The guiding role played by the French government in regulating cultural and artistic affairs is often regarded, within the family of western democracies, as a curiosity. 1 The institution of an official policy for culture was consecrated in 1959, in the early days of the Fifth Republic, by the creation of a Ministry for Cultural Affairs headed by André Malraux.

The fledgling administration was, however, able to draw on a long history with a substantial body of theory and legislation, and could count on a large number of cultural institutions concentrated in and around Paris. 2 Malraux had been the target of much criticism, especially from adult educationists [l'Education populaire] who disputed the elitist nature of his policies. 3 But a principal debate was triggered by the action of the left-wing government beginning in the early 1980s. Our contribution is intended to clarify the pattern of this debate from the 1980s to the present day, and to describe the main developments that have shaped France"s cultural policy. 4 A digression into historiography will help us understand how the French academic world has viewed the issues. Cultural policy studies, often government-sponsored, for a long time limited themselves to examining or assessing the socio-economic aspects of past and future policy. 5 Interest in the history of cultural policies and institutions is comparatively recent in France. 6 The social history of cultural habits is the field of action preferred by "modernist" historians. 7 Scholars of the "contemporary" period who claim for themselves the cultural history label are usually recycled political historians. [START_REF] Rioux | Pour une histoire culturelle[END_REF] The Ministry of Culture"s History Committee, headed since 1993 by Augustin Girard and Geneviève Gentil, has done much to promote research in the field by holding conferences and seminars, compiling oral records, and issuing a score of publications. The effect has been to extend the field to all the social science disciplines. [START_REF] Poirrier | For a description of the History Committee"s action, see "L"historien, la culture et les institutions[END_REF] The Cultural Imperative: A Policy Under Debate, 1981-1993: Goal-Oriented Policy When the Left came to power in 1981, a threefold break with the past occurred. The major change was the doubling of the Ministry of Culture"s budget. [START_REF]The budget[END_REF] Secondly, this change of scale was fittingly embodied in the public mind in the person of the new Minister of Culture, Jack Lang, who had the unswerving support of the President. Thirdly, the intentional linkage of culture with the economy amounted to a Copernican revolution in Socialist thinking. While every policy department benefited from these quantitative and qualitative changes, creative activities and the performing arts received more attention than the heritage sector [le patrimoine]. Further evidence of the Presidential role in cultural policy was provided by the grand-scale works program [les Grands Travaux], essentially devoted to heritage institutions. [START_REF] Ory | La décision en politique culturelle: l"exemple de la présidence Mitterrand[END_REF] The original 1959 Decree was officially amended for the first time under Mitterrand.

The text promulgated on 10 May 1982 considerably altered the tasks allotted to the Ministry of Culture:

The Ministry of Culture serves the following purposes: to let all French citizens develop their inventive and creative abilities, freely exercise their talents and receive their choice of artistic training; to preserve national and regional heritage and the heritage of the various social groups for the common benefit of the entire community; to foster the creation of works of art and of the mind and offer them the widest possible audience; to help propagate French culture and art in an unrestricted interchange of all the world"s cultures.

The idea of culture for all was replaced by that of individual self-expression through creativity, each type of culture being respected. The intentional broadening of the culture field brought recognition-although not everyone approved-to cultural activities that had previously been considered unimportant. Jazz, pop music, break dancing, and cartoon strips all became worthy objects of government cultural support. Two remarks need to be made here. This cultural relativism was not invented in the 1980s. In the heritage field [le patrimoine], for example, the Ministry of Culture had begun to include "ethnological" and "industrial" heritage in the mid-1970s. [START_REF] Poirrier | Heritage and Cultural Policy in France under the Fifth Republic[END_REF] Furthermore, the nod given to the cultural activities listed above did not basically modify the system for defining cultural legitimacy. As Vincent Dubois points out, "Quite a distance separated the "everything is culture" line and any real undermining of the established cultural ranking system." [START_REF] Dubois | Une politique pour quelle(s) culture(s)[END_REF] Still, the growing role and independence of the various cultural occupations helped to strengthen the legitimacy of newly promoted activities and instill in them their own ways of doing things. Government, moreover, was only one factor contributing to a broader transformation of the cultural landscape. Also at work were mass education, the decline of a humanities education that had provided the food for the "cultivated mind," and the rising power of the media as a standard of reference for art and culture.

The linking of culture with the economy was an equally central element. At a Mexico City conference in 1982, Jack Lang outlined his hopes for this new alliance. Commentators have put the accent on the speech"s Third World and anti-American aspects while glossing over the Minister"s new line of reasoning. Whereas Augustin Girard"s ideas on cultural industries" role in democratizing culture had raised a storm of protest on the Left, and especially among French Communists, in 1978, [START_REF] Girard | Industries culturelles[END_REF] Jack Lang could now calmly list the positive effects expected from the new collaboration: Economy and culture. The issue is the same. I should like to mention two seemingly contradictory facts in this connection. The first is that creation in culture and the arts suffers from a system of multinational financial domination that needs to be fought. The second, which seems to contradict the first, is that the international crisis can be solved by artistic and scientific creativity and invention. … There is another aspect: creativity can drive an economic recovery. … To quote Nietzsche, "Art should not be a bauble hung up here and there to look pretty." Art and creation must, on the contrary, occupy a central, and not just ornamental or decorative place in our society. [START_REF] Repr | [END_REF] Cultural policy, through support for cultural industries [les industries culturelles], took on an economic and industrial dimension. Its central administration assumed a double charge: on the one hand, it was the Ministry for artists and artistic institutions and occupations; on the other hand, it was the Ministry for cultural industries.

Lang versus Malraux

Comparison with the 24 July 1959 Decree defining the assignments of the Ministry for Cultural Affairs under André Malraux is instructive. It reveals how much government theory had changed in the space of twenty years:

The Minister in charge of cultural affairs has the task of giving the largest possible number of French citizens access to the great works of mankind, and of France in particular; of ensuring the widest possible audience for our cultural heritage; and of fostering the creation of those works of art and mind which enrich that heritage.

The implementing paragraphs of this initial Decree consecrated the ideology of equality. The all-important intention was to democratize and give equal access to culture, a policy which formed part and parcel of the welfare state. Equal treatment for all was to be a matter of practice, not only words. The goal of the welfare state was to make cultural assets equally available to everyone. That goal was expressed in two parallel policies: giving citizens at large access to works of culture; and extending welfare benefits to the artistic community.

Cultural policy was, in addition, molded by the Gaullist Republic"s ideal of modernization.

The state was required to play a driving role of guidance, incentive, and regulation. It would be a mistake, however, to think of this policy as an over-arching and carefully thought-out product of Gaullist ambition. The setting-up of the Ministry for Cultural Affairs resulted also in part from De Gaulle"s desire to keep André Malraux inside his Cabinet. In Malraux"s eyes, only art had the power to rally people within a society dominated by cold reason and machines. Having access to culture meant putting works of art and the public in direct touch with one another, in an act of revelation and communion. The stricture requiring artistic creation of the highest level shut out the adult education movement"s didacticism and amateurism. Lastly, the support given to the avant-garde was evidence of the intention to weaken the influence of the university crowd. [START_REF] Urfalino | L'Invention de la politique culturelle[END_REF] Despite its merits, the Ministry for Cultural Affairs was a fragile construction. It had been set up in difficult conditions, with a small budget and persistent feuding between the Finance and Education Ministries. The Foreign Ministry kept a tight rein over France"s external cultural policy and would continue to do so. The events of May 1968 severely shook the Ministry for Cultural Affairs, and it found itself under attack from the Left and Right.

Leftists derided the fiction of cultural democracy, while order-lovers denounced government support for artists considered as subversive. The bond between creative artists and the Ministry was torn asunder. Furthermore, at a time when mass consumption and mass culture were gathering momentum, the Ministry for Cultural Affairs was not paying the kind of attention to cultural industries [les industries culturelles] demanded by the situation. The rift with adult education [l'Education populaire] confirmed an impression of culture being reserved for an elite, out of touch with the cultural habits of ordinary French people. [START_REF]La Culture de masse en France: de la Belle Epoque à aujourd'hui[END_REF] The Ministry, still strongly centralized and opposed to decentralization, was thinly present across the country, even though the few cultural centers [les maisons de la culture] that had been built were beginning, not without difficulty, to establish a partnership with the larger towns. The 1970s were turbulent times for the Ministry. Georges Pompidou, who did not want to be hemmed in by his strictly Presidential functions, became a new actor on the scene. [START_REF] Grohens | Culture et action chez Georges Pompidou[END_REF] In spite of the instability of the era, the Ministry"s structure was given permanent form. The idea of "cultural development" provided a response to the destabilization fomented by the events of May 1968. The Duhamel ministry (1971-73) attempted to modernize cultural policy by working on an inter-Ministerial basis, obtaining a substantially bigger budget and acknowledging the importance of local governments. "Cultural development," [le développement culturel] the third component of Prime Minister Jacques Chaban-Delmas"s "New Society," expressed a philosophy in sharp contrast with André Malraux"s cherished doctrine of cultural action [l'action culturelle]. While the goal of democratizing "culture" remained, two fresh approaches modified the concept itself. The anthropological sense of the word replaced the usual meaning of "high" culture; and the admission that many paths lead to democratization replaced the "esthetic shock" principle. The state, though, continued to occupy a leading role. Its task was to foster the propagation of culture and refuse market-economy rules. Although the Duhamel ministry"s policy sowed seeds for the future, it was seriously affected by the political misfortunes of Jacques Chaban-Delmas. [START_REF]Les Affaires culturelles au temps de Jacques Duhamel: 1971-1973[END_REF] The more economically liberal philosophy that prevailed after 1974 led to a winddown in state funding, and President Valéry Giscard d"Estaing showed less passion for cultural policy. The public service vocation of the Ministry lost strength as liberalism took hold. The period was notable for the growth of urban cultural policies, varyingly in line with state directives. By the end of President Giscard"s seven-year term, the state"s guardianship of culture had greatly weakened, and it found itself confined essentially to heritage [le patrimoine] protection. Cultural policy, much to the discontent of cultural professionals, was no longer a government priority. Meanwhile, the opposition, and especially the revitalized Socialist Party, was offering a different sort of program. The Socialist culture platform had two highly political purposes: first, to pry the people working in the cultural field away from the still-strong influence of the French Communist Party and, second, to rally these people behind Presidential candidate François Mitterrand, at a time when the polemic over "totalitarianism" had estranged the intellectuals from the Socialists. [START_REF] Gremion | Ecrivains et intellectuels à Paris: une esquisse[END_REF] At the end of Giscard"s term, the culture-state was a mere shadow of the "juggernaut" excoriated by its foes. Under the Ministers Françoise Giroud, Michel d"Ornano, and Jean-Philippe Lecat, the "system" dreamt of by Malraux was quietly but purposefully dismantled.

