Thomas Pellard

Ryukyuan perspectives on the Proto-Japonic vowel system

 gives the location of the different dialects mentioned in this study. Varieties for which I have first-hand data coming from my own fieldwork (Sakamine, Yuwan, Koniya, Ōgami, Yonaguni) are indicated in italics. In addition, I have used

Introduction

The Ryukyuan languages are a family of (at least) five endangered languages spoken in the Ryūkyū Islands, an archipelago stretched between Kyūshū and Taiwan and naturally delimited by the Kuroshio current (Figure 1). They form a sister branch to Japanese, and both derive from a common ancestor, Proto-Japonic (PJ). Ryukyuan can be divided into a Northern branch that includes Amami and Okinawan, and a Southern branch comprising Miyako, Yaeyama and Yonaguni [START_REF] Pellard | Ōgami: Éléments de description d'un parler du Sud des Ryūkyū[END_REF].

Traditionally, Japanese historical linguistics has been virtually synonymous with philology, and the reconstruction of PJ has thus exclusively relied on the evidence from the Old Japanese (OJ) texts of the 8 th c. ce. Comparative data from the Ryukyuan languages is nevertheless of great importance for this topic, though this importance is still too often underestimated.

In particular, it is known since [START_REF] Hattori | Ryūkyū to kokugo no on'in hōsoku[END_REF]Hattori ( , 1978Hattori ( -1979) that the Ryukyuan vowels do not straightforwardly correspond to the Old Japanese ones, and a careful comparison of Ryukyuan and OJ requires reconstructing more vowels in Proto-Japonic than usually posited. Based on data I collected in the field and from existing sources, 1 I adduce new comparative evidence from several Ryukyuan languages that add further support to the reconstruction of six vowels in Proto-Japonic, and I propose to reconstruct a new diphthong, namely *oi.

The reconstruction of the PJ vowel system 2.1 The four-vowel hypothesis (4VH)

The application of the method of internal reconstruction to the vowel alternations and vowel crasis phenomena of OJ has led many scholars to reconstruct, more or less independently, only four vowels in PJ [START_REF] Miller | The Japanese language[END_REF][START_REF] Matsumoto | Kodai nihongo boin soshiki kō: Naiteki saiken no kokoromi[END_REF][START_REF] Ōno | On'in no hensen[END_REF][START_REF] Whitman | The phonological basis for the comparison of Japanese and Korean[END_REF][START_REF] Martin | The Japanese language through time[END_REF], represented in Figure 2. The other vowels of OJ and later Japanese are said to be later developments of diphthongs constituted of the above four primary vowels: i 2 < PJ *ui, *əi; e 1 < *ia, *iə; e 2 < *ai, *əi; o 1 < *ua, *au, *uə. This reconstruction is however problematic since it does not take into consideration any comparative data from the Ryukyuan languages, and in fact is not able to account for it, as will be shown below.

The PJ six-vowel hypothesis (6VH)

Previous works on the comparative reconstruction of Ryukyuan and Japanese (Hattori 1978-1979[START_REF] Thorpe | Ryūkyūan language history[END_REF][START_REF] Serafim | Reflexes of proto-Korea-Japonic mid vowels in Japonic and Korean[END_REF], 2008[START_REF] Pellard | Proto-Japonic *e and *o in Eastern Old Japanese[END_REF][START_REF] Pellard | Ōgami: Éléments de description d'un parler du Sud des Ryūkyū[END_REF] have led to the the addition of two mid vowels, *e and *o, to the 4VH (Figure 3). For several decades, this hypothesis has been mostly rejected, and the 4VH has remained, and still does for some scholars, the common doxa in Japanese historical linguistics. It is only recently, more than thirty years after Hattori's (1978Hattori's ( -1979) ) commanding comparative study of Ryukyuan and Japanese, that it has begun to gain general acceptance. 2 The fact few examples were given in support of this hypothesis, with Proto-Ryukyuan (PR) and PJ reconstructions and correspondences sometimes given without clear explanation, probably explains why the 6VH has been rejected so far. The following sections will propose new and straightforward Ryukyuan evidence 3 for the 6VH and counter-evidence to [START_REF] Whitman | The phonological basis for the comparison of Japanese and Korean[END_REF]'s criticisms. 4 2 See Frellesvig and Whitman (2008a), [START_REF] Vovin | Koreo-Japonica: A critical study in the proposed language relationship[END_REF].

3 This reconstruction is also partly supported by data from Eastern Old Japanese (EOJ, Azuma uta, Sakimori uta) and some archaic dialects like Hachijō [START_REF] Hino | Nihon sogo no boin taikei: Jōdai Tōgoku hōgen shiryō ni yoru saikō. Nihongo keitōron no genzai/Perspectives on the origins of the Japanese language[END_REF][START_REF] Pellard | Proto-Japonic *e and *o in Eastern Old Japanese[END_REF]. See also [START_REF] Miyake | Philological evidence for *e and *o in pre-old Japanese[END_REF] for a summary of the subject. 4 Whitman has since accepted the reconstruction of *e and *o (Frellesvig and Whitman 2008a i 1 ⟨ *i > *i > ʔ i, N :: ʔ i, ʲi, N :: ɿ, ɯ, s, N, ∅ :: ɿ, N, ∅ :: i, N, ∅ *e > *e > ʰɨ, i :: ʰi, i :: i :: i :: i [START_REF] Whitman | The phonological basis for the comparison of Japanese and Korean[END_REF] made the important remark that in all the examples proposed by Hattori (1978Hattori ( -1979) ) and [START_REF] Thorpe | Ryūkyūan language history[END_REF],5 *e always appears before a sonorant or a voiced/prenasalized obstruent followed by a high vowel, while *ə never does. He thus proposed to account for *e by the following fronting rule:

(1) PJ *ə > PR *e, OJ i 1 / C [+voiced] V [+high] There are nevertheless counter-examples to [START_REF] Whitman | The phonological basis for the comparison of Japanese and Korean[END_REF] fronting hypothesis, such as to 2 ri 'bird' or no 2 ri 'seaweed', that do not have alternating shapes and thus should simply be reconstructed as *təri and *nəri (Miyake 2003:94). Such words cannot be reconstructed with a final *əi, as shown by their Ryukyuan reflexes (ex: Miyako-Ōgami tuɯ, su-nuɯ, see section 4). Moreover, Arisaka's (1934b) second law states *ə and *u do not coexist in a disyllabic root, so there is little chance these words come from *tərui and *nərui. Anyway, *tərui and *nərui would have shifted to tiri and niri in OJ according to Whitman's fronting rule. Hattori's (1978Hattori's ( -1979) and [START_REF] Thorpe | Ryūkyūan language history[END_REF] examples thus remain valid.

Table 2 OJ < PJ > PR > Amami :: Okinawan :: Miyako :: Yaeyama :: Yonaguni u ⟨ *u > *u > ʔ u, N :: u, N :: u, N, ∅ :: u, N, ∅ :: u, N, ∅ *o > *o > ʰu :: u :: u :: u :: u PJ *u also has a special fronted reflex after coronals in most Ryukyuan varieties, which can be seen in the reflexes of PJ *usu 'mortar', while *o is always u, as in *kusori 'medicine' (Table 7).

Another example is PJ *ori 'melon', which can be reconstructed from the comparison 'melon' OJ uri :: Ōgami uɯ (cf. 'sell' OJ ur-i :: Ōgami ʋʋ).

PJ diphthongs: Where have all the *oi gone?

From morphophonemic alternations occuring in OJ, it is possible to reconstruct two different PJ sources of OJ i 2 :

. *ui, for OJ cases where OJ i 2 alternates with u, e.g. tuku-yo 1 'moon(light)' ∼ tuki 2 'moon' or sugus-'pass (transitive)' ∼ sugi 2 'pass (intransitive)';

. *əi, where i 2 alternates with o 2 , like in ko 2 dati 'stand of trees' ∼ ki 2 'tree' or oko 2 s-'raise' ∼ oki 2 'rise'. As first noticed by Arisaka (1934a), and then fully demonstrated by Hattori (1932),7 the Shuri dialect preserves this distinction, but this is also the case with many other Ryukyuan varieties8 (Table 8). The fact that Ryukyuan faithfully preserves a distinction already lost in Japanese in the 8 th c. and only recoverable through internal reconstruction suffices to establish beyond reasonable doubt the two have split at least before the beginnings of Japanese history. In Ryukyuan, PJ *ui and *i merge as PR *i but stay distinct from the merged reflexes of PJ *əi and *e (as well as *ai and *ia, PR *e). On the other hand, in OJ, the primary vowels *i and *e generally merge as i 1 (see however subsection 5.2) and stay distinct from i 2 , the merged reflex of the diphthongs *ui and *əi. These evolutions are summarized in Figure 4.

The PJ system however surprisingly lacks a diphthong *oi. I propose that in fact some of the *ui and *əi usually posited should rather be reconstructed as *oi, and that *oi merges with *ui and not *əi in Ryukyuan to give PR *i.

From *ui to *oi

OJ u ∼ i 2 apophonic stems usually have Ryukyuan cognates that exhibit a similar *u ∼ *i alternation, and are thus reconstructed as *ui in PJ. See for instance *padui 'shame' (Table 9) or *mui 'body, flesh' (Table 10).

However, some u ∼ i 2 apophonic nouns, traditionally reconstructed as *ui, have an alternating shape (hifukukei) that should be reconstructed with a stem-final *o rather than *u, as is clear from their Ryukyuan reflexes. They should thus be reconstructed with a diphthong *oi in PJ. This is the case of 'yellow' (Table 11), which appears in OJ as ku-∼ ki 2 but has a reflex *ko-in PR. This *ko is reconstructed on the basis of the aspiration or lenition of the velar in Northern Ryukyuan, and the lack of fricativization in Southern Ryukyuan, to be compared with the reflexes of *ku exemplified by PJ *kumo 'cloud'. This reconstruction is further confirmed by the existence of the later Japanese form kogane, the standard form since the Middle Japanese period.

The verb 'exhaust' (OJ tukus-∼ tuki 2 ) should be reconstructed as *tukosfor the same reasons as 'yellow'. The cognate of OJ tukus-in Miyako-Hirara is tsɿkus-, which implies *tukos-since *ku > f(u) (cf. 'make' tsɿfuɿ :: OJ tukur-). Moreover, the Amami-Boma (Tokunoshima) form tsɨkijuŋ confirms the PR reconstruction *tuki for the transitive form, which ultimately derives from PJ *tukoi (cf. Boma uɨjuŋ 'rise' < PR *oke < PJ *əkəi).

