Sagot Benoît
email: benoit.sagot@inria.fr

Alpage

DeLex, a freely-avaible, large-scale and linguistically grounded morphological lexicon for German

Keywords: Morphological Lexicon, German Morphology, Lexicon Development

We introduce DeLex, a freely-avaible, large-scale and linguistically grounded morphological lexicon for German developed within the Alexina framework. We extracted lexical information from the German wiktionary and developed a morphological inflection grammar for German, based on a linguistically sound model of inflectional morphology. Although the developement of DeLex involved some manual work, we show that is represents a good tradeoff between development cost, lexical coverage and resource accuracy.

Introduction

Contrarily to, e.g., syntax or semantics, inflectional morphology involves finite datasets over which acceptability judgments are most often clear-cut. It is no surprise that morphology is therefore one of the key areas where collaborative work is being carried out, which involves computational linguistics, formal linguistics, linguistic typology, and descriptive linguistics. This constitutes one of the main objectives of recent work for renewing the formalization and implementation of inflectional morphology within Alexina, an NLP framework for developing and encoding morphological and syntactic lexicons (Sagot, 2010), whose initial morphological formalism is described in (Sagot, 2005). 1 In particular, we have adapted and integrated within Alexina a formal model of inflectional morphology, named parsli (Walther, 2011; Walther, 2013b), which we shall sketch below. The result is both an extension of Sagot's (2005) morphological formalism and an implementation of parsli. It comes with various companion tools dedicated to quantitative morphological analysis. This new version of Alexina's morphological layer, described and named Alexina parsli in (Sagot and Walther, 2013), has been mostly used until now for acquiring new insights into specific linguistic issues within morphology or at the iterface between morphology and other components of grammar (Sagot and Walther, 2013; Walther, 2013a). In this paper, we want to show how this morphologically and typologically motivated framework can improve the theoretical soundness and the pratical maintainability of a morphological lexicon and its associated morphological description (or morphological grammar), as well as the speed of the development of such a language resource. In this paper, we describe the recent development of DeLex, a new Alexina morphological lexicon for German. Apart 1 Alexina lexicons have been developed for a fair amount of languages, among which French (the Lefff ), Spanish (the Leffe ), Polish, Slovak, Persian, Kurdish or English. Other Alexina resources have been created by adapting existing freely-available resources. More recently, partial resources covering only (part of) one category have been developed for quantitative linguistics purposes, e.g., for Latin, Maltese or Khaling (Sino-Tibetan).

from DeLex, and to our best knowledge, there has been surprisingly no freely available morphological lexicon for German until recently, as pointed out by Adolphs (2008). The only available lexicon is apparently the medium-scale lexicon distributed with the morphisto morphological analyzed (Zielinski and Simon, 2009) (18,624 entries among which 17,749 "base stems", i.e., mostly lemmas but also irregular inflected forms).2,3 Moreover, German morphology is not strictly concatenative. It involves various kinds of non-canonical phenomena, in the sense of (Corbett, 2003), three of them being particularly widespread: stem alternation, syncretism and overabundance (in the sense of (Thornton, 2011), see below). We shall come back to this below in more detail, but this overall picture makes it the ideal testbed for validating Alexina's new morphological formalism through the development of a new large-scale resource: parsli, the underlying formal model, was specifically designed to address non-canonical phenomena. Our implementation within Alexina is now compatible with nonconcatenative phenomena. And we believe the resulting resource is the first freely-available large-scale morphological lexicon for German.

A few words on the Alexina parsli model of inflectional morphology

As mentioned above, the morphological level of the Alexina formalism has recently been the focus of important collaborative efforts between computational and formal morphologists (Sagot and Walther, 2011; Sagot and Walther, 2013), resulting in the adaptation and implementation of the parsli model of inflectional morphology (Walther, 2011; Walther, 2013b), while retaining most features from the existing morphological framework (Sagot, 2005), thus resulting into Alexina parsli . New Alexina parsli features include, among others:

• an arbitrary number of realisational levels (e.g., stem level, thematic level, first exponence level, second exponence level); realisation rules, which are clustered in zones or in tables are level-specific (e.g., a German adjective could be modeled as having a nonmarked, a comparative and a superlative stem which use the same exponence zone as the base stem as far as gender/number/case suffixes are concerned);

• realisation rules within a zone or table can be structured in blocks in the sense of (Stump, 2001);

• generic morphological operations that can be defined in a morphological grammar, which allow for modeling vowel alternations, reduplications, and other non-concatenative operations;

• mechanisms for modeling stem alternations, including suppletive stems;

• a mechanism for modeling form suppletion;

• structured morphological tags (flat morphological feature structures) and support of feature structure unification throughout morphological descriptions;

• lexical entries can specify explicitely information about deficiency (i.e., feature structures for which they do not have inflected forms, contrarily to most entries for the same category).

A few words on German morphology

German morphology is not stricly concatenative, in particular because nominal and verbal inflection involves vowel alternations (ablaut and umlaut) at the stem level, leading to stem allomorphy. (e.g., Baum 'tree', Bäume 'trees', or fahren 'drive', (er) fährt '(he) drives', (er) fuhr '(he) drove'). In addition, overabundance is massive, in particular within nominal and adjectival paradigms, both at the stem and at the exponence levels. For instance, at the exponence level, many masculine and neuter nouns can bear the suffix -s or -es for the GEN.SG, and/or the null suffix or the suffix -e for the DAT.SG (e.g., Mann(e)s 'man gen.sg ', Mann(e) 'man dat.sg ').

The stem level also shows overabundance for these nouns, specifically for the plural stem. As a result, the manual development of a morphological grammar for German was simplified and speeded up thanks to notions defined in parsli and implemented in Alexina parsli . For example, our morphological grammar involves two realisational levels for adjectives and nouns, namely one stem level and one exponence level, as mentioned above. Indeed, all variation within adjectival inflection lies at the stem level, i.e., in the way the comparative and superlative stems are built. It can resort to morphonology and/or morphological irregularities, such as stem suppletion (gut 'good', besser 'better' best-'best'), stem-related deficiency (alkoholfrei 'alcool-free' has no comparative or superlative) or stem overabundance (frei 'free', freier 'freer', freistor freiest-'freest'). Verbs involve an additional exponence level, which uses the unique adjectival exponence zone for inflecting the past participle. Note that the way this exponence zone is defined and used by past participles prevents them from having comparative and superlative forms.

In addition, the interface between morphology, syntax and the lexicon involves at least two massive and problematic phenomena: compounding and verbs with separable particles. Compounding is a massive phenomenon in German, particularly for nouns (e.g., Tageslicht 'day light').

Our goal being to create an inflectional lexicon, constuctional processes need not be modeled in DeLex.

However, one could consider modeling compound components (e.g., licht) as belonging to the set of forms associated with their corresponding lexical entries, leaving composition rules to another level of morphological modeling. On the other hand, many compounds are lexicalized, or at involve at least strong conventional preferences, thus suggesting that being the result of a morphological process and being a reasonable lexical entry should be considered as two distinct properties (Gaeta and Ricca, 2009). In this work, we decided to leave these difficult theoretical issues aside and rely on a purely empirical strategy: we shall include in DeLex all lexical entries found in the German Wiktionary, be they compounds or not. Moreover, our inflectional grammar will not generate compound components, as we consider that compoud creation rules belong to another layer of morphological modeling, which should be able to take as an input inflectional lexical entries (be it in the form of a stem or of an inflected form). In other words, DeLex does not provide any direct and dedicated way to analyse compounds it does not already contain, but could be used as a source of lexical information for developing a compound analysis module. The last phenomenon we shall briefly discuss is that of verbs with a separable (prefixal) particle. Given a simple verbal entry, such as geben 'give', one can find derivationaly related entries that also include one (sometimes several) prefixal elements, which can be of various kinds. Among them, the most challenging are separable particles. For example, aufgeben 'give up' involves a separable particle, auf, which is can be associated with the preposition auf. 4 It is said to be separable because some cells in aufgeben's paradigm contain a form without that particle (e.g., gebe 'I give (up)') but require it to be at a well-defined other place in the clause (Ich gebe niemals auf 'I never give up'). Prefixal realization rules at work in other cells insert the corresponding morph between the separable particle and the stem (e.g., past participle aufgegeben). In this work, we consider the result of all realisation rules that are applied to the stem involved -including the extraction of the separable particle as an independent "word" -, together with an information associated with the inflected entry that make explicit the fact that this particle is to be found elsewhere in the clause, as an inflected form of a verb with a separable particle. In other words, both aufgegeben and gebe will be inflected entries for the lexical entry aufgeben in DeLex, but the latter will also specify the need for an auf elsewhere in the clause. 5,6

