Clément Deleau

Thidsanu Apiphatnaphakul

Han Cheng Seat

Frederic Surre

Usman Zabit

Franck Carcenac

Pierre-Franc ¸ois Calmon

Thierry Bosch

Olivier Bernal
email: olivier.bernal@laas.fr

Towards Integrated Optical Feedback FM-to-AM Conversion in Silicon Nitride for Displacement Sensing Applications

Keywords: Mach Zehnder interferometer, self-mixing, optical feedback interferometry, laser sensors, silicon nitride, integrated photonics

In this paper, we demonstrate the potential of silicon nitride integrated edge filters to perform frequency to amplitude (FM-to-AM) conversion of optical feedback interferometric (OFI) signals to achieve better signal-to-noise ratio than the typical AM OFI signals. Compared to already existing OFI FM-to-AM conversion techniques such as edge filters based on gas cells, and free-space or fiber Mach Zehnder interferometer (MZI), integrated photonic processing of OFI signals can offer greater compactness and design flexibility. In addition, compactness can also indirectly improve noise performances by facilitating temperature control of the chip and allowing a higher immunity to parasitic mechanical vibrations. The FM OFI detection is performed with an integrated MZI with a 2 cm imbalanced pathlength implemented on a silicon nitride process. The achieved OFI FM-to-AM conversion factor is 0.83 (GHz) -1 and the noise equivalent displacement is 16 nm for a 1 kHz bandwidth.

I. INTRODUCTION

Optical feedback interferometry (OFI), also referred as the self-mixing (SM) effect in laser diodes (LDs), has been widely investigated in recent decades as it results in a self-aligned and cost-effective sensing system for displacement, vibration or velocity measurements for instance [START_REF] Donati | Developing self-mixing interferometry for instrumentation and measurements[END_REF], [START_REF] Taimre | Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing[END_REF]. The resolution of OFI-based displacement sensors depends on the employed signal processing techniques [START_REF] Magnani | Self-mixing vibrometer with real-time digital signal elaboration[END_REF]- [START_REF] Naveed | Real-time adaptive spectral filter based parasitic vibration cancellation in self-mixing laser sensor signals[END_REF].

In OFI, a portion of the laser beam can be backscattered from a target placed away from the laser and can thus reenter the active laser cavity. This directly affects the optical fields within the LD cavity. It results in the modulation of both amplitude (AM) and frequency (FM) of the optical field (see Fig. 1 a)) as described by Lang and Kobayashi equations [START_REF] Lang | External optical feedback effects on semiconductor injection laser properties[END_REF]. The OFI information embedded in the optical power modulation (and thus AM channel) is usually preferred to the FM channel as it is directly available by the LD monitoring photodiode or simply by monitoring a fraction of the optical ANR-20-CE42-0010 PICSONDE Fig. 1. a) Typical Fiber OFI sensing setup using a laser diode and a piezotransducer (PZT) as a target, showing in particular the AM channel retrieved from the monitoring photodiode. b) OFI frequency modulation (FM) signal embedded in the laser frequency is converted into amplitude modulation (AM) by using the fringe edge of a Mach Zehnder interferometer (MZI) used as an optical filter. power emitted by the LD. However, it was shown in [START_REF] Norgia | Exploiting the fm-signal in a laser-diode smi by means of a mach-zehnder filter[END_REF] that the SNR of FM OFI signals is approximately two orders of magnitude better than the AM one. As a result, FM OFI signals are thus of utmost importance to enhance OFI sensitivity not only for low-noise small displacement applications but also for non-cooperative diffusive remote target surfaces resulting in low back-scattered OFI signal.

Recently, OFI FM-to-AM conversion has been achieved using the edge of optical filters. The conversion principle is illustrated in Fig. 1 b). In [START_REF] Contreras | Edge filter enhanced selfmixing interferometry[END_REF], an acetylene gas cell is used as the converting filter. The steep edge of an optical absorption profile of the cell provides an OFI FM-to-AM conversion factor of approximately 2.2 (GHz) -1 with a typical absorption line depth of 8 dB. To improve further both the sensitivity and the flexibility of the filter, other approaches have then ©2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. emerged based on free-space and fiber-based Mach Zehnder interferometers (MZI) [START_REF] Norgia | Exploiting the fm-signal in a laser-diode smi by means of a mach-zehnder filter[END_REF], [START_REF] Norgia | High-sensitivity vibrometer based on fm self-mixing interferometry[END_REF]. As a result, the LD wavelength can be more easily tuned to one of the periodic signatures of the MZI. In addition, the sensitivity related to the steepness of the response can also be adjusted via the optical path length imbalance between the two MZI arms. The achieved sensitivity is 19 (GHz) -1 [8] so approximately 10 times better than the gas-cell based filter and the noise equivalent displacement (NED) that is achieved is 1.3 pm/

√

Hz. Here, we propose to implement an integrated MZI on silicon nitride (Si 3 N 4 ) process (Fig. 2) to perform the OFI FM-to-AM conversion since silicon nitride has low propagation loss and a lower thermo-optic coefficient (2.45•10 -5 RIU/ • C) compared to silicon photonics (1.8•10 -4 RIU/ • C) at 1550 nm [START_REF] Tang | Temperatureinsensitive mach-zehnder interferometer based on a silicon nitride waveguide platform[END_REF]. The aim is to demonstrate the possibility to fully integrate this conversion on chip, to ease the tuning of the filter and to improve further the resilience regarding temperature and parasitic mechanical vibrations compared to fiber-based implementation [START_REF] Norgia | Exploiting the fm-signal in a laser-diode smi by means of a mach-zehnder filter[END_REF]. The implemented path-length imbalance is set to 2 cm that corresponds to a 0.83 (GHz) -1 sensitivity.

This paper is organized in two main parts. First, in Section II, an overview of the FM OFI signal and the proposed integrated OFI FM-to-AM conversion are presented. In Section III, the experimental setup and results are shown followed by a conclusion in section IV.

II. INTEGRATED OFI FM-TO-AM DEMODULATION

A. FM OFI overview

As previously mentioned, in the case of the AM channel, the OFI induces fluctuations in the optical output power (OOP) of the LD, denoted P (t), given by [START_REF] Donati | Developing self-mixing interferometry for instrumentation and measurements[END_REF]:

P (t) = P 0 [1 + m cos (Φ F (t))] , (1)
where P 0 is the emitted optical power under free-running conditions, m is the modulation index and Φ F (t) is the laser output phase in the presence of feedback. Φ F (t) is related to the laser output phase without feedback Φ 0 (t) = 4πD (t) /λ 0 with D (t) the laser-target distance, by [START_REF] Donati | Developing self-mixing interferometry for instrumentation and measurements[END_REF], [START_REF] Taimre | Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing[END_REF]:

Φ 0 (t) = Φ F (t) + C sin (Φ F (t) + arctan α) , (2)
where α is the linewidth enhancement factor of the laser and C the optical feedback factor. Depending on C, the laser can operate in different regimes. SM sensing is generally performed under weak feedback regime (C < 1), moderate feedback regime (1 < C < 4.6), or strong feedback regime (C > 4.6). However, a moderate feedback regime is usually preferred as the apparently simple saw-tooth shaped SM fringes belonging to such a regime [START_REF] Bernal | Classification of laser self-mixing interferometric signal under moderate feedback[END_REF] intrinsically provide motion direction indication and require simplified SM fringe detection processing [START_REF] Magnani | Self-mixing vibrometer with real-time digital signal elaboration[END_REF]. Regarding FM OFI channel, (2) can also be expressed as follows:

[ν 0 (t) -ν F (t)] τ ext = C 2π sin (2πν F (t) τ ext + arctan α) , (3)
where ν 0 is the unperturbed LD frequency, ν F is the LD frequency, and τ ext is the external round trip time of flight. From (3), the variation of the LD frequency ∆ν = ν F -ν 0 is strongly constrained as:

|∆ν| ≤ C 2πτ ext . ( 4
)
The design of the optical filter should consider this maximum span to avoid folding. Note that fringe-locking to the half fringe [START_REF] Norgia | Exploiting the fm-signal in a laser-diode smi by means of a mach-zehnder filter[END_REF] can extend this range further and thus might allow the design of an even steeper filter.

B. The OFI FM-to-AM conversion

As previously mentioned, the OFI FM-to-AM conversion is performed using a Mach Zehnder interferometer (MZI) shown in Fig. 2. The output power P out can be expressed as:

P out = P in 1 + cos 2πn (ν) ∆d path c ν , (5)
where P in is the injected light power in the MZI, ∆d path the MZI path difference length, n the refractive index of the optical path and c the speed of light. As mentioned in [START_REF] Norgia | Exploiting the fm-signal in a laser-diode smi by means of a mach-zehnder filter[END_REF], the maximum sensitivity S is achieved at the middle of the transmission power response of the MZI. It is given as:

S = 1 P in ∂P ∂ν = 2πn g (λ) ∆d path c , (6)
where λ is the LD wavelength and n g the group index defined as n g (λ) = n (λ) -λ ∂n(λ) ∂λ . In the case of integrated MZIs, it might be more convenient to express S as function of the free spectral range FSR(λ) defined as follows:

F SR (λ) = λ 2 n g (λ) ∆d path . (7)
Consequently, (6) becomes:

S = 2π λ 2 cF SR . ( 8
)
In the case of open-loop OFI approach, similar to bandwidth definition, it might be useful to define the range δλ dyn for which the sensitivity is reduced by a factor of √ 2. It can be shown from ( 6) that δλ dyn can then be approximated by:

δλ dyn ≈ λ 2 4n∆d path . ( 9
)
©2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. This conversion range should ideally cover not only the LD wavelength variation induced by OFI but also the wavelength variation induced by current modulation that might be required for dithering [START_REF] Bernal | Sub-λ/2 displacement sensor with nanometric precision based on optical feedback interferometry used as a non-uniform event-based sampling system[END_REF]. There is thus a trade-off between sensitivity and conversion range for open-loop configurations.