Public Media: A Defeat for the Ministry of Culture

The public media scene also went through a revolution in the 1980s. After Canal + (encrypted pay-TV) and two private channels (La Cinq and TV6) had been created and Channel One was privatized in 1986, competition among the several channels intensified. The Ministry of Culture was unable to impose its views. Jack Lang spoke out vigorously, but in vain, against the handing over of La Cinq to the Seydoux-Berlusconi consortium. The film industry also opposed this choice, which ushered in a television service subordinated to the rule of money. Silvio Berlusconi, the Italian TV magnate, was seen as the incarnation of lowgrade television consisting of variety programs, game shows, and sitcoms. What the Minister of Culture did obtain was a European TV channel focused on culture. The idea he proposed in 1984 grew into "La Sept" (February 1986), later to become Arte (Association relative à la télévision européenne). Arte, a Franco-German cultural channel broadcasting on the open airwaves since 1992, was something of an odd-man-out in a media world now dominated by commercial thinking. Certain commentators deplored the institution of a cultural "ghetto," which would serve as a perfect pretext for commercial abuses by the other public channels. [START_REF] Wolton | Télévision culturelle: "l"apartheid" distingué[END_REF] In 2003, the philosopher Catherine Clément, author of a report for the Minister of Culture, was obliged to acknowledge a state of affairs in which commercial pressures had been aggravated over the previous twenty years. Television had become French people"s main cultural practice, but programs labeled as "cultural" were consigned to non-prime time-"at night or in summer." [START_REF] Clément | La Nuit et l'été: rapport sur la culture à la télévision[END_REF] French intellectuals-using the term in its broad sense of members of the intellectual professions-had for a long time had mixed feelings towards the audiovisual media, especially television. They were divided between condemning the propagation of a mass culture that watered down real culture [la culture légitime], and wanting to use the media for popularizing that same culture. [START_REF] Hélène | Médias audiovisuels et intellectuels[END_REF] In academia, the cultural purism of French scholars had blocked the inclusion of these new issues in social science studies. British cultural studies, initiated outside of university circles, had been much bolder in exploring the subject.

The Lang ministry had nevertheless thrown all its weight into binding the film and television industries together. This singular combination had been in the works since the early 1970s. Under the Act of 29 July 1982, television was constrained to accept a certain quota of French movies, and the way in which the different channels could broadcast outside-produced movies was defined in detail. The creation of the pay-view channel Canal + in 1985 provided the movie industry with a solid source of funds. Canal +, in exchange for an exceptionally generous operating charter, was to occupy a key position in French movie production for more than ten years. Television became the cinema industry"s main funder and production outlet. Movie theater screenings were looked upon only as showcases and launch pads. In 1997, the creation of the TPS satellite bouquet spelled the end of the Canal + monopoly. A third party, the communication industry, with its trans-frontier outlook and strategies, was about to upset an already delicate balance. [START_REF] Creton | Le Cinéma à l'épreuve du système télévisuel[END_REF] Criticism from the Intelligentsia:

From Cultural Pessimism to the Contemporary Arts Crisis

The main charges made by the intelligentsia bore on cultural relativism. The publication of Alain Finkielkraut"s La défaite de la pensée in 1987 unleashed a storm of controversy. The philosopher and essayist lamented the decline of culture. His polemic was, in fact, an attack on cultural laxity. It is true that the policy conducted by Jack Lang since 1981 and continued in its general outlines by François Léotard in 1986 under the "cohabitation" government was not the principal subject of debate. Alain Finkielkraut was nonetheless of the opinion that the policy had, by incorporating fields hitherto ignored by the Ministry (rock music, fashion, advertising, etc.), done much to dilute culture in a sort of everything-goes cultural broth. [START_REF] Finkielkraut | La Défaite de la pensée[END_REF] Bernard-Henri Lévy, in his Eloge des Intellectuels (1987), wrote in a similar vein. Lévy, the standard-bearer of the nouveaux philosophes, mentioned the "sick spirit in culture." The Ministry of Culture was accused of compounding the sickness by giving respectability to the "minor" fringes of culture. [START_REF] Lévy | Eloge des intellectuels[END_REF] Perhaps it was a sign of the times but, the same year, the French translation of The Closing of the American Mind by American scholar Allan Bloom was published. The book criticized the United States education system which, in incorporating minority culture, was responsible for muddying values. The more or less simultaneous publication of the three books-and the stir created by the first twoprovided theoretical ammunition for launching attacks against the Ministry. The subject of the dispute was not, however, multi-culturalism. In this respect, the French debate differed considerably from the one going on in the United States and Canada.

In 1990, the Ministry published a survey on the French population"s "cultural practices" that rekindled and expanded the debate. [START_REF] Donnat | Les Pratiques culturelles des Français[END_REF] The survey conclusions, which revealed the persistence of material and symbolic barriers restricting access to "high" culture, were interpreted as proof that cultural democratization had failed. Minister Jack Lang was shaken by the findings and by the way the survey was exploited, considering that the government had committed itself to an extremely bold policy over the previous ten years. Sociological studies, which had for a long time been a valuable adjunct to democratization policies, were now feeding the skepticism of the foes of the state as cultural arbiter.

In the Fall of 1991, the publication of Marc Fumaroli"s L'Etat culturel gave a new twist to the controversy. The French press, soon followed by television and radio, gave wide coverage to the argument generated by the author"s ideas. Two of France"s most eminent intellectual magazines, Esprit and Le Débat, gave space to partisans on both sides of the fence. The public at large, especially after all the media treatment, also showed interest in the subject. The book remained on top of the specialized press" best-seller lists for many weeks.

It needs to be said that the author of L"Etat culturel is a member of one the most honorable of all French academic institutions, the Collège de France. Fumaroli, a historian of literary and artistic forms in modern Europe, has held the Chair of Rhetoric and Society in sixteenth and seventeenth-century Europe since 1986, and is the author of a series of works respected by the international scholarly community; they include L'Age de l'éloquence (1980) and Héros et orateurs, rhétorique et dramaturgie cornéliennes (1990). Little known by the public at large for his academic work, Marc Fumaroli chose to tackle the subject of cultural policy in contemporary France by writing not a scholarly treatise but a lively broadside. His academic standing indeed carried more weight than the style he chose to employ. He is considered, wrongly, by many critics as a reputable specialist on the issue under discussion.

His argument should not, however, be dismissed lightly. Cultural policy, he argues, has transformed the state into a mass supplier of "popular recreation" and "consumer products." This makes the welfare state, using public money, a competitor in the culture market. The conflict of interest inherent in an everything-goes culture stems from confusing cultural action with a form of tourism. Culture of this kind, spread by a growing army of cultural bureaucrats, takes on the proportions of a modernist religion. The nationwide music festival held annually on 21 June [la Fête de la Musique], the Fureur de Lire, commemorative celebrations (notably the Bicentenary of the French Revolution in 1989), along with the Presidential "Grands Travaux," are manifestations of state-inspired manipulation, the state being a neo-Leviathan of culture serving a political party and an ideology. What is not said is that this "culture state" [l'Etat culturel] is the product of policies initiated well before the Socialists came to power in 1981. As a worthy disciple of the political philosophy of Tocqueville and Raymond Aron, Marc Fumaroli assumes the role of preacher for a "liberal" state that would equip our leisure and consumer societies with a certain number of firewalls-principally an education system and some legal and tax safeguards. Culture in this scheme of things would become an essentially personal matter.

