A,B* , ALESSIA ANDREOTTI C,D , MARIO BRESSAN E Laura Rampazzi
email: laura.rampazzi@uninsubria.it

D , CRISTINA CORTI A , OANA CUZMAN D Maria Perla Colombini C

Nicola D'alessandro E

Lolita Liberatore F

Lorenzo Palombi

Valentina Raimondi

Barbara Sacchi

Lucia Tonucci H

Silvia Vettori B

Elisabetta Zanardini A

Giancarlo Ranalli

An interdisciplinary approach to a knowledge-based restoration: the dark alteration on Matera Cathedral (Italy)

Keywords: Dark alteration, FT-IR, LIF, Py-GC-MS, calcium oxalate film, biocleaning Highlights

An interdisciplinary analytical campaign was carried out on the exterior walls of the Santa Maria della Bruna and

Sant'Eustachio Cathedral in Matera. Large areas of these walls have become darkened and the main objective was to evaluate the state of conservation of the stone material (a very porous, organogenic limestone called "Pietra di Matera"), and to suggest the best strategy for the current restoration. Several techniques were used in situ and ex situin laboratory analyses: X-ray diffraction, infrared spectroscopy, ion chromatography, pyrolysis/gas chromatography coupled with mass spectrometry, colour change measurements, laser-induced fluorescence together with biological techniques. Ex-situ and in situ cleaning tests were also carried out on the stone surface.

The results showed the presence of chlorophyll and bacteria on the surface, together with sulfation and calcium oxalate films as the main decay phenomena. In addition, the determination of saccharide and egg residues suggest both biological activity and past conservative treatments as the cause of oxalate films. Data obtained from the analyses proved to be very useful for the conservation work; a complex plan of restoration was adopted, including both traditional and innovative techniques (such as biocleaning, bacterial-gel and a laser system) together with a final evaluation of several protective methods.

Introduction

Matera Cathedral was built in a Romanesque-Apulian style in the 12 th century in the old part of Matera (Italy), called "I Sassi", which is an agglomeration of dwellings made of the local rock (tufa) from the Gravina torrent valley. Matera lies at the western edge of a UNESCO World Heritage site.

The exterior walls of the cathedral are made from the local organogenic limestone. The lower part of the walls (up to 6-8 meters) have become darkened since at least the beginning of the 20 th century (Figure 1). This is of major concern because of the serious aesthetic damage to one of the most important landmarks in Matera. No historical information could explain the alteration, for example whether it originated from past surface protection treatments or to unknown phenomena affecting the stone inside.

Conservation work undertaken from 2014 to 2016 was a unique opportunity to examine the state of conservation of the surfaces, to provide new insights into the dark alteration and to propose a cleaning strategy. The characterization of the stone surfaces of monuments was very important in order to identify the decay phenomena, which could be both biological and chemical, and indirectly provides information on the conservation history of the building [START_REF] Vv | Science and Art: A Future for Stone[END_REF].

Due to the importance of the monument and the age of the dark alteration, an interdisciplinary team of experts in conservation science with complementary and multiple expertise and skills, including biologists, chemists, geologists, and physicists, was created in 2015 in order to plan an extensive analytical campaign, consisting of both in situ and ex situ analyses and cleaning tests (Table 1). A multi-technique approach prior to a conservation work is fundamental in order to consider and study the decay phenomena from various points of view. This strategy is also a good practice to reveal the artistic techniques, the provenance of materials, and the testing of restoration materials as evident in the recent literature [START_REF] Genestar | Different decay patterns observed in a nineteenth-century building (Palma, Spain)[END_REF][START_REF] Rosado | Pink! Why not? On the unusual colour of Évora Cathedral[END_REF][START_REF] Pinna | Brownish alterations on the marble statues in the church of Orsanmichele in Florence: what is their origin?[END_REF][START_REF] Lettieri | Surface characterization and effectiveness evaluation of anti-graffiti coatings on highly porous stone materials[END_REF][START_REF] Crupi | Multi-technique investigation of Roman decorated plasters from Villa dei Quintili (Rome, Italy)[END_REF][START_REF] Ion | Caoxite-hydroxyapatite composition as consolidating material for the chalk stone from Basarabi-Murfatlar churches ensemble[END_REF][START_REF] Ricca | Multianalytical approach applied to the provenance study of marbles used as covering slabs in the archaeological submerged site of Baia (Naples, Italy): The case of the "Villa con ingresso a protiro[END_REF]. In addition, the cleaning procedure represents a particularly delicate phase in terms of the monitoring, selection, and preventive evaluation of the effects produced by the intervention. Safety, both for the restorer and the environment as well as for the work of art, is of primary importance for deciding the best strategy.

During the first step of the campaign (Table 1), a non-invasive monitoring of the surfaces, based on the use of laser induced fluorescence (LIF), was planned in order to map the presence of biodecay phenomena and to verify a selection of cleaning methods, including a biocleaning treatment. The sampling of small fragments from the most representative altered areas was also performed and the samples were characterized by traditional and innovative techniques, in order to highlight signs of ongoing decay, which include inorganic and organic components and microorganisms. X-ray diffraction (XRD) and optical microscopy (OM) were used to highlight the mineralogical phases and to evaluate the stratigraphy of the mortar, Fourier transform infrared spectroscopy (FT-IR) was used to identify inorganic compounds, ion chromatography (IC) was used to quantify water-soluble salts, pyrolysis-gas-chromatography coupled with mass The cleaning strategy was studied in Step 2 (Table 1), evaluating bio-and chemical-methods both in ex situ tests on stone specimens and in in situ experiments, also considering ablation by laser. The in situ tests were monitored by evaluating the colour changes of the surface. In addition, various protective agents were tested and applied in situ, evaluating the results with non-invasive reflectance FT-IR and Py-GC-MS on micro-fragments.

The results from Steps 1 and 2 were used to define the best cleaning method and the best protective agent. This was carried out during Step 3, and was completed in 2017. The data concerning the choice and monitoring of the protective agents are discussed in detail elsewhere [START_REF] Sacchi | A non-destructive multi-analytical approach for the conservation of the Matera Cathedral (Italy), Submitt[END_REF]. Only some aspects are mentioned in the conclusions of this paper, in order to emphasize the importance of a preliminary analytical campaign in protecting the stone surface. Eight samples (C1-C7, OG) of "Pietra di Matera" and mortar (C4 sample) were taken from the stone surfaces of the southern wall and from the façade, both as powder and fragments scraped off the surface using a scalpel (Table 2).

Selective sampling of C4 and C6 sample was carried out by a micro-lancet under an optical microscope, in order to analyse the three and two layers of the stratigraphy, respectively. The samples of powders for the biological analyses (B1-B4) were scraped off the surface with a sterile chisel (Table 2). Figure 1 shows the sampling points.

"Pietra di Matera" specimens

A block of 36 x 22 x 25 cm (about 40 kg) was sampled at 1-1.5 meters from the ground on the left external facade wall of the Cathedral and removed. It was cut horizontally by a dry mechanical saw into thinner specimen blocks (36 x 22 x 3.5 cm, each): the external blocks corresponding to the exposed surface were used for ex situ chemical and biocleaning treatments. The hole created by the removal was filled by another block of an aged piece of tufa rock of the same size [START_REF] Alfano | The bioremoval of nitrate and sulfate alterations on artistic stonework: The case-study of Matera Cathedral after six years from the treatment[END_REF]. The LIF technique exploits a laser source, which is emitted at an appropriate wavelength and which induces a fluorescence emission from the surface of interest. The fluorescence emitted by the irradiated surface is analysed using a spectrometer in order to detect possible fluorescent compounds present on the surface: for example, photosynthetic pigments such as chlorophyll, which may indicate a biological photoautotrophic patina colonising the surface [START_REF] Raimondi | The fluorescence lidar technique for the remote sensing of photoautotrophic biodeteriogens in the outdoor cultural heritage: A decade of in situ experiments[END_REF].

LIF spectra were acquired using a portable, optical fiber-coupled LIF spectrometer, which was developed in-house at IFAC-CNR. The prototype was specifically designed for an in situ operation and is battery-powered. The laser source emits at 405 nm, while the fluorescence spectrum is acquired in the spectral range of between 450 nm and 930 nm with a spectral sampling interval of 0.5 nm. The hand-held measuring head is optically coupled to the spectrometer and detects the fluorescence signal emitted from a 5mm diameter area.

Fourier transform infrared spectroscopy (FT-IR)

The samples scraped off the surface were analyzed as KBr (Sigma-Aldrich FTIR Grade) pellets by an FTIR spectrophotometer BioRad Excalibur Series FTS 3000, DTGS detector, in transmission mode (4000 to 400cm -1 , 4cm -1 resolution, 16 scans). Samples C5 and C7 were analysed with a portable Bruker Optics ALPHA FTIR spectrometer equipped with SiC Globar source and DTGS detector, collecting 24 scans in ATR mode, with 4cm -1 resolution in the 4000-400 cm -1 range. The spectra were analysed with OPUS 7.2 software.

Ion chromatography (IC)

Ion chromatography analyses were performed with a Dionex ICS-1000 Instrument, equipped with a suppressed conductivity detector. For cation analyses, an IonPac® CS12 4x250 mm Analytical Column -which is specifically conceived for the analysis of alkali metals, alkaline earth metals and ammonium -and an IonPac® CG12 4x50 mm Guard Column. The eluent was methane sulfonic acid 20 mM and the suppressor was a Cation Self-generating Suppressor 300x4mm (CSRS® 300). For anion analyses, an IonPac® AS 4A 4x250 mm Analytical Column and an IonPac® AG 4A 4x50 mm Guard Column were used. The eluent was an aqueous solution of Na 2 CO 3 (1.8mM) and NaHCO 3 (1.7mM) and the suppressor was an Anion Self-generating Suppressor 300x4mm (ASRS® 300). ICS-1000 operation is remote controlled using Chromeleon software (version 6.7 SP1) which also provides data acquisition and data processing. For the chromatographic analyses, a given amount of water was added to a weighted amount of the sample. The solution was then stirred for 24 hours, decanted, and filtered with hydrophilic PTFE filters (pore size 0.45 μm).