III. EXPERIMENTAL RESULTS

A. Experimental set-up

The experimental set-up described in Fig. 2 is shown in Fig. 3. The light source is a pigtailed laser diode WSLD-1550-020m from Wavespectrum emitting 8 mW of power at λ=1550 nm. This emitted light is then injected into two branches using a fiber coupler with a splitting ratio of 80/20 as shown in Fig. 2: (1) the measurement branch which connects the laser to the target and (2) the OFI FM-to-AM conversion photonic chip. Here, a speaker is used as a target. The OFI FM-to-AM converted output of the photonic chip is monitored by the photodiode PDA50B-EC from Thorlabs.

The laser-target distance is currently approximately 2.5 m, which corresponds to τ ext ≈ 24 ns. Based on (6), the maximum variation ∆ν that can be observed in moderate feedback regime is approximately 30 MHz.

The integrated photonic circuit is fabricated using a 300 nm Si 3 N 4 process with a refractive index n Si3N4 ≈ 2 (Fig. 2). An MZI, with an imbalanced path length of 2 cm (inset of Fig. 3) was designed and inmplemented. This 2 cm MZI has been characterized and reported using the set-up shown in Fig. 1.The light from the fiber is injected into the integrated photonic circuit via integrated grating couplers using a fiber array with a 8 • polishing angle.

B. MZI characterization

Fig. 4 a) shows the optical transmission spectrum of the 2 cm imbalanced MZI using an APEX206 optical spectrum analyzer. As expected, the FSR is approximately 62 pm, which corresponds to a sensitivity of ≈ 0.83 (GHz) -1 . Note that the displayed MZI spectrum takes also into account the injection loss into the integrated photonic circuit via the grating couplers (≈ 11 dB/coupler) and also the propagation loss (≈ 1 dB/cm) in the Si 3 N 4 waveguide.

C. OFI FM-to-AM noise performances

Fig. 4 b) shows a typical OFI signal obtained after OFI FMto-AM conversion using the integrated MZI. The photodiode was set with a gain of 20 dB. The NED can also be estimated as expressed in [START_REF] Norgia | Exploiting the fm-signal in a laser-diode smi by means of a mach-zehnder filter[END_REF]:

N ED = λ 2 V RM S V pp , (10)
where V pp is the peak-to-peak amplitude of the OFI signal and V RM S the OFI RMS noise. Here, the currently achieved NED is approximately 16 nm for 1 kHz bandwidth.

IV. CONCLUSION AND DISCUSSION

We have demonstrated the possibility to fully integrate an OFI FM-to-AM conversion for OFI applications. The sensitivity is 0.83 (GHz) -1 for a 2 cm imbalanced MZI. The NED is approximately 16 nm, which is approximately more than two orders of magnitude larger than [START_REF] Norgia | Exploiting the fm-signal in a laser-diode smi by means of a mach-zehnder filter[END_REF]. Nevertheless, the sensitivity could be enhanced in future design with longer imbalanced MZI and shorter laser-target distance for instance. In addition, noise performances can also greatly benefit from better injection performance by using edge couplers [START_REF] Marchetti | Coupling strategies for silicon photonics integrated chips[END_REF] instead of grating couplers and reducing the waveguide propagation loss down to a few dB/m as in [START_REF] Liu | High-yield, wafer-scale fabrication of ultralow-loss, dispersion-engineered silicon nitride photonic circuits[END_REF].

The advantages of the integrated OFI FM-to-AM conversion are its compactness, its ability to be electrically tuned and also to be thermo-regulated more easily than fiber based approaches. To achieve higher sensitivity, other type of filters based on integrated photonic filters can also be envisioned. This integrated OFI FM-to-AM conversion paves the way to more compact OFI FM-to-AM converters compared to fiberbased approaches.

ACKNOWLEDGMENT

The authors would like to thank A. Lecestre for her technical support. All fabrication and characterization of the integrated photonic MZI were supported by both ANR-20-CE42-0010 PICSONDE in cooperation with ACOEM, Thierry MAZOYER and LAAS-CNRS micro and nanotechnologies platform member of the French RENATECH network.

Fig. 2 .

 2 Fig. 2. Schematic diagram of one of the three OFI FM-to-AM optical feedback interferometry systems that includes one laser diode (LD), two photodiodes (PD), one fiber array, the Integrated Photonic OFI FM-to-AM filter based on a Mach Zehnder interferometer (MZI) and a speaker as remote target.

Fig. 3 .

 3 Fig. 3. Experimental setup with the fiber array and the integrated photonic chip including the 2 cm imbalanced MZIs shown in the inset (microscope photograph).

Fig. 4 .

 4 Fig. 4. a) Optical transmission spectrum of the 2 cm imbalanced MZI b)OFI output signal obtained after OFI FM-to-AM conversion using a) MZI for a speaker (target) vibrating at 100 Hz.


Cédric Hugrée

Yasmine Siblot

France: The dynamics of internal changes within a persistent class structure

Keywords:

Introduction

Surveys conducted in France since the beginning of the Covid-19 pandemic have revealed the profoundly unequal effects of this social and health crisis, on three levels. Firstly, there are cumulative inequalities between social classes, whether in health, work and employment, living conditions, or school [START_REF] Bajos | Les inégalités sociales au temps du Covid-19 [Social inequalities in the time of Covid-19[END_REF]. Secondly, the crisis highlights the impact of local disparities: The effects of the crisis are heightened in the working-class outer cities (Mariette & Pitti, 2020a[START_REF] Mariette | Covid-19 en Seine-Saint-Denis (2/2) : Comment le système de santé accroît les inégalités[END_REF], and are manifested differently in urban, peri-urban, or rural zones, and in regions. Thirdly, these class inequalities are intrinsically linked to gender relations, in both the domestic sphere and the workplace, and ethno-racial relations, in terms of access to healthcare, exposure to surveillance procedures, and racism.

These three aspects are central to the sociology of social class in France, a field in which recent studies have done a great deal to shed light on the effects of the pandemic. Here we follow these three threads to present the contributions and new insights offered by this field of research: its reassertion of the persistence of a class structure, notwithstanding the changes within it; the importance of taking into account territorial differences and scales of analysis, as well as movement flows and mobility; the need to articulate social class relations with other social relations in order to grasp the internal divisions and the processes of reproduction. We emphasise, across these three threads, the centrality of public institutions and the state in enhancing or attenuating class and power relations.

1.

The persistence of class structures in 21 st -century French society

In France, academic debate in the 1980s and 1990s was marked by the challenge to Marxism.

As in many other countries, the theory of the "end of social class" became popular. A number of sociologists interpreted the changes in employment and living conditions as a process of "middleisation" of French society, arguing that this resulted in a blurring of the material and symbolic boundaries separating the working classes from the middle and dominant classes.

But following a series of studies that documented renewed inequality, the issue of social class "returned" in the early 2000s [START_REF] Bouffartigue | Le retour des classes sociales : Inégalités, dominations, conflits [The return of social classes: Inequalities, dominations, conflicts[END_REF]. Though it takes a number of diverse approaches, this sociology is strongly influenced by the work of Pierre Bourdieu [START_REF] Coulangeon | The Routledge Companion to Bourdieu's "Distinction[END_REF]. His representation of the social space as a hierarchy determined by the amount and composition of capital remains pertinent. This approach became popular in particular thanks to sociologists' increasing use of the National Institute of Statistics and Economic Studies (INSEE)'s Occupations and Socio-occupational Categories (Professions et categories socio-professionnelles, PCS) classification, which enabled Bourdieu and his team to link social status to lifestyles. Forty years later, there are still many studies that draw on an updated version of this classification [START_REF] Desrosières | How to make things which hold together: Social science, statistics and the State[END_REF] to document the dynamics of class inequality. The widespread use of this tool has without doubt contributed to the relative consensus, well beyond the sociology that claims its inheritance from Bourdieu, around a representation of the social space as being divided into three broad areas. The dominant classes are usually identified with the label "cadres et professions intellectuelles supérieures" (translated here as: managers and higher intellectual professions), the middle classes with that of "professions intermédiaires" (intermediate professions), and the working classes with "ouvriers" (manual workers) and "employés" (service workers). "Artisans commerçants et chefs d'entreprise" (shopkeepers, self-employed tradespeople, and business owners) as well as "agriculteurs exploitants" (farmers) may be allocated to one or other of these three main social classes depending on the size of their business.

