of the plasmon resonance in the visible and in the SWIR region. They also have intense longitudinal resonances which is located in the SWIR and hot spots at the tips. [START_REF] Navarro | electron tomography and single-particle optical response of twisted gold nano-bipyramids[END_REF][START_REF] Chateau | From gold nanobipyramids to nanojavelins for a precise tuning of the plasmon resonance to the infrared wavelengths: experimental and theoretical aspects[END_REF][START_REF] Chateau | Beyond the Concentration Limitation in the Synthesis of Nanobipyramids and Other Pentatwinned Gold Nanostructures[END_REF] Moreover, they are stable and can easily be incorporated in a sol-gel matrix such as previously demonstrated. 37,[START_REF] Lundén | Dispersion and self-orientation of gold nanoparticles in sol-gel hybrid silica -optical transmission properties[END_REF] The synthesis of the hybrid sol-gel monolith, co-doped with dye 1 and various concentrations of AuBPs, is described here and the materials are optically fully characterized.

Figure 1. Structure of the functionalized azabodipy dye 1 and SEM images of the plasmonic nanostructures AuBPs@1730nm (the outline of one of the particles is enhanced with a dotted line to show the bipyramidal shape).

RESULTS AND DISCUSSION.

The Aza-Bodipy and the plasmonic NPs. The Azabodipy 1 has an intense and broad absorption band in the visible (Figure 2). In the NIR a broad transition centered at 740 nm with a cut-off wavelength of 865 nm and a red-tail close to 1000 nm is observed, as previously described in ref. 45. The compound also exhibits two-photon absorption properties in the SWIR band between 1100 and 1600 nm, with a maximum at 1150 nm and a less intense band at around 1500 nm. [START_REF] Bouit | Two-Photon Absorption-Related Properties of Functionalized BODIPY Dyes in the Infrared Range up to Telecommunication Wavelengths[END_REF] Detailed photophysical studies indicate that the contribution of the charge transfer transition from the dialylamino-donor group to the central azabodipy core, acting as an electron withdrawing moiety, is responsible for the position and intensity of the nonlinear response. [START_REF] Lundén | Dispersion and self-orientation of gold nanoparticles in sol-gel hybrid silica -optical transmission properties[END_REF] The molecule is also known to present straightforward optical limiting properties in solution and in a sol-gel matrix. [START_REF] Château | Efficient hybrid materials for optical power limiting at telecommunication wavelengths[END_REF][START_REF] David | High-Performance Optical Power Limiting Filters at Telecommunication Wavelengths: When Aza-BODIPY Dyes Bond to Sol-Gel Materials[END_REF][START_REF] Pascal | Unraveling the Two-Photon and Excited-State Absorptions of Aza-BODIPY Dyes for Optical Power Limiting in the SWIR Band[END_REF] The AuBPs are prepared according to the reported procedure. [START_REF] Chateau | Beyond the Concentration Limitation in the Synthesis of Nanobipyramids and Other Pentatwinned Gold Nanostructures[END_REF] Few of the particles present small star-shaped excrescences at the tips, which is a common defect encountered for very elongated Au bipyramids, but as shown in Figure 2, this does not affect the plasmon absorption bands significantly. The overall background absorbance of the bipyramids below 1000-1200nm is slightly increased as shown by the comparison between the spectra of defect free bipyramids from Chateau et al. compare to the one used in this study. 37 The AuBPs are functionalized with a thiolated silicone oligomer (Figure 3), 37,[START_REF] Chateau | Controlled surface modification of gold nanostructures with functionalized silicon polymers[END_REF] to improve the dispersability and stability of the particles in the hybrid silicon precursor sol. The chosen AuBPs have a main Surface Plasmon Resonance (SPR) absorption centered at 1730 nm and an aspect ratio close to 10 (mean particle size of 20 x 205nm). Thus, the AuBPs has no or very low SPR absorption in the spectral range 1200 -1500 nm (the range of interest, ROI) where 1 is NLO active (Figure 2). Consequently, in the ROI, the linear absorption of all components is relatively low. Sol-Gel monoliths. In order to study the influence of the AuBPs on the NLO properties of dye 1, both particles and dyes need to be homogeneously dispersed in the transparent sol-gel matrix. The procedure that was successfully used in the visible with platinum acetylide dyes and gold nanoparticles cannot be applied straightforward here since the dye is pH sensitive. 37,[START_REF] Château | Efficient hybrid materials for optical power limiting at telecommunication wavelengths[END_REF] Thus the procedure is modified and a hybrid sol is prepared from the citric acid catalyzed hydrolysis of a mixture of methyltriethoxysilane (MTEOS) and vinyltriethoxysilane (VTEOS) in a 50:50 molar ratio (Scheme 1). The presence of VTEOS limits the formation of micropores that may induce oxygen trapping or diffusion and consequently loss of optical properties. [START_REF] Zieba | Preparation of Functional Hybrid Glass Material from Platinum (II) Complexes for Broadband Nonlinear Absorption of Light[END_REF] Chromophore / Functionnalized Au NPs (THF) The use of a tertiary amine is more convenient than the previously reported trimethoxysilane, which induces hydrogen formation and therefore possible trapping of gas bubbles during the gelification. [START_REF] Château | Efficient hybrid materials for optical power limiting at telecommunication wavelengths[END_REF] This substitution avoids bubbles being trapped in the final solid material, which represents an important improvement of the initial process.

The co-doped monolithic xerogels are obtained by dissolving dye 1 in a minimal volume of THF and mixing with the sol. Then, the suspension of gold NPs in THF and the gelation agent are added.

The monolithic material is obtained after 2 days aging at 45°C in a mold. Xerogels with AuBP but without dye were also prepared later to verify the impact of the AuBP on the OPL performance.