Developing DeLex

The efficient development of this morphological grammar has been realised together with the extraction of lexical data and continuous validation of both the grammar and the lexicon via the paradigms they generate. More precisely, it can be described as a three-step process, starting with the German Wiktionary. 7 First, we have developed a dedicated rule-based tool that automatically extracts German lexical entries from a dump of the German Wiktionary, which can be considered as a semi-structured, noisy and incomplete lexical dataset. [START_REF] Peter | Acquiring a poor man's inflectional lexicon for German[END_REF] We limited the use of this tool to nominal, verbal, adjectival and adverbial lexical entries. 9 For each of them, we extracted the citation form as well as the partial morphological paradigms that can be found within Wiktionary articles (see Figure 1). We converted this into triples of the form (citation form, 10 inflected form, morphological tag), in which morphogical tags are manually specified based on Wiktionary cell names (e.g., "Befehl du" is converted into IMP.SG). In some cases, including for the case of the example shown in Figure 1, the partial paradigms we extracted exhibit overabundance: for some cells within the paradigm, such as imperative singular (IMP.SG) in this case, two distinct forms are possible (mach and mache, 'do'). In the German Wiktionary, this overabundance can be expressed using brackets (mach(e)), explicitely within one cell (geh!, gehe! 'go'), or as belonging to two distinct parts of the paradigm that realize the same set of feature structures (e.g., when a noun has two distinct plural stems, both stems lead to a distinct plural subparadigm). In a second step, and for each of the four open-class categories, we applied an algorithm that automatically builds inflection classes to the set of triples extracted 5 In fact, the situation is even more complex. A verb such as rollerbladen behaves like a simple verb as far as particle separation is concerned (Ich rollerblade 'I rollerblade') but as a verb with a separable particle when inflectional prefixation is involved (past participle: rollergebladet). This is correctly modeled in DeLex. 6 The other major class of prefixal elements is that of non-separable particles/prefixes, which prevent any inflectional prefixation to apply: compare geben 'give' and gegeben 'given' with vergeben 'forgive' and vergeben 'forgiven.' 7 http://de.wiktionary.com 8 This extraction task is non-trivial as each page covers one citation form, which can represent several lexemes (i.e., lexical entries) in various languages (not only German).

9 Inventories of closed class lexical entries (determiners, pronouns, etc.) have been extracted directly from the TIGER treebank (Brants et al., 2002; Smith, 2003), ignoring hapaxes, then adapted and extended manually. We do not provide details in this abstract for space reasons, and focus on the Wiktionarybased extraction of open-class lexical entries.

10 "Citation form" is more correct than "lemma", as a lemma is rather an equivalence class of inflected forms, whereas the citation form here is one of the inflected forms of a lemma, chosen as its representative.

during the first step. 11 The underlying objective is not to build DeLex's final inflection classes (recall that the triples extracted during the first step only cover parts of the paradigms). Rather, it aims at identifying sets of lexical entries that share a common inflectional behavior. This automatic inflection class construction algorithm works as follows. First, it performs on each partial paradigm (i.e., on each set of triples that share the same part-of-speech and the same citation form) the following actions:

• it identifies the longest initial substring of the citation form (machen, in our example) that is also a (nonnecessarily initial) substring of all inflected forms of the (partial) paradigm (mach in our example);

• it considers this substring as the stem;12

• it considers that, for each inflected for in the (partial) paradigm, the initial substring to the left, resp. right, of the stem is its prefix, resp. suffix, and associates the resulting (prefix, suffix) pair to the triple's morphological tag;

• it sorts (prefix, suffix) pairs according to the corresponding morphological tag (alphabetically);

• it considers the result of this sort as a signature of the inflection class that has generated the (partial) paradigm at hand, i.e., the inflection class for the corresponding lexical entry.

Once this has been achieved on all (partial) paradigms, we generate a morphological description (i.e., morphological grammar) that defines all extracted inflection classes, provided they are associated with at least 3 lexical entries. Indeed, inflection class signatures contain all the information needed for defining the inflection classes' realization rules.

Inflected forms for lexical entries requiring inflection classes used by less than 3 lexical entries are listed explicitely and are given the special inflection class 0 which generates no (additional) inflected form. For the third step of DeLex's construction, we performed two costly manual task,13 namely:

• the development of a complete morphological description of German, which is not restricted to the partial paradigms extracted from the German wiktionary, and which is linguistically motivated, based on the Alexina parsli formalism mentioned above. This description relies for example on an explicit modeling it allows for explicitely specifying suppletive stems within the lexicon, which is for instance necessary for several comparatives and superlatives and for some verbs; it also contains constraints associated with inflection classes and with rules, in order to restrict the use of an inflection class or realisation rules to compatible stems;

• the conversion of the lexicon extracted automatically in the previous steps, in order to generate a lexicon that relies on these manually defined inflection classes.

This allowed us to produce a lexicon which is satisfying both from a linguistic and from a coverage point of view. It also allowed us to identify and correct many errors within the German Wiktionary. This is the case for example whenever the inflected forms generated by our lexicon for a given lexical entry were different from what had been extracted from Wiktionary, and whenever some inflection classes of some realization rules could not be applied for a given cell and a given lexical entry because of the abovementioned constraints. The result of this three-step process is illustrated in Figure 4, which displays both the lexical entry for our running example, the verb machen 'do', together with excerpts of the morphological description that are relevant for inflecting this entry.

The resulting lexicon: DeLex

The DeLex is already freely-available for download under the LGPL-LR licence, both in the form of an intensional lexicon (lexical entries and inflectional grammar) and in the form of an extensional lexicon (inventory of inflected forms associated with morphological information, directly usable in tools such as part-of-speech taggers, unknown word analysis modules for parsers, and many others). In this figure, an extensional entry includes an inflected form, a category, a morphological tag, and the lemma's citation form and inflection class.

15 At the URL https://gforge.inria.fr/frs/ ?group_id=482, one can find the alexina-tools package, which is required for inflecting DeLex (version 1.5 and higher is required), as well as the delex package itself, whose version 0.1 is described here. The delex package also contains the resulting extensional morphological lexicon, i.e., entries of the form (inflected form, category, citation form) (see file delex.mlex).

Lexical entry:

machen v-std:fs:ge:0

Explanation:

• "v-std" is the inflection pattern (or improperly the inflection class) of verbs that only have one stem for all cells (regular, most frequent case);

• "fs" specifies one of the possible behaviors of the schwa at the stem-suffix interface;

• "ge" specifies that the past participle is realized with the prefix ge-

• "0" specifies that, beyond the standard -e imperative singular form (mache), an alternative suffixless form est possible (mach).

Morphological description (or morphological grammar):

• Several morphological operations, which are not readily avaible in the formalism, are defined explicitely. For example, the operation remove sp removes the separable particle from a verb, provided it has one (e.g., aufmachen → machen).

• Various morphographemic rules are defined, such as the rule given below, which indicates that a suffix starting with t-, when place at the right of an element ending in -t, leads to the insertion of an extra -e-.

<sandhi source="t t" target="t et"/>

• The inflection pattern below specifies at the level 1 how the stem is built for each cell within regular verbal paradigms, and at the level 2 how affixal exponents should be computed.

<pattern name="v-std" cat="v"> <subpattern> <realzone level="1" table="v-stems"/> <realzone level="2" table="v-infl"/> </subpattern> </pattern>
• The stem construction table v-stems is not very informative, as verbs which use it have only one stem for all cells.