1.1 The dynamics of social class in France in the 21 st century A comparison of the class structure of contemporary French society with that presented by Bourdieu in the 1970s shows substantial changes in the arena of social status (see Table 9.1). At the beginning of the 2010s, two predominantly male groups showed a marked decline: There were fewer farmers and agricultural workers, and half as many unskilled manual workers as forty years earlier. The number of skilled manual workers had also fallen, although not as steeply. But above all, thanks to the boom in the lowest-skilled service sector jobs, the group of service workers, the vast majority of whom are women, had expanded to overtake that of manual workers. There were thus very real changes within the social groups at the bottom of the social space, but these did not point to an erasure of the boundaries with other classes. The substantial growth in employment in intermediate professions and higher managerial occupations helped to maintain and renew the social distance between classes. Because of a new dynamic of increasing wealth among the richest and more intense competition for access to the highest educational qualifications on the one hand, and the continuation of various forms of domination on the other, French society remains a class society. Rather than indicating the blurring of class boundaries, the statistical data reveal a dynamic of reproduction of inequalities. The economic repercussions of the 2008 financial crisis and the 2020 health crisis have accentuated forms of collective downgrading, prolonging the destabilisation of the Wage-Earning Society that began in the 1980s [START_REF] Castel | From Manual Workers to Wage Labourers: Transformation of the Social Question[END_REF]. In terms of economic capital, the apparent stasis in the inequality of salaries and living standards masks a dynamic whereby the richest are becoming wealthier, thanks to assets, and the income of the poorest is falling (INSEE, 2021). On the one hand, the richest 1% are the only group to have seen their income rise since 2013, thanks to an unprecedented rise in income from assets (particularly transferable securities) and a reduction in taxes. In 2018, the estimated average gross assets of unskilled manual workers' households were €15,000, compared to more than €500,000 for the households of those in the liberal professions. On the other hand, poverty is now concentrated among the unemployed and, for those who are employed, among the selfemployed and manual and white-collar employees on precarious (short-term, part-time, and temporary) contracts. In 2018 more than one in three unemployed people (38%) and more than one in five self-employed people (18%) were living below the poverty line, set at €1,063 per month. France is still one of the least unequal countries in Europe, thanks to its economic redistribution policies, although inequality in pre-tax income is very high. However, the dynamic of inequality aligns with the international trend whereby only the highest incomes capture the benefits generated by the growth in fixed and movable assets [START_REF] Piketty | Capital in the Twenty-First Century[END_REF][START_REF] Piketty | Capital in the Twenty-First Century[END_REF].

In terms of cultural capital, particularly educational capital, the situation appears very different from that of the late 1970s. Since the mid-1980s, a proactive education policy has set the objective of 80% of each generation achieving the baccalauréat, certifying completion of secondary education. Thirty-five years later this goal has been achieved, thanks in part to the creation of the "vocational baccalauréat." The youngest manual workers and service workers are consequently significantly more qualified than the oldest [START_REF] Poullaouec | An unquiet mobilization: Working-class families and school in French contemporary society[END_REF]. Among manual workers and white-collar workers under 40, there are often as many who hold the baccalauréat or higher education qualifications as those who hold only certificates from short vocational training or no qualifications. However, behind these profound shifts in access to qualifications, the structure of educational inequality persists. The intermediate professions and managerial and higher intellectual professions include overwhelmingly more people with higher qualifications and all of these social groups have intensified their use of higher education. At the beginning of the 2010s, 75% of children of those in managerial and higher intellectual professions had obtained a bachelor's degree or higher qualification, compared to only 20% of the children of unskilled manual workers, 31% of the children of skilled manual workers, and 35% of the children of service workers [START_REF] Lemaire | Les parcours dans l'enseignement supérieur : devenir après le baccalauréat des élèves entrée en sixième en 1995[END_REF].

Lastly, health inequalities, particularly in terms of mortality, also demonstrate the processes of reproduction of class inequality. While the life expectancy of all social groups has increased since the 1970s, this has not been a uniform development across the social space. Social inequality in relation to mortality primarily affects men and manual workers: Female managers are still the least affected [START_REF] Blanpain | Les hommes cadres vivent toujours 6 ans de plus que les hommes ouvriers [Male managers live 6 years longer than male workers[END_REF]. Between the period of 1976-1984 and the period of 2009-2013, the life expectancy at 35 for senior managers rose from 47.5 to 53 years for women, and from 41.7 to 49 years for men. Among manual workers, it rose from 44.4 to 49.8 years for women, and from 35.7 to 42.6 years for men. Furthermore, the disability-free life expectancy was ten years longer for male managers than for male manual workers, and eight years longer for female managers than for female manual workers.

France's tripartite social space

Debates about the categorisation and boundaries of social groups have pervaded the sociology of class in France since the 1970s: "classe ouvrière" or "classes populaires," "petite bourgeoisie" or "classes moyennes," "classes dominantes" or "supérieures," "bourgeoisie" or "élites." 1 These debates were revived with the new turn in sociology in the 2000s. At the bottom of the social space, a number of studies analysed the restructuring and fragmentation of the dominated groups. At the top of the social space, the new distinctions went hand in hand with a persistence of the boundary between managerial ("cadres") and non-managerial occupations, and the expanding role of international capital in the creation of dominant positions. Between these two poles of the social space, there was much discussion of the theory of the "loss of status" of the new "middle classes." Ultimately, notwithstanding the great plurality of approaches, a consensus emerged with the recognition of three main social classes in France.

Meanwhile they appear to be more internally diverse than they had been in the 1970s, because of various social evolutions, and because the attention to this internal diversity has grown among sociologists: Some consider this diversity in terms of differences between class fractions whereas others focus on micro-classes.

A number of studies have reopened the question of the boundary between the dominated and the middle classes. These studies documented the decline of the "working class" (in the singular) and the concomitant emergence of subaltern jobs outside of industry that are centred on interaction with the public. Olivier Schwarz (2011) proposes the term "working classes" (in the plural) as a way to acknowledge this restructuring and the diversity that it implies while also encompassing within the same term social groups that are in a similar position (service and manual workers), distinguished by their limited economic resources and their distance from established forms of legitimate culture. This reflection has influenced a number of studies [START_REF] Siblot | Sociologie des classes populaires contemporaines[END_REF], which show that although there is a marked gender segregation among them in the labour market, manual and service workers have in common their strong conjugal and family ties. These studies emphasise how the extension of education affects ways of seeing the social world and modes of consumption, and demonstrate the obsolescence of a sharp cultural separation between dominated groups and the middle classes. They also show that forms of class domination persist at work, in education, in living conditions, and in politics. While social and symbolic boundaries between classes remain, they have also been analysed with regard to the specific resources that the different sectors of the working classes possess. Thanks to the creation of a "vocational baccalauréat" in 1985, increasing numbers of the new generations of manual workers hold a secondary education certificate. As a result, a new level of educational qualification is now expected of manual workers, whereas for the older generations having a high school education had been a strong symbolic marker that distinguished technicians from manual workers [START_REF] Misset | Une nouvelle « élite des réprouvés » ? Les bacheliers professionnels industriels devenus ouvriers qualifiés [A new "reprobate elite"? Industrial baccalaureate graduates who have become skilled workers[END_REF].

Initially seen as a set of positions with indistinct boundaries, in France the group "cadres supérieurs" ("higher managerial occupations") was identified through a symbolic and political unification around the figure of the graduate working in a large company [START_REF] Boltanski | The Making of a Class: Cadres in French Society[END_REF]. In the early 2000s the specificity of this social group was once again called into question. Between the 1980s and the 2000s, internal changes were observed in this group, while a social distance from other employees persisted [START_REF] Bouffartigue | Cadres, classes moyennes : Vers l'éclatement[END_REF]. Managerial staff today are younger, more highly qualified, include more women, and are still the best paid employees. The data on this group indicate a shift in its centre of gravity toward engineers and commercial managers, and a gradual convergence between the dominant fractions. Managers in private companies are increasingly similar to one another, while those in the arts now appear closer to the intermediate professions in information technology, arts, and theatre.

This division of the French social space into three parts is still subject to debate. Recent research has indeed revealed splits that run through the social space, and a blurring of class boundaries in some practices. For example, there is a quite sharp contrast between the social characteristics of the self-employed, whatever their level of education, and employees. Moreover, in France the fact of working in the private or the public sector still underpins a division across classes in terms of contrasting relationships to public institutions, voting patterns, and cultural practices [START_REF] Hugrée | Differences between public-and private-sector employees following the managerial turn in European states[END_REF]. Finally, in comparison with the 1970s the disparities in modes of consumption between working-class and middle-class households, and between these two and the dominant classes, are much less marked (Sinthon, 2019).

Social mobility, migrations, and changes of scale

Beyond the question of the porosity of class boundaries, analysis of the social structure in France has been profoundly reinvigorated by the recognition of the social and spatial dynamics that run through and reshape classes and relations between classes: The analysis of social and spatial mobility improves the understanding of class structure on the national but also on the international, especially European, scales.

Social mobility: Crossing the boundaries

The study of social mobility is still dominated in France by statistical analyses that quantify changes in the level of social fluidity in French society over the long term, following the work of John Goldthorpe [START_REF] Vallet | Intergenerational mobility and social fluidity in France over birth cohorts and across age: The role of education[END_REF]. Recently a series of studies has connected the analysis of intergenerational trajectories to the institutions-not only educational but also sports or cultural institutions-that make them possible, and has included other approaches and methods [START_REF] Pagis | Observer les mobilités sociales en train de se faire. Microcontextes, expériences vécues et incidences socio-politiques [Observing social mobilities in the making. Micro-contexts, lived experiences and socio-political implications[END_REF].