Overall, seven monoliths containing dyes 1 (5 wt%) and the NPs at different concentrations were prepared (Table 1). After demolding, dark monolithic discs of 1.2 cm diameter and 0.3 cm thickness are obtained (Figure 4). Optical Characterization of monoliths. High optical quality of the prepared glasses (Mlt-1-7) is required to undertake NLO measurements, since any surface or bulk inhomogeneity (aggregate, phase segregation…) immediately results in strong linear absorption or scattering of the incident laser. The monoliths are therefore polished to high optical quality and their optical transmission is verified (Figure 5 and Figure 6, left). All dye doped materials are opaque below 1000 nm, corresponding to the red-tail of the absorption of dye 1 present in the material (5 wt%). In the ROI (1200-1600 nm), all dye doped materials except Mlt-4 present a very good transparency with an average transmission of 75-90 %.

However, the overall linear transmission tends to decrease within ROI with an increase in AuBPs concentration (Figure 5). This can be attributed to the tail of longitudinal SPR band and also from its broadening due to a partial aggregation of NPs at the highest concentrations. Substraction of the matrix/dye spectrum from Mlt-3 reveals the clear presence of the AuBP longitudinal SPR band (Figure S1) with a red-shift and a visible broadening compared to the AuBP in water. This confirms the presence of dispersed AuBP in the co-doped xerogels, as complete aggregation would result in no visible SPR peak, but also indicates an influence of the dye 1 on the quality of the dispersion. On the other hand, the xerogels doped only with AuBP (Figure 6) show the transverse SPR band (at 510 nm) and a slightly blue-shifted longitudinal SPR band (at 1605 nm), compared to the particles in solution at 510 and 1630 nm respectively (see Figure S2). This slight blue-shift indicates a small alteration of the tips of the nanoparticles but no significant degradation of the nanoparticles caused by the incorporation into the sol-gel matrix. The thin LSPR band also indicates minimal aggregation compared to the dye-doped materials. The reduction in transmission with an increased fluence originates from laser induced damage (even though the glasses were moved in a direction perpendicular to the laser beam after each shot). This allows to exclude the direct role of NPs in the OPL efficiency of the chromophores in the co-doped systems and to discuss it in terms of plasmonic coupling between both moieties.

Figure 7 compares the OPL data of the two doped glasses, the reference material, Mlt-1, (with a dye concentration of 56 mM) and Mlt-3 (with a concentration of 56 mM and 0,08 nM for dye 1

and AuBPs, respectively). For all incident wavelengths (1200, 1300, 1400, 1550 nm), an important improvement of the OPL response is observed, with a significant decrease of the clamping level compared to the reference (glass material doped solely with the dye). The strongest effect occurs at 1550 nm (Figure 7): an optical limiting clamping of the transmitted intensity at 30 J is measured against 50 J for the sample without any NPs, while the plasmon resonance presents the highest value among the four incident studied laser wavelengths. This observation illustrates the relevant role in optical limiting properties of plasmonic NPs. The role of the concentration of NPs is also evaluated. Monoliths with concentrations of 56 mM in dyes and various concentrations of NPs are optically evaluated (Figure 8). In the OPL measurements, on the different glasses, it appears that the clamping level decreases from 35 J to 20 J when the NPs concentration increases (Figure 8 center). However, when normalized OPL is drawn considering the linear transmission (Figure 8 left), obviously the most efficient enhancement appears to be at the lowest concentration in NPs (Figure 8 right). This observation is due to the increase in linear absorption at 1550 nm related to the plasmon intensity, screening out the nonlinear absorption effect, while the overall effect combining both linear and nonlinear absorption appears as more efficient. This is consistent with results previously observed for visible wavelengths. 37 Gold bipyramids synthesis. Gold pentatwinned seeds are prepared as described in the literature. [START_REF] Chateau | Beyond the Concentration Limitation in the Synthesis of Nanobipyramids and Other Pentatwinned Gold Nanostructures[END_REF] A growth solution is prepared by dissolving 8 mg of CTAB (for a LSPR around 1700) in 12 ml of CTAC solution (140 mM in water) and adding 120 μl of HAuCl4 (25 mM in water).

Then 90 μl of aqueous silver nitrate (5 mM) are added, followed by 120 μl of 8-hydroxyquinoline (0.4 M solution in ethanol). The growth solution turns light yellow. After 2 minutes, 80 μl of gold seeds are then quickly introduced into the growth solution. The mixture is gently stirred for ten seconds and directly put into an oven at 40-45°C for 20 minutes. Then, 80 μl of HQL (0,4 M in ethanol) is added to the mixture and left for 45 minutes at 40-45°C. Finally, 1 ml of a 10% TWEEN80 solution is then added to improve the colloidal stability of the particles and the mixture is kept at 45 °C for an additional 45 minutes. The resulted particles are purified by centrifugation at 6000 rpm and redispersed into 12ml of 3mM CTAC to yield a suspension at 0,25 mM in Au 0 and 0,16 nM of AuBP. The final particles are then functionalized and transferred to THF (maintaining the same concentration) just before use, according to the literature. [START_REF] Chateau | Controlled surface modification of gold nanostructures with functionalized silicon polymers[END_REF] MTEOS/VTEOS sol synthesis. An equimolar mixture of MTEOS (50 ml) and VTEOS (53 ml) is hydrolyzed using a 15g/l citric acid solution in water. The hydrolysis molar ratio water/precursor is h=11. The mixture is stirred overnight at 25°C. The resulting mixture is evaporated under vacuum to remove the ethanol formed during hydrolysis, allowing the decantation of a hydrophobic phase, rich of hydrolyzed and condensed MTEOS. This phase is dissolved in diethyl ether (150 ml) to remove residual water after phase separation. The resulting organic phase is washed thoroughly four times with water (30 ml each time) to remove traces of citric acid and thereby reach pH above 4.5. Then diethyl ether is evaporated and replaced by THF. Evaporation of the solvents gives a dried solid residue with 45 % in mass. This prepared sol can be stored at -24°C and kept for several months.

