Bettina Migge

Donald Winford

Margie Linger

Renata Mcbean

The origin and development of possibility in the creoles of Suriname 1

Keywords: Possibility, Creoles of Suriname, contact-induced change

In this paper we discuss the origin and development of the expression of possibility in the creoles of Suriname. We first describe the systems of possibility in Sranan and three Maroon creoles (Ndyuka, Pamaka, and Saamaka), drawing on data elicited from informants, conversational data as well as the published literature. We examine several modal elements, namely sa, kan, man, poy, whose distribution differs across the different varieties and also over time. Our analysis reveals that the system of possibility in Sranan is organized quite differently from that of the Maroon creoles. To explain these facts, we trace the development of this area of grammar by drawing on historical data from the early Sranan and Saamaka texts, and by exploring possible influence from the Gbe substrate languages as well as Dutch. We argue that the overall structure of this subsystem in the Maroon creoles was broadly modelled on Gbe while the rather different system found in Sranan Tongo is due primarily to influence from Dutch, and to internal 1 Most of the data on the contemporary creoles of Suriname and on the Gbe varieties come from fieldwork conducted in Benin and Suriname between 2002-2004 as part of the project "The influence of West African Languages on the tense/mood/aspect (TMA) systems of two Surinamese creoles". We would like to hereby gratefully acknowledge the funding of the National Science Foundation (NSF Grant #BCS-0113826). We would also like to thank informants in Benin and Suriname/French Guiana for generously giving their time and H. Capo, D. Gagnon, J. Essegbey and B. Sanna for helping with the collection of the data, and H. Capo, J. Essegbey

Introduction

The creoles of Suriname raise some intriguing questions concerning the interplay of internally and externally motivated language change in the formation and development of creole grammar. While the Surinamese creoles appear to descend from a common source, they still display noticeable differences that raise questions regarding their origins and developments that have not yet been fully explored. One subsystem that shows a fair amount of variation across the different Surinamese creoles is the expression of possibility. It involves four elements -sa, kan, man, poy -whose distribution varies across the different varieties and also over time. Moreover, the different elements do not appear to have emerged as the result of the same processes. This suggests that the development of this area of grammar represents an instance of gradualism in the emergence of the grammar of the creoles of Suriname (Arends and Bruyn 1995: 111;[START_REF] Arends | Towards a gradualist model of creolization[END_REF][START_REF] Arends | The origin of the Portuguese element in Surinam Creoles[END_REF][START_REF] Singler | Theories of creole genesis, sociohistorical considerations, and the evaluation of evidence: The case of Haitian creole and the relexification Hypothesis[END_REF][START_REF] Singler | On the use of sociohistorical criteria in the comparison of creoles[END_REF].

In this paper we examine the expression of possibility in Sranan Tongo (henceforth Sranan) and the Maroon creoles (Ndyuka, Pamaka, and Saamaka), drawing on data elicited from informants, conversational data as well as the published literature. Our analysis reveals that the system of possibility in Sranan is organized quite differently from that of the Maroon creoles. The most striking difference has to do with the different meanings that the modal sa conveys in the creoles. The Maroon creoles, (Pamaka, Ndyuka and Saamaka) employ this modal in a wider range of functions than Sranan does. For instance, Sranan distinguishes sa from modal kan, which expresses both epistemic and deontic possibility, and also from mag (< Dutch mag), which expresses permissibility. By contrast, the Maroon creoles subsume all of these types of possibility under sa, suggesting that they employ this auxiliary as a marker of potential mood. To explain these facts, we trace the development of this area of grammar by drawing on historical data from the early Sranan and Saamaka texts, and by exploring possible influence from the Gbe substrate languages as well as Dutch. We argue that the overall structure of this subsystem in the Maroon creoles was broadly modelled on Gbe. On the other hand, we argue that the rather different system found in Sranan is due primarily to influence from Dutch, and to internal developments.

The paper is organized as follows. Section 1 compares the systems of possibility and the various modal elements that are used to express it in the contemporary creoles. Section 2 describes the distribution and uses of these elements in early Sranan and Saamaka documents, and compares them with their modern counterparts, with a view to determining similarities and differences in the expression of possibility. Section 3 focuses on the influence of Dutch in the emergence of the Sranan system of possibility. Section 4 examines the uses of sa in the early Sranan and Saamaka documents, while section 5 compares the uses of sa in Sranan with the uses of Dutch zullen. Section 6 argues that the subsystem of possibility in the Maroon creoles, and the uses of sa in particular, were heavily influenced by varieties of Gbe. The final section summarizes the findings and discusses their implications.

Possibility in the modern creoles of Suriname

Possibility includes notions such as ability, present likelihood, and permissibility, which respectively represent cases of dynamic, epistemic and deontic modality. Cases of dynamic modality simply assert what seems to be a factual statement, and do not involve the opinion or attitude of the speaker, except that the statement is true (Palmer 1986:102).

Epistemic possibility "indicates the extent to which the speaker is committed to the truth of the proposition, for instance, when s/he conveys the sense that a proposition may possibly be true" (Bybee et al. 1994: 177). Finally, deontic possibility refers essentially to permission, which involves a deontic source that may be either the speaker or some other person or institution that creates the permission.

Dynamic possibility in the creoles of Suriname

Dynamic possibility includes mental or learned ability and physical ability, as well as the broader concept of root possibility, which "is not restricted to the internal condition of ability, but also general external conditions" (Bybee et al 1994:178) .

All of the Surinamese creoles distinguish learned ability from physical ability. The former is expressed by a construction in which the verb sabi/sá 'to know' selects an activity verb. It MOD COP that rain FUT come today night 'It may rain tonight (= It's possible it will rain tonight)' Some Sranan speakers seem to employ the modal auxiliary kan to convey epistemic possibility, but this appears to be quite rare. Only one of our informants allowed for the following:

(18) SN John kan de na oso nownow.

John MOD be-there LOC house nownow 'John may be at home now.'

It must be noted that Sranan also uses sa in an epistemic function, as in the following sentence:

(19) SN Jan sa de na oso nou(nou).

John MOD be-at LOC house nownow 'John should be at home now (i.e. it's probable that John is at home now).'

This at first glance suggests that Sranan uses sa in the same way as the Maroon creoles. But Winford (to appear) demonstrates that the epistemic sense of sa in Sranan is somewhat different from that in its sister creoles.

In the former, sa conveys the speaker's assessment of the probability of a situation being true, based on her/his assumptions, expectations and weighing of options in relation to the situation in question. This is in fact quite close to the epistemic use of English must, that is, the sense of strong probability. Closely related to this is the use of sa in assertions like the following which express strong expectation on the speaker's part:

[Context: John's father has died in Holland, and John is in Suriname. He hates flying, but still:]

(20) SN Fu di na en p'pa dede, a sa go na p'tata.

For that FOC his father dead he MOD go LOC Holland 'Since it's his father that died, he shall/should go to Holland.'

In such cases, sa conveys the sense that the other person will perform the future action because he is under some kind of moral or social obligation to do so. By contrast, in the Maroon Creoles, the informants always insisted that sa in cases like these always expresses possibility, and never claimed that it could convey strong probability or the sense of an obligation. 4 The similarities between the Maroon creoles and Sranan in relation to the use of sa, then, are limited, or partial. We discuss the use of sa in Sranan in more detail in section 5.