The revitalisation of the sociology of class began with a more fine-grained quantification of social trajectories, based on a more detailed version of the Occupational and Sociooccupational classification (PCS). This approach reveals the "small social shifts" [START_REF] Lahire | La culture des individus[END_REF] that are still the most common intergenerational movements in France. For example, in France in the 2010s, most of the children of skilled manual workers became manual workers, but some also gained service positions. A significant number of these are jobs in the public sector: More children of skilled manual workers than of unskilled manual workers become service workers in the public sector (13%), such as cleaners in schools or police service staff, and more of them enter the female-dominated intermediate occupations in the public sector (8%, as primary school teachers and nurses) than the male-dominated groups of technicians or work supervisors. Using a micro-class approach, recent works underline the strength of professional reproduction, while documenting its variability and the forms of "micro-mobility" [START_REF] Jayet | The microclass analysis of social mobility: An application to French data[END_REF].

This fine-grained quantification of the social shifts between generations, which takes seriously the influence of gender and the relative position in the class of origin on social trajectories, makes for a fertile dialogue with ethnographic studies. Field studies also help to shed light on the cultural and symbolic effects of small shifts and on the diversity of pathways via which individuals cross social boundaries. Observation of trajectories over the long term above all offers an opportunity to better grasp the amplitude of movements, on both material and symbolic levels, their reversibility, and the meaning individuals ascribe to them. From this point of view, analysis of the trajectories of "class exiles" (transfuges de classe) too often reduces upward movements by children of the working classes to an experience of uprooting, whereas repeated crossing of social boundaries generates daily accommodations between their original environment and the space they arrive in [START_REF] Lagrave | Se ressaisir. Enquête autobiographique d'une transfuge de classe féministe [Getting it together: An autobiographical investigation of a feminist class exile[END_REF].

This return to biographical and longitudinal analyses has led to a broadening of analysis beyond the educational and work spheres: Many "discreet" shifts within the labour market appear to be less individual than familial or even collective when they are articulated with strategies around housing and assets, and with domestic practices, as localised studies show.

Changing scale: Localising class boundaries

The development of monographs focused on class relations within an urban district or a rural municipality, along with consideration of housing and sociability, have resulted in growing attention to scales of analysis and local variations in class structure. National-scale studies tend to erase these specificities, and the many studies conducted in large conurbations, particularly the greater Paris region, tend to generalise the results thus produced, whereas in fact there are substantial regional differences in class structure.

Studies of rural areas have developed a "localised approach to social relations" (Bruneau, Laferté, Mischi, & Renahy, 2018), in a critical extension of Bourdieu's analysis of the "effects of place" that draws on research in history and geography and on the practice of ethnography. This approach reveals the geographical variability of the internal composition of social classes, and of the forms of relations between them. In the rural region of eastern France explored by Bruneau et al., the position of the dominant classes is based primarily on economic capital, but also on local prestige (business owners, liberal professions, "gentrified" farmers). The involvement of the "technical and cultural lower middle class" (teachers, technicians) in local social and political life varies widely with generation: The youngest among them work there but live elsewhere, reinforcing the local social divides. In this local space, the families of manual workers who have social capital linked to their strong local roots (which the researchers call "capital of autochthony") and limited economic assets (timber industry and railway workers) coexist with very poor households (stigmatised as "charity cases"). The power relations are starkly evident but may be tempered by inter-class practices such as hunting in local associations.

This localised approach to the rural space in its turn illuminates research on urban spaces. Thus, research on the greater Paris region reveals the diversity of configurations of social space but also the importance of the scale of analysis. At the level of municipalities or arrondissements, notwithstanding the continued rise in the proportion of people from the middle and dominant classes, and the fall in that of the working classes, especially manual workers, there is still a tendency for classes and fractions of classes to coexist. The strongest level of social segregation appears in the areas where the richest fractions of the dominant classes live [START_REF] Préteceille | La ségrégation sociale a-t-elle augmenté ? La métropole parisienne entre polarisation et mixité [Has social segregation increased? The Parisian metropolis between polarisation and diversity[END_REF]. These processes of segregation and neighbour relations between classes are thus highly variable. Two studies describe contrasting forms of class power relations in the Île-de-France region. In the more recently constructed neighbourhoods of the West outer city, close to the Défense business district, the financial fractions of the dominant classes form a tight enclave, explicitly creating a material and symbolic separation between themselves and the working classes, particularly the poor and racialised communities living in social housing in neighbouring municipalities [START_REF] Cousin | Entre-soi mais chacun chez soi : L'agrégation affinitaire des cadres parisiens dans les espaces refondés [Amongst themselves but each at home: The affinity aggregation of Parisian executives in reconfigured spaces[END_REF]. By contrast, in the municipalities of the East outer city that are currently being gentrified, the cultural fractions of the middle and dominant classes, who are locally the dominant class fractions, are reshaping the space so that it conforms to their tastes and aspirations, and increases in market value. At the same time, they seek to "facilitate cohabitation" with the working classes who have lived for a long time in these former industrial towns, by, for example, campaigning for better education conditions in state schools [START_REF] Collet | Rester bourgeois. Les quartiers populaires, nouveaux chantiers de la distinction [Staying bourgeois : Working-class neighbourhoods[END_REF].

Migrations: Crossing spatial and social boundaries

Adopting a frame of analysis complementary to the localised approach, other studies have moved in the opposite direction in their scale of analysis. By broadening the focus beyond national borders, they have sought to understand the effects of international migrations on the power relations and boundaries between classes: Combining the sociology of social class and the sociology of migration, these studies bring new insights to the analysis of the national social space in France. This change of scale presents a serious conceptual and methodological challenge, given that the sociology of class is shaped by the national context owing to the classification it is based on and the role of the state in the reproduction (or mitigation) of inequality. The concepts of "international capital" (Wagner, 2020) and "migrating social capital" (Bréant, Chauvin, & Portilla, 2018) help to construe the "globalisation of social classes," both "top-down" and "bottom-up."

Studies of the dominant classes were the first to take this approach [START_REF] Pinçon | Sociologie de la bourgeoisie[END_REF][START_REF] Wagner | La mondialisation des classes sociales [The globalisation of social classes[END_REF]. The richest fractions of the French dominant classes (the upper bourgeoisie and aristocracy) have long had an international capital that is at once embodied (fluency in foreign languages, transnational sociability), material (homes abroad or in internationalised vacation resorts such as the Côte d'Azur, belonging to highly selective clubs such as the Union Interallié), and institutionalised (education at private or public international schools such as the Lycée international in Saint-Germain-en-Laye). Alongside these established forms of social capital there are competing forms associated with corporate globalisation that are held by the senior executives of large companies: international qualifications in finance and management from French Grandes Écoles following training in English, and working as an expatriate in transnational business spaces such as Dubai, where national and racial hierarchies are such that the status of these French executives is higher than that of their peers from economically and symbolically dominated countries (Le Renard, 2019).

While the middle and working classes are less mobile than the internationalised fractions of the dominant classes, some of their members also cross-national borders by emigrating. But in the case of France the principal dynamic in these intermediate and dominated regions of the social space has been one of substantial immigration since the late 19 th century. These flows have been subject to stringent restrictive policies since the 1970s. However, they continue to exist and produce complex social mobilities that accompany spatial mobility, where the position in the national and social space of origin is combined with that in the national and social space of settlement [START_REF] Sayad | La double absence[END_REF]. The proportion of immigrants is thus higher among the working classes (the majority of the men being manual workers and the women service workers); they are largely concentrated in low-skilled sectors (construction, personal care, cleaning) even though some of these migrants come from the middle class or upper fractions of the working classes in their country of origin.

The social space in France is still powerfully structured by the nation-state, but this research helps to situate classes and fractions of classes in relations that incorporate a transnational dimension, and to think, at least partly, in terms of "international class relations" (Bidet, in Bréant, Chauvin, & Portilla, 2018).

Articulating social relations: Internal boundaries within classes, reproduction, and accumulation

Alongside the return to research on social class, during the 2000s a number of French studies discovered or rediscovered the analysis of gender relations and racial divisions, contributing to the development of intersectional approaches. Broadening the understanding of modes of domination, these studies drew attention to the way that, in different practices and social scenes, class relations are articulated with other social relations. This enriches the relational analysis of the social space through a new scrutiny of the foundations of material and symbolic boundaries between fractions of classes, and sheds further light on processes of reproduction and accumulation.