Sol-gel monolith preparation. Typically, in a PFA mold, 17 mg of dye 1 is introduced and dissolved in 100-150µl of THF. After complete dissolution, 700 mg of the previously prepared MTEOS/VTEOS sol are introduced, followed by the desired amount of AuBP suspension in THF (0, 75, 150 or 300µl for 0, 0.04, 0.08 and 0.16 nM of AuBP in the final material respectively) and completed to around 1ml volume with THF. 30 µl of N,N-dimethylaminopropyltrimethoxysilane is added to the mixture in the mold. The mold is closed with a screw cap, that has a small hole (x mm), and manually stirred for 30s and then kept undisturbed at room temperature until gelation occurs (typically around 20-30 min). The gelified material is put in an oven at 45°C for 2 days to yield the final xerogel.

Polishing of the glass materials. A Struers polishing equipment (Struers LaboPol-25) was used.

The final polished materials had a thickness of 1± 0.05 mm.

Optical characterization. Transmittance spectra were recorded with a CARY 5000 UV-Vis-NIR spectrophotometer. The OPL measurement set-up, which has been described earlier, [START_REF] Thesis) Lundén | Nonlinear materials for optical power limiting: characterization and modelling[END_REF] is based on a f/5 system. The laser beam (source an EKSPLA NT342C OPO, emtting 5 ns pulses) is expanded by a 3x Keplerian telescope. The beam passes through a pinhole filter with 25 μm diameter, placed between two f=20 cm lenses, for beam clean up, leading to a pellicle beam splitter that sends a small part of the pulse energy to the reference detector. The remaining pulse energy is expanded by a 2.66x Keplerian telescope and passed through a 2 cm aperture creating an approximately tophat beam, which reaches a 2.5 x Keplerian telescope that forms the f/5 system. A simple planospherical lens of focal length 1m is used to focus the transmitted beam onto the signal pulse energy detector. The system is aligned at 1550 nm with the help of a digital video camera and a small aperture placed after the plano-spherical lens. For other wavelengths the 1550 nm alignment is used while transposing the front lens of the f/5 telescope to maximize the transmission through the last aperture. The beam diameter, for all the used wavelengths, was measured with the 10-90%

knife-edge method. The beam diameter varied between 16 and 25 µm depending on the wavelength.

ASSOCIATED CONTENT

Figure 2 .

 2 Figure 2. Overlay of normalized 1PA in dichloromethane (black) with 2PA (black squares) in carbon tetrachloride spectra of dye 1 16 and AuBP@1730 (blue) in water.

Figure 3 .

 3 Figure 3. Schematic representation of the thiolated silicone oligomer designed for functionalization of the AuBPs.9l

Scheme 1 .

 1 Scheme 1. Overview of the process used for the preparation of the hybrid materials.

Figure 4 .

 4 Figure 4. Different monoliths (from left to right, diameter 13 mm): Mlt-1 doped with dye 1 (5%

Figure 5 .

 5 Figure 5. Transmission spectra of the dye doped monoliths. The spectral region of interest (ROI) is marked, dashed line.

Figure 6 .

 6 Figure 6. Left: Linear transmission of glass materials Mlt-5 to Mlt-7. Right: Optical limiting of

Figure 7 .

 7 Figure 7. Optical Limiting of Mlt-1 (black dots) and Mlt-3 (blue dots) at 1550, 1400, 1300 and

Figure 8 .

 8 Figure 8. Transmission spectra, OPL and normalized OPL (respectively from left to right) of

Table 1 .

 1 Composition of the monoliths doped with chromophore 1 and gold NPs.

		Chromophore	Gold nanobipyramids
	Material	Concentration	AuBP
	reference
		%wt	mM	LSPR (nm) a)	[AuBP] (nM)
	Mlt-1 (reference)	5%	56	n/a	n/a

a) LSPR measured in water dispersions

ACKNOWLEDGMENT

Hampus Lundén is thanked for participation in the OPL measurements. SP acknowledges Région Auvergne Rhône-Alpes for financial support (Project HYPOLASE).

Supporting Information. Absorption spectra of AuBP and sol-gel materials are available as Supporting Information.

AUTHOR INFORMATION

Corresponding Author

Prof. S. Parola, Univ. Lyon, Ecole Normale Supérieure de Lyon, Université Claude Bernard Lyon 1, CNRS UMR5182, Laboratoire de Chimie Lyon, 46 allée d'Italie, F69364 , France. E-mail: stephane.parola@ens-lyon.fr Dr. C. Andraud, Univ. Lyon, Ecole Normale Supérieure de Lyon, Université Claude Bernard Lyon 1, CNRS UMR5182, Laboratoire de Chimie Lyon, 46 allée d'Italie, F69364 , France. E-mail: chantal.andraud@ens-lyon.fr

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. SYNOPSIS (Word Style "SN_Synopsis_TOC").

