Marcel Heinz

Ralf Lämmel

Mathieu Acher

Discovering Indicators for Classifying Wikipedia Articles in a Domain: A Case Study on Software Languages

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INTRODUCTION

Wikipedia is a very large-scale, continuous community effort to collect and organize (informal) knowledge in almost all domains. In general, the quality of the articles and the mere principles of collection and organization challenge automated procedures in the quest of extracting and structuring Wikipedia knowledge [START_REF] Stvilia | Information quality work organization in wikipedia[END_REF][START_REF] Do | Exploiting the wikipedia structure in local and global classification of taxonomic relations[END_REF][START_REF] Wu | Automatically refining the wikipedia infobox ontology[END_REF][START_REF] Rebele | YAGO: A multilingual knowledge base from wikipedia, wordnet, and geonames[END_REF]. Recovering articles describing topics in a certain domain is a reoccurring problem [START_REF] Wang | Wikipedia2Onto. Building Concept Ontology Automatically, Experimenting with Web Image Retrieval[END_REF][START_REF] Dong | Learning to discover subsumptions between software engineering concepts in wikipedia[END_REF][START_REF] Chen | Building a domain knowledge base from wikipedia: a semi-supervised approach[END_REF]. This paper presents a supervised machine learning approach for recovering Wikipedia articles relevant to an ontological class.

We conduct a case study on software languages defined as a set of digital artifacts, for which syntax, type system, semantics, and pragmatics can be (in)formally defined, documented, and implemented [START_REF] Combemale | SLEBOK: The Software Language Engineering Body of Knowledge (DS 17342)[END_REF]. For the purpose of this research, we assume that Wikipedia's notion of "computer language" -a notion that is used all across computer science and beyond -is essentially equivalent to the notion of "software language"; see also [START_REF] Favre | Guest Editors' Introduction to the Special Section on Software Language Engineering[END_REF] for a discussion.

An important barrier is that, in a large number of cases, a single Wikipedia article covers multiple subjects. Looking at the very first sentences of the article about MATLAB (see Figure 1), several topics are mixed together (e.g., user interfaces, numerical computing, and software languages). The category graph is impacted as well (e.g., linear algebra and array-programming languages are both mentioned). There are also two infoboxes-one about MATLAB the language, the other about MATLAB the software.

Moreover, Wikipedia's category graph cannot be directly used for classification within the domain [START_REF] Mirylenka | Bootstrapping domain ontologies from wikipedia: A uniform approach[END_REF]; categories serve purposes other than classification, for example, collecting articles related to a common topic; see https://en.wikipedia.org/wiki/Category: Java_(programming_language). Wikipedia's guidelines partly explain why multiple topics appear in an article: there should not be a new article, when the description benefits from explaining two or more subjects in the context of an existing article 1 . Such rules and editing practices, though reasonable, complicate classification, i.e., automated identification of articles (instances) and underlying categories (classifiers). For some domains, like animals [START_REF] Wang | Wikipedia2Onto. Building Concept Ontology Automatically, Experimenting with Web Image Retrieval[END_REF], categories that represent scientific classification are consistently used and provide a decision ground for still identifying relevant articles. In the domain of software languages such crucial features do not exist for articles or are used inconsistently. Alas, the recovery of relevant articles becomes looking for needles in a hay stack.

Significance of the multiple-subject problem. We initially explored articles that link to (subcategories of) the category 'Computer languages' and search for the top ten frequent nouns ('NN' tag recovered with part-of-speech tagger). We noticed that nouns from the music and astronomy domain are dominant. Fig. 2 reports the number of articles for the top ten most frequent nouns in the category tree of 'Computer languages' tree that we cut off at a depth of seven. Within the category tree, we observed subjects that belong to other domains, such as 'songs', 'stars', 'al- Research question and contributions. Overall, our case study is an instance of the problem of extracting domain-specific knowledge from Wikipedia [START_REF] Chen | Building a domain knowledge base from wikipedia: a semi-supervised approach[END_REF][START_REF] Mirylenka | Bootstrapping domain ontologies from wikipedia: A uniform approach[END_REF][START_REF] Suárez-Figueroa | The neon methodology framework: A scenariobased methodology for ontology development[END_REF]: How can we classify Wikipedia articles by their relevance to a given domain when relevant articles are rare and multiple main topics are covered by articles? Our objective is essentially to detect members (here: Wikipedia articles) of a certain class (here: software languages).

We develop a seed-based learning methodology for identifying articles relevant to a domain-specific class while leveraging the available limited ground truth (based on Github and TIOBE) and identifying indicators by inspecting a learnt decision tree that include URL patterns, summary text, infoboxes, list articles and category graph. We then present a case study which, in itself, results in the most comprehensive corpus on software languages available today. The results are evaluated by domain experts through a survey.

SEED-BASED LEARNING

We now detail how we exploit fragments of ground truths and instrument a decision tree classifier. The datasets including plotted decision trees are available online2 . Figure 3 provides an overview of our approach. i.) For training, we recover articles describing elements that appear in trustworthy external resources. ii.) Based on such seed, we define the scope in which we want to isolate relevant articles and label 4000 randomly sampled articles from this scope for additional training data. iii.) From the training data, we build a feature matrix with categorical values that state whether a structural feature, such as the programming language infobox template, is present. iv.) We configure a binary classifier that decides whether an article is relevant.

Seed Matching

As first step, we have identified and reused two trustworthy external data sets: both act as seeds for recognizing relevant articles. GitHub presents statistics on which languages are used for any repository. The complete list of languages that are recognized can be extracted 3 . The TIOBE index presents statistics on how often software languages (mainly programming languages) are mentioned on the web. The list includes variations of names 4 .

Identifying each language on Wikipedia from the lists is challenging. We encountered favorable and problematic cases. To our favor, most names can be matched with article titles by leaving out annotations (e.g. '(programming language)') or by using Wikipedia redirects. However, manual reviewing remains crucial to make sure that languages are matched correctly, e.g., 'Red' is matched with 'Red (programming language)' and not with the color. In some cases, we succeeded by varying the writing style for the name, e.g., by unfolding acronyms, such as EBNF. In less favorable cases, we had to rely on Wikipedia's search engine to detect mentions in the text of existing articles. Only if the existing article describes the seed language as its main topic in the summary, we take it as the recalled article. To ensure availability of the necessary features for classification, we exclude stub articles from further analysis.

The resulting merged seed contains 327 recalled seed languages where 110 overlap. 158 seed languages are only casually mentioned in articles and 99 seed languages remain unrecognized.

Category Scope Exploration

We narrow the scope of our analysis to articles that are reachable by links to (subcategories of) chosen upper categories. Technically, we have manually identified common upper categories so that the seed is linked to them. At first, we hypothesized that all relevant languages link to 'Computer languages', but we found 'Augmented BackusNaur form' as a seed member that links to 'Formal languages', where 'Computer languages' is a subcategory. Moreover, for seed articles such as 'CSV', we noticed the upper category 'Computer file formats' that is disconnected from 'Formal languages'. Such formats are categorized in a different manner, but they do conform to the definition of software language. Our experiments show that all seed articles are linked to (subcategories of) two disconnected roots in the category tree at a maximum depth of eight.