• The exponence table v-infl is the same for all verbs, although a few rules only apply on strong verbs and other for other verbs. We provide below a simplified version of this table, which covers indicative present forms for all non-strong verbs whose stem does not end in -s, -z, -ß or -x. Note the first rule, which removes the separable particle when relevant (their pattern indicate spge or spnoge instead of the ge in the pattern mentioned above).

<table name="v-infl"> <form block="1" operation="remove sp" except="inf" var="spge|spnoge"/> <form block="4" suffix="" features="short" var="ns"/> <form block="4" suffix="" features="short" except="ind.pres.1.sg" var="hs"/> <form block="4" suffix="" features="short.ind.pres.2|short.ind.pres.3.sg" var="fs"/> <form block="4" suffix="e" features="ind.pres.1.sg.long" var="hs|ns"/> <form block="4" fail="1" var="hs|ns"/> <form block="4" fail="1" features="long.ind.pres.2.sg|long.ind.pres.3.sg" var="fs"/> <form block="4" suffix="e" features="long"/> <form block="4" fail="1"/> <form block="5" suffix="" features="1.sg"/> <form block="5" suffix="st" features="2.sg"/> <form block="5" suffix="t" features="ind.pres.3.sg"/> <form block="5" suffix="n" features="1.pl"/> <form block="5" suffix="t" features="2.pl"/> <form block="5" suffix="n" features="3.pl"/>

</table>

Overabundance -a frequent phenomenon in German -is dealt with using an ad hoc 'length' feature, which can take one of the following values: short and long. In any case, realization rules are tried in their order of appearance within each block, until one is found that can be applied or that explicits a failure (fail="1").

Figure 4: The lexical entry for machen 'do' in DeLex, together with excerpts of the morphological description relevant for building its full paradigm. Everything involving participial forms has been discarded for the sake of readability.

Evaluation of DeLex

6.1. Recall on the TIGER corpus A first way to evaluate DeLex is simply to assess its coverage on a gold annotated corpus. We have extracted all words from the TIGER corpus (Brants et al., 2002; Smith, 2003) and performed a look-up for each of them in DeLex. We have discarded all tokens involving only digits and punctuation signs and have allowed lowercase look-ups whenever case-sensitive loop-up failed. Because DeLex does not contain any named entities, the raw coverage figure computed this way, namely 88.9%, is a lower bound of DeLex's coverage. Unsurprisingly, the most frequent TIGER tokens unknown to DeLex are named entities. 16 On the other hand, ignoring all tokens involving at least one capital letter provides an upper bound of the coverage, as many tokens involved named entities will be discarded, but all common nouns as well (in German, common nouns are capitalized). This upper bound is as high as 97.8%.

A maybe better estimate can be obtained by ignoring all tokens in TIGER that are annotated as "NE" (named entities). The resulting figure is 93.1%.

Comparison with the morphisto lexicon

Another way to evaluate DeLex is to compare it with the above-mentioned morphisto lexicon (Zielinski and Simon, 2009). The morphisto lexicon is an inventory of various lexical elements, among which "base stems." As far as nouns, adjectives and adverbs are concerned, almost all morphisto "base stems" are in fact citation forms in the same sense as for DeLex. Therefore, they can be compared with the one another. DeLex contains 39,795 unique nominal citation forms, whereas morphisto contains 9,165 nominal "base stems." Among them, 6,476 are also DeLex citation forms, whereas 2,481 are not. The latter are mostly proper names (Laos, Nike), but also nouns that are indeed missing from DeLex. On the other hand, 33,319 DeLex nominal citation forms are not among morphisto's "base stems." DeLex contains 6,568 unique adjectival citation forms, whereas morphisto contains 3,212 adjectival "base stems." 2,123 are shared by both resources, 1,089 morphisto adjectival "base stems" are unknown to DeLex, and 4,445 DeLex citation forms are not morphisto "base stems." Because the definition of what an adverb is or should be is not necessarily the same between DeLex and morphisto, we have looked for morphisto adverbial "base stems" within adverbial entries as well as within entries in all DeLex closed categories (i.e., excluding nouns, verbs and adjectives). Out of the 706 morphisto adverbial "base stems," 545 are known to DeLex as citation forms, whereas 161 are not.17

As far as verb forms are concerned, morphisto "base stems" are not citation forms any more. They are really stems, associated with an inflection class or a partial inflection class, or inflected forms associated with the corresponding morphological tag. We have therefore replaced all stems by one of the inflected forms that would be generated by the (partial) inflection class (e.g., we replaced the stem brat associated with the tag VVPres1+Imp by the form brate). Together with explicitely listed inflected forms from morphisto, we thus obtained a set of 4,243 unique verb forms. Out of these forms, 2,892 are known to DeLex as inflected forms. Out of the 1,041 other forms, the vast majority of them are infinitives, from rare verbs yet unknown to DeLex. Overall, DeLex has therefore a much larger coverage than the morphisto lexicon, but there is still a significant room for improvement for DeLex. As mentioned in the conclusions of this paper, corpus-based extension of DeLex is among our first priorities for future work.

Lexicon-based improvement of a POS tagger

Another way to assess the quality and usefulness a lexicon is to perform a task-based evaluation. We have performed POS-tagging experiments for quantifying whether using DeLex as an external lexicon within a statistical POStagger leads to improvements over the same POS-tagger when only trained on the training corpus. For these experiments we have used MElt (Denis and Sagot, 2012), a statistical POS tagging system that relies on Maximum-Entropy Markov Models trained on an annotated corpus and optionaly an external lexicon. MElt has been trained on data for French, for which is is state-of-the-art, and several other languages. We have trained MElt on a slightly modified 18 version of the TIGER corpus. More precisely, the corpus we used contains 50,499 sentences (888,533 tokens), from which we selected the first 45,000 sentences (788,448 tokens) for training and the remaining 5,499 sentences (100,859 tokens) for evaluation. The unknown token rate in the evaluation subcorpus (rate of words in the evaluation subcorpus that do not occur in the training subcorpus) is 9.2%. Our modifed version of the corpus relies on a 65-tag tagset. We have trained MElt on the training subcorpus, both without and with DeLex as an external lexicon (respectively MElt nolex de and MElt de ). 19 Results, displayed in Table 1, show that using DeLex allows for a significant increase in POS tagging accuracy, in particular on unknown words.

Conclusion and perspectives

The next steps of DeLex's development shall consist in large-scale unknown word extraction from various corpora and (semi-)automatic lexical entry creation, following (pronominal adverb, e.g., dafür 'for that') and PROWAV (adverbial interrogative or relative pronoun, e.g., warum 'why'). 18 Our modification only consists in assigning to punctuation tokens a tag identical to the token itself, instead of one of the three punctuation tags. 19 We extracted POS and morphological tags within DeLex and converted them into morphosyntactic categories compatible with our modified version of the TIGER corpus. previous comparable efforts on Spanish (Molinero et al., 2009) or French (Sagot et al., 2013). This should also allow us to automatically identify errors in the current version of DeLex, as non-covered forms of covered lemmas can only be the result or erroneous or missing lexical information and/or of errors in the inflectional grammar. Apart from extending and improving DeLex, we have already planned or initiated work using DeLex for various purposes such as quantitative morphology, acquisition of derivational lexical information, advanced tokenization and statistical parsing. Moreover, DeLex has already started to be used for extraction derivational morphlogical relations (Baranes and Sagot, 2014). Beyond derivation, dealing with the German constructional morphology in a linguistically sound and NLP-ready lexicon is a challenging task, that we would like to study in the future.

Figure 1 :

 1 Figure 1: Illustrating the first step of DeLex's construction on an example: extracting lexical entries and partial paradigms from the German Wiktionary

Figure 3 :

 3 Figure 3: A few (simplified) adjectival extentional entries. In this figure, an extensional entry includes an inflected form, a category, a morphological tag, and the lemma's citation form and inflection class.

Table 1 :

 1 Evaluation of the MElt de POS-tagger, which uses DeLex as an external lexicon, and comparison with its variant MElt nolex de which does not use DeLex. The tagset involved contains 65 distinct categories.