The reconstruction of 'moon' (OJ tuku-∼ tuki 2 ) as PJ *tukoi proposed by Thorpe (1983:355) is less secure. Though Thorpe is very confident9 about his proposal, he doesn't clearly explain his reasoning, and the Yaeyama-Hateruma form sɿkeŋ is the only evidence he adduces. This is perhaps based on the fact that Hateruma e comes from *oe ('burn' PJ *mojai > PR *moe > meː) or *ae ('front' PJ *mape > PR *mae > meː), but never *ue. The evolution of the Hateruma form might then be reconstructed as follows:

(2) PJ *tuko-jo > *tuko-je > *tsɿkoe > sɿkeŋ (with prothetic -ŋ) (cf. Ishigaki *tuko-jo > *tuko-je > tsɿkui)

From *əi to *oi and *ə to *o

Traditionally, all instances of OJ i 2 ∼ o alternations have been interpreted as involving an o 2 < *ə, and thus automatically reconstructed as *əi. However, several problems arise from this mechanistic reconstruction. First, cases where no o 1/2 distinction is attested could theoretically go back to an earlier *o rather than *ə. Moreover, in some cases this o coexists with a in the same root, which according to Arisaka's (1934b) I must admit that all the examples where I propose to correct the reconstruction from *əi to *oi contain a labial consonant before the vowel. It could thus be the case that PJ *əi merɡes with *ui after a labial to give PR *i rather than the usual PR *e:

(3) PJ *əi > PR *i / C [labial] There is however at least one counter-example to this generalization: the PR *o(p)pe 'big' form (Yuwan xɨː :: Koniya ɸɨː :: Nakijin ɸupi :: Shuri ʔuɸi) is undoubtedly related to OJ opo 'big' ∼ opi 2 'grow'. Since this form has a reflex with *e in PR and that nearly every (C)oCo form in OJ have an o 2 in both syllables, it can be safely reconstructed as PJ *əpəi. This form thus exemplifies a case where PJ *əi corresponds to PR *e and not *i after a labial.

The distributional constraint observed might also be just a coincidence: *o usually gets raised to u in OJ, which leaves few possibilities for i 2 ∼ o 1 alternations. There are not so many cases of i 2 ∼ o alternations anyway, and only a handful of examples remain after all cases clearly involving an o 2 are eliminated. Not surprisingly, most remaining examples involve syllables where OJ has no o 1/2 distinction: po, bo, mo, wo, which all have a labial initial. Some examples with unclear o 1/2 attestation after a non-labial consonant are found, but they seem to have no cognates in Ryukyuan.

Anyway, 'blue' should be reconstructed as *awo in order to comply with Arisaka's (1934b) Hattori (1978Hattori ( -1979) ) reconstructs *ü in a handful of words only, most of which are problematic. First, the reconstruction of 'all' as *müina ∼ *mürə is only valid if the two OJ words mi 2 na and mo 2 ro 2 are indeed etymologically related, which remains unsure. Then, 'finger' (Hattori's *əjübəi) is a word with a complex history, and Hattori didn't clearly explain his motivations for this reconstruction (Hattori 1978(Hattori -1979:(9)116):(9)116). Finally, Hattori's *püi 'fire' and *pü-dai/*pü-dee 'lightning' are reconstructed with *ü since they have a reflex *i in PR instead of the expected *e, but this can be explained by my own hypothesis as cases of *oi.

On the other hand, Frellesvig and Whitman's (2008a) *ɨ is reconstructed on the main basis of OJ-internal evidence, and there seems to be no supporting comparative evidence from Ryukyuan, EOJ or Hachijō (Table 16). While this hypothesis is interesting, well-founded, and aims at explaining a few otherwise irregular vowel alternations, I feel there are still too few good examples. There is little merit in reconstructing an extra vowel in PJ over recognizing a small number of irregular forms.

Vowel raising in OJ

There are several cases of e (1) and o 1 in OJ that cannot be explained away as originating in earlier diphthongs, and these thus constitute exceptions to the vowel raising process that affected OJ. Examples of unraised e (1) in OJ include for example pe 1 ra 'moldboard, spatula', ke 1 pu 'today', pe 1 ta 'near the shore', sake 1 b-'shout', kape 1 r/s-'return', uke 1 ra 'Atractylodes japonica', ter-'shine', etc. Instances of unraised o 1 can be seen in ko 1 pi 2 'love', ko 1 ga-'burn', to 1 ma 'woven rain-cover', so 1 ra 'sky', to 1 ra 'tiger', yo 1 wa-'weak', mo 1 zu 'shrike', ko 1 mo 'eelgrass', etc.

Several explanations have been proposed to account for such cases of unraised *e and *o in OJ. One of these states a phonotactic constraint stipulating final position blocks raising (Frellesvig and Whitman 2008a). This hypothesis does not however explain the cases where an unraised vowel is found in a non-final syllable, such as those quoted above. Hattori (1978Hattori ( -1979) ) proposed on the other hand it is vowel length that blocks raising, and that unraised *e and *o should be reconstructed as long *ee and *oo in PJ. This is however at odds with the general trend of vowel changes, since long vowels generally tend to rise (Labov 1994:116). If vowel length is to be reconstructed, Hattori's hypothesis should be reversed and raised vowels reconstructed as long (this might account for their relative paucity). Other solutions could include reconstructing those unraised vowels as diphthongs (*ia and *ua or *uə), but the lack of alternating shapes for these deprives us of a solid basis to do so. Reconstructing still extra vowels (*ɛ, *ɔ?) would be quite unparsimonious. The explanation could lie into a combination of two or more of the above hypotheses.

Conclusions

A detailed examination of the correspondences between Old Japanese and the different Ryukyuan languages leads to a revision of earlier hypotheses about the Proto-Japonic vowel system. The most important improvements are the reconstruction of the two mid vowels *e and *i and of the diphthong *oi, as well as the reinterpretation of several cases of *ə as *o. The Ryukyuan vowel systems are in a way more archaic than the OJ system, since they preserve a distinction between the vowels *i/*e, *u/*o, *ui/*əi as well as *oi/*əi, which had already merged by the time of the first OJ texts. This is not to say we can dispense with the OJ data, which reveals several distinctions absent from Ryukyuan (*i vs. *ui/*oi, *e/*ia/*iə vs. *ai/*əi).

Though the Ryukyuan data supports the reconstruction of a six-vowel system for PJ, it offers no solid evidence for a seventh vowel, which is anyway supported by too few examples. The reconstruction of the PJ vocalism however still suffers from several problems to be solved by future research. The most important question to be answered concerns the details of the evolution of the PJ vowels in OJ, namely the exact conditions of mid vowel raising. Further work is also needed in order to find more examples of PJ *e, which are still not very numerous, and more evidence for a distinction between *oi and *əi after labials in Ryukyuan. Many other details of the PJ vocalism are still obscure, like the distributional constraints described by Arisaka's laws.

A large-scale comparison of Ryukyuan and Japanese, followed by the application of the methods of internal reconstruction, is without doubt the key to a better understanding of the PJ vowel system.

  FIGURE 1 The Ryukyuan languages

  FIGURE 2 The four-vowel hypothesis

  FIGURE 3 The 6-vowel hypothesis

). 3 The PJ mid vowels *e and *o 3.1 PJ *e

  The reconstruction of PJ *e has been the most controversial one. There are indeed few examples, and further examples of *e can only be discovered by a minutious examination of the Ryukyuan data. The supporting correspondences are summarized in Table 1.

	TABLE 1 Reflexes of PJ *i and *e
	OJ < PJ > PR > Am. :: Ok. ::	Mi.	:: Ya. :: Yo.

  illustrates a minimal pair PJ *piru 'daytime' vs. *peru 'garlic', which are homophoneous in OJ but are distinguished in most varieties of Ryukyuan. Other examples where I propose to reconstruct *e include PJ *erə 'color', which can be reconstructed from the correspondence OJ iro 2 :: Ōgami iɾu (cf. 'put in' OJ ire :: Ōgami ɯɾi < PJ *irV), and PJ *neNkə 'get muddy', from the correspondence OJ nigo 2 r-:: Yuwan nɨɡuɾɨɾ-(cf. 'grasp' OJ nigi 1 :: Yuwan niɡij-< PJ *niNkir-).

TABLE 2

 2 PJ *i vs. *e: PJ *peru 'garlic' The

			'daytime' 'garlic'
	OJ Amami	Koniya	pi 1 ru çiɾ	pi 1 ru ɸɨɾ
	Amami Okinawa Nakijin p ʔ iɾuː Yoron pju:	piɾu pʰiɾuː
	Miyako	Ōgami	psː-ma	piɯ
	Yaeyama Ishigaki pɿːɾɿ Yonaguni ts ʔ uː	piŋ çiɾu
	PR		*piru	*peru
	3.2 PJ *o

reconstruction of *o is less problematic and is supported by many examples of a correspondence OJ u :: PR *o. As listed in Table 3, PJ *u tends to drop or assimilate to a contiguous nasal, contrary to *o, whose reflex is regularly u. These two different reflexes are exemplified by PJ *mukap-'to face' vs. *moNki 'wheat' (Table 4), *mukaNtV 'centipede' vs. *moko 'bridegroom' (Table 5), *uma 'horse' vs. *omi 'sea' (Table 6). 6 TABLE 3 Reflexes of PJ *u and *o

TABLE 4

 4

		PJ *u vs. *o (1): PJ *moNki 'wheat'
			'to face' 'wheat'
	OJ		mukap-mugi 1
	Amami	Sakamine mutɕui	muɲi
	Okinawa Shuri	ŋkeː	mudʑi
	Miyako	Ōgami	ŋkai	mukɯ
	Yaeyama	Ishigaki	ŋkai	muŋ
	Yonaguni		ŋkai	muŋ
	PR		*mukae *mogi

TABLE 5

 5

		PJ *u vs. *o (2): PJ *moko 'bridegroom'
				'centipede'	'bridegroom'
	OJ			mukade (MJ) mo 1 ko 1 > muko
	Amami	Sakamine mukadʑi	muku, mukka
	Okinawa Shuri	ŋkadʑi	muːku
	Miyako	Ōgami	ŋkati	muku
	Yaeyama	Ishigaki	ŋkadza	muku
	Yonaguni			ŋkadi	muɡu
	PR			*mukade	*moko
	TABLE 6 PJ *u vs. *o (3): PJ *omi 'sea'
				'horse'	'sea'
	OJ			(m)uma umi 1
	Amami		Sakamine ma	ʔumi
	Amami		Yuwan	m ʔ a	ʔuŋ
	Amami		Yoron	uma	uŋ
	Okinawa Nakijin	m ʔ aː	ʔumi
	Yaeyama	Ishigaki	mma	iŋ ∼ umɿ
	Yonaguni		mma	iŋ ∼ unnaɡa
	PR			*uma	*omi
	TABLE 7 PJ *u vs. *o (4): PJ *kusori 'medicine'
				'mortar' 'medicine'
	OJ			usu	kusuri
	Amami		Yuwan	ʔusɨ	k ʔ usui
	Amami		Koniya	ʔusɨ	kusuɾ
	Amami		Sakamine usu	sui
	Okinawa Nakijin	ʔuɕi	kʰusui
	Miyako		Ōgami	us	ffuɯ
	Yaeyama	Ishigaki	usɿ	ɸuɕiɾɿ
	Yonaguni		utɕi	ts ʔ uɾi
	PR			*Usu	*kusori