5. Summary of the findings

To sum up, several differences are found among the Creoles.

Saamaka employs sa to express all forms of possibility, including ability, root possibility, permissibility and epistemic possibility, in both positive and negative contexts. Ndyuka and Pamaka, on the other hand, use sa to express all these notions in positive contexts, but employ poy and man, respectively, in negative constructions, as well as in a few positive constructions involving physical ability. We suggest that, in the Maroon creoles, all types of possibility are subsumed under a category of Potential mood, expressed by sa. This accords with previous analyses in the literature. Thus, Huttar and Huttar (1994:513) describe sa in Ndyuka as conveying intention, uncertainty and potential mood, that is, the sense of 'be able.' Similarly, Rountree (1992:44) describes sa in Saamaka as meaning 'may, might, can', and says it indicates possibility or ability. The various interpretations of sa as expressing a particular type of possibility are largely dependent on the discourse context. We further suggest that, in Ndyuka and Pamaka, poy and man respectively convey dynamic and deontic possibility in negative contexts. It may be reasonable to say that their core meaning is ability. Finally, in Sranan, we can identify at least three distinct modals of possibility, kan for root possibility, man for physical ability, and mag for permission. Epistemic possibility is generally expressed by adverbial kande or the expression A kan de taki, and not by a modal, though some speakers appear to use kan in this sense. But changes seem to be in progress in Sranan that appear to be leading towards making man the marker of physical ability, kan the marker of possibility and mag the marker of permission in both negative and positive contexts. In addition, Sranan differs from the Maroon creoles in that it uses sa to convey the sense of probability, and in some contexts, expectation or obligation.

The differences among the creoles raise two sets of questions.

First, have these differences always been present or did they emerge later, as each creole evolved separately? If they emerged later, what was the original system like? Second, what are the reasons for the differences in the modern systems? Did they come about due to different kinds of input, different kinds of contact-induced changes and/or different kinds of language-internal changes?

In the next section, we investigate the ways in which the various modal elements surveyed above are used in the early Surinamese creole texts, in order to determine whether their functions there are similar to those found in the modern creoles.

The uses of the modals of possibility in the early documents

Our investigation is necessarily limited to earlier Sranan and Saamaka, since no early documents are available for the Eastern Maroon varieties Pamaka and Ndyuka. While there are a few documents from the late 17 th century, e.g., court records (van den Berg 2000), and early 18 th century, e.g., the Herlein fragment of 1718, most of the linguistically substantial documents date from the mid to late 18 th century, that is at least 70 years after the projected emergence of the early plantation varieties in the later 17 th century. 5 Most of these documents were written by Europeans whose exposure to the language and its different varieties varied. For discussions of the background of the authors see Arends (1995), [START_REF] Bruyn | Grammaticalization in Creoles: The development of determiners and relative clauses in Sranan[END_REF], [START_REF] Van Den Berg | A Grammar of Early Sranan[END_REF].

Our investigation of the early documents from Sranan and Saamaka reveals similarities to, as well as some important differences from, the modern varieties. The most significant difference concerns the modal sa, which is used in the early texts primarily to express futurity and related modal notions to be discussed below, but is not used to convey possibility, as far as we can tell. In general, possibility seems to be expressed primarily by kan in both early Sranan and Saamaka. We will discuss the expression of possibility in the following section, and then consider the uses of sa. ). Poli is not used in modern Saamaka, but, as we have seen, its modern form, the modal verb poy, conveys ability in negative contexts in modern Ndyuka.

3. Summary

To sum up, both early Sranan and Saamaka express possibility with the modal kan and other elements such as mag (Sranan) and poli (Saamaka). Sa, as far as we can tell, is not part of the system of possibility in the early documents. We also found clear continuities between early and modern Sranan in the use of the modals of possibility. By contrast, the system represented in the early Saamaka texts differs sharply from that of modern Saamaka, where sa is the primary means of expressing possibility.

In the following sections, we attempt to explain the different developments in the expression of possibility across the contemporary creoles. First, we argue that the modern Sranan system of possibility, centered around modals kan and mag, is modelled largely on that of Dutch, with the exception of modal man, which is due to both substrate influence and internal developments. Second, we attempt to explain the significant differences between Sranan and the Maroon creoles in their use of the modal sa. We argue that the uses of sa in both early and modern

Sranan have been strongly influenced by the uses of Dutch zullen 'shall, should.' On the other hand, we argue that the use of sa as a marker of Potential mood in the contemporary Maroon creoles may have been due

to the influence of Gbe languages, which had markers of Potential mood with uses similar to those found in the Maroon Creoles.

Dutch influence on possibility in Sranan.

Most of the uses of kan attested in the early Sranan texts are identical to those found in contemporary Sranan. The meanings it conveys in both forms of the language include root possibility, ability and permission. Its use to convey epistemic possibility is marginal at best in modern Sranan, and seems to be a recent development. name can surely obstructed be 'Jan may be held up.' (Coppen et al. 2004: 18

•5•4•4•ii•a, ex. 1)
The close similarities in meaning and function between kan in Sranan and kunnen in Dutch suggest that the former was modelled on the latter.

However, given that the main superstrate input to the creoles of Suriname, English, has a phonetically and semantically similar form, can, it is equally likely that the meanings of kan initially emerged due to influence from English and were subsequently only reinforced by Dutch influence.

Finally, we saw that, in early Sranan, permission could also be expressed by mag, which is clearly a borrowing from Dutch, compare (23) with its translation (29b) and ( 29a).

(29) a. DU Je mag hier niet roken.

You MOD here NEG smoke 'You are not allowed to smoke here.' (Coppen et al. 2004: 18•5•4•4•iii•d, ex. 36 The future marker o in the contemporary creoles most likely developed from (de/tan) go via a cross-linguistically well-attested path of grammaticalization [START_REF] Bybee | The evolution of grammar. Tense, aspect, and modality in the languages of the world[END_REF]) and phonological reduction (Migge and Goury in press).

The emergence of this new future marker as a competitor to sa is quite relevant to our discussion, since it may explain why sa gradually became associated less with the temporal notion of later time reference, and more with the modal meanings it now conveys. Van den Berg (2007: 192) suggests that (de) go expressed predictive future, prospective future, and intention. She also suggests that (de) go conveyed a greater degree of commitment to the truth, while sa became associated with overtones of uncertainty (p. 193-4).

With regard to early Saamaka, Schumann, and following him, Reimer, provide some insight into the difference between the competing markers of futurity. They argue that tan + go is the main or common future-marking strategy in 18 th century Saamaka. Riemer also notes that "…sa can be used instead of tanngo, e.g. mi sa go, mi sa worko;

but actually sa in this case is Town-language (Sranan), because in Saramaccan it means only: sollen [German: 'shall, should']." (Perl's translation of Riemer 1779 in Arends and Perl 1995:374).

