Svetla Koeva
email: svetla@ibl.bas.bg

Cvetana Krstev
email: cvetana@matf.bg.ac.yu

Duško Vitas
email: vitas@matf.bg.ac.yu

Morpho-semantic Relations in Wordnet -a Case Study for two Slavic Languages

Keywords: global wordnet, morpho-semantic relations, derivational relations

In this paper we present the problem of representing the morphosyntactic relations in wordnets, especially the problems that arise when wordnets for languages that differ significantly from English are being developed on the basis of the Princeton wordnet, which is the case for Bulgarian and Serbian. We present the derivational characteristics of these two languages, how these characteristics are presently encoded in corresponding wordnets, and give some guidelines for their better coverage. Finally, we discuss the possibility to automatically generate new synsets and/or new relations on the basis of the most frequent and most regular derivational patterns.

Introduction

The aims of this paper are to present the current stage of the encoding of morphosemantic relations in Bulgarian and Serbian wordnets, briefly to sketch the derivational properties of Slavic languages based on the observations from Bulgarian and Serbian, to discuss the nature of morpho-semantic relations and its reflection to the wordnet structure and to analyze the positive and negative consequences of an automatic insertion of Slavic derivational relations into it.

The wordnet is a lexical-semantic network which nodes are synonymous sets (synsets) linked with the semantic or extralinguistic relations existing between them [START_REF]WordNet: An Electronic Lexical Database[END_REF], [START_REF] Miller | Five Papers on WordNet[END_REF]. The wordnet structure also includes semantic and morpho-semantic relations between literals (simple words or multiword expressions) constituting the different synsets. The representation of the wordnet is a graph. The cross-lingual nature of the global wordnet is provided by establishing the relation of equivalence between synsets that express the same meaning in different languages [15].

The global wordnet offers the extensive data for the successful implementation in different application areas such as cross-lingual information and knowledge management, cross-lingual content management and text data mining, cross-lingual information extraction and retrieval, multilingual summarization, machine translation, etc. Therefore the proper maintaining of the completeness and consistency of the global wordnet is an important prerequisite for any type of text processing to which it is intended.

The structure of the paper outlines the underlined goals. In the following section we present a short analysis of related work. In the third section, we briefly describe the properties of Slavic derivational morphology based on examples from Bulgarian and Serbian and their reflection into the wordnet structure. The forth section explains how the morpho-semantic relations are encoded in Bulgarian and Serbian wordnets respectively. We then discuss the manners to incorporate the (Slavic) derivational relations into the wordnet structure and some limitations of their automatic insertion. Finally, we raise some problematic questions connected with the presented study and propose future work to be done.

Related work

Wordnets have been developed for the most of the Slavic languages -Bulgarian, Serbian, Czech, Russian, Polish, Slovenian, and some initial work has been done for Croatian. Wordnets for three Slavic languages (Czech -started with the EuroWordNet (EWN), Bulgarian and Serbian) have been developed in the scope of the Balkanet project (BWN) [START_REF]Design and Development of a Multilingual Balkan Wordnet (BalkaNet IST-2000-29388)[END_REF], [START_REF] Stamou | BALKANET: A Multilingual Semantic Network for the Balkan Languages[END_REF] and later on continue developing as nationally funded projects 1 or on the volunteer basis.

Originally, the Princeton wordnet (PWN) is designed as a collection of synsets that represent synonymous English lexemes which are connected to one another with a few basic semantic relations, such as hyponymy, meronymy, antonymy and entailment [START_REF]WordNet: An Electronic Lexical Database[END_REF], [START_REF] Miller | Five Papers on WordNet[END_REF]. This same structure has basically been mirrored in most of the wordnets developed on the basis of PWN. The structural difference of Slavic languages which show many similar features has induced the enrichment of wordnets with new information. Added information is mostly related to the inflectional and derivational richness of a language in question. For instance, information related to inflectional properties has been added to all lexemes in Bulgarian [START_REF] Koeva | Bulgarian Wordnet-Structure and Validation in[END_REF] and Serbian [START_REF]Design and Development of a Multilingual Balkan Wordnet (BalkaNet IST-2000-29388)[END_REF] wordnets, and for Serbian some rudimentary semantic relations that can be inferred from the derivational connectedness, for instance derived-pos (for possessive adjectives) and (for gender motion) derived-gender [START_REF]Design and Development of a Multilingual Balkan Wordnet (BalkaNet IST-2000-29388)[END_REF] has been added too. On the other hand, the recognized importance of PWN, and global wordnet in general, for various NLP applications has initiated the major additions and modifications of PWN itself.

The existence of derivational relations that exhibit a fairly regular behavior and that connect lexemes that belong to the same or to the different categories seemed to many as a good starting point for the substantial wordnet enrichment. We will present the most interesting approaches. All these approaches rely on the fact that if there is a derivational relation between two lexemes belonging to different synsets then most probably there is a kind of semantic relation between the synsets to which the lexemes belong.

The automatic enrichment of wordnet on the basis of the derivational relations has been proposed and used for the Czech wordnet [START_REF] Pala | Derivational Relations in Czech Wordnet[END_REF]. The basic and most productive derivational relations in Czech have been included in a Czech morphological analyzer and generator, and semantic labels were added to the derivational relations.

The sharing of semantic information across wordnets has been proposed by [START_REF] Bilgin | Morphosemantic Relations In and Across Wordnets -A Study Based on Turkish[END_REF]. Namely, if wordnets for several languages are connected to each other (for instance via Interlingual index (ILI) [15], as has been done for wordnets developed in scope of EuroWordnet and Balkanet projects), then semantically related synsets in a source language for which the connection has been established on the basis of the derivational relatedness of some of the lexemes can be used to connect the synsets in a target language whose lexemes may not exhibit any derivational relation.

The method to improve the internal connectivity of PWN has been proposed in [START_REF] Miller | Morphosemantic links in Wordnet[END_REF]. The existing synsets have been manually connected on the basis of the automatically produced list of pairs of lexemes that are (potentially) derivationally, and therefore also semantically, connected. In this paper we will try to show why we find the last approach the most appropriate for Bulgarian and Serbian.

Slavic derivation in wordnet structure

The derivation is highly expressive in all Slavic languages. Some of the most frequent and regular derivational mechanisms in Bulgarian and Serbian are given in the Table 1. The status of the derivational mechanisms listed is not the same. Some of them represent the more or less frequent models which are not applicable to every lemma that has certain syntactic or semantic property, while the other models can always be applied. For instance, the pattern Verb → Noun representing a profession is one of the numerous derivational pattern in Bulgarian (уча → учител) and Serbian (učiti → učitelj), while the pattern Verb → Verbal noun is a general rule that can be applied to all imperfective verbs in the two languages. Similarly, a possessive adjective exists for every animate noun [START_REF] Vitas | Regular derivation and synonymy in an e-dictionary of Serbian[END_REF]. We call this phenomenon a regular derivation since in some respect it enhances the notion of inflectional class.