TABLE 8

 8

		PJ *ui and *əi in Ryukyuan
		*ui		*əi
		'moon'	'pass'	'tree'	'rise'
	OJ Am. Yuwan	tuki 2 (∼ tuku-) (∼ sugus-) sugi 2 tsɨki sɨɡi-	ki 2 (∼ ko 2 ) (∼ oko 2 s-) oki 2 kʰɨː xɨːɾ-
	Ok. Nakijin ɕitɕiː	ɕidʑiːɾuŋ	kiː	ɸukiɾuŋ
	Ok. Shuri	tsitɕi	sidʑijuŋ	kiː	ʔukijuŋ
	Mi. Ōgami	ksks	sɯki	kiː	uki
	Ya. Ishigaki tsɿkɿ	sɿɡiɾuŋ	kiː	ukiɾuŋ
	PR	*tuki	*sugi	*ke	*oke

TABLE 9

 9

					PJ *hadui 'shame'
				'shameful'	'shame'	'corner'	'arm'
	OJ			hadukasi	hadi (MJ) kado (MJ) ude
	Amami	Yuwan	hatsɨkaɕa	hadʑi	kadu	ʔudɨ
	Amami	Yoron	pantɕikaɕaŋ padʑi	hadu	udi
	Okinawa Nakijin	padʑikaɕeŋ	padʑiː	haduː	ʔudiː
	Miyako	Ōgami	pakskaskam		katu	uti
	Yaeyama Ishigaki	patsɿka-	padzɿ	kadu	udi
	Yaeyama Hateruma pasɿkahaŋ		kadu	udʑi
	PR			*pazu	*pazi	*kado	*Ude
			TABLE 10 PJ *mui 'body, flesh'
					'chest'	'flesh' 'thing' 'eye'
	OJ Amami	mune Sakamine munɪ	mi 2	mono 2 me 2 muŋ miː
	Amami	Yoron		niː	miː	munu	miː
	Okinawa Nakijin		niː	miː	munuː miː
	Miyako	Ōgami		mmi-fkɯ miɯ	munu	miː
	Yaeyama Ishigaki	nni	mɿː	munu	miː
	Yonaguni			nni	miː	munu	miː
	PR				*mune	*mi	*mono *me
			TABLE 11 PJ *koi 'yellow'
				'yellow'	'tree'	'gold'	'cloud'
	OJ Amami	Koniya	ki 2 (∼ ku) ki 2 (∼ ko 2 ) kugane k ʔ iː kʰɨː kʰuɡani k ʔ umu kumo 1
	Amami Okinawa Nakijin Yoron	kinsaŋ tɕ ʔ iːɾuː	ɕiː ∼ çiː kiː	huɡani ɸuɡaːni k ʔ umuː kumu
	Miyako	Ōgami	ksː-		kiː	kukani	fumu
	Yaeyama Ishigaki	kɿː		kiː	kuɡani	ɸumu
	Yaeyama Hateruma kɿŋkɿː	kiː	kuɡani	ɸumoŋ
	PR			*ki		*ke	*kogane *kumo

  third law is almost never possible for an o 2 . But the most conclusive piece of evidence comes from Ryukyuan: some examples have reflexes with PR *i, while OJ i 2 ∼ o 2 < PJ *əi usually corresponds to PR *e, like 'tree' OJ ki 2 ∼ ko 2 -:: PR *ke. Reconstructing these exceptional cases not as *əi but *oi permits to solve all these problems.Such examples include 'be, sit', 'fire', 'get drained' and 'indigo', which I respectively reconstruct as PJ *woi, *poi, *poi and *awoi (instead of the traditional *wəi, *pəi, *pəi and *awəi, cf. Martin 1987) on the basis that these have a vowel *i in PR. OJ makes no distinction between o 1 and o 2 after w and p, and there is thus nothing that could prevent us from reconstructing an *o instead of an *ə in these. Tables12 to 15give the reflexes of the above words in Ryukyuan and contrast them with examples of PR *e in similar environments.

  third law, which states OJ a and o 2 (< *ə) seldom co-exist in the same morpheme. Evidence for a seventh vowel in PJ remains scarce. Two proposals for a seventh, high central vowel exist, but they have been made on the basis of completely different kinds of evidence.

	5 Further developments and remaining issues
	5.1 A seventh PJ vowel?	'sit'	'be'	'get drunk'
	OJ			wi	wor-	wep-
	Okinawa Shuri	jijuŋ	wuŋ	wiːjuŋ
	Miyako	Ōgami	pɯɾ-uɾ-	piː
	Yaeyama Ishigaki bɿɾuŋ uŋ	biŋ
	PR			*wir-*wor-*we
			TABLE 13 PJ *poi 'fire'
				'fire' 'moldboard, spatula'
	OJ Okinawa Nakijin	pi 2 p ʔ iː	pe 1 ra pʰiɾaː
	Yaeyama Ishigaki	pɿː	piɾa
	Yaeyama Hateruma pɿː Yonaguni tɕ ʔ iː	piɾa çiɾa
	PR			*pi	*pera
			TABLE 14 PJ *poi 'get drained'
				'get drained' 'moldboard, spatula'
	OJ Amami Okinawa Nakijin Koniya	pi 2 çiɾ-p ʔ juŋ	pe 1 ra ɸɨɾa pʰiɾaː
	Miyako	Ōgami	psː	piɾa
	Yaeyama Ishigaki	pɿsuŋ	piɾa
	Yaeyama Hateruma pɿsuŋ	piɾa
	PR			*pi	*pera
			TABLE 15 PJ *awoi 'indigo'
				'blue' 'indigo' 'front'
	OJ Amami Koniya ʔoːsa ʔeː awo awi	mape 1 məː
	Amami Yoron	oːsaŋ ai	meː
	Miyako Hirara oːkaɿ aɿ	mai
	PR			*aU	*ai	*mae

TABLE 16

 16

	F & W's PJ *ə and *ɨ
		'seaweed' 'this'
	F&W's PJ OJ Amami Okinawa Shuri Koniya Miyako Ōgami PR	*mə mo muː muː muː *mU	*kɨ ko 2 kʰuɾ kuɾi kuɾi *kuri

Published inBjarke Frellesvig & Peter Sells. 2013. Japanese/Korean Linguistics 20. Stanford: CSLIPublications, p. 81-96. isbn: 9781575866383.

PJ *eNtu'which', *keNtu 'wound', *memeNsu 'earthworm', *meNtu 'water', *peru 'garlic', *peNsi 'elbow'. Serafim (1999), quoted in[START_REF] Miyake | Philological evidence for *e and *o in pre-old Japanese[END_REF], proposed several examples of *e appearing in other environments, but the comparative data underlying his reconstructions remains unkwnown, and my own data does not agree with most of his reconstructions.

See Pellard (2008) for more details on this last reconstruction.

Arisaka's paper appeared after Hattori's, but Hattori acknowledged it was Arisaka's discovery.

In Southern Ryukyuan, *ui-ending polysyllabic verb stems have been reshaped by analogy with the *əi-ending ones, and the *ui/*əi distinction is thus well-preserved in nominals only.

"This is why one can legitimately pick out a word like sïkeN 'moon' in the remote Hateruma language as convincing evidence for *ko in JR *tukojo 'moon(night)' millenia earlier"(Thorpe 1983:355)

Acknowledgments

I express my gratitude to all the Ryukyuan speakers whose help made this study possible. This study was supported by a Grant-in-Aid for Scientific Research (22•01000) from the Japanese Society for the Promotion of Science.

Note on and transcriptions

Old Japanese syllable distinctions are marked with subscript numbers, with 1 indicating the kō-rui series and 2 indicating the otsu-rui. PR *U indicates it is not possible to decide whether the vowel should be reconstructed as *u or *o, and *N stands for an unknown nasal.

The various Ryukyuan data taken from existing sources have been converted to a broad phonetic notation, that mostly follows the principles of the IPA, except for the symbol [ɿ], that marks a sound rather freely alternating between an alveolar approximant and a fricative, and for [ ʔ ], which marks non-ejective glottalization. Contextual devoicing and tones are left unmarked.


Philippe Polomé
email: polome@gate.cnrs.fr

Forest Owners Motivations for Adopting Programs of Biodiversity Protection

Keywords: Non-indutrial private forest owner, biodiversity program, motivation crowding-out, adoption decision C64, H41, Q23, Q28, Z13 Non-indutrial private forest owner, biodiversity program, motivation crowding-out, adoption decision JEL Codes. D64 Altruism, Philanthropy, H41 Public Goods, Q23 Forestry, Q28 Government Forestry Policy, Z13 Economic Sociology

Introduction

This paper focuses on measuring the role of non-industrial private forest owners' motivations in adopting programs of biodiversity protection in France. There are a number of public voluntary programs that include biodiversity protection to some extent, even though they have other objectives as well. These include wood products sustainable forest management certification (PEFC-FSC), several types of forest management plans and the famous Natura 2000 program in which owners voluntarily commit to specific norms of forest management. Forest owners can also take part in non-public voluntary programs of forest management with varying degrees of biodiversity protection. These include various professional organizations such as cooperatives and syndicates (unions), environmental NGOs and the "ProSilva" organization promoting an alternative form of forest management.

Generally, behaviors of biodiversity protection may be termed prosocial in the sense that their social benefits exceed those of the supplier of the protection, or have a positive effect on society. Adoption of prosocial behaviors has been shown to depend more on social motives than on economic ones; for example, Freeman (1996) [START_REF] Richard B Freeman | Working for nothing: The supply of volunteer labor[END_REF] shows that having been asked by a person is a more powerful motive than the economic one. Oliver (2013) [START_REF] Oliver | Behavioural public policy[END_REF] reviewed a number of applications of nudges that illustrate the importance, for designing public policy, of understanding the motives of adoption of such programs.

Self-determination theory, developed by psychologists Ryan and Deci (e.g. [START_REF] Richard | Intrinsic and extrinsic motivations: Classic definitions and new directions[END_REF]) is a general theory that argues that individual may have extrinsic and intrinsic motivations. Extrinsic ones lead to what is termed "controlled" actions, that is, actions motivated by a pressure that is perceived as external to the individual, and can be more or less accepted ("integrated") by that individual. Such pressures are akin to penalties or rewards and encompass economic or financial incentives (penalties or rewards) to include threats, peer pressures and feelings of moral obligation. Intrinsic motivations, on the other hand, lead to actions that are called "autonomous", in the sense that the individual carries them out for themselves, without any perceived external pressure. The satisfaction (or utility) of an action is deemed higher when it is motivated intrinsically than by an extrinsic motivation. Thus, for a forest owner, adopting any particular program may be motivated extrinsically (e.g. because of financial incentives or because she thinks that it would be illegal not to adopt or because she is asked to adopt by a friend) or intrinsically because she identifies with this particular program. The diversity of motives in silvicultural and harvesting behaviors is well documented for non-industrial private forest owners, e.g. Karpinnen (1998) [START_REF] Karppinen | Values and objectives of non-industrial private forest owners in finland[END_REF] or [START_REF] Kendra | Motivations and behaviors of new forest owners in virginia[END_REF] [START_REF] Kendra | Motivations and behaviors of new forest owners in virginia[END_REF].