		TOKENS	ACCURACY
		NB.	%	MElt de MElt nolex de
	Known words	91,603 90.8%	97.8%	97.5%
	Unknown words	9,256	9.2%	90.6%	88.8%
	All words	100,859	100% 97.2%	96.7%

Distributed under the Creative Commons

3.0 BY-SA licence.3 For instance, the free dictionary in the Unitex platform[START_REF] Paumier | De la reconnaissance de formes linguistiques à l'analyse syntaxique[END_REF], contains 300,000 word forms that constitue 10% of the CISLEX lexicon[START_REF] Langer | CISLEX, an electronic dictionary for German. Its structure and a lexicographic application[END_REF].

Not only preposition-like elements can serve this role, see for example a noun-like element in heimkommen 'come home' (Heim 'home' [noun] + kommen 'come') or an adjective-like element in freigeben 'set free, release' (frei 'free' [adj.] + geben 'give'). Such elements are virtually always separable.

We use the term "inflection class" with its usual sense here. In parsli, the adequate term would be here "inflection pattern."

As said, we try and identify sets of lexical entries that share a common inflectional behavior. Clearly, lexical entries involving more than one stem will be treated suboptimally. This will be fixed later on by manually defining stem realization classes and merging sets of lexical entries that have the same behavior at the exponence level.

A few dozens of hours for a non-native speaker of German with a good linguistic knowledge of this language and a B1/B2 proficiency level, with access to online dictionaries and other online resources.

One should note that, in German, adjectives can often be used as adverbs, which explains why there are only 904 adverbial entries: such entries only cover non-adjectival adverbs, such as schon 'aldready,' zugleich 'similarly,' etwa 'approximately, around,' and so on.

In decreasing frequency order, the three most frequent unknown tokens are SPD (party name), Bonn (city), M. (almost always part of Frankfurt a. M., abbreviation for Frankfurt am Main, the city of Francfort). Note also some non-words such as ap or rtr, typical of article signatures.

Apart from ADV (standard adverbs), the most frequent DeLex (i.e., TIGER) category for morphisto adverbial base stems are PTKVZ (separable verbal particle), PROROAV

Acknowledgements

This work was partly funded by the French ANR national grant EDyLex (ANR-09-CORD-008) and the French "Investissements d'avenir" project PACTE.


Valérie Hanoka
email: valerie.hanoka@inria.fr

Benoît Sagot
email: benoit.sagot@inria.fr

YaMTG: An Open-Source Heavily Multilingual Translation Graph Extracted from Wiktionaries and Parallel Corpora

Keywords: Translation Graph, Multilingual Lexicon, Wiktionary

. We detail the translation extraction process for 21 wiktionary language editions, and provide an evaluation of the translations contained in YaMTG.

Introduction

Large-scale general-purpose multilingual translation databases are useful in a wide range of Natural Languages Processing (NLP) tasks. This is especially true for research efforts targeted to under-resourced languages. Translation databases can be used for adapting existing resources in other languages. This has been applied for example for the development of wordnets in languages other than English (see e.g., de Melo and Weikum ( 2009)). There is thus a real need in NLP for open-source multilingual lexical databases that compile as many translations as can be found on any freely available resource in any language. In this paper, we introduce YaMTG (Yet another Multilingual Translation Graph), a new open-source heavily multilingual translation database (over 664 languages represented). While developing YaMTG, we have put the emphasis on the three following features: 1. Our resource is intended for general purpose; 2. We tried to to find a right balance between 'large-scale' and 'quality', thus leaving a limited amount of noise in the data;

3. No information is inferred from the data in order to avoid noise propagation.

4. The resource is easily accessible, usable and extensible.

The remainder of this paper is organised as follows. We first overview the literature on related work (Section 2). Next, we describe in Section 3 how we extracted automatically a preliminary large-scale set of translation and synonym pairs from 21 wiktionary language editions and from the OPUS parallel corpora [START_REF] Tiedemann | News from OPUS -A Collection of Multilingual Parallel Corpora with Tools and Interfaces[END_REF]. We detail the translation extraction process for 21 wiktionary language editions, and provide an evaluation of the extracted translation and synonym pairs. Sections 4 and 5 are then respectively dedicated to the construction of the translation and synonym graph proper and to its filtering, in order to reduce the level of noise in the graph. Finaly, we provide in Section 6 quantitative data about the final (filtered) graph YaMTG, together with an overall evaluation of YaMTG and an assessment of the quality of our filtering step.

Related work

Research focused on the creation of heavily multilingual word-based translation databases tends to use collaboratively constructed lexicons such as wiktionaries. By doing so, the community is taking advantage of the 'the wisdom of crowds,' thus overcoming the lack of expert-built lexicons for some languages [START_REF] Meyer | Wiktionary: a new rival for expert-built lexicons? exploring the possibilities of collaborative lexicography[END_REF]. [START_REF] Etzioni | Lexical translation with application to image search on the web[END_REF] built TransGraph, a multilingual sensedistinguished dictionary extracted from wiktionaries and other dictionaries. It covers 100 languages, with 3 languages having over 100,000 words. This translation graph was later reshaped by [START_REF] Mausam | Compiling a massive, multilingual dictionary via probabilistic inference[END_REF] with the creation of the PanDictionary. It covers over 1000 languages, whose translations are still emanating from wiktionaries and other dictionaries. Yet, the resource doesn't seem to be openly available as we write. The PanLex project [START_REF] Baldwin | Panlex and lextract: Translating all words of all languages of the world[END_REF] To our knowledge, it is the most complete heavily multilingual translation database. However, we were not able to find an evaluation of the quality of the data. Some researchers have produced multilingual translation data as a side effect of other goals. The success of the Princeton WordNet (Fellbaum, 1998) as a generic language resource spurred the creation of similar resources in other languages. Many researchers in non-English speak-ing countries launched various projects, sometimes involving several teams and languages, all aimed at creating new wordnets in their respective languages [START_REF] Hamp | Germaneta lexical-semantic net for german[END_REF][START_REF] Farreres | Using wordnet for building wordnets[END_REF][START_REF] Pianta | Multiwordnet: developing an aligned multilingual database[END_REF][START_REF] Diab | The feasibility of bootstrapping an arabic wordnet leveraging parallel corpora and and an english wordnet[END_REF][START_REF] Sagot | Building a free french wordnet from multilingual resources[END_REF]. A list of resources which follows the wordnet design can be found on the Global WordNet Association website. 3 Researchers undertook several actions to establish relationship between wordnets in different languages. The EuroWordNet project [START_REF] Vossen | EuroWordNet: a multilingual database with lexical semantic networks[END_REF] devolved the manual creation and linkage of new wordnets in 8 european languages upon several institutions over many years. [START_REF] Atserias | The meaning multilingual central repository[END_REF] Among the resources mentioned above, the most interesting ones in terms of language coverage [START_REF] Baldwin | Panlex and lextract: Translating all words of all languages of the world[END_REF][START_REF] De Melo | Towards a universal wordnet by learning from combined evidence[END_REF][START_REF] Navigli | Ba-belNet: The automatic construction, evaluation and application of a wide-coverage multilingual semantic network[END_REF]) make use of a specific translation graph architecture which links translations to meaning nodes rather than translations between them. This data structure tends to amplify noise in the translation data in case of a erroneous link between a term in a language and a meaning node.

3 http://globalwordnet.org/?page_id=38

Compared to these works, YaMTG contains less lexical items (900,000 vs. 1.5M for UWN, 12M for PanLex and 44M for BabelNet 2.0; see Table 1 for details). This is because it was designed using fewer sources, keeping its development cost low. We plan to include more translations to YaMTG in order to increase it's coverage soon. However, only PanLex covers more languages than YaMTG.

Translation Extraction

Wiktionaries 3.1.1. Extraction

As described by [START_REF] Meyer | Wiktionary: a new rival for expert-built lexicons? exploring the possibilities of collaborative lexicography[END_REF], Wiktionary is a multilingual online dictionary encoding linguistic knowledge in multiple languages. There exists independent wiktionaries for each language. Those are called language editions. In each language edition, a wiki page corresponds to a character string which can represent one or more lexical items. These items can either be in the editions' language, in other languages or both. Language editions dumps are made available by the Wikimedia Foundation at http://dumps.wikimedia.org/. We extracted translations from the dumps4 available in September 2013.