Formally, regular derivation is performed by derivational operators that significantly influence the structuring of the lexicon of Slavic languages. The analysis of this phenomenon is given in [START_REF] Vitas | Morphologie dérivationnelle et mots simples: Le cas du serbo-croate, Lingvisticae Investigationes Supplementa 24 (Lexique, Syntaxe et Lexique-Grammaire / Syntax[END_REF], [START_REF] Vitas | Restructuring Lemma in a Dictionary of Serbian[END_REF] on the examples of processing of possessive and relational adjectives, amplification and gender motion in various English-Serbian and Serbian-English dictionaries. Moreover, the derivational potential is, as a rule, connected to the specific sense of a lemma (see sections 5 and 7).

Current state of morpho-semantic relations in Bulgarian and Serbian wordnets

Eight semantic relations between synsets are represented (in a correspondence with the Princeton wordnet) in Bulgarian [START_REF] Koeva | Bulgarian Wordnet-Structure and Validation in[END_REF], [START_REF] Koeva | Bulgarian WordNet -development and perspectives[END_REF] and Serbian wordnets [START_REF]Design and Development of a Multilingual Balkan Wordnet (BalkaNet IST-2000-29388)[END_REF].These relations are: hypernymy, meronymy (three subtypes are registered among others recognized), subevent, caused, be in state, verb group, similar to and also see (also see in PWN actually encodes two different relations: between verbs and between adjectives, the former one being a kind of morpho-semantic relation between literals roughly corresponding to Slavic verb aspect while the second one is a semantic relation of similarity between synsets). Three extralinguistic relations between synsets are encoded as well: usage domain, category domain and region domain. The wordnet structure includes also semantic and morpho-semantic (derivational) relations among literals belonging to the same or to the different synsets. Semantic relations between literals are: synonymy and antonymy (in Bulgarian and Serbian wordnets antonymy links synsets); derivational are: derived, participle, derivative in Bulgarian, and derived-pos, derived-gm, and derived-vn in Serbian.

Encoded morpho-semantic relations

The morpho-semantic relations in Bulgarian and Serbian wordnets link synsets although they derivationally apply to the literals only (single word and multi-word lemmas). On the other hand, morpho-semantic relations express different kinds of semantic relations which hold between synsets. Neither the derivational links between the exact literals, nor labels [START_REF] Pala | Derivational Relations in Czech Wordnet[END_REF] for the respective semantics relations operating between synsets are encoded so far in Bulgarian and Serbian wordnets. The subsumed morpho-semantic relations are briefly presented below (some statistical data are shown in Table 2):

Derivative is an asymmetric inverse intransitive relation between derivationally and semantically related noun and verb. For example the Bulgarian literal водя from the synset {насочвам:1, насоча:1, водя:4, напътвам:1, напътя:1, направлявам:1} (the corresponding English synset is {steer:1, maneuver:1, maneuver:2, manoeuvre:2, direct:11, point:4, head:5, guide:1, channelize:1, channelise:1} with a definition 'direct the course; determine the direction of traveling') is in derivative relation with the noun водач from the synset {водач:3} (the corresponding English synset is {guide:2} with a meaning 'someone who shows the way by leading or advising').

Derived is an asymmetric inverse intransitive relation between derivationally and semantically related adjective and noun. For example the literal меден from the Bulgarian synset {меден:1} (the English equivalence {cupric:1, cuprous:1} with a definition 'of or containing divalent copper') is in a derived relation with the literal мед from the synset {мед:2, Cu:1} (in English → {copper:1, Cu:1, atomic number 29:1}). A productive derivational process rely Slavic nouns with respective relative adjectives with general meaning 'of or related to the noun'. For example, the Bulgarian relative adjective {стоманен:1} defined as 'of or related to steel' has the Serbian equivalent {čelični:1} with exactly the same definition. Actually in English this relation is expressed by the respective nouns used with an adjectival function (rarely at the derivational level, consider wooden↔wood, golden↔gold), thus the concepts exist in English as well and the mirror nodes should be envisaged.

Participle is an asymmetric inverse intransitive relation between derivationally and semantically related adjective denoting result of an action or process and the verb denoting the respective action or process. Consider играя from {играя:7} (the English equivalent {play:1} with a definition 'participate in games or sport') which is in a Participle relation with the literal игран from {игран:1} denoting '(of games) engaged of' for the English counterpart {played:1}. All Bulgarian verbs produce participles (the number of participles varies from one to four depending on the properties of the source verb) which are considered as verb forms constituting complex tenses or passive voice. On the other hand, a big part of the Bulgarian participles acts as adjectives with separate meaning. The similar relations between a verb and its participles hold for Serbian.

It can be seen that the actual derivational relations are established between particular literals although the synsets are formally linked (the actual semantic relation between synsets which marker is the derivation itself is not labeled). The English derivative, derived, and participle relations are automatically transferred to Bulgarian wordnet. As they are language specific and obviously there is no one to one mapping between English and Bulgarian the expanded links are manually validated. A specification whether a given morpho-semantic relation exists in English only is declared in a synset note (SNote).

The relation eng_derivative has been also automatically transferred to Serbian although the corresponding derivational relation may hold in Serbian as well but need not (see the Serbian example in section 5). The new relations derived-pos, derived-vn, and derived-gender have been introduced in Serbian wordnet to relate possessive and relative adjectives, verbal nouns and female (or male) doublets, assigned mainly to the Balkan specific or Serbian specific synsets.

Not-encoded morpho-semantic relations

The general observations are that not all existing derivative, derived, and especially participle links are marked in Bulgarian and Serbian wordnets. The main reason originates in the language specific character of the word-building in view of the fact that an exact correspondence with the PWN has been mostly followed in the expand wordnet model. As a result a lot of language-specific derivational relations (that can be described in terms of derivative, derived, and participle relations) remain unexpressed in Bulgarian and Serbian wordnets. For example the literals from the Bulgarian synset {метален: In Serbian, for instance, the adjective synset {zamisliv:1} (English equivalent is {conceivable:2, imaginable:1, possible:3} with a definition 'possible to conceive or imagine') is not linked with the verbal synset {zamisliti:2y, koncipirati:1b} (in English {imagine:1, conceive of:1, ideate:1, envisage:1} with a definition 'form a mental image of something that is not present or that is not the case'), although relation derived, or some more specific, would be appropriate.

Language-specific morpho-semantic relations

There are systematic morpho-semantic differences concerning derivational mechanisms between English and Slavic languages [START_REF] Krstev | Towards the Global Wordnet[END_REF]. Some of the most productive derivational relations in Slavic languages are briefly presented here: namely verbal aspect pairs, gender pairs, and diminutives.