Sol-gel NIR


Karim Tabia
email: tabia@cril.fr

Tabia Karim

Cril

Updating Probability Intervals with Uncertain Inputs

HAL is

Introduction

Probability intervals are compact, intuitive and convenient means for representing imprecise probabilities. The theory of imprecise probabilities [Levi, 1980;Walley, 1991] is a unifying uncertainty theory particularly suited for encoding and reasoning with imprecise or ill-known information. It is used to reason with multiple expert information [Nau, 2002], perform sensitivity analysis [Bock et al., 2014], make decisions with partial information [Antonucci et al., 2007], etc. Imprecise probabilities are often associated with a robust Bayesian interpretation [Berger et al., 1994] assuming that the probability measure representing the actual beliefs exists and it is unique but it is unknown due to lack of knowledge. Imprecise probabilities are specified in the form of sets of probability measures, credal sets [Levi, 1980;Walley, 1991] or using other representations such as intervalbased probabilities [START_REF] De Campos | [END_REF], probabilistic logic programs [Lukasiewicz, 2001], belief functions, etc. As for updating probability intervals, it should be noted that there are several conditionings depending on the associated interpretation of probability intervals [START_REF] Moral | Updating uncertain information[END_REF]. In this paper, we limit ourselves to the three most common types of conditioning that are the robust Bayesian conditioning [START_REF] Moral | Updating uncertain information[END_REF], Dempster conditioning [Shafer, 1976] and the geometric one [START_REF] Suppes | [END_REF]. Given an initial belief set, one may learn new information which can be in the form of a hard evidence or in the form of uncertain or soft evidence (e.g. unreliable observation). The focus of the paper is updating1 a set of probability intervals with new information also expressed by probability intervals. In the classical point-wise probabilistic setting, Jeffrey's rule [Jeffrey, 1965] generalizes the standard probabilistic conditioning to the case where the inputs are uncertain. This conditioning rule has been studied in many uncertainty settings (for instance, see [START_REF] Dubois | A synthetic view of belief revision with uncertain inputs in the framework of possibility theory[END_REF] for the possibilistic setting and [Smets, 1993;[START_REF] Tang | Generalized jeffrey's rule of conditioning and evidence combining rule for a priori probabilistic knowledge in conditional evidence theory[END_REF]Ma et al., 2011] for Dempster-Shafer theory). We place ourselves in the framework of Jeffrey's rule where the new information is pervaded with uncertainty and this latter bears on a partition of the set of possible worlds. We therefore take for granted the principles underlying Jeffrey's rule and we do not focus on the foundations and justifications of such principles. The focus is on extending Jeffrey's rule for updating probability intervals with uncertain inputs. The main contributions of the paper are :

1. Extending Jeffrey's rule of conditioning to update a prior credal set with a new uncertain and imprecise input also in the form of a credal set ;

2. Casting Jeffrey's rule principles (success and probability kinematics) in the probability intervals setting and show that the proposed rules capture such principles ;

3. Proposing Jeffrey's rule counterparts for two alternative conditionings (Dempster and geometric) of probability intervals.

Preliminaries

Probability Intervals

Let us in the following denote by Ω={ω 1 ,..,ω n } the set of elementary possible states and denote by ω i ∈Ω a given state. Sets of states φ, ψ⊆Ω are called events. An interval-based probability distribution (IPD for short) is defined as follows: Definition 1 (Interval-based probability distribution). Let Ω be the set of possible states. An interval-based probability distribution P is a function that maps every state ω i ∈Ω to a closed interval

P (ω i )=[l i , u i ]⊆[0, 1].
In an IPD P , every state ω i ∈Ω is associated with a probability interval P (ω i )=[l i , u i ] where l i (resp. u i ) denotes the lower (resp. upper) bound of the probability of ω i . In the following, P (ω i ) (resp. P (ω i )) denotes the upper (resp. lower) bound of P (ω i ).

In order to be reachable and not empty, the bounds should satisfy the following constraints:

ωi∈Ω l i ≤ 1 ≤ ωi∈Ω u i ∀ω i ∈ Ω, l i + ω j =i ∈Ω u j ≥ 1 and u i + ω j =i ∈Ω l j ≤ 1
We call a model or a member of IPD P any point-wise (or single-valued) probability distribution p s.t.

∀ω i ∈Ω, p(ω i )∈P (ω i )=[l i ,u i ].
Accordingly, the semantics associated with an IPD is the set of its models, namely all probability measures that comply with the probability intervals. A credal set K on Ω compactly encoded by an IPD P denotes the closed convex set of point-wise probability distributions p that are models of P . Namely,

K = {p | l i ≤ p(ω i ) ≤ u i , ∀ω i ∈ Ω} (1)
A commonly used way to encode a credal set K is the vertex-based representation. This is done by specifying a finite set of standard probability distributions representing the extreme points of K. An extreme point (also called vertex) p of a credal set K is a probability distribution such that it is impossible to find two different probability distributions p 1 ∈K and p 2 ∈K such that p=α*p 1 +(1-α)*p 2 with α∈]0, 1[. Enumerating the set of extreme points ext(P ) underlying an IPD P requires solving a linear program [START_REF] De Campos | [END_REF]]. An IPD P can be completely represented by the extreme points of the credal set K underlying P .

Reasoning with interval probabilities can either be done at the semantic level using the credal set induced by probability intervals or directly manipulating intervals and using interval arithmetics. Reasoning using a credal set K underlying an IPD P comes down to exploring all the models of that credal set2 . For instance, marginalizing a credal set K(X, Y ) over two subsets of variables X and Y is done as follows:

K(X) = { Y p(X, Y ) : p ∈ K(X, Y )} (2)
For credal sets, the most common form of conditioning on an event φ⊆Ω is defined as follows :

K(ω i |φ) = {p(ω i |φ) : p ∈ K and p(φ) > 0}
(3) This is often referred to as cautious or robust Bayesian conditioning (see [START_REF] Moral | Updating uncertain information[END_REF] for other forms of conditioning probability intervals). For computational reasons, reasoning on K is done on ext(K) which provides an equivalent representation. Indeed, inference on a credal set K is equivalent to inference on its extreme points [Walley, 1991;[START_REF] De Campos | [END_REF]. For instance, marginalizing a credal set K(X, Y ) over two subsets of variables X and Y is done as follows :

K(X) = CH({p(X) : p ∈ ext(K(X, Y )}), (4
) where CH stands for the convex hull operator. Conditioning of Equation 3 comes down to :

K(ωi|φ) = CH({p(ωi|φ) : p ∈ ext(K) and p(φ) > 0}). (5)
In the following, we briefly recall Jeffrey's rule in the standard probabilistic setting.