Intra-class gender and race divisions: The role of work and domestic space

Since the 1970s, studies based in materialist feminism have demonstrated the significant role of gender inequality in the domestic sphere and the workplace. This inequality is still fundamental, but is nevertheless socially differentiated according to class, and its modalities have altered to some extent. For example, gender segregation at work is particularly marked among subaltern service workers (Table 9.2): It is among these groups that the majority of the most heavily female-dominated occupations are found (cleaners, care workers, secretaries) as well as the majority of the most male-dominated occupations (manual workers in construction and transport). The naturalisation of roles and skills that structures recruitment and staff management practices is quite explicit here [START_REF] Avril | Saisir les dynamiques de genre en milieu populaire depuis la scène du travail subalterne [Grasping gender dynamics in working-class settings from the stage of menial labour[END_REF]. But for both women and men, this gender segregation goes alongside poor chances of acceding to intermediate, and still less dominant, occupations, because their skills are not recognised and such occupations demand qualifications that involve lengthy studies. While there is more mixing in the middle and upper levels of the workforce, it remains limited (as evidenced by the categories of technicians and nurses), and gender hierarchies play out through specialisations (human resources versus finance; gynaecology versus surgery) and through the persistence of major inequalities in careers despite the spread of "market feminism" in private companies [START_REF] Pochic | Féminisme de marché et égalité élitiste ? [Market feminism and elitist equality[END_REF]. This socio-occupational hierarchy structures women's social space in such a way that it is much wider at the base and smaller at the top than that of men: women are mainly concentrated among service workers and intermediate professions offering few possibilities of ascension; men are often manual workers, but also technical intermediate professionals and managers, and have better careers. This imbalance is intensified by domestic practices and family socialisation that are differentiated by class and gender. The majority of the household tasks is undertaken by women in all social groups. These tasks weigh particularly heavily, however, on female manual and service workers, who are often forced to leave their jobs when they have young children because they have less control over their work hours and are less well paid [START_REF] Bentoudja | Être parent : des cadres aux ouvrières, plus de conséquences sur l'emploi des femmes[END_REF], while middle-and dominant-class households delegate part of their domestic work to unskilled service workers. The gender inequalities internal to the different classes thus reinforce class inequalities between women: In 2010, female white-collar employees, female manual workers, and unskilled female white-collar employees did 2 hours 35 minutes more domestic work per day than the men in their category, but also 1 hour 5 minutes more than women in managerial positions in the private sector [START_REF] Brousse | Travail professionnel, tâches domestiques, temps « libre » : quelques déterminants sociaux de la vie quotidienne [Professional work, domestic tasks, "free" time: Some social determinants of daily life[END_REF].

Sociohistorical studies have provided long-term and varied documentation of the segregating and discriminatory effects of the heritage of colonialism and migration policies on immigrants and their descendants, both men and women. While they affect all social groups, these processes are particularly evident among the working classes, especially in certain sectors of the economy. Ethnic and racial management of the workforce was long a characteristic of industrial labour, for example in the mining [START_REF] Perdoncin | Tous au charbon : Inégalités de carrières et mobilités ouvrières dans la récession (Nord-Pas-de-Calais, 1945-1990)[END_REF] and automotive sectors [START_REF] Pitti | Catégorisations ethniques au travail. Un instrument de gestion différenciée de la main-d'oeuvre [Ethnic categorisations at work. A tool for differentiated workforce management[END_REF]. Immigrant workers, mainly of Algerian and Moroccan origin, in these sectors were assigned to the least skilled jobs, on the most precarious contracts, without any prospects of promotion. This mode of management indirectly contributes to the reproduction of the most stable fractions of the manual workforce and reinforces the internal divisions among subaltern employees. The hierarchy between stable and precarious employees can be intensified by racial ascriptions, as in the heavily male-dominated construction sector, where there are legal constraints weighing on (sometimes undocumented) foreigners, and where stereotypes linked to (real or supposed) origin structure the division of work between "French," "Portuguese," "Arabs," and "Mamadous" (a common given name used generically as a pejorative term for West Africans) [START_REF] Jounin | Chantier interdit au public. Enquête parmi les travailleurs du bâtiment [Construction site closed to the public: A survey of construction workers[END_REF]. In the highly female-dominated sectors of personal care, such as home care assistant for dependent elderly people, these modes of management are less evident. But divisions involving social resources and racial ascriptions emerge between service workers. On the one hand, non-immigrant women from stable working-class backgrounds (former retail assistants and salespeople) who are rooted in the local space limit their involvement in care work with the most dependent elderly; on the other hand, some immigrant women who are better qualified but in precarious situations take on this work to enhance their position through closer proximity to the intermediate professionals who supervise their work [START_REF] Avril | Les Aides à domicile. Un autre monde populaire [Domestic help: Another working class world[END_REF].

Exploration of the way class relations are articulated with race and gender relations reveals situations where forms of power are aggregated, but also more ambivalent combinations: It is therefore essential to grasp these articulations in order to understand the dynamics that are reshaping the social space, as well as the processes of reproduction.

3.2 Social reproduction: New forms of selection in a restructured education system Within an education system that is formally unified up to the end of secondary education, there are in fact numerous options and tracks that ensure the reproduction of educational inequalities at all levels of the education system.

To begin with, as soon as they show evidence of early difficulty in basic learning, children from the working classes, especially when they are migrants or have immigrant parents, are quickly directed toward vocational education. This educational sector remains separate from general education and recruits children almost exclusively from working-class backgrounds.

For both boys and girls, the vocational track is strongly influenced by gender stereotypes [START_REF] Palheta | La domination scolaire. Sociologie de l'enseignement professionnel et de son public[END_REF].

However, the children of immigrants by no means all have the same experience of educational inequality [START_REF] Ichou | Les enfants d'immigrés à l'école. Inégalités scolaires, du primaire à l'enseignement supérieur[END_REF]. As for French-born people, the parents' qualifications and their social assets (prior to migration) are the key resources in their educational career. This is partly why the children of immigrants from West Africa and Turkey, whose parents are low-skilled, are more likely to experience early educational inequality than those of immigrants from South-East Asia and China, who more often come from families who had financial or cultural resources before migration. More broadly, minorities tend to be more segregated at school than in their residential environment in France, owing to practices of parents avoiding schools in working-class areas.

At the other end of the educational scale, the Preparatory Courses for Grandes Écoles (CPGE) 2 that give access to the elite higher education of the grandes écoles are the most elitist educational institutions. The recently declared proactive policy to "open" these places to students from schools in "deprived" neighbourhoods comes up against recurrent campaigns by the elites who studied there. When questions of the social openness and racial diversity of the French elites became a political issue, these elite schools created small local social accessibility programmes while at the same time raising their entrance requirements and manoeuvring to avoid introducing quotas for scholarships at each institution [START_REF] Pasquali | Héritocratie : Les élites, les grandes écoles et les mésaventures du mérite (1870-2020)[END_REF].

The various French higher education institutions thus contrast strongly with one another in their recruitment in terms of social, educational, and migration background. The professional tracks that recruit students from working-class backgrounds, such as post-secondary vocational courses, sit alongside elitist institutions like the CPGE. Until the 1990s, in French universities only healthcare-related courses (medicine, dentistry, pharmacy) had the right to strictly select their students. The introduction in 2018 of the Parcoursup program, which manages the applications and allocation of students leaving secondary school to begin higher education courses, has extended forms of selection for university entrance, and reinforced the disparities between tracks.

Accumulation of assets: The strictly limited effect of public policies

The changes in housing policy since the late 1970s, which facilitated access to property ownership and investment in property to the detriment of the construction of public housing, reshaped the social and symbolic gaps between classes and class fractions, but without eliminating inequalities in the accumulation of assets.

Although the number of households who own their homes is lower in France than in most EU countries, incentive policies introduced since the 1970s have democratised access to home ownership. While still lower than the proportion of middle-and dominant-class households that own their own home, the proportion of working-class households who own or are buying their home has continued to rise since that time, with the exception of immigrant households, where the proportion of home-owners has fallen steadily since the early 2000s [START_REF] Domergue | Les déterminants du statut d'occupation des logements et de la mobilité résidentielle des ménages immigrés [Determinants of housing tenure and residential mobility of immigrant households[END_REF]. But for the working classes, virtually the only the property they can afford to buy is concentrated in peri-urban zones, using loans that cover almost the entire value of the property. Members of these households therefore find themselves geographically distant from their jobs, and with the part of their budget that is spent on housing unavailable for other expenses. This access to property has thus led to forms of disillusionment [START_REF] Lambert | Tous propriétaires !" L'envers du décor pavillonnaire[END_REF] that vary depending on social trajectory and previous housing experience. Here, class boundaries intersect to some extent with racial discourses. In her study, Anne Lambert found that the majority of the families between which tensions arose lived in the least well-appointed part of the housing development; they had come mainly from large social housing estates and originated in North Africa, sub-Saharan Africa, and South-East Asia. Although they might seem socially close to the upwardly mobile or stable fractions of the working classes, the members of the white lower middle class on the new estate adopted a racial register as a way of symbolically distinguishing themselves. This micro-segregation thus opens material and symbolic divides within the working classes, but also with other social classes.

The urban renewal policy of the early 2000s sought to transform the boundary between tenants on the large social housing estates and the owners of small houses on the outskirts of the city and in rural areas. This policy offered the opportunity to buy property in social housing estates.

In the suburbs of Lyon, studied by Pierre [START_REF] Gilbert | Devenir propriétaire en cité HLM : Petites promotions résidentielles et évolution des styles de vie dans un quartier populaire en rénovation [Becoming a homeowner in a low-income housing project: Small residential developments and changing lifestyles in a working-class neighbourhood undergoing renovation[END_REF], this offer did not attract the middleclass buyers expected by the social housing landlords. It was in fact members of the stable fractions of the working classes who applied for these state-sponsored sales. By becoming home-owners in a social housing estate, these households gained a little mobility in housing while preserving the social relations essential to their social lives. Their trajectory, and the work this new housing required of them, led them to invest intensively in their private sphere and to keep the neighbourhood and those residents deemed less respectable at a distance. Thus, while urban renewal modified the population of inner-city estates by slowing down the departure of stable or upwardly mobile groups, the fact that they lived alongside poorer groups did not lead to the two becoming socially closer.