Aspect pairs

The verb aspect is a category that occurs in all Slavic languages, its nature is very sophisticated. Generally speaking, the verb aspect in Slavic languages can be descried as a relation between the action and its bound (limit) regardless of the person, speaker and speech act. The perfect aspect verbs express integrity and completeness, while the imperfect aspect verbs -lack of integrity or a process (duration, recurrence). Each Slavic verb is either perfective or imperfective; there are a number of verbs that are bi-aspectual and act as both imperfective and perfective. Most verbs form strict pairs where perfective and imperfective members form a derivational relation between two lexemes expressing generally the same meaning. The Bulgarian verbs are classified as: imperfective (perfective correspondent exists), perfective (imperfective correspondent exists), bi-aspectual, imperfective tantum (perfective correspondent does not exist), perfective tantum (imperfective correspondent does not exist). In Bulgarian wordnet the aspect pairs are introduced in one and the same synset with an LNote (literal note) describing the respective aspect. For example {съчинявам:2 LNOTE: imperfective, съчиня:2 LNOTE: perfective, пиша:4 LNOTE: imperfective, написвам:2 LNOTE: imperfective, напиша:2 LNOTE: perfective} (an equivalent of the English synset {write:1, compose:3, pen:1, indite:1} with a definition 'produce a literary work'). Similarly, in Serbian wordnet the aspect pairs are introduced in a same synset. For instance in a synset {zamišljati:2x, zamisliti:2x, dočaravati:2x, dočarati:2x, predočavati:1, predočiti:1} (in English {visualize:1, visualise:3, envision:1, project:9, fancy:1, see:4, figure:3, picture:1, image:1} with a definition 'imagine; conceive of; see in one's mind'), LNOTE element corresponding to each literal describes inflectional and derivational properties of each verb, e.g. LNOTE content for the imperfective verb zamišljati is V1+Imperf+Tr+Iref+Ref, while LNOTE content the perfective correspondent zamisliti is V162+Perf+Tr+Iref+Ref [START_REF] Krstev | Combining Heterogeneous Lexical Resources[END_REF]. In most cases, however, perfective verbs derived from the imperfective by prefixation express different meaning and are not in the same synset, for example the perfective verb uraditi 'do, perform' and its imperfective correspondent raditi are not in the same synset.

Gender pairs

The gender pairing is systematic phenomenon in Slavic languages that display binary morpho-semantic opposition: male → female, and as a general rule there is no corresponding concept lexicalized in English. The derivation is applied mainly to nouns expressing professional occupations, but also to female (or male) correspondents of nouns denoting representatives of animal species. For example, Bulgarian synset {преподавател:2, учител:1, инструктор:1} and Serbian synset {predavač:1} that correspond to the English {teacher:1, instructor:1} with a definition: 'a person whose occupation is teaching' have their female gender counterparts {преподавателка, учителка, инструктурка} and {predavačica} with a feasible definition 'a female person whose occupation is teaching'.

There are some exceptions where like in English one and the same word is used both for masculine and feminine in Bulgarian and Serbian, for example {президент:1} which corresponds to the English synset {president:3} with a definition: 'the chief executive of a republic', and as a tendency the masculine noun can be used referring to females. Following the PWN practice the female counterparts are encoded in Bulgarian and Serbian wordnets as hyponyms of the corresponding synset with the male counterpart. For example {актриса:1} (English equivalent {actress:1} with a definition 'a female actor') is a hyponym of {актьор:1, артист:} (corresponding to the English synset {actor:1, histrion:1, player:3, thespian:1, role player:2} expressing the meaning 'a theatrical performer'). It might be foreseen of introducing a new relation describing the female -male opposition of nouns in Slavic languages as has already been done for Serbian.

Diminutives

Diminutives are standard derivational class for expressing concepts that relate to small things. The diminutives display a sort of morpho-semantic opposition: big → small, however sometimes they may express an emotional attitude too. Thus the following cases can be found with diminutives: standard relation big → small thing, consider {стол:1} corresponding to English {chair:1} with a meaning 'a seat for one person, with a support for the back' and {столче:1} with an feasible meaning 'a little seat for one person, with a support for the back'; small thing to which an emotional attitude is expressed. Also, Serbian synset {lutka:1} that corresponds to the English {doll:1, dolly:3} with a meaning 'with a replica of a person, used as a toy' is related to {lutkica} which has both diminutive and hypocoristic meaning. There might be some occasional cases when this kind of concept is lexicalized in English, {foal:1} with a definition: 'a young horse', {filly:1} with a definition: 'a young female horse under the age of four', but in general these concepts are expressed in English by phrases.

For the moment the diminutives are included in Bulgarian and Serbian wordnets only in the rare case when the English equivalent is lexicalized. On the other hand, almost from every concrete noun a diminutive (in some cases more than one lexeme) can be derived. Consequently a place for the diminutives in the wordnet structure has to be provided.

The nature of morpho-semantic (derivational) relations

One of the most important features of the morpho-semantic relations is that being derivational relations between literals (i.e. assistant is a person that assists; participant is the person that participates etc.) they express also regular semantic oppositions holding between synsets [START_REF] Miller | Morphosemantic links in Wordnet[END_REF]. The derivational relation linking assist and assistant from the respective synsets {help:1, assist:1, aid:1} 'give help or assistance; be of service' and {assistant:1, helper:1, help:4, supporter:3} 'a person who contributes to the fulfillment of a need or furtherance of an effort or purpose' implies a kind of semantic relation over synsets formulated in [START_REF] Pala | Derivational Relations in Czech Wordnet[END_REF] as an agentive relation existing between an action and its agent.

Given morpho-semantic relation may be realized by different derivation mechanisms. Consider the literals from the Bulgarian synset {певец:2, вокалист:1} (in English {singer:1, vocalist:1, vocalizer:2, vocaliser:2} with a definition 'a person who sings'), the former one певец is derived with the suffix -ец from the literal пея constituting the synset {пея:1} (the English equivalent {sing:2} with a definition 'produce tones with the voice'}, while the second one вокалист is derived with the suffix -ист from the literal вокализирам belonging to the synset {вокализирам:1} (in English {vocalize:2, vocalise:1" with a definition 'sing with one vowel').

On the other hand, different derivational mechanisms might correspond to different semantic relations. For example in Bulgarian, as well as in English the verb чeта from the synset {чета:3; прочитам:2; прочета:2} corresponding to the English synset {read:1} with a definition 'interpret something that is written or printed' has the following derivates among others:

-the noun четене from the synset {четене:1} ↔ {reading:1}, with a definition: 'the cognitive process of understanding a written linguistic message'. The derivation transforms the verb into a verbal noun. The respective relation between synsets is formulated as an action relation in [START_REF] Pala | Derivational Relations in Czech Wordnet[END_REF].

-the noun читател from the synset {читател:1} ↔ {reader:1}, with a definition: 'a person who enjoys reading'. The derivational relation links the source verb with a noun build by an affixation. The respective relation between synsets expresses a property over the underlying action.

In some cases when the source literal has more than one meaning the exact correspondences with the derivates can be traced. Consider the verb чeта from the synset {чета:1, прочитам:1, прочета:1} equivalent with the English synset {read:3} with a definition 'look at, interpret, and say out loud something that is written or printed'. Its verbal noun derivative четене from the synset {четене:1; поетическо четене:1; рецитал:1} (in English {recitation:2, recital:3, reading:7}) expresses a meaning which is related with the meaning of the source 'a public instance of reciting or repeating (from memory) something prepared in advance'. As the source derivates counterpart in two different synsets (equivalent to {read:1} and {read:3}), this presupposes the corresponding difference in the meanings of the resulting derivatives. Thus the same derivational mechanism might indicate for different semantic oppositions if it targets graphically equivalent literals expressing different meaning (the observed difference in the semantic oppositions remains undistinguished). It is natural that the synsets {read:1} and {read:3} are related with a verb group relation.