"The arts are not dishes that can be divided into an infinite number of equal portions," he writes. "They are the rungs of an ascent; to be desired, not meted out." In other words, the main enemy of liberal democracy is mass culture. Marc Fumaroli"s book may be seen as a weather-vane, symptomatic of the uncertainty concerning the place and conception of culture in today"s societies. It also expresses the will to defend the fine arts and the humanities against the ravages of mass culture. The work thus belongs to a long-established polemical tradition, even though its media exposure was exceptional. Indeed, the ideas are not new, and Fumaroli himself aired most of them in a 1982 article in the magazine Commentaire. But their extensive popularization in France and abroad gave them an added impact. Although some historians seriously question the sequence of events recounted by Fumaroli, certain media reproduced his thesis without comment. The novel element was the fierce contestation of the Ministry of Culture"s very legitimacy. Marc Fumaroli refrains from demanding the complete dismantling of the Ministry, but he argues for its being limited to the heritage field [le patrimoine]. The success of L"Etat cultureltwenty-three thousand copies sold in the original printing and over twenty thousand in the paperback edition-probably amplified by the disputes over the Bibliothèque Nationale de France and the run-up to the March 1993 parliamentary elections, revivified an already simmering debate. The book"s success was further evidence that the cultural policy propelled by Jack Lang was losing steam, and this at a time when the second seven-year term of François Mitterrand was marred by a spate of scandals. Lastly, the confirmation in 1991 of a darkening in the economic and social forecast made it difficult to defend a cultural policy based on support for creative artists.

In 1992, L'Utopie française: essai sur le patrimoine, by Jean-Michel Leniaud, with a preface by Marc Fumaroli, proved that the debate was still very much alive. [START_REF] Leniaud | L'Utopie française: essai sur le patrimoine[END_REF] With the Presidential elections only a few months away, books by Henry Bonnier-Lettre recommandée à Jack Lang et aux fossoyeurs de la culture-and Zadig, a pen-name for a group of intellectuals belonging to right-leaning university clubs-L'Implosion françaisegave extreme and somewhat over-simplified form to the ideas of Marc Fumaroli. [START_REF] Bonnier | Lettre recommandée à Jack Lang et aux fossoyeurs de la culture[END_REF] The goal was perfectly clear: suppression of the Ministry of Culture. Zadig promptly listed the authors who were required reading: "It is vital to culture that those who tomorrow will hold the destiny of France in their hands should have read and pondered Steiner, Bloom, Lussato, Finkielkraut, Fumaroli, on the return of true culture."

As 1993 dawned, a critique not only from the nominal Left but from within the cultural administration fold itself, joined the chorus of protests from the "liberals" close to what was then the political opposition. Michel Schneider, a senior civil servant, Director of Music and Dance at the Ministry of Culture from 1988 to 1991, an authority on Schumann and Glenn Gould, conceded to Marc Fumaroli a number of points: the dissolution of Art in "Culture"; the political exploitation of the arts by a Ministry turned towards the "creators"; the disproportionate attention paid to media use. In his La comédie de la culture, Schneider countered the free-choice solutions proposed by Marc Fumaroli with the necessity for public service, i.e., the obligation for a democratic nation to provide education in the arts in order to attenuate unequal access to works of culture. This vital education function would round out a Ministry of Culture limited to two other duties: heritage protection and the democratic propagation of art. The book-regarded by some as a settling of scores-received plentiful coverage in the national and regional press. It also was a resounding public success, with over thirty-two thousand copies sold. [START_REF] Schneider | La Comédie de la culture[END_REF] A year later, Culture et contre-cultures, by the jurist Jean-Louis Harouel, revisited the subject, but in a broader framework. Harouel wrote of the "decline" in Western culture dating from the advent of avant-gardism up until the alleged clash with the lower forms of "counter-culture." The author tied up his argument with the conclusion that the Ministry of Culture should be scrapped and replaced with a Ministry for Heritage flanked by a small-scale national arts fund. [START_REF] Harouel | Culture et contre-cultures[END_REF] A second line of criticism, again well-publicized by the media, had to do with the Grands Travaux. Jean-François Revel attacked them as "the President"s private playground" and held them up as an example of what his essay denounced as "inefficient absolutism." [START_REF] Revel | L'Absolutisme inefficace ou contre le présidentialisme à la française[END_REF] What struck many observers were the personal nature of Grands Travaux policy decisions, their attachment to a political agenda, and their staggering cost. The three factors kept the controversy alive. Once his decision was taken, the President kept a close eye on each project"s progress.

The opponents of the Grands Travaux castigated the fortunes spent on the different programs. Le Canard Enchaîné called them Pharaonic and derided the choices of "Tontonkhamen." [START_REF]the popular nickname for François Mitterrand[END_REF] Over a period of ten years, the "Grand Works" ended up costing roughly thirty-four billion francs. More than the initial outlay, it was the operating budget needed for running the new cultural facilities which was to cause a problem in the following years.

408

Although a few projects were located in the provinces, local officeholders berated a policy which, as always, mostly benefited Paris. The Grand Louvre, later acclaimed as a model of its kind, was at first the subject of heated During François Mitterrand"s second seven-year term, it was the new Bibliothèque Nationale de France [BNF] which set passions aflame. This time, the opposition came less from political factions, as had been the case with the Grand Louvre, than from the scientific community. The supervising team, the architect, Dominique Perrault, and the prospective scientific users argued at length over the functions and shape of the future building. The researchers organized themselves into a pressure group, with the journal Le Débat playing an active role, and obtained significant alterations to the original plans. The dispute reshaped the intellectual landscape, one of its side-effects being to harmonize the positions on cultural matters of the magazines Commentaire and Le Débat. Two different conceptions of the future library, which many commentators believed to be incompatible, underlay the divergence: it was conceived, on the one hand, as a public library offering a sort of agora for cultural interaction, and, on the other hand, as a new Bibliothèque Nationale with a primarily heritage mission catering to scholars. This twin purpose, set out in the Decree of 3 January 1994 defining the BNF"s tasks, was to prove extremely difficult to manage institutionally in terms of everyday running.

Cost of the

Beginning in 1991, another polemic hatched up by the magazines Esprit and Télérama on the "crisis in contemporary art" had an influence on the Ministry of Culture"s fine arts policy. The nub of this very "in-house" French debate turned on an attack against avant-garde aesthetics. Government policy was cited as evidence for the prosecution; the state was accused of aesthetic partiality. It has always been tempting to belittle "official art."

The neo-liberals Marc Fumaroli and Yves Michaud believed that the state should retire from the field and allow more independence to other institutions. Philippe Dagen was one of the few protagonists to state that government support for the fine arts was still inadequate. He did not deny that a certain official academicism existed, but he felt that it was of minor importance and was not sure that the market would fill the void if the state pulled out. [START_REF] Dagen | La Haine de l'art[END_REF] The debate took a more political turn in November 1996 when an issue of the review Krisis, sympathetic to the far-right, edited by Alain de Benoist, expatiated on the subject. Upholders of contemporary art, such as the magazine Art Press, edited by Catherine Millet, or Le Monde"s art critic, Philippe Dagen, quickly assimilated the attack on contemporary art with conservative reaction and fascism. Admittedly, the political context of the day, marked by the gains of the National Front in town council elections, tended to make parties on all sides exaggerate their positions. [START_REF] Michaud | The arguments used in the dispute are[END_REF] The three controversies-cultural relativism debates (Fumaroli), the Grands Travaux, and the "crisis of contemporary art" controversy-crisscrossing from time to time, were instrumental in casting doubt on cultural policy"s purpose and legitimacy. But their impact, outside a small coterie of intellectuals and specialists, should not be overstated. The artists themselves had no say in the proceedings. Polls bore out the fact that the great majority of the French population did not contest the government"s cultural policy. That policy, especially from 1993 on, was basically one of continuity. The fact remains that the principles used to adjust cultural policy to a gloomier financial outlook and the onslaughts of globalization were grounded partly in argumentation heard during the 1980s.

Towards a New Cultural Policy Model, 1993-2003:

Cultural Policy and "Social Fracture"

The return to power of the parliamentary Right in 1993 ushered in a new period.

Cultural policy was definitely no longer a government priority, and Presidential backing was less in evidence after Jacques Chirac"s election in 1995. The trimming of the budget was felt all the more keenly in that the overheads of the Grands Travaux monuments and institutions made heavy inroads into the Ministry of Culture"s finances. The desire for a new cultural policy model was illustrated by the idea of "reformulation" [la refondation] elaborated by Jacques Rigaud for Minister Philippe Douste-Blazy in 1996. Discussion had shifted to mending the "social fracture." Cultural policy at the approach of the twenty-first century seemed destined to take a more "liberal" turn.