Optical microscopy (OM)

Reflected light microscopy Polished cross-sections were obtained from fragments of collected samples. A Nikon Eclipse E600 Microscope was used, which was connected to a high-resolution digital camera and controlled by NIS Elements software for image analyses.

Transmitted light microscopy Petrographic observation of a 30 μm section was performed to obtain the main texturalcompositional parameters of the C4 sample. An optical transmitted light polarized microscope, Zeiss AxioScope A.1 was used, with parallel and crossed nicols, and with 2.5X, 5X and 10X of magnification. The images acquired were processed with Axiovision software.

Pyrolysis gas chromatography mass spectrometry(Py-GC-MS)

A pyrolyser operating in constant temperature mode was used (multi-shot pyrolyzer® EGA/PY-3030D (Frontier Lab), which was connected to a split/splitless injector of a 6890N Network GC System gas chromatograph (Agilent Technologies, Palo Alto, CA, USA), coupled with a 5973 Mass Selective Detector (Agilent Technologies, Palo Alto, CA, USA) single quadrupole mass spectrometer. The mass spectrometer operated in the electron impact (EI) positive mode (70 eV).

The pyrolysis conditions employed for the analysis of the samples were: furnace temperature 500°C, interface temperature 300°C. In the pyrolysis cup, 2 µl of hexamethyldisilazane (HMDS) was added to a few micrograms of each sample. The gas chromatograph was equipped with an HP-5MS fused silica capillary column and the chromatographic parameters are reported in the literature [START_REF] Orsini | Analytical pyrolysis of proteins in samples from artistic and archaeological objects[END_REF].

X-ray diffraction (XRD)

In order to identify the mineralogical composition, XRD analysis was performed. A PANalytical X'Pert PRO X-ray diffractometer with Cu anticathode (λ = 1.54 Å) was used, under the following conditions: current intensity of 30 mA, voltage 40 kV explored 2Θ range between 3° -70°, step size 0.02°, time to step 50 s and scan speed of 0.04°/s. The instrument was equipped with X'Celerator multirevelatory and High Score data acquisition and interpretation software.

Biological analyses

The samples were examined in transmitted and UV light with an optical microscope (Nikon Eclipse E600 with an UV-2A filter). A small aliquote of the collected powder (10 mg) was sown on three different nutrient media: Potato Dextrose Agar for micromycetes (PDA, Difco), Plate Count Agar, for heterotrophs such as fungi and bacteria (PCA, Difco), and BG 11 M liquid medium for algae and cyanobacteria [START_REF] Stanier | Generic Assignments, Strain Histories and Properties of Pure Cultures of Cyanobacteria[END_REF]. Another nutrient media, Nutrient Agar (NA, Difco) was used to isolate bacteria developed on PCA. The developed colonies were observed using the UV-light.

2 Step 2 2.2.1 Ex situ and in situ cleaning tests Ex situ chemical cleaning

In order to evaluate the cleaning of the dark area, advanced oxidation processes (AOPs) were considered [START_REF] Parsons | Introduction[END_REF][START_REF] Azbar | Application of Advanced Oxidation Processes (AOPs) to Wastewater Treatment. Case Studies: Decolourization of Textile Effluents, Detoxification of Olive Mill Effluent, Treatment of Domestic Wastewater[END_REF]. These processes are designed to produce hydroxyl radicals which are highly efficient at destroying organic compound residues. AOPs therefore constitute a crucial technology for the degradation of non-easily removable organic compounds. They combine, for example, ozone (O 3 ), ultraviolet irradiation (UV), hydrogen peroxide (H 2 O 2 ) and/or (photo)catalysts (e.g., Fe ions, TiO 2 , polyoxometalates) thus providing a powerful solution for the decrease or removal of residues [START_REF] Bressan | Improved Combined Chemical and Biological Treatments of Olive Oil Mill Wastewaters[END_REF][START_REF] Liberatore | Chemical and biological combined treatments for the removal of pesticides from wastewaters, Water, Air[END_REF][START_REF] Garrel | Photosensitized degradation of cyclohexanol by Fe(III) species in alkaline aqueous media[END_REF][START_REF] D'ambrosio | Water-Soluble Transition-Metal-Phthalocyanines as Singlet Oxygen Photosensitizers in Ene Reactions[END_REF][START_REF] Tonucci | Mild Photocatalysed and Catalysed Green Oxidation of Lignin: A Useful Pathway to Low-Molecular-Weight Derivatives[END_REF].

For example, the entire surface of the "Pietra di Matera" specimen was split out into vertical and horizontal regions.

Several oxidation reagents were tested on the stone surface: H 2 O 2 , commercial Oxone®, TiO 2 , UV light irradiation, H 2 O 2 +UV light and TiO 2 +UV light. H 2 O 2 was used at 35%; Oxone was dissolved in a water solution (0.1 M concentration of active oxygen such as KHSO 5 ). A mixture of TiO 2 was obtained by sonicating 0.75 g anatase and 0.25 g rutile in 10 mL deionized water for 10 min. For the photochemical reactions, a photochemical multiray apparatus (Helios Italquartz; Milan, Italy) was used, which contained 10 UV lamps of 15 W power each of which emitted light at around 254 nm. All chemicals were purchased from Sigma Aldrich.

Microorganisms, media and ex situ biocleaning

On the basis of previous results, P. stutzeri strain A29 (DISTAM-DISTAAM Strain Collections, University of Milan -University of Molise, Italy), was used for the biocleaning [START_REF] Ranalli | Biotechnology applied to cultural heritage: Biorestoration of frescoes using viable bacterial cells and enzymes[END_REF][START_REF] Lustrato | Fast biocleaning of mediaeval frescoes using viable bacterial cells[END_REF]. In order to optimize the growth conditions for the cells to be applied to the altered stone surface, the biodegradation activity was assessed: the strain was incubated at 28°C for 24-36 hours, on mineral medium M9 and on the same medium supplemented by organic substances, such as linseed oil or egg (C.T.S. srl, Altavilla Vicentina, Italy) at 0.5% v/v and 0.5% w/v, respectively.

Suspensions containing exponentially-growing bacteria, approximately 10 8 CFU ml -1 were obtained by inoculating 10 ml of an overnight broth-culture into 1,000 ml of fresh broth medium (in a 3,000 ml total volume flask) and incubating it in a shaker (200 rpm) for 24 hours at 28°C. The cells were centrifuged at 7,000 rpm for 10 minutes at 4°C, washed twice with a phosphate-buffered saline solution, and re-suspended in sterile distilled water, at pH 6.9. The final cell concentration was above 10 8 cells ml -1 , corresponding to an O.D. 560 nm of about 1.6; pH was about 7.0 and the solution conductivity was about 947.1 μs cm -1 ; the cells were stored at 4 °C before bio-applications.

In these biocleaning tests, laponite was adopted as delivery system instead of agar gel which needs a higher temperature before solidification [START_REF] Bosch-Roig | Biocleaning of Cultural Heritage stone surfaces and frescoes: which delivery system can be the most appropriate?[END_REF]. In duplicate, laponite gel (control) and laponite gel including bacteria (laponite biogel, at 9.0 % w/v in the suspension bacterial cells) were employed. A 0.8 -1.0 cm thick continuous layer of P. stutzeri A29 cells entrapped in the laponite delivery system was applied onto the ex situ noble tufa rock surface using a sterile spatula. The entire piece was not covered in order to favour the aerobic conditions; the application times were optimised and ranged from 12 -24 hours, with a room temperature of 25ºC ± 2ºC.

Laser cleaning

The Nd:YAG Q-Switch laser group (C.T.S. srl, Altavilla Vicentina, Italy) had an integrated cooling system and microprocessor control panel for the laser beam frequency and power (wavelength, 1064 nm; energy, from 0 to 350 mJ; pulse duration, 10 ns; spot diameter 1-12 mm; power, 5 mW). The system was equipped with a 3 m optical fiber, easy to-use and particularly suitable for in situ works.

In situ chemical and biological treatments

In situ chemical cleaning tests on the outer wall of Matera cathedral were carried out adopting two different cleaning agents: H 2 O 2 at 35% and AB57. The AB57 includes inorganic salts (ammonium bicarbonate, sodium bicarbonate, EDTA disodium salt, quaternary ammonium salt, carboxymethyl cellulose) in aqueous solutions with other reagents and inert materials for the preparation of cleaning mixtures and poultice to apply to stone surfaces and frescoes (C.T.S. srl, Altavilla Vicentina, Italy).

The area selected for the in situ bioapplication was part of the external wall of the right-hand side of the cathedral and was about 2 m 2 . Areas with the same size (25x40 cm) were identified as follows: i) Laponite delivery system enriched with P. stutzeri A29 cells (biogel); ii) Laponite carrier alone as a control using only the delivery system without bacterial cells (control 1, gel). Finally, in the second control test of an untreated area, no carrier was used for comparison. For each area tested, about 0.3 kg of the delivery system was used, which had been stored in a sterile plastic box; the application and distribution were facilitated by sterile flat plastic spatulas, which made it possible to apply a thin layer of gel averaging 0.5 cm in depth.

Representative areas from around 25 cm 2 to around 1,000 cm 2 were selected and bio-application tests were performed as follows:

i) Japanese paper, 9.0 g/m 2 , natural, white, un-buffered (Klug Conservation, Walter GmbH & Co. KG., Immenstadt, Germany) was applied to protect the surface;

ii) Laponite and laponite-bacterial gels without or with suspension of P. stutzeri viable cells (biogel), respectively, were obtained by manually mixing components with a plastic spatula, for 5 min;

iii) The delivery systems (laponite-gel or laponite-bacteria gel, biogel) were stored at 4°C or immediately applied to the selected areas; iv) Japanese paper, sterile water and the delivery system as negative controls (only laponite gel) were performed on each area. No other facilities were adopted (such as a plastic net or a plastic film to reduce water evaporation).