This suggests that the use of sa as a marker of later time reference survived somewhat later in early Sranan than in Saamaka. If so, it might be attributed to Dutch influence. Van den Berg (2007:196) Unfortunately, Schumann himself provides no translations of these sentences, nor explanation of what sa meant in ningretongo, since we have to assume it was also used in that variety. Whatever the facts might have been, it is at least clear that there was a growing distinction between sa and de/tan go in later 18 th century Sranan and Saamaka. Moreover, as we saw, sa developed very different meanings in the two creoles. In the following sections, we attempt to explain the differences in term of stronger and continuing Dutch influence on the use of sa in Sranan, and stronger Gbe influence on its use in the Maroon creoles.

Dutch influence on sa in Sranan

We suggested earlier that sa derives from Dutch zullen, though English shall cannot be ruled out as an alternative source. The meanings and uses of sa in modern Sranan in many respects parallel those of its Dutch (and/or English) source. Both can be used to convey dynamic, epistemic and deontic modality.

When used to refer to future events, both sa and zullen express dynamic modality, that is, they express what seems to be a factual statement that does not involve the attitude or opinion of the speaker (Palmer 1986:102). In cases where the speaker is the agent of the future action, they both convey commitment, which may be construed either as a promise (38), or a threat (39), depending on the circumstances.

( (Wendelaar and Koefed 1988:68) There also seems to be some sense, at least in Sranan, that the agent is under some obligation to act. This may be related to the fact that zullen, like its counterparts in other Germanic languages (English shall, German sollen, Scandinavian sk), originally conveyed obligation, and that modal sense still underlies its use today.

This undertone of obligation is also relevant to the fact that both sa and zullen can be used in both epistemic and deontic senses, to convey probability and requirement (a command) respectively. 9 The former use is It is interesting that a Sranan and Dutch informant both expressed the view that sa seems odd here, since no one can command the weather, except God, as the Sranan informant noted.

To sum up, the use of sa to express dynamic modality in modern Sranan parallels its use in early Sranan, though it may have lost its earlier temporal character. The other uses of sa in contemporary Sranan, to express epistemic and deontic modality, may have developed more recently, though we cannot be sure that they did not exist in early Sranan.

Finally, all three modal meanings of sa in modern Sranan seem to have been modeled on the meanings of zullen.

Gbe Inflence on possibility in the Maroon Creoles.

We argue in this section that the system of possibility in the Maroon creoles was shaped primarily by influence from Gbe languages. This explains the use of sa in these creoles, as well as several other strategies they use to express possibility. Some of the latter strategies are also found in Sranan.

Gbe influence on sa.

We offer two possible scenarios that might explain how substrate influence shaped the meanings and uses of sa in the Maroon creoles.

According to one scenario, argued for in [START_REF] Winford | Substrate influence on the emergence of the TMA systems of the Surinamese creoles[END_REF] The correspondences between the Gbe markers of possibility and sa suggest that the latter may have been modeled on the former.

6.2

Gbe influence on other aspects of possibility.

In We suggest that the uses of man (in Sranan) and man and poy (in Pamaka and Ndyuka) were modelled on these modals (cf. [START_REF] Migge | Tracing the origin of modality in the creoles of Suriname[END_REF].

Another similarity between the creoles and Gbe languages is that they all express epistemic possibility in negative contexts by means of a periphrastic construction translateable as 'it can be that.'

  should be sleeping now.' (Margot van den Berg, pc April 2008) This use of present forms of zullen to express probabilty in Modern Dutch seems to be somewhat archaic. Generally, speakers seem to prefer to use a past form of zullen e.g, zoud(en) 'should' or the modal moet CHECK in this function (Margot van den Berg, pc, April 2008).Finally, in certain contexts, both sa and zullen have deontic force, and convey the sense of a command, as in the following: rain tonight' (= I command it to rain).(Margot van denBerg, pc April 2008)

  addition to sa, other devices used to express possibility in the Surinamese creoles also have counterparts in Gbe. For instance, like the creoles, several Gbe varieties (Aja, Xwela, Xwla) employ different auxiliaries to express possibility in negative contexts from those used in positive contexts. 10 (53) a. Xwla NyOÚnuÙ xeÙOÙ uÚn •oÚ OÙNkuÚsiÙ oÙ, uÚn kpeÚ-ji ÚaÚ woman DEM NEG have eye NEG NEG can FUT kpOÚn miÙ oÙ. see us NEG 'This woman is blind. She cannot see us.' b. Xwela IviÚ lOÚ goÚ kpego laÚ nO xye •eÚ hwedoÚ child DET NEG can FUT stay here LOC night eÙ. NEG 'The child cannot stay here tonight.' 10 Other Gbe varieties (Gen, Gun, Waci, Maxi) employ the same modal in both positive and negative contexts like Saamaka (cf. Migge 2006: 40ff). The auxiliaries used to express negative possibility derive from a construction expressing physical strength, or the sense of having achieved manhood (cf. Migge 2006: 49). (54) Aja sE)--Nu be hard/be strong-body Xwla/Waci kpé-é-ji reach-its-summit/top Maxi kpé-e-ewu! reach-its-body 'be strong' (Capo, p.c. 2003/Aboh p.c. 2004)

(

  55) a. Waci «Ú ti!NÚ nyi! b«Ú Ko@fiÚ ýuÙ-naÙ nu! cu! sa)Ú. it can COP that name eat-HAB thing all before 'It is possible that Kofi was greedy before' (Capo, p.c. under the modal sa, which became a marker of Potential mood. Ndyuka and Pamaka also employed poy and man respectively, to express the sense of ability under negation. The use of sa in the Maroon creoles contrasts strongly with its use in Sranan, where it conveys meanings quite similar to those of Dutch zullen. Finally, Gbe seems to have provided a model for other devices used to express epistemic possibility, such as the expression 'it can be that.' Our study has shown that the grammars of the Surinamese creoles continued to evolve over time, in accordance with Arends' gradualist hypothesis. The combined effects of Superstrate and substrate influence, and internal factors, interacting with each other in different ways and at different times, all contributed to the creation and development of the grammar of possibility in the Surinamese creoles. The changes that occurred in the systems of possibility over time present a challenge to any attempt to reconstruct the historical processes that led to the present state of affairs. Our investigation leaves many questions unanswered. For instance, what circumstances led to the more pervasive influence from Gbe languages on the Maroon creoles? Were there greater similarities between the earlier forms of Sranan and the earliest forms of the Maroon creoles? What similarities and differences were there between ningretongo and bakratongo, and how did each variety affect the later development of Sranan? What part did internal

  All of the modern modals, kan, man and, more rarely mag, can be found in the early Sranan texts. Kan and mag are used in much the same way as in modern Sranan, while man functions more like a noun, though it is clearly the source of the modern modal.Kan closely resembles both the English modal can and the singular form of its Dutch counterpart, kunnen (Cf: ik kan, jij kan, zij/hij kan Kan is also used in the same meanings in negative contexts (22b), where man would be expected in modern Sranan. The use of man (also spelt In early Saamaka, only kan and poli, the reflex of modern poy, are attested. Kan is not attested in the Saamaka dictionaries, but we found one example in Wietz' translation of the gospel and four instances of kan in Our conversational and elicited data from the Maroon creoles suggest that kan is not generally used in the contemporary Maroon creoles outside of cases of code-mixing and codeswitching with Sranan (cf.[START_REF] Migge | Codeswitching and Social Identities in the Eastern Maroon community of Suriname and French Guiana[END_REF] and the phrases a (no) kan taki… 'It is (not) possible that…'.