After the 1993 parliamentary elections, the Ministerial policy pursued by Jacques

Toubon was quick to follow the path traced by his predecessor Jack Lang. The first cohabitation, from 1986 to 1988, had to a large degree remained faithful to the policy in place since 1981. François Léotard, who advocated a "retiring state," to use Michel Crozier"s term, had sought to push cultural policy in a more rightwards direction. The emphasis was placed back on heritage; education in the arts was promoted; and "cultural development" lost its priority status. In 1993, the Ministry of Culture adopted three guiding principles: territorial development (one of the avowed hobbyhorses of Prime Minister Édouard Balladur"s government), intensified projection of French culture abroad. and education and promotion of culture for all. The debate in the Fall of 1993 over Europe"s right to "cultural exception" forged broad agreement among the public and politicians alike. The Minister of Culture drew attention, over and above the technical issues and the trade imbalance between Europe and the U.S.-in the latter"s favor-to broader questions of symbolic importance and the defense of a cultural identity under threat:

We do not want our spirits to be stifled, our eyes blinded, our industries enslaved. We want to breathe freely, to breathe our own air, the air that has vivified every culture in the world, and could tomorrow be lacking to mankind. On both sides of the Atlantic and the Pacific, let us mobilize for this battle for survival. [START_REF] Toubon | Laisser respirer nos âmes![END_REF] The return of the Right, in other words, picked up the thread of the earlier concerns of the Left. There was no longer any question of a financial upturn or a priority proclaimed by the government. Moreover, the absence of Presidential backing, combined with the traditional stinginess of the Ministry of Finance, weakened the Ministry of Culture"s position. In retrospect, the two major advances of the 1980s could be better appreciated: the budget increases won for the Ministry by Jack Lang, and the backing, sometimes distant but unfailing, of President François Mitterrand.

The 1995 Presidential campaign was not concerned with cultural policy. Philippe Douste-Blazy, the Centrist Minister of Culture in Prime Minister Alain Juppé"s cabinet, ordered a remodeling of the Ministry"s policy. The deliberations were the first step towards establishing a new type of policy for culture that would take account together of the advances made in the previous decade, the criticisms that had been expressed, and the international economic situation as affected by rising globalization. The Rigaud Report submitted to the Minister in October 1996 reasserted the validity of the French system of cultural action as a public service. The state"s role as an instrument for government action, in consultation with local bodies, was firmly asserted. The Ministry"s financial muscle should be restored, and the Report noted the Ministry of Finance"s mistrust, if not downright hostility, towards government spending on culture. There were several suggestions for reorganizing the Ministry of Culture"s central administration. Education in art and culture was termed a "national cause." Another priority was the harmonization of state policies concerning cultural industries. [START_REF] Rigaud | Pour une refondation de la politique culturelle[END_REF] The "remodeling" process was brutally interrupted in the Spring of 1997, when President Jacques Chirac chose to dissolve National Assembly. 39

Restoring the Republican Social Pact

Catherine Trautmann, Minister of Culture and Communication in the new cabinet headed by Socialist Lionel Jospin, tailored her action to the pattern devised by her

predecessor. An administrative overhaul was accompanied by a "reconstruction" budget (15.1 billion French francs in 1998). Further deconcentration and the signing of contracts with local bodies showed that the Ministry was willing to take part in the reform of the state"s structure.

A "Charter on the Public Service Duties of the Performing Arts" stressed the need to write public service clauses into contracts covering public aid for cultural institutions. The Charter was a genuine attempt to break with past practices. In the same vein, a charter signed in June 1999 with the main adult education federations endeavored to end the divorce, dating back to Malraux"s time, between cultural action and further education. 40 The new partnership satisfied two of the Ministry of Culture"s primary objectives: to strengthen education in arts Education. [START_REF] Lismonde | Les Arts à l'école: le Plan de Jack Lang et Catherine Tasca[END_REF] The Ministry of Culture, having noted the nationwide proliferation of locally based artistic endeavors that owed little to institutions or the market, set about alerting public authorities to the scale of the phenomenon and introducing measures to support these ventures, since they promised to play a major role in local cultural life and the development of art activities. [START_REF] Lextrait | Les valeurs de l"art: entre marché et institutions[END_REF] The media scene continued to be occupied by the war between haters and admirers of cultural policy à la française. [START_REF]Maryvonne de Saint-Pulgent[END_REF] In the field, the prevailing discussion had turned to the new problems, nearly all relating to globalization, facing French cultural policy. The sharp debate over paying for borrowing in public libraries, which had the book trade divided, showed how much the public cultural services were being buffeted by the commercial logic of increasingly trans-national cultural industries-television in particular-and the globalized habits of individual cultural consumption. [START_REF] Mollier | Où va le livre?[END_REF]

From Cultural Exception to Cultural Diversity

During international trade talks, France did not budge from its position and shaped the As we can well see, in these times of uncertainty, when traditional borders are becoming blurred, when there are justified fears that the rule of the market will replace the rule of governments, when people no longer feel that they are in charge of their agenda, the need to assert one"s identity is making a strong comeback all over the world. It is like a vital instinct, since everyone feels that the snuffing out of traditions, languages, cultures and knowledge is akin to throwing away chances, or closing doors that we shall never be able to open again. Let us then be vigilant. Uniformization, when it acts like a steamroller, has the backlash effect of encouraging ghetto attitudes and fanaticism. Affirming diversity is quite the opposite: it is one the keys to assuaging contemporary fears. … That is why not everything can be a tradable good, and why not everything can be abandoned to market rule.

Until the official opening of the Presidential election campaign, the battle of views swirled.

On 9 March, certain artists, incensed by business mogul Jean-Marie Messier"s statements, took advantage of the Victoires de la Musique ceremony to make their opinions known.

Messier had quoted as an example of cultural diversity the inclusion of "rebels" such as Noir references to the limits of cultural democratization. Cultural institutions were blamed for having deepened the cultural divide between the "cognoscenti" and the "people" because of their highbrow policies.

The appointment of the former Curator of the Georges Pompidou Center, Jean- National Assembly that reform of patronage and the status of foundations was a necessity, "expressing, alongside deconcentration and decentralization, the government"s trust in an assumption of responsibility by civil society, individuals, business institutions and foundations, since it aims at associating their commitments more closely with the action of the state." He drew attention to France"s backwardness in this respect, comparing the situation to that in the United States where patronage contributed roughly two hundred and seventeen billion euros, or 2.1 percent of GDP, as against only 0.09 percent in France.

These first measures produced mixed reactions. Debate focused on two topics. The first was the question of non-salaried performing artists. [START_REF]Unemployment insurance thus indirectly funds the performing arts sector. 48 Periodicals have reflected this fear: "Quelle culture de la décentralisation?[END_REF] Employers and unions asked questions of the government concerning the special unemployment insurance scheme applying to performing arts professionals. At stake was the survival of a large section of the country"s cultural activity. The Minister of Culture, who expressed his attachment to the scheme, failed to convince labor representatives opposed to the employers" demands that he had the capacity to act. The second topic concerned the decentralization process set in motion by Prime Minister Jean-Pierre Raffarin. The Prime Minister, a proponent of a "decentralized France," hoped to strengthen the cultural prerogatives of the regions. Art and culture professionals, especially in the performing arts, criticized a possible divestment by the state at the very moment when many urban cultural systems were being threatened by the reluctant attitudes of certain local authorities. 48 Local officeholders themselves appeared afraid of a transfer of expenditures in their disfavor and a retreat by the state towards institutions having exclusively national stature. Some voices pointed out that the government"s program remained silent about other projects in the works, such as "intercommunality" (partnerships among several Communes), particularly in rural districts, of "home areas" and "settled communities." The hesitancy could also have been a sign of the local authorities" difficulty in assimilating and practicing the new rules of the game imposed by the growing localization of government action. [START_REF] Saez | L"action des collectivités territoriales en matière culturelle[END_REF] The Ministry of Culture"s own staff began to worry. The "culture" interunion group at the Ministry of Culture issued a strike call for 24 April 2003 to protest what it called a "scheduled dismantling of the Ministry" by a "bogus decentralization which does not pursue any goal of improving and consolidating public service."

The government continued, however, to present state commitment to art and culture as a fundamental requirement of democracy. The primary objective remained the popularization of cultural habits. It was equally important to guarantee maximum freedom for the citizen in his choice of cultural activities. To that end, the state has proclaimed its determination to support artists, while ensuring them absolute freedom in their work of creation and dissemination. A seeming contradiction may be noted here. The defense of cultural diversity in the international arena has been accompanied domestically by an avowed "liberal" shift, which many observers have seen as a serious threat to the public service cultural ideal.

For nearly half a century now, government cultural policies in France seem, despite certain changes in trajectory, to have followed a continuous path. There can be no denying their role in regulating the national culture scene; yet this can in no way be likened to a "straitjacketing" of the country"s art and culture. The small size of official budgets, the growing influence of world-scale cultural industries, and highly eclectic cultural habits have prevented the state from becoming the string-puller of France"s cultural life. Arguing over cultural policy has never really stopped since the early 1980s, from the condemnation of the all-is-culture credo to criticism of decentralization, experienced by art and culture professionals as a whittling away of state-guided policy. In truth, these disputes have had little impact on the nature of public policies as practiced, and the public in general has always seemed quite happy with them. At the beginning of the twenty-first century, one trend seems to be emerging: criticism is giving way to defense of state policy, recognized as a bulwark against the erosion of cultural identities and the resurgence of populist philistinism.