Feature Extraction

Next, we describe which features we extract from the content types discussed in Section 1. To counter lack of decisive features and to reduce dimensionality, we exclude stub articles and only consider features that appear in at least ten articles. For now, we also exclude articles at a distance to the upper categories that is higher than 8. As a resulting dataset, the extraction returns an article-feature matrix with 104, 186 × 46, 173 entries with label '1' for present and '0' otherwise. Then, the set of 4000 randomly sampled articles and the seed form the training set. To avoid memory issues and enable loading the whole article-feature matrix, we use sparse matrices.

In the evaluation, we discuss that irrelevant articles in the training set are actually members of many other domain classes and have many different features. Hence, the resulting decision tree decides by features present in seed articles. In fact, GitHub and TIOBE seeds have a more general interest: Seed articles provide representative features for recognizing relevant articles. We show which features of seed articles provide a representative positive indication and hence may be found in a fit decision tree.

Infobox Templates. In the scope, there exist 864 distinct infobox template names. 263 seed articles use an infobox template. We found the templates on 'programming language' (215), 'file format' (32), 'technology standard' (3) and 'software license' [START_REF] Stvilia | Information quality work organization in wikipedia[END_REF]. Especially, the templates 'programming language' and 'file format' provide a strong positive indication, but they do not exist for every relevant article.

URL Pattern. We extract all words separately in braces from every article's title as they provide a semantic annotation for disambiguation. Only around a third of the seed articles has such an annotation. The words in braces appearing more than once with their frequency are: 'language' (125), 'programming' (114), 'software' (6), 'stylesheet' (3) and 'markup' (2). 'programming' always appears together with 'language'.

Lemmas in Summary. From the summary, we extract all words, filter them by a stop word list and apply lemmatization on the remainder. We recovered 5449 lemmas from seed articles while 457, 164 distinct lemmas can be recovered from all articles. We enumerate the top five lemmas from the seed: 'language' (301), 'programming' (253), 'use' (216), 'develop' (120), 'design' (117). Lemmas are often used for topic models [START_REF] Sarencheh | Automatic algorithm for extracting an ontology for a specific domain name[END_REF]. They are essential to distinguish the domain of software languages and co-occurring topics in articles from other unrelated domains such as natural languages.

Dependency Pattern. Following [START_REF] Mirylenka | Bootstrapping domain ontologies from wikipedia: A uniform approach[END_REF], we identified multiple kinds of relationships that can be extracted from the first sentence. Hypernyms provide a reliable feature as most articles on Wikipedia begin with a sentence containing 'is a'. Based on the dependency graph, we extract the hypernym [START_REF] Flati | Two Is Bigger (and Better) Than One: the Wikipedia Bitaxonomy Project[END_REF], if the sentence does not start with 'A'. This allows us to ignore subset relationships as in 'A programming language is a formal language[...]'. To reach higher coverage, we infer instantiation from relationships such as partof as in 'Perl 6 (also known as Raku [START_REF] Wang | Wikipedia2Onto. Building Concept Ontology Automatically, Experimenting with Web Image Retrieval[END_REF]) is a member of the Perl family of programming languages', where 'languages' is the extracted hypernym. In the seed, we found 'language' (235), 'format' ( 16) and 'dialect' [START_REF] Mirylenka | Bootstrapping domain ontologies from wikipedia: A uniform approach[END_REF] as the most frequent hypernyms. The hypernym 'language' and 'dialect' are also recovered when analyzing articles on natural languages.

Wikipedia List Entries. As an internal resource, we consider Wikipedia lists of things [START_REF] Kuhn | Type inference on wikipedia list pages[END_REF]. Such lists collect names of entities of a certain kind and optionally provide additional information, e.g., 'List of dog breeds'. First, we recover the list of lists by searching for 'list of' in the title of articles in the scope. Binary Features state in which lists an article is linked. In the seed, we found 267 articles linked in such lists, such as the 'List of programming languages'.

EVALUATION

We investigate the results from evaluating a classifier based on labels by experts (RQ1). Since we chose to use a decision tree classifier, we can present technical insights on how decisions are made and explain the effects of using a representative seed (RQ2). We conclude with presenting how many software language articles and categories are estimated in the scope (RQ3).

Precision & Recall (RQ1)

We provide qualitative insights on How well does the classifier perform beyond the seed?

To gain an evaluation set that is as objective as possible, we conducted a survey in which 31 domain experts from our research groups and external collaborators participated.

In each question in the survey, participants decided whether a presented article explicitly describes a software language as one primary topic. We received 990 articles labelled by at least two experts. In order to gain additional insights on problems with decision making, we emphasized the possibility of commenting on each question. For 43 articles, experts did not agree. The articles present border cases, for which experts are not sure. For example, articles do not directly refer to logic formulae as a software language, but such formulae are often digitally encoded, follow a syntax, well-formedness rules (thus, a type system) and have semantics and pragmatics. For the future of SLEBOK [START_REF] Combemale | SLEBOK: The Software Language Engineering Body of Knowledge (DS 17342)[END_REF], such border cases can be used to further investigate on the borders of the term software language. A refinement of definitions would then again lead to better expert labels to, for now, problematic articles. Such refinement hopefully leads to a better categorization on Wikipedia itself.

As long as experts cannot reliably decide for an article whether a software language is described, a machine cannot as well. As a consequence, we exclude these articles. Since only ~4% is excluded, the threat to validity is reasonably low. We took the remaining expert labels as the evaluation set and explored several configurations with a k-best feature selection and synthetic oversampling with SMOTE [START_REF] Chawla | SMOTE: synthetic minority over-sampling technique[END_REF]. With k = 23, the learned classifer performs with an f1score of 0.7, balanced accuracy of 0.9, recall of 0.81 and specificity of 0.99.

Indicator Discovery (RQ2)

For imbalanced datasets, a classifier usually overfits towards the major class unless countermeasures are taken [START_REF] Chawla | SMOTE: synthetic minority over-sampling technique[END_REF]. This problem is countered in two ways. i.) By adding the seed as an addition to the random sample to our training set, the rate of relevant articles in it is not representative which can be seen as a form of manual oversampling. ii.) Actually, we isolate articles relevant to a single class from articles relevant to many other classes (e.g., songs, stars, software). Feature selection based on Gini index or entropy pick features according to their score in discriminating classes, in our case relevant versus not relevant. Consequently, the features frequently appearing in seed articles are preferred, since the irrelevant articles have a huge amount of different features and thus single features recall a lesser percentage. At last, synthetic oversampling with SMOTE further improves results. This peculiarity motivates further inspection of the learned decision tree as an answer to what features indicate articles on software languages.

The templates 'programming language' and 'file format' provide a strong positive indication, but they do not exist for every relevant article. While the hypernym 'language' also provides strong indication, it cannot be used alone. Oth- erwise, natural language related articles are confused to be relevant. Lemmas, lists and URL pattern help in discriminating relevant articles from articles with overlapping features. Below, we list the most important indicators in the decision tree categorized by their source (see Section 2.3).

Infobox Templates: file format, programming language; URL Pattern: programming, language; Lemmas: syntax, code, programming, compiler, design, general-purpose, language, support, compile, object-oriented, use; Dependency Pattern: language; Wikipedia List Entries: List of programming languages, List of programming languages by type, List of file formats, List of C-family programming languages, List of object-oriented programming languages.