	DET lady say if Schuchardt's edition of Schumann's 1778 Saamaka dictionary and in master please he MOD go
	2. 1. Possibility in the early Sranan Tongo texts lange da boto. early or modern Saamaka. Wietz' translation of the gospel poli conveys a sense of ability.
	PREP DET boat Mag also appears, though rarely, to express permission in early
	'The lady says you can go on [with] her boat.' the Maroon Letters from Saamaka. They express ability (24a) and root Sranan. (25) a. ESM wi no poli va brokko di tranga watra aki.
	(Van Dyk c1765 in Arends and Perl 1995: 214) (23) ESN zomma no mag jarri hem da ogeri possibility (24b). we NEG can PREP break DET strong water here kwitti
	person NEG MOD carry him that terrible very (24) a. Joe no sabbi da moeffe va mi bieka joe no 'We cannot fight against the current here.'
	na mi hay. you NEG know DET mouth for me because you NEG (Schumann 1778 in Schuchardt 1914: 57)
	LOC my eye kan jerri gweette gweet.
	can hear at.all	at.all
	(22) ESN Mi no	man va hoppo datti.
	I	NEG	man for lift	that
	'I am not strong enough to lift that (lit.: I am not man for lift that).'
	'I cannot lift that.'
	(Schumann 1783: 107 cited in van den Berg 2007: 216).
		I NEG have teeth more I NEG MOD chew
	5 The historical data for Sranan come from the following sources: the Court Records from
	1667 until 1767 (in van den Berg 2000); Herlein 1718, van Dyk c1765, Nepveu 1770 (in 'I don't have any teeth anymore, I cannot chew.' In (22) man is a noun meaning '(male) person'. The modern modal
	Arends and Perl 1995); the Saramaccan Peace Treaty in Sranan (1762) (Arends 2004 and
	van den Berg 2004); Schumann's dictionary of Sranan (1783) (Kramp 1983); the Neger-Engelsch -Nederlands Woordenboek from Focke 1855. The historical data for Saamaka (Schumann 1783 in Kramp 1983: 150) auxiliary emerged from this noun due to a process of reanalysis and
	come from a Saramaka dictionary and grammatical sketch by Riemer (1779) (in Arends
	and Perl 1995), Schuchardt's edition of Wietz' edition of Schumann's Saamaka c. ESN Da misi takki offe mastra plessi a kan go grammaticalization, and substrate influence (Van den Berg 2001, Van den
	dictionary (1778), the translation of the story of the Apostles by Wietz 1805 (in
	Schuchardt 1914) and the Saamaka Maroon Letters (Arends and Perl 1995).

'I/you/she/he can'), suggesting that it originally derived from either one or possibly from both of them. In the early Sranan texts kan is used in ways quite similar to its modern counterpart, to express root possibility (21a), (physical) ability (21b) and permission (21c). (21) a. ESN lange monni mi kan kissi alle zanti na engelsze konderi. with money I MOD get all thing LOC English country 'I can get anything with money in England.' (Van Dyk c1765 in Arends and Perl 1995: 153) b. ESN mi no ha tiffi morro, mi no kann kau. mann) to express ability is first attested in the court records in 1707 (Van den Berg 2007: 215). It is also attested in Schumann's 1783 Sranan dictionary where it also expresses negative physical ability but selects a non-finite VP headed by the preposition va (22). 'People were not allowed to carry him. It's a terrible sight.' (Van Dyk c1765 in Arends and Perl 1995: 198) We found no examples of kan being used to express epistemic possibility, but instances of this are rare in the early texts anyway WHAT'S YOUR REFERENCE?, so no firm conclusion can be drawn. However, the adverb zomtem 'perhaps' is sometimes used, with sa, to express possibility (see below). This parallels the use of kande (in combination with future o and sa) in modern Sranan. In general, then, the system of possibility in early Sranan is quite similar to what it is in modern Sranan. 2.2. Possibility in the early Saamaka documents. 'And you don't know my language becaue you can't understand it at all.' b. na hem ala somma kan findi boene liebi va teh go. in him all person can find good life for until go 'In Him everybody can find eternal life.' (Maroon Letters from Saramaka 1790 in Arends and Perl 1995: 286) Kan is not used in the modern Maroon Creoles. 6 The modal poy derives from the earlier form, poli, which in turn derives from pode [podi] 's/he can,' the third person singular form of the Portuguese verb poder 'can' (Smith 1987). In the examples attested in 6 In most examples it occurs together with the negative marker and selects a non-finite VP headed by the preposition va. However, there is one example in Wietz where it is used without the negation marker and another one in which va is absent (cf. Wietz 1805: Schuchardt 1914: 29, 16

  The various uses of kan are very similar to those of kunnen in Dutch, as illustrated in the following

			suit		come so can he with go
	examples, taken from van Dyk's (1765) early Dutch (EDU) translations of 'The lady says if it suits my master he can come along.'
	early Sranan.	(Van Dyk 1765, in Arends and Perl 1995: 214)
		First, kunnen expresses root possibility, including ability in both
	languages. Third, in modern Dutch, and presumably also in early Dutch, kunnen
	(26) a. ESN Mastra da homan no kan wakke expresses the notion of epistemic possibility.
	(28)	Jan	master the woman NEG can walk kan wel verhinderd zijn.
				rasi	brokke kwetti.
				buttocks break	quite
		b. EDU Meester dat Vrouws-Perzoon kan niet loopen,
				master that woman-person	can not walk
				haar Billen binnen stukkend.
				her buttock in	pieces
				'The woman can't walk, master. Her buttocks are
				completely ruined.'
				(Van Dyk 1765, in Arends and Perl 1995: 179)
	Second, kunnen also conveys the notion of permission (compare 21c).
	(27) EDU De Jufvrouw zeid als het myn Heer
			DET lady	say if	it	my	master
			geleegen komt zoo kan hy meê gaan.

  Van den Berg (2007:188) confirms that later time reference is expressed by (variants of) the preverbal marker sa, and/or by temporal adverbs denoting later time reference, in all of the early Sranan sources.These findings are in keeping with the views expressed by Schumann (1783:146), who observes that, in early Sranan, sa conveys future time reference, as well as the sense of (German) sollen 'shall, should.' With regard to earlier Saamaka, however, Schumann and, following him, Riemer sugggest that sa means only: sollen (Perles translation of Riemer 1779 inArends and Perl 1995:374).The early documents also show that, by the late 18 th century, the motion verb go, usually in combination with progressive markers de (Sranan) and tan (Saamaka) had emerged as a means of conveying futurity, thus entering into competition with sa.