The semantic part of the morpho-semantic relations is not language specific, language specific are the derivational mechanisms of lexicalization. There are several English derivatives of the literal paint from {paint:3} with a definition 'make a painting of':

En 1.{paint:1} -'a substance used as a coating to protect or decorate a surface (especially a mixture of pigment suspended in a liquid); dries to form a hard coating' En 2. {painter:1} -'an artist who paints' En 3. {painting:1, picture:2} -'graphic art consisting of an artistic composition made by applying paints to a surface' En 4.{painting:2} -'creating a picture with paints' Neither of the corresponding Bulgarian equivalents: Bg 1.{боя:2} Bg 2.{живописец:1, художник:1} Bg 3.{картина:3} Bg 4,{живопис:1} are derivatives of the Bulgarian synset equivalent to {paint:3} -{рисувам:2; нарисувам:2}. Nevertheless the same semantic oppositions exist in Bulgarian although they are not marked with any semantic or morpho-semantic relations.

In Serbian some of the related synsets to {naslikati:1 LNOTE: V101+Perf+Tr+Iref} (equal to {paint:3}) include derivatives, while the other do not (e.g. {boja:2x, farba:1x}). The derivative relation is transferred from English to Serbian wordnet, but the name of the relation has not been changed in order to indicate that the origin of the relation is English, and that it may hold for Serbian but need not, as shown by the same example. Sr 1. {boja:2x, farba:1x} Sr 2. {slikar:1} Sr 3. {slika:1} Sr 4. {slikarstvo:1} This means that the derivational relations in a particular language might be successfully used not only for the detecting of a given semantic opposition. Moreover they can be exploited for the identification of the corresponding semantic relations in other languages where lexicalization is expressed by different mechanisms. Thus we have to make clear distinction between the derivation as a literal relation (asymmetric, inverse, and intransitive) and the semantic oppositions between synsets for which the derivation itself might be a formal pointer.

Approaches to cover Slavic specific derivations in wordnet

There are several possible approaches for covering different lexicalizations resulting from derivation in different languages [START_REF] Krstev | Towards the Global Wordnet[END_REF], [START_REF] Stamou | BALKANET: A Multilingual Semantic Network for the Balkan Languages[END_REF]:

-to treat them as denoting specific concepts and to define appropriate synsets (gender pairs in Bulgarian and Serbian; relative adjectives in Bulgarian and Serbian); -to include them in the synset with the word they were derived from (verb aspect in Bulgarian and in most of the cases in Serbian); -to omit their explicit mentioning (diminutives in Bulgarian);

-to provide source literals with flexion-derivation description encompass these phenomena as well.

Treating morpho-semantic relations such as verb aspect, relative adjectives, gender pairs and diminutives among others in Slavic languages as relations that involve language specific concepts requires an ILI addition for the languages where the concepts are presented (respectively lexical gaps in the rest). This solution takes grounds from the following observations:

-Verb aspect pairs, relative adjectives, feminine gender pairs and diminutives denote an unique concept;

-Verb aspect pairs, relative adjectives, feminine gender pairs and diminutives are lexicalized with a separate word in Bulgarian, Serbian, Czech and other Slavonic languages;

-Relative adjectives, feminine gender pairs and diminutives in most of the cases belong to different category or different inflectional class comparing to the word from which they are derived (there are some exceptions in the difference of the category, like diminutives that are derived from neuter nouns in Bulgarian).

Although the new wordnets do not compare yet with PWN's coverage, the former are continuously extended and improved so that a balanced global multilingual wordnet is foreseen. For that reason the task of proper encoding of different levels of lexicalization in different languages is becoming more and more important in the view of the various Natural Language Processing tasks. The Slavic languages possess rich derivational morphology which has to be involved into the strict one-to-one mapping with the ILI.

Automatic building of derivational relations in Bulgarian and Serbian

The derivational relations for literals that already exist in wordnet can be interpreted in terms of derivational morphology, e.g., the noun teacher is derived from the verb teach and so on. Wordnet already contains a lot of words that are produced by the derivational morphology rules: verbal nouns are linked with verbs, etc. In order to make explicit the morpho-semantic relations that exist already it would be necessary to include more links. On the other hand, a special attention has to be paid on the language specific derivational relations (some of them valid for big language families as Slavic languages). Several problems can be formulated following the observations and analyses presented in this study:

It is necessary to distinguish the pure derivation form the semantic relations which meaning is presupposed by the derivation itself. Concerning Bulgarian and Serbian wordnets this will be reflected particularly in the proper encoding of the derivational links between exact literals as it has been done in PWN; in the identification of derivational relations between literals already encoded in wordnets (comparing with PWN or exploiting language-specific derivational models), and in the introducing of language specific derivations in their appropriate place in the wordnet structure providing the exact correspondence with other languages.

In more general plan a theoretical investigation is needed to describe the nature of the semantic relations to which derivations are formal pointers. Ones a consistent classification is provided the respective semantic relations might be identified in the wordnets on the basis of the derivational ones in a particular language Several tasks may be done semi-automatically: to link literals instead of synsets with derivational relations; and to identify synsets where the potentially derivationally related literals appear. Bellow we provide some observations why the complete automation is not appropriate; although the derivational regularities are in most of the cases well established.

Although derivation is in many cases regular in the sense that it yields predictable results, it cannot be freely used for generation since it can lead to over-generation; namely, one could generate something which exists in a language system but does not exist in language usage. For instance, in Bulgarian and Serbian an abstract noun can be regularly derived (with a suffix -ост; -ost) from a descriptive adjective X meaning 'the quality of something that has the characteristic X', and a prefix ( -не, -ne; -без, -bez, etc.) can be used to produce both the adjective and a noun with the opposite meaning. One such example in Serbian is osećajan 'be able to respond to affective changes' → osećajnost 'the ability to respond to affective changes' → bezosećajan 'not being able to respond to affective changes' → bezosećajnost 'the inability to respond to affective changes'. However, if the same pattern is applied to the adjective sličan 'marked by correspondence or resemblance' → sličnost 'the quality of being similar' → ?nesličan 'not similar' → ?nesličnost 'the quality of being dissimilar', two last lexemes in a sequence though easily understood are not lexicalized.

In a context of a wordnet production it is not sufficient to produce new synsets, for instance by applying the regular derivational mechanisms. It is equally important to place the generated synsets in the already existent network consisting of various relations. For instance, in Serbian the nouns sposobnost, vidljivost and popustljivost are regularly generated from the adjectives sposoban 'having the necessary means or skill to do something', vidljiv 'having the characteristics that make it visible' and popustljiv 'easily managed or controlled'. However, the produced nouns have three different hypernyms: osnovna karakteristika {quality:1}, svojstvo {property:3}, and osobina {trait:1}. The correct placement of newly generated synsets in an existent network is not straightforward.

It has been noted (in section 5) that many senses of some words are distinguished by their different derivational capabilities. For instance, Serbian verb polaziti has five different meanings according to the Serbian explanatory dictionary, and one submeaning of the second presented meaning is 'to go somewhere regularly and often to perform some duty'. That meaning is the only one from which the noun polaznik 'someone who attends a school or a course' can be derived by the agentive relation (realized by a suffix -ik).