Article and Category Relevance (RQ3)

We give quantitative insights as an answer to the question: How many articles and categories remain relevant for the domain of software languages? Table 2 summarizes the degree of the reduction per root category based on the configuration from Section 3.1. Based on the predicted reduction, we find 2797 more articles in the scope than there are already in the seed. That is, we significantly augment the identification of software languages.

For the root category 'Computer file formats', the classifier predicts a lot of irrelevant articles. When inspecting infobox templates used in these articles, we observe a topic mix with, for example, software, websites and companies. Here, a threat remains that the articles do not directly address formats in several articles in a recognizable way. In comparison to 'Formal languages', a higher percentage of categories is estimated as relevant. Accordingly, we observe that the subcategory tree is more consistently maintained.

Inspecting single categories backs up subjective hypotheses based on manual inspection of the usefulness of specific categories. For example, out of 22546 articles that are (transitive) members 'Statistical data types', only 52 are classified as relevant. The number of articles indicated to be relevant in a category becomes a more objective estimation for its usefulness.

Table 2 provides a summary when inspecting all categories based on the assumption that the number of relevant articles hints at the relevance of categories. Out of 21822 categories below 'Formal languages', 353 categories contain seed members and 1339 categories remain relevant.

Only 6% of transitive subcategories under 'Formal languages' are estimated to be useful within the scope.

RELATED WORK

Previous attempt: In [START_REF] Lämmel | Method and Tool Support for Classifying Software Languages with Wikipedia[END_REF], the category graph is pruned by manually excluding categories not serving for classification of software languages in a common sense, subject to a small suite of criteria. In contrast to our work, the approach relies on manual selection of categories instead of automatically classifying articles by combining several indicators. Domain ontology: Wikipedia is a frequent target for knowledge discovery. To the best of our knowledge, no approach tries to detect articles describing a class from an inconsistently maintained domain, where relevant articles are rare and require to combine multiple indicators. For deriving domain ontologies, various distinct approaches can be found. The approaches range from supporting manual crafting [START_REF] Völkel | Semantic wikipedia[END_REF][START_REF] Vrandecic | Wikidata: a new platform for collaborative data collection[END_REF] to unsupervised crafting from text [START_REF] Sarencheh | Automatic algorithm for extracting an ontology for a specific domain name[END_REF]. Wang et al [START_REF] Wang | Wikipedia2Onto. Building Concept Ontology Automatically, Experimenting with Web Image Retrieval[END_REF] extract a domain ontology for animals from Wikipedia based on the category graph, article graph and section structure in articles. For the animals domain, most pages are maintained well and describe exactly one concept in a consistent order. Mirylenka et al [START_REF] Mirylenka | Bootstrapping domain ontologies from wikipedia: A uniform approach[END_REF] extract subset, membership, part-of, sub-topic and other relationships by analyzing the varying nature of subcategorization. Dong et al [START_REF] Dong | Learning to discover subsumptions between software engineering concepts in wikipedia[END_REF] learn subsumptions from articles describing domain concepts based on Hearst pattern. In a related approach [START_REF] Chen | Building a domain knowledge base from wikipedia: a semi-supervised approach[END_REF], the concept set is learned by matching articles with Stackoverflow tags. Based on the aricle-and category graph, the concept set is expanded and further relationships are learned. Related works discover knowledge from different structural features, such as the title [START_REF] Mirylenka | Bootstrapping domain ontologies from wikipedia: A uniform approach[END_REF][START_REF] Zarrad | Title-based approach to relation discovery from wikipedia[END_REF], text [START_REF] Flati | Two Is Bigger (and Better) Than One: the Wikipedia Bitaxonomy Project[END_REF][START_REF] Presutti | Uncovering the Semantics of Wikipedia Pagelinks[END_REF], an article's section structure [START_REF] Wang | Wikipedia2Onto. Building Concept Ontology Automatically, Experimenting with Web Image Retrieval[END_REF], links to other articles [START_REF] Chen | Building a domain knowledge base from wikipedia: a semi-supervised approach[END_REF][START_REF] Presutti | Uncovering the Semantics of Wikipedia Pagelinks[END_REF], infoboxes [START_REF] Wu | Automatically refining the wikipedia infobox ontology[END_REF], lists [START_REF] Kuhn | Type inference on wikipedia list pages[END_REF], etc. External knowledge: The usefulness of WordNet varies depending on the coverage of terms in a domain. While WordNet is a known assistant in more common knowledge domains, low coverage is reported in more specific domains [START_REF] Do | Exploiting the wikipedia structure in local and global classification of taxonomic relations[END_REF][START_REF] Wang | Wikipedia2Onto. Building Concept Ontology Automatically, Experimenting with Web Image Retrieval[END_REF]. Our experience confirms it for software languages. Various knowledge graphs exist. DBpedia Live [START_REF] Morsey | Dbpedia and the live extraction of structured data from wikipedia[END_REF] mirrors Wikipedia while refining it with more ontological knowledge. YAGO [START_REF] Rebele | YAGO: A multilingual knowledge base from wikipedia, wordnet, and geonames[END_REF] is a knowledge graph derived from Wikipedia, WordNet and Geonames that explicitly focuses only on structured knowledge aspect that is then again linked by Dbpedia as well. Wikidata [START_REF] Vrandecic | Wikidata: a new platform for collaborative data collection[END_REF] is widely manually crafted and tries to reach a clean knowledge graph from scratch. A type encompassing the term software language is not maintained by any of the mentioned resources.

CONCLUSION

In a domain where only a small set of positively labeled articles can be used as a ground truth, we did find strong indicators for classifying 3000+ articles as relevant for software languages. We showed that a learned decision tree classifier provides reasonably high recall and low false-positive-rate and allows one to inspect isolated articles inside Wikipedia. While learned random forests might provide higher accuracy, we are specifically interested in higher interpretability of decision trees. 31 domain experts thoroughly evaluated our classifier by labelling over 990 Wikipedia articles.

A natural follow up of this work is a collaborative engagement with experts to further extract and improve finegrained classification knowledge into a high-quality domain ontology that will help students and professional developers to better understand software languages. We are in the process of repeating the experiments for the domain of software (in a broad sense). The general problem is the same: We are confident that our learning methodology can identify the body of domain knowledge within Wikipedia. There are also specific challenges ahead due to particularities of the domains (for example, 'Computing platforms' as a category is disconnected from 'Software').

1Figure 1 .

 1 Figure 1. General structure of a Wikipedia article and some indicators.

Figure 2 .

 2 Figure 2. Top 10 most frequent nouns in articles below 'Formal languages' category.

Figure 3 .

 3 Figure 3. The ingredients of the approach. The seed in the training data influences decisions.

Table 1

 1

	illus-

4

https://www.tiobe.com/tiobe-index/ programming-languages-definition/#instances

Table 1 . Number of seed articles per depth.

 1

		0	1	2	3	4	5 6-8
	Formal languages	2	6 85 136 70 16	7
	Computer file formats 8 22 11	7	0	0	3

Table 2 . How many articles and categories exist in total

 2 , identified by the seed, and classified as relevant.