	In the early Sranan and Saamaka texts, sa expresses future time b. ESM Mi sa da yu kangra. 1995:341)
	) reference in contexts specially marked for future time reference (30a and I MOD give you kangra
	b. EDU … Mogt hy niet b) and in constructions that lack such overt reference (31a and b). 8 door Menschen 'I will/shall use the kangra method to determine whether or
	MOD Sa also conveys certain modal overtones, such as commitment or he NEG through people not you are guilty (of some crime).' (Schumann 1778 in
	Begraaven worden. (30) a. ESN Pikien morre wi za intention on the part of the speaker, or the sense of weak obligation or drinki koffi. Schuchardt 1914: 76)
	bury small more we ??? requirement. We regard these as secondary meanings of sa, which are FUT drink coffee
	'… was he not allowed to be buried by people/humans.' 'We'll drink coffee in a few moments/a little while.' contextually determined. Commitment typically takes the form of a In both early Saamaka and Sranan we also find uses of sa where it seems
	(Van Dyk 1765, in Arends and Perl 1995: 198) (Van Dyk c1765 in Arends and Perl 1995: 150) promise (32) or a threat (33): to express a sense of weak obligation or requirement.
	b. ESM mi bribi,	tide	bakkraman	sa	doro.
	4. (32) a. ESN Mastra tanki I believe today European.person FUT arrive fo joe mino zal doe morre. (34) a. ESM mi sa go jusnu?
		'I believe that the people of Paramaribo will come back I MOD go just now
		today.' 'Should I go immediately?' (Schumann 1778 in Schuchardt 1914: 61)
	(31) a. ESN Ke (Riemer 1779 in Arends and Perl 1995: 341) joe za habi plyziri noeffe. he IMF go pass over
	INTERJ you FUT have pleasure enough b. ESN Da bassia takki mi za go na mastra 'It will/is going to overflow.'
		'You'll have a lot of fun.' when I go LOC plantation I MOD come look you one.time DET manager say I MOD go LOC master (Schumann 1783 in Kramp 1983: 45)
	(Van Dyk c1765 in Arends and Perl 1995: 144) 'When I go to the field/plantation, I'll come to see you as fo locke koekeroe worke. b. ESN Mi go meki brifi.
	b. ESM ju well.' for look kitchen sa poli tulle sanni. (Schumann 1778 in Schuchardt 1914: 113) work I FUT make letter
		you FUT spoil all 'The manager said that I should go to the master to do work things 'I'll/am going to write a letter.'
	'You will spoil everything.' (33) a. ESN a doe wan trom morre mi za kiele da homan. in the kitchen.' (Schumann 1783 in Kramp 1983: 69)
	(36) ESM	(Schumann 1778 in Schuchardt 1914: 97) she do one time more I MOD kill the woman (Van Dyk c1765 in Arends and Perl 1995: 185) mi tann go worko.
		'If she does it one more time, I shall kill the woman.' I IMF go work
		(Van Dyk c1765 in Arends and Perl 1995: 179) 'I will work.' (Riemer 1779 in Arends and Perl

Sa in the early texts

We now examine the use of sa in the early texts and compare them with its use in the modern varieties. Our examination reveals that sa is used in quite similar ways in early Sranan and Saamaka. It's primary use is to express futurity, though it also conveys certain modal notions. In the early Sranan (ESN) texts, sa is spelt variously as zal,

zel, za, ze, zey, zoe, saa, sal and sa (van den Berg 2007: 199)

.

7

It appears mostly as sa in the early Saamaka documents. Recent analyses (Winford to appear, Migge and Goury in press) suggest that it derives from zal, the first and third person singular form of the Dutch auxiliary zullen 'must', but others link it also to English shall (cf. Van den Berg 2007: 188). master thanks for you I.NEG MOD do more 'Please, master, (I promise) I won't do it again.' (Van Dyk c1765 in Arends and Perl 1995: 179) b. ESM teh mi go na plantasi, mi sa komm luku ju wanten.

(35) a. ESN a de go passa abra.

  , the broad use of sa in the Maroon creoles is modelled closely on the potential future marker found in Western Gbe languages. Several Gbe varieties, e.g., Ewegbe varieties such as Wacigbe and Anlogbe, as well as Xwelagbe and Xwlagbe, employ a marker of futurity (lá, á) which can be used to express notions such as epistemic possibility, root possibility, permissibility, and so on. Essegby (to appear) argues that these markers convey the sense of potential mood in Ewe. For instance, (l)à can be used with stative predicates to refer to a possible state of affairs in the present:This range of meanings closely resembles that expressed by sa in the Maroon Creoles. In all these cases, what seems to be expressed by the modal auxiliary is the sense of something that is possible, or a potential situation -a sense which may be interpreted in the various ways described above. These similarities suggest that sa in the Maroon creoles might have been modelled after the Potential future marker of Western Gbe varieties.An alternative scenario to explain substrate influence on the expression of possibility in the Maroon creoles is offered in[START_REF] Migge | Tracing the origin of modality in the creoles of Suriname[END_REF].She argues that sa is modelled on the Gbe modals of possibility like teŋu, tiŋ, tєn, sixu, etc, which express a range of meanings quite similar to that of sa in the Maroon Creoles, including physical ability (49), root possibility (50), permission (51), and epistemic possibility (52). (49) Xwla e jO asu. e tEn kO kilo kOnwewi. he be man he MOD lift kilo 100 'He is very strong, i.e. acts like a man. He can lift 100 kilos.'

		you-POT-read book that easily
		'You can read that book easily.'
	In this case, Essegby comments that "this sentence does not make
	predictions about a state of affairs to come. Instead, it suggests that the
	addressee is in a position to read the book without difficulty" (To appear,
	p. 11). Our elicited data also contain examples in which notions like
	ability (47), and permissibility (48) are expressed by (l)a or its cognates.
	(47) Wací	´! s´N NtO ya!		kO	kilo ka!v´ kuwu'i!
		he strong very	he-FUT lift	kilo 40x2+20
	(44) Anlo ¶evi--a--wo 'He is very strong; he can lift a hundred kilos.' A$--NO alO dO--m	fifia
		child-DEF-PL POT-AUX:NPRES sleep sleep-PROG now
	'The children may be sleeping now.' (48) Xwela nE! o hE!n O, o la! lE le!le! (50) Xwla e tEn luter na EmE--bu ya.	gO
	It may also be used to refer to a future possibility: if you want PART you FUT for amuse with he can fight PREP person-other EMPH
	(45) Anlo	m--a--Ãu a!--yi xi! ha! lE. 'He can fight for someone else.' fofo!--nye gbO!
	1SG-POT POT-go father-1SG place your friend PL (51) Xwla evi O tEn nO fi xwesa xeO.
		'I may move to my father's place.' 'If you want to, you can play with your friends.' (E11b) child DET MOD stay here night this
		(Essegby to appear:16) 'The boy may stay here tonight.'
	Third, the Potential marker can also be used to convey the sense that it is (52) Xwla Jan tEn le OÙxweÚmE siÙEÚ.
	possible for someone to do something: Jean can COP house	now
	(46) Anlo	á--xlé 'Jean may be at home now.' agba-lê	mábbeee	(2004 p. 10)

In the Maroon Creoles, sabi also selects a small number of nouns that imply a certain activity, e.g. buku implies the activity of reading, to express mental ability[START_REF] Migge | Tracing the origin of modality in the creoles of Suriname[END_REF]

35).3 Unless otherwise indicated, the examples come from the authors' own field work.

In the Maroon creoles obligation is expressed by mu or musu (fu).