Jeffrey's Rule of Conditioning

Jeffrey's rule [Jeffrey, 1965] extends the classical probabilistic conditioning to the case where the new information is uncertain. It allows to update an initial probability distribution p into a posterior one p given the uncertainty bearing on a set of mutually exclusive and exhaustive events λ={λ 1 ,..,λ n } (namely, λ is a partition of Ω). In this setting, the new input is in the form (λ i , α i ), i=1..n where α i denotes the new probability of λ i . Jeffrey's rule lies on the two following principles: Success principle (P1).

∀λ i ∈ λ, p (λ i ) = α i (6) After the update operation, the posterior probability of each event λ i must be equal to α i as required in the new inputs. The uncertain inputs are seen as constraints or an effect once the new information is fully accepted. Probability kinematics principle (P2).

∀λ i ∈ λ, ∀φ ⊆ Ω, p(φ|λ i ) = p (φ|λ i ).

(7) This principle aims to ensure a kind of minimal change by ensuring that the posterior distribution p should not change the conditional probability degrees of any event φ given the uncertain events λ i . Jeffrey's rule assumes that in spite of the disagreement about the events λ i in the prior distribution p and the posterior one p , the conditional probability of any event φ⊆Ω given any uncertain event λ i should remain the same in the original and the revised distributions.

Given a probability distribution p encoding the initial beliefs and new inputs in the form (λ i , α i ) for i=1..n, the updated probability degree of any event φ⊆Ω is obtained as follows:

p (φ) = λi α i * p(φ ∩ λ i ) p(λ i ) . ( 8
)
The posterior distribution p obtained using Jeffrey's rule always exists and it is unique [START_REF] Chan | [END_REF]. Note that in Jeffrey's rule, the events λ i should be somewhat possible in the prior distribution (namely, ∀λ i ∈λ, p(λ i )>0).

Motivating Example

To illustrate and motivate the interest in expressing and updating uncertain information in the form of probability intervals, let us take the example of the polls generally carried out in the elections and suppose that they are about presidential elections that are going to take place in a given country. Suppose for simplicity that there are only four candidates (denoted C 1 , C 2 , C 3 and C 4 ) and that in this country the political polarization is Left wing and Right wing. Assume now that a first poll of voter preferences for the four candidates on a small sample yielded the following results in Table 1 (a) (the estimates are in the form of probability intervals to account for the margin of error). Assume now that a second poll on a much larger sample and where the question is no longer Which candidate are you going to vote for? but Do you prefer a left wing candidate or a right wing one? To simplify, we assume that candidates C 1 and C 2 are classified in the left wing and the other two in the right wing. Also, the estimates (see Table 1 (b)) are given in the form of probability intervals to take into account the margin of error. It fully makes sense in this case to update the initial distribution (results of the first poll) with new information (results of the second poll) as this latter is more confident since it is carried out on a much larger sample. It is important to note that the initial information to update is an interval-based probability distribution P and the new input uncertain is also an interval-based probability distribution P new over a partition of the set of candidates. This update task is fully in the spirit of Jeffrey's rule (the new inputs are uncertain information bearing on a partition of Ω and the aim is to give priority to the new inputs). In this example, the task is not a simple belief merging but aims at giving priority to new inputs as in belief revision tasks. This is exactly in line with Jeffrey's rule in the standard probabilistic setting. But this latter handles only point-wise probability distributions and not interval-based ones. There is to the best of our knowledge no counterparts to Jeffrey's rule in the interval-based setting.

Updating Credal Sets with Uncertain Inputs

Before addressing updating probability intervals with uncertain inputs, let us first focus on updating credal sets with uncertain inputs using Jeffrey's rule. The idea is to use Jeffrey's rule to update every distribution p from the prior credal set K with every distribution p new from the new input credal set K new . In order to apply Jeffrey's rule on any prior distribution p from K, we assume that ∀p∈K, ∀λ i ∈λ, p(λ i )>0.

This section answers two fundamental questions: i) does updating a credal set with another credal result in a credal set and ii) can this update be done by manipulating only the vertices of the credal sets as in the case of conditioning with hard evidence. The following are the first main results of this paper (proofs are provided as supplementary material): Proposition 1. Let K be the closed convex set to update. Let p new be a single probability distribution over the partition λ and encoding the new information. Let K be such that

K = {p : p = p ⊕ p new ; p ∈ K}.
Then K is a convex credal set.

In Proposition 1, p⊕p new denotes conditioning a pointwise probability distribution p with new inputs encoded in the form of a probability distribution p new over a partition of Ω using Jeffrey's rule. This result states that using Jeffrey's rule to update each member of a convex credal set K with the same input distribution p new results in a convex credal set K . Hence, we generalize the result that conditioning a credal set with a hard evidence gives a convex credal set. Proposition 2. Let K be the closed convex set to update and let p∈K be a single probability distribution. Let K new be a credal set over the partition λ and encoding the new information at hand. Let K be such that

K = {p : p = p ⊕ p new ; p new ∈ K new }.
Then K is a convex credal set.

Proposition 2 states that updating using Jeffrey's rule a single probability distribution p over Ω with each member of credal set K new over a partition λ of Ω results in a convex credal set. Let us now generalize to the case where both K and K new are arbitrary convex sets. Proposition 3. Let K be the closed convex set to update and K new be a closed convex set over the partition λ on Ω encoding the new information. Let

K = {p : p = p ⊕ p new ; p ∈ K and p new ∈ K new }.
Then K is a convex credal set.

Proposition 3 states that using Jeffrey's rule to update each member of a prior credal set K with each member of credal set K new results in a convex credal set K .