  Source : Philippe Poirrier, « French Cultural Policy in Question, 1981-2003 » dans Julian BOURG (Dir.), After the Deluge, New Perspectives on Postwar French Intellectual and Cultural History, Lanham Md., Lexington Books, 2004, p. 301-323. French Cultural Policy in Question, 1981-2003 Philippe Poirrier

  controversy. The plans of American architect Ieoh Pei for the Louvre Pyramid stirred intense debate. Among those taking sides were former Minister of Culture, Michel Guy, and the authors (Bruno Foucart, Antoine Schnapper, and Sébastien Loste) of Paris mystifié, published in 1985. The favorable opinion expressed by the curators of the Louvre in 1984 was a deciding factor. With the agreement of the Mayor of Paris, Jacques Chirac, and the unwavering support of François Mitterrand, the team led by Emile Biasini was able to carry the project through to completion. The violent debate over the daring construction of a glass pyramid at the heart of the old Louvre palace buildings was soon forgotten when its public success became apparent.

  and culture, and to encourage amateur artistic activity. On the political far-right, populism was again on the rise. Catherine Trautmann, like Douste-Blazy before her, took action to assert the state"s role in preserving cultural pluralism. The authority of the Minister, who had tried to escape from interest-group domination, was irremediably sapped, however, by pressure from the performing arts lobbies, abetted by the national press.Catherine Tasca, who took over from Catherine Trautmann following a Jospin cabinet shuffle in March 2000, geared her action to three main principles: promoting cultural diversity, improving access to culture, and advancing cultural decentralization. The political background was also of great importance-with an eye to the next elections, the Minister had to defuse the rebellion of the performing arts movement leaders. The Ministry of Culture"s participation in the new round of decentralization demanded by the government was expressed through a set of "cultural decentralization protocols." One of the protocol"s goals was to establish a clear apportionment of responsibilities between central and local government. The late parliamentary term also saw the conclusion of several long-awaited legislative measures, including those on rescue archeology, museums, and the art market. The intention to support education in the arts, which politicians had been talking about since the early 1990s, led to closer cooperation between the Ministry of Culture and the Ministry of

Grands Travaux 34

	Name	Architect(s)	Cost in billions	Year
			of French francs
	Musée d"Orsay	Gae Aulenti	1.3	1986
	Parc de la Villette	Bernard Tshumi	1.3	1986
	Musée des Sciences	Adiren Fainsilber	5.4	1986
	Institut du Monde Arabe Jean Nouvel	0.424	1987
	Opéra Bastille	Carlos Ott	2.8	1989
	Arche de la Défense	Paul Andreu,	3.175	1989
		Otto von Spreckelsen
	Ministry of Finance	Paul Chemetov,	3.7	1989
		Borja Huidobro
	Cité de la Musique	Christian de Portzamparc 1.1	1994
	Muséum	Paul Chemetov,	1	1994
		Borja Huidobro
	Grand Louvre	Ieoh Pei	5.7	1995
	Bibliothèque de France	Dominique Perrault	7.8	1995
	Total		34.239

  European Union"s attitude. In 1998, in negotiations with the Organization for Economic Cooperation and Development over a Multilateral Investment Agreement, France pleaded for what it called "cultural exception." The Jospin government decided to oppose any inclusion the policy of Canal +, part of Messier"s empire, which was a vital partner in French movie production.The official reaction "at the state"s highest level" shows just how important this question was. Prime Minister Lionel Jospin, who had already raised the point in presenting his greetings to the press on 15 January 2002, took the opportunity six days later when inaugurating the Palais de Tokyo site for contemporary creation to restate the government"s position. He implicitly condemned Jean-Marie Messier"s remarks:Our vision in no way implies retiring into our national shell; on the contrary, it betokens a desire to reach out to the world. Through culture, we celebrate all cultures. Their diversity is mankind"s priceless heritage.

President Chirac, for his part, formally confirmed France"s position regarding cultural diversity, on February 12th:

  Jacques Aillagon, as Minister of Culture and Communication in the Jean-Pierre Raffarin Cabinet was on the whole well-received by arts professionals. The text setting out the Minister"s terms of reference indicated that there would be no break with previous policy.Still, the reduction in the culture budget (2,490,72 million Euros, or a 4.3 percent cut) was perceived in culture circles as a worrying sign. Several priorities were set, backed by official

reports: heritage and decentralization (Bady Report), violence on television (Kriegel Report), and cultural content on television (Clément Report). The Ministerial agenda displayed right from the first months an intention to encourage private patronage and allow greater independence to the major cultural institutions. A symposium held in March 2003 by the French-American Foundation and the Centre Français des Fondations under the auspices of the Minister of Culture and the Minister for the Budget and State Reform-"Culture Funding and Management in the United States and France: New Synergies and Interdependences of Private and Public Sources"-produced arguments for a shift towards liberalism in France"s cultural policy model. In April 2003, the Minister of Culture and Communication told the

of a culture clause and refused to take part in the negotiations. In 1999, on the eve of the World Trade Organization [WTO] meetings in Seattle, France strove in the European Union to retain its "cultural exception." The Commission"s mandate, approved by member states on 26 October 1999, endorsed the French position: "The Union will, during the forthcoming WTO negotiations, take care to guarantee, as in the Uruguay Round, the possibility for the Community and its member states to preserve and develop their capacity to determine and implement their cultural and audiovisual policies so as to conserve their cultural diversity." Catherine Trautmann added, "Cultural exception is the legal tool; cultural diversity is our objective." This semantic shift enabled a frail consensus to be reached within the European Union. "Cultural diversity" continued to be defended. On 15 October 2001, President Chirac in his speech at the opening of the Thirty-First UNESCO General Conference solemnly reiterated France"s position: "The answer to culture-crushing globalization is cultural diversity. Diversity based on the conviction that each people has a unique message to transmit to the world, and that each people can enrich humanity by contributing its share of beauty and its share of truth."

The shift towards cultural diversity was not universally appreciated by art and culture professionals. The exact meaning of the phrase remained highly ambiguous. On 17 December 2001, when taking control of USA Networks, Jean-Marie Messier, then CEO of Vivendi-Universal, declared in New York: "The French cultural exception is dead … and domestic French anxieties are archaisms." The declaration quickly had culture professionals up in arms; they denounced what they saw as an attack on the French system for funding movie-making. Politicians of every stripe-from the Communist Party to the far-right-also castigated Messier"s pronouncement. Everyone was wondering in private what would now be Désir and Zebda in the catalogue of the music major he had just acquired. From the mid-1990s, the government received support from the main professional organizations as well as from the associations opposed to globalization. The sociologist, Pierre Bourdieu, who had become a new icon of intellectual activism, on several occasions warned against reducing culture to the status of merchandise, "To reintroduce commercialism into systems crafted, piece by piece, … is to endanger mankind"s noblest achievements: art, literature, and even science." [START_REF] Bourdieu | Questions aux vrais maîtres du monde[END_REF] Jacques Chirac, following his reelection as President in 2002, returned more than once to the theme of defending cultural diversity, increasingly presented as the backbone of France"s cultural policy. France also did its best to change the forum of discussion from the WTO to UNESCO. 46 On 2 February 2003, President Chirac, speaking at the Second International Meeting of Cultural Professional Organizations, proposed "establishing [cultural] diversity as a principle in international law" and called for the "adoption by the international community of a world convention on cultural diversity." France also wanted "promotion of and respect for cultural diversity" to be included in the future Treaty on European Unity.

A Model in Search of Its Bearings

The election campaign in the Spring of 2002 was not notable, however, for the public attention accorded to cultural policy, although all the candidates mentioned it in their platforms. The qualification of the populist leader, Jean-Marie Le Pen, for the second round of the Presidential elections triggered an "anti-Fascist" reflex on the part of the art and culture world. In the following weeks, the discussions on the condition of French society included

Notes

Our gratitude goes to the Ministry of Culture"s History Committee for financing the translation, and to Augustin Girard and Julian Bourg for their thoughtful revision of the French text.


Christyves Chevallier
email: christyves.chevallier@supelec.fr

Frédéric Genty

Nicolas Fressengeas

Joel Jacquet

Robust design of subwavelength grating mirror for mid-infrared VCSEL application

est destinée au dépôt

Robust design of subwavelength grating mirror for mid-infrared VCSEL application Christyves Chevallier * † , Frédéric Genty * † , Nicolas Fressengeas † and Joel Jacquet * †

I. INTRODUCTION

Subwavelength grating mirrors (SGM) made of high index contrast material system can be designed to exhibit very high performances at the 0-order of diffraction. With only a few layers, SGMs can achieve a reflectivity as high as 99.9 % and bandwidths larger than 100 nm together with a strong polarization selectivity between TM and TE mode [START_REF] Mateus | Ultrabroadband mirror using low-index cladded subwavelength grating[END_REF], [START_REF] Chevallier | Optimized sub-wavelength grating mirror design for mid-infrared wavelength range[END_REF]. Such characteristics are very interesting for integration in VCSEL structures where SGM can advantageously replace classical Bragg mirrors. Currently, the wavelength of emission of mid-IR electrically-pumped VCSEL is limited close to 2.6 µm since the necessary semiconductor Bragg mirrors become as thick as 11 µm impairing the electro-thermo-optical properties of the device [START_REF] Ducanchez | Mid-infrared GaSb-based EP-VCSEL emitting at 2.63 µm[END_REF], [START_REF] Bachmann | Single-mode electrically pumped GaSb-based VCSELs emitting continuous-wave at 2.4 and 2.6 µm[END_REF]. With a total thickness of less than 2 µm, the use of SGMs as top mirror should allow VCSEL operation at larger wavelengths.