			Articles			Categories
		T otal Seed Relevant T otal Seed Relevant
	Formal Languages	101641	301	2897 21822	353	1339
	Computer File Formats	6116	46	745	235	18	79

https://github.com/softlang/wikionto/

https://github.com/github/linguist/blob/master/lib/ linguist/languages.yml

Acknowledgements. This research was partially funded by the ANR-17-CE25-0010-01 VaryVary project.


Jérémy Riporto

Mathias Urbain

Yannick Mugnier

Volodymyr Multian

Florian Riporto

Kévin Bredillet

Sandrine Beauquis

Christine Galez

Virginie Monnier

Yann Chevolot

Volodymyr Gayvoronsky

Luigi Bonacina

Ronan Le Dantec
email: ronan.le-dantec@univ-savoie.fr

Second harmonic spectroscopy of ZnO, BiFeO 3 and LiNbO 3 nanocrystals

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Nonlinear optical frequency conversion in nanocrystals and nanostructures provides coherent and widely tunable signals for applications in various fields such as nanophotonic systems and bio-imaging. The intensity of the emitted harmonics is reduced at the nanoscale so that different strategies, mostly based on optically resonant structures, have recently emerged to compensate for this scale law. For instance, metal-based nanostructures [START_REF] Kauranen | Nonlinear plasmonics[END_REF][START_REF] Pu | Nonlinear Optical Properties of Core-Shell Nanocavities for Enhanced Second-Harmonic Generation[END_REF][START_REF] Zhang | Toward Surface Plasmon-Enhanced Optical Parametric Amplification (SPOPA) with Engineered Nanoparticles: A Nanoscale Tunable Infrared Source[END_REF] and Mieresonant nanoparticles [START_REF] Timpu | Second-Harmonic Enhancement with Mie Resonances in Perovskite Nanoparticles[END_REF][START_REF] Shcherbakov | Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response[END_REF][START_REF] Smirnova | Multipolar second-harmonic generation by Mie-resonant dielectric nanoparticles[END_REF] evidence strong enhancements of the nonlinear optical effects and offer very appealing perspectives in terms of improved conversion efficiency. Nonlinear optical spectroscopy has thus become essential in order to accurately assess the magnitude of these frequency-dependent enhancements. In addition to nanostructures, the study of the intrinsic nonlinear response of bare nanocrystals is also of great importance to better quantify and understand the above mentioned resonances. Regarding bulk crystals, the wavelengthdependence of their nonlinear susceptibilities have been scarcely studied and data are often limited to discrete wavelengths [START_REF] Shoji | Absolute scale of second-order nonlinear-optical coefficients[END_REF] due to the lack in the past of readily available tunable excitation sources and because of material transparency issues. Among spectrally resolved Second Harmonic (SH) studies, an increase in the susceptibility has been demonstrated so far for zinc-blende II-VI semiconductors [START_REF] Wagner | Dispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnS[END_REF], GaAs [START_REF] Bergfeld | Second-Harmonic Generation in GaAs: Experiment versus Theoretical Predictions of χxyz ( 2 )[END_REF] and ZnO [START_REF] Pedersen | Second-harmonic generation from ZnO nanowires[END_REF][START_REF] Lafrentz | Second harmonic generation spectroscopy of excitons in ZnO[END_REF] when the SH frequency is tuned to the electronic transition energies. These studies provide experimental data for comparison with theoretical models which still need to be refined.

To this end, we here report on the development of a tunable Hyper Rayleigh Scattering (HRS) setup to quantitatively access the orientation-averaged second-order susceptibility  (2)  of three different Harmonic Nanoparticles (HNPs), namely Zinc Oxide (ZnO), Bismuth Ferrite (BiFeO 3 or BFO) and Lithium Niobate (LiNbO 3 or LN). HNPs are inorganic oxide nanocrystals displaying a very rich nonlinear response because of their non-centrosymmetric crystal structure. Both even and odd harmonics originate from the nanocrystal volume (when its size is typically above 10-20 nm) and among the different accessible nonlinear processes, SH generation has already proven to be very useful in bio-imaging because of unique properties including coherent and narrow spectral emissions, excitation wavelength tunability, and photostability [START_REF] Staedler | Harmonic nanocrystals for biolabeling: a survey of optical properties and biocompatibility[END_REF][START_REF] Pantazis | Second harmonic generating (SHG) nanoprobes for in vivo imaging[END_REF][START_REF] Slenders | Image Correlation Spectroscopy with Second Harmonic Generating Nanoparticles in Suspension and in Cells[END_REF]. When HNPs are used as exogenous optical contrast agents, sensitivity and selectivity can also be increased when the second-and third-order signals are simultaneously collected in demanding imaging applications [START_REF] Dubreil | Multi-harmonic Imaging in the Second Near-Infrared Window of Nanoparticle-Labeled Stem Cells as a Monitoring Tool in Tissue Depth[END_REF][START_REF] Ramos-Gomes | Bismuth Ferrite Second Harmonic Nanoparticles for Pulmonary Macrophage Tracking[END_REF].

Contrary to individual nanocrystal imaging, ensemble measurements from HRS are more prone to give quantitative information as initially demonstrated with the determination of molecular hyperpolarizabilities [START_REF] Clays | Hyper-Rayleigh scattering in solution[END_REF]. This technique has later been extended to probe the second harmonic scattering (SHS) of larger objects such as metallic nanoparticles [START_REF] Vance | Enormous Hyper-Rayleigh Scattering from Nanocrystalline Gold Particle Suspensions[END_REF], quantum dots [START_REF] Jacobsohn | Size Dependence of Second Harmonic Generation in CdSe Nanocrystal Quantum Dots[END_REF] and biological structures [START_REF] Deniset-Besseau | Measurement of the Second-Order Hyperpolarizability of the Collagen Triple Helix and Determination of Its Physical Origin[END_REF]. SHS spectroscopy from ensemble measurements was then first developed by Campo et al. [START_REF] Campo | Highly sensitive setup for tunable wavelength hyper-Rayleigh scattering with parallel detection and calibration data for various solvents[END_REF] ten years ago with the design of a widely tunable (excitation from 600 to 1800 nm) HRS set-up to study the hyperpolarizability dispersion of various molecules [START_REF] Campo | Accurate Determination and Modeling of the Dispersion of the First Hyperpolarizability of an Efficient Zwitterionic Nonlinear Optical Chromophore by Tunable Wavelength Hyper-Rayleigh Scattering[END_REF][START_REF] Campo | First Hyperpolarizability Dispersion of the Octupolar Molecule Crystal Violet: Multiple Resonances and Vibrational and Solvation Effects[END_REF]. So far, SH spectroscopy from individual nanocrystals and nanostructures was only carried out in multiphoton microscopy [START_REF] Linnenbank | Second harmonic generation spectroscopy on hybrid plasmonic/dielectric nanoantennas[END_REF][START_REF] Winter | The Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors-A Single Dot Study[END_REF] with the severe limitation that calibration of the focusing and collection parameters from diffraction-limited objects is not trivial, especially because the intensity at 2 strongly depends on the nanocrystal orientation [START_REF] Schwung | Nonlinear optical and magnetic properties of BiFeO3 harmonic nanoparticles[END_REF]. SHS spectroscopy from colloidal nanocrystal suspensions is here developed from a recent tunable commercial source.