It has already been stated in [START_REF] Miller | Five Papers on WordNet[END_REF] that even derivation that seems very predictable can show very unpredictable behavior. Some derivational mechanisms in Bulgarian and Serbian are very predictable, like production of possessive adjectives that are produced from (mostly) animate nouns. As a consequence possessive adjectives are not listed in traditional Serbian dictionaries. The production of verbal nouns from imperfective verbs is also regular and produces a predictive meaning, the act of doing something. The verbal nouns are however, listed as a separate entries in Bulgarian and Serbian dictionaries. Besides the predicted meaning they often acquire the additional meaning. For instance, the verbal nouns учение in Bulgarian, učenje in Serbian and pečenje in Serbian are derived from imperfective verbs уча, učiti 'to study' and peći 'to roast'. Besides the predicated meanings 'the act of studying' and 'the act of roasting' they have acquired in Serbian the additional meaning, 'doctrine' and 'roast meat', respectively. In the case of other derivational mechanism it can be more difficult to establish the meaning of the derived word. For instance, adjectives pričljiv and čitljiv in Serbian are derived respectively from the verbs pričati 'to talk' and čitati 'to read' using the same suffix -iv. Both verbs are imperfective and can be used both as transitive and intransitive: Marko priča priču 'Marko tells the story', Marko puno priča 'Marko speaks a lot', Puno ljudi čita knjigu 'A lot of people read the book', Marko puno čita 'Marko reads a lot'. The meaning of the adjective pričljiv is derived from the itransitive usage of a verb (namely, Marko puno priča implies Marko je pričljiv 'Marko is talkative'), while the adjective čitljiv is derived from the transitive usage (here Puno ljudi čita knjigu implies Knjiga je čitljiva 'The book is easy to read').

The complexity of the issue of automation is best illustrated by the derivation of gender pairs in Serbian, since they exhibit all the previously mentioned problems. If we consider the derivation of female counterparts for the nouns of professions we encounter the following situations:

-The female counterpart morphologically does not exist: for instance, sudija 'judge' is therefore used for both men and women; -The female counterpart morphologically exists but is never used, vojnik 'solider' vs. *vojnica and žena vojnik '(woman) soldier';

-The female counterpart exists and is exclusively used for women performing that profession or function: kelner 'waiter' and kelnerica 'waitress';

-The female counterpart exists but the male noun is also sometimes used for women: profesor '(man or woman) professor' and profesorka '(woman) professor';

-The female counterpart exists but it does not mean quite the same as a noun it was derived from: sekretar 'secretary' is treated as someone performing an highly responsible function, as opposed to sekretarica '(woman) secretary' who is performing the low-level tasks in an organization; -The female counterpart exists but it has acquired a different meaning, so it is not used to denote a woman performing certain function: saobraćajac 'traffic cop' vs. saobraćajka 'car accident'.

Conclusions and future work

We have briefly presented the current stage of the encoding of morpho-semantic relations in Bulgarian and Serbian wordnets. Grounding on the derivational properties of Slavic languages we provided some observations over the sophisticated nature of morpho-semantic relations and presented some examples proving the negative consequences from a purely automatic insertion of Slavic derivational relations into the wordnet structure. We believed we added additional evidences supporting the approach presented in [START_REF] Miller | Morphosemantic links in Wordnet[END_REF] namely the utilization of a semi-automatic identification or insertion of morpho-semantic relations. Such an approach would significantly facilitate the wordnet development although a manual connection on the basis of the automatically produced lists of suggested pairs has to be provided.

Further development of both Bulgarian and Serbian wordnets is narrowly connected with an investigation towards the theoretical grounds of the nature of morpho-semantic relations. At first stage the encoding of derivational relations between exact literals instead of synsets is foreseen. Another important task is the introducing of Slavic language specific derivations in a uniform way providing at the same time ILI correspondences. The accomplishment of these tasks will also reflect in the successful implementations of approaches based on cross-lingual information extraction, retrieval, and data mining, multilingual summarization, machine translation, etc.

Table 1 .

 1 Some of the derivational mechanisms in Bulgarian and Serbian

	Relation	Bulgarian	Serbian	English

Table 2 .

 2 Statistical data for the encoded morpho-semantic relations in Bulgarian and Serbian wordnets.

	Number of	BG WN	SR WN	PWN 2.0
	Synsets	29,136	13,612	115,424
	Literals	56,223	23,139	203,147
	Relations	53,144	18,210 4	204,948
	Derived	1,696	314	1,296
	Derivative	8,920	83 5	36,630
	Participle	212	0	401

http://dcl.bas.bg/bulNet/general_en.html

There is actually a whole list of perfective verbs that correspond to the imperfective verb учити: izučiti, naučiti, obučiti, preučiti (se), podučiti, poučiti, priučiti, proučiti.

Today obsolete.

Without extralinguistic relations: category and region, and relation eng_derived.

Includes relations: derived-pos, derived-vn, and derived-gender.


Christophe Finot
email: christophe.finot@u-bourgogne.fr

Alexej Sysoliatin

Stefan Wabnitz

Nonlinear parametric resonances in quasiperiodic dispersion oscillating fibers

Keywords: Dispersion oscillating fiber, modulation instability, four-wave mixing

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Modulation instability (MI) is a nonlinear process that has been widely investigated in various fields of physics including plasmas, hydrodynamics and optics, to cite a few. In the presence of a high power continuous wave (CW), MI leads to the emergence and amplification of gain sidebands in the wave spectrum. In nonlinear fiber optics, such a process has been demonstrated in fibers with anomalous, constant group velocity dispersion (GVD) [START_REF] Tai | Observation of modulational instability in optical fibers[END_REF], as well as in normal GVD fibers by enabling the fulfillment of the nonlinear phase-matching condition through either fourth order dispersion [2; 3], birefringence or a multimodal structure [4; 5; 6]. However, the efficiency of such parametric processes may highly suffer from unwanted longitudinal fluctuations of the fiber parameters, with a rapid drop of the gain as well as a broadening of its bandwidth. Ultimately, parametric gain may fully disappear in the presence of fiber fluctuations [4; 5; 6], which leads to the requirement of sophisticated devices or fiber designs [2; 3; 6; 7; 8].

The efficiency of pulse reshaping processes may also be seriously impaired by stochastic fluctuations of the fiber parameters [START_REF] Kuo | Dispersion-stabilized highlynonlinear fiber for wideband parametric mixer synthesis[END_REF].