In this work, we design a Si/Si 3 N 4 SGM for a VCSEL application at 2.65 µm. First, the optical properties that VCSEL mirrors must exhibit are defined and quantitatively expressed through a quality factor. The dimensions of the SGM structure are then adjusted by an optimization algorithm to maximize the quality factor and meet VCSEL requirements. Finally, the SGM robustness is enhanced through a precise study of the tolerance on the physical dimensions, to define an error margin that the physical growth process must respect.

II. MIRROR DESIGN

Several structures have been designed based on different material combination such as Si/SiO 2 , Si/Si 3 N 4 or GaAs/AlO x . We present here a Si/Si 3 N 4 -based SGM as shown in Fig. 1. The mirror structure is made of a silicium grating (n = 3.435) on top of a Si 3 N 4 layer as the low index sublayer (n = 1.99). The SGM is combined to a Si/Si 3 N 4 quarter wavelength pair in order to increase the total reflectivity [START_REF] Chevallier | Optimized sub-wavelength grating mirror design for mid-infrared wavelength range[END_REF]. Contrary to Bragg mirrors, SGM design rules to achieve high reflectivity are not straightforward. The values of the etched

GaSb Substrate Λ Air Si λ/4n λ/4n n =3.853 λ 0 T g T L Si n =3.435 λ 0 n =1.99 λ 0 L e L f Si N 3 4
Si N length L e , the filled length L f , the grating thickness T g and the sublayer thickness T L are thus adjusted thanks to an optimization process to match the expected properties.

The design of mirror for VCSEL application requires to precisely define the optical properties to exhibit. Due to the thin gain region of the VCSEL structure, the reflectivity of the mirror should be as high as 99.5 % for the largest possible bandwidth in order to create a cavity of sufficient quality for laser emission. During the conception, a 99.9 % reflectivity threshold has been chosen to ensure a fabrication security margin. Moreover, contrary to Bragg mirrors, SGM can show a polarization selectivity due to their 1D symmetry. By choosing a minimum reflectivity of 99.9 % for transverse magnetic (TM) mode and a maximum reflectivity of 90 % for transverse electric (TE) mode, laser effect should be possible only in TM polarization and thus should increase the stability of the emitted beam.

The mirror is simulated by the rigorous coupled wave analysis method (RCWA) [START_REF] Moharam | Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach[END_REF], [START_REF] Rathgen | mrcwa 20080820[END_REF] which computes reflection spectra for TM and TE modes. This method allows us to solve Maxwell equations in a faster way than finite difference based methods such as Finite Difference Time Domain (FDTD) but makes the assumption of a perfectly squared grating with an infinite number of periods. Through a numerical analysis of the computed reflection spectra, the quality of a design is quantitatively evaluated by the use of a quality factor Q [START_REF] Chevallier | Optimized sub-wavelength grating mirror design for mid-infrared wavelength range[END_REF]: This quality factor is written to take into account the VCSEL requirements with a polarization selectivity performed by selecting the range of wavelengths λ around λ 0 for which R T M is higher than 99.9 % and R T E is lower than 90 %. The centering of the bandwidth around λ 0 = 2.65 µm is made by computing a gaussian weighted average of the R T M values on the N points of the bandwidth [λ 1 , λ 2 ]. Finally the influence of the width of the stopband is expressed by multiplying the latter average by the bandwidth itself ∆λ = |λ 2 -λ 1 | and normalized by λ 0 .

Q = ∆λ λ 0 1 N λ2 λ=λ1 R T M (λ)g(λ) (1)

III. OPTIMIZATION

The optimization of the mirror performance is made by adjusting the grating parameters to maximize the quality factor Q. However, with its numerous local maxima, its non linearity, its discontinuity and an unknown derivative, optimizing Q can only be done using a global maximization method. An optimization algorithm [START_REF] Kroshko | OpenOpt 0.27[END_REF], the Differential Evolution algorithm [START_REF] Feoktistov | Differential Evolution, ser. Optimization and Its Applications[END_REF], has been used to automatically find the best set of parameters which maximizes the mirror quality.

The Si/Si 3 N 4 -based structure presented in Fig. 1 has been optimized under constraints which aim to limit the etching process difficulties by using large grooves (L f and L e > 500 nm) with a squared shape (L e /T g > 0.9). The resulting optimum grating parameters are L e = 829 nm, L f = 522 nm, T g = 899 nm, T L = 896 nm. This mirror design exhibits a 218 nm large bandwidth with R T M > 99.9% together with a high polarization selectivity by keeping R T E < 90% and meets all previously defined VCSEL requirements.

IV. ROBUSTNESS ENHANCEMENT

In order to evaluate the feasibility of the design, a tolerance study has been performed on the dimensions of the grating. The tolerance of one parameter is defined as the allowed variation range of this parameter for which the mirror is enough performant for a VCSEL application with R T M > 99.9 % and R T E < 90 %. The tolerance computation of the structure described in Fig. 1 shows a very robust design with several percent of tolerance on each optimized length.

The map of the variation range of L e , L f , the period Λ and the fill factor F F = L f /Λ (Fig. 2), indicates the area of the (L e , L f ) couples allowed for the design. We only present here the tolerance study of the parameters related to the grating etching which is a critical step of the fabrication process. The optimum point (829, 522) with the largest bandwidth found by the optimization algorithm is located at the edge of the tolerance area. Thus, even if tolerances of 8.2 % on L e and 4.6 % on L f are allowed, the accuracy of the fabrication process is very critical with a L f optimum value equals to the upper tolerance limit. So, in order to enhance the robustness of the grating with respect to the error of fabrication which could be made on L e and L f , a non optimum point located at the centre of the tolerance area has been chosen.

The resulting design (784, 537), with grooves larger than 500 nm and large tolerance values (5.5 % for L e and 8 % for L f ), exhibits a 186 nm large bandwidth and seems to be well adapted for integration in a 2.65 µm operating VCSEL device.

V. CONCLUSION

In this work, the design of a Si/Si 3 N 4 subwavelength grating mirror exhibiting a 186 nm large 99.9 % high reflectivity bandwidth is presented. The mirror structure has been automatically optimized thanks to the definition of a VCSEL mirror quality factor maximized by an optimization algorithm. The resulting design tolerance has been evaluated and the grating parameters were adjusted to enhance the robustness with respect to the error of fabrication. This mirror, with a thickness of less than 2 µm, should allow laser operation of VCSEL devices at 2.65 µm.

3 4 Fig. 1 .

 341 Fig. 1. Scheme of the SGM optimized for λ 0 = 2.65 µm exhibiting a 218 nm large bandwidth with Le = 829 nm, L f = 522 nm, Tg = 899 nm, T L = 896 nm.

Fig. 2 .

 2 Fig. 2. The tolerance map represents the variation range for the (Le, L f ) couples while keeping Tg and T L at their optimum value. The center of the largest incircle of the polygon (⋆) indicates the most tolerant point with respect to the fabrication errors on Le and L f .

ACKNOWLEDGMENT

The authors thank the French ANR for financial support in the framework of Marsupilami project (ANR-09-BLAN-0166-03) and IES and LAAS (France), partners of LMOPS/Supélec in this project. This work was also partly funded by the InterCell grant (http://intercell.metz.supelec.fr) by INRIA and Région Lorraine (CPER2007).


Christophe Chesneau
email: christophe.chesneau@gmail.com

Wavelet linear estimation of a density and its derivatives from observations of mixtures under quadrant dependence

Keywords: Mathematics Subject Classification. 62G07, 62G20. Keywords. density estimation, mixtures, quadrant dependence, wavelets

The estimation of a density and its derivatives from a finite mixture under the pairwise positive quadrant dependence assumption is considered. A new wavelet based linear estimator is constructed. We evaluate its asymptotic performance by determining an upper bound of the mean integrated squared error. We prove that it attains a sharp rate of convergence for a wide class of unknown densities.

Introduction

The following mixture density model is considered: we observe n random variables X 1 , . . . , X n such that, for any i ∈ {1, . . . , n}, the density of X i is the finite mixture:

h i (x) = m d=1 w d (i)f d (x), x ∈ [0, 1],
where

• (w d (i)) (i,d)∈{1,.
..,n}×{1,...,m} are known positive weights such that, for any i ∈ {1, . . . , n}, m d=1 w d (i) = 1,

• f 1 , . . . , f m are unknown densities.

For a fixed ν ∈ {1, . . . , m}, we aim to estimate f ν and, more generally, its r-th derivative f (r) ν from Pairwise Positive Quadrant Dependent (PPQD) X 1 , . . . , X n .