This article is organized as follows: after a description of the experimental setup and theoretical background in section 2, calibration is detailed in section 3.1 from suspensions of LN nanocrystals. The relative, orientation-averaged susceptibility of LN is first calibrated against chloroform and compared with literature values of bulk crystals. Rescaling in the full excitation range is then derived from the well-known response of para-nitroaniline molecules at 1064 nm. In section 3.2 and 3.3, SHS spectroscopy is applied to ZnO and BFO suspensions to quantitatively assess their orientation-averaged second-order susceptibility. Validity of the calibration procedure and the wavelength-dependence of the absolute susceptibilities are finally discussed in section 4.

Experimental procedures and data analysis

As illustrated in Fig. 1, SHS spectroscopy is here based on a widely tunable (680 to 1300 nm) femtosecond laser (Insight X3 Spectra-Physics). The incident power is controlled thanks to an achromatic half-wave plate (Thorlabs, 690-1200 nm) associated with a vertical Glan-laser polarizer (Thorlabs). A near-infrared achromatic lens (Newport, f = 100 mm) mounted on a motorized translation stage is placed after a long pass filter (Thorlabs FELH0650) and is used to focus the incident beam at the center of a UV-fused quartz cuvette filled with the nanocrystal suspensions. The second harmonic signal is collected perpendicularly to the laser beam through two fused silica lenses (Thorlabs, f = 50 mm). An uncoated BK7 Dove prism (Thorlabs) is inserted between these two lenses so as to rotate the (horizontal) focal point image and to increase the signal [START_REF] Van Steerteghem | Third-Harmonic Scattering for Fast and Sensitive Screening of the Second Hyperpolarizability in Solution[END_REF] on the vertical entrance slit of a spectrometer (Andor Shamrock 193) coupled to a Peltier-cooled CCD camera (Andor iDus 401). A short pass filter is also placed in front of the spectrometer to remove scattering of the fundamental beam. This short pass filter is either composed of two FGB37 Thorlabs colored glass filters in the 700-800 nm excitation range or a single Thorlabs FESH0700 filter in the 800-1300 nm range. Full automation of the whole set-up ensures reproducible experimental conditions and allows scanning the excitation wavelength during a spectroscopic measurement. At each wavelength, the incident power is set to the required value and the dispersion pre-compensation is optimized to keep constant the pulse width within the sample. For an incident power of 500 mW, the wavelength-independent intensity at the focal point is calculated at 1 GW/cm 2 and a typical 1 min integration time is then necessary to collect the SHS spectra at each excitation wavelength. In case of residual background or multiphoton photoluminescence, a multi-peak fitting procedure was also implemented to only assess the SH signal. In previous works, we have adapted the HRS technique to measure at 1064 nm the orientation-averaged second-order susceptibility of several harmonic nanoparticles [START_REF] Staedler | Harmonic nanocrystals for biolabeling: a survey of optical properties and biocompatibility[END_REF][START_REF] Le Dantec | Ensemble and Individual Characterization of the Nonlinear Optical Properties of ZnO and BaTiO3 Nanocrystals[END_REF][START_REF] Joulaud | Characterization of the nonlinear optical properties of nanocrystals by Hyper Rayleigh Scattering[END_REF]. In the following, the formalism at a single excitation wavelength is briefly reminded. The second harmonic signal scattered by a nanoparticle (np) suspension (neglecting the solvent contribution) can be expressed as:

  4 2 2 2 22 G np p eff n I N t t I       (1)
G includes experimental parameters and fundamental constants related to the electric dipole radiation, N np is the concentration of nanoparticles and I  the incident intensity. The macroscopic internal field factor that is calculated at the fundamental and SH frequencies is defined at  as:

  2 2 2
3( ) 2( ) ( )

s s np t n n n    



(2) for a spherical nanoparticle [START_REF] Jackson | Classical Electrodynamics[END_REF] with s n  and np n  the refractive indexes of the solvent and nanoparticle, respectively. 2 eff    is the effective squared hyperpolarizability averaged over all possible orientations. The term "effective" is here introduced to account for the volumedependent response of the nanocrystal (SH fields radiated from each crystal cell build up coherently within the dipolar approximation) and to make a clear distinction with the usual (microscopic) molecular hyperpolarizability. The orientation-averaged second-order susceptibility  (2)  is then calculated from the effective hyperpolarizability and nanoparticle volume V np as [START_REF] Le Dantec | Ensemble and Individual Characterization of the Nonlinear Optical Properties of ZnO and BaTiO3 Nanocrystals[END_REF]:

  2 2 eff np V   (3)
In practice, quantitative information can only be derived after calibration of the experimental setup. Herein, the external reference method is applied with para-nitroaniline (pNA) molecules dissolved in methanol. Neglecting again the solvent contribution, the unknown experimental constant G of eq. 1 can be removed since the HRS signal from a pNA solution is given by: where f  and f 2 are the usual Lorentz-Lorenz field factors and  pNA  = (6/35) ½  33 ( 33 =25.9.10 -30 at 1064 nm [START_REF] Huyskens | Solvent dependence of the first hyperpolarizability of pnitroanilines: Differences between nonspecific dipole-dipole interactions and solute-solvent H-bonds[END_REF]). After plotting I 2 as a function of the pNA concentration and of the concentration of nanoparticles,  eff  is then obtained from the ratio of the two corresponding slopes. Note that N np is estimated by weighing the residual powder after evaporation of the suspension used in the SHS measurements. The average nanoparticle volume is also determined within the suspension by using Dynamic Light Scattering (DLS) so that  (2)  can be finally calculated from eq. 3. The experimental protocol, applied conventions and different limitations of this technique are detailed in ref. [START_REF] Joulaud | Characterization of the nonlinear optical properties of nanocrystals by Hyper Rayleigh Scattering[END_REF] which describes the same experiments at a constant 1064 nm excitation wavelength.

  4 2 2 2 22 G pNA pNA pNA I N f f I       (4)
For this work and to ensure consistency with our previous reports, we have applied this protocol to retrieve the absolute  (2)  values of the different HNPs at 1064 nm. However, pNA cannot be used on the whole spectral range as a reference since its hyperpolarizability is known exclusively at 1064 nm. The ideal reference should be transparent at the fundamental and second harmonic frequencies besides having a high non-resonant nonlinear susceptibility and limited hindrance by multiphoton fluorescence. Calibration over broad excitation range is thus a major issue. In ref. [START_REF] Campo | Highly sensitive setup for tunable wavelength hyper-Rayleigh scattering with parallel detection and calibration data for various solvents[END_REF], the dispersion of a large range of solvent has been measured against chloroform. Considering its transparency range and non-resonant response, chloroform was here first used as a reference to evaluate dispersion of the LiNbO 3 secondorder susceptibility that was found in good agreement with literature values. Lithium niobate nanocrystal suspensions are then employed as a new external reference material to better match with the above criteria because of the large material transparency range, the scattering of strong SH signals and the absence of any stray luminescence.