The present paper focuses on the formalization of prosocial behaviors, developed by [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] [2] and (2006) [START_REF] Bénabou | Incentives and prosocial behavior[END_REF]. These authors have a particular interest in economic (extrinsic) motives that they differentiate not only from the intrinsic motives (that they term public good and altruism) but also from the more social motives, with interpretations that include social network and self-image. 1 In the present paper, economic motives (some of the extrinsic motives) include financial compensations or penalties; fiscal exemptions; risk hedging; time saving; acquiring management information; future or putative use. For many forest owners in the population of interest, the forest property is relatively small and forest income are negligible (Table (3)); economic motives might then not be very powerful and the other types of motives might be relatively important. Social motives (the rest of the extrinsic motives) include reputation (networks, family, peer-pressure); self-image (moral satisfaction, warm glow); desire to belong to a group or differentiate from it; socialization; reciprocity.

Ethical (the intrinsic motives) include the willingness to work for quality of the environment; well-being of the community; bequest or patrimony values; attachment to the forest. Bénabou and Tirole (2005, p.8) [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] refers to the intrinsic motives as "altruism and public goods". However, these concepts are interpreted in very different ways by different forests owners, as became apparent in stakeholders discussions and questionnaire tests. For example, [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] state on p.8 that the agent "... may care about the overall level of a public good...". Such overall level is confusing when addressing forest owners as there is not a single dimension of the forest. Some owners will consider that the level is wood production, others that it relates more to a closerto-nature state of the forest, while still others see any reference to a non-private level as a form of intrusion into what they see as their sovereign property. For these reasons, intrinsic motives have been termed "ethical" here. Depending on the program under consideration, they include notions of attachment (to the forest) or of mastery (over forest practices), the environmental quality of the forest, or definite ideas on how a forest should be managed.

In some cases, the difference between social and ethical motives may appear thin, e.g. moral satisfaction and altruism, to name a case. The difference resides in the notion of extrinsic vs. intrinsic motivation following selfdetermination theory, that is, whether the adoption is "controlled" (motivated by a pressure perceived as external to the individual) or "autonomous" (the individual chooses the adoption "for itself"). That corresponds to the difference between the "moral satisfaction" and "public good and altruism" in [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] [START_REF] Bénabou | Incentives and prosocial behavior[END_REF], although, ultimately, the difference is one of degrees rather than a well-defined border. Indeed, self-determination theory argues that any motivation for any particular action can be gradually internalized ("appropriated"), that is, the motivation can become more intrinsic as the individual identifies more with it.

These three categories of motives may act positively or negatively for any program, depending on the forest owner, e.g. economic incentive may be considered attractive for one, thus be declared as a motive for adopting a program; while it may be insufficient for another, and thus be cited for non-adoption of a program. [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] formalize the idea that the two kinds of extrinsic motivations -economic and socialmay be opposed and that everyone prefers that her actions (adoptions of programs in the present case) appear intrinsically motivated. Consequently, when a program becomes financially compensated, some of the adopters of this program might be concerned that their social network believes they adopt the program out of their interest for the money. While if it had not been financially incentivized, it would have been clear that they were adopting for intrinsic reasons. This "social credit" could then lead to an improved social image or better social networks.

Whether this desire for improved social image is in turn motivated economically (e.g. accessing better prices through a network) or by other motives, is beyond the scope of the present paper.

Such crowding-out of the economic motives on the social ones reduces adoption for a range of (possibly small) economic incentives. This is one of the key ideas in [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] [START_REF] Bénabou | Incentives and prosocial behavior[END_REF], but similar crowding-out effects are abundantly documented, see e.g. Frey and Oberholzer-Gee (1997) [START_REF] Frey | The cost of price incentives: An empirical analysis of motivation crowding-out[END_REF], [START_REF] Gneezy | A fine is a price[END_REF] [10], [START_REF] Nyborg | Does public policy crowd out private contributions to public goods?[END_REF] [START_REF] Nyborg | Does public policy crowd out private contributions to public goods?[END_REF], or Oliver (2013) [START_REF] Oliver | Behavioural public policy[END_REF]. From a public policy viewpoint, the existence of crowding-out questions the efficiency of the policy, particularly for environmental policies, since it means that economic incentives are sometimes wasted as they reduce adoption (or have a lower-than-expected effect) because they crowd-out social motivation. In Bénabou and Tirole, crowding-out can only occur when individuals are socially motivated; if they are only motivated intrinsically, then there is no ground for a crowding-out. But if it can be shown that forest owners' adoptions are (at least in part) socially motivated, then it may be the case that public policy should focus more on social reward of a policy than on economic reward. [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] state that visibility of the economic motives is essential: if the economic incentive (reward or penalty) is not visible to a third party, then economic motives cannot be inferred from the forest owner's actions, and therefore, there is not crowding-out. In the present forestry case, economic incentives are fairly visible for some programs, because they correspond to legal aspects (e.g. fiscal deductibility of management plans), or because they are associated to a label (certifications) that presumably allows the owners to sell at a higher price. Some other programs do not have an explicit economic return, e.g. adhesion to the ProSilva association or adoption of Natura 2000 practices; or even may apparently lack any economic incentive, e.g. in the case of adhesion to an environmental NGO. Ultimately, visibility is a perception; it is very difficult to assess whether all forest owners have the same beliefs on the economic return of any program.

The main focus of the present paper is to test whether the Bénabou and Tirole construct might apply to a sample of forest owners, and to test whether crowding-out effects are already at work with the current biodiversity programs. It also to identify potential pitfalls and leverages of forest biodiversity policies. The paper is organized as follows. Section 2 presents the cross-section sample of forest owners in a French Regional Natural Park, then details the programs that these owners can choose to adopt, the motives that have been revealed in the survey and finally a socio-economic descriptions of the forest owners, of their properties, and of the spatial relationships between them. Section 3 presents the Bénabou and Tirole model formally, then interprets it in term of the econometric analysis of the adoption decisions and finally presents the econometric results. Section 4 concludes.

Survey and Data Description 2.1 Sample

The survey was administered during the first quarter of 2015 on the nearly 50 000 non-industrial private forest owners of the Parc Naturel Régional des Ballons des Vosges (PNR),2 using the quotas method, by telephone following a prior letter that the respondents had received a few days earlier. The letter summarized the programs of interest, presented in the next section.

Table [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] shows that a large majority of forest owners have tiny, less than one ha, properties, as is quite typical in many French regions. Because there was an interest in these small owners, the minimal size of the property to qualify for the sample was set at 0.05 ha (500 m 2 ), reaching a compromise between representativity in terms of property sizes and number of owners. Further, the 4 ha point is relevant as it is the minimum for fiscal exemption of forest management expenses or investments. The 10 ha point is the minimum to adopt a "Simple Management Plan" described in the next section. The resulting sample has four classes of about 130 respondents each, but for the fifth class, the properties of more than 10 ha, it was deemed that a sample of about 100 was the best that could be achieved. The final sample size was 627. The sample was established on the basis of the properties in the cadastre of the PNR, but some owners also had properties elsewhere, or has sold some properties recently, explaining the differences between cadastre and stated surfaces.

Programs

In the survey, private forest owners were asked about whether they had adopted each of the following programs. 3 These programs are a grouping of several, possibly exclusive, programs, that are documented on the "forest owners portal" http://www.foretpriveefrancaise.com/les-services-aux-adherents-27775.html, the "directory of the private forest" http://www.foretpriveefrancaise.com/annuaire and in Weiss (2003) [START_REF] Weiss | Le regroupement des petits propriétaires forestiers en vue d'une gestion concertée[END_REF]. They were established with groups of stakeholders, bearing in mind that interview length must be kept short (about 20 minutes) and that the introduction letter had to be small (1 page).

These programs constitute the owner's choice set, that is all the possible forest-related programs feasible in the area of the PNR. This choice set may or may not be completely known by all the owners. Owners may certainly participate in other social or environmental actions, but the following list is bound together by the forest property and by the fact that the programs are in some ways competing for the owner's resources, in particular time. The programs are the following.

• Any (wood) certifications of sustainable management: PEFC, FSC,...

• Any forest management plan. These are good practices plans, that are intended for sustainable forest management. There are three plans, with different charters, that are directed to different types of forest owners or property. The "Simple Management Plan" (Plan simple de gestion) may be adopted when property size reaches 10 ha and becomes mandatory when it reaches 25ha. The "Good Forest Practices

Code" (Code des bonnes pratiques sylvicoles) is directed towards smaller properties. The "Standard Management Rule" (Règlement type de gestion) is directed to forests owners who own less than 25ha, and for whom the property is managed by a third party (expert or organization).

• Any forest owners management-related organization, e.g. cooperative (common management of their stakeholders' plots) or union (represent, inform and advice their members, can enter a "Simple Management Plan" on behalf of their members, may have environmental goals, or may be constituted only for very specific tasks). 37 such structures are available for owners of the PNR, and several forms of owners associations outside of these structures exist. They have a strong social component following Darses, Garcia, and Stenger (2011) [START_REF] Darses | Drivers of cooperation for environmental goods: Evidence from a national survey on french private forest owners[END_REF] and [START_REF] Weiss | Le regroupement des petits propriétaires forestiers en vue d'une gestion concertée[END_REF] [START_REF] Weiss | Le regroupement des petits propriétaires forestiers en vue d'une gestion concertée[END_REF].

• Pro Silva (French branch, "Association Futaie Irrégulière" -AFI) association of forest owners promoting a continuous and irregular form of management, closer to a natural cover.

• Natura 2000 are sites; for properties that belong to these sites, management must be submitted to impact assessment or the owner may choose to adopt the Natura 2000 charter. The owner may further commit through a Natura 2000 contract, which is more demanding than the charter.

• Any NGO for the defense of the environment (WWF, "France-Nature-Environnement"...) that presumably has bearing on forest management practices.

That amounts to seven programs since Natura 2000 is actually two programs : Charter and Contract. The survey showed several striking facts, some of which where unexpected. First, relatively few respondents, 22%, participate in any program. Second, very few respondents (2 out of 627) participate in more than one program. 4 Third, on 627 observations, there is only one adoption of the Natura 2000 charter and two adoptions of the Natura 2000 contract; these response rates have led to dropping these observations altogether. Consequently, the Natura 2000 program cannot be examined other than through understanding the motives of non-adoption. Fourth, few respondents adopted a certification, a management plan or a management organization; since these programs may be perceived as complements, they have been grouped into one, called "conventional management" for want of a better name as in the sample region, the "ProSilva" management is possibly more conventional than the other practices. These results led to redefine three "programs": ProSilva, Conventional management, Environmental NGO. The figures for adoption are presented in Figure (1) below.