At different lengths of time from the bioapplication of viable bacterial cells (after 12 and 24 hours respectively), the biogel layers were removed by a soft brush from the surface of the stone wall and the areas were subjected to three gentle manual washes using a soft sponge soaked in distilled water. Samples from all the cleaned and control areas were removed after contact, and stone samples were collected for chemical and microbiological analyses. Three replicates (30 x 20 cm) were carried out for all biocleaning and control tests. During the bioapplication, the outdoor environmental temperature was recorded.

Monitoring of cleaning tests Capillarity water absorption tests

To estimate the water uptake coefficient on the tuff surface of selected area, the contact sponge method, a nondestructive in situ methodology, was used [START_REF]Beni culturali Materiali lapidei naturali ed artificiali -Misura della capacita di assorbimento di acqua mediante spugna di contatto[END_REF]. The UNI Normal procedure requires using a sponge with a known density (Spontex type Calypso) soaked in water, weighed, and placed on the surface of the material for one minute with a given pressure, then weighed again [START_REF] Tiano | Valutazione in situ dei trattamenti protettivi per il materiale lapideo: proposta di una nuova semplice metodologia[END_REF].

Colorimetric analysis

Colour measurements (Miniscan with light D65; Hunter Lab, Bergamo, Italy), were taken to evaluate any chromatic changes in the paint of the biocleaned surface on several previously selected areas, over the short and medium term. The

Results and Discussion

3.1

Step 1: In situ and ex situ analyses

LIF investigations

The LIF spectra were mainly acquired on selected ashlars of the southern wall and of the façade of the cathedral, focusing on those characterised by dark and clear areas. As an example, one of the examined ashlars is shown in Figure 2a.

For each ashlar, several LIF spectra were acquired in various spots of both the upper (clear) and lower (dark) part of the ashlar (Figure 2a). The spectra acquired on the dark part of the ashlar (blue-line spectra in Figure 2b) showed two main fluorescence contributions centred at about 510 nm and 610 nm. The spectra measured on the clear part of the ashlar (red-line spectra in Figure 2b) showed a similar spectral distribution to the one detected in the dark part (broad bands at about 510 nm and 610 nm), along with the typical fluorescence peak of chlorophyll at about 685 nm. In order to separate the fluorescence contribution of chlorophyll from the one due to the stone substrate, all the spectra were processed by interpolating the spectrum in the 640 nm -800 nm spectral range (Figure 2c): the interpolated spectrum (blue line) was subtracted from the original one (red line) in order to determine the chlorophyll fluorescence spectrum (green line), which shows the typical main peak at about 685 nm and a shoulder at about 740 nm. that could be attributed to chlorophyll. All the spectra measured in the clear part of the ashlar were processed by applying the procedure described in Figure 2c, and the retrieved chlorophyll fluorescence spectra are shown in Figure 2d. These results suggest the presence of an extended phototrophic biological patina in the upper part of the ashlar.

In order to further investigate the possible presence of a phototrophic patina also in the dark part of the ashlar, additional measurements were carried out after mechanically removing the very first layer of the dark patina and moistening the area with water. The aim was to expose and activate any phototrophic organisms underneath the dark patina. Also after this procedure, the LIF spectra showed no peak that could be attributed to chlorophyll.

The LIF technique was also used to study a plastered portion of the façade (Figure 3). The surveyed area included a plastered surface (Area A in Figure 3a), an area of stone ashlar from which the plaster was mechanically detached immediately before the measurements (Area B in Figure 3a), and an area with a large gap due to a previous plaster detachment (Area C in Figure 3a). Figure 3b shows the LIF spectra acquired in the three different areas, while Figure 3c shows the same spectra normalised in intensity to the corresponding standard deviation in order to highlight their spectral distribution. Figure 3b shows that Area B (plaster removed before the measurement) has a high fluorescence efficiency, significantly greater than in the other two areas. Figure 3c shows that Area C has a very similar spectral distribution to that observed on the dark areas of other ashlars (Figure 3b, blue spectra), with two main fluorescence contributions at about 500-510 nm and 600-610 nm.

The spectra acquired in Area B are remarkably similar to the spectra that were acquired on a stone specimen obtained from an ashlar quarried out of an external wall of the cathedral (spectrum not shown in this paper). This suggests that the layer of plaster had in some way protected the stone from external agents, natural and/or artificial, which caused the darkening of large areas of the outer surface of the cathedral.

Chemical and mineralogical investigations Optical microscopy

On the basis of observations in transmission mode, the bulk of C4 sample was constituted by an air-hardened calcitic lime binder with a homogeneous appearance and a micritic texture. The binder/aggregate ratio was 2:1. The grain size of the aggregate was bimodal (250-350 μm and 700-800 μm) and the shape of the grains was from sub-angular to sub rounded. Its composition was characterized by fragments of rock (i.e. "Pietra di Matera"), fossiliferous remains and micritic limestone. The porosity was due to the shrinkage cracks.

Samples C2 and C3, observed in reflection mode, were affected by a biological colonization producing a red fluorescence under UV light, due to chlorophyll. C6 sample was observed in stratigraphy which revealed a superficial layer (about 200 μm thickness) that had been altered, and a layer constituted by sound carbonatic stone. Biological colonization was not identified.

X-ray diffraction

XRD analyses revealed the presence of whewellite (CaC 2 O 4 •H 2 O, calcium oxalate monohydrate) in the samples coming from the facade, with the exception of the mortar C4. Gypsum was detected in C1 (from southern wall) and in C6 and C7 (from façade). Calcite and quartz are present in all analyzed samples (Table 3).

Infrared spectroscopy

FT-IR spectra proved very effective in identifying the inorganic component of the samples. In all the samples, the presence of calcium carbonate in the mineralogical form of calcite, was suggested by an asymmetric C=O stretching band around 1420 cm -1 and by the absorbance at 873 cm -1 (out-of-plane bending vibration) and 712 cm -1 (in-plane bending vibration) [START_REF] Derrick | Infrared Spectroscopy in Conservation Science[END_REF]. Alkaline nitrate and silicates were also present, as shown by the absorbance at 1384 and for the Si-O stretching band at 1033 cm -1 , respectively [START_REF] Wilson | Clay mineralogy: spectroscopic and chemical determinative methods[END_REF].

The spectra of samples C1, C5 and C7 presented the characteristic sharp peaks of gypsum (CaSO 4 • 2H 2 O), of hydroxyl stretching bands (3545 and 3408 cm -1 ) and bending vibrations (1689 and 1622 cm -1 ), for the S-O asymmetric stretching modes at 1145 and 1116 cm -1 , and bending mode at 669 cm -1 [START_REF] Derrick | Infrared Spectroscopy in Conservation Science[END_REF]. Organic matter was found as traces in almost all the samples, with peaks of around 2982 and 2875 cm -1 , which could be ascribed to C-H signals, although FT-IR analysis cannot give a complete diagnosis without the support of more sensitive techniques. The results suggest that the dark alteration was mainly due to the presence of calcium oxalate films. Since the first famous occurrence on the Parthenon at the end of the 19th century [START_REF] Liebig | Ueber den Thierschit[END_REF], such films have been frequently observed in the Mediterranean basin on stone materials (mortars, mural paintings and sculptures) and various natural stones (mainly marble), and under several climatic conditions. The occurrence of calcium oxalate films has been well documented [START_REF] Genestar | Different decay patterns observed in a nineteenth-century building (Palma, Spain)[END_REF][START_REF] Pinna | Brownish alterations on the marble statues in the church of Orsanmichele in Florence: what is their origin?[END_REF][START_REF] Vv | International Symposium on the Oxalate Films: Origin and Significance in the Conservation of Works of Art[END_REF][START_REF] Vendrell-Saz | Are patinas of Mediterranean monuments really related to the rock substrate?[END_REF][START_REF] Garcia-Vallès | Interaction of rock and atmosphere: patinas on Mediterranean monuments[END_REF]. These films, whose color ranges from ochre to black, are practically insoluble in water, and thus are stable and difficult to remove. The issue regarding the formation of these films has involved scientific communities [START_REF] Realini | The Oxalate Films in the Conservation of Works of Art[END_REF][START_REF] Vv | International Symposium on the Oxalate Films: Origin and Significance in the Conservation of Works of Art[END_REF] who have debated whether the origin is biological or chemical. A biological explanation would be that calcium oxalate comes from the biofilms on the stone surfaces, which produce oxalic acid that reacts with calcium carbonate from the substrate or from deposits [START_REF] Vendrell-Saz | Are patinas of Mediterranean monuments really related to the rock substrate?[END_REF][START_REF] Garcia-Vallès | Interaction of rock and atmosphere: patinas on Mediterranean monuments[END_REF]. On the other hand, oxalic acid could originate from the oxidation of natural organic materials, such as oil, egg and milk, applied during previous conservation work on the surface [START_REF] Cariati | Calcium oxalate films on stone surfaces: experimental assessment of the chemical formation[END_REF][START_REF] Rampazzi | Analytical investigation of calcium oxalate films on marble monuments[END_REF]. Such conservation was carried out to artificial age new architectural elements, or to protect against natural decay, or for maintenance.

Ion chromatography

All samples from the south wall (samples C1, C2 and C3) were analyzed by Ion Chromatography (Table 4): samples C1 and C2 contained similar amounts of chloride (Cl -), nitrate (NO 3 -) and sodium (Na + ). These ions, instead, were lower in sample C3. Sulfates (SO 4 2-) showed a significant amount only in sample C1. Finally, decreasing amounts of potassium (K + ) and calcium (Ca 2+ ) ions were found going from sample C1 to C2 to C3. These results suggest the presence of gypsum in sample C1, from the blackened area, and of soluble salts (probably sodium and potassium nitrate and chloride) in samples C1 and C2, from areas where the surface has clearly been contaminated by an external agent.

Sample C3 was less affected.

Samples C5 and C6 from the façade were also analyzed by ion chromatography. Sample C5 contained a small amount of chloride, sodium and potassium, whereas nitrate, sulfate, calcium and magnesium were present in large amounts. The powder was probably contaminated by gypsum and soluble salts (presumably calcium and magnesium nitrate and/or organic nitrates). Sample C6 on the other hand, contained sodium, nitrate and sulfate in large amounts. Chloride, potassium and calcium were also present in significant amounts, highlighting that gypsum and soluble salts (particularly nitrates, presumably calcium and/or organic ones) were also present in this powder.