Results

SHS spectroscopy of LiNbO 3 suspensions

Lithium niobate is a well-known ferroelectric material with high and well-characterized SH coefficients [START_REF] Shoji | Absolute scale of second-order nonlinear-optical coefficients[END_REF]. Regarding nanoparticle suspensions with low size and shape polydispersity, they can be prepared by using the solvothermal approach developed by Mohanty et al [START_REF] Mohanty | Synthesis and piezoelectric response of cubic and spherical LiNbO3 nanocrystals[END_REF]. Well-crystallized and phase-pure LiNbO 3 nanoparticles were thus obtained with an average size at 50 nm as imaged by Transmission Electron Microscopy (Fig. 2). When dispersed in Ethanol, suspensions are very stable and show a very low polydispersity with a typical mean diameter measured at about 70 nm from DLS measurements. LN    is then determined at 1064 nm from the reference solution of pNA molecules dissolved in methanol. Finally, the absolute value of the orientation-averaged susceptibility of LN is rescaled in the full spectral range. Fig. 3 illustrates typical emission spectra for both LN suspensions and chloroform obtained from a constant power set at 500 mW, i.e an incident intensity at 1 GW/cm 2 . Only pure SH scattering is observed for LN suspensions in ethanol while chloroform spectra show an additional, small red-shifted peak associated to a hyper-Raman scattering band [START_REF] Xu | Hyper-Rayleigh and Hyper-Raman Scattering Background of Liquid Water in Two-Photon Excited Fluorescence Detection[END_REF]. Note also that the SH signal from chloroform is very low comparatively to LN suspensions and that the integration time had to be increased to 5 minutes. In spite of the lower response of chloroform, the relative susceptibility of LN could be derived from the measured SH intensities and by using the following relationship deduced form eq. 1 and eq. 4 in the case of a chloroform solution:

                   
             (5)
Dispersion of the chloroform hyperpolarizability  chloro () was taken from reference [START_REF] Campo | Highly sensitive setup for tunable wavelength hyper-Rayleigh scattering with parallel detection and calibration data for various solvents[END_REF] and field factors f  and t  were calculated at the fundamental and SH frequencies from the refractive indexes of chloroform, ethanol [START_REF] Kedenburg | Linear refractive index and absorption measurements of nonlinear optical liquids in the visible and near-infrared spectral region[END_REF] and LN [START_REF] Zelmon | Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol% magnesium oxide -doped lithium niobate[END_REF]. The resulting relative, orientation-averaged susceptibility of LN is depicted in Fig. 3.b and shows a non-resonant response as expected for lithium niobate nanocrystals. (b) The relative, orientation-averaged susceptibility of LN (blue circle, normalized at =1200 nm) is compared to the d33 coefficient measured in ref. [START_REF] Shoji | Absolute scale of second-order nonlinear-optical coefficients[END_REF] (red triangle for congruent LiNbO3 and black triangle for 5%MgO:LiNbO3). Note that the left and right scales are proportional to make relevant the comparison. The blue line is a fit with the two-level model (eq. 6) described in the text. measured at three discrete wavelengths by Shoji et al. [START_REF] Shoji | Absolute scale of second-order nonlinear-optical coefficients[END_REF] is also shown in Fig. 3b. It is worth to mention that the orientation-averaged susceptibility of LN is given in Table 1 according to the individual components of its susceptibility tensor and that the contribution of the d 33 coefficient is the predominant one according to literature values [START_REF] Shoji | Absolute scale of second-order nonlinear-optical coefficients[END_REF][START_REF] Klein | Absolute non-linear optical coefficients of LiNbO3 for near stoichiometric crystal compositions[END_REF]. Comparison between d 33 and

  2 Re LN l  
 is thus relevant. Therefore, the measured susceptibility frequency dispersion from a LN colloidal suspension is found in good agreement with bulk crystal values, which is consistent with the already demonstrated absence of any size effect for the SH response of various inorganic nanocrystals in this size range [START_REF] Joulaud | Characterization of the nonlinear optical properties of nanocrystals by Hyper Rayleigh Scattering[END_REF][START_REF] Knabe | Spontaneous polarization in ultrasmall lithium niobate nanocrystals revealed by second harmonic generation[END_REF]. In Fig. 3b, experimental values were also fitted with a simple two-level model which is valid far from resonance conditions [START_REF] Oudar | Hyperpolarizabilities of the nitroanilines and their relations to the excited state dipole moment[END_REF]:

         4 22 0 2 2 2 2 2 , , 4
eg eg eg             (6)
 eg stands for the electronic transition frequency,  for the fundamental frequency and  were used as free parameters to adjust the experimental data leading to a transition wavelength of  eg = 286 nm which is close to the experimental absorption edge of LN crystals measured at 305 nm [START_REF] Bhatt | Urbach tail and bandgap analysis in near stoichiometric LiNbO3 crystals[END_REF]. The as-obtained smooth dispersion curve is then assumed in the following sections to be representative of the relative, orientation-averaged susceptibility of LN.

Finally, the absolute value of the orientation-averaged susceptibility of LN is obtained at 1064 nm with suspensions of varying concentrations and after calibration of the SHS set-up with pNA solutions as discussed in section 2. We found an experimental value of [START_REF] Klein | Absolute non-linear optical coefficients of LiNbO3 for near stoichiometric crystal compositions[END_REF]). This further supports the experimental calibration route here developed and the use of LN suspensions as an almost ideal reference because of its background-free and high non-resonant SH response. Rescaling of

  2 Re LN l   
in the full excitation range is then easily achieved to derive dispersion of the absolute orientationaveraged susceptibility. Corresponding data for LN are plotted below in Fig. 5 for a direct comparison with the dispersion curves then obtained for ZnO and BiFeO 3 nanocrystals.

           
1 D, mean diameter obtained from the DLS size-distribution in number. 2 Orientation-averaged susceptibilities are here expressed assuming the Kleinman's condition and for the current experimental configuration, i.e. a vertical input polarization and no analyzer in the detection arm.

SHS spectroscopy of ZnO suspensions

ZnO is a wide band-gap semiconductor with a non-centrosymmetric crystal structure.

Comparatively to LN and BFO, SH and even third harmonic properties of ZnO films and nanostructures including nanowires and nanoparticles have already been the subject of reviews and several papers [START_REF] Larciprete | Second harmonic generation from ZnO films and nanostructures[END_REF][START_REF] Johnson | Near-Field Imaging of Nonlinear Optical Mixing in Single Zinc Oxide Nanowires[END_REF]. Nanoparticle suspensions of ZnO dispersed in ethanol were prepared from commercial nanopowders (NanoAmor Inc., commercial size 90-200 nm) as recently described [START_REF] Multian | Averaged third-order susceptibility of ZnO nanocrystals from Third Harmonic Generation and Third Harmonic Scattering[END_REF]. SHS spectroscopy is then performed with an excitation wavelength varying from 700 to 1300 nm as shown in Fig. 4a and4b. Above 760 nm, only the SH signal is detected whereas an additional multiphoton-excited luminescence peak centered at 377 nm gradually appears when the excitation is shifted towards shorter wavelengths. Measured spectra are thus fitted with multiple Gaussian functions to separate the SHS intensity from the band-gap photoluminescence contribution [START_REF] Pedersen | Second-harmonic generation from ZnO nanowires[END_REF]. Fig. 4c shows the normalized SHS intensity ratio I ZnO ()/ I LN () which is used to derive the relative susceptibility with eq. 5 (modified by substituting chloroform with LN and LN with ZnO). A strong resonance (corrected from any photoluminescence contribution) occurs at a fundamental wavelength of 750 nm with a 10fold enhancement compared to a non-resonant excitation at 1300 nm. This resonance will be discussed later.