Motives

For each of those programs, when a respondent answered that she adopted it, she was queried on her motives.

Table [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] presents all the motives that have been invoked for adopting or not a certain program. Some motives are not relevant for some programs. The motives proceed from several meetings with stakeholders, but, for each program, there was the possibility for each respondent to state one or several other motives. Those were recorded, but were all anecdotical, that is, did not occur more than twice over the whole survey. Respondents also could always answer that they did not know or could also refuse to answer -though they were not prompted for such answers.

Apart from the Economic-Social-Ethical motives, there can be more circumstantial motives that respondents sometimes evoke. These are linked to a lack of information, e.g. the respondent was not aware of the program, or those who adopted out of curiosity. More generally, in contexts other than those of the forest owners in the present sample, there could be more technical motives -e.g. one could not adopt a Natura 2000 charter if her forest was not in a Natura 2000 forest. Such technical motives were not evoked by the respondents of the present survey. Finally, some respondents inevitably answer that they do not know their motivations of adoption, or that they have none. 2, respondents may have from zero to four motives to adopt a program, but also zero to four motives not to adopt a program. In the econometric model of the next section, each motive is modeled by a dichotomous variable, that is defined for each respondent. When a respondent does not reveal any of the four motives (economic, ethical, social, information), the value of these four dichotomous variables are set to zero.

To assign a value to each of these dichotomous variables, each respondent could be asked to state the motive for adopting or not a particular program. However, at the conception phase of the questionnaire, it was quickly realized that it would be very tedious, and certainly unreliable, to ask respondents why they would not adopt a particular program. Instead, respondents were asked whether they adopted any of the programs, and in case they would answer negatively (no program was adopted -78% of the respondents), they were queried for the motives for not adopting. Similarly, respondents, who answered that they adopted some program(s), were asked to state the motives for not adopting any of the other programs in a single question, not program by program.

For each of the three previously defined programs -ProSilva, Conventional management, Environmental NGOthese questions provide data on motives for those who adopted, but also for those who did not (whether they did not adopt any program, or they adopted another). In the very few instances in which a respondent has adopted more than a program, these motives have been merged. This process of construction of the motives data safeguards against a possible endogeneity of motives in the adoption decision, that would occur if only the motives for adopting were recorded.

The intensity of participation in the program could also be used to test crowding-out effect. Carpenter-Myers (2010) [START_REF] Carpenter | Why volunteer? evidence on the role of altruism, image, and incentives[END_REF] found that intensity did not follow the same determinants as adoption. Intensity could be captured in the part of the property that is allocated to a program, for those programs for which that is meaningful, in the complexity of the program, and also in the length of time of the commitment. Regarding the former two, it was conjectured that such information would be too unreliable to be useful. The length of the commitment is captured in the questionnaire. Respondents who stated that they adopted a program were queried upon the date of adoption and whether the program was still followed. In case it was not, the person was asked the motive for abandonment. These are identical to the motives for not adopting. Since only four respondents rescinded adoption, this measure of intensity does not appear usable in the present sample.

Exogenous Variables

The owner's characteristics and the property characteristics that are susceptible to play a significant role in the probability of adopting a program were collected in the survey according to e.g. Delacote, Garcia, Stenger (2014) [START_REF] Delacote | Private forest owners' participation behavior related to an incentive conservation program: a case study of natura 2000 contracts in france[END_REF]. The more salient traits of the sample are described in Table [START_REF] Cameron | Microeconometrics Methods and Applications[END_REF]. The questionnaire allowed for the collection of some spatial and social characteristics, i.e. whether the owner lives in the same postal code as the oldest parcel of her property (presumably of highest attachment), whether the owner knows the neighboring woods and/or their owners. Regarding social interactions, 32% consider that networking is useful to acquire professional competences and 30% to acquire professional contacts. Regarding management, 67% consider that climate change is a serious environmental threat (used to measure the degree of awareness of environmental issues) and 24% consider that abandonment is a good ecological management.

Exogeneity of the variables of Table (3) refers to the decision of adoption of programs; although it is difficult to reject the idea that there may be some common components between adoption decision and some of the variables of Table 3, they are deemed negligible.

Characteristics of the area and of the programs certainly play a role on adoption. However, they are independent of the owners' individual characteristics and of the social networks in which they participate; they are also fairly uniform across the Parc. Therefore, their absence should not affect the results in significant ways.

Models and Econometric Results

Bénabou and Tirole ( 2006) [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] present their model in a time-like context: when economic incentives are introduced, they may crowd-out social motives. From an empirical point of view, individual time-series data of incentive changes are very difficult to create or as that would require a panel purposefully designed. Experimental data might be an alternative, but it is also possible to interpret a cross-section in a long-term sense.

Individual characteristics in the sample allow for a control; that is, each respondent is seen as an experimental unit that has undergone a number of treatments. These are his/her individual characteristics and those of his/her property. Econometric estimation sorts out the effect of each of those "treatments" so that each individual can be examined "other things equal", that is, as if they were identical experimental units. More intuitively, the econometric model uses the stated motives for each program and infer, for each respondent, a probability of adoption. When two motives are evoked at the same time, elicitation of a crowding-out effect boils down to estimating whether the probability of adoption is smaller than it would be if the two motives simply added up.

Economic Model

As indicated, the Bénabou and Tirole ( 2006) [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] model presents the possibility of crowding-out between monetary and social incentives; that means that if a forest owner is socially motivated, introducing monetary benefits (and thus possibly motives) may reduce her interest in the program, for a range of benefits. Bénabou and Tirole consider that this is due to the fact that the social motive is in fact not fully genuine but caused by a desire for a good reputation, and that for the prosocial behavior to have its full reputational utility (for the respondent), it must also appear disinterested. When the respondent does not appear disinterested, she does not enjoy such a large reputational boost as if she did. For Bénabou and Tirole, this appears to be a maintained hypothesis, as there is no formal empirical basis to assert its occurrence in a population. This hypothesis can be tested using the present sample. It will turn out not to be verified.

Formally, Bénabou-Tirole's equations ( 1), ( 2) and (3), in [START_REF] Bénabou | Incentives and prosocial behavior[END_REF], can be summarized by writing the individual utility of an effort a (discrete or continuous) of participation to the public good, that is adoption of a program, 5 as

(n a + n y y) a + x (g a g y ) C (a) (1)
where n a and n y are the owner's intrinsic motive ("valuation", utility) from participating in the program (the "public good" incentive of Bénabou and Tirole 6 ) and extrinsic economic motive ("valuation" of money), respectively; x is the visibility of the behavior -that is, how much of the owner's program adoption can be "seen" by her social network or others; g a is the utility derived from having others observe the adoption and g y is the disutility from having others observe the monetary reward y derived from the adoption. g y appears with a negative sign to underline that agents would prefer to appear disinterested, so that the social incentive is x (g a g y ) (a "reputational" motive for Bénabou and Tirole). C (a) is the cost of effort a, in term of disutility. The model assumes such functional form and does not test it, but the scope of the model is not empirical.

In the case of forest owners, utility is not observed, but only whether the owner adopts any particular program, that is, adding an error term e to Equation 1:

adopt i f (n a + n y y) a + x (g a g y ) C (a) > e not adopt else (2)
None of these quantities, a, n a , n y , x, g a , g y , y or C (•), can be observed or inferred easily. It might be considered that y (monetary reward) or x (visibility of the monetary incentive) are publicly known, but it is apparent from the survey results that owners' subjective views of the monetary return of any particular program are highly variable, especially for non-adopters who sometimes hold erroneous beliefs on programs. The effort a is not observable as adoption depends on several hard-to-observe factors, as indicated earlier.

However, the significance of the different motives n a , n y , g a , and of the crowding-out effect, g y , in the adoption decision may be tested in a discrete choice model. The social incentive g a exists only if the social motive has a significant effect on the probability of adoption; similarly for the economic incentive n y with the economic motive, and the "public good" incentive n a and the ethical motive. Crowding-out exists only if the impact of the social motive, on the probability of adoption, is reduced by the economic motive. Such a reduction can be tested in a discrete choice model by the sign of the coefficient of the cross-product of the two motives: the presence of the two motives together (and hence of the two incentives, at least subjectively for the respondent) leads to a lower probability of adoption than the added effects of the two motives. This opens the way to small empirical generalizations of the crowding-out effect to the other pairs of motives : economic-ethical, social-ethical.

Discrete Choices Models

The outcome of interest are discrete, without any particular order between the programs. Non-adoption of all the programs is complementary to adoption of at least one program, but not to the adoption of any particular program, so that the sets of significant regressors need not mimic each others. Two models may be estimated.

The first is a probit model on the probability of adopting "no program". This is expressed formally in the classical probit as:

Pr {No adoption} = Z xb • f (t) dt (3)
where f is the standard normal density, x is a vector of explanatory variables including the motives, and b is the vector of unknown coefficients of these explanatory variables, reflecting their quantitative impacts on the probability of adoption.

Probit models could also be estimated for each of the three programs (ProSilva, Conventional Management, Environmental NGO), but that would assume a zero correlation between the three. Since it appears that virtually no respondent is involved in more than one program, there is a strong negative (unconditional) correlation between the adoption decisions, which may remain even when conditioned on regressors. The second model of interest is then a multivariate probit, 7 which is based on a multi-integral extension of the probit probability 3, see e.g. Cameron and [START_REF] Cameron | Microeconometrics Methods and Applications[END_REF] [3] p.517.

Estimation is classically by maximum likelihood for probit, and maximum simulated likelihood for the multi-variate probit, see e.g. Cameron and Trivedi (2005) chapter 12 [START_REF] Cameron | Microeconometrics Methods and Applications[END_REF]. 8 The methodology is encompassing, starting from a large set of regressors and gradually reducing to avoid multicollinearity issues and test whether groups of dropped regressors are not jointly significant. Polynomial of the regressors have not been used, except for the motives products, as that would have entailed too large a regressors set and there was no a priori theory for non-linear effects of such variables.

The motives are modeled in Probability (3), or in its multivariate equivalent, by a set of three dichotomous variables (some respondents indicate no motive for adoption so that all three dichotomous variables may be zero simultaneously). Interaction between motives is modeled by the three cross-products of the motives, see Table [START_REF] Cappellari | Multivariate probit regression using simulated maximum likelihood[END_REF]. More precisely, consider two motives, A and B; each dichotomous variable reflects, through its coefficient, the direct effect of the motive on the probability. Such direct effect is due, for a particular respondent, both to the presence of the motive in her "psychological" profile, e.g. whether the respondent cares about an economic incentive, and to the perceived presence of the corresponding (extrinsic) incentive, e.g. whether the respondent believes an economic benefit is associated to a given program. The cross-product of the two dichotomous variables reflects the joint effect of the two motives on the probability. If its sign is negative, the joint effect is a crowding-out: the combined presence of the two motives is less than the sum of the effect of the two separate motives. If the sign is positive, the two effect are complementary and their joint presence reinforces each other:

the combination of the two motives leads to more than the sum of the parts.