In the text we will continue to use sa to refer to these orthographic variants, but the textual examples will, of course, preserve the original orthographic variant.

In the former contexts sa appears to be optional (cf.Van den Berg 2007: 188-9).

Though we found no clear examples of the epistemic and deontic uses of sa in the early Sranan documents, this does not necessarily mean that they did not exist at the time.

Nov. 2003)

All of these similarities in the expression of possibility in Gbe and the Maroon creoles suggest that substrate influence played an important role in the emergence of this area of the creoles' grammar. The differences between Sranan and the Maroon creoles seem to be due to the fact that Dutch had stronger influence on the former, while Gbe had stronger influence on the latter.

Conclusion.

To sum up, the emergence and development of possibility in the Surinamese creoles followed quite different paths in Sranan as opposed to the Maroon creoles. A variety of factors contributed to these differences, the most significant of which appears to have been the degree to which Dutch as opposed to the Gbe languages influenced the respective creoles.

The strong influence of Dutch in Sranan, evident even in the early texts, led to a system in which kan expressed most types of root possibility, supplemented by and mag for permission. Later internal developments, possibly aided by Gbe influence, led to the emergence of man as a marker of physical ability. Gbe influence on the Maroon creoles, on the other hand, led to a system in which all types of possibility were subsumed


N Fressengeas

N Khelfaoui

C Dan

D Wolfersberger

G Montemezzani

H Leblond

M Chauvet

Roles of resonance and dark irradiance for infrared photorefractive self-focusing and solitons in bi-polar InP:Fe

Keywords: numbers: 42, 70, Nq, 42, 65, Tg

This paper shows experimental evidence of photorefractive steady state self-focusing in InP:Fe for a wide range of intensities, at both 1.06 and 1.55µm. To explain those results, it is shown that despite the bi-polar nature of InP:Fe where one photocarrier and one thermal carrier are to be considered, the long standing one photocarrier model for photorefractive solitons can be usefully applied. The relationship between the dark irradiance stemming out of this model and the known resonance intensity is then discussed.

I. INTRODUCTION

Photorefractive self-focusing and spatial solitons have been up to now the object of more than a decade of growing interest, both experimentally and theoretically [1,2,3,4]. During this decade, solitary waves induced by the photorefractive effect have been shown to interact with each other showing specific behaviors upon collision [5] like birth and fusion [6]. When a spatial solitary wave propagates in a photorefractive medium, it induces an optical waveguide that does not disappear instantly when the beam is turned off [7,8], leading to optically induced optical components such as junctions [9], couplers [10] and even optically induced soliton arrays and photonic lattices [11,12,13].

The attraction towards photorefractive solitons and the waveguides they induce owes a lot to their potentialities in optically inducing photonic circuitry in the bulk of a material, on the one hand, and to their reconfiguration properties, on the other hand. The former relies on the material property of retaining the waveguide once the beam is shut off, like, for instance, LiNbO 3 [14]. The latter, on the contrary, relies on the possibility to rapidly screen out an applied field with a laser beam, to give rise to a waveguide, and to erase this waveguide as rapidly. Up to now and to our knowledge, the characteristic build-up and erase times are on the order of the second in the widely used Sr x Ba 1-x Nb 2 O 6 [15] or sillenite Bi 12 TiO 20 [16], or up to the minute in LiNbO 3 [17], all of these experiments having been conducted at visible wavelengths to which most of the cited materials are sensitive. Shorter response times around the millisecond have been obtained in semiconductor CdZnTe [18] and around the nanosecond for visible ultra-high intensities in Bi 12 TiO 20 [19].

If one wants to target optical networking as a potential application of the dynamically reconfigurable photorefractive soliton technology, build-up and reconfiguration times below the millisecond is to be achieved ideally at infrared wavelengths compatible with optical fiber networks. Thanks to their high carrier mobility and despite their low electro-optic coefficient, photorefractive semiconductors such as InP:Fe might respond to this demand. Charge transport, in this material, has a bi-polar nature with photoexcited holes co-existing with thermally excited electrons. This yields a peculiar resonance of the photorefractive Two Wave Mixing (TWM) gain as a function of intensity [20,21,22,23].

Owing to its bipolar transport character, the physics behind spatial soliton propagation in InP:Fe is more complicated than in other materials. It has already been the subject of theoretical analysis [24]. In this work, we present a series of recent investigations on photorefractive self-focusing in InP:Fe at infrared wavelengths and we propose a model that takes into account the transport of both charge carriers. Unlike in previous work [24,25,26], the experimental results in our crystal samples and the corresponding theoretical prediction do not show any clear evidence of an influence of the intensity resonance of the bi-polar photorefractive effect, thus indicating that photorefractive self-focusing behaves to a large extent like in the case of the well-known singlecarrier model [27]. Hint is finally provided that the reason why qualitatively different behaviors are observed in different samples might be a varying oxydo-reduction state.

The bi-polar photorefractive resonance mechanism was first evidenced experimentally by Mainguet and Idrissi et. al. [20,21] and then defined by Picoli et. al. [22,23] as the mean intensity at which the Two Wave Mixing (TWM) gain is greatly enhanced. It corresponds to the intensity at which the intensity dependent hole generation rate is equal to the temperature dependent electron generation rate. The resonance intensity is thus expected to depend on the temperature and has been evidenced so. It has also been shown that this TWM enhancement corresponds to a π/2 phase shift between the index grating and the optical intensity grating, and to a corresponding enhancement of the index modulation amplitude.

As a preliminary characterization of our samples, we evidenced and measured the resonance intensity in the samples used in our experiments. The iron content of the various samples we tested varied from 8 × 10 16 cm -3 to 2 × 10 17 cm -3 , as measured by Secondary Ion Mass Spectroscopy. Similar behavior were observed in all samples. The results presented in this paper were obtained with a 2 × 10 17 cm -3 Fe doped sample.

The resonance intensity has been measured with a standard TWM setup [20] as described in [28], for various temperatures, and particularly at 20 • C, the temperature at which further self-focusing experimental results have been obtained. The measured resonance intensity at 20 • C is 300mW/cm 2 at a wavelength of 1.06µm and 3W/cm 2 at a wavelength of 1.56µm. This wavelength disparity is explained by the wavelength dependent photoexcitation cross sections.

The experiments were conducted independently at both 1.06µm and 1.56µm wavelengths, the latter because it corresponds to a telecom wavelength and the former for InP:Fe large sensitivity to it. The 1.06µm wavelength is delivered by a solid state diode pumped Nd:YAG laser whereas the 1.56µm beam is produced by a laser diode.

The beam is focused down to a 30µm waist onto the entrance face of a temperature regulated InP:Fe sample. The beam propagates along the < 110 > direction and is linearly polarised along < 110 > with an electric field of 10kV/cm applied along < 001 >. The crystal dimensions are 5 × 5 × 12mm 3 along the directions < 001 >< 110 > and < 110 >, respectively.

The steady state beam profile on the sample output face is observed through a microscope objective and a silicon CCD camera. The camera is sensitive enough for the 1.06µm wavelength but requires to be equipped with a luminescent phosphorus converter to achieve 1.56µm observations. The observed image is then gamma corrected to account for the conversion non-linearity.