Progressively excluded from property ownership in the centre of France's major cities, the working-classes households who own their own homes rarely see their property rise in value, unlike dominant-class households. The latter usually amass various forms of assets (property and movable assets, personal and business assets), and create entire family strategies for the accumulation of wealth. The transfer of these assets within the family usually works to the advantage of men and the detriment of women [START_REF] Bessière | The Gender of Capital. How Families Perpetuate Wealth Inequality[END_REF]. Bessière and Gollac show how the institutions that govern the division and transfer of assets, and the legal professionals who organise it, authorise and foster legal and family arrangements that deepen gender inequality and maintain class inequalities. Men of the dominant classes systematically benefit more from the formal and informal rules governing the division of all forms of assets on the occasion of separation, succession, or transfer of a business. Thus, among those who possess financial capital, it is family organisation and the mechanisms of the law at the time of transfer that explain the recent growth in inequality of assets between women and men.

* * * While it is still entirely appropriate to distinguish three main divisions of the social space in France, the outlines and internal composition of these have altered radically since the 1970s.

Varying the scale of analysis by taking both national and local approaches, taking into account the effect of migration and social mobility, and integrating the way gender and race relations combine with class relations into a relational analysis make it possible to get a better understanding of the dynamics of power relations, but also of the internal divisions within classes.

One crucial contribution from recent studies on social class in France has also been to note that, in the first quarter of the 21 st century, public institutions and the state still play a central role in processes of reproduction and accumulation: While public services or the existence of an employment sector may mitigate inequalities, education and economic policies, along with migration and family policies, are far from challenging the power relations that structure class society.

At the political level, the question of inequalities between social classes is certainly often present, as during the Yellow Vests movement in 2018 and 2019 or through the claims of the workers worn out by the pandemic since 2020. But the classist scheme seems to have lost its mobilisation strength as parties and trade unions no longer make it the central mode of defining their action and as various forms of social and symbolic domination become more obvious [START_REF] Hugrée | Classes[END_REF].

1 Few of these terms have an exact equivalent in English. As discussed in what follows, in France there is a significant difference between "classe ouvrière" and "classes populaires," which the terms "working class" and "working classes" attempt to render. Similarly, the English term "middle class" is not an equivalent to the French term "classes moyennes," which is regularly the focus of disputes around definition both among academics and in the public sphere. Finally, the term "dominant class" used by Mills to define the "classes supérieures" or "classes dominantes" is sometimes preferred to "ruling class," which is closer to "classe dirigeante."

2 Classes Préparatoires aux Grandes Écoles: intensive two-year post-secondary course preparing for entrance to the elite higher education institutions.

Table 9 .

 9 1 Changes in various social groups in the working population

	1982	2007	2019

Note: In 2019, 19.6% of persons employed were manual workers. Scope: France including overseas territories except Mayotte, population of households, population in employment Source: INSEE, Labour Force Surveys, long series on the labour market

Table 9 .

 9 2 Main socio-occupational groups among women and men (2020) (%)

	Women	Men	All

  The grandes écoles (Polytechnique, Sciences-Po Paris, École des Hautes Etudes Commerciales, École Normale Supérieure…) play a central role in the selection, training, and reproduction of economic, political, administrative, and intellectual French elites.


Yoann Jus
email: yoann.jus@cetim.fr

Amr Nassef

Hubert Lejeune

Cedric Alzard

Raphaël Turpin

Guy Lebret

Saïd Moussaoui

Franck Plestan

Digital

Digital twin of a hydraulic loop: application to fault detection and parameter estimation

Keywords: Jumeau numérique, Machine Learning, Détection de défauts, Estimation de paramètres, Boucle fluidique

  L'archive ouverte pluridisciplinaire

Thème visé : « Jumeaux numérique et calibration de modèle »

Jumeau numérique d'une boucle hydraulique : application à la détection de défauts et estimation de paramètres

Or Digital twin of a hydraulic loop: application to fault detection and parameter estimation Y. Jus 1 , A. Nassef 2 , C. Alzard 1 , R. Turpin 1 , H. Lejeune 1 , G. Lebret 2 , S. Moussaoui 2 , F. Plestan 2 1 Cetim, 74 route de la Jonelière -CS 50814-44308 Nantes Cedex 3, France 2 Ecole Centrale Nantes, LS2N UMR CNRS 6004, F-44321 Nantes, France

Introduction

Hydraulic systems are used in several industrial applications, including manufacturing, automobiles and energy for example. Monitoring the condition of hydraulic equipment is essential to maintain high productivity and reduce the costs of system process. Best practices have been evolving from reactive maintenance to predictive maintenance, with intelligent fault diagnosis, using recent advances in industrial Internet of Things, artificial intelligence, edge computing, etc. Beyond design, the digital twins offer advantages in predictive maintenance and provides a new paradigm for fault diagnosis. The CETIM's "Fluid & Sealing Technologies" department proposes to present its R&D work to illustrate the concept of a digital twin of a fluidic loop. Digital twin includes the virtual space, physical space and the interaction between the two. The virtual model mimics its physical entity in terms of geometry, behaviors, and rules, it can reflect, predict, and verify the performance of physical entity.

Our physical and simulated hydraulic system

For this purpose, a closed hydraulic loop has been instrumented. It is equipped with a pump, a heat exchanger, a tank, a regulation valve and three piping sections. The hydraulic loop diagram is shown in Figure 1. Its function is simply to provide the flow, pressure and/or temperature requested by the operator. Several control devices have been added to generate some defects artificially in the future. Our hydraulic system operates as follows. The pump, driven by the variable speed electric motor, controls the working fluid through the circuit. A control valve is used to regulate the fluid flow. A heat exchanger, cooled by a cooling unit, is placed in parallel to the control valve to regulate the temperature. The loop is pressurized by a tank connected to the site's compressed air circuit. Some components have been designed to generate malfunctions: heating collars, one manual valve placed at pump suction (to generate cavitation) and one placed upstream of the exchanger (to degrade its performance). Finally, an air pressure regulator allows to disturb the pressure level of the whole loop.

Methodology

The objective of the study is to identify process faults (pipe clogging, exchanger performance degradation, etc.). These defects are manifested by a difference between the process variables (pressure, flow, and temperature) evaluated by the model and measured on the physical loop. If the deviation exceeds a certain threshold, an alarm is activated (see Figure 2). To estimate a state change, a methodology based on Machine Learning models has been developed. A large set of data for each default is typically required, which is hard to obtain from any physical hydraulic system for an exhaustive combination of faults and control actions. Instead, a simulation model of the system was used with Simcenter Flomaster software. In a first approach, transient phenomena have not been considered. The 1D model of the hydraulic loop is shown in Figure 3. The fault location strategy is based on two decision trees. Indeed, some parameters are strongly correlated (pump rated flow and pump rated head). After investigating this coupling, we find out that there some operating points where this coupling is weak, which allows the second decision tree algorithm to distinguish between them easily.

Once the defect is located, a Support Vector Machine model is used to estimate the new characteristic of the component. An estimation model exists for each of the 6 components of the loop. Figure 4 illustrates the complete structure of the proposed machine learning algorithm.

Results and discussion

The database contains 6 930 samples: 77 operating point (control point) x 6 hydraulic loop parameters (component vector) x 15 different values for each. We used 80% of the database to train the model and 20% to test the model. Figure 5 shows the results when the proposed algorithm was tested against the operating points with which it was trained. Another database was created with 300 new different operating points to verify the relevance of our approach. Two options were available to run the test: use the nearest operating point's database or use the four surrounding operating points. For example, Figure 6 shows a situation when the test sample is at the operating point (star) of pump speed=80% (800 rpm) and the valve opening is equal to 33%. The selected operating point (black circle) in the database corresponds to pump speed = 80 % and valve opening =30%. The four surrounding points are included in the red box. The table above shows the first decision tree's result using the two approaches.

Conclusion and future work

This first exploratory work should be continued because the proposed algorithm can estimate only a single change (and not a succession of state of the system). Indeed, one parameter changes have been considered while all the other parameters remain the same as their nominal (known) values in the Machine Learning model construction. Once the model will be reliable, it can be tested with experimental data from the loop. Another idea is to mix the simulated data with a small amount of the experimental data (hybrid approach) to study whether the performance of the model is improved or not.

Thanks to its ability to estimate the system state changes, the proposed algorithm is the first step to have a live digital replica of the hydraulic loop.

Figure 1 :

 1 Figure 1: Hydraulic loop diagram (left) and physical bench (right)

Figure 2 :

 2 Figure 2: Architecture of the proposed fault diagnosis system

Figure 3 :

 3 Figure 3 : Case study 1D Flomaster model

Figure 4 :

 4 Figure 4 : Full structure of the proposed solution

Figure 5 :

 5 Figure 5 : Detection and Estimation Result of proposed algorithm

Figure 6 :

 6 Figure 6 : Operating points map and results at different operating point than trained ones


Kristiaan Kerstens

Jens Krüger

Zhiyang Shen

Localized technological change

Keywords: Technological change, Learning, Convexity, Nonconvexity I

. Then, we succinctly sketch the economic theory and implications of such a notion of localized technological change in detail through the work of Acemoglu, Antonelli, and others. Finally, we provide attention to the small rather recent literature that has provided an operational definition of this notion of localized technological change and its implications using nonparametric production frontier estimations rooted in activity analysis. We summarize this small empirical literature and sketch some avenues for future work.

Introduction

The origin of the concept of localized technological change perhaps dates back to the rather famous [START_REF] Atkinson | A New View of Technological Change[END_REF] article. This article does not provide a formal definition, but heuristically contrasts technological progress that is shifting the entire production function outwards with a local notion of technological progress that is locally shifting a part of the production function outwards or different parts to a different extent. It explicitly refers to activity analysis as the origin of the neo-classical production function: "as the number of production processes increases (in an activity analysis model), the production possibilities can be more and more closely approximated by a smooth, differentiable curve. But the different points on the curve still represent different processes of production, and associated with each of these processes there will be certain technical knowledge specific to that technique. […] that if one brings about a technological improvement in one of the blue-prints this may have little or no effect on the other blue-prints." (p. 573).