Wiktionaries are organised as a set of formatted pages, whose title may correspond to an article describing one or more lexical entries. The page content of the different language editions displays different structures for the linguistic information of the lexical entries. [START_REF] Sérasset | Dbnary: Wiktionary as a lmf based multilingual rdf network[END_REF] plans to build and provide open-source access to an a languageindependent automatic extractor for a comprehensive set of linguistic information encoded in different wiktionary language editions. It is possible to custom a set of language specific triggers (Figure 1) for the identification and extraction of translations and synonyms from wiktionary pages. These triggers are instantiated by regular expressions. For instance, English edition triggers are: • All_swiches_OFF: ^=

• Ignore:

• Source_Language: space="preserve">{?{?-?(.[^-]*)-?}?}?

• Implicit_translations:

^*#\s*\(([^)]*)\)\s*\[\[.*
Using this customized triggers for every article page in a wiktionary edition enable the extraction of raw translation data. Figure 2 shows an example of raw data extracted from the Hindi edition with the set of Hindi triggers presented above.

Noisy elements (e.g., punctuation characters) are then removed. Languages names (e.g., हिन् दी5 ) and language codes (e.g., 'en' for English, 'gu' for Gujarati) are converted in ISO 639-2 (alpha-3) language codes. 6 The information is put in a normalised format (

Evaluation of the extraction process

Table 3 presents We chose not to identify this trigger because, as it is also used to mark related terms (συγγενικά). Therefore, it would bring much noise in the data. We evaluated the wiktionary extraction process for the 21 languages edition we processed by randomly selecting 3150 translation pairs (150 extracted from each language edition) and manually checking their adequacy against the corresponding wiktionary. Table 4 presents the result of this evaluation with an error analysis. Accuracy scores range from 78% (Spanish) to 97.3% (Dutch). We identified 4 major error classes:

1. translations whose source or target language hasn't been successfully identified;

2. definitions mistaken for translations; Most instances of class 1 to 3 errors will be discarded by the filtering step described in Section 5. Type 4 ('miscellaneous') errors are the most detrimental, as most of them will remain in the data.

Table 4 indicates a mean score of 87.6% for the unfiltered extraction. However, this score is biased because smallest languages edition are overrepresented. Weighting the scores by the size of the extraction (Table 3) leads to a more reliable score of 89.4%.

Opus

OPUS [START_REF] Tiedemann | News from OPUS -A Collection of Multilingual Parallel Corpora with Tools and Interfaces[END_REF]) is a collection of translated texts from the web aimed at providing the community with a publicly available parallel corpus in a wide variety of languages. We used the word alignments issued on October 9, 2013 to add new translations links in the graph, retaining only alignments with a frequency at least 10. With this threshold, we retained 576131 terms (or lexical items) for 3598227 translations in 31 languages from OPUS translations. As OPUS translations are prone to alignment errors, their quality may not be as good as desired. Yet they contains a substantial amount of inflected forms which are not necessarily present in other sources. We rely on the filtering steps (Section 5) to avoid retaining the worst translations.

Building the Translation Graph

Previous steps resulted in more than 18M translations or synonyms in 4,324 languages.7 Note that while wiktionary translations are mainly lemmas, OPUS translations consists in a lot of inflected terms' translations. We kept them in the graph, but it is easy to get rid of them as we kept translations origins in the data. Discarding translations originating from OPUS only thus removes inflected terms. All these translations contains a lot of noise such as errors identified in Section 3.1.2 plus alignment flaws arising in OPUS data.

Filtering

Unfiltered translations gathered in previous steps need to be filtered in order to decrease the noise. We removed many translations matching one of the following criteria:

• They are 'too long' (usually Section's 3.1.2 class 3 errors, i.e., definitions mistaken for translations) or 'too short'8 (usually stopwords);

• They contain punctuations and/or numbers;

• Their charset does not correspond to their declared language;

• They appear in less than X other translations;

• They appear in more than Y other translations (when too many translations are available for the same word, it turns out it is very often the result of an error).

The first three filtering criteria are designed to avoid errors from class 1 to 3 as identified in Section 3.1.2 The two last criteria were rather designed to get rid of types 4 errors (miscellaneous errors) and OPUS misalignment. We empirically chose X = 3. Concerning Y , we assumed that it may be somehow related to the number of languages present in the graph. After applying the first filtering criteria, the graph displays terms (lexical items) in 3,739 languages. For those languages, only 468 appears 100 times or more. We assumed that a term should reasonably not have more than an average of 2 translations for all of these 468 languages.

In other words, we decided that a term should not be involved in more than Y = 936 translations.

Results and evaluation

Graph properties

The filtered graph retains nearly a tenth of the terms and 15% of the languages present in the unfiltered graph. Yet, 32.3% of the translation/synonym links, or edges, are retained (cf. Table 6). 51.2% of these remaining translations are extracted solely from OPUS, 42,1% from wiktionaries only, whereas 3.2% were extracted from both OPUS and at least one of the 21 wiktionaries we processed. Table 6: Number of terms, translations and synonyms before and after applying filters.

Figure 3 displays the degree distribution in the graph. This distribution resembles that of a small-world graph, a type of graph often encountered when modeling real-world data.

Small-world graphs have tightly interconnected node clusters, and a shortest mean path length that is similar to that of a random graph that has the same number of nodes and the same number of edges. We checked the 'small-world-ness' of our translation graph against the definition of that notion as presented by [START_REF] Humphries | Network `small-world-ness': a quantitative method for determining canonical network equivalence[END_REF].9 It turns out that our translation graph indeed qualifies as a small-world graph.

Figure 4 illustrates the distribution of translation or synonymy pairs w.r.t. the languages at both ends of the pair.

path length in G (resp. G rand ). Moreover, let n ∆ G the number of triangles in G and n ( 2)G the number of paths of length 2 in G.

We then define the transitivity-based clustering coefficient in G as We evaluated the quality of the final graph by looking over 1,639 edges selected as follows:

C ∆ G = 3n ∆ G /n ( 2)G.
• 1,439 edges which were evaluated in Section 3.1.2 and still present in the final graph. This will allow us to evaluate the reliability of the filtering described in Section 5;

• 200 more edges randomly chosen in the translation graph, in order to assess its overall accuracy in a non biaised way.

These edges cover 3,332 terms in 72 languages.

Overall evaluation

We estimated the overall accuracy of the links in the filtered graph based on a manual evaluation of the 200 randomly selected pairs. The overall accuracy result is 91% (see Table 7). Amongst the 18 incorrect links in this random evaluation dataset (9%), 4 are incorrect because one of the terms is a small-world graph whenever LG ≥ LG rand and C ∆ G ≫ C ∆ G rand . 10 This figure is lower than one would expect given the results displayed in Figure 4. This is an artifact related to the low number of pairs randomly selected for evaluation, resulting in a even lower number of errors among pairs extracted from wiktionaries only (8 errors). As we shall see below, the accuracy of translation pair extracted from wiktionaries is closer to 97%.

have been assigned a wrong language, 2 still contain noise and 13 are miscellaneous errors (mostly alignment errors from OPUS). Note that all wikipedia definitions mistaken for translations were successfully removed by the filter.

Filter's accuracy

Amongst the 3,150 translation links evaluated in Section 3.1.2, 1,439 (45.3%) were still present in the final graph. Table 4, restricted to links extracted from wiktionaries, compares the figures before and after filtering. It displays an average accuracy score of 95.8% for the filtered extraction. Weighting the scores by the proportion of translations/synonyms from the different language editions present in the final graph (Table 8) gives us a final accuracy score of 97.2% for translations extracted from wiktionaries. On average, the filter managed to improve accuracy scores by nearly 8 points the (raw and weighted).