The output beam profile is observed at steady state for various intensities, at a temperature of 20 • C. The best result giving the strongest focusing is shown in figure 1(b) for both 1.06µm and 1.56µm wavelengths. Varying the input intensity reveals that the steady-state self- focusing phenomenon vanishes at low and high intensities and reaches an optimum focusing for an intermediate intensity, when the output beam width reaches a minimum. This is quantified on figure 2 where the normalized horizontal output beam diameter is plotted as a function of the input beam peak intensity for an applied field of 10kV/cm. Linear diffraction is observed at steady state for small and large intensities. A minimum output beam diameter is reached for an intermediate intensity of approximately 210mW/cm 2 at 1.06µm and 7.5W/cm 2 at 1.56µm. A noticeable characteristic of both curves in figure 2 is that the upside down bell-shape is attracted by the dotted line when it approaches it, i.e. when the output beam diameter approaches the input one. This evidences a domain of stability of the beam width (from 1 to 4W/cm 2 at 1.06µm and from 5 to 50W/cm 2 at 1.56µm). The horizontal beam width associated to these domains roughly corresponds to the input beam width. This may suggest that the beam evolution is attracted towards a solitary wave solution

We note that the width stability intensity range is much wider at 1.56µm than at 1.06µm: this is due to the fact that in the former case, the intensity that fullfills soliton propagation conditions is much nearer to the minimum of the reverse bell-shaped curve. Furthermore, one has to keep in mind that the beam intensity decreases with propagation, owing to absorption, -2cm -1 at 1.06µm and 0.5cm -1 at 1.56µm-so that the soliton propagation conditions cannot be fullfilled precisely all the way through the crystal, thus strengthening the need for stability if a spatial soliton is to propagate.

As in many other materials, self-focusing in InP:Fe comes together with beam bending, as is evidenced by figure 1 and particularly 1(c). The steady-state beam shift at the output face with respect to the propagation axis, along the same horizontal direction as the applied field, is reported on figure 3. We observe that beam bending is negligible for sufficiently low and high intensities, that is, when self-focusing is absent. Furthermore, the maximum beam shift is obtained precisely at the intensity at which self-focusing is most efficient.

The curves reported on figure 2 resemble the photorefractive soliton existence curve -giving the soliton width as a function of the beam peak intensity to dark irradiance ratio-as predicted earlier [27,29] in the framework of a single-photo-carrier model. The dark irradiance plays a central role in this theory since it is the ratio of the beam peak intensity to the dark irradiance which determines the steady state soliton width. The dark irradiance is the equivalent uniform optical irradiance which should be shone on the crystal to generate as much photocarriers as thermal excitation does. In other words, shining the dark irradiance on a given sample doubles its conductivity.

If we assume that this one-photo-carrier modelwhich will be named model I in the following-describes our experiments in InP:Fe, we can use its predictions to deduce the dark irradiance in InP:Fe at both the wave- lengths we have used. According to it, the minimum width soliton which corresponds to the maximum selffocusing effect is obtained for a beam peak intensity equal to 3 times the dark irradiance, as confirmed recently [30] in a (1+1)D model. Using this hypothesis, the dark irradiance in our sample can then be evaluated, from figure 2, to approximately 70mW/cm 2 for the 1.06µm wavelength and 2.5W/cm 2 for 1.56µm, at a temperature of 20 • C; the accuracy of the value measured at 1.06µm is low because of InP:Fe strong absorption at this wavelength. Let us note here that, in both cases, the estimated dark irradiance and the resonance intensity are on the same order of magnitude.

At this stage, one may wonder why model I should describe photorefractivity in InP:Fe since both holes and electrons are known to play a significant role [22,23] in this semiconductor -holes are photo-excited and electrons are thermally generated-whereas model I assumes only one carrier, both photo and thermally excited. One may hastily think that electrons can simply be neglected in our experiments. This is however not the case since both the dark irradiance and the resonance intensity are included in our measurement range: they respectively correspond to the equality of the concentrations on the one hand and of the generation rates on the the other hand, of holes and electrons.

Therefore, since model I assumptions do not match with our experiments, we now need to ascertain by an independent measurement that the values of the dark irradiance we have deduced from figures 2 and 3 are in-deed correct. We have thus shone a uniform intensity on our InP:Fe sample and measured its photo-conductivity. This allowed us to deduce a 50mW/cm 2 dark irradiance at 1.06µm and a 2W/cm 2 one at 1.56µm, confirming the previous evaluation at this wavelength. The slight discrepancy is accounted for by the relative precision with which figure 2 allows the determination of the dark irradiance on the one hand and on the other hand by the difficulty to shine a strong and uniform light on the whole sample for the independent measurement. These recent experimental results evidence the fact that the dark irradiance plays a crucial role in photorefractive self-focusing in InP:Fe, as it does in previously developed models, despite the greater complexity of the photorefractive mechanisms in InP:Fe. However, as previously explained, model I does not allow to explain our observations, nor do previously published ones [24,25,26], as will be detailed later on. Therefore, we will now tackle the theoretical problem of showing how and why the dark irradiance plays a significant role in InP:Fe.

III. SPACE CHARGE FIELD AND DARK IRRADIANCE

Let us start with the general two-carrier band transport model [20,24] reduced to one dimension x, which we will consider at steady state in the following. It is described by the set of equations below:

∂E ∂x = e ǫ (N D -N A + p -n -n T ), (1a)
j n = eµ n nE + µ n k B T ∂n ∂x , (1b)
j p = eµ p pE -µ p k B T ∂p ∂x , ( 1c
)
∂n ∂t = e n n T -c n np T + 1 e ∂j n ∂x , (1d)
∂p ∂t = e p p T -c p pn T - 1 e ∂j p ∂x , ( 1e
)
∂n T ∂t = e p p T -e n n T -c p pn T + c n np T , (1f)
N T = n T + p T , ( 1g
)
where E is the electric field, n and p are the electron and hole densities in the respective conduction and valence bands, n T = Fe 2+ is the density of ionized occupied traps, p T = Fe 3+ is the density of neutral unoccupied traps, j n and j p are the electron and hole current densities, respectively. N T , N D and N A are the densities of iron atoms, the shallow donors and the shallow acceptors, respectively. The charge mobilities are given by µ n for electrons and µ p for holes, the electron and hole recombination rate are respectively c n and c p , T is the temperature and k b is the Boltzmann constant. The dielectric permittivity is given by ǫ while e is the elementary charge.The electron and hole generation rates e n and e p depend on both thermal and optical emission as described by

e n = e th n + σ n I ≈ e th n , (2a)
e p = e th p + σ p I ≈ σ p I (2b)

where the thermal contribution to the emission rate coefficient is e th and the optical cross section of the carriers is given by σ, I is the spatially dependent intensity of light.