The orientation-averaged susceptibility is then determined as before at 1064 nm from pNA solutions and the as-obtained experimental value

  2 1064 5.3 / ZnO pm V   
is found very consistent with previous measurements carried out with the same commercial ZnO nanopowder but with a YAG laser source [START_REF] Le Dantec | Ensemble and Individual Characterization of the Nonlinear Optical Properties of ZnO and BaTiO3 Nanocrystals[END_REF]. Afterwards, the relative susceptibility is rescaled to obtain dispersion of the absolute, orientation-averaged susceptibility of ZnO which is also plotted in Fig. 5. suspensions in ethanol as a function of the excitation wavelength and after normalization at =1300 nm. A strong resonance is observed at 750 nm.

SHS spectroscopy of BiFeO 3 suspensions

Bismuth Ferrite (BFO) is a promising room-temperature multiferroic material with a very high and rich nonlinear optical response. Its SH efficiency has been measured at 1064 nm and is known to vary between 158 and 440 pm/V according to the synthesis route that can yield to more or less phase-pure, monocristalline and monodisperse nanocrystals [START_REF] Schwung | Nonlinear optical and magnetic properties of BiFeO3 harmonic nanoparticles[END_REF][START_REF] Clarke | Preparation from a revisited wet chemical route of phase-pure, monocrystalline and SHGefficient BiFeO3 nanoparticles for harmonic bio-imaging[END_REF]. In this study, BFO nanoparticles were prepared through a simple co-precipitation route leading to stable BFO suspensions in ethanol with a mean typical DLS size at 90-100 nm [START_REF] Tytus | Preparation and Preliminary Nonlinear Optical Properties of BiFeO3 Nanocrystal Suspensions from a Simple, Chelating Agent-Free Precipitation Route[END_REF]. This chelating agent-free chemical synthesis has however some drawbacks since the absence of size and morphology control has been clearly evidenced from TEM observations (Fig. 2) and a decrease of the orientation-averaged susceptibility is thus expected [START_REF] Joulaud | Characterization of the nonlinear optical properties of nanocrystals by Hyper Rayleigh Scattering[END_REF]. After preparation of a batch of BFO powders from the co-precipitation route, the same procedure previously described for ZnO is again applied for the determination of the relative and absolute orientation-averaged susceptibilities of BFO. Note that a background-free and pure SHS signal is detected over the entire excitation range (data not shown). The absolute value of

  2 1064 BFO  
 is found to be 120 pm/V at 1064 nm, which is significantly higher than the one of LN and ZnO. The frequency-dependent orientation-averaged susceptibility of BFO is also compared in Fig. 5 with the ones of LN and ZnO nanocrystals.

Discussion

Validity and limitations of the calibration procedure

Because of the wide excitation range and the high number of optical parameters that should be adjusted for the whole experimental set-up, a series of sanity checks has been performed to ensure robustness of the measurement protocol. First, the SHS signal was optimized with a horizontal translation of the excitation lens and found to be maximal for each wavelength when the excitation beam is focused at the center of the sample cuvette. Interestingly, we did not observe any shift of this position in the entire excitation range thanks to the achromatic focusing lens. A well-defined and constant excitation intensity I  is thus maintained at the sample. Second, we confirmed that the response of the orientation-averaged susceptibility of ZnO is reproducible by varying several experimental parameters including the incident power (250 mW or 500 mW), the focal length of the excitation lens (50 mm or 100 mm) and the position of the excitation beam after shifting it from the center of the sample cuvette. In each case, the wavelength-dependent intensity ratio I ZnO ()/I LN () was measured and found constant. Measurement of the relative, orientation-averaged susceptibility is thus very reproducible provided that the experimental conditions are kept constant for the successive acquisition of the sample and reference data.

On the other hand, quantitative assessments of the absolute susceptibility at 1064 nm have some limitations which are detailed in ref [START_REF] Joulaud | Characterization of the nonlinear optical properties of nanocrystals by Hyper Rayleigh Scattering[END_REF]. As reminded in section 2, we indeed assume perfectly spherical nanoparticles with a narrow size-distribution and for each material, the number density of nanoparticles in suspension is supposedly known with enough accuracy. Deviations from these ideal conditions can be minimized though after reduction of the nanoparticle size-and shape-polydispersity but from our experience, the orientation-averaged susceptibility can be slightly underestimated compared to the literature values of bulk crystals. We estimate the overall experimental reproducibility of the absolute measurements at about ±15% [START_REF] Joulaud | Characterization of the nonlinear optical properties of nanocrystals by Hyper Rayleigh Scattering[END_REF].

Finally, we point out here that SHS signals are assumed to be proportional to the concentration of nanoparticles (eq. 1) but deviation may occur for highly concentrated samples, simply because of absorption and linear scattering of the fundamental and SH photons (both effects are more pronounced at the shortest wavelengths). Working in the low-concentration regime is thus a prerequisite and ideally a series of intensity-concentration measurements should be carried out at each wavelength. To avoid these time consuming experiments, our approach consisted in repeating a second wavelength-dependence measurement of the normalized SHS intensity ratio but with a more diluted sample. For ZnO (and BFO), the ratio I ZnO ()/I LN () was found constant demonstrating that the the linear regime assumed in eq.1 holds for each wavelength. For the different suspensions of nanoparticles of this work, we noticed that the mass concentrations should be typically below 0,1 mg/mL and under these conditions, absorption and linear scattering of the fundamental and SH photons could not be evidenced.

Dispersion of the absolute susceptibilities

Dispersion of the absolute, orientation-averaged nonlinear susceptibility of ZnO, LN and BFO nanocrystals is plotted in Fig. 5 and from bottom to top according to their quantitative value. In the case of ZnO, the sharp resonance observed at the excitation wavelength of 750 nm ( 2 = 375 nm) is consistent with the exciton resonance near the bandgap [START_REF] Pedersen | Second-harmonic generation from ZnO nanowires[END_REF][START_REF] Prasanth | Resonance enhancement of optical second harmonic generation in a ZnO nanowire[END_REF]. A 7fold enhancement can be noticed comparatively to the off-resonance experimental value measured at 4 pm/V. Interestingly, dispersion of the absolute susceptibility measured here is very similar to the relative frequency-dispersion obtained by Perdersen et al [START_REF] Pedersen | Second-harmonic generation from ZnO nanowires[END_REF] from SHG reflection on both single crystals and nanowires. For LN, the absolute susceptibility is stronger than that of ZnO and it does not present defined resonances as already discussed. The absolute nonlinear susceptibility of BFO is well higher than the other two materials in the whole excitation range and the resonance observed at 880 nm ( 2 = 440 nm) is close to the BFO bandgap [START_REF] Clark | Band gap and Schottky barrier heights of multiferroic BiFeO3[END_REF]. Noticeably, the resonance (3-fold enhancement from the 1300 nm excitation wavelength) is less pronounced and much broader comparatively to the one of ZnO. Fig. 5: Experimental absolute values of the orientation-averaged second-order susceptibility of BFO, LN and ZnO nanocrystals (blue dots). Each dispersion curve is rescaled in the 700-1300 nm excitation range from the quantitative assessment made at 1064 nm from the external reference method applied with pNA molecules dissolved in methanol (red square). For LN, the two level model is also shown (green dashed line). For each nanomaterial, the Miller's rule calculated from the linear complex susceptibility is indicated with a red dashed line.