To the contrary, whenever the longitudinal variations of the fiber parameters are quite large, periodic and controlled such as in loss (or nonlinearity) [START_REF] Kibler | Impact of structural irregularities on high bit rate pulse compression techniques in photonic crystal fibre[END_REF], dispersion [START_REF] Matera | Sideband instability induced by periodic power variation in long-distance fiber links[END_REF] or polarization [12; 13] managed fiber transmission or laser systems, new deterministic MI or parametric resonance (PR) sidebands appear. In this work, we shall use the terms of MI or PR as fully equivalent in our description of the sideband spectrum generation process. As a matter of fact, the periodic oscillation of the fiber dispersion leads to the quasi-phase-matching (QPM) of the nonlinear fourwave mixing (FWM) process. As a result, unequally spaced MI sidebands can emerge [START_REF] Murdoch | Quasi-phase matching in an optical fiber with periodic birefringence[END_REF]. A renewed experimental and theoretical interest in these studies has been recently stimulated by the availability of fibers presenting a longitudinal and periodic modulation of their dispersion properties [12; 15]. Recent experimental works have confirmed the QPM-induced MI process in the normal GVD regime of microstructured dispersion oscillating fiber (DOF) around 1 m [16; 17], as well as of non-microstructured highly nonlinear DOF at telecom wavelengths [START_REF] Droques | Experimental demonstration of modulation instability in an optical fiber with a periodic dispersion landscape[END_REF].

However, one may wonder how in this case deviations from a strictly periodic evolution of the fiber parameters may affect the QPM-MI spectrum. Can it be beneficial to the enlargement of the sideband bandwidth to use a chirped DOF, in a manner that is similar to the approach commonly used in the context of chirped QPM quadratic crystals? Or is the longitudinal variation of the parameters of the fiber always detrimental to the parametric resonance gain? To answer this important question, in this work we numerically study the impact of a linear longitudinal evolution of the main parameters of a DOF. Therefore our analysis is organized as follows. In a first section, we describe our numerical model and the specific DOF under investigation. Next, by using systematic simulations and an approximate approach that is based on the Floquet theorem and the associated linear stability analysis, we separately discuss the influence of each parameter, i.e., the spatial dispersion oscillation period, the value of the average dispersion, and finally the amplitude of the dispersion oscillations.

Model and situation under investigation

The evolution of the optical field  in an optical fiber can be described by the nonlinear Schrödinger equation (NLSE) that includes both the Kerr nonlinearity  and the longitudinally varying second-order dispersion  2 (z)

2 2 2 2 () | | 0 2 zt z i           (1)
where  is the complex electrical field, z is the propagation distance and t is the reduced time.

MI or parametric resonance induced by the longitudinal variations of chromatic dispersion was theoretically investigated before in a wide range of configurations, ranging from sinusoidal profiles with a spatial period of a few tens of meters [18; 19], up to dispersion-managed systems with periods of several kilometers [16; 18]. Let us consider a fiber dispersion profile that evolves with distance z according to the following sinusoidal rule

b 2 (z) = b 2av + b 2amp sin 2p z / L ( ) , (2)
where  is the spatial period of the dispersion oscillation,  2av is the average dispersion of the fiber, and  2amp is half of the peak-to-peak amplitude of the dispersion variation. In the presence of sinusoidal longitudinal GVD variations, and when pumped by a continuous wave of power

where J p is the Bessel function of order p.

In this contribution, we analyze by means of extensive numerical simulations the spectrum of parametric resonances in a quasi-periodic DOF. Specifically, we consider a linear variation with propagation distance z of each of the three parameters in the right hand side of Eq. [START_REF] Wong | Continuous-wave tunable optical parametric generation in a photonic-crystal fiber[END_REF]. By expressing any of these parameters as Q, we set

( ) 1 2 z Q z Q L         (5)
Here , respectively.

Inspired by the highly nonlinear DOF that was experimentally used in [START_REF] Droques | Dynamics of the modulation instability spectrum in optical fibers with oscillating dispersion[END_REF], we consider here a DOF with the following average values:  =20 m, 2av  = 2 ps 2 .km -1 and 2amp  = 2 ps 2 .km - 1 . We set the overall fiber length equal to 400 m, and its nonlinear coefficient = 10 W -1 .km -1 .

The DOF is pumped by a CW pump with the average power P = 4 W at the telecommunication wavelength =1550 nm. Examples of the resulting dispersion profiles for the different cases of linear variation of either one of the  2av , or  2amp parameters are illustrated in Fig. 1. We numerically solved the NLSE (1) by the standard split-step Fourier algorithm, including a weak input white noise seed: the results were averaged over 12 independent noise shots. We consider here separately the impact of a longitudinal variation of either  2av , or  2amp . We also took advantage of the LSA based on the Floquet theorem, which proved to be a very powerful tool for the analysis of the evolution of the MI gain spectrum in DOFs [18; 19; 23]. Clearly the Floquet theorem is strictly valid for a periodic DOF only: nevertheless, it may still be used as an approximate tool to compute the sideband spectrum in quasi-periodic DOFs as well. Indeed, in Ref. [17; 21; 24; 25] we have shown that the Floquet method may still be used to obtain an approximate averaged description of the MI gain spectrum in the presence of fiber loss (or gain).

The way to compute the gain profile g(, Q) by the Floquet method is to consider first a constant value of Q, obtain its corresponding PR spectrum, and then compute the overall MI spectrum g F by simply averaging over the probability distribution of the Q parameter

0 ( ) ( , ) ( ) ( , ) L Q F Q d d g g Q pdf Q Q g Q Q       (6)

Impact of the spatial period variation

We start our study by investigating the influence of a spatial dispersion oscillation period that varies linearly with the propagation distance (see Fig. 1(a)). The resulting spectra (or Arnold resonance tongues [START_REF] Finot | Gain sideband splitting in dispersion oscillating fibers[END_REF]) are plotted in Fig. 2(a), and highlight the rapid fall of the gain that is experienced by the various sidebands when || is increased. We may also note that the unequally spaced and initially narrow spectral sidebands continuously broaden and eventually may overlap.

The splitting and fan-out of the resonant sideband frequencies is strikingly reminiscent to the Stark splitting of the electron resonances or spectral lines of atoms and molecules in the presence of an applied electric field [26; 27]. Moreover, Fig. 2(a) also shows that an oscillatory pattern develops in the sideband amplitudes within the resonance tongues. In Fig. 2(b), we evaluate the tolerance  p on , that we defined as follows:  p = max - min where  max and  min are the values of  leading to a decrease of 10 dB of the maximum gain of the p th sideband.  p rapidly decreases with the order of the gain sideband, from a value of 0.26 for the first sideband down to a value below 0.05 for the 6 th sideband. As a practical consequence, we may infer that even very small deviations from a strictly periodic dispersion oscillation structure may hamper the development of high-order gain sidebands, and that a tight drawing precision is therefore required for generating a large set of spectral lines. Details of the spectra obtained for  = -0.1 and  = 0.1 are provided in Fig. 3(a1), and compared with the gain spectrum that obtained in perfectly periodic conditions. We may note here that the sign of  has a crucial influence on the sideband spectrum, as can also be observed by noting the tilt of the spectrum in Fig. 2(a). In other words, according to the direction of propagation along the chirped DOF, the sideband spectrum may exhibit a shift towards either lower or higher frequencies whenever the dispersion oscillation period grows larger or smaller, respectively.