Let us now present a brief survey related to this problem under various configurations. On the one hand, when X 1 , . . . , X n are independent, the estimation of f ν has been considered in e.g. [START_REF] Maiboroda | Estimators of components of mixtures with varying concentrations, Ukrain[END_REF], [START_REF] Hall | Nonparametric estimation of component distributions in a multivariate mixture[END_REF] and [START_REF] Pokhyl'ko | Wavelet estimators of a density constructed from observations of a mixture[END_REF]. The estimation of f (r) ν has been recently studied by Prakasa [START_REF] Rao | Wavelet linear estimation for derivatives of a density from observations of mixtures with varying mixing proportions[END_REF]. This is particularly of interest to detect possible bumps, concavity or convexity properties of f ν . On the other hand, when X 1 , . . . , X n are identically distributed i.e. h = h 1 = . . . = h n , the estimation of h for associated X 1 , . . . , X n (including PPQD) has been investigated in e.g. [START_REF] Cai | Efficient estimation of a distribution function under quadrant dependence[END_REF], [START_REF] Dewan | A general method of density estimation for associated random variables[END_REF], [START_REF] Masry | Multivariate probability density estimation for associated processes: Strong consistency and rates[END_REF] and Prakasa [START_REF] Rao | Wavelet linear density estimation for associated sequences[END_REF]. The estimation of h (r) has been considered by [START_REF] Chaubey | Wavelet based estimation of the derivatives of a density with associated variables[END_REF]. However, to the best of our knowledge, the combination of these two complex statistical frameworks i.e. the estimation of f

(r) ν , including f ν , under PPQD conditions is a new challenge.
Such a problem occurs in the study of medical, biological and other types of data. The most common situation is the following: for any i ∈ {1, . . . , n}, X i depends on an unobserved random indicator I i taking its values in {1, . . . , m}. Applying the Bayes theorem, the density of X i is h i defined with w d (i) = P(I i = d) and f d the conditional density of X i given {I i = d}. We refer to [START_REF] Maiboroda | Estimators of components of mixtures with varying concentrations, Ukrain[END_REF] and the references there in. Naturally, in some situations, X 1 , . . . , X n are not independent and this motivates the study of various dependence structures as the PPQD one. Further details and applications on the concept of associated random variables can be found in [START_REF] Roussas | Positive and negative dependence with some statistical applications, Asymptotics, nonparametrics and time series[END_REF], Prakasa [START_REF] Rao | Associated sequences and related inference problems[END_REF] and [START_REF] Sancetta | Strong law of large numbers for pairwise positive quadrant dependent random variables[END_REF].

To estimate f (r) ν , several methods are possible as kernel, spline, wavelet, . . . (see e.g. Prakasa [START_REF] Rao | Nonparametric functional estimation[END_REF]Rao ( , 1999)), [START_REF] Härdle | Wavelet, Approximation and Statistical Applications[END_REF] and [START_REF] Tsybakov | Introduction à l'estimation non-paramétrique[END_REF]). In this study, we focus our attention on the multiresolution analysis techniques and, more precisely, the wavelet methodology of [START_REF] Pokhyl'ko | Wavelet estimators of a density constructed from observations of a mixture[END_REF] and Prakasa [START_REF] Rao | Wavelet linear estimation for derivatives of a density from observations of mixtures with varying mixing proportions[END_REF]. We construct a linear wavelet estimator and explore its asymptotic performance by taking the mean integrated squared error (MISE) and assuming that f (r) ν belongs to a Besov ball. We prove that, under some specific assumptions, it attains the same rate of convergence as the one obtained in the independent case. This paper is organized as follows. Assumptions on the model and some notations are introduced in Section 2. Section 3 briefly describes the wavelet basis on [0, 1] and the Besov balls. The linear wavelet estimator and the results are presented in Section 4. Section 5 is devoted to the proofs.

Assumptions

Additional assumptions on the model are presented below. The integers r and ν refer to those in f (r) ν .

Assumption on f 1 , . . . , f m . Without loss of generality, for any d ∈ {1, . . . , m}, we assume that the support of f d is [0, 1] (our study can be extended to another compact support).

We suppose that there exists a constant C * > 0 such that, for any d ∈ {1, . . . , m},

f (r) d (x) ≤ C * . (2.1)
We suppose that, for any d ∈ {1, . . . , m} and v ∈ {0, . . . , r},

f (v) d (0) = f (v) d (1). (2.2)
Assumption on the weights of the mixture. We suppose that the matrix

Γ n = 1 n n i=1 w k (i)w (i) (k, )∈{1,...,m} 2 satisfies det(Γ n ) > 0.
For the considered ν (the one which refers to the estimation of f (r) ν ) and any i ∈ {1, . . . , n}, we set

a ν (i) = 1 det(Γ n ) m k=1 (-1) k+ν γ n ν,k w k (i), (2.3)
where γ n ν,k denotes the determinant of the minor (ν, k) of the matrix Γ n . Then a ν (1), . . . , a ν (n) satisfy

(a ν (1), . . . , a ν (n)) = argmin (u 1 ,...,un)∈∩ m d=1 U ν,d 1 n n i=1 u 2 i , (2.4)
where

U ν,d = (u 1 , . . . , u n ) ∈ R n ; 1 n n i=1 u i w d (i) = δ ν,d
and δ ν,d is the Kronecker delta.

Technical details can be found in [START_REF] Maiboroda | Estimators of components of mixtures with varying concentrations, Ukrain[END_REF].

We set

z n = 1 n n i=1 a 2 ν (i).
(2.5)

For technical reasons, we suppose that z n < n.

Assumptions on X 1 , . . . , X n . We suppose that X 1 , . . . , X n are PPQD i.e. for any (i, ) ∈ {1, . . . , n} 2 with i = and any (x, y)

∈ [0, 1] 2 , P(X i > x, X > y) ≥ P(X i > x)P(X > y).
This weak kind of dependence has been introduced by Lehmann (1966).

Examples of PPQD variables can be found in [START_REF] Sancetta | Strong law of large numbers for pairwise positive quadrant dependent random variables[END_REF].

We suppose that, for any (i, ) ∈ {1, . . . , n} 2 , there exists a constant C > 0 such that sup

(x,y)∈[0,1] 2 |h i, (x, y) -h i (x)h (y)| ≤ C, (2.6)
where h i, is the density of (X i , X ).

We suppose that there exists a constant C > 0 such that

n i=1 i 3 i-1 =1 a 2 ν (i) + a 2 ν ( ) C ov (X i , X ) ≤ Cnz n , (2.7)
where a ν (1), . . . , a ν (n) are (2.3) and z n is (2.5).

This assumption seems important to obtain "suitable" asymptotic properties in the estimation of f (r) ν from PPQD X 1 , . . . , X n .

Wavelets and Besov balls

Throughout the paper, we work with the wavelet basis on [0, 1] described below. Let N be an integer such that N > r + 1, and φ and ψ be the initial wavelet functions of the Daubechies wavelets dbN . In particular, these functions are compactly supported and belong to C r+1 . Set

φ j,k (x) = 2 j/2 φ(2 j x -k), ψ j,k (x) = 2 j/2 ψ(2 j x -k).
Then there exists an integer η satisfying 2 η ≥ 2N such that, for any ≥ η, the collection

B = {φ ,k (.), k ∈ {0, . . . , 2 -1}; ψ j,k (.); j ∈ N-{0, . . . , -1}, k ∈ {0, . . . , 2 j -1}},
with an appropriate treatment at the boundaries, is an orthonormal basis of L 2 ([0, 1]) (the set of square-integrable functions on [0, 1]) and, for any v ∈ {0, . . . , r}, (φ j,k ) (v) (0) = (φ j,k ) (v) (1). Details can be found in [START_REF] Cohen | Wavelets on the interval and fast wavelet transforms[END_REF].

For any integer ≥ η, any h ∈ L 2 ([0, 1]) can be expanded on B as

h(x) = 2 -1 k=0 α ,k φ ,k (x) + ∞ j= 2 j -1 k=0 β j,k ψ j,k (x), x ∈ [0, 1],
where

α j,k = 1 0 h(x)φ j,k (x)dx, β j,k = 1 0 h(x)ψ j,k (x)dx. (3.1)
A function h belongs to B s 2,∞ (M ) if and only if there exists a constant M * > 0 (depending on M ) such that (3.1) satisfy

sup j≥η 2 2js k∈Λ j β 2 j,k ≤ M * .
We refer to [START_REF] Meyer | Ondelettes et Opérateurs[END_REF].

Estimator and results

Assuming that

f (r) ν ∈ B s 2,∞ (
M ), we define the linear estimator f (r) by

f (r) (x) = 2 j 0 -1 k=0 α(r) j 0 ,k φ j 0 ,k (x), x ∈ [0, 1], (4.1)
where α(r)

j 0 ,k = (-1) r n n i=1 a ν (i)(φ j 0 ,k ) (r) (X i ), (4.2)
a ν (1), . . . , a ν (n) are (2.3), j 0 is the integer satisfying

1 2 n z n 1/(2s+2r+1) < 2 j 0 ≤ n z n 1/(2s+2r+1)
and z n is defined by (2.5).

The definitions of α(r) j 0 ,k and j 0 , which take into account the PPQD case, are chosen to minimize the MISE of f (r) .

Note that f (r) is close to one considered by (Prakasa Rao, 2010, eq. (4.5)) in the independent case. Further details on derivatives density estimation via wavelet can also be found in [START_REF] Chaubey | Wavelet based estimation of the derivatives of a density with associated variables[END_REF] and [START_REF] Hosseinioun | Nonparametric estimation of the derivatives of a density by the method of wavelet for mixing sequences[END_REF].

Theorem 4.1 below investigates the MISE of f (r) when f

(r) ν ∈ B s 2,∞ (M ).
Theorem 4.1 (Upper bound for f (r) ) Let X 1 , . . . , X n be n random variables as described in Section 1 under the assumptions of Section 2. Suppose that f (r) ν ∈ B s 2,∞ (M ) with s > 0. Let f (r) be (4.1). Then there exists a constant C > 0 such that

E 1 0 ( f (r) (x) -f (r) ν (x)) 2 dx ≤ C z n n 2s/(2s+2r+1)
.