The following proposition states that the extreme points of the updated credal set K correspond to the updated set of extreme points of K with the new input using Jeffrey's rule. Proposition 4. Let K be the closed convex set to update and let ext(K) be its set of extreme points. Let p new be a single probability distribution over the partition λ and encoding the new information. Let K be the updated credal set following Proposition 1. Then

ext(K ) = {p : p = p ⊕ p new ; p ∈ ext(K)}
is the set of extreme points of K . Lemma 1. Let K be the closed convex set to update and let ext(K) be its set of extreme points. Let p new be a single probability distribution over the partition λ and encoding the new information. Let K be the updated credal set following Proposition 1. Then

K = CH({p : p = p ⊕ p new ; p ∈ ext(K)}).
In fact, given that from Proposition 1 K is convex (namely, it contains every convex combination of its extreme points) and the fact that CH(ext(K )) is by definition equal to all the convex combinations of the finite set ext(K ) computed following Proposition 4 then K can be recovered using only such extreme points.

Proposition 5 and Lemma 2 generalize conditioning credal sets based on extreme points to the case where the new input is also a credal set.

Proposition 5. Let K be the closed convex set to update and K new be a closed convex set over the partition λ encoding the new information. Then

ext(K ) ⊆ {p : p = p ⊕ p new ; p ∈ ext(K), p new ∈ ext(K new )}
This means that an extreme point of K is necessarily a combination using Jeffrey's rule of an extreme point of K with an extreme point of K new as illustrated in Example 1. But it is not true that every combination of p⊕p new such that p∈ext(K) and p new ∈ext(K new ) is an extreme point of K . This is due to the union operation of convex sets. Indeed, in the case where the union of convex sets is convex, the extreme points of the resulting convex set (here ext(K )) is included in the union of sets of extreme points of the starting sets. Lemma 2. Let K be the closed convex set to update and let ext(K) be its set of extreme points. Let p new be a single probability distribution over the partition λ over Ω and encoding the new information. Let K be the updated credal set following Proposition 3. Then K = CH({p : p = p⊕pnew; pnew ∈ ext(Knew), p ∈ ext(K)})

Up to now, we showed that updating a credal set K with another uncertain input in the form of a credal set K new over a partition of Ω results in a credal set K and amounts to updating the extreme points of K with those of K new using Jeffrey's rule. Let us now use such findings to update prior probability intervals P with new uncertain inputs also in the form of probability intervals P new .

Updating Probability Intervals with Uncertain Inputs

Let us first place Jeffrey's conditioning rule principles in the context of probability intervals then provide update methods complying with such principles.

Let P be the IPD encoding the prior beliefs. Let P new be the IPD encoding the new information in hand. P new is of the form ((λ 1 , [l 1 , u 1 ]),..,(λ n , [l n , u n ])). This notation means that

P new (λ i )=[l i , u i ].
As in Jeffrey's rule, the set λ={λ 1 ,..,λ n } is a partition of Ω and we assume that the events λ i are somewhat possible in the prior beliefs P (namely, we assume that ∀λ i ∈λ, P (λ i )>0). The properties to satisfy stated in terms of interval-based probabilities are:

• Success principle (IP1) : It ensures that in the updated IPD P', the new information is fully accepted. Namely,

∀λ i ∈λ, P (λ i )=P new (λ i ).
This principle enforces posterior intervals to be equal to the input intervals P new (λ i ). • Probability Kinematics principle (IP2) : The objective here is to ensure that the intervals of P are modified as little as possible to accept the new information while respecting the probability kinematics principle in order to avoid affecting non relevant information.

∀λ i ∈ λ, ∀φ ⊆ Ω, P (φ|λ i ) = P (φ|λ i )
Obviously, in the case of IPDs where lower bounds coincide with upper bounds (namely, if the initial belief set consists in a point-wise distribution over Ω and the new inputs are a point-wise distribution over the partition λ, the principles IP1 and IP2 collapse to P1 and P2 of Jeffrey's rule of Section 2.2. The success principle IP1 may be questionable but it may be a desired property in some applications such as in [Skulj, 2006]. In order to keep in Jeffrey's rule line, we just rephrase P1 in the context of probability intervals. Now given an IPD P encoding the current information and new information P new , there are basically two possible ways to update P with P new : a credal-set based method applying at the semantic level and an interval-based one manipulating directly the bounds of the IPDs. In this paper, we rely on the former for robust conditioning while the latter is used for Dempster and geometric conditionings.

Updating Probability Intervals via Robust Conditioning on the Underlying Credal Sets

Our objective here is to update a belief set consisting of a prior IPD P with new inputs P new also provided in the form of an IPD over a partition λ of Ω. The belief update here consists in updating the credal set K underlying P (containing all the models of P ) by the credal set K new underlying P new using Jeffrey's rule. Indeed, one direct way to update a set of probability measures is to apply Jeffrey's rule on each member of the set where the new input is a single probability measure over the partition λ of Ω.

Following Lemma 2, we can define the update operation using only the extreme points of K and K new , namely update each p∈ext(K) with each p new ∈ext(K new ) using Jeffrey's rule then recover the credal set K using the convex hull operator. Once K computed, the IPD P is computed as lower and upper bounds from K . Hence, the credal-based update method is defined as follows: Definition 2. Let P be IPD to update and P new be the new input IPD over a partition λ of Ω. Let K be the updated credal set computed according to Lemma 2 on K and K new underlying respectively P and P new . P is an IPD on Ω such that ∀ω i ∈Ω,

P (ω i ) = [inf p ∈ext(K ) (p (ω i )), sup p ∈ext(K ) (p (ω i ))].
Example 1. Let us assume in this example that the current beliefs about a given problem over two binary variables A and B are given by the IPD P (AB). In Assume now that we have new uncertain inputs given in probability distribution P new (B) as follows:

B P new (B) b 1 [.7, .8] b 2 [.2, .3]
In order to update P to absorb P new using Definition 2, we update K with K new using Lemma 2. Note that K has two extreme points p 1 =(.70, .05, .1, .15) and p 2 =(.50, .25, .1, .15) and K new has also two extreme points, namely p new1 =(.7, .3) and p new2 =(.8, .2). p 1 will be updated into p 1 =(.65, .05, .12, .18) and p 1 =(.75, .05, .08, .12) and p 2 will be updated into p 2 =(.47, .23, .12, .18) and p 2 =(.53, .27, .08, .12). Hence K =CH({p 1 , p 1 , p 2 , p 2 }).