The econometric model does not restrict a priori the sign of the coefficients of such cross-products, or their statistical significance; so that estimation is really a test of the [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] theory: if the sign of the coefficient of the social-economic cross-product turns out statistically non-significant, or significantly larger than zero, than the present survey turns out an evidence against the theory. Conversely, if it turns out significantly smaller than zero, then the present paper offers one piece of evidence in favor of the theory. The other two cross-products, economic-ethical and social-ethical, do not belong to the [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] [1] theory, and so their signs cannot be interpreted as a test of the theory.

Results

Table [START_REF] Cappellari | Multivariate probit regression using simulated maximum likelihood[END_REF] presents the results of the estimation the probit model (3) on no adoption, and its corresponding multivariate equivalent for the three programs (ProSilva, Conventional Management, Environmental NGO). There is a significant crowding-out between the economic and ethical motives, but not between the economic and social motives. The Bénabou and Tirole (2006) [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] model suggests only the latter and not the former. The crowding-out effect can only be shown for the Prosilva and for the Conventional management programs, as there were too few observations for which both motives were evoked for the Environmental NGO program. The ethical motive, in the context of the present survey, refers to feelings of attachment to the forest and concerns about the mastery of own practices. This crowding-out effect is then likely driven by a feeling of incompatibility between economic incentives and such attachment or mastery, possibly associated with the selling of a part of one's control over one's property.

As indicated, unconditionally, adoptions of the programs are strongly negatively correlated to each other, as evidenced by the fact that nearly no respondent has adopted more than one program. In the Multivariate probit model of Table [START_REF] Cappellari | Multivariate probit regression using simulated maximum likelihood[END_REF], the conditional (residual) correlation between adoption decisions turns out to be not significantly different from zero for the correlation between ProSilva and Environmental NGO, significantly positive between the two management programs (ProSilva and conventional) and significantly negative between Conventional management and Environmental NGO. Therefore, once conditioned on the individual characteristics and motives, and the forest characteristics, the correlations show a certain antagonism between environmental goals and conventional management goals, but that is not the case with ProSilva. Globally, the hypothesis that the three correlation coefficients are jointly zero can be rejected at about the 1% confidence level.

Finally, Table [START_REF] Carpenter | Why volunteer? evidence on the role of altruism, image, and incentives[END_REF] presents the variables that turned out non-significant for the same models. One of the unexpected results is that none of the spatial regressors is significant. characteristics have the expected effects, in particular, programs are adopted primarily on larger properties. Unsurprisingly, economic incentives (extrinsic motivations) have a clear positive effect on the probability to adopt a management program (conventional or ProSilva), but not to adhere to an environmental NGO. Social motivations significantly impact adoption of all three types of program positively, while ethical motivations have a clear positive effect on adhesion to an environmental NGO. Non-adoption primarily depends on economic and social motives, not on ethical ones.

There appears to be a significant crowding-out between economic and ethical motivations, for the adoption of both types of management programs (conventional and ProSilva), so that the probability of adopting such programs will decrease when economic incentive increase over a certain range and ethical motivation remains constant. Similarly, owners are more likely not to adopt a incentivized program if they perceive that their control over their property is lessened (positive and significant sign of the cross-product of the ethical and economic motives in the decision not to adopt any program).

Such conclusions on individual motivations for adopting programs lead to questioning of the use of ecosystem services valuation from a policy point of view. Valuation leads naturally to the idea that forest owner might be financially compensated for these services. Although that is undoubtedly an important motivation following the results presented in the present paper, and therefore, it might be expected that paying for ecosystems services will increase their supply, it remains that these economic incentives crowd-out the ethical motivations.

Therefore, it is not immediately clear that because ecosystem services values can be estimated, they should be translated to economic incentives for the supplier of such services. As already seen in several circumstances, e.g. Oliver (2013) [START_REF] Oliver | Behavioural public policy[END_REF], such crowding-out might induce a decrease of ecosystem services.

Contrarily to the maintained hypothesis in [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] [START_REF] Bénabou | Incentives and prosocial behavior[END_REF], there is no crowding-out between the economic and the social motives. There is also no crowding-out between the ethical and the social motives.

Therefore, social motives are in a particular position for being used as leverages of public programs. That would constitute a future research avenue, but one could imagine that programs that enhance the social reward of the forest owners who adopt them might be more economically efficient, that is, lead to more adoption for the same level of public spending -or less public expenses for the same level of adoption. That leads to programs that would allow owners to interact between themselves and with the administration, and thus advocates for a more bottom-up approach. Further, programs, that focus on the social motivations through the use of nudges, see e.g. Oliver (2013) [START_REF] Oliver | Behavioural public policy[END_REF], might also achieve the same outcome at a lower cost. The strong negative correlation between adoption decisions lead to recommend avoiding creating new programs and instead modify existing ones.

  figure reflects the different motives for the each program, and for not adopting any program. The large number of respondents who did not adopt any program led to a separate analysis.

Figure 1 :

 1 Figure 1: Motivations by Program -n=627

Table 1 :

 1 Surface (S) and owners classes

	Pop.	Surface (ha) Forest area (ha)	<1 10 675	1< S <4 16 364	4< S <10 10< S <25 10 908 7 499	25< 15 651 61 097 Sum
		Owners (#)	38 486	8 580	1 828	505	185	49584
	Sample	Surface (ha) Planned (#) In cadastre PNR (#)	.05 -.74 .75 -1.99 2 -3.99 128 128 128 136 132 131	4 -9.99 128 129	>10 90 88	600 627
		Stated (#) -60 miss	109	157	114	102	74

Table 2 :

 2 Detailed motives and grouping

		Adoption	Non-adoption
		economic interest	lack of economic interest
	Economic	time savings fiscal advantages compensation for game damages insurances	time consuming too complicated commitments are too long or uncertain owner's practices already include the program
		owner believes program is legally mandatory	owner believes program is not legal in his/her case
	Social	recommended by a friend/family/colleague... do as the other owners make an example	lack of commitment of the other owners the program does not have social recognition lack of dialogue with forest owners on this program
		desire to enter a professional network
		attachment to his/her forest	(retaining mastery of practices)
	Ethical	desire to bequest a better forest improving the woods personal belief	respondent wishes no intervention in his practices program is incompatible w/ owner's ecolog. or manag. actions control by administration is too uncertain or complicated
		protect certain species
	Info.	information need (technical, manag...) curiosity	lack of information on the program
	Following Table

Table 3 :

 3 Descriptive statistics of the owners and the properties -627 observations

	Owner	Prop.	Forest (owner's opinion)	Prop.
	Man	81%	Last intervention before 2010 48%
	Aged 66+	59%	Used by third party	71%
	Completed high school	34%	Has remarkable feature	34%
	In non-professional association	29%	Large or very large trees	27%
	Visits property at least 1/year	70%	Family management	90%
	Lives close to oldest parcel	50%	No wood sold	76%
	Has own environmental actions 14%	Firewood production	62%
	Hunts	8%	In a protected area (not PNR)	2%
	Knowns neighboring properties 73%	Associated loan	2%
	Forest income negligible	88%
	Will bequest forest	76%	Mostly inherited	75%

Table 4 :

 4 Probability to adopt any one of the three programs -or nothing -n=600

				Multivariate Probit				Probit
		Pr {ProSilva} Pr {Env.NGO} Coef. p Coef. p	Pr {C.M.} Coef. p	Pr {nothing} Coef. p
	Forest property
	Property surface	+	***	0		+	***	-	***
	Large or very large trees	+	º	0		0		-	*
	Mostly inherited Managed by 3 rd party	0 +	**	-0	*	0 +	***	0 -	.1 ***
	Firewood production	0		0		-	*	0	.1
	In a protected area (not PNR) Owner	0		+	*	0		0	.4
	Age	0		+	**	+	º	-	***
	Education	0		+	**	0		-	º
	Gender	0		-	*	0		0	.4
	In non-professional association	0		+	*	+	*	-	º
	"network useful for prof. competences"	+	**	0		0		0	.1
	"network useful for prof. contacts"	0		+	*	+	*	0	.1
	"abandonment not good ecol. manag."	+	*	0		0		-	*
	Has own environmental actions		Not significant at >20%		-	*
	Will bequest forest	+	*	0		0		0	.4
	Visits property often Motivation	0		+	*	+	*	-	*
	Economic	+	***	+	*	+	***	-	***
	Ethical	0		+	**	+	*	0
	Social	+	***	+	***	+	***	-	***
	Economic ⇥ Social Economic ⇥ Ethical: crowding-out Social ⇥ Ethical Information	0 -0 0	**	too few obs. 0	0 -0 +	** **	0 + 0 -	** º
	Significance levels p : º 10%; * 5%; ** 1%; *** 0.1% -C.M. Conventional Management

Table 5 :

 5 Regressors that can safely be said not to impact adoption -n=600There are fewer program adopters than expected, about 22% of the sample. Out of 627 observations, only two owners adopt two programs, 139 adopt one program. The probabilities of adopting any of the three programs (ProSilva, Environmental NGO, Conventional management) were estimated using a multivariate probit model to account for the strong negative correlation between adoption decisions. The owners' and properties'

	p-values

[START_REF] Bénabou | Incentives and prosocial behavior[END_REF] [2] use the term "reputational" for these motives, e.g. p.3. For a more neutral interpretation, the present paper uses the term "social" instead.

Regional natural area in the North-East of France.

As indicated, the owners had previously received the list of programs by post, on which each program was briefly summarized.

Regarding the econometric model presented in the sequel, this means that adoption of a program cannot be used as a regressor for estimating the probability of adopting another program as that would lead to a complete determination, that is 100% of those who adopt a program do not adopt another. Furthermore, the use of such regressor would lead to endogeneity issues since both adoptions would be correlated.

[START_REF] Tirole | Motivation intrinsèque, incitations et normes sociales[END_REF] [START_REF] Tirole | Motivation intrinsèque, incitations et normes sociales[END_REF] suggests to normalize the effort a to 1 (participate) or 0 (does not), but in an empirical context, the efforts of different persons might differ, so that normalization is not feasible.

As noted earlier, such "public good" dimension is ambiguous as different respondent tend to have a different opinions of what the public good is.

Sometimes referred to as "Multinomial probit", see Cameron and[START_REF] Cameron | Microeconometrics Methods and Applications[END_REF] [START_REF] Cameron | Microeconometrics Methods and Applications[END_REF]. The well-known multivariate (or multinomial) logit is also ruled out as this model imposes a fixed correlation between decisions.

The Stata Package has been used. For multivariate probit, the user-developed mvprobit package has been used, see[START_REF] Cappellari | Multivariate probit regression using simulated maximum likelihood[END_REF] [START_REF] Cappellari | Multivariate probit regression using simulated maximum likelihood[END_REF].