In order to gain a simple qualitative understanding of the observed evolution of the resonance tongues, we computed the corresponding sideband spectrum as it is predicted by the Floquet linear stability analysis: Fig. 3(b) summarizes the evolution of the gain g() for a strictly periodic DOF. In agreement with the analytical expression for the sideband position given in Eq. ( 3), the central frequencies of the gain sidebands are strongly affected by changes of the oscillation period . This explains the observed broadening of the average gain profile resulting from Eq. ( 6) (see Fig. 3(c)), as well as the dramatic decrease of the gain values. Quite unexpectedly, the approximate Floquet method approach may indeed qualitatively well reproduce the numerically observed gain drop that is summarized in Fig. 2(b). Nevertheless, the Floquet method approach is unable to capture some of the numerically observed features, such as the previously discussed spectral asymmetry of the sideband spectrum obtained as the direction of propagation is reversed in the linearly chirped DOF. Clearly, the main a limitation inherent in using Eq. ( 6) is that the average gain is a sum of positive contributions: the overall gain may intrinsically only increase upon propagation. Hence it is not possible to reproduce with Eq. ( 6)

the large spectral oscillations of the gain sidebands: as it can be seen in Fig. 3(a2), instead of strong oscillations, a kind of plateau is predicted. This means that the sideband splitting and associated rapid amplitude oscillations with frequency of the resonance tongues are due to the coherent constructive (or destructive) interference among the sidebands that are generated at different points along the chirped DOF. Moreover, Eq. ( 6) leads to the same result irrespective of the sign of : consequently, it fails in predicting the asymmetric spectral evolution that is obtained by the full numerical simulation of the NLSE.

Impact of the average dispersion variation

Let us investigate next the influence on the Arnold resonance tongues of a longitudinal change of the average DOF dispersion (see Fig. 1(b)). To this end, we carried out a numerical study similar to that of section 3. The corresponding results are summarized in Fig. 4 and Fig. 5. As it can be seen in Fig. 4, in this case the sideband splitting and fan-out as the amplitude of the average dispersion excursion grows larger is now symmetric with respect to the direction of variation of  (or  2av ). On the other hand, we may also note from Figs.4-5 several features that are common with the trends observed in the presence of a chirped spatial period . Namely, the broadening to the gain sidebands results owing to the change of  p according to  2av as predicted by Eq.( 3).

Moreover, the peak sideband gain is strongly sensitive to the presence of a chirp: its amplitude drops and exhibits strong frequency oscillations. Finally, an overlap of the different gain sidebands is also observed in Fig. 4 and Fig. 5. Once again, in spite of some limitations owing to the fact that the Floquet method is only rigorously applicable to a perfectly periodic DOF, the approximate approach that is based on Eq. ( 6) may provide a useful insight on the evolution of the gain bandwidth or the range of . But even if the difference between DOF with constant parameters and the aperiodic DOF under study seem relatively small (see Fig. 1(b) plotted for = 0.5), Eq. ( 6) is unable to capture the rapidly oscillating spectral pattern that is observed when the gain sideband spectrum is substantially broadened. Once again, the sideband splitting and amplitude oscillations within the Arnold resonance tongues result from the coherent (or phasesensitive) summation of sidebands generated at different points of the linearly chirped DOF. predicted by the use of the approximate approach based on Eq. ( 6). Same colormap as Fig. 4.

Impact of the amplitude of the dispersion oscillation

In this last section, we shall examine the impact of a linear variation of the amplitude of the fiber dispersion oscillation on the shape and amplitude of the Arnold resonance tongues. As we shall see, in this case the evolution of the sideband spectrum is strongly dependent on the value of the (constant) average dispersion  2ave . Let us consider first the case with  2ave = 2 ps 2 /km. The corresponding results, obtained from the numerical integration of the NLSE, are summarized on Fig. 6(a). As it can be seen, in this case the linear chirp of the oscillation amplitude has a quite limited impact on the sideband spectrum, contrary to the two previous cases discussed in sections 2 and 3. Remarkably, Fig. 6(a) shows that may vary over a relatively large range, without significantly affecting the peak gain values. Indeed, Fig. 6(b) confirms that  may be as high as 3 and even higher, without decreasing the gain in the first QPM by more than 3 dBs. We may also note that the gain bandwidth does not broaden significantly, even when deviations from constant  2amp as large as = 2 are imposed (see Fig. 1(c) for an illustration of the dispersion evolution for = 2). Finally, Fig. 6(a) shows that the spectrum deformation is fully symmetric as the sign of  is changed. This means that in this case the same spectrum results at the output of the chirped DOF, irrespective of the propagation direction. Now, a relatively good qualitative and quantitative match is obtained between the numerical spectra and the approximate analysis relying on the Floquet LSA as it has been summarized in Fig. 7. The key point is that, contrary to the previous cases, the variation of the amplitude of the dispersion oscillations  2amp should in principle have no influence, on the basis of Eq. ( 3), on the position of the MI sidebands. As we shall see now, the impact of the longitudinal evolution of  2amp is dramatically different whenever the average dispersion is reduced down to  2ave =1ps 2 /km. The corresponding results of our systematic simulations are reported in Fig. 8(a): as can be seen, the resonance tongues display several new features. First of all, at low || values, a new sideband that is induced by FWM between the pump and the first QPM sideband is observed. The corresponding generation efficiency drops down as soon as the gain of the first QPM sideband decreases.

However, an interesting and novel property of the MI or PR spectrum is that for selected sidebands, the linearly chirped oscillation amplitude leads to a substantial increase of the spectral gain. This is for example the case for the 4 th sideband: the corresponding gain is increased by more than 20 dBs with respect to the case of a strictly periodic DOF (see Fig. 8(c)). This gain enhancement may be explained with the help of Eq. ( 4): with  = 0 and for the parameters under study, the gain predicted by Eq. ( 4) for the 4 th sideband vanishes. To the contrary, whenever  ≠ 0, nonzero gain is experienced at different stages of the propagation along the fiber, so that the overall gain is finite. The numerically observed gain enhancement of Fig. 8(c) is well reproduced with the help of Eq. ( 6): the corresponding LSA results are presented in Fig. 9(b). Results from the numerical integration of the NLSE (solid curves) are compared with predictions from Eq. ( 6) (dotted curve).

On the other hand, in a manner similar to the cases described in Fig. 2 and Fig. 4, the use of the Floquet approach based on the incoherent summation of Eq.( 6) is not able to capture a very interesting spectral feature that is observed for the higher-order QPM sidebands in Figs.

8(b) and 9(a). As can be seen, instead of a spectral sideband with a single peak, a splitting into two separate sub-peaks occurs for sidebands of order p>1. Moreover, for each of these sidebands the frequency spacing between the two sub-peaks grows larger as || increases. The observed sideband splitting is analogous to the Zeeman splitting of the spectral lines in a gas in the presence of a strong magnetic field owing to the two states of the spin of the electron [START_REF] Stark | Beobachtungen über den Effekt des elektrischen Feldes auf Spektrallinien. I. Quereffekt (Observations of the effect of the electric field on spectral lines I. Transverse effect)[END_REF].

A sideband splitting of the resonance tongues was previously theoretically reported for a strictly periodic DOF in Ref. [START_REF] Zeeman | Doublets and triplets in the spectrum produced by external magnetic forces[END_REF], and later experimentally demonstrated in Ref. [START_REF] Finot | Gain sideband splitting in dispersion oscillating fibers[END_REF], and its potential implications for all-optical signal processing were further analyzed in Ref. [START_REF] Feng | Experimental demonstration of gain sideband splitting in dispersion oscillating fibers[END_REF].