The updated distribution is given in P of Table 3. 2.

Table 2 and 3 show that the new input beliefs P new (B) are fully accepted in the posterior IPD P (see the marginal distribution P (B) computed from the updated distribution P (AB)). Moreover, one can also check that

P (A|B)=P (A|B) ∀a i ∈D A , ∀b i ∈D B .
It is obvious that in case the IPDs P and P new underlying credal sets K and K new each consisting of only one single probability distribution then updating P with P new following Definition 2 is equivalent to updating using Jeffrey's rule. Lemma 3. If ∀ω∈Ω, P (ω)=P (ω) and ∀λ i ∈λ, P new (λ i )=P new (λ i ) then the posterior distribution P computed following Definition 2 amounts to using Jeffrey's rule to update the unique point-wise distribution p∈K underlying P with p new ∈K new underlying P new .

In the general case, Proposition 6 states that updating using Definition 2 ensures that principles IP1 and IP2 are satisfied. Proposition 6. Let P be IPD to update. Let the new information be the IPD P new over the partition λ of Ω. Let P be the posterior IPD computed from P and P new following Definition 2. Then P satisfies IP1 and IP2.

Alternatives to Robust Conditioning

Robust conditioning of probability intervals based on updating at the credal set level suffers from some drawbacks. For instance, it manipulates extreme points of credal sets underlying the IPDs while the number of such extreme points for an IPD with n states can be up to n! [Wallner, 2007]. More over, it requires that the events λ i must be somewhat possible in the prior beliefs (namely, ∀λ i ∈λ, P (λ i )>0) which may be seen as a strong constraint. The alternative then is to update directly the bounds of the intervals of the IPD to accommodate the input P new and use conditionings that do not require such strong assumptions. For instance, Dempster conditioning [Shafer, 1976] directly manipulates the bounds of the prior IPD P to condition on a new evidence ψ⊆Ω and it is defined as long as P (ψ)>0. It is defined as follows:

P (φ||ψ) = P (φ ∩ ψ) P (ψ) (9)
The lower bound can be easily obtained by duality (namely, P (φ||ψ)=1-P (φ C ||ψ)=1-P (φ C ∩ψ) P (ψ) . Here φ C denotes the complement of φ in Ω. Proposition 7 extends Jeffrey's rule to probability intervals using Dempster conditioning. Proposition 7. Let P be the IPD to update. Let the new information be the IPD P new on a partition λ of Ω. Let P be the posterior IPD computed from P and P new as follows: ∀φ⊆Ω,

P (φ) = [1- λ i P (φ C ||λi) * P new (λi), λ i P (φ||λi) * P new (λi)]
Then P satisfies principles IP1-IP2.

Another common definition of conditioning of probability intervals is the one of geometric conditioning [START_REF] Suppes | [END_REF] defined as follows:

P (φ|Gψ) = P (φ ∩ ψ) P (ψ) (10)
The upper bound is obtained by duality

P (φ| G ψ)=1 -P (φ C | G ψ)=1-P (φ C ∩ψ) P (ψ)
. Proposition 8. Let P be the IPD to update. Let the new input be the IPD P new on a partition λ of Ω. Let P be the posterior IPD computed from P and P new as follows: ∀φ⊆Ω,

P (φ) = [ λ i P (φ|Gλi) * P new (λi), 1- λ i P (φ C |Gλi) * P new (λi)]
Then P satisfies principles IP1-IP2.

It is easy to check that in case where ∀ω∈Ω, P (ω)=P (ω) and ∀λ i ∈λ, P new (λ i )=P new (λ i ) then updating following Propositions 7 and 8 collapses to updating using Jeffrey's rule in the standard setting (see Equation 8).

Jeffrey's rule generalizes the standard probabilistic conditioning to the case where the inputs are uncertain. This conditioning rule has been studied in many uncertainty settings such as possibility theory [START_REF] Dubois | A synthetic view of belief revision with uncertain inputs in the framework of possibility theory[END_REF] and Dempster-Shafer theory [Smets, 1993;Ma et al., 2011]. In [START_REF] Benferhat | [END_REF], it is claimed that this rule can successfully recover most of belief revision rules such as natural and drastic belief revision. In [Skulj, 2006], the author use Jeffrey's rule to update a single probability distribution in order to obtain the desired neighborhood of events of interest expressed only in terms of interval probabilities. In [Yue and Liu, 2008], the authors dealt with updating imprecise knowledge in the framework of probabilistic logic programming. This work is mostly dealing with revising probabilistic logic programs and the proposed extension coincides with Jeffrey's rule and Bayesian conditioning only when the updated probabilistic logic program induces a single probability distribution. In [Marchetti and Antonucci, 2018], the authors propose a family of generalized adjustment operators tailored to update convex sets of probability measures with inconsistent piece of evidence. In case where the imprecise knowledge is compactly encoded by means of belief graphical models called credal networks [Cozman, 2000], there is only one work [da Rocha et al., 2008] dealing with updating a credal network with soft evidence. Note that the new here is not a set of probability intervals but consists in likelihood ratios that can be cast into a single point-wise probability distribution. The authors in [Ma and Liu, 2011] deal with belief merging and focus on the influence of the strengths of inputs. The unique operator complying with the proposed postulates was proved to be a merging operator. Updating sets of probability measures is not a new topic [Grove and Halpern, 1998;Levi, 1980;Walley, 1991;Tabia, 2018]. However, most of this work update sets of probability measures with hard evidence while the focus of the current work is updating sets of probability measures with new inputs both expressed by means of probability intervals. There are essentially two types of generalizations of Jeffrey's rule in the literature. The first one concerns rather generalizations or counterparts of this conditioning rule in some uncertainty theories such as belief functions [Smets, 1993] or possibility theory [START_REF] Dubois | A synthetic view of belief revision with uncertain inputs in the framework of possibility theory[END_REF] (note that such settings do not generalize the one of probability intervals and credal sets.). The second type of generalizations concerns rather the relaxation of the requirements of Jeffrey's rule. For example, in [Smets, 1993] it is no longer necessary that the new inputs be a partition of the set of possible states. Moreover, in [Smets, 1993;Wagner, 1992] the authors deal with the generalization of the conditionings (including Dempster and the geometric ones) of belief functions where both the prior information and new inputs are encoded by means of basic probability assignments over 2 Ω (the powerset of Ω).