A key point of this work is that we assume that nanoparticle nonlinear susceptibility is representative of the bulk one. Indeed, we had already verified this assumption at a fixed wavelength of 1064 nm [START_REF] Le Dantec | Ensemble and Individual Characterization of the Nonlinear Optical Properties of ZnO and BaTiO3 Nanocrystals[END_REF][START_REF] Joulaud | Characterization of the nonlinear optical properties of nanocrystals by Hyper Rayleigh Scattering[END_REF] and it is related to the noncentrosymmetric crystalline structure of the nanoparticles. In this framework, SH signal originates from the nanoparticle volume with negligible surface contribution [START_REF] Knabe | Spontaneous polarization in ultrasmall lithium niobate nanocrystals revealed by second harmonic generation[END_REF]. The very distinct dispersion curves obtained here for the three nanomaterials are thus very similar to their bulk response. It is therefore interesting to compare these experimental dispersion curves with the empirical Miller's rule [START_REF] Miller | Optical second harmonic generation in piezoelectric crystals[END_REF] stating that the ratio between the linear and non-linear susceptibilities is constant in the transparent region, i.e.

                2 1 1 1 2 , , 2
           .
This rule is usually applied to estimate dispersion of the nonlinear susceptibility from   1  . It was even suggested that a similar relationship still hold if the SH frequency lies in the material absorption band [START_REF] Seres | Dispersion of second-order nonlinear optical coefficient[END_REF]. In the latter case, the real and imaginary parts of the linear susceptibility are to be taken into consideration to derive the nonlinear susceptibility through

        21 2 , , 2        (neglecting the dispersion of     1
 in the transparency range). We applied this approach by using the complex linear susceptibilities from the literature values of ZnO [START_REF] Srikant | On the optical band gap of zinc oxide[END_REF], LN [START_REF] Zelmon | Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol% magnesium oxide -doped lithium niobate[END_REF] and Fundamental wavelength (nm)

ZnO BFO [START_REF] Choi | Optical anisotropy and charge-transfer transition energies in BiFeO3 from 1.0 to 5.5 eV[END_REF], and then plotted the corresponding curves in Fig. 5. Superimposition of the experimental (blue dots) and calculated curves (dashed red line) is first fixed in the transparency region, namely 1200 -1300 nm. As shown in Fig. 5, the estimated susceptibilities are similar in terms of shape and resonance wavelengths but this rule underestimates the amplitude of the resonance for ZnO and BFO. Notably, it does not even correctly model the experimental response of LN, which is transparent at both fundamental and second harmonic wavelengths. The validity of the Miller's rule has already been questioned [START_REF] Shoji | Absolute scale of second-order nonlinear-optical coefficients[END_REF] in the literature and more sophisticated approaches such as ab initio methods have then been developed. However, they still need refinements since the experimental

  2 
dispersion is still poorly described in most cases [START_REF] Luppi | A review of recent theoretical studies in nonlinear crystals: towards the design of new materials[END_REF]. We believe that our experimental estimations can be useful for the development of new nonlinear susceptibility models.

5-Conclusion

A new SHS spectroscopy setup was developed from an easily-operable commercial laser source. Careful attention has first been paid to the calibration and measurement protocol in the 700-1300 nm excitation range. Because of the sensibly lower HRS signals collected from typical solvents like chloroform, colloidal suspensions of LN are proposed as an appropriate choice for the external reference method. To support the use of LN suspensions, the relative orientation-averaged susceptibility of LN was first calibrated against chloroform and then compared with literature values of bulk crystals. The good agreement achieved demonstrates that LN nanocrystals are well appropriate for the calibration of SHS spectroscopy because of the high and background-free SH signals and non-resonant response of lithium niobate. This calibration route was then applied to derive dispersion of the absolute, orientation-averaged susceptibility of ZnO and BFO nanocrystals. Distinct and specific nonlinear susceptibilities were found for each nanomaterial with resonances arising from the electronic transition near the bandgap energy for ZnO and BFO. The nanocrystal response is very similar to the bulk crystal one because SHG originates from the nanoparticle volume as previously demonstrated for an excitation wavelength at 1064 nm [START_REF] Joulaud | Characterization of the nonlinear optical properties of nanocrystals by Hyper Rayleigh Scattering[END_REF]. This experimental approach allows to quantitatively assessing the intrinsic material second order nonlinearity and can help to refine new susceptibility models besides offering a precise tool to characterize resonances of engineered nanostructures.

Fig. 1 :

 1 Fig. 1: Sketch of the SHS spectroscopy setup. HWP: achromatic Half-Wave Plate, VP: Vertical Glan-laser Polarizer, LPF: Long Pass Filter, DP: uncoated BK7 Dove Prism and SPF: Short Pass Filter.

Fig. 2 .

 2 Fig. 2. Representative TEM images of ZnO, LiNbO3 and BiFeO3 samples As previously introduced, the calibration protocol is based on a two-step procedure with first, the determination of a relative value of the frequency-dependent susceptibility

Fig. 3 .

 3 Fig. 3. (a) Representative scattering spectra from chloroform and LN suspensions in ethanol.(b) The relative, orientation-averaged susceptibility of LN (blue circle, normalized at =1200 nm) is compared to the d33 coefficient measured in ref.[START_REF] Shoji | Absolute scale of second-order nonlinear-optical coefficients[END_REF] (red triangle for congruent LiNbO3 and black triangle for 5%MgO:LiNbO3). Note that the left and right scales are proportional to make relevant the comparison. The blue line is a fit with the two-level model (eq. 6) described in the text.

Fig. 4 :

 4 Fig. 4: (a) Two-photon scattering spectra from ZnO suspensions for an excitation wavelength below 760 nm. The two-photon absorption induces a photoluminescence peak at 377 nm that is superimposed to the SHS contribution. (b) Only the SHS signal is detected in the 760-1300 nm excitation range. (c) Experimental ratio of the SHS signals IZnO and ILN from ZnO and LN

Table 1 : Mean nanocrystal size and experimental values of

 1

			  2	 measured at 1064 nm.   2  	 is

also given according to the individual, non-zero elements of the second-order susceptibility matrix

		D 1 (nm)	  2    14 06 (pm/V)	Squared orientation-averaged susceptibilities from individual tensor elements 2
	LN	73	24			  2 LN		2		6 35		2 33		8 21		2 22		92 105		2 31		32 105	31 33  
	ZnO	134	5.3					  2 ZnO	2  	6 35		2 33		92 105		2 31		32 105	31 33  
	BFO	90	120		  2			2	6		2				8		2		92		2	32
					BFO					35		33				21	22		105	31	105	31 33

Acknowledgements

The authors thank the CLYM platform for access to TEM equipment

Funding

French-Swiss Interreg V program (project NANOFIMT), French Agence Nationale de la Recherche (project RACINE), French Contrat Plan État Région (project E-TIME).