Conclusions

YaMTG is an Open-Source Heavily Multilingual Translation Graph, freely available online.11 Its aim is mainly to provide a ready-to-use translation graph. Further versions including other translation sources are under development and will be released soon. The version of YaMTG presented in this paper contains 5.8 million translation or synonymy pairs (edges in the graph) covering 664 languages. Those translations were extracted from 21 wiktionary language editions and from the OPUS Parallel Corpora, and then filtered using generic and graph-based heuristics. We evaluated the quality of the initial set of translations extracted from our various source, the precision of our filtering step and the accuracy of its output, namely YaMTG. We estimated overall quality of the translation/synonymy links within YaMTG as approximatively 91%. If one only considers translation/synonymy links extracted from wiktionaries, this figure reach as much as 97.2%. In the future, these scores could be improved with a twofold strategy:

• Adding new translations to the unfiltered graph. The filtering step removes lexical items which are not present in at least X other translations. Thus, satisfying translations which tend to occur less in translation databases are more likely to be removed by the filter. More pairs in the unfiltered graph should lead to less overfiltering.

•

  Translation_ON: {{trans-top.* • Translation_OFF: {{trans-bottom}} • Synonyms_ON: ====Synonyms==== • Synonyms_OFF: {{checksyns}} • All_swiches_OFF: ^\s*$ • Ignore: {{trans-[^tb].+}}

Figure 1 :

 1 Figure 1: Language independent set of triggers, switches and side effects for translation and synonymy extraction from wiktionaries.

  the proportion of wikipedia entries from which translations or synonyms have been extracted for 21 languages edition, together with the number of raw (unfiltered) translation pairs extracted. The biggest wiktionary editions (English and French) have an extraction for roughly a quarter of their non-empty entries (respectively 18.4% and 17.8%). The best extraction rates (90.1% and 80.9%) have been achieved for the Portugese and Romanian editions. Greek and Polish are the two edition having worst extraction rates (11.7% and 11.4%). For Greek, that poor score seem to be mainly due to implicit translations which are not extracted from articles in another language. More precisely, we retain only one trigger for implicit translations (^#\s*\[\[.+\]\].?$). Yet many implicit translations are marked by the pattern ^* \[\[.

Figure 3 :

 3 Figure 3: Degree distribution in the filtered graph YaMTG.

Figure 4 :

 4 Figure 4: Language matrix for the 25 most frequent languages in the graph (darker means more translation or synonymy pairs).

Table 1 :

 1 Comparison of the basic figures for the main multilingual translation resources and YaMTG

	Multilingual Resource # Lexical Entries # Languages
	YaMTG	881,643	664
	UWN	1,481,412	419
	PanLex	12,000,000	1353
	BabelNet 2.0	44,490,880	50

Table 2

 2

	Figure 2: Fragment of the raw extraction of translation for
	the Hindi wiktionary.
	Source language	Target language	Wiktionary
	Lang	Term	Lang	Term	edition (lang.)
	hin hin hin hin hin hin hin hin hin hin	आसमान आसमान आसमान आसमान आसमान आसमान आसमान आसमान आसमान आसमान	hin hin hin hin eng guj ben spa fra fas	hi hi hi hi hi hi hi firmamento hi अम् बर आकाश गगन नभ sky આકાશ আকাশ ciel hi آﺳﻤﺎن hi

), keeping track of the origin of the translation at the end of each line (e.g., #WY_hi stands for Hindi wiktionary; see Table

2

).

Table 2 :

 2 Fragment of the normalized translation extracted from the Hindi wiktionary.

Table 4 :

 4 Evaluation figures for the unfiltered extraction of translations/synonymy from 21 wiktionary editions (sample size for each language: 150), and for the filtered graph.

				Before Filtering				After Filtering
	Lang.		Erroneous		Good		Erroneous		Good	Kept
	edition	Misc. lang. def. noise	raw	%	Misc. lang. def. noise	raw	%	%
	bg	8	1	1	4	136 90.7	3	0	1	0	93 95.9	64.7
	cs	4	0	19	5	122 81.3	0	0	2	1	16 84.2	12.7
	da	2	0	3	2	143 95.3	1	0	0	0	65 98.5	44
	de	4	1	0	2	143 95.3	1	0	0	0	46 97.9	31.3
	el	4	0	5	2	139 92.7	1	0	2	1	65 94.2	46
	en	1	5	0	4	140 93.3	1	1	0	0	111 97.4	76
	es	5	0	15	13	117	78	0	0	0	1	46 97.9	31.3
	fr	4	0	8	0	138	92	2	0	3	0	95	95	66.7
	hi	9	3	3	4	131 87.3	1	0	0	0	42 97.7	28.7
	hu	3	9	1	1	136 90.7	1	2	0	0	56 91.8	40.7
	it	0	4	14	2	130 86.7	0	1	0	0	58 96.7	40
	ja	2	10	26	15	97 64.7	1	3	8	0	38 71.7	35.3
	nl	0	0	0	4	146 97.3	0	0	0	0	68	100	45.3
	pl	3	3	0	2	142 94.7	0	0	0	0	68	100	45.3
	pt	5	3	18	4	120	80	2	0	0	0	89 97.8	60.7
	ro	0	0	13	3	134 89.3	0	0	0	0	116	100	77.3
	ru	10	1	4	2	133 88.7	1	0	0	0	95	99	64
	sk	5	7	12	0	126	84	2	0	0	0	39 95.1	27.3
	sv	4	4	9	0	133 88.7	2	3	0	0	69 89.6	51.3
	tr	2	11	4	0	133 88.7	0	1	1	0	83 96.5	57.3
	vi	9	0	20	0	121 80.7	0	0	0	0	31	100	20.7
	Total	84	62 175	69 2760	-	19	11	17	3 1389	-	-
	%	2.7	1.9	5.6	2.2	87.6	-	1.3	0.8	1.2	0.2	96.5	-	-
		Wiktionaries		OPUS	Union
	Terms		5,246,987		487,251	7,903,646
	Transl./Syn. links	14,449,087 3,631,229 18,080,316
	Languages		4,324		32	4,324

Table 5 :

 5 Number of unfiltered translations and synonyms links in wiktionaries and OPUS and the union of both

Before filtering After Filtering (YaMTG)

	Terms	7,903,646	881,643
	Transl./Syn.	18,080,316	5,842,279
	Languages	4,324	664

Table 7 :

 7 Evaluation figures for the final (filtered) translation/synonymy graph, i.e., YaMGT based on a randomly selected sample of 200 edges.

http://panlex.org/tech/plrefs.shtml (consulted in

February 2014) 2 http://www.panlex.org/

They can be downloaded directly at http:// dumps.wikimedia.org/*L*wiktionary/latest/ *L*wiktionary-latest-pages-articles.xml. bz2, replacing the '*L*' part in the URL by the ISO-639-1 language code of the desired language edition.

हिन् दी is the Hindi term for 'Hindi.'

List available at http://www.loc.gov/standards/ iso639-2/php/code_list.php

As a standard of comparison, the Ethnologue association states living languages number more than 7,105 (http://www. ethnologue.com/).

This filter does only concern less than 3 letters words written in Cyrillic, Greek and Latin script.

[START_REF] Humphries | Network `small-world-ness': a quantitative method for determining canonical network equivalence[END_REF] define a graph G as being small-world by constrast with an Erdős-Rényi random graph G rand that has the same number of nodes and edges than G in the following way. Let us call LG (resp. LG rand ) the average shortest

http://alpage.inria.fr/~hanoka/yamgt. html

 a). Note that a same translation link (or edge) may have been extracted from more than one sources (e.g., wiktionary language editions) and can be therefore counted more than once (see subtable b).

• Similarly, playing with the thresholds X and Y (see section 5) may improve the quality of the graph. But increasing X requires to have a sufficient amount of translations.


Kevin Elis
email: kevin.elis@recherche.enac.fr

Alexandre Chabory
email: chabory@recherche.enac.fr

Jérôme Sokoloff
email: jerome.sokoloff@laplace.univ-tlse.fr

3D interaction of Gaussian beams with dichroic surfaces for the modeling of quasi-optical systems

Keywords: component-Gaussian beam, dichroic surface, spectrum matching, quasi-optics

Full Paper-In quasi-optical systems, dichroic surfaces are used to split, filter, or mix signals. Such surfaces cannot be modeled by existing Gaussian-beam methods. In this article we propose a technique to model the interaction of a Gaussian beam (GB) with a dichroic surface based on a matching in the spectral domain. For a GB illumination of the surface, the reflected and transmitted fields are approximated by one reflected and one transmitted Gaussian beams. Their characteristics are determined by means of the spectral matching through a second-order approximation of the dichroic surface response when excited by plane-waves. Numerical experiments and comparisons with reference solutions show the efficiency of the method in terms of accuracy and computation time.