The proof of Theorem 4.1 uses a moment inequality on (4.2) and a suitable decomposition of the MISE. Let us mention that the obtained rate of convergence is exactly the optimal one related to the independent case i.e. (z n /n) 2s/(2s+2r+1) (see (Prakasa Rao, 2010, Theorem 6.1 and Remark 6.1)).

Note that Theorem 4.1 can be extended to other kinds of associated X 1 , . . . , X n as Negative Associated (NA), Pairwise Negative Quadrant Dependence (PNQD), . . . . This is due to the Newman inequality (Newman, 1980, Lemma 3) used in the proof of Theorem 4.1 which still holds in these cases.

Remark that f (r) is not adaptive with respect to s. Adaptivity can perhaps be achieved by using a non-linear wavelet estimator as the hard thresholding one. This approach works in the independent case (see [START_REF] Pokhyl'ko | Wavelet estimators of a density constructed from observations of a mixture[END_REF] Theorem 4)). However, the proof of this fact uses technical tools as the Bernstein and the Rosenthal inequalities and it is not immediately clear how to extend this to the PPQD case.

Proofs

In this section, we consider the density model described in Section 1 under the assumptions of Section 2. Moreover, C denotes any constant that does not depend on j, k and n. Its value may change from one term to another and may depends on φ.

Proposition 5.1 Let X 1 , . . . , X n be n random variables as described in Section 1 under the assumptions of Section 2. For any k ∈ {0, . . . , 2 j 0 -1}, let α (r)

j 0 ,k = 1 0 f (r)
ν (x)φ j 0 ,k (x)dx and α(r) j j 0 ,k be (4.2). Then there exists a constant C > 0 such that E((α (r)

j 0 ,k -α (r) j 0 ,k ) 2 ) ≤ C2 2rj 0 z n n .
Proof of Proposition 5.1. Proceeding as in (Prakasa Rao, 2010, eq. (4.6)), it follows from (2.4), r integrations by parts, (2.2) and, for any v ∈ {0, . . . , r},

(φ j,k ) (v) (0) = (φ j,k ) (v) (1), that E(α (r) j 0 ,k ) = (-1) r n n i=1 a ν (i)E((φ j 0 ,k ) (r) (X i )) = (-1) r n n i=1 a ν (i) 1 0 (φ j 0 ,k ) (r) (x)h i (x)dx = (-1) r m d=1 1 0 f d (x)(φ j 0 ,k ) (r) (x)dx 1 n n i=1 a ν (i)w d (i) = (-1) r 1 0 f ν (x)(φ j 0 ,k ) (r) (x)dx = 1 0 f (r) ν (x)φ j 0 ,k (x)dx = α (r) j 0 ,k . Therefore E((α (r) j 0 ,k -α (r) j 0 ,k ) 2 ) = V(α (r) j 0 ,k ) = 1 n 2 n i=1 n =1 a ν (i)a ν ( )C ov ((φ j 0 ,k ) (r) (X i ), (φ j 0 ,k ) (r) (X )) ≤ 1 n 2 n i=1 a 2 ν (i)V((φ j 0 ,k ) (r) (X i )) + 1 n 2 n i=1 n =1 =i |a ν (i)||a ν ( )||C ov ((φ j 0 ,k ) (r) (X i ), (φ j 0 ,k ) (r) (X ))|. (5.1)
Let us bound the first term in (5.1). For any i ∈ {1, . . . , n}, using (2.1) which implies sup

x∈[0,1] h i (x) ≤ C * and (φ j 0 ,k ) (r) (x) = 2 j 0 /2 2 rj 0 φ (r) (2 j 0 x -k), we have V((φ j 0 ,k ) (r) (X i )) ≤ E(((φ j 0 ,k ) (r) (X i )) 2 ) = 1 0 ((φ j 0 ,k ) (r) (x)) 2 h i (x)dx ≤ C * 2 2rj 0 1 0 (φ (r) (x)) 2 dx ≤ C2 2rj 0 . Therefore 1 n 2 n i=1 a 2 ν (i)V((φ j 0 ,k ) (r) (X i )) ≤ C2 2rj 0 1 n 2 n i=1 a 2 ν (i) = C2 2rj 0 z n n .
(5.2)

Let us now investigate the bound of the covariance term in (5.1) via two different approaches.

Bound 1. By a standard covariance equality and (2.6), for any (i, ) ∈ {1, . . . , n} 2 with i = , we have

|C ov ((φ j 0 ,k ) (r) (X i ), (φ j 0 ,k ) (r) (X ))| = 1 0 1 0 (h i, (x, y) -h i (x)h (y))(φ j 0 ,k ) (r) (x)(φ j 0 ,k ) (r) (y)dxdy ≤ 1 0 1 0 |h i, (x, y) -h i (x)h (y)||(φ j 0 ,k ) (r) (x)||(φ j 0 ,k ) (r) (y)|dxdy ≤ C 1 0 |(φ j 0 ,k ) (r) (x)|dx 2 .
Moreover, since (φ j 0 ,k ) (r) (x) = 2 (2r+1)j 0 /2 φ (r) (2 j 0 x -k), by the change of variables y = 2 j 0 x -k, we obtain

1 0 |(φ j 0 ,k ) (r) (x)|dx = 2 rj 0 2 -j 0 /2 |φ (r) (x)|dx.
Therefore

|C ov ((φ j 0 ,k ) (r) (X i ), (φ j 0 ,k ) (r) (X ))| ≤ C2 2rj 0 2 -j 0 .
(5.3) Bound 2. Since X 1 , . . . , X n are PPQD, it follows from (Newman, 1980, Lemma 3) that, for any (i, ) ∈ {1, . . . , n} 2 with i = , 2r+3) .

|C ov ((φ j 0 ,k ) (r) (X i ), (φ j 0 ,k ) (r) (X ))| ≤ sup x∈[0,1] |(φ j 0 ,k ) (r+1) (x)| 2 C ov (X i , X ). Since (φ j 0 ,k ) (r+1) (x) = 2 (2r+3)j 0 /2 φ (r+1) (2 j 0 x-k) and sup x∈[0,1] |φ (r+1) (x)| ≤ C, we have sup x∈[0,1] |(φ j 0 ,k ) (r+1) (x)| 2 ≤ C2 j 0 (
Therefore

|C ov ((φ j 0 ,k ) (r) (X i ), (φ j 0 ,k ) (r) (X ))| ≤ C2 j 0 (2r+3) C ov (X i , X ).
(5.4) Combining (5.3) and (5.4), for any (i, ) ∈ {1, . . . , n} 2 with i = , we obtain

|C ov ((φ j 0 ,k ) (r) (X i ), (φ j 0 ,k ) (r) (X ))| ≤ C min(2 j 0 (2r+3) C ov (X i , X ), 2 2rj 0 2 -j 0 ). (5.5) It follows from (5.5) that 1 n 2 n i=1 n =1 =i |a ν (i)||a ν ( )||C ov ((φ j 0 ,k ) (r) (X i ), (φ j 0 ,k ) (r) (X ))| = 2 n 2 n i=2 i-1 =1 |a ν (i)||a ν ( )||C ov ((φ j 0 ,k ) (r) (X i ), (φ j 0 ,k ) (r) (X ))| ≤ 1 n 2 n i=2 i-1 =1 a 2 ν (i) + a 2 ν ( ) |C ov ((φ j 0 ,k ) (r) (X i ), (φ j 0 ,k ) (r) (X ))| ≤ C(E + F ), (5.6)
where

E = 1 n 2 2 2rj 0 2 -j 0 2 j 0 -1 i=2 i-1 =1 a 2 ν (i) + a 2 ν ( ) and F = 1 n 2 2 j 0 (2r+3) n i=2 j 0 i-1 =1 a 2 ν (i) + a 2 ν ( ) C ov (X i , X ).
We have

E ≤ C 1 n 2 2 2rj 0 2 -j 0 2 j 0 n i=1 a 2 ν (i) = C2 2rj 0 z n n .
(5.7)

Using (2.7), it comes

F ≤ 1 n 2 2 2rj 0 n i=0 i 3 i-1 =1 a 2 ν (i) + a 2 ν ( ) C ov (X i , X ) ≤ C2 2rj 0 z n n .
(5.8)

Putting (5.1), (5.2), (5.6), (5.7) and (5.8) together, we obtain E((α (r)

j 0 ,k -α (r) j 0 ,k ) 2 ) ≤ C2 2rj 0 z n n .
This ends the proof of Proposition 5.1.

Proof of Theorem 4.1. We expand the function f .

Therefore

E 1 0 ( f (r) (x) -f (r) ν (x)) 2 dx ≤ C z n n 2s/(2s+2r+1)
.

The proof of Theorem 4.1 is complete.

  ν (x)ψ j,k (x)dx.We have, for any x ∈ [0, 1],f (r) (x) -f (r) r) (x) -f (r) ν (x)) 2 dx = A + B,Using Proposition 5.1 and the definition of j 0 , we obtain A ≤ C2 j 0 2 2rj 0 z n n

Acknowledgment.

This work is supported by ANR grant NatImages, ANR-08-EMER-009.