Financial support from the research program "Biodiversité, gestion forestière et politiques publiques" of the "Ecosystèmes Forestiers" Public Interest Group (http://www.gip-ecofor.org/) and from the "Ministère de l'Environnement, de l'Energie et de la Mer -MEDDE" are gratefully acknowledged.


Victorien Raulot
email: raulot@lsp.u-strasbg.fr

Philippe Gérard

Bruno Serio

Manuel Flury

Bernard Kress

Patrick Meyrueis

Modeling of the angular tolerancing of an effective medium diffractive lens using combined finite difference time domain and radiation spectrum method algorithms

A new rigorous vector-based design and analysis approach of diffractive lenses is presented. It combines the use of two methods: the Finite-Difference Time-Domain for the study in the near field, and the Radiation Spectrum Method for the propagation in the far field. This approach is proposed to design and optimize effective medium cylindrical diffractive lenses for high efficiency structured light illumination systems. These lenses are realised with binary subwavelength features that cannot be designed using the standard scalar theory. Furthermore, because of their finite and high frequencies characteristics, such devices prevent the use of coupled wave theory. The proposed approach is presented to determine the angular tolerance in the cases of binary subwavelength cylindrical lenses by calculating the diffraction efficiency as a function of the incidence angle.

Introduction

Subwavelength diffractive optical elements (SWDOE) refer to diffractive optical elements (DOE) that have features sizes smaller than the wavelength of the illumination light. These elements have been studied for several years [START_REF] Farn | Binary Grating with increased efficiency[END_REF]. They have a growing interest because they can achieve high diffraction efficiency by using only two levels for fabrication [START_REF] Farn | Binary Grating with increased efficiency[END_REF]. Their "superwavelength" counterparts are diffractive elements that have lateral features larger than the wavelength of illumination. To validate our approach that we propose, we present here the simulation of a subwavelength diffractive cylindrical lens. We demonstrate its ability to focus an incident beam light as a function of the incidence angle.

The diffraction efficiency of a digital scalar DOE is related to the number of phase levels. For example, a 2-levels element, designed by using the scalar theory, can achieve a diffraction efficiency of 40.5% in both first orders. 4-levels diffractive components can reach 81% [START_REF] Kress | Digital Diffractive Optics: An introduction to planar diffractive optics and related technologies[END_REF]. The phase quantification which fits the ideal phase function is theoretically better if more levels are fabricated. Therefore, a higher diffraction efficiency becomes possible for a multilevel DOE. This kind of multilevel diffractive element is usually fabricated by multiexposures photolithography technique and requires successive expositions using a high optical precision alignment system. For instance, a 2 N -levels element involves N masking steps. Due to both these successive steps and the reducing of the size of the critical feature sizes of the successive masks, their lateral alignments become more and more critical, since a small misalignment between masks seriously affect the quality of the multilevel DOE [START_REF] Sinzinger | Diffractive Optics[END_REF].

A great interest of binary SWDOE is accounted for their fabrication method. Indeed, one single step of photolithography is required, avoiding the alignment process between masks while maintaining very good diffractive efficiencies [START_REF] Lalanne | Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff[END_REF][START_REF] Gombert | Subwavelength-structured antireflective surfaces on glass[END_REF][START_REF] Yu | Fabrication of multilevel phase computergenerated hologram elements based on effective medium theory[END_REF][START_REF] Miller | Synthesis of a subwavelength-pulsewidth spatially modulated array illuminator for 0.633 µm[END_REF][START_REF] Chen | Diffractive lens fabricated with mostly zeroth-order gratings[END_REF][START_REF] Prather | Design and application of subwavelength diffractive lenses for integration with infrared photodetectors[END_REF]. Prism-like gratings [START_REF] Lalanne | Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff[END_REF], antireflective surfaces [START_REF] Gombert | Subwavelength-structured antireflective surfaces on glass[END_REF], computer generated holograms [START_REF] Yu | Fabrication of multilevel phase computergenerated hologram elements based on effective medium theory[END_REF], array illuminator [START_REF] Miller | Synthesis of a subwavelength-pulsewidth spatially modulated array illuminator for 0.633 µm[END_REF] or diffractive lenses [START_REF] Chen | Diffractive lens fabricated with mostly zeroth-order gratings[END_REF][START_REF] Prather | Design and application of subwavelength diffractive lenses for integration with infrared photodetectors[END_REF][START_REF] Mait | Design of binary Subwavelength diffractive lenses by use of zeroth-order effective-medium theory[END_REF] designed by effective medium theory have been introduced in the past years. These elements are made up of subwavelength structures that define an effective medium. By varying the lateral feature size, the index of refraction of the effective medium, defined by the subwavelength structures, varies and creates a SWDOE corresponding to a multilevel element. In this case, light "sees" a medium containing an effective modulation of the refractive index. This modulation corresponds to a zeroth order grating which has a high efficiency in the zeroth order, and changes the phase of the incident light [START_REF] Lalanne | On the effective medium theory of subwavelength periodic structure[END_REF]. The fill factors of the structures, composing the effective medium, induce a change of the phase of the incoming light. One dimensional [START_REF] Farn | Binary Grating with increased efficiency[END_REF][START_REF] Prather | Design and application of subwavelength diffractive lenses for integration with infrared photodetectors[END_REF] and two dimensional [START_REF] Lalanne | Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff[END_REF][START_REF] Yu | Fabrication of multilevel phase computergenerated hologram elements based on effective medium theory[END_REF][START_REF] Lalanne | Blazed binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings[END_REF] surface modulations type have been successfully proposed in the literature in order to create different fill factors and, therefore, different phase modulations.

This paper focuses on a method that proceeds with one dimensional subwavelength DOEs simulations. One of their interests is to avoid the optical alignments of the successive photolithographic masks, while keeping high diffraction efficiency. Our novel approach to simulate subwavelength elements is based on a rigorous propagation of the light through a dielectric structure, using what we called a combined "Finite Difference Time Domain" -"Radiation Spectrum Method" (FDTD -RSM method) algorithms [START_REF] Taflove | Computational Electrodynamics: The Finite-Difference Time-Domain Method[END_REF][START_REF] Gerard | Towards a full vectorial and modal technique for the analysis of integrated optics structures: the Radiation Spectrum Method[END_REF]. This method enables us to describe the vectorial behavior of light over a long distance. In particular, a f/6 subwavelength diffractive lens has been designed by pulse width modulation and has been studied by the FDTD-RSM method that we propose.

Proposed combined FDTD-RSM numerical method principles

In our proposed approach, we use the FDTD method to calculate the propagation of light through the dielectric structure in the near field and the RSM to simulate the propagation of the light through free space preserving the vectorial aspects of the electromagnetic fields.

The Finite Difference Time Domain is a very well known vector-based method used to simulate the propagation of light by discretising Maxwell's equation on a grid in space and time [START_REF] Taflove | Computational Electrodynamics: The Finite-Difference Time-Domain Method[END_REF]. To achieve our calculations, we use a FDTD solver called Meep, which is an open software package with subpixel smoothing for increased accuracy, developed at Massachusetts Institute of Technology [START_REF] Farjadpour | Improving accuracy by subpixel smoothing in FDTD[END_REF]. Since the studied geometry has only one dimension, we choose to use one longitudinal coordinate for field propagation and one transverse coordinate for the definition of the profile. An advantage of the FDTD simulation is that a large range of different geometries can be simulated, if the lattice is correctly chosen. However, this method, well suited for near field calculations, is very time-consuming. The need to calculate every component of the complex electric and magnetic fields for each space and time grid point limits its use to local scale.

Since the aim is to simulate the propagation of light from the front of the lens to its focal point, which is localized far from the center of the lens, a modal method called Radiation Spectrum Method (RSM) [START_REF] Gerard | Towards a full vectorial and modal technique for the analysis of integrated optics structures: the Radiation Spectrum Method[END_REF] is used. The RSM is a modal Beam Propagation Method (BPM). It is based on the rigorous analysis of the radiation mode spectrum, and has been mainly used in guided optics (e.g. for the reflection calculation at the end of a strongly guiding optical waveguide [START_REF] Khalil | Extension of the radiation spectrum method for the reflection calculation at the end of a strongly guiding optical waveguide[END_REF]). The code source enables the determination of both electric and magnetic fields components in the free propagation space resulting from any electromagnetic excitation. Its main advantages concern both the calculation time, the radiation modes sampling conditions and the window edge reflecting problem.

Therefore, in order to make a complete simulation of a SWDOE, jointly concerning near and far fields, the modeling and simulation problem is divided into two parts with appropriate solutions. The first one uses FDTD to evaluate the electromagnetic fields near the diffractive component. The second one describes the propagation in the free space by the RSM up to the focal point of the lens, what considerably reduces the modeling and simulation computing time.

Binary subwavelength design

One characteristic of a binary subwavelength grating under normal incidence is that the only diffracted electromagnetic wave is the zeroth order [START_REF] Lalanne | On the effective medium theory of subwavelength periodic structure[END_REF]. In this case, the wave only observes a phase modification related to the geometrical parameters of the grating. In general cases, analytic methods are usually preferred to evaluate the phase difference introduced by this kind of grating [START_REF] Lalanne | Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff[END_REF][START_REF] Mait | Design of binary Subwavelength diffractive lenses by use of zeroth-order effective-medium theory[END_REF][START_REF] Lalanne | On the effective medium theory of subwavelength periodic structure[END_REF]. The drawback is that this kind of methods is limited to basic layout.

In this paper, we propose a numerical method to evaluate the phase difference introduced by a binary subwavelength grating based on FDTD. Thanks to the possibility to solve periodic boundaries conditions problems by Meep, the propagation of a monochromatic wave is calculated over a single period of the grating. From a practical point of view, we consider a TE polarized plane wave which is generated about 5 wavelengths units (roughly 3 µm) below the grating. The TE polarization stands for an electric vector that is perpendicular to the incident plane. Then, the wave propagation is computed through the grating until the steady state is reached. Finally, the phase difference is estimated for all the points located within distance ranging from 1 to 5 wavelength units away from the grating (between 0.6 and 3 µm), and is calculated by estimating the arguments of the electric complex field.