Regarding iterative revision, it is well-known that Jeffrey's rule is not commutative (since the new inputs are fully accepted, then updating first with (λ i ,[l i , u i ]) then with (λ i ,[l i , u i ]) will be different from updating first with (λ i ,[l i , u i ]) followed by updating with ((λ i ,[l i , u i ])). Among the questions still open, there is that of the uniqueness of the proposed extensions (the paper only answers the question of the existence of extensions) as well as their optimality in terms of minimizing the belief change. As mentioned earlier, the objective of probability kinematics principle is to ensure some kind of minimal change. In the standard probabilistic setting, Jeffrey's rule was shown to provide the optimal solution in the sense of a distance measure between a prior distribution p and the posterior one p [START_REF] Chan | [END_REF]. One can directly extend such a distance measure to credal sets as follows : Proposition 9. Let K and K be two credal sets over the same states space Ω. Let D(K, K ) = max( max Then D is a distance measure.

The distance measure D combines Hausdorff distance (measuring distances between sets) with a distance for pointwise probability distributions specifically designed to bound belief change yielded by Jeffrey's rule in the standard probabilistic stetting [START_REF] Chan | [END_REF]. Then there remains to show that updating a prior credal set K with new inputs in the form of credal set K new following Lemma 2 guarantees that the posterior credal set K is optimal in the sense of distance D(K, K ) of Proposition 9. These two questions of optimality and uniqueness of the solution are part of the avenues for future work. Another important open question concerns certain strong requirements of Jeffrey's conditioning in particular that the new information relates only to a partition of the set of possible states and the one requiring that the events λ i ∈λ be somewhat possible in the prior beliefs. We have proposed to use Dempster conditioning for this purpose but there are other tracks and other conditionings that address this problem such as the conditioning based on adjustment [Marchetti and Antonucci, 2018]. This latter is supposed to absorb even new inconsistent inputs (for instance in case of updating the probabilities of an event λ i which could be impossible in the prior belief set in the case where P (λ i )=0). In addition to these open questions, other avenues of future work will concern computational complexity and study of extensions of Jeffrey's rule with Pearl method of virtual evidence [START_REF] Pearl | Judea Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference[END_REF] in the context of imprecise belief graphical models [Cozman, 2000]. Another avenue will be extending Jeffrey's rule to the case where the probabilistic beliefs are compactly encoded by probabilistic constraints as in conditional logic programs.

Table 1 :

 1 Example of probability intervals encoding initial information (a) and new uncertain inputs (b).

	Candidate	Estimate
	C1	[0.24, 0.30]	Wing	Estimate
	C2	[0.22, 0.26]	Lef t	[0.49, 0.53]
	C3	[0.14, 0.17]	Right [0.47, 0.51]
	C4	[0.30, 0,36]		(b)
		(a)

Table 2 :

 2 Table 2, we have the marginal distribution of A (namely, P (A)), the one of B (namely, P (B)) and the conditional distribution of A given B (namely, P (A|B)). Example of an initial IPD P and the underlying marginal and conditional distributions.

			A	P (A)
	A B	P (AB)	a1 [.60, .80]	A B	P (A|B)
	a1 b1 [.50, .70]	a2 [.20, .40]	a1 b1 [.67, .93]
	a2 b1 [.05, .25] a1 b2 [.10, .10]	B	P (B)	a2 b1 [.07, .33] a1 b2 [.40, .40]
	a2 b2 [.15, .15]	b1 [.75, .75]	a2 b2 [.60, .60]
			b2 [.25, .25]

Table 3 :

 3 Updated beliefs of the distribution given in Table

			A	P (A)
	A B	P (AB)	a1 [.59, .83]	A B	P (A|B)
	a1 b1 [.47, .75]	a2 [.17, .41]	a1 b1 [.67, .93]
	a2 b1 [.05, .27] a1 b2 [.08, .12]	B	P (B)	a2 b1 [.07, .33] a1 b2 [.40, .40]
	a2 b2 [.12, .18]	b1	[.7, .8]	a2 b2 [.60, .60]
			b2	[.2, .3]

In this paper, we use interchangeably the terms update, revision and conditioning even if some authors associate different meanings to these belief change processes. See[Dubois and Prade, 1994] for a survey of belief revision and updating rules.

Alternative approaches consist for instance in selecting the most informative model (in the sense of information entropy for example) of K to draw inferences as it is done in[Lukasiewicz, 2001].

Acknowledgements

This research has received support from the ANR CROQUIS project, grant ANR-21-CE23-0004 of the French research funding agency ANR.