I. INTRODUCTION

In mm-wave radiometry, quasi-optical systems are used to guide, filter, split, or mix the input signals. Such systems are generally constituted by the combination of different elements, e.g. mirrors, lenses, dichroic filters, or horn antennas. The number and the size of these elements render their modeling difficultly amenable by classical methods [START_REF] Bondo | Physical optics analysis of beam waveguides using auxiliary planes[END_REF] in acceptable computation times. For such systems, even physical optics may become computationally costly. The use of Gaussian beams (GBs) constitute a possible alternative [START_REF] Chabory | Application of Gaussianbeam based techniques to the quasi-optical systems of radiofrequency radiometers[END_REF]. Such methods are generally divided in two parts. In the first one, the incident field is expanded into a finite sum of GBs. In the second part, the beams are tracked through the system. The interaction of a GB with many elements of quasi-optical systems is well-known. Indeed, in [START_REF] Deschamps | Ray techniques in electromagnetics[END_REF], Deschamps has demonstrated that the fields reflected and/or transmitted at a dielectric/metallic moderately-curved interface illuminated by an incident GB can be approximated by Gaussian beams. In contrast, dichroic surfaces cannot be modeled by such methods because they are in some cases structures which period may not exceed a fraction of wavelength [START_REF] Munk | Frequency Selective Surfaces: Theory and Design[END_REF], [START_REF] Winnewisser | Characterization and application of dichroic filters in the 0.1-3-THz region[END_REF].

To deal with periodic structures, one generally assumes a plane-wave illumination. The response can then be computed with the help of the method of moments coupled with a Bloch-Floquet modal description [START_REF] Mittra | Techniques for analyzing frequency selective surfaces -a review[END_REF]. For the modeling of a complete quasi-optical system, one has to integrate this response [START_REF] Yu | 3D design software for quasi-optical systems based on integrated methods[END_REF].

In this article, we propose a solution to model the interaction of a GB with a dichroic surface. The method consists in assuming that the fields reflected and transmitted by the interface are GBs. This is realized with a general analytical formulation of GBs that includes the fundamental GB [START_REF] Goldsmith | Quasioptical Systems: Gaussian beam, quasioptical propagation and applications[END_REF]. The characteristics of the reflected and transmitted beams are found by means of a matching in the spectral domain. This involves an approximation of the dichroic surface response for plane-waves illuminations. This article is organized as follows. Details about the configuration are presented in Section II. In Section III we briefly present GBs. Next in Section IV, we explain the spectrum-matching method. Numerical experiments are performed to confront the results with reference methods in the last section.

II. CONFIGURATION

The configuration is presented in Fig. 1. We associate a reference frame (0 i , xi , ŷi , ẑi ) with the incident GB. The dichroic surface is of arbitrary shape and thickness, and can be associated with a coordinate system(0 d , xd , ŷd , ẑd ). This dichroic surface is either made of periodic elements or dielectric layers. In order to describe the orientation of the different reference frames, we use the classic definition of Euler angles such as presented in Fig. 2.

The dichroic surface response is assumed to be already known, for example in a tabulated form. In practice, this response can be computed numerically using a method based on Floquet modes, for example via [START_REF] Mittra | Techniques for analyzing frequency selective surfaces -a review[END_REF].

III. GAUSSIAN BEAMS

A. Definition

Gaussian beams are analytical solutions of the approximated wave equation upon the paraxial approximation. This requires that the field is only weakly diverging with respect to its main propagation direction. The maximal angle of divergence generally admitted is of 20 o [START_REF] Goldsmith | Quasioptical Systems: Gaussian beam, quasioptical propagation and applications[END_REF].

In this article, we use a formulation as general as possible in order to describe accurately the reflected and transmitted fields. The transverse electric field of a GB in z i = 0 is so that

E t ,i (x i , y i , 0) = u i (x i , y i , 0) a i x xi + a i y ŷi (1)
with a i x and a i y the coefficients associated with the x i and y i polarization, respectively. Besides u i represents the scalar GB expression in z i = 0, and is given by

u i (x i , y i , 0) = exp -j k 2 ρ i -ρ i0 t Q i0 ρ i -ρ i0 -j β t i ρ i -ρ i0 .
(

) 2
with k the wavenumber and

ρ i = x i y i , ρ i0 = x i0 y i0 and β i0 = β xi0 β y i0 .
Q i0 represents the complex curvature matrix of the GB. When diagonal, this matrix can be related to the sizes and the positions of the waists of the GB. From these expressions, in its reference frame the beam is characterized by five parameters, i.e. two complex amplitudes (a i x , a i y ), a complex curvature matrix Q i0 , a spectral shift β i0 and a spatial shift ρ i0 .

B. Paraxial formulation

Assuming a propagation of the beam towards z i ≥ 0, the field can be expressed as

E i (x i , y i , z i ) = ∞ ∞ Ẽi (k xi , k y i )e -j k t i r i dk xi dk y i (3) with k t i = k xi k y i k zi the wavevector, r t i = x i y i z i and, Ẽi (k xi , k y i ) = ũi (k xi , k y i )e i (k xi , k y i ). (4)
In this expression,

e i = a i x xi + a i y ŷi -k xi /k zi a i x + k y i /k zi a i y ẑi , ( 5
) ũi = j 2π k det (Q i0 ) exp j 1 2k k ρi -β i0 t Q -1 i0 k ρi -β i0 + j ρ t i0 k ρi , (6)
with

k ρi = k xi k y i .
If

β zi0 = k 2 -β 2 xi0 -β 2 y i0
< k, the GB spectrum is centered on a propagative plane wave. We associate with this GB a new coordinate system (0 β , xβ , ŷβ , ẑβ ) such as

  x β y β z β   = P i→β   x i -x i0 y i -y i0 z i   , (7)
where P i→β is a 3 × 3 transformation matrix from the incidence reference frame to a new coordinate system with Euler angles (φ 0 , θ 0 , 0) as illustrated in Fig. 3.

Fig. 3: Reference frame of the incident GB.

Note that

φ 0 = angle β xi0 , β y i0 θ 0 = acos β zi0 /k . ( 8
)
In this coordinate system, the asymptotic evaluation of (3) via the steepest descent path method [START_REF] Felsen | Radiation and Scattering of Waves[END_REF] yields the classical paraxial formulation of GB

E i (x β , y β , z β ) =u i (x β , y β , z β ) a βx xβ + a βy ŷβ - j k a βx ∂u i ∂x + a βy ∂u i ∂y ẑβ , (9)
with

u i (x β , y β , z β ) = det Q β (z β ) det Q β0 exp -j kz β . exp -j k 2 ρ t β Q β (z β )ρ β , (10)
and

Q -1 β (z β ) = Q -1 β0 + z β I, (11)
inside which I is the identity matrix. In the new coordinate system, the beam curvature matrix Q β0 and coefficients (a βx , a βy ) are given by

  a βx a βy 0   = P i→β   a i x a i y -cos φ 0 a i x -sin φ 0 a i y   , ( 12
)
and

Q β0 = R -1 t Q -1 i0 R -1 (13) with R = 1 cos θ 0 cos φ 0 -cos θ 0 sin φ 0 sin φ 0 cos θ 0 cos φ 0 , (14)
Note that an alternative formulation can be obtained that only depends on the far-field approximation by applying the stationary phase method to (3).

IV. METHOD

A. Principle

We suppose that the incident GB yields one transmitted and one reflected GBs. Because the method is similar for both cases, only the study of the transmitted field will be exposed in details. The transmitted GB must be as close as possible to the transmitted field. Therefore, the goal is to determine its reference frame and parameters.