We were looking to simulate a 2 N phase levels surface spatial modulation in order to design the layout of a subwavelength lens which acts as a 2 N levels digital lens. For a dielectric grating in a transmission configuration, the curve giving the phase difference as a function of the fill factor presents a strictly positive slope [START_REF] Lalanne | On the effective medium theory of subwavelength periodic structure[END_REF]. Then, the maximum phase difference is reached for an aspect ratio value equal to 1 (i.e. a plane surface corresponding to the height level of the grating). The height of the grating h and the maximum phase difference φ max are linked therefore by Eq. ( 1):

max 1 2 1 h n λ ϕ π = ⋅ ⋅ - ( 1
)
where n is the refractive index of the medium at the wavelength λ. The diffraction efficiency of the zeroth order is defined by the ratio between the electromagnetic energy (calculated by the flux of the Poynting vector) before the grating, and the energy in the zeroth order. For simulations presented in Fig. 1 below, the considered refractive index was 1.46, corresponding to the index of the fused silica at the 632.8 nm designed wavelength that we selected (He-Ne laser). Using the FDTD method, we calculated the main parameters of the zeroth order gratings. An eight phase levels diffractive structure simulated by the effective medium method was targeted, for which the higher phase difference was 7/8 x 2π rad. According to Fig. 1, diffraction efficiency of the zeroth order remains high for gratings with period smaller than 600 nm. In order to obtain the optimum feature size for the subwavelength structure, the phase difference is plotted to link the phase level of the scalar DOE to the fill factor of the subwavelength grating, for a given period. The following graph (Fig. 2) shows the phase difference ∆φ versus the fill factor obtained by FDTD simulations, for several periods. These results are compared to the value computed from relation [START_REF] Kress | Digital Diffractive Optics: An introduction to planar diffractive optics and related technologies[END_REF] given by zeroth order Effective Medium Theory (EMT) [START_REF] Lalanne | On the effective medium theory of subwavelength periodic structure[END_REF][START_REF] Born | Optics of crystals[END_REF] where f is related to the fill factor of the grating and n to the refractive index:

( ) 2 1 2 1 fn f n ϕ π + - ∆ = - (2)
Fig. 2. Phase Difference versus the fill factor of the 1203.5 nm high grating in fused silica for several periods for a TE polarization wave, using FDTD.

For periods smaller than the third of the wavelength, a good agreement between the phase difference computed by the FDTD method and the zeroth order EMT approximation is observed. But the zeroth order EMT is only valid for grating periods smaller than the wavelength.

These results are taken into account to determine eight fill factors corresponding to the required phase differences of an 8-levels diffractive structure. However, our aim is to propose a design that can fit fabrication technologies. The period of 600 nm is only considered, even if the diffraction efficiency of the zeroth order is higher in the case of a 100 nm period grating, what is hard to achieve experimentally. This period leads us to consider the following fill factors and grating sizes, reported in the following Table 1:

Lens design method

The proposed method to design and simulate the behavior of SWDOE is explained in detail in this section. Our aim is to design a diffractive lens operating with subwavelength structures in order to use the coding scheme of the phase.

Diffractive lens layout design

The classical formulae for plano-convex lens, linking the radius of the curvature of the dioptre R, the refractive index n and the focal length f l , is used (Eq. ( 3)):

( )

1 1 1 l n f R = -⋅ (3)
The profile of the superwavelength diffractive lens is estimated by "cutting" the phase function into 2π slices with a straight forward design. The resulting profile defines a simple Fresnel lens. Once the 2π modulation has been reached, the resulting profile may be quantified into N phase levels to obtain a N-levels diffractive lens. In fact, the diffractive lens introduces a phase modulation described by a relief surface to the incident wave. As we observed in the previous section, zeroth order gratings can be considered as DOEs which modulate the incident phase. Our aim is now to change the relief modulation into a new structure which contains only binary subwavelength gratings. Figure 3 summarizes the process to design the subwavelength lens.

Electromagnetic wave propagation simulation validating the proposed approach

In order to simulate the electromagnetic behavior of the lens, the combined FDTD -RSM approach that we propose is employed, taking into account all the components of the electric and magnetic fields. Indeed, scalar simulation is not accurate enough because of the subwavelength characteristics of the element.

First, the near field simulation is done with Meep, in order to calculate the propagation of light over a few micrometers after the dielectric structure of the lens. To ensure a sufficient resolution, several samplings are tested and the results compared to each other. Once this propagation is calculated, the components of the fields after the dielectric structures are retrieved in order to become the source of the RSM program.

To calculate the propagation of light through the free space, we use the RSM method. From any electromagnetic source, the guided and radiation modes of a dielectric medium are estimated using this algorithm. In our case, only the radiation modes are considered because the propagation is done through the free space.

The diffraction efficiency η of the lens is then estimated as the flux of the Poynting vector through a surface that corresponds to the diameter of the Airy disk at the focal point. For a lens diameter D, a focal length f, a focal point located at the y f coordinate, at a working wavelength λ and at final step of the simulation t f , the diffraction efficiency is calculated by Eq. ( 4):

0 0 0 0 0 0 0 ( , , ) ( , , ) ( , , ) ( , , ) f N
u f f u f f u N v v v

E x y y t t H x y y t t E x y y t t H x y y t t

η = = = = × = = = = = × = = ∑ ∑ (4)
0.66 .

1.22 . . , 0 , , .

u RSM f f RSM f f x u x u N N D D x λ λ - = + ∆ ≤ ≤ = ∆ 0 0 . , 0 , 2 v FDTD FDTD D D x v x v N N x - = + ∆ ≤ < = ∆
where × denotes the cross product of the electric E and magnetic H vector fields, x u the u th spatial discrete coordinate x, at the focal plane, and x v the v th spatial discrete coordinate x, at the incident plane, ∆x FDTD and ∆x RSM stand respectively for the sampling steps for FDTD and RSM computation.

Angular tolerance analysis in a specific case

In this section, we present the simulation of a 6 mm focal length focussing subwavelength lenses, described in the previous section. In order to simulate rigorously the behaviour of such lenses, spatial, temporal and spectral resolutions of each simulation steps have to be small enough. Nevertheless, in order to optimize the calculation time, these values should not be too low. Different samplings for the FDTD method were tested to ensure the convergence of simulations for a computational cell size of 1 mm by 5 µm with perfectly matched layer boundaries condition. The obtained fields were compared in order to find the minimum correct sampling. A 20 nm resolution step was found as an enough value to solve properly the propagation of electromagnetic waves through the dielectric subwavelength lenses for the FDTD step, because the electromagnetic fields values were the same for smaller resolutions (mean square error less than 10% for each point). The RSM method has been implemented using a sampling step of 1.3 µm toward Y direction, what defined 800 points in the spatial domain, and therefore, 800 modes in the spectral domain so as to respect the Shannon's sampling theorem. The sampling in the X direction has been 100 µm and 150 reconstructions have been calculated to draw the Poynting vector field around the focal point. In this case, this value is high enough because the total energy is mostly preserved during the propagation, excluding losses due to the divergence out of the calculation window (more than 90% of the incident energy is conserved). Two different implementations of subwavelength lenses were designed and their optical tolerancings were studied. The first one uses 8 phase levels-simulated structures (considering the phase function previously presented in Fig. 2). The second one is a 4 levels simulated lens for which another phase function was calculated, because the maximum desired phase difference was not the same (3π/2 vs 7π/4). As a consequence, the high level value of the gratings composing this 4 levels simulated lens dropped from 1203.5 nm to 1031.6 nm.

In order to determine the angular tolerance of the proposed lenses, the diffraction efficiency was considered as a function of the incident angle of illumination. Figure 5 presents the comparison results: Figure 5 confirms that the diffraction efficiency of the 8 levels simulated lens is higher than the 4 levels simulated one. Thanks to the scalar theory, it is known that diffraction efficiency is a function of the number of levels. This relation is also verified, because the subwavelength structures are defined thanks to the complete classical diffractive profiles.

The diffraction efficiencies follow the same curve for the angular tolerance whereas the number of phase levels has changed. 65% and higher diffraction efficiencies are obtained for angles smaller than 25°. For angles higher than 40°, diffraction efficiency drop to less than 30%. As the angle increases, no significant differences are observed between the 8 levels and 4 levels cases. So increasing the number of levels is not a rewarding strategy, because the phase sampling is not improved in the case of subwavelength structures. Indeed, at worst, a 8 levels simulated lens requires 1.2 µm high gratings whose line width can be 54 nm thick (see Table 1) although a 4 level simulated lens needs only 1 µm with a 150 nm line thick. That means an aspect ratio of about 22 in the first case versus about 7 in the second case. Therefore, 4 levels seem to be a good compromise between technological constraints and optical performances with angular tolerancing.

Conclusion

SWDOEs have a great interest since they offer the possibility to reduce the fabrication process steps to one single photolithographic step, avoiding optical alignment issues. Thanks to the strong improvement of nanotechnology processes, SWDOEs could be used into micro-electro-opto-mechanical-systems (MEOMS). Therefore, specific methodologies are necessary to design on demand optic functions based on their specificities. This paper presents a new rigorous vector analysis approach fitting these requirements. The proposed algorithm enables to simulate the propagation of light over subwavelength, aperiodic and finite optical elements. The Finite Difference Time Domain algorithm is used to simulate the propagation in the near field condition, and the Radiation Spectrum Method enables the computation of the propagation in the far field. This approach has been used to study a subwavelength diffractive cylindrical lens design, implemented to simulate 4 and 8 levels of phase.

Nevertheless, because 8-levels simulated lens is more difficult to fabricate (because of a higher aspect ratio of the subwavelength gratings for which the width of the line can be smaller), 4-levels simulated lens seems to present the best compromise between good diffraction efficiency and technological constraints.

Fig. 1 .

 1 Fig. 1. a) Phase difference and b) Diffraction efficiency as a function of the period and the fill factor of a 1203.5 nm height grating in fused silica without substrate for TE polarization at 632.8 nm.

Fig. 3 .

 3 Fig. 3. Subwavelength diffractive lens designed from blazed lens Subwavelength diffractive lenses designed from N-levels superwavelength lens are called N-levels simulated subwavelength lens.

Figure 4 Fig. 4 , 4 .

 444 Fig. 4, we present amplitude of electric field E inside the dielectric structure and the recovery plane used for the RSM simulation. The Poynting vector is then carried out, step by step, with the RSM to analyse the energy propagation. A detail in the focal plane is observed on the Fig. 4. The diameter of the lens is here 1 mm with a 6 mm focal length. Computation time length of this simulation is approximately 2 hours for FDTD and several minutes for the RSM on Dual Core Pentium(R) D CPU 3.40 GHz Computer model.

Fig. 4 .

 4 Fig.4. Near electric field propagation modeling in an inhomogeneous medium by the FDTD method on the left (the subwavelength lens is located inside the dotted black line) and far field Poynting vector simulation by RSM on the right. The plane A corresponds to the incident plane for the FDTD and the plane B to the recovery plane for the FDTD and also the incident plane for RSM.

Fig. 5 .

 5 Fig. 5. Angular tolerance of subwavelength diffractive lenses

Table 1 . Fill factors and widths of the line in the gratings allowing to realize an 8 levels simulated diffractive element. The grating is 1203.5 nm high, for an incidence light having a TE polarization and a normal incidence.

 1

	Phase Difference	0	π/4	π/2	3π/4	π	5π/4	3π/2	7π/4
	Fill Factor	0	0.09	0.18	0.27	0.39	0.53	0.75	1
	Width of the line (nm)	0	54	108	162	234	318	450	600