Learning is indeed a major source of technological knowledge and competence, but it does not concern the whole range of techniques available on each isoquant as implicitily suggested by the Arrovian intuition [START_REF] Arrow | The Economic Implications of Learning by Doing[END_REF]. As a matter of fact, it enables to improve only the specific techniques practiced at each point in time. Consequently firms, that rely on the tacit knowledge acquired by means of learning processes to innovate, can introduce new technologies only in the proximity of the techniques in place at each point in time. [START_REF] Stiglitz | Learning to Learn, Localized Learning and Technological Progress[END_REF] spells out the link with the theory of learning. Technological progress is localized because the learning is localized. Learning by doing and learning by learning are examples of economies of specialization that provide a kind of externality. Stiglitz (1987: p. 125 and 133-134) stresses the necessity to analyze these nonconvexities and the resulting imperfect competition and multiple equilibria.

II.

Localized Technological Change: Conceptual and Theoretical Developments

The notion of localized technological change plays an important role in the microeconomic and industrial economics framework developed by [START_REF] Antonelli | The Economics of Localized Technological Change and Industrial Dynamics[END_REF][START_REF] Antonelli | The Economics of Innovation, New Technologies and Structural Change[END_REF][START_REF] Antonelli | Localized Technological Change: Towards the Economics of Complexity[END_REF][START_REF] Antonelli | The Knowledge Growth Regime: A Schumpeterian Approach[END_REF] in a series of books, complemented by some articles. The purpose of his work is to develop a unifying framework in which the Schumpeterian notion of technological innovation as a creative response and the model of localized and directed technological change can be integrated. In a Schumpeterian view firms are in an out-of-equilibrium state of product and factor markets and are inducing changes by innovation. Firms either react in an adaptive way by moving on the existing map of isoquants, or creatively by the introduction of technological changes as a creative response to mismatches between planned and actual conditions of factor and product markets. The latter option is associated with knowledge externalities and if effective learning processes are available, that make the generation of new technological knowledge possible at low costs with the consequent introduction of changes in the input-output space.

The analysis of this creative response based upon localized technological change presupposes the central role of knowledge as an enabling factor. In the localized technological change approach to this creative response, knowledge plays a double role: (i) it is a necessary condition for the creative response to happen, and (ii) it shapes the direction of technological change. The introduction of biased technological change directed at increasing the output elasticity of the factor that is locally abundant has strong effects in terms of augmented total factor productivity and competitive advantage. The intensive use of a production factor that is locally cheaper with respect to other factor markets where competitors are based is a major source of competitive advantage.

The stock of technological knowledge capitalized as a component of the wide concept of capital is the current abundant factor and a new source of competitive advantage for advanced countries. The dynamics of factor cost equalization together with the globalization of factor markets has made capital available to industrializing countries at cost and conditions close to advanced countries. However, the relative distribution of the stocks of technological knowledge is far more asymmetric in that it is far more abundant in advanced countries because of the unique availability of quasi-public knowledge and the high quality of knowledge governance mechanisms. This leads to persistent differences of advanced and developing countries which are in the focus of macroeconomic growth theory. [START_REF] Acemoglu | Localised and Biased Technologies: Atkinson and Stiglitz's New View, Induced Innovations, and Directed Technological Change[END_REF] relates the notion of localized technological change to the work of induced innovation and directed (factor-biased) technological change [START_REF] Acemoglu | Directed Technical Change[END_REF][START_REF] Acemoglu | Equilibrium Bias of Technology[END_REF] in the context of aggregate growth. At the center of this work is a production function of the form 𝑌 𝐹 𝐴 𝐿, 𝐴 𝐾 where 𝐴 and 𝐴 are distinct technology parameters which augment the production factors labor 𝐿 and capital 𝐾 differentially and therefore expand the production possibilities in different directions. Thus, the frontier function advances unevenly in the direction of labor if 𝐴 grows faster than 𝐴 and advances in the direction of capital if 𝐴 grows faster than 𝐴 . Therefore, the frontier function needs not evenly expand in all input directions.

If an innovation leads to an advancement of the frontier function at a particular capitallabor ratio and also improves the frontier function at neighboring capital-labor ratios as pretended in Figure 2 of [START_REF] Atkinson | A New View of Technological Change[END_REF] but leaves the frontier function at more distant capitallabor ratios unaffected then this is associated with the notion of directed technological change with differential changes of 𝐴 and 𝐴 as pointed out by Acemoglu (2015, p. 444).

The basic concept of directed technological change is exploited by Acemoglu in a vast number of papers, part of which is reviewed in the second part of [START_REF] Acemoglu | Localised and Biased Technologies: Atkinson and Stiglitz's New View, Induced Innovations, and Directed Technological Change[END_REF]. Particularly interesting is the analysis of [START_REF] Acemoglu | Productivity Differences[END_REF] where they show that technologies developed in advanced countries are less suitable to the factor endowments of developing countries. The reason is the absence of incentives for advanced countries to invent technologies adapted to the factor endowments of developing countries because of the weak institutions for property rights enforcement there. This explains the large and persistent productivity differences between both groups of countries.

III. Localised Technological Change: Empirical Work

Turning back to micro-economic empirical work in this section we build upon axiomatically founded nonparametric deterministic frontier-based technologies (see [START_REF] Färe | Production Frontiers[END_REF], [START_REF] Hackman | Production Economics: Integrating the Microeconomic and Engineering Perspectives[END_REF]). The use of these methods allows in principle to define precise measurement tools for measuring the notion of global and localized technological change. This work is based on the definition of discrete-time productivity indices (e.g., the Malmquist) and indicators (e.g., the Luenberger) which can be decomposed into technical change (i.e., movements in the production frontier) and efficiency change (i.e., movements towards or away from the production frontier). The literature is immense and continues expanding. To give some flavour of this literature, we make a selection of the few studies we are aware of that explicitly appeal to the notion of local technological change. [START_REF] Bernard | Technological Leadership and Variety: A Data Envelopment Analysis for the French Machinery Industry[END_REF] examine the innovators and technological change in French machinery industry using a nonparametric approach under a convex production technology. They argue that the technological knowledge and local technological change both have impacts on the evaluation of firms and the innovators can promote technological progress in a particular field. [START_REF] Cantner | Localized Technological Progress and Industry Structure: An Empirical Approach[END_REF] also apply a nonparametric convex approach to analyze the innovations in German industries. They empirically measure technical efficiency and the shift of production technology for machinery, electronics and chemical corporates. Their results indicate that the technological progress is localized. Naturally, the assumption of heterogeneity among industries prevails which is opposite to the homogeneous production functions advocated by neoclassical theory. Similarly, [START_REF] Westermann | Localised Technological Progress and Intra-Sectoral Structures of Employment[END_REF] analyze the technological efficiency and labour costs in manufacturing sectors using a convex production frontier. They find a positive relation between innovative firms and employment growth. They argue that although the effect of labour costs is generally negative, the companies with increasing labour costs have an insignificant impact on technologically leading industries. The technological changes vary across 12 manufacturing sectors.

Also the transition of global and local technological changes is informally discussed in literature. For example, [START_REF] Bonaccorsi | Technological Frontiers and Competition in Multi-Technology Sectors: Micro-Evidence from the Aero-Engine Industry[END_REF] investigate the technological progress in the commercial jet aero-engine industry. The technological discontinuities and a convergence progress of technological frontier are observed. This implies overall evolution (global technological change) can be aggregated by incremental improvements (local technological change). [START_REF] Timmer | Localized Innovation and Productivity Growth in Asia: An Intertemporal DEA Approach[END_REF] investigate the labor productivity growth across Asian countries. Their results show that the technology innovation is localized in agricultural and manufacturing sectors. In the agricultural sector, the technological progress usually appears when the capital intensity is high while the innovation has a more significant impact on labour productivity change in manufacturing sector. López-Pueyo and Mancebón (2010) explore the labour productivity changes in information and communication technologies sector in advanced countries. They argue that the localized innovation is an additional driving force of worldwide productivity growth divergence across countries. Their results imply that labour productivity growth is mainly due to technical change and capital intensification.

Besides the assumption of convexity for the production technology, some investigations are conducted within a nonconvex approach. [START_REF] Los | Localized Innovation, Localized Diffusion and the Environment: An Analysis of Reductions of CO2 Emissions by Passenger Cars[END_REF] evaluate technological change and carbon emissions by passenger cars using a nonparametric nonconvex model. Their results show that the rates of technological change and diffusion have a significant disparity between segments of the car market. They argue that better policies should be derived on the diffusion of best-practice technology.