In order to keep spectrum with Gaussian amplitudes, we make the matching of the incident/transmitted spectrum in the reference frame of the incident beam. Besides, the spectrum matching is realized on the transverse components of the spectrum. For the incident GB, these components can be expressed as

Ẽt,i (k xi , k y i ) = ũi (k xi , k y i ) a i x xi + a i y ŷi (15)
For the transmitted field to be approximated by a GB, its transverse components must be approach by a similar form, i.e.

Ẽt,tr (k

xi , k y i ) = ũt (k xi , k y i ) a t x xi + a t y ŷi (16)
The principle of the method can be divided in two parts. We find the expression of the transmitted spectrum from the incident spectrum and the response of the dichroic surface.

Then, this expression is matched to (16).

B. Expression of the transmitted spectrum

Each plane wave impinging the dichroic plate is expanded in parallel and orthogonal polarizations. For the incident spectrum (4), this yields

Ẽ i = ũi (e i .ê ), Ẽ ⊥ i = ũi (e i .ê ⊥ ). ( 17
) with ê⊥ = ŝi × nd ∥ ŝi × nd ∥ , ê = ê⊥ × ŝi , ŝi = k i /k, (18)
and nd the normal to the dichroic plate. Making use of the dichroic surface response, we end up with the orthogonal and parallel components of the transmitted spectrum

Ẽ t r Ẽ ⊥ t r = T , T ,⊥ T ⊥, T ⊥,⊥ Ẽ i Ẽ ⊥ i , ( 19
)
where T , , T ,⊥ , T ⊥, , T ⊥,⊥ are the transmitted coefficients associated with both polarizations. Finally, the transmitted spectrum is expressed as

Ẽtr = Ẽ t r ê + Ẽ ⊥ t r ê⊥ . (20)
Using ( 17), ( 19) and ( 20), the transverse components of Ẽtr can be formulated as

Ẽt,tr = ũi (k xi , k y i )S(k xi , k y i ), (21)
with

S = S x (k xi , k y i )x i + S y (k xi , k y i )ŷ i . ( 22
)

C. Spectrum matching

To match ( 16) with (21), we need the following approximation of S(k xi , k y i )

S ≈ S a = s x0 xi + s y 0 ŷi exp k t ρi ρ s + k t ρi Q s k ρi ( 23
)
where (s x0 , s y 0 ) are complex coefficients, ρ s is a complex vector, and Q s is a complex symmetric matrix. (s x0 , s y 0 ) are obtained by matching S with S a at the maximal value of the incident spectrum, i.e. k xi = β xi0 and k y i = β y i0 . Then, we project (23) on s x0 xi + s y 0 ŷi . We end up with

S. s * x0 xi + s * y 0 ŷi |s x0 | 2 + |s y 0 | 2 ≈ exp k t ρi ρ s + k t ρi Q s k ρi . ( 24
)
This approximation must be valid in the interval where the incident spectrum is significant. In order to obtain these coefficients, we perform a point matching at nine different plane-waves: one at k xi = β xi and k y i = β y i and two at half of the maximal power of the incident GB spectrum for φ = (0 • , 45 • , 90 • , 135 • ).

D. Identification

From ( 21) and ( 23), we find

Ẽt,tr ≈ ũi (k xi , k y i ) exp k t ρi ρ s + k t ρi Q s k ρi . s x0 xi + s y 0 ŷi . (25)
We insert ( 6) and ( 25) in (21) and we obtain

Q -1 t r 0 = Q -1 i0 -j 2kQ s β t r 0 = Im Q -1 t r 0 -1 Im Q -1 i0 β i0 + kRe ρ s ρ t r 0 = ρ i0 -Im ρ s + β t r 0 Re Q -1 t 0 -β i0 Re Q -1 i0 k a t r x a t r y = s x0 s y 0 det Q t r 0 exp j 1 2k β t i0 Q -1 i0 β i0 det Q i0 exp j 1 2k β t t r 0 Q -1 t r 0 β t r 0 ( 26
)
We have completely specified the transmitted GB. The reflected beam can be determined in a similar way. Its reference frame is chosen as the image of the incident reference frame. Finally, either the paraxial or the far-field formulation of GBs ( 9) can be employed to compute the fields at any point of space.

V. NUMERICAL CONFRONTATIONS WITH REFERENCE METHODS

A. Configuration

For validation purpose, we consider a dichroic surface constituted by a thick infinite dielectric slab of relative permittivity ε r = 4 and thickness 2λ 0 placed at z i = 0.1m. The Euler angles between the incident beam reference frame and the dichroic plate reference frame are (0 • , 45 • , 0 • ). The frequency is set at 100GHz. The configuration is presented in Fig. 4.

B. Methods

We confront the Gaussian beam matching (GBM) with two reference methods. Because the dichroic surface is only made by large dielectric interfaces, we can compare our results with the GB tracking method (GBT). This method is classically used in GB techniques [START_REF] Chabory | Application of Gaussianbeam based techniques to the quasi-optical systems of radiofrequency radiometers[END_REF].

The second one is based on the radiation of equivalent currents. They are obtained by modeling the dichroic surface by means of the reflexion and transmission coefficients.

This method is denoted "RTC" and is classically used to model dichroic surfaces in quasi-optical systems [START_REF] Yu | 3D design software for quasi-optical systems based on integrated methods[END_REF]. Contrarily to GBT, this method is exact but slow.

C. Incident field

The incident beam parameters are

a i x a i y = 1 0 , β i0 = 0 0 , ρ i0 = 0 0 (27)
and

Q -1 i0 = j kW 2 0 2 -z i0 0 0 j kW 2 0 2 -z i0 , ( 28
)
with W 0 = 1.5λ 0 and z i0 = 0.1m the sizes and positions of the beam waist. In other words, the incident field is a fundamental GB polarized according to x i with a waist placed at the dielectric slab.

D. Results

In Fig. 5, we show the total near-field obtained with GBM. In Fig. 6, we display the difference of this result with the reference solution RTC. We see that GBM yields very low errors, that mainly originate in the paraxial approximation. This is confirmed by the far-field patterns displayed in Figs. 7 and8. Beam parameters are modified when passing through the surface, e.g. the spatial and angular shifts are of

φ r e f 0 θ r e f 0 ≈ 0 -2.2 • and ρ r e f 0 ≈ 1.7λ 0 , (29)
for the reflected GB, and of

φ t r 0 θ t r 0 ≈ 0 -0.3 • and ρ t r 0 ≈ -1.12λ 0 , (30)
for the transmitted GB. The complex curvature matrices of both GBs are significantly modified. Moreover, due to the tracking process, GBT yields several transmitted and reflected GBs, which is not the case with GBM. This explain why GBM outperforms GBT in terms of computation time in this simulation. Note also that we observe in the far-field pattern that GBT does not model the angular shift.

VI. CONCLUSION

In this paper, we have proposed a method to treat the interaction of a Gaussian beam (GB) with a dichroic surface. This has been achieved by assuming that the transmitted and reflected fields are also GBs. The method uses general analytical formulations for the GBs that depend either on a paraxial or far-field approximation. The beam parameters have been obtained by means of a matching in the spectral domain.

To validate these formulations, numerical tests have been performed for a thick dielectric slab. We have presented comparisons with two reference methods in the far-field and near-field zones. We have observed results that show that the modifications of the beam when passing through the surface are correctly estimated, notably the spatial and angular offsets. This method can be used with any incident field represented by a sum of GBs. Thus it can be used to model a complete quasi-optical system that includes dichroic surfaces.

Simulations of a complete quasi-optical system including periodic dichroic surfaces will be realized in future works.

Fig. 1 :

 1 Fig. 1: Configuration.

Fig. 2 :

 2 Fig. 2: Euler angles.

Fig. 4 :

 4 Fig. 4: Configuration.

Fig. 5 :Fig. 6 :

 56 Fig. 5: Normalized total electric field obtained with GBM (dB): (a) plane φ = 0 • , (b) plane φ = 90 • .

Fig. 7 :

 7 Fig. 7: Normalized far field (dB), reflected (left) and transmitted (right). Obtained with GBM, GBT, and RTC.

Fig. 8 :

 8 Fig. 8: Difference of GBM and GBT with RTC in the far field (dB).

ACKNOWLEDGMENT

The authors would like to thank the CNES and the ENAC for supporting this work under a PhD.