Pyrolysis gas chromatography mass spectrometry (Py-GC-MS)

The pyrolysis performed on the sample with derivatization reaction in situ [START_REF] Orsini | Analytical pyrolysis of proteins in samples from artistic and archaeological objects[END_REF] led to the characterisation of the origin of the organic materials in the thick crust and patina of the façade.

All the samples analyzed presented a variable protein content. The hexadecanenitrile and octadecanenitrile markers of egg are shown in the pyrogram of sample C3, reported in Figure 4b as an example [START_REF] Orsini | Analytical pyrolysis of proteins in samples from artistic and archaeological objects[END_REF].

The results obtained from the samples collected from the ashlar were particularly interesting, which were investigated by LIF techniques in three different areas (C1, C2, and C3). The analyses highlighted the presence of saccharide material in the clear part of the ashlar (C2 and C3) but not in the darkest part (C1). Figure 4b for sample C3 also shows the markers of saccharide material (glycerol, levoglucosane, arabinitol, ribitol, mannitol, inositol). This material is most likely ascribable to the presence of plant cells; in fact, an in situ investigation by laser-induced fluorescence revealed chlorophyll. Natural resins, waxes or siccative oil were absent. Finally, the analysis of sample OG, which was assumed to be an ochre mortar that had been applied by the restorers, suggested that it was in fact a siloxane-based protective for masonry. Figure 4c reports all the siloxane compounds identified, together with the markers of a trace of saccharide material.

Biological investigations

The powders of four samples, collected from the south wall (B1, B2, B3) and the façade (B4), were observed in epifluorescence with an optical microscope. They showed a red fluorescence due to the chlorophyll of photoautotrophs, and a yellow, orange or blue fluorescence ascribable to the presence of other unidentified organic products (Figure 5).

Sample B4 had a lower microbial contamination than the other samples characterised by a greater biodiversity, especially compared to sample B3 (Table 5). The fungal strain Alternaria sp. was present in almost all the samples, except B4. In the nutrient media BG 11 M used for the development of phototrophic microorganisms, the coccoid cyanobacteria of the family of Xenococcaceae, with spherical mucilaginous colonies were dominant in all the samples, except for sample B4 where the presence of phototrophs was insignificant. Light and deep pink pigmented bacteria (cocci and rod-shaped) were observed, above all in samples B1, B2 and B3 in the heterotrophic nutrient media (PCA). Some of these bacterial colonies showed a pink (B1 sample) or yellow (B2 sample) autofluorescence.

Laser cleaning

Laser technology was used on a limited area of the carved surface of the cathedral that had been altered by hard layers of dark films that were still present even after the previous cleaning treatments cited above (Figure 6). The laser cleaning proved to be: i) selective, the laser radiation was absorbed by the superficial dirt without affecting the substrata; ii) adjustable, the beam power, frequency and laser spot diameter could be selected according to the operator's needs; iii) highly effective without the need to pre-consolidate the stone.

In situ chemical and biological treatments

The in situ chemical (AB57 mixtures-poultice and H 2 O 2 ) and biological (laponite biogel) treatments showed similar results in terms of ∆E* (Table 7). The chromatic changes were over the value of 3-which is the threshold to be perceptible to the naked eye-and the dark alteration appeared less visible (Figure 7). In addition, when laponite biogel (laponite + bacteria) was used, the average chromatic change ∆E* values were higher (4.75 + 1.40) compared to the use of the laponite alone (4.1 + 1.28), but with no statistical significance. However, in both cases, the wide range of SD values suggested that the irregular and not homogeneous surface of the stone with large pits, empty spaces, due to loss of materials, exfoliation and decay of the surface material, greatly affected the final results. In the case of the chemical cleaning methods, ∆E* values measured after the treatments with H 2 O 2 or AB57 poultice were higher respect to the laponite but with no statistical significance. Table 7 shows the water absorption test values by contact sponge (mean values +SD in g), before and after the various tests on the stone surface of the left gate close to the column.

The data from the treated areas are significantly different from the reference data (control/blank). In the case of the chemical cleaning, the highest value was obtained with AB57, demonstrating the high cleaning performance over all the treated surface.

In case of the application of laponite gel and laponite biogel bacteria, the water uptake values showed higher SD values probably due to the irregularity of the surface and the presence of lacunes. Finally, the water capillary absorption values after the H 2 O 2 treatment did not appear significantly different from those previously cited.

Monitoring of the cleaning tests Water absorption

The capillary water absorption coefficient on the tufa surface of selected areas was evaluated by the contact sponge method. The levels of ammonium oxalate and ammonium phosphate, which are commonly used for this kind of stone, were determined along with the more recent siloxane emulsion with the medium/high molecular weight synthetic protective agents, Pulvistop, Idroblock and Hydrophase Acqua. The best results were obtained with Hydrophase acqua [water uptake 1.58 mg/(min*cm 2 )], compared to the other protective agents tested [START_REF] Sacchi | A non-destructive multi-analytical approach for the conservation of the Matera Cathedral (Italy), Submitt[END_REF] and to the control area [water uptake 90.05 mg/(min*cm 2 )]. The tests and the monitoring of the surfaces by non-invasive FT-IR spectroscopy and Py-GC-MS confirmed the efficacy of Hydrophase acqua, which showed the best performance in terms of its capacity to penetrate the stone, resistance to UV irradiation, and absence of decay [START_REF] Sacchi | A non-destructive multi-analytical approach for the conservation of the Matera Cathedral (Italy), Submitt[END_REF].

Conclusions

A systematic investigation of the external walls of Matera cathedral by biological, chemical and also non-invasive techniques, identified the dark alteration as a calcium oxalate film.

In terms of the debate regarding the chemical and biological origin of calcium oxalate, our study revealed that the two mechanisms may actually have occurred simultaneously on the stone surface. In fact, the results revealed the presence of biological activity (cyanobacteria, pigmented heterotrophic bacteria, bacilli and cocci type bacteria) as well as residues of proteinaceous material [START_REF] Cariati | Calcium oxalate films on stone surfaces: experimental assessment of the chemical formation[END_REF][START_REF] Rampazzi | Analytical investigation of calcium oxalate films on marble monuments[END_REF][START_REF] Bugini | Unveiling the Use of Creta in Roman Plasters: Analysis of Clay Wall Paintings From Brixia (Italy)[END_REF], which seem to indicate that past conservation treatments had been applied.

As egg was most abundant in the blacker area, the presence of a darker and thick crust on the bottom of the walls, may be due the easier accessibility for such conservation work over time. In sample C4, the sequence of layers suggests past conservation work, when the surface of the mortar degraded to gypsum was covered by a mixture of calcite and organic compounds, which was then transformed into calcium oxalate. Where the mixture (known as beverone) of natural organic compounds applied on the stone surface in form of a thin film was not applied, the sulfates, which were more abundant compared to the calcium oxalate, may have prevented the growth of microorganisms. LIF measurements in fact showed a competitive behaviour between the presence of phototrophic biodeteriogens and the dark alteration of the stones: chlorophyll was present almost exclusively in areas without a dark alteration. The formation of calcium oxalate may therefore be exclusively chemical.

The analytical campaign was effective in managing and guiding the cleaning tests carried out on the stone specimens treated with different cleaning methods. Using AB57, followed by washing with deionized water, was the most effective method. The test with an Nd:YAG laser was also effective, however it was carried out only on the carved elements because the cleaning time proved to be too long due the large size of the cathedral. In addition, Hydrophase acqua was found to be the best protective agent.

Our multi-technique approach was thus useful for the whole conservation work, and in particular the removal of the dark alteration. Figure 8 shows the final appearance of the cathedral, which is one of the symbols of Matera and which was proclaimed the European Capital of Culture for 2019.

Fig. 1 .

 1 Fig. 1. Facade of Matera Cathedral (Matera, Italy) with the dark alteration in the lower part of the structure (up to 6-8 m) and details of the niche with St. Eustache statue (above). The provenance of the samples (bottom).

  chromatic coordinates L* (lightness axis), a* (red/green axis), and b* (yellow-blue axis) of the wall surfaces were determined according to the CIE L*a*b* standard colour system[START_REF] Sacchi | A non-destructive multi-analytical approach for the conservation of the Matera Cathedral (Italy), Submitt[END_REF][START_REF] Ranalli | Biotechnology applied to cultural heritage: Biorestoration of frescoes using viable bacterial cells and enzymes[END_REF].

Fig. 2 .

 2 Fig. 2. LIF analysis of the ashlars of the southern wall: (a) Photographic image of one of the examined ashlars featuring the characteristic dark patina. (b) LIF spectra acquired in various spots of the ashlar, both in the upper and clear part of the ashlar (red lines) and in the lower and dark part of the ashlar (blue lines). (c) Procedure applied to the acquired LIF spectra (red line) in order to separate the fluorescence contribution of chlorophyll (green line) from that of the stone substrate (blue line). (d) Fluorescence of chlorophyll retrieved from the LIF spectra acquired in various spots of the dark area. All the LIF spectra were normalised in intensity to their standard deviation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3 .

 3 Fig. 3. LIF measurements on a plastered area of the cathedral's façade. (a) Photographic image of the examined area: three areas are highlighted in the photo: a plastered area (Area A), an area where the plaster was deliberately detached mechanically (Area B), and an area corresponding to a former gap in the plaster (Area C). (b) LIF spectra measured in the three different areas. (c) LIF spectra measured in the three different areas and normalised in intensity to the relative standarddeviation.

Fig. 4 .

 4 Fig. 4. (a) Transmission FT-IR spectrum of the external layer of sample C4, showing the presence of whewellite (3487, 3435, 3341, 3068, 1623, 1315, 780, and 674 cm -1 ), calcite (1436, 874, and 712 cm -1 ), alkalin nitrates (1384 cm -1 ), traces of organic substances (2870-2980 cm -1 ); (b) pyrogram of sample C3, with the most significant compounds identified: 1: glycerol, 2: levoglucosane, 3: arabinitol, 4: ribitol, 5: hexadecanenitrile, 6: mannitol, 7: palmitic acid, 8: octadecanenitrile, 9: inositol; (c) pyrogram of sample OG with the siloxane compounds marked (#) and in the upper part, the extract ion m/z 217, highlighting the markers of a saccharide source (glycerol, levoglucosane and other minors labelled with '°').