There is a strand of macroeconomic empirical work focusing on the nonparametric measurement of (total factor) productivity change by using a Malmquist index. Key articles in this strand include Färe et al. (1994), [START_REF] Ray | Productivity Growth, Technical Progress, and Efficiency Change in Industrialized Countries: Comment[END_REF], [START_REF] Kumar | Technological Change, Technological Catch-Up, and Capital Deepening: Relative Contributions to Growth and Convergence[END_REF], [START_REF] Henderson | Human Capital and Convergence: A Production-Frontier Approach[END_REF], [START_REF] Krüger | The Global Trends of Total Factor Productivity: Evidence from the Nonparametric Malmquist Index Approach[END_REF][START_REF] Krüger | Long-Run Productivity Trends: A Global Update with a Global Index[END_REF] among others. These contributions exploit the decomposition of the Malmquist index mentioned above. The magnitude of components can be different for different positions in the input-output space which fundamentally makes the measurement of productivity become local. [START_REF] Caselli | The World Technology Frontier[END_REF] investigate directed technological change in a production function setting with skilled and unskilled labor as inputs in addition to capital. They rely on a parametric CES production function with calibrated parameters. Their main finding is that developing countries are relatively more efficient in using unskilled labor whereas advanced countries are more efficient in using skilled labor. Although this analysis is static the efficiency differences have their heritage in differential long-run rates of technological advancement. This finding can be confirmed by [START_REF] Krüger | Revisiting the World Technology Frontier: A Nonparametric Directional Distance Function Approach[END_REF] using nonparametric methods of efficiency measurement without specifying a functional form. [START_REF] Jones | The Shape of Production Functions and the Direction of Technical Change[END_REF] and Krüger (2016) provide related analyses.

IV. Towards More Operational Definitions and Concepts

In this section, we focus on two representative strands in the literature that shed some light on the formal definition of global and local technological change.

To the best of our knowledge, the article of [START_REF] Kerstens | Total Factor Productivity Growth and Convergence in the Petroleum Industry: Empirical Analysis Testing for Convexity[END_REF] Global technical progress is defined as resulting from efficient observations at two time periods relative to a convex frontier that do experience positive technological change between these two time periods relative to the same convex frontier. Local technical progress is defined as resulting from efficient observations at two time periods in terms of a nonconvex frontier that are both inefficient with respect to a convex frontier and that that do experience positive technological change in terms of a nonconvex frontier between these two time periods. The same authors also defined weaker versions of the same definitions whereby an observation sis only efficient in one of the two time periods considered.

Both these definitions are illustrated with the help of Figure 1. We observe two production frontiers in a single input and single output space: one pertains to time period t, while the other is related to the next time period t+1. The convex technology is drawn as a dash-dotted line. The nonconvex staircase-shaped technology is indicated as a full line. Observe that the production frontier in period t+1 has shifted mainly upwards and to the left, though observation 1 is in common with both period t and period t+1. Observe that for the convex technology only observations 1, 3, 5 and 7 are situated on the frontier: all other observations are technically inefficient. Observe that for the nonconvex technology all the observations from 1 to 7 belong to the frontier: no observation is technically inefficient. Now, while observations 3, 5 and 7 satisfy the conditions for global technical progress, observations 2, 4 and 6 meet the conditions for local technical progress. Thus, in this Figure 1 more observations satisfy the conditions for local technological change than the ones for global technological change. But, it should be realised that this is not a mathematical necessity: it simply depends on the empirical structure of the sample.

The empirical results in [START_REF] Kerstens | Total Factor Productivity Growth and Convergence in the Petroleum Industry: Empirical Analysis Testing for Convexity[END_REF] for a sample of 933 oil fields over a 50year time horizon in the Mexican Gulf indicate that about 62.8% more observations satisfy the conditions for local technological change compared to global technological change. We are also aware of two other articles applying this same method proposed by [START_REF] Kerstens | Total Factor Productivity Growth and Convergence in the Petroleum Industry: Empirical Analysis Testing for Convexity[END_REF] to distinguish between global and local technological changes. First, [START_REF] Barros | How Scale and Ownership Are Related to Financial Performance? A Productivity Analysis of the Chinese Banking Sector[END_REF] analyze the productivity growth for 14 Chinese commercial banks during the period 2003-2010 using a Luenberger productivity indicator based on both convex and nonconvex technologies under the assumption of variable returns to scale. It turns out from their Table 4 that Second, [START_REF] Fujii | How Enterprise Strategies are Related to Innovation and Productivity Change: An Empirical Study of Japanese Manufacturing Firms[END_REF] use exactly the same methodology as in the previous article to analyse 16 Japanese manufacturing sectors over the 2008-2010 period. The relative importance of global and local technological change varies across sectors and also across the two-year time periods. In brief, there is no clear pattern to discern.

A major problem with this approach is that it exploits differences between convex and nonconvex technologies to define some notion of local technological change. But, this approach fails to recognize that both in the case of convex and nonconvex technologies one may conceptualize both global and local technological change. This needs to be remedied in future work.

There is a second strand of literature based on nonparametric frontier methods that can offer some perspective on the problem of local technical change. When computing productivity indices and indicators like the Malmquist index or the Luenberger productivity indicator, it is not always possible that the efficiency measures can be achieved. The corresponding (linear) programming problems can be infeasible.

To grasp the intuition behind this case, let us study Figure 2. Again, we observe two production frontiers in a single input and single output space for a time period t and the next time period t+1. We only consider convex technologies drawn as a full line. The frontier in period t is spanned by 2 observations and the frontier in period t+1 is spanned by 3 observations. Observe that the two frontiers have one point in common which lets them conform to the basic idea of [START_REF] Atkinson | A New View of Technological Change[END_REF]. We see that the technology in period t+1 has locally moved outwards to the left towards point 1 t+1 and upwards towards the point 3 t+1 .

It is intuitively clear that if we measure the distance between point 3 t+1 and the frontier in period t to assess the change in productivity, then this distance cannot be achieved when we measure in the input orientation (meaning cost minimization). A similar reasoning can be made with respect to point 1 t+1 . If we want to measure the distance between point 1 t+1 and the frontier in period t to assess the change in productivity, then this distance cannot be achieved when we measure in the output orientation (meaning revenue maximization). In mathematical programming language: the corresponding (linear) programming problems are infeasible as indicated by the double arrowed dashed horizontal (vertical) line emanating from point 3 t+1 (1 t+1 ).

The precise conditions under which the efficiency measures being part of the Malmquist index or the Luenberger productivity indicator become infeasible have been analysed in [START_REF] Briec | Infeasibilities and Directional Distance Functions: With Application to the Determinateness of the Luenberger Productivity Indicator[END_REF]. However, in the literature there has been a tendency to mask the problem of infeasibility by making stronger assumptions on the technology, e.g. imposing constant returns to scale. The extent of these infeasibilities has been reported in a multitude of studies, see e.g. [START_REF] Kerstens | Comparing Malmquist and Hicks-Moorsteen Productivity Indices: Exploring the Impact of Unbalanced vs. Balanced Panel Data[END_REF]. V.

The Regional Dimension

Within innovation economics there is another strand of literature which considers local agglomerations of industries within a certain geographical area (regional clusters) as another dimension of localization (see [START_REF] Antonelli | The Economics of Localized Technological Change and Industrial Dynamics[END_REF][START_REF] Antonelli | Collective Knowledge Communication and Innovation: The Evidence of Technological Districts[END_REF], [START_REF] Boschma | Proximity and Innovation: A Critical Assessment[END_REF], inter alia). The firms within the agglomeration benefit from external spillover effects realized by information exchange and also by local labor flows (e.g. by scientists changing employers). In general, this is easier within a limited geographical area than across more distant regions. The benefits of local agglomerations are that they lower adjustment costs and generate complementarities by mutual fertilization of ideas and easing the establishment of user-producer relationships. This fosters technological diffusion by spillover effects which are largely confined to the firms within an agglomeration. In the end, this leads to a faster accumulation of knowledge, massive increasing returns, more innovations and new products. This mainly describes Marshallian (or Marshall-Arrow-Romer, MAR) externalities due to [START_REF] Marshall | Principles of Economics[END_REF], [START_REF] Arrow | The Economic Implications of Learning by Doing[END_REF] and [START_REF] Romer | Increasing Returns and Long-Run Growth[END_REF] which are related to the agglomeration of firms within the same industry or technological area. Another form of externalities due to [START_REF] Jacobs | The Economy of Cities[END_REF] emphasizes the importance of the variety of the industrial structure in an ag-glomeration and knowledge spillovers across firms in different industries. More recently, [START_REF] Frenken | Related Variety, Unrelated Variety and Regional Economic Growth[END_REF] have introduced the concept of related variety with its explicit focus on inter-industry knowledge spillovers (see [START_REF] Content | Related Variety and Economic Development: A Literature Review[END_REF] for a recent survey).

VI. The Way Ahead

While some limited progress has been made to distinguish between global and local technological change in the literature, it is rather surprising that up to now nobody has managed to come up with universally accepted formal definitions for global and local technological change that are valid for general nonparametric technologies under a wide variety of assumptions.

  offers the first attempt to provide an operational definition for the notion of localized technological change. These authors utilize a primal Luenberger productivity indicator based on deterministic, nonparametric production frontiers to measure global and local technical change and decompose it into technical efficiency change and production frontier change.

Figure 1 :

 1 Figure 1: Global and Local Technological Change under Nonconvex and Convex Technologies

  overall most banks in the sample experience global technical change rather than local technical change: now 42.8% more observations satisfy global technological change rather than local technological change. There is only one among 17 periods where local technological change is slightly bigger than global technological change.

Figure

  Figure 2: Technological Change under Convex Technologies: Infeasibilities

Future

  work could furthermore focus on accumulating more evidence on the empirical incidence of global and local technological change, on developing dual cost function-based measurement of global and local technological change, on transforming the infeasibility into a measurable indication of local technological change, among others.