Because of the periodicity of the dispersion oscillations, in those cases the Floquet analysis could be successfully applied to reproduce the sideband splitting effect. In the case of a linearly chirped DOF, a proper extension of the Floquet method should be developed, similarly to the case of quasi-periodic (i.e., containing several incommensurable frequencies) parametric forcing [START_REF] Finot | On the influence of the pump shape on the modulation instability process induced in a dispersion oscillating fiber[END_REF].

However the development of this new mathematical tool is beyond the scope of the present numerical study. . In [START_REF] Murdock | On the Floquet problem for quasiperiodic systems[END_REF], Mussot et al analyzed analytically and experimentally the emergence of new sidebands in the spectrum obtained after propagation in a fiber with a complex dispersion oscillating profile characterized by two different spatial frequencies. We do not see here straightforward connections to the splitting highlighted in the fiber we investigate so that additional elements may be involved in the emergence of those peaks located at the edges of the gain sidebands. In the present case, the sideband splitting results from a coherent interference among the resonant waves that are generated at different points of the chirped DOF. The longitudinal evolution of the sideband amplitudes is illustrated on Fig. 10(a): as can be seen, the MI or PR gain at the mid-point of higher-order (i.e., with p>1) sidebands may experience a non-monotonic behavior. The initial growth is followed by a gain decrease at the mid-point of each parametric resonance. Correspondingly, two gain peaks emerge with a frequency spacing that grows progressively larger with the propagation distance.

We thus performed a systematic study of the evolution of the resonance tongues as a function of the value of the average dispersion. The corresponding results are summarized on Fig. 10(b), and reveal that the spacing between the sub-sidebands also increases with the ratio  2amp / 2av , so that two well-separated peaks may result, as the relative amplitude of the dispersion oscillations grows larger. Fig. 10(b) reveals that the sideband splitting is not necessarily restricted to higher order QPM sidebands: for average dispersion 2av <0.5 ps 2 /km, a splitting of the first-order QPM sideband may be observed, too. Output spectrum according to the value of  2av for  = 2.

Conclusion

In this work we have studied how a longitudinal and linear evolution of the parameters of a dispersion oscillating fiber may affect the spectrum of quasi-phase-matched modulation instability or parametric resonance. We found that even slight deviations of a constant spatial period significantly affect the output sideband spectrum, in a manner that is sensitive to the direction of use of the DOF. Moreover, variations of the average DOF dispersion also impact the MI gain, leading to a noticeable sideband broadening, and the development of additional amplitude modulations in the sideband gain. To the contrary, as long as the ratio of sideband amplitude fluctuations to average dispersion  2amp / 2av remains moderately high, even large changes of the amplitude of the dispersion fluctuations do not significantly affect the main properties of the spectrum. On the other hand, in the regime of sufficiently high  2amp / 2av ratios, a new type of spectral sideband splitting was observed. In all of the above cases, a simple but approximate approach based on the averaging of Floquet spectra obtained by a linear stability analysis was found to be useful, and it could permit to qualitatively reproduce most of the observed spectral sideband features. However, the sideband splitting effect could not be captured by the averaged Floquet approach, since it results from a coherent constructive or destructive interference among the resonant waves generated at different points along the linearly chirped DOF.

Figure 1 :

 1 Figure 1 : Evolution of the longitudinal dispersion profile of a DOF with sinusoidal dispersion modulation (grey line), or for a quasi-periodic DOF with linear evolution of either: the spatial period (panel a, black line,  = 0.1); the average dispersion (panel b, black line,  = 0.5); the amplitude of the dispersion fluctuations (panel c, black line,  = 2).

Figure 2 :

 2 Figure 2 : (a) Evolution of the MI gain spectrum (or Arnold resonance tongues) according to the amplitude of the linear change of the spatial period. Results are obtained from the numerical integration of the NLSE. (b) Evolution of  at -10dB according to the QPM sideband number p: results from numerical simulations (black triangles) are compared with results from the Floquet LSA (grey circles)

Figure 3 :

 3 Figure 3 : (a) Output spectrum obtained from numerical simulations (panel a1) without linear evolution of  (black line), for  = -0.1 (grey curve) and  = 0.1 (light grey curve). Result from Eq. (6) are plotted on panel (a2) for || = 0.1 (grey line). (b) Evolution of the QPM MI spectrum g() according to  for a strictly periodic DOF, as

Figure 4 :Figure 5 :

 45 Figure 4 : (a) Evolution of the MI gain spectrum or resonance tongues vs. the amplitude of the linear variation of the average dispersion. Results are obtained from the numerical integration of the NLSE. (b) Evolution of  at -10dB vs. the QPM sideband order: results from numerical simulations (black triangles) are compared with results from the Floquet LSA (grey circles)

Figure 6 :

 6 Figure 6 : (a) Evolution of the MI gain spectrum or resonance tongues vs. the strength of the variation of the dispersion oscillation amplitude; here 2av = 2 ps 2 /km. Results are obtained from the numerical integration of the NLSE. (b) Evolution of  at -3dB according to the order of the QPM sideband: results from numerical simulations (black triangles) are compared with approximate results from the Floquet LSA (grey circles).

Figure 7 :

 7 Figure 7 : (a) Output spectrum obtained from numerical simulations (panel a1) and from Eq. 6 (panel a2) for a constant  2amp (black curve), and for  = 2 (grey curve); here 2av = 2 ps 2 /km. (b) Evolution of the QPM MI spectrum according to  2amp for a strictly periodic DOF as predicted by the Floquet LSA. (c) Evolution of the output spectrum according to || as predicted by the use of the approximate approach based on Eq. (6). Same colormap as Fig. 6.

Figure 8 :

 8 Figure 8 : (a) Evolution of the MI gain spectrum as a function of the value of the linear amplitude change of the dispersion fluctuations: here  2av = 1 ps 2 /km. Results are obtained from the numerical integration of the NLSE. (b) Details of the output spectrum recorded for =2. (c) Evolution of the maximum value of the sideband gain vs.  for various sidebands (black, dark grey and light grey are for the first, second and fourth QPM sideband, respectively).

Figure 9 :

 9 Figure 9 : Same as Fig. 7 but for  2av = 1 ps 2 /km. Same colormap as Fig. 8.

Figure 10 :

 10 Figure 10 : (a) Longitudinal evolution of the QPM MI gain spectrum in a fiber with  = 2 and  2av = 1 ps 2 /km. (b)

  Q stands for either  2av , or  2amp ; Q is the average value of the quantity under study, and  characterizes the rate of evolution of Q. Positive values of  indicate that the value of the Q parameter grows larger along the propagation distance. The ratio of the peak-to-peak amplitude of fluctuation Q over Q is ||. The values of the parameters Q 0 and Q L at the input

	and output of the fiber are therefore 0 QQ   1 2   	and	1  2    QQ L 

Acknowledgements

We thank Julien Fatome and Kamal Hammani for stimulating discussions. We acknowledge the financial support of the Conseil Regional de Bourgogne (Pari Photcom), the funding of the Labex ACTION program (ANR-11-LABX-01-01), and the Italian Ministry of University and Research (grant no. 2012BFNWZ2).