Fig. 5 .

 5 Fig. 5. Autofluorescence examples of the powder samples B1 (above, left), B2 (above, right), B3 (bottom, left) and B4 (bottom, right).

Fig. 6 .

 6 Fig. 6. Laser cleaning of a limited area on the tufa stone surface of the Matera Cathedral.

Fig. 7 .

 7 Fig. 7. In situ biocleaning test performed on the lateral wall of Matera cathedral; laponite biogel bacteria, after 48 h of bioapplication, before (left) and after (right) the soft biogel removal.

Fig. 8 .

 8 Fig. 8. Matera Cathedral before (left) and after (right) the conservation work reported here.

Table 1

 1 Description of the analytical campaign

	2. The experimental
	2.1 Step 1
	2.1.1 Sampling

Table 2

 2 Description of the samples

	Sample label Provenance	Description
	C1	Southern wall dark area, powder
	C2		ochre area, powder
	C3		ochre area, fragment
	C4	Façade	ochre area, fragment of mortar
	C5		dark area, powder
	C6		ochre area, fragment
	C7		ochre area, fragment
	B1	Southern wall powder (close to sample C1)
	B2		powder (close to sample C2)
	B3		powder (close to sample C3)
	B4	Façade	powder (close to sample C4)
	OG		ochre painted area, fragment
	2.1.2 In situ and ex situ analyses

Laser-induced fluorescence (LIF)

Table 3 .

 3 Results of XRD analysis of the samples

		Sample label	Composition
	C1		calcite, gypsum, alkaline nitrate, silicates, traces of quartz
	C2		calcite, quartz, alkaline nitrate, silicates, traces of clay minerals
	C3		calcite, quartz, alkaline nitrate, silicates
		inner layer	calcite, traces of quartz
	C4	intermediate layer calcite, gypsum
		external layer	calcite
	C5		calcite, gypsum, silicates, traces of whewellite and quartz
		bulk	calcite, gypsum, whewellite, traces of quartz
	C6
		external layer	gypsum, whewellite
	C7		calcite, gypsum, whewellite, silicates
	OG		calcite, quartz, whewellite, traces of goethite

Table 4 :

 4 IC sample analyses. All the reported amounts (mmol) refer to 100 mg of sample.

		Li +	Na +	K +	Mg 2+	Ca 2+	Cl -	NO3 -	SO4 2-
	C1		1.43•10 -2	0.79•10 -2		6.03•10 -2	1.17•10 -2	2.35•10 -2	5.33•10 -2
	C2		1.23•10 -2	0.31•10 -2		5.93•10 -2	1.15•10 -2	2.13•10 -2	0.09•10 -2
	C3		0.13•10 -2	0.19•10 -2		2.79•10 -2	0.24•10 -2	0.39•10 -2	0.04•10 -2
	C5		0.31•10 -2	0.09•10 -2	1.76•10 -2	3.79•10 -2	0.55•10 -2	1.92•10 -2	7.24•10 -2
	C6	0.16•10 -2	1.27•10 -2	0.42•10 -2		6.29•10 -2	1.09•10 -2	4.34•10 -2	7.60•10 -2

Table 5 .

 5 The microbial count/presence observed on the three nutrient media; CFU (colony forming units): +/-= scarce, + = present, ++ = frequent, +++ = abundant.

	Nutrient media		Microbial count (CFU/100 mg)
		Sample B1	Sample B2	Sample B3	Sample B4
	PDA (fungi)	1 F	2 F	3 F	1 F
	PCA (fungi and bacteria)	2 F	1 F	5 F	0 F
		41 B	37 B	52 B	1 B
	BG11M (phototrophs)	++	+++	+++	+/-

Table 7 .

 7 Colour changes (∆E*) and capillary water absorptions g + SD results in in situ sandstone artwork of Matera Cathedral, with chemical or bio-

	cleaning agents, measured at 48 h, after application.
	Area	Treatments	∆E*	Water capillary
				absorption
	1-3	Control/Blank	0.70 + 0.10	0.6 + 0.1
	4-5, 8-10	AB57 mixtures and	4.65 + 0.50	3.4 + 0.3
		poultice
	6	Laponite gel	4.1 + 1.28	2.5 + 1.4
	7	Laponite biogel	4.75 + 1.40	2.9 + 1.9
	11	H2O2	4.95 + 0.70	3.0 + 0.6

ACKNOWLEDGMENTS

The authors wish to thank Biagio Lafratta, Soprintendenza Beni Architettonici e Ambientali della Basilicata, Italy; S.E.

Mons. Salvatore Ligorio Archbishop of Potenza-Muro Lucano-Marsiconuovo; Mons. Lionetti, Arcidiocesi di Matera-Irsina; Impresa D'Alessandro Restauri srl; Grazia De Cesare. The authors would like to dedicate this manuscript to the memory of Mario Bressan.

Step 2 3.2.1 Ex situ and in situ cleaning tests Ex situ chemical and biological cleaning

Preliminary laboratory cleaning treatments were applied and tested on a block of "Pietra di Matera" extracted from the cathedral, with UV light at 254 nm for 72 h, or without exposure (at 24 h), as previously described. Table 6 summarizes the ∆E* values, after treatment by chemical and biological agents.

Studies using advanced oxidation processes (AOPs) for the removal, reduction and/or mineralization of organic pollutants have shown good results in soils, water and wastewater treatments. AOPs can work as an alternative or complementary method in traditional treatment cleaning in several contexts. In addition highly reactive free radicals,

reactions, act as the main oxidant, producing simple, relatively harmless inorganic molecules [START_REF] Parsons | Introduction[END_REF]. Under optimum operation conditions, (i.e, adequate contact time), it is possible to mineralize the residual contaminant to CO 2 and H 2 O, which means that AOPs can be defined as "environmentally friendly" [START_REF] Azbar | Application of Advanced Oxidation Processes (AOPs) to Wastewater Treatment. Case Studies: Decolourization of Textile Effluents, Detoxification of Olive Mill Effluent, Treatment of Domestic Wastewater[END_REF]. The colorimetric measurements showed that the most effective chemical treatments were H 2 O 2 + UV, Oxone and TiO 2 + UV, respectively. However, the area treated with oxone was ruined in some points, therefore this oxidant was discarded.

Only UV radiation had no effect compared to the control (blank). In addition, the area treated with TiO 2 appeared to be "coated" with white pigment, which remained the same also after a washing with deionized water. Conversely, when treated with H 2 O 2 , all the dark alteration was bleached, and the cleaning efficiency was much greater when H 2 O 2 was combined with UV exposure (72 h). H 2 O 2 is a green and environmentally friendly reagent because the only by-products are water and molecular oxygen. Furthermore, H 2 O 2 is not an acidic compound and can also be utilized on limestone.

After 48 h from the biological treatment, the water had spontaneously evaporated, and the gel was manually removed with a soft brush. The results from colorimetric measurements showed the efficacy of the biocleaning agents when laponite biogel enriched with P. stutzeri viable cells was applied for 48 h.


Adrien Lalisse

Abeer Al Mohtar

Minh Chau Nguyen

Rémi Carminati

Jérôme Plain

Gilles Tessier
email: gilles.tessier@sorbonne-universite.fr

Quantitative Temperature Measurements in Gold Nanorods Using Digital Holography

Keywords: thermoplasmonics, digital holography, photothermal holography, gold nanostructures, optical modeling, thermal modeling

Temperature characterization and quantification at the nanoscale remain core challenges in applications based on photoinduced heating of nanoparticles. Here, we propose a new approach to obtain quantitative temperature measurements on individual nanoparticles by combining modulated photothermal stimulation and heterodyne digital holography. From full-field reconstructed holograms, temperature is determined with a precision of 0.3 K via a simple approach without requiring any calibration or fitting parameters. As an application, the dependence of temperature on the aspect ratio of gold nanoparticles is investigated. A good agreement with numerical simulation is observed.

INTRODUCTION

The excitation of electronic oscillations, plasmons, in non-perfect ohmic metals is necessarily associated to heat generation through Joule effect. Metal nanostructures can therefore be used to generate strong optical near-field enhancements, but also local heat sources. Historically, sub-diffraction optical confinement kick-started research in plasmonics. More recently, the possibility to create ultra-local heat sources has also found a lot of innovative applications, namely in photothermal cancer therapy [START_REF] Lal | Nanoshell-Enabled Photothermal Cancer Therapy: Impending Clinical Impact[END_REF][START_REF] Cherukuri | Targeted Hyperthermia Using Metal Nanoparticles[END_REF][START_REF] Huang | Plasmonic Photothermal Therapy (PPTT) Using Gold Nanoparticles[END_REF] nanosurgery, [START_REF] Urban | Single-Step Injection of Gold Nanoparticles through Phospholipid Membranes[END_REF] drug delivery, [START_REF] Zograf | All-Optical Nanoscale Heating and Thermometry with Resonant Dielectric Nanoparticles for Controllable Drug Release in Living Cells[END_REF][START_REF] Timko | Remotely Triggerable Drug Delivery Systems[END_REF][START_REF] Han | Drug and Gene Delivery Using Gold Nanoparticles[END_REF] photothermal imaging, [START_REF] Cognet | Photothermal Methods for Single Nonluminescent Nano-Objects[END_REF] photoacoustic imaging, [START_REF] Copland | Bioconjugated Gold Nanoparticles as a Molecular Based Contrast Agent: Implications for Imaging of Deep Tumors Using Optoacoustic Tomography[END_REF] magnetic recording, [START_REF] Challener | Heat-Assisted Magnetic Recording by a near-Field Transducer with Efficient Optical Energy Transfer[END_REF] nanochemistry, [START_REF] Cao | Plasmon-Assisted Local Temperature Control to Pattern Individual Semiconductor Nanowires and Carbon Nanotubes[END_REF] thermonics, [START_REF] Wang | Thermal Memory: A Storage of Phononic Information[END_REF] optofluidics, [START_REF] Donner | Plasmon-Assisted Optofluidics[END_REF] and temperature shaping. [START_REF] Liu | Reconfigurable Temperature Control at the Microscale by Light Shaping[END_REF] However, despite the recent progress of the thermoplasmonics field, quantitative temperature measurements at the nanoscale remain a real challenge. To access this local information, scanning probes and far-field optical techniques have been developed. The near-field techniques, initiated in 1986 by Williams and Wickramasinghe, [START_REF] Williams | Scanning Thermal Profiler[END_REF] rest upon the use of a sharp AFM-type tip associated to a temperature sensor. Although their spatial resolution is unrivalled, nearfield techniques are inherently invasive and difficult to interpret quantitatively due to the complexity of tip-sample thermal interactions. Far-field optical techniques can circumvent these limitations. While infrared-based methods offer limited spatial resolution, fluorescence-based methods are widespread today: fluorescence polarization anisotropy [START_REF] Baffou | Mapping Heat Origin in Plasmonic Structures[END_REF] , fluorescence intensity [START_REF] Carlson | Local Temperature Determination of Optically Excited Nanoparticles and Nanodots[END_REF][START_REF] Ross | Temperature Measurement in Microfluidic Systems Using a Temperature-Dependent Fluorescent Dye[END_REF] , luminescent molecular thermometers [START_REF] Brites | Thermometry at the Nanoscale[END_REF] or fluorescence spectroscopy [START_REF] Vetrone | Temperature Sensing Using Fluorescent Nanothermometers[END_REF][START_REF] Jaque | Luminescence Nanothermometry[END_REF] can provide fast, accurate and spatially resolved temperature measurements. However, these measurements can suffer from photo-or thermo-bleaching and can be sensitive to the local fluorophore environment (pH, ions and viscosity), their main drawback is that they are intrinsically invasive: fluorophores must be incorporated to the sample, which is straightforward in biological applications, but much less so in plasmonics. Recently, Raman spectroscopy has emerged as a major tool for quantitative temperature measurements in silicon [START_REF] Zograf | Resonant Nonplasmonic Nanoparticles for Efficient Temperature-Feedback Optical Heating[END_REF] , or gold [START_REF] Carattino | Gold Nanoparticles as Absolute Nanothermometers[END_REF] that was first demonstrated by Hugall et al. [START_REF] Hugall | Demonstrating Photoluminescence from Au Is Electronic Inelastic Light Scattering of a Plasmonic Metal: The Origin of SERS Backgrounds[END_REF] for distributed structures and Xie et al. [START_REF] Xie | Thermometry of Plasmonic Nanostructures by Anti-Stokes Electronic Raman Scattering[END_REF] for individual nanostructures. Techniques to enhance the signal, thus reduce the read-out time, have been proposed through the introduction of plasmonic [START_REF] Pozzi | Evaluating Single-Molecule Stokes and Anti-Stokes SERS for Nanoscale Thermometry[END_REF] or nonplasmonic nanoparticles [START_REF] Zograf | Resonant Nonplasmonic Nanoparticles for Efficient Temperature-Feedback Optical Heating[END_REF] , or a combination of both [START_REF] Milichko | Metal-Dielectric Nanocavity for Real-Time Tracing Molecular Events with Temperature Feedback[END_REF] . However, Raman spectroscopy still requires careful calibrations and relatively long integration times to measure the temperature of a single point. While peerless as nanoscale-thermometers, these techniques are not adapted to imaging.

Methods relying on the temperature dependence of the refractive index with temperature are entirely non invasive, fast, and sensitive enough to study single objects [START_REF] Boyer | Photothermal Imaging of Nanometer-Sized Metal Particles Among Scatterers[END_REF] . In some cases, they have the advantage of being able to probe the temperature of the particle rather than that of its environment [START_REF] Baffou | Thermal Imaging of Nanostructures by Quantitative Optical Phase Analysis[END_REF] , and can be used to localize single nanoparticles or molecules [START_REF] Cognet | Photothermal Methods for Single Nonluminescent Nano-Objects[END_REF][START_REF] Gaiduk | Room-Temperature Detection of a Single Molecule's Absorption by Photothermal Contrast[END_REF] . However, using them to deduce the temperature quantitatively remains a challenge. n(T) can be considered linear over a few tens of Kelvins for most metals or dielectrics [START_REF] Berto | Quantitative Absorption Spectroscopy of Nano-Objects[END_REF] , for higher temperature an iterative numerical approach can be followed, as proposed by Baffou et al. [START_REF] Baffou | Thermal Imaging of Nanostructures by Quantitative Optical Phase Analysis[END_REF] . In simple structures, e.g. bulk materials or flat thin films, measuring the refractive index, usually through related properties such as reflectivity [START_REF] Rosencwaig | Theory of the Photoacoustic Effect with Solids[END_REF][START_REF] Tessier | Photothermal Techniques[END_REF] or optical path difference [START_REF] Baffou | Thermal Imaging of Nanostructures by Quantitative Optical Phase Analysis[END_REF] is therefore a straightforward way to measure temperature. In the present work, we propose the use of heterodyne digital holography to image thermally-induced scattering modulations. Combined with a simple analysis, it provides fast (tens of seconds), sensitive (0.3 K) calibration-free (no adjustable parameter), noninvasive (no contrast agent) full field imaging (1024 2 pixels) of high-frequency (up to the MHz) modulated temperature components on individual nanostructures. A specially designed and fabricated array of nanoellipsoids with varying aspect ratios allowed us to validate the method by comparison with numerical simulations, based either on a finite difference time domain electromagnetic method coupled to thermal simulations (Lumerical), or on the boundary element method. This extensive analysis validates heterodyne digital holography as a reliable single nano-object thermal imaging technique.

EXPERIMENTAL MATERIALS

Sample design and fabrication

Plasmonic nanostructures offer an elegant way to control absorption and temperature at small scales. Whenever local optical enhancement is the main target, in e.g. plasmonic detection, heating is usually detrimental to the preservation of the plasmonic structure and/or the analyte, and should be minimized. Conversely, most thermoplasmonic applications demand efficient heat generation. Recently, we have proposed [START_REF] Lalisse | Quantifying the Efficiency of Plasmonic Materials for Near-Field Enhancement and Photothermal Conversion[END_REF] criteria to assess the ability of a given plasmonic system to enhance local or optical thermal phenomena. Resonances can indeed be tuned by changing the size, morphology, and composition of metallic nanostructures. A given volume of a chosen metal can therefore be shaped in order to maximize or minimize heat generation, depending on the application. In this study, we chose a simple visible-resonating plasmonic system as reference: a flat gold cylinder, with a radius of 50 nm and a thickness of 20 nm, i.e. a volume of 1.6×10 5 nm 3 , or approximately 9.2×10 6 gold atoms. The shape of the disc was then modified in order to investigate the temperature reached by various single structures of similar volume. Exploring all possible forms is of course impossible, and we restricted the present study to a class of shapes which can be fabricated relatively easily: cylinders with elliptic bases. As illustrated in Figure 1 (a) this family of objects is described by its aspect ratio AR=a/b where a is the long axis and b the short axis of the ellipse. We fabricated several identical series of objects ranging from discs (AR=1) to rod-like elongated ellipses (AR=9) in steps of AR=0.5. The nanostructures were fabricated on a 1 mm thick float glass substrate using e-beam lithography (EBL). The substrate was sonicated several times in deionized water and detergent for 15 minutes. Remaining particles were removed from the surface with deionized water (5 minutes), acetone (5 minutes), ethanol (10 minutes) and isopropanol (IPA) (15 minutes). A polymethylmethacrylate (PMMA) electron-sensitive resist was then spin-coated on the cleaned glass substrate and annealed for 15 minutes at 170 °C. A conductive resist layer (Espacer 300Z) was also spin-coated on top of the PMMA to prevent charging effects during the EBL exposure. The areas of interest were then exposed in an EBL machine (RAITH e-line) under a 15 kV voltage. After exposure, the conductive resist layer was removed in deionized water, and the exposed PMMA was revealed in a cold mixture of methyl isobutyl ketone (MIBK) and isopropanol (IPA). An evaporator was then used to deposit 3 nm of titanium dioxide as an adhesive layer, and 20 nm gold. To dissolve the remaining PMMA and lift-off metal from the unexposed regions, the sample was put in acetone for 4 hours. Finally, the sample was rinsed with IPA and exposed to an O 2 plasma before Scanning Electron Microscopy (SEM), and holographic characterizations. As can be seen in Figure 1 (b), the fabricated objects are close to the targeted geometry. However, SEM revealed that the narrowest and most elongated structures (AR>5) were too thin to allow ideal fabrication, and exhibited either discontinuities (these were discarded) or were slightly wider than expected (b above the targeted value) due to the limited resolution of the EBL system. Dark-field optical spectroscopy on individual objects [START_REF] Brasiliense | Electrochemistry of Single Nanodomains Revealed by Three-Dimensional Holographic Microscopy[END_REF] revealed a good agreement (± 10 nm on the resonance wavelengths) between the calculated and measured spectra, confirming a good precision in the fabrication, particularly regarding the length a of the structures. Scanning Electron Microscopy carried out on selected structures (see Figure 1) indicated a fabrication precision of ±5 nm on the length a of the ellipses.

Spectroscopic characterization

Nanostructures were illuminated using unpolarized tungsten-halogen light and an air Dark-Field (DF) condenser, while the scattered light was collected by a lower numerical aperture microscope objective (60X, NA=0.7), and sent either to a fast camera (Andor Zyla CMOS) for direct DF imaging, or towards a line-imaging spectrometer (400-800 nm, Inspector V8E Specim) coupled to a second camera (Andor Ixon CCD). The measurement arm was equipped with an analyzer. After calibration with a Hg lamp, a reference spectrum I ref accounting for source, transmission and detection efficiencies was acquired. Relevant nano-objects were identified with the imaging camera and positioned at the same location to ensure identical illumination conditions. Scattering I scat and background noise I bgd were measured simultaneously, at the nanostructure position and in a neighboring dark region, to derive normalized spectra I()= (I scat -I bgd )/I ref for each individual nano-object [START_REF] Brasiliense | Electrochemical Transformation of Individual Nanoparticles Revealed by Coupling Microscopy and Spectroscopy[END_REF] . Scattering spectra were modelled using MNPBEM [START_REF] García De Abajo | Retarded Field Calculation of Electron Energy Loss in Inhomogeneous Dielectrics[END_REF] for a polarization along the short axis of the structures. Here, plasmonic resonances along the long axis were not modelled since they are essentially in the infrared, beyond the sensitivity range of our system. Figure 1 (c) shows the simulated spectra for aspect ratios AR varying from 1 to 4 which exhibit a Lorentzian shape characteristic of plasmon resonance, and a blue shift of the resonance at larger AR, as the short axis dimension decreases With the exception of the AR = 1.5 structure, the simulated and experimental spectra (Figure 1. (c) and (d) respectively) follow a similar blueshift and decrease of the scattering with AR. Experimental spectra, however, are significantly broader, which is consistent with the roughness and irregularities observed by SEM (Figure 1 (b), typically of the order of 5%. As illustrated by the AR=1.5 structure in the spectra of Figure 1 (d), some nanoobjects inevitably deviate from the intended shape. A systematic SEM control of the nanoparticles being impractical, we chose to mitigate such deviations by averaging multiple objects of identical intended geometry. As discussed below, these morphological variations, although limited, are the main source of variations in the observed temperatures, and far outweigh the measurement noise.

Figure 1. (a) Schematic of the sample geometry. (b) Typical SEM images of four nanorods of different aspect ratios (AR). The radius of the AR=1 structure is 50 nm. (c) Scattering spectra simulated using a Boundary Elements Method for AR 1 to 4, for a polarization perpendicular to the long axis of the structures, and (d) scattering spectra measured for a polarization perpendicular to the long axis of the structures on individual structures using white, dark field illumination. N.B.: spectra and SEM images were not measured on the same samples.

EXPERIMENTAL METHODS

Description of physical mechanisms

In order to probe temperature, we have chosen to rely on far-field measurements of the light scattered by the nanostructures. The modulated optical power ( sent to the sample induces a local temperature modulation ( ), which changes the refractive index of the object and its surroundings , and in turn modulates the optical scattering , a process which can be summarized as:

.

(1)

However, thermo-optical coefficients are typically on the order of 10 -4 to 10 -5 K -1 , and optical scattering by single nano-objects is very weak, which leaves us with the challenge of measuring small changes in a small signal. This can only be achieved in a modulated regime, taking advantage of lock-in type techniques. In order to avoid high order effects, we have chosen to dissociate the thermal excitation and the probe laser: The structures are heated using a modulated, high power laser, and a low power continuous probe laser is then used to measure the resulting variations in the scattered field. The modulated excitation induces both continuous and modulated temperature rises in the sample.

The DC component of temperature decays as around the heated particle, and the superimposed AC component has a similar behavior at low modulation frequencies. At higher frequencies however, in the "thermal waves" approach [START_REF] Rosencwaig | Theory of the Photoacoustic Effect with Solids[END_REF] , the modulated compo-nent of the temperature spreads over shorter lateral distances described by the thermal diffusion length , where D is the thermal diffusivity. In e.g. glass ( ≈8×10 -7 m 2 .s -1 ), is therefore confined within tens of micrometers around the structure above modulation frequencies of the order of the kHz. The modulated part of the scattered intensity, , therefore carries information on the local temperature of the nanostructure. At 1kHz, the constant component of temperature is typically 1 order of magnitude lower than the modulated part of the temperature distribution for the structures studied in this work. Using the formalism proposed by Berto et al. [START_REF] Berto | Time-Harmonic Optical Heating of Plasmonic Nanoparticles[END_REF] , we also verified that the contribution of neighboring nanostructures to the modulated temperature increase was low enough to be neglected, and could be compensated if necessary. Nevertheless, the µm-range distance over which light interacts with this hot region can be much larger than the nm-range of the nano-objects, which ensures good signal-to-noise ratios [START_REF] Berciaud | Photothermal Heterodyne Imaging of Individual Metallic Nanoparticles: Theory versus Experiment[END_REF] . Here, we propose to use a spatially resolved measurement of in order to retrieve the local temperature variations . Holography is able to image both the amplitude and the phase, and thus gives access to optical reconstructions of the scattered light modulation in the plane of the scattering structure, or above it. However, most cameras are limited to frame rates of a few tens of Hertz, particularly in low-light conditions, and cannot directly measure kHz modulations. For this reason, we use a heterodyning scheme in which the probe illumination is modulated in order to create a low-frequency beating with the relevant modulation of , as detailed below (Figure 2).

Experimental setup

The experimental holographic setup is based on a Mach-Zehnder interferometer, schematically depicted in Figure 2 (c). The probe light source is a monomode laser diode at a wavelength , (maximum power P max = 130 mW) separated by a polarizing beam splitter (PBS) into reference and object arms polarized along the long axis of the nanostructures. The sample is illuminated through a prism in a total internal reflection configuration so that the illumination beam is reflected away and only the scattered light is collected by a microscope objective (x100, NA=0.8) which provides diffraction-limited spatial resolution at 785 nm, typically over a 30x30 µm field of view. A beam splitter combines the scattered and reference light. The component of the scattered field which is parallel to the polarization of the reference interferes with the reference on the camera (SCMOS, Andor Zyla 5.5). A heterodyne digital holography configuration [START_REF] Gross | Digital Holography with Ultimate Sensitivity[END_REF][START_REF] Lalisse | Optimisation Thermique de Nanostructures Plasmoniques: Conception, Modélisation et Caractérisation[END_REF][START_REF] Suck | Digital Heterodyne Holography for Plasmonic Nanostructures[END_REF] allows the implementation of a spatial [START_REF] Cuche | Spatial Filtering for Zero-Order and Twin-Image Elimination in Digital off-Axis Holography[END_REF] and temporal filtering of the interference pattern [START_REF] Suck | Imaging the Three-Dimensional Scattering Pattern of Plasmonic Nanodisk Chains by Digital Heterodyne Holography[END_REF] . For this purpose, the reference and object arms are phase-modulated by Acousto-Optical Modulators (AOM1 and AOM2, respectively), at frequencies and , both in the 80 MHz-range. If the sample is not heated, the interference pattern on the camera bears a modulation at the beating frequency . Choosing closely spaced frequencies, e.g. = 80 MHz and = 80 MHz -3 Hz therefore induces a hologram beating at low frequency, = 3 Hz. Using a 4 × = 12 Hz camera frame rate and 4-bucket acquisition, the slowly beating hologram can be demodulated and reconstructed in order to obtain images of the scattered optical field with an improved signal to noise ratio since low-frequency noises are filtered out. Heterodyne holography can provide shot-noise limited measurements [START_REF] Gross | Digital Holography with Ultimate Sensitivity[END_REF] . In particular, when the imaging plane is at infinity, holography allows to reconstruct images while the information is spread over the whole imaging sensor. In the case of sparse scatterers, this allows to increase the illumination without saturating the few camera pixels imaging the object. When operating close to the saturation level of a pixel with a full-well capacity N= 3×10 4 e-, 4-buckets demodulated images yield a SNR which varies as 350 for each pixel and can be improved by spatial (binning) and/or temporal averaging. splitter; M: mirror; OD: optical density; HWP: half-wave plate. The 532 nm modulated heating beam induces refractive index changes in the nanostructure, and therefore a modulation of the scattering of the probe beam (785 nm), from which temperature modulations are deduced. The red illumination and reference beams, as well as the green heating beam are polarized perpendicularly to the plane of the figure, and parallel to the long axis of the nanostructures.

Heterodyning also allows the measurement of high frequency components induced by heating with a powerful (P max = 2 W) singlemode laser at = 532 nm, sinusoidally-modulated in amplitude by AOM3 at = 1 kHz. Accordingly, the frequency detuning of the reference and object arms has to be set to = 1003 Hz in order to bring the component which is originally modulated at = 1 kHz down to a frequency of = 3 Hz, which can be properly demodulated by the slow camera. Using this heterodyning technique, the components of the scattered optical field which are modulated at the heating frequency = 1 kHz are retrieved. The same setup is therefore able to selectively retrieve holograms and images of the DC component of the scattered light I DCscat (for = 3 Hz) or of its AC component I ACscat , which is thermally modulated ( = 1003 Hz) by simply modifying the detuning .

Signal processing and experimental temperature determination

In the system considered in Figure 2 (b), the scattered field received at the detector can be written as:

(2)

Here is the dielectric function of the nanorod, is the dielectric function of the medium in which the nanorod is immersed, and with the optical frequency of the probe beam. The Green's function describes the optical response of the environment of the nanorod (including the substrate). The integral involves the total field and is extended over the volume V of the nanorod. The heating beam slightly changes through its dependence on temperature, which in turns changes the amplitude of the scattered field. To first order, the change in the dielectric function can be written (3) Where is the room temperature and . Likewise, the change in the field inside the nanorod is:

(4)

Due to the intensity modulation of the heating beam, is modulated at a frequency . The heterodyne holographic signal is proportional to the interference term where is the reference field used to build the interferogram, and is the modulated part of the scattered field at the detector (here * stands for the complex conjugate). The latter is obtained by inserting Eqs. ( 3) and (4) into Eq. ( 2):

] ( ′, 0 ′ 3 ′ (5)
Recovering from the signal requires the solution to a complex inverse problem. Here we follow a different approach, and use the simplest relevant model providing a direct estimate of from the data, in the particular case of small metallic nanorods. For weakly scattering objects, the second term in the right-hand side of Eq. ( 5) can be neglected. To see this, we start by considering the Born approximation, that corresponds to single scattering inside the nanorod. This approximation amounts to replacing the field in the integrals by the external field in absence of the nanorod. The second term in the right-hand side of Eq. ( 5) vanishes in this simple case since is not affected by external heating. The gold nanorods considered in this study cannot be treated in the Born approximation. Nevertheless, it can be seen that for optically small nanorods (with respect to the illumination wavelength), we can still expect the first integral in Eq. ( 5) to give the leading contribution to the signal. Consider the particular case of a sphere. In the quasi-static approximation, the internal field is homogeneous and well approximated by (6) where is a point inside the sphere, so that (7) Using the expressions above for the fields in the integrals in Eq. (5) shows that the second integral can be neglected provided that the condition is fulfilled. Note that in practice, even a weak inequality in this condition is suffi-cient to provide a fair estimate, as justified by the good agreement between the measurements and full numerical simulations shown below. We conclude that for small nanorods satisfying this condition, we can write (8) Inside a thermally small object, the temperature field is considered to be homogeneous. This occurs when the conduction time across the object (L being the size of the object and is the thermal diffusivity) is smaller than the characteristic heating time , or in other words when , which is satisfied in our experiment. In this case become independent on position. One can define the ratio of the modulated (AC) and the unmodulated (DC) signals as (9) From two measurements with and without modulated heating, we can therefore obtain the change in temperature inside the object as (10) 4 RESULTS

Numerical methods

In order to model and validate the experimental results, two types of numerical optical simulations were used and compared: the Boundary Element Model (BEM) method [START_REF] García De Abajo | Retarded Field Calculation of Electron Energy Loss in Inhomogeneous Dielectrics[END_REF] , using the freely available Matlab-based package MNPBEM, and the Finite Difference Time Domain (FDTD) method, using the commercially available solver Lumerical solutions Inc. 45 For each aspect ratio, the structures under study are considered as thermally and optically independent from neighboring objects, with substrate and structure materials, dimensions and illumination matching the performed experiments. For FDTD, the area surrounding the structures is discretized by a non-uniform conformal variant mesh with a meshing size of 1 nm. The incident laser beams are modeled as plane waves. The simulation area was terminated by Perfectly Matched Layers (PMLs) to suppress spurious reflection from boundaries.

Scattering cross-section

The measurement of the I DCscat hologram without heating the sample reveals the scattering efficiency of the structures at the wavelength = 785 nm. In order to obtain quantitative cross-section values, we used the average of measurements carried out on identical structures as reference, and used the dispersion of these results to evaluate the uncertainty. The 3D scattering pattern of the same structure was then modelled using FDTD method, taking into account the objective´s collection angle for a numerical aperture of 0.8. Taking this calculated value and the average measurement for references, measurements for all aspect ratios were normalized to obtain quantitative scattering cross-section values. Subsequent measurements were carried out on individual sets of nanostructures using the same illumination power, collection N.A., and therefore the same normalization factor. Possible variations in collection efficiency due to changes in the shape of the scattering pattern for AR≠1 were not considered, and therefore contribute to possible discrepancies between the model and the experiment. Figure 3 shows the experimental results obtained for the structures of varying aspect ratio described in Figure 1 along with the corresponding numerical simulations. As expected, the scattering crosssections display a decrease with the aspect ratio, as the resonance shifts away from the probe wavelength (785 nm) into the infrared. Figure 3. Experimental (dotted line) and modelled (continuous line) scattering cross sections as a function of nanostructure aspect ratio for a polarization along the long axis of the nanostructures. R eq denotes the targeted radius of the AR= 1 disc. Experimental measurements are given for R eq = 45 and 50 nm to illustrate deviations induced by the 5 nm fabrication precision.

Temperature measurements

As described above, the modulated component of the temperature increase can be evaluated using heterodyne measurements of I ACscat to derive the values of the dimensionless ratio defined in Eq. ( 9), a process which does not require the use of the model for calibration.

Figure 4. Experimental temperature variations along with the corresponding error, compared to FDTD and BEM -based modelling, for an irradiance = 2.36×10 8 W/m 2 . Note that no parameter is adjusted in the simulations. Experimental error bars correspond to the standard deviation over measurements carried out on 4 independent samples.



The value of for gold at the probe wavelength (785 nm) was deduced from experimental data 46 , = 1.2×10 -4 K -1 at = 785 nm.

Note that for borosilicate glasses, is relatively constant with wavelength and is 2 orders of magnitude lower in the visible at 3×10 -6 K -1 , and can therefore be neglected [START_REF] Jewell | Thermooptic Coefficients of Some Standard Reference Material Glasses[END_REF] . Varying irradiance conditions caused by the position of nanostructures with respect to the Gaussian heating beam were taken into account and corrected to allow comparisons at equivalent irradiance = 2.36×10 8 W/m 2 , for a polarization of the heating beam parallel to the long axis of the nanorods.Figure 4 shows the experimental temperature measurements obtained in gold nanorods, determined using Eq. ( 10), as a function of the structure aspect ratio. For each of the 17 aspect ratios and for 2 equivalent radii (45 and 50 nm), 4 different structures were fabricated and characterized. A standard deviation analysis on these 17×2×4= 136 measurements yields uncertainties of 20% as plotted in Figures 3 and4. The measurements shown here were obtained using temporal averaging over n=100 series of 4 images, i.e. = 33 s at = 12Hz, which improves SNR by a factor = 10 as compared to a single 4-image series ( =333 ms). Experimentally, we measured the repeatability of the measurement on 9 different structures. We obtained a noise which varies in good agreement with a (or ) law, with (0.333s) = 1.6 K, (6.7s) =0.6, (16.7s) = 0.35 K, and (33s) =0.3 K, corresponding to n = 1, 20, 50 and 100 respectively. This clearly indicates that the temperature fluctuations observed in Figure 4 are not caused by measurement uncertainty, but merely by deviations from the targeted geometry during the fabrication, which is consistent with both SEM and spectroscopic measurements. To compare the performances of optical methods potentially limited by photon noise (table 1 in reference [START_REF] Van De Broek | Temperature Determination of Resonantly Excited Plasmonic Branched Gold Nanoparticles by X-Ray Absorption Spectroscopy[END_REF] ), we therefore propose to consider as a criterion to be minimized. With , our method performs similarly to liquid crystals and TRSTM ( ), which are physically and thermally invasive methods, and notably better than Raman spectroscopy (100 K.s 1/2 ). Finally, keeping in mind that holography measures typ.

=1024 2 pixels at once, a criterion arguably gives a clear advantage to such full-field methods in terms of temperature sensitivity per point and unit of time. In order to quantify accurately temperatures at the nanoscale and verify the experimental approach, the results are compared to two numerical simulations. In these two simulations, radiative transfer and ballistic transport are considered to be negligible compared to heat diffusion. In the first approach, we calculated the absorption cross section, using the FDTD method. This cross section was then used to calculate the heat source term , i.e. the absorbed power with = 2.36×10 8 W/m 2 the irradiance used in the experiments. The 3D temperature map was then calculated using the Heat Transport solver (HEAT) from Lumerical, 45 which solves the heat diffusion equation using a Finite Elements Method and a meshing of the system. The temperature at the bottom of the substrate was set at 300 K, and the effective boundary conditions between solids and air were set as convective. By varying the heating frequency down to values in the Hz range, we verified that steady-state heating and high frequency modulation of the heating (1 kHz) yield very similar temperatures, which is expected since the involved spatial scales (nm-range) are well below the thermal diffusion lengths, even at = 1 kHz. The inset in Figure (3) shows the 3D temperature map obtained for two nanostructures with aspect ratios 1 and 9. It confirms the validity of the uniform temperature approximation within the particle, which was used to determine the temperature experimentally and in the following MNPBEM numerical approach. This second approach is based on calculations performed with the freely available MNPBEM14 software package, [START_REF] García De Abajo | Retarded Field Calculation of Electron Energy Loss in Inhomogeneous Dielectrics[END_REF] in which the absorption cross section is determined by a boundary element method, then under the uniform temperature approximation, the temperature elevation in the structures can be calculated using (11) With , the effective thermal conductivity of the medium ( and being, respectively, the thermal conductivities of the substrate and air), and the Laplace radius of the numerically meshed nanostructure. [START_REF] Baffou | Thermoplasmonics Modeling: A Green's Function Approach[END_REF] Here, one can define the so-called Laplace matrix as for , and for , where, is the position vector of the mesh element and is the grid constant.

Then is obtained by summing the elements of the inverted matrix:

(12)

CONCLUSION

The comparison of experimental and calculated temperatures is generally satisfactory, considering the fact that individual nanostructures are investigated. However, some discrepancies appear for aspect ratios of 6 and above. Despite the high precision of electron beam lithography, some deviations from the required dimensions are unavoidable, and were quantified using SEM images at typically 5 nm. This is most significant for the thinnest and elongated structures in which metal continuity can be interrupted, and where the targeted elliptic shape is distorted. Another source of error is the approximation made in Eq. ( 8), where the second integral term in Eq. ( 5) induced by the modulation of the internal field is neglected. Whenever the condition 0+22] is fulfilled, however, this approximation provides a simple way to derive quantitative temperatures which are in remarkably good agreement with experiments considering that no adjustable parameter is used.

Using a far-field optical microscopy technique based on heterodyne detection of the modulated scattered electric field, we have proposed a methodology to determine quantitatively the temperature variations of single nanostructures under modulated optical excitation. Since the method is holographic, it also gives access to the optical path difference, although this information was not exploited here. In the DC regime, phase information has already proven its ability to yield sensitive temperature measurements [START_REF] Baffou | Thermal Imaging of Nanostructures by Quantitative Optical Phase Analysis[END_REF] : its exploitation is a promising way to further improve sensitivity in modulated regimes.

Figure 2 .

 2 Figure 2. (a) Reconstructed hologram of an array of nanostructures, with increasing aspect ratio (left to right) and size (top to bottom), (b) Principle of heterodyne photothermal holography. (c) Schematic of the experimental setup. AOM: acousto-optic modulator; BS: beam

Funding sources

French Agence Nationale de la Recherche (ANR NATO, Grant ANR-13-BS10-0013-03).

ACKNOWLEDGMENT

The fabrication and electron microscopy were performed at nano'mat -UTT, Serguei Kochtcheev and Jérémie Béal are acknowledged for their assistance. We thank Guillaume Baffou for his inputs at early stages of this work and for his critical reading of the manuscript.