The approximations in both above equations are valid in the case of InP:Fe as stated above, with thermally excited electrons and photo-excited holes. The need for a large applied electric field leads us to neglect carrier diffusion when compared to drift, diffusion which is accounted for by all the terms which contain k B T . Solving the differential system (1) requires defining the boundary conditions. To that aim, we will consider that the input beam is small with respect to the sample dimensions and thus that its influence is limited to its surroundings. Therefore, the boundary values for E, n, p, n T , p T , j n and j p are the values reached at steady state in the dark. As was previously done [20,22,23,28], they can be deduced from system (1), setting to zero all derivatives and optical intensity. Following, if the 0 subscript identifies the boundary condition: E 0 = V /d where V is the applied voltage and d the crystal width; n 0 = e th n n T0 /c n p T0 and p 0 = e th p p T0 /c p n T0 where n T 0 = N D -N A and p T 0 = N T -n T 0 are the ionised and nonionised iron densities in the dark [22,23]; j n = eµ n n 0 E 0 and j p = eµ p pE 0 .

The time evolution of the electric field E is determined as follows. The time derivative of Eq. (1a) is simplified using Eqs. (1d,1e,1f) to yield

∂ 2 E ∂t∂x = -1 ε ∂ (j n + j p ) ∂x . (3)
Making use of Eqs. (1b,1c) and integrating with respect to the space variable x, it becomes

∂E ∂t = -e ε [(µ n n + µ p p) E. -(µ n n 0 + µ p p 0 ) E 0 ] (4)
At steady state, the evolution equations (1d), (1e), (1f) and (4) reduce to

(µ n n + µ p p) E -(µ n n 0 + µ p p 0 ) E 0 = 0, (5a) e n n T -c n n (N T -n T ) + µ n n ∂E ∂x + µ n E ∂n ∂x = 0, (5b) e p (N T -n T ) -c p pn T -µ p p ∂E ∂x -µ p E ∂p ∂x = 0. (5c)
To further simplify this differential system, the light intensity spatial variations will be assumed small enough so that the resulting free carriers spatial variations remain small, and the derivatives ∂n/∂x and ∂p/∂x can be neglected in equations (5b) and (5c). In the same spirit, we also assume that the field derivative ∂E/∂x in the same equations can be neglected, precisely ∂E/∂x ≪ e th n N T /µ n n 0 . The left-hand-side of Eq. (1a) becomes zero within this approximation.

Light intensity and temperature will again be considered not so large so that free carriers density remain small with respect to ionised iron density: n ≪ n T and p ≪ n T , so that equation (1a) reduces to n T = n T 0 , and consequently from Eqs. (5b) and (5c) we get n = n 0 and p = (σ p Ip T 0 ) / (c p n T 0 ). The assumptions made are valid for intensities small against c p n 2 T 0 /σ p p T 0 , itself greater than the TW/cm 2 , and at temperatures orders of magnitude above room temperature. Within these approximations, we obtain the electric field E dependency upon the local intensity I:

E = I d I d + I E 0 . ( 6
)
This expression is strictly the same as the one found before in the framework of model I [27,29] when diffusion is neglected; only the expression of the dark irradiance I d differs (the approximation below being valid in the case of InP:Fe):

I d = e th p p 2 T 0 c n µ p + e th n n 2 T 0 c p µ n σ 0 p p 2 T 0 c n µ p + σ 0 n n 2 T 0 c p µ n ≈ e th n n 2 T 0 c p µ n σ 0 p p 2 T 0 c n µ p . (7)
Let us point out here that this has implications beyond the above equation: all the conclusions that have been driven, in model I, from an equation identical to (6), still hold, if the above value of I d is accounted for. There is for instance no need to derive again the soliton existence curve since the involved equations are the same. Therefore and using the predictions model I, our observations of self-focusing in InP:Fe are partially explained. Indeed the presence of self-focusing over a broad range of intensity is predicted and no electron-hole resonance effect is observed. The beam bending is however not yet accounted for, which is one reason to further extend and refine our model.

IV. RESONANCE INTENSITY

As was said before, we felt the need to establish the theoretical analysis presented in the previous sectionwhich we will now call model II-because we could not satisfactorily explain our observations using the theory available in the literature [24,26] for one particular reason: this theory -which we will call model III-predicts the existence of a resonance intensity around which the self-focusing phenomenon reverses itself; while we have successfully measured the resonance intensity in our samples, we were never able to measure the predicted reversal from focusing to defocusing.

The resonance intensity around which a self-focusing behavior reversal is predicted by model III is precisely the same resonance intensity at which TWM gain is maximum. As was said before, the resonance intensity is also the intensity at which the generation rates of holes and electrons are equal:

I r = e th p p T0 + e th n n T0 σ 0 p p T0 + σ 0 n n T0 ≈ e th n n T0 σ 0 p p T0 . (8)
At resonance, the imaginary part of the space charge field reaches a maximum while the real part -the in phase component-reaches zero. Following, the real part changes sign around I r [22,26]. The link with selffocusing can then be thought of as follows: the in phase component of the space-charge field -the real partis a local photorefractive response, responsible for selffocusing or defocusing, whereas the out-of phase one -the imaginary part-is responsible for beam bending. This phenomenological interpretation has been confirmed by an experiment [26]. Later on, analytical derivations and numerical simulations [24] came to the same predictions. As stated before, however, we were never able to reproduce the experiment, for a reason we will now try to give hints of.

First of all, in the case of InP:Fe, there is a close and fairly simple relationship between I d and I r :

I d I r = µ n c p µ p c n × n T 0 p T 0 . (9)
We have evaluated the above ratio to 0.65 with values issued from the literature [31,32,33], a numerical value which is fully compatible with those we have measured at 1.56µm, the discrepancy for the measures at 1.06µm being accounted for by the strong absorption impairing the I d estimation from figures 2 and 3.

The above ratio (9) evidences the ratio Fe 2+ / Fe 3+ in the dark (n T 0 /p T 0 ). Therefore, the oxydo-reduction state plays a significant role, as well as the doping of the samples, in the ratio of the two characteristic intensities. Therefore, a different doping but also a different Fe 2+ / Fe 3+ ratio might be the cause of the discrepancy between our experimental results, which do not exhibit a crucial role for the resonance intensity, and those previously reported.

However, our theoretical analysis, model II, does not either bring the resonance intensity forth. A more universal model is in development in order to predict a doping dependent behavior and thus unify model II and model III.

V. CONCLUSION

In the light of recent experimental results, we have shown that despite more complex photorefractive charge transport mechanisms, steady-state photorefractive selffocusing and solitons in InP:Fe can be described by the same model as was done for the simpler one-free-carrier model, provided diffusion is neglected. We have evidenced in the process the role of the dark irradiance. Our model does not, however, evidence any crucial role for the resonance intensity, a role which is to be the subject of future studies.

FIG. 1 :FIG. 2 :

 12 FIG.1:(Color online) Steady state output beam profile for a peak intensity of 190mW/cm 2 at 1.06µm (top) and a peak intensity of 8.7W/cm 2 at 1.56µm (bottom), for no applied field (a) and a 10kV/cm horizontally applied field (b). The images total width is 200µm. The non Gaussian shape of the 1.56µm beam in absence of the applied field is due to the multimode character of the laser itself. The curves (c) are the horizontal beam profiles (a.u.) at beam maximum.

FIG. 3 :

 3 FIG.3: Output beam horizontal shift as a function of input beam peak intensity at 1.06µm (top) and 1.56µm (bottom) wavelengths.

VI. ACKNOWLEDGEMENTS

The authors wish to thank the Lorraine